Physics of ionizing shock waves in magnetic fields

PHYSICS REPORTS (Review Section of Physics Letters) 84, No. 1(1982) 1—84. North-Holland Publishing Company

PHYSICS OF IONIZING SHOCK WAVES IN MAGNETIC FIELDS M.A. LIBERMAN and AL. VELIKOVICH Institute for Physical Problems, USSR Academy of Sciences (Moscow, V-334, Vorobyovskoye shosse 2) Received August 1981

Contents: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Introduction Formulation of the problem. Basic equations Magnetic structures of ionizing shock waves Evolutionarity conditions and the number of boundary conditions for ionizing shock waves Precursor ionization. Ionization stability The additional relation and magnetic structures of shock waves Limiting regimes Transverse shock waves (qualitative theory) Shock structure of a transverse ionizing shock wave

3 5 10 19 26 31 34 37 41

10. Shock structure and ionization relaxation 11. Formation of a magnetic structure of a GD shock propagating across the magnetic field 12. Magnetic structures of normal ionizing shock waves 13. Self-similar piston problem 14. Plasma heating in normal ionizing shock waves 15. Normal ionizing shock waves. Numerical simulation of flow ahead of a conducting piston 16. Switch-off shock waves References

45 48 54 59 64 72 79 83

Abstract: This paper presents the theory of ionizing shock waves in a magnetic field. Depending on the shock type (which is determined by the relation between the gas outflow velocity, at the shock front, and local values of the characteristic fast and slow magnetosonic speeds and Alfvén speed), the evolutionarity conditions of a shock wave either imply additional boundary conditions, apart from those which follow from the conservation laws and continuity equations, or no additional boundary conditions are implied for a shock wave of a given type. Generally, an additional relation determining the shock front structure is required in the latter case. This additional relation is a consequenceof the stability condition for an ionizing shock front structure, i.e. the ionization stability condition. The ionization stability condition meansthat the ionization wave induced in a neutral gas by an ionizing shock wave moves, relative to the unperturbed gas, at a velocity equal to that of the shock compression front, irrespective of the mechanism of ionization transfer in the neutral gas by a transverse electric field induced by the shock wave. The explicit form of the additional relation in question can be obtained from the ionization structure of the shock front leading edge. In the simplest case where precursor ionization processes (photoionization, etc.) are negligible the additional boundary condition implies that the upstream electric field in the neutral gas should be equal to the breakdown field. For a sufficiently high intensity of the ionizing shock wave the solutions obtained are reduced to those for magnetohydrodynamic shock waves. Structures of ionizing shock fronts have been considered for different orientations of the magnetic field relative to the shock front plane: transverse, normal and switch-off shock waves. The paper presents, in particular, solutions for self-similar problems and numerical solutions for non-stationary problems on the formation of the shock front structure of transverse and normal ionizing shock waves. Solution of these problems permits a comprehensive description of flows, which arise in inverse Z-pinch type devices and coaxial electromagnetic shock tubes. The paper also presents a detailed analysis of experimental investigations of the shock structure of ionizing shock waves.

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1982 North-Holland Publishing Company

PHYSICS OF IONIZING SHOCK WAVES IN MAGNETIC FIELDS

M.A. LIBERMAN and A.L. VELIKOVICH Institute for Physical Problems, USSR Academy of Sciences, Moscow, USSR

I

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

3

1. Introduction Extensive investigation of ionizing shock waves began in the fifties when powerful electromagnetic shock tubes had been created, which made it possible to study such shocks under laboratory conditions. Let us remind ourselves that an ionizing shock wave represents a rather strong shock, which propagates in a gas of zero conductivity, so that the downstream gas is ionized and becomes conducting. In the absence of magnetic fields such shock waves are called gas-dynamic (GD) shocks. Shock front structures, ionization rates, the origin of primary electrons, and so on, are problems, which are of considerable interest in the latter case. These problems have been studied fairly well; for details we refer the reader to a famous book by Zel’dovich and Raizer [1]. An important specific feature of ionizing shock waves in a magnetic field lies in the fact that an upstream electric field, parallel to the shock front plane, is induced in the neutral gas. This electric field is essential only for ionizing shock waves in a magnetic field and it does not arise in GD-shocks; in a similar problem of a shock wave in a plasma (MHD-shock) both upstream and downstream electric fields are zero in a flow-fixed coordinate system. The existence of an upstream electric field in an ionizing shock wave implies an uncertainty in the parameters of the downstream plasma. In other words, the boundary conditions, which follow from continuity equations for fluxes of mass, momentum and energy and from Maxwell’s equations, do not determine completely the downstream state in a magnetic field. This fact was discovered fairly long ago; we find a first indication of the problem in ref. [2]. An excellent review of theoretical and experimental investigations in this field until 1969 can be found in the review by Chu and Gross [3], which also gives references to previous papers. For instance, in the simplest case of a transverse shock wave (magnetic field parallel to the shock front) the flow is completely determined by four variables: the density j5, velocity t3, temperature T, and magnetic field H. In this case the conservation laws give only three algebraic relations (boundary conditions), which connect upstream and downstream values of these variables. (For MHD-shocks there is an additional relation, which follows from the fact that the upstream electric current outside the shock front is zero and leads to the magnetic-field-freezing condition* ü0H0 = 132H2. For ionizing shock waves the upstream electric current is zero due to zero gas conductivity.) Obviously, if an ionizing shock wave is not very strong, it does not alter the transverse magnetic field. On the contrary, for a very strong shock the energy spent for gas ionization can be neglected, so that such a shock almost does not differ from a magnetohydrodynamic one, that is the change in the magnetic field at the shock front corresponds to density compression. It can be said that the magnetic structures of ionizing shock waves are intermediate between those of GD and MHD shock waves, depending on the intensity. In mathematical respects shock waves represent discontinuous solutions of the appropriate system of equations, which describes the state of an ideal (dissipationless) medium. These differential equations are invalid at the discontinuity surface. Considering the discontinuity surface, that is idealizing the problem, we neglect in the equations dissipative terms containing high-order derivatives. That is the reason why differential equations, in which dissipation is not taken into account, do not determine jumps at the discontinuity surface, so that boundary conditions at the discontinuity surface should be added to the system of equations, to obtain an unambiguous solution of the problem. The number of these conditions should correspond to the number of variables determining the shock wave. * In magnetohydrodynamics the condition of magnetic field freezing, together with the condition of continuity of the tangential electric field at the shock front, make it possible to eliminate the electric field from the equations, by expressing it in terms of local values of the velocity and the magnetic field. This also implies, in particular, that MHD shocks are always plane-polarized, while this is not the case for ionizing shock waves.

:

4

MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

In the problem considered the system of basic equations includes Maxwell’s equations, equations of motion and heat conduction for plasma components, and the ionization kinetics equation. It is very important, in principle, to know what boundary conditions for this system of equations correspond to physically realized ionizing shocks. This question will be discussed in detail below. Going over, as usual, to a moving coordinate system, in which the shock front is at rest, let us consider a one-dimensional problem (all the variables are functions of only one coordinate x; the x axis is normal to the front), namely the problem of the magnetic structure of a plane front of a steady-state ionizing shock wave. It should be noted that while solving the problem of the actual flow of the plasma in an ionizing shock, one has to understand clearly the idealization, which underlies the mathematical theory of the shock front structure, and the assumptions of the steady-state character of the shock front structure. The conventional treatment is as follows. Equilibrium states of the gas (plasma) far upstream and downstream are identified with singular points of the system of differential equations. Equating all terms containing derivatives to zero, we obtain a system of algebraic equations, which relate the initial and final equilibrium states of the gas upstream and downstream. These algebraic equations the Rankine—Hugoniot relations represent integral conservation laws and are boundary conditions at the discontinuity surface, with which the shock front is identified. The structure of a shock front, which actually has a finite width, is represented in this case by an integral curve connecting the singular points of the basic system of differential equations. It is natural to expect that physically realized solutions should uniquely correspond to definite structures. The above procedure is simply realized for ordinary gas-dynamic shock waves. In this case the shock front steadiness and the possibility to describe it in the framework of a one-dimensional model are local characteristics of the flow in the vicinity of a given section of the front and follow from the single condition that all characteristic dimensions of the system should be much larger than the front width, which is of the order of the mean free path of atoms or molecules even for shocks of a moderate intensity. For scales, which are larger than the mean free path, the shock front can be considered as a discontinuity surface of zero thickness, and the shock front structure problem can be solved, irrespective of the shock dynamics. A slightly more complicated situation arises for shock waves in plasma, even when there is no external electromagnetic field. For instance, the electrostatic potential discontinuity across the shock front cannot be found from the conservation laws and depends on the front structure. Generally, the long-range electromagnetic fields lead to the fact that the one-dimensional character of the shock front becomes not a local, but a global feature of the flow, and this poses much stricter limitations on the possibility of using the model of a steady-state plane one-dimensional shock front. In the case of ionizing shock waves in a magnetic field there are, as was noted above, two additional degrees of freedom related to the magnitude and direction of the upstream induction electric field in the neutral gas. This requires a more exact definition of the upstream “equilibrium state of the gas” for x x~Even the formulation of the problem of an ionizing shock wave in a magnetic field in the form adequate to physical experiments becomes closely related to the shock structure problem and to the problem of the ionizing shock front dynamics. In other words, to solve the problem, one has to go beyond the framework of a purely mathematical study of the integral curve field for the system of differential equations. Obviously, the steady-state solutions for ionizing shocks do exist only if the upstream electric field induced in the neutral gas is less than the breakdown field. This conclusion is not trivial because it implies that the solution of the ionizing shock problem requires an analysis of the internal structure of the shock when ionization kinetics processes are taken into account. This analysis yields an additional relation for the electric field magnitude, which is not a consequence of the conservation laws. —

—

—~ —

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

5

Let us note that such a situation for shock fronts, which alter the state of the medium, is not new in physics, detonation theory being a classical example. As was shown by Zel’dovich in 1940 [4], the internal structure of a wave, with due account for chemical reaction kinetics, and the flow as a whole should be considered to justify an additional relation (e.g., the Jouguet condition) for the conservation laws. A similar situation occurs when the combustion front velocity is determined (Ya.B. Zel’dovich, 1944) [5].

2. Formulation of the problem. Basic equations Let us consider a plane shock front, which propagates in a cold non-conducting gas with a magnetic field. The shock intensity is assumed to be sufficiently high, so that the downstream gas is ionized and its conductivity is high. Formulation of the problem and the shock front of electromagnetic structure is of primary importance in the physics of ionizing shock waves in an electromagnetic field. Therefore, we first disregard comparatively well studied details of the shock front structure, such as the structure of the zone of ionization relaxation or temperature relaxation of electrons and heavy species, and so on [1, 6]. The characteristic scale of the magnetic field variation at the shock front is, as a rule, much larger than the scales of viscosity, thermal conductivity, etc., and this justifies the possibility of a formulation, in principle, of the magnetic structure problem in terms of the hydrodynamics of an ideal liquid of finite conductivity. The effects of deviations from thermodynamic equilibrium, which are responsible for more subtle details of the shock front internal structure, will be considered below for the most important particular cases of the magnetic field orientation relative to the plane of the front. Suppose that a steady-state shock front structure exists. The zero upstream gas conductivity o~omeans that the change in the upstream magnetic field in the initial state is negligible, that is, the upstream magnetic Reynolds number of the gas should be small. (2.1)

2,

Rm = 4iroovL/c

where L is the characteristic length scale, e.g., the shock-tube length. The necessary condition is, obviously, that the upstream gas conductivity must change negligibly over characteristic times. This condition leads to an important restriction on the electric field upstream of an ionizing shock, namely to the condition of ionization stability of the neutral gas [7,8]. Let us go over to a shock front-fixed coordinate system. In this system the neutral gas comes into the shock front, which is at rest, at x = and the ionized gas comes out of the shock front at x = + The x axis is chosen normally to the front surface. All the parameters characterizing the gas are functions of only one coordinate, x, and all the processes are described by ordinary differential equations. In the unperturbed upstream flow of the neutral gas, at x = we have for the velocity and magnetic and electric field intensities — ~,

~.

— ~,

=

{i5~ 0,~

vz~},

o

{1~o,1?yo, 1z~},

E0 = {E~0,E~0,E~0} .

(2.2)

In a uniform flow of the ionized gas, which comes_out of Ek, thewhich shockhave fronttoatbex found = + °~,the velocity and 5k, Hk and in the solution of magnetic and electric fields take certain values i the problem. The electric and magnetic fields satisfy Maxwell’s equations, which in the steady-state case take the

6

MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

form curl 1!

~

(2.3)

~

curlE=0.

(2.4)

The electric current j in eq. (2.3) is /-

1~

—\

JcrI1~E+—VXH)~

(2.5)

where cr is the local gas conductivity. In the equilibrium downstream flow at x = + all derivatives vanish, and the condition of zero electric current, with due account for a finite (but high) gas conductivity, leads to the relation, (2.6)

Ek+—vkXHk——O,

which connects the downstream electric and magnetic fields. Let us choose a coordinate system where the electric field vector lies in the (xz) plane, then Vk = {i5xk, Vyk, 0},

~

=

{1~i~k, H~k,0}.

(2.7)

The equation divH=0

(2,8)

implies that the normal component of the magnetic field is continuous, i.e., H~0 Hxic = const. Assuming the gas and downstream plasma to be quasineutral everywhere, we find from the equation divE=O that E~is continuous, and it follows from (2.6) and (2.7) that we may take, without loss of generality, E~=0.

(2.9)

At the same time, we find from (2.4) that the electric field transverse components are continuous, i.e. EYO=EYk=O,

E~o=E~k.

(2.10)

The one-dimensional gas flow, which is steady in a shock front-fixed coordinate system, is completely determined by nine variables: the density ~ the gas temperature T, the three components of the velocity 13, and the transverse components of the electric and magnetic fields E5 = {E~,E~}and H~= {H~,H~}. For steady flow equations the boundary conditions, which imply the absence of

MA Liberman and A L Velikovich Physics of ionizing shock waves in magnetic fields

7

“upstream” and “downstream” gradients, i.e. that all hydrodynaniic variables tend to initial (at x = — and final (at x = + co) equilibrium values, represent the conservation laws for the fluxes of mass, momentum and energy [,5i5.~]=0,

(211)

[~+p+_~_I.?2l=O,

(2.12)

L

J

81T

(213) (2.14)

[,~x(~2+w)+~Ex1~]0,

where w

=

e + p/5 is the enthalpy per unit mass of the gas, p is the pressure, and [F]

F0

—

Fk.

Let N denote the density of the heavy species in the gas, i.e. the total density of neutral atoms and ions (N = N0 + Ni), let n = Ne be the density of the electrons, and let a = n/N be the degree of ionization. We have then p— nT~+NT,

(2.15)

w

(2.16)

and (aEion+—~+~aie)/ma,

where Ctofl is the ionization energy of atoms or molecules, and y is the specific heat ratio. Let us also consider the equation of the gas ionization kinetics va)n—qr+ñ.

(2.17)

Here i’~is the frequency of atomic ionization by electron impact, ~a represents the electron losses, which are linear in n, and q~,represents the recombination losses which are proportional to a higher power of the electron density (associative, triple, and other types of recombination). The last term in eq. (2.17) describes precursor ionization in a shock wave caused by photoionization of the upstream cold gas due to shock front radiation or to diffusion of electrons from the shock front. To formulate the problem of an ionizing shock in a magnetic field and to find the shock front structure, one has to solve the system of ordinary differential equations in ninevariables ~5,ii, T, H~,E~,a, Te with boundary conditions at the shock front, which is considered in this case as a zero-thickness discontinuity surface. The ionizing shock problem is a fairly specific one, because the conservation laws (2.11)—(2.14) together with the continuity equations (2.10) for the tangential electric field give only seven boundary conditions for nine variables. The ionization kinetics equation (2.17) does not lead to additional relations, since in equilibrium states the degree of ionization is determined by the local values of the

M.A. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magneticfields

8

density and temperature of the gas or plasma. Thus, in the classical formulation the ionizing shock problem is characterized by two arbitrary parameters which we are free to choose. From a physical standpoint this is related to the fact that the upstream induced electric field E5 does not lead to an electric current due to the zero upstream gas conductivity. Let us remind ourselves that in the case of a shock in an ionized gas (magnetohydrodynamic shock wave) a relation similar to (2.6) holds in the upstream flow, which connects the electric field at x = with the local values of the velocity and magnetic field. Hence, the electric field can be eliminated for an MHD shock, and there are seven boundary conditions for the remaining seven variables. We shall consider the theory of ionizing shock waves in gases when the magnetic Prandtl number is negligibly small, i.e. when —

(2.18) 2/4iru is the magnetic viscosity. It can easily be seen that where i’ is the kinematic Pm =waves c the parameter Pm ~ 1 is viscosity small forand shock in gases at densities exceeding 1013 cm3 and for a downstream temperature less than or of the order of several electron-volts. It follows from shock wave experiments [9, 10] that the gas-dynamic structure tends to the MHD structure, with increasing shock front velocity, for negligibly small Pm values. It will be shown below that though Pm grows rapidly (as the fourth power of the temperature) with the shock front velocity, the intensity of ionizing radiation from the shock front increases much more rapidly (exponentially), and that is the reason why the transition from an ionizing shock to a shock with the MHD structure takes place for Pm~1. Apparently, it is reasonable, from a physical standpoint, to consider only collision ionizing shock waves with Pm 1, since at gas densities less than 1013 cm3 shock waves are, in fact, collisionless, and ionization is caused by other physical processes. The limit of small magnetic Prandtl numbers means that the magnetic field diffusion length, i.e. the scale characteristic of magnetic field variations, is much larger than the scales corresponding to relatively small viscosity, thermal conductivity, etc. In a mathematical respect we consider in this case the shock front magnetic structure in the zeroth-order expansion in Pm 1. The shock front magnetic structure is then described by eqs. (2.3), (2.4) and the equations of hydrodynamics (2.11)—(2.14) where the effects of viscosity and thermal conductivity, which are of importance only inside infinitely thin internal gas-dynamic shocks, are neglected. Though it is most natural, from a physical standpoint, to choose the magnitude and direction of the shock wave electric field as two free parameters to be determined, for calculations it is more convenient to use the electric field in a shock front-fixed coordinate system and the z-component of the upstream magnetic field. This set of parameters is obtained if the z-axis in the coordinate system chosen is directed along the electric field, that is Pm’z”p/pm~1,

-~

‘~

EYO—EYk—0,

E~o=EZk—=E~.

(2.19)

In this case the z-components of the downstream magnetic field and velocity are zero, as can be seen from (2.7), but H~ 0and E~still remain arbitrary parameters, and the problem is either to find their values or a relationship between them. To simplify the equations, the y-component of the upstream velocity can be eliminated with the help of a Galilean transformation. We have then

MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

=

9

{i5~~, 0, Vz~}.

It is convenient to analyse the equations using dimensionless variables that will be denoted by the same symbols, but without bars: v~=i5~/i5~0, VyVyIVxo,

H~H~IH~, HzHzIf~x,

V~V~/V~o,

TT/T0,

ppI~o,

TeTe/To,

E~cE~Ii3~oH~.

Obviously, H~= 1 in the dimensionless system, and eqs. (2.3), with due account for (2.5), take the form (2.20) dH~ 1 ~

(2.21)

2/4’rrov~ois the scale characteristic of magnetic field diffusion. where c Not 4, to =complicate the presentation with irrelevant details concerning energy losses for gas ionization, let us neglect the corresponding term in expression (2.14) for the specific enthalpy. Below, in the calculation of particular magnetic structures and in the comparison with experimental results for transverse and normal shocks both the energies of the ionization and of the dissociation of the molecular gas are taken into account, Eliminating the density with the help of eq. (2.11), which is in the dimensionless variables

pv,, = 1,

(2.22)

we obtain the following form of eqs. (2.12)—(2.14) in the dimensionless variables: (2.23) v~ H~/M~ —

=0 —H~0/M~0,

v~,

—

HZ/M~O=

0,

vx+vY+vzlvzo+(l)M2[T+2aTel]+M2_O,

(2.24) (2.25)

(2.26)

where the sonic Mach number is 2 M0

p 0v~o/yp0,

(2.27)

and the Alfvén Mach number is M~0= 40i3~0/1~.

(2.28)

10

MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

Since the conductivity, and hence the parameter 4,, are functions of the degree of ionization of the gas a and the electron temperature Te, equations (2.20) and (2.21) relate the change in the hydrodynamic variables p, v, Te, H to the ionization kinetics equation. It is this fact, which makes it possible to obtain an additional relation for the electric field. At the same time, since the ratio dF4/dH~does not depend on 4, and, being a function only of the hydrodynamic variables, holds everywhere except x = co, both the type of downstream singular points on the phase plane (Hr, H~)and the form of the field of integral curves in phase space (v~,H~,H~)are determined only by structural relations stemming from eqs. (2.20)—(2.26) irrespective of the ionization equation. Let us first consider, therefore, what solutions for ionizing shock waves may follow from eqs. (2.20).—(2.26). —

3. Magnetic structures of ionizing shock waves We now analyse the qualitative form of solutions of system (2.22)—(2.26) [11], using the approximation Pm ~ 1. Elimination of the density p, temperature T and transverse components of the velocity v~and v~with the help of (2.22)—(2.25) yields the equation of a surface in the 3-dimensional phase space (vi, H~,Hi): F(v~,1-4, H~)~4v~

—

v~[s+~~— 2M~0(H~+ H~

—

H~0—H~o)]

2 + 2E. 14—H~ + 1+

—

H~H~0 (HyHyo)

0=~

—

(3.1)

(the value y = ~is taken in eqs. (2.23)-.(2.28), to make the formulae less cumbersome). Equation (3.1) can be rewritten in the equivalent form [12] +2H~=

[H~

-

—

H~(v~)}2 + H~

H~(v~)]

0+ 4M~0(v~1)(v~-VGD) at) -

(3.2)

where LI)’

—

ESI%1 ~o + 1 ~ 12Vx

2 aO

,

(3.3)

23 VGD

.

(3.4)

The magnetic structure of an ionizing shock wave can be represented in phase space by a curve lying on the surface F = 0. This surface comprises, in particular, the points corresponding to the initial (x = co, v,, = 1, FI~= H~0,H~= H~0)and final (x = +ca) states of the gas, and also, as can be seen from (3.2), the point with the coordinates —

VXVGD,

H~=H~0, H2=H~0.

(3.5)

M.A. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

It follows from eq (3 2) that v~= const planes intersect the surface F the hyperbola H2=0,

=

11

0 on circles with centres on

H~=H~(v~).

In the sections .FI~= const. and H2 = const. the surface F = 0 has the form of third-order curves, which usually consist of several separate branches. The surface F = 0 itself may also contain more than one connected component. Let us note that for E. + H~0/M~0 = 0, i.e. for H(v2) = 0, the variables H~ and H2 appear in eq. (3.2) only in the combination H~+ H~,i.e. the surface F = 0 is invariant with respect to rotations about the v~axis. In the analysis to follow we shall need expressions for local Mach numbers, as calculated relative to characteristic velocities at a given point: 112, (3.6) M13x/aMovx(T+aTe)~

Ma =

13x/Ca =

(3.7)

MaoV~2.

It follows from eq. (3.7) that the circles v~= const. on the surface F = 0 also correspond to constant values of the local Alfvén Mach number. Expressing the gas temperature T in terms of v~,H~,H 2 with the help of (2.26), we obtain from (3.6) and (3.1)

3v~(1_~).

~

(3.8)

A paraboloid of revolution, whose equation is easily obtained from (3.8) ~ /

‘3

LI 2 ~j-2 LI 2 IlyT152JTIyOE120

LI 2

2 M corresponds to a constant value of the current Mach number M in the phase space. It is clear that only values M2> 0 can have a physical meaning. Therefore, the curves representing magnetic structures should lie inside the paraboloid (3.9), but with zero right-hand side (i.e. inside the surface M2 = :~).A region of the surface F = 0 lying outside the above paraboloid can naturally be called the non-physical region of the surface. Certain general properties of the surface F = 0 can easily be derived if the consideration is confined to a neighbourhood of the point 0. In fact, we can study the same surface, using variables, which are made dimensionless with respect to their values at any given point. Obviously, eq. (3.1) retains its form in this case, the set of the parameters (M 0, Mao, I4~,H20, E5) only being changed. In particular, for given values of .1-4 and H2 two points, in general, belong to the surface F = 0: the point 0 and the point (3.5). Using eq. (3.8), we find the local sonic Mach number M’ at the point (3.5):

_:L_1i+......:~~..__ 3v2 k. 5M~ V~

2M~0

Ml2=~~.

It follows from (3.4) and (3.10) that

——

39

—

(3.10)

MA. Liberman and A.L. Velikovich. Physics of ionizing shock waves in magnetic fields

12 1

2

1

A.(~

5’..IVJ

~—1,

2,

1

1

VGO~—k,

A.(.

[-.IVIt),

1
1/\/5
(3.11)

1
4
M 0<1/\/5.

Thus, the surface F = 0 is uniquely divided into two regions, in one of which the local sonic Mach number M is everywhere more than 1 (the supersonic sheet of the surface F = 0), and in the other region M is everywhere less than 1 (the subsonic sheet). For given .14, H2 the point corresponding to larger v~values lies on the supersonic sheet, and that corresponding to smaller values on the subsonic sheet. It can be seen from eq. (3.11) that the non-physical region of the surface F = 0 is part of its supersonic sheet. The supersonic and subsonic sheets of this surface may represent individual connectivity components, but may intersect along the line M = 1, the so-called sonic line formed due to the intersection of the surface F = 0 with the paraboloid (3.9) corresponding to M = 1. If the surface F = 0 is not degenerate at a given point (i.e. one of the derivatives ~FtoH~,t9F/t9H~is not equal to zero), the vector normal to the surface at this point is parallel to the plane (Hr, He). If all the three derivatives vanish, the surface F = 0 becomes irregular at this point of the sonic line, and the normal to it is not defined. This violation of the regularity is possible at certain points of the sonic line, and also along the whole sonic line (see fig. 1) for a special choice of the parameters (e.g., for E. + H~0/M~0 = 0, M~0 = or ~, M~ cc). Thus, the sonic line represents a set of critical points when the surface F 0 is projected onto the (Hr, H2) plane, and its projection is the discriminant curve of eq. (3.1). Obviously, in the neighbourhood of any point of the projection of the surface F = 0 onto the (Hp, H2) plane that does not belong to the discriminant curve, the velocities v~± and v,~corresponding to the super- and subsonic —*

/

0

‘3

I’ll /‘1~I

0.5

‘p

0.25

0.5 /0

is

sub

11y~I1~)space in the degenerate case. The curves are symmetrical with respect Fig.rotations to 1. Integral about curves the v~ for axis eqs.because (3.13), (3.14) the relation on the E, surface + H (3.1) in (vi, 50/M~0= 0 holds. On the sonic line M 1 the regularity of the surface is violated due to a special choice of the parameters: M0—* a~,Ml0 = 8/5.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

13

sheets of the surface may be expressed uniquely in terms of H~and H2 with the help of eq. (3.1), which is considered as a quadratic equation in v~. Let us express v,, in terms of H~and H2, and change the variables: d~=dx/41.

(3.12)

As a result, we obtain a system of equations, which describe the magnetic structure of an ionizing shock at the (Hr, H2) plane: dH~/d~ = E. + H~0/M~0 + (v2 — dH2/d~= (v~—

1/M~0)

1/M~0)

(3.13)

1I~,

H2.

(3.14)

Actually, we have, of course, two systems of equations: one for the supersonic sheet of the surface F = 0 (v~= v2÷(H5,H2)) and the other for the subsonic sheet (v2 v2(H~,H2)). Both these systems of equations coincide at points of the discriminant curve on the (11~,H2) plane. Downstream, 4, has a certain finite value. Integration of (3.12) yields ~ + cc as x + cc, Therefore the singular point k of the basic system of equations is a singular point of the system (3.13)—(3.14), at which the right-hand sides of these equations must vanish. The integral curve representing the shock front magnetic structure should come into this point as — + cc, On the contrary, as x — — cc the right-hand sides of equations (2.20)—(2.21) vanish together with 1/4,. If 1/4, drops sufficiently fast as x — cc, so that the integral -~

—~

—*

(3.15)

~(_cc)=

converges, the state of the gas upstream of an ionizing shock is characterized by a non-singular point of the system (3.13)—(3.14). This case is of special importance to us, because if the integral (3.15) diverges, on the (Hr, H2) plane and on the surface F = 0, the corresponding magnetic structures of MHD shocks have been studied in rather great detail (see [13, 14]). It follows from (3.13) and (3.14) that the state of the gas downstream, as x — + cc, is the point of intersection of the line described by the equations H2 =0,

~

=0

(3.16)

with the surface F = 0, i.e. the intersection of the curve F(v2, H~,0) = 0 on the (v2, .FI~)plane with the line (3.16) representing a hyperbola on this plane, which is characterized by the zero electric field E + (lIc)i5 x H in the gas-fixed coordinate system (equilibrium condition in a conducting gas for x = +cc). The hyperbola (3.16) can naturally be called the “zero field hyperbola”. Equations (3.7) and (3.16) imply that Ma> 1 on one branch of the hyperbola and Ma < 1 on the other branch. Substitution of (3.16) into (3.1) yields a fourth-power equation in v~.Hence, there are no more than four singular points, which may represent the state of the gas downstream. Like the case of ordinary MHD-theory [15],we assign the numbers from 1 to 4 to these points that increase with the decreasing velocity. It may be shown, as is done in magnetohydrodynamics, that

14

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

S(1)

i3~(2) i5~(3)>i5~(4),

(3.17)

S(2) ~ S(3)

(3.18)

T(2)

<

S(4),

~t(3)~ T(4),

Cf(1)~13~(1), Ca(2)

i3

2(2)

c~(3) 13~(3)~ i52(4) ~

(3.19)

(3.20) cf(2),

(3.21)

Ca(3),

(3.22)

c~(4),

(3.23)

where S is the entropy and Cf and c~are the velocities of fast and slow magnetosonic waves, respectively. We shall classify ionizing shocks in a magnetic field in accordance with the type of a singular point corresponding to the state downstream [11].Thus, ionizing shocks of types 1, 2, 3, 4 are possible a priori. As will be shown below, only ionizing shock waves of types 2, 3, 4 can exist. Relations (3.17)—(3.23) become equalities only in degenerate cases corresponding to the confluence of two or more singular points. As can be seen from (3.20)—(3.23), this is the case where the downstream velocity is equal to one of the characteristic velocities. Points 2 and 3 may coincide, in particular, only if = Ca, i.e. Ma = 1, downstream. It can be seen from (3.7) and (3.16) that this also requires E. + H~/M~0 = 0, i.e., in this degenerate case a singularity arises on the circle v2M~0= 1, instead of isolated singular points, both on the surface F = 0 and on the (Hp, H2) plane. As was already mentioned, in this case the integral curves in (v2, H~,H~)space are symmetrical with respect to rotation about the v~axis (fig. 1), and the integral curves on the (14, H2) plane represent segments of straight lines passing through the origin. The confluence of points 3 and 4 means that i32 = c. in the downstream flow. Such a flow is called Chapman—Jouguet flow, in analogy with detonation waves. The Chapman—Jouguet flow is usually characterized by the fact that the curve F(v~,14, 0) = 0 is tangent to the zero field hyperbola (3.16) on the plane H2 = 0, and this is quite natural from geometrical considerations: as the two points of intersection approach each other, we obtain, in the limit, one point of contact. The exception is the degenerate case where the direction of the tangent to the curve at the only common point with the hyperbola is not defined. This is possible only if at that point 14 0, i.e. only for switch-off shocks. (Equation (3.16) implies that in this case also E. + H5IM~0= 0.) The singularity similar to that, which exists on the sonic line of fig. 1, occurs here at one point on the v~axis (see below section 15, figs. 31b and 33b). We may conclude now that magnetic structures of ionizing shock waves are represented by integral curves of equations (3.13)—(3.14), which tend to one of the singular points 1, 2, 3 or 4 as x + cc, Families of integral curves of these equations may be constructed both on the (14, H2) plane (superand subsonic integral curves separately) and on the surface F 0. The integral curves can naturally be assigned a certain direction corresponding to an increase in ~ and also in the entropy S (this direction is indicated by arrows in figs. 1—3). For given 14 and H2 the conservation laws permit a transition from the supersonic sheet to the subsonic one. As far as the ionizing shock wave structure is concerned, this —*

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

15

//~,~// Fig. 2. Integral curves for eqs. (3.13), (3.14) in the (fI~,H~) plane for M 0-~~, M~0= 5/3, H~5 = 0.2, H~0= 0.3, E~+H50/M~5= 0.04. The top of the figure corresponds to the supersonic sheet of the surface (3.1), and the bottom to the subsonic sheet. The vertical arrow indicates the isomagnetic shock from the supersonic sheet to the subsonic one.

Fig. 3. Integral curves for eqs. (3.13), (3.14) on the (Hp, H~)plane corresponding to the subsonic sheet of the surface (3.1) for M~= 0.2, M~0= 0.8, H,5 = 0.4, H~0= 0.3, E~+H~0/Mi0 0.05.

transition corresponds to an internal isomagnetic shock, which is identical to an ordinary gas-dynamic shock wave in a partially ionized plasma. In the approximation Pm = 0 the magnetic field does not change within this shock, but the entropy increases. The transition from the supersonic sheet of the surface F = 0 to the subsonic one corresponds to physically realized isomagnetic shocks only in the case where the point on the supersonic sheet, from which the transition occurs, does not belong to the non-physical region of this surface, as it follows from (3.5), (3.10) and (3.11). Therefore, the magnetic structure of a supersonic ionizing shock wave (M0> 1) can be treated as follows. As ~ increases from ~(_cc), the phase point corresponding to the state of plasma in (vi, 14, H2) space starts moving from the non-singular point 0 on the supersonic sheet of the surface F = 0 along the integral curve of the system of equations (3.13)—(3.14) in the direction of increasing ~ (the direction indicated by arrows in figs. 1—3). The phase trajectory may terminate on the supersonic sheet via approaching one of the singular points as ~ -~+ cc, or the transition onto the subsonic sheet via the isomagnetic shock takes place at a certain point of the integral curve. In this case we jump, generally speaking, to the non-singular point on the subsonic sheet, and the phase point has to continue moving along that integral curve, on which it found itself while approaching one of the singular

16

MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

points on the subsonic sheet as ~ + cc.* A case is also possible where the isomagnetic shock comes into the singular point on the subsonic sheet, thereby completing the shock front structure. The magnetic structure of a subsonic ionizing shock wave is represented by a portion of the integral curve on the subsonic sheet from the non-singular point 0 at ~ = ~(— cc) to one of the singular points for —*

+cc.

~

Let us note an obvious property of the system of equations (3.13)—(3.14): if H2 = 0 for one of the points of the integral curve, the entire integral curve lies in the plane H2 = 0, i.e. the integral curves of this system of equations are represented by segments of the straight line H2 = 0 on the (14, H2) plane between the singular points and the intersection of this straight line with the discriminant curve. In particular, if 1I~~ = 0, then H2 = 0 for the whole shock front structure. Thus, the condition H20 0 distinguishes the class of oblique ionizing shock waves among the general class of skew shock waves [11]. MHD shocks, considered as a limiting case of ionizing shocks when the intensity of the latter increases infinitely, represent oblique shock waves, since MHD shocks are plane-polarized. Let us now consider the behaviour of integral curves of the system (3.13)—(3.14) in the vicinity of singular points. To this end, we linearize the equations near the singular point of type k and seek a solution in the form HY=HYk+Ae’~,

ff2=fi~~~•

We find then the eigenvalues q1.2 and the corresponding eigenvectors 2k+Hyk Vxk, 0~={1,0}; q1 1 1/Ma

01.2

on the (1-4, H2) plane: (3.24)

q 211/Mak,

(3.25)

Q2{O,1}.

The type of the singular point k is determined by the signs of q1 and q2. It can be seen from (3.20)—(3.23) that q2 >0 for points 1 and 2, and q2 <0 for points 3 and 4. By virtue of (3.1) the condition q1 >0 can be rewritten in the form M~k>1+ H~kM~/(M~ -1).

(3.26)

Since the squared speeds of the fast and slow magnetosonic waves, c~and c,~,are determined in the equilibrium state k2—a2(k)] respectively as the large and small, roots of the quadratic equation = c2c~(k)H~k [c2_ c~(k)][c we can write for the Mach numbers corresponding to these speeds: ,

~J_...Jf-..L±1+H~k±1L.L~ 1+H~k\2 M~. 21M~

M~k

—

~

M~k I

~ 1112 M~M~kj

(3.27)

328

,

(

.

)

It can easily be shown with the help of (3.28) that (3.26) is equivalent to the condition *

It should be noted that the motion along the above integral curve on the supersonic sheet does not lead us to the non-physical region of the

surface F = 0. To prove this, it is sufficient to show that limM2,O,(d/d~)(1/M2)> 0 and, as was noted above, the derivative may be calculated at the point 0. It can be shown with the help of (3.1), (3.8), (3.13) and (3.14) that the inequality in question does hold.

MA Liberinan and A L Velikovich Physics of ioni..ingshock waves in magnetic fields

V

> C~(k)

Vxk

>

5k

c~(k),

for Mk

>

17

1,

for Mk <1.

Since c~< a 1) and point 4 is always subsonic (M4 < 1). Points 2 and 3 may be both super- and subsonic, and we distinguish between these cases by assigning subscripts super and sub to the numbers of these points. All the above considerations can be summarized as a table. Table i Singular point corresponding to the state, downstream

~ 4

Sign of qi

Sign of

+

+

—

+

+

+

—

—

+

—

—

—

Type of the singular point on the (H,, H~)plane Unstable node as ~-+~ Saddle Unstable node as ~-+~ Stable node as ~ - + ~ Saddle Stable node as ~ —~+ ~

As 4 + cc neither of the integral curves approaches the unstable node; on the contrary, the stable node is the centre of attraction for all the integral curves, which pass sufficiently close to it. Only two integral curves (two saddle separatrices) approach the singular point of the saddle type as ~ —~ + cc, The field pattern of the integral curves near the singular points is shown in fig. 4. The singular curve, the circle Ma = 1, corresponds to a degenerate case where the points 2 and 3 coincide, but this case does not deserve special investigation, since the symmetry of the problem permits one to reduce it to the one-dimensional problem (see fig. 1). A degenerate singular point of the saddle-node3sub type andcorresponds 4; the two to the Chapman—Jouguet flow. This pointregions, arises inin the pointscurves approach the separatrices of the saddle-node delimit oneconfluence of which of all singular the integral degenerate singular point as 4 + cc (similar to the node), and in the other region only one integral curve, a separatrix, approaches this point as ~ —~+ cc, while the others move away from it as ~ —~+ cc (similar to the saddle). Let us consider the behaviour of integral curves near the singular points 1, 2, 3, 4, downstream. The neighbourhood of point 1 is not shown in fig. 4, because point 1 cannot represent the state downstream. In fact, we cannot come into this point as ~ + cc, while moving from point 0 on the supersonic sheet along whatsoever integral curve. The following possibilities exist for type 2 shock waves. Two integral curves corresponding to the eigenvector 01 (see table 1 and fig. 4), i.e. the integral curves, for which H 2 = 0, enter point 2su~ras + cc, Therefore the corresponding magnetic structure is represented on the (14, H2) plane by a segment of the linereached H2 = 0 at from the supersonic sheetbelonging (in this case H20subsonic = 0) to point 2sub straight cannot be all aspoint ~ —~ 0 + cconalong integral curves to the sheet 2super. Point (therefore type 2 subsonic ionizing shock waves can not exist). This point may be reached if one moves along an integral curve on the supersonic sheet from the point 0 to the point (H~= 14 (2~Ub), H 2 = 0) 2sub via an isomagnetic shock. It follows from the above from which one pass considerations thatmay in this caseinto H point 2 is also zero, i.e. the magnetic structure of a supersonic ionizing shock —~

—~

—*

—*

18

M.A. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields ~rx

Cf

__________

-

z/~cjI

Ca

~ ~

}P~.. &~‘ ~

Z&~53

L~

~

____

1r

2-cs

c I’~

::~t;:;

Cs

~

~

Fig. 4. The number and types of waves diverging from the shock front of an evolutionary ionizing shock wave, and the behaviour of integral curves for eqs. (3.13), (3.14) in the (H,, H~) plane in the neighbourhood of a downstream singular point. The general properties of an ionizing shock wave are determined by relations between upstream and downstream flow velocities, 0~oand i5~, respectively, and the characteristic velocities (the upstream velocity of sound in a non-conducting gas, the downstream velocities of fast and slow magnetosonic waves and the Alfvén wave, C1, C,, Ca) in the plasma.

can be represented by a segment of the straight line H2 = 0, lying on the supersonic sheet, followed by the isomagnetic shock. Here the magnetic field at the shock front changes completely ahead of the isomagnetic shock. Thus, the existence of the structure poses the following restrictions on type 2 ionizing shock waves: such shocks must be supersonic and correspond to the class of oblique shocks, i.e. H20 = 0 for them. Let us note that ionizing shock waves of this type include, in particular, transverse ionizing shocks, for which H~= 0 and the above dimensionless equations are, strictly speaking, invalid. Yet, in the limit H~—*0,and consequently ~ we find that Ma9cc downstream, i.e. Ma> 1, so a singular point of type 2 corresponds to the state downstream. Generally, it can easily be seen from (3.1) and (3.16) that for Mao cc the state downstream may be represented only by a type 2 singular point, i.e. ionizing shocks of types 3 and 4 exist only in a limited range of variation of Mao. Let 3super, us now consider typecorresponding 3 shocks. As magnetic ~ — + cc allstructures integral curves, which passbysufficiently to the enter it. The are represented segmentsclose of integral point on the supersonic sheet from the point 0 to the point ~ curves shock waves of this type are, generally speaking, skew ones, but in a particular case H 3super are 20 =to0 oblique ionizing shocks of into type the point also possible. As ~ cc, two integral curves corresponding the eigenvector 02 come 3sub (see table 1 and fig. 4). Therefore, type 3sub subsonic ionizing shocks are possible. For the structure to exist in this case, the point 0 must lie on one of the two integral curves mentioned above; the magnetic structure is then represented by a segment of this integral curve from point 0 to the point 3sub’ The structure of a type 3sub supersonic ionizing shock consists of a segment of the integral curve on the supersonic sheet, an isomagnetic shock from the supersonic sheet onto one of the integral curves, which —*

—~

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

19

comes into the point 3sub on the subsonic sheet, and a segment of this curve from the end of the isomagnetic shock to the point 3sub• As can be seen from fig. 2, H along these curves, so a 3sub, which corresponds to 2 the 0class of both oblique shocks, should supersonic ionizing shock wave of type contain an isomagnetic transition from the supersonic sheet directly to the point 3sub, similarly to a supersonic ionizing shock wave of type 2sub~ Finally, all the integral curves, which pass close to point 4 as ~ cc, enter it. Ionizing shock waves of type 4 may be both super- and subsonic, and also skew and oblique. Figures 2 and 3 show integral curves in the (14, H 2) plane for M0 —p cc, M~0= ~, H,o = 0.2, H20 = 0.3, E. + I-I~0/M~0 = 0.04 (fig. 2) and M~ = 0.2, M~0 = 0.8, H,~., = 0.4, H20 = 0.3, E. + H~0/M~0 = 0.05 (fig. 3). Figure 2 shows both the supersonic (top) and subsonic (bottom) sheets of the surface F = 0, as projected onto the (14, H2) plane, because the initial (x = cc) point 0 lies on the supersonic sheet and the end point (x = + cc) may belong either to the supersonic sheet (3super) or to the subsonic one (4). Figure 3 presents only the projection of the subsonic sheet onto the plane (14, H2), because the origin 0 belongs to this sheet and only magnetic structure 0 4 can exist. Let us note that the neighbourhood of the singular point 1 on the supersonic sheet (fig. 2) corresponds to the non-physical regioninofthe thedirection surface 2 increases F = 0, since the point 0 lies at the boundary of this region (M0 cc), and 1/M indicated by arrows as M2 cc, It can be seen from figs. 2, 3 that in the framework of the theory considered the following structures are possible, in principle: 0 3super and 0—* a b —*4 for conditions of fig. 2 and 0 —*4 for conditions of fig. 3. Hence, the theory should be able to answer the following questions: which of the singular points, 3super or 4, corresponds to the state downstream for conditions of fig. 2? If it is point 4, where is the isomagnetic shock beginning (point a) located on the integral curve 0 3super? When is the transition 0 —*4 actually possible for the conditions of fig. 3? General relations, which follow from the conservation laws and Maxwell’s equations, are insufficient to answer these questions. Ionization at the shock front should be considered in explicit form, that is, we have to choose a physically adequate model for ionization. It is important here that non-equilibrium ionization is, as a rule, essential for shock waves. In other words, it is insufficient to assert that the gas is ionized when passing through the shock front, it is necessary to determine what processes are responsible for ionization, and in which part of the shock front ionization becomes essential, and so on. Before considering processes of ionization and its structure and deriving an additional relation for answering the above questions, let us find out how many boundary conditions are, in general, necessary to formulate and solve the shock wave problem. -+

—

—~

—~

—~

—*

~

4. Evolutionarity conditions and the number of boundary conditions for ionizing shock waves Let us consider the restrictions imposed by stability requirements on possible structures of ionizing shock waves in a magnetic field. Let W = (j5, ~ O~,~, S, fl~,H 2,...) be a vector, which characterizes a complete set of the variables of the problem (in this case these are the hydrodynamic variables and the transverse components of the electric and magnetic fields). In the coordinate system where the shock front is at rest its steady-state structure is given by a certain function W = Wo(x). The conventional method of studying stability lies in applying a small perturbation tW W=W0(x)+e~e’°’ 1(x), e—*0

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

20

to the solution Wo(x). Substitution of this expression into the equations, which describe non-stationary gas flow, yields, in first approximation in ~, a system of linear equations in the components of the vector W1(x). Solving these equations under certain boundary conditions (they may correspond, for instance, to incoming waves of a given amplitude or to a localization of the perturbation near the shock, i.e. W1(±cc)= 0, etc.), we find the spectrum of eigenvalues of the problem, i.e. a set of frequencies {wj. If all Im w, 0, the shock structure Wo(x) is stable with respect to perturbations of this kind, otherwise it is unstable. In practice, however, the determination of the spectrum of eigenvalues {~,} is a very complicated problem even for a simple form of the function Wo(x) which is, incidentally, not always known*. Let us consider the shock front as a discontinuity surface of zero thickness, that is, assume that its width ~i and the time T needed for the gas to flow through the shock are much less than all the characteristic scales of the problem. This refers, in particular, to frequencies and wavevectors of simple linear waves, which may exist upstream and downstream of the shock front wr~1 and

(4.1)

k~i<<1.

But in this case we eliminate dissipation and dispersion and the problem includes no parameters with the dimensions of length and time, so that the spectrum of the perturbations, which corresponds to the given boundary conditions, cannot be determined. At the same time, we can study a linear response of the shock to small perturbations, because such perturbations in a homogeneous medium, both upstream and downstream, can be represented as a superposition of a finite number of linear simple waves. The stability condition thus obtained for the shock front, considered as the zero thickness discontinuity surface, is known as the evolutionarity condition of a shock wave [14, 17]. In this case the perturbation takes the form W— Wo(x)

=

~

e0W,,f~[x (c. —

(4.2)

+ i3~)t]

where i5~is the flow velocity in a given equilibrium state and summation is performed over all types of waves, which may propagate against the flow background (different signs of the phase velocities C~in the gas-fixed coordinate system, i.e., different terms in the sum, correspond to waves propagating in opposite directions). Here f~is an arbitrary scalar function, which varies over a scale larger than ~, En —*0 is the amplitude of the wave of a given type, and the relations between the components of the vector W,. are uniquely determined by the wave type. In the zeroth-order approximation in & = co’r and o~ = ku = wr/(c~+ i5~)(w is the perturbation frequency) the system of equations describing the shock structure is reduced to a system of algebraic relations boundary conditions at the shock front which connect the perturbations upstream and downstream. It is important here that the parameters ~ and Ox should be small, i.e., that different scales of length and time should exist. From the standpoint of the “external” scales A and 1/w the shock front is an infinitely thin discontinuity, so that a perturbation of the flow parameters on one side of it is immediately transferred to the other side, and the values of these parameters on both sides of the discontinuity are related by the boundary conditions. From the standpoint of the “internal” scales zi —

*

—

In certain cases the character of the spectrum {w,} can be determined from rather general considerations [18].

MA Ltherman and AL Velikovich Physics of ionizing shock waves in magneticfields

21

and r the perturbations are infinitely slow small variations of the equilibrium states far upstream and downstream (x = ~ cc), and the steady-state shock structure is rearranged in accordance with changes in the boundary conditions. Waves, which may propagate upstream and downstream, can be divided into incident waves and waves leaving the shock front. Upstream, the incident (from x = — cc) waves are those, for which the phase velocity, with due account for the wave transfer by the flow, c~+ i3x~is positive; downstream they are those waves, for which the phase velocity c,, + i3x 5 is negative. All other waves are going away from the shock front. A special case is possible where the flow velocity upstream or downstream is equal to one of the characteristic velocities. For instance, switch-on and switch-off MHD shock waves, as well as Chapman—Jouguet ionizing shock waves are special ones. The theory considered here cannot be directly applied to the special shock waves, because the parameter Ox cannot be small, as a matter of principle, if c,, = — i3~. The initial unperturbed flow contains a discontinuity between two equilibrium states. The amplitudes of the waves upstream and downstream are related by the boundary conditions. The perturbations are a linear combination of the incident and outgoing waves. The incident amplitudes can be specified arbitrarily, these waves correspond to perturbations, which arrive at the shock front from ±cc, For the amplitudes of the waves going away from the shock front we obtain a system of linear equations whose number is less by one than the number of boundary conditions at the shock (it will be denoted by j). Writing the continuity equation (2.2) in the form 15o(~Xo—D)=,5k(~Xk—D),

where D is the shock front velocity (D = 0 in the shock-fixed coordinate system), we obtain 7xo 1o 150 t5D = OPk V,~k+ &5~k OD, — Pk

(4.3)

0150’ Vxo+ &

where 6,5~,t5i5~ (s = 0, k) are small changes in the density and velocity upstream and downstream. Each of these quantities is a component of the vector (4.2) and represents a sum over different waves. Eliminating the shock speed perturbation t3D with the help of (4.3), we come to a system of (j — 1) linear equations, which relate the amplitudes of incident and outgoing waves. Let us note that the coefficients of this system of equations do not depend on the frequency w, i.e. these equations alone do not permit one to determine the spectrum corresponding to given boundary conditions and to reach conclusions about the shock structure stability. (It can easily be seen that a continuum {~}, Im w = 0, ~ hr corresponds to these perturbations; frequencies with Im w ~ 0 are also formally possible, but the corresponding solutions are not travelling waves of a finite amplitude for — cc < x < + cc.) The existence and uniqueness of the solution of the problem of the interaction of a shock front with small perturbations in the form of sufficiently long waves incident onto the shock front from ±cc, that is, the unique solution of the system of equations for the amplitudes of the outgoing waves, is precisely the evolutionarity condition of a shock wave, i.e., the necessary condition of its stability. If the system of equations does not have a unique solution, non-trivial solutions do exist for zero-amplitude incident waves, i.e., the shock front may spontaneously radiate waves, and hence it is~absolutelyunstable. If the system of equations is overdetermined (there are no solutions), the shoëk response to a small perturbation is not small: in this case the shock wave may spontaneously decay into several discontinuities of a finite amplitude. The shock wave evolutionarity requires that the rank of the matrix A for the coefficients of the

a

MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

22

unknown amplitudes appearing in the system of (j 1) equations in the m amplitudes of the outgoing waves should be equal to m: rank A = m. It is rather difficult to verify this condition, and the analysis is usually confined to a simpler requirement: the number of equations in the amplitudes of the outgoing waves should be equal to the total number of the outgoing waves (i.e. the matrix A is assumed square and non~degenerate*).One has to verify, however, that the structure of the matrix is not a block one, i.e. that the system of equations for the amplitudes of the outgoing waves cannot be decomposed into several independent systems of equations, as may be the case where the perturbations of certain variables (and they alone) are transferred by waves of certain types (and only by them). If this is the case, the evolutionarity condition should be formulated separately for each set of wave types. In such an approach the evolutionarity study appears to be very simple: one has not even to write down a fairly cumbersome system of equations for the amplitudes of the outgoing waves. What is needed is only to count the number of outgoing waves and to compare it with j 1 (let us remind that j is the number of relations at the shock). An ionizing shock wave in a magnetic field is characterized by nine variables. In an unperturbed neutral gas the upstream flow is described by gas-dynamic equations in the variables j5, 13, S and Maxwell’s equations in the transverse components of the fields E and H. Nine types of waves may propagate in this region. Two acoustic waves, one of which propagates in the laboratory coordinate system upstream and the other downstream, transfer perturbation of jS and z5~. Perturbations of t3,~, i5~ and S (together with perturbation of ~5corresponding to the entropy wave) are transferred independently along with the flow. Two electromagnetic waves transfer perturbations of E~and H 2 (one upstream and the other downstream), two other waves transfer perturbations of E2 and H,. Thus, the upstream outgoing waves include: two electromagnetic waves propagating upstream (and also one acoustic wave for subsonic ionizing shocks). The downstream flow is described by the equations of magnetohydrodynamics, i.e., the same nine variables should satisfy seven differential equations and two algebraic relations, which express the components of the vector E1 in terms of ü and H1, because in the downstream conducting region —

—

E+xli=o. Therefore, seven types of waves may propagate downstream. Perturbations of the entropy and the corresponding perturbations of the density are transferred, as 15xk before, flow. Two fast and andalong ±Cs +with 15xk, the respectively) transfer two slow magnetosonic waves (whose velocities are ± Cf + perturbations of j5, v, i5~,14. Two Alfvén waves with phase velocities ±Ca + 13xk transfer perturbations of v 2 and H2. Among these waves, four propagate necessarily away from the shock front (the entropy wave and three waves whose velocities are directed downstream). The total number of outgoing waves depends on the relation between the downstream velocity and the characteristic velocities, i.e., on the shock type. It is a simple matter to verify (see section 3) that the type k of an ionizing shock wave is defined as k

1 + (the number of downstream characteristic velocities, which exceed v~k).

Thus, for an ionizing shock of the type *

k

we have (8—

k)

(4.4)

types of waves, which go downstream away

One of the rare degenerate cases, which are of interest, is a switch-on MHD shock wave, which is, as was already mentioned, a special one.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

23

from the shock front. Taking into account waves going away upstream from the shock front, we find the total number of outgoing waves for a k type ionizing shock: The number of outgoing waves

=

Till10— —

k k

for supersonic shocks, for subsonic shocks.

(45)

Let us consider equations in the amplitudes of the outgoing waves for the case of supersonic ionizing shocks. Upstream, there are two outgoing electromagnetic waves, polarized perpendicular to each other. Let E1 denote the perturbation amplitude for FI~in the first wave and E2 the perturbation amplitude for H2 in the second wave. We find then for the waves going upstream away from the shock front: 0140

=

0E20 =

E1,

OH20 =

—

=

E2.

(4.6)

In the downstream region the perturbation amplitudes for 14 in the fast magnetosonic waves propagating downstream and upstream are denoted by F÷and F, respectively, and in the slow magnetosonic waves by S÷and S; the perturbation amplitudes for H2 in the Alfvén waves are denoted by A+ and A, and, finally, the entropy perturbation amplitude in the entropy wave is denoted by S~. We have then for the waves downstream: LP(F+, F, S÷,S, S0),

OPk =

OSk=So,

OVzk

(4.8) F_, S+, S.),

(4.9)

LY(F+, F, S+, S),

(4.10)

&5~k= LX(F+,

=

(4.7)

= L2(A+, A_),

(4.11) (4.12)

=

A+ + A,

(4.13)

where L~,L~,L~,L2 are linear forms whose coefficients can easily be found. Yet these coefficients are not essential here, it is only important that they all are not zero. Since we are interested only in the matrix A, we may not retain in the system of equations for the amplitudes of the outgoing waves the terms, which describe perturbations corresponding to incident waves. Let us linearize the relations at the shock, taking into account that only four electromagnetic variables (4.6) are perturbed upstream, and, generally, all the seven variables (4.7)—(4.14) downstream. The amplitudes F±,A±,S÷and S0 always correspond to outgoing waves, but it depends on the shock type, which of the amplitudes F., A, S. correspond to these waves. Performing simple calculations, we obtain from the conservation law for the x-component of the momentum: H~0E1+Hz0E2+ L1(F+, F, S÷,S, S0)= 0;

(4.14)

M.A. Liberman and AL. Velikovich. Physics of ionizing shock waves in magnetic fields

24

from the conservation law for the y-component of the momentum: —H2E5

+

L2(F±,F~,S+, S_) = 0;

(4.15)

from the conservation law for the z-component of the momentum: —H~E2+L3(A~,A_) = 0;

(4.16)

from the energy conservation law: 2i3~0H,0E1+ (i3x0.A~zo—

i3201 )E2 + L4(F+, F-, S+, S—, S~)= 0;

(4.17)

and from the continuity equations for the electric field tangential component: —E2+L5(A±,A_)=0,

(4.18)

E1 + L6(F±,F_, S±,S_, S0) = 0.

(4.19)

As before, L1, , L6 are linear forms whose particular form is not essential. As can be seen from equations (4.14)—(4.19), this system cannot, in general, be decomposed into systems of equations for groups of variables. The same is true for subsonic ionizing shocks. The basic system of relations at the shock front yields six equations for the amplitudes of outgoing waves. Substituting this number into (4.5), we find: 10— k = 6, i.e. k = 4. This means that the evolutionarity condition holds for type 4 supersonic ionizing shocks, provided that no additional relations exist, which connect the upstream and downstream values of the variables. A two-parameter family of various structures, with the corresponding two-dimensional region on the plane of the parameters (H20, E2), should exist for a given shock wave velocity. If the upstream and downstream variables are connected by one relation, in addition to those used in deriving eqs. (4.14)—(4.19), the number of equations for the amplitudes of outgoing waves will be seven. We obtain then from (4.5) that type 3 supersonic ionizing shocks and type 4 subsonic ionizing shocks correspond to this case. Here, there is a one-parameter family of various structures for a given shock wave velocity, and a certain curve on the plane (H20, E2) corresponds to this family. Finally, if there are two additional relations, the number of amplitudes of outgoing waves is 8. Type 2 supersonic ionizing shocks and type 3 subsonic ionizing shocks correspond to this case. Here, for a given shock front velocity its structure and the discontinuities of all the variables at the shock front are completely determined by the boundary conditions. This situation of the evolutionarity of ionizing shocks of various types is schematically shown in fig. 4. Since the flow is characterized by nine variables, the total number of boundary conditions (including the additional relations) should not exceed 9, that is the number of outgoing waves cannot be more than 8. This means that type 1 supersonic ionizing shocks and type 2 subsonic ionizing shocks are not evolutionary and hence cannot exist; in section 3 the same conclusion was reached using structural arguments. It is not, of course, by chance that the results obtained due to different approaches are identical; there is a close relationship between the evolutionarity of a shock and the existence of its structure. Let us note an interesting particular case where the matrix A of the system (4. 14)—(4. 19) has a block . . .

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

structure Suppose we consider an oblique shock wave,

25

e v20 = H20 = 0 The~ionly eqs (4 16) and (4.18) contain the amplitudes E2, A÷,A. Two of them (E2 and A+) correspond to outgoing waves. If downstream fi~
(4.20)

5[H~0H22—H20H~2]=0.

Taking into account that H20 = Hz2 = 0, we find with the help of (4.16), (4.18) the additional relation in question: H~2E2 fl~~0(A+ + A) = 0.

(4.21)

—

In this case we have four equations (4.14), (4.15), (4.17) and (4.19) for five amplitudes F±,S÷,S_, So, E1 of the outgoing waves, i.e., another additional relation containing E1 should exist. Let us note that relation (4.21) does not hold for ionizing shocks of types 3 and 4, because oblique shocks of these types are included into 1- and 2-parameter families of skew shocks, respectively, so that the perturbed shock wave is not, generally speaking, an oblique one. Let us now make some general remarks concerning the evolutionarity conditions of ionizing shock waves in magnetic fields. Firstly, we note that the assumption of the possibility of propagating upstream electromagnetic waves is too idealized, if a particular experimental setup or a numerical simulation problem are considered. In fact, during the experiment the electromagnetic wave is reflected many times from the shock front and far end of the shock tube, i.e. the problem is a quasi-stationary one from the standpoint of electrodynamics. This fact may be taken into account if we assume, for instance, a one-dimensional model for a coaxial electromagnetic shock tube with R0~1 ~ ~ R1~,i.e. assume that the gas flows between two conducting planes perpendicular to the z axis, so that v2 = 0 and H2 = 0 [19]. All the conclusions of the present paper can easily be extended to this case, if we take into account that, due to boundary conditions: 1) Alfvén and electromagnetic waves do not propagate, 2) the upstream etectric field is an eigenvalue characterizing the flow as a whole, 3) all the shock waves are oblique and, in particular, any difference between type 2 and 3 ionizing shocks disappears [20]. Secondly, the ordinary evolutionarity conditions may be used, if the upstream conductivity is considered exactly zero, i.e., perturbations of the conductivity (or the degree of ionization) should vanish or decay for x = cc, In the theory this is achieved due to the condition of upstream ionization stability [20]. While studying the evolutionarity of special shocks, let us note that the dispersion relation for waves in the flow w(k) (c~+the 5~)k is, in fact, inthek first term in w(k) for smallin k.k,For special 3~= 0= and term linear vanishes, so the thatexpansion the next of term, quadratic should be shocks into ~,, + account, i taken that is we have to include dissipation explicitly [21]. In this case we shall be interested in Chapman—Jouguet ionizing shocks, which are the limiting case of normal ionizing shock waves. In the gas-fixed coordinate system the dispersion law of “dissipative” waves, which damp rapidly on —

—

26

MA. Liberrnan and AL. Velikovich, Physics ofionizing shock waves in magnetic fields

moving away from the shock front, is w+ik2DO

(4.22)

where D is a positive constant. Two dissipative waves exist for a given w, one propagating in the positive direction (decaying with increasing x), and the other in the negative direction, i.e. downstream. This means that one of the waves is going away from the shock front. Thus, the same number of outgoing waves corresponds both to the Chapman—Jouguet ionizing shock and to a type 3 shock (one of the waves is dissipative), and if the structure of type 3 shock is a limiting case of type 4 structures this shock wave is not an evolutionary one: there are “too many” outgoing waves, they may be radiated spontaneously. Non-evolutionarity of this kind is of a peculiar character. It can easily be seen that the dispersion relation (4.22) corresponds to the diffusion equation. The boundary conditions require, in general, that a slow MHD rarefaction wave should be adjacent, downstream, to the Chapman—Jouguet shock. The “head” of the rarefaction wave represents a weak discontinuity and broadens with time in a diffusive manner, if dissipation is taken into account. Since the “tail” of the Chapman—Jouguet shock cannot be separated from the “head” of the rarefaction wave, the non-evolutionarity implies diffusion broadening, proportional to t”2, of the entire region round the Chapman—Jouguet ionizing shock. In other words, such a shock is not decomposed into other shocks or discontinuities, but its “width” u increases with time as ~hf2, The situation is exactly the same for rotational, contact, tangential and weak discontinuities, which do not have a steady-state width and broaden with time as t”2. In the class of discontinuities they are special ones (the velocity on both sides of the discontinuity coincides with one of the characteristic velocities), and they are also nonevolutionary if dissipative waves are taken into account. This does not mean that the study of such discontinuities is devoid of sense. In a self-similar problem, for instance, the distances between the waves and discontinuities of various types increase proportionally to t, so that the relative width of regions occupied by broadening discontinuities decreases as f112, i.e. these discontinuities may still be considered as “narrow” despite their broadening. —

5.

Precursor ionization. Ionization stability

It was shown above that the boundary conditions for ionizing shock waves cannot be determined without an explicit analysis of ionization at the shock front. The most essential point here is: how does a non-zero conductivity arise, i.e., what is the solution for the ionizing shock structure in the upstream flow? Ionization in shock waves proceeds mainly in the region of shock compression and heating. This does not mean, however, that the gas flowing into this region is perfectly neutral. A wide region of precursor ionization is known to exist upstream of a strong ionizing shock; this ionization is caused by the diffusion of hot electrons from the shock and by the photoionization of the cold gas due to the radiation of the plasma heated by the shock. The induced upstream electric field heats precursor electrons leading to breakdown. As compared to the wide region of precursor ionization, the shock compression front may be considered as a zero-thickness discontinuity surface at x = 0, on which there is a shock of the hydrodynamic variables (velocity, density, temperature, etc.). In the wide precursor region the hydrodynamic variables change little and the degree of ionization a ~ 1. This region is of most interest to us,

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

27

because here the gas becomes conducting, i.e., its conductivity changes from 0 for x -* cc to small, but finite values. In the region where the gas has a finite conductivity the actual dependence of the conductivity (and the degree of ionization) on the x coordinate and other hydrodynamic variables is not very important. Magnetic structures of ionizing shocks may be studied qualitatively in the phase space even if the exact form of this dependence is not known (section 3). In the precursor region the situation is the opposite, to a certain extent. The most essential equation here is the ionization kinetics equation, and it is necessary first of all to determine the degree of ionization as a function of the coordinate. The study of the precursor region is simplified, since for small values of the degree of ionization a all the equations can be linearized with respect to a and the system of equations describing the ionizing shock structure is reduced to one equation, linear in a, for given flow velocity, electric and magnetic fields, etc. Thus, we may obtain profiles of the degree of ionization (and conductivity) in the precursor region for constant shock wave characteristics, which are considered at this stage as free parameters. In the next-order approximations magnetic field compression and variation of the hydrodynamic variables can be found from the above determined conductivity profile, which makes it possible to solve the problem in a self-consistent manner. Let us first consider conditions, under which a steady-state precursor layer may arise upstream. If precursor ionization is caused by photoionization of the neutral gas due to radiation from the shock front, we can write for ,i in (2.17) up to factors, which vary slowly as compared to the exponential dependence, —

ñ=

S~NOO~,JI exp(xJl~h),

(5.1)

where N0 is the neutral gas density upstream, o~,is the photoionization cross-section, ‘ph = l!NoOph ~S the mean free path of an ionizing quantum, S0h is the flux density of ionizing quanta with an energy of hw J radiated from the shock front with the temperature Tk, and J is the neutral atom ionization potential. We have for S~h 2D I J1 S~h 41T2c2h3 j exP[_.--j[l exp(—T)J. —

Here r is the optical thickness of the radiating gas, and its temperature 7’~ J. Since compression of the gas, the change in its velocity, and three-particle recombination at a small degree of ionization can be neglected in the precursor region (x cc), we obtain the following equation linearized with respect to a ~ 1 in the upstream flow: ‘~

—*

ôa

-

9a

a

-

1ionO (11

(IX

,~

—

a 0

5ph

/x\ \4ph/

(5.2)

~

where a 0 = S0h(Tk)/NOi3XO and u~0~= t~~0/(v~0~ Pa) is the characteristic length of shock ionization in the upstream flow. Let us note that the frequency of ionization due to electron impact ~ion(Te), and also the quantity u~0~ o(Te), depend significantly on the temperature of the electrons in the precursor layer, this temperature being determined by the energy balance of the electrons in the upstream induction electric field E~generated by an ionizing shock in a magnetic field. 0

—

28

M.A. Liberman and AL. Velikovich, Physics ofionizing shock waves in magneticfields

The solution of eq. (5.2) with the initial condition a(x, 0) = 0 is a(x, t) =

a

0

( ph”

~ iono)

—

ion0

~‘)i5xot]

—

(5.3)

i}.

ph

It can be seen from (5.3) that a steady-state solution for t

—~

cc

does exist only if (5.4)

L1~on(Teo)‘ph.

This inequality can be rewritten in the form (55)

PphO,

Pa

where = Noo~~hi5Xo. Thus, the physical meaning of condition (5.5)— the existence of a steady-state solution is a restriction on the magnitude of the upstream electric field in the neutral gas, i.e., the absence of electric breakdown. In fact, if (5.5) does not hold, an ionization wave is formed upstream, whose phase velocity, relative to the shock compression front, is ?52O(l0h/u~0~ 1). If precursor ionization is caused by the diffusion of electrons from the shock front, that is, if 2n/ox2, (5.6) ii = D,, a ~ph

—

—

the condition similar to (5.4) can be represented in the form Llion(Teo)

L 0,

(5.7)

where LD = De/i5xo is the characteristic length of electron diffusion. Estimating the temperature of the electrons in the upstream flow, proceeding from the energy balance for the heating of the electrons in the upstream electric field (Es) in the gas-fixed coordinate system, and energy losses for shock ionization (Te0 J), we obtain 2 E*2 ~ion(ieo)J. (5.8) e mepeo ‘~

Using (5.8), condition (5.6) can be written in the form E~EJ(1+PphIPa)”2~E*,

(5.9)

where =

(Jme~eova/e2)”2.

Thus, a steady-state solution for ionizing shock waves in a magnetic field can exist only, if the upstream electric field induced in the neutral gas is less than the breakdown field, determined by condition (5.5) or (5.9). The condition expressed by inequalities (5.5), (5.7) and (5.9) may naturally be called the “condition of ionization stability of the neutral gas in the upstream region” [7].

MA. Liberman and AL. Velikovich, Physics ofionizing shock waves in magnetic fields

29

In a magnetic field the main role in the formation of a precursor layer is played by the photoionization of the gas due to radiation. Since such a layer is especially important for the formation of the structure of ionizing shock waves, let us consider its structure in more detail. Suppose that a shock wave radiates ionizing quanta like an infinite plane surface at x = 0. Denoting the spectral density of radiation of e(w), we obtain for the number of ionizing quanta in the range (w, w + dw), absorbed per unit time per unit volume of the upstream flow at a distance x from the shock front ~~2~rce(w)E(x\~ dV hWl~h

510 lph)

2\

where the cross-section of photoionization of neutral atoms with an ionization potential J is

f

0

for

h~’.o
0ph = const. for /1w J. The exponential integral in (5.10) is defined as usual: —

1~ph

E~(z)=Jeztt_n dt.

Integrating (5.10) over the frequency yields the following expression for the density of photoelectrons produced in the precursor layer per unit time: pie = jo(D)

E

(5.11)

2(” X/lph),

where

jo(T)

/1

f

e(w)dw Ct)lph

(5.12)

f/It

The particular form of the radiation spectral density depends both on the kind of gas and on experimental conditions. It determines the constant factor jo(T) in the expression for the rate of photoelectron production in the precursor layer, but this is not essential for the problem considered. In particular, if the shock front radiates like a black body with a temperature equal to the downstream temperature Tk (the index k characterizes the shock type), then e(w) = eo(w, Tk) =

2

3

4ir c [exp(hoi/Tk)—

1]

.

(5.13)

Substitution of (5.13) into (5.12) yields for Tk <
(5.14)

MA. Liberman and AL. Velikovich. Physics of ionizing shock waves in magnetic fields

30

(Note that the condition Tk J should hold for the theory to be valid in the approximation Pm ~ 1.) The factor R in eq. (5.14) is introduced to take into account the finite optical thickness of the radiating shock front and to correct for the difference between the integral radiation density and the corresponding radiation density of the black body. Had the radiating layer optical thickness been taken into account in greater detail, the form of eq. (5.14) would have changed more significantly. However, for a steady-state problem such a detailization is almost impossible and is not necessary. For instance, for a shock wave ahead of a piston the thickness of the radiating layer, i.e. the region between the piston and shock front, depends on time (in a self-similar problem, e.g., it grows linearly). Taking the optical thickness phenomenologically into account with the help of the factor R can be substantiated rigorously, if during the time of interest the radiating layer remains either optically thick (R 1) or optically thin (R ~ 1). A strong source of ionizing radiation moving along with a shock, e.g., the current sheet of a magnetic piston in shock wave installations, can be taken into account by assuming R 1. Let us consider how the ionization equation is solved in the precursor layer. Linearization of (2.17) for a ~1 and x-*—cc yields ‘~

~‘

~

dx

(5.15)

N0v20

lphJ

For x cc the quantity K = ~ ~a)/Z320 is a constant depending on the upstream electric field. The solution of (5.15) satisfying the boundary condition a(— cc) = 0 exists for —~ —

—

K1Ph<1

(5.16)

and is of the form a(x)

Ce~(2

—

jo(f’k) OVxO

{KlPhE2(—-~-)+ ph

ph

E1(_~~_)e1(2Ei[~_~~_ (1— KlPh)]}~ —

ph

(5.17)

ph

where C is an arbitrary integration constant. The sign of the coefficient K determines the possibility of multiple production of electrons in the upstream flow. If K <0, the electron density perturbation in this region decays with time (in a steady-state solution it decays with increasing X); if K > 0, the perturbation grows (increases with x). Thus, electrostatic breakdown takes place for K> 0, and in the upstream flow with the electric field, corresponding to K >0, the gas is unstable relative to small perturbations in the electron density. If we confine ourselves to the situation where the equilibrium state of the gas for x cc is stable (i.e. K <0), then the condition a(— cc) = 0 implies that C = 0 should be taken in eq. (5.17). The physical meaning of the latter requirement is quite clear: it is a shock front, which has to be the source of ionization. A particular solution with C 0, increasing as x cc, describes the decay of ionization (with increasing x) produced by an external, i.e., different from the shock front, source of electron density perturbations. Assuming C = 0, we require the absence of ionization in the upstream flow due to the action of external sources. Strictly speaking, we would have to confine ourselves to the range of pre-breakdown electric fields (K < 0), since the equilibrium state instability for x = cc corresponds to stronger fields, so that the phase velocity of small perturbations of ionization in the upstream flow may take on any value (even an infinite one), depending on the perturbation profile, and the structures of ionizing shocks appear to be -‘~ —

—*

—

—

MA. Liber,nan and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

31

non-unique even for given boundary conditions [7,22]. Nonetheless, under certain experimental conditions steady-state ionizing shocks may exist for a higher electric field, so that K > 0. The fact is that if the characteristic time of the development of an electron avalanche in the electric field exceeds the time of the shock propagation in the installation, the time for the gas breakdown due to electric field is insufficient [22]. The ionization wave, induced by the shock wave and moving together with it, is essential here; its phase velocity is determined by the perturbation profile of the degree of ionization in the upstream flow, which arises due to photoionization. There are no other sources of ionization, under actual experimental conditions, that could produce a comparable perturbation of a (let us note that this “small” perturbation is fairly large, as a rule, 10~—10~ such a considerable ionization cannot be a result of accidental circumstances). In order to include these shock waves into consideration, one has also to take here C = 0, thereby distinguishing the shock front as the only source of ionization*. Rewriting (5.16) as a restriction on the upstream electric field, we obtain the conditions for ionization stability in the precursor region in the form C =0 and KIPh <1.

(5.18)

Let us note that in the framework of the model considered, in which the precursor layer is singled out in the shock structure, the ionization stability condition is exact, and any modification of the theoretical model does not change the meaning of the ionization stability condition in the form of (5.18).

6. The additional relation and magnetic structures of shock waves The selection of the precursor ionization region in the structure of an ionizing shock wave is arbitrary, to a considerable extent. It may be said that the precursor ionization region occupies the whole portion of the shock front for a (x) 4 1. The ionization structure of this upstream portion of the shock front is determined by the solution of the linearized ionization equation (2.17) and is given, for ionizing shock waves in a magnetic field, by eq. (5.17), which takes into account both photoionization and ionization due to the Joule heating of the electrons. This selection of the ionization precursor region considerably simplifies the problem and makes it possible to obtain a solution in closed analytical form. In the general case, where such an approach cannot be used, we come to a fairly complicated multi-dimensional eigenvalue problem for a system of integro-differential equations. The solution of this problem, which implies simultaneous solution of eqs. (2.20) and (2.21) together with the ionization equation (2.17) and the equation for the transfer of the ionizing radiation, can hardly be possible in the general case. Certainly, the above condition of ionization stability is always valid. Let us divide, arbitrarily of course, the shock front into the regions of precursor ionization and shock compression as follows. If the shock structure contains the isomagnetic shock, its beginning will be Let us note that this situation is similar to that in the theory of burning with an analogous problem of ignition of a steady-state combustion wave for x = —w (the so-called “cold boundary problem”). If the rate of combustion reactions in a cold mixture is assumed to be arbitrarily small, but finite (it seems natural at first sight), we obtain arbitrarily long structures of waves of burning, which proceed arbitrarily slowly, and the burning rate remains undetermined. Of course, solutions of this kind do not have a physical meaning, because the initial assumptions of the problem are not actually valid for them— first and foremost the assumption that the problem is one-dimensional, To obtain a physically reasonable result, one has to employ a model, in which the reaction rate is exactly zero up to a certain temperature T* characteristic of burning; then the burning rate may be calculated and is in excellent agreement with experiment [5].The idea of this model can alsobe formulated as follows: the combustion wave itself is the only source of the heat, which produces combustion.

—

MA. Liberman and AL. Velikovich. Physics of ionizing shock waves in magnetic fields

32

considered as a natural boundary between the precursor region, to which the supersonic part of the magnetic shock structure belongs, and the shock compression region. In the case where the shock structure does not contain the isomagnetic shock, the shock front may be divided into the precursor ionization region and the shock compression region in any physically reasonable way. For instance, we may take as the end of the precursor ionization region a point, inside the shock front, which corresponds to the maximal gradient of the density or temperature, or a point, at which the heat exchange due to collisions of electrons with heavy particles is comparable with the Joule heating of the electrons, etc. Suppose that the shock front has been divided into the precursor ionization region and the shock compression region, i.e. a point a0 is selected on the integral curve coming out of the point 0, such that all the points of the integral curve from 0 to a0, whose entropies are less than S(a0), may belong to the precursor region. Using the condition a 4 1, we can take* in the precursor region =

.~0/a,

~

=

const.

(6.1)

Employing eqs. (2.20) and (2.21), let us write down the following equality: 2+ (dH~)2 ]~1/2 62 dx f~(E~+H~ (dH~) 2+ 2H~(v~ l/M~ 2J (.) 0/M~0) 0)(E~+ H~0/M~0)+ H~(v~1/M~0) Substitution of expression (6.1) for il into (6.2) and integration yields the following equation, which contains the electric field E. as a parameter: — -

-

-

0

p

ff (dH~)2 + (dHj2 J l(Es + HyoI.M~o)2+ 2H~,(v~1/IvI~

1/2

2J

—

0)(E.+ H~0IM~0) + H~(v5 1IM’~0) —

—

1

—

4’o

~ d ~ a X) X, ‘~

(6.3)

where H 5 denotes a two-dimensional magnetic field vector (Hp, Ha). The integration on the left-hand side of (6.3) is performed along the integral curve of the system of equations (3.13)—(3.14) from the point 0 to the point p, the end of the precursor region on the integral curve. The point p lies between 0 and a0, if the structure of the shock front contains the isomagnetic shock, and coincides with a0, if there is no isomagnetic shock. Let us note that there is a functional relationshipbetween H~and H~along the integral curve, and the variable H5 (or H~)may be chosen as the integration variable in (6.3) for a portion of the integral curve where dH5/dx~0 (or dH~ldx~ 0). In eqs. (6.2) and (6.3) v~is assumed to be expressed in terms of H5 and H~on the same part of the sheet of the surface F(v~,H~,H~)= 0 where the origin 0 lies. The integral on the right-hand side of (6.3) can be calculated by virtue of (5.17)

J

0

-k--LI10

a(x) dx

~k(MO, Mao, H50, E~) jo(Tk)1Ph g(Klh)

(6.4)

0V~0 JO

“Expression (6.1) for 4 is obviously exact for x -. — where a -.0. It is a fairly simple matter to take into account the dependence of 4 on compression, electron temperature, etc. [7],yet in this case it would mean too high an accuracy, because the value of ~ varies within one order of magnitude throughout the precursor ionization region.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fleu,c

where 11ln(1—~) 2C ~2 ~3 is a function of the downstream plasma temperature only, For fixed values of M0, Mao, H50 and E5, that is, it depends solely on the shock type, in accordance with the singular point type k = 2, 3 or 4. Since the ionizing radiation intensity increases with downstream plasma temperature, it follows from (3.19) that 4k

(6.5)

4~2~3~4’4.

Let us now show how the above formulated problems may be solved with the help of relation (6.3). For a type 2 shock the evolutionarity condition implies, as was already mentioned, two additional boundary conditions. It was shown in section 3 that shock waves of types 2sub and 2super are plane-polarized, i.e., the first additional condition is H~ 0= 0 (see also (4.20) and fig. 4). The second additional condition, i.e., the condition for the electric field magnitude is given by relation (6.3) for k = 2 and H5(,p) = H5(2). Denoting, for brevity, the left-hand side of (6.3) by ~Ik(Mo,Mao, H50, E~,p), we obtain for a type 2 shock the following equation, which determines E5: =

4.2(E5).

(6.6)

In the case of a type 3 supersonic shock the evolutionarity conditionsingular implies point one additional 3sub type shock wave, the downstream is a saddleboundary (see fig. condition (see the fig. only 4). If integral this is acurve along the entropy growth direction comes into this singular point. 4), and hence The point a, the beginning of the isomagnetic shock, is then the only point of the integral curve, leaving the supersonic sheet (H 5, H~)at the point 0, which is transferred by the isomagnetic shock into an 3sub~ The only additional relation is in integral lower coming into the point this casecurve givenon bythe (6.3) for k(subsonic) = 3 and psheet = a~. Similarly, for a supersonic shock wave of type ~ the only additional relation required is given by (6.3) for k=3 andp= a 0.

For a subsonic type 3 shock wave the evolutionarity condition implies two additional boundary 3sub is a saddle (see fig. 4), the first additional boundary condition is conditions. Since thethe singular point that the point 0 on subsonic sheet should lie on the only (in the entropy growth direction) integral curve, which comes into point 3sub, i.e., such a shock is a plane-polarized one: H~ 0= 0. The second additional boundary condition is given by (6.3) for k = 3 and p = a0. The evolutionarity condition for a supersonic type 4 ionizing shock wave implies no additional boundary conditions, other than relations which follow from conservation laws and Maxwell’s equations. Limitations which are possible in this case and which are in the form of inequalities, are related to the following fact. First, eq. (6.3) for k 4 must have a solution, which determines on the integral curve, coming out of the point 0, the beginning of the isomagnetic shock, the point a (p = a). The solution does not exist if, for instance, the upstream electric field exceeds the gas breakdown threshold for a given set of the parameters. Second, the isomagnetic shock should transfer the above point a into a two-dimensional region of attraction of the singular point 4 on the subsonic sheet (point 4 is a stable node).

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

34

One additional boundary condition, which is necessary for the evolutionarity of a subsonic type 4 shock wave, is relation (6.3) for k = 4 and p = a0. Let us now show how eq. (6.3) makes it possible to determine what type of a shock transition is realized for a given set of the parameters (e.g., for the case of fig. 2), if this type is not unambiguously determined by the integral curve field pattern. The shock transition type can always be determined uniquely for supersonic ionizing shock waves in the gas-dynamic limit and for subsonic ionizing shock waves. We note that the integrand on the left-hand side of eq. (6.3) is always positive, i.e., ~,(p) increases monotonically, as the point p goes away from the origin 0. Hence, the existence, say, of the solution for a supersonic type 4 shock wave implies that the inequality (6.7) should hold at the point p’ (most remote from point 0) of a portion of the integral curve coming out of 0, which is transferred by the isomagnetic shock into the point 4 attraction region. Then the function ~l’(p)may be equal to çb,~ at one of the intermediate points between 0 and p’ (the equality will necessarily hold if the isomagnetic shock transforms the entire portion O—p’ of the integral curve into the point 4 attraction region). Let us consider, as an example, the integral curves presented in fig. 2. As can be seen a priori, solutions are possible here, which correspond to both structures 0 3super 3super, and 0 i.e., a p’= b a4. Let a0 be the boundary of the shock compression region on the integral curve O’* 0. A type 4 solution is possible if çli(a0)> ~, and the existence of a type 3su~r solution implies that tfr(a0) Since (according to eq. (6.5)), the situation of fig. 2 is characterized by the existence of a solution for a type 4 shock wave if ~(a0)> 4)~,or for a type ~ shock wave if ~1t(a0)= 4~ <4)~,or neither of the solutions exists if 4~ 4!F(a0) < The type of shock wave, for which the solution exists for any set of parameters, can be determined in a similar way. “~

~4>

—~

—*

—~

/3

7. Limiting regimes Let us consider the application of the above general theory to the limiting cases of “weak” and “strong” ionizing shock waves. Obviously, any ionizing shock wave heats the gas considerably, i.e., it is strong in this sense. Let us refine, therefore, that by “weak” we mean an ionizing shock wave whose intensity is still insufficient for going over to the MHD-limit of the theory, i.e., the downstream flow corresponds to one of the MHD singular points, while the upstream flow does not. The limiting case of “weak” ionizing shock waves will be called the gas-dynamic limit of the theory. In the same sense, an MHD shock may be considered as a strong ionizing shock wave, and for the former both the upstream and downstream equilibrium states correspond to MHD singular points. To go over to the gas-dynamic limit, one has to assume the effect of both mechanisms of precursor ionization negligible, i.e., to take formally R 0, E~ cc• Then the right-hand side of eq. (6.3) vanishes in the whole range of variation of the parameters (H~0,E~)admitted by structural restrictions. The zero value of the integral on the left-hand side of eq. (6.3) means that p = 0, i.e., non-trivial shock structures corresponding to supersonic—supersonic and subsonic—subsonic transitions are not possible in this limiting case, and the point 0 lies on the supersonic sheet and coincides with the isomagnetic shock beginning. Thus, in the gas-dynamic limit an ionizing shock wave starts with a gas-dynamic shock, in —~

—*

:

M.A. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

35

which the gas becomes conducting and the downstream state corresponds to one of the subsonic singular points. It can easily be seen that the gas-dynamic limit yields just the number of additional relations, which is determined by the evolutionarity conditions. For instance, in the case of supersonic type 2s~b ionizing shock waves structural restrictions lead to one additional relation H~0= 0 and to the 2sub

requirement that the isomagnetic shock go over supersonic on the subsonic sheet (see section 3).should Together withfrom the the condition p = sheet 0 we exactly obtain tothepoint second additional relation: .11y2 = H 50. Thus, the transverse magnetic field can change neither on the supersonic sheet (the property of the gas-dynamic limit) nor on the subsonic sheet (due to structural restrictions), hence it remains unchanged in such a shock. This conclusion can be applied, in particular, to transverse shocks in the gas-dynamic 3sub limit. ionizing shock waves the only additional condition is that the isomagnetic For must supersonic shock go overtype from the point 0 to the only integral curve on the subsonic sheet, which (for a given sign of H~ 3sub for ~ There are no additional relations for supersonic 0)enters thewaves, singular type 4 ionizing shock the point only restriction is that the isomagnetic shock must go over from the point 0 into the two-dimensional point 4 attraction region on the subsonic sheet. No other types of shock transitions are possible in the GD limit. The questions posed at the end of section 3 are not difficult to answer in the gas-dynamic limit. A type 4 shock wave corresponds to the conditions of fig. 2, and the point a coincides with the point 0. An ionizing shock wave without isomagnetic shock (fig. 3) is not possible in this limit. The limiting case corresponding to a finite breakdown field E~ of the neutral gas with a low photoionization level (R 0) will be called the electrostatic breakdown limit. It can be seen from (5.5) that in this case the breakdown field is determined from Paschen’s law p~= Pa. Under these conditions photoionization creates only primary electrons, while their multiple production in the precursor layer is caused only by the heating in the electric field. Let us remind ourselves that the right-hand side of (6.3) holds for E
4)k

4)k

-~

-,

—*

[(E 5 + ~)2

+ 2Hyo(1

—

+

+

H~0(1

—

1)2]

1/2

(7.1)

It can be seen that in this limit we go over to the MHD régime, if the parameter E~is decreased formally. For E~—*0 the upstream electric field, which is proportional to the left-hand side of (7.1), also vanishes. For oblique type 2 shock waves relation (7.1) gives the law of transition to the MHD régime when Mao increases at constant E~.Let us remind ourselves that if Mao increases infinitely, only ionizing shock waves of this type are possible (see section 3). For these shocks E5 + H50~= E*/Mao.

(7.2)

36

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

Let us note that the transition to the MHD régime, as Mao~*x, occurs slowly: the dimensionless upstream electric field decreases as 1/Mao. Consider now the effect of the precursor ionization intensity. The right-hand side of eq. (6.3) grows infinitely as R The corresponding increase in the left-hand side of this equation is possible only if at least one of the limits of integration approaches the singular point of the integrand. If the interval of integration remains finite under this limiting process, the integral on the left-hand side of (6.3) may not diverge at the upper limit, because the point a0 (the end of the precursor layer) is chosen so that it may —~* ~.

not correspond to the downstream flow, i.e. it may not coincide with the MHD singular point k. Thus, the infinite growth of the left-hand side of (6.3) is possible only if the point 0 approaches one of the singular MHD points which just means the transition to the MHD limit. The integral may diverge at the upper limit only if the trivial transition k k is the MHD limit of a type k ionizing shock wave for chosen values of the parameters. Then both points 0 and a0 tend to the singular point k as R If precursor ionization is taken into account the transition from the gas-dynamic limit to the MHD limit occurs in a narrower velocity range, as compared to that of the electrostatic breakdown limit. This can be explained by a strong (exponential) dependence of the ionizing radiation flux onthe thenarrower downstream will temperature. The larger the numerical pre-exponential factor in the expression for be the transition to the MHD limit. We have from (5.14) and (6.4), within order of magnitude: —*

—~ ~°.

4)k,

4)k(T)=~F(T~~)exp(_~~-),

(7.3)

where F is a dimensionless quantity dependent on the initial state parameters of the gas and shock, and, as a rule, it may be assumed In F 1. The expression in the exponential index also exceeds unity, so that the right-hand side of (7.3) varies rapidly with a slight change in the shock velocity when the change in in F is smali. The dimensionless temperature downstream of a type k shock wave is given by ~“

Tk

=

Vxk (Mao){[1

Vxk

(Mao)]2 M~

2k —

0 ~(Ht —

H~ 3)},

(7.4)

where f~ = 8irpo/R~. The transition to the MHD régime, i.e., the transition from cbk 1 to cIk 1 takes place (for In F ~ 1) mainly in the velocity range, which is determined from (7.3)—(7.4) with a double logarithmic accuracy: 4

~-1kGD~IflFlnlflF) ~ ~°

~
aO<

T’ ~ J/T0 \ k.MHD~lnFlnlnF)~

~‘

75 -

where Tk.GD(MaO) and Tk.MHD(MaO) are functions of the Alfvén Mach number of a type k shock for a fixed initial state of the gas, which determine the dimensionless downstream temperature in the gas-dynamic and MHD limits, respectively. In the formal limiting process J/T0—* ~, ln F—* ~, J/T0 in F = const. the transition from the gasdynamic to the MHD limit takes place in an exact range of Mao values determined by (7.5), and GD and MHD regimes occur outside this range. —

M.A. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

37

8. Transverse shock waves (qualitative theory) Shock waves, in which the magnetic field is parallel to the plane of the front, are called transverse shock waves. As was shown in section 3, they refer to type 2, i.e., only supersonic transverse shock waves may exist and they are determined by two additional boundary conditions. The first of these conditions, as follows from section 6, implies that transverse shock waves are plane-polarized, that is the vectors of the upstream and downstream magnetic fields and the normal to the shock front lie in the same plane. This means that the magnetic field does not change its direction for a transverse shock. The second additional boundary condition should determine the transverse electric field in a shock wave. To use the general relations derived in previous paragraphs, one has to make a limiting process H~—*0, introducing a new notation Mao/Hyo Mao, or to employ a new Alfvén Mach number in the upstream flow, defining it as “~

M~0= 4v~o/H~o. Let us choose the y axis along the magnetic field direction in a coordinate system where the shock front is at rest and write H = H5!H50 ,

v~=

i3,Jv~o,

E~= cE~ii3~0H~0.

The subscripts 0 and 2 correspond to the upstream (x —so) and downstream (x +co) state, respectively. The phase surface equation (3.1) is reduced in this case to the equation of a phase curve on the (vi, H) plane: —*

2— 1)+~~-(H— 1)= 0.

—*

(8.1)

F~(v~,H)—4(v~1)(v~ vGD)+2~~(H —

—

Equations (3.13) and (3.14) yield one equation: dH/d~= Hv~+ E 5.

(8.2)

The condition of a zero electric field in the downstream ionized flow implies that H2v~2+E~=0.

(8.3)

In the gas-dynamic limit (a “weak” shock) the magnetic field remains unchanged, i.e., H2 = 1. Thus, the boundary value for the electric field is ES=—v~2=—voD.

(8.4)

Since for all supersonic ionizing shock waves, including “weak” ones, M0 ~ 1, we have: VGD = (M~+ 3)/4M~ ~, so that the upper boundary of the dimensionless electric field in a transverse ionizing shock wave is: E~= The upstream electric field vanishes in the MHD limit. Taking into account that — ~.

M.A. Liberman and AL. Velikovich, Physics of ionizing shock waves in magneticfields

38

H = 1 and v~= 1 as x

-+

—

so,

we obtain from (8.3) the lower boundary for the electric field in the

shock-fixed coordinate system: E~=

—

1.

Thus, E. values in the range (8.5)

correspond to evolutionary transverse ionizing shock waves. The value of v,~2for a given E~can be determined from (8.1) for H = EJv~,and the root of the equation obtained does exist in the range v.,~2 1 provided relation (8.5) holds. Let us note that the equality E5 = 1 is valid only for Mao> 1, and the corresponding solution is an MI-ID shock. If Mao < 1, the null solution (i3~2= 5~o,T2 = T0, etc., line “null” in fig. 7) corresponds to E. = 1. This solution, in itself, cannot, of course, describe an ionizing shock wave. Yet, ionizing shocks are possible, which are arbitrarily close to it, because for M0 ~ 1 the temperature jump T2/T0 M~(1 v~2)across the shock front may be sufficiently large for any v~2< 1. A typical form of the phase curve (8.1) and the magnetic structure of a transverse ionizing shock wave on the phase plane (vt, H) are presented in fig. 5. As can be seen, the whole magnetic field compression at the shock front takes place upstream of the gas-dynamic shock, if the shock structure contains an isomagnetic gas-dynamic shock, i.e., if M2 < 1. In this case a particular form of eq. (6.3) for a transverse shock wave will be —

—

—

—

~‘

H~(E~,M~o)

J

0

dH if Hvx(H;Es,Mao)+E5 LI10 j a x)dx,

8.6

where v~(H;E5, Mao) is the larger root of eq. (8.1) which is quadratic in v~. A particular choice of the point H~(E~, Ma0) is not essential here, it is only important that it should not coincide with the point 2 for the integral on the left-hand side of (8.6) to converge at the upper limit. If the shock structure contains an isomagnetic shock, we should take H~= H2, as was mentioned in section 6, and the integral in (8.6) will converge. The region of variation of the parameters, which corresponds to the isomagnetic shock in the shock front structure, is shown on the (Mae, E) plane in fig. 6. The boundary of this region is determined by eqs. (8.1), (8.3) and the condition M2 = 1 [23].As can

A

_

/ Fig. 5. The magnetic structure of a with an isomagnetic shock (M2 < 1)

/,~

transverse ionizing shock wave

zHM

/

2

2.763

Al

Fig. 6. The region between the rays GD and MHD (E = 0) in the (Mao, E) plane corresponds to transverse ionizing shock waves. The isomagnetic shock is absent in shock waves whose parameters lie below the sonic line M2 = 1.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

39

be seen from fig. 6, the isomagnetic shock is always present in the shock front structure of a transverse shock wave for Mao > 2.76. Using (6.4) and (5.14), we can rewrite the right-hand side of eq. (8.6) in the form 4i2(E0, Mao) = Ff(4-) g[~ ~ (i

—

i~~)],

(8.7)

where F—R

ir

~ Fl

2e2J3 a

C PfleNOVTeUeOffph

~ is the cross-section of elastic collisions between electrons and atoms, VTe is the thermal velocity of the electrons. The argument of the function g(~)in (8.7) is expressed in terms of the electric field E = (1 + E~) Mao and the dimensionless breakdown field E~,and A is a parameter of dimension length, which takes into account electron diffusion losses. Let us remind ourselves that in accordance with (5.9) or (5.16) the electric field E~ corresponding to the “breakdown” is determined by the condition KlPh = 1. Let us now analyse the character of the solutions of eq. (8.6). In the gas-dynamic limit, i.e., for E 5—+ H2—* 1 too. Denoting the left-hand side of eq. (8.6) by cli’2(Es, Ma0), we have 0e0

—~,

cli2(E5, Mao)

—

~(H2 1) —*0

for H2

—

In the MHD limit, that is for E5—* .I’2(E~,Mao): 1, Mao>1)

—

1

--*

1.

(8.8)

(E —*0) and Ma0> 1, we have the following asymptotic behaviour for

M~.0—1 ln (M~0 1)[H~ 1, Maij)

—

h1~

(8.9)

Thus, for Mao> 1 the function cli2(E5, Ma0) varies from zero to infinity if E5 varies from to —1, while the function 4)2(E5, Mao) remains finite if E~> ~Mao or grows infinitely at E = (E*IMao 1) if E~< 4Mai3. Hence, the solution of eq. (8.6) does exist and is unique for —~

—

—1

E. ~ min{—~,E*/Mao

—

1}.

(8.10)

If the shock front ionizing radiation is negligible, that is in the electrostatic breakdown limit, photoionization creates only primary electrons, and the upstream ionization, in the precursor region, is maintained at a certain level due to the heating of electrons in the induction electric field. In the case of weak photoionization R —*0 should be taken in (8.7). In this case 4)(E5, Mao)~~* 0 everywhere except the point E5 = (E*/Maø 1) where ~/.~2(E5E*/Mao 1, Mao) so~Thus, in the electrostatic breakdown limit —

—

--*

40

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

the electric field E~in the shock-fixed coordinate system is for Mao
—~

(8.11)

Es(Mao) =

for Mao>M*, where M~= ~ is the dimensionless shock front velocity, at which the breakdown of the upstream neutral gas becomes possible. If this mechanism of precursor ionization is also eliminated, i.e., if E~ ~ is assumed, we arrive at the gas-dynamic limit of the theory. In this limiting case transverse shock waves have only a trivial magnetic structure, i.e., E5 = and H2 = 1, for all Ma0 [12]. On the contrary, in the limit of low —*

—~

breakdown threshold, for E~—*0, we come to the MHD limit where E. = 1 [6]. The case of infinitely strong precursor ionization is also reduced to the MHD limit. In fact, for R so the right-hand side of eq. (8.6) is infinite, so that the solution of eq. (8.6) for E. tends to such a value, at which 4112(E5, Mao) has a singularity, i.e., E5—o 1. The solution of eq. (8.6) for Ma0 < 1 deserves special consideration, because in this case the form of the function cli2(E5, Mao) for E5—* 1 depends significantly on the choice of the point H~(E~, Mao). In this case the integration interval is of the same order of smallness as the integrand denominator, so that a singularity of type (8.9) cannot arise and cli2(E~,Mao) remains finite for E. 1 if the upper limit of integration in (8.6) approaches the singularity not faster than the integration interval converges to a point —

—p

—

—

—* —

0
1*0.

It is obvious, at the same time, that since both transitions to the MHD limit of the theory, R oc and E~—+0, should yield the same results, the division (correct from the physical standpoint) of the shock structure into the precursor region and shock compression region for Ma0 < 1 should provide a singularity in the behaviour of clh2(E~,Ma0) for E~—*—1, which is necessary for the solution to exist. Let us now consider how the inclusion of finite intensity of ionizing radiation affects the rate, at which the solution of eq. (8.6) goes over from the GD limit to the MHD limit with increasing shock front velocity. It was noted in previous papers [6, 24] that the finite intensity of the photoionization of the upstream neutral gas should be taken into account to explain the transition of a transverse ionizing shock from the GD régime to the MHD régime in experiments [9]. It was shown in section 7 that the width of the GD MHD transition, with respect to either the shock front velocity or the dimensionless velocity in Ma0 units, depends on the value of the parameter F, the transition interval usually being rather narrow due to a large value of F ~ 1. For instance, we obtain F = 1.7 x io~R for= typical values a shock wave in hydrogen: J 13.6 eV, 2, 0e0 2, N of the parameters 3, VTe = 108for cm/s. = 10 cm cm 0 = 1016 cm Figure 7 shows plots of the quantity --*

—

10t5

H 52II-1~~o— 1= 1 p2/po

(812)

E~+ V~2 1—v~

2

as a function of Mao for different values of R

.

=

1; 0.1; 0.01 (curves 1,2,3) and R —*0 (curve

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

/‘IHD

12/41

41

____________

0L~!0~T Fig. 7. Transition from the gas-dynamic to the MHD régime with increasing intensity of transverse ionizing shock waves. For curves 1, 2, 3 the ratio of the density flux of plasma radiation from behind the shock front to the density flux of the radiation of a black body of the same temperature is taken to be 1, 0.1 and 0.01, respectively. The curve “breakdown” is plotted in the electrostatic breakdown limit where this ratio tends to zero.

“breakdown”), respectively. The curves of fig. 7 have been obtained from a numerical solution of eq. (8.6) together with eq. (8.1) for j,maca

r;

—

.1..)

,

IVA

*

—

w,

“Itph

_.i

2..’~

1.11

0)/l —1 ionltph — I A(

— ~ —

where ma is the atomic mass and Ca = ‘1yo (4irpo)~2.As can be seen from the figure, in the electrostatic breakdown limit, for R -*0, the shock structure changes from the GD limit (H 2 = H0 = 1) to the MHD limit in a slow, power-like manner, in accordance with eq. (8.11). If the finite photoionization of the upstream neutral gas is taken into account, the transition from the GD régime to the MHD régime proceeds in a more or less narrow range of the shock velocity variation. In this case a large value of the parameter F assures such a narrow range even if the radiation intensity varies considerably (within several orders of magnitude), as compared to the black body radiation. It follows from (7.5) that this GD—MHD transition takes place in the range TGD~lFllF)~Ma0~TMHD~JFILF -~( ~ “-at —1 ~“ ~T/T0 <

.

813

Substituting numerical values corresponding to the conditions of fig. 7 into (8.13), we obtain the following limits of the variation of the dimensionless shock velocity for the GD—MHD transition: 3.67 SMao2~4.36

for R

=

1;

4.00 ~ Mao ~ 4.65

for R

=

0.1;

5.03

for R

=

0.01.

4.42 ~ Mao

S

9. Shock structure of a transverse ionizing shock wave The shock structure of a transverse ionizing shock wave has been investigated in a number of experimental works [9,25, 26,271. The most thorough investigation of magnetic and electric shock structure has been performed in ref. [9]. In particular, the authors of that work have studied in detail the transition from the GD régime to the MHD régime as the shock front velocity increased. The

42

MA. Liberman and AL. Velikovich. Physics of ionizing shock waves in magnetic fields

installation they used an inverse Z-pinch in an external axial magnetic field seems to be the most apt one for studying transverse shock waves. This installation provided a good reproducibility and permitted a reliable resolution of the electric and magnetic fields of the shock front and current sheet of the discharge current due to their orthogonality. In this case the shock wave arises ahead of the current sheet of the discharge current near the Z-pinch axis, and the current sheet itself plays the role of a piston, which pushes the shock wave in the radial direction outwards from the Z-pinch axis. The velocity of such a piston in the above experiments was almost constant with a linear increase in the discharge current, and the shock front could be considered almost fiat. In ref. [9] experiments were performed with molecular hydrogen at an initial pressure of 0.1 and 0.25 torr, and an initial temperature ~ = 300 K. In earlier works shock waves in helium were also studied. In the experiments with hydrogen the authors studied in detail the transition from the GD régime, characterized by a shock velocity of less than 5 x 106 cm/s, to the MHD régime, which started at shock velocities ~ ~ 8 x 106 cm/s. In the gas-dynamic régime magnetic field compression at the shock front was not observed, i.e., H2/H0 = 1, while in the MHD régime magnetic field compression was —

—

maximal, i.e. H2/H0 = /52/~o. Intermediate magnetic field compression degrees, from 1 to /52//50, corresponded to intermediate shock front velocities from 5 x 106 cm/s to 8 x 106 cm/s. The analytical solution of eq. (8.6) together with eq. (8.1), which is necessary to interpret experimental results, is rather difficult to obtain in the general case. Yet fairly simple analytical relations, which give satisfactory agreement with experiment, may be obtained in the electrostatic breakdown limit [22]. In this case we actually neglect the dependence of the neutral gas breakdown field on the photoionization and assume, as an additional boundary condition, that the upstream electric field is equal to the breakdown field, in accordance with (8.11). Using dimensional variables, we obtain for the upstream electric field induced in the immovable neutral gas (9.1) Denoting the threshold value of the breakdown field by E~,we may write the additional boundary condition (8.6), in the electrostatic breakdown limit, in the form (9.2)

calculating E~,one has to bear in mind that the upstream electric breakdown is a quasi-steady process (unlike the Paschen breakdown), i.e., the breakdown condition is determined from the requirement that the electron density in the upstream gas should increase considerably for a time needed for a shock wave to pass through the installation. Denoting the installation size by L, we may write the electric breakdown condition in the form [22]: While

PiVa+PA+.Lln(~),

(9.3)

where is the ionization frequency, ~a is the frequency of recombination losses due to the creation of negative ions, v~is the frequency of diffusion losses at the shock tube walls, and fleO and ~ei are the ii~

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

43

densities of primary electrons and electrons directly ahead of the viscous shock, respectively; here 2 1, so that the upstream conductivity o~(neo)may be assumed to be zero. Rm 41T0(fleo)i5oLIC Substituting the conductivity of a weakly ionized gas a~

~(neo)

=

e2neoreo

where r~= l/Nocrcov-re, into the expression for Rm, one can easily see that for typical shock velocities 10~—10~ cm/s and a degree of ionization a ~ 10_5_106 Rm is also much less than 1, provided the installation size is not too large, L ~ 100 cm. The breakdown threshold is eventually determined from the condition that the ionization frequency should be equal to the maximum term on the right-hand side of eq. (9.3). Under the conditions considered, even slight precursor photoionization leads to the fact that ionization proceeds mainly due to electron impact at a frequency NoVTe[

=

T~fr)]

2(J + 2T~)exp(—J/i’e), where o~(e) is the impact ionization cross-section for an electron energy e, and J is the ionization potential of the atoms. If the production of negative ions with a cross-section o~adominates among electron losses, we obtain for the breakdown threshold

E

Q Il________ —

Po

2

0a0e0

To\~eYa

~1/2 S I

(9.4)

,

/

where Po and T0 are the upstream pressure and temperature, respectively, and Ya =

ln[(0t~)) de

0a L]

E’—J

If the electron losses are mainly due to diffusion at the walls of the installation with a characteristic diffusion length A, then =

J 8 A~ (e2)

1/2

(9.5)

,

where VA

=

~

In the case where E~is determined by a finite time of shock wave propagation through the installation we have —

—

J fNocreoi5o flei\112f 8me \h/4 i~—~ ~ e \ L fleO/ \ITJYL/

E*L——t

,

9.6

44

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

where =

In A ~ln ln A, —

A= ~ \rrm~J

3/2

j

L ds

ia—J

N 0L

Vo lfl(fleilfleo)

3, i5 Assuming N0 = 1016 cm 0 = 5 x 10~cm/s, L = 20 cm and A = 5 cm, we obtain for hydrogen, in accordance with (9.3), (9.4) and (9.5), the following values of the breakdown field: E*a 0.4 V/cm, = 2 V/cm, E+L = 8 V/cm. Hence, for the experimental conditions of [9] we have to take E~ = E~L= 8 V/cm. Let us rewrite (9.2) in the form v2H2/voHo = 1 cE*/50H0.

(9.7)

—

As can be seen from (9.7), the less cE~/f50f~, the more the magnetic field compression approaches the value reached in an MHD shock wave, H2/H0 = v0/v2. In fig. 8 the curve is calculated from (9.7) and the points are experimental values from ref. [9]. As can be seen from the figure, even in the electrostatic breakdown limit calculations for the magnetic shock structure agree satisfactorily with experiment. The results of electric field measurements depend on the position of electric probes and on whether the shock wave reaches its steady-state régime. The fact is that while being formed ahead of a piston as a gas-dynamic shock, in which the magnetic field does not vary, such a shock wave induces the maximum possible upstream transverse electric field (for M0 ~ 1) =

0.75i50Ho1c.

For characteristic values of ~o iO~cm/s and H0— several kOe the induced electric field may reach at the initial moment hundreds of V/cm. In a neutral gas such an electric field intensity leads to breakdown, and the breakdown wave (ionization wave) propagates ahead of the shock front. This

‘a

0

/

0 ~--___-

MILD

~ v/ 4o 60 80 100 /20 /40 c 7cm Fig. 8. The ratio vsHsJvoHo as a function of v~IIois a transverse ionizing shock wave. The curve is plotted by eq. (9.7) for the experimental conditions of ref. [9]; the experimental points are from ref. 19]

MA. Liberman and AL. Velikovich, Physics of ionizingshock waves in magnetic fields

45

phenomenon of breakdown ahead of the front of a transverse shock wave has been observed experimentally in argon [28]. As we alreadymentioned, in a steady-state shock wave the upstream electric field is equal to the breakdown field E~.The downstream electric field in the laboratory coordinate system is (9.8) Taking into account (9.2) and vo/v2 =

4, we obtain for M0 ~‘ 1

3(E~ i50HIC).

(9.9)

If the shock front structure contains an isomagnetic shock, the main shock of the density and velocity takes place, as ~ rule, in the latter one. Since the magnetic field is compressed entirely ahead of the isomagnetic shock, the electric field shock should be observed at this shock [6,8].

10. Shock structure and ionization relaxation It is well known that disregard of energy losses due to ionization of the gas does not affect the qualitative picture of magnetic shock structure. Yet such terms are important for a correct calculation of the magnetic shock structure as a whole. If energy losses due to ionization of atoms are taken into account, the equation similar to (2.26) for a transverse shock wave at the point 2 takes the form 10~a v~2—

1+

[T2(1+ a2)

—

1] +

+ 6f9

2E5

2=

0,

(10.1)

where a2 is the degree of ionization downstream and = fIT0. An additional term corresponding to the last term of (10.1) will appear then in the phase trajectory equation (8.1): ~IOfl

5

2—1)+—---~-(H—1)— 5 2 2E 60b0n Mao

FS(vX,H,a)rm4(vX—1)(vX—vGD)+2 Map (H ~

a=0,

(10.2)

that is, we arrive at the equation of a surface (10.2) in the (vi, H, a) phase space. With the inclusion of energy losses due to ionization the critical Mach number for the formation of an isomagnetic shock in the shock front structure will be M~cr’a’a14[15(1+(?2~2)

a2)

(T)2

1o(1+~)2]_1,

(10.3)

rather than Al’ 2 = 1 [6]. If M2> M2 Cr, the shock structure is completely determined by Joule dissipation, while for M2 < M2 Cr an internal viscous isomagnetic shock will necessarily be formed in the shock structure. Inelastic processes may affect significantly the magnitude of density and velocity shocks, and hence the magnetic field in a shock wave. In an idealized case for M0 ar 1 the ultimate compression of an ideal

MA. Liberman and AL, Velikovich, Physics of ionizing shock waves in magnetic fields

46

gas is

/52//50

=

4. If ionization is taken into account, compression for a shock wave with a structure close

to that of an MHD shock (cE~/i5oHo~ 1) will be 2+8(K+1)(4K+1)1”2 f1~_~_~S+4K + 1(1+±.+5+4K\ V0p28(K+1)l. M~ 2M~ 0 R M~ 2M~0J M~0 ~ v2_

1

(10.4)

Here K cx2J/2T2(1 + a2) is a parameter characterizing the effect of ionization on compression in a shock wave. The effect of ionization is negligible in the case of a weak shock wave for a2 ~ T2/J ~ 1 and for a very strong shock wave, for f/ T2 ~ 1. The former case almost always corresponds to the GD limit, while the latter one corresponds to the MHD limit of the theory. In other words, energy losses due to ionization have to be taken into account in a quantitative study of the most important intermediate case. As the gas density decreases, the parameter K grows infinitely and gas compression in a shock wave may be arbitrarily high. Figure 9 presents the compression w = N0JN2, the dimensionless temperature (9= Ti/TO, the degree of ionization a, and the degree of dissociation ~ as functions of the sonic Mach number for a transverse ionizing shock wave in hydrogen, as calculated in the MHD limit. Energy losses due to both ionization and dissociation of molecular hydrogen were not taken into account in the calculations [6]. The solid line corresponds to hydrogen for T0 = 300 K, Po = 1.5 X 106 torr, j3~= 8irp0/H~= 8.8; the dashed line corresponds to Po 0.1 torr, f3~= 3 x iO~.Figure 10 shows the results of similar calculations for a shock wave in argon [6]. Since the frequency of impact ionization depends exponentially on the temperature, it is convenient to distinguish between two cases: for strong shock waves the characteristic scale of impact ionization is less than the scale of all other processes almost throughout the entire shock front, while for moderately strong shocks the scale of impact ionization is larger than any scale characteristic of dissipative —

OeJ

___

8

20

4o

60

&7

Fig. 9. Jumps of the temperature i9 = T2/T0, velocity = ~ ionizingofshock degree ionization wave in a, hydrogen and dissociation as a function s~ in anof MHD the sonic transverse Mach number M (point 0 coincides here with point 1) for T 0 = 300 K, = 1.5 x 10~6 torr, $~ = 8.8 (solid lines); po = 0.1 torr, ~ = 3 x ~ (dashed line).

~

~‘

20

40

60

~

80

Fig. 10. Jumps in the temperature 19 = 1’2/To and velocityt°~ wand = i3~2!6~0, singly and doubly also thecharged dimensionless ions, pt~t densities and v, ofand neutral the degree atoms of v ionization a downstream of an MHD transverse ionizing shock wave in argon 3, T as functions of the sonic Mach number M1 for N0 = i0’~cm 0= 300 K, $~ = 0.1. Dashed line, the corresponding parameters when energy losses due to ionization are not taken into account; dash-anddot line, when only the first ionization is taken into account.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

47

processes. In the latter case the shock structure is similar to a well known Zel’dovich—von Neumann model for a detonation shock wave. The front of such a shock wave consists of a more or less narrow region of compression and heating (dissipation zone) followed by a much wider region of ionization relaxation, where dissipation is negligibly small, the density, velocity and magnetic field vary slowly, and the temperature may drop due to gas ionization. Though relative changes in the magnetic field and the compression in the ionization relaxation region are small as compared to those in the shock compression region, the total changes in the density and magnetic field in the ionization relaxation region for K ~ 1 are major ones. The presence of the ionization relaxation region in the shock structure is not related to specific features of an ionizing shock wave in a magnetic field. The width of the relaxation region has been measured in a number of shock wave experiments in argon for different initial gas pressures [29, 30], and in mixtures of argon with krypton [31,32]. The rise time r of the ionization in the gas is usually measured in the experiments. Denoting the coefficient of ionization by electron impact as ~ we obtain for the characteristic ionization length -

(10.5)

,

a10~N0v0 whence we find for the characteristic ionization rise time zL,~ v2lvo V2 a10,, A convenient parameter is PoT, which depends weakly on the initial pressure Po (To

=

300 K), (10.7)

= ?0(~2/~o)

a10,,

1/

;~./ ,o3~ // ‘Ii --

1/

6 2~/~/~/~#

8 /2

/0

/2

/8

Fig. 11. Ionization relaxation time for shock waves in argon without a magnetic field. Points, experiment; dash-and-dot various measurements and numerical calculations compiled in ref. [30J;the dashed line is plotted from eq. (10.7).

and solid lines, the results of

:

48

MA. Liberman and AL. Velikovich. Physics of ionizing shock waves in magnetic fields

Figure 11 shows the dependence of p0T on the downstream reciprocal temperature, as calculated when energy losses due to ionization are not taken into account. Here experimental points, dash-and-dot and solid curves represent the results of measurements and numerical calculations [30].As can be seen, even a simple approximate formula (10.7) yields a good agreement with experiment (dashed line in fig. 11).

11. Formation of a magnetic structure of a GD shock propagating across the magnetic field The solution of the problem on the magnetic shock structure in the electrostatic breakdown limit is a simple and convenient approximation. This approximation makes it possible to obtain an analytical solution, though not of a high accuracy, but describing satisfactorily the experimental results. To obtain a better accuracy, one has to take into account the effect of photoionization on the shift of the neutral gas breakdown threshold, that is to solve simultaneously eqs. (8.1) and (8.6). At the same time, it is important to know, while analysing experimental results, whether the shock wave in a shock tube is a steady-state or a non-stationary one. Experimental results for the velocity of a piston and a GD shock (obtained by piezoelectric transducers) are often insufficient to reach unambiguous conclusions. That is why a non-stationary problem of the formation of a transverse ionizing shock wave ahead of a moving piston is of interest with relation to simulation and interpretation of shock wave experiments [7,33]. Suppose that at an initial moment, t = 0, a gas-dynamic ionizing shock wave arises ahead of a piston moving across the magnetic field. The shock front width is of the order ofsothethat atomic mean freefield path Ia, 2 ~ 1, the magnetic does and the corresponding magnetic Reynolds number is Rm = 41T01)la/C not vary throughout the shock front of a GD shock wave. In accordance with Faraday’s law, an electric field is induced ahead of a conducting shock front moving in a magnetic field. For instance, in the case of the inverse Z-pinch we have

2irrE 0

—~~J

H(r’) r’ dr’.

(11.1)

Considering an infinitely thin shock, we obtain for a flat shock front eq. (9.1), instead of (11.1), i.e., (11.2)

For t = 0 when the magnetic field in the GD shock is not compressed, the upstream induced electric field is maximal, (11.3)

Here ü~is the shock front velocity. If the induced field (11.3) is less than the breakdown field, the shock wave remains gas-dynamic for t >0, in accordance with (8.11). If E*(t = 0) exceeds the breakdown threshold, an ionization wave will propagate ahead of the gasdynamic shock front (see section 5). A finite conductivity of the gas upstream of the GD shock front leads to magnetic field compression ahead of it, thereby decreasing the upstream

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

49

induced electric field. This process will continue until a steady-state transverse ionizing shock wave arises whose shock structure contains an internal isomagnetic shock instead of the initial GD shock. In other words, a steady-state ionizing shock wave is formed when the upstream electric field drops to such a value that the ionization wave phase velocity is exactly equal to the shock compression front velocity. To solve the non-stationary problem numerically, let us consider the flow ahead of an ideally conducting piston. Let the piston start moving in a cold neutral gas at a velocity U0 in the direction x >0. The initial unperturbed magnetic field in the gas is parallel to the z axis. At t = 0 a gas-dynamic ionizing shock is formed ahead of the piston; the parameters of the shock (i.e. the upstream and downstream state of the gas) are determined by ordinary Hugoniot relations for a shock wave in the absence of a magnetic field. Equations, which describe a non-stationary flow, represent Maxwell’s equations; considering the problem as a quasi-stationary one (in the electrodynamic sense), i.e., neglecting displacement currents, we obtain from these equations*:

~t

ox

Ox \4inr Ox

~114

J~

equations of continuity, equations of motion and heat transfer for the gas as a whole and the electron component: IllS 19t

OX

M a(Nv) ~

‘

(PVINV2+NT+ flTe+~) z= 0,

(11.6)

ot,,

~

~flTe)+~(~flTe)+

flTe9~ 16’~2

(~~) Q.l~

J~e+T~~ñPh.

(11.7)

(11.8)

The ionization equation (11.9)

~

should be added to eqs. (11.4)—(11.8). Here cr is the gas conductivity, M is the mass of the heavy particles (ions, atoms and molecules), T is their temperature, N is the total density, and O~is the heat absorbed by ions and atoms due to elastic collisions with electrons. For molecular hydrogen the specific heat ratio is y~= ~ and for atomic hydrogen, as well as for ions, Y2 = In the ionization equation the rate of impact ionization and triple recombination is 3, (11.10) ~.

~e’a aenNo—f3en *

In this section we do not use the bar notation for dimensional variables.

:

50

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

where N0 is the density of neutral particles. The constants of impact ionization ae and recombination /3~ are related, in accordance with the detailed balancing principle, by 2exp(_~~). (11.11) 2~ (2~~Te)S/ The values of ae and 13e used in the calculations were chosen according to experimental data available for the coefficient of triple recombination and impact ionization cross-sections with due account for (11.11). The last term in (11.9) describes diffusion losses of electrons (DeIA2)fl where Dc = (irTel8me)112/ueoNo and A is a characteristic diffusion length, which is of the order of the distance between the metallic walls of the installation. The term corresponding to photoionization and photorecombination may be presented in the form

~F=

flph =

a,~.N

(11.12)

2, 0 b~n —

where

I J\ I8Te\112 g 3 ~ b,. ~ ~im~c2Te~0~ 0 J Here o~ is the photoionization cross-section near the threshold h~ J and T~is the corresponding 8irJ2

=

—~-~-

effective radiation temperature defined so that the radiation density at the frequency v

=

f/h is

U(~~)(T~)= U~(T), where U~(T~) is defined by Planck’s formula. By analogy, the effective upstream temperature T~ 1of the electrons heated due to radiation is defined as = U,,.(T), where the value = v~corresponds to a frequency at which the heating of electrons due to radiation from the shock front is maximal. The energy acquired by ions, atoms and molecules in elastic collisions with electrons is ii

Q4~~(TeT)fl~c~.

(11.13)

The conductivity a is related to the magnetic viscosity 2

mac2

~m

by (11.14)

c

where “8Te~’1122ir’e3”2

/8T~\11~

I Veff =

~N 0

(~—) 1Tm~

ae0

+

n

(k—)

‘irm~

—3

Te

A

is the effective frequency of electron collisions (A is the Coulomb logarithm). Equations (11.4)—(l1.9) were solved numerically under the following initial and boundary conditions. At a time t = 0 an ideally conducting flat piston is pushed into an immovable neutral gas (molecular hydrogen) at a velocity U0. For t >0 a gas-dynamic shock exists upstream of the shock front that divides the flow into regions 1 and 2; in region 1 the gas is assumed to be non-dissociated and in region 2 completely dissociated. Later on, the boundary conditions at the shock are determined by Hugoniot relations for a gas-dynamic shock

*1

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

51

wave n1(v1— U)= n2(v2— U),

(11.15)

M1N1(v5— U)= M2N2(v2— U),

(11.16)

2+N

M1N1(v1— U)

2+N 1T1+fl1TeM2N2(V2

3+ N

M1N1 (v1 — U)

:

U)

2T2+fl2Te,

(11.17)

3 + ~N

1T1(v1

—

U) + Ed(Vl — U) = M2N2 (v2 — U)

2T2(v2U), —

(11.18)

where subscripts 1 and 2 refer to the flow upstream and downstream of the GD shock, respectively; U is the shock velocity; M1, N1 and M2, N2 are the masses and densities of molecules and atoms (including those ionized) upstream and downstream of the shock, respectively; e~is the dissociation energy of molecular hydrogen. Here we take into account that the scale of the electron thermal conductivity exceeds considerably the GD shock width, so that the temperature of the electrons upstream and downstream of the gas-dynamic shock is the same. To solve the problem, we have developed a special calculation procedure, which is a combination of the K.S. Godunov technique and Lax—Wendroff’s type scheme with explicit consideration of the gas-dynamic shock; for details see [33]. The numerical solution of the problem has been performed for conditions close to those of Stebbins and Viases’s experiment [9]. The magnitude of the transverse magnetic field was varied from 0 to 4400 Oe. The characteristic length A of electron diffusion losses in the transverse direction was assumed A = h/ir where h is the height (along the z axis) of the pinch cylindrical tube. Since in the experiment [9] h = 15 cm, we took A = 5 cm. In numerical simulations the piston velocity was chosen so that the velocities of the shock waves formed ahead of the piston corresponded to those observed in [9] at an initial pressure of 0.25 torr and a voltage across the capacitor battery of 25 kV. Table 2 lists typical results of the numerical simulation. Here H0 is the unperturbed upstream magnetic field, U0 is the piston velocity, U is the shock front velocity, and x = 0 corresponds to the place where the shock wave is formed. The last two columns of the table list calculated and measured [9] values of the magnetic field compression in the shock wave. Figure 12 shows typical x—t diagrams for a gas-dynamic shock wave for different values of the transverse magnetic field. It can be seen from fig. 12 that the GD shock velocity is almost constant (deviation from the mean value does not exceed 8—10%), as was observed in ref. [9]. Figure 13 shows the results of the numerical simulation of signals of magnetic and electrical probes that give H, H and the electric field in the laboratory coordinate system for various positions of the probes. These results Table 2

H0, Dc

U0, cm/gs

U(x = 0), cm/ps

U(x = 14cm). cm/ps

(H2JH0)

(H2IH0) mess

420 850 1280

4.6 4.7 4.8

6.0 6.13 6.26

6.40 6.55 6.72

5.73 4.28 3.20

1720

4.9

6.81

2.47

2140

5.0

6.39 6.52

5.50 4.19 3.16 2.44

7.08

2.07

2.29

:

52

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields ~us)

0 4 Fig. 12. Typical x—t diagrams for a gas-dynamic shock. I: H

3

0=

420

/2

Dc, U0 =

4.6 cm/ps;

.27c~n)

2: H0 = 3490 Oe, U0 = 6cm/ps;

3:

H0 = 4390 Oe,

U0 = 7 cm/p.s.

~

R~(/0’Oe)

1

/

3/ 2

/

~2

2

(b)

57x(c~

______________________

5 //(f0~Oe)

H

0

1

Fig. 13. Signals of “probes” (a, tI~b, 1~,arb. units; c,

2s~’,.s)

5

6

7

E

119) located at 2, 6 and 10cm from the place of the formation of a gas-dynamic shock: H0 = 1720 Oe, U0 = 4.9 cm/p.s.

Fig. 14. lop: Velocity profile for a transverse Ionizing shock wave for H0 = 1280 Oe, Us = 4.8 cm/p.s. t = I p.s. Bottom: Magnetic field profile for the same values of the shock wave parameters.

reproduce almost exactly the oscillograms of the corresponding signals presented in ref. [9]. This makes it possible to suggest that the above non-stationary problem of the formation of a steady-state magnetic and electric structure of a transverse shock wave describes completely the experimental situation. Typical profiles of velocity and magnetic field in the shock wave are shown in fig. 14. Figure 15 presents typical profiles of the electron and ion temperature, and also the radiation temperature T~in a shock wave; and fig. 16 shows profiles of the degree of ionization of hydrogen. Figure 17 illustrates how the formation of a steady-state shock front structure in a transverse ionizing

: : :

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

5

7(eV)

2

-f

0 2 4 717&(eV)

-2

-t

0

Fig. 15. Temperature profiles for H

2

5 = 1720 Oe, U0 = 4.9 cm/p.s. 1: t = 0.5 p.s. ~

U = 6.8 cm/p.s; 3: t

=

4

5-27c~a)

a

x(c~”)

6

3.27 cm, U

6.69 cm/p.s; 2: t

=

1.0 p.s.

x,,,,

=

6.58 cm,

1.5 p.s, x,,~,0,,= 9.9 cm, U = 6.77 cm/p.s.

3

3~.

2

__

0 2 4 6 X(cm) Fig. 16. Profiles of the degree of ionization of hydrogen in a shock wave for the same values of the parameters as in fig. 15. -/

0

4 1 fZ X(c~) Fig. 17. The ratio (HS/Hs)/(H2/Ho)MHD as a function of the ionizing shock front coordinate. 1: H0 = 4390 Oe, Us = 7cm/ps; 2: H5 2140 Oe, Uo = 5 cm/p.s; 3: H5 = 1280 Oe, Uo 4.8 cm/ps; 4: H5 = 420 Oe, Uo = 4.6 cm/p.s.

shock wave (as time goes on) leads to an increase in the magnetic field compression, which approaches a constant value characteristic of a steady-state shock wave. In this figure the abscissa is an equivalent parameter — the distance from the shock front to the place of its formation at t = 0. Data presented in fig. 17 permit, in particular, an estimate of the upstream breakdown field of the neutral gas. In fact, neglecting the shift of the breakdown threshold due to photoionization, we obtain for hydrogen at Po = 0.25 ton a breakdown threshold value, which coincides within 10—12% with that used in ref. [9]. The time after which the steady-state structure in a shock wave is achieved, i.e., the time for the establishment of a constant magnetic field compression (I-12/H0) (see fig. 17), is determined by the non-linear stage of ionization and strongly depends on the precursor photoionization level. It is obvious,

54

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

a 2 3 Fig. 18. Compression of the magnetic field in a transverse ionizing shock wave, propagated at x function of the initial magnetic field. Experimental points are from ref. [9].

=

14cm from the place of the formation, as a

at the same time, that the higher the electric field induced upstream of the initial gas-dynamic shock, the smaller this time (all other conditions being the same); this does agree with the data of fig. 17. Figure 18 shows the calculated values of the magnetic field compression in a steady-state transverse ionizing shock wave as a function of the initial magnetic field, and also data of the corresponding measurements [9].

12. Magnetic structures of normal ionizing shock waves Ionizing shock waves are called normal ionizing shocks if the upstream magnetic field is normal to the shock plane. The coordinate system used has been defined above (section 2). In this system the upstream flow velocity is also normal to the shock plane and is parallel to the magnetic field direction: = 0, H~ 0= 0. The latter condition means that normal ionizing shock waves fall into the class of oblique shocks. Let the following consideration be confined to the limiting case of strong supersonic ionizing shock waves (M0 1, = ~)which corresponds to most of the experimental conditions. Putting FI~ H, we obtain for this case from eqs. (3.1) and (3.13) ~‘

VGD

(12.1)

and dH/d~= E. + H(v~ 1/M~0). —

(12.2)

The state downstream is determined by the intersection of the curve (12.1) in the (H, v~)plane with the zero field hyperbola H=l/M~.

(12.3)

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

55

Let us note that the system of equations for the boundary conditions always has the following solution: E~= 0, Hk = 0, Vxk = This solution corresponds to a purely gas-dynamic ionizing shock wave, in which the flow is everywhere parallel to the magnetic field and therefore does not interact with it. Simple calculations show that a gas-dynamic shock wave is of type 2 for Mao >2 and of type 4 for Mao <2. This solution is a special one, to a certain extent: it corresponds simultaneously both to the gas-dynamic and MHD limits. While considering structures with E. 0, we see that eqs. (12.1)—(12.3) are invariant with respect to the transformation H H, E. E~.Equation (12.2) implies that ~.

—* —

—* —

dHId~=E. for H = 0, that is, if E~>0 we always remain within the region H > 0, while moving along the integral curve from H = 0. It is sufficient, therefore, to confine the analysis to the case E~>0, H > 0. We find from the last condition and eq. (12.3) that at a point k downstream 1/M~o—vXk>0, that is, Mak <1, and hence only a singular point of type 3 or type 4 may correspond to the state downstream for E~>0. It is a simple matter to show that for Ma0 >2 a branch of the curve F~= 0, containing the point 0 (H = 0, v~= 1) in the (H, v~)plane, lies entirely above the horizontal asymptote of the hyperbola (2.13) and therefore does not have common points with the hyperbola for E~>0. Hence, this case is characterized only by a purely gas-dynamic ionizing shock wave of type 2, for which the relations E~=0,

H~2=0

(12.4)

hold in addition to the main system of boundary conditions. Thus, for Mao> 2 the above arguments make it possible to single out the only permissible type of normal ionizing shock waves, namely purely gas-dynamic ionizing shock waves of type 2, for which an appropriate number of additional boundary conditions does exist. It remains to consider the case 0 0.” If Mao <2, the system of equations for the boundary conditions does have a solution for 0sE~E~~.

(12.5)

The lower boundary of the interval (12.5) corresponds to the MHD limit. In this limit the shock wave represents a purely gas-dynamic slow MHD shock for Mao < 1 and a switch-on MHD shock for Let us emphasize that here, like in the general case, type 2 evolutionary shock waves correspond to isolated points, type 3 shock waves to

curves (1-parameter families), and type 4 shock waves to regions (2-parameter families) in the (H~o,E,) parameter plane. Since only normal ionizing shock waves are considered here, we select from the above manifolds submanifolds of a dimension, which is less by one than that ofthe manifolds (a point on a curve, a curve in a region). Therefore, the electric field E, must be determined uniquely for type 3 ionizing shock waves, while type 4 shocks must constitute a 1-parameter family.

56

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

1
HP(E~,M~s,k)

J

~I’~k(Es, Mao)

E5+ H[v~(H;E5,M5o)— 1/M~0]= 4~’~s, Mao),

(12.6)

a

0.35~0.3

0.2

—

•N\clP

—

/

/

~

a

-~--—--

0.1/

Fig. 19. The region between the line CJ and the ray MHD (E = 0) on the plane (Mao, E) corresponds to normal ionizing shock waves. In shock waves whose parameters lie below the sonic line M3 = 1 the Isomagnetic shock is absent,

0 Fig. 20. Magnetic structures of normal ionizing shock waves of types 3 and 4 for M~>1.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

57

where k = 3, 4 and v~(H;E~,Mao) is, as before, the larger root of the equation (12.1) which is quadratic in v~.For k = 3 let H~(E5,Mao) H~(E5,Mao, 3) stand for the magnetic field corresponding to the end of the precursor region for a type 3 shock wave (point a0 in the notation of section 6). For k = 4 let us denote by Ha (E5, Mao) H0(E~,Mao) the magnetic field at the beginning of the isomagnetic shock (point a) contained in the shock structure of a type 4 shock wave. The function 4k(E~,Mao) is defined by relation (6.4). Under the conditions of fig. 18 we have H~(ES,Mao) = H3(E~,Mao) and Ha(Es, Mao) < H~(E~, Mao). If the singular point 3 is a supersonic one (this corresponds to the parameter region below the line M3 = 1 in fig. 19), the latter inequality remains valid but, as in section 8, H~(ES,Mao) > H3(E5, Ma0) so that the integral on the left-hand side of (12.6) converges. The integrand in (12.6) is positive and due to the above arguments H~(E5,Mao) Ha(Es, Mao), i.e., t14(E5, Mao) < t~t3(E~, Mao) (in both cases the equality is possible only if E~= ~ This means that a shock wave of only one type can exist for given values of E. and Mao: =

either

<~(/3=

<414,

i.e., k = 3,

or

4i3>i~i4—.4~4>q53,i.e., k=4 41k have been omitted for brevity). Ionizing shock waves of both types (arguments of the functions can exist simultaneously only~‘k, if the following condition holds tI’ 3(E~’(Ma0),Mao) = 4~3(E~(Mao), Mao),

(12.7)

that is, for the Chapman—Jouguet flow. Let us note that a purely gas-dynamic solution for a type 4 shock wave (E~= 0, H4 = 0) cannot be obtained from eq. (12.6), since both sides of eq. (6.3) vanish identically for this solution and the basic relation (6.3) does not hold. Therefore, this solution should be considered as an additional one to solutions of eq. (12.6). The behaviour of the function ç13(E5, Mao) can be analysed in the same way as in section 8, and it is, in general, similar to the behaviour of lIJ2(E~,Mao). In particular, if Ma0> 1 and the1”2for upperMao> limit 1.503), of the integral (12.6) doessingularity not vanishofascLfE. —~0 (e.g., H3(0, Mao) = [~(M~0 1)(4 — M~0)] the maininlogarithmic 3(E~,Mao) for E. —*0 is expressed by the right-hand side of (8.9), in which H~(0,Mao) should be substituted for H~ 1, Mao) and E. for E. + 1. In order that both ‘imits E~ 0 and R ~ lead to the same result for Mao < 1 (transition to the MHD régime), the following condition must hold, as in section 8. —

—~

—*

tIi3(E~_*0,Mao< 1)’SK.

(12.8)

The form of the solutions of eq. (12.6) is shown in fig. 21. In this figure the functions çfr3(E5, Mao), 413(E5, Mao), 414(E5, Mao) are presented for a fixed Mao> 1 for various values of the parameter E = MaoEs for E~> MaoE~~~ (fig. 21a) and E~
58

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

C

li~_ E~. L~,

~

£

0

E4a,a.

E

3

Fig. 21. The functions iJi3(E~,Mae), çb3(E~,Mao), 4~4(E,,Mao) for constant Ma~and variable E type 4 shock waves, and the value E = E3 to type 3 shocks, a) for E~ E5.

La =

~

E

E,Mao. The interval 0

E
waves may exist in the entire range 0 0.358, see fig. 19), the inequality E
if E* ~

(12.9)

Transition to the MHD limit occurs, as a rule, in a similar way both for E —*0 and R —÷cx• In both cases the range of E~values permissible for type 4 shock waves converges to the point E. = 0, and E3 tends to the same value (see fig. 21). Out of these two solutions, however, only one corresponds to a non-trivial evolutionary MHD shock transition for each value of Mao. For 1
—*

—*

:

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

59

infinite number of integral curves representing such structures; any of these structures can be obtained by an appropriate choice of limiting process.

13. Self-similar piston problem In order to illustrate more clearly the existence of various types of ionizing shock waves and the character of transition to the MHD limit, let us consider the following examples. (1) The problem of a rigid ideally conducting piston. At t = 0 a flat ideally conducting piston is moved into a stationary neutral gas at a velocity U along the initial magnetic field, normal to the piston plane, and at a velocity V in the transverse direction to result in the formation of a normal ionizing shock wave. In the unperturbed gas the transverse electric field is zero; the electric field E corresponding to the normal ionizing shock wave is transferred by an electromagnetic wave propagating in the upstream region. The gas velocity, relative to the piston (or the gas density), and the electric field transverse component must vanish at the piston surface in a piston-fixed coordinate system. (2) The problem of a magnetic piston. The discharge current flowing along an immovable nonconducting wall creates at t = 0 a transverse magnetic field and ionizes the gas near the wall. The magnetic field pushes the conducting gas forward to result in the formation of an ionizing shock wave. The normal component of the velocity (or the gas density) must vanish at the wall surface, and the transverse magnetic field takes up a given value H0. Both problems will be solved in the electrostatic breakdown approximation. Normal ionizing shock waves are assumed, as in section 12, to be strong and supersonic ones (M0 a~1). In the discussion which follows the velocities U and V are considered as normalized with respect to the axial Alfvénic velocity. Let us note that here the ionizing shock wave moves in the positive x-direction, unlike the case of the shock waves considered previously. To take this fact into account, it is sufficient to make the substitution H H in all the equations of section 12. The following assumptions are made [17]in constructing the solution of a self-similar problem, which does not involve parameters with the dimension of length or time. A slow MHD shock wave or a slow MHD centred rarefaction shock wave can follow a type 3 ionizing shock wave, while a type 2 ionizing shock wave can be followed only by the latter type of shock*. Neither of those waves can follow a type 4 ionizing shock. Either a~vacuumregion, in which the electric and magnetic fields do not vary, or a region of uniform flow where the gas is at rest relative to the piston, can be adjacent to the concLcting piston. In the magnetic piston problem either a vacuum region or a region of uniform flow, where the axial component of the velocity is zero, can be adjacent to the non-conducting wall. Matching solutions of this type, we obtain the solution of the above-formulated problem. Let us first solve the problem of a conducting piston. The dimensionless components of the flow velocity downstream of an ionizing shock wave of type k are —* —

U’MaO(lVxk),

V’=

.EsMao2

V~k

aO

,

(13.1)

* In the general case, a type 2 ionizing shock wave can also be followed by a rotational discontinuity which tums the magnetic field transverse component. But type 2 normal ionizing shock waves are purely gas-dynamical ones with a zero downstream magnetic field, so that there is no need to consider rotational shocks. Similarly, in this case a slow MI-ID shock wave, which follows a purely gas-dynamic shock, degenerates into a null (trivial) shock transition.

60

MA. Liberman and AL. Velikovich. Physics of ionizing shock waves in magnetic fields

where Mao, E~,V~kare the parameters of an ionizing shock wave propagating in an unperturbed gas. The flow velocity downstream of a slow MHD shock wave, which follows a type 3 ionizing shock wave, is rr~,_ TT’i. ~A’ —

(.1

1/2(1

lvi a3Vx3~

F —

V~4)

V” = V’ + v//~(H~ — H3)/M~.

(13.2)

Here v~3and H3 characterize the type 3 ionizing shock wave, and M~3,v~, H~the slow MHD shock wave in the shock-fixed coordinate system. For type 2 ionizing shock waves (they are purely gas-dynamic) v...2 ~, E. = 0, Mao 2. We find from (13.1) that points, which describe flows in the downstream region, lie on the ray (U ~, V~= 0) in the (U, V~)plane. If a point lying in the (U, VI) plane and corresponding to the conducting piston velocity belongs to that ray, a purely gas-dynamic ionizing shock wave arises ahead of the piston and is separated from it by a uniform flow region. This result remains unchanged in the MHD limit. Moreover, the following shocks are possible in this limit: purely gas-dynamic shock waves (a limiting case of type 4 ionizing shocks), which refer, according to the MHD classification, to slow ones and for which v~4= ~, E. 0, 0< Mao < 1, i.e., in the (U, I V~) plane the segment (0< U <~, V~= 0) corresponds to these shocks; and also switch-on (fast) MHD shock waves (a limiting case of type 3 ionizing shocks), for which in the downstream region U’

=

M’ao

“~

i/Mao

112, 1
=

‘~

Iv~ 64 HO

SW3RC\

2-

a

SW5R

64

64—

/

/5 U Fig. 22. Regions in the piston velocity space (U, vj) corresponding to different types of flow ahead of a piston in the MHD limit of the theory. Notations: GD, purely gas-dynamic shock wave; SW, MHD switch-on shock wave; M, slow MHD shock wave; R, slow MHD rarefaction wave; C, cavitation. The index indicates the type of the downstream singular point.

0.5

0.75

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

61

ionizing shock wave, whose limiting case is the MHD shock). The region below the curve (13.3) in the (U, I VI) plane, labeled as SW3M4, corresponds to the presence of two shock waves ahead of the piston. If the point corresponding to the piston velocity belongs to this region, the switch-on MHD shock wave (SW3) propagates in the unperturbed gas, the slow MHD shock wave (M4) follows the former shock and propagates in the gas compressed by it, and next there is the uniform flow region. Switch-off shock waves are the strongest ones among slow MHD shock waves (M4); the interval (~ < U <~, I VI = 0) in the (U, I VI) plane corresponds to the configuration “switch-on shock—switch-off shock”. This interval also formally corresponds to non-evolutionary gas-dynamic transitions 1 4~* A slow MHD rarefaction shock wave (R) may follow ionizing shock waves of types 2 and 3. The equations, which describe it, can be written in the form [36] —*

2(1—g) ~g dr 1—q2r5

‘134

(13.5)

= Aq”2,

dV 1 ~i~=_A(i_q~5)

1/2

(13.6)

where q = c~Ia2,r = (a2Ic~)”5= (~NT/(H~/8ir))°”5, A = 3ak/rkcao (the subscript k = 2 or 3 characterizes the state downstream of the ionizing shock wave, the subscript 0 corresponds to the unperturbed state). Equations (13.4)—(13.6) can be integrated over r from Tk to 0. For r = 0 the density of the gas vanishes (cavitation). Denoting the transverse magnetic field and the velocity components at the point of cavitation by H~,U~,~ respectively, we find from the continuity of the tangential electric field that the velocity components of a conducting piston separated from the point of cavitation by a vacuum region should satisfy the relation V— V~~+H~(U— U~)=0,

(13.7)

which describes a straight line in the (U, IVI) plane. Thus, a line obtained by integration of eqs. (13.4)—(13.6), which passes through the point (U = ~, I VI = 0) in that plane, bounds a region of piston velocities, which lead to a purely gas-dynamic type 2 ionizing shock wave ahead of the piston and a slow MHD rarefaction shock wave downstream (region GD 2R). Points of cavitation form a line C in the plane. Behind the line C the boundary of the region GD2R is continued by the straight line (13.7), which bounds the region GD2RC where the “tail” of the rarefaction shock wave is separated from the piston by a vacuum region. Since purely gas-dynamic type 2 normal shock waves correspond simultaneously to the gas-dynamic and MHD limits, the character of the flow in the regions GD2R and GD2RC, and the boundaries of these regions in the (U, I VI) plane do not depend on the gas parameters (see figs. 22—24). Figure 22 illustrates the solution of the problem of a conducting piston in the MHD limit. The region SW3R lies in the (U, I VI) plane between the line SW3 and the boundary of the region GD2R. If the point corresponding to the piston velocity lies in the region SW3R, an MHD switch-on shock wave * Note that normal ionizing shock waves, however close to MHD switch-on shock waves, are not special ones (see section 5), i.e., they can be considered as evolutionary ones.

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M.A. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

followed by a slow MHD raref action wave arises ahead of the piston. The region SW3RC is located behind the line C (the rarefaction wave is separated from the piston by a vacuum region). Figure 23 corresponds to the gas-dynamic limit (E > 0.358) of the problem of a conducting piston. Only type 4 ionizing shock waves can exist in this limit (region 14 in the (U, I VI) plane). The boundary of the parameter region corresponding to shock waves of this type, line CJ in fig. 19, gives the line Icj in the (U, VI) plane. If a point of the plane corresponding to the piston velocity lies on this line,1c.~ a Chapman—Jouguet ionizing shock wave arises ahead of the piston. Points lying above the line represent flows with a Chapman—Jouguet ionizing shock wave followed by a slow MHD rarefaction shock wave (region IciR); the latter can be separated from the piston by a vacuum region (region I~. 5RC). The most interesting intermediate case (E~ E in fig. 19) the gas-dynamic limit does not take place. According to (12.9), a type 3 ionizing shock wave (line 13) followed by a rarefaction shock wave (region 13R) or a slow MHD shock wave (region 13M4) can arise ahead of the piston. As E~—*0, the line 13 tends to the line SW3, and fig. 24 goes over to fig. 22. The solution of the three-dimensional self-similar problem of a conducting piston can be found if figs. 22, 23, 24 are rotated about the horizontal axis. Under the conditions of fig. 23 the region 14 goes over to the 3-dimensional region I4~the curve 13 to the 2-dimensional surface of revolution 13, and the 1-dimensional manifold (ray GD2) corresponds, as before, to type 2 ionizing shock waves. The difference in the dimensions of those regions corresponds to the different number of additional boundary conditions for ionizing shock waves of types 2, 3 and 4. Ionizing shock waves of type 4 require no additional relations: if certain restrictions in the form of inequalities hold true (the point cor-

lvi

lvi 6D~,

G~

4/ 3

Fig. 23. The same as in fig. 22, but in the case where the gas-dynamic limit holds true (E~>0.358). Notation: ID, Chapman—Jouguet ionizing shock wave; 14, type 4 ionizing shock wave,

Fig. 24. The same as in figs. 22, 23, but in the case where the gas-dynamic limit does not hold true in a certain region of the shock wave parameters in the (Mao, E) plane (see fig. 19); in this case E~= 0.15, i.e. this region in fig. 19 lies below the line Ci above the straight line E~= 0.15. 13 is type 3 ionizing shock wave.

a a

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

63

responding to the piston velocity in the velocity space (U, V~,V~)must lie within the region 14), only a shock wave of this type arises ahead of the piston. Type 3 ionizing shock waves require one additional relation, so that for this type of ionizing shock waves only to arise ahead of the piston the dimension of the manifold, to which the point in the velocity space should belong, is less by one. A similar situation occurs for type 2 ionizing shock waves (2 additional relations, 1-dimensional manifold). Let us emphasize, however, that ionizing shock waves of types 2 and 3 are not certain special cases, which are realized only if the components of the piston velocity satisfy certain conditions. Rotation of the 2-dimensional regions GD2R, GD2RC, 13M4, I3R, I3RC results in 3-dimensional regions of the (U, V~,V~)space, and if the point corresponding to the piston velocity lies within one of those regions,

an ionizing shock wave of type 2 or 3 arises ahead of the piston and its shock front is separated from the piston by an MHD flow of the corresponding type. To solve the problem of a magnetic piston, eqs. (13.4)—(13.6) are integrated until either the axial velocity or gas density vanishes. In the former case the “tail” of a slow MHD rarefaction shock wave is separated by a weak discontinuity from the region of uniform flow, which is adjacent to the immovable wall. In the latter case there is a vacuum region between the “tail” and the wall. In either case, the region, in which the magnetic field is uniform, is adjacent to the wall. It is this uniform magnetic field H0, which is identified with the field of a magnetic piston that pushes an ionizing shock wave. The rarefaction wave is necessary in order that the boundary condition at the immovable wall is satisfied (let us recall that in the downstream region the gas has a velocity in the direction of the shock wave propagation; only a rarefaction shock wave can diminish the velocity or density of the gas to zero). As can easily be seen from fig. 24, a rarefaction shock wave can follow only a Chapman—Jouguet Mao

7/ / I

—

MA/B

/

~

~

I

Fig. 25. The Alfvén Mach number for a normal ionizing shock wave ahead of a magnetic piston as a function of the ratio of the driving magnetic field H~to the axial field H~.The transition from the Chapman—Jouguet to the MHD flow corresponds here to the transition from the gas-dynamic to the MHD limit with decreasing dimensionless breakdown threshold E~.Curves 1, 2, 3, 4 and Ci are plotted for E~= 0.3, 0.25, 0.15, 0.05 and E5>0.358, respectively.

64

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

ionizing shock wave in the gas-dynamic limit. If the conditions of the gas-dynamic limit do not hold true and type 3 ionizing shock waves can arise, the Chapman—Jouguet flow does not take place in the corresponding interval of the values of the driving magnetic field H0. Thus, a specific role of the Chapman—Jouguet ionizing shock wave is manifested only in the gas-dynamic limit. Accordingly, the Chapman—Jouguet condition [37] cannot be considered, as is sometimes stated, to be a universal condition relation determining the type of the flow in normal ionizing shock waves. This condition corresponds, firstly, to a certain form of the initial- and boundary-value problem and, secondly, to a limited interval of intensities of ionizing shock waves (the gas-dynamic limit). The transition from the Chapman—Jouguet flow to the MHD flow in the problem of a magnetic piston is illustrated in fig. 25, which presents the dependence of the dimensionless shock front velocity Mao on the magnetic field H0, divided by the axial magnetic field H~,for different values of the dimensionless electrostatic breakdown threshold E~.As was mentioned above, for E~> 0.358 (the gas-dynamic limit) the Chapman—Jouguet flow arises ahead of the wall (line CJ in fig. 25). Curves 1, 2, 3, 4 have been plotted for E~= 0.30, 0.25, 0.15 and 0.05, respectively. As can be seen, the lower the threshold E~,the more significant is the deviation from the predictions of the Chapman—Jouguet theory and the closer is the flow to the MHD limit (line MHD).

14. Plasma heating in normal ionizing shock waves A number of experiments have been performed to study the heating of a plasma and the shock front structure of normal ionizing shock waves. The major part of experimental investigations with normal ionizing shock waves has been conducted either on coaxial electromagnetic shock tubes or on an installation representing a certain (but not essential) modification of them. The electric and magnetic shock front structures were first measured, with a relatively good reproducibility, by Heywood [39].In this work Heywood measured, for the first time, the upstream electric field as a function of the shock front velocity. Later on, similar measurements as well as the measurements of magnetic shock front structure with the help of magnetic probes were performed by Miller [40].Levine [41]has carried out a detailed experimental study of electric and magnetic fields for normal ionizing shock waves. The experiments were performed in a coaxial electromagnetic shock tube in a wide range of the initial gas pressure and axial magnetic field. He also obtained data concerning the downstream densities and temperatures. The data obtained in the above-mentioned works convincingly show that we actually deal with switch-on ionizing shock waves and that such shock waves do induce the upstream transverse electric field. At this stage the experimental data on the shock front structure of normal ionizing shock waves seemed to agree well with the calculations of Gross, Levine and Geldon [42] based on a theory using the Chapman—Jouguet hypothesis. Nonetheless, even at that time discrepancies were observed between the measurements and calculations. Here it is worth-while to make some remarks concerning the treatment of experimental data and their comparison with theoretical calculations. Measurements of the “switch-on” transverse component of the magnetic field indicate qualitatively that we actually deal with a switch-on shock wave, but the experimental accuracy available does not permit one to decide between different variants of the theory, since any variant implies that the magnetic field transverse component is switched on as Mao increases, passes through a maximum, and vanishes at a certain shock front velocity. As far as the measurements of the upstream electric field are concerned, it is difficult to speak about

: :

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MA Liberman and A L Velikovich Physics of Ionizing shock waves in magnetic fields

65

complete agreement with the calculations [42].In fact, at a high shock front velocity (i5~o~ 7 x 106 cm/s) the shock is in no way a switch-on one (such velocities correspond to the “purely gas-dynamic” region, i.e., Mao 2, if energy losses due to dissociation and ionization are not taken into account), and the electric field is zero in any variant of the theory. At lower velocities (fi~o~ 5 x 10~cm/s) the above calculations differ considerably from the electric field measurements. Calculated and measured values of E agree only in a narrow range 5 x 106 ~ ~ 7 x 10~cm/s; the reasons will become clear from the discussion to follow. Detailed investigations of normal ionizing shock waves in hydrogen and helium for various initial pressures and in a wide range of shock front velocity have been performed by a team of Australian physicists at the Sydney University [43—51]. A Supper II cylindrical electromagnetic shock tube, which is a certain modification of a coaxial electromagnetic shock tube, has been used in the experiments [50]. The results obtained in the experiments differ considerably from the values predicted by the theories of gas-dynamic and magnetohydrodynamic switch-on shock waves, and also by the theory based on the Chapman—Jouguet hypothesis [37]. The experimental results show that the gas-dynamic limit of the theory of ionizing shock waves does not hold true under the above experimental conditions. In fact, in this limit the shock front structure must begin with a viscous GD subshock, in which the temperature and density of atoms and ion5 increases over a distance of the order of the atomic mean free path. In helium for an initial pressure Po = 0.1 torr and shock front velocity about 10~cm/s the characteristic time of the change in the density and temperature must be 0.1 p~sfor a scale of the order of 1 cm. In the experiment, however, a wide region from 20 to 30 cm is observed, in which the degree of ionization and temperature increases gradually. Let us note here that the width of this region is of the order of the free path of ionizing quanta, so that precursor ionization does play a decisive role in the above experiments, in accordance with the theory presented in previous sections. The theory of fast* (type 3) normal ionizing shock waves in the electrostatic breakdown approximation has been considered by Liberman [38]. In this case the additional boundary condition implies that the upstream electric field must be equal to the threshold breakdown field. Unlike the case of transverse ionizing shock waves (sections 8, 9), the present situation is more complicated due to the non-scalar form of Ohm’s law (see below). One of the consequences of this fact is a strong Joule overheating of the ions and atoms, as compared to that of the electrons, at the leading part of the shock front (for a ~ 1). So the problem of calculating the electric breakdown threshold is rather difficult to solve. In the paper by Liberman [38] a semi-phenomenological formula for the electric breakdown field has been suggested, and the calculations of the shock front structure revealed quite a satisfactory agreement between theory and experiment. The possibility of a considerable overheating of the heavy species in normal ionizing shock waves, due to strong friction between ions and atoms as a result of the high value of recharge cross-section under ion-neutral collisions, was first pointed out by Cowling [52]. These effects have been studied theoretically in ref. [53] particularly for ionizing shock waves in a magnetic field. Overheating of ions and atoms, as compared to that of electrons, has been observed in experiments with normal ionizing shock waves [47—49]. Since the precursor region does not contain the viscous subshock, in which the heavy species are * It is this type of shock waves or their limiting case, the Chapman—Jouguet shock wave, which is formed in an electromagnetic shock tube, because in this case shock waves are formed ahead of the current sheet of the discharge current representing a rarefaction shock wave.

66

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

predominantly heated, the overheating observed for atoms and ions can naturally be attributed to the fact that the ions are heated in the electric field, the conditions of the electric current flow through the plasma being such that the Joule heating of the ions exceeds significantly that of the electrons. Strong heat exchange between the ions and neutral particles results in a predominant heating of the heavy species as a whole (i.e., T, = Ta as T). The simplest model involving a scalar conductivity is invalid here, so that the equations of motion for both the heavy species and the electrons should be solved independently, to elucidate the character of plasma heating in the precursor region for a ~ 1.~ Let us consider the asymptotic behaviour of the shock front structure for x —~ ~ when a -*0 and H~—*0, H~—*0. The shock front structure will be calculated in first-order approximation in a <~1. In this case a will be considered as a small parameter of the first order, taking into account that H~= 0(a) and = 0(a). It follows from the ionization equation that a decreases exponentially as x —

—p

a

~=-

—

const. exp(x/&),

(14.1)

where & is a characteristic scale due to both photoionization of the gas and ionization in the electric field. As we shall see below, in practice photoionization dominates due to weak Joule heating of the electrons for x —~ ~, so we may take z.L = l/Noffph. Let us introduce a new variable, putting —

d~= a dx.

(14.2)

Near the point ~ = 0, corresponding to x

=

—

~,

all derivatives are of order unity. Using the change of

variables (14.2), we obtain from Maxwell’s equations — V~= ~h dH~Jd~,

—

V~ V~=

4h

dH5/d~,

(14.3)

(14.4)

where cH~

— 4ireNov~o The conservation equations for the transverse component of the plasma momentum yield

—

v’~=

(H~/M~0v~,)[1 + 0(a)], —

v~— v’~= (H~/M~o — v~)[1 +

0(a)]

.

(14.5) (14.6)

(From now on we neglect the inertia of the electrons.) These relations show, in particular, that the atoms are set into motion when the transverse component of the magnetic field is switched on. Strictly speaking, we cannot use here a three-fluid hydrodynamics since the ion—atom relaxation times for momentum and energy are less than the ion—ion relaxation times. Yet, since the solutions of the kinetic equations cannot be obtained in a simple form, the solutions of the equations of hydrodynamics obtained below should be considered as those describing the process qualitatively.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

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From the equations of motion for the electrons we have, apart from terms which are small in a: (14.7)

Es+Hyv;ke(v~v~),

(14.8)

where ke

= 1/IleTea,

lie

eHx/mec.

Adding the equations of motion for the electrons and ions, we find da v~1~ a(v v~) dx M~0dx — /.la

da v’~

—

dx

‘

a(v~— v~)

idH~

M~0dx —

Zla

whence we obtain, using (14.1), 149)

~1 M~o\~d~4a1

(

1 (dHZ~HZ\ ~1 M~o’~d~~1a)’

(4.

1 10

v~

where L1a

VxOT~a,

LI

It follows from eqs. (14.5)—(14.10) that upstream of a normal ionizing shock wave, for x ~, where a —*0, H~ —*0 and H~—*0, the transverse velocities of the electrons and ions do not vanish, due to a non-zero upstream induced electric field, and are, respectively: —* —

=

(k~+iXk~+1)ES;

Vy=k2+1;

V~=

(k~+1Xk~+1)ES~

(14.11)

vz=k2+lEs,

(14.12)

where k1 = l/f1~r,

~ = 1/(~~ + v0)

i2~= e/m1c,

~

= VXO//ia.

Using (14.11) and (14.12), we can write Ohm’s law in the form* * Similar calculations for a transverse shock wave yield field, which arises because of plasma polarization [8).

~

a

ooEj, i.e. in this case the conductivity is a scalar quantity due to an axial electric

68

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

(14.13)

where the following notation is introduced:

=

00

=

°°

keki(keki + 1) (k~+ 1)(k~+ 1)~

—

O~A=

°~°

k~(k~ke) (k~+ 1)(k,~+ 1)’

e2n Tea me

In the limit a —*0 expression (14.13) coincides with that obtained for Ohm’s law in ref. [54]. It follows from eqs. (14.9)—(14.12), in accordance with the results of experimental investigations, that H~and H~decrease monotonically in the upstream region, without oscillations, unlike the case of an MHD switch-on shock wave in a magnetized plasma [36].One can easily find that for x —~ —

H~ kjke — kike + 1

1414

fleTea

.~

1 + (u1eTea)(11i~)~

It follows from (14.14) that for a typical relation I1eTea~’ 1>

(1~T

there is a rotation of the plane of polarization (i.e., the plane, which contains the magnetic field vector) in the shock front structure of a normal ionizing shock wave. Of course, real experimental conditions do not correspond to the idealized one-dimensional model, since the walls of a shock tube distort the idealized pattern of free flow of the heavy species in the transverse direction. Let us now calculate, within the model considered, plasma heating in a normal ionizing shock wave, Since the characteristic length of ionization ha exceeds considerably the scale LI a of heat exchange between the ionic and neutral species of the plasma (for helium, e.g., LIj/ia i7ph/tjia 10~),we can write T 1 = Ta as T. Taking into account then that Va = 0(a) for x —~— ~o, we obtain the following equation for the temperature [16]: —

- dT ~NaVxo~~

2

/ 4T

SflNal)

\1/2_

-2

OiamavxO(k2+

E~

1)(k~+1)’

(14.15)

Here ó~8is the 2cross-section for helium). of ion—atom collisions averaged over the distribution of the ions (&iaAssuming = 1 x iO~’~ kecm ~ k, ~ 1 in (14.15) and neglecting the weak dependence of ó~aon the temperature, we obtain after integration with due account for (14.1) fl

=

Na~!~~

1/2

~

T112.

(14.16)

Thus, we may conclude from (14.16) that the temperature of ions and atoms in the precursor region

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

69

decreases exponentially with a characteristic scale 2NaOph.

lI

LITha/2

For helium N 3, so we find LIT 30cm for ~ea = 5 x 10- 16 cm2 and = 5 x 1018 cm2, 5 = 3.3 x i0~ which agrees well with the cm measurements of ref. [48]. The upstream electric field measured in that same work is E 0 = 300 V/cm for i5~~ = 1.1 x 10~cm/s and H~= io~ Oe. Substituting these values into (14.16), one can plot the dependence n(x) from the values measured for T(x). As can be seen from fig. 26, the dependence n(x) thus calculated is in good agreement with experimental results. It is a simple matter to verify, using similar calculations, that direct Joule heating of electrons in the precursor region is small, and the electrons acquire most heat due to collisions with neutral atoms. The equation for the electron temperature will be ffph

~n~0~j~=

8V2Nafl (in

~a(T_

e)

whence, taking into account that T

1~(~-~-’~) Uea /

T~= 13\

~‘

(14.17)

Te),

Te and using (14.1) and (14.16), we obtain

lT2.

(14.18)

mavxocrphj

IT

In deriving (14.18) we assumed that the cross-section of elastic electron—atom collisions, averaged over the velocities, does not depend27oncompares T~.For helium the ofmeasured of ~eafrom are almost 2 [55].Figure the values T~(x),asvalues calculated (14.18), constant, with the ö~ea = 5 x 10_b results. cm experimental The values of T(x) in the calculations by eq. (14.18) have been taken from the measurementsof ref. [49].As can be seen from fig.27, the discrepancy between the experimental points and theoretical curve becomes noticeable only for a 0.1 and Te 6 eV where the density and temperature of the electrons are sufficiently high, so that energy losses due to impact ionization become significant. 7eV

17/0 -____

‘.1

.‘~o4. 30

3•

S

/

20 2 10 1.

S

1eV

~ ~ ~

/

_~_._~—+---q-—

5~3

~‘I/

o7~

~ ~

~

~

o

Jjisec 4 ~jiseC Fig. 26. The electron density profile n in the precursors region of the structure of a normal ionizing shock wave (solid line), as calculated from measured Tvalues (dashed-dotted line) using eq. (14.16). Experimental points and the interpolation of experimental data for T (dasheddotted line) are from ref. [47].

~usec~ ~jeseC Fig. 27. Profile of the electron temperature T~in the precursor region of the structure of a normal ionizing shock wave (solid line), as calculated from measured T values (dotted line) using eq. (14.18). Experimental points and the interpolation of experimental data for T (dotted line) and n (dashed line) are from ref. [49].

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MA. Liberrnan and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

We see thus that in accordance with the general concept presented above the region of precursor ionization is a determining one for the shock front structure of normal ionizing shock waves. It is this region where the conductivity becomes non-zero, thereby governing the qualitative character of the shock front structure as a whole. Let us make some other remarks concerning the comparison of experimental characteristics of normal ionizing shock waves with the theory based on the Chapman—Jouguet hypothesis. Calculations for a Chapman—Jouguet shock wave give more or less satisfactory results only_for the shock front velocity and upstream electric field as a function of the driving magnetic field H0 (the pressure of a magnetic piston). Yet, as was already said at the beginning of this section, these characteristics are not significant. In fact, we see from fig. 25 that the shock front velocity varies within a narrow interval, for a given H0, as we go over from the gas-dynamic limit of the theory (line CJ in fig. 25) to the MHD limit. The difference of the electric field from the value corresponding to the Chapman—Jouguet flow is, obviously, a small quantity of the second order (since ~ = Emax) in the deviation from the parameters of the Chapman—Jouguet flow. The downstream density and temperature exhibit a much stronger dependence on the shock wave type. For instance, for a normal ionizing shock wave in helium the experiments [47] performed at Po = 0.12 torr, H~= 2500 Oe, i3~0= 6.2 x 1063cm/s gave following and T = 5 the eV, while the values of the downstream electron density and temperature: fle = 5 x iO’~ cm calculation of these parameters for the corresponding Chapman—Jouguet shock wave yielded n~(CJ)= 3 x 10~~ cm3, T(CJ) = 6 eV. For Po = 0.1 torr, FI~= 10~Oe and v~o 1.1 x iO~cm/s the experimental values are fle = 3 X iO” cm3 and 4.5 x 1015 cm3, Te = 15 eV and 10 eV, as compared to the calculated values: ne(CJ) = 1.3 x 1016 cm3, Te(CJ) = 31 eV. Therefore significant discrepancy has also been observed in experiments with hydrogen [44]. This discrepancy can hardly be explained by the nonequilibrium character of the plasma, because the shock front profiles observed show that the relaxation processes have enough time to be completed. Thus, calculations based on the Chapman—Jouguet hypothesis reveal a considerable discrepancy with experimental results. Let us note that in the experiment [49] a normal ionizing shock wave, whose structure and shocks of the temperature and electron density do not correspond to the Chapman— Jouguet hypothesis, is not separated from the magnetic piston, i.e., in this sense the shock wave behaves like a Chapman—Jouguet ionizing shock wave. An ionizing shock wave in the above experiment differs from shock waves in preliminarily and partially ionized plasma, which arise in the same installation under the action of a discharge current pulse of the same magnitude. The higher the level of preliminary ionization, the faster the propagation of such shock waves; and the magnetic field profiles exhibit two well-defined steps, the first corresponding to the shock wave and the second to the magnetic piston. In particular, for the initial degree of ionization a 0 = 0.06 an MHD switch-on shock wave is similar, in its structure, to an ionizing shock wave, but the velocity of the former is 40% higher. Thus, in the above experiment a normal ionizing shock wave can, naturally, be referred to type 3, since it resembles strongly, in structure and boundary conditions, an MHD switch-on shock wave. At the same time, the intensity of the former shock wave is insufficient for this shock to be separated from the magnetic piston and to propagate forward, ionizing the neutral gas due to the shock’s own radiation. In this case, as was noted in the paper by Liberman [38], a normal ionizing shock wave moves together with the piston and behaves, to a certain extent, like a Chapman—Jouguet ionizing shock wave, though the conditions of the applicability of the Chapman—Jouguet hypothesis are not satisfied (a wide zone of precursor ionization exists in the upstream region), so that the shock wave is not a Chapman—Jouguet shock wave, this being clearly evidenced by the experimental results for the density and temperature. Despite this fact, the transverse electric field (which can be estimated by eq. (14.16) when expressed in terms of the observed

MA. Liberman and AL. Velikovich, Physics of ionizingshock waves in magnetic fields

71

n and T values) amounts to about 50 V/cm, which is close to the Chapman—Jouguet value (E~3= 57 V/cm). The apparent contradiction between the results of the observation of the structure and dynamics of a shock wave is fairly simple to resolve: the flow of plasma in an ionizing shock wave is neither completely steady-state nor quite one-dimensional. For instance, the transverse electric field in a cylindrical shock tube, directed along its radius, cannot be uniform over the tube cross-section: it drops to the centre and increases near the edges. As was already mentioned, the walls also restrict the transverse motion of the heavy species, etc. Therefore, qualitative, rather than quantitative conclusions of the above general theory can be applied to the discussed experiments. There is, probably, no sense in improving this theory, using a more realistic description of plasma, that is, to derive an equation, similar to (6.3), which involves Ohm’s law (14.15) instead of (2.5). The fact is that the form itself of Ohm’s law, which has to be used, is determined by the flow geometry in the installation considered. Let us illustrate this dependence for a quasi-one-dimensional model of plasma flow between two conducting planes; this model corresponds to a coaxial electromagnetic shock tube with R0~1— R1~I~ R1~(see section 4). Assuming that the conducting walls prevent the heavy species from moving in the z-direction (hence, v~= = 0), we obtain from (14.4): V~~4hdH~/dS~.

(14.19)

It follows from the equations of motion of the electrons in the zeroth order in a ~ 1 that Vez_keVey,

(14.20)

EsVey_keVez.

Equations (14.9) and (14.12) remain unchanged for x —~ together with (14.19) we find

—

- — k1(k~+1) E5.

~,

i.e., in the upstream flow, and using them

(14.21)

Comparison of (14.21) and (14.12) shows that the transverse velocity of the ions is small, as compared to that of the electrons: 2“~ 1. (14.22) (m~/m5)” Thus, it is the electrons, which contribute mainly to the conductivity, so that in expression (14.13) for Ohm’s law the quantities

Iv~/v(= ke/ki

k2 OeJOOk2+1~

k 0eA00k2+1

(14.23)

corresponding to the electron conductivity alone should be substituted for o-~and OA~ Hence, if the motion of the heavy species in one of the transverse directions is restricted by the walls, a flow arises, which differs significantly from that occurring in the idealized one-dimensional problem described above. Whereas in the one-dimensional problem the electric current in a magnetized plasma (k~~ 4 1) is transferred by ions in the z-direction and by both ions and electrons in the y-direction, in

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MA. Liberman and AL. Velikovich. Physics of ionizing shock waves in magnetic fields

the case under consideration the ionic current is negligibly small. Though in this case the ions still slip relative to the atoms, as follows from (14.21), this slip does not lead to heating of the heavy species for x—*—~,because (v~)2is smaller by a factor me/ma than the quantity given by (14.11), and the right-hand side of eq. (14.15) will be of the same order of magnitude. In this case the Joule heat released in collisions of the heavy particles will be comparable (by an order of magnitude) to the Joule heat released by the electrons. Yet unlike the latter, the former heat is distributed over a larger number of heavy particles and does not lead to their overheating, as compared to the electrons, since in the precursor region a 4 1. Thus, a viscous gas-dynamic subshock is the only possibility for overheating of ions and atoms in comparison with electrons. However, the experiments available have not revealed such a subshock at the leading part of the shock front. We see, thus, that a quantitative interpretation of experiments with normal ionizing shock waves is a rather non-trivial problem. If we solve a one-dimensional problem, it is impossible to take into account simultaneously two factors, which affect significantly the dynamics and structure of normal ionizing shock waves, namely the presence of the walls and slipping of ions relative to neutral particles leading to overheating the heavy species. The solution of a two-dimensional problem, though numerical, still encounters considerable difficulties. It is reasonable, therefore, to analyse in detail all the information gained in solving one-dimensional problems. Possibly, an acceptable compromise will be found, which will permit shock wave experiments to be interpreted quantitatively within the framework of the one-dimensional approximation, taking into account (in a model way) specific features of the installation — coaxial or cylindrical electromagnetic shock tube, or an inverse Z-pinch. Such problems are solved, as a rule, by computer simulation techniques, but the corresponding calculations still remain rather labour-consuming. One of the simplest examples is considered in the next section.

15. Normal ionizing shock waves. Numerical simulation of flow ahead of a conducting piston In the light of the above considerations it is interesting, of course, to perform a detailed analysis, not restricted to the narrow framework of a self-similar problem, of the formation and propagation of a normal ionizing shock wave in an electromagnetic shock tube. Obviously, such an analysis can only be performed by a numerical simulation of the process. The problem of the formation of an ionizing shock wave ahead of a magnetic piston, the discharge current sheet playing the part of such a piston in an electromagnetic shock tube, corresponds most closely to actual experimental conditions. Such calculations should be performed for a particular experimental installation, in which the magnetic piston parameters are assumed to be specified. In this case the flow ahead of a magnetic piston can be found by the simultaneous solution of the system of equations of plasma dynamics and radiation transfer, and the experimental installation itself is described either by the appropriate electrical-circuit equations or by a certain model, which determines the distribution and structure of the fields and currents in the vicinity of the magnetic piston. The simultaneous solution of the electrical-circuit equations and the equations of plasma dynamics and radiation transfer encounters calculational difficulties. From the standpoint of calculations, it is a simpler matter to solve a problem involving the simulation of a magnetic piston, yet the data available in the literature are insufficient to formulate such a problem. Therefore, we confine ourselves to a simple example, which makes it possible to investigate the formation of the magnetic structure of a normal ionizing shock wave by the numerical solution of the basic equations, describing this process,

:

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

73

without using any particular models. To this end, we shall treat the problem of the formation of a normal ionizing shock wave ahead of a rigid conducting piston [56]. Let us consider a neutral gas at rest with an axial magnetic field H~.*At t = 0 a rigid conducting piston is moved into the gas at a velocity v~,and a transverse magnetic field Hd is given at the surface of the piston. We shall consider an ionizing shock wave, which arises ahead of such a piston. The flow will be assumed quasi-one-dimensional (see sections 4 and 14; note that we have taken here H~= v~,,= 0, H~0 and v~ 0), and since we analyse a model problem, we shall not take into account the slipping of the ions relative to the electrons, that is we shall use Ohm’s law in the form (3.12). Of course, to describe the problem more realistically, for instance in the numerical simulation of the magnetic piston problem, one has to use an appropriate generalization of eqs. (14.22) and (14.23). Let us note the difference in the formulation of this problem, as compared to the self-similar problem considered in the preceding section. In the self-similar problem only the velocity of an ideally conducting piston is given, while the magnetic field near it is determined as an eigenvalue, which characterizes the flow as a whole. The corresponding transverse electric field, which is also an eigenvalue, is transferred into the neutral gas, upstream of the shock wave, by the electromagnetic wave. Most of the experimental conditions are characterized by a quasi-one-dimensional and quasistationary, from the viewpoint of electromagnetic theory, flow pattern [19] (see section 4), in which electromagnetic waves do not propagate. However, the transverse electric field in the self-similar problem of a piston retains its role of an eigenvalue. In the numerical simulation such a formulation of the problem would require an increase in the amount of calculations far beyond the possibilities now available. Therefore, another way has been chosen: the rigid piston is assumed to be not ideally conducting i.e. j,, ôH~/ôx~ 0 near it. After eliminating the ideal conductivity of the piston, we can specify the transverse magnetic field on it and simulate the formation of a normal ionizing shock wave. The numerical simulation has been performed for helium. We integrated the continuity equation (11.5), the equation of conservation of the x-component of the plasma momentum (11.60), the equation of heat transfer for the electron component (11.8), the ionization kinetics equation (11.9), and also the equation of conservation of the z-component of the plasma momentum, —

—

8 8/ H~H~\ M-~NvZ+1I\MNvXvZ— ~ )=0.

(15.1)

The equation of conservation of plasma energy has been taken in the form H

____

f

+~- NVX(M ~“

t’~+~T)+flVx(~Te+J)+~—~(v~i-r-_v~H~ —

=

0.

(15.2)

The equation of magnetic field diffusion is taken in the form ~+f(HVxHxVz_Z1m~)”r0. *

(15.3)

Actually, for the numerical calculation procedure to be stable, the magnetic field in the neutral gas is assumed not exactly axial, but a small

transverse component is introduced I~o/I-?~,0.1—0.2.

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MA. Liberman and A.L. Velikovich, Physics of ionizing shock waves in magnetic fields

The notation is the same as in section 11, but here M is the mass of a helium atom, H H~,y = ~ let us remind that the bar notation is not used for the dimensional variables. In the calculations we took into account the possibility of double ionization of helium by introducing the effective ionization potential J = Jeff (see [1]), but in most of the variants investigated, as well as in the experiments performed, helium is practically singly ionized, so that J = Jeff J1 = 24.5 eV. The calculation procedure is similar to that described in section 11. 3.At t = 0 a gas-dynamic Theisinitial conditions Po = 0.1 torr, T0 = 300goes K, N0 = 3.22 iO’~ cm~ shock formed ahead of are: the piston and thereafter away fromx it. The velocity of the gas-dynamic shock U is identified with the shock front velocity. The flow pattern ahead of a piston is formed as follows. At the initial moment the electron concentration rapidly increases, mainly due to photoionization, in the region between the piston and the gas-dynamic shock. The magnetic field decreases monotonically with increasing x (on moving away from the piston). But as soon as the degree of ionization becomes appreciable (10_3_10_2), the profile of H changes. From that moment on the magnetic field begins to grow on moving away from the piston, this fact indicating that the magnetic field in the shock wave is “switched-on”. When the gas-dynamic shock front goes away over a distance of 60 cm from the place where the shock wave was formed, the temperature profile exhibits a minimum, which becomes deeper as time goes on. This indicates that a rarefaction wave is formed between the piston and the shock wave. The rarefaction wave is followed by a “plateau” where the magnetic field is almost constant. The transverse magnetic field at the plateau can be identified, to a certain extent, with the driving magnetic field H 0 in the problem of a magnetic piston. As in the latter case, we have a region with a uniform magnetic field separated by the rarefaction shock wave from the ionizing shock wave. However, in this case the driving magnetic field is not given a priori, but is determined in solving the problem and remains unchanged in time and space after the formation of the rarefaction wave, this being precisely the condition of the self-similar problem of a magnetic piston (see section 13). Let us note that the rarefaction region is formed only at sufficiently high velocities of a rigid piston corresponding to H0 ~ 0.8—0.9 kOe. At lower piston velocities the rarefaction region is not formed and a strong hydrodynamic coupling is retained between the flow regions near the piston and downstream of the shock front. Such a character of the flow corresponds to the above-mentioned general concepts (at low velocities an ionizing shock wave cannot be separated from the piston, at high velocities it is separated and moves into the upstream region [38]; see section 14). However, it is reasonable to compare the results of the numerical simulation with the theory of ionizing shock waves only for high shock velocities, because if an ionizing shock wave moves together with the piston, it is the piston and not the shock front, which mainly determines the flow parameters in the downstream region. Figure 28(a, b) shows the profiles of the temperatures T and Te, and the densities n and N for H~= 10 kOe, H0 = 1 kOe, U = 6 cm/~.l.s.The time is measured from the moment when the gas-dynamic shock passes a probe mounted at a distance I = 180 cm from the place where the shock wave is formed. As can be seen, the jump of T at t = 0 takes place within the gas-dynamic shock. This means that our numerical simulation corresponds to the conditions of the applicability of the gas-dynamic limit in the ionizing shock wave theory. Since a rarefaction wave exists in the downstream region, we have to obtain a solution close to the Chapman—Jouguet flow. It was noted above (in section 14) that such a flow is not characteristic of actual experimental conditions. In this case the numerical simulation leads to the gas-dynamic limit because the main source of photoionizing radiation (the region near a rigid piston) is far from the gas-dynamic shock front and cannot produce a sufficient ionization ahead of it, while in the rarefaction wave, which plays here the part of a magnetic piston, the temperature is low (see fig.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

75

(a)

/o--—————— ci /,‘i&e

(a)

__~

1Ocrn~

~L.

EkV/en~

tids

Fig. 28 (a) Profiles of temperatures T and T~(b) profiles of densities N and n for a normal ionizing shock wave obtained for H~= 10 kOe, H~=lkOe, U= 6cm/ps, 1=180cm.

Fig. 29. (a) Profiles of H; (b) profiles of E for the same conditions as in fig. 28 for I = 60cm and I = 180 cm.

28a) and the radiation from this region does not ionize the gas ahead of the shock front either. Under actual experimental conditions, the temperature near a magnetic piston is several electron-volts, and its ionizing radiation is rather strong. The shock front and the region of the rarefaction wave are clearly distinguished in fig. 28(a, b). In the rarefaction wave the density N and temperatures T and T~drop, while the magnetic field increases (its profiles for the same conditions, for I = 60 cm and I = 180 cm are shown in fig. 29a). As can be seen from figs. 28—29, the shock wave is not separated from the raref action wave; in particular, the profile of H does not contain two steps characteristic of a flow, in which the shock wave is separated from the magnetic piston. Figure 29 indicates that the corresponding structures are steady-state ones: as the shock front moves from I = 60 cm to I = 180 cm, the region ahead of the raref action wave retains a constant width (about 40 cm), the profiles of H and E becoming sharper and approach a steady-state shape. This agrees with the general theory of attaining a steady-state structure of shock waves [1,57], according to which the steady-state structure is formed during a time required for a shock wave to travel a distance exceeding several times the shock front width. The formation of a steady-state profile is shown in greater detail in fig. 30 where for H5 = 6 kOe, H0 = 2.5 kOe, U = 6.7 cm/gis this profile is shown for three positions of the probe (1 = 60 cm, 120 cm, 180 cm). The sharp jump in E in figs. 29, 30 corresponds to the rarefaction wave. The value of E on the plateau is identified with the measured electric field, which corresponds in a self-similar problem to the electric field in the uniform flow region near the rear wall of the shock tube: E4. Thus, we may conclude that the numerical simulation under the present conditions describes the formation of a steady-state normal ionizing shock wave, which is not separated from the magnetic piston, its shock front structure beginning with a gas-dynamic shock. In the idealized self-similar problem described in section 13 such a flow would correspond exactly to the Chapman—Jouguet hypothesis. We see, however, that the profiles of Te, T, n, N, E, H shown in figs. 28—30 can hardly be identified with the structure of a Chapman—Jouguet ionizing shock wave followed by a slow MHD rarefaction wave (and the shock wave must be considered as a narrow discontinuity in comparison with the raref action wave). In this case the widths of the shock compression zone and rarefaction region are of the same order, and the relaxation processes are completed even in the rarefaction region (fig. 28). Dashed lines in figs. 28—30 show, respectively, the values of T and N downstream of the

76

MA. Liber,nan and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

~,

kV/cnt

Q8

/~fQc,n 80cm 6~2

Fig. 30. Profiles of E for H,

=

6 kOe, H~=

2.5

kOe, U = 6.7 cm/~s,I = 60cm, 120cm and 180 cm.

Chapman—Jouguet shock wave of the same velocity, and also the corresponding magnetic and electric fields at the plateau. It can be seen that the values of H4 and E4 agree with the Chapman—Jouguet hypothesis, while the compression and temperature downstream differ considerably from those corresponding to the Chapman—Jouguet ionizing shock wave of the same velocity (see lines CJ in figs. 28—30). Let us recall that the temperature jump in the Chapman—Jouguet ionizing shock wave (line CJ in fig. 28a) corresponds to the completion of temperature relaxation, rather than to the temperature maximum. The “downstream temperature” about 5 eV, which is approximately half T~5,corresponds to the temperature profile shown in fig. 28a. The compression (see fig. 28) appears to be higher than in 3), the Chapman—Jouguet shock wave, but the downstream density of the electrons is lower (fl~j 1016 cm since the Chapman—Jouguet hypothesis predicts a higher downstream temperature, at which helium is doubly ionized. In practice, however, the plasma heated by a shock wave cools down in the rarefaction wave so rapidly that the degree of ionization in the plateau region (a 1) appears to be even higher than the equilibrium one. The jump of the magnetic field in the Chapman—Jouguet shock wave (H~= 1.24 kOe) is close to the experimental one; the magnetic field downstream of the rarefaction wave, identified with H 0, should be 1.28 kOe, line CJ in fig. 27a. Calculating the ratio of the velocity of the flow outgoing from the shock front, U — v, to the characteristic velocity E. corresponding to the slow magnetosonic wave in a two-fluid plasma, one can directly verify that the solution obtained is close to that for the Chapman—Jouguet flow. This ratio cannot be less than unity, otherwise the rarefaction wave would overtake the shock front, but such a flow ahead of the magnetic piston cannot take place. For the Chapman—Jouguet ionizing shock wave (U — v)/e. = 1. If (U v)/E,> 1, which is characteristic of type 3 ionizing shock waves, the plasma flows away from the shock front more rapidly than the “head” of the rarefaction wave propagates in the plasma, so that the shock wave is separated from the rarefaction wave, though if (U v)!c, — 1) 4 1 the separation for short periods of time is too small to notice. Table 3 lists the values of the ratio (U v)/E. —

—

—

Table 3 H~,kOe H~.kOe~~

1

3 6 10

1.38 1.6 1.72

1.5

1.14 1.25 1.35

2.5

2

_________ 1.06 1.1 1.14

1.04 1.06 1.07

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

77

for different H5 and H0. As can be seen, the values presented differ not too much from unity, and this means that the dynamics of shock waves considered is described satisfactorily by the Chapman—Jouguet hypothesis. Thus, a numerical solution of the problem of the formation of a normal ionizing shock wave ahead of a rigid piston has shown that a flow arises, whose leading part contains an ionizing shock wave pushed by a cold (not radiating at frequencies v f/h) magnetic piston. In this case the conditions of the applicability of the gas-dynamic limit of the ionizing shock wave theory are satisfied artificially, and in accordance with this theory the ionizing shock wave formed must be the Chapman—Jouguet shock wave. It was shown, that normal ionizing shock waves do have much in common with Chapman—Jouguet shock waves; this refers first of all to their dynamics and to the induced electric fields. Such characteristics of shock waves as compression, temperature and the downstream degree of ionization, which depend more strongly on the flow type, differ considerably from the predictions of the theory based on the Chapman—Jouguet hypothesis. This is caused mainly by a finite rate of relaxation processes in shock waves, owing to which a shock wave cannot be considered as a narrow discontinuity preceding a wide rarefaction region; on the contrary, in this case the width of the rarefaction wave is less than the shock front width. Therefore, the calculations performed can be considered, to a certain extent, as the next stage after studying the flow, on the basis of the Chapman—Jouguet hypothesis, in an idealized problem of a piston when energy losses due to ionization and dissociation are taken into account [42, 511. In this case the conditions of the applicability of this hypothesis are satisfied (unlike the actual experimental situation, see section 14). Yet, the flow between the gas-dynamic shock and magnetic piston is simulated on the basis of the basic equations, without additional assumptions that the ionizing shock front is narrow and that it is precisely a centred slow MHD raref action wave, which exists in the downstream region. Let us compare the results of the calculations with experimental data. Figure 31 presents the dependence of the shock front velocity on the driving magnetic field, for H5 = 10 kOe and 12 kOe, as obtained according to our calculations (solid lines) and to the Chapman—Jouguet hypothesis (dashed lines), and also experimental data of ref. [46]. Similarly, in fig. 32 solid and dashed curves show the function E4(H0); experimental points are taken from that same work. As can be seen, the results of the numerical simulation agree satisfactorily with experiment, though one can hardly say that the agreement is much better than with the theory based on the Chapman—Jouguet hypothesis. Let us remind ourselves that the agreement is reached despite the fact that the calculated compression and temperature profiles do not resemble the experimental ones; this justifies the above statement on the weak dependence of the velocity of the shock front and electric field in it on the details of the shock front structure. The above calculations and the analysis of the experimental data available performed in section 14 permit us to teach the following conclusions. Firstly, it has been shown that the idealized model for plasma flow, used in the theory of ionizing shock waves, describes satisfactorily the flow character, at least qualitatively. In this case the conditions of the applicability of the gas-dynamic limit hold true, and as the theory predicts, normal ionizing shock waves formed have much in common with the Chapman—Jouguet shock waves. Secondly, we have established significant quantitative discrepancies between the characteristics of shock waves considered and those which follow from the Chapman—Jouguet hypothesis, though the latter is applicable. These discrepancies can be attributed to the fact that the width of the shock front is comparable to the size of the rarefaction region. This should be taken into account, while comparing the results of the idealized theory with experiment. Thirdly, it has been shown that the agreement of the velocities of normal ionizing shock waves and the induced electric fields, as determined in the Chapman—Jouguet model and in the above numerical

:

78

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields /

/

/

cm/p

/

/

:1__

// /

°2r

o’f2k~

/

II

1

2

~,k0e

////

_

04[)e

x—

1

2

/1,4-dc

Fig. 31. The shock front velocity of a normal ionizing shock wave f,o

Fig. 32. The downstream electric field E

as a function of the pushing magnetic field H5. Solid lines, numerical calculations; the dashed line corresponds to the Chapman—Jouguet shock wave, experimental points are from ref. [46].

4 as a function of the driving magnetic field H5. Solid lines, numerical calculations; the dashed line corresponds to the Chapman—Jouguet shock wave, experimental points are from ref. [46].

calculations, with experiment does not mean that this hypothesis describes correctly the basic physical processes in normal ionizing shock waves. The point is quite the reverse, to a certain extent: the shock wave characteristics, which depend weakly on the type of processes forming the shock front structure, are usually calculated successfully. Therefore, the agreement between calculation and experiment does not imply that those processes are taken into account appropriately in the calculations. Experimental results (see section 14) show that among those physical processes the main part is played by photoionization of the gas in the upstream flow due to radiation from behind the shock front and by friction between the ions and neutral particles, which leads to overheating of the heavy species. It is that flow region, which appears to be most essential, whose effect is considered negligibly small in the gas-dynamic limit. A correct description of normal ionizing shock waves by numerical simulation techniques requires, firstly, the use in the equation of magnetic field diffusion Ohm’s law in the form, which is adequate to the problem, and, secondly, an appropriate treatment of the discharge current region magnetic piston the radiation from which, as was shown in the calculations, contributes mainly to the ionization in the upstream flow. (The shock front is, as a rule, optically thin and its own radiation is weak.) Whereas the first problem is relatively simple to solve (see section 14), the solution of the second problem requires detailed information on the experimental installation where the shock waves are simulated. A slight modification of the above-considered numerical calculation procedure will permit the solution of this important problem. —

—

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magneticfields

79

16. Switch-off shock waves In this section we shall consider briefly one more type of “special” ionizing shock waves, namely the so-called switch-off shock waves. In magnetic hydrodynamics a switch-off shock wave is a slow one, and, for instance, in terms of plasma heating such a shock wave is the strongest one among all slow shock waves. This leads, in particular, to some interesting features of such shock waves. Ionizing switch-off shock waves are a natural analogue of such shock waves in a preliminarily ionized gas. In other words, an ionizing switch-off shock wave is a shock wave, such that in it the upstream magnetic field is at some angle to the shock front plane, and the downstream magnetic field is normal to the shock front plane. It is obvious, thus, that switch-off shock waves are plane-polarized. It will be shown below that supersonic switch-off shock waves of types 3 and 4, as well as subsonic ones of type 4, are possible. The evolutionarity condition implies in this case one additional boundary condition for type 3 supersonic shock waves and type 4 subsonic shock waves. Type 4 supersonic shock waves require no additional boundary conditions [23]. As in section 6, let us describe the equation of a phase trajectory in the (vi, H) plane and Ohm’s law in dimensionless variables: /

dH

i

1

(5

F~~(v5, H)

4(v5 — 1)(v5 —

HI

1\

VGD) + \

—~ JVI

~. rj2 ‘~I1

aO/

[J2 iTL~j

A,f2 IVI aO

=

0,

(16.1)

(16.2)

It can easily be seen that for switch-off shock waves the electric field in the shock-fixed coordinate system is zero, while the dimensionless upstream electric field in the laboratory coordinate system is 2~oIH. (16.3) E = MaoIl 1/M The zero right-hand side of eq. (16.3) in the (v 5, H) plane corresponds to two straight lines (degeneration of the hyperbola) —

H=0,

v5=1/M~0.

(16.4)

The intersection of the phase trajectory with the straight line H = 0 corresponds to the solution for switch-off shock waves (in the (vi, H) plane). Typical magnetic structures for supersonic switch-off shock waves (in the limit M0 —~ce) are shown in fig. 33 for: A10<~(a, b); ~<1W’~0<1 (c, d); 1 2. As can be seen from fig. 33, switch-off shock waves can only be of types 3 and 4 for 0< Mao <2 (the downstream singular points are located below the straight line v5 = 1/M~0). A peculiar character of the curves, which present in the (v5, H) plane the structure of the Chapman—Jouguet shock wave, is a specific feature of switch-off shock waves. In all other cases the Chapman—.Jouguet shock wave corresponds to the tangency between the zero-current hyperbola dH/dx = 0 and the phase trajectory F(v~,H) = 0. In this case the supersonic type 3 singular point can coincide with the subsonic type 4 singular point (fig. 33b) only on the sonic curve, i.e., for M3 = M4 = 1,

:

80

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields 7p

X

(b)

(a)

—

2

-——

—

(d)

(C)

iiio~5/:~iir:~

(f)

(e)

ft:~:

_

Fig. 33. Magnetic structures of supersonic switch-off ionizing shock waves in the limit; (a) M~ M~o<2/5; (c) 2/5< M~o<1, H0
which implies t9F/8v5 = 0.

(16.5)

Points 3 and 4 coincide on tne straight line H = 0, along which also ~1F/ÔH=

0.

(16.6)

If conditions (16.5) and (16.6) hold true simultaneously, the regularity of curve (16.1) at the point 3 = 4 is violated. At this point the direction of the vector tangent to the curve (16.1) suffers a discontinuity (fig. 33b). Three equations (16.1), (16.5) and (16.6) in the variables Vx, H, H0, Mao determine the Chapman—Jouguet line (CJ) in the plane of the parameters (Mao, H0) (see fig. 34). It can be seen from fig. 33 that supersonic switch-off ionizing Chapman—Jouguet shock waves can exist only if M~0 The Chapman—Jouguet condition corresponds, as usual, to a maximum value of the electric <~.

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magneticfields

~j

81

~

“70 / 2 I

I 7

I

~

(E”.E~)

I-

t.~

0.5— ci ‘/215

/

2Mg,

Fig. 34. Regions in the parameter (M,,~,H

5) plane where supersonic type 3 ionizing shock waves (curve I~)and type 4 shock waves (region L~)exist in the electrostatic breakdown limit, for M5-~ ~ and E~= 0.6.

field E in the gas-fixed coordinate system: E E~,i.e., for M~0<~ the solutions can exist only if Fl0 < .H~0cj(M’ao). It can easily be shown, using (16.1) and (16.4), that switch-off type 3 shock waves (H3 = 0) exist only for M80 < 1 (see fig. 33c, d), if the following condition is satisfied: 2. (16.7) H0 Hmax(.M~ao)= [~(1— .M’~0)(4— .M’~0)]” The corresponding region in the (Mao, H 0) plane is presented in fig. 34. Here, as in section 6, the additional boundary conditions are determined by eq. (6.3), in which E5 should be taken to be zero and the integration contour changed appropriately. A general analysis of the solutions of these equations, that shows, which of the solutions, type 3 or type 4, exists for given values of Mao and H0 (and if it is a type 4 solution, where the point a, the beginning of the isomagnetic shock, is located), is completely similar to that performed in section 6. Relation (6.3) implies one additional boundary condition for type 3 supersonic ionizing shock waves and none for type 4 shock waves. For R ce or E~—*0 only MHD switch-off shock waves can exist, and Mao 1. Let us consider, as an example, magnetic structures for switch-off shock waves in the electrostatic breakdown limit (R 0), in which the additional relations can be written in the explicit form —*

—*

}

E3=E~, O_’E4
(16.8)

where E~can be determined, for instance, from (5.8). Denoting, in this case, the characteristic Mach number as M~= E~,we can rewrite condition (16.8) in the form

MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

82

F

Homin~

aoM~

AK2 L1VJaOI.

1

,Hocj(Mao)I.

(16.9)

J

Switch-off shock waves of type 4 lie inside the region determined by (16.9), and type 3 switch-off shock waves lie on its boundary (13 in fig. 34). As can be seen from fig. 34, type 3 shock waves do not exist for large M~(for M~>1.14, to be exact). In the inverse limiting case, E~= M~”40, Mao = 1 can be singled out for each finite H0 with the help of (16.9) that corresponds to switch-off shock waves in magnetohydrodynamics. Let us consider subsonic switch-off ionizing shock waves (Mao < 1) in the electrostatic breakdown limit. Here Mao for a given initial state cannot be considered as a parameter independent of VGD, as was the case above, -2 — v~ a 2 — ~ oi-’m~ 6 Mo~— C~ oH~/4i~0

AK2 lvi aO

.~ ~

—

16 10

I

(subscript 1 indicates that now the parameter f3~is not the ratio of the gas pressure to the magnetic one in the initial state: /3~= 8irNoTo/(R~+ 1’~~)).The parameters Mao, H 0, f3~characterize a subsonic switch-off ionizing shock wave. The magnetic structures in the (vi, H) phase space are shown in figs. 35(a) and (b) for M~0<~ and in fig. 35(c) for ~< M~< 1. Switch-off shock waves do not exist for M~0>1. As can be seen from fig. 35, subsonic switch-off shock waves are of type 4 and represent rarefaction waves for M~0< ~ or compression waves for M~0> Like the case of supersonic shock waves, the derivative on the curve (16.1) suffers a discontinuity at the point 3 = 4 (see fig. 35(b)), corresponding to the Chapman—Jouguet condition. The only condition required for the evolutionarity of subsonic ionizing shock waves is, in this case ~.

E4=E~.

(16.11)

____ _____ (a)

(b)

.~81

to

~

/~o

~

/

‘~

J

:~‘~ ~

Fig. 35. Magnetic structures of subsonic switch-off ionizing shock waves: (a) M~0 <2/5; (b) Chapman—Jouguet shock wave for M~o< 2/5; (c) 215
Fig. 36. Curves in the parameter (Ma5, Ho) plane corresponding to subsonic switch-off ionizing shock waves in the electrostatic breakdown limit; (a) ~ < 12/25; (b) 12/25 <$~ < 16/25, E,,, > E,2 (c) 16/25<$~<6/5; (d) 6/5
MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

83

Figure 36 shows the curves in the (Ma0, H0) plane, along which (16.11) is satisfied and subsonic switch-off ionizing shock waves can exist. Figures 36(a), (b), (c) and (d) correspond to f3.~< 12/25; 12/25 < $~< 16/25; 16/25 E~2) the region of the existence of switch-off shock waves splits into two parts. As might be anticipated, only an MHD switch-off subsonic shock wave remains in the limit E~—*0: Mao(Ho) = 1 (/3~>6/5).

Subsonic switch-off ionizing shock waves are similar, in many respects, to slow combustion waves. They do not contain the density shock, as in a gas-dynamic shock wave, and the gas is heated in such waves due to the energy of the magnetic field corresponding to the switch-off transverse component. 2 1 the pressure of a These are transverse slow (subsonic, sub-Alfvénic), in no way weakgas-dynamic ones: for H0pressure strong waves upstream field may balance thebut high downstream ~‘

N 4T—H~o/8ir, whence N H~0

2.3

x 103 (106 crn3)

1/2

1/2

(13.6~v) Oe.

The fact that Chapman—Jouguet subsonic switch-off ionizing shock waves can exist means that such shock waves could exist, under certain conditions, in electromagnetic shock tubes. At present, we have no information on experiments with switch-off ionizing shock waves. Moreover, the technical aspects of such experiments have not so far been understood completely. We believe such experiments to be of great interest. The authors are grateful to Prof. I.M. Lifshitz for his stimulation and support and to Prof. Ya.B. Zel’dovich for his useful comments on the manuscript. References [1] Ya.B. Zel’dovich and Yu.P. Raizer, Physics of shock waves and high-temperature hydrodynamic phenomena (Academic Press, 1966). [2] AG. Kulikovskii and GA. Lyubimov, Doki. Akad. Nauk SSSR 129 (1952) 52 [Soy.Phys. Doklady 4 (1953) 1185). [31C.K. Chu and R.A. Gross, in: Advances in plasma physics, Vol. 2, eds. A. Simon and W.B. Thompson (Interscience, 1969). [4] Ya.B. Zel’dovich, Zh. Eksperim. i Teor. Fiz. 10 (1940) 542. [5] Ya.B. Zel’dovich, Theory of combustion and detonation of gases (in Russian; Moscow, Izdatel’stvo AN SSSR, 1944). [61 AL. Velikovich and M.A. Liberman, Zh. Eksperim. I Teor. Fiz. 73 (1977) 891 [Soy.Phys. JETP 46 (1977) 469]. [7] MA. Liberman and AL. Velikovich, Plasma Phys. 20 (1978) 439. [8] AL. Velikovich and MA. Liberman, Usp. Fiz. Nauk 129 (1979) 377 [Soy. Phys. Uspekhi 22 (1979) 843]. [9] CF. Stebbins and G.C. Viases, J. Plasma Phys. 2 (1968) 633. 110] S.H. Robertson and Y.G. Chen, Phys. Fluids 18 (1975) 44. 1111 B.P. Leonard, J. Plasma Phys. 10 (1973) 13. [12]B.P. Leonard, J. Plasma Phys. 7 (1972) 133. [13]J.E. Anderson, Magnetohydrodynamic shock waves (M.I.T. Press, 1963). (14] A.I. Akhiezer, IA. Akhiezer, R.V. Polovin, AG. Sitenko and K.N. Stepanov, Plasma Electrodynamics (Pergamon, Oxford, 1975). (15] P. Gennain, Rev. Mod. Phys. 32 (1960) 951. [16] M. Jaffin, Phys. Fluids 8 (1965) 606. 1171 A. Jeffrey and T. Taniuti, Nonlinear wave propagation (Academic Press, 1964). (18] 0.1. Barenblatt and Ya.B. Zel’dovich, Priki. Mat. Mekh. 21(1957) 856.

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MA. Liberman and AL. Velikovich, Physics of ionizing shock waves in magnetic fields

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