On the formation of shocks in warm magnetized plasmas

23 November 1998 PHYSICS LETTERS A

Physics Letters A 249 (1998) 93-98

ELSEVIER

On the formation of shocks in warm magnetized plasmas * Anders odblom

’

Institute for Electromagnetic Field Theory, Chalmers University of Technology and EURATOM-NFR Association, S-412 96 Gdteborg, Sweden

Received 19 December 1997: accepted for publication 26 August 1998 Communicated by M. Porkolab

Abstract An exact analytical solution of the nonlinear ideal magnetohydrodynamical equations for the formation of either perpendicular or parallel shock waves in a warm, isothermal or adiabatic, magnetized plasma is presented. A condition for whether or not an initially smooth wave profile of arbitrary amplitude steepens and forms a shock within a finite time is outlined, and the functional dependence of the shock formation length on the wave amplitude is given. @ 1998 Elsevier Science B.V.

1. Introduction Stationary shocks occur, for example, in space plasmas as the result of the interaction of the supersonic solar wind with celestial bodies. The bow shock around a planet is formed when the solar wind is forced to flow around the obstacle, and the shock is established as a balance between the kinetic and the magnetic pressure contributions, see for example Ref. [ 11. Another, formally very different case, is encountered in laboratory plasmas, when the plasma interacts with material surfaces. The kinetic energy is in this case transferred into the electrostatic potential energy of the stationary shock (= sheath), cf. e.g. Ref. [ 21. The macroscopic properties of fully established shocks can be expressed in terms of jump conditions, which are based on conservation laws, across the shock [3,4]. For the internal structure of the shock, as well as for the shape of nonlinear solitary waves, *This work has been supported by the European Communities under an association contract between EURATOM and Sweden. ’ E-mail: [email protected].

deviations from quasi-neutrality are important. Analytical solutions for solitons propagating in a direction which is either parallel, or perpendicular, to the magnetic field lines are given in Refs. [5,6] for dusty adiabatic plasmas. The solitons are stationary in the reference frame moving with the wave. Similarly, the structure of a stationary sheath layer, where a parallel shock (i.e., the shock normal is parallel to the magnetic field lines) is formed, is given in Ref. [ 71, and is generalized to an oblique sheath in Ref. [ 81. In the present paper we give exact analytical solutions to the nonlinear magnetohydrodynamical (MHD) equations for the propagation of a perturbation in a warm plasma either across or along the magnetic field lines, respectively. It is stressed that the analysis applies not only to small amplitude waves. In fact, the solutions are valid for perturbations with arbitrary amplitudes. The initially smooth wave profile is shown to either smooth out further, or to steepen and eventually form a shock within a finite time. Conditions are outlined for when a shock is formed. An expression for the shock formation

0375-9601/98/$ - see front matter @ 1998 Elsevier Science B.V. All rights reserved. PII SO375-9601(98)00678-l

94

A. &%lotn/Physics

length is given, and neglect of collisionality is justified as long as the shock formation length is shorter than the dissipative scale length. Analytical solutions are given for an isothermal plasma, and the results are generalized to include the effects of an adiabatic equation of state. The analysis includes solutions for parallel (k 11Be) shock waves where the dynamics is governed by the electrostatic contribution introduced by the pressure gradient term, and, conversely, the formation of perpendicular (k I Be) shocks, including electrostatic and electromagnetic contributions, for which an elegant solution was given recently for a cold plasma [ 91.

Letters A 249 (1998) 93-98

shocks considered in Ref. [ 91. Note that the pressure gradient term in Eq. (2) does not generate any magnetic field provided that the pressure is determined by an adiabatic equation of state, p = p(p) , since ~3,B cx vp x vp. In order to illustrate the similarities of the dynamical behavior of the plasma fluid velocity and that of the magnetic field it is noted that the magnetic force term can be rewritten in the form jxB=--

BVB

+ (B.V)B

CL0

PO

’

We adopt the following ideal MHD model for the nonlinear evolution of the perturbed state, where the convective inertia term plays an essential role in the dynamics of the formation of the shock,

which explicitly shows the convective character of the magnetic field. The formation of shock waves in a magnetized plasma with an external constant magnetic field BO is considered. Without loss of generality the planar shock is assumed to propagate in the f-direction, where all spatial variation is across the shock front, i.e. u = v,(x) P, and B = i?[Bo + B, (x) 1. The dynamical evolution is given by the following equations,

CY,p+v.pv=o,

&p + &,pu = 0,

(4)

&B+&Bu=O,

(5)

2. Formation

of shocks in isothermal

p&u + p(v - V)v

= j x B - Vp ,

plasmas

(1)

where p, v, j and B are the density, the fluid velocity, the current density and the magnetic field, respectively. p denotes the electron pressure, and we consider initially an isothermal plasma with cold ions. The effects of deviations from constant electron temperature is considered in the next section. The generation of magnetic fields and currents are given by Maxwell’s equations in the MHD limit, viz. V x E = -&B and V x B = poj, where the electric field E is governed by Ohm’s law, E+uxB+%(Vp-jxB)=O.

(2)

The ratio of the pressure term to the motional term in Eq. (2) scales as ( p;/LP)cS/u sin 8, where USB/uB = cos 0, and L, = [d(lnp) /dx] -’ denotes the scale length of the pressure variation, p; = cS/O is the ion Larmor radius, 0 = eB/M is the ion cyclotron frequency, and c, = (T/M) 1/2 is the ion-acoustic velocity. The electrostatic contribution to the electric field is important if either 0 is small, i.e. for nearly parallel shocks, or for small amplitude waves u < c,. Furthermore, for cS = 0 we recover the cold electromagnetic

where the normalization p/a 4 p, (Bo + BI ) /Bo -+ B, vX/vA -+ v, c,/vA + c, and v,t -+ t has been applied, and the Alfven velocity VA = Bo/(,u~//il)‘/~. From Eq. (4) and (5) we find directly that p = PB, where p is an arbitrary constant. Thus, the plasma is frozen into the magnetic field. The role of p is described by the relative ratio between the terms for the pressure-gradient and the magnetic force, respectively, in the momentum equation (6), pact_----. B2/2po

2/w P

(7)

Thus, the momentum of the shock is essentially governed by the Vp, i.e. the shock is electrostatic, when p2c2 > p, whereas it is electromagnetic in the opposite limit. Alternatively, the electrostatic (electromagnetic) limit corresponds to high (low) plasma p, cf. Eq. (7). It is worth to note that when the density is low, then the plasma p is large, which is an aspect of the frozen-in condition, p c( B. The constant of proportionality, p, can be eliminated from Eqs. (4)-( 6)

A. &Mom/Physics

Letters A 249 (1998) 93-98

by a renormalization of the magnetic field, B = B//l, where the bar-notation, for simplicity, is omitted in the subsequent text. Note, however, that the constant B reenters in the dimensional results. The characteristic feature of the shock is that the nonlinear convective term drives the evolution and the formation of the shock. We therefore look for solutions where not only p = p(B) but also u = u(B), or equivalently B = I3(u), which yields &u+

[u-t

(d,,lnB)-‘]d,u=O,

a,o + [u + d,,B( 1 + c2/B)]&u

= 0.

plasma. In the opposite limit where the pressure gradient term is neglected in the momentum equation, we obtain the formation of the cold perpendicular electromagnetic shock waves considered in Ref. [ 91. The solutions of Eqs. (4) -( 6) in the limits given in Eq. ( 12) read in the electrostatic case p = po exp( *v/c), and for the electrostatic potential ec$/T = flu/c, where ,B is an arbitrary constant, and in the cold electromagnetic case B = ( vo * ~)~/4, respectively. Consider as an example an initial pulse in normalized units of the form

(8) p(x,O)=$

However, since there can be only one equation describing the evolution of the velocity, we can from Eqs. (8) identify the convective velocity ui as

(9) From Eq. (9) we find the solutions At first, dB du=

for both u and ur .

l

(10)

&&

which integrated

gives the solution for the velocity,

&u=ua+2c[+2c[+cln

5-l I 5+1’ I

(11)

-+vhh=(3ufvo)/2, The first limit in Eq. mation of electrostatic magnetized plasma, or magnetic field lines in

B-+0, c-0.

(

(12)

(12) corresponds to the forshock waves either in an unshocks propagating along the a homogeneously magnetized

,

l-2arctan.r T

>

(13)

which is a convenient choice for further analytical and numerical treatment. This pulse evolves according to the relation x = (ufdm)t+tan

[;(I-:)],

(14)

where v = v(p) is given in Eq. ( 11) . The nonlinear convective derivative leads to a selfsteeping of the wave, and the wave breaks at t = tr , i.e. when dp( tr )/dx = 00 or 11 G -dxo/dvr, where xa = x( p, t = 0), which for the initial profile equation ( 13) yields t1 = f

%.@-Gm

T/PO

cos2[ +r( 1 - 2p/pa)

where 5 = dm = dm, and ve is a constant of integration to be determined by the initial condition. Moreover, the use of Eq. (10) in Eq. (9) gives the convective velocity vr as 01 = v f ct. Thus, Eqs. (8) reduce to the equation I&U+ ur &v = 0, and the general solution of this nonlinear equation is of the form v( x, t) = u( n-vi r). By tracing the contributions from the pressure gradient term and the j x B-force term in Eq. (6)) we obtain the following limits for the convective velocity, v, -+UfC,

95

]

2c2 + 3p/p2

(15) The solution with the positive sign in Eq. ( 11) (and Eq. (15)) always breaks after some time, whereas the other solution may smooth out, or steepen, depending on the amplitude of the wave. Indeed, the solution with the minus sign breaks only if -(c/3) 2 6 p < -2(~/3)~/3, cf. Eq. (15) and Figs. 1, 2. The breaking of the wave starts at t = t* e min[ tl ] , and at the density p*. For large amplitude waves, po > ( cP)~, a first estimate based on Eq. ( 15) of p* yields p* = po/2. However, a more accurate result can be found from the fact that at, (p* ) /ap = 0, and dt, /dp cx 4n-p( 2c4@ +5c2p2p+ 3p2) cos( ~p,‘po) (4c4@ + 6c2p2p + 3p’)po sin( rp/po). Thus, in the limiting cases of very large or very small amplitude waves, respectively, we find that p* is given by the solution to tan6 = f(l) where f = 45 and f = 2J, respectively, and 5 = ~p*/po, with the

96

A. tidblom/Physics

Letters A 249 (1998)

93-98

I

I

l.

I

:

Q

p_

; : : : ;

0.8

0.6

$

s

4

0.4

20 O,2 10

t* 4

-0,8

-0,6

-0,4

-0,2

I

0 -40

0

-20

0

20

Fig. I. The wave breaking time, tl. in EQ. ( 15) versus density p in a case where -(cP)~ < po < -2(cp)*/3 (po = -0.9, c = 1, p=

I).

solutions 5 z 1.39 and LJ x 1.16. Thus, approximate solutions for the shock formation time read t* N (2/3)dmsinm2 [ = 1.44jpI/&j when p2c2, and t* z (l/c) sinm2 [ x 1.38/c when IPI @C P 2c2. The shock formation length, Ax* = 01 (p*) t*, can be estimated as p >

Ax* = 2.87 + vote ,

60

40

X

Density p

Fig. 2. The formation malized to pcl in the (po = -0.9, c = 1, t* x 42 (solid) and

4

I

of a shock in the density p_ profile norcase where -(cp)’ < po < -2(cp)*/3 p = 1, LQ = 0). The profiles are taken at at t = 0 (dashed), cf. Fig. I.

I

I

I_ _- - - - _ - _ _-*,r:‘-

Lurge

__/. -.

amplrtude estimate

____.-•--- . - _- J_-

p >> p2c2 ,

= 0.85 + 1.381n lpo/ (Pc)~~ Small amplitude estimate

+uot* ,

JpJ< p2c2.

(16)

The shock formation length of the p--solution fulfills Ax: (p. = -c2) x -6.2 assuming va = 0 and p = 1, and Ax? -+ -co when po -+ -2~~13. The good agreement of the asymptotic value, Eq. ( 16)) and the exact numerical solution for the formation length is illustrated in Fig. 3. From Eq. (16) it is concluded that for an initial profile of the form given in JZq. (13) all p+-waves will sooner or later form a shock, and it is only a matter of propagation distance before breaking sets in. Small amplitude waves propagate a long distance before the wave front is steepened, IAx* 0: Iln[po/(fl~)~]\ >> 1, a situation considered in Fig. 4. However, large amplitude waves form shocks within a distance comparable to the scale length of the initial pulse, cf. Fig. 5. Note that only one of the backward/forward-propagating waves (p+) forms a shock in Figs. 4, 5, whereas the

-6’

’

I

I

I

I

I

0

2

4

6

8

10

Amplitude, p,

Fig. 3. The shock formation length, Ax;, as a function of the wave amplitude, po, in comparison with asymptotic forms, Eq. (16). for c= 1, p= 1, qj=O.

other wave (p_) profile smoothes out. However, an example of the case where the p--wave breaks is shown in Fig. 2, and for the same set of parameters the p+-wave has already reached the breaking point at a much earlier instant as concluded from Fig. 1. Guided by the results in Figs. 2, 4, 5, a simple but correct condition for wave breaking can be formulated: an

A. Cidblorn/Physics

(a)

’

Letters

A 249 (1998)

91

93-98

I

(4 : : :

P_ I_ : :

P+

: 0,2

0 -40

-20

20

0

40

60

-15 -40

-20

0

20

40

60

’ -15

I

I

I

I

I

-10

-5

0 x

5

10

X

Fig. 4. The formation of shocks in (a) the density p profile normalized to po. and (b) the wave velocity 01 profile for a small amplitude wave (po = 10-9, c = 1. ZQ = 0, fi = 1). The profiles are taken at r; z 1.38 (solid) and at I = 0 (dashed).

initial profile evolves into a shock provided that the rear part of the wave moves in the direction of, and with a higher velocity than, the slower leading edge. The magnitude and the direction of the magnetic field generated by the evolving shock is significantly different in the limiting cases shown in Figs. 2, 4, 5 (c = 1, ug = 0, j3 = 1) . The perturbed magnetic field compared to the background field is in the three cases given by -2 < Bl/Bo < -1 (Fig. 2), Bl/Bo NN-1 (Fig. 4), and Bl/Bo > -1 (Fig. 5).

15

Fig. 5. The formation of shocks in (a) the density p profile, and (b) the wave velocity ~11profile for a large amplitude wave (~0 = 10, c = 1, r~ = 0, p = 1). The profiles are taken at I; N 0.44 (solid) and at t = 0 (dashed).

3. Effects of non-isothermal

plasma

In the present section is considered the effects of non-isothermal plasma conditions. The adiabatic equation of state reads Y

p=po

(> L PO

7

(17)

where y = 1 in an isothermal plasma. The modified momentumequation (6) reads dtu+ud,u+Bp-‘d,B =

98

A. &Mom/Physics

-~~p~-~&,p, where c = cs/u~ and cf = ypa/po. Accordingly, Eq. (10) now takes the form dB du=

*

B

(18)

1 + c~BY-~

and the solution

for the velocity

C-l/(Y--2)Fy(c2/(Y--2)B),

reads fv

Letters A 249 (1998) 93-98

both of them. pends on the wave. Shocks the rear parts leading edge,

Whether of not a shock is formed relative convective velocity within are formed if the convective velocity of the wave is in the direction of and with higher speed.

dethe of the

= uo +

Where

Acknowledgement (19)

The general solutions to the integral in Eq. (19) can only be found numerically, but analytical solutions for some special y’s (y = 3/2,2, and 513) can be found in Ref. [lo] where the expansion into a magnetized vacuum of a laser-produced plasma is studied.

Overall inspiring and stimulating discussions about shock waves with Dr. S. Krasheninnikov, Plasma Science and Fusion Center, MIT, and the helpful discussions with Professors D. Anderson and M. Lisak, Chalmers University of Technology, are gratefully acknowledged.

References 4. Conclusions An exact analytical solution for the formation of shock waves propagating perpendicular or parallel to an external magnetic field in a warm magnetized plasma is presented. The analysis is based on the nonlinear ideal MHD equations which include electrostatic and magnetic contributions to the dynamics of the evolving wave in an isothermal or an adiabatic plasma. It is shown that large amplitude waves steepen and form shocks within a distance comparable to the initial scale length of the wave, whereas small amplitude shocks are formed after much longer distances of propagation. Shocks may be formed for either one of the backward/forward-propagating waves, or for

Ill R. Bingham, Plasma Physics: an Introductory Course, R. Dendy, ed. (Cambridge Univ. Press, Cambridge, 1993).

121 F.F. Chen, Introduction to Plasma Physics and Controlled Fusion, Vol. 1: Plasma Physics (Plenum Press, New York, 1984). (31 A. Burgi, J. Geophys. Res. 96 (1991) 17689. [41 Yu.A. Shchekinov, Phys. Lett. A 225 (1997) 117. to Plasma Physics r51 W.B. Thompson, An Introduction (Addison-Wesley, Reading, MA, 1962) pp. 86-95. t61 P Meuris, F. Verheest, Phys. Lett. A 219 (1996) 299. [71 D. Bohm. The Characteristics of Electrical Discharges in

Magnetic Fields, A. Guthrie, R.K. Wakerling, eds. (McGrawHill, New York, 1949). [f31 R. Chodura, Phys. Fluids 25 (1982) 1628. 191 L. Stenflo, A.B. Shvartsburg. J. Weiland, Phys. Lett. A 225 (1997) 113. (101 D. Anderson, M. Bonnedal, M. Lisak, Phys. Ser. 22 ( 1980)

507.