Inductive interaction of electroconductive bodies with ionospheric and interplanetary plasma

Acta Astronautica Vol. 8, pp. 47-56 Pergamon Press Ltd., 1981. Printed in Great Britain

0094-5765/8110101-0047/$02.00/0

Inductive interaction of electroconductive bodies with ionospheric and interplanetary plasmat A. V. G U R E V I C H

AND N. T. P A S C H E N K O

Intercosmos Council, U.S.S.R. Academyof Sciences, LeninskyProspect 14, 117901 Moscow V-71, U.S.S.R. (Received 15 April 1980)

Abstract--The hypersonic hyperalfv~nplasma flow around electroconductivebodies is considered and electromagnetic inductive plasma-body interaction is analyzed. The possible applications of results obtained in this analysis to body motion in ionosphere and interplanetary plasma are discussed. Introduction IT ts of interest to consider plasma flow with frozen magnetic field around the electronconductive bodies. This is an interesting problem both for magnetohydrodynamics itself (Kulikovski, 1962; Kadomtsev, 1976) and for application to the problems of body motion in ionosphere and interplanetary plasma. There are two main contributions to interaction of plasma and streamlined body. The first contribution is a result of particle collisions with the body surface, it leads to usual hydrodynamical friction and particle redistribution around the body. The second one is determined by the electric field induction on the streamlined body. The induced electric field generates a surface current which, in turn, induces electric field perturbations transferred into plasma along magnetic force lines. The electric field around the body is changed and this effects on plasma velocity. So, besides hydrodynamicai action there is an electromagnetic interaction bwtween incident plasma flow and streamlined body--so called inductive interaction. The perturbations arising in plasma result in effective braking force, lift and lateral force. The present paper is devoted to investigation of this interaction which is of purely electromagnetic nature. General relation The model problem of steady-state plasma flow around electroconductive plate is considered in the frame of ideal magnetohydrodynamics (Kulikovskij,

tPaper presented at the XXXth Congress of the InternationalAstronautical Federation, Munich, F.R.G., 17-22 September 1979. 47

48

A . V . Gurevich and N. T. Paschenko

1962; Landau and Livshits, 1962) div 15t5= 0

div/~ = 0,

l

rot [~/3] = 0, --

--

g V g = - ~ grad p + [rot BB]/47r15 = 0

(1)

g grad p15-~ = 0. Here, p, IS, v are pressure, density and plasma velocity respectively, /~ is the magnetic field. Let unperturbed plasma flow be written in the following form: f=v0~l,

/~0=B0~3,

15=p0,

p=P0.

(2)

The plate is supposed to be infinitely thin, it is located along incident flow with normal ti = n2~E+n3~3 (Fig. 1), so it does not induce purely hydrodynamical perturbations and in the absence of inductive interaction the unperturbed flow (2) is not changed. Let us assume that the plate has a small anisotropic surface conductivity

where EH is the Hall and Ee the Pedersen integral surface conductivity. The electric field induced by plasma flow generates a surface current on the plate

[ = ~: ~, = - ~ 1 [e g],,

(3)

and this results in plasma flow and magnetic field perturbations which we can write in the form = go + Voa,

B = Bo + MaBob,

15 = Po + Mpop.

I

es--n

i i

et = e t

Fig. 1. Electroconductive plate with normal ti.

(4)

Inductive interaction of electroconductive bodies

49

Here ~, /7, p are the non-dimensional perturbations of velocity, magnetic field and density respectively, M = Vo/Co, Ma = VO/VA, and

are the sound velocity and Alfv~n vel6city of unperturbed flow. Consider the case of small surface electroconductivity, leading to plasma flow perturbations induced by conductive plate small enough as to allow to restrict oneself by consideration of the linearized problem. In this case eqns (1) can be written in the form

o3.1 aX l

1 (o~bl ab3"~ M A \-~3

1 ap

tgX l /l "~ M "~X l ~" 0 '

eu2 + l__l_(eb3 8 b 2 ~ + l S P = O , CgXl MA\aX2 aX3! M ax2 au3 1 ap aXl + M ~ = O '

abl ax,

1 aul M~ax3

Ob3 1 au3 M ap =0, ~Xl MA c9X3 Ma axl

0,

ab2 ~Xl

1 au2 = 0,

(5)

M A aX3

abl + ab2 + ob3 -~1 ~ -~3 = O,

ap , 1/au,. ou + aUq=o. ex, '- Xi' Tx, *

ex,:

One can pose boundary conditions using a continuity of normal magnetic field and tangent electric field components, a hydrodynamic flow condition for plasma velocity at the plate and using the fact that the discontinuity of tangent magnetic field component is determined by surface current. This leads to the following relations: {[ti/~]}~ = [,

{[tDi]}, = O,

(~/ff)[~= O.

(6)

Here {a}, means the magnitude of discontinuity at the plate considered, and [ = 41rf/cMABo. These boundary conditions with the condition of incoming waves absence are sufficient to resolve uniquely the equation system (5). Following the previous analysis (Gurevich et al., 1978) consider the case of hypersonic, hyperalfv~n plasma flow around the plate and let the following condition hold M >>Ma >> 1.

(7)

Under this condition the equations system (5) is split into three subsystems describing Alfv~n, fast and slow magnetoacoustic waves, the last two being

50

A.V. Gurevich and N. T. Paschenko

interrelated by boundary conditions. For Alfvbn waves we have: OU 1 OXl

1 Obl= O, Obl

1 Oul= O, OXI MA OX3

MA OX3

{b,}~ =fp(x,, x2),

{u,}~ = O.

(8)

For fast magnetoacoustic waves:

&u2++ l_l_{Ob3+ &b2+]=0 '

Ob2+

OX1

3XI

MA \ OX2

OX3 }

Obz++--=O,

p+

OX2

1 &uz +=0, m m a OX3

Ma + =-~--b3 ,

(9)

{(nzb3 +- n3b2+)}3 =fH(Xl, X2),{U2+}~ = O. For slow magnetoacoustic waves:

3u3-_{ 1 3p-=o, &xt

M 8x3

Op1 8u3-_0, ~ -~ M 8x3

(n2u2++n3u3-)s = 0 ,

{/13 }s = 0 .

(10)

Here: fp = 4rrY.pvAn3[c,

fn = 4zrEl-lVAn3/c.

We can see that in the approximation accepted the Alfv~n waves are generated by Pedersen current component, and magneto-acoustic w a v e s - - b y Hall current component. Note that in the case of plate normal to magnetic field n2 = 0 (/~0-LS), the slow magnetoacoustic wave is not generated at all.

Alfv~u waves To begin with, consider the wave generated by Pedersen current component. From (8) we obtain an equation for bl MA 2 02bl OXl2

02b3= 0, 3x3 2

{b,}.~= fp(xl, x2),

cJX3Js

The solution of eqn (11) has a form:

1 f fe(X~o, X2o), (n2x3 -F n3x3) > 0, bl = 2 ~ - fp(X;o, x20), (n2x2 + n3X3) < O,

(12)

Inductive interaction of electroconductive bodies

51

where: X~o = Xl +--MA(n2X2 + n3x3)/n3,

x20 = x2,

/p(Xlo, x20)= 4~'VAn3~;,p(Xlo, X20)/C.

So, as one can see from 02), the magnetic field perturbations generated by Pedersen current near the point (xl0, x20, - n2x2o/n3) at the plate S are transferred along the characteristics of eqn (ll) passing through this point: x3 = - - ~ x2 +- M-M--~ (x' -

x2= x2°"

For velocity perturbations we obtain from (8) - bl, ul =

bl,

(n2x2-1- n3x3) > O, (n2x2 + n3x3) < O.

Let us determine the current structure in plasma: ] = 4--~ rot/~ = +-V°Bcn3rot ~

p

(x~0, x20)~l,

where sign " + " is for (n2x2 + n3x3) > 0, and " - "-for (n2x2 + n3x3) < 0. The main current is concentrated at the discontinuity surface-at characteristics passing through plate edges. If plate boundary is determind by the equation F ( x l , x2) = 0,

n3x2+

n3x3 = O,

so the AIfv~n characteristics passing through boundary edges form two cylindrical surfaces F(x~o, x20) = 0, where x~o, x20 are determined by (12). The surface current flows along the cylindrical surfaces: =

is - volBol

2c

[n3l ~ (XTo, X2o)~l, "~p

(13)

where 8t - 8 the function for the surface F (Vladimirov, 1976; Gelfand, 1958). Flow perturbation patterns in the case of constant plate conductivity are shown in Fig. 2. The low-intensive discontinuity is separated from the plate edges and propagates with the velocity VA. The perturbations are located only inside the region restricted by discontinuity surfaces. They are generated by the surface current/~ flowing on the discontinuity surfaces. This current closes the electric current in plasma (4) which is generated at the conductive plate by induced electric field P. = -[~Bllc.

52

A.V. Gurevich and N. T. Paschenko

Fig. 2. Perturbation pattern--Alfv6n waves. One can apply the linear approximation under the condition B~ ~ B0 which gives: P2 = 4"rrY~pVo/C2 "~ 1.

(14)

This condition has a simple physical meaning: diffusion velocity C2/4"rrEp of magnetic field B0 must be much more than plasma flow velocity. For finite values of parameter pt ,~ l, the flow pattern is similar to the above considered one, the shock wave of finite amplitude being formed instead of low-intensive discontinuity. At p >> 1 (superconductive case), the magnetic field is pushed out of the plate and flow pattern is fully changed. Fast magnetoaeoustie waves

Consider now the waves generated by Hall current component. It is convenient to introduce scalar potential function ~o(x~, x2, x3) and to determine the perturbations b2÷, b3+, u2+ by the following relations:

b2+ - Ox3'

b3+ = - Oxz' u2+ = Ma~-~xl"

It follows from (9):

02~0 02~0 02~0= O, MA2 ~ -- c3X22 OX32 n2 -~x2 + n3

~

= $

=0.

f ~, $

(15)

Inductive interaction o/electroconductive bodies

53

The solution of eqn 0 5 ) is represented in the form (Vladimirov, 1976): ~0=-

C2

, dyIdy2~n(yt, y2)O(xl-y,-MA[(x2-y2)2

+ (x3 + n2y2/n3)21112{(x, - yO2 - Ma2[(x2 -- y2)2 + (x3 + nEyJn3)2]'/2}, where s' is a projection of s at the plane (xl, x2) and O(z)-Heaviside function. Then we have:

K=

r o t ~oel,

] = c MABo 4~r

0_9_~

1,42 "~- M A OXl,

rotrot ~1,

or

I , - c MA ~ Bo w -,,p, A - e MA ~ B o v- l T0~o x ,, where

3X3

For the case of a plate stretched along ~, we have

X.(x,, x2)= G.8(x2)O(Xl), =~

U

In [xdMAr

-

(XI2/MAEr

2 --

l) 1/2]O ( X ,

-- MAr),

where g = (X22 -Jr- X32) 1/2.

The electric current lines inside the Mach cone are described by the condition: xl = Const. These current lines are closed at the cone surface along which the surface current flows. The electric field lines and magnetic field perturbations are shown in Fig. 3. Note that at n2 # 0 there is a velocity c o m p o n e n t normal to the plate and the slow magnetoacoustic wave necessary exists.

54

A.V. Gurevich and N. T. Paschenko

Fig. 3. Electric current and magnetic field lines.

Slow magnetoacoustic waves

The equation system for slow magnetosound wave is integrated in the same way as system (8) with the boundary condition obtained after the solution for the fast magnetoacoustic wave: u3-= n2

+~

n3 u2 [

w(xl, x2).

The solution has the form: W(X]'0, X20), (n2x2+ n3x3) > 0, X~o, X2o~ S, u3- = w(xlo, x20), (n2x2+ n3x3) < 0 , Xlo, X20 ~ S, O, X~o, x 2 ~ S , U3,

P=

--u3,

(n2x2+n3x3)>0,

(n2x2+n3x3)
where X~o = xl ~ M(n2x2 + n3x3)/n3,

X2o = Xz.

In the approximation acepted this solution describes the perturbations of hydrodynamical parameters. To apply the linear approximation one has to satisfy the following condition (similar to the condition (14)): P2 = 4~rEHvo/c 2"~ 1. Note that full separation of perturbations (Alfv~n wave generation by Pedersen currents only, and magnetoacoustic waves--by Hall currents) takes place only for high Mach numbers Ma and M in the first approximation with respect to ~ = lIMA. When we take into account the next series terms, Aifv~n and magnetoacoustic perturbations are interrelated, and Pedersen current generates both Alfv~n and magnetoacoustic waves.

Inductive interaction of electroconductive bodies

55

Forces acting on a body and energy dissipation

To estimate drag and plasma energy absorption which is converted into Joule heating it is convenient to write the equations in the form of conservation laws (Kulikrovsky, 1962; Landau and Livshits, 1962) div

fl=0,

dive=0,

(16)

where lfI is the tensor of impulse flux density, ~ is the energy flux density. The force P from plasma to the body is equal to the tensor fl flux through an arbitrary closed surface Q around the body s. In the reference connected with the streamlined plate we have for the force surface density: f = - [I-Bo][C, f[ = v o B ~ ] ~ J c 2,

T - - £[~o/~o]/C,

f~ = - v o B ~ F~H]c 2,

f~ = B~3(YnB~2 - Y p B ~ O / c 2.

So we see that the drag force F1 is generated by the magnetic field component B~ normal to the plate and Pedersen conductivity Ep, the lateral force-F2--by Hall conductivity Yn and B~3, for lift normal and tangent components of magnetic field are necessary. In the same way the absorbed energy is determined = Ep B~ 2 Vo2/C2.

It is fully converted into Joule heating and depends on Pedersen conductivity and normal magnetic field component only. Inductive plasma-body interaction considered here may be of importance for the interplanetary plasma flow around natural and artificial bodies. Flow patterns of the type shown in Fig. 2, with two "tails", are observed in the solar wind flow around planets. It takes place both for planets without own magnetic field (e.g. the Moon) and for planets with small magnetic field (Venus). The processes similar to ones considered here may essentially contribute to "tail" formation. To investigate the large (in comparison with Larmor radius) body motion in Earth ionosphere where magnetic field is not small, one needs to treat the other limit case--high Mach numbers and low Alfv~n numbers. For this case there is no such clear splitting into Alfv~n and fast and slow magnetoacoustic waves. The flow structure is more complicated because of elliptical character of governing equations but the preliminary analysis shows that also in this case the inductive interaction effects essentially on the aerodynamical flow around the body and contribution of inductive interaction into the total drag may exceed the aerodynamical drag of space vehicle and the braking force because of inductive interaction may determine the life time of large space vehicle in the orbit. To obtain more exact flow pattern around the body moving in ionosphere more detailed analysis is required.

56

A . V . Gurevich and N. T. Paschenko

References Gelfand I. M. and Shilov G. E. (1958) Obobschenniye funktsiyi i deistviya nad nimi. Fizmatgiz, Moskva. Gurevich A. B., Krylov A. L. and Fedorov E. N. (1978) JETF (J. Exp. Theoret. Fiz.) 75, 6(12), 2132-2140. Kadomtsev B. B. (1976) Kollektivniye yavleniya v plazme. Nauka, Moskva. Kulikovski A. G. and Lyubimov G. A. (1962) Magnitnaya gidrodynamika. Fizmatigiz, Moskva. Landau L. D. and Livshits E. M. (1962) Elektrodynamika sploshnichsred. Nauka, Moskva. Vladimirov V. C. (1978) Obobscheniye [unktsiyi v matematicheskoifizike. Nauka, Moskva.

Appendix Nomenclature /~ 6 c Co P

F(x~) f P f J M

MA ri p t7 ti

VA

magnetic field perturbation magnetic field light velocity sound velocity electric field coordinate vector energy plate boundary non-dimensional surface current force on the body surface current non-dimensional surface current Mach number Alfv/~n Mach number normal to the body pressure energy flux density perturbation velocity Alfv~n velocity

x~ coordinates 8 8-function ~ small parameter O(z) Heaviside function fI tensor impulse flux density p density ~,, electroconductivity ¢ scalar potential function

Subscripts 0 s t P H i

initial and boundary values surface values tangent values Pedersen component Hall component components along coordinate axes

Primes refer to the values in reference systern normal to the body, " + " and " - " refer to the upper and low sides of the plate.