Damping of waves in the current sheet in the geomagnetic tail by radiation

Planet.

Space Sci. 1969,

Vol.

17, pp. 1291 to 1296.

Pergamon

Press.

Printed

in Northern

Ireland

DAMPING OF WAVES IN THE CURRENT SHEET IN THE GEOMAGNETIC TAIL BY RADIATION Department

J. W. DUNGEY of Physics, Imperial College, London, S.W.7, England (Receioed 27 January 1969)

Abstract-In the preceding paper it is seen that the velocity distribution in the current sheet will give energy to waves, but that the overall stability problem involves the radiation of energy from the sides of the sheet. For waves propagating parallel to the current, the problem is simple, radiation occurs when a Cerenkov condition is satisfied and then the rate of radiation is large, so that the waves are probably damped. For waves propagating perpendicular to the current the problem is more complicated and there is typically an evanescent wave as well as a radiated wave. The rate of radiation may then be much smaller. INTRODUCTION

Reconnection of field lines at current sheets appears to be an important phenomenon in plasma physics, particularly in relation to the geomagnetic tail, but the problem still outstanding concerns the intensity and nature of electromagnetic noise in such a sheet (Axford, 1967; Friedman and Hamberger, 1968). The general problem is discussed in an accompanying paper (Dungey and Speiser, 1969) and it is seen that the model derived from Speiser’s (1965) trajectory calculations, neglecting noise, leads to a velocity distribution in the sheet which should be highly unstable. This distribution exists only in the very thin sheet, however, and the condition for wave growth may be very different from that in a If wave energy is radiated out from uniform plasma with the same distribution function. the sides of the sheet, this energy must be supplied by the amplification mechanism in the sheet and so waves grow less easily in a sheet than in the corresponding uniform plasma. In this paper the radiation from the sides of the sheet is investigated by means of a simplified model of slab form. The magnetic field outside the sheet is in they direction on one side and the opposite direction on the other side. ENERGY

RADIATED

OUT OF THE SHEET BY ELECTROSTATIC THE z DIRECTION

WAVES IN

The sheet is taken to be the slab --a < x < CIand in this slab the magnetic field is taken to vanish and the distribution function to be independent of all three position coordinates. The waves in the slab are then plane waves and we consider only the polarisation for which the only non-zero disturbances is in E, so that its space time dependence is like exp i(ot + kz) and it does not vary withy or X. Outside the slab, x < --a or x > a, we assume a uniform magnetic field B in they direction and a cold plasma of uniform density p. The changes at the edges of the slab x = &u are of course discontinuous steps in this simplified model. The waves outside the slab must satisfy the cold plasma dispersion equation and must neither diverge at infinity nor propagate in from infinity (there is no attenuation in a cold plasma, but the slightest attenuation causes waves propagating inward to diverge at infinity). The waves must satisfy continuity relations corresponding to Maxwell’s equations at x = fa. Consequently the waves outside the slab must vary like ezkz and must not vary with y. The latter condition is the origin of the great simplification of the case under 1291

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J. W. DUNGEY

consideration as it means that the propagation vector outside the slab is perpendicular to the steady magnetic field. The continuity relations require that the iangential components of the electric disturbance and the normal component of the magnetic disturbance be continuous, the latter vanishing for electrostatic waves in the slab. Then the electric disturbance E outside the slab is perpendicular to the undisturbed magnetic field B and, since in addition low frequencies are important, it is permissible to approximate the cold plasma dispersion equation by the hydromagnetic dispersion equation. Because k and E are perpendicular to B, only the fast mode is excited in this case and its dispersion is given simply by ~2 = @A2

(1)

k2 = k,” + k,2

(2)

with A = Alfvtn speed = (47rp)-lj2B and k, being the same as inside the slab. (1) and (2) must be used to determine kS2, thus k,” = (co/A)~ - k,2

(3)

and so k, is real if w/k, > A

(4) and imaginary otherwise. Equation (4) states that the AlfvCn speed is less than the phase speed of the waves in the slab. Thus (4) is a Cerenkov condition; if (4) is satisfied k, is real and energy is radiated out of the sides of the slab. Otherwise k, is imaginary and the evanescent wave must be chosen and no power is radiated. If the wave is growing, energy is required to build up the energy of the evanescent wave, but the rate of energy supply is proportional to the growth rate. Consequently, when (4) is not satisfied, the stability condition and often also the growth rate are not greatly affected by the thinness of the slab and should be approximately the same as in the corresponding uniform plasma. When (4) is satisfied, the energy radiated must be calculated from the x component of the Poynting vector c(E A b),/4rr, where b is the magnetic disturbance. For the present case of the fast mode propagating perpendicular to B, b is in they direction and ioby = c(kJ& - k,E,) = ckaE,/k, using the fact that k . E vanishes for the fast mode. Then using (1) the power flux in the x direction is

The factor k/k* shows that P is very large near the critical Cerenkov condition, and (5) shows that this is due to b, being large. The latter is also true when (4) is just not satisfied and neglect of the thinness of the slab is then not so clear as when (4) is far from satisfied. Apart from the factor k/k, (6) shows that the energy E2/8n in the slab will all beradiated in a time &Au/c2 and this is likely to be a very short time showing that the damping due to radiation is strong and probably prevents the wave from growing. R~~TION

BY ELECTROSTATIC

WAVES IN THE y DIREC~~N

This case is much more complicated than that with electrostatic waves in the z direction, because there is necessarily from the continuity conditions a non-zero E, outside the slab, but B is in the y direction, and the hydromagnetic approximation then makes E, vanish. It is therefore necessary to use the full dispersion equation for a cold plasma. In general

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two elliptically polarised modes are excited outside the slab and their superimposed fields satisfy the ~olltinuity conditions. If k, is real for either of these modes energy will be radiated. Using the full dispersion equation the condition fork, to be real involvesnot only co/ky, the phase velocity in the slab, but also e$I,, where Q2, is the angular gyrofrequency for protons, though it may be assumed that the most interesting frequencies are considerably smaller than !Z?,. The cold plasma dispersion equation for frequencies much less than the plasma frequency and phase velocities much less than the speed of light may be written, (CX-

If

Y)(CX-

c+

Y)=&Y

(7)

where x = (U&4)2

p = m,/m, ci 1839 and C = kg2(k,” + kg2)-‘. It may be noted that when k, is real, C = cos2 8, where 0 is the angle between the direction

of propagation and B, and so real k, requires 0 < C < 1. Also X, Y and ,U are always positive. From (1) the condition for C to be real (necessary but not sufficient for k, to be real) is (2XY-Xf 1 - (,u+ 1)Y)2>4X(1 - Y). (8) Now if CDg LID,,u Y is small and so Y is very small. For small Y the right hand side of (8) is negative for X > 1, and (8) is therefore satisfied for any small value of Y and X > 1. When X < 1 and Y is very small (8) may be approximated by l-X>pY.

(8a)

This may be verified by substituting ,UY for 1 - X on the right hand side of (8), when it is seen that the right hand side of (8) contributes a correction of ~(,I.JX)~/~ Y to the right hand side of (8a) and this correction is only about 3 per cent. For X < 1 and for frequencies high enough to violate (Sa), both roots of (7) for C are complex and both waves are evanescent, so that no energy is radiated. To elucidate the behaviour when X < 1 and @a) is satisfied note that (a) for X = 0 (7) is satisfied by

(b) for Y = 0 (7) is satisfied by C = 0 or X-l

(IO)

(c) for Y = 0 and X = 1, (7) is satisfied by any value of C. Since the value given by (9) is small and positive for Y < (y + 1)-l, it follows that one value of C satisfies 0 < C < 1 in the region defined by X < 1 and (8a). Consequently a real value of k,, leading to radiation, occurs for one of the modes when (8a) is satisfied. The condition X < 1 corresponds to the condition for the wave to be evanescent in the case of electrostatic waves in the I direction, but the difference has now appeared that the frequency must be large enough to violate (Sa). Very low frequencies violate (8a) when X is nearly 1,

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J. W. DUNGEY

however, and the more interesting question concerns conditions for X > 1. For X > 1 and small Y an appropriate approximation is obtained by adding terms of first order in Y to (10) giving

c, = -

Y

or

1-x

C+.=X-‘(l+&)

(11)

of which the first is negative, but the second satisfies 0 < C < 1 for small enough Y. The exact maximum frequency for the existence of a radiating mode is obtained by putting C = 1 in (7), resulting in X=l+(pY)+Y or approximately p Y = (X - 1)2 so that this limitation is unimportant for values of X substantially greater than 1. The important result is that a radiating mode exists for low frequencies and high phase velocities. The amplitude of the radiating wave is not, however, related to the amplitude in the slab in the simple way that was found for the case of electrostatic waves in the z direction. The fact that an evanescent wave is also excited is very important and the relations between the amplitudes must be discussed, before it is possible to estimate the power radiated. POLARISATIONS AND RELATIONS BETWEEN AMPLITUDES

The polarisation is given by simple equations in terms of the circularly polarised components El = E, + iE,,

Using a suffix e = fl

E_, = Es -

E,, = E,.

iE,,

or 0, the polarisation is determined by the three equations E = e

(k. 0% k2 -

(12)

02/we2

where k, is defined in the same way as E, and (wJA)~ = e2 + e(p Y)* -

Y.

(13)

The dispersion equation (7) can be obtained by eliminating E, from (12). For the waves needed here k*, = k, sin 8, k, = k, = k cos e and this formalism may be used when sin 8 is complex.

cos e

E, =

1 - (ro/kwJ2

(E, sin

e+

Then (12) with e = 0 is E,

cos e)

(14)

while (13) gives (co/kqJ2 = - CX/ Y.

Hence one quantity defining the polarisation is l-c+cx/r -4 p = g. = 2/(C(l-

(1%

C)) -

The other can be taken to be

q=

EZ

co2(W_12 - W12)

E, = 2&+32W_,2

after a little algebra.

-

wy1q

+ w-,2) = p Y +

CX(PY)” CX(1 -

Y) - (1 - Y)2

(16)

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Approximations for low frequencies (small Y) may be obtained with the help of (11). The first value in (11) corresponds to the evanescent wave and leads in the first approximation to pe 55 ( Y(1 - x))-+

(17)

(18)

and

showing that l& is large and in quadrature with Ek and E, is small. The second value in (11) corresponds to the radiating wave and has

x + ( !

y-1

PvN
(19)

and 4r = W?-*

(20)

showing that I& is much larger than E, and E, is larger than E8. Now the amplitude of the radiating wave may be related to the amplitude of the wave inside the slab by applying the continuity of E, and E, at the slab surface, expressed by

(20 where I?,,, is the vaIue inside the slab, and %

-+-%e = p&&r

-I- p,q,E,.B = 0.

(22)

Eliminating Evsegives G,, = (1 - p~~~~~q~)-‘E~.~. The approximations

(23)

(17), (IQ, (19) and (20) give

(p,q,)=

X-l

(A,%) PX”y ’

(24)

which is large for small Y, and consequen~y Ek,, is much smaller than E,,,. Then Es,, = p,q,E,., = peq&.~

(25)

and from (17) and (18)

Consequently Es., is large compared to Eva,, but small compared to E,,,. From Maxwell’s equations wb=ck

A E

(27)

and taking the vector product with E wE A b/c = ,Pk - (k . E)E

(38) from which the Poynting vector can be obtained. The rate of radiation is given by the x component of the Poynting vector and the x component of (28) is, o(E A b&/c = (Eg2 + Es2)k, - k,E,E*.

(29)

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J. W. DUNGEY

Now for small Y the approximations (19) and (20) show that the term Ez2k, dominates the right hand side of (29) and (25) may be used to evaluate this term. To assess the value of the Poynting flux and to compare with electrostatic waves in the z direction the quantity computed was R = (p,q,)2 tan 8,.,which gives the ratio of Es2,.kzto Eg2,k,. The computations were performed for X = 10 and several values of Y: these included some quite large values of Y, since as the program was written with exact formulae, no extra effort was needed. First the roots of (7) for C were computed and identified. Thenp and q were computed from (15) and (16) for both roots. Finally tan 8, was obtained from C, and the values of R are shown in Table 1. The time for the energy in the slab to be radiated is X”aA/2Rc2, which TABLE1 1O’Y 106R

1 I

2 26

3 55

4 91

5 131

6 183

8 283

10 391

20 893

104Y 10BR

30 1236

40 1432

50 1522

60 1543

80 1482

100 1358

200 754

300 397

400 161

is X*/R times the corresponding time in the simpler case of electrostatic waves in the z-direction. The Table shows that this radiation time can be much greater than in the case of electrostatic waves in the z-direction. Consequently it will be necessary to calculate more carefully the rate of input of energy from the particles to these waves. Acknowledgement-This work was sponsored by the Air Force Cambridge Research Laboratories Contract No. AF 51(052)-927 through the European Office of Aerospace Research, OAR.

under

REFERENCES AXFORD, W. I. (1967). The interaction between the solar wind and the magnetosphere. In Physics cf Geomagnetic Phenomena (Ed. S. Matsushita and W. H. Campbell), p. 1243. Academic Press, New York. DUNGEY, J. W. and SPEISER,T. W. (1969). Electromagnetic noise in the current sheet in the geomagnetic tail. Planet. Space Sci. 17, 1285. FRIEDMAN,M. and HAMBERGER,S. M. (1968). On the neutral point region in Petschek’s model of magneticfield annihilation. Astrophys. J. 152, 667. SPEISER,T. W. (1965). Particle trajectories in model current sheets, 1: analytical solutions. J. geophys. Res. 70, 4219.