Resonance absorption of propagating fast waves in a cold plasma

Planet. Space Sci., Vol. 38, No. 8, pp. 1017-1030, Printed in Great Britain.

1990 0

00324633/90 53.00+0.00 1990 Pergamon Press plc

RESONANCE ABSORPTION OF PROPAGATING FAST WAVES IN A COLD PLASMA JOSEPH

Department

V. HOLLWEG

Space Science Center, Institute for the Study of Earth, Oceans and Space, University of New Hampshire, Durham, NH 03824, U.S.A. ; and of Applied Mathematics, University of St Andrews, St Andrews, KY16 9SS, Scotland (Received infinalform

26 March 1990)

Abstract-Resonance absorption of MHD surface waves has received considerable attention recently, but rather little attention has been paid to the absorption of propagating waves impinging on a “surface” in which the plasma and magnetic field may change. Here we examine in some depth a very simple but instructive problem : the plasma is cold, the magnetic field is uniform, and the density in the “surface” varies linearly from zero at the left end to some finite value at the right end, beyond which the density is constant. We consider two cases : (1) the plasma is a vacuum everywhere to the left of the surface, or (2) the plasma density jumps to a very large value to the left of the surface. Case (1) may correspond to coronal conditions, while case (2) may mimic the magnetosphere with the dense region at the left corresponding to the plasmasphere. The goals of the paper are to study the parametric behavior of the absorption coefficient numerically, and to provide several useful analytical approximations. We find that the parametric dependence of the absorption is far richer than implied by the single curve appearing in Fig. 2 of Kivelson and Southwood (1986, J. geophys. Res. 91,4345), although we do recover that curve as a limiting case in which the waves are essentially WKB at the right end of the “surface” and a dimensionless parameter Kz (defined in Section 5) is moderately large. In case (1) we find that the absorption coefficient is always less than about SO%, but in case (2) the absorption can approach 100%. Thus the boundary condition at the left critically affects the results. We also find that the thickness of the surface affects the parametric dependence of the absorption coefficient. For example, a thin surface yields an absorption coefficient scaling as CI-*, where a2 [defined in equation (4)] is a measure of the steepness of the density ramp. On the other hand, if the surface is thick enough so that the waves are essentially WKB as they start down the density ramp, then the absorption scales as tl- 8’3 [case (l)] or a- 4’3 [case (2)] for large c?. Our numerical results are presented in a format which reveals the dependence of the absorption on the propagation direction of an incident wave. The absorption depends on the angle of incidence with respect to the surface, maximizing at moderately large angles of incidence [around 70” in case (l)]. The absorption depends also on the angle between the magnetic field and the plane of incidence [there is a broad maximum around 30” in case (l)]. Finally, along the way we offer two other analytical results : (1) we show that the mathematical discontinuity in the Poynting flux which occurs in the present steady-state analysis is precisely equivalent to the rate at which energy is pumped into the resonant layer as calculated by Hollweg and Yang (1988, J. geophys. Res. 93, 5423) using a simple harmonic oscillator model ; (2) we show that a convenient approximation scheme used by us for calculating the absorption of propagating waves in another context (Hollweg, 1988, Astrophys. J. 335, 1005) has a useful domain of validity.

1. INTRODUCTION

In solar physics, resonance absorption of MHD surface waves has been a candidate for coronal heating since the idea was introduced by Ionson (1978). Since then, various aspects of the resonance absorption process have been studied, including initial value problems (Hollweg, 1987a; Lee and Roberts, 1986), dissipative effects (Davila, 1987 ; Einaudi and Mok, 1985; Hollweg, 1987b; Mok and Einaudi, 1985), and the effects of plasma compressibility with dissipation (Grossman and Smith, 1988; Hollweg and Yang, 1988 ; Poedts, 1988 ; Poedts et al., 1989a,b,c). Emphasis has been placed on the Alfvenic surface mode, which exists in a low-beta plasma such as the solar corona, because it carries energy along the ambient

magnetic field at a rate sufficient to replenish the coronal losses due to heat conduction and radiation. Resonance absorption of surface waves has also received much attention in the magnetospheric (e.g. Chen and Hasegawa, 1974b,c; Goertz and Smith, 1989 ; Inhester, 1986; Kivelson and Southwood, 1986; Lanzerotti et al., 1973; Smith et al., 1986; Southwood, 1974; Southwood and Hughes, 1983; Southwood and Kivelson, 1986) and fusion plasma (e.g. Allan et al., 1985, 1986a,b, 1987a,b; Appert et al., 1984; Chen and Hasegawa, 1974a; Grossman et al., 1973 ; Grossman and Tataronis, 1973 ; Hasegawa and Chen, 1974; Pritchett and Canobbio, 1981; Tataronis, 1975 ; Tataronis and Grossmann, 1973) literature. See also the monograph by Hasegawa and Uberoi (1982). 1017

1018

J.V. HOLLWEG

Surface waves have the property that the energy is confined to the vicinity of the surface ; the wave amplitude declines exponentially away from the surface. Thus the component of the wavevector normal to the surface is imaginary on both sides of the surface. Roberts (1986) has given an excellent summary of the various properties of surface waves. Resonance absorption can also occur when a propagating wave impinges on a “surface” which separates two regions having different plasma properties and/or different magnetic fields. In that case the eomponent of the wavevector normal to the surface is real on at least one side of the surface. In solar physics, a problem of current interest is the possible resonance absorption of propagating sound waves {the solar “pmodes”) impinging on a sunspot (Hollweg, 1988). This problem is motivated by recent observations (Braun et al., 1987, 1988) which show that the incoming p-mode power is greater than the outgoing power by as much as a factor of two. The conclusion of our study was that significant resonance absorption can occur for some of the incoming p-modes, and that the absorbed waves might reappear as the “running penumbral waves” seen in the photosphere and low chromosphere. See also Lou (1990). Hollweg (1988) used a simple approximation [Hollweg and Yang, 1988 ; see their equation (431 for calculating the absorption coefficient. A standard reflection/transmission problem was first solved for a discontinuous sunspot boundary in order to obtain the total pressure perturbations at the boundary. Under the assumption that the same total pressure perturbations exist when the boundary is thin rather than discontinuous, the absorbed energy was easily calculated. This approximation required a thin sunspot boundary, inside which the Aifven speed gradient was large. The absorption coefficient then scales as (AlfvCn speed gradient)-‘, and the absorption coefficient is small. One of the goals of this paper is to show that this approximation scheme can indeed be valid and useful, at least for the simple case to be examined here in depth. The resonance absorption of propagating waves has been considered also for the magnetosphere by Kivelson and Southwood (1986), Southwood and Kivelson (1986), and Zhu and Kivelson (1988). They were concerned with propagating fast waves in a cold plasma, and their potential for exciting the observed ULF field line resonances in the magnetosphere. Kivelson and Southwood (1986 ; seetheir Fig. 2) give an absorption coefficient which scales as (Alfv&n speed gradient)- zi3 even when the absorption coefficient is

small; at first sight, this result appears to conflict with the results of Hollweg and Yang (1988) and Hollweg

RG. I.SC~ATICILLUST~TIONO~~DENSITY PROFILE. The AlfvCnresonance occurs at p = y_ and equation (1) has a turning point at y = yl > y,. In y < 0 we either take the density to be zero (the vacuum case} or infinite (the wall case).

(1988). The papers by Kivelson and Southwood and by Zhu and Kivelson show also an explicit dependence of the absorption coefficient on the component of the wavevector perpendicular to the ambient magnetic field but lying on the plane of the surface, k, [see equation (7b)], but the dependence on the parallel component, k,, [see equation (4) et seq], is obscure ; at first sight this is difficult to reconcile with Hollweg and Yang (1988) and Hollweg (1988) where the absorption also depends on k,, . The goal of this paper is to reconcile these apparent conflicts, and to provide a complete discussion of the resonance absorption of propagating fast waves for the simple but instructive case considered by Kivelson and Southwood. We shall emphasize approximate analytical results whenever possible, and we shall compare them with exact numerical solutions. The problem to be considered is illustrated in Fig. 1, which shows the spatial dependence of the ambient plasma density, p,, ; the y-direction is perpendicular to the ambient magnetic field, B,, which will be taken to be spatially uniform. (In y < 0 we will either take p. = 0 or p. = m.) The plasma is cold, which is a reasonable approximation in the solar corona where the thermal pressure is roughly two orders-of-magnitude smaller than the magnetic pressure. We assume that the region y > a contains an incoming fast wave propagating to the left, and an outgoing reflected wave propagating to the right. Resonance absorption occurs somewhere (see below) in 0 < y < a, and the reflected wave energy is less than the incoming energy. The energy deficit will be translated into an absorption coefficient. (We will always deal with situations where there are no propagating waves in y < 0, and the energy deficit represents a real loss due to absorption, rather than transmission.) We will find that in general the absorption co-

1019

Resonance absorption of propagating fast waves in a cold plasma

efficient depends on both k,! and k,, on the steepness of the density ramp in 0 < y <: a, and on the thickness a ; the “coupling parameter” used by Kivelson and Southwood (1986) is not the only important parameter. If the ramp is steep and a small, then the absorption coefficient will scale as c(-*, where uZ is proportional to the density gradient [see equation (4)] ; this corresponds to the results of Hollweg (1988) and Hollweg and Yang (1988). On the other hand, if the density ramp is steep but a is large enough so that the waves are nearly WKB just inside y = a, then the absorption coefficient scales as a-q3 or a-w3 depending on the boundary condition at y = 0; the 01~~‘~ dependence is the one given by Kivelson and Southwood (1986). The importance of the boundary at y = 0 is one of the surprising results of this work. We will also find the new result that the absorption coefficient can be quite large, at times approaching 100%. We note in advance that the excellent series of magnetospheric papers by Allan et al. (1985, 1986a,b, 1987a,b) also include resonance absorption of propagating waves, but in a cylindrical geometry. Although the different geometries and boundary conditions make quantitative comparison between their results and ours difficult, it appears that the general features of the parametric dependences of the absorption coefficient are the same in their works and ours. However, their work does not include the series of analytical approximations which will be one of the central focuses of this paper.

There is a singularity where A = 0, leading to the resonance absorption. In y > a, equation (1) yields the usual dispersion relation for the fast mode in a homogeneous medium : wz = (k:+k;,)v;, where k,, is the y-component y > a. InO
(3)

of the wavevector in

&/vi: = c?=y.

(4)

point where w2 = kfvi, (A = 0) is at we see that ym < yAand the singularity is to the left of the turning point so that the incoming waves have to tunnel into the singularity. [In fact, by applying a local dispersion analysis to equation (31) of HY, it is readily shown that the waves have to tunnel into the Alfvenic singularity (A = 0) even when the cold plasma assumption is relaxed. Thus absorption at the Alfvenic singularity can only occur where the magnetic field or density vary sufficiently to trap or reflect waves propagating in a region of higher density or lower field strength. However, a similar statement cannot be made about the cusp singularity (see HY) where a local dispersion analysis does not apply.] If we define v = y-yu, we have Equation

(1) has a tuning

i.e. at y = k~/a* E y,. The sing~a~ty y = k!;/a= ESy,. Since k,f = k:B&/B&

A = u2v;v/k,2

(5)

and equation (1) becomes 2. BASIC EQUATIONS

6P”-GP’/v+a2(v-A)

As in Hollweg and Yang (1988 ; hereafter “HY”) to which the reader is referred for details, we consider the ambient magnetic field to lie in the x--z plane. The ambient plasma is stationary and varies only in the y-direction. Without loss of generality we take the wavevector along the x-direction, so that all fluctuating wave quantities vary as f(y) exp (i&x-id& and a/& = 0. After linearizing and taking the plasma to be cold, we obtain the following equation for the total pressure fluctuations, 6P [see equation (31) of HY] :

where vA is the Alfven speed (vi = &47rp0) and A = co2/k,2- B&/(4np,)

(2a)

A = w’/kx”- &. .

(2’3

or

SP = 0,

where the prime indicates differentiation to v and A = yn --y=

(6)

with respect

(7a)

or VW

since kf = k,“-- k,f. Equation (6) is equivalent to equation (8) in Kivelson and Southwood (1986). It has also been studied by Chen and Hasegawa (1974a) for surface waves in the limit that kf >>k,f, and by many other workers. Equation (6) is also very closely related to the equation governing the absorption of laser light propagating into a plasma target (Forslund et al., 1975 ; Friedberg et al., 1972 ; Uberoi, 1989). As emphasized by Allan et al. (1987b), the absorption coefficient is sensitive to the distance A the waves have to tunnel to reach the singularity. It is convenient to define dimensionless variables

1020

J. V. HOLLWEG

y

u2’3v

I

(8)

but from equation

SV, = -iw(p,Ak,2)-’

and L E a2’3h

(9a)

or

Equations (S,)

L = kf/d3

(9b)

The quantity L was called the “coupling parameter” by Kivelson and Southwood. Equation (6) becomes 6P”---GP’/Yf(Y-L)6P= where respect We indices

(30) of HY we have

0,

(10)

the prime now indicates differentiation with to Y. develop a Frobenius series around Y = 0. The are 0 and 2. The latter yields a regular series 6P, = Y2xR,Y”,

(11)

where the summation is for n = 0, 1,2,. . . We find R, = 0, R, = R,L/8, and the recursion relation: R, = (LR,_,-R,_3)/(n2+2n).

(12)

We will later find it useful to note that

where F is the Gamma function. The singular series is

function

6P, = C(GP,/Ro)ln

(13)

and J is a Bessel

Y-t x&Y”,

(14)

where C is a constant and again n = 0, 1,2.. We find S, = 0 and C = S,L/2. We take S2 = 0 since any terms involving S2 can be absorbed into the series given by equation (11). The recursion relation is then (n2-2n)S,

= LS,_,-S,_,-L(n-l)R,_,/R,.

(15)

We will later find it useful to note that 6P,(L = 0) = So3--2’3r(1/3)YJ_2,3(2Y3’2/3).

J.POYNTING

The average Poynting (S,)

(16)

FLUX

flux in the y-direction

that B,

&?P/8Y) BP).

(19)

Now the logarithm in equation (14) jumps by in at Y = 0, and thus (S,,) is discontinuous at Y = 0. This discontinuity in the Poynting flux is the resonance absorption. Its magnitude is lA(SJ

= 27~~lwJk:S;(aB,))~.

(20)

We now show that this is equivalent to the energy flux density into the resonant layer at Y = 0 derived by HY using an entirely different method. Differentiate equation (5) with respect to y and evaluate the result at y = y,, i.e. at Y = 0, where o’/k: = vi,: (dv,,ldy),=,

=

-(~B,)2(8np,)-‘lwk,l-‘,

where pm = p,,(y = y,). and (21) then yields

=

Combining

equations

4klS~(&z14,)2 +wldv,xlW Y=O

(21) (20)

(22)

since kzB& = k:Bi. Noting that So = 6P( Y = 0), we see that equation (22) is precisely equivalent to equation (45) of HY. The derivation in HY used a simple physical model in which an ensemble of harmonic oscillators was driven by total pressure fluctuations. The oscillators which were driven near their resonant frequencies absorbed the bulk of the energy, and the y-integral of the energy absorption was given by equation (22) above. Note that the energy absorption calculated by HY was distributed in y, with the maximum energy absorption occurring at the resonant field line ; in contrast, the absorption calculated here occurs discontinuously at y = yw. The two results can be reconciled by noting that the thickness of the energyabsorbing layer in HY scaled as t- ‘, and the result obtained here then corresponds to the limit t + co in HY.

4. CONDITIONS

where 6E is the wave electric field, given by

Recalhng

= -4aoa-2’3B;2((iY-’

is

= (c/47c)(6Ez6BX-6E,6Bz),

6E = -6V

(18)

(5) (8) and (18) then give

I&VI

bP,(L = 0) = 2R,3-“31-(2/3)YJ2,3(2Y3’2/3),

asPjay.

x B,/c.

= 0, we obtain

(S,,) = (6 V,B, - 6B/4x)

(17a)

=
(17b)

or

ATy=O

We now proceed to use these results to analytically calculate energy absorption coefficients for some special cases. In most of these calculations we will assume that we can apply boundary conditions at y = 0 (i.e. at Y = - kIF/a4’3= Y,) by using only the first few terms in the series (11) and (14). By inspecting these series, we see that this requires 1Yi1 << 1 and Y$L << 1. To the extent that k,,, = k,(y = yJ can be defined as a local effective wavenumber from equation

Resonance absorption of propagating fast waves in a cold plasma (I), the second condition can be rewritten as ]k,y,f ’ << 1 and it is thus a measure of the degree to which the boundary conditions at y = 0 affect the results. Alternatively, the two inequalities above can be combined into Y&L - Y,) CC1 which can be rewritten as ]k,,y,]* CC1, where k$, ZE[k,(y = O)]’ = -k,’ to the extent that k,, can be defined as a local effective wavenumber from equation (1). This is again a measure of the degree to which the boundary conditions affect the results. We will consider two cases :

5. ABSORY-ITON

1021

AT A THIN “SURFACE”

In this section we consider the case where a is small so that Y, ELY(y = a) = ~?‘~a+ Y, CC1. We also take ] Y,] << I. In that case we use equations (26) or (30) in the entire region 0 < y < a. The absorption coefficient is obtained by matching equations (26) or (30) to incident (r) and reflected (R) plane wave solutions in y > a. We write SP(y > a) = Zexp [-i]k,,](y-a)]

+Rexp

The vacuum case

The region y < 0 is first taken to be a vacuum. Although obviously highly idealized, this might correspond to a wave propagating out of a dense coronal region, such as an active region loop, toward more rarefied surroundings. We then can take

[+ilk,,I(y--@)I.

The quantity ]k,,] is not inde~ndent variables. From equation (4) we have K;, = a/-K,‘,

(31)

of the other (32a)

where we have defined

(23)

SKv < 0) = exp (Mu).

(We have set 6P(y = 0) = 1 without loss of generality.) Now at y = 0 we require continuity of 6P and &SP/pjay.This gives R 0 z (L- Y,)“*/(2Y0)-((L&/2)

In Y,

and

+ I Y,/2 - ~/41

(24)

and S * CC1 -Y&C-

Yo)“2/2.

(25)

(For self-consistency, the term in square brackets in (24) should be retained only in calculating adP/a Y.) In the vicinity of Y = 0, we then have 6P x I +(L-

Y,)“*(Y2-

Note that the existence of a propagating wave in y > a requires a’ > (L- Y,), i.e. a’ > K:. Also, in terms of these variables the location of the resonance is given by yoja = (K,‘- L)/a’.

(32b)

The coefficients I and R are obtained by matching

Y;)/(zY(J +(L/2)Y2 In (Y/Y,).

(26)

6P and t%Pjay at y = a.

The wall case

The vacuum case

As an opposite extreme, consider the case where the plasma density becomes very large in y < 0. This is essentially the case covered in Kivelson and Southwood (1986), where y = 0 represents the plasmapause ; see also Allan et al. (1987a). In the limit of infinite density, the boundary condition at y = 0 is

It is sufficient to retain the leading real and imaginary parts of equation (26). In 0 < y < a we take

(asPjay),,, 0 = 0.

(27)

Again taking BP(y = 0) = 1 gives R, z - (L/2) In Y,,+ [ Y,/2 - L/4]

(28)

So B 1,

(29)

where the term in square brackets is retained only in calculating asp/a Y. In the vicinity of Y = 0 we then have SP x 1+(L/2)Yz

6P=

(33a)

and &?P/aY E IK,IY/Y,-idYH(Y),

(33b)

where H is the unit step function. After matching on to equation (3 1) we obtain 2R z 1 --E+i/s 21% 1 +&-ii/S,

(34a) (34b)

where s = ~~Y~~(Y~)/~~,

I

WW

c = (2nk,k,/a~o>2(pQA>,,,H(Y,)/lk,lI WW

In (Y/Y,) +[Y2(Yo/2-L/4)-

1

Y3/3].

(30)

and

1022

J. V. HOLLWEG

(36b)

In this case we use equations Matching at y = a, we obtain

The energy absorption coefficient is c, = 1 -(R]2/]1)*.

(37)

(31).

2R z 1-a+ig

(44a)

21% l+.s-ig

(44b)

9 = ~‘~a/wy:,,l.

(46)

and

We obtain CA z 4&S2/[14sZ(1 -#].

(38)

A shortcut method

In our calculation of the resonance absorption of p-modes by a sunspot (Hollweg, 1988) we employed a simple approximation which did not explicitiy involve a differential equation such as (6) and its attendent Frobenius series. We instead solved a standard reflection/transmission problem at a truly discontinuous boundary, and deduced the total pressure fluctuations at the boundary. Equation (22) [or equation (45) of HY] was then used to calculate the absorbed energy for the case where the boundary is thin but not discontinuous. This approximation is only valid if the total pressure fluctuations are nearly the same in the discontinuous and smooth cases, and it only works if the absorption coefficient is small. We now wish to apply this approximation to the present case, and compare the results with equation (38). We put the discontinuous boundary at y = 0. In y > 0 we take 6P(y > 0) = exp (-ilk,,ly)+Rexp

(+ilkylly). (3%

(The amplitude of the incident wave has been taken to be unity without loss of generality.) In y < 0 we have ~P(Y < 0) = Tew

hatching

(30) and

6P and (poA)-

(IkAy).

’ &?Pjay

(40)

at y = 0 gives

T = Zis,/(l +is).

(41)

16P21,,o = 4s2/(1 ss2).

(42)

Thus

Inserting this result into equation (20) or (22) gives the absorbed energy if the boundary is thin. The incident energy flux density is given by equations (17b) and (18). Dividing the absorbed energy by the incident energy gives c, % 4&S2/(1fs2).

(43)

Comparing equations (38) and (43) shows that the shortcut method is indeed equivalent to the Frobenius series method as long as E, and thus CA, is small.

where

This completes the list of approximations which can be used when a is small. Note that in both the vacuum case and the wall case the absorption coefficient basically scales as E. From equation (35a) we see that this means that the absorption is basically proportional to a, if all quantities in y > a are held fixed. Similarly, equation (35b) shows that the absorption scales as a-*.

6. THE STEEP-THICK

APPROXIMATION

We again assume that I Y,I is small for the reasons given in Section 3. Thus equations (24) and (25) can be used for R. and So in the vacuum case, while equations (28) and (29) are used in the wall case. We will also assume that L is small, which is compatible with the assumption on Y0 since L/I YO]= kf/k,f. However, we now assume that a (or equivalently a’) is large enough so that Y, >> 1; it is easily shown that this is equivalent to assuming that the wavelength is small at y ;2: a so that the waves obey the WKB approximation there. The essence of our approximation follows from inspection of equation (10). We take 6P( Y >>L) x 6P(L = 0)

(47)

if L << 1. This also follows if we compare the magnitudes of the individual terms in the series (11) and (14). We can therefore use equations (13) and (16) to approximate the behavior of 6P in the vicinity of Y,. The vacuum case

From equations (13), (16), (24) and (2.5) we can easily show that the J,, sotution dominates the J_ 2,3 solution at large Y, and we shall therefore ignore the I t,3 solution. Since Jz13 has a sinusoidal behavior at large Y, we essentially have a standing wave, which can be

regarded as a superposition of the incoming and reflected waves. The amplitude of the incoming wave is half the amplitude of the standing wave. Thus I W,f s (3”“/2?n ~~~)~~~~3)~~~~~~~ Yli+

(48)

(the subscript “i” denotes the incoming wave). The incoming Poynting flux can be found from equation (19):

T%e absorbed energy is given by eq:quation (20) with S, zz I, and the ratio of absorbed to incoming energy gives the abso~tion coe~cient :

We now use equations (13), (t6) and (47) to evaluate Sf( Y >>L). We find SP{Y>> Lf = 3-~?$J-(fi’3)Y(L,,

-‘&&

(53)

where the Bessel function argument is 2 Y”‘/3. This is again a standing wave, which can be split into incoming and reflected pieces. Using equation (20) for the absorbed energy, we now find

We will see below that this carresponds to the smallL part of Fig. 2 of Kivelson and Southwood. We should note that equation (54) applies both to the vacuum and wail cases, and presumably to other cases as weIf, because the detaiIs of the boundary condition at Y = Y0 (Le. at y = 0) were not used. It was only required that I YOfbe large and L small. The correctness of this statement will be verified via exact numerical results in the next section.

From equations (23), (I6), (28) and (291, we carr easily show that the d’_z,s solution is now the dominant one at farge Y. Proceeding as before, we now obtain CA ‘x 3’~3r2~~~(l~3)]2~.

(51)

la this case C, depends only on the ‘%cvupling parameter” L. In dimensional variables, CA 1x a--4i3. This q~a~itativ~Iy corresponds to the behavior of C, exhibited in Fig. 2 of Kivelson and Southwood (1986). However, at small L the slope of the C, vs L curve in their Fig. 2 does not agree with the slope predicted by equation (5i) above. From numerical solutions, which wifi be discussed beIow, we have determined ihat Fig. 2 of Kiveison and Southwood does not correspond to small v&es of 1Y,l , as was assumed in deriving equation (51). It instead corresponds to large values of K,” = L- Y,, If L is smalt, then j Fo;31 is iarge, and we can develop an a~~roximaiion for C, as foollows: We first examine the behavior of SP at Y,, where Yuis large but negative; we still assume that L is small (and positive). Proceeding as we did with equation [47), we can still use equations (13) and (16) to approximate the behavior of &P( Y,), except that J+ zi3 must be reptaced by r* 213and Ymust be replaced by tI’“i. However, to satisfy any finite boundary conditions at Y, it is necessary to eliminate the exponen~a~ly growing terms associated with Jkjr213. This is accomplished by taking

In the previous two sections we prodded several approximate forms for the wave absorption caefficient. fn this section we cafculate the absorption coeffrcent exactly, using numerical methods. We wiIl thereby assess the validity of the analytica restits, and investigate the behavior of the absorption coefficient in parameter regimes where we were unable to find analytical approximations. The numerical scheme is straightforward. We calcuhte &P(Y) by summing the series fl I) and (14) ; 500 terms were used in doubIe precision on a Prime 9955. The coefficients ,Ro and & were dete~ined by requiring, at y = 0, 6P = 1 and either ~~F/ay = l&J (vacuum case) or ~~Ft~~ = 0 (wall case). The quantities Iand R in equation (31) are then determined by requiring ~ontin~t~ of 6B and &%‘!aY at Y = 6, and the abso~t~on eoe&zient is then given by equation (37). AI1 calculations are done in the dimensionless variables Y, L, K,, etc. No approximations are used apart from the truncation of the series which has 51 negligible effect on the answers. Our basic approach is to first pick the tbicknass, fz’, of the region with the density ramp. We then pick the magnitude, &, of the wavevector in the x-z plane, subject to the constraint lu,” < ct’. Finally, we imagine Be to rotate with respect to the x-direction so that L takes on a11 possible values 0 < L < K:, while 1Y,l SimultaneousIy varies according to f Yof = Kz-L.

1024

J. V. HOLLWEG

ks,l/k,=

(a)

2.02 125

_._

0.0

ky,/k,=

_..

0.2

0.1

0.3

2.24 1.25

0.4

0.5

0.125

(b)

kyl/kx=

;.,I;

::::-I 0.096

-

tIIk1 0.025 0.000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 L 2.THE ABSORPTION COEFFICIENTAS A FUNCTION OF L FORTHEZVACUUMCASEWITHd = 0.255. The exact numerical results are the solid curves and the dotted curves are from the “thin surface” approximation (38). In (b) the values of K: are equally spaced between the indicated values. The values of k,,/k, for the four curves in each panel are obtained from equation (32a). The location of the resonance may be found from equation (32b); the resonance approaches y = 0 as L increases to K,'at the right end of each curve. FIG.

The vacuum case Consider first absorption at a thin “surface”. Figure 2 displays C, vs L for a’ = 0.255. We have chosen eight values of K,’ ; Fig. 2b emphasizes that the absorption is very sensitive to K,’ when K,’ z a’ and k,,/k, becomes small [equation (32a)]. The solid curves are the numerical results, while the dotted curves represent approximation (38). The analytical result is generally accurate to within 10 % . It yields an accurate estimate of the value of L leading to maximum absorption ; and it reproduces the exact result that the absorption is zero for parallel (L = 0) and perpendicular (L = K:, i.e. Y0 = 0) propagation ; this occurs because E oc L and s cc (K,‘- L). Figure 3 uses the same format for a somewhat thicker “surface”, i.e. a’ = 0.6. The qualitative behavior of CA is the same as in Fig. 2. Note from Fig. 3a that approximation (38) is quite good for modest values of ky,/kx, even when C, is as large as 30 % The approximation begins to fail, however,

0.0

0.1

0.2

0.3 L

0.4

0.5

0.6

FIG. 3. SAME AS FIG. 2 BUT FOR A THICKER “SURFACE", a' = 0.6.

The thin surface approximation (38) becomes inaccurate only at the largest values of K:.

when the incident waves graze the X-Z plane at y = a; see Fig. 3b. (The maximum absorption coefficient in this case is about 30%, occurring for 0.5 < K,’ 6 0.55.) Consider next the absorption at a thick “surface”. Figure 4 displays CA vs L for a’ = 9. (We have verified that the results are insensitive to CI’as long as n’ >> 1 and the waves are essentially WKB at y = a. In fact, if a’ 3 2 it turns out that the results are not very different from those appearing in Fig. 4.) Figure 4a is for 0.02 < K,’ ,< 0.06. The solid curves are the numerical results and the dotted curves represent approximation (50). We see that our approximation is valid only for very small values of K, and thus for small values of C,. Approximation (50) has a very small domain of validity. There are several reasons for this. One of the steps in deriving (50) was dropping 6P, compared to 6P, at large Y. If we trace back through the derivation, we find that this requires 0.731K,I/(K~- L) >> 1. If K, = 0.06 and L = 0, this ratio is only 3; we believe that this accounts for the inaccuracy of the approximation at the smaller values of L. On the other hand, the ratio becomes large as L approaches K,’ ; we then expect the approximation to improve as is indeed the case in Fig. 4a. As a rule-of-

Resonance 0.000

i

I .o

0”

(a)

’ ,:,” “‘,.,’

(b)

/

0.0

absorption

T2

wstope

=

[r

of propagating

1025

FIG. 5. SAME AS FIG. 2 BUT FOR AN INTERMEDIATE VALUE OF a’(=1.3) WHICH DOES NOT YIELD CONVENIENT ANALYTICAL APPROXIMATIONS.

3113

(1/3)sin

fast waves in a cold plasma

7rr/312

0.6 0.4 0.2 0.0

,I!

,b.0

0.5

1.0

1.5

2.0

L FIG. 4. SAME AS FIG. 2 BUT FOR A THICK LAYER WITH a' = 9. The waves are nearly WKB at y = a. In (a) the dotted curves are from approximation (50), which fails at even very low values of CA. In (b) the dashed line is from equation (54), which provides a good approximation to the initial slope of the heavier curve. The heavier curve is the one appearing in Fig. 2 of Kivelson and Southwood (1986). It is an asymptotic result valid for 2.5 < K,’ -cca’, and it is not sensitive to the boundary conditions at y = 0. In all cases k,,/k, can be determined from equation (32a).

thumb, we have found that (50) is accurate to within about 25% as long as K, 6 0.4 and L 3 0.8OK:. However, approximation (SO) is totally inadequate if K,” 2 0.4, because approximation (47), which was used in deriving (50) only works if L << 1. Figure 4b displays C, vs L for larger values of K,. The heavier curve represents an asymptotic behavior which results when 2.5 6 K: cc a’. We believe that this curve is identical to the curve appearing in Fig. 2 of Kivelson and Southwood (1986). At the left side of the curve L is small and 1Y,l is large if K,’ is large. Approximation (54) is then appropriate, which appears as the dashed line in Fig. 4b. As a rule-ofthumb, (54) is accurate to within about 15 % if L < 0.1 and K,’ 2 1.5. We have unfortunately not succeeded in obtaining analytical approximations for other portions of the heavier curve. Finally, Fig. 5 displays C, vs L for an intermediate value of “surface” thickness, viz. a’ = 1.3. At larger values of K:, CA exhibits the general behavior found in Figs 2b and 3b.

The wall case Figure 6 displays C, vs L for a thin “surface”, a’ = 0.255. The solid curves are the numerical results, while the dotted curves represent approximation (45). The approximation tends to be quite good for all values of K, [except where L is very close to K,’ and the resonance is very close to the wall as indicated by equation (32b)], and we have chosen to display only two representative comparisons of (45) with the exact

0.0-

z (u

d I,

-0

0.05

0.00

.s

0.10

0. I5

0.20

I .o

4

v

0.8 0.6

0.4 0.2

o.ov 0.00 0.05 ’

I

0.10

0.15

0.20

0.25

I

0.30

L FIG. 6. SAME AS FIG. 2 BUT FOR THE WALL CASE.

The solid curves are the exact numerical results. The two dotted curves provide representative comparisons of approximation (45) with the exact results.

1026

J. V.

kg/k, ul d II -0

o.or 0.0

i

0.8,

0”

HOLLWEG

=2.24 I .25 0 80

I 0. I

0.2

0.3

0.4

0.5 0.0

I

0.5

I .o

I .5

20

L

FIG. 8. SAMEASFIG. 4 BUTFORTHEWALLCASE. The dashed line represents approximation (51), which is useful if K,' 6 0.1. The heavier curve is identical to the heavier curve in Fig. 4.

those appearing in Fig. 8.) The dashed line represents approximation (51). We can see from the figure that 0.0” 1 the approximation is very good as long as K, is small ; 0.0 0.1 0.2 0.3 0.4 0.5 0.6 recall that (51) was derived by assuming L 1Yoj = (K, - L) << 1. We have found that (51) is a approximation if K, 6 0.1, in which case FIG. I. SAMEAS FIG. 6 BUT FOR A THICKER“SURFACE”, reasonable u’ = 0.6. (5 1) overestimates C, by less than 15 % . The thin surface approximation (45) becomes inaccurate As was the case in Figs 6 and 7, the absorption can only at the largest values of K,'. approach 100 % ; we have found that this occurs if 0.25 6 K: 6 0.5. The heavier curve in Fig. 8 represents the asymptotic behavior which results when 2.5 6 K,’ cc a’. This results. A surprising result is that the absorption curve is identical to the heavier curve in Fig. 4b, and coefficient can approach 100%) in contrast to the its behavior near L = 0 is given by approximation vacuum case where the absorption coefficient never (54). It is noteworthy that the heavier curve in Fig. 8 exceeds about 50 % . The boundary condition at y = 0 is asymptotically approached from above in the wall has a major impact on the maximum absorption. We case, while it is approached from below in the vacuum believe that this behavior is a consequence of 16Pl at case (Fig. 4b). However, this curve is independent the resonance being larger in the wall case than in the of the boundary condition at y = 0 because K: is vacuum case, for a given incident energy flux ; we have essentially a measure of the distance that the waves not shown that this is true in general, but it is true for have to tunnel from the turning point toy = 0, and if a “thin” surface, as can be seen by carrying out an Kz is large the waves never get to “feel” the boundary analysis parallel to equations (39)-(42) for both the condition at y = 0. wall and vacuum cases. Finally, Fig. 9 shows the behavior of C, for an Figure 7 gives the results for a thicker “surface”, intermediate surface thickness, viz. a’ = 1.3. Absorpa’ = 0.6. As expected, approximation (45) is less satistion coefficients close to 100 % are possible in this case factory in this case, particularly when K, z a’ and too. the incident waves graze the x-z plane at y = a. It should be noted that approximation (45) does not give the correct result that C, = 0 when L = K:. 8. DISCUSSION This is in contrast to approximation (38) for the In solar, magnetospheric, and fusion plasma vacuum case, which does reproduce the correct physics, much attention has been given to the resbehavior of C, when L = Kz. onance absorption of MHD surface waves, since that Consider next the absorption at a thick “surface”. process is a prime candidate for plasma heating. HowFigure 8 displays C, vs L for a’ = 9. (We have verified ever, the resonance absorption of propagating waves that CA is insensitive to a’ as long as a’ >> 1. In fact, impinging on a “surface” has received little attention. taking a’ 2 2.5 yields results not very different from

Resonance absorption of propagating fast waves in a cold plasma 1.0

0.2

0.0 ~ 0.0

0.2

1.4

FIG. 9. SAG ASFro.6 BUT FOR AN IN~~D~~ VALCZ OF ff'(=1.3) WHICH DOES NOT YIELD CONVENIENT ANALYTICAL APPROXIMATiONS.

In solar physics we know of only one such analytical calculation, viz. the resonance absorption of sound waves impinging on a sunspot, and that calculation was restricted to a thin boundary and to cases where the absorption coefficient was small ; the behavior of the absorption coefficient under more general conditions still needs to be examined, although a numerical study has been carried out by Lou (1990). The sunspot problem is quite involved, however, since the plasma beta (the ratio of thermal to magnetic pressure) is not small, and thermal as well as magnetic effects come into play. In this paper we have begun an examination of the general behavior of resonance absorption of propagating waves by considering a much simpler case. We have thrown out thermal effects by taking the plasma to be cold, and we have considered a linear density profile as in Fig. 1. Fast mode MHD waves propagate into the density ramp from y > a, and are resonantly absorbed somewhere in the ramp. If the region y < 0 is taken to be a vacuum, then we may regard our results as roughly representative of what might happen to coronal fast waves propagating toward a region of much lower density ; two possible examples are waves trapped inside corona1 active region loops (e.g. Edwin and Roberts, 1982; Gordon and Hollweg, 1983) or waves propagating toward a coronal hole boundary. The trapped wave case has been suggested as a possible explanation of pulsations observed in Type IV radio events (Roberts et al., 1984). We expect, however, that the large absorption coefficients found here may severely limit the ability of some of the trapped modes to persist long enough to be observed. On the other hand, the “sausage” modes discussed by these authors in effect have k, = 0 if the magnetic field is aligned along the coronal loop, and those modes will not suffer absorption.

1027

The situation in Fig. 1 has also been applied to the magnetosphere by Kivelson and Southwood (1986). The magnetospheric case has been crudely modelled here by changing the boundary condition at y = 0 to correspond to a very dense region (the plasmasphere) in y < 0. We have referred to this situation as the “wall case”. In general we have found that the boundary condition at y = 0 can change the behavior of the resonance absorption dramatically. For example, in the vacuum case the absorption coefficient never exceeds about 50%, but in the wail case the absorption coeihcient can approach 100 % . We have also found that in both the vacuum and wall cases the parametric dependence of the absorption coefficient is far richer than implied by Fig. 2 of Kivelson and Southwood. In particular, their “coupling parameter” L is not the only quantity controlling the absorption coefficient. We have provided several analytical results which give the absorption coefficient when the “surface” is thin ; see equations (38) and (45). Comparison with exact numerical results has shown that these approximations have a useful domain of validity. We have also provided some analytical results for the opposite situation of a ramp which extends to large y values where the waves are essentially in the WKB limit ; see equations (50), {Si) and (54). Here our approximations are less useful, but we did succeed in analytically deriving the small-L behavior of Fig. 2 of Kivelson and Southwood, which was found to be insensitive to the boundary conditions at y = 0 because of the long distance the waves have to tunnel when K, is large. Two other useful analytical results were obtained along the way. In Section 3 we showed that the discontinuity in the y-component of the Poynting tlux which results here ma~ematically from the logarithm in equation (14) is precisely equivalent to the energy flux into the ‘~energy-containing” layer obtained by HY using the physical picture of driven harmonic oscillators. We also found that the “shortcut method” presented in Section 5 for a thin “surface” yielded a useful approximation to the exact results. This result is reassuring, since a similar shortcut method was used by us in investigating the resonance absorption of pmodes by sunspots. We should also mention that we derive some additional reassurance from the fact that the approximate treatment of the sunspot problem yielded the same general dependence of CA on the propagation direction of the incident waves as we have found from the exact solutions in this paper ; the geometrical effects are discussed explicitly below. In Figs 2-9 we have presented our results in a form

1028

J. V. HOLLWEG

which emphasizes the dependence of the absorption coefficient on the propagation direction of the incident waves in y > a. For example, suppose that w, a, and r+,(y) are given. Then ~1~is a constant given by equation (4), i.e. 0~’= [w/~~(u)]*/a, and the dimensionless variables used in Figs 2-9 are then readily converted to physical variables. In particular, picking a specific value for a’ is equivalent to fixing the magnitude of the wavevector in y > a, since k: + k-z, = cc2a.Now let us look at Fig. 2 as a specific case. We see that C, depends on two angles. The first is the angle of incidence with respect to the x-z plane. This is fixed for each choice of K:, since

Finally, we would like to note that the general approach of this paper can be used also to investigate the excitation and damping of the surface modes. Consider first the shortcut method used in Section 5 for the vacuum case. We put a discontinuous boundary at y = 0 and solve the standard reflection/ transmission problem, but we take lk,,l = ire in equation (39), with K > 0. The first term in (39) can then be regarded as an evanescent wave launched by an imaginary antenna somewhere in y > 0. The second term in (39) is then the response of the discontinuity. The response follows from equation (41), since l+R = T:

(,‘~/k,)~ = a%/k,2- 1

R = (is- l)/(is+ 1).

or

(55)

The response is infinite when is = - 1, i.e. (kyl/kJz

= at/K;-- 1.

The absorption coefficient is small at very small or very large angles of incidence, and maximizes at moderately large angles of incidence around 70” or so. The absorption vanishes when the angle of incidence is zero because k,, and thus ]A($,)/, is zero [equation (2011, and the absorption vanishes when the angle of incidence is 90” because a fast wave propagating parallel to the “surface” cannot impart any total pressure perturbations to the surface. Now suppose that the angle of incidence is fixed by picking K,. The second angle affecting C, is the angle between the magnetic field and the plane of incidence. This angle varies with L, since L a kf = k,‘- k,f. The absorption vanishes when L = 0 because ]A(.!$)] a L [equation (20)] ; and the absorption vanishes when L = K,’ because k,, is then zero and the resonance condition cannot be satisfied. The absorption exhibits a broad maximum when the angle between k, and B, is around 30” or so. Inspection of Figs 2-5 shows that these statements are fairly representative of the vacuum case. It is also worth noting that the same general dependence of C, on the propagation direction of the incident wave was found in our study of the interaction ofp-modes with sunspots (Hollweg, 1988). This gives us some confidence that the approximations used there have validity. Figures 69 show that the wall case is different in some respects from the vacuum case. The angle of incidence leading to the largest absorption coefficients is more sensitive to a’ ; and there is a tendency for the absorption to maximize at larger angles between k, and B, than was found in the vacuum case. All of these dependences agree qualitatively with the results of Allan et al. (1985, 1986a,b, 1987a,b), who used a cylindrical geometry in a magnetospheric context.

+oAh
= - lkxl (~oA),>o-

(56)

This is precisely the dispersion relation for the surface wave [see equation (61) of HY], and the infinite response results because the antenna is perfectly coupled to the undamped surface wave. To eliminate the inanity, we replace the discontinuity by a thin “surface”. From equation (34) we find (57) and the resonant denominator is never zero. An interesting result can be obtained if we examine the behavior of (57) as a function of w, while holding k,, k,, and k, fixed, but retaining the w-dependence of K. If the “surface” is thin, then IRI’ will show a narrow resonant peak very close to the frequency for which is = - 1. (A related resonance peak can be found in Fig. 2.11 of Poedts, 1988.) The half-width at halfmaximum, Ao is given by ](l +is)/is] = nkf’Y&,

(58)

where Y, = a(1 -k,$$,,,h2) and 1)~~= VA(a). If the peak is narrow we can represent the left-hand side of (58) by the first term in its Taylor series about is = - 1. After considerable algebra we find A.0 =

nk:a(d

- k$&. 1)2

~3(2k-+k;llkl)

’

(59)

where w and K must be solutions of equation (56). Now it is a general property of driven resonant systems that (Aw)r,,

= 1,

(60)

where rr,, is the amplitude decay time for the undriven system. Thus equation (59) gives the decay rate, Z&

Resonance absorption of propagating fast waves in a cold plasma of the surface mode due to resonance absorption. We have verified that equation (59) agrees with the decay rate which one obtains using the physically-motivated procedure in HY, and the reader is referred to that paper for a discussion of the physics involved in surface wave decay due to resonance absorption. The above simple example illustrates that the techniques of this paper can be used to deduce, indirectly, the decay rates of the surface modes, if we simply allow k,,, to be imaginary and look at the widths of the resonant peaks. We have exploited this technique in another study (Hoiiweg, 1990), with particular emphasis on the decay rates of modes on thick “surfaces”, a regime which has been inadequately studied in the past. substantial part of this work was carried out while the author was a guest of the solar physics

Acknowledgements-A

group at the University of St Andrews. The hospitality of B. Roberts, E. Priest, and A. Hood is gratefully acknowledged, as is partial support from the Science and Enginee&g Research Council of the U.K. The author is grateful for useful conversations with M. Goossens, P. Isenbgrg, M. Lee, S. Poedts, B. Roberts, andC. Uberoi. The referee’scomments have also substantially improved the paper. This work was also supported in part by the NASA Solar-Terrestrial Theory Program under Grant NAGW-76, and by NASA Grant NSG-7411.

REFERENCES

Allan, W., Poulter, E. M. and White, S. P. (1986b) Hydromagnetic wave coupling in the magnetosphere-plasmapause effects on impulse-excited resonances. Planet. Space Sci. 34, 1189. Allan, W., Poulter, E. M. and White, S. P. (1987a) Hydromagnetic wave coupling in the magnetosphere-magnetic fields and Poynting fluxes. Planet. Space Sci. 35, 118 1. Allan, W., Poulter, E. M. and White, S. P. (1987b) Structure of magnetospheric MHD resonances for moderate “azimuthal” asymmetry. Pforret. Space Sci. 35, 1193. Allan, W., White, S. P. and Poulter, E. M. (1985) Magnetospheric coupling of hydroma~etic waves-initial results. Geop!rys. Res. Lett. 12,287. Allan, W., White, S. P. and Pouiter, E. M. (1986a) Impuiseexcited hydromagnetic cavity and field-line resonances in the magnetosphere. Planet. Space Sci. 34,371. Appert, K., Vaclavik, J. and Villard, L. (1984) Spectrum of low-frequency, nonaxisymmetric oscillations in a cold, current-carrying plasma column. Phys. Fluids 27,432. Braun, D. C., Duvall, T. L., Jr. and Labonte, B. J. (1987) Acoustic absorption by sunspots. Astrophys. J. 319, L27. Braun, D. C., Duvall, T. L., Jr. and Labonte, B. J. (1988) The absorption of high-degree p-mode oscillations in and around sunspots. Astraphys. J. 335, 1015. Chen, L. and Hasegawa, A. (1974a) Plasma heating by spatial resonance of AIfv&n wave. Phys. FZzxids17, 1399. Chen, L. and Hasegawa, A. (1974b) A theory of long period magnetic pulsations 1. Steady state excitation of field line resonance. J. geophys. Res. 79, 1024. Chen, L. and Hasegawa, A. (1974~) A theory of long-period

1029

magnetic puhations 2. Impulse excitation of surface eigenmode. J. geaphys. Res. 79, 1033. Davila, J. M. (1987) Heating of the solar corona by the resonant absorption of AlfvCn waves. Astrophys. J. 317, 514.

Edwin, P. M. and Roberts, B. (1982) Wave propagation in a magnetically structured atmosphere III: the slab in a magnetic environment. Solar Phys. 76, 239. Einaudi, G. and Mok, Y. (1985) Resistive Alfven normal modes in a non-uniform plasma. J. plasma Phys. 34, 259. Forslund, D., Kindel, J. M., Lee, K., Lindman, E. L. and Morse, R. L. (1975) Theory and simulation of absorption in a hot plasma. Phys. Rev. A 11,679. Friedberg, J. P., Mitchell, R. W., Morse, R. L. and Rudsinski, L. I. (1972) Resonance absorption of laser light by plasma targets. Phys. Rev. Lett. 28,795. Goertz, C. K. and Smith, R. (1989) The thermal catastrophe _ model of substorms. J. geophys. Res. 94,6581. Gordon. B. E. and Hollwea, J. V. (1983) Coliisional damninrr of surface waves in thesolar corona. Astrophys. J. i66, 373.

Grossmann, W., Kaufmann, M. and Neuhauser, J. (1973) Damping of Alfvkn and magneto-acoustic waves at high beta. Nucl. Fusion 13,462. Grossmann, W. and Smith, R. (1988) Heating of solar coronal loops by resonant absorption of Alfvtn waves. Astrophys. J. 332,476.

Grossmann, W. and Tataronis, J. (1973) Decay of MHD wave by phase mixing, II. Theta pinch in cylindrical geometry. 2. Phys. 261, 217. Hasegawa, A. and Chen, L. (1974) Plasma heating by Alfvenwave phase mixing. Phys. Rev. Let?. 32,454. Hasegawa, A. and Uberoi, C. (1982) The Alfven wave. Rep. DOE,TZC 11197, Tech. Inform. Cent., U.S. Dept of Energy, Washington, D.C. Hollweg, J. V. (1987a) Resonance absorption of magnetohydrodynamic surface waves : physical discussion. Astrophys. J. 312, 880. Hollweg, J. V. (1987b) Resonance absorption of MHD surface waves : viscous effects. Astrophys..J. 320, 875. Hollweg, J. V. (1988) Resonance absorntion ofsolar n-modes by su
Kivelson, M. G. and Southwood, D. J. (1986) Coupling of global magnetospheric MHD eigenmodes to field line resonances. igeophys. Rex 91, 4345. Lanzerotti. L. J.. Fukunishi. H.. Haseeawa. A. and Chen. L. (1973) Excitation of the plasmap&se’at ultralow frequencies. Phys. Rev. Z&t. 31,624. Lee, M. A. and Roberts, B. (1986) On the behavior of hydromagnetic surface waves. Astrophys. J. 301, 430. Lou, Y.-Q. (1990) Viscous ma~etohydrod~~ic modes and p-mode absorption by sunspots. Astraphys. J. 350, 452.

Mok, Y. and Einaudi, G. (1985) Resistive decay of Alfven waves in a non-uniform plasma. J. plasma Phys. 33, 195.

1030

.J. V. HOLLWE~

Poedts, S. (1988) Verhittung van de corona van de zon door resonante absorptie van magnetische golven. Ph.D. thesis. Katholieke Universiteit. Leuven. Poedts, S., Goossens, M. and Kerner, W. (1989b) Numerical simulation of coronal heating by resonant absorption of Alfven waves. Solar Phys. 123,83. Poedts, S., Goossens, M. and Kemer, W. (1989~) Coronal loop heating by resonant absorption, in Physics of Magnetic Flux Ropes (Edited by Russell, C. T.). AGU Monograph (in press). Poedts, S., Kemer, W. and Goossens, M. (1989a) Alfven wave heating in resistive MHD. J. plasma Phys. 42,27. Pritchett, P. L. and Canobbio, E. (1981) Resonant absorption of Alfven waves. Phys. F&ids 25,2374. Roberts, B. (1986) Ma~etohydrodyn~ic waves, in Mar System magnetic Films (Edited by Priest, E. R.). D. Reidel, Hingham, MA. Roberts, B., Edwin, P. M. and Benz, A. 0. (1984) On coronal oscillations. Astrophys. J. 279, 857. Smith, R. A., Goertz, C. K. and Grossmann, W. (1986)

Thermal catastrophe in the plasma sheet boundary layer. Geophys. Res. L,e&. 13, 1386 Southwood. D. 3. (19741 Some features of field line resonances in the magnetosphere. Planet. Space Sci. 22,483. Southwood, D. J. and Hughes, W. J. (1983) Theory of hydromagnetic waves in the magnetosphere. space Sci. Rev. 35, 301.

Southwood, D. J. and Kivelson, M. G. (1986) The effect of parallel inhomogeneity on magnetospheric hydromagnetic wave coupling. J. geophys. Res. 91, 6871. Tataronis, J. A. (1975) Energy absorption in the continuous spectrum of ideal MHD. J. plasma Phys. 13, 87. Tataronis, J. A. and Grossman, W. (1973) Decay of MHD waves by phase mixing, I. The sheet pinch in plane geometry. %. P!zys. 261, 203. Uberoi, C. (1989) Resonant absorption of Affven compression~‘surface waves. J. geop~ys. Res. 94,6941. Zhu, X. and Kivelson, M. G. (1988) Analytic formulation and quantitative solutions of the coupled ULF wave problem. J. geaphys. Res. 93,8602.