Exact and approximate methods for Alfvén waves in dissipative atmospheres

Wave Motion 17 (1993) 101-112 Elsevier

101

Exact and approximate methods for Alfvkn waves in dissipative atmospheres L.M.B.C. Campos Institute Superior Tecnico, 1096 Lisboa Codex, Portugal

Received 20 September 1991

The Alfvtn wave equation in a dissipative atmosphere has been solved exactly (Campos [l-3]) or using the phase mixing approximation (Heyvaerts and Priest [4], Nocera, Leroy and Priest [5]). In the present paper, we compare the phase mixing approximation, as it appears the reference above (which we designate henceforth HP), with an exact solution of the same model problem (Section 1). It is shown that: (Section 2) the dissipative Alfven wave equation in HP is correct for the magnetic field perturbation, but not for the velocity perturbation; (Section 3) HP makes assumes implicitly that both the static viscosity and resistive diffusivities are constant, and omits restrictions on the external magnetic field, so that we redefine the atmospheric model which is the background for the wave propagation and dissipation; (Section 4) the phase mixing ‘ansatz’, when combined with the principle of superposition, which must hold for linear waves, is incompatible with Fourier analysis; (Section 5) the exact solution for dissipative Alfven waves in an atmosphere, demonstrates the existence of a critical level, separating regions of dominant viscous and resistive dissipation; (Section 6) none of these properties is apparent in the phase mixing approximation, which can be compared with the exact solution if three restrictions are made: (i) high- or low-altitude i.e. far from the critical layer; (ii/iii) high-frequency and weak damping. Even under these restrictions no satisfactory agreement is found.

1. Introduction

Alfven waves were ‘discovered’ theoretically (Alfven [6]), and soon ‘applied’ to problems, such as heating of the solar atmosphere (Alfven [7]), for which it is essential to account for dissipation. In these early researches the propagation speed and damping rates were treated as constant, i.e. the atmosphere was treated as an homogeneous medium; the need to lift this restriction was, however, appreciated (lot. cit.). A similar approach has been revived recently, adding wave forcing, as the ‘RLC-analogy’ (Ionson [S, 93, Kuperus, Ionson and Spicer [lo]), but again ignoring the variation of propagation speed and damping rates with altitude, this time without acknowledging the need to overcome such restrictions. Awareness of this difficulty existed before, e.g. Osterbrock [l l] reconsidered dissipation of AlfvCn waves, including viscosity as well as Ohmic resistance, and applied the results to the solar atmosphere represented by homogeneous layers, whose properties ‘jump’ at the interfaces. A more suitable model would consider continuous changes of propagation speed with altitude, and it has been developed, neglecting dissipation, by several authors (Ferraro and Plumpton [ 121, Zugzda [ 131, Hollweg [ 143, Leroy [ 151, Campos [ 161, Parker [17], Schwartz, Cally and Be1 [18]). These results have been used again to assess the feasability of heating of the solar atmosphere by AlfvCn waves (Hollweg [ 19,201, Leroy [21]); however, it can be argued that for that application dissipation should be taken into account when calculating the waveforms, for nondissipative waves cannot heat the atmosphere. The preceding theories of non-dissipative Alfven waves in atmospheres may be relevant for such conditions, e.g. oscillations in solar spicules (Campos [22]) and sunspot umbras (Campos [23]), where damping is weak. 01652125/93/$06.00

0 1993 - Elsevier Science Publishers B.V. All rights reserved

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L. M. B. C. Campos / Methods for Alfib

waves

In order to provide an adequate physical basis for the assessment of atmospheric heating by dissipation of AlfvCn waves, the AlfvCn wave equation with viscous and resistive terms, should be solved taking into account that both propagation speed and damping rates may vary with altitude. The first solutions of this kind were published simultaneously, using exact methods (Campos [ 11) or the phase mixing approximation (Heyvaerts and Priest [4]). The first solution, obtained for uniform vertical magnetic field and constant magnetic diffusivity (Campos [l]), has been extended to include viscosity as well as Ohmic resistance, and to uniform oblique magnetic fields (Campos [2]); the case of viscous and resistive AlfvCn waves in an atmosphere under uniform or non-uniform horizontal magnetic field has also been considered by means of exact solutions (Campos [3]). The phase mixing approach has been reconsidered as an approximate method (Nocera, Leroy and Priest [5]) and compared with the case of an external magnetic field which is a linear function of distance (Steinholfson [24]). The latter case is limited to short distances, and thus it may be of interest to compare phase mixing with an exact theory applying to an atmosphere of infinite extent. The simplest way to do this is to re-consider the problem in HP, to solve it exactly in a suitable atmospheric model, and then to compare with the phase mixing approximation. This comparison is made more difficult by the existence of some inconsistencies in HP: (i) the dissipative Alfven wave equation is given for the velocity perturbation, but it is incorrect for the latter, since it misses out some terms, although it turns out to be correct for the magnetic field perturbation; (ii) it is assumed implicitly in HP that the static viscosity and magnetic diffusivity are constant, but other physical conditions of the background state need to be specified for an exact solution e.g. in an atmosphere; (iii) HP also omits some restrictions on the form of the external magnetic field, and all these points must be clarified, before a meaningful comparison of the phase mixing approximation with exact results can be made. The present paper is organized as follows: (Section 1) in the introduction we discuss the motivation for the comparison of the phase mixing approximation with exact solutions for Alfven waves in atmosphere, and the way in which this task is approached; (Section 2) the AlfvCn wave equation in HP are checked as particular case of the wave equation for an external magnetic field orthogonal to the direction of stratification (which we call altitude), and allowing for the Alfven speed, external magnetic field, viscous and resistive diffusivities to be arbitrary functions of this altitude; (Section 3) the test case for comparison is an isothermal atmosphere, under an uniform horizontal magnetic field, with constant static viscosity and magnetic diffusivity; (Section 4) the phase mixing approximation, when combined with the principle of superposition, which must be satisfied by linear waves, leads to a representation of the wave field which is incompatible with the Fourier integral; (Section 5) the latter is used to solve the wave equation exactly in terms of hypergeometric functions, showing that there is a critical level separating regions of dominant dissipation either by viscosity or electrical resistance; (Section 6) the phase mixing result ignores the existence of the critical level, and can be compared to the exact solution only far from it, and for weakly damped waves of high-frequency. Comparison with the exact solution far below or above the critical level leads to a dependence of the wavenumber on altitude which is in contradiction with that assumed in HP. The only possible explanation is then that the phase mixing representation does not exist in the form assumed in HP, for the particular problem considered here; in another case, with external magnetic field not constant but a linear function of distance (Steinholfson [24]) the comparison of the exact solution with the phase approximation appears to be more favourable. Thus we conclude that the phase mixing approximation, as presented in HP, is not a reliable method of study of dissipative Alfvtn waves in atmospheres. It remains open whether: (a) the type of phase mixing approximation in HP could apply reliably to other types of waves; (b) Alfven waves in dissipative atmospheres could be studied by the approaches to phase mixing distinct from that in HP.

L.M. B. C. Campos / Methodr for Alfvt% waves

103

2. Dissipative Alfvbn wave equation for external magnetic field transverse to stratikation Since Alfven waves are transversal, and hence incompressible, they are described by the induction and momentum equations alone, and dissipation of linear waves is not affected by thermal conduction or radiation, but only by electrical resistance and incompressible viscosity. The former appears in the induction equation : ~H/CU+VX(HXV)=-VX

{(c2/4npa)VxH},

(1)

where H is the magnetic field, V the velocity of the fluid, p the magnetic permeability, c the speed of light in vacua, and cr the Ohmic electrical conductivity, which may be non-uniform, i.e. depend on the coordinate x in the direction of stratification, so that the resistive diffusivity (2a) may be non-uniform: x(x)=c2/4xpo(x),

(2a, b)

v(x) = <(x)/P(x),

the viscous diffusivity or dynamic viscosity v may also be non-uniform, if either the static viscosity 5, or mass density p, or both, are non-uniform. The viscous stresses involve two static viscosities, incompressible 4 and compressible 4: V)6g

Tg=caK/axj+f(V.

(34

of which the latter will not affect Alfven waves; the divergence of the viscous stress tensor: aT~/axj=5a2v,/ax,‘+ag/axjaV,/axj+a{4(V.

V)}/ax,,

W)

appears on the 1.h.s. of the. momentum equation : ~{av~at+(V~V)V~+Vp+(~/4~)H~(V~H)+f=~V2V+(V~~V)V+V{~(v~V)},

(4)

balancing the inertia force, pressure gradient, magnetic force, and any other external force, e.g. gravity f=pg. We assume exactly the same geometry as in HP, viz. an external magnetic field in the z-direction, transverse to the x-direction of stratification, and velocity and magnetic field perturbation in the y-direction, transverse to both the external magnetic field and direction of stratification: H(x, t) = B(x)e, + h(x, z, t)e,,

V(x, 1) = 0(x, 2, t)e,.

(5a, b)

Since the velocity and magnetic field perturbations depend only on transverse coordinates the waves are imcompressible and satisfy V . H=O. The induction equation (5) becomes linear on both wave variables: ah/at-~

avia

= x azh/a2+ a(x ah/ax)/ax,

(6)

and it also restricts the external magnetic field to satisfy: d(X dB/dx)/dx

= 0.

(7)

In HP it is assumed implicitly ([4], eq. (6)), that the magnetic diffusivity is constant in (6), and the restriction (7) is not mentioned. The momentum equation is linear on the velocity perturbation but not in the magnetic field perturbation, i.e. must be linearized for the latter: aqat - (jd3/4~p) ah/az = ( c/p) a%/a22+ p-1 a( 4 av/ax)jax,

(8)

L. M. B. C. Campos / Methods for AlfvCn waves

104

and it also restricts the background state to satisfy: fQ+@*/8rr)/ax=f/p.

(9)

In HP it is assumed implicitly ([4], eq. (5)) that the static viscosity is constant in (8), and the restriction (9) is unmentioned, perhaps because a background pressure can be found which satisfies it. In the induction (6) and momentum (8) equations, all coefficients may be functions of stratification, viz. the mass density p, external magnetic field B, Alfvtn speed: A(x) = JGFGCG

B(x),

(10)

magnetic diffusivity x and incompressible static 5 or kinematic v viscosity. We can write in operator form the induction (6) : (apt-x

i3*/az*-- alax x a/ax)h = B aqaz,

(11)

and momentum (8) :

{a/at - v a*/az* - p-1 a/ax 4 a/a+=

(k/B)

ah/az,

(12)

equations, and without further restrictions eliminate between them, viz. : B a%/az* = {a/at-~ = {a/at-x

for the velocity perturbation,

a*/az* - a/ax x a/ax} ah/az a*/az* - a/ax x a/ax}(Bp*){a/at

- v a*/az* - p-1 a/ax 4 a/axjv,

(13)

or:

(A*/B)a*h/az* = {a/at = {a/at -

v

a*/az* - p-1 a/ax 5 a/ax) av/az

v a*/az* - p-1 a/ax 5 ajaxpi-‘{a/at

- x a*/az* - a/ax x a/ax)h,

(14)

for the magnetic field perturbation.

3. Atmosphere in viscous and resistive magnetohydrostatic equilibrium

Thus we have obtained the viscous and resistive AlfvCn wave equation, for an external magnetic field transverse to the direction of stratification, both for the velocity: a% $-A

*a% i a azz__-4 a% p azZat-pdaxat

_-a*v

a% _-__ ‘aZ*at

B a

a pa0

paxxGZG

(15) and for the magnetic field:

=_%~~_~~~a’a,~_~a,~_~~~~~~, p ax ax B ax ax pax az ax p ax perturbations,

were all coefficients may be functions of x.

ax B aZ

(16)

L. hf. B. C. Campos / Methods for A&?n waves

In HP the velocity perturbation

105

is stated to satisfy:

a*u/at* - A2 a*u/az* = (v + x)(a3v/axZ at + a’?J/az’ at),

(A)

where for weak damping the products of small diffusivities are neglected. This assumption leads to the omission of the r.h.s. of (15) and (16), but even in this case, the velocity and magnetic field perturbations satisfy different wave equations : a*t~/af* - A*

a*o/az’ =(X+ v)(a30/a2at+a%/az* at) + we+ (~ip)[(~pi~)‘+~(pi~)‘i) Wax at + w~mw~~

wat,

a*h/at* - A* a*h/az2 =(X+ qa3h/a2 at+a3h/az* at)+ { 5 f/p + wd( 5(1iwwm

(17) + ( t~/p)(

i p3y + f }a*h/axat (18)

where prime denotes derivative with regard to x. In HP it was implicitly assumed that the magnetic diffusivity x and static viscosity 5 were constant; for an inhomogeneous medium the mass density p, and kinematic viscosity v (26) would not be constant, and thus (17) does not coincide with (A), i.e. HP have missed in the wave equation for the velocity perturbation the terms: ~X(B/LWB)’

aWax at + XW)(~~W

au/at;

(A’)

these terms are not negligible ‘a priori’, because they are comparable to terms retained in (A). On the other hand, the wave equation for the magnetic field perturbation (18), in the case of uniform static viscosity and magnetic diffusivity, has in addition to (A), the terms: ( 2ypp(i/Bya*h/axat + (
i

pyah/at,

(A”)

which vanish only for a uniform external magnetic field B=O; since, with the latter restriction, the wave equation in HP applies to magnetic field instead of the velocity perturbation, we use the former in the following comparison of the phase mixing approximation in HP with the exact solutions obtained here. For a constant magnetic diffusivity, the condition (7) implies that the external magnetic field B" = 0 is either a constant or a linear function of x. The latter case has been considered elsewhere (Steinholfson 1985), and since the external magnetic field diverges as x + foe, it is not relevant to an unbounded atmosphere. Thus we consider here the case of uniform external magnetic field: B(X) = Bo,

xw=xo,

5(x> = 50 3

Wa, b, 4

as well as constant resistive diffusivity and static viscosity. In HP no external force field or cause for the stratification is given; in the absence of external force f =0, the condition (8) requires the total, gas plus magnetic, pressure to be constant. For a magnetic field a linear function of distance (Steinholfson [24]), this would lead to a gas pressure which is a quadratic decreasing function of x2; this is rather peculiar, since it would also apply to the mass density, in an isothermal medium. Here we consider the magnetic field to be constant, and the external force to be gravityf= -pg, so that (9) specifies magneto-hydrostatic equilibrium, with the mass density decaying exponentially on the scale height: p(x) = p0 emxlL,

L=RT/g,

QOa,b)

where T is the temperature and R the gas constant. It follows that the kinematic viscosity (2b) grows exponentially on the same scale height: v(x) = v. ez’L,

vo = SO/PO,

@la, b)

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L. M. 8. C. Campos / Methods for AIfvCn waves

and the Alfvin speed on twice that scale: A(x) = a eXlzL,

a=&%%&,

(22a, b)

the wave equation (18) for the magnetic field (18) : a*h/at* - a2 eziL

a*h/az* = (x0 + v. e”L)(a3h/aX2 at + a%/aZ at),

can be solved exactly (Section 5), for comparison with the phase mixing approximation consider next.

(23) in HP, which we

4. Incompatibility of phase mixing with Fourier integral representation In the phase mixing approximation pseudo-plane wave :

in HP the solution of the wave equation is sought in the form of a

h,(x, z, t) = W(x, z; w) exp{i[k(x)z - at]},

09

where the wavenumber is a function of the transverse coordinate: k(x)

E

o/A(x)

= (w/a)

exp(-x/21).

In HP the phase mixing representation approximations, respectively : k-1

ajaz-4 1,

(24)

(B) is combined with the weak damping and the strong phase mixing

(z/k) dk/dx B 1.

w

Since the properties of the medium depend only on x, we use a Fourier representation magnetic field perturbation is given by:

ss I

+a,

+a0

h(x, z, t) =

in z, t, i.e. the

dk G(x;

do

-co

k, o)

exp{i(kz-

cot)},

(25)

-CO

where G is the magnetic field perturbation spectrum, for a wave of frequency w and transverse wavenumber k at position x. The existence of the Fourier integral (25) is well known for (i) ordinary functions which are absolutely integrable on the real line, and locally of bounded fluctuation (Titchmarsch [25]) ; (ii) generalized functions defined with regard to base functions which are of rapid decay or ‘very good functions’ (Lighthill [26]). No existence theorem is quoted in HP for the phase mixing representation (B), and it is inconsistent with the Fourier integral representation (25), because the latter implies: Ik-’

a/aZI = Ii]= 1,

(z/k)

dk/dx = 0,

(26~ b)

which contradicts (C). Since the existence of the phase mixing representation (B) in general is open to question, we try to check whether it leads to a result compatible with an exact solution, under some kind of restriction. The substitution of (B) in the wave equation (23) E (A), together with (C), leads HP to: W(x, z; w) = W(x, 0; w) exp{ -k’*kz3(x

+ v)/6w),

(W

L.M. B.C. Campos / Methodr for A@&

waves

107

which we call the phase mixing spectrum. It applies to arbitrary frequencies, and because the present problem is linear, can be superimposed in a Fourier integral to specify the magnetic field perturnation: +cO h(x, z, t) = W(x, z; o) e-j”’ do. (27) s -co Comparing (27) with (25) it is clear that the phase mixing spectrum is related to the Fourier spectrum by: +UJ W(x, z; w) = dk G,(x; k, o) e**; (28) s -m we use G, to distinguish the spectrum calculated from the phase mixing approximation: +a, W(x, z ; o) emikrdz, G,(x; k, w) = (1/2x) I --oo from that G obtained from the exact solution (Section 5). The former is obtained substituting (D) into (29), we obtain: +a0 W(x, 0; o) cos(kz + bz3/3) dz, G,(x; k, w) = (l/x) I --oo

(29)

(30)

where : b = ik”k( v + x)/2w.

(31)

The Airy (30) integral (Abramowitz and Stegun [27]): G,(x; k, w) = W(x,O; a)b-“3

Ai(b-“3k),

(32)

leads to a spectrum like: G,(x; k, w) w (k’2k)-“3 Ai(e-i”‘3{2w/(

v +x)}

“3(k/k’)2’3);

(33)

since phase mixing applies to high frequencies and weak damping, the argument of the Airy function is large, and we may use the asymptotic approximation (lot. cit.) : Ai(n)=(1/2&)t7-1’4

exp{-(2/3)17-3’21P

+O(llrl)L

(34)

to simplify the wave spectrum to: G*(x; k, o) w (k’k)-“2 exp{ ($/3)(i

- l){ (x + v)/2w} 1’2k/k’},

(35)

which will be compared with the exact solution (Section 6), which we obtain next.

5. Exact solution at all altitudes including critical level The exact solution of the wave equation (23) for the magnetic field perturbation is similar to that for the velocity perturbation (Campos [3]), and we outline only the essential steps here. Substituting (25) we obtain an ordinary second-order differential equation for the magnetic field perturbation spectrum: {l+(s/fi)

e-“L}L2G”-{K2(1-i~)+(K2~/S-if22/~)e--r’L)G=0,

(36)

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L. M. B. C. Campos / Methods for AlfvGn waves

involving four dimensionless parameters, namely : K=kL,

l2 s @L/a,

6 = vow/a2,

.5s Xow/a2,

(3%

b, c, 4

the dimensionless wavenumber K transverse to the stratification, the dimensionless frequency R, and the viscous 6 and resistive E damping. The coefficient of G” shows that there is a singularity placed at c = 1 if we choose the new independent variable (38a), which is minus the ratio of magnetic (18a) and viscous (2 1a) diffusivities : c=-(c/6)

e-“‘“=-x(x)/v(x),

G(x; k, 0) = (“4 (0,

(38a, b)

we also perform a change of dependent variable (38b), where the constant II is chosen so that, after substitution of (38a, b) in (36), the whole equation can be divided by c. This is possible only if u satisfies: v=Km,

u2-K2(1+ia)=0,

(3%

in which case 4 satisfies an hypergeometric

b)

equation:

with parameters (Forsyth [28]) determined by: y=1+20,

a+P=2u,

a/l = -i(n2/.s-

K26),

(4la, b, c)

where the latter two equations (41a, b) can be solved: a,P=v*t,

IyJ_IE,

(42a, b)

for a, p. The solution of (40) is : (43) where C, D are arbitrary constants of integration, to be determined subsequently. Substituting (43) in (38b) we obtain the magnetic field perturbation spectrum: G(x>x,;k,w)=C~“F(o+yl, +D<-“F(o-

o-y/;

1+2~;-(~~/v~)e-“‘~)

y, u+ w; l-20;

-(xO/vO) ePXIL),

(44)

which is valid for ]c] < 1, i.e. for resistive diffusivity smaller than the viscous diffusivity (45a) :

(4% b)

x(x) < vc4

viz. in the high-altitude range above the critical level (45b). Thus we can use solution (44) to determine the asymptotic wave field at high-altitude: G(x-+ oo; k, co)-4 *“-exp(fux/L)

=exp{&kxJm},

(46)

showing that it consists of decaying and diverging components, respectively the first and second terms of (44). We impose a dissipation condition (Yanowitch [29], Yanowitch and Lyons [30], Campos [ 1, 3 1, 32]),

L.M. B.C. Campos / Methods for A&n

109

waves

requiring that the dissipation of energy per unit time, due to electrical resistance, integrated over a column of fluid, be finite: +oO

(c2/167r2a)lG(x; k, w)l dx< 00;

E,= s

(47)

--co

the coefficient in (47) is c2/167c20 = px/4x where (26) the Ohmic diffusivity is constant (19b), so that (47) is met only if the wave field vanishes at high-altitude G(x; k, w) + 0 at x + co. This condition, or the finiteness of (47), is met (46) by the first, but not the second, term of (44), so that we must set D=O. The remaining constant of integration C is determined from the initial velocity spectrum G(0; k, o) at altitude x= 0; if this lies below the critical layer x,>O, which is the case for x0> vo, then the initial condition can be applied after we extend the solution to x 1 in (38a), and thus we must use the variable ]l/cl< 1 in the hypergeometric solution (Caratheodory [33]) : Jxa, Pi Yi c) = {UY)W +

- a)lWWY

interchange(a,

k,

w)=Cei”a~V{T(l

+

1-

Y, a; a +

1-

Pi l/c)

/I) ;

this leads to the magnetic field perturbation G(x
- a)}(-T)-“F(a

(48)

spectrum below the critical level:

+2o)T(-21y)/T(u-

yl)T(l+

II- w)>

x F( ly - u, w + u, 1 + 2 ry ; -( vo/xo) e”‘“) + interchange(f

v).

(49)

The constant of integration C is determined from the initial spectrum, using (49) if x, > 0 : G(O;k,w)=Cei”“(-~o/vO)~(~(lI-2u)~(-2y)/~(u-~)~(1+u-~)) xF(yl-

u, y+ u; 1+2yl; -v~/~~)+interchange(fyl),

(50)

and using (44) if x, < 0 : G(0; k, w) = C(-x,,/v,,)“F(u+

I, u- I; 1+2u; --x0/t+,).

(51)

The expression (49) holds asymptotically in the deep layers x + -co, where c + cc in (38a), so that l/c 4 0 and the hypergeometric functions in (49) reduce to unity; the two terms in (49) then scale as: G(x +-Co)-c

*W-exp(rylx/L)-exp{Fkx,/~}.

(52)

Since the medium is infinitely dense as x + - co, both solutions can coexist, and they represent upward and downward propagating waves : G(x+ -co; k, o)-exp{Fkxu(cos

0-i

sin 6)}

(53)

where : u=ll + (0/#)2]“2,

20 = -arctan(w/Xdc2)

;

(5% b)

this may be approximated: (WXd2)2 6 1,

G(x+ -a~; k, w)-exp{Fkz&iwz/2~&},

(55)

and either expression shows that the critical level is of reflecting layer type (Campos [32, 34, 35]), because there are upward and downward propagating waves below it.

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L.M. B.C. Campos / Methods for Alfvin waves

6. Comparison for weakly damped, high-frequency far from the reflecting layer

We have obtained the exact wave field at all altitudes, using separate expressions above (44) and below (50) the critical layer; we can also obtain a single expression valid everywhere, if we note that c < 0 in (38a), so that I<- I]> jr] and I[/({- l)] < 1, so that we use this as variable in the hypergeometric function: (56)

Substituting in (44) we obtain the magnetic field perturbation G(x; k, w) = C{ 1 +(x0/v,,) F(x)=F(o+v,

u-w;

spectrum at altitudes:

e-X’L>-W{1 + (vO/xO) eX’L}-“F(x),

1+20;~~/(~~+v~e”‘~)),

(57a) (57b)

which can be used to check all preceding results, viz. : the constant of integration C is determined from the initial wave field, always, by: G(0; k, w)= C(VO)‘(XO)“(VO+XO)-‘-ULV),

(58)

which coincides with (50) for xe > 0 and (5 1) for x, < 0; (ii) the singularity of the wave equation (40) occurs at a complex altitude: x* = L log( -x0/ vo) = x, + ircL,

T(z*) = 13

(59)

and thus the wave field should be finite at the critical level; (iii) this is confirmed by the fact that y - a - p = 1 in (41a, b), so that (Bromwich [36]) the hypergeometric series (44) and (49) converge on the radius of convergence ](I = 1, which includes the critical level, the wavefield being given there by: G(x,; k, ~)=C2-“-~F(1/2)=C2-“-~F(o+yl,

o-y/;

1+2u; l/2),

(60)

(iv) asymptotically at high- or low-altitude the hypergeometric function (57b) tends to a constant, and the wave field scales as (46) as x + + co, and as (52) as x + -co. Since the phase mixing solution gives no account of the existence of a critical level, and consists of a single exponential (D), we can compare it with the exact solutions (44, 50, 57a, b) only when the latter reduce to a single exponential, viz. in the high- or low-altitude asymptotic limits, far from the critical level. Since phase mixing applies to weak damping of high-frequency waves, we further simplify the high (46): G(x+co;k,o)-exp{‘f(l+i)(kx/a),/g},

(61)

and low-altitude (52) : G(x+

-co;

k,

w)-exp{F(i-

1)x,/m},

(62)

solutions. The high-altitude solution would coincide with (35) the phase mixing result the wavenumber would satisfy : k’= &(2i/3)(a/ox)dw,

(63)

which would imply it would have a logarithmic singularity at x=0 k(x) N log x,

(k’k)-1’2 m J;s

log-“2x

(64a, b)

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L.M. B. C. Campos / Methods for Arfo& waves

which would extend to the amplitude factor; the low-altitude solution (62) would coincide with the phase mixing result (35) if: k’lk = (J2/3wx)

{(~0 + x&o} I’*,

(65)

leading to a linear wavenumber and singular amplitude factor: k-x,

(&k)-l/*NX-‘/*.

(f-% b)

Both wavenumbers (64a) and (66a) are inconsistent with that (24) assumed in HP, if we use the latter (24) in (35) we obtain: G(x; k, ~)~e~‘*~exp{(,/?/3){(1

-i)/2L}{(v0+x0)/2w}“*)

(67)

which does not coincide either with (61) or (62). The amplitude factor could be absorbed in (D), but not the phase factor. Thus we did not find satisfactory agreement between the presented here, and the phase mixing approximation as used in HP. The possibility remains comparison could be made in another way, if there is a physical justification for proceeding

W(x, 0; w) in exact theory open that the differently.

Acknowledgment The author acknowledges the hospitality of the Max-Planck Institut fur Aeronomie where this work was done as Alexander von Humboldt Scholar.

References [ll L.M.B.C. Campos, “On Theor. Appl. 2, 861-891 121L.M.B.C. Campos, “On [31 L.M.B.C. Campos, “On

viscous and resistive dissipation of hydrodynamic and hydromagnetic waves in atmospheres”, J. WC. (1983). oblique Alfvtn waves in viscous and resistive atmospheres”, J. Phys. A21, 291 l-2930 (1988). the dissipation of atmospheric Alfven waves in uniform and non-uniform magnetic fields”, Geophys.

Astrophys. Fluid. Dyn. 48, 193-215 (1989).

[41 J. Heyvaerts and E.R. Priest, “Coronal heating by phase mixed Alfven waves”, Astron. Astrophys. 117, 220-234 (1983). 151L. Nocera, B. Leroy and E.R. Priest, “Phase mixing of propagating Alfv&n waves”, Astron. Astrophys. 133, 387-494 (1984). [61 H. Alfven, “On the existence of electromagnetic hydrodynamic waves”, Ark. Mat. Astron. Fys. A29, l-7 (1942). [71 H. Alfv&n, “Granulation, magneto-hydrodynamic waves and the heating of the solar corona”, Month. Not. Roy. Astron. Sot. 107,21 l-222 (1947).

[81 J.A. Ionson, “Resonant electrodynamic heating of stellar coronal loops: an LRC circuit analogy”, Astrophys. J. 254, 318-332 (1982).

[91 J.A. Ionson, “A unified theory of electrodynamic coupling in coronal magnetic loops: the coronal heating problem”, Astrophys. .I. 276, 357-376 (1984).

WI M. Kuperus, J.A. Ionson and D.S. Spicer, “On the theory of coronal heating mechanisms”, Ann. Rev. Astron. Astrophys. 79, 7-20 (1981).

1111 D.E. Osterbrock, “Heating of the solar atmosphere, plages and corona by magnetohydrodynamic

waves”, Astrophys. J. 134,

347-374 (1961).

WI V.C.A. Ferraro and C. Plumpton, “Hydromagnetic waves in an horizontal stratified atmosphere”, Astrophys. J. 129, 459479 (1958).

[131 Y.D. Zugzda, “Low-frequency oscillatory convection in the strong magnetic field’, Cosmic Electrodyn. 2, 267-279 (1971). r141 J.V. Hollweg, “Supergranulation-driven Alfven waves in solar chromosphere and related phenomena”, Cosmic Electrodyn. 2, 423-444 (1972).

USI B. Leroy, “Propagation of waves in an atmosphere in the presence of a magnetic field. II. Reflection of AlfvCn waves”, Astron. Astrophys. 81, 136-146 (1980).

112

L.M. B. C. Campos / Metho& for Alfvt?n waves

[16] L.M.B.C. Campos, “On magneto-acoustic-gravity waves standing or propagating vertically in an atmosphere”, J. Phys. ,416, 417437 (1983). [17] E.N. Parker, “Alfvkn waves in thermally stratified fluid”, Geophys. Astrophys. Fluid Dyn. 29, l-12 (1984). [ 181 S.J. Schwartz, P.S. Cally and N. Bel, “Chromospheric and coronal AlfvCn oscillations in non-vertical magnetic fields”, Solar Phys. 92, 81-98 (1984). [ 191 J.V. Hollweg, “Alfvkn waves in the solar atmosphere”, Solar Phys. 56, 307-333 (1978). [20] J.V. Hollweg, “Alfvkn waves in the solar atmosphere. II: Open and closed flux tubes”, Solar Phys. 70, 25-66 (1981). [21] B. Leroy, “Propagation of waves in an atmosphere in the presence of magnetic field. III: Alfv&n waves in the solar atmosphere”, Astron. Astrophys. 97, 245-250 (1981). [22] L.M.B.C. Campos, “On a theory of solar spicules and the atmospheric mass balance”, Month. Not. Roy. Astron. Sot. 205, 547574 (1984). [23] L.M.B.C. Campos, “On oscillations in sunspot umbae and wave radiation in stars”, Month. Nor. Roy. Astron. Sot. 241, 215229 (1989). [24] R.S. Steinholfson, “Resistive wave dissipation on magnetic inhomogeneities: normal modes and phase mixing”, Astrophys. J. 295, 213-219 (1985). [25] E.C. Titchmarsch, Theory of Fourier Integrals, Oxford Univ. Press (1935). [26] M.J. Lighthill, Fourier Series and Generalized Functions, Cambridge Univ. Press (1959). [27] M. Abramowitz and I. Ste.gun, Handbook of Mathematical Functions, Dover, New York (1965). [28] A.R. Forsyth, Treatise of Di$erential Equations, 6th ed., MacMillan, New York (1927). [29] M. Yanowitch, “The effect of viscosity on vertical oscillations of an isothermal atmosphere”, Can. J. Phys. 45,2003-2008 (1967). [30] M. Yanowitch and P. Lyons, “Vertical oscillations in a viscous and thermally conducting isothermal atmosphere”, J. Fluid Mech. 66,273-288 (1974). [31] L.M.B.C. Campos, “On waves in non-uniform, compressible, ionized and viscous atmospheres”, Solar Phys. 82,355-368 (1983). [32] L.M.B.C. Campos, “On waves in gases. Part II. Interaction of sound with magnetic and internal modes”. Rev. Mod. Phys. 58, 363-463 (1987). [33] C. Caratheodory, 1935 Theory of Functions, Birkhauser Verlag, Base1 (1935), reprinted Chelsea, New York (1972). [34] L.M.B.C. Campos, “On vertical hydromagnetic waves in a compressible atmosphere under an oblique magnetic field”, Geophys. Astrophys. Fluid. Mech. 32, 217-272 (1985). [35] L.M.B.C. Campos, “On the properties of hydromagnetic waves in the vicinity of critical levels and transition layers”, Geophys. Astrophys. Fluid Dyn. 40, 93-132 (1988). [36] J.T.A. Bromwich, Znjinite Series, Longman, New York (1926).