alfvén-magnetosonic waves interaction in the solar corona

A L F V I ~ N - M A G N E T O S O N I C W A V E S I N T E R A C T I O N IN T H E

S O L A R C O R O N A

K. M U R A W S K I

Department of Mathematical Sciences, The Mathematical Institute, University of St Andrews, St Andrews, KY16 9SS, Scotland

(Received 2 April, 1991; in revised form 21 January, 1992)

Abstract. The nonlinear propagation of the Alfv6n and magnetosonic waves in the solar corona is investigated in terms of model equations. Due to viscous effects taken into account the propagation of the fast wave itself is governed by Burgers type equations possessing both expansion and compression shock solutions. Numerical simulations show that both parallely and perpendicularly propagating fast waves can steepen into shocks if their amplitudes are in excess of some sizeable fraction of the Alfv6n velocity. However, if the magnetic field changes linearly in the perpendicular direction, then formation of perpendicular shocks can be hindered. The Alfv6n waves exhibit a tendency to drive both the slow and fast magnetosonic waves whose propagation is described by linearized Boussinesq type equations with ponderomotive terms due to the Alfv6n wave. The limits of the slow and fast waves are investigated.

I. Introduction

Although the problem of the propagation of linear magnetohydrodynamic waves in an inhomogeneous medium is of great interest in solar physics (e.g., Roberts, 1984; Lee and Roberts, 1986; Roberts, 1991 ; Steinolfson et aL, 1986; Nocera et al., 1986, Nocera and Priest, 1986; Hollweg, 1987; Hollweg and Yang, 1988; Berton and Heyvaerts, 1987; Grossmann and Smith, 1988; Cally, 1991; Murawski and Roberts, 1992), it has not been yet investigated in sufficient detail. Moreover, the propagation of MHD waves has been studied mostly in the case where the Alfvdn waves decouple from magnetosonic ones. See, e.g., Mann (1988) and Som et al. (1989).

Few analytical calculations relating both the Alfv6n and the magnetosonic waves have been attempted. There are one-dimensional numerical simulations made by Hollweg, Jackson, and Galloway (1982). They conclude that shear Alfv6n wave in solar magnetic- flux tube (imbedded in an inviscid but inhomogeneous plasma) can drive sound waves which eventually dissipate into shocks. It is qualitatively suggested that Alfv6n waves may heat the corona indirectly by driving the slow wave, with some of the properties of spicules. Sakai and Sonnerup (1983) have derived model equations which describe the long dispersive Alfvan wave driving sound wave. On the other hand, the equation governing the evolution of the fast wave envelope modulated by a slow wave driven by the ponderomotive force has been derived for a sausage wave travelling along a magnetic photospheric slab with rigid walls by Sahyouni, Zhelyazkov, and Nenovski (1988). Model equations describing dispersive Alfv6n-magnetosonic wave interaction have been derived by Shukla, Feix, and Stenflo (1988). Obliquely (to an ambient magnetic field) propagating random Alfven waves can convert their energy to magnetosonic waves that can be further Landau damped (Hamabata and Namikawa, 1990). Coupling between

Solar Physics 139: 279-297, 1992. �9 1992 Kluwer Academic Publishers. Printed in Belgium.

280 K. MURAWSKI

magnetosonic waves and tearing wave has been described in terms of model equations by S akai and Washimi (1982). Coupled, nonlinear Schr0dinger equations governing the

interaction of sausage and kink surface waves in a plasma slab have been derived by

Vladimirov, Stenflo, and Wu (1991). In most above-mentioned cases the derived equations are some generalizations of the

Zakharov equations describing the Langmuir wave and slow density plasma response (e.g., KarlickS~ and Jungwirth, 1989). A recent source of references on these equations

can be found in Murawski, Infeld, and Ziemkiewicz (1991). The Zakharov equations,

however, do not take viscosity into account. The purpose of this paper is to describe in terms of model equations the coupling

between Alfv6n and magnetosonic waves which are driven by the former. We are not

going to treat resonant interaction between Alfv6n waves. This process can lead to

creation of new waves (see Wentzel, 1974). Actually, the Alfv6n wave can drive both

the slow and fast wave. Due to the small value of the sound speed in comparison to

the Alfv6n one, one should expect weak coupling in the Alfv6n-slow wave interaction and a strong one for the Alfv6n-fast wave interaction. This process of driving of the fast

wave can even be enhanced by inhomogeneities in a magnetic field. The paper is organized as follows. The next section presents the fundamental set of

equations for the Alfv6n and magnetosonic waves. A dispersion relation for the viscous,

nondispersive Alfv6n wave is derived in Section 3. Burgers type equations describing the Alfv6n wave propagation are presented in Sections 4 and 5. A case of linear polarization is discussed in more detail. Equations which govern the Alfv6n-

magnetosonic wave interaction are derived in Section 6. The final part of the paper

contains a short summary.

2. Fundamental Equations

Let us consider a viscous compressible plasma with infinite conductivity described by

the equations of magnetohydrodynamics (e.g., Priest, 1982):

#, + 7 . (pV) = 0, (2.1)

1 p[V, + (V.V)V] + Vp = - (g x B) x B + r/og2V + (r h + �89

# (2.2)

B, = V x (V x B) , (2.3)

7 . B = O, (2.4)

P~ P + (V-V) p =r/o Y'. 7 - 1 ~ i.~=1

(v, xk + v ~ , - 2a,kV'V)V,~k, (2.5)

where p is the plasma density, V -= [u, v, w] the velocity (V 1 = u, V 2 = v, V 3 = w), p the pressure, # the magnetic permitivity, B - [a, h, b] the magnetic induction, 7 the ratio of

ALFVI~N-MAGNETOSONIC WAVES INTERACTION IN THE SOLAR CORONA 281

specific heats, and qo >> t/l the dynamic and bulk viscosity coefficients, respectively. The index t denotes partial differentiation with respect to time. In the interest of mathematical tractability, in the above equations two important effects have been neglected: electron thermal conduction (Ruderman, 1991) and Braginskii viscosity vector simplification (e.g., Nocera et al., 1986; Nocera and Priest, 1986).

We introduce a Cartesian coordinate system with z-axis parallel to the undisturbed inhomogeneous magnetic field Bo(x ). In what follows we assume that all variables depend on x and z only. For simplicity a dependance on y is artificially suppressed although the generalization is direct. We consider a horizontally inhomogeneous atmosphere; the undisturbed state is characterized by V = 0, Po = const., p =po(X),

Bo : [0, 0, Bo(X)].

3. Dispersion Relation for the Alfv6n Waves

To derive model equations which govern the Alfv6n wave propagation in the viscous plasma we need to know the dispersion relation in the homogeneous medium. In this way linearizing Equations (2.1)-(2.5) around the homogeneous undisturbed state we see that the equations for the Alfv4n wave decouple and we get the following dispersion relation for parallel propagation (e i(kz+ cot)):

09 2 = j f 2 k 2 + i r/~~ k 2 c o . (3.1) Po

In the limit of long wavelength (k ~ 0) waves we obtain

co i ~/o - - = +V A + - k. (3.2) k 2po

Note that essentially this expression is similar to this one for the inviscid and dispersive Alfv6n wave (with the Hall term in the induction Equation (2.3) included) (see, e.g., Ruderman, 1987). The important difference is that there is the imaginary unit i, corresponding to a dissipation.

A linear long Alfvhn wave reduces its amplitude with a linear damping rate (imaginary part of co) equal to (tlo/2po)k a. For k r 0, the solution of Equation (3.1) is

co = 2 po - gk2J I (3.3)

Hence, for k--, oe we get

co = i t/oo k2 or co = 0. (3.4) Po

Because of the non-zero real parts of co, only very long waves for which k is less than kc - 2VA/(qO/PO) are oscillatory in time and there are only damped (co t > 0) solutions for k > k~.

282 K. MURAWSKI

4. The Long Fast Wave Propagating Parallel to the Magnetic Field

Let the dimensionless wave amplitude be equal to ~ ~ 1. We consider long waves which propagate nearly parallel to the ambient magnetic field B o. In order for the dissipation to compete with the nonlinearity, it is necessary that the correction in the dispersion relation for the long wavelength waves be of the order of ~2. Thus, for the problem in question (k ---> 0) it is necessary to introduce the following stretched variables:

= g2(z - Vat ) , "c = e4t, { = e3x. (4.1)

The dissipation coefficient can be defined as the ratio of a transverse length scale to a characteristic wavelength. The refraction is proportional to ~3.

The scaling (4.1) is essentially the same as for the homogeneous medium (Mjolhus and Wyller, 1986) and follows from (3.2) because the fast wave possesses the same type of dispersion relation as the Alfvdn wave does. We assume here that VA depends weakly on x, or in other words it is locally constant. The corona is in fact highly structured across the ambient magnetic field and this assumption can be only partially justified. By this way we can solve analytically the problem and provide the insight into more complex phenomena which can be modelled by numerically solving the full set of MHD equations (2.1)-(2.5). For this moment, the problem is too advanced and we limit ourself to the assumptions (4.1) which also say (due to the scaling of x-coordinate) that weakly obliquely propagating waves are considered.

We use the following expansion:

f = fo + efl + g2f2 + . . ' , (4.2)

where now ~ describes the weakness of nonlinearity (e.g., Murawski, 1986) and allow the density to be varied in time. Note, however, that the Alfv6n wave propagation can be described in a similar way. The nonlinear coefficient ~ can be defined as the ratio of a characteristic wave amplitude to a transverse length scale. The collection of terms at g3 (e.g., Jeffrey and Kawahara, 1982) as a compatibility condition gives us coupled

two-dimensional Burgers equations

+ v s v i< v 4 ,o (VA . = . . . . . c s ) u l ~ O, (4.3) 4 " 2Bo - ~Po

VA (V2 v , ) ~ - V3 tl~ (V 2 - c2)v,r162 = 0 (4.4) - + - '

.oJ- t / l ~ < t./l ~ d~ ' (4.5)


where

v l --- Ul +

Note that Bo/V A is a constant. Similar equations have been derived by Ruderman (1987) and Mjolhus and Wyller (1986). They, however, have taken resistivity instead of viscosity into account. See also Malara and Elaoufir (1991) for a derivation of equations describ-

ing obliquely propagating MHD waves in an ideal nondispersive plasma. Equations (4.3)-(4.5) are difficult to solve analytically. Because there is a lack of

corresponding discussion in the literature, we will simply consider some one-

dimensional cases.

4.1. ONE-DIMENSIONAL CASE

In the one-dimensional case ~{ = 0 which implies 7/= 0 and we get coupled Burgers equations (the subscript 1 is dropped)

u~ + fl(V2u)r - ~ur162 = 0, (4.6a)

v~ + fl(V 2 v)~ - c~vr162 = 0, (4.6b)

where the nonlinear fl and dissipative ~ coefficients are defined as

~o VA 2/)o 4(V 2 - c 2)

Note that ~ does not depend on { but fl does. The dissipation is not spatially-dependent,

whereas the nonlinearity is. Due to this, a wave steepening is also {-dependent. It has a direct consequence in creating a wave front. Let us imagine that at z = 0 a wave profile

is uniform with respect to {. In time this profile is smoothed by dissipation and steepened

by nonlinearity. But the last one depends on {. So, the wave front is now modulated

in the ~ direction - there are some humps and valleys. It is thought that the waves with

higher amplitudes propagate faster. Two parts of the wave corresponding to different values of { can propagate with different speeds. Their phases can be different (phase- mixing effect). The wave progressively looses the 'nice' front it had at ~ = 0. Because

the phase mixing is due to the nonlinearity we call this phenomenon the nonlinear phase mixing. On the other hand, the linear phase mixing is due to a dependence of the linear

speed V A on x.

4.2. CONSERVATION LAWS FOR THE COUPLED BURGERS EQUATIONS

Equation (4.6) can be written in the differential conservation form

V i . c _ [ 3 ,4 _ 5~VJ-e 2c~(uuee + wee) = 0

or in the equivalent integral form

- o o f (uu:~ + v~r d~.

l o w

(4.7)

(4.8)

284 K. MURAWSKI

The first term in this equation describes the energy. So, we see that energy is decreased in time with the nonlinear damping rate equal to 1/6 I dz.

Equation (4.6) can also be rewritten in the form

u~ + ( ~ V ~ u - ~ur = o ,

v~ + ( ~ v l v - ~vr = o ,

(4.9a)

(4.9b)

which says that momentum along the z-axis is conserved in the reference moving frame

with a speed V A.

4.3. L I N E A R POLARIZATION

Now we restrict ourselves to linear polarization. So,

u = V~ cos ~b, v = V• sin q~, (4.10)

where (p is a constant polarization angle. In this case we obtain a modified Burgers

equation (with a cubic nonlinear term)

V• + 3/~V 2 V • e V • = O. (4.11)

From this equation we can get the nonlinear dispersion relation

co = (V A + 3~V~)k + iotk 2 ,

in which the phase velocity depends also on the squared wave amplitude. So, waves with larger amplitudes move faster. Compare this relation with the linear dispersion (3.2).

Looking for stationary solutions of (4.11) we require

V j _ = V ( ~ = - ~ - s z ) , s > 0 , (4.12)

and integrating (4.11) with respect to ~. we get the following ordinary differential

equation:

:~Vr = fiV 3 - s V + l , l = c o n s t . , (4.13)

which is very convenient for phase analysis (e.g., Murawski, 1987). For e > 0 and fl < 0 there are no physical solutions. In the corona, however, both coefficients are greater than

zero. In this case the phase analysis leads to the conclusions that we should expect finite solutions for - lm < l <lm, where l,, =- 2s/3 x / s / - - ~ . Under this constraint for each value of the free parameter 1 there are two shocks characterized by

(1) expansion shock, where a < V < b and increases with ~, (2) compression shock, where b < V < c and decreases with ~,

where a < b < c are the roots of the equation made from the right-hand side of (4.13). For l =lm, b = c and the expansion shock solution is given by

log V - a a - b fi V T b + V - b - (b - a) 2 -e ~ + const. (4.14)


Otherwise the shock solutions are described by

l o g ] V - c ] + l o g ] V - b ] + l o g ] V - a ] = - fl ~+const . (4 .15) c 2 - ( b + a ) c +ab ( b - a ) ( b - c ) ( a - b ) ( a - c )

Let us now write down Equation (4.11) in the laboratory reference frame ( ~ and 8r are replaced by e-4(0~ + V A c3z) and e 2 (~z, respectively) to get

VA_ ̀ -t- VA VA_ z "~ 3fle2V 2, V• - e V • z = O.

But V• is a first-order correction to the real velocity U (see (4.2)), U = eV• + . . ' .

Using dimensionless variables,

X U ~ VAU, t ~ - - t , z ~ X ( z - VAt),

VA

we can finally write

U, + fl*U2Uz - ~*Uzz = 0, (4.16)

where ~*= ~/(XVA) and /7* = 3flV A. Both z and t are measured in the moving coordinate frame with the Alfv6n speed V A. But all variables are unscaled. For typical coronal conditions (Priest, 1982) (VA = 2 x 106ms -1, c s = 2 • 10Sins 1, ~/o=�89 c m - l s - i , p o = 1.5 x 10 lSgcm 3,X= 108m, a n d B = 10 G), we get that ~*

and/7* are approximately equal to 43- and - 8 x 10-s, respectively. Hence, we deduce

that the fast waves in the corona are weakly damped by the viscosity. However, they can be dissipated very efficiently by means of resonant absorption (Hollweg et al., 1990;

Poedts, Goossens, and Kerner, 1990; Okretic and Cadez, 1990). Other means of

damping are described in Sections 5 and 6.

4.4. NUMERICAL RESULTS

Let us now ask the question how long it takes for shocks to form in the corona. Guided

by this point we consider an initial value problem for Equation (4.16). Initially, at t -- 0, there is a localized disturbance driven by a flare, spicule or granule, buffeting in a

homogeneous (x-independent) corona. This disturbance can be modelled by

U(t = O, z) = U o e - a2z2 (4.17)

To solve the problem we use the flux-corrected transport (FCT) technique (Boris and

Book, 1976). The method has also been described by Schnack and Killen (1980). Typical numerically obtained results are displayed in Figure 1 for a variety of values

of an initial wave amplitude Uo and a wave width a 2. The switch-on shocks are evident in the top two panels of Figure 1. We have verified that both Uo and a 2 play an important role in determining whether shocks do or do not form. A wider initial disturbance corresponding to a smaller value of a 2 moves faster and therefore it needs less time to convert itself into a shock. E.g., for U o = 0.5 and a 2 = 10 this time is less than 10. See the second top panel of Figure 1. Another factor which influences shock formation is

286 K. MURAWSK1

1 . 0 ~t=O~ Uo = 1 a 2 = 10

U t=2.5 t=5

O0 t 0 . 5 - / ~ Uo=O.5 a 2=10

t=lO U

0.0 i i i I J I

0'l~/t=O ~ - ] ~ Uo=0.1 a z=lO

f \

o 0 0.0 z 1.0

Fig. 1. Snapshots of U(z) for a parallel propagating fast wave (as a solution of the modified Burgers equation) which initially consists of a single pulse: U(t = 0, z) = Uo e - a2~2. The three panels correspond to different values of the wave amplitude Uo and the wave width a 2. Switch-on shocks are present in the upper

two panels. Note that all parameters are dimensionless.


the wave amplitude Uo. Larger amplitudes lead to shocks quicker. For example, the

disturbance with Uo = 1 converts into a shock at a time less than 5. See the top panel

of Figure 1. The waves in the lowest panel of Figure 1 move through the corona without forming

a shock because their amplitudes are too small. For smaller values of Uo, the dissipation becomes more competetive than the nonlinearity and the wave profile spreads out

reducing its amplitude. E.g., for U 0 = 0.1 no shocks are observed (the bottom panel of

Figure 1). On the whole, it appears that the fast waves can steepen into shocks if their amplitudes

are in excess of half of the Alfv6n velocity VA, which is never the case in the corona.

5. The Fast Mode Propagating Perpendicular to the Magnetic Field

In the previous section, we applied the expansion method to derive model equations for a fast wave propagating nearly parallel to the applied magnetic field. Due to weak refraction (see scaling for x in (4.1)) taken into account, we were not able to discuss

perpendicular wave propagation, which can be crucial for wave damping, due to the

inhomogeneity in this direction. Turning our interest here, in this case, we develop the

reductive Taniuti-Wei method (Taniuti and Wei, 1968; Murawski, 1986) to obtain the inhomogeneous Burgers equation in the limit of long-wavelength waves. For this purpose

we expand the quantities into the series

f = fo(~) + ~ e'~f,, (5.1) / 4 = i

and due to the spatial inhomogeneity introduce the following coordinate stretching:

z = e - d r , ~ = e 2 x , c ~ = c , + VA 2. (5.2)

Here ~ corresponds to the time measured in a moving coordinate frame with the velocity

c A. ~ is a consequence of the inhomogeneity in the x-direction. In the case of the time

inhomogeneity, c A = cA(t ), the corresponding stretching would be ~ = e ~ (dx - c A dt), "c = g2t.

Note also that whereas for the fast wave propagating vertically the inhomogeneity was assumed to be weak, here due to (5.2) no such assumption is given: c A can arbitrarily

depend on x. Substitution of the expansion (5.1) and (5.2) into (2.1)-(2.5) leads at e 2 to the

inhomogeneous Burgers equation

u.,. + fiuu, - ~u,, = -�89 (lnCA).U . (5.3)

where the nonlinear fi (this coefficient looks similar for that one derived by Roberts

288 K. MURAWSKI

(1985) in the slow-mode case) and dissipative ~ coefficients are defined as

3th + 4t/~ V*2 ,/~ v ' z [ ( 7 + 1)c~ + 3 V ~ l / 2 c 4 (5.4) 6po c3 X *

and the following dimensionless variables were introduced:

bl 1 = V ~ u , ~ = - - x - g , ~ = X * x . (5 .5) V * A \ c A

V* and X* denote the constant typical coronal values described in Section 4.

Equation (5.3) has been written in a similar way as Equation(4.16). So, now all

variables are unscaled but the time is measured in the coordinate moving frame with

the cusp speed c A . Note that x and t in the Burgers equation (5.3) are reversed from the usual form. This

is a consequence of applying coordinate stretching (5.2) which is the only way of deriving

a model equation in the inhomogeneous medium. Equation (5.3) possesses also one

more useful property. It allows us to study directly an evolution in space, a driven wave

(e.g., Wilson, 1981) excited at x = 0 say, by the way that u ( x = O, t) = f ( t ) .

Equation (5.3) shows that if viscosity is dropped (we can do this only formally because we violate the method of the derivation of Burgers equation), the linear terms on the left-hand side give UzCA = const., implying that wave energy flux density is conserved.

For the homogeneous Burgers equation, both the viscous c~ and the nonlinear/3

coefficients are positive. So, there is a solution

which represents an expansion shock moving to the right. In dimensional variables, its

amplitude and velocity can be expressed as e (V A//3) and c A (1 + (e / 1 - s )), respectively. Here ~ - (CA/VA). Due to the nonlinearity, the velocity of the wave is larger than in the

linear case. The correction is represented by the second term in the expression for the

velocity and is approximately proportional to e.

5.1. NUMERICAL RESULTS

Equation (5.3) has been solved numerically in the time interval - �89 < t < 1.2 using the

FCT technique developed by Boris and Book (1976). As has been suggested by Hollweg, Jackson, and Galloway (1982) the Alfv6n waves

propagating in the tubes can steepen into shocks in the chromosphere. When they pass through the transition region, they produce large pulses in the x-direction and have

amplitudes of 60 km s - 1. Such pulses can be modelled by

u(x = O, t) = u o e-a2r . (5.7)

First, a homogeneous case (c A va CA(X)) has been studied. Some of the results are


0.3- uo =0.25 a z=lO

x=2 = =

0 .0 t I - T - - 0 ,1-

u o = O . 1 a 2 = I 0 0 x=O

x=4

u =8

0 . 0 t

uo=O.O1 a 2=100

U

0.0 tO t

Fig. 2. Same as Figure 1, except the snapshots are made for a perpendicularly propagating fast wave satisfying the homogeneous Burgers equation. The switch-on shocks are seen in the top two panels.

290 K. MURAWSKI

presented in Figure 2. They are very similar to those obtained for the modified Burgers equation. A switch-on shock is presented in the two top panels of Figure 2. Because Burgers equation (5.3) has a quadratic nonlinear term (the modified Burgers equation has a cubic one), lower amplitudes are needed for a switch-on shock to take place. Compare the bottom panel of Figure 1 and the second top panel of Figure 2.

Fig. 3.

J ,0=0.25 a2=10 b = l

0.2- x=l

U X=2

X=3

0.1

0.0 I I i

/ / ~ u o = 0 . 2 5 a 2 = 1 0 b = 2 . 5

/ \ 0.2 x=l

u

x=2 0.1 x=3

0.0 . . . . . . . . . i i x

uo = .25 =10 b =5 0 a 2

0.2-

u

0.1 x=2

0 , 0 i - I - -

0 . 0 t tO

Same as Figure 2, except the snapshots are made for a fast wave satisfying the inhomogeneous Burgers equation.


To study the effect of the inhomogeneity on the initial disturbance (5.7) the case of

a linearly increasing magnetic field has been taken into consideration. Thus, the magnetic

field has been described by

~o(X) : B;~(1 + bx) , (5.8)

and the equilibrium condition

Po + __B~ = const. (5.9) 2#

has been also taken into account. The results indicate that there is a switch-on shock for sufficiently small values of the

coefficient b. Both for b = 1 and b = 2.5 shocks are created. See the two top panels of

Figure 3. No such behaviour is observed for b = 5 (the bottom panel of Figure 3). In all cases the waves reduce their amplitudes due to the inhomogeneities. For example at x = 2 the shock reduces its amplitude to about 0.15 (the top panel of Figure 3) whereas

the same wave passing through a homogeneous medium has an amplitude slightly less

than 0.25 (the top panel of Figure 2).

6. Magnetosonic Waves Driven by Long Alfv6n Waves

The Alfv~n wave exhibits a tendency to drive both the slow and fast waves; the waves are driven by centrifugal - p ( V . V ) V and the Lorentz # 1(7 x B) x B forces. So, let

us consider a mix of these waves. Because in the corona the sound speed is very small, the Alfv~n fast-wave interaction is very strong. In other words, because in the corona v f~ V A, the process of driving the fast waves is more efficient than the corresponding

one for the slow wave. Due to phase mixing we can expect that in the upper corona the

fast waves can propagate obliquely to the magnetic field. In the following parts of the

paper we consider long Alfv~n waves modulated by a magnetosonic wave response. Due

to a lower-order effect, the self-interaction of the magnetosonic waves is neglected.

To study the long Alfv6n-magnetosonic wave interaction, the physical quantities are divided into the following parts:

f ( x , z, t) : fo(X) + efl + ~2f2 + " " , (6.1)

where fo(X) is the undisturbed state, f l and f2 describe the longAlfvdn and magnetosonic waves, respectively. Because the magnetosonic waves are driven by the Alfv6n wave

(which is incompressible) we should expect that the former ones are lower in magnitude. Thus f l describes the quantities

f l : ul r 0 , 131 -~ 0, a I • 0 , hj -~ 0. (6.2)

Other first-order quantities are taken to be zero. Due to the dispersion relation for the Alfv6n wave we use the same variables, stretching as in Section 4 (see (4.1)). From

292 K. MURAWSKI

(2.1)-(2.5) in e, we get

B o B o a 1 - U 1 , hi - v 1 �9

VA

Note that Bo/V A does not depend on x. Both x- and y-components of the magnetic field

and velocity are in anti-phase independent of x. Collecting terms in ~3 we get equations describing the Alfvdn wave propagation and

written here in the laboratory reference frame ( ~ , #~, and ar have been replaced by

~ + VA Oz, ~ , and ~ , respectively):

~0 1 2 V2)x -[- ~ 0 [(b2b/1)z + b 2 U l z ] + ldlt ~- V A b l i z ?PO btlzz ~- ~(ul "}- VA

1 (VApzUl~ _ Pzx) = 0 , (6.3)

VA l ) l tq - V A V l z 2D~ l ) l z z - } - ~ o [(b2Vl)z -'}- b2~)lz] -I--

Va if" I[(W2Vl)z q- W2Vlzl -- ~ 0 P21)Iz = 0 . (6.4)

Note that the Alfv6n wave variables are coupled to the vertical magnetic field, velocity,

density, and pressure. The process of interaction is thus much more complicated than

in the case of the Zakha rov equations. There are fewer terms in Equation (6.4) as a consequence of the artificial assumption

~y = 0. Because of the symmetry with respect to x and y, it is straightforward to write

down equations for the case of 0y r 0 and B o = Bo(y). We consider the case Oy = 0 in

the interest of simplicity only, however being aware that an inclusion of inhomogenei t ies

in the both directions is more realistic. The magnetosonie waves are described by equations which are obtained by considering

terms in ~2:

bl2tt C2bl2x x C L b t 2 x ~lbl2xxt V2b l2zz ~0 2 _ ~ 2 W 2 x z t . . . . . . bl2zzt -- C s W2xz = Po

= _ 5(ull 2 + v~)xt - (7 - 1) rio (~Ulx4 2 -I- U2z + V2x ~- l)21z)x, (6.5) P0

v 2 , - V~v2= qo (v2x x + v2=) t = 0 , (6.6) Po

2 -- ~1 W2zzt -- - - W2t t -- C s W2zz ~0

Po 2 __ ~ 2 b l 2 x z t = W2xxt -- C s bl2xz

= _ 5(ula 2 + v ~ ) ~ - (7 - 1) r/~ (~Ulx4 2 + U5 .jr V2x + 1)2z)z, PO

(6.7)

ALFVfZN-MAGNETOSONIC WAVES INTERACTION IN THE SOLAR CORONA 293

where

3:71 + 4:70 3rh + :70 C2 ~ C. 2 + m 2 , ~1 ~ , 1~2 ~

3po 313o

Additionally, we must include equations describing an evolution of b2, P2, and P2 with

which the Alfv~n wave is coupled:

b2, + (BoURL = O,

P2t ~- PO("2x q- W2z) = O,

P2t C2102t + POxU2 (~ __ 4 2 - = 1):7o(~U,x + u21~ + V~x + v2~).

(6.8)

(6.9)

(6.1o)

Note the ponderomotive terms at the right-hand side of (6.5) and (6.7) and lack of any one in (6.6). This is a consequence of the assumption c~y = 0. The y-component of the

velocity moves on its own characteristic - there are no ponderomotive terms. Thus we see that the flow is driven due to the existence of gradients in this direction, described

by the first terms on the right-hand side of (6.5) and (6.7).

6.1. SLOW WAVE LIMIT

Let us consider the case when the slow wave amplitude is larger than the fast one, e.g.,

b2, u2, v2 ~ w2 and time changes in the driven wave are negligible with respect to the

Alfv6n time, ~, ~ VA Cgx- According to Barnes and Hollweg (1974) it happens when the Alfv6n wave propagates nearly along the magnetic field. Moreover, from (6.5) and (6.7) it follows that the magnitude of the driven velocity is proportional to the gradient of the square of the Alfv6n wave amplitude. If this gradient is negligible in the x-direction, the

vertical motion wil have larger amplitude; the slow wave will be driven. Oppositely, in the case of large gradients in the x-direction, the fast wave will be driven. These gradients

are due to the Alfv6n wave motion. But the gradient in the horizontal direction is also

a consequence of the inhomogeneity applied. Thus, we should expect that the slow wave will be driven in the case of the Alfv6n waves propagating nearly along the magnetic field

and very weak inhomogeneities. Otherwise, the fast wave-driven process will be com-

petetive to the slow wave one. Then the AlfvOn wave is described by

:70 1 2 "It + VA"lz 7 Ulzz + 4(Ul + F~)x +

2po

VA 1 + �89 + w2.1z] - ~ p2.xz + --2po P ~ = o , (6.11)

~0 1 1 VA 1)1' + VAFIz -- ~ l)lzz + 2(W21)l)z "+" 2W2Ulz 2DO P2VIz = 0. ( 6 . 1 2 )

294 K. MURAWSKI

The slow wave propagation is governed by

2 -- ~)1 W2tt -- Cs W2zz W2zzt -- - - t9o

1 2 W2xxt = --~(H 1 q- 92)zt --

- ( ~ - 1 ) r/~ 4 2

Po

Additionally, the equations for P2 and P2 take the form

P2t c2p2t (7 4 2 _ = _ 1)r/o(SU,x + U21z + v 2 + v 2 ) ,

P2t + poW2~ = O.

(6.13)

(6.14)

(6.15)

Consider a case of the long Alfv6n wave driving only vertically propagating slow waves

(~x = 0), and look for approximate solutions. Now, Equations (6.11)-(6.15) will be rewritten in the case of the homogeneous field. Neglecting in them some terms propor-

tional to t/o due to the weak influence of the viscosity on the long Alfvdn wave, and

considering the case of linearly polarized (91 = 0) Alfvdn wave, we can get a shock wave with an amplitude w o and moving on the background 4 g V A - [(c 2 - C2)/(2C~1)] WO,

4 C s - w o + w otan - - . (6.16) w2 = 3 VA 2Cal

So, the slow wave can propagate in the form of shocks gaining energy from the Alfv6n wave (the right-hand side of (6.13)) and being damped by the viscosity (terms Wz~zt and

Wxx, in (6.13)).

6.2. FAST WAVE LIMIT

In this case, an amplitude of the slow wave is smaller than the fast one, e.g., w 2 ~ u2, 92 and the wave propagates on time scales which are larger than the acoustic time scales,

cs ~z ~ ~t. This happens when the Alfv6n wave propagation is at larger angles (Barnes and Hollweg, 1974) and has a stronger inhomogeneity in the horizontal direction than

in the case of the slow wave limit discussed above. This wave will also be better driven

in the case of a low-fi plasma. So, for the most coronal conditions we should expect that this process would be dominant.

The Alfv~n wave equations are

~0 1 2 VA gl t + VAHlz - - ~ 0 Ulzz + ~(H1 + 9~)x + - ~ - [ (b2Hl )z + b2Ulz] -

2/~o

V A 1 p e u l z + _ ~ P 2 x O, (6.17)

2po 2po

v~ VA Ult.4_ VAUI z ? 0 l ) l z z+ [ (b2Vl)z + b2/) lz] _ p2/)lz = 0 " ( 6 . 1 8 )

2po

ALFVEN-MAGNETOSONIC WAVES INTERACTION IN THE SOLAR CORONA 295

The fast magnetosonic wave propagation is governed by the following equations:

2 __ C 2 A x l A 2 x __ ~ l b t 2 x x t V 2 b l 2 z z ~10 b l2 t t - - CA b/Z~c_ v - - _ _ _ b l 2 z z t :

Po

I 9 ~ ( u ; + v ~ ) x , (7 1) ~0 ~ 2 . . . . (~<x + G + G + G ) u , Po

(6.19)

U2t, __ b , 2 U 2 z z __ ~10 (/)2x.v q- V 2 z z ) t = O ,

Po (6.20)

Equations for b2, Pc, and P2 take the following form:

P2t + poU2x = 0 ,

2 4 2 /d2 Pzt Cs P Z t q- POxU2 (Y - - q- U 2 x q- l z ) ' _ = 1)t/o(SUlx u2z +

G, + (BoU2L : O.

(6.21)

(6.22)

(6.23)

The right-hand side of (6.19) describes the forcing which is exerted by the Alfv6n wave on the fast wave. This forcing is both x- and z-dependent. Thus different parts of the fast wavefront can gain different amounts of energy. The second term on the right-hand side describes some energy losses due to the viscosity during the energy transfer.

The left-hand side of (6.19) describes the linear fast wave itself. Its self-interaction leading to nonlinear terms has been neglected. Similarly for the slow wave, we notice terms corresponding to the viscous damping (Uz~xt and u2=t) and the phase mixing (U2x x

and U2zz) which due to the existence of large gradients in the corona can play a crucial role in a process of damping of the fast waves.

7. Summary

In order to apply the theory of waves in flux tubes to the coronal atmosphere, it is necessary to assess the importance or otherwise of damping. The nonlinear propagation of Alfv6n and magnetosonic waves in the solar corona has been investigated taking into account both viscous and inhomogeneous effects.

The long fast (also Alfv6n) waves propagate according to Burgers type equations which have both compression and expansion shock wave solutions. The linear Alfv6n wave propagating in a homogeneous medium is viscous damped with the damping rate dependent on the wave vector k. Only long-wavelength waves for which k < k c are oscillatory in time.

The Alfv6n waves drive the slow and fast magnetoacoustic waves. The driving mechanism is dependent on the propagation angle of the Alfv6n wave. For parallel propagation mostly a slow wave is driven (Barnes and Hollweg, 1974) whereas larger values of the propagation angles correspond to a driven fast wave. The slow and fast magnetosonic waves are further damped by the viscosity (the waves are compressive).

296 K. MURAWSKI

The numerical calculations performed for the fast waves propagating both perpendicular and parallel to the ambient magnetic field have shown that these waves can steepen into shocks if their amplitudes are in excess of some sizeable fraction of the main Alfv6n velocity. But even such waves can be affected by the inhomogeneous magnetic field and other inhomogeneities not discussed in this paper (e.g., by random inhomogeneities, Murawski and Roberts, 1992). It has been found that if this magnetic field increases linearly with x, no switch-on shocks can be observed, and instead wave amplitudes subside.

It should be kept in mind that the process of the Alfv6n-magnetosonic wave interaction is undoubtly more complicated than the theoretical idealization of this paper. One of the possible ways of a generalization of this work is to solve numerically the initial value problem for the full set of MHD equations. Work in this direction is in progress and will be published elsewhere.

Acknowledgements

The author wishes to express his sincere thanks to the members of the Solar Theory Group at St Andrews, especially Dr Bernard Roberts, for their encouragement and help, to Dr Luigi Nocera for his correspondence, to Dr Colin Steele for reading the manuscript, and to the SERC for its financial support. The author would also like to thank an unknown referee for many valuable suggestions which allow for improvement of the knowledge and even to understand the problems considered.

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alfvén-magnetosonic waves interaction in the solar corona

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