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Astron. Astrophys. 326, 1241-1251 (1997)

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3. Governing equations and solutions

The driven waves are studied with the standard set of linearized equation of ideal MHD:

[EQUATION]

An index '0' denotes an equilibrium quantity, while an index '1' denotes an Eulerian perturbation. Since the equilibrium quantities depend on z only we can Fourier analyze the perturbed quantities with respect to the ignorable spatial coordinates x and y and put them proportional to [FORMULA] with [FORMULA], and [FORMULA] the components of the horizontal wave vector. The present paper is concerned with the steady state of driven waves excited by an incoming wave with prescribed frequency [FORMULA] so that the time dependency of all perturbed quantities is given by [FORMULA]. Hence the perturbed quantities are written as:

[EQUATION]

The Eqs. (5) can be reduced to two coupled ordinary differential equations for the normal component of the Lagrangian displacement [FORMULA] and for the Eulerian perturbation of the total pressure [FORMULA]:

[EQUATION]

The coefficient functions D, [FORMULA], and [FORMULA] are given by

[EQUATION]

The Alfvén frequency [FORMULA], sound frequency [FORMULA], and cusp frequency [FORMULA] are defined as

[EQUATION]

The set of ordinary differential equations has two mobile regular singularities at the positions where [FORMULA], this is at the positions where

[EQUATION]

Since [FORMULA] and [FORMULA] are functions of z, the relations (8) define two continuous ranges of frequencies referred to as the slow continuum and the Alfvén continuum.

3.1. Solutions in the uniform layers

The solutions to the ordinary differential equations (6) can be obtained in closed analytic form for a uniform equilibrium state. For a uniform plasma the coefficient functions D, [FORMULA] and [FORMULA] are constants and it is straightforward to rewrite the set of ordinary differential equations of first order as a single ordinary differential equation of second order for P:

[EQUATION]

Depending on the sign of [FORMULA] the solutions are spatially propagating waves ([FORMULA]) or spatially evanescent or exponentially growing waves ([FORMULA]).

The object is to study how magnetosonic waves that are generated in the underlying uniform plasma propagate upward toward the nonuniform plasma layer and are partially absorbed there. This means that we have to choose [FORMULA], [FORMULA], and [FORMULA] so that [FORMULA] is positive. In that case we define the square of the [FORMULA] component of the wave vector [FORMULA] as

[EQUATION]

The frequencies [FORMULA] and [FORMULA] are the cut-off frequencies for fast and slow magnetosonic waves in a uniform plasma. Their squares are given by

[EQUATION]

[EQUATION]

In the lower uniform plasma layer the solution of Eq. (9) represents a superposition of two waves propagating in opposite directions along the [FORMULA] axis

[EQUATION]

where all coefficients are constant and evaluated at [FORMULA].

We do not want to consider leaky waves that transport energy toward infinite and restrict our analysis to waves that are evanescent in the upper uniform layer. This means that we have to choose [FORMULA], [FORMULA] and [FORMULA] so that [FORMULA] is negative there and can be written as

[EQUATION]

where [FORMULA] is real and positive. The solutions that vanish in the limit [FORMULA] are:

[EQUATION]

with all the coefficients evaluated at [FORMULA]. The integration constants are choosen as to provide a unit total pressure perturbation at [FORMULA].

Let us return to the expression for [FORMULA]. It is well-known that [FORMULA] for a uniform plasma. As a consequence we have two frequency windows for propagating magnetosonic waves in a uniform plasma, namely

[EQUATION]

and

[EQUATION]

The first window corresponds to propagating slow magnetosonic waves, the second to propagating fast magnetosonic waves. Fast magnetosonic waves are practically isotropic under solar conditions [FORMULA]. Slow magnetosonic waves, however, are largely anisotropic. A clear manifestation of this anisotropy of slow magnetosonic waves is that the phase velocity and the group velocity have z -components of different sign so that an upward directed phase velocity corresponds to a downward directed group velocity.

The product of the [FORMULA] components of the group velocity [FORMULA] and the phase velocity [FORMULA] can be written as

[EQUATION]

A simple analysis shows that

[EQUATION]

Since the group velocity of a wave is related to the corresponding energy flux, expressions (16) show the sense of the energy transport with respect to the wave front motion, taken along the [FORMULA] axis: the orientations are the same for fast waves and opposite for slow waves.

In an attempt to make the geometry of the wave propagation more visible, we introduce two propagation angles [FORMULA] and [FORMULA] related to the wave vector and the magnetic field as:

[EQUATION]

[EQUATION]

Here [FORMULA] is the angle between the wave vector [FORMULA] and the direction of the nonuniformity and [FORMULA] is the angle between the magnetic field and the horizontal wave vector.

Substituting expressions (17) into (10), we can solve Eq. (10) for the absolute value of the wave vector [FORMULA] as

[EQUATION]

[EQUATION]

where [FORMULA] is the angle between the magnetic field and the wave vector. In Eq. (18) the plus sign stands for the fast and the minus sign stands for the slow magnetosonic waves respectively.

In our computations we first fix the values of the driving frequency [FORMULA] and of the propagation angles [FORMULA] and [FORMULA]. We then compute the absolute value of the wave vector from Eq. (18) depending on the type of the incoming wave (slow or fast). Using the value of k and angles [FORMULA] and [FORMULA] we can compute the components of the wave vector. Therefore we study the behaviour of an incoming wave with a prescribed wave frequency and with prescribed propagation angles [FORMULA] and [FORMULA].

3.2. Solution in the nonuniform layer

The solutions of the ordinary differential equations that govern the linear motions cannot be obtained in closed analytic form in a nonuniform plasma. Numerical integration is required there. In addition, the prime purpose of this paper is to study the resonant damping of incoming magnetosonic waves by coupling to local resonant waves. In order for this phenomenon to occur the frequency of the incoming wave has to be within either the Alfvén or the slow continuum. Hence for the situation that is of interest to us the ideal MHD equations are singular. In ideal MHD the solutions for resonant waves are singular indicating that they can not be used for the description of resonant waves.

The inclusion of non-ideal effects removes the singularity from the equations but at the same time raise the order of the system of ordinary differential equations. A correct description of resonant wave behaviour requires the use of dissipative MHD. SGH(1991) and GRH(1995) show that numerical integration of the dissipative MHD equations can be avoided for weakly dissipative plasmas with very large values of the magnetic and viscous Reynolds numbers. For resonant MHD waves the dissipative terms are only important in a narrow layer around the ideal resonant position. Outside this narrow layer the MHD waves are accurately described by the equations of ideal MHD. The analysis by SGH(1991) and GRH(1995) leads to a simple numerical scheme for the accurate computation of resonant MHD waves as was shown by Stenuit et al (1995).

For a fixed wave frequency [FORMULA] and wave vector [FORMULA] the location of the possible resonances (cusp resonance alone or Alfvén and cusp resonance together) and the width of the dissipative layers are determined before hand. The solutions in the nonuniform plasma outside of the dissipative layers are obtained by numerical integration of Eqs. (6) starting from initial values for P and [FORMULA] at [FORMULA]. These values are given by the analytical solutions (12) evaluated at the boundary of the region 1 as both, P and [FORMULA], should be continuous at [FORMULA]. The numerical integration of the two ordinary differential equations of first order of ideal MHD does not require any special treatment and can be done by a simple Runge Kutta Merson scheme. In the vicinity of a resonant point the jump conditions (see SGHR method from Sect. 4) are applied to connect the solutions accross the dissipative layer. Integration of Eqs. (6) is continued up to the boundary of region 2 at [FORMULA].

Once the solutions are obtained at [FORMULA], say [FORMULA] and [FORMULA] we can relate them to the analytical solutions (11) by requiring their continuity at [FORMULA] which yields the following expressions for the amplitudes [FORMULA] and [FORMULA]:

[EQUATION]

which are related to the waves with [FORMULA] and [FORMULA] respectively.

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© European Southern Observatory (ESO) 1997

Online publication: April 8, 1998
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