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Astron. Astrophys. 339, 215-224 (1998)

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3. Governing equations and solutions

A linear magnetosonic wave with a prescribed frequency [FORMULA] and wave vector [FORMULA] is launched from the uniform domain [FORMULA] (region 2) and it propagates towards the nonuniform layer, in the negative [FORMULA]direction. When the wave enters the nonuniform layer, its characteristics change depending on the wave parameters and on the values of the basic state quantities.

In what follows, we restrict the investigation to those waves that cannot propagate through region 1 i.e. to the waves for which region 1 is nontransmitting.

Fluid motions driven by these incoming waves are described by the standard set of linearized equations of ideal MHD:

[EQUATION]

where the subscripts `0' and `1' denote the equilibrium quantities and their Eulerian perturbations respectively.

As the equilibrium quantities depend on z only, the perturbed quantities can be Fourier analyzed with respect to the ignorable spatial coordinates x and y by taking them proportional to [FORMULA]. Since we treat a stationary steady state of driven motions excited by an incoming wave with prescribed frequency [FORMULA], the time dependency of all perturbed quantities is given by [FORMULA]. The perturbed quantities have therefore the following form:

[EQUATION]

Eqs. (6) can be reduced to two coupled ordinary differential equations for the normal component of the Lagrangian displacement [FORMULA], where [FORMULA], and for the Eulerian perturbation of the total pressure [FORMULA]:

[EQUATION]

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

[EQUATION]

Here [FORMULA][FORMULA], [FORMULA] [FORMULA] and [FORMULA][FORMULA] are the Alfvén, the sound and the cusp frequency respectively, while [FORMULA][FORMULA] is the component of the wave vector parallel to the plane of the layer.

The set of ordinary differential equations (7) has two mobile regular singularities at positions where [FORMULA] vanishes in (8):

[EQUATION]

Since [FORMULA] and [FORMULA] are functions of z, the relations (9) define two continuous ranges of frequencies referred to as the slow continuum and the Alfvén continuum respectively.

The existence of these singularities means that the initial assumption of ideal MHD is not valid inside domains localized around each of the resonant points. Dissipative processes cease to be negligible there, they have to be taken into account. The Alfvén and the cusp singularity lead to resonant phenomena of absorption and over-reflection of MHD waves. The details of how dissipation is taken into account in the domains with singularities, are given later in the text.

Eqs. (7) with coefficients (8) indicate that the inclusion of the equilibrium flow [FORMULA] causes only a replacement of frequency [FORMULA] by a Doppler shifted frequency [FORMULA] in the corresponding equations for the linear waves superimposed on a static equilibrium. Thus, when the incoming wave of frequency [FORMULA] enters the transitional layer with the flow, it behaves there in the same way as would a wave of frequency [FORMULA] when the flow is absent. Formally speaking, we are now dealing with a static model in which the wave frequency suffers a jump from [FORMULA] to [FORMULA] at [FORMULA]. Moreover, the frequency [FORMULA] can be negative, contrary to the frequency [FORMULA] which is always positive. Such a viewpoint simplifies the analysis of the wave propagation characteristics and it will be used in our paper.

The formal frequency change at [FORMULA] is shown in Fig. 1: the shifted frequency [FORMULA] can be larger, equal or smaller than the initial frequency [FORMULA], depending on whether the shift [FORMULA] is negative, zero or positive respectively.

[FIGURE] Fig. 1. A schematic wave frequency `change' from [FORMULA] to [FORMULA] at the boundary separating two regions where plasma is moving and at rest. The superscript of [FORMULA] indicates whether the value of [FORMULA] is positive, zero or negative.

To visualize the various possibilities, we give schematic profiles of the characteristic frequencies in Fig. 2. The shaded areas in Fig. 2 designate the frequency domains where waves cannot propagate as their amplitudes are evanescent. However, in the nonuniform layer a wave can partially tunnel through a shaded, nontransparent section separates two propagating domains.

[FIGURE] Fig. 2. Schematic profiles of characteristic wave frequencies of the considered model: [FORMULA], [FORMULA], [FORMULA] and [FORMULA] are the local Alfvén, cusp, lower and upper cutoff frequency respectively. At [FORMULA], the frequency of the incoming (fast/slow) wave formally changes from [FORMULA] to [FORMULA] located within one of ten indicated intervals related to typical conditions for wave propagation inside the layer. The solid and the dashed parts of horizontal arrows indicate propagating and evanescent waves respectively. Locations of Alfvén and slow resonances are marked by solid and empty squares respectively.

The positive frequency of the incoming fast/slow magnetosonic wave is represented by a solid arrow in region 2. As mentioned above, the effect of the flow corresponds to a formal frequency jump at [FORMULA] from [FORMULA] of the incoming wave to a new value [FORMULA] related to one of the white horizontal arrows in Fig. 2, depending on the value of [FORMULA]. The solid and the dashed parts of these arrows indicate propagating and evanescent waves respectively. The locations of Alfvén and slow resonances are marked by solid and empty squares respectively.

The frequency intervals labeled from 1 to 10 in Fig. 2, represent typical cases of interest in our further treatment. For example, if [FORMULA] is in domain 2, we have a wave that propagates over some distance as it enters the layer, then tunnels, reaches the location of the Alfvén resonance and tunnels further up to the next propagating domain. The wave then propagates towards the location of the cusp resonance and becomes evanescent from thereon.

3.1. Solutions in uniform regions

The solutions to Eqs. (7) can be obtained in closed analytic form for a uniform equilibrium state since the coefficients (8) i.e. D, [FORMULA] and [FORMULA], are constants then. In this case, it is straightforward to write Eqs. (7) as a single ordinary differential equation of the second order for P:

[EQUATION]

The solutions of Eq. (10) have an exponential [FORMULA]dependence of the form [FORMULA] where the square of [FORMULA] is given by:

[EQUATION]

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

[EQUATION]

which yields the following frequency ordering:

[EQUATION]

3.1.1. Region 1; [FORMULA]

As wave transmission through the nonuniform layer is not considered in this paper, we restrict the analysis to waves that are evanescent in region 1. The choice of [FORMULA], [FORMULA] and [FORMULA] should then provide negative values for [FORMULA] in Eq. (11) and the required evanescent solution of Eq. (10) is

[EQUATION]

The integration constant in (13) is fixed by the normalization condition [FORMULA].

The corresponding solution for [FORMULA] follows from Eq. (7) as

[EQUATION]

3.1.2. Region 2; [FORMULA]

In region 2, where the equilibrium state is static with [FORMULA], the waves propagate in both directions towards and away from the nonuniform layer. This means that the choice of the values for [FORMULA], [FORMULA] and [FORMULA] ([FORMULA]) have to yield positive values for [FORMULA] in expression (11).

The solution of Eq. (10) is here a superposition of two waves propagating in opposite directions along the [FORMULA]axis:

[EQUATION]

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

The corresponding solution for [FORMULA] follows from Eq. (7) as

[EQUATION]

There are two frequency windows for propagating magnetosonic waves in a uniform plasma, that follow from Eq. (11) for [FORMULA]. They are

[EQUATION]

for slow magnetosonic waves and for fast magnetosonic waves respectively.

A particular property of a slow magnetosonic wave is that the z-components of its phase velocity, [FORMULA][FORMULA], and of its group velocity, [FORMULA][FORMULA], have opposite signs. This is evident from their product written as:

[EQUATION]

[EQUATION]

A simple analysis then shows that

[EQUATION]

Since the group velocity of a wave is related to the corresponding energy flux, expressions (17) indicate the sense of energy transport with respect to the motion of the wave front, 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 to the magnetic field as:

[EQUATION]

Here [FORMULA] is the angle between the wave vector [FORMULA] and the direction of the nonuniformity or the angle of normal incidence, while [FORMULA] is the azimuthal angle or the angle between the magnetic field and the parallel wave vector [FORMULA][FORMULA].

Substitution of expressions (18) into Eq. (11) yields the absolute value of the wave vector:

[EQUATION]

where

[EQUATION]

is the angle between the magnetic field and the wave vector. The plus and the minus sign in Eq. (19) stands for fast and slow magnetosonic wave respectively.

3.2. The nonuniform layer

To solve the system of Eqs. (7), a numerical integration is required in the nonuniform layer. This is done by a simple Runge Kutta Merson scheme starting from values of [FORMULA] and [FORMULA] at [FORMULA].

As P and [FORMULA] should be continuous at both boundaries [FORMULA] and [FORMULA], the initial values [FORMULA] and [FORMULA] are obtained from the analytical expressions (13) and (14) for P and [FORMULA] in region 1, taken at [FORMULA]:

[EQUATION]

Numerical integration of Eqs. (7) is then performed up to the dissipative layer, containing the resonant point, where the ideal MHD approximation becomes unapplicable. The SGHR method developed by Sakurai, Goossens, Hollweg and Ruderman which is described in details in the review article by Goossens & Ruderman (1996), gives the analytical dissipative solutions for P and [FORMULA] valid in plasma around the resonance. The locations of the end points of this dissipative a layer are estimated from dissipative properties of the medium for each resonance separately.

For our purposes, however, we are not interested in the exact solutions within the dissipative layers and all we need is to know how to connect the ideal solutions between the end points of the layer. This is easily done by the SGHR method which provides us with relevant connection formulae for both P and [FORMULA].

In the case of the slow (the cusp) resonance the connection formulae are (Goossens, Hollweg & Sakurai 1992):

[EQUATION]

where [FORMULA] and all equilibrium quantities in Eq. (20) are taken at [FORMULA].

The second relation in Eq. (20) represents the conservation law for Eulerian perturbation of total pressure inside the dissipative layer.

The extent of the dissipative layer follows from the asymptotic analysis by Goossens, Ruderman & Hollweg (1995) who estimate it as

[EQUATION]

where

[EQUATION]

and the isotropic electric resistivity [FORMULA] taken as the dominant dissipative process.

The jump conditions (20) are then applied to connect the numerical values of P and [FORMULA] across the interval [FORMULA] around the cusp singularity.

In the case of the Alfvén resonance, the connection formulae are (Goossens, Hollweg & Sakurai, 1992):

[EQUATION]

where [FORMULA].

The corresponding dissipative layer is located at

[EQUATION]

where

[EQUATION]

while the jumps (22) connect the solutions for P and [FORMULA] across the interval [FORMULA] around the Alfvén singularity at [FORMULA] (Goossens, Ruderman & Hollweg, 1995).

Once the dissipative layer with the singular point has been crossed, the numerical integration is carried on until the boundary of region 2 is reached at [FORMULA] and the related values of [FORMULA] and [FORMULA] are obtained.

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© European Southern Observatory (ESO) 1998

Online publication: September 30, 1998
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