3. Governing equations and solutions
A linear magnetosonic wave with a prescribed frequency and wave vector is launched from the uniform domain (region 2) and it propagates towards the nonuniform layer, in the negative 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.
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 . Since we treat a stationary steady state of driven motions excited by an incoming wave with prescribed frequency , the time dependency of all perturbed quantities is given by . The perturbed quantities have therefore the following form:
Here , and are the Alfvén, the sound and the cusp frequency respectively, while is the component of the wave vector parallel to the plane of the layer.
Since and 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 causes only a replacement of frequency by a Doppler shifted frequency in the corresponding equations for the linear waves superimposed on a static equilibrium. Thus, when the incoming wave of frequency enters the transitional layer with the flow, it behaves there in the same way as would a wave of frequency when the flow is absent. Formally speaking, we are now dealing with a static model in which the wave frequency suffers a jump from to at . Moreover, the frequency can be negative, contrary to the frequency 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 is shown in Fig. 1: the shifted frequency can be larger, equal or smaller than the initial frequency , depending on whether the shift is negative, zero or positive respectively.
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.
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 from of the incoming wave to a new value related to one of the white horizontal arrows in Fig. 2, depending on the value of . 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 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, and , 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:
The solutions of Eq. (10) have an exponential dependence of the form where the square of is given by:
The frequencies and are the cut-off frequencies for slow and fast magnetosonic waves in a uniform plasma. Their squares are given by
which yields the following frequency ordering:
3.1.1. Region 1;
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 , and should then provide negative values for in Eq. (11) and the required evanescent solution of Eq. (10) is
The integration constant in (13) is fixed by the normalization condition .
3.1.2. Region 2;
In region 2, where the equilibrium state is static with , the waves propagate in both directions towards and away from the nonuniform layer. This means that the choice of the values for , and () have to yield positive values for in expression (11).
where all coefficients are constant and evaluated at .
There are two frequency windows for propagating magnetosonic waves in a uniform plasma, that follow from Eq. (11) for . They are
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, , and of its group velocity, , have opposite signs. This is evident from their product written as:
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 axis: the orientations are the same for fast waves and opposite for slow waves.
Here is the angle between the wave vector and the direction of the nonuniformity or the angle of normal incidence, while is the azimuthal angle or the angle between the magnetic field and the parallel wave vector .
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 and at .
As P and should be continuous at both boundaries and , the initial values and are obtained from the analytical expressions (13) and (14) for P and in region 1, taken at :
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 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 .
In the case of the slow (the cusp) resonance the connection formulae are (Goossens, Hollweg & Sakurai 1992):
where and all equilibrium quantities in Eq. (20) are taken at .
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
and the isotropic electric resistivity taken as the dominant dissipative process.
The jump conditions (20) are then applied to connect the numerical values of P and across the interval around the cusp singularity.
In the case of the Alfvén resonance, the connection formulae are (Goossens, Hollweg & Sakurai, 1992):
while the jumps (22) connect the solutions for P and across the interval around the Alfvén singularity at (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 and the related values of and are obtained.
© European Southern Observatory (ESO) 1998
Online publication: September 30, 1998