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Astron. Astrophys. 357, 1073-1085 (2000)

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3. Alfvén wave eigenmode equation

Instead of the number density [FORMULA], it is convenient to introduce the Boltzman-like potentials

[EQUATION]

where [FORMULA] for ions and [FORMULA] for electrons, and [FORMULA] is a constant (in a uniform plasma, one can take the equilibrium number density [FORMULA] for [FORMULA] ). Then, by separating background and perturbed magnetic field [FORMULA] [FORMULA] [FORMULA], we can rewrite the Eqs. (4)-(6) as

[EQUATION]

[EQUATION]

The function [FORMULA] is the charge force per unit mass and unit charge, which includes the linear electric force, the collisional friction force, the pressure gradient force, and the nonlinear force:

[EQUATION]

The velocity difference is expressed through the current (the quasineutrality condition [FORMULA] is used):

[EQUATION]

and the nonlinear force is

[EQUATION]

Note that the functions [FORMULA]s in (7)-(10) contain both linear and nonlinear parts.

Multiplying [FORMULA] we find

[EQUATION]

Inserting this into [FORMULA], we get an equation for the velocity in the plane [FORMULA] [FORMULA] as

[EQUATION]

where [FORMULA] is the unit vector in the direction of equilibrium magnetic field (that is [FORMULA], etc.), [FORMULA] is the cyclotron frequency.

For the motions that are slower than the cyclotron gyration of the particles, as the low-frequency MHD waves, the electron and ion velocities in the plane, normal to [FORMULA], may be found in a drift approximation:

[EQUATION]

The velocity of the particles along the magnetic field follows the equation

[EQUATION]

Using (14) and (13), we eliminate the particle velocity from the continuity Eq. (9):

[EQUATION]

or, inserting the explicit expression for [FORMULA],

[EQUATION]

here [FORMULA]. Taking into account the quasineutrality condition, [FORMULA], and neglecting coupling to the fast magnetosonic waves (via terms with vector products), we rewrite the continuity Eq. (18) for electrons and ions separately:

[EQUATION]

[EQUATION]

where [FORMULA] and [FORMULA] denote the following nonlinear terms:

[EQUATION]

and

[EQUATION]

One more equation is obtained from the parallel component of Ampére's law:

[EQUATION]

or, explicitly,

[EQUATION]

where the electron skin length [FORMULA] and the nonlinear term

[EQUATION]

The set of Eqs. (19), (20) and (22) constitute a basis for the investigation of linear and nonlinear properties of short-scale Alfvén waves. Further simplification may be achieved by expressing the electromagnetic fields through electromagnetic potentials. A usual choice is [FORMULA] and [FORMULA], such that

[EQUATION]

and

[EQUATION]

However, here it is convenient to introduce an effective potential [FORMULA] for the field-aligned inductive (solenoidal) part of electric field,

[EQUATION]

so that [FORMULA] is related to the parallel component of the potential [FORMULA] through the equation

[EQUATION]

The potential part of the electric field is described by the usual potential [FORMULA]:

[EQUATION]

This choice enables us to express both [FORMULA] and [FORMULA] in terms of [FORMULA]. Namely, insertion [FORMULA] from (22) into (19) gives [FORMULA] in terms of [FORMULA] (note that only the inductive part of the electric field enters Ampére's law):

[EQUATION]

Furthermore, when we apply the inverse operator [FORMULA] to (19) we get a formal solution with respect to [FORMULA]:

[EQUATION]

Inserting this into (20) we get

[EQUATION]

where [FORMULA]. We can now remove [FORMULA] from Eq. (28) by means of (26):

[EQUATION]

When we multiply this equation by [FORMULA] and treating the right-hand part as a perturbation, we get the eigenmode equation

[EQUATION]

where the collisional term is expressed in terms of [FORMULA],

[EQUATION]

The differential order of this equation may be reduced by dropping the term [FORMULA], which has a small effect on the Alfvén wave branch in a low-[FORMULA] plasma:

[EQUATION]

In the above expressions [FORMULA] and the ion skin-length [FORMULA]

Eq. (31) is the key equation of the present paper. It is the (second-order in [FORMULA]) nonlinear Alfvén wave eigenmode equation in a non-uniform plasma. It contains the nonzero ion-gyroradius correction ([FORMULA]), the electron inertia correction ([FORMULA]), and the collisional dissipative term ([FORMULA]) in the linear (left-hand side) part. The nonlinear (right-hand side) part contains second-order terms coming from the electron and ion continuity equations ([FORMULA], [FORMULA]) and from the parallel component of Ampére's law ([FORMULA]).

The Alfvén mode Eq. (31) is linearly decoupled from the fast (slow) mode equation because the fast (slow) wave has much higher (lower) frequencies than the Alfvén wave for the same wavenumber and for large propagation angles. The nonlinear part of (31) contains the beatings of the pump AW with the fluctuations corresponding to different modes, including fast, Alfvén and slow modes. However, the secondary waves are effectively enhanced from the background noise by the beatings which are in resonance with the excited waves. Due to the separation of the oblique Alfvén mode from the other modes in the ([FORMULA]) space, the resonance can occur with the low-wavenumber fast wave or with the low-frequency slow wave. These resonances are less efficient than the resonance among oblique Alfvén waves, which are close in the ([FORMULA] ) space. There are two other important properties of the fast and slow waves which make it difficult for them to be excited by the three-wave resonant interaction with Alfvén waves in the solar corona. The growth of the fast wave is suppressed by: (1) strong refraction of the isotropic ([FORMULA]) fast wave in a nonuniform plasma, quickly bringing the wave out of the resonance; (2) by ion temperature anisotropy [FORMULA], very probable in the corona in view of recent SOHO observations. The growth of the slow wave is suppressed by its strong Landau damping unless [FORMULA], which is unlikely to be the case in the corona. Thereto, direct comparison shows that the increment (56) is much larger than the increment of decay into slow and fast waves (Yukhimuk et al. 1999).

On the contrary, the Alfvén mode is less sensitive to the above mentioned stabilizing factors and has a highly anisotropic dispersion making it possible for the wave triads to remain in resonance even in a nonuniform plasma. That is why our focus is on the excitation of highly oblique Alfvén waves by a highly oblique pump Alfvén wave set up by phase mixing due to plasma nonuniformity. The situation can be different with the parallel-propagating pump AW.

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© European Southern Observatory (ESO) 2000

Online publication: June 5, 2000
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