# MHD Waves in Magnetically Confined Plasmas

## THE LINEARIZATION PROCEDURE AND MAIN TYPES OF MHD WAVES

We start from Eqs. (4.1)—(4.7) and use (4.13) for dividing plasma variables into equilibrium and per-

*S J*

turbed quantities (denoted by *S*) where all the perturbed quantities satisfy 5 к 1, that is, — s 1,

*Jo*

etc. The equilibrium terms are balanced owing to the plasma equilibrium, and now we consider only the terms linear in <5. The linearized ideal MHD equations take the form:

For clarifying the physics of the perturbations, we introduce a vector of plasma displacement from the equilibrium, $, related to <5V via

so (5.1) and (5.3) take the form

where we used the vector analysis equation

and considered for simplicity a slab geometry case with

By substituting these expressions for *8**p,**8**B* in the remaining Eqs. (5.2) and (5.4), we obtain
where

is square of ion sound speed, and is squared Alfven velocity.

Equation (5.10) describes linear MHD perturbations of homogeneous ideal conducting plasma. This equation for the single vector variable I gives three scalar equations for three different types of MHD waves [5.1]. Consider the first two “compressible” types of waves, in which *8**, _{z} Ф* 0 and div £j_ ф 0. The displacement parallel to the equilibrium magnetic field is described by the parallel projection of (5.10):

and equation for div^_{x} is obtained from the divergence of the perpendicular projection of (5.10):

where Д_{х} = divV_{x}. We see that (5.13) and (5.14) for *8**,-* and div^_ are coupled. However, if we con- sider the limit — « 1, Eq. (5.14) decouples from *8**,-* and reduces to

Equation (5.15) describes compressional Alfven (CA) wave, for which the magnetic pressure Д?/8л determines the “returning” force that acts perpendicular to *B _{0}.* The displacement

*8*

*,*parallel to

_{z}*B*

*is described by*

_{0 }

and the wave equation (5.16) describes ion sound wave existing in plasma even without *B _{0}.*

However, if *cfv* is not small, the two different compressible waves couple. In this case, the ion sound wave is modified by the magnetic pressure and becomes slow magneto-acoustic (SM) wave. The coupled equations give for *8*,., div£_{x} =«= exp (-/ft» + *ik ■ r)* the following dispersion relation between wave frequencies and wave vectors:

FIGURE 5.1 Plasma displacement *%* in compressional Alfven (CA), shear Alfven (SA), and slow magnetoacoustic (SM) wave.

In contrast to the compressional waves, the third type of MHD weaves, the so-called shear Alfven (SA) wave, is incompressible, = 0 and div£_{±} = 0. For such waves, the main MHD equation (5.10) simply becomes

which coincides with the well-knowm equation for oscillations of a string. The “returning” force for SA waves is the tension of magnetic field lines, which act similar to the strings.

The three different types of MHD w'aves derived above are schematically shown in Figure 5.1. Figure 5.1 show's that CA and SM waves have “returning” forces as the magnetic and the kinetic pressure, respectively, w'hile SA wave has the “returning” force as the tension of magnetic field lines.

## SHEAR ALFVEN WAVES IN INHOMOGENEOUS PLASMA AND THE WAVE CONTINUUM DAMPING

Among all MHD waves in plasma, SA w'ave constitutes the most significant part of the MHD spectrum and is probably the best studied [5.2]. In SA wave the fluid displacement vector *£,* and perturbed electric field 8*E* are perpendicular to the magnetic field *B _{0},* while the wave propagates along

*Bo*wfith frequency and parallel wave-vector determined by

However, wave packets wfith dispersion relation (5.19) are not structurally stable in inhomogeneous plasmas. Let us consider plasma with inhomogeneous density, *V _{A} = V_{A}* (r), and finite magnetic shear, that causes a radial dependence in parallel wave-vector,

*k*(r), and assume that a w'ave- packet of SA type is created by some means at

_{n}= k_{n}*t*=

*t*

*as Figure 5.2 show's. If the wave-packet satisfies SA dispersion relation (5.19) in each point along the axis*

_{0}*r,*and both Alfven velocity and parallel wave-vector do vary along

*r*too, then every individual “slice” of the wave-packet at different

*r*will propagate wfith a different phase velocity (due to the /'-dependence in Alfven velocity) and in a different direction (due to the r-dependence in the parallel w'ave-vector).

FIGURE 5.2 A wave-packet of SA type extended over r-variable in inhomogeneous plasma. The arrows show the directions of *B _{0},* which change along

*r*due to the assumed magnetic shear.

After some time, the slices of the wave-packet with have their phases shifted significantly enough for spreading the wave-packet along magnetic field. The life-time of the wave-packet of SA waves is limited by the “phase mixing”

In addition to the initial value problem above, it is instructive to consider a boundary value problem with SA wave launched by an external antenna into an inhomogeneous cold plasma with density gradient, *n*_{0}* = n** _{0}* (x),

*P*0,

_{u}=*B*

_{0}*= B*as shown in Figure 5.3.

_{{)}e_{z},An externally excited electromagnetic wave with perturbed electrostatic potential

*(x)exp(ik _{y}y* -

*icot)*is described in our case by equation

where

Equation (5.22) has a zero bracket at the high-order derivative term at the point *x = x** _{0}* of the local Alfven resonance layer, where frequency of the externally launched wave matches the local SA frequency.

The equation in the vicinity of this point can be simplified to:

FIGURE 5.3 Wave is launched with frequency *ш* and wave vector *k _{n}* in a slab geometry from low-density plasma edge into inhomogeneous plasma with density gradient.

which after integration gives

and expansion in the vicinity of the point *x = x _{0}: *
gives the following solutions:

The singularity at *x = x** _{0}* shows that the energy of the launched wave peaks at

*x*=

*x*while the term

_{0},*in*shows that resonant absorption of the wave energy occurs at this point, the continuum damping of the wave launched.

Due to the large values of the continuum damping, excitation of SA wave by super-Alfvenic energetic ions was not considered for a while as a major threat to the burning plasma operation. However, this situation changed in the 1980s w'hen global Alfven eigenmodes (GAEs) and toroidal Alfven eigenmodes (TAEs) were reported.