Gravitational Instability

Let us consider rather simple problem of stability of a plane plasma in a field of constant external force, exp., gravitational force. There are some kinds of MHD instabilities can be classified as gravitational instabilities, i.e., flute instability, Rayleigh-Taylor instability, interchange instability, etc. Detailed analysis can be found in reference [6], pp.362-374. Here, I just want to show some basic results.
Assume the perturbation only developes in x direction. For any perturbations distributed on (y,z) plane, they can be expressed in the form of periodic functions of coordinates y and z or their sum.
;
; (61)
In deriving the displacement equation ([6] eq.(10.67)), it is assumed that unperturbed field and are constant, which means
(Of course this is not true in practice. But we can introduce an effective force corresponding to the gradient of the field in future.)
Next show how to prove displacement equation ([6] eq.(10.67) in Gaussian units):
(62)
Use vector idendity: (63)
and eq.(36), the last two terms in right side eq.(41) ([6] - (10.57)) become

Use eq. (35)([6] - (10.55) and (10.65):
,
the 1st term in right side of eq.(41)([6] - (10.57)) becomes:

The eq.(41) ([6]-(10.57)) reduces to :
(64)
Notice that if define , then
displacement equation eq.(62)
Where satisfies Laplace equation:
(65)
where k is wave constant in (y,z) plane. It's quite difficult to solve eqs.(62) and (65), if not impossible. But we are more interested in conditions of plasma stability. This can be shown in the dispersion relation of and k. One can resort to [6] pp.364-366 for reference. Here is just the result (in MSK):
(66)
Where G is gravitational force acted on unit volume directed in -x direction, and each term is evaluated at the boundary.
Equation (66) enable one to find the conditions of stability of the plasma boundary, the oscillation frequency of the perturbation, and the instability increment. As noted above, the stability region corresponds to the real values of , that is, to the positive right-hand side of Eq.(66). Therefore the absolute stability of the plasma boundary is only possible if the external force is nonexistent or is directed from the boundary into the plasma. With an outward-directed force the possibility of instabilities with different depends on the relationship between the negative term of Eq.(66) and the positive terms describing this stabilizing factor.
Flute Instability

When the magnetic fields outside and inside are parallel ( ), the positive terms reduce to zero at . Therefore the plasma boundary is unstable with respect to perturbations extending along the magnetic field. This is so-called flute instability. From eq.(66), its increment rate is given by: (67)
From eq.(66), it shows that magnetic shear is a stabilizing factor. With shear, however, for sufficient small k, there still exist possibility of instability. But the minimum value of k is limited by the plasma's dimension: . Then we get the boundary stability condition in the extreme shear case :
(68)
L is characteristic length of the plasma. Figure 1 shows picture of flute instability.
Rayleigh-Taylor Instability

Rayleigh-Taylor instability occurs when a heavy fluid is placed over a light one; the role of the heavy fluid is played by the plasma, and the role of the light one, by the magnetic field. Figure 2 shows the idea of Rayleigh-Taylor instability.
The physical mechanism of instability of the plasma boundary in the field of an external force for the limiting cases of a high and a low plasma pressure is discussed in the reference [6] pp.367-370 and in many other books.
In high pressure case, the instability is totally driven by the work done by the external force which gives the same increment rate as eq. (67). The instability extends along the magnetic field as a flute instability. With a magnetic shear, the perturbation increase is impeded by the distortion of the magnetic field. The distortion of the field lines give rise a tension restore force:
(69)
See figure 3 for each term. R is curvature radius, which is . A sufficient large can suppress the instability.
In low pressure case, we can consider the problem from a microscopic view. Look at figure 4 for clarification. Under the force G, ions and electrons drift in opposite directions. In the presence of flute type perturbations the drift results in accumulation of ions on one side of the flute and electrons on the other. The electric field of polarization causes a drift of the plasma as a whole in the direction of increasing perturbation. The increment rate can be found from the displacement equation:
(70)
The increment rate is the same as eq.(67). For sufficient long wavelength (small k), the growth rate is given by [1]:
(71)
The above we assume there exist an external force (i.e, gravitational force) which if directed from plasma to the boundary, can cause flute type instability extending along the magnetic field. For the laboratory plasma, gravitational force can be neglected. But the inhomogeneity due to gradient and curvature of the magnetic field can present an effect force which acts as gravitational force. For example, in high plasma, magnetic pressure gradient acts on the boundary as an external force. If this force directed along the normal to the plasma surface, that is, if the magnetic field decreases away from the plasma, an instability may arise. Therefore, in order to stabilize the plasma, we need to place the plasma in the minimum field region. In low case, the centrifugal force caused by the magnetic field curvature and the diamagnetic force cause by the plasma pressure gradient([6]) can cause instability. If the lines of magnetic force are convex outward(fig.5a), the plasma is unstable; if the lines are concave outward (fig.5b), the plasma is stable.
Interchange Instability

The development of the flute instability in a plasma with can be ascribed to interchange of the flux tubes filled with plasma. The magnetic field strength and the shape of the lines of force remain constant, but the plasma volume and pressure in the tubes change.The energy accumulated in the plasma-magnetic field system changes accordingly. Instability may appear, caused by the interchange of the flux tubes for different magnetic field configurations. Because of this geometric picture of the process, this type of instability has come to be called interchange instability. Let's derive the stability condition for interchange instability. Let us assume that a line of magnetic force has 'good' curvature at one plasce B and 'bad' curvature at another place A (fig.6a). Then the directions of centrifugal force at A and B are opposite, as is the charge seperation. Let us here consider perturbations in which the magnetic flux of region 1 is interchanged with that of region 2 and the plasma in the region 2 is interchanged with the plasma in the region 1 (interchange perturbations, fig.6b). It is assumed that the plasma is low-beta so that the magnetic field is nearly identical to the vacuum magnetic field. Any deviation from the vacuum field will increase the energy of the disturbed field.
The energy of the magnetic field inside a magnetic tube is
, (72)
where l is length taken along a line of magnetic force ans S is the cross section of the magnetic tube. As the magnetic flux is constant, the energy is
. (73)
The change in the magnetic energy due to the interchange of the fluxes of region 1 and 2 is
(74)
If the exchanged fluxes and are the same, the energy change is zero, so that perturbations resulting in are the most dangerous.
The kinetic energy of a plasma of volume V is
, (75)
where is the specific-heat ratio. As the perturbation is adiabatic, is conserved during the interchange process. The change in the plasma energy is
. (76)
Setting
and ,
we can write as
. (77)
Since the stability condition is , the sufficient condition is .
Since the volume is ,
the stability condition for the interchange instability is written as
. (78)
Usually the pressure decreases outward ( , so that the condition becomes
(79)
in the outward direction. The integral is to be taken only over the plasma region.
One can define the average agnitude of the magnetic field by
.
When the total length of the magnetic-field line is denoted by L, the stability condition is . Therefore this condition is called average minimum-B condition. The magnetic configuration shown in fig.7 satisfies this requirement.