26. EXAMPLES OF APPLICATION OF THE ENERGY PRINCIPLE
In this Section we will present several examples of the application of the Energy
Principle for determining stability of a magneto-fluid system that illustrate the power of
the method. These examples are all based on the simple case of a fluid/vacuum interface,
as shown in the figure.
In the fluid, the magnetic field vanishes, B0  0 , and the pressure is constant. In the
vacuum, the magnetic field is finite, B̂ 0  0 , and the pressure vanishes.
The potential energy in the fluid is
 WF 
1
2
dVp0     0 ,

2F
(26.1)
since all other terms vanish; according to the Energy Principle, this term is stabilizing.
However,  WF can be minimized by choosing a displacement for which     0 . We
conclude that     0 is always stabilizing when p0  0 .
The jump in total pressure across the interface S is
2 ¨
2
©
™p  B0 °  B̂0  p .
™0
°
0
™
2 0 °Æ 2 0
´
(26.2)
The first term on the right hand side comes from the vacuum and the second term comes
from the fluid. Therefore
©
 B̂02 
B02 ¨°
™
 ™p0 
,
°
™
2 0 °Æ  2 0 
´
(26.3)
since p0  constant . Then, if     0 , the total potential energy of the system is
W  WS  WV ,

 B̂02 
2
1
1
2
dS
n̂


n̂



dV   Â .




2S
 2 0  2 0 S
(26.4)
1
As noted in Section 25, the vacuum contribution is always stabilizing. Stability is
therefore determined by the sign of n̂  B̂02  B̂02 / n :

If B̂02 / n  0 , then W  0 and the system is stable.

If B̂02 / n  0 , then W  0 and the system can be unstable. The details
depend on the balance between  WS and  WV .
A vacuum magnetic field that decreases away from the fluid is destabilizing.
Conversely, and magnetic field that increases away from the fluid is stabilizing. The
latter is an example of minimum-B stabilization.
Now consider the integrand in the surface term in more detail. A useful identity is

1 
  B̂ 2   B̂  B̂  B̂    B̂
2 
.


In the vacuum, 0 Ĵ    B̂  0 , so  B̂2 / 2  B̂  B̂ . We want to compute B̂ at the
interface S . The geometry is shown in the figure.
Point C is the center of curvature of the surface. The vector R points from the
surface to the center of curvature, and â r  R / R . In this geometry,
  b̂  
B̂   â r 
 B̂b̂ ,
 r R  
 â r b̂
B̂ b̂b̂ B̂ b̂â r


B̂ ,
r
R 
R
since b̂ /    â r . Then
2

B̂ b̂b̂ B̂ b̂â r
B̂  B̂  B̂b̂   â r b̂


r
R 
R



B̂ ,

b̂ B̂
B̂ 2
B̂
 â r
,
R 
R
so that

1
n̂  B̂ 2  n̂  B̂  B̂
2

,
n̂  b̂ B̂
B̂ 2
B̂
 n̂  â r
.
R

R
(26.5)
But n̂  b̂  0 because B̂ lies in the surface S , and n̂  â r   n̂  R / R . Therefore
1
B̂ 2
n̂  B̂ 2  n̂  R  2 .
2
R
(26.6)
The surface energy  WS is then
WS 
1
4 0
 dS n̂  
S
2
n̂  R 
B̂ 2
,
R2
(26.6)
so that stability may be determined by the sign of n̂  R :

n̂  R  0 is stabilizing. This is called (creatively!) good curvature; the center
of curvature lies inside the vacuum, as shown in the figure.

n̂  R  0 is destabilizing. The is called bad curvature; the center of
curvature lies inside the fluid, as shown in the figure.
3
In order to find the total perturbed potential energy, we need to determine  WV .
This requires that we consider the effect of displacements of the surface S when the
equilibrium surface is flat (i.e., R   ). We work in a Cartesian coordinate system with
x normal to the surface, and y and z in the plane of the surface. The magnetic field in
the vacuum is in the z-direction. As before, the field vanishes in the fluid and the
pressure is constant. The configuration is shown in the figure.
In this case, n̂  ê x , B̂0  B̂0 ê z , and n̂    x . We also assume that the system extends
i k y k z
infinitely far in the y- and z-directions, so that on S we can write    e  y z . This is
x
0
the vertical displacement of the surface.
In the vacuum J  0 , so the perturbed vector potential in the vacuum satisfies
    Â  0 .
(26.7)
% ikr , k  k ê  k ê  k ê ,   ik , and using vector identities, this
With   Ae
x x
y y
z z
becomes


% k k  A
%0 .
k 2A
(26.8)
% ikr , where B  ik  A
%.
The perturbed vacuum field is B̂1  Be
On S ( x  0 ), A must satisfy the boundary condition
4
%  n̂   B̂ .
n̂  A
0
(26.9)
In light of the above discussion, this becomes
Az  0 ,
(26.10)
Ay  x B̂0S ,
(26.11)
and
%, the
where B̂0S is the equilibrium vacuum magnetic at the surface. Since B  ik  A
perturbed magnetic field only depends on the component of A that is parallel to k . We
% 0 . In light of Equation (26.10), this requires
are therefore free to set k  A
Ax  
ky
kx
A%y ,
(26.12)
and Equation (26.8) becomes
% 0 .
k 2A
(26.13)
A non-trivial A therefore requires k  0 , or k x  i , where   k  k . After some
algebra, the solution for the perturbed vacuum field that is bounded at infinity is found to
be
%  x ei ky y kz z ,
(26.14)
B̂  Be
2
2
2
y
2
z
1
where
B
iA%y
kx
ê k k
x z x
 ê y kz ky  ê z 2

,
(26.15)
where k x  i .
The magnetic energy density in the perturbed vacuum field is therefore, again after
some algebra,
wV 
1
B̂1  B̂1*e2 x ,
2 0
1 % 2 2 2 x

Ay kz e
,
0
or, using Equation (26.11)
wV 
1
0
kz2x2 B̂02s e2 x .
(26.16)
The vacuum contribution to the potential energy is therefore
5
WV   wV dV ,
V


1
0
 k  B̂
1
 0
2 2
z x
2
0s

dydz  dxe2 x ,
0

S
kz2x2 B̂02s

dS .
(26.17)
If     0 ,  WF  0 (see Equation (26.1)), and the total perturbed potential energy
of the system is
W  WS  WV ,

1
4 0
2
 dSx
S
B̂02
1

x  0

S
kz2x2 B̂02s

dS .
There, instability occurs if B̂02 / x  0 and kz2 /   0 .
(26.18)
The latter condition is
equivalent to kz  0 , i.e., k  B̂ 0 S  0 . This means that, on S , the wavefronts of the
perturbation are parallel to the equilibrium magnetic field. These perturbations do not
bend the field lines. The displacements to the surface can slip through the magnetic field
without increasing the magnetic energy. The situation is sketched in the figure.
If B̂02 / x  0 , the upward moving tip of the perturbation enters a region where
B̂02 /  0  p0 . The fluid in the tip thus expands and feels a “restoring force” in the
direction of x . Instability ensues.
While the conclusions drawn here regarding stability (or lack thereof) derive from
the simple problem of an unmagnetized, uniform pressure plasma in contact with a
unidirectional vacuum magnetic field, the concepts of incompressible perturbations,
minimum-B, good and bad curvature, and field line bending are sufficiently general that
they are applicable in other, more complex situations.
6