19. BEHAVIOR OF SMALL DISPLACEMENTS IN IDEAL MHD
We have just studied MHD equilibrium, states in which the forces are exactly in
balance. We now begin studies of states that are slightly displaced from equilibrium.
We assume that the fluid and magnetic field are in an equilibrium state, F  0 . Now
let a fluid element be displaced by a small amount  from its equilibrium position. That
is, a fluid element with equilibrium position r is displaced slightly to a new position
r  r   , as shown in the figure.
After this displacement, the system is no longer in equilibrium, i.e., F  0 . Since F  0
when   0 , we can write
F  F  .
(19.1)
(The curly bracket notation is standard here.) Further, for small displacements we expect
F to be a linear function of  .
Now suppose that   F  0 . Then the displacement and the force are in opposite
directions. The force tends to restore the system to its original equilibrium position; it
opposes the displacement. In this case we might expect the system to oscillate about its
equilibrium position, so that the system can be said to be stable (of course, this must be
proven). On the other hand, when   F  0 , then the force is in the same direction as the
displacement. The force tends to amplify the displacement and drive the system further
from its equilibrium position. In this case we might expect the original displacement to
grow in time, so that the system can be said to be unstable (again, this must be proven).
A third possibility is that   F  0 , so that the force and the displacement are orthogonal.
The system is then said to be neutrally stable. In ideal MHD, these are the only
possibilities. (For example, when   F  0 the restoring force could be so large as to
amplify the oscillations, leading to growing oscillations. This case is called overstable.
We will see that this is not allowed in ideal MHD.)
We can be more specific about these concepts. Let the subscript (..)0 denote
equilibrium quantities. If the equilibrium is stationary ( V0  0 ) and the displacements
are small, we can ignore the quadratic term V  V , and the equation of motion is
0
V
F .
t
(19.2)
1
We introduce the displacement vector  , defined by
V

.
t
(19.3)
Then the equation of motion is
0
2 
 F  .
t 2
(19.4)
In light of the discussion of the previous paragraph, we anticipate that Equation (19.4)
will be a wave equation, and that F will be a second order spatial differential operator.
We are thus motivated to represent the time dependence of the displacement  in terms
of wave-like solutions as
 r,t    r ei t  c.c. ,
(19.5)
where  r  is now a complex function, and c.c. denotes the complex conjugate. (This is
required to make the physical displacement real. We will often leave it out of the ensuing
formulas, but it is always implied. Alternatively, one can interpret the behavior of the
physical displacement as being represented by the real part of the formulas.) Introducing
the ansatz (19.5) into (19.4), we have
 0 2   F  .
(19.6)
Since F is linear in  , we can write symbolically F   F   , so that F is now
represented as a tensor (or matrix). Then
 0 2   F  
(19.7)
is the linear equation of motion for small displacements from stationary equilibrium.
Since  is a as yet undetermined function of space and time, F   is called a functional
of  . It is called the ideal MHD force operator. (Heuristically, we can see that if
  F   0 , this suggests that  2  0 and the motion is oscillatory; and, if   F   0 ,
this suggests that  2  0 ,   i , and the motion will have exponentially growing
behavior. However, this still needs to be proven.) Again, we emphasize that formulas
such as Equation (19.7) only have physical meaning when combined with their complex
conjugate, or their real part is taken.
Equation (19.7) is a linear system of the form A x  x , where A ( F) is a linear
operator (matrix, tensor, or differential), x ( ) is a vector, and  (  2 ) is a
constant. The problem is to find non-trivial solutions for x (i.e., x  0 ). This special
important problem is called an eigenvalue problem. Non-trivial solutions are possible
only for certain special values of  that are roots of the equation det A  I x   0 .
These special values of  are called the eigenvalues of A , and the corresponding nontrivial solutions x are called the eigenvectors of A .
Equation (19.7) can be written as the homogenous system
2
F    I   0
2
0
(19.9)
.
This suggests that Equation (19.9) has non-trivial solutions for  only for special values
of the (negative of the square of the) frequency  2 that satisfy
det  F  0 2I  0 .
(19.10)
The frequencies  2 are the eigenvalues of F , and the corresponding displacements are
the eigenvectors.
We anticipate that, in differential form, F will contain the vector operator  . For
the important special case of an infinite (or periodic) system, we can let   ik , where
k is the wave vector in the direction of wave propagation with amplitude k  2 /  .
(Here  is the wavelength, not to be confused with the eigenvalue of the previous
paragraph.) Equation (19.8) then becomes a set of linear algebraic equations whose roots
can be written as
 2   2 (k) .
(19.11)
This is called the dispersion relation for the system under investigation. The roots
(19.11) are also called the “characteristic oscillations”, or the “normal modes”
If the system is not spatially periodic, or the substitition   ik cannot be made for
any other reason, then Equation (19.9) is an ordinary differential equation that must be
solved subject to the proper boundary conditions.
The procedure for studying the behavior of small displacements from stationary
equilibrium in ideal MHD is therefore:
1. Find the functional form F  .


2. Determine the eigenvalues of F  0 2I    0 . This may require solving
an ordinary differential equation, or the algebraic equation
det  F  0 2I  0 .
3. Examine the behavior of the eigenvalues of this equation with respect to their
implications for oscillatory of exponentially growing behavior.
In Section 2 we defined introduced the concept of the adjoint, F† , of an operator F .
The adjoint has the property that if u  F  x , then u*  F†  x* , where (..)* denotes the
complex conjugate. If F is a matrix, then Fij†  Fji* . If F  F† , then F is said to be selfadjoint. An important property of a self-adjoint operator is that it satisfies
 dVu
*
 F  v   dVv*  F  u .
(19.12)
(More generally, (u,F  v)  (v,F  u) , where (x, y) denotes an inner product in function
space.)
We will soon prove that the ideal MHD force operator is self-adjoint. This has
important consequences for the behavior of small oscillations in ideal MHD. First, let  i
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be two eigenvectors of F corresponding to the eigenvalues  i2 and  2j , respectively,
i.e.,
F   i   0 i2  i ,
(19.13)
F   j   0 2j  j .
(19.14)
and
The complex conjugate of Equation (19.14) is
F     F  
*
j
*
j
 

  0  2j  j ,
(19.15)
since F is assumed to be self-adjoint. Now dot  j with Equation (19.13), and  i with
Equation (19.15), subtract, and integrate over all space:
 

j

 

 F   i   i  F   *j dV   0  i2   2j    i   j dV .


(19.15)
The left hand side vanishes by Equation (19.12), since F is assumed to be self-adjoint.
Therefore,
0  i2   2j     i   j dV  0 .



(19.16)
There are two non-trivial possibilities for satisfying Equation (19.16). First, let
2
i  j . Then, since  i   i   i  0 because  i is a non-trivial solution of Equation
(19.9), we require
 i2   i2  ,

(19.17)
i.e., the eigenvalues of F are real. Therefore, in ideal MHD the normal modes are either
purely oscillating or purely growing (or damped). Overstable modes are impossible. If
 i2  0 , then  i  i , and the displacement evolves according to  ~ eii t , so the
normal modes are pure oscillations. If  i2  0 , then  i  i i , the displacement evolves
according to  ~ e  t , and one of the normal modes exhibits pure exponential growth.
The second possibility for satisfying Equation (19.16) is that i  j and  i2   2j .
Then

i
 *j dV  0 ,
(19.18)
so that the eigenvectors of F are orthogonal. Further, they can be normalized so that

i
  *j dV   ij ,
(19.20)
in which case they are said to be orthonormal.
Finally, it can be shown that the eigenvectors  i form a complete set. By this we
mean that any “piecewise continuous” function (r) can be approximated “in the mean”
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arbitrarily closely by a linear combination of the eigenfunctions  i . That is, for any
reasonably behaved displacement (r) , we can write

(r,t)   ak  k (r,t) ,
(19.21)
k 1
where the ak are complex numbers called expansions coefficients. Dotting Equation
(19.21) with  *j and integrating, we have

*
*
  j  dV   ak   j  dV ,
k 1
so that, in light of Equation (19.20),
a j   *j  dV .
(19.22)
Therefore, the behavior of any arbitrary displacement can be obtained by knowing the
behavior of the eigenvectors  k . If we find the eigenvectors and eigenvalues, we will
know the behavior of the system.
Recall that the time dependence of the eigenvectors is given by  k (r,t)   k (r)ei k t ,
so that

(r,t)   ak ei k t  k (r) .
(19.23)
k 1
Therefore, if all of the quantities  k are real (  k2  0 ), then the system exhibits
oscillatory behavior about its equilibrium position; it is stable. However, if one of the
 k is imaginary (  k2  0 ), then the system exhibits exponential deviation from its
equilibrium position. The existence of a single unstable eigenvector renders the entire
system unstable.
Now we just need to find the functional form of F  , and prove that it is selfadjoint!
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