12. THE WÖLTJER INVARIANTS OF IDEAL MHD, TOPOLOGICAL
INVARIANCE, AND MAGNETIC HELICITY
We now return to ideal MHD, so that E  V  B  0 . The magnetic flux through and
closed circuit C is
   B  n̂dS ,
(12.1)
S
where S is any surface bounded by C . Since   B  0 , we can write B    A , where
A is the vector potential. Then the flux can also be written as
     A  n̂dS  —
 A  dl .
S
(12.2)
C
Now consider the volume defined by all field lines passing through the curve C .
This volume V defines a flux tube. The flux  within is constant because B is
everywhere tangent to its boundary. We know that, since   B  0 , the tube thus defined
either closes on itself or fills space ergodically. Any finite volume V0 contains an infinite
number of such flux tubes.
Now consider the following integral:
Kl   A  BdV ,
(12.3)
Vl
where Vl is the volume of the l th in V . The flux tube will move about with the fluid
velocity V . As it does, Equation (12.3) changes according to
dK l
B
d
 A

 
 BdV  A 
 A  B dV  .
 t

dt
t
dt
Vl
(12.4)
The last term is evaluated as
d
d
dV  dx1dx2 dx3  V1dx2 dx3  V2 dx2 dx3  V3dx1dx2 ,
dt
dt
 V  n̂dS .
(12.5)
Then using Faraday’s law, we have
dK l
 E    BdV   A    E BdV   A  BV  n̂ dS , (12.6)
dt Vl
Vl
Sl
where  is the scalar potential. Now
  A  E  E    A  A    E ,
(12.7)
so that the second integral can be written as
 A    E BdV   E  BdV     A  EdV
Vl
Vl
,
Vl
1
  E  BdV   A  E n̂dS .
Vl
(12.8)
Sl
Similarly, the first integral can be rewritten as
   BdV     BdV
Vl
,
Vl
  B  n̂dS  0 ,
(12.9)
Sl
because   B  0 and B  n̂  0 on Sl by definition since Vl is a flux tube. Therefore
dK l
 2  E  BdV   A  E n̂dS   A  BV  n̂ dS .
dt
Vl
Sl
Sl
(12.10)
Now invoking ideal MHD, E  V  B , Equation (12.10) becomes
dK l
   A  BV  n̂   A  V B  n̂  dS  0 ,
dt
Sl
(12.11)
since both B  n̂ and V  n̂ vanish on Sl . Therefore K l  constant for each and every
flux tube in the system. The K l are called the Wöltjer invariants. They depend on
E  V  B (ideal MHD), and B  n̂  V  n̂  0 on Sl . Of the latter two equalities, the
first is a property of the flux tube, and the second is also a consequence of ideal MHD
(the flux tube moves with the fluid).
It is possible to give a physical interpretation of the Wöltjer invariants. Consider the
linked flux tubes, shown in the figure.
Flux tube C1 contains flux 1 . Flux tube C 2 contains flux  2 . The Wöltjer invariant
for tube C1 is
K1   A  BdV .
(12.12)
V1
For this flux tube, we have
2
BdV  (B1ê1  B2 ê 2  B3ê 3 )dx1dx2 dx3 ,
 ê1dx1 B1dx2 dx3   ê 2 dx2 B2 dx1dx3   ê 3dx3 B3dx1dx2  ,
 B  n̂dS dl ,
(12.13)
so that Equation (12.12) becomes
K1   B  n̂dS —
 A  dl .
S1
(12.14)
C1
The first integral is just 1 , the flux contained within tube C1 . From Equation (12.1), the
second integral is the flux enclosed, or linked, by the curve C1 , which is  2 if the tubes
have “right hand” linkage,  2 if the tubes have “left hand” linkage, and 0 if the tubes
are not linked. For now we write
K1  12 .
(12.15)
K 2  2 1  K1 .
(12.16)
Similarly,
If the tubes are linked N times, we have K1  K 2  N 2 1 . The same results are
obtained for a single knotted flux tube, as shown in the figure.
The Wöltjer invariants are thus a direct measure of the linkage, or topology, of the
flux tubes. Since the K l are constant in ideal MHD, it means that the topology of the flux
tubes cannot change and is preserved for all time. This property is called topological
invariance. (It is really just another way of saying that the magnetic field is co-moving
with the fluid.) It is a result of the ideal MHD Ohm’s law, E  V  B  0 , and places a
very strong constraint on the allowable motions of the fluid.
Now consider a fixed volume V of fluid (no longer a flux tube). The volume
integral
K M   A  BdV
(12.17)
V
3
is called the magnetic helicity associated with the volume V . We remark that the
integrand A  B contains the vector potential, and hence depends on the choice of gauge.
Letting A   A   , we have
K M   A   BdV ,
V

 A    BdV
,
V
  A  BdV     BdV ,
V
V
 K M      B dV ,
V
 KM  —
  B  n̂dS .
(12.18)
S
Therefore, K M  K M only of the surface integral vanishes. There are many practical
cases where this is true. Examples are periodic boundary conditions, or perfectly
conducting boundaries. (However, if the geometry is not simply connected, as in a torus,
the flux within the fluid may link some external flux, and this must be taken into account.
We will discuss this more when we take on MHD relaxation.)
Nonetheless, in future topics we will find it useful to have a definition of magnetic
helicity that is manifestly gauge invariant. This can be obtained by defining
KM 0 
 A  A  B  B dV
0
0
,
(12.19)
V
where B0    A 0 is a reference field, to be defined. Letting A   A   , it is easy to
show that
K M 0  K M 0  —
  B  n̂  B0  n̂dS .
(12.20)
S
If we then choose the reference field such that B0  n̂  B  n̂ on S , K M 0 will be gauge
invariant. (This holds true if we also introduce A0  A 0   .) A straightforward
calculation then shows that
dK m0
 2  E  B  E0  B0 dV ,
dt
V
(12.21)
where E0  A 0 / t . Then if E  V  B and E0  V  B0 , K M 0 remains constant
for all time. This is called the generalized magnetic helicity. It is conserved in ideal
MHD.
Since generalized helicity is conserved, it is tempting to interpret the integrand A  B
as a helicity density. This can be misleading. From the discussion of this Section, it is
clear that helicity only has physical meaning as a volume integral. Attempts to assign
some physical meaning to the local quantity A  B have not led to significant insights.
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