Equilibrium
Susan Lea
January 2007
1
Equilibrium and stability
It seems easy to con¤ne a collisionless plasma: we arrange the geometry so
that the particle drifts are harmless. But collisions give rise to di¤usion which
cause the plasma to escape rapidly. Even ignoring collisions, things may not be
as good as they seem. Random clumps of charge may form ("charge bunching") creating E-¤elds and the resulting E-cross-B drifts. Random motions of
charges create ¢uctuating currents producing magnetic ¤elds, grad-B and curvature drifts. And on and on. We need to analyze the stabilty of each plasma
con¤guration carefully.
1. First ¤nd an equilibrium con¤guration.
2. Check to make sure itzs a stable equilibrium.
Stable means that small perturbations of the system away from the equilibrium state are damped and the system returns to its original state.
Some instabilities are worse than others. "Explosive" instabilities arise when
one disturbance is able to gain energy at the expense of another, and grow
rapidly as a result. (eg plasma oscillations in a drifting plasma where the
fast and slow waves may grow together.) Absolute instability (which involves a
perturbation gowing at one spot) are worse than convective instabilities. Hydromagnetic (low-frequency) instabilities are very destructive to con¤nement.
It is not always easy to ¤nd an equilibrium state.
2
General considerations.
In an equilibrium we may set the explicit time derivative (@=@t) terms to zero.
Letzs begin with ~v = 0 and ~g = 0 too. Then the equation of motion (MHD eqn
7) simpli¤es.
~ = ~j £ B
~
rP
(1)
The pressure gradient force is balanced by Lorentz force on the plasma currents.
Where does the current come from? We can see this by crossing equation (1)
~
with B.
³
´
~ £B
~
~j £ B
~
~
rP £ B
=
= ~j
B2
B2
1
Thus ~j is just the diamagnetic current (¢uid notes eqn 16) induced by the
pressure gradients themselves.
In the form I labelled Astrophysical MHD, ~j has been eliminated and we
have
µ
¶
2
1 ³~ ~ ´ ~
~ P + B
r
=
B¢r B
(2)
2¹0
¹0
If right hand side is zero, magnetic pressure balances gas pressure. Magnetic
¤eld con¤nes the plasma. When the RHS is not zero, ¤eld line tension also
contributes to con¤nement.
~ are perpendicular to rP
~ : current
These equations show that both ~j and B
¢ow and ¤eld lines lie on constant pressure surfaces, or, put another way, if T
is constant n is constant along ¤eld lines.
Examples of equilibrium states:
Earthzs or neutron starzs magnetosphere, solar corona, CNF experiments.
From equation (2), we see that the sum of particle plus magnetic pressures
is constant: as one decreases the other increases. Decrease of B is caused by
diamagnetic currents. The ratio
P
´¯
P m ag
is a signi¤cant indicator of plasma behavior. Most of what wezve done so far
~ for example). High-¯
applies to low-¯ plasmas (any system with uniform B
plasmas occur often in astronomy where ¯ > 1 (eg a cluster of galaxies) and
in CNF (¯ < 1 but not ¿ 1): But pulsars are very low ¯ because B is so
enormous! We can have regions of very high ¯ locally, but averaged over a ¤nite
region, ¯ is always < 1 for con¤nement. .
3
An example of equilibrium: The z¡pinch.
2
In this con¤guration, we have a current along the axis of the cylinder, and
magnetic ¤eld lines wrap around in the µ direction. The plasma extends from
r =¡
0 to r = a:
!
!
¡
j = j bz; (r < a) ;
j = 0; (r > a); with no dependence on z, µ:
Assume a steady state. Then the MHD equations give:
MHD eqn 8
!
¡
1 @
!
r ¢ (n¡
v ) = 0 =)
(r½v r) = 0
(3)
r @r
Equation (1)
!
¡
! ¡
¡
!
@P
rP = j £ B =) ¡jz Bµ =
(4)
@r
Ohmzs law (MHD eqn 14):
¡
!
!
¡
!
¡
! ¡
! ¡
!
¡
1 ³¡
! ´
1 ³¡
! ´
!
E+¡
v £B =´ j +
j £ B ¡ r Pe = ´ j +
rPi
en
en
This has components:
E r ¡ vz Bµ =
1
en
µ
@Pi
@r
¶
(5)
(6)
Eµ = 0
(7)
E z = ´j z
(8)
The z¡component of electric ¤eld is needed to drive the current along the axis.
Finally we add Amperezs law:
¡ ¡
!
!
!
¡
1 @
r £ B = ¹0 j =)
(rBµ ) = ¹0 jz
r @r
(9)
We can integrate this equation if jz is constant for r < a:
Bµ =
¹0
jz r
2
(10)
r<a
For r > a; the current is zero, so rBµ = constant. Requiring the ¤eld to be
continuous at r = a; we have
Bµ =
¹ 0 a2
jz
2
r
(11)
r>a
Next we integrate equation (4) to ¤nd P :
¡j z Bµ
=
=
So:
³¹
´
@P
0
= ¡j z
jz r
@r
2
1 2
¡ j z ¹0 r
2
1
P = ¡ j 2z ¹0 r 2 + C
4
3
r<a
r<a
(12)
For r > a, since j = 0; and there is no plasma, P = 0: Thus the constant
C = 14 jz2 ¹0 a2 and:
¡
¢
1
P = j z2 ¹0 a2 ¡ r 2
r<a
(13)
4
Thus we can determine the size of the column in terms of the central pressure
P 0 and the current density it carries:
s
P0 2
a=
(14)
¹0 jz
The total current is:
I = ¼ a2j z = ¼
Ãs
P0 2
¹0 j z
!2
P0
jz = 4¼
= 2¼a
¹0 jz
s
P0
¹0
(15)
Note: for r < a (equations 13 and 10)
´2
¡
¢ 1 ³ ¹0
¡
¢ 1
B2
1
1
1
= j z2 ¹0 a2 ¡ r 2 +
j z r = jz2 ¹0 a2 ¡ r 2 + ¹0 jz2 r2 = jz2 ¹0 a2
¹0
4
¹0 2
4
4
4
(16)
which is a constant.
Using }Astrophysical MHD} , equation (30) in MHD notes, we would ¤nd
the gradient of the total (gas plus magnetic) pressure to be:
µ
¶
!
¡
B2
1 ³¡
! ¡
!´ ¡
!
r P+
=
B¢r B
(17)
2¹0
¹0
P+
Since the only variations are in the radial direction, we have:
µ
¶
@
B2
1 Bµ @ ³ b´
1 Bµ2 @ ³b´
1 B2µ
P+
br =
Bµ µ =
µ =¡
br
@r
2¹0
¹0 r @µ
¹0 r @µ
¹0 r
(18)
Thus:
P+
B2
2¹0
Z ³
1
¹0 ´ 2
1
r2
jz rdr = ¡ jz2 ¹0
+ constant
(19)
¹0
2
4
2
´2
1 ³ ¹0
1
= ¡
jz r + constant = ¡
B 2 + constant (20)
2¹0 2
2¹0 µ
= ¡
Thus
P+
B2
= constant
¹0
(21)
as we obtained above. Thus we see that ¤eld line tension helps to con¤ne the
plasma.
4
1
0.8
0.6
r
0.4
0.2
0
0.2
0.4
P
0.6
0.8
1
Black - gas pressure. Green- magnetic pressure
The problem with this con¤guration is that it is unstable. Suppose we put a
kink in the cylinder. Then at the outside of the kink, the ¤eld lines are pushed
apart, which means that the ¤eld is weaker. On the inside of the kink the ¤eld
lines are closer together, which means the ¤eld is stronger. Thus the plasma
drifts outward where there is less con¤ning ¤eld, and the kink grows.
Another instability, called the sausage instability, also o ccurs. Imagine perturbing the cyinder as shown below. The same kind of ¤eld perturbations occur,
and the sausage gets fatter where it is already fat, and skinnier where it is skinny.
5
Another kind of pinch, called the µ¡pinch , may be constructed. In this con¤guration the magnetic ¤eld is along the cylinder and the current is azimuthal.
One of these models may be relevant to astrophysical jets.
A combination of these two ideas, plus rolling the cylinder up into a torus,
gave rise to the tokamak.
4
Accreting neutron star magnetospheres
The accreting plasma exerts a ram pressure that is balanced by magnetic pressure at the magnetosphere boundary:
³
´
³
´
2
¡
¢
~ ~v = 1 r
~ ½v 2 ¡ v r½
~ ¡ ½~v £ r
~ £ ~v
½ ~v ¢ r
2
2
so our equation of motion, in equilibrium, is:
µ
¶
³
´
½v 2
B2
v2 ~
1 ³¡
! ¡
!´ ¡
!
~
~ £ ~v
0=r P+
+
¡ r½
¡
B ¢ r B ¡ ½~v £ r
2
2¹0
2
¹0
~ is ¼ 0 except at the
If the ¢ow is irrotational, the last term is zero. r½
boundary, where v ! 0; and a similar result holds for B: Thus,integrating
across the magnetopause boundary, we have
P +
½v 2
B2
+
= constant
2
2¹0
6
On one side the ram pressure dominates, and on the other the magnetic pressure.
Thus, for a dipole ¤eld (recall that the dipole dominates at large distance from
the source)
¯
¯
½v 2 ¯¯
B 2 ¯¯
B¤2 ³ r¤ ´6
=
=
(22)
2 ¯o ut sid e
2¹0 ¯ ins ide
2¹0 r
The velocity is almost free-fall
v2 ¼
2GM ¤
r
and the mass accretion rate is
M_ = 4¼r2 ½v
thus
r
_
M
2GM ¤
½v =
2
4¼r
r
The gravitational potential energy of the infalling matter is converted to radiation, so the luminosity is
2
_
L=M
Thus equation (22) becomes
GM ¤
Lr¤
=) M_ =
r¤
GM ¤
p
Lr¤ 2
B 2 ³ r¤ ´ 6
p
= ¤
5=2
¹0 r
4¼r
GM ¤
and thus the magnetopause radius is given by
B 2 r5 p
7=2
rM = ¤ ¤ GM ¤
¹0 L
or
µ 2 5 ¶2=7
B¤ r¤
1=7
rM =
(GM ¤ )
¹0 L
Using typical values, we get
¡ 8 ¢4=7 ¡ 4 ¢10=7
¡
¢1=7
10 T
10 m
¡11
rM = ³
m3 =kg ¢ s2 £ 2 £ 1030 kg
´2=7 6:7 £ 10
4¼ £ 10 ¡7 N/A2 £ 10 30 J/s
=
=
=
1: 885 9 £ 10 6 ³
T4=7 ¢ m10=7
m3=7
´2=7
s2=7
N/A2
(J/s)2=7
4=7
4=7
(N/A ¢ m)
¢ m13=7
(N)
¢ m9=7
2 £ 106 ³
= 2 £ 10 6
´2=7
2=7
2=7
(N) (N ¢ m)
N/A2
(J)2=7
2 £ 106 m = 2000 km
The stabilty of the magnetosphere was investigated by Arons and Lea. 1976
Astrophysical Journal, 207,914. and 1976 Astrophysical Journal, 210,792
7