Magnetohydrodynamics
S.M.Lea
January 2007
Magetohydrodynamics (MHD) is the study of plasma motions in the low frequency
approximation, ! ¿ ! p ; ! c . That means that we use the plasma approximation (ne ' ni )
while allowing for non-zero electric ®elds and also currents that depend on the small
difference between ne and ni : MHD is a very useful tool for studying bulk motion of
plasmas and MHD waves in astrophysical and lab situations.
1 The equation of motion
We may write an equation of motion for each species, including the momentum transfer
due to collisions. For the ions:
·
´ ¸
@~vi ³
~ ~v i = nieE
~ + eni~vi £ B
~ ¡ rP
~ i + P~ie + ni M~g
M ni
+ ~v i ¢ r
(1)
@t
with a similar equation for the electrons:
·
´ ¸
@~v e ³
~
~ ¡ ene~ve £ B
~ ¡ rP
~ e + P~ei + ne m~g
mne
+ ~ve ¢ r ~ve = ¡ne eE
(2)
@t
The term P~ei is the rate of momentum transfer from ions to electrons (Diffusion notes
equation 21):
~ ie
P~ei = n2 (~vi ¡ ~ve ) e2 ´ = ¡ P
(3)
If we add the two equations, we can eliminate this term:
h ³
´
³
´ i
@ (M ~vi + m~ve )
~ ~v i + m ~v e ¢ r
~ ~v e = en (~vi ¡ ~ve )£ B¡
~ r
~ (P i + P e )+n (M + m)~g
n
+n M ~vi ¢ r
@t
Now we de®ne an average velocity using the total momentum density ~p :
~p
= (M + m)~v = M~vi + m~ve
(4)
n
The total pressure is the sum of the partial pressures,
P = Pi + Pe ;
the current is
and the mass density is
~j = ne (~vi ¡ ~ve ) ;
½ = n (M + m)
1
(5)
We these de®nitions, the equation of motion becomes:
h ³
´
³
´ i
@~v
~ ~vi + m ~v e ¢ r
~ ~v e = ~j £ B
~ ¡ rP
~ + ½~g
½
+ n M ~vi ¢ r
@t
(6)
The convective derivative terms may be neglected for motions that are subsonic and
sub-Alfvenic. In fact, we may replace these terms 1 with an equivalent term in ~v (see
problem set 11)
³
´
@~v
~ ~v = ~j £ B
~ ¡ rP
~ + ½~g
½
+ ½ ~v ¢ r
(7)
@t
We complete the set with the two continuity equations, which add to give
@½ ~
+ r ¢ (½~v ) = 0
(8)
@t
and subtract to give the charge conservation equation:
@½q
~ ¢ ~j = 0
+r
(9)
@t
We also need an equation that gives us the time evolution of the current. So multiply
equation (1) by m; equation (2) by M and subtract:
h³
´
³
´ i
@ (~vi ¡ ~ve )
~ ~vi ¡ ~ve ¢ r
~ ~ve
M mn
+ nM m ~v i ¢ r
@t
~ + en (m~v i + M ~ve ) £ B
~ ¡r
~ (mP i ¡ M Pe ) ¡ (M + m) P~ ei(10)
= en (m + M ) E
Using the de®nition (5) of current, and expressing the collision term (3) in terms of
resistivity, we have
à !
h³
´
³
´ i
n @ ~j
~ ~vi ¡ ~ve ¢ r
~ ~ve
Mm
+ nM m ~v i ¢ r
e @t n
~ ¡ e´ (M + m) n~j + en (m~vi + M ~v e ) £ B
~ ¡r
~ (mP i ¡ M Pe )
= e½E
where (using eqn 4)
m~vi + M ~v e
= M ~v i + m~v e + (m ¡ M ) (~vi ¡ ~ve )
½
m¡M ~
=
~v +
j
n
ne
To proceed, we make use of the fact that m ¿ M ; so that ½ ' nM; for example, and assume
Ti is not À Te :
à !
µ
¶
h³
´
³
´ i
m @ ~j
~ ~v i ¡ ~v e ¢ r
~ ~v e
~ ¡ e´½~j + en ½ ~v + m ¡ M ~j £ B
~ +r
~ (M P e)
½
+ ½m ~vi ¢ r
= e½E
e @t n
n
ne
³
´
~ + ~v £ B
~ ¡ e´½~j ¡ M ~j £ B
~ + M rP
~ e
= e½ E
1
Strictly we must also re-evaluate the pressure in terms of random motions about the MHD velocity ~
v . Pe and Pi
are evaluated with respect to the electron and ion mean velocities ~
ve and ~
vi : Of
course these three velocities are almost the same in the plasma approximation.
2
Divide out the density:
à !
h³
´
³
´ i
³
´
~
m @ ~j
~ ~vi ¡ ~ve ¢ r
~ ~ve = e E
~ + ~v £ B
~ ¡ e´~j ¡ 1 ~j £ B
~ + rPe
+ m ~vi ¢ r
e @t n
n
n
(11)
Now compare the magnitude of terms on the left with those on the right:
à ! µ
¶
m @ ~j
1~
~ » mB » 1
=
j£B
e @t n
n
e¿
! c¿
where ¿ is the time-scale over which physical properties such as ~j and n are changing.
This ratio is small provided that time scales of interest are long compared with the electron
cyclotron period.
m @
e @t
à !
³ ´
~j
= e´~j
n
»
1=2 n1=2 (" kT )3=2
m
m 16¼"20 (kTe )3=2
"1=2
0
e
0 m
»
»
e2 ´n¿
e2 n¿ Zi e2 m 1=2
e¿
e3 n3=2
»
n¸ 3D
ND
»
! p¿
! p¿
This ratio is small provided the time scales we are considering are long compared with N D
plasma oscillation periods.
(For future reference note that
m
ND
»
(12)
2
e ´ n¿
¿ !p
implies
"0 m ! p ¿
1
´» 2
=
)
(13)
e n¿ "0 N D
"0 ! p N D
The convective derivative term is of the same order as the time derivative term
h³
´
³
´ i
~ ~v i ¡ ~v e ¢ r
~ ~v e en¿ » vj en¿ = v ¿ » 1
~vi ¢ r
j
Len j
L
Thus we may neglect all terms on the left hand side in the slow (MHD) approximation,
which we may now state more speci®cally as:
ND
¿ À
!p
and
1
¿ À
!c
Then the current equation (10, 11) becomes:
³
´
~ + ~v £ B
~ = ´~j ¡ 1 ~j £ B
~ ¡ rP
~ e
E
(14)
en
This is Ohmzs law. There are several changes from the version we are used to seeing. The
~ term merely generalizes to the total electromagnetic force driving current. The
~v £ B
~
additional terms on the right side are the Hall current term ~j £ B=en
and a term that depends
3
on the electron pressure gradient. This is a diffusion term.
The Hall current term may be compared with the other terms in the equation. Remember
~ £ B=¹
~
that from the Maxwell equations, j » r
0 » B=L¹0 : Thus
Hall curent term
B
eB
m"0
c2
! c c2 1
»
=
= !c 2
=
2
~ term
enLv¹0
m ¹0 "0 e nLv
! p Lv
! p v2 ! p¿
~v £ B
and so the Hall current term may be neglected if time scales are suf®ciently long:
¿ À
! c c2
!2p v 2
(15)
This conditions is often satis®ed, especially in atstronomical applications where time scales
are very long and ! c =! p is small. .
Now letzs compare the grad P term with the resistivity term:
P
nkT e m L¹0
»
Len´j
Lm e2 n´ B
We already calculated the middle ratio (eqn 12), so:
2
2
nev2th,e ND
vth,e
P
ne2 m vth,e
ND
!p
»
=
=
ND
Len´j
"0 c2 B ! p
"0 m eB c2 !p
c2 ! c
This grad P term may be neglected when:
2
vth,e
!p
ND ¿ 1
c2 ! c
i.e. most of the time, since vth,e =c is usually small..
Finally we can even neglect the resistivity when
´j
´B
1 ´
=
=
¿1
vB
Lv¹0 B
Lv ¹0
and so the ratio
Lv¹0
RM =
(16)
´
must be very large. This number is called the magnetic Reynolds number.
Equivalently, using relation 13 and L = v¿ , this condition is
³ c ´2
´j
´B
1
c2
1
=
=
=
=
¿1
vB
Lv¹0 B
vL¹0 "0 ! p N D
vL! p N D
v
! p ¿ ND
³ c ´2 1
¿ À
(17)
v
!p N D
For example, in interstellar space, with ! p » 6 £ 10 4 rad/s and v » 10 km/s, we would
need
µ
¶2
3 £ 108 m/s
1
1
¿ À
= :0075 s
4
4
10 m/s
6 £ 10 /s 2 £ 106
which is always true.
Ohmzs law is often used to ®nd the plasma velocity.
4
2 Diffusion in fully ionized plasmas
Now we are ready to study diffusion in fully ionized plasmas. In a steady state, zero ~g
plasma, equation (7) becomes
~ = rP
~
~j £ B
(18)
Here we see that the magnetic force on the (usually small) plasma currents balances the
pressure gradient. From Ohmzs law (14), neglecting the terms in parentheses,
~ + ~v £ B
~ = ´~j
E
~ to solve for ~v :
We cross with B
³
´
~ £B
~ + ~v £ B
~ £B
~
E
~£B
~ ¡ ~v? B 2
E
=
~
´~j £ B
=
~
´ rP
and thus
~£B
~
~
E
´ rP
¡
B2
B2
~ £B
~ drift, while the second is the diffusion term. The particle -ux
The ®rst term is the E
due to diffusion is:
~
´k (T i + Te ) rn
F~ ? = n~v ? = ¡n
2
B
which is Fickzs law with a diffusion coef®cient
´ k (T i + Te )
D? = n
(19)
B2
The dependence on B (D / 1=B2 ) is the same as we found in partially ionized plasmas
(diffusion2 notes eqn 18). The temperature dependence is different, however. Since
´ / T ¡ 3=2 ; D / T ¡1=2 and decreases as T increases. In partially ionized plasmas
D / T 3=2 and increases as T increases. The difference arises from the temperature
dependence of the coulomb cross section. At high temperatures there are fewer collisons
and so less motion of the guiding centers. Another important difference is that this diffusion
coef®cient depends on the plasma density (because plasma particles are colliding with
themselves). Equation (19) describes so-called classical diffusion.
~ £B
~ drift.
There is no transverse mobility! Instead, we have the E
~v ? =
2.1
Solutions to the diffusion equation
2.1.1 Time dependence
We write D = 2nA where A = ´kT =B 2 and we have assumed Te = T i: Then the
continuity equation is
´
@n ~
@n ~ ³
~
+ r ¢ (n~v ) = 0 =
+ r ¢ ¡2nArn
@t
@t
or, if A is constant:
@n
= Ar2 n2
@t
5
a non-linear, partial differential equation for n: Letzs try to ®nd a solution by separating:
n (t; r) = T (t) R (r) : Then
1 dT
A
1
= r2 R2 = ¡
T 2 dt
R
¿
where as usual we have assumed that n is decreasing in time and thus chosen a negative
separation constant. We may solve the temporal part to get
1
1
t
=
+
T
T0
¿
T0
T =
1 + T 0 t=¿
so the density decreases as 1=t at long times.
2.1.2 Steady state solutions
With a source or sink we can ®nd steady state solutions:
¡Ar 2n2 = S
If the sink is recombination, then
¡Ar2 n2 = ¡®n2
and we can solve this linear equation for n2 : In 1-D, setting f = n2 ;
d2 f
®
= f
2
dx
A
with solution
µ r
¶
®
2
n = f = f 0 exp ¡
x
A
The plasma density decreases over a scale length
r
A
L»2
®
2.2
Bohm diffusion
Experimentally we often ®nd D / 1=B rather than the 1=B 2 dependence predicted by
classical diffusion. An empirical formula for this Bohm Diffusion is
kT
DB =
(20)
16eB
Since DB is independent of n it leads to an exponential decay in time, which is catastrophic
for containment.
Bohm diffusion corresponds to having a collision frequency of order the cyclotron
frequency. Recall (diffusion notes eqn 22)
mº
m eB
B
´= 2 »
=
2
ne
ne m
ne
6
where we put º = !c ,and thus
kT
kT
=
2
B
eB
~ £B
~ drift dominates the collisional diffusion.
The same dependence results if E
E
F »n
B
and
e©m ax » kT e
so
µ
¶
kT
kT n
F »n
»
» DB rn
eBL
eB L
The Bohm diffusion coef®cient is often much larger than the classical diffusion coef®cient.
The con®nement time may be estimated as
D » n´
L2
D
where L is a characteristic length scale. The con®nement time scales inversely with D:
Chen calculates values of D (classical and Bohm diffusion) for a 100 eV plasma in a 1
T ®eld, and shows that DB is four orders of magnitude larger than the classical diffusion
coef®cient, leading to con®nement times of the order of seconds rather than hours.
¿ »
3 Astrophysical MHD
3.1
Magnetic ®eld evolution
To make the MHD equations really useful, we need to include an equation governing the
!
¡
evolution of B : This comes from Maxwellzs equations. The Ampere-Maxwell law is
!
¡
¡
!
!
¡
!
¡
@E
r £ B = ¹0 j + ¹0 ² 0
(21)
@t
The last term is the displacement current term, and in low-frequency motions, we can
! ! ¡
¡
!
neglect this term. From Ohmzs law, if the resistivity ´ is small, then E ¼ ¡
v £ B and so :
³ ¡
´
!
¡ ¢
³ v ´2
¹0 ²0 @ E =@t
vB= ¿ c2
¼
¼
(22)
! ¡
¡
!
B=L
c
r£B
which we expect to be small in almost all situations.
The other Maxwell equation we need is Faradayzs law:
Now take the curl of Ohmzs law:
!
¡
¡ ¡
!
!
@B
r£E =¡
@t
7
(23)
"
á
á
! ¡
!!
! !#
¡ ¡
!
! ¡
! ³¡
!´ ¡
¡
! ³ ¡
!´ 1 !
¡
j £B
!
¡
rP e
!
r£E +r£ v £B =r£ ´ j +
r£
¡r £
e
n
n
~ and for ~j :
Substitute in for curl E
à ¡
!
¡
! ¡
!!
@B ¡
! ³¡
!´ ¡
¡
!
r£B
1 ¡
!
!
¡
!
¡
+r£ v £B =r£ ´
+ (Hall term) + 2 rn £ rPe (24)
@t
¹0
en
! ¡
¡
!
If T is constant, then rnk rPe and the last term is zero. Then if we neglect the Hall
! ¡
¡
!
term, and use the Maxwell equation r ¢ B = 0, the RHS of equation (24) becomes:
´ 2¡
!
r B:
¹0
Now letzs investigate the LHS. To see what it means, integrate over a surface S with
bounding curve C: Then:
Z
Z h
Z ³
³
@
! ¡
¡
!
!´i ¡
¡
!
@
!´ ¡
¡
!
!
¡
!
¡
~
B ¢ dA ¡
r £ v £ B ¢ dA =
©B ¡
v £B ¢d`
@t S
@t
S
C
The last term may be rewritten as :
Z
! ³¡
¡
!´
¡
B¢ !
v £d `
C
which we recognize as the rate at which -ux is swept up by the surface moving at velocity
!
¡
!
¡
v : Thus the LHS is just d B =dt; allowing for the effects of a moving frame. Thus this
!
¡
equation (24) is a diffusion equation for B ; with diffusion coef®cient ´=¹0 : The ®eld
diffuses in a characteristic time
¿ diff ¼ L2 ¹0 =´:
(25)
Thus if resistivity is small, the diffusion time becomes very long (the usual case in
astrophysics).
We may de®ne the magnetic Reynolds number:
v¿ diff
(26)
L
where L and v are characteristic length and velocity scales for the -ow.
Diffusion
dominates when RM < 1 (the usual lab situation) but convection of the ®eld with the -uid
dominates when RM > 1 (the usual astrophysical situation).
RM =
!
¡
What happens to the magnetic ®eld energy as B changes? Field lines are moving
through the plasma, induced currents -ow and we have ohmic heating. The current has
magnitude:
¯¡
! ¯¯
¯! ¡
¯r £ B ¯
B
j=
»
¹0
¹0 L
8
where L is a characteristic length scale for variations in the plasma. Thus energy is
dissipated at a rate:
´j 2 = ´
Thus the ®eld energy is dissipated in a time:
µ
B 2 =2¹0
B
¹0 L
¶2
1 ¹0 2
L ¼ ¿ diff
2 ´
´ (B=¹0 L)
This is also the timescale for ®eld to diffuse into a plasma.
¿ diss =
2
=
(27)
Letzs look at the highly conductive case (´ ¼ 0), neglecting the Hall current term. Then
eqn (24) becomes:
!
¡
@B ¡
! ³! ¡
!´
¡r£ ¡
v £B =0
!@t¡
¡
!
Expanding the curl, and using r ¢ B = 0;
!
¡
@ B ³¡
!´ ¡
¡
! ¡
! ³¡
! !´ ³¡
! ¡
!´ !
+ !
v ¢r B+ B r¢¡
v ¡ B¢r ¡
v =0
@t
We may combine the ®rst two terms, since the second is the convective derivative. We
! !
¡
use the continuity equation to eliminate r ¢ ¡
v in the third term, to get:
µ
¶ ³
!
¡
dB ¡
!
1 d½
! ¡
¡
!´ !
+B ¡
¡ B¢r ¡
v =0
dt
½ dt
Now we may combine the ®rst two terms:
á
!! ³
d
B
! ¡
¡
!´ !
½
¡ B¢r ¡
v =0
(28)
dt ½
!
Thus if the second term is zero ( ¡
v is constant along ®eld lines) then B _ ½: This is an
example of -ux freezing. We may imagine the ®eld lines being swept along with the -ow.
We may now insert the simpli®ed LHS back into equation 24, again neglecting the Hall
current term:
á
!! ³
d B
! ¡
¡
!´ !
´ 2¡
!
½
¡ B¢r ¡
v =
r B
(29)
dt ½
¹0
3.2
Magnetic pressure
We may now go further to eliminate the current density from the MHD equations. We use
Maxwellzs equations to write:
á
! ¡
!!
! ¡
¡
!
r£B
!
¡
1 h ³¡
! ¡
!´ ¡
! ¡
! i
j £B =
£B =
2 B ¢ r B ¡ rB 2
¹0
2¹0
Then the momentum equation (7) becomes:
9
µ
¶
!
d¡
v
1 ³!
¡ ¡
!´ ¡
! ¡
! B2
!
½
=
B¢r B¡r
+ p + ½¡
g
dt
¹0
2¹0
We may interpret the B 2 =2¹0 term as magnetic pressure, while the
³¡
! ¡
!´ ¡
!
B ¢ r B term
1
¹0
is due to ®eld line tension. Notice that the Alfven speed is then given by v2A ¼ P mag =½;
while the ordinary -uid sound speed is given by vs2 ¼ P gas =½:
!
¡
!
We now have a consistent set of MHD equations in ½; ¡
v ; and B :
µ
¶
!
d¡
v
1 ³¡
! ¡
!´ ¡
! ¡
! B2
!
½
=
B¢r B¡r
+ p + ½¡
g
(30)
dt
¹0
2¹0
!
¡
@ B ³¡
!´ ¡
¡
! ¡
! ³¡
! !´ ³¡
! ¡
!´ !
+ !
v ¢r B+ B r¢¡
v ¡ B¢r ¡
v =0
(31)
@t
³
d
!´
¡
!
½+ ¡
v ¢r ½=0
(32)
dt
An equation of state linking P and ½ completes the set.
3.3
Using the MHD equations: Alfven waves
!
! = 0 and ¡
We may linearize this set of equations in the usual way, with ¡
v
B0 = constant.
0
Eqn 32:
! ¡
¡
!
! !
¡
k ¢v
1
¡i!½1 + i k ¢ ¡
v1 ½ = 0 =) ½1 = ½
(33)
!
Eqn 31 :
³¡
³¡
¡
!
! !´ ¡!
! ¡!´ !
¡i! B1 + i k ¢ ¡
v1 B 0 ¡ i k ¢ B 0 ¡
v1 = 0
(34)
or:
á
!
Ã
!
! ¡
! ¡!
¡
! ¡!
¡
!
k ¢v
k ¢ B0 ¡
1
!
B1 =
B0 ¡
v1
(35)
!
!
Eqn 30:
Ã
¡
! ¡! !
¡
!
³¡
!
¡
B1 ¢ B0
! ¡
! ´ B1
¡
!
¡i!½v1 = ¡i k P 1 +
+ i k ¢ B0
(36)
¹0
¹0
From the equation of state:
P1 =
dP
½ = vs2 ½1
d½ 1
(37)
So then equation 36 becomes:
!
! = ¡i ¡
¡i!½¡
v
k
1
Use equation 33 to eliminate ½1 :
Ã
¡! ¡
!!
¡!
³¡
B
¢
B
! ¡
!´ B1
1
0
2
vs ½1 +
+ i k ¢ B0
¹0
¹0
10
(38)
¡ ¡
!
! ¡
!!
¡!
! ¡
³¡
k
¢
v
B
¢
B
! ¡
!´ B1
1
1
0
vs2 ½
+
+ i k ¢ B0
!
¹0
¹0
!
¡
Now dot equations 35 and 39 with k :
!
! = ¡i ¡
¡i!½¡
v
k
1
Ã
³¡
! !´
¡i!½ k ¢ ¡
v1 = ¡ik2
Ã
vs2 ½
¡ ¡
!
! ¡
!!
¡ ¡!
!
! ¡
³¡
k ¢v
B1 ¢ B0
! ¡
!´ k ¢ B1
1
+
+ i k ¢ B0
!
¹0
¹0
(39)
(40)
and
á
!
á
! ¡
! ¡
!!
´
k ¢!
v1 ³ ¡
! ¡
!´
k ¢ B 0 ³¡
! ¡
k ¢ B0 ¡
k ¢!
v1 = 0
(41)
!
!
~ is perpendicular to ~k: Then equation 40 simpli®es:
Thus the perturbation to B
Ã
! ¡
¡
! ¡
!!
! ¡
¡
¡
! ¡!
³¡
´
! ¡
!
! ¡
k2
B1 ¢ B0
k ¢v
1
!
2 k ¢ v1
2 B1 ¢ B0
k ¢ v1 =
vs ½
+
= k 2 vs2
+
k
!½
!
¹0
!2
!½¹0
Gathering up terms, we have:
¶
¡! ¡
!
³¡
´µ
! ¡
k 2 v2s
!
2 B1 ¢ B0
k ¢v
1
¡
=
k
(42)
1
!2
!½¹0
Now we can put this back into equation 39:
Ã
¡
! ¡! ¡! ¡
!! ³
¡!
2
!
¡
v
½
B
¢
B
B
¢
B
! ¡
¡
!´ B1
1
0
1
0
!
¡
s
2
¡!½v1 = ¡ k
k
+
+ k ¢ B0
! (1 ¡ k 2 vs2 =! 2 )
!½¹0
¹0
¹0
!
¡
¡! ¡
!
¡
!
k
B 1 ¢ B 0 ³¡
! ¡!´ B1
= ¡
+
k
¢
B
(43)
0
(1 ¡ k 2 vs2 =! 2 ) ¹0
¹0
!
¡
Now cross equations 43 and 35 with k :
¡ ¡!
!
k ¢ B1 =
¡
!
¡
!
³¡
³¡
! !´
! ¡
!´ k £ B1
¡i!½ k £ ¡
v1 = i k ¢ B0
¹0
and
á
!
Ã
! ¡
! ¡! ! ³
¡
! ³¡
´
! ¡
¡
!
k ¢v
! ¡!´
k ¢ B0
! ¡
¡
1
!
k £ B1 =
k £ B0 ¡
k £v
1
!
!
Combining (44) and (45), we have:
á
!
! ¡
! ¡!
¡
! ¡!
¡
k ¢!
v1 ³ ¡
! ¡!´ ³¡
! ¡!´ ³¡
! ¡!´ k £ B1
k £ B1 =
k £ B0 + k ¢ B0
k ¢ B0
!
! 2 ½¹0
and using equation 42
0
1
³¡
! ¡
! ´2
Ã
!
¡! ¡
!
³¡
´
³¡
k
¢
B
0
! ¡! B
! ¡!´
k2 B 1 ¢ B 0
C
k £ B1 @ 1 ¡
k £ B0 = 0
A¡
2
2
2
2
2
! ½¹0
! ½¹0 (1 ¡ k v s =! )
11
(44)
(45)
(46)
(47)
¡!
Now if we look at propagation along B0 ; the second term is zero. Since we have already
~ 1 = 0; ~k £ B
~ 1 cannot also be zero unless B
~ 1 is zero, and so we have:
found that ~k ¢ B
³¡
´
! ¡! 2
k ¢ B0
2
!2 =
= k 2 vA
(48)
½¹0
which is the dispersion relation for Alfven waves. As we found in our previous discussion
of Alfven waves, these waves are transverse waves, and we can liken them to waves on
strings.
¡!
¡!
For propagation across B0 ; dot equation 35 with B0
Ã
á
¡! ¡
!!
! ¡
!!
¡! ¡
!
k2
B1 ¢ B0
k ¢ B0 ³¡
! !´
2
B1 ¢ B0 =
B0 ¡
B0 ¢ ¡
v1
2
2
2
2
(1 ¡ k vs =! ) ! ½¹0
!
! ¡!
¡
Since k ¢ B0 = 0 in this case,
µ
¶
³¡
! ¡!´
k2
B20
B1 ¢ B0
1¡
=0
(1 ¡ k 2 vs2 =! 2 ) ! 2 ½¹0
¡
! ¡!
If B1 ¢ B0 6= 0;then:
µ
¶
!2
k2 v 2s
1¡
= v 2A
k2
!2
!2
2
= vA
+ v 2s
(49)
k2
These are magnetosonic waves. (cf Chen eqn 4-142, p 144, in the limit c À v A , and
plaswav notes eqn 65)
3.4
MHD power generators
The MHD power generator is used to convert thermal energy to electric energy. In this
device, a weakly ionized gas expands down a channel with a magnetic ®eld across it. Its
operation may be understood using Ohmzs Law (14):
³
´
~ = ´~j ¡ 1 ~j £ B
~ ¡ rP
~ e ¡E
~
~v £ B
en
and the equation of motion (7)
³
´
@~v
~ ~v = ½ d~v = ~j £ B
~ ¡ rP
~
½
+ ½ ~v ¢ r
@t
dt
where we have neglected gravity.
Initially the pressure gradient drives the plasma along the generator, perpendicular to
~ force generates current across the
the applied magnetic ®eld. As ~v increases, the ~v £ B
~
~
generator. The resulting j £ B force opposes the pressure gradient, and a steady state -ow
results. In a steady state, the momentum equation gives:
1 dv 2
d
½
=
(kinetic energy density) » j B
2 dz
dz
12
Thus the kinetic energy of the -ow is converted to electric current. To extract most of the
-ow energy, we want a channel of length L where
L»
But from Ohmzs law:
j»
1 ½v 2
2 jB
vB
´
and thus
1 ½v
L»
´
2 B2
The device is ef®cient because the losses are surface effects (friction, heat conducted out of
the region) while the energy conversion is a volume effect.
1 ~
~ is important to consider in the design of a good MHD generator.
The Hall term en
j£B
¯
¯
2
¯ 1
¯
¯ ~j £ B
~ ¯ = ´ jB = ´j B e n ¿
¯ en
¯
en´
en m
(equation 22 in diffusion2 notes, ¿ is the collision time).
¯
¯
¯ 1
¯
¯ ~j £ B
~ ¯ = ´j eB ¿ = ´ j! c ¿
¯ en
¯
m
Thus the Hall term in Ohmzs law is of order !c ¿ times the resistivity term. If ! c ¿ is not
too small, then the current -owing across the channel creates, through the Hall term, a Hall
~ H along the channel. The Hall electric ®eld drives a current ~j H = E
~ H =´
electric ®eld E
opposite ~v ; which in its turn causes a current opposite the original current. The magnitude
of this current may be estimated approximately as follows:
1
EH »
jB = ´j! c ¿
en
Thus
jH » j! c ¿
2
E H 2 » ´ (! c ¿ ) j
and therefore
2
j 0 = (! c ¿ ) j
When !c ¿ is not small, it is important to prevent the Hall current from -owing.
The gas used is usually air, CO 2 ; or argon with a small amount of easily ionized gas
to provide the ions. This might be 0.1-1% of an alkali metal vapor such as sodium or
potassium. The tempertature is 2-3000 K and the ®eld is around 1.4 T (similar to the ®eld
in a typical loudspeaker). The magnetic reynolds number R M is typically ¿ 1 in these
devices, which means that -ow dominates. Thus the physics is more astrophysical than it is
like a typical fusion experiment. Fusion devices tend to have high RM :
13