Plasmas as -uids
S.M.Lea
January 2007
So far we have considered a plasma as a set of non-intereacting particles, each following
its own path in the electric and magnetic ®elds. Now we want to consider what other views
of a plasma might be useful.
1 Plasma as a material medium
1.1
Permeability
In a magnetic medium, the individual particles of the medium (atoms, molecules etc)
contribute magnetic moments m
~ and the integral over all the individual magnetic moments
~ . This can be viewed as arising from a |bound current} ~jB
gives rise to a magnetization M
where
~ £M
~ = ~j B
r
Amperezs law takes the form
³
´
~
~ £B
~ = ¹0~j = ¹0 ~j f + ~j B + ¹0 "0 @E
r
@t
or equivalently
³
´
³
´
~
~ £B
~ =r
~ £¹ H
~ +M
~ = ¹ ~j f + ~jB + ¹ "0 @ E
r
0
0
0
@t
or
~
~ £H
~ = ~jf + "0 @ E
r
@t
~ = ÂmH
~ so that
In a LIH material, we assume that M
~ =H
~ (1 + Âm ) = ¹H
~
B
But in a plasma, the magetization is
~ =
M
X
~¹i
i
where ¹ is the magetic moment of a gyrating particle,
¹=
2
1 mv?
2 B
1
and thus we would have
1
B
Thus both  m and ¹m would have a nasty dependence on B: Thus the plasma is not a linear
material (nor is it isotropic) and thus it not useful to view a plasma as a magnetic material.
M /
1.2
Dielectric properties
~ and E
~ for a material is a frequency dependent relation.
In general the relation between D
The materialzs response depends on the time-dependence of the applied electric ®eld. This
is true for plasmas too,
First wezll look at the |almost static limit} (! ¿ ! c ). The polarization is the sum of all
the electric dipoles in a medium:
X
P~ =
~p i
i
and the polarization gives rise to a |bound charge density}
~ ¢ P~
½B = ¡ r
Gausszs law is
Now let
½ + ½B
~ ¢E
~ = ½ = f
r
"0
"0
³
´
~ = "0 E
~ + P~
D
³
´
~ ¢D
~ =r
~ ¢ "0 E
~ + P~ = ½f
r
Now again in an LIH material, we expect
~
P~ = Â E
so that
e
so that
~ = "0 (1 + Â ) E
~ = "E
~
D
e
In a plasma we have seen that a time-dependent applied ®eld (with ! ¿ ! c ) gives rise to a
polarization current: ("Motion" notes equation 17)
~
~jp = ½ @E
2
B @t
~
which will affect B through Amperezs law:
Ã
!
~
~
½ @E
@E
~
~
~
r £ B = ¹0 j f + 2
+ ¹0 "0
B @t
@t
Thus this bound current can be combined with the displacement current to give:
Ã
!
~
~
½
@
E
@
E
~ £B
~ = ¹0~jf + ¹0
r
+ "0
B2 @t
@t
Ã
!
~
@E
= ¹0 ~jf + "
@t
2
with dielectric constant
½
(1)
B2
~ it appears that this could be a useful description of a plasma.
Since " is independent of E;
Convince yourself that charge conservation allows us to write ½B in terms of the
polarization current, and hence we also get the correct form for Gaussz law as well.
When we study plasma waves we will be able to ®nd additional expressions for " valid
in different frequency regimes.
" = "0 +
2 The -uid picture
When collisions between the plasma particles become important, we can no longer
regard them as independent particles, but instead we ®nd it useful to look at small volumes
containing many particles and consider average properties of the particles in each small
volumew this is the -uid point of view. When dealing with a plasma, we must recognize
that particles can in-uence each other through the long-range electromagnetic forces in the
system, and so we must regard the term |collision} rather generally.
The -uid properties of interest are: the density ½; pressure P; and velocity ~v : The
equations governing the motion of a -uid are:
1. Conservation of mass
The mass within an arbitrary, ®xed volume of the plasma can change only by -ow of
plasma into or out of that volume:
Z
Z
Z
d
~ ¢ (½~v ) dV
½dV = ¡
½~v ¢ ^
n dA = ¡
r
dt V
S
V
and since this must be true for an arbitrary volume V; we have the differential equation:
@½ ~
+ r ¢ (½~v ) = 0
(2)
@t
Equation (2) is called the continuity equation.
2Momentum equation
This equation is just Newtonzs second law applied to the plasma. The relevant forces
are the Lorentz force, and pressure forces. (We can add gravity when appropriate.) The
acceleration of a mass element dm = ½ dV is given by
³
´ Z
d~v
~ + ~v £ B
~ ¡
dm
= dq E
Pn
^ dA
dt
S
where the last term is an integral over the differential volume, and represents the effect of
neighboring -uid on our element. (Recall that pressure is the normal force per unit area.)
The charge dq = nedV : Thus, applying the divergence theorem to the last term, we have
Z
³
´
d~v
~ + ~v £ B
~ dV ¡ rP
~ dV
½ dV = ne E
dt
3
and so we obtain the differential equation:
³
´
d~v
~ + ~v £ B
~ ¡ rP
~
½
= ne E
(3)
dt
Because the charge of the particles appears in this equation, we will need one equation for
the ions and a different equation for the electrons.
3Energy equation
The equations involve the ®ve variables ½; P and ~v ; but with one vector and one scalar
equation (for a total of 4) we are one short of a complete set. The third equation we need is
an energy equation. Often we can avoid a full energy equation by using an equation of state,
that is, a known relation between the pressure P and the density ½: Two common choices
are:
The ideal gas law with a constant temperature T (isothermal plasma)
P = nkT
(4)
or the adiabatic equation of state
P / ½°
(5)
where ° = (2 + N ) =N and N is the number of degrees of freedom of our system. For an
unmagnetized gas of ions and electrons, N = 3 and ° = 5=3:
2.1
Formal derivation of the -uid equations
2.1.1 More about the distribution function
An important link in the jump from considering individual particles to viewing the plasma as
a -uid is the distribution function f (~r; ~v ; t). It tells us how many particles are where, going
how fast, in which direction.
number of particles with position vectors ~r to ~r +d~r and velocities in range ~v to ~v +d~v is f (~r ; ~v; t) d3 ~r d3~v
For many purposes the distribution function gives more information than we need. Thus we
obtain desired quantities as averages. For example, we may obtain the density by summing
up over all possible velocities:
Z
n (~r; t) = f (~r; ~v ; t) d3~v
(6)
where the integral is over the complete velocity space, and the average velocity of the
particles at position ~r is
R
Z
~v f (~r ; ~v ; t) d3~v
1
~u (~r; t) = R
=
~v f (~r ; ~v ; t) d3~v
(7)
f (~r; ~v ; t) d3 ~v
n
Sometimes the distribution function is normalized by dividing out the density:
1
f^ (~r ; ~v) = f (~r; ~v )
n
4
so that
Z
f^ (~r ; ~v ) d3 ~v = 1
We have already seen one example of a distribution function: the Maxwellian:
µ
¶
³ m ´3=2
mv 2
f (~r ; ~v) = n (~r)
exp ¡
2¼kT
2kT
(8)
where here the dependence on the space and velocity variables is separable. This is an
isotropic velocity distribution, since f does not depend on the direction of the vector ~v but
only on its length. Particles in a magnetic mirror may have a loss-cone distribution, which
~ This
has a de®ciency of particles with velocity vectors pointing along the direction of B:
is an anisotropic distribution. (See Figure 7.7, pg 232). Additional examples are shown on
pages 230-232 of the text.
2.1.2 The Boltzmann and Vlasov equations
The Boltzmann equation is a mathematical statement of the fact that particles cannot
disappear. The number of particles in a region of space can change only if they move
somewhere else. The number of particles in a given region of velocity space can change
only if (a) they move to a new velocity (they are accelerated) or (b) a collision knocks them
into a new region of velocity space. This physical principle is stated mathematically as
µ ¶
@f
@
df
~
= ¡~v ¢ rf ¡ ~a ¢
f+
@t
@~v
dt due to collisions
where
~ = dx @f + dy @f + dz @f
~v ¢ rf
dt @x
dt @y
dt @z
@
and similarly for ~a ¢ @~
f; or equivalently
v
µ ¶
df
df
=
dt
dt due to collisions
where the total time derivative is
df (~r; ~v ; t)
@f
~ + ~a ¢ @ f
=
+ ~v ¢ rf
dt
@t
@~v
and we have written the shorthand expression
@
~v
´r
@~v
as the gradient in the velocity space. Thus Boltzmannzs equation says that the distribution
function can be changed only by collisions.
Next we replace ~a with the force acting:
µ ¶
@f
F~
@
df
~
+ ~v ¢ rf +
¢
f =
@t
m @~v
dt due to collisions
5
In a plasma, the relevant force is the Lorentz force:
µ ¶
³
´
@f
~ + q E
~ + ~v £ B ¢ @ f = df
+ ~v ¢ rf
@t
m
@~v
dt due to collisions
(9)
and in this form, equation (9) is called the Vlasov equation. Setting the right side to zero,
we obtain the collisionless Vlasov equation. This equation is very powerful in predicting
the plasma behavior w it contains a lot of information, and we will look at some of its
conseqeunces later. But often we donzt need that much power, so we average over the
velocity distribution.
2.1.3 The zeroth moment
To obtain the nth moment, we multiply equation (9) by ~v n and integrate over the velocity
space. We start with n = 0 :
Z
Z
Z ³
Z µ ¶
´ @
@f
q
df
~
~
dV + ~v ¢ rf dV +
E + ~v £ B ¢
f dV =
dV
@t
m
@~v
dt due to collisions
Immediately we see that the right side is zero, since every particle knocked out of a volume
element dV by a collision must be knocked into another, and both dV s are in the integrated
volume.
On the left side, the ®rst term is:
Z
@
@n
f dV =
@t
@t
where we used equation (6). To evaluate the second term, note that ~r and ~v are independent
variables. We are integrating over ~v but differentiating with respect to ~r: Thus, using
equation (7):
Z
Z h
Z
i
~
~
~
~
~ ¢ (n~u )
~v ¢ rf dV =
r ¢ (~v f ) ¡ f r ¢ ~v dV = r ¢ ~v f dV ¡ 0 = r
To evaluate the third term we use the divergence theorem in the velocity space,
Z
Z
Z
@
~ ¢ @ f dV = E
~ ¢
~¢
E
f dV = E
n
^ v f dAv
@~v
@~v
S1
The surface integral is at in®nity in the velocity space, and f ! 0 there. In fact
R f must go to
zero at least as fast as 1=v 4 to ensure that the total plasma energy is ®nite. ( v 2 f v 2 dvd-v
is ®nite.) Thus the third term is zero.
The fourth term is:
¾
Z
Z ½
@
@
@
(~v £ B) ¢
f dV =
¢ [(~v £ B) f ] ¡ f
¢ (~v £ B) dV
@~v
@~v
@~v
Z
=
f (~v £ B) ¢ n
^ dAv
~ contains only vy and v z ;
¢ (~v £ B) is identically zero, since the ®rst component of ~v £ B
and not v x; and similarly for the other terms. The surface integral is zero because f ! 0
@
@~v
6
suf®ciently fast at in®nity. Thus we have
@n ~
+ r ¢ (n~u ) = 0
@t
and we have derived equation (2).
(10)
2.1.4 The ®rst moment.
Now we multiply the Vlasov equation (9) by m~v before integrating:
¸
µ ¶
Z
Z ³
Z ·³
Z
´
´
@f
~ f dV +q ~v E
~ + ~v £ B ¢ @ f dV = m~v df
m ~v
dV +m ~v ~v ¢ r
dV
@t
@~v
dt due to collisions
The term on the right is the total change of momentum due to collisions, which is zero,
provided that we are integrating over all the colliding particles. We can include more than
one species of particle by writing this term as P ab = momentum transferred to species a by
species b:
Now for the left side. The ®rst term is:
Z
@
@
m
~v f dV = m (n~u )
@t
@t
The second term is hardest, so letzs leave it aside for the moment. The third term is, in index
notation:
½ ·
µ ·
Z ½
³
´ ¸ ¾
³
´ ¸¶¾
@
@
~
~
v i E j + ~v £ B
f ¡f
vi E j + ~v £ B
dV
@v j
j
@vj
j
We convert the ®rst term to a surface integral, as before, and it vanishes, leaving
Z ½
·
·
³
´ ¸
³
´ ¸¾
@v i
@
~
~
¡
f
E j + ~v £ B
+ f vi
E j + ~v £ B
dV
@vj
@vj
j
j
½
·
¸
¾
Z
³
´
~
= ¡
f ± ij E j + ~v £ B
+ 0 dV
j
³
´
~
where Ej is independent of vj and ~v £ B
contains the two components of ~v other than
j
vj : Finally we have
¡
Z
h
³
´i
h
³
´i
~
~
f E i + ~v £ B
dV = ¡n E i + ~u £ B
i
i
Now we tackle the second term, again using index notation:
Z
Z
@f
@
v ivj
dV =
vi vj f dV
@x j
@x j
where the integral is over the velocity space. Now we write each v i in terms of the average
-uid velocity ~u (which is a function of ~r but not of ~v ) and the difference w
~ between ~v and ~u :
vi = u i + wi
so that
v ivj = ui uj + u iw j + uj wi + wi wj
7
and
Z
@
viv j f dV
@x j
Z
@
=
(u iu j + ui wj + u j w i + w iw j ) f dV
@xj
· Z
¸
Z
Z
@
@
@
=
(ui uj n) +
ui wj f dV + uj
wi f dV +
wi wj f dV
@xj
@xj
@x j
¡R
R
¢
The second term is zero by de®nition of w
~ : f wj dV = f (vj ¡ u j ) dV = nu j ¡ nu j = 0
We express the last term using the stress tensor
Z
Pij ´ m wi wj f dV
so that
m
Z
vi vj
@f
dV
@x j
=
=
@
@
(ui u j n) +
Pij
@xj
@xj
@
@
@
mu i
(u j n) + mnu j
ui +
Pij
@xj
@xj
@x j
m
Thus the ®rst moment equation is:
³
´
³
´
³
´
@
~ ¢ n~u + mn ~u ¢ r
~ ~u + mr
~ Pe ¡ qn E
~ + ~u £ B
~ = Pab
m (n~u) + m~u r
@t
We can simplify this using our ®rst relation (10):
µ
¶
µ
¶
³
´
³
´
@n
@~u
@n
~ ~u + r
~ Pe ¡ qn E
~ + ~u £ B
~
m ~u
+n
+ m~u ¡
+ mn ~u ¢ r
=
@t
@t
@t
·
¸
³
´
@~u ³ ~ ´
~ Pe ¡ qn E
~ + ~u £ B
~
mn
+ ~u ¢ r ~u + r
=
@t
or
³
´
d~u
~ + ~u £ B
~ ¡r
~ Pe + Pab
mn
= qn E
dt
where the total time derivative
d~u
@~u ³ ~ ´
=
+ ~u ¢ r ~u;
dt
@t
the second term being the convective derivative.
When the velocity distribution is isotropic, Pij is diagonal:
Z
f wi wj d3 ~v = P ±ij
and the temperature is de®ned by
3nkT = m
Z
wiw if d3~v = mn < w 2 >
Then the rms velocity about the mean is
p
3kT
< w2 > =
m
8
P ab
P ab
(11)
(12)
a familiar result. Then we write
P ij
0
P
=@ 0
0
0
P
0
1
0
0 A
P
where P = nkT : This reduces our equation to the form:
³
´
d~u
~ + ~u £ B
~ ¡ rP
~ + Pab
mn
= qn E
(13)
dt
and we have derived equation (3).
The second moment gives the energy equation. We wonzt need it, so we wonzt derive
it here. (See http://www.physics.sfsu.edu/~lea/courses/grad/-uids.PDF page 2 if you are
interested.)
3 Fluids, plasmas and distribution functions
Wezll be using equations (10) and (11) extensively, so letzs review how we got them.
We needed to assume some properties of the distribution function, speci®cally that
f (~v ) ! 0 as v ! 1 at least as fast as v 4 : This is necessary to ensure that the total energy
is ®nite. The Maxwellian distribution satis®es this, since it goes to zero exponentially, i.e.
faster than any power.
The Maxwellian distribution is the solution to the equation:
µ ¶
df
=0
dt due to collisions
i.e. a Boltzmann equation where the right side dominates. That means that collisions are
very important.
How important are collisions in plasmas? The mean free path (|mfp") between
collisions, computed using the Coulomb force (see e.g. Spitzerzs lovely book |Physics of
fully ionized gases}) can often be very large, larger than the length scale of the plasma
system. This is especially true in Astrophysics. Yet we see phenomena that are intrinsically
-uid phenomena, like shock waves. Satellites in far Earth orbit have obtained very clear
evidence of a shock wave where the solar wind meets the Earthzs magnetic ®eld, for
example. In a plasma the mfp is restricted because the charged particles are forced to gyrate
~ and this usually makes them good -uids. Additional collisional-type interactions
around B;
occur when the plasma particles interact with plasma waves. Thus we have good reason to
believe that we can successfully apply the -uid equations to most plasmas.
4 Using the equations to compute plasma drifts
We will begin by using the plasma -uid equations to compute drifts under the
assumptions:
1. The drifts are slow. Wezll make this assumption quantitative in a minute, and
9
2. The drifts are constant in time.
These assumptions are consistent with the kind of behavior wezve come to expect from
the individual particle motions.
First letzs look at what we mean by |slow}. The momentum equation, (3) is
µ
¶
³
´
d~u
@~u ³ ~ ´
~ + ~u £ B
~ ¡ rP
~
½
=½
+ ~u ¢ r ~u = nq E
dt
@t
and if the drift is time-independent, the partial derivative with respect to time vanishes,
leaving
³
´
³
´
~ ~u = nq E
~ + ~u £ B
~ ¡ rP
~
½ ~u ¢ r
³
´
~ ~u
~u ¢ r
´ ~
q ³~
~ ¡ rP
E + ~u £ B
m
½
With L being a length scale for our system, the order of the terms is
=
LHS
v2
L
RHS, ®rst term, second term ! c
and
µ
E
;v
B
¶
c2
RHS, third term s
L
where cs is the sound speed in the -uid (see below). Thus the LHS is much less than the
third term on the right provided the drift is highly subsonic, v ¿ cs: The LHS is much less
than the second term on the right provided that the drift speed is much less than the particlezs
orbital speed times L=rL: This suggests that we can safely neglect the quadratic term in v
on the left side. Then in a steady state we have
´ ~
q ³~
~ ¡ rP
0=
E + ~u £ B
m
½
~
~ to get
where only components of ~u perpendicular to B contribute. Now dot with B
0=
~
q ~ ~
~ ¢ rP
E¢B ¡B
m
½
(14)
~ to get
and cross with B
0
=
0
=
³
´
´ ~
q ³~
~ + ~u £ B
~ £B
~ ¡ rP £ B
~
E£B
m
½
´ ~
q ³~
~ ¡ ~uB 2 ¡ rP £ B
~
E£B
m
½
~ = 0 since only perpendicular components of ~u appear in this relation.
where we took ~u ¢ B
10
Next solve for ~u :
~ £B
~
~
~
E
m rP
B
¡
£ 2
2
B
q ½
B
~ £B
~ drift. The second term is new. It is called the
The ®rst term is our old friend the E
diamagnetic drift.
~
~
rP
B
~v diama gnetic = ¡
£ 2
(15)
nq
B
or, for an isothermal plasma:
~
~
kT rn
B
~vdiam agnetic = ¡
£ 2
q n
B
Since the charge appears explicitly, ions and electrons go in opposite directions, and so there
is a diamagnetic current:
~
~ £ B
~jD = ne (~v i ¡ ~v e) = ¡k (T i + Te ) rn
B2
~
B
~
= k (T i + Te ) 2 £ rn
(16)
B
In the -uid picture we have lost a lot of detail. We did not get any of the |®nite Larmor
radius effects}, but we also get something new. The diamagnetic drift is a purely -uid
phenomena that arises because in a given region, more particles move in one direction than
the opposite direction if there are gradients in the density, as shown in the picture below.
The drift is independent of the particlezs mass, because the particlezs thermal speed does
depend on m (v / m ¡1=2 ) but the Larmor radius is proportional to m +1=2 ; so the slower
particle samples more of the density gradient. The two effects exactly cancel.
~u =
11
Now letzs go back and see what we can learn from the parallel component of the
momentum equation (14). Again letzs specialize to an isothermal plasma. Then:
Ã
!
~
~
q ~ ~
rP
q ~
kT rn
~
~
0 =
E ¢B ¡B ¢
= B ¢ ¡ rÁ ¡
m
½
m
m n
~
B
~ (qÁ+kT ln n)
¢r
m
and thus, for electrons with q = ¡e,
= ¡
eÁ¡kT ln n = constant along a ®eld line
or, taking the exponential of both sides:
n = n0 exp
µ
eÁ
kT
¶
(17)
which is the Boltzmann relation.
Remember that we obtained equation (14) by neglecting the acceleration in the
momentum equation. Because of the mass dependence on the right, the acceleration of the
electrons is much larger than the acceleration of the ions. The electrons move rapidly until
equation (17) is satis®ed. The ions do not have time to move.
5 The plasma approximation
The plasma is quasi-neutral: that means that the electron and ion densities are almost
equal everywhere in the plasma (assuming the ions are protons, or, at least, singly ionized).
When an electric ®eld exists in the plasma, the electrons move rapidly to neutralize the
®eld. The ions follow, more slowly. Generally, for low frequency motions, we do not use
~ : Rather, we use the equation of motion to get E
~ ; and then use
Poissonzs equation to get E
Poissonzs equation to compute the small difference between ni and ne: Our derivation of
the Boltzmann relation (17) is an example of this procedure.
~ 6= 0 is called the plasma
The assumption that ne ¼ ni (and jne ¡ ni j ¿ ne ) with E
approximation. It is valid for frequencies that are low in a sense that we shall describe more
precisely later.
12