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Dr. Peter T. Gallagher
Astrophysics Research Group
Trinity College Dublin
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o  Fluid approach describes bulk properties of plasma. We do not attempt to solve
unique trajectories of all particles in a plasma. This simplification works very well
for majority of plasmas, despite gross simplifications made.
o  Fluid theory follows directly from moments of the Boltzmann equation (Lecture 7).
o  Each of the moments of the Boltzmann equation is a transport equation describing
the dynamics of a quantity associated with a given power of v.
"n
+ # $(nu) = 0
"t
Continuity of mass or
charge transport
%"u
(
mn' + (u # $)u* = qn(E + u + B) , $ #P + Pij
& "t !
)
# 1
" # 1 2&
n mu ( + ) * %n m u 2u
%
'
$ 2
!"t $ 2
Momentum
Transport
&
m 2 , "f /
(' + nq E * u = 2 2 u .- "t 10 du
coll
Energy
Transport
!
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o  Simplest set of macroscopic equations can be obtained by simplifying the
momentum transfer equation and neglect thermal motions of particles.
o  Here, set kinetic pressure tensor to zero, i.e., P = mn <ww> = 0 as w = 0.
o  Remaining macroscopic variables are then n and u, described by
"n
+ # $(nu) = 0
"t
%"u
(
mn' + (u # $)u* = qn(E + u + B) + Pij
& "t
)
!
o  Collision term Pij can be approximated by by an “effective” collision frequency.
!
o  Assumed that collisions cause a rate of decrease in momentum:
Pij = "mnveff u
!
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o  Alternative set of macroscopic equation is obtained by truncating energy
conservation equation.
" pxx pxy pxz %
$
'
o  Consider pressure tensor: P = mn ww = $ pyx pyy pyz '
$p
'
# zx pzy pzz &
o  Components represent transport of momentum. Diagonal elements represent
pressure, while off-diagonal
represent shearing stresses.
!
o  In warm-plasma model, only consider diagonal pressure elements, so
" #P = "p
o  That is, viscous forces are neglected. We then have
"n
+ # $(nu) = 0
"t
%"u
(
mn' + (u # $)u* = qn(E + u + B) , $p + Pij
& "t
)
!
!
!
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o  The previous system of equations does not form a closed set, since scalar pressure
is now a third variable. Usually determined by energy equation
o  If plasma is isothermal, assume equation of state of form:
p = nkBT
and
"p = kBT"n
o  Holds for slow time variations, allowing temperatures to reach equilibrium.
!
o  If plasma does not exchange energy with its surrounds, assume it is adiabatic:
p n-! = constant
and
"p
"n
=#
p
n
where ! is the ratio of the specific heats at constant pressure.
!
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o  Note, the energy equation can be written
"[1/2nm < w2 >]
+ # $ 1/2nm < w2 > u + (P $ #)u + # $ q = Pij
"t
(
)
where q is the heat flow vector. For electrons, commonly used approximation for q is
!
q = K"T
where K is the thermal Spitzer conductivity.
! is 1/2m <ww> = 3/2 k T and using p = n k T
o  As average energy of plasma
B
B
=> 3/2 p = 1/2nm<ww>. Energy equation can then be written
"(3/2 p)
+ # $ ( 3/2 pu) % p# $ u + # $ q = Pij
"t
o  The quantity 3/2pu represents the flow of energy density at the fluid velocity.
!
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o  Consider plasma of two species; ions and electrons, in which fluid is fully ionised,
isotropic and collisionless. The charge and current densities are
" = ni qi + ne qe
j = ni qi v i + ne qe v e
o  Using v = u, complete set of electrodynamics equations are then (j = i or e)
%"v
(
! m j n j ' j + (v j # $)v j * = +$p j + q j n j (E + v j , B)
& "t
)
"n j
+ $ #(n j v j ) = 0
"t
-0 $ # E = .
"B
$,E = +
"t
$ #B = 0
1
"E
$ , B = j+ -0
µ0
"t
/
pj = Cjnj j
!
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o  Since a fluid element is composed of mane individual particles, expect drifts
perpendicular to B. But, the grad (p) term results in a fluid drift called diamagnetic
drift.
o  Consider momentum equation for each species:
%"v
(
mn' + (v # $)v* = +$p + qn(E + v , B)
& "t
)
(1)
o  Consider ratio !
of terms (1) to (3):
(2)
(3)
(1) mni#v$ #
"
"
(3)
qnv$B
#c
o  Here we have used " /"t = i# . If only consider slow drifts compared to time-scale
of the gyrofrequency, can set (1) to zero.
!
!
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o  Therefore can write
0 " #qn(E + v$ % B) # &p
where v " B = (v# + v || ) " B = v# " B
!
o  Taking the cross-product
!
0 = qn[E " B + (v# " B) " B]$ %p " B
o  Using the identity (A " B)" C = (C # A)B $ (C # B)A we can write
!
0 = qn[E " B + (v# $ B)B % (B $ B)v#]% &p " B
o  As v⊥ is !
perpendicular to B, v⊥. B = 0. Therefore
!
E # B %p # B
$
B2
qnB2
= v E + vD
v" =
!
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o  In previous equation is
and
vD = "
#p $ B
qnB2
!
vE =
E"B
B2
E ! B drift
is diamagnetic drift.
o  The vE drift is same as for guiding centres, but there is now a new drift, called the
! diamagnetic drift. Is in opposite directions for ions and electrons.
o  Gives currents in plasma that reduce magnetic field in plasma. More ions moving to
left in shaded area that to the right (Inan & Golkowski, Page 111).
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o  Diamagnetic drift first measured in Q-machines
o  See http://www.physics.uiowa.edu/xplasma/Qmachine.html
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o  Consider z component of fluid equation of motion:
%"v
(
"p
mn' z + (v # $)vz * = +qnEz +
& "t
)
"z
o  The convective term can be neglected as is is much smaller than "vz /"t
"n
! "p
= k BT
o  Using p = n kB T or
we can write
"z
"z
!
"vz q
$k T "n
= Ez # B
"t m
mn "z
!
o  This shows that the fluid is accelerated along B under the combined electrostatic
and pressure gradient forces.
!
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o  Taking the limit as m -> 0 and q = -e and Ez = "#$ /dz we have
"# $k T "n
eEz = e = B
dz
n dz
!
o  Electrons are so mobile that their heat conductivity is almost infinite.
!
o  Assuming isothermal electrons and taking ! = 1, we can integrate to get
e" = kBT ln(n) + C
o  We can therefore write
n = n0 exp(e" / kBT )
!
o  This is called the Boltzmann relation or Boltzmann factor for electrons.
!
o  Implies that electrons have a tendency to move rapidly in response to and external
force (i.e., electrostatic potential gradient).