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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 Continuity of mass or + # $(nu) = 0 "t charge transport
%" u ( Momentum mn' +(u #$)u* = qn(E + u + B), $ #P+ Pij & "t ! ) Transport
" # 1 2& # 1 2 & m 2, "f / Energy %n mu ( + ) *%n m u u ( + nq E *u = 2 u . 1 du !" t $ 2 ' $ 2 ' 2 - "t 0coll 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
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
! C-$/A+,-./-*B%&;,* 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 ' $ ' # pzx 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), $p + Pij &' "t )* !
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C-$/A+,-./-*B%&;,* 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 p n-! = constant and = # p n where ! is the ratio of the specific heats at constant pressure.
! D2/=,2E;&*F";$G1*FH'-)%"* o Note, the energy equation can be written
2 "[1/2nm < w >] 2 + # $ 1/2nm < w > 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.
! o As average energy of plasma is 1/2m
"(3/2 p) + # $(3/2 pu) % p# $u + # $q = Pij "t o The quantity 3/2pu represents the flow of energy density at the fluid velocity.
!
@%/=,;#;*.;#*%<*#I%AJ'2&*;H'-)%".* o Consider plasma of two species; ions and electrons, in which fluid is fully ionised, isotropic and collisionless. The charge and current densities are
" = niqi + neqe
j = niqivi + neqeve o Using v = u, complete set of electrodynamics equations are then (j = i or e)
%" v j ( ! m jnj' +(v j #$)v j * = +$pj + qjnj (E + v j , B) & "t )
"nj + $ #(njv j ) = 0 "t
-0$ #E = . "B $ , E = + "t $ #B = 0 1 "E $ , B = j+-0 µ0 "t / j pj = Cjnj
! >,'2&.*&$2K.*=;$=;"&2(',-$*#%*L* o Since a fluid element is composed of mane individual particles, expect drifts perpendicular to B. But, the grad (p) term results in a fluid drift called diamagnetic drift. o Consider momentum equation for each species: %" v ( mn +(v #$)v = +$p + qn(E + v , B) &' "t )*
(1) (2) (3) o Consider ratio !of terms (1) to (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 v = $ " B2 qnB2
= v E + vD
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E " B o In previous equation is v = E ! B drift E B2 #p $ B and v = " is diamagnetic drift. D qnB2 ! 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).
M2-/-G";)(*M$2K*2"*NA/-(02";.* o Diamagnetic drift first measured in Q-machines
o See http://www.physics.uiowa.edu/xplasma/Qmachine.html >,'2&*&$2K.*=-$-,,;,*#%*L* o Consider z component of fluid equation of motion:
%" vz ( "p mn +(v #$)vz = +qnEz + &' "t )* "z o The convective term can be neglected as is is much smaller than "vz /"t
! "p "n o Using p = n k T or = kBT we can write B "z "z !
"vz q $kBT "n = Ez # ! "t m mn "z o This shows that the fluid is accelerated along B under the combined electrostatic and pressure gradient forces. !
>,'2&*&$2K.*=-$-,,;,*#%*L* o Taking the limit as m -> 0 and q = -e and E z = " #$ / dz we have
"# $k T "n eE = e = B z 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).