Fundamentals of Plasma Physics

40 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD

The second identity is obtained by dotting the equation of motion withuσand is

nσmσ

[

∂

∂t

(

u^2 σ

2

)

+uσ·

(

∇

(

u^2 σ

2

)

−uσ×∇×uσ

)]

= nσqσuσ·E−uσ·∇Pσ−Rσα·uσ

or

nσ

d

dt

(

mσu^2 σ

2

)

=nσqσuσ·E−uσ·∇Pσ−Rσα·uσ. (2.32)

Inserting Eqs. (2.31) and (2.32) in Eq.(2.30) gives the energy evolution equation

N

2

dPσ

dt

+

2 +N

2

P∇·uσ=−∇·Qσ+Rσα·uσ−

(

∂W

∂t

)

Eσα

. (2.33)

The first term on the right hand side represents the heatflux, the second term gives the
frictional heating of speciesσdue to frictional drag on speciesα, while the last term on
the right hand side gives the rate at which speciesσcollisionally transfers energy to other
species. Although Eq.(2.33) is complicated, two important limiting situations become ev-
ident if we lettbe the characteristic time scale for a given phenomenon andlbe its char-
acteristic length scale. A characteristic velocityVph∼l/tmay then be defined for the
phenomenon and so, replacing temporal derivatives byt−^1 and spatial derivatives byl−^1
in Eq.(2.33), it is seen that the two limiting situations are:

1. Isothermal limit – The heatflux term dominates all other terms in which case the
   temperature becomes spatially uniform. This occurs if (i)vTσ>>Vphsince the ratio
   of the left hand side terms to the heatflux term is∼Vph/vTσand (ii) the collisional
   terms are small enough to be ignored.


2. Adiabatic limit –The heatflux terms and the collisional terms are small enough to
   be ignored compared to the left hand side terms;this occurs whenVph >> vTσ.
   Adiabatic is a Greek word meaning ‘impassable’, and is used here to denote that no
   heat isflowing.
   Both of these limits make it possible to avoid solving forQσwhich involves the third
   moment and so both the adiabatic and isothermal limit provide a closure tothe moment
   equations.
   The energy equation may be greatly simplified in the adiabatic limit by re-arranging the
   continuity equation to give

∇·uσ=−

1

nσ

dnσ

dt

(2.34)

and then substituting this expression into the left hand side of Eq.(2.33) to obtain

1

Pσ

dPσ

dt

=

γ

nσ

dnσ

dt

(2.35)

where

γ=

N+ 2

N

. (2.36)

Equation (2.35) may be integrated to give the adiabatic equation of state

Pσ

nγσ

=constant; (2.37)

this can be considered a derivation of adiabaticity based on geometry and statistical me-
chanics rather than on thermodynamic arguments.