Geoscience Reference
In-Depth Information
Let us assume that the number concentration of the particles of each sort
is conserved in collisions. Then
∂f α
∂t
d V =0 .
(1.16)
coll
Substitution Q ( α ) ( V ) = 1 into (1.15) yields the equation of conservation of
the particle numbers
∂N α
∂t
·
( N α v ( α ) )=0 ,
+
(1.17)
and Q ( α ) ( V )= m α V leads to the momentum equation
m α N α ∂v ( α ) j
∂t
) v ( α ) j = N α F ( α ) j
∂x k P ( α ) jk + R ( α ) j .
+( v ( α )
(1.18)
Here
F ( α ) = Z α e E + 1
B ] + m α g ,
c [ v
×
where g is the acceleration in non-electromagnetic fields (e.g., accelera-
tion of gravity), for neutrals Z α =0. P ( α ) jk is the pressure tensor with
elements
P ( α ) jk = P ( α ) kj = m α N α
( V j
v ( α ) j )( V k
v ( α ) k )
.
(1.19)
The last term in the right-hand side of (1.18) is the frictional drag
R ( α ) = m α V
v ( α ) ∂f α
∂t
d V .
(1.20)
coll
Equation (1.18) is the equation of the momentum conservation and it is
an analogue of the corresponding hydrodynamic equation. An equation simi-
lar to the hydrodynamic energy conservation law can be obtained by taking
Q ( α ) ( V )= m α V 2 / 2 in (1.15). These equations become completed just with
additional assumptions about the pressure tensor P ( α ) jk and the friction force
R ( α ) . Strictly speaking, they should be obtained from the initial kinetic equa-
tions [2].
In the simplest approximation, pressure tensors are supposed to be isotropic
P ( α ) jk = P α δ jk , where P α is pressure (scalar), δ jk is the unit tensor ( δ jk is the
Kronecker symbol ). A more common approximation is often used in plasma
applications with diagonal tensors P ( α ) jk but non-equal components P ( α ) xx ,
P ( α ) yy , P ( α ) zz . Finally, it is possible to take into account viscous stresses for
the pressure tensor. Then P ( α ) jk = P α δ jk
τ jk , where τ jk is a tensor of vis-
cous stresses. A 'minus' sign reflects the fact that the viscous stresses cause
momentum losses caused by an internal friction.
 
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