38 Chapter 2. Derivation offluid equations: Vlasov, 2-fluid, MHD
Equation (2.25) defines pressure for a three-dimensional isotropic system. However, we
will often deal with systems of reduced dimensionality, i.e., systems with just one or two
dimensions. Equation (2.25) can therefore be generalized to these other cases by introduc-
ing the generalN-dimensional definition for scalar pressure
Pσ=mσ
N∫
v′·v′fσdNv′=mσ
N∫ ∑N
j=1vj′^2 fσdNv′ (2.26)wherev′is theN-dimensional random velocity.
It is important to emphasize that assuming isotropy is done largely for mathematical
convenience and that in real systems the distribution function is often quite anisotropic.
Collisions, being randomizing, drive the distribution function towards isotropy, while com-
peting processes simultaneously drive it towards anisotropy. Thus, eachsituation must be
considered individually in order to determine whether there is sufficient collisionality to
makefisotropic. Because fully-ionized hot plasmas often have insufficient collisions to
makefisotropic, the oft-used assumption of isotropy is an oversimplification which may
or may not be acceptable depending on the phenomenon under consideration.
On expanding the derivatives on the left hand side of Eq.(2.24), it is seen that two of
the terms combine to giveutimes Eq. (2.19). After removing this embedded continuity
equation, Eq.(2.24) reduces to
nσmσduσ
dt=nσqσ(E+uσ×B)−∇Pσ−Rσα (2.27)where the operatord/dtis defined to be theconvective derivative
d
dt=
∂
∂t+uσ·∇ (2.28)which characterizes the temporal rate of change seen by an observermoving with the mean
fluid velocityuσof speciesσ.An everyday example of the convective term would be the
apparent temporal increase in density of automobiles seen by a motorcyclist who enters a
traffic jam of stationary vehicles and is not impeded by the traffic jam.
At this point in the procedure it becomes evident that a certain pattern recurs for each
successive moment of the Vlasov equation. When we took the zeroth moment, an equation
for the density
∫
fσdvresulted, but this also introduced a term involving the next higher
moment, namely the mean velocity∼
∫
vfσdv. Then, when we took the first moment to
get an equation for the velocity, an equation was obtained containing a term involving the
next higher moment, namely the pressure∼
∫
vvfσdv. Thus, each time we take a moment
of the Vlasov equation, an equation for the moment we want is obtained, but because of the
v·∇fterm in the Vlasov equation, a next higher moment also appears. Thus, moment-
taking never leads to a closed system of equations;there is always be a “loose end”, a
highest moment for which there is no determining equation. Some sort of ad hoc closure
procedure must always be invoked to terminate this chain (as seen below,typical closures
involve invoking adiabatic or isothermal assumptions). Another feature of taking moments
is that each higher moment has embedded in it a term which contains completelower
moment equations multiplied by some factor. Algebraic manipulation can identify these
lower moment equations and eliminate them to give a simplified higher moment equation.