Fundamentals of Atmospheric, Ionospheric, and Magnetospheric Plasma Dynamics - The Earth's Ionosphere: Plasma Physics and Electrodynamics - page 28

molecules that statistical techniques can be used to define such quantities as the
temperature, density, and mean flow velocity per each species. Such a volume
element is macroscopically small, even though microscopically it contains many
particles colliding in a random fashion. We use this approach almost exclusively
in the text. Schunk and Nagy (2000) discuss this and other approximations for
tenuous fluid behaviors.

2.1.1 Conservation of Mass
Conservation of mass dictates that the flux of material into or out of a volume
through its surface must be equal to the rate of increase or decrease of mass
inside the volume. The mathematical statement in integral form relates the mass
density ρ and the fluid velocity U through

∂ρ/∂ tdV =− ρ U · d a (2.1)

v

where the surface and d a is normal to the surface and

directed outward. The differential form of (2.1) can be derived from Gauss's
theorem, which relates the surface integral on the right-hand side to a volume
integral of the divergence of

encloses the volume V ,

ρ U —that is,

∂ρ/∂ =− ρ · =− ∇· (ρ )

tdV U d a U dV

v v

and thus,

∂ρ/∂ =−∇· (ρ )

t U (2.2)

This constitutes the continuity equation for the neutral atmosphere with mass
density

and velocity U .
Expanding the divergence operator, (2.2) can also be written

ρ

∂ρ/∂ t + U ·∇ ρ + ρ( ∇· U ) = 0 (2.3)

or, equivalently,

d ρ/ dt + ρ( ∇· U ) = 0 (2.4)

where the total time derivative

d / dt = ∂/∂ t + U ·∇ (2.5)