Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 400

where the mass density ρ( x i , t ) is taken to be a function of both the co-ordinates,
x i , and time, t . The surface , moving with the fluid, sweeps through the density

field, increasing the enclosed mass at the rate

S

ρv k n k d S , (6.87)

S

for surface velocity components (v 1 ,v 2 ,v 3 ) and unit outward normal vector n , with
components ( n 1 , n 2 , n 3 ). Since the density is also a function of time, the total rate
of change of the enclosed mass is then

∂ρ
∂ t d

V+ ρv k n k d S . (6.88)

V S

Applying the theorem of Gauss to the surface integral, the total rate of change of
the enclosed mass is

∂ρ

(ρv k ) d

∂ t + ∂

V . (6.89)

∂ x k

V

Since the surface moves with the fluid, no mass enters or escapes the enclosed
volume, and conservation of mass implies that the integral (6.89) vanishes. The
volume is arbitrary and hence the integrand of (6.89) must vanish everywhere,
giving

∂ρ
∂ t +

∂
∂ x k

(ρv k ) = 0 (6.90)

as the law of mass conservation. In symbolic notation, conservation of mass yields
the equation of continuity ,

∂ρ
∂ t +∇·

(ρ v ) = 0. (6.91)

Expanding the divergence, the equation of continuity becomes

∂ρ
∂ t +

v ·∇ ρ + ρ ∇· v = 0. (6.92)

The first two terms on the left side measure the time rate of change of density,
moving with the fluid, or the substantial time derivative

Dt ≡ ∂

D

∂ t +

v ·∇ . (6.93)

The equation of continuity is then expressible as

D Dt + ρ ∇·

v = 0, (6.94)