Rotating fluids and the outer core - Earth Dynamics: Deformations and Oscillations of the Rotating Earth - page 401

or

Dt logρ +∇·

D

v = 0. (6.95)

Thus, departure from the solenoidal flow condition (6.2) of the ideal fluid arises
from changes in density from local flow pressure fluctuations and from transport
through the stratified core fluid. In the Earth's outer core, we will find the former
negligible at long periods but the latter very important for motions with substantial
radial component.
Now consider the equation for linear momentum. Again, we take a surface S

V

moving with the fluid and enclosing a volume . The linear momentum of the

enclosed fluid is

ρv i d V . (6.96)

V

Let f i be the body force per unit mass and τ ji be the stress field. The total force
acting on the fluid within S is then

τ ji n j d S+ ρ f i d V . (6.97)

S V

Applying the theorem of Gauss to the surface integral, the total force acting is
expressible as

∂τ ji

∂ x j + ρ f i d

V . (6.98)

V

The rate of change of the linear momentum enclosed by the surface S is expressible

as the sum of the rate of change within S and the rate at which it is being swept up

as S moves through the density and velocity fields, and is thus

∂
∂ t (ρv i ) d

V+ (ρv i )v j n j d S . (6.99)

V S

Again, applying Gauss's theorem to the surface integral, the total time rate of
change of linear momentum becomes

∂
∂ t (ρv i ) ∂ x j ρv i v j d

∂

+ V . (6.100)

V

Equating this to the total force (6.98), for arbitrary V , yields the equation of

motion as

∂ x j ρv i v j

∂τ ji
∂ x j .

∂
∂ t (ρv i )

+ ∂ = ρ f i + (6.101)

The left side may be expanded as
ρ ∂v i

∂ x j + v i ∂ρ ∂ x j ρv j ∂ t + ρv j ∂v i

∂

∂ t +

. (6.102)