62
DYNAMICAL OCEANOGRAPHY
Conservation of heat and salt is formulated as
ρC p
DT
dt
= F T + Q T ,
(3.30a)
ρ
DS
dt
= F S + Q S ,
(3.30b)
where F T (Wm −3 )andF S (kg m −3 s −1 ) represent the mixing due to turbulent
motions as in (3.20). The terms Q T (Wm −3 )andQ S (kg m −3 s −1 ) represent the
internal sources/sinks of heat and salt.
The unknown quantities in the equations are the velocity field v, the pressure p,
the density ρ, the temperature T and the salinity S, a total of 7. The last equation
needed to close the system of equations is the equation of state
ρ = ρ(T,S,p),
(3.31)
which was discussed at length in chapter 1 (section 1.3).
3.2.2. Spherical coordinates
In traditional fluid mechanics on the sphere, the equations of motion are usually
given in spherical coordinates (r, ϑ, ϕ). In oceanography we work with longitude
φ = −ϕ, latitude θ = π/2 − ϑ and the velocity components in zonal, meridional
and vertical direction are represented as v =( u, v, w). The equations of motion
(3.29) can be written in coordinates (φ, θ, r), using (3.8)), as
Du
dt
+
uw
r
−
uv
r
tan θ − 2Ω (v sin θ − w cos θ)=
−
1
ρr cos θ
∂p
∂φ
+ F
φ
I ,
(3.32a)
Dv
dt
+
wv
r
+
u 2
r
tan θ +2Ωu sin θ =
−
1
ρr
∂p
∂θ
+ F
θ
I ,
(3.32b)
Dw
dt
−
u 2 + v 2
r
− 2Ωu cos θ =
−
1
ρ
∂p
∂r
− g + F
r
I ,
(3.32c)
Dρ
dt
+ ρ(
∂w
∂r
+
2w
r
+
1
r cos θ
(
∂(v cos θ)
∂θ
+
∂u
∂φ
)) = 0,
(3.32d)
ρC p
DT
dt
= F T + Q T ,
(3.32e)
ρ
DS
dt
= F S + Q S ,
(3.32f)
ρ = ρ(T,S,p),
(3.32g)