Governing Equations

Governing Equations

The governing equations are the unsteady, incompressible, three-dimensional (3D) primitive equations with the Boussinesq, and hydrostatic approximation. These equations in the spherical coordinate (λ, ϕ, z) and time t can be written as

Continuity equation:

1
Rcosϕ
( ∂u
∂λ
+ ∂(vcosϕ)
∂ϕ
)+ ∂w
∂z
= 0,
(G.1)

Horizontal momentum equations:

∂u
∂t
+ L u − (f + u tanϕ
R
) v = − 1
ρ₀R cosϕ
∂p
∂λ
+ D_m u + ∂
∂z
(A_u ∂u
∂z
),
(G.2)

∂v
∂t
+Lv + ( f + u tanϕ
R
) u = − 1
ρ₀R
∂p
∂ϕ
+D_m v + ∂
∂z
( A_v ∂v
∂z
) ,
(G.3)
Hydrostatic equation:

∂p
∂z
=−ρg,
(G.4)
Equation of state:

ρ = ρ(S,T,p),
(G.5)

Conservation of scalar (potential temperature and salinity)

∂T
∂t
+L T = D_h T+ ∂
∂z
(K_T ∂T
∂z
),
(G.6)

∂S
∂t
+L S = D_h S+ ∂
∂z
(K_S ∂S
∂z
),
(G.7)

where u, v, and w are the velocity components in the longitudinal λ, latitudinal ϕ, and vertical z directions, respectively; p is pressure and p = p_s + p_b, where p_s represents the surface pressure and baroclinic pressure p_b is

p_b=g

⌠
⌡

0

z

ρdz;

(G.8)

ρ is the in-situ density and ρ₀ is the reference density; T and S are potential temperature and salinity, respectively; A_u and A_v are the vertical eddy viscosity coefficients for horizontal momentum equations while K_T and K_S are the vertical eddy diffusivity for temperature and salinity equations, respectively; f is the Coriolis parameter; g is the gravitational acceleration; the convection operator is

L =

Rcosϕ

∂

∂λ

∂

∂ϕ

+ w

∂

∂z

;

(G.9)

and the horizontal diffusion operator is

D_m(h) =

A_m(h)

R²

(

cos² ϕ

∂²

∂λ²

−tanϕ

∂

∂ϕ

∂²

∂ϕ²

(G.10)

where A_m and A_h are the horizontal eddy viscosity and diffusivity, respectively.

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