DERIVATION OF THE BAROTROPIC STREAMFUNCTION EQUATION

The horizontal momentum equation in z-coordinates is:

∂_{t}u + (u · ∇)u + u∇ · u + ∂_{z}(wu) + fk̂ × u + ∇P = ∂_{z}τ + F

Where τ is the wind stress and F is all other forcings, including bottom, horizontal and vertical friction.

For our purposes, we write this as

∂_{t}u + fk̂ × u + ∇P = −∂_{z}(wu) + F̃

where F̃ contains the wind stress, friction and the nonlinear advection terms.

We now vertically integrate the momentum equations to derive the barotropic equation. The vertically averaged (barotropic) horizontal velocity is defined as:

ū = \frac{1}{H} ∫_{0}^{H} udz

We assume there is a rigid lid at z = 0, which exerts a pressure, P_{TOP} on the fluid below and that the vertical velocity is zero at both z = 0 and z = H.

The vertically integrated momentum equations can be written as:

∂_{t}ū + fk̂ × ū + ∇P_{TOP} = F̄ − \frac{1}{H} ∫_{0}^{H} ∇P_{M}dz  (1)

The rigid lid assumption, and no-normal-flow boundary conditions at the side and bottom of the ocean, guarantee the non-divergence of the barotropic flow field. Thus we can write ū in terms of a stream function, with

Hū = −∂_{y}ψ, Hv̄ = ∂_{x}ψ

The pressure of the rigid lid, P_{TOP}, can be eliminated from the momentum equation by taking the curl of Equation 1. This gives:

∂_{t}(∇ · \frac{1}{H} ∇ψ) + J(ψ, \frac{f}{H}) = ∇ × (F̄ − \frac{1}{H} ∇E)  (2)

where the Jacobian, J is defined as:

J(f, g) = (∂_{x}f)(∂_{y}g) − (∂_{y}f)(∂_{x}g)

We have recently rewritten Equation 2 as:

∂_{t}(H∇ · \frac{1}{H} ∇ψ) + HJ(ψ, \frac{f}{H}) = ∇ × HF̄ + H∇(\frac{1}{H}) × (HF̄ − ∇E)  (3)

and discretized this form of the equation. The last term on the righthandside is the cross gradient topographic forcing term, which we expect to be small (as per Mark, Vladimir and Sasha’s observations).

TWO SPECIAL CASES

1. Homogeneous ocean In the case of a homogeneous ocean, with ρ = ρ_{0}, E = gH^{2}/2 and so J(E, \frac{1}{H}) = 0.

The stream function therefore satisfies:

∂_{t}(∇ · \frac{1}{H} ∇ψ) + J(ψ, \frac{f}{H}) = ∇ × \frac{F̄}{H}

I.e., ψ is the integral of the curl of the wind stress along lines of constant f/H.

2. Flat bottomed ocean

In the case of a flat bottomed ocean,

∂_{t}(∇^{2}ψ) + β∂_{x}ψ = ∇ × F̄

where

β = ∂_{y}f

ISLANDS

The no-normal flow boundary conditions for the flow field translate to a Dirichlet boundary condition, ψ = C, on the boundary. If we have a closed basin (simply connected), then the value of the constant doesn’t matter, and we take it to be zero. When we have a periodic domain or islands, then the constant can be different on each separate land mass. These constants must be found as part of the solution process - continuity of pressure is the key. Since Equation 2 is linear in ψ, we can write

ψ = ψ_{0} + # of islands ; ∑ ; m=1 T_{m}ψ_{m}  (4)

where ψ_{0} is the solution of the non-homogeneous problem with ψ = 0 on all land masses and ψ_{m} is the solution of the homogeneous problem with ψ_{m} = 1 on the mth land mass and ψ_{m} = 0 on all other land masses. Note that we also have

ū = ū_{0} + ∑_{m} T_{m}ū_{m}.

Suppose we have an elliptic solver, which solves for the ψ_{m}. We then need only to find the coefficients, T_{m}. We rewrite Equation 1 as:

∂_{t}ū + fk̂ × ū + ∇P_{TOP} = Ḡ

where

Ḡ = F̄ − \frac{1}{H} ∫_{0}^{H} ∇P_{M}dz.

By Green’s theorem in a plane,

∮_{∂I} ∇P_{TOP} · n̂ds = ∫ ∫_{I} ∇ × ∇P_{TOP}dxdy = 0

where ∂I is any closed curve. Therefore

∮_{∂I}(∂_{t}ū + fk̂ × ū) · n̂ds = ∮_{∂I} Ḡ · n̂ds

Define

A_{km} = ∮_{∂I_{k}}(∂_{t}ū_{m} + fk̂ × ū_{m}) · n̂ds.

Then, for each island, k,

∑_{m} A_{km}T_{m} = ∮_{∂I_{k}} Ḡ · n̂ds − A_{k0}

We solve this linear system for the T_{m}, and then compute ψ from Equation 4.

PRESSURE GRADIENT FORCE

x, y : horizontal coordinates (longitude, latitude)
z : vertical coordinate (depth)
u = (u, v)^{T} : horizontal velocity
w : vertical velocity
p : pressure
Θ (or T) : potential temperature
S : salinity
ρ : in-situ density, ρ(T, S, z)
H : total ocean depth, H(x,y)

From the hydrostatic approximation:

ρg = −\frac{∂p}{∂z}

we get the formula for scaled pressure, P:

P (x, y, z) = \frac{p}{ρ_{0}} = P_{TOP} + P_{M}

where P_{TOP} is due to the rigid lid on the ocean surface and P_{M} is due to the weight of the water above the point (x, y, z),

P_{M} = \frac{−g}{ρ_{0}} ∫_{0}^{z} ρ(x, y, z)dz.

For the barotropic/baroclinic split, we need the vertically integrated pressure gradient force:

∫_{0}^{H} ∇P dz = H∇P_{TOP} + ∫_{0}^{H} ∇P_{M}dz

After manipulation of the integral, including an integration by parts, the vertically integrated ∇P_{M} term can be written as:

∫_{0}^{H} ∇P_{M}dz = \frac{−g}{ρ_{0}} ∫_{0}^{H}(∇ ∫_{0}^{z} ρdz′)dz = H∇E_{0} + ∇E

where

E = \frac{−g}{ρ_{0}} ∫_{0}^{H} ρzdz

and

E_{0} = \frac{−g}{ρ_{0}} ∫_{0}^{H} ρdz

Note that

∇E = \frac{−g}{ρ_{0}} (ρbotH∇H + ∫_{0}^{H} (∇ρ)zdz)