Model Formulation

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Primitive Equations

The governing equations consist of the following momentum, continuity, temperature, salinity, and density equations:


\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} + w \frac{\partial u}{\partial z} - fv = -\frac{1}{\rho_o} \frac{\partial P}{\partial x} + \frac{\partial}{\partial z}\left(  K_{m} \frac{\partial u}{\partial z}\right) + F_u


\frac{\partial v}{\partial t} + u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} + w \frac{\partial v}{\partial z} +fu = -\frac{1}{\rho_o} \frac{\partial P}{\partial y} + \frac{\partial}{\partial z}\left(  K_{m} \frac{\partial v}{\partial z}\right) + F_u


\frac{\partial P}{\partial z} = -\rho g

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} +   \frac{\partial w}{\partial z}  = 0


\frac{\partial \theta}{\partial t} + u \frac{\partial \theta}{\partial x} + v \frac{\partial \theta}{\partial y} + w \frac{\partial \theta}{\partial z}  = \frac{\partial}{\partial z}\left(  K_{h} \frac{\partial \theta}{\partial z}\right) + F_u


\frac{\partial s}{\partial t} + u \frac{\partial s}{\partial x} + v \frac{\partial s}{\partial y} + w \frac{\partial s}{\partial z}  = \frac{\partial}{\partial z}\left(  K_{h} \frac{\partial s}{\partial z}\right) + F_u

ρ = ρ(θ,s)

where "x","y", and "z" are the east, north, and vertical axes in the Cartesian coordinate system; "u","v", and "w" are the x, "y", "z" velocity components; "θ" is the potential temperature; "s" is the salinity; "ρ" is the density; "P" is the pressure; "f" is the Coriolis parameter; "g" is the gravitational acceleration; Km is the vertical eddy viscosity coefficient; and Kh is the thermal vertical eddy

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