Governing Equations
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The governing equations consist of the following momentum, continuity, temperature, salinity, and density equations:
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,and 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 viscosity. Fu, Fv, Ft, and Fs represent the horizontal momentum, thermal, and salt diffusion terms. The total water column depth is , where H is the bottom depth (relative to z = 0) and ζ is the height of the free surface (relative to z = 0). p = pa + pH + q is the total pressure, in which the hydrostatic pressure P satisfies
Failed to parse (syntax error): \frac{\partial p_H}{\partial z} & = & -\rho g \Rightarrow p_H = \rho_o g \zeta + g \int_{z}^{0}\rho dz'\\ a & = & b
The surface and bottom boundary conditions for temperature are:
Failed to parse (unknown function\begin): \begin{eqnarray} \frac{\partial T}{\partial z} = \frac{1}{\rho c_p K_h} \left[ Q_n \left(x,y,t\right) - SW \left( x,y,\zeta,t \right) \right], \text{at} \; z = \zeta \left( x,y,t \right)