# Governing Equations

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

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

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

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

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

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

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

$\rho = \rho \big ( \theta,s \big )$

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.
Illustration of the orthogonal coordinate system: x: eastward; y: northward; z: upward.
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

$\frac{\partial p_H}{\partial z} = -\rho g \Rightarrow p_H = \rho_o g \zeta + g \int_{z}^{0}\rho dz'$

The surface and bottom boundary conditions for temperature are:

$\frac{\partial T}{\partial z} = \frac{1}{\rho c_p K_h} \left[ Q_n \left(x,y,t\right) - SW \left( x,y,\zeta,t \right) \right], \text{at} \; z = \zeta \left( x,y,t \right)$

$\frac{\partial T}{\partial z} = - \frac{A_H \tan \alpha }{K_h} \frac{\partial T}{\partial n} , \text{at} \; z = -H\left( x,y \right)$

where $Q_n \left(x,y,z \right)$ is the surface net heat flux, which consists of four components: downward shortwave, longwave radiation, sensible, and latent fluxes, $SW \left(x,y,0,t \right)$ is the shortwave flux incident at the sea surface, and cp is the specific heat of seawater. Ah is the horizontal thermal diffusion coefficient, α is the slope of the bottom bathymetry, and n is the horizontal coordinate (Pedlosky, 1974, Chen et. al., 2004b).

## References

• Pedlosky, Joseph, 1974. Longshore currents, upwelling and bottom topography. Journal of Physical Oceanography, 4, 214–226.