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.
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).


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