Governing Equations
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The surface and bottom boundary conditions for temperature are: | The surface and bottom boundary conditions for temperature are: | ||
<math> | <math> | ||
+ | \begin{array} | ||
a & = & b\\ | a & = & b\\ | ||
c & = & d | c & = & d | ||
+ | \end{array} | ||
</math> | </math> | ||
- | \begin{ | + | \begin{array} |
\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{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)\\ | ||
a & = & b | a & = & b | ||
- | \end{ | + | \end{array} |
</math> | </math> |
Revision as of 15:24, 10 November 2011
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
The surface and bottom boundary conditions for temperature are: Failed to parse (PNG conversion failed; check for correct installation of latex, dvips, gs, and convert): \begin{array} a & = & b\\ c & = & d \end{array}
\begin{array} \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)\\ a & = & b \end{array} </math>