Governing Equations

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

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

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

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

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

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

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

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

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; K_m is the vertical eddy viscosity coefficient; and K_h is the thermal vertical eddy viscosity.

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

F u

,

F v

,

F t

, and

F s

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 = p a + p H + 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 $c p$ is the specific heat of seawater. $A h$ 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).

The Palace of Westminster

References

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

Governing Equations

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