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| [[Primitive Equations]] | | [[Primitive Equations]] |
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- | == Primitive Equations ==
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- | The governing equations consist of the following momentum, continuity, temperature, salinity, and density equations:
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- | <math>
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- | \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
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- | </math>
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- | <math>
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- | \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
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- | </math>
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- | <math>
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- | \frac{\partial P}{\partial z} = -\rho g
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- | </math>
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- | <math>
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- | \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0
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- | </math>
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- | <math>
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- | \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
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- | </math>
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- | <math>
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- | \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
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- | </math>
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- | <math>
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- | \rho = \rho \big ( \theta,s \big )
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- | </math>
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- | 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 [http://en.wikipedia.org/wiki/Coriolis_frequency Coriolis parameter]; ''g'' is the gravitational acceleration; ''K<sub>m</sub>'' is the vertical eddy viscosity coefficient; and ''K<sub>h</sub>'' is the thermal vertical eddy viscosity.
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- | [[File:FVCOM_coordinate_system.png|Illustration of the orthogonal coordinate system: x: eastward; y: northward; z: upward. ]]
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- | <math>F_u</math> and 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
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