Model help:ROMS: Difference between revisions

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a) Momentum balance in the x-directions, respectively
a) Momentum balance in the x-directions, respectively
::::{|
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|width=530px|<math> {\frac{\partial u}{\partial t}} + \vec{v} \nabla u - fv = -{\frac{\partial \phi}{\partial x}} + F_{u} + D_{u}  </math>
|width=530px|<math> {\frac{\partial u}{\partial t}} + \vec{v} \nabla u - fv = -{\frac{\partial \phi}{\partial x}} - {\frac{\partial}{\partial z}} \left ( \overline{u'w'} - \nu {\frac{\partial u}{\partial z}} \right ) + F_{u} + D_{u}  </math>
|width=50p=x align="right"|(1)
|width=50p=x align="right"|(1)
|}
|}
b) Momentum balance in the y-directions, respectively
b) Momentum balance in the y-directions, respectively
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::::{|
|width=530px|<math> {\frac{\partial v}{\partial t}} + \vec{v} \nabla u - fv = -{\frac{\partial \phi}{\partial y}} + F_{v} + D_{v}  </math>
|width=530px|<math> {\frac{\partial v}{\partial t}} + \vec{v} \nabla v + fu = -{\frac{\partial \phi}{\partial y}} - {\frac{\partial}{\partial z}} \left ( \overline{v'w'} - \nu {\frac{\partial v}{\partial z}} \right )+ F_{v} + D_{v}  </math>
|width=50p=x align="right"|(2)
|width=50p=x align="right"|(2)
|}
|}
c) Advective-diffusive equations for temperature
c) Advective-diffusive equations for a scalar concentration field C(x,y,z,t) (e.g. salinity, temperature, or nutrients)
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::::{|
|width=530px|<math> {\frac{\partial T}{\partial t}} + \vec{v} \nabla T = F_{T} + D_{T} </math>
|width=530px|<math> {\frac{\partial C}{\partial t}} + \vec{v} \nabla C = {\frac{\partial}{\partial z}} \left ( \overline{C'w'} - \nu_{\theta} {\frac{\partial C}{\partial z}} \right )+ F_{C} + D_{C} </math>
|width=50p=x align="right"|(3)
|width=50p=x align="right"|(3)
|}
|}
d) Advective-diffusive equations for salinity
e) Equation of state
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|width=530px|<math> {\frac{\partial S}{\partial t}} + \vec{v} \nabla S = F_{S} + D_{S} </math>
|width=530px|<math> \rho = \rho \left (T,S,P \right ) </math>
|width=50p=x align="right"|(4)
|width=50p=x align="right"|(4)
|}
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e) Equation of state
f) Vertical momentum equation (assume that the vertical pressure gradient balances the buoyancy force)
::::{|
::::{|
|width=530px|<math> \rho = \rho \left (T,S,P \right ) </math>
|width=530px|<math> {\frac{\partial \phi}{\partial z}} = {\frac{- \rho g}{\rho_{o}}} </math>
|width=50p=x align="right"|(5)
|width=50p=x align="right"|(5)
|}
|}
f) Vertical momentum equation
g) continuity equation for an incompressible fluid
::::{|
::::{|
|width=530px|<math> {\frac{\partial \rho}{\partial z}} = {\frac{- \rho g}{\rho_{o}}} </math>
|width=530px|<math> {\frac{\partial u}{\partial x}} + {\frac{\partial v}{\partial y}} + {\frac{\partial w}{\partial z}} = 0 </math>
|width=50p=x align="right"|(6)
|width=50p=x align="right"|(6)
|}
|}
g) continuity equation for an incompressible fluid
h) Reynolds stresses (an overbar represents a time average and a prime represents a fluctuation above the mean)
::::{|
::::{|
|width=530px|<math> {\frac{\partial u}{\partial x}} + {\frac{\partial v}{\partial y}} + {\frac{\partial w}{\partial z}} = 0 </math>
|width=530px|<math> \overline{u'w'} = - K_{M} {\frac{\partial u}{\partial z}} </math>
|width=50p=x align="right"|(7)
|width=50p=x align="right"|(7)
|}
|}
2) Vertical boundary conditions
a) top boundary condition ( z = ζ (x,y,t ))
::::{|
::::{|
|width=530px|<math> K_{m} {\frac{\partial u}{\partial z}} = \tau_{S}^x \left (x,y,t \right ) </math>
|width=530px|<math> \overline{v'w'} = - K_{M} {\frac{\partial v}{\partial z}} </math>
|width=50p=x align="right"|(8)
|width=50p=x align="right"|(8)
|}
|}
::::{|
::::{|
|width=530px|<math> K_{m} {\frac{\partial v}{\partial z}} = \tau_{S}^y \left (x,y,t \right ) </math>
|width=530px|<math> \overline{C'w'} = - K_{C} {\frac{\partial C}{\partial z}} </math>
|width=50p=x align="right"|(9)
|width=50p=x align="right"|(9)
|}
|}
2) Vertical boundary conditions
a) top boundary condition ( z = ζ (x,y,t ))
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::::{|
|width=530px|<math> K_{T} {\frac{\partial T}{\partial z}} = {\frac{Q_{T}}{\rho_{O} c_{P}}} + {\frac{1}{\rho_{O} c_{P}}}{\frac{d Q_{T}}{dT}} \left ( T - T_{ref} \right ) </math>
|width=530px|<math> K_{m} {\frac{\partial u}{\partial z}} = \tau_{S}^x \left (x,y,t \right ) </math>
|width=50p=x align="right"|(10)
|width=50p=x align="right"|(10)
|}
|}
::::{|
::::{|
|width=530px|<math> K_{S} {\frac{\partial S}{\partial z}} = \left (E - P \right ) S </math>
|width=530px|<math> K_{m} {\frac{\partial v}{\partial z}} = \tau_{S}^y \left (x,y,t \right ) </math>
|width=50p=x align="right"|(11)
|width=50p=x align="right"|(11)
|}
|}
::::{|
::::{|
|width=530px|<math> w = {\frac{\partial \zeta}{\partial t}} </math>
|width=530px|<math> K_{C} {\frac{\partial C}{\partial z}} = {\frac{Q_{C}}{\rho_{O} c_{P}}} </math>
|width=50p=x align="right"|(12)
|width=50p=x align="right"|(12)
|}
|}
b) bottom boundary condition (z = -h(x,y))
::::{|
::::{|
|width=530px|<math> K_{m}{\frac{\partial u}{\partial z}} = \tau _{b}^x \left (x,y,t \right ) </math>
|width=530px|<math> w = {\frac{\partial \zeta}{\partial t}} </math>
|width=50p=x align="right"|(13)
|width=50p=x align="right"|(13)
|}
|}
b) bottom boundary condition (z = -h(x,y))
::::{|
::::{|
|width=530px|<math> K_{m}{\frac{\partial v}{\partial z}} = \tau _{b}^y \left (x,y,t \right ) </math>
|width=530px|<math> K_{m}{\frac{\partial u}{\partial z}} = \tau _{b}^x \left (x,y,t \right ) </math>
|width=50p=x align="right"|(14)
|width=50p=x align="right"|(14)
|}
|}
::::{|
::::{|
|width=530px|<math> K_{T}{\frac{\partial T}{\partial z}} = 0 </math>
|width=530px|<math> K_{m}{\frac{\partial v}{\partial z}} = \tau _{b}^y \left (x,y,t \right ) </math>
|width=50p=x align="right"|(15)
|width=50p=x align="right"|(15)
|}
|}
::::{|
::::{|
|width=530px|<math> K_{S}{\frac{\partial S}{\partial z}} = 0 </math>
|width=530px|<math> K_{C}{\frac{\partial C}{\partial z}} = 0 </math>
|width=50p=x align="right"|(16)
|width=50p=x align="right"|(16)
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|width=50p=x align="right"|(17)
|width=50p=x align="right"|(17)
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|}
c) Horizontal boundary conditions (Eastern and western boundary)(considering biharmonic friction)
::::{|
::::{|
|width=530px|<math> \tau_{b}^x = \left ( \gamma_{1} + \gamma_{2} \sqrt{u^2 + v^2} \right ) u </math>
|width=530px|<math> {\frac{\partial}{\partial x}} \left ( {\frac{h \nu}{mn}}{\frac{\partial^2 u}{\partial x^2}} \right ) = 0 </math>
|width=50p=x align="right"|(18)
|width=50p=x align="right"|(18)
|}
|}
Horizontal boundary conditions (Northern and southern boundary)
::::{|
::::{|
|width=530px|<math> \tau_{b}^y = \left ( \gamma_{1} + \gamma_{2} \sqrt{u^2 + v^2} \right ) v </math>
|width=530px|<math> {\frac{\partial}{\partial y}} \left ( {\frac{h \nu}{mn}}{\frac{\partial^2 u}{\partial y^2}} \right ) = 0 </math>
|width=50p=x align="right"|(19)
|width=50p=x align="right"|(19)
|}
|}
c) Horizental boundary conditions (Eastern and western boundary)
3) Vertical transformation equations
a) vertical coordinate transformation (been available in ROMS since 1999)
::::{|
::::{|
|width=530px|<math> {\frac{\partial}{\partial x}} \left ( {\frac{h \nu}{mn}}{\frac{\partial^2 u}{\partial x^2}} \right ) = 0 </math>
|width=530px|<math> z \left (x,y,\delta, t \right ) = S \left (x,y,\delta \right )+ \zeta  \left (x,y,t \right ) [1 + {\frac{S \left (x,y,\delta \right )}{h \left (x,y\right )}}] </math>
|width=50p=x align="right"|(20)
|width=50p=x align="right"|(20)
|}
|}
Horizental boundary conditions (Northern and southern boundary)
::::{|
::::{|
|width=530px|<math> {\frac{\partial}{\partial y}} \left ( {\frac{h \nu}{mn}}{\frac{\partial^2 u}{\partial y^2}} \right ) = 0 </math>
|width=530px|<math> S \left (x,y,\delta \right ) = h_{c} \delta + [h \left (x,y\right ) - h_{c}] C \left (\delta \right) </math>
|width=50p=x align="right"|(21)
|width=50p=x align="right"|(21)
|}
|}
 
b) vertical coordinate transformation (been available in ROMS since 2005)
::::{|
|width=530px|<math> z \left (x,y,\delta,t \right ) = \zeta \left (x,y,t\right ) + [\zeta \left (x,y,t\right ) + h \left (x,y \right )] S \left (x,y,\delta \right ) </math>
|width=50p=x align="right"|(22)
|}
::::{|
|width=530px|<math> S \left (x,y,\delta \right ) = {\frac{h_{c} \delta + h \left (x,y \right ) C \left ( \delta \right )}{h_{c} + h \left (x,y\right )}} </math>
|width=50p=x align="right"|(23)
|}
<div class="NavFrame collapsed" style="text-align:left">
<div class="NavFrame collapsed" style="text-align:left">
   <div class="NavHead">Nomenclature</div>
   <div class="NavHead">Nomenclature</div>
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| -
| -
|-
|-
| F<sub>u</sub>,F<sub>v</sub>,F<sub>T</sub>,F<sub>S</sub>
| F<sub>u</sub>,F<sub>v</sub>,F<sub>C</sub>
| forcing terms
| forcing terms
| -
| -
|-
|-
| f
| f(x,y)
| coriolis parameter
| coriolis parameter
| -
| -
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| m/s<sup>2</sup>
| m/s<sup>2</sup>
|-
|-
| h
| h(x,y)
| bottom depth
| bottom depth
| m
| m
|-
|-
| ν
| ν,ν<sub>θ</sub>
| horizontal viscosity
| molecular viscosity and diffusivity
|
|-
| κ
| horizontal diffusivity
|  
|  
|-
|-
| K<sub>m</sub>,K<sub>T</sub>,K<sub>S</sub>
| K<sub>m</sub>,K<sub>C</sub>
| vertical viscosity and diffusivity
| vertical eddy viscosity and diffusivity
| -
| -
|-
|-
| P
| P
| total pressure, approximately -ρ<sub>O</sub> gz
| total pressure, approximately -ρ<sub>o</sub> gz
|  
|  
|-
|-
| Φ
| Φ(x,y,z,t)
| dynamic pressure, Φ = (P/ρ<sub>O</sub>)
| dynamic pressure, Φ = (P/ρ<sub>o</sub>)
|  
|  
|-
|-
| ρ + ρ(x,y,z,t)  
| ρ<sub>o</sub> + ρ(x,y,z,t)  
| total in situ density
| total in situ density
|  
|  
|-
|-
| S
| S(x,y,z,t)
| salinity
| salinity
|  
|  
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|  
|  
|-
|-
| T
| T(x,y,z,t)
| potential temperature
| potential temperature
|  
|  
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|  
|  
|-
|-
| ζ
| ζ(x,y,t)
| surface elevation
| surface elevation
|  
|  
|-
|-
| E
| Q<sub>C</sub>
| evaporation
| surface concentration flux
| -
|-
| P
| precipitation
|
|-
| γ<sub>1</sub>
| linear bottom stress coefficient
| -
|- 
| γ<sub>2</sub>
| quadratic bottom stress coefficient
| -
|- 
| Q<sub>T</sub>
| surface heat flux
| -
| -
|-
|-
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| -
| -
|-
|-
| T<sub>ref</sub>
| S(x,y,δ)
| surface reference temperature
| a nonlinear vertical transformation functional (in the vertical transformation equations)
|  
|-
|-
|-
| ζ(x,y,t)
| time-varying free-surface(in the vertical transformation equations)
|-
|-
| h(x,y)
| unperturbed water column thickness (in the vertical transformation equations)
|-
|-
| δ
| fractional vertical stretching coordinate (ranging from -1 <= δ <= 0)
|-
|-
| C(δ)
| nondimensional, monotonic, vertical stretching function (ranging from -1 <= C(δ) <= 0)
|-
|-
| h<sub>c</sub>
| a positive thickness controlling the stretching
|-
|-
|}
|}


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==Notes==
==Notes==
<span class="remove_this_tag">Any notes, comments, you want to share with the user</span>
1 In the Boussinesq approximation, density variations are neglected in the momentum equations except in their contribution to the buoyancy force in the vertical momentum equation.
 


<span class="remove_this_tag">Numerical scheme</span>




Revision as of 15:48, 25 July 2011

The CSDMS Help System

ROMS

ROMS is a community model shared by a large user group around the world, with applications ranging from the study of entire ocean basins to coastal sub-regions.

Model introduction

ROMS incorporates advanced features and high-order numerics, allowing efficient and robust resolution of mesoscale dynamics in the oceanic and coastal domains. The model solves the free surface, hydrostatic, primitive equations of the fluid dynamics over variable topography using stretched; terrain-following coordinates in the vertical and orthogonal, curvilinear coordinates in the horizontal. This allows enhancement of the spatial resolution in the regions of interest.

Model parameters

Parameter Description Unit
Input directory Path to input directory -
Input File name Name of input file -
Site prefix Site prefix for input/output files -
Case prefix Case prefix for input/output files -
ROMS application title - -
C-preprocessing flag - -
Variable information filename - -
Runinterval timestep multiplied by timestep size (In cases where input is read from a file rather than GUI second
Parameter Description Unit
Number of nested grids - -
Number of nodes in I-direction number of I-direction interior rho-points -
Number of nodes in J-direction number of J-direction interior rho-points -
Number of vertical levels - -
Number of sediment bed layers - -
Number of active tracers usually 2; e.g., temperature and salinity -
Number of inactive passive tracers - -
Number of cohesive (mud) sediment tracers - -
Number of non-cohesive (sand) sediment tracers - -
Number of titles in I-direction used for parallel mode -
Number of titles in J-direction used for parallel mode -
Parameter Description Unit
Number of timesteps - -
Timestep size - seconds
Number of barotropic steps - -
Starting perturbation or iteration - -
Ending perturbation or iteration - -
Max number of 4DVar outer loop iterations - -
Max number of 4DVar inner loop iterations - -
Number of stochastic optimals interval divisions - -
Number of eigenvalues for GST analysis - -
Number of eigenvectors for GST analysis - -
Parameter Description Unit
Model restart flag - -
Switch to recycle restart records True (T) or False (F) -
Number of time-steps between restart records - -
Number of time-steps between stations records - -
Number of time-steps between floats records - -
Number of time-steps between info diagnostics - -
Switch to create (T) or append (F) files True (T) or False (F) -
Number of time-steps between history records - -
Number of time-steps between creation of new history file - -
Starting averages timestep - -
Number of time-steps between averages records - -
Number of time-steps between creation of new averages file - -
Starting diagnostics timestep - -
Number of time-steps between diagnostics records - -
Number of time-steps between creation of new diagnostics file - -
Number to recycle TLM time records True (T) or False (F) -
Number of time-steps between creation of new TLM file - -
Switch to recycle ADM time records True (T) or False (F) -
Number of time-steps between ADM records - -
Number of time-steps between creation of new ADM file - -
Number of time-steps between 4DVAR adjustment of SFF Number of time-steps between 4DVAR adjustment of surface forcing flux -
Number of time-steps between 4DVAR adjustment of OBF Number of time-steps between 4DVAR adjustment of open boundary fields -
Switch for GST restart Generalized Stability Theory restart switch (T or F) -
Maximum number of iterations for GST Maximum number of iterations for Generalized Stability Theory -
Checking point interval for GST Checking point interval for Generalized Stability Theory -
Parameter Description Unit
Relative accuracy of Ritz values in GST Relative accuracy of Ritz values in Generalized Stability Theory -
Harmonic horiz. diffusion coeff. for tracers (I-dir, NLM) Harmonic horizontal diffusion coefficient for tracers in NONLINEAR model in I-direction m2/s
Harmonic horiz. diffusion coeff. for tracers (J-dir, NLM) Harmonic horizontal diffusion coefficient for tracers in NONLINEAR model in J-direction m2/s
Biharmonic horiz. diffusion for tracers (I-dir, NLM) Biharmonic horizontal diffusion coefficient for tracers in NONLINEAR model in I-direction m4/s
Biharmonic horiz. diffusion for tracers (J-dir, NLM) Biharmonic horizontal diffusion coefficient for tracers in NONLINEAR model in J-direction m4/s
Harmonic horiz. diffusion coeff. for tracers (I-dir, ADJ) Harmonic horizontal diffusion coefficient for tracers in ADJOINT model in the I-direction m2/s
Harmonic horiz. diffusion coeff. for tracers (J-dir, ADJ) Harmonic horizontal diffusion coefficient for tracers in ADJOINT model in the I-direction m2/s
Biharmonic horiz. diffusion for tracers (I-dir, ADJ) Biharmonic horizontal diffusion coefficient for tracers in ADJOINT model in I-direction m4/s
Biharmonic horiz. diffusion for tracers (J-dir, NLM) Biharmonic horizontal diffusion coefficient for tracers in ADJOINT model in J-direction m4/s
Harmonic horiz. viscosity coeff. (NLM) Harmonic horizontal viscosity coefficient in NONLINEAR model m2/s
Biharmonic horiz. viscosity coeff. (NLM) Biharmonic horizontal viscosity coefficient in NONLINEAR model m4/s
Harmonic horiz. viscosity coeff. (ADJ) Harmonic horizontal viscosity coefficient in ADJOINT model m2/s
Biharmonic horiz. viscosity coeff. (ADJ) Biharmonic horizontal viscosity coefficient in ADJOINT model m4/s
Background vert. mixing coeff. for active tracers (NLM) Background vertical mixing coefficient for active tracers in NOLINEAR model m2/s
Background vert. mixing coeff. for passive tracers (NLM) Background vertical mixing coefficient for passive tracers in NOLINEAR model m2/s
Background vert. mixing coeff. for active tracers (ADI) Background vertical mixing coefficient for active tracers in ADJOINT model -
Background vert. mixing coeff. for passive tracers (ADI) Background vertical mixing coefficient for passive tracers in ADJOINT model -
Background vert. mixing coeff. for momentum (NLM) Background vertical mixing coefficient for momentum in NOLINEAR model m2/s
Background vert. mixing coeff. for momentum (ADJ) Background vertical mixing coefficient for momentum in ADJOINT model -
Parameter Description Unit
Turbulent closure parameter 1 - m2/s
Turbulent closure parameter 2 - m2/s
Turbulent closure parameter 3 - m2/s
Turbulent closure parameter 3 - m4/s
K-epsilon parameter for GLS K-epsilon parameter (Generic Length Scale closure) -
Turbulent kinetic energy exponent for GLS Turbulent kinetic energy exponent (Generic Length Scale closure) -
Turbulent length scale exponent for GLS Turbulent length scale exponent (Generic Length Scale closure) -
Min value of specific turbulent energy for GLS Min value of specific turbulent energy (Generic Length Scale closure) -
Min value of dissipation for GLS Min value of dissipation (Generic Length Scale closure) -
Stability coefficient (GLS, closure indep.) Stability coefficient (Generic Length Scale closure) -
Shear production coefficient (GLS, closure indep.) Shear production coefficient (Generic Length Scale closure) -
Dissipation coefficient (GLS, closure indep.) Dissipation coefficient (Generic Length Scale closure) -
Buoyancy production coefficient, minus (GLS, closure indep) Buoyancy production coefficient, minus (Generic Length Scale closure) -
Buoyancy production coefficient, plus (GLS, closure indep.) Buoyancy production coefficient, plus (Generic Length Scale closure) -
Constant Schmidt number for turb.KE diffusivity (GLS, closure indep.) Constant schmidt number for turbulent kinetic energy diffusivity (Generic Length Scale closure) -
Constant Schmidt number for turb. "psi" field (GLS, closure indep.) Constant schmidt number for turbulent generic statistical field, "psi" (Generic Length Scale closure) -
Charnok surface roughness - -
Roughness from wave amplitude - -
Roughness from wave dissipation - -
Graig and Banner wave breaking coefficient - -
Parameter Description Unit
Momentum stress constant 1 constant used in momentum stress computation m/s
Momentum stress constant 2 constant used in momentum stress computation -
Momentum stress constant 3 constant used in momentum stress computation m
Momentum stress constant 4 constant used in momentum stress computation m
Height of measurement for air humidity (bulk flux) Height of atmospheric measurement for air humidity (bulk flux) m
Height of measurement for air temperature (bulk flux) Height of atmospheric measurement for air temperature (bulk flux) m
Height of measurement for winds (bulk flux) Height of atmospheric measurement for winds (bulk flux) m
Min depth for wetting and drying - m
Jerlov water type for shortwave radiation depth scale Jerlov water type used to set vertical depth scale for shortwave radiation depth scale -
Deepest level to apply surf. momentum stress as a body force - -
Shallowest level to apply surf. momentum stress as a body force - -
Mean water density - kg/m3
Brunt-Vaisala frequency - 1/s2
Time-stamp for model initialization - days
Reference time origin for tidal forcing - days
Model reference time for output NetCDF units attribute - yyyymmdd.dd
Nudging/relaxation time scale 1 - days
Nudging/relaxation time scale 2 - days
Nudging/relaxation time scale 3 - days
Nudging/relaxation time scale 4 - days
Nudging/relaxation time scale 5 - days
Factor between passive and active open BCs Factor between passive (outflow) and active (inflow) open boundary conditions -
Linear equation of state density parameter - kg/m3
Linear equation of state temperature parameter - Celsius
Linear equation of state salinity parameter - PSU
Linear equation of state temperature coeff. - 1/Celsius
Linear equation of state salinity coeff. - 1/PSU
Slipperiness parameter: 1.0 (free slip) or -1.0 (no slip) - -
Switch to use point sources/sinks for temperature True (T) or False (F) -
Switch to use point sources/sinks for salinity True (T) or False (F) -
Switch to use point sources/sinks for intert tracer 1 True (T) or False (F) -
Parameter Description Unit
Choice of vertical transformation equation Choice of vertical transformation equation for terrain-following coordinates (1 or 2) -
Choice of vertical stretching function Choice of vertical stretching function for terrain-following coordinates (1,2 or 3) -
Surface control parameter for terrain-following coords Surface control/stretching parameter for terrain-following coords -
Bottom control parameter for terrain-following coords Bottom control/stretching parameter for terrain-following coords -
Width of surface or bottom layer which requires higher-res stretch - m
Parameter Description Unit
Model name Name of the model -
Author name name of the model author -

Uses ports

This will be something that the CSDMS facility will add

Provides ports

This will be something that the CSDMS facility will add

Main equations

1) Equations of motion

a) Momentum balance in the x-directions, respectively

[math]\displaystyle{ {\frac{\partial u}{\partial t}} + \vec{v} \nabla u - fv = -{\frac{\partial \phi}{\partial x}} - {\frac{\partial}{\partial z}} \left ( \overline{u'w'} - \nu {\frac{\partial u}{\partial z}} \right ) + F_{u} + D_{u} }[/math] (1)

b) Momentum balance in the y-directions, respectively

[math]\displaystyle{ {\frac{\partial v}{\partial t}} + \vec{v} \nabla v + fu = -{\frac{\partial \phi}{\partial y}} - {\frac{\partial}{\partial z}} \left ( \overline{v'w'} - \nu {\frac{\partial v}{\partial z}} \right )+ F_{v} + D_{v} }[/math] (2)

c) Advective-diffusive equations for a scalar concentration field C(x,y,z,t) (e.g. salinity, temperature, or nutrients)

[math]\displaystyle{ {\frac{\partial C}{\partial t}} + \vec{v} \nabla C = {\frac{\partial}{\partial z}} \left ( \overline{C'w'} - \nu_{\theta} {\frac{\partial C}{\partial z}} \right )+ F_{C} + D_{C} }[/math] (3)

e) Equation of state

[math]\displaystyle{ \rho = \rho \left (T,S,P \right ) }[/math] (4)

f) Vertical momentum equation (assume that the vertical pressure gradient balances the buoyancy force)

[math]\displaystyle{ {\frac{\partial \phi}{\partial z}} = {\frac{- \rho g}{\rho_{o}}} }[/math] (5)

g) continuity equation for an incompressible fluid

[math]\displaystyle{ {\frac{\partial u}{\partial x}} + {\frac{\partial v}{\partial y}} + {\frac{\partial w}{\partial z}} = 0 }[/math] (6)

h) Reynolds stresses (an overbar represents a time average and a prime represents a fluctuation above the mean)

[math]\displaystyle{ \overline{u'w'} = - K_{M} {\frac{\partial u}{\partial z}} }[/math] (7)
[math]\displaystyle{ \overline{v'w'} = - K_{M} {\frac{\partial v}{\partial z}} }[/math] (8)
[math]\displaystyle{ \overline{C'w'} = - K_{C} {\frac{\partial C}{\partial z}} }[/math] (9)

2) Vertical boundary conditions a) top boundary condition ( z = ζ (x,y,t ))

[math]\displaystyle{ K_{m} {\frac{\partial u}{\partial z}} = \tau_{S}^x \left (x,y,t \right ) }[/math] (10)
[math]\displaystyle{ K_{m} {\frac{\partial v}{\partial z}} = \tau_{S}^y \left (x,y,t \right ) }[/math] (11)
[math]\displaystyle{ K_{C} {\frac{\partial C}{\partial z}} = {\frac{Q_{C}}{\rho_{O} c_{P}}} }[/math] (12)
[math]\displaystyle{ w = {\frac{\partial \zeta}{\partial t}} }[/math] (13)

b) bottom boundary condition (z = -h(x,y))

[math]\displaystyle{ K_{m}{\frac{\partial u}{\partial z}} = \tau _{b}^x \left (x,y,t \right ) }[/math] (14)
[math]\displaystyle{ K_{m}{\frac{\partial v}{\partial z}} = \tau _{b}^y \left (x,y,t \right ) }[/math] (15)
[math]\displaystyle{ K_{C}{\frac{\partial C}{\partial z}} = 0 }[/math] (16)
[math]\displaystyle{ -w + \vec{v} \nabla h = 0 }[/math] (17)

c) Horizontal boundary conditions (Eastern and western boundary)(considering biharmonic friction)

[math]\displaystyle{ {\frac{\partial}{\partial x}} \left ( {\frac{h \nu}{mn}}{\frac{\partial^2 u}{\partial x^2}} \right ) = 0 }[/math] (18)

Horizontal boundary conditions (Northern and southern boundary)

[math]\displaystyle{ {\frac{\partial}{\partial y}} \left ( {\frac{h \nu}{mn}}{\frac{\partial^2 u}{\partial y^2}} \right ) = 0 }[/math] (19)

3) Vertical transformation equations a) vertical coordinate transformation (been available in ROMS since 1999)

[math]\displaystyle{ z \left (x,y,\delta, t \right ) = S \left (x,y,\delta \right )+ \zeta \left (x,y,t \right ) [1 + {\frac{S \left (x,y,\delta \right )}{h \left (x,y\right )}}] }[/math] (20)
[math]\displaystyle{ S \left (x,y,\delta \right ) = h_{c} \delta + [h \left (x,y\right ) - h_{c}] C \left (\delta \right) }[/math] (21)

b) vertical coordinate transformation (been available in ROMS since 2005)

[math]\displaystyle{ z \left (x,y,\delta,t \right ) = \zeta \left (x,y,t\right ) + [\zeta \left (x,y,t\right ) + h \left (x,y \right )] S \left (x,y,\delta \right ) }[/math] (22)
[math]\displaystyle{ S \left (x,y,\delta \right ) = {\frac{h_{c} \delta + h \left (x,y \right ) C \left ( \delta \right )}{h_{c} + h \left (x,y\right )}} }[/math] (23)

Notes

1 In the Boussinesq approximation, density variations are neglected in the momentum equations except in their contribution to the buoyancy force in the vertical momentum equation.



Examples

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See also: Help:Images or Help:Movies

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References

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