An Unstructured Grid, FiniteVolume, Three-Dimensional,
Primitive Equations Ocean
Model: Application to Coastal
Ocean and Estuaries
CHANGSHENG CHEN, HEDONG LIU,
And ROBERT C. BEARDSLEY
Summary by Charles Seaton,
all formulae and figures taken from paper unless otherwise specified
Primitive equations
Momentum (u component)
Momentum (v component)
density
continuity
Advection of temperature
Advection of salinity
Density as a function of S and T
Eddy viscosity
•
Mellor and Yamada 2.5 turbulence closure scheme
Turbulent kinetic energy (TKE)
Vertical shear – source for TKE
Density (in)stability – source or sink for TKE
TKE dissipation – sink for TKE
Law of the Wall (E?)
Distance from bed and surface
Eddy viscosity (continued)
Stability functions
Function of density, TKE and length scale
Boundary conditions
Surface boundary for u and v is a function of wind shear
Surface boundary condition for w
Bottom boundary for u and v is a function of bottom stress
Bottom boundary condition for w is a function of bathymetry
Temperature has surface heat flux and shortwave radiation sources
Salinity has surface precipitation and
evaporation
Solid horizontal boundaries have 0
velocity and S and T advection normal
to the boundary
(from manual)
Vertical grid
Sigma coordinates (level depths normalized as a fraction of
total depth (bathymetry + free surface height)
Horizontal diffusion term
(horizontal diffusion is restricted
to single layer)
Sigma layers can be uniform or parabolic
2D depth averaged equations
• Solution for sea surface elevation is determined using
depth averaged velocities
• Other variables (u,v,w,S,T,etc) are solved in 3D using
the sea surface elevation from the 2D calculations
• “mode splitting” 2D (“external”) and 3D (“internal”) modes
are calculated on different timesteps
2D continuity equation
Unstructured grids
NBi(4)
NBi(3)
Ni(1)
Nbi(2)
NBi(1)
NBE(1)
Ni(3)
E
NBE(3)
NBi(5)
NBi(6)
NBE(2)
Ni(2)
2D External mode
Integrated continuity equation
• Numerically integrated using modified 4th order Runge-Kutta
• Accuracy is 2nd order, as formulation sets final weights of
steps 1 and 2 to 0
• Depth averaged velocity and surface elevation are calculated
simultaneously for each sub-step
Standard 4th order Runge-Kutta
From http://www.physics.orst.edu/~rubin/nacphy/ComPhys/DIFFEQ/mydif2/node6.html
Modified 4th order Runge-Kutta
N+1 incorporates initial value and 3rd estimate
2D Numerical method
P2m+1
P2m-1
p2m
k = 1:4
= (1/4, 1/3, 1/2, 1)
Figure not from paper
3D Numerical method
1st order upwind advection scheme (other schemes available)
1 level momentum function
All the complexity
mid-level velocity
Next timestep is a function of mid-level velocities
Creates a tri-diagonal matrix
Merging the internal and external modes
Vertical velocity is calculated to merge results of 2D and 3D modes
Vertically integrated form
Valid if:
Distribute error in u and v throughout water column
before calculating w
(Functions from manual)
Bohai Sea
Improved resolution
of features
Tidal wave propagation
Temperature structure
Satilla River
Tidal performance
General velocity structure
Not clear why there is no
velocity in the streams in ECOM
Detailed velocity structure
More complex eddy structure
in FVCOM