1. CHANGSHENG CHEN, HEDONG LIU, And ROBERT C. BEARDSLEY
An Unstructured Grid, Finite-Volume, 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

2. Primitive equations Momentum (u component) Momentum (v component)
density
continuity
Advection of temperature
Advection of salinity
Density as a function of S and T

3. Eddy viscosity Turbulent kinetic energy (TKE)
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

4. Eddy viscosity (continued)
Stability functions
Function of density, TKE and length scale

5. 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)

6. 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

7. 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

8. Unstructured grids E NBi(4) NBi(3) NBi(5) Ni(1) Nbi(2) NBi(6) NBi(1)
NBE(1)
NBE(2) E
Ni(3)
Ni(2)
NBE(3)

9. 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

10. Standard 4th order Runge-Kutta
From
Modified 4th order Runge-Kutta
N+1 incorporates initial value and 3rd estimate

11. 2D Numerical method k = 1:4 = (1/4, 1/3, 1/2, 1) Figure not from paper
p2m
P2m-1
P2m+1
2D Numerical method
k = 1:4
= (1/4, 1/3, 1/2, 1)
Figure not from paper

12. 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

13. 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)

14. Bohai Sea

15. Tidal wave propagation
Improved resolution of features
Tidal wave propagation
Temperature structure

16. Satilla River
Tidal performance

17. General velocity structure Detailed velocity structure
Not clear why there is no velocity in the streams in ECOM
More complex eddy structure in FVCOM