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

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 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

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 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)

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 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

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

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

Satilla River Tidal performance

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