1. – Equations / variables – Vertical coordinate – Terrain representation – Grid staggering – Time integration scheme – Advection scheme – Boundary conditions – Map projections – Dynamics parameters WRF Height-Coordinate Dynamical Solver

2. Conservative variables: Inviscid, 2-D equations in Cartesian coordinates Flux-Form Equations in Height Coordinates where we have removed a hydrostatic base state

3. Pressure terms directly related to Flux-Form Equations in Height Coordinates Acoustic mode separation Recast Equations in terms of perturbation about time t Linearize ideal gas law about time t:

4. Flux-Form Equations in Height Coordinates

5. Moist Equations in Height-Coordinate Model Moist Equations

6. Height-Coordinate Model, Terrain Transform Vertical coordinate transformation Redefine variables and likewise for perturbation variables

7. Height-coordinate model, grid staggering z W W UU x V V UU x y C-grid staggering horizontalvertical

8. Height-Coordinate Model, Time Integration 3 rd Order Runge-Kutta time integration advance Amplification factor

9. Time-Split Leapfrog and Runge-Kutta Integration Schemes

10. Phase and amplitude errors for LF, RK3 Oscillation equation analysis

11. Phase and amplitude errors for LF, RK3 Advection equation analysis 5 th and 6 th order upwind-biased and centered schemes. Analysis for 4  x wave.

12. Acoustic Integration in the Height Coordinate Model Forward-backward scheme, first advance the horizontal momentum Second, vertically-implicit integration of the acoustic and gravity wave terms

13. Advection in the Height Coordinate Model 2 nd, 3 rd, 4 th, 5 th and 6 th order centered and upwind-biased schemes are available in the WRF model. Example: 5 th order scheme where

14. For constant U, the 5 th order flux divergence tendency becomes Advection in the Height Coordinate Model (cont.) The odd-ordered flux divergence schemes are equivalent to the next higher ordered (even) flux-divergence scheme plus a dissipation term of the higher even order with a coefficient proportional to the Courant number.

15. Runge-Kutta loop (steps 1, 2, and 3) (i) advection, p-grad, buoyancy using (ii) physics if step 1, save for steps 2 and 3 (iii) mixing, other non-RK dynamics, save… (iv) assemble dynamics tendencies Acoustic step loop (i) advance U,V, then W, (ii) time-average U,V,W End acoustic loop Advance scalars using time-averaged U,V,W End Runge-Kutta loop Other physics (currently microphysics) Begin time step End time step WRF Model Integration Procedure

16. Height Coordinate Model: Boundary Condition Options 1.Specified (Coarse grid, real-data applications). 2.Open lateral boundaries (gravity-wave radiative). 3.Symmetric lateral boundary condition (free-slip wall). 4.Periodic lateral boundary conditions. 5.Nested boundary conditions (not yet implemented). Lateral boundary conditions Top boundary conditions 1.Rigid lid. 2.Gravity-wave radiative condition. 3.Absorbing upper layer (increased horizontal diffusion). 4.Rayleigh damping upper layer (not yet implemented). Bottom boundary conditions 1.Free slip. 2.Various B.L. implementations of surface drag, fluxes.

17. Height Coordinate Model: Coordinate Options 1.Cartesian geometry (idealized cases) 2.Lambert Conformal 3.Polar Stereographic 4.Mercator

18. 3 rd order Runge-Kutta time step Acoustic time step Divergence damping coefficient: 0.1 recommended. Vertically-implicit off-centering parameter: 0.1 recommended. Advection scheme order: 5 th order horizontal, 3 rd order vertical recommended. Height Coordinate Model: Dynamics Parameters Courant number limited, 1D: Generally stable using a timestep approximately twice as large as used in a leapfrog model. 2D horizontal Courant number limited: