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Equations / variables Vertical coordinate Terrain representation Grid staggering

WRF Mass-Coordinate Dynamical Solver. Equations / variables Vertical coordinate Terrain representation Grid staggering Time integration scheme Advection scheme Boundary conditions Map projections Dynamics parameters. Flux-Form Equations in Mass Coordinate.

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Equations / variables Vertical coordinate Terrain representation Grid staggering

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

  2. Flux-Form Equations in Mass Coordinate Hydrostatic pressure coordinate: hydrostatic pressure Conserved state variables: Non-conserved state variable:

  3. Flux-Form Equations in Mass Coordinate Inviscid, 2-D equations without rotation: Diagnostic relations:

  4. Flux-Form Equations in Mass Coordinate Introduce the perturbation variables: Note – likewise Momentum and hydrostatic equations become:

  5. Flux-Form Equations in Mass Coordinate Acoustic mode separation: Recast Equations in terms of perturbation about time t Linearize ideal gas law about time t Vertical pressure gradient becomes

  6. Flux-Form Equations in Mass Coordinate Small (acoustic) timestep equations:

  7. Moist Equations in Mass-Coordinate Model Moist Equations: Diagnostic relations:

  8. Mass-Coordinate Model, Terrain Representation Lower boundary condition for the geopotential specifies the terrain elevation, and specifying the lowest coordinate surface to be the terrain results in a terrain-following coordinate. Vertical coordinate: hydrostatic pressure

  9. y W V U U U U W V x x Height/Mass-coordinate model, grid staggering C-grid staggering z horizontal vertical

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

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

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

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

  14. Acoustic Integration in the Mass Coordinate Model Forward-backward scheme, first advance the horizontal momentum Second, advance continuity equation, diagnose omega, and advance thermodynamic equation Finally, vertically-implicit integration of the acoustic and gravity wave terms

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

  16. Advection in the Height/Mass Coordinate Model For constant U, the 5th order flux divergence tendency becomes 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.

  17. WRF Mass-Coordinate Model Integration Procedure Begin time step 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 m, Q, then w, f (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) End time step

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

  19. Mass/Height Coordinate Model: Coordinate Options • Cartesian geometry (idealized cases) • Lambert Conformal • Polar Stereographic • Mercator

  20. Mass Coordinate Model: Dynamics Parameters 3rd order Runge-Kutta time step Courant number limited, 1D: Generally stable using a timestep approximately twice as large as used in a leapfrog model. Acoustic time step 2D horizontal Courant number limited: Divergence damping coefficient: 0.1 recommended. External mode damping coefficient: 0.05 recommended. Vertically-implicit off-centering parameter: 0.1 recommended. Advection scheme order: 5th order horizontal, 3rd order vertical recommended.

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