Regional Weather and Climate Modeling Techniques: Finite Difference Equations Explained

1. Atmospheric Science 820.01 Regional Weather and Climate Modeling: Finite Difference Equations, Vertical Coordinates & Model Grids Keith Hines Byrd Polar Research Center Sources: Wikipedia A.J. Broccoli class notes (http://www.envsci.rutgers.edu/~broccoli/) Peter Lynch class notes (http://mathsci.ucd.ie/met/msc/NWP/MAPH-P313.html) Chris Bretherton class notes (http://www.atmos.washington.edu/2002Q2/547/) Mesoscale Meteorology and Forecasting (Ray) Climate System Modeling (Trenberth) An Introduction to Three-Dimensional Climate Modeling (Washington and Parkinson)

2. Atmospheric modeling Finite difference Equations Vertical Coordinates Model Grids

3. Solving the atmospheric equations on an Earth atmosphere grid Numerical Weather Prediction solves for dynamics, kinematics, physics (radiation, latent heat, surface exchange) and hydrology

4. Finite Differencing Space Time

5. Use continuous functions rather than grid points? can use multiple sine and cosine waves As we add more waves to the sum more advanced shapes can be represented Finite Element or Galerkin Method

6. Use continuous functions rather than grid points Cartesian Coordinates: 2D function can be expressed as a sum of as a series of waves (Fourier Series) Differencing with sums of functions is often called “exact” although truncation errors still exist due to the truncated (rather than infinite) series Spherical Coordinates: Use spectral techniques with sums of associated Legendre functions: θ is colatitude, λ is longitude, anm and bnm are normalized harmonic coefficients Rnm and Snm, are fully normalized spherical harmonics. Pnm (cos θ) is an Associated Legendre Function.

7. Grid Points – still popular for limited-area models Δ y Represent the atmospheric variables over a rectangular grid. spherical coordinates

8. How do we solve our differential equations? x0 - 2Δx F(x0) x0 + 2Δx Set up a grid F(x)  F(x0 - Δx) F(x0 + Δx)

9. Approximate derivative with finite difference

10. What is the error in finite difference? Error is order “h” What if we combine two unbalanced differences? Finite difference approx. from right and left sides We obtain a centered one Centered differences is 2nd order

11. What if we want a higher order difference scheme? 4th Order Solution: Use more points Above scheme uses 4 adjacent points

12. Numerical Techniques What do we want in our numerical models? • Accuracy (obvious) • Balance (not dominated by “noise”) • Conservation of key physical properties • Numerical Stability (model survives run) • Efficiency (obvious)

13. (early numerical weather prediction) Quasi-Geostrophic Model geostrophic horiz. velocity geostrophic vorticity (curl) Vorticity Equation Thermodynamic Equation Physical Mode + Computational Mode Omega Equation (forces balance between V and T) Or Synoptic Feature + Sound Waves + Gravity Waves How do we achieve balance in a primitive-equation model?

14. How do we test for computational stability? Can test with simple (linear) advection equation C is advective speed u is a scalar periodic waves sines and cosines Impose simple wave solution Can solve advection equation analytically

15. Could test forward (Euler) time scheme Or test centered (leapfrog) scheme t0 - 2Δt F(t0) t0 + 2Δt F(t)  F(t0 – Δt) F(t0 + Δt)

16. Imposed wave solution Courant Number: Critical for Computational stability u = A eiωt μ < 1 for stability (CFL Condition) smaller timestep more stability

17. Courant Number: Critical for Computational stability Imagine a wave being advected by a speed “c” in a model with grid spacing and time step If a wavelet is advected more than in there can be instability.

18. u = A eiωt Courant Number: Critical for Computational stability μ < 1 for stability (CFL Condition) smaller timestep more stability Diffusion Equation Stability criteria Keep your timestep small!

19. Centered space difference (second order) Leapfrog scheme Apply centered differencing in time. Requires 2 previous values to calculate next value. (advection equation)

20. False Negatives Red curve “real” solution Blue curve numerical solution Numerical solution can cause scalars to improperly become negative

22. Upstream advection scheme x0 - 2Δx F(x0) x0 + 2Δx U  downstream F(x0 - Δx) F(x0 + Δx) advection - U(x0) [ F(x0) - F(x0 - Δx) ] / Δx for U > 0 - U(x0) [ F(x0 + Δx) - F(x0) ] / Δx for U < 0 WRF has an upstream advection scheme, but much more complicated

23. Time Integration k1 k4 k2 ,k3 4 Steps to do one step

24. 3rd Order Runge-Kutta time steps used in WRF 3rd order accuracy used with fast acoustic time steps “physics” are outside RK3

25. Vertical Coordinate Sigma = P / Ps Z or P Intersects topography Terrain following dz/dt, dσ/dt, dp/dt u, v, P, T, q dz/dt, dσ/dt, dp/dt

26. Modified sigma-coordinate Vertical Velocity Model Levels u, v, T Vertical Velocity terrain-following

27. Numerical Techniques: Grids

28. Arakawa Horiz. Staggered Grid Categories Grid A (simplest) Grid B Grid C Grid D Grid E Has advantages for GW dispersion T, P, q (physical scalars) u, v (velocity) Treat gravity waves to simulate desired synoptic waves

29. Map Projections Mercator Lambert Conformal Polar Stereographic

30. Map Scale Factor

31. Map Parameters &geogrid s_we = 1, e_we = 74, s_sn = 1, e_sn = 61, geog_data_res = '10m','2m', dx = 30000, dy = 30000, map_proj = 'lambert', ref_lat = 34.83 ref_lon = -81.03 truelat1 = 30.0, truelat2 = 60.0, stand_lon = -98.0 geog_data_path = '/data3a/mp/gill/DATA/GEOG' opt_geogrid_tbl_path = 'geogrid/' From WRF Preprocessing System (WPS)

32. What is next after WRF? The Hexagon Global Grid? http://www.mmm.ucar.edu/wrf/users/workshops/WS2009/presentations/6-06.pdf