1. Regional Weather and Climate Modeling:
Atmospheric Science
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 (
Peter Lynch class notes (
Chris Bretherton class notes (
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
Rn m and Sn m, are fully normalized spherical harmonics.
Pn m (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. We obtain a centered one
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
u is a scalar
C is advective speed
periodic waves
sines and cosines
Impose simple wave solution
Can solve advection equation analytically

15. F(t)  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. u = A eiωt Courant Number: Critical for Computational stability
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. Keep your timestep small!
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. Leapfrog scheme Centered space difference (second order)
Apply centered differencing in time.
Requires 2 previous values to calculate next value.
(advection equation)

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

22. WRF has an upstream advection scheme, but much more complicated
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 Z or P Sigma = P / Ps 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. From WRF Preprocessing System (WPS)
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 =
ref_lon =
truelat1 = ,
truelat2 = ,
stand_lon =
geog_data_path = '/data3a/mp/gill/DATA/GEOG'
opt_geogrid_tbl_path = 'geogrid/'
From WRF Preprocessing System (WPS)

32. The Hexagon Global Grid?
What is next after WRF?
The Hexagon Global Grid?