1. Reporter: Prudence Chien
The Overamplification of Gravity Waves in Numerical Solutions to Flow over Topography
Reporter: Prudence Chien
Reference:
Reinecke, P.A., and D. Durran, 2009: The Overamplification of Gravity Waves in Numerical Solutions to Flow over Topography. Mon. Wea. Rev., 137, 1533–1549.

2. Outline Discrete flow over topography Group-velocity analysis
More realistic atmosphere structures
Conclusions

3. Discrete flow over topography a. The discrete Boussinesq system
Governing Equations

5. Horizontal wavenumber: k
Vertical wavenumber: l
Oscillation frequency: ω
Semidiscrete dispersion relationship
Continuous dispersion relationship
In the limit of good horizontal and vertical resolution
implying that

6. b. Flow over topography

7. Continuous solution
(Nonhydrostatic)
Discrete solution

8. Continuous solution
(hydrostatic)
Discrete solution
△x’= △x’=1.35

9. c. Pressure drag
Non-
hydrostatic
Hydrostatic

10. Group-velocity analysis a. Continuous and discrete group velocities
In continuous case, horizontal and vertical group-velocity:
2nd-order is unable to accurately approximate the correct nonhydrostatic group velocities for any value of δ’ between 0.5 and 2.5.
The 6th-order scheme perform well, even at the shortest horizontal wavelengths.

11. b. Angle of propagation In the continuous solution δ ‘=1.55, Θ=50 °
δ ‘=8.63, Θ=80 ° for 4△x-wide mountain

12. c. Higher-order finite differences on the staggered mesh
At intermediate resolution the 6th-order scheme often generates larger errors than 4th-order scheme.

13. At very fine resolutions
One way to improve the solution obtained using sixth-order advection is to employ a fourth-order approximation of the derivatives on the staggered mesh:
6-4 scheme:
6th-order advection with 4th-order pressure gradient and divergence.

14. d. Unstaggered meshes
No wave energy propogates upstream for λx/△x > 4

15. --Prototypical westerly flow
More realistic atmospheric structures a. Time-dependent linear numerical model
--Prototypical westerly flow
△x’ = △x’ = △x’ = 0.17

16. b. Real case Using COAMPS model △x= 27km, 9km, 3km
△t = s, 10.89s, 3.3s
70-member ensemble simulation

18. 2-order
4-order

19. Downslope wind speeds at AGL 10m
2nd-order
4th-order
Obs.
2nd-order
4th-order

20. 2nd-order
4th-order

21. Conclusions
2nd-order FD method produces significant errors in the wave field.
Nonhydrostatic wave: 30%-40% overamplification.
Hydrostatic wave: 7% underamplification.
In nonhydrostatic solution the group-velocity vector for the dominant wave forced by the coarsely resolved topography doesn’t point sufficiently downstream.

22. Group-velocity errors are mostly due to C-grid staggering.
Numerical errors will be present when the orographically forced gravity waves are poorly resolved.
Nonhydrostatic effects
Numerical scheme
Avoid discretization errors:
Remove the poorly resolved wavelengths from the topographic forcing
Use 4th-order advection
Group-velocity errors are mostly due to C-grid staggering.
Ensemble mean vertical velocities forced by the topography are larger in the 2nd–order solution than in 4th-order solution.