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.

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

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

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

b. Flow over topography

Continuous solution (Nonhydrostatic) Discrete solution

Continuous solution (hydrostatic) Discrete solution △x’=0.67 △x’=1.35

c. Pressure drag Non- hydrostatic Hydrostatic

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.

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

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

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.

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

--Prototypical westerly flow More realistic atmospheric structures a. Time-dependent linear numerical model --Prototypical westerly flow △x’ = 0.67 △x’ = 0.67 △x’ = 0.17

b. Real case Using COAMPS model △x= 27km, 9km, 3km △t = 35.937s, 10.89s, 3.3s 70-member ensemble simulation

2-order 4-order

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

2nd-order 4th-order

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.

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.