1. Terrain-following coordinates over steep and high orography
Oliver Fuhrer

2. Outline Terrain-following coordinates Two idealized tests
Atmosphere at rest
Constant advection
Possible remedy: SLEVE
How to specify SLEVE parameters?

3. Overview Terrain-following coordinates have many advantages...
rectangular computational mesh
lower boundary condition
stretching → boundary layer representation
But also many disadvantages...
additional terms
horizontal pressure gradient formulation (Janjic, 1989)
metric terms (Klemp et al., 2003)
truncation error (Schär et al., 2002)

4. Ideal test case I 2-dimensional Schaer et al. MWR 2002 topography
Gal-Chen coordinates
∆x = 1 km, Lx = 301 km
∆z = 500 m, Lz = 22.5 km
∆t = 25 s
Constant static stability frequency N = 0.01 s-1
Tropopause at 11 km
Rayleight sponge (> 12 km)
l2tls = .true.
irunge_kutta = 1
irk_order = 3
iadv_order = 5
lsl_adv_qx = .true.
lva_impl_dyn = .true.
ieva_order = 3

5. Sensitivity: Topography height
h = 0 m
h = 1 m
h = 3 m
h = 10 m
wmax = 2.8e-5 m/s
wmax = 2.8e-5 m/s
wmax = 2.8e-5 m/s
wmax = 3.0e-5 m/s
h = 30 m
h = 100 m
h = 300 m
h = 1000 m
Crash!!!
wmax = 4.0e-5 m/s
wmax = m/s
wmax = m/s
wmax > 80 m/s

6. Sensitivity: Overview

7. Explanation ?

8. Ideal test case II 2-dimensional Schaer et al. MWR 2002 topography
Gal-Chen coordinates
∆x = 1 km Lx = 301 km
∆z = 500 m Lz = 22.5 km
∆t = 25 s
Constant static stability frequency N = 0.01 s-1
Tropopause at 11 km
Rayleight sponge (> 12 km)
l2tls = .true.
irunge_kutta = 1
irk_order = 3
iadv_order = 5
lsl_adv_qx = .true.
lva_impl_dyn = .true.
ieva_order = 3

9. Sensitivity: Topography height
h = 0 m
h = 1 m
h = 3 m
h = 10 m
wmax = m/s
wmax = m/s
wmax = m/s
wmax = m/s
h = 30 m
h = 100 m
h = 300 m
h = 1000 m
Crash!!!
wmax = 0.02 m/s
wmax = 0.07 m/s
wmax = 0.4 m/s
wmax > 80 m/s

10. Sensitivity: Overview

11. Explanation Advection of intensive quantity
Theoretical analysis of trunction error

12. Explanation Upstream advection Centered advection
Error term solely due to transformation
Linear in h0 and u

13. Solution Gal-Chen SLEVE zmin = 16.5 m SLEVE2 zmin = 13.3 m

14. SLEVE2 formulation Gal-Chen coordinate SLEVE2 coordinate
Many parameters! How to choose?

15. Criteria from numerics...
Local truncation error is a function of metric terms...
...and their derivatives

16. Gal-Chen grid

17. SLEVE2 grid

18. Optimal grid? Optimize cost function
We have neglected cross-derivatives
Using n=2 for simplicity
Choose weights i inspired from SLEVE2 values

19. Optimized grid

20. SLEVE2 grid

21. Optimize SLEVE2
Choose h1 like a smoothed hull of topography
Adapt decay parameters to match “optimal” grid
We can optimize parameters of SLEVE2 using “optimal” grid
SLEVE2 vs. Optimized

22. Optimize SLEVE2
We can optimize parameters of SLEVE2 using “optimal” grid
Optimized SLEVE2
SLEVE2 vs. Optimized

23. Conclusions
COSMO shows considerable truncation errors in presence of steep and high topography
Both for atmosphere at rest and constant advection test cases
Better vertical coordinate transformations (SLEVE2) may reduce the amplitude at higher levels
But they have many parameters to choose and effects near the land-surface can be critical
Grid-optimization with numerically motivated cost function...
may be a tool to choose parameters intelligently
may assist in formulation of new vertical coordinates
also relies on some tuning

24. Outlook Test optimized SLEVE2 coordinate with idealized test cases
Implement optimized SLEVE2 coordinate operationally
Investigate source of error for atmosphere at rest case?
Behaviour of COSMO with respect to specific grid deformations could help determine weight’s of cost function
Extend study to different time and space differencing scheme’s