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A Modified Atmospheric Nonhydrostatic Model on Low Aspect Ratio Grids

A Modified Atmospheric Nonhydrostatic Model on Low Aspect Ratio Grids. Wen- Yih Sun 1,2 Oliver M. Sun 3 and Kazuhisa Tsuboki 4. 1.Purdue University, W. Lafayette, IN. 47907 USA 2.National Central University, Chung-Li, Tao- yuan , 320, Taiwan

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A Modified Atmospheric Nonhydrostatic Model on Low Aspect Ratio Grids

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  1. A Modified Atmospheric Nonhydrostatic Model on Low Aspect Ratio Grids Wen-Yih Sun1,2 Oliver M. Sun3 and Kazuhisa Tsuboki4 • 1.Purdue University, W. Lafayette, IN. 47907 USA • 2.National Central University, Chung-Li, Tao-yuan, 320, Taiwan • 3. Woods Hole Oceanographic Institute, Woods Hole, MA 02543-1050, USA • 4. Nagoya University, Nagoya, 464-8601. Japan

  2. 2. Basic Equations • (2.1) • (2.2) • (2.3) • (2.4) • (2.5) • (2.6)

  3. Linearized Equations • (2.7) • (2.8) • (2.9) • (2.10) • (2.11)

  4. Following Hsu and Sun (2001), we obtain .

  5. The solutions of differential in time and difference in space are assumed: (semi-analytic= reference) We obtain the dispersive relationship: includes sound wave and internal gravity wave

  6. Let define : If , It consists of the acoustic wave: and the gravity waves:

  7. 3. Eigenvalue of finite difference equations Modified Forward-backward

  8. Finite Difference form for HE-VI (Horizontal Explicit-Vertical Implicit) , =1

  9. Eigen value for HE-VI with =1

  10. (a) (b) Fig. 1: Frequency of gravity wave from FB with t=1.49s, g=10 m s-2, C=300m s-1, x=1000m, z=500m, and So=1.0-5m-1, (solid line) and (dash line); (b) and with t=2.98s; and (c) and with t=5.96s.

  11. Fig. 2: (solid line) with t=1.179x10-2 sx=z= 5m, and (dot line) is from -2.5x10-8 to 2.x10-9s-1 with contour interval of 5x10-9 s-1. (FB)

  12. (a) (b) Fig. 3: (a) with x=1000m, z= 500m, t=2.98s; contour interval of (dash line) is 5x10-8s-1; (b) Same as (a) except x= z= 5m,  t=1.179 x10-2s, contour interval of dot lines is 1x10-7s-1.

  13. (a) (b) Fig. 4: (a): Initial x-component wind for Kelvin-Helmholtz instability (b) initial background (solid line) and perturbation ’ (color) for 4.1.

  14. Fig. 5: (a): Simulated at t=320s with tb= 0.4s from (ts=0.01s, =0.01s, black), (ts=0.02s, green), (ts=0.04s, red), and (ts=0.02s, blue), contour interval is 0.5K;

  15. (b) Fig. 6: (a) Initial background (black) and perturbation ’ (color) for 4.2-4.3;(b) Simulated ’ at t=240s from FB with =1 (black), =4 (green), =16 (red), and HE-VI scheme (blue) with ts=0.0025s, tb=0.1s, and x= z=5m. Contour interval is 0.005K.

  16. Fig. 6: (a) Initial background (black) and perturbation ’ (color) for 4.2-4.3

  17. Fig. 6(b) Simulated ’ at t=240s from FB with =1 (black), =4 (green), =16 (red), and HE-VI scheme (blue) with ts=0.0025s, tb=0.1s, and x= z=5m. Contour interval is 0.005K

  18. Fig. 7: (a) Simulated ’ at t=240s from FB with =1, ts=0.005s (solid line); =4, ts=0.01s (long dash); and =16 ts=0.02s (short dash). Contour interval is 0.005K;

  19. Fig. 7b: ’ from HE-VI with ts=0.005s (dot line) and ts=0.01s (solid line). Contour interval is 0.005K.

  20. Initial condition for KH-wave, w(black, solid), Ub(red, dash), and  (green. dot) Simulated  Short KH-wave at t=240s, Forward-back(dts=.005,black, solid), Implicit (dts=.01s red, dash), Mod. Forward-B (dts=.01s blue. dot)

  21. Simulated vertical velocity w Short KH-wave at t=240s, Forward-back(dts=.005,black, solid), Implicit (dts=.01s red, dash), Mod. Forward-B (dts=.01s blue. dot)

  22. Simulated w Short KH-wave at t=480s, Forward-back(dts=.005,black, solid), Implicit (dts=.01s red, dash), Mod. Forward-B (dts=.01s blue. dot)

  23. (a) Fig. 8 (a) Initial  for FB with =1, ts = 0.008s, x= z=5m& tb= 0.12s. interval is 0.05K; (b) simulated  and v at t=6 m; (c) t=12 m; (d) at t=18 min.

  24. (d) Fig. 8: simulated velocity v and ’at (c) t=12 min; (d) at t=18 min.

  25. (a) (b) Fig. 9: Simulated v and at t=18 min: (a) with =16, ts = 0.03s; (b) with HE-VI, ts = 0.008s;

  26. (c) (d) (c)(ts = 0.03s)- (ts = 0.008s), contours from -0.03K to 0.06K with interval of 0.01K; (d) (ts = 0.008s)- (ts = 0.008s), contours from -0.3 to 0.2 with interval of 0.5K.

  27. (a) Fig. 10: Simulated w at t=6000s with x= 400m, z=125m and tb=10s, (a) (ts = 0.25s, black); (ts= 0.5s, green); (ts= 1.0 s, red); and (ts= 1.0s, blue);

  28. (b) (b) - (red) and - (blue), the contour interval is 0.02ms-1.

  29. Table 1: Variation of change of the total mass with respect to its initial value as function of time for MFB (=16, ts= 0.04s) and HE-VI (ts= 0.02s).

  30. 5. Summary • Modified nonhydrostatic model (MNH) with (between 4 and 16) applied to the continuity equation can suppress the frequency of acoustic waves very effectively with insignificant impact on gravity waves, which enables to use a longer time step . • MNH is simple, in which many numerical schemes can be easily incorporated. • The eigenvalues and nonlinear model simulations of Kelvin-Helmholtz instability, mountain wave, and thermal bubble when FB is applied to MNH (i.e., MFB) show that MFB can reproduce the results of the original NH very accurately and efficiently. • It is also found that the simulations from the HE-VI are consistent with those from FB if the time interval ts is very small, or the time variation of horizontal gradient is not as important as the vertical gradient within ts. Otherwise, HE-VI simulations can depart from the FB significantly. • MFB can use a higher-order scheme in space to simulate LES, and turbulence, etc., which requires a fine-resolution in both horizontal and vertical directions.

  31. Simulations of Lee vortices by WRF & NTU-Purdue Model NTU-Purdue Model use modified semi-implicit scheme with =4 discussed in (E) Model configuration and experiment design –idealized mountain Grid points : 600 x 600 x 50 in x, y, z directions dx = dy = 1 km dz ~ 300 m Free-slip lower boundary condition U = 4 m s-1 N2 = 10-4 s-2 Mountain peak = 2 km Half width lengths: 5 and 10 km (tilted by 30 degrees). Fr = V/Nh = 0.2

  32. Surface streamlines after 10h NTU-Purdue WRF

  33. Surface streamlines after 20h NTU-Purdue WRF x

  34. Sea level pressure after 10h Interval 0.05 hPa Interval 5 Pa NTU-Purdue WRF

  35. Sea level pressure after 20h Interval 5 Pa Interval 0.05 hPa NTU-Purdue WRF

  36. Procedure • The Purdue/ NTU model was initialized with the El Paso 12Z sounding. • Model results are shown for 2 - 4 hours of model forecasts. • Also shown are Meso-g and WSMR Surface Atmosphere Measuring System (SAMS) wind and pressure observations, the White Sands Profiler and other data. • For 01-25-2004, both the model and observations show strong downslope winds, and a stationary wave train extending downwind from the mountains, and apparently a hydraulic jump. For 01-19-2004, both the model and observations show blocked flow in the lee of the Organ mountains and downslope flow to the north of the WSMR post area.

  37. NTU/Purdue WSMR Model Domain 201 x 201 Grid @ 1km Dx, Dy

  38. El Paso sounding for 12 Z January 25, 2004 Scorer Parameter } l = 3.5 x 10-4 } l = 5.7 x 10-4

  39. Observed and Modeled WSMR Pressure Perturbations for 0800 January 25, 2004 Color Lines: Purdue/NTU NH Model Pressure Perturbations (pascals) 0 to -200 Dark Bold Numbers: Observed Pressure Perturbations -175* -139 -149 -103 -91 -111* 0* -27 Grid number west to east direction E

  40. WSMR 10 m wind field 0800 January 25, 2004 Hydraulic Jump/ Trapped Lee Waves 201x201 grid 1km Grid spacing. This plot shows the central part of the model domain Wind Barbs: Purdue/NTU NH Model Dark Arrows: Observed Winds Color contours show the model terrain in m asl. Grid number west to east direction

  41. WSMR 10 m wind field 0800 January 25, 2004 Hydraulic Jump/ Trapped Lee Waves Wind Vectors: WRF NH Model Red Arrows: Observed Winds 199x199 grid 1 km grid spacing. This plot shows the central part. Color contours show the model terrain in m asl

  42. (a) wind inWhite Sands after 4-hr integration, (dx=dy=2km, and dz=300m). initial wind U= 5 m/s; (b) Streamline (white line) and virtual potential temperature (background shaded colors) at z=1.8km, warm color (red) indicates subsidence warming on the lee-side, and cold (blue) color adiabatic cooling on the windward side of the mountain.

  43. 0700-0800 + 0800-0900 # + # + # + # + # + # + # + # + # # # + # + # 0730 + + # 0800 + # + + # 0830 # + + 0900 vertical velocity m/sec WSMR Wind Profiler Vertical Velocities NTU/Purdue model vertical velocities (connected lines) Wind Profiling Radar observed vertical velocities (+ and # ) with 0.5 m/ sec error bars. The vertical scale is altitude in meters asl WSMR Profiler 21000 NTU/Purdue Model Vertical Velocities at 0730 0800 0830 0900 18000 15000 12000 9000 6000 Average terrain height upwind 3000 Elevation of Basin Floor

  44. Conclusions * Both theory & models show that modified forward-backward scheme & semi-implicit scheme with =16 are accurate for short waves in nonhydrostatic equations. * For an the idealized mountain, the simulations from both models are similar at 10-hrs integration, but difference becomes significant after 20-hrs integration. It is also noticed that a stagnation exists at the windward side in the NTU-Purdue model but not in the WRF. * The WSMS simulations show in the lee of the Organ Mountains, the WRF simulated wind mainly comes from the South without the formation of lee-vortices on the lee side. On the other hand, the wind simulated from the NTU-Purdue model mainly descends from the Mountains and forms the lee-vortices, which are in good agreement with observations. The vertical wind profile from the NTU-Purdue is also consistent with the observed wind according to the ARL wind profiler.

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