Wen-Yih Sun 1,2 Oliver M. Sun 3 and Kazuhisa Tsuboki 4 1.Purdue University, W. Lafayette, IN USA 2.National Central University, Chung-Li, Tao-yuan, 320, Taiwan 3. Woods Hole Oceanographic Institute, Woods Hole, MA , USA 4. Nagoya University, Nagoya, Japan

2.Basic Equations

Linearized Equations

. Following Hsu and Sun (2001), we obtain

The solutions of differential in time and difference in space are assumed: (semi-analytic= reference) We obtain the dispersive relationship:

Let define  : and the gravity waves:

3. Eigenvalue of finite difference equations Modified Forward- backward

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

Eigen value for HE-VI with  =1

(a) (b)

(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.

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

(b) Fig. 6: (a) Initial background (black) and perturbation  ’ (color) for ; (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.

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

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

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;

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

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)

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)

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)

(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.

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

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

(c)( d)

(a)

(b)

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). Time= 0 s80 s160 s240 s320 s [M 16 (t)-M(0)] /M(0) x x10 -7 [M im (t)-M(0)] /M(0) x x x10 -6

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.

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 N 2 = s -2 Mountain peak = 2 km Half width lengths: 5 and 10 km (tilted by 30 degrees). Fr = V/Nh = 0.2 Simulations of Lee vortices by WRF & NTU-Purdue Model NTU-Purdue Model use modified semi-implicit scheme with  =4 discussed in (E)

Surface streamlines after 10h WRFNTU-Purdue

x Surface streamlines after 20h NTU-PurdueWRF

Sea level pressure after 10h WRFNTU-Purdue Interval 0.05 hPa Interval 5 Pa

Sea level pressure after 20h WRFNTU-Purdue Interval 5 Pa Interval 0.05 hPa

Procedure The Purdue/ NTU model was initialized with the El Paso 12Z sounding. The Purdue/ NTU model was initialized with the El Paso 12Z sounding. Model results are shown for hours of model forecasts. Model results are shown for hours of model forecasts. Also shown are Meso-  and WSMR Surface Atmosphere Measuring System (SAMS) wind and pressure observations, the White Sands Profiler and other data. Also shown are Meso-  and WSMR Surface Atmosphere Measuring System (SAMS) wind and pressure observations, the White Sands Profiler and other data. For , 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 , 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. For , 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 , 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.

NTU/Purdue WSMR Model Domain 201 x 201 1km  x,  y

El Paso sounding for 12 Z January 25, 2004 l = 3.5 x l = 5.7 x } } Scorer Parameter

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

WSMR 10 m wind field 0800 January 25, 2004 Hydraulic Jump/ Trapped Lee Waves Color contours show the model terrain in m asl. Grid number west to east direction 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

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

(a) wind in  White 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.

vertical velocity m/sec # # # # # # # # # # # # # # # # # 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 Wind Profiler Vertical Velocities NTU/Purdue Model Vertical Velocities at WSMR Profiler Average terrain height upwind Elevation of Basin Floor

* 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. Conclusions