1. 中央大學大氣科學系 1 Transient Mountain Waves in an Evolving Synoptic-Scale Flow and Their Interaction with Large Scales Chih-Chieh (Jack) Chen, Climate and Global Dynamics Division National Center for Atmospheric Research Dale R. Durran and Gregory J. Hakim Department of Atmospheric Sciences University of Washington April 24, 2007

2. 中央大學大氣科學系 2 Outline Background and Motivation Background and Motivation Methodology and Experimental Design Methodology and Experimental Design Results Results mesoscale response mesoscale response large-scale response large-scale response Summary Summary

3. 中央大學大氣科學系 3 Mountain Waves Queney (1948) idealized 2D mountain constant N and U linear stationary hydrostatic, non-rotating h = 1 km a = 10 km U = 10 m s -1 pressure drag

4. 中央大學大氣科學系 4 Momentum Flux and Pressure Drag Breaking U pressure drag “action at a distance” a sink for momentum HL

5. 中央大學大氣科學系 5 Gravity Wave Drag Parameterization Current parameterizations assume the wave are in steady state with the large-scale flow. Relatively little research has be devoted to mountain waves in a slowly evolving flow. Relatively little research has be devoted to mountain waves in a slowly evolving flow. Suppose the waves develop and decay over a period of two days? Does transience matter on this time scale? Suppose the waves develop and decay over a period of two days? Does transience matter on this time scale? Do the current GWD parameterizations do a good job in capturing the “true” response? Do the current GWD parameterizations do a good job in capturing the “true” response? Determine momentum flux carried by the waves Determine level of wave overturning Apply a decelerating force at that level

6. 中央大學大氣科學系 6 Transient Mountain Waves Bell (1975) Bell (1975) Bannon and Zhender (1985) Bannon and Zhender (1985) Lott and Teitelbaum (1993) Lott and Teitelbaum (1993)

7. 中央大學大氣科學系 7 Transient Mountain Waves Lott and Teitelbaum (1993) : maximum mean flow  : period : half width of mountain U = U(t) 2D configuration large-scale dynamics unspecified

8. 中央大學大氣科學系 8 Goals of the study To study characteristics of transient mountain waves embedded in a slowly evolving large-scale flow. ( ) momentum flux distribution time evolution of pressure drag Does transience matter? What is the impact of these disturbances on the large-scale flow? global momentum budgets spatial response Can a current GWD parameterization scheme capture the actual spatial flow response?

9. 中央大學大氣科學系 9 Methodology numerical model following Durran and Klemp (1983) and Epifanio and Durran (2000) nonlinear and nonhydrostatic nonlinear and nonhydrostatic f-plane approximation (f = 10 -4 s -1 ) f-plane approximation (f = 10 -4 s -1 ) Boussinesq approximation Boussinesq approximation parameterized subgrid-scale mixing parameterized subgrid-scale mixing terrain-following coordinates terrain-following coordinates

10. 中央大學大氣科學系 10 Construction of the Synoptic-scale Flow Desirable features: At least one ascending/descending phase for the mean wind At least one ascending/descending phase for the mean wind At least one stagnation point at the ground At least one stagnation point at the ground Dynamics well understood without mountain Dynamics well understood without mountain We have chosen: A nondivergent barotropic flow with uniform stratification (constant N 2 A nondivergent barotropic flow with uniform stratification (constant N 2 ). The streamfunction includes a sinusoidal square wave in both x and y. The streamfunction includes a sinusoidal square wave in both x and y.

11. 中央大學大氣科學系 11 Construction of Synoptic-scale Flow

12. 中央大學大氣科學系 12 Initial Condition Ingredients doubly periodic

13. 中央大學大氣科學系 13 Boundary Conditions Periodic in x and y Periodic in x and y Upper boundary is a rigid lid with scale-selective sponge layer Upper boundary is a rigid lid with scale-selective sponge layer 1.Fourier transform flow fields. 2.Zero short-wavelength Fourier coefficients. 3.Inverse transform back to physical space to obtain the “large-scale” flow. 4.Rayleigh damp perturbations about this large-scale flow.

14. 中央大學大氣科學系 14 Domain Setup and Model Resolution  x = 6 km 300  x  y = 6 km 300  y H = 16 km  z = 150~500 m sponge layer H = 16 km

15. 中央大學大氣科學系 15 u’ and  forced by h = 250 m

16. 中央大學大氣科學系 16 Horizontal Group Velocity Is Doppler Shifted by the Synoptic Flow Dispersion relation for 2D gravity waves For stationary waves at Horizontal group velocity of mountain wave packet launched at time

17. 中央大學大氣科學系 17 How Does the Domain Averaged Momentum Flux Vary with Time and Height?

18. 中央大學大氣科學系 18 Hypothetical z-t Momentum Flux Distribution under linear theory: t z  /2 - - + +

19. 中央大學大氣科學系 19 Momentum Flux Forced by h = 250 m constant U 10 m/s constant U 20 m/s

20. 中央大學大氣科學系 20 Vertical Group Velocity Increases with the Speed of the Synoptic Flow Dispersion relation for 2D gravity waves For stationary waves at Vertical group velocity of mountain wave packet launched at time

21. 中央大學大氣科學系 21 WKB Ray Tracing for U = U(t) U increasing with time t = t 1 t = t 2 t = t 3 U decreasing with time t = t 4 t = t 5 t = t 6

22. 中央大學大氣科學系 22 Ray Path Diagram: x-z plane

23. 中央大學大氣科學系 23 Ray Path Diagram: z-t plane

24. 中央大學大氣科學系 24 Conservation of Wave Action Wave action density changes when neighboring rays converge or diverge

25. 中央大學大氣科學系 25 Momentum Flux Changes Along a Ray Ways to change momentum flux change wave action (convergence or divergence of neighboring rays) change intrinsic frequency and/or local wavenumbers And for hydrostatic Boussinesq gravity waves:

26. 中央大學大氣科學系 26 Change of intrinsic frequency k increases k decreases x y Accelerating Phase x y Decelerating Phase

27. 中央大學大氣科學系 27 Momentum Flux Forced by h = 125 m model outputWKB solution

28. 中央大學大氣科學系 28 Momentum Flux for Higher Mountain h = 250 mh = 500 mh = 1 km

29. 中央大學大氣科學系 29 Pressure Drag Evolution in steady-state framework, drag U h = 125 m

30. 中央大學大氣科學系 30 Nonlinearity and Past History higher  lower 

31. 中央大學大氣科學系 31 Large-Scale Flow Response global momentum budgets global momentum budgets spatial response spatial response

32. 中央大學大氣科學系 32 Momentum Budget Perspective

33. 中央大學大氣科學系 33 Forcing for Zonal Mean Flow h = 1.5 km

34. 中央大學大氣科學系 34 Global Response for h = 1.5 km

35. 中央大學大氣科學系 35 Zonally-averaged fields

36. 中央大學大氣科學系 36 Zonally-averaged fields at 30 h

37. 中央大學大氣科學系 37 Spatial Response 1.The dynamics of the large-scale flow is well known in the absence of a mountain. 2. We may define “difference fields” as

38. 中央大學大氣科學系 38 Difference fields t = 25 hours z = 1.5 km

39. 中央大學大氣科學系 39 Difference fields z = 1.5 km

40. 中央大學大氣科學系 40 Difference fields t = 50 hours z = 1.5 km

41. 中央大學大氣科學系 41 Difference fields t = 50 hours z = 3.5 km

42. 中央大學大氣科學系 42 Can the flow response be explained by balanced dynamics? PV difference is inverted by using geostrophic balance as the balance constraint.

43. 中央大學大氣科學系 43 u difference vs balanced u t = 50 hours z = 1.5 km

44. 中央大學大氣科學系 44 u difference vs balanced u t = 50 hours z = 3.5 km

45. 中央大學大氣科學系 45 Implication of PV Inversion What is the effect of GWD? Can we recover the spatial response by using a GWD parameterization scheme?

46. 中央大學大氣科學系 46 GWD Parameterization Experiment Assuming: Gravity wave drag is deposited in the mountainous region (area = ) only.

47. 中央大學大氣科學系 47 GWD Parameterization Experiment 18 km (exact) t = 50 hours z = 3.5 km (GWD Exp.) -6 m/s -2 m/s

48. 中央大學大氣科學系 48 Large scale flow response h = 125 m 0.01 m/s h = 250 m 0.02 m/s h = 500 m 0.04 m/s h = 1 km 0.08 m/s h = 1.5 km 0.16 m/s

49. 中央大學大氣科學系 49 Summary Transience matters! Transience matters! On a time-scale of 2 days, transience renders the steady-state solution irrelevant. For quasi-linear regime (h<=125m): For quasi-linear regime (h<=125m): Larger momentum fluxes in the accelerating phase. Larger momentum fluxes in the accelerating phase. Largest momentum fluxes are found in the mid and upper troposphere before the time of maximum cross-mountain flow. Largest momentum fluxes are found in the mid and upper troposphere before the time of maximum cross-mountain flow. Low-level convergence of momentum flux produces an surprising acceleration of low-level cross-mountain flow during the accelerating phase. Low-level convergence of momentum flux produces an surprising acceleration of low-level cross-mountain flow during the accelerating phase. In an accelerating flow, wave packets tend to accumulate above the mountain, enhancing wave activity aloft. In an accelerating flow, wave packets tend to accumulate above the mountain, enhancing wave activity aloft. The momentum flux distribution may be understood using WKB ray tracing theory. The momentum flux distribution may be understood using WKB ray tracing theory. The instantaneous drag is given by the steady linear solution. The instantaneous drag is given by the steady linear solution.

50. 中央大學大氣科學系 50 Summary Continued For moderately nonlinear regime(250 m <= h <= 1000 m): For moderately nonlinear regime(250 m <= h <= 1000 m): Nonlinearity reinforces the low-level mean flow acceleration. Nonlinearity reinforces the low-level mean flow acceleration. A higher drag state is present during the accelerating phase. A higher drag state is present during the accelerating phase. In particular, the drag is not determined by the instantaneous value of the nonlinearity parameter (  =Nh/U). In particular, the drag is not determined by the instantaneous value of the nonlinearity parameter (  =Nh/U). For highly nonlinear regime (h>= 1250 m): For highly nonlinear regime (h>= 1250 m): Severe wave dissipation hinders vertical propagation of wave packets and thus no low-level momentum flux convergence is found. Severe wave dissipation hinders vertical propagation of wave packets and thus no low-level momentum flux convergence is found. The pressure drag reaches a maximum at t = 27.5 hour. The pressure drag reaches a maximum at t = 27.5 hour. A board region of flow deceleration extends far downstream from the mountain with patches of flow acceleration north and south of it. A board region of flow deceleration extends far downstream from the mountain with patches of flow acceleration north and south of it. Despite the small scales of PV anomalies generated by wave breaking, PV inversion recovers most of the actual response. Despite the small scales of PV anomalies generated by wave breaking, PV inversion recovers most of the actual response. The experiment with a “perfect” conventional GWD parameterization fails to produce enough flow deceleration/acceleration. The experiment with a “perfect” conventional GWD parameterization fails to produce enough flow deceleration/acceleration.

51. 中央大學大氣科學系 51 Questions?