1. May 2005 ICAM - MAP 1 Mountain-Wave Momentum Flux in an Evolving Synoptic-Scale Flow Chih-Chieh Chen, Dale R. Durran and Gregory J. Hakim Department of Atmospheric Sciences University of Washington

2. May 2005 2ICAM - MAP Transient Mountain Waves Lott and Teitelbaum (1993) : maximum mean flow  : period : half width of mountain U = U(t) 2D configuration large-scale dynamics unspecified

3. May 2005 ICAM - MAP 3 How does the domain-averaged momentum flux vary with time and height? Lott and Teitelbaum: when  1, “roughly the momentum flux remains that predicted by the stationary theory.”

4. May 2005 4ICAM - MAP Nonlinearly Balanced Barotropic Synoptic- Scale Flow doubly periodic

5. May 2005 5ICAM - MAP Hypothetical z-t Momentum Flux Distribution under linear theory: t z  /2 - - + +

6. May 2005 6ICAM - MAP Horizontally Averaged Momentum Flux (h = 250 m) Constant U of 10 ms -1 Constant U of 20 ms -1

7. May 2005 7ICAM - MAP 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

8. May 2005 8ICAM - MAP Conservation of Wave Action Wave action density changes when neighboring rays converge or diverge

9. May 2005 9ICAM - MAP 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:

10. May 2005 10ICAM - MAP Comparison of Momentum Fluxes from the Model and the WKB Reconstructon (h=125 m) model outputWKB solution

11. May 2005 11ICAM - MAP Change of intrinsic frequency k increases k decreases x y Accelerating Phase x y Decelerating Phase

12. May 2005 12ICAM - MAP Influence of Confluence and Difluence (h=125 m) Translating square waveUniform westerly flow

13. May 2005 13ICAM - MAP Momentum Flux for Higher Mountains h = 250 mh = 500 m h = 1 km Linear (h 2 ) contour scaling

14. May 2005 14ICAM - MAP Pressure Drag Evolution From steady-state linear theory: drag U min(Nh/U)=0.0625min(Nh/U)=0.50min(Nh/U)=0.125min(Nh/U)=0.25min(Nh/U)=0.375

15. May 2005 15ICAM - MAP Summary On a time-scale of 2 days, transience renders the steady-state solution almost irrelevant. In an accelerating flow, wave packets tend to accumulate above the mountain, enhancing wave activity aloft. Large-scale confluence and difluence also affect the momentum flux. Momentum flux distribution: 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.

16. May 2005 16ICAM - MAP Summary Continued For almost-linear flows: The momentum flux distribution may be understood using WKB ray tracing theory. The instantaneous drag (but not the momentum flux aloft) is given by the steady linear solution. For nonlinear flows The instantaneous drag is not determined by the instantaneous value of Nh/U.

17. May 2005 17ICAM - MAP Average Momentum Flux Profile over One Period Average Momentum Flux Profile over One Period

18. May 2005 18ICAM - MAP Group Velocity of Mountain-Wave Packets Dispersion relation for 2D gravity waves For stationary waves at Vertical group velocity of mountain wave packet launched at time

19. May 2005 19ICAM - MAP Construction of the Synoptic-Scale Flow doubly periodic

20. May 2005 20ICAM - MAP Momentum Flux and Pressure Drag Breaking U pressure drag a sink for momentum HL

21. May 2005 21ICAM - MAP Ray Paths and Momentum Flux in the z-t plane