Mountain-Wave Momentum Flux in an Evolving Synoptic-Scale Flow

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 ICAM - MAP

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

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

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

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

7. U increasing with time U decreasing with time t = t2 t = t3 t = t4 t = t1 t = t6 t = t5 WKB Ray Tracing for U = U(t) ICAM - MAP

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

9. And for hydrostatic Boussinesq gravity waves: 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 ICAM - MAP

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

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

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

13. Momentum Flux for Higher Mountains Linear (h2) contour scaling h = 250 m h = 500 m h = 1 km ICAM - MAP

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

15. 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. ICAM - MAP

16. 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. ICAM - MAP

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

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

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

20. <uw> <uw> <uw> <uw> pressure drag H L Momentum Flux and Pressure Drag Breaking a sink for momentum U ICAM - MAP

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