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

May ICAM - 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

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.”

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

May ICAM - MAP Hypothetical z-t Momentum Flux Distribution under linear theory: t z  /

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

May ICAM - 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

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

May ICAM - 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:

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

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

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

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

May ICAM - 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

May ICAM - 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.

May ICAM - 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.

May ICAM - MAP Average Momentum Flux Profile over One Period Average Momentum Flux Profile over One Period

May ICAM - 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

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

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

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