May 2005 ICAM - MAP 1

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

Chih-Chieh Chen,Dale R. Durran and Gregory J. Hakim

Department of Atmospheric SciencesUniversity of Washington

May 2005 2ICAM - MAP

Transient Mountain WavesTransient 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 2005 4ICAM - MAP

Nonlinearly Balanced Barotropic SynoptiNonlinearly Balanced Barotropic Synoptic-Scale Flowc-Scale Flow

doubly periodic

May 2005 5ICAM - MAP

Hypothetical Hypothetical z-tz-t Momentum Flux Distribution Momentum Flux Distribution

under linear theory:

t

z

/2

- - + +

May 2005 6ICAM - MAP

Horizontally Averaged Momentum Flux Horizontally Averaged Momentum Flux ((hh = 250 m) = 250 m)

Constant U of 10 ms-1

Constant U of 20 ms-1

May 2005 7ICAM - MAP

WKB Ray Tracing for U = U(t)U increasing with time

t = t1t = t2t = t3

U decreasing with time

t = t4t = t5t = t6

May 2005 8ICAM - MAP

Conservation of Wave ActionConservation of Wave Action

Wave action density changes when neighboring rays converge or diverge

May 2005 9ICAM - MAP

Momentum Flux Changes Along a RayMomentum 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 2005 10ICAM - MAP

Comparison of Momentum Fluxes from the Model and the WKB Reconstructon (h=125 m)model output WKB solution

May 2005 11ICAM - MAP

Change of intrinsic frequencyChange of intrinsic frequency

k increases k decreases

x

y

Accelerating Phase

x

y

Decelerating Phase

May 2005 12ICAM - MAP

Influence of Confluence and Difluence (h=125 m)Translating square wave Uniform westerly flow

May 2005 13ICAM - MAP

Momentum Flux for Higher MountainsMomentum Flux for Higher Mountains

h = 250 m h = 500 m h = 1 km

Linear (h2) contour scaling

May 2005 14ICAM - MAP

Pressure Drag Pressure Drag EvolutionEvolution

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 2005 15ICAM - MAP

SummarySummary• 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 2005 16ICAM - MAP

Summary ContinuedSummary 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 2005 17ICAM - MAP

Average Average Momentum Momentum Flux Profile Flux Profile over One over One

PeriodPeriod

May 2005 18ICAM - MAP

Group Velocity of Mountain-Wave PacketsGroup 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 2005 19ICAM - MAP

Construction of the Synoptic-Scale Construction of the Synoptic-Scale FlowFlow

doubly periodic

May 2005 20ICAM - MAP

Momentum Flux and Pressure DragMomentum Flux and Pressure Drag

Breaking

U

pressure drag<uw>

<uw>

<uw>

<uw>

a sink for momentum

H L

May 2005 21ICAM - MAP

Ray Paths and Momentum Flux in the Ray Paths and Momentum Flux in the z-t z-t planeplane