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

中央大學大氣科學系中央大學大氣科學系 11

Transient Mountain Waves in an Transient Mountain Waves in an Evolving Synoptic-Scale Flow and Evolving Synoptic-Scale Flow and Their Interaction with Large ScalesTheir Interaction with Large Scales

Chih-Chieh (Jack) Chen,Chih-Chieh (Jack) Chen,Climate and Global Dynamics DivisionClimate and Global Dynamics Division

National Center for Atmospheric ResearchNational Center for Atmospheric Research

Dale R. Durran and Gregory J. HakimDale R. Durran and Gregory J. HakimDepartment of Atmospheric SciencesDepartment of Atmospheric Sciences

University of WashingtonUniversity of Washington

April 24, 2007April 24, 2007


OutlineOutline

Background and MotivationBackground and Motivation

Methodology and Experimental DesignMethodology and Experimental Design

Results Results mesoscale responsemesoscale response large-scale responselarge-scale response

SummarySummary


Mountain WavesMountain 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-1pressure drag


Momentum Flux and Pressure DragMomentum Flux and Pressure Drag

Breaking

U

pressure drag

<uw>

<uw>

<uw>

<uw>“action at a distance”

a sink for momentum

H L


Gravity Wave Drag ParameterizationGravity Wave Drag Parameterization

Current parameterizations assume the waveCurrent parameterizations assume the waveare in steady state with the large-scale flow.are in steady state with the large-scale flow. Relatively little research has be devoted to Relatively little research has be devoted to

mountain waves in a slowly evolving flow.mountain waves in a slowly evolving flow. Suppose the waves develop and decay over a Suppose the waves develop and decay over a

period of two days? Does transience matter on this period of two days? Does transience matter on this time scale? time scale?

Do the current GWD parameterizations do a good Do the current GWD parameterizations do a good job in capturing the “true” response?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


Transient Mountain WavesTransient Mountain Waves Bell (1975)Bell (1975)

Bannon and Zhender (1985)Bannon and Zhender (1985)

Lott and Teitelbaum (1993)Lott and Teitelbaum (1993)


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


Goals of the studyGoals of the studyTo 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?


MethodologyMethodology

numerical model following numerical model following Durran and Klemp Durran and Klemp (1983)(1983) and and Epifanio and Durran (2000)Epifanio and Durran (2000)

nonlinear and nonhydrostaticnonlinear and nonhydrostatic

ff-plane approximation (-plane approximation (ff = 10 = 10-4-4 s s-1-1))

Boussinesq approximationBoussinesq approximation

parameterized subgrid-scale mixingparameterized subgrid-scale mixing

terrain-following coordinatesterrain-following coordinates


Construction of the Synoptic-scale FlowConstruction of the Synoptic-scale Flow

Desirable features: Desirable features: At least one ascending/descending phase for the mean windAt least one ascending/descending phase for the mean wind At least oneAt least one stagnation stagnation point at the ground point at the ground Dynamics well understood without mountainDynamics well understood without mountain

We have chosen:We have chosen: A A nondivergent barotropicnondivergent barotropic flow with flow with uniform uniform

stratificationstratification (constant (constant NN22). The streamfunction includes a sinusoidal The streamfunction includes a sinusoidal

square wave in both square wave in both xx and and y.y.


Construction of Synoptic-scale FlowConstruction of Synoptic-scale Flow


Initial Condition IngredientsInitial Condition Ingredients

doubly periodic


Boundary ConditionsBoundary Conditions Periodic in Periodic in xx and and yy Upper boundary is a rigid lid with scale-selective Upper boundary is a rigid lid with scale-selective

sponge layersponge 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.


Domain Setup and Model ResolutionDomain Setup and Model Resolution

x = 6 km300 x

y = 6 km300 y

H = 16 kmz = 150~500 m

sponge layerH = 16 km


u’u’ and and forced by forced by hh = 250 m = 250 m


Horizontal Group Velocity Is Doppler Shifted Horizontal Group Velocity Is Doppler Shifted by the Synoptic Flowby the Synoptic Flow

Dispersion relation for 2D gravity waves

For stationary waves at

Horizontal group velocity of mountain wave packet launched at time


How Does the Domain Averaged Momentum Flux Vary with Time and Height?


Hypothetical Hypothetical z-tz-t Momentum Flux Momentum Flux DistributionDistribution

under linear theory:

t

z

/2

- -+ +


Momentum Flux Forced by Momentum Flux Forced by hh = 250 m = 250 m

constant U10 m/s

constant U20 m/s


Vertical Group Velocity Increases with the Vertical Group Velocity Increases with the Speed of the Synoptic FlowSpeed of the Synoptic Flow

Dispersion relation for 2D gravity waves

For stationary waves at

Vertical group velocity of mountain wave packet launched at time


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

t = t1t = t2t = t3

U decreasing with time

t = t4t = t5t = t6


Ray Path Diagram: Ray Path Diagram: x-zx-z plane plane


Ray Path Diagram: Ray Path Diagram: z-t z-t plane plane


Conservation of Wave ActionConservation of Wave Action

Wave action density changes when neighboring rays converge or diverge


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:


Change of intrinsic frequencyChange of intrinsic frequency

k increases k decreases

x

y

Accelerating Phase

x

y

Decelerating Phase


Momentum Flux Forced by Momentum Flux Forced by hh = 125 m = 125 m

model output WKB solution


Momentum Flux for Higher MountainMomentum Flux for Higher Mountain

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


Pressure Drag EvolutionPressure Drag Evolutionin steady-state framework, drag U

h = 125 m


Nonlinearity and Past HistoryNonlinearity and Past History

higher lower


Large-Scale Flow ResponseLarge-Scale Flow Response

global momentum budgetsglobal momentum budgets spatial responsespatial response


Momentum Budget PerspectiveMomentum Budget Perspective


Forcing for Zonal Mean FlowForcing for Zonal Mean Flow h = 1.5 km h = 1.5 km


Global Response for h = 1.5 kmGlobal Response for h = 1.5 km


Zonally-averaged fieldsZonally-averaged fields


Zonally-averaged fields at 30 hZonally-averaged fields at 30 h


Spatial ResponseSpatial 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


Difference fieldsDifference fieldst = 25 hours z = 1.5 km


Difference fieldsDifference fieldsz = 1.5 km






Can the flow response be explained by balanced dynamics?

PV difference is inverted by using geostrophic balance

as the balance constraint.


u difference vs balanced uu difference vs balanced u

t = 50 hours z = 1.5 km


u difference vs balanced uu difference vs balanced u

t = 50 hours z = 3.5 km


Implication of PV InversionImplication of PV Inversion

What is the effect of GWD? Can we recover the spatial response by using

a GWD parameterization scheme?


GWD Parameterization ExperimentGWD Parameterization ExperimentAssuming:Gravity wave drag is deposited in the mountainous region (area = ) only.


GWD Parameterization Experiment GWD Parameterization Experiment 18 km

(exact)

t = 50 hours z = 3.5 km

(GWD Exp.)

-6 m/s -2 m/s


Large scale flow responseLarge scale flow response

h = 125 m0.01 m/s

h = 250 m0.02 m/s

h = 500 m0.04 m/s

h = 1 km0.08 m/s

h = 1.5 km0.16 m/s


SummarySummary Transience matters!Transience matters!

On a time-scale of On a time-scale of 2 days2 days, transience renders the steady-state, transience renders the steady-statesolution irrelevant.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 Largest momentum fluxes are found in the mid and upper

troposphere before the time of maximum cross-mountain flow.troposphere before the time of maximum cross-mountain flow. Low-level convergence of momentum flux produces an Low-level convergence of momentum flux produces an

surprising accelerationsurprising acceleration of low-level cross-mountain flow during of low-level cross-mountain flow during the accelerating phase. the accelerating phase.

In an accelerating flow, wave packets tend to In an accelerating flow, wave packets tend to accumulateaccumulate above above the mountain, enhancing wave activity aloft.the mountain, enhancing wave activity aloft.

The momentum flux distribution may be understood using WKB The momentum flux distribution may be understood using WKB ray tracing theory. ray tracing theory.

The instantaneous drag is given by the steady linear solution.The instantaneous drag is given by the steady linear solution.


Summary ContinuedSummary 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 vIn particular, the drag is not determined by the instantaneous v

alue of the nonlinearity parameter (alue of the nonlinearity parameter (=Nh/U=Nh/U).).

For highly nonlinear regime (h>= 1250 m):For highly nonlinear regime (h>= 1250 m): Severe wave dissipation hinders vertical propagation of wave pSevere wave dissipation hinders vertical propagation of wave p

ackets and thus no low-level momentum flux convergence is foackets and thus no low-level momentum flux convergence is found.und.

The pressure drag reaches a maximum at The pressure drag reaches a maximum at tt = 27.5 hour. = 27.5 hour. A board region of flow deceleration extends far downstream froA board region of flow deceleration extends far downstream fro

m the mountain with patches of flow acceleration north and soum the mountain with patches of flow acceleration north and south of it.th of it.

Despite the small scales of PV anomalies generated by wave breDespite the small scales of PV anomalies generated by wave breaking, PV inversion recovers most of the actual response.aking, PV inversion recovers most of the actual response.

The experiment with a “perfect” conventional GWD parameteThe experiment with a “perfect” conventional GWD parameterization fails to produce enough flow deceleration/acceleration.rization fails to produce enough flow deceleration/acceleration.


QuestionsQuestions??

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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