A Parameter Sweep Experimenton Quasi-Periodic Variations

of a Circumpolar Vortexdue to Wave-Wave Interaction

in a Barotropic Model

Yasuko Hio and Shigeo YodenDepartment of Geophysics, Kyoto University, Japan

1. Introduction

Hio and Yoden (2004; JAS, 61, 2510-2527)“Quasi-periodic variations of the polar vortex in the Southern H

emisphere stratosphere due to wave-wave interaction”Animations of the potential vorticity

– NCEP/NCAR reanalysis dataset– 8th~27th in October 1996

PV map Total Traveling component

Wave 1 Wave 2

Wave-wave interactions between– stationary Wave 1

propagated from the troposphere (e.g. Hio & Hirota, 2002)– eastward traveling Wave 2

generated by instability of mean zonal flow (e.g. Manney, 1988)approaches

– data analysis of NCEP/NCAR reanalysis dataset– numerical experiment with a barotropic model

In this study, we do further numerical experiments flow regimes

dependence on the parameters– height of sinusoidal surface topography h0

– width of eastward zonal mean jetstationary, periodic, quasi-periodic, and irregular solutions

transitionsperiodic sol. quasi-periodic sol. as h0 smalldominant triad interactions for each of these solutions

2. Model and Numerical ProcedureA dynamical model of 2D non-divergent flow on the earth with zonal-flow forcing and

dissipationpotential vorticity equation

A “stratospheric” model zonal flow forcing: Hartmann (1983)

barotropically unstable profile

surface topography: Taguchi and Yoden (2002)only in the Southern Hemisphere

experimental parametersB : jet width r = h0/H : topographic heightfixed parameters: U ０ =240m/s, Φ ０ =55oS

numerical schemesspectral model (Ishioka & Yoden, 1995): T85 (128x256)time integrations: 4th-order Runge-Kutta method

0q

B-r dependence

r =0 (Ishioka & Yoden, 1995)No wave Steady wave: constant eastward propagation with Vacillation: + periodic variation of wave structureas B small

r =0Stationary wave Periodic Vacillation or Irregular (time constant) (not steady)

3. Flow regimes

3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 9.0 10 11 12 14 16 18

0 V V V V V V S S S S S S S S N N N N

0.02 V V P P P P P P P P P Sta Sta Sta Sta Sta Sta Sta

0.04 P P P P P P P P P P P Sta Sta Sta Sta Sta Sta Sta

0.06 P P P P P P P P P P P Sta Sta Sta Sta Sta Sta Sta

0.08 I I I P P P P P P P P Sta Sta Sta Sta Sta Sta Sta

0.10 I I I I I P P P P P P Sta Sta Sta Sta Sta Sta Sta

Br

narrower jet widerh

igh

er

to

pog

raph

y

r dependence at B = 4 time mean and variable range

zonal mean zonal flowamplitudes of Wave 1 and Wave 2: stationary & traveling

at the transition point P V – time variation of U ~ 0– time variation of the amplitude of traveling waves ~ 0

r r rstationary wavecomponent

V P I

time variationstime series of U harmonic dials of W2 and

W1:Re[W2 ] - Im[W2 ]

r =0 larger r

U

Wave 2

Wave 1

Vacillation Periodic Irregular

?no W1

Re[W2 ]

Im[W2 ]

stationarywave

travelingwave

Transition from periodic solution to vacillationharmonic dials around r =0.15

– periodic solution: synchronized variation of traveling waves– vacillation: traveling waves have not fixed phase relation for smaller topographic heights

» unsynchronized traveling waves + modulation of U

Vacillation Periodic

power spectra (at 65.1oS)changes in the predominant frequencies

Vacillation Periodic Irregular

Zonal mean PV

|W2|

W2 at a fixed longitude

power spectrar dependence of the predominant frequencies and

power

Periodic variation of Uis synchronized withthe traveling Wave 2with frequency

Periodic variation with another frequencyappears at rb , and increases its poweras r decreases

rb

Periodic variation with frequency ispredominant for small r

Power spectra of zonal mean PV

(1) (2)Vacillation Periodic

4. Transitionsdiagnosis on wave-wave interactions

Fourier decomposition of the PV equationzonal wavenumber s = 0, 1, and 2source and sink ~ 0

wave-mean flow interaction and wave-wave interaction

low-order “empirical mode expansion” of meridional profile composites of stationary and traveling waves

– Ex. a periodic solution (B =4, r =0.02)U Stationary W1 Traveling W1 Traveling W2

(1) topographic effect on vacillation around r =0pure vacillation at r =0

+ topographically forced W1

stationary W1 x traveling W2 traveling W1

stationary W1 x traveling W1 mean-flow variation

traveling W2 x mean-flow variation traveling W2

r dependence of the power of each component

(2) transition from periodic solution to vacillationperiodic solution nearr =rb

modulation of traveling W2 ( Hopf bifurcation)

traveling W2 x traveling W2 mean-flow variation

stationary W1 x traveling W2 traveling W1

r dependence of the power of each component

　

5. Conclusionsparameter sweep experiment on quasi-periodic variations of a circumpolar vortex in the stratosphere with a barotropic model

6 flow regimes depending on

– topographically forced stationary waves (S1 )

– traveling waves (T2 ) generated by the barotropic instability of mean zonal flow (U )diagnosis of wave-wave interactions with a low-order “empirical mode expansion” of the PV equation

topographic effect on vacillation around r =0 was clarifiedtransition from periodic solution to vacillation for smaller r

– in the periodic solutions, variations of U and S1 synchronize with periodic progression of T2

– in the quasi-periodic vacillation, on the other hand, variations of U and amplitudes of S1 and T2 are independent of the progression of T2