Yasuko Hio and Shigeo Yoden Department of Geophysics, Kyoto University, Japan

1. A Parameter Sweep Experimenton Quasi-Periodic Variationsof a Circumpolar Vortexdue to Wave-Wave Interactionin a Barotropic Model Yasuko Hio and Shigeo Yoden Department of Geophysics, Kyoto University, Japan

2. Wave 1 Wave 2 1. Introduction • Hio and Yoden (2004; JAS, 61, 2510-2527) “Quasi-periodic variations of the polar vortex in the Southern Hemisphere 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

3. 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 jet • stationary, periodic, quasi-periodic, and irregular solutions • transitions • periodic sol.  quasi-periodic sol. as h0 small • dominant triad interactions for each of these solutions

4. 2. Model and Numerical Procedure • A dynamical model of 2D non-divergent flow on the earth with zonal-flow forcing and dissipation • potential vorticity equation

5. A “stratospheric” model • zonal flow forcing: Hartmann (1983) • barotropically unstable profile • surface topography: Taguchi and Yoden (2002) • only in the Southern Hemisphere • experimental parameters • B : jet width r = h0/H : topographic height • fixed parameters: U０=240m/s, Φ０=55oS • numerical schemes • spectral model (Ishioka & Yoden, 1995): T85 (128x256) • time integrations: 4th-order Runge-Kutta method

6. B 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 r 0 V V V V V V S S S S S S S S N N N N V V P P P P P P P P P 0.02 Sta Sta Sta Sta Sta Sta Sta P P P P P P P P P P P 0.04 Sta Sta Sta Sta Sta Sta Sta P P P P P P P P P P P 0.06 Sta Sta Sta Sta Sta Sta Sta I I I P P P P P P P P 0.08 Sta Sta Sta Sta Sta Sta Sta I I I I I P P P P P P 0.10 Sta Sta Sta Sta Sta Sta Sta 3. Flow regimes • B-r dependence • r =0(Ishioka & Yoden, 1995) No wave • Steady wave: constant eastward propagation with Vacillation: + periodic variation of wave structure as B small • r =0 Stationary wave  PeriodicVacillationorIrregular (time constant) (not steady) narrower  jet  wider higher  topography

7. V I P r r r stationary wave component • r dependence at B = 4 • time mean and variable range • zonal mean zonal flow • amplitudes 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

8. r =0 larger r U Wave 2 Wave 1 no W1 ? Irregular Vacillation Periodic Im[W2 ] • time variations • time series of U • harmonic dials of W2 and W1: Re[W2 ] - Im[W2 ] stationary wave Re[W2 ] traveling wave

9. Transition from periodic solution to vacillation • harmonic 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

10. Zonal mean PV |W2| W2 at a fixed longitude Periodic Vacillation Irregular • power spectra (at 65.1oS) • changes in the predominant frequencies

11. (1) (2) Periodic variation with frequency is predominant for small r Periodic variation of U is synchronized with the traveling Wave 2 with frequency Periodic variation with another frequency appears at rb , and increases its power as r decreases rb • power spectra • r dependence of the predominant frequencies and power Power spectra of zonal mean PV Vacillation Periodic

12. 4. Transitions • diagnosis on wave-wave interactions • Fourier decomposition of the PV equation • zonal wavenumber s = 0, 1, and 2 • source and sink ~ 0 • wave-mean flow interaction and wave-wave interaction

13. U Stationary W1 Traveling W1 Traveling W2 • low-order “empirical mode expansion” of meridional profile • composites of stationary and traveling waves • Ex. a periodic solution (B =4, r =0.02)

14. (1) topographic effect on vacillation around r =0 • pure 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

15. r dependence of the power of each component

16. (2) transition from periodic solution to vacillation • periodic solution nearr =rb • modulation of traveling W2 ( Hopf bifurcation) • traveling W2 x traveling W2  mean-flow variation • stationary W1 x traveling W2  traveling W1

17. r dependence of the power of each component

18. 5. Conclusions • parameter 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 clarified • transition from periodic solution to vacillation for smaller r • in the periodic solutions, variations of U and S1synchronize 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