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POTENTIAL VORTICITY AND ENERGY ASPECTS OF THE MJO THROUGH EQUATORIAL WAVE THEORY

POTENTIAL VORTICITY AND ENERGY ASPECTS OF THE MJO THROUGH EQUATORIAL WAVE THEORY. Master’s Thesis Defense Matthew T. Masarik Colorado State University Atmospheric Science Department.

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POTENTIAL VORTICITY AND ENERGY ASPECTS OF THE MJO THROUGH EQUATORIAL WAVE THEORY

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  1. POTENTIAL VORTICITY AND ENERGY ASPECTS OF THE MJO THROUGH EQUATORIAL WAVE THEORY Master’s Thesis Defense Matthew T. Masarik Colorado State University Atmospheric Science Department Advisor: Prof.Wayne Schubert - CSU Atmospheric Science Department Co-advisor: Prof.Tom Vonder Haar - CSU Atmospheric Science Department Committee Member: Prof.David Randall - CSU Atmospheric Science Department Committee Member: Prof.Richard Eykholt - CSU Physics Department Wednesday, November 15th 2006

  2. Committee Advisor:Prof. Wayne Schubert Co-advisor:Prof.Tom Vonder Haar Committee Member:Prof.David Randall Committee Member:Prof.Richard Eykholt ACKNOWLEDGEMENTS • Friends • Family • Schubert Research Group, • Brian McNoldy, Jonathan Vigh, Paul Ciesielski, and Rick Taft • Funding Sponsor DoD Center for Geosciences/Atmospheric Research at Colorado State University under Cooperative Agreements DAAD19-02-2-0005, and W911NF-06-2-0015, with the Army Research Laboratory.

  3. Primitive Equation Model: • Governing equations • Model forcing • Solutions • Potential Vorticity (PV) Aspects: • PV principle • Idealized PV principle • Invertibility principle • Energy Aspects: • Total energy principle • Parseval relation • Energy dependence OVERVIEW • Motivation • Madden-Julian Oscillation (MJO) phenomena • Conclusions

  4. MOTIVATION • MJO not satisfactorily explained • Complex phenomena: • * multi-scale structure, • * intra-seasonal time scale •  Look at one piece of the puzzle. • Inspired by: mid-latitude balanced theory (QG, SG) • PV in the tropics? • New approach…

  5. MJO PHENOMENA • A composite, mean MJO lifecycle seen in OLR (W/m²) anomalies. enhanced convection supressed convection * Animation credit: Dr. Adrian Matthews

  6. MJO PHENOMENA • A composite, mean MJO lifecycle seen in OLR (W/m²) anomalies. enhanced convection supressed convection * Animation credit: Dr. Adrian Matthews

  7. Planetary Scale Circulation field ~10,000-20,000 km Planetary Scale Circulation field ~10,000-20,000 km Mesoscale Cloud complexes ~100 km Synoptic Scale Convective core ~2,000-4,000 km MULTI-SCALE STRUCTURE

  8. Upper Level Anti-cyclones Equatorial broad zonal flow Equatorial zonal wind burst Convective core ~2,500-3,000 km 20° HORIZONTAL STRUCTURE • Anomalous MJO-filtered OLR and circulation from ERA-15 reanalysis, 1979-93. (Kiladis et al. 2005)

  9. max u’≈ 4.0 ms-1 max u’≈ -7.0 ms-1 Center of OLR minimum max u’≈ 4.5 ms-1 max u’≈ -1.5 ms-1 VERTICAL STRUCTURE • MJO-filtered OLR and zonal wind anomalies at the equator (ERA-15 reanalysis). (Kiladis et al. 2005) E W W E

  10. Equatorial β-plane • Vertical coordinate: z = ln(p0/p) • Linearize about resting basic state • α = coefficient for cooling/friction • Γ = basic state static stability • Q = diabatic source term • ∂v/∂t retained • Longwave approximation neglects ∂v/∂t term (Gill, 1980) PRIMITIVE EQUATION MODEL • Governing equations, vertical struture • Forcing function • Frame of reference, solutions

  11. Observed heating profile Peak heating level ≈ 395 hPa u,v,Ø vertical structure Prescribed heating profile * Fig. courtesy of Paul Ciesielski SEPARATION OF VERTICAL STRUCTURE • Vertical structure functions: Z1(z) = Z(z) and Z1'(z) = Z'(z). • The curve labeled Q/cp is the 120-day mean vertical profile of heating rate for the western Pacific warm pool during TOGA-COARE. Adapted from (Johnson and Ciesielski, 2000)*.

  12. DIABATIC FORCING STRUCTURE • Eastward propagating deep convection • ξ = x – ct • c = 5 m/s • Q0/cp = 12 K/day • a0 = 1250 km • b0 = 450 km • y0 = 0, 450 km

  13. ZONAL COORDINATE TRANSFORM • Zonal distance variable, ξ = x - ct • Steady state translating at constant speed c • Reference frame propagating with convective forcing • Primitive equation generalization of the simplest MJO model involving the 1st baroclinic mode response to a large-scale moving heat source under the long-wave approximation. (Chao, 1987)

  14. SOLUTIONS / WAVE COMPONENTS FOR Y0 = 0

  15. SOLUTIONS / WAVE COMPONENTS FOR Y0 = 450

  16. Primitive equation potential vorticity anomaly: • Potential vorticity principle: Moving equatorial heat source: • Produces two PV streamers • Lower Troposphere: +PV in NH, -PV in SH • Upper Troposphere: Oppositely signed anomalies POTENTIAL VORTICITY ASPECTS • PV principle • Idealized PV principle (analytical solution) • Invertibility principle

  17. RIGID LID RIGID LID Θ = 345 K Θ = 345 K RESTING BASIC STATE -PV Θ = 335 K Θ = 335 K decreased static stability Θ = 325 K Θ = 325 K Θ = 315 K Θ = 315 K MASS PUMP Θ = 305 K Θ = 305 K +PV increased static stability Θ = 295 K Θ = 295 K DIABATIC PV GENERATION

  18. (shallow water PV anomaly) DimensionlessKelvin wave frequency (spectral space PV anomaly) PV VIEW OF THE KELVIN WAVE

  19. 10 10-6s-1 -5 10-6s-1 PV WAKE: Y0 = 0 AND Y0 = 450 6 10-6s-1 -6 10-6s-1

  20. Solution IDEALIZED PV PRINCIPLE • Insight into: (1) βv term, (2) PV wake magnitude. • PV principle • Consider only dissipation and convective forcing • Assume translating steady state.

  21. Measure of convective region passage time. Measure of convective overturning time. Number of convective overturnings during the passage of the convective region. WAKE MAGNITUDE PARAMETER • Large PV anomaly: * Zonally long * Slow moving * Highly convective

  22. βv not included 4 10-6 s-1 6 10-6 s-1 βv included βV INFLUENCE  βv tends to make PV anomaly stronger, broader in westward, north-south extent. • Idealized PV 68% of correct strength, does not extend far enough westward or poleward.

  23. Divergence equation • Approximate u,v by rotational components • Linear balance relation • Assuming βy varies slowly compared to ψ and, EQUATORIAL BALANCE RELATIONSHIP

  24. PV anomaly • Express vorticity in terms of streamfunction • Use balance relation • Invertibility principle INVERTIBILITY PRINCIPLE

  25. CIRCULATION FROM INVERTED PV

  26. Steady state energy balance • Generation – small, localized heating. • Dissipation – wave propagation of energy to far-field, removed by radiation and friction. ENERGY ASPECTS • Total energy principle • Parseval Relation • Energy dependence on forcing parameters

  27. Dimensional frequency of diabatic forcing Dimensional natural frequencies , where (m/a) = dimensional zonal wavenumber c = constant phase speed of forcing PARSEVAL RELATION • PARSEVAL’S THEOREM - The energy contained in a waveform f(x) integrated over all physical space (x) is equal to the energy contained in the spectrally transformed waveform F(k) integrated over all of its wavenumber components.

  28. NORMALIZED ENERGY VS. LAT. DISPLACEMENT

  29. NORMALIZED ENERGY VS. PHASE SPEED

  30. NORMALIZED ENERGY VS. ZONAL WAVENUMBER

  31. TOTAL RESPONSE PER ZONAL WAVENUMBER

  32. Analytical solutions to the primitive equation generalization of the simplest MJO model under the long-wave approximation. • PV dynamics approach to MJO: • Invertibility principle • Wake magnitude parameter CONCLUSIONS New contributions -

  33. Response to MJO-like convection: • Inertia-gravity near forcing, • Rossby to west, • Kelvin to east • Wake flow is essentially balanced and derivable from the PV field. • βv strengthens, increases extent of PV. CONCLUSIONS

  34. PV anomaly magnitude depends on the ratio, • = passage time / convective overturning time • Response energy distribution across low zonal wavenumbers suggests long-wave approximation solutions probably in good agreement with exact solutions using our prescribed forcing parameters. • Resonance of natural and forcing frequencies affects response energy distribution across: • Equatorial wave type • Zonal wave number CONCLUSIONS

  35. RECENT CSU ATMOS. DEPT. MJO RESEARCH • Benedict, J. J., and Randall, D. A., 2006: An analysis of the MJO guided by TRMM Rainfall data. J. Atmos. Sci., Submitted for publication. • Pakula, L., Gabriel, P., and Stephens, G. L.: On the forced response of waves on a β-plane. Submitted for publication. CITED WORK • MJO animation: Dr. Adrian Matthews, School of Environmental Sciences, University of East Anglia, http://envam1.env.uea.ac.uk/~e058/ • Kiladis, G. N., K. H. Straub, and P. T. Haertel, 2005: Zonal and vertical structure of the Madden-Julian Oscillation. J. Atmos. Sci., 62, 2790-2809. • Johnson, R. H., and Ciesielski, P. E., 2000: Rainfall and radiative heating rates from the TOGA-COARE atmospheric budgets. J. Atmos.Sci., 57, 1497-1514. • Schubert, W. H., and Masarik, M. T., 2006: Potential vorticity aspects of the MJO. Dynamics of Atmospheres and Oceans, 42, 127-151.

  36. ^ MASTERS PRESENTATION ^ v EXTRA SLIDES v

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