Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 15 3 2008 10.5194/npg-15-469-2008 http://www.nonlin-processes-geophys.net/15/469/2008/ http://www.nonlin-processes-geophys.net/15/469/2008/npg-15-469-2008.html http://www.nonlin-processes-geophys.net/15/469/2008/npg-15-469-2008.pdf 469 487 2008-06-23 Breeding and predictability in the baroclinic rotating annulus using a perfect model R. M. B. Young young@atm.ox.ac.uk P. L. Read Atmospheric, Oceanic and Planetary Physics, Department of Physics, University of Oxford, UK We present results from a computational study of predictability in fully-developed baroclinically unstable laboratory flows. This behaviour is studied in the Met Office/Oxford Rotating Annulus Laboratory Simulation – a model of the classic rotating annulus laboratory experiment with differentially heated cylindrical sidewalls, which is firmly established as an insightful laboratory analogue for certain kinds of atmospheric dynamical behaviour. This work is the first study of "predictability of the first kind" in the annulus experiment. We devise an ensemble prediction scheme using the breeding method to study the predictability of the annulus in the perfect model scenario. This scenario allows one simulation to be defined as the true state, against which all forecasts are measured. We present results from forecasts over a range of quasi-periodic and chaotic annulus flow regimes. A number of statistical and meteorological techniques are used to compare the predictability of these flows: bred vector growth rate and dimension, error variance, "spaghetti plots", probability forecasts, Brier score, and the Kolmogorov-Smirnov test. These techniques gauge both the predictability of the flow and the performance of the ensemble relative to a forecast using a climatological distribution. It is found that in the perfect model scenario, the two quasi-periodic regimes examined may be indefinitely predictable. The two chaotic regimes (structural vacillation and period doubled amplitude vacillation) show a loss of predictability on a timescale of hundreds to thousands of seconds (65–280 annulus rotation periods, or 1–3 Lyapunov times). Barnes, J R.: Midlatitude Disturbances in the Martian Atmosphere: A Second Mars Year, J. Atmos. Sci., 38, 225–234, 1981. Brier, G W.: Verification of forecasts expressed in terms of probability, Mon. Weather Rev., 78, 1–3, 1950. Cai, M., Kalnay, E., and Toth, Z.: Bred Vectors of the Zebiak-Cane Model and Their Potential Application to ENSO Predictions, J. Climate, 16, 40–56, 2003. Collins, M. and James, I N.: Regular baroclinic transient waves in a simplified global circulation model of the Martian atmosphere, J. Geophys. Res. – Planet, 100, 14 421–14 432, 1995. Collins, M., Lewis, S R., Read, P L., and Hourdin, F.: Baroclinic Wave Transitions in the Martian Atmosphere, Icarus, 120, 344–357, 1996. Corazza, M., Kalnay, E., Patil, D J., Yang, S C., Morss, R., Cai, M., Szunyogh, I., Hunt, B R., and Yorke, J.: Use of the breeding technique to estimate the structure of the analysis "errors of the day", Nonlinear Proc. Geoph., 10, 233–243, 2003. Evans, E., Bhatti, N., Kinney, J., Pann, L., Peña, M., Yang, S.-C., Kalnay, E., and Hansen, J.: RISE Undergraduates find that Regime Changes in Lorenz's Model are Predictable, B. Am. Meteorol. Soc., 85, 520–524, 2004. Francisco, G. and Muruganandam, P.: Local dimension and finite time prediction in spatiotemporal chaotic systems, Phys. Rev. E, 67, 066 204, 2003. Früh, W G. and Read, P L.: Wave interactions and the transition to chaos of baroclinic waves in a thermally driven rotating annulus, Philos. T. Roy. Soc. A, 355, 101–153, 1997. Gilmour, I.: Nonlinear model evaluation: ι-shadowing, probabilistic prediction and weather forecasting, D. Phil, thesis, University of Oxford, UK, 1998. Gilmour, I., Smith, L A., and Buizza, R.: Linear Regime Duration: Is 24 Hours a Long Time in Synoptic Weather Forecasting?, J. Atmos. Sci., 58, 3525–3539, 2001. Hide, R.: Some experiments on thermal convection in a rotating liquid, Q. J. Roy. Meteor. Soc., 79, 161, 1953. Hide, R. and Mason, P J.: Sloping convection in a rotating fluid, Adv. Phys., 24, 47–100, 1975. Hignett, P., White, A A., Carter, R D., Jackson, W D N., and Small, R M.: A comparison of laboratory measurements and numerical simulations of baroclinic wave flows in a rotating cylindrical annulus, Q. J. Roy. Meteor. Soc., 111, 131–154, 1985. Houtekamer, P L. and Derome, J.: Prediction Experiments with Two-Member Ensembles, Mon. Weather Rev., 122, 2179–2191, 1994. Judd, K.: Nonlinear state estimation, indistinguishable states, and the extended Kalman filter, Physica D, 183, 273–281, 2003. Kalnay, E.: Atmospheric Modeling, Data Assimilation and Predictability, Cambridge University Press, 341 pages, 2003. Kalnay, E., Corazza, M., and Cai, M.: Are bred vectors the same as Lyapunov vectors?, in: Proceedings of a Symposium on Observations, Data Assimilation, and Probabilistic Prediction, part of the 82nd AMS 2002 Annual Meeting, Orlando, Florida, 13-17 January 2002, 173–177, 2002. Kalnay, E., Peña, M., Yang, S.-C., and Cai, M.: Breeding and predictability in coupled Lorenz models, in: Proceedings of a Seminar held at ECMWF on Predictability of Weather and Climate, Reading, UK, 9-13 September 2002, pp. 29–34, 2003. Leovy, C B.: The General Circulation of Mars: Models and Observations, Adv. Geophys., 28, 327–346, 1985. Lorenc, A C., Bell, R S., and Macpherson, B.: The Meteorological Office analysis correction data assimilation scheme, Q. J. Roy. Meteor. Soc., 117, 59–89, 1991. Lorenz, E N.: Deterministic nonperiodic flow, J. Atmos. Sci., 20, 130–141, 1963. Lorenz, E N.: Climate Predictability, in: The physical bases of climate and climate modelling, Vol 16 of GARP Publication Series, 132–136, World Meteorological Organisation, 1975. Massey, F J.: The Kolmogorov-Smirnov Test for Goodness of Fit, J. Am. Stat. Assoc., 46, 68–78, 1951. Newman, C E., Read, P L., and Lewis, S R.: Investigating atmospheric predictability on Mars using breeding vectors in a general-circulation model, Q. J. Roy. Meteor. Soc., 130, 2971–2989, 2004. Patil, D J., Hunt, B R., Kalnay, E., Yorke, J A., and Ott, E.: Local Low Dimensionality of Atmospheric Dynamics, Phys. Rev. Lett., 86, 5878–5881, 2001. Peña, M. and Kalnay, E.: Separating fast and slow modes in coupled chaotic systems, Nonlinear Proc. Geoph., 11, 319–327, 2004. Press, W H., Flannery, B P., Teukolsky, S A., and Vetterling, W T.: Numerical Recipes in Fortran 77, Cambridge University Press, Cambridge, UK, 2nd edn., 1992. Primo, C., Rodr\'iguez, M A., López, J M., and Szendro, I.: Predictability, bred vectors, and generation of ensembles in space-time chaotic systems, Phys. Rev. E, 72, 015 201(R), 2005. Pu, Z.-X., Kalnay, E., Parrish, D., Wu, W., and Toth, Z.: The Use of Bred Vectors in the NCEP Global 3D Variational Analysis System, Weather Forecast., 12, 689–695, 1997. Read, P L.: A combined laboratory and numerical study of heat transport by baroclinic eddies and axisymmetric flows, J. Fluid Mech., 489, 301–323, 2003. Read, P L., Bell, M J., Johnson, D W., and Small, R M.: Quasi-periodic and chaotic flow regimes in a thermally driven, rotating fluid annulus, J. Fluid Mech., 238, 599–632, 1992. Read, P L., Lewis, S R., and Hide, R.: Laboratory and numerical studies of baroclinic waves in an internally heated rotating fluid annulus: a case of wave/vortex duality?, J. Fluid Mech., 337, 155–191, 1997. Read, P L., Collins, M., Fruh, W.-G., Lewis, S R., and Lovegrove, A F.: Wave Interactions and Baroclinic Chaos: A Paradigm for long Timescale Variability in Planetary Atmospheres, Chaos Soliton. Fract., 9, 231–249, 1998. Read, P L., Thomas, N P J., and Risch, S H.: An Evaluation of Eulerian and Semi-Lagrangian Advection Schemes in Simulations of Rotating, Stratified Flows in the Laboratory. Part I: Axisymmetric Flow, Mon. Weather Rev., 128, 2835–2852, 2000. Smith, L A. and Gilmour, I.: Accountability and Internal Consistency in Ensemble Formation, in: Proceedings of a Workshop held at ECMWF on Predictability, Reading, UK, 20-22 October 1997, pp. 113–127, 1999. Toth, Z. and Kalnay, E.: Ensemble Forecasting at NMC: The Generation of Perturbations, B. Am. Meteorol. Soc., 74, 2317–2330, 1993. Toth, Z. and Kalnay, E.: Ensemble forecasting at NCEP, in: Proceedings of a Seminar held at ECMWF on Predictability, Reading, UK, 4-8 September 1995, vol 2, pp. 39–60, 1996. Toth, Z. and Kalnay, E.: Ensemble Forecasting at NCEP and the Breeding Method, Mon. Weather Rev., 125, 3297–3319, 1997. Toth, Z., Kalnay, E., Tracton, S M., Wobus, R., and Irwin, J.: A Synoptic Evaluation of the NCEP Ensemble, Weather Forecast., 12, 140–153, 1997. Tracton, M S. and Kalnay, E.: Operational Ensemble Prediction at the National Meteorological Center: Practical Aspects, Weather Forecast., 8, 379–398, 1993. Wei, M., Toth, Z., Wobus, R., and Zhu, Y.: Initial perturbations based on the ensemble transform (ET) technique in the NCEP global operational forecast system, Tellus, 60A, 62–79, 2008. Yang, S.-C., Cai, M., Kalnay, E., Rienecker, M., Yuan, G., and Toth, Z.: ENSO Bred Vectors in Coupled Ocean-Atmosphere General Circulation Models, J. Climate, 19, 1422–1436, 2006. Young, R M B. and Read, P L.: Flow transitions resembling bifurcations of the logistic map in in simulations of the baroclinic rotating annulus, Physica D, in press, 2008.