Lewis, S.R.

Martinez-Alvarado, O.

Moroz, I.M.

Read, P.L.

English

Open Research Online (The Open University)

Open Research OnlineThe Open University’s repository of research publicationsand other research outputsAtmospheric predictability of the martian atmosphere:from low-dimensional dynamics to operationalforecasting?Conference or Workshop ItemHow to cite:Read, P.L.; Lewis, S.R.; Moroz, I.M. and Martinez-Alvarado, O. (2006). Atmospheric predictability of themartian atmosphere: from low-dimensional dynamics to operational forecasting? In: Second Workshop on Marsatmosphere modelling and observations, 27 Feb - 3 Mar 2006, Granada, Spain.For guidance on citations see FAQs.c© [not recorded]Version: [not recorded]Link(s) to article on publisher’s website:http://adsabs.harvard.edu/cgi-bin/nph-bib query?bibcode=2006mamo.conf..155R&amp;db key=AST&amp;data type=HTML&amp;format=&amp;high=44aa7ebb1d26986Copyright and Moral Rights for the articles on this site are retained by the individual authors and/or other copyrightowners. For more information on Open Research Online’s data policy on reuse of materials please consult the policiespage.oro.open.ac.ukATMOSPHERIC PREDICTABILITY OF THE MARTIAN ATMOSPHERE: FROMLOW-DIMENSIONAL DYNAMICS TO OPERATIONAL FORECASTING?.P. L. Read, Dept. of Physics - AOPP, University of Oxford, Oxford, UK (p.read1@physics.ox.ac.uk), S. R. Lewis,Dept. of Physics & Astronomy, CEPSAR, The Open University, UK, I. M. Moroz, Mathematical Institute, Universityof Oxford, Oxford, UK, O. Martinez-Alvarado, Mathematical Institute, University of Oxford, Oxford, UK.IntroductionSince the work of Barnes (1980, 1981), it has been evi-dent that Martian mid-latitude meteorology is commonlydominated by large-scale baroclinically unstable tran-sient waves. Viking Lander data showed that such wavesare strongly modulated by the seasonal cycle, apparentlydisappearing altogether during northern summer. Dur-ing seasons around northern winter, however, they aretypically very active and dominated by relatively lowwavenumbers (m = 1-3). Unlike their terrestrial coun-terparts, however, Barnes (1980, 1981) found that Mar-tian baroclinic transients were typically highly regularin their behaviour in time, with near-periodic signalsat the VIking Lander sites persisting for months andwith temporal frequency spectra (see Fig. 1) exhibitingstrong peaks at periods of 2-8 days. The impressionwas therefore that Mars’s atmosphere is typically morecoherent, and therefore perhaps intrinsically more pre-dictable, than that of the Earth.Subsequent modelling and observations have largelyconfirmed this impression. In particular, Collins et al.(1996) showed in simple model experiments that, in theabsence of thermal tidal forcing, baroclinic waves in anotherwise reasonably realistic circulation would developinto extremely regular, spatially monochromatic wavepatterns progressively moving around the winter pole,with occasional jumps in wavenumber as the seasonalcycle advanced. Comparison of successive years of suchsimulations even gave the impression of hysteretic be-haviour in wavenumber transitions, much as found in theregular regime of laboratory experiments on baroclinicwaves in rotating, cylindrical containers (e.g. Hide andMason (1975)). Such extremely regular behaviour inthe non-diurnal Mars general circulation model (GCM)simulations was significantly more coherent than the ob-servations of Barnes (1980, 1981), although more real-istic episodic variability was restored when simulationswere re-run with the diurnal cycle included. Such resultssuggest that much of Mars’ synoptic variability may owemuch to an interaction between a relatively small num-ber of freely evolving, global, baroclinically unstableatmospheric modes and the periodically-forced diurnaltide.The simple spatial structures associated with sucha simple spectrum of near-normal modes of the atmo-sphere with a periodic forcing would strongly suggestthat synoptic-scale meteorology on Mars may be gov-erned by a relatively low-dimensional attractor. This isin significant contrast to the Earth, for which the likelyattractor dimension is probably quite high. The conse-quences for atmospheric predictability in a regime dom-inated by a low-dimensional attractor (chaotic or not)could be profound, with correspondingly important im-plications for the practical ability of atmospheric mod-els to forecast Martian meteorology from a given initialstate. In this presentation, we will review various stud-ies exploring the possible nature of the low-dimensionalMartian meteorological ‘attractor’ in simplified mod-els, and recent attempts to characterize more realisticatmospheric circulation systems in Mars GCMs and as-similated data (see Lewis et al., this volume). Mostrecently, attempts have been made by Newman et al.(2004) to determine quantitatively the growth of pertur-bations in realistic Martian circulation patterns, based onthe ‘breeding vector’ method (Kalnay, 2004). This leadson naturally to the possibility of ensemble forecastingexperiments, and the presentation will conclude by giv-ing an overview of recent results and future prospectsof making detailed synoptic forecasts of Martian mete-orology.1 Low-order POD modelsGiven a system whose behaviour is governed by thenonlinear interaction of a limited number of discrete,orthogonal modes, various approaches have been devel-oped to formulate simplified nonlinear models of suchsystems directly from timeseries of realisations from thefull system. A statistical analysis of suitable timeseriescan be used to decompose the spatio-temporal ‘signal’as the superposition of time-varying amplitudes of a setof distinct, orthogonal patterns. Depending upon thenorm used, such modes may represent distinct contribu-tions e.g. to the variance of the timeseries, or its kineticor total energy. The approach uses singular value de-composition of covariance or energy matrices, and isoften referred to as ‘Proper Orthogonal Decomposition’(Holmes et al., 1996) or POD.Subsets of such POD modes can be used as the basisfor a low-dimensional dynamical system, by substitutioninto the original (or approximated) equations of motion,together with suitable projections of physical forcingand dissipative processes. The resulting model, in theform of a closed set of nonlinear ordinary differentialequations, is referred to as a ‘POD-Galerkin’ model.Whitehouse et al. (2005a,b) conducted a study in1 LOW-ORDER POD MODELSFigure 1: Temporal spectra of baroclinic transients found byViking Lander 2 (after Barnes 1980; 1981).the context of Mars, in which a POD analysis was per-formed on data from a series of simple GCM simulationsbased on the work of Collins and James (1995). The lat-ter model comprised a low-resolution (T21) simulationof a Mars-like circulation forced by thermal relaxationtowards a reasonably realistic background temperaturedistribution, but without topography or diurnal or sea-sonal cycles. The resulting dominant POD modes werefound to take the form of latitudinally-confined wavetrains with sinusoidal variation in longitude of rela-tively low wavenumber. An example of such a dominantmode, projected onto four separate vertical eigenmodes,is shown in Fig. 2.Subsequent POD-Galerkin models based on rela-tively small subsets of such modes (Whitehouse et al.,2005b), in conjuction with the quasi-geostrophic equa-tions of motion in a spherical atmosphere, were found toFigure 2: First POD mode in an analysis of a wavenumber m=3, southern hemisphere winter flow regime of Collins & James(1995) on a Mars-like planet with flat topography, obtained byWhitehouse et al. (2005a). Fields are shown projected ontoeach of the four principal vertical eigenmodes of the verticalstructure equation (see Whitehouse et al. (2005a) for furtherdetails).be remarkably successful in reproducing the behaviourof the dominant wavenumbers in the original (primitiveequation) model flow. Moreover, by varying the forcingand dissipation parameters, a bifurcation analysis of thePOD-Galerkin model could be carried out which pre-dicted the existence of various wave regimes, includinga mixed-wavenumber state characterized by coexistenceof two distinct wavenumbers (e.g. see Fig. 3).Figure 3: Bifurcation diagram obtained in a POD-Galerkinmodel based on the POD analysis of the simple Mars-likecirculations of Collins & James (1995) by Whitehouse et al.(2005b). The figure plots normalized wave amplitude for eachof two dominant zonal wavenumber components as a functionof the Rayleigh friction dissipation parameter. Stable solu-tions are labelled S and unstable U.More recently, Martinez-Alvarado et al. (2005) hasbegun a comparable study using more realistic dataderived from a data assimilation of the Mars GlobalSurveyor Thermal Emission Spectrometer (MGS TES)dataset (Smith et al. (2001); see also Lewis et al., thisvolume). In this case, the derived POD modes includestrong contributions from large-scale modes representa-2 PREDICTABILITY AND ‘BREEDING VECTORS’tive of the diurnal tide in addition to those from baro-clinic transients (see Fig. 4).Figure 4: First and third POD mode derived from a north-ern winter assimilation of Martian atmospheric data from theMGS TES instrument (Martinez-Alvarado et al. 2005), plot-ted as a map of temperature at a nominal altitude of 20 km.In work currently in progress, POD-Galerkin modelswill be derived using these modes as a basis within thefull meteorological primitive equations. The latter arenecessary because the very large topography on Mars,and the strong non-geostrophic thermal tides, rendersthe quasi-geostrophic approximation inapplicable to thisproblem.2 Predictability and ‘breeding vectors’The existence of a low-dimensional attractor govern-ing the meteorology of Mars might be expected to havesome influence on the predictability of the atmosphere.The intrinsic predictability of an instantaneous atmo-spheric state is commonly quantified by use of a varietyof perturbation techniques in which the growth of a smallperturbation is monitored, either in linearized (as ‘singu-lar vectors’) or fully developed form (as ’bred vectors’;see Kalnay (2004)). Newman et al. (2004) computedbred vectors within a set of numerical simulations ofMars climate using the Oxford MGCM under variousseasonal conditions. Depending upon the amplitude ofthe initial perturbation, Newman et al. (2004) found that,for a number of seasons, the typical exponential growthrate of perturbations was actually negative (i.e. pertur-bations actually decayed slowly with time over a 30 solinterval; see Fig. 5). This would imply a near-absence ofchaotic behaviour during this season, implying a highlyforecastable flow. Only during seasons around NH win-ter was the mean growth rate significantly greater thanzero, implying an actively chaotic atmosphere. The peakgrowth rate corresponds to a mean perturbation doublingperiod of around 2.1 sols, comparable to the Earth’s tro-pospheric meteorology (Kalnay, 2004).0.30.20.10.0Growth rate (sol-1)300250200150100500Ls (o)Figure 5: Growth rate (in sol−1 of the dominant breeding vec-tor in the Viking (relatively dusty) scenario Mars GCM sim-ulations of Newman et al. (2004), averaged over sols 3-30 ofeach of 12 seasons throughout the Martian year.The bred vector method also enables the structure ofthe most rapidly growing perturbation. During NH win-ter, Newman et al. (2004) found that, after 30 sols, themode exhibiting the most rapid sustained mean growthresembled the dominant large-scale baroclinic mode ofinstability. Fig.6 shows some examples of horizontalstreamfunction maps at two vertical levels in the loweratmosphere for seasons 11 and 12 (Ls = 300 − 360◦),which clearly show modes dominated by m = 1 − 2 athigh northern latitudes.Newman et al. (2004) found some sensitivity inthe structure and behaviour of the bred vectors to theamplitude of the initial (random) perturbations, withsome very small amplitude perturbations initially grow-ing rapidly but then saturating before reaching the am-plitudes obtained in the computations shown in Figs 5and 6.REFERENCESFigure 6: Normalized streamfunction perturbation at the endof a 30 sol breeding run in the Viking (relatively dusty) sce-nario Mars GCM simulations of Newman et al. (2004). Mapsare shown for seasons 11 and 12 (NH winter) at altitudes of5km and 16km.3 Future workIn ongoing research, the subsequent evolution of atmo-spheric circulation patterns emerging from finite ampli-tude perturbations based on bred vectors is being inves-tigated (a) from stand-alone GCM simulations, and (b)from GCM forecasts initialized from atmospheric statesinitialized using MGS TES observations during the pe-riod 1999-2004 (see Lewis et al., this volume). Thispresentation will conclude with a report on preliminaryensemble predictions from this study.ReferencesBarnes, J. R., 1980. Time spectral analysis of mid-latitude disturbances in the Martian atmosphere. J.Atmos. Sci. 37, 2002–2015.Barnes, J. R., 1981. Midlatitude disturbances in the Mar-tian atmosphere: A second Mars year. J. Atmos. Sci.38, 225–234.Collins, M., James, I. N., 1995. Regular baroclinic tran-sient waves in a simplified global circulation model ofthe Martian atmosphere. J. Geophys. Res. 100 (E7),14421–14432.Collins, M., Lewis, S. R., Read, P. L., Hourdin, F.,1996. Baroclinic wave transitions in the Martian at-mosphere. Icarus 120, 344–357.Hide, R., Mason, P. J., 1975. Sloping convection in arotating fluid. Adv. Phys. 24, 47–100.Holmes, P., Lumley, J. L., Berkooz, G., 1996. Turbu-lence, coherent structures, dynamical systems andsymmetry. Cambridge University Press, Cambridge.Kalnay, E., 2004. Atmospheric modelling, data assimi-lation and predictability. Cambridge University Press,Cambridge.Martinez-Alvarado, O., Moroz, I. M., Lewis, S. R.,Read, P. L., 2005. Reduced order models of the Mar-tian atmospheric dynamics. ENOC-2005, Eindhoven,The Netherlands, pp. 1–9.Newman, C. E., Read, P. L., Lewis, S. R., 2004. In-vestigating atmospheric predictability on Mars us-ing breeding vectors in a general-circulation model.Quart. J. R. Meteor. Soc. 130, 2971–2989.Smith, M. D., Pearl, J. C., Conrath, B. J., Christensen,P. R., 2001. Thermal Emission Spectrometer results:Mars atmospheric thermal structure and aerosol dis-tribution. J. Geophys. Res. 106 (E4), 23929–23945.Whitehouse, S. G., Lewis, S. R., Moroz, I. M., Read,P. L., 2005a. A simplified model of the Martian atmo-sphere — part 1: a diagnostic analysis. Nonlin. Proc.Geophys. 12, 603–623.Whitehouse, S. G., Lewis, S. R., Moroz, I. M., Read,P. L., 2005b. A simplified model of the Martian atmo-sphere — part 2: a POD-Galerkin analysis. Nonlin.Proc. Geophys. 12, 625–642.

Atmospheric predictability of the martian atmosphere: from low-dimensional dynamics to operational forecasting?

http://oro.open.ac.uk/4814/1/Read_Granada2006.pdf

Atmospheric predictability of the martian atmosphere: from low-dimensional dynamics to operational forecasting?

Abstract

Similar works

Full text

Available Versions

Open Research Online (The Open University)