Research ArticleA Gyrostatic Low-Order Model forthe El Ntildeino-Southern Oscillation

Alexander Gluhovsky12

1Department of Earth Atmospheric and Planetary Sciences Purdue University West Lafayette IN 47907 USA2Department of Statistics Purdue University West Lafayette IN 47907 USA

Correspondence should be addressed to Alexander Gluhovsky aglupurdueedu

Received 30 July 2016 Accepted 13 December 2016 Published 15 January 2017

Academic Editor Dimitri Volchenkov

Copyright copy 2017 Alexander Gluhovsky This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

The paper demonstrates that similar to other important models of atmospheric phenomena beginning with the celebrated Lorenzmodel of the Rayleigh-Benard convection the Vallis low-order model (LOM) of the El Nino-Southern Oscillation admits agyrostatic form and discusses how gyrostatic LOMs may offer a general framework for deriving effective physically sound modelsfor atmospheric dynamics and time series analysis Any such model has a quadratic integral of motion (interpreted as someform of energy) which eliminates unphysical behaviors that have often plagued atmospheric LOMs and paves way for developingHamiltonian LOMs Restricting LOMs to a gyrostatic form also helps to design LOMs of optimum size and provides a modularconstruction of LOMs using gyrostats as elementary building blocks

1 Introduction

Atmospheric and climate dynamics exhibit complex behaviorreflected in a hierarchy of models from ldquosimplerdquo nonlinear 3-mode ordinary differential equations (ODEs) to full-fledgedclimate-system models involving nonlinear partial differ-ential equations (PDEs) The need for such hierarchy wasargued by Held [1] who noted that ldquowe typically gain someunderstanding of a complex system by relating its behaviorto that of other especially simpler systemsrdquo and asked ldquowhatdoes it mean after all to understand a system as complexas the climate when we cannot fully understand idealizednonlinear systems with only a few degrees of freedomrdquo

An effective way to deal with formidable mathematicaldifficulties posed by the PDEs of fluid dynamics via approx-imating them with finite systems of nonlinear ODEs (theso-called low-order models (LOMs)) has been established inpioneering work by Kolmogorov (described in [2 3]) Lorenz[4 5] and Obukhov [6 7] LOMs are commonly derivedfrom the PDEs via the Galerkin method fluid dynamicalfields are expanded into infinite series in time-independentbasis functions (commonly Fourier modes) then the seriesare truncated and substituted into the PDEs yielding a finite

system of ODEs (the LOM) for the time evolution of thecoefficients in the truncated expansions

Both Lorenz and Obukhov insisted that LOMs shouldretain conservation properties of the original PDEs Arbitrarytruncations in the Galerkin method however lead to modelsthat may lack the fundamental physical properties of theoriginal equations such as energy conservation (here andthroughout the paper understood as conservation in the limitof no damping and forcing) For example the celebratedLorenz model [4] of the two-dimensional Rayleigh-Benardconvection (RBC)

= 120590 (119910 minus 119909) = minus119909119911 + 119903119909 minus 119910 = 119909119910 minus 119887119911

(1)

has revolutionized understanding of randomness in naturebut attempts to extend it to larger more realistic modelsof atmospheric dynamics have sometimes led to LOMsexhibiting unphysical behaviors The problem was addressed[8 9] by developing the so-called gyrostatic LOMs or G-models (see Section 2 in particular the simplest such model

HindawiComplexityVolume 2017 Article ID 6176045 4 pageshttpsdoiorg10115520176176045

2 Complexity

in a forced regime proved equivalent to the Lorenzmodel (1))and via extended Nambu or Lie-Poisson formalisms [10ndash17]

RBC is the most carefully studied example of nonlinearsystems exhibiting self-organization and transition to chaosIt also promotes understanding of many real-world fluidflows in the atmosphere (being the principal mechanism ofmesoscale shallow convection) liquid core of the Earth andastrophysics

The El Nino-Southern Oscillation (ENSO) is also oneof the most important and longest-studied phenomena asit affects the global atmospheric circulation and thereforetemperature and precipitation across the globe What isdemonstrated in this paper is that thewell-knownVallis LOM[18 19] of ENSO

= 119861 (119879119890 minus 119879119908)2Δ119909 minus 119862 (119906 minus 119906lowast)

119908= 119906 (119879 minus 119879119890)2Δ119909 minus 119860 (119879

119908minus 119879lowast)

119890= 119906 (119879119908 minus 119879)2Δ119909 minus 119860 (119879

119890minus 119879lowast)

(2)

is a G-model as wellG-models are defined and briefly described in Section 2

and Section 3 presents the derivation of the G-model formfor the Vallis model and a general discussion of G-models asa novel tool in atmospheric dynamics and time series analysiswith conclusions in Section 4

2 G-Models

The basic G-model is the Volterra gyrostat [20 21] a classicalsystem which admits various mechanical and fluid dynami-cal interpretations and can be written [22] as

1= 11990111988321198833+ 1198871198833minus 1198881198832

2= 11990211988321198833+ 1198881198831minus 1198861198833

3= 11990311988321198833+ 1198861198832minus 1198871198831

(3)

where 119901 + 119902 + 119903 = 0 Note that unlike linear friction termslinear terms in (3) (linear gyrostatic terms) do not affect theconservation of energy nor the conservation of phase spacevolume

The simplest Volterra gyrostat (119903 = 119887 = 119888 = 0 in (3)) in aforced regime that is with added constant forcing and linearfriction

1= minus119883

21198833

minus12057211198831+ 119865

2= 119883

31198831minus 1198833minus12057221198832

3= 119883

2minus12057231198833

(4)

was proved [8] to be equivalent to the Lorenz model (1) (forthis reason we call it the Lorenz gyrostat) In (4) and othersbelow separate Volterra gyrostats are shown within verticalbars variables are denoted by 119883

119894 friction coefficients by 120572

119894

forces by 119865 or 119865119894 and the overdot means differentiation with

respect to dimensionless time 120591It was also found [23 24] that effective LOMs for

atmospheric circulations and turbulence could be developedas systems of coupled gyrostats (3) For example the followingsystem of two coupled Lorenz gyrostats (4) in a forced regime

1= minus119883

21198833

minus11988341198835

minus12057211198831+ 119865

2= 119883

31198831minus 1198833

minus12057221198832

3= 119883

2minus12057231198833

4= 119883

51198831minus 1198835

minus12057241198834

5= 119883

4minus12057251198835

(5)

provides an analog of the Lorenz model the three-dimensional RBC where the Lorenz gyrostats describethe dynamics in two perpendicular planes [25]

Finally similar to the Arnoldrsquos definition of the 119899-dimensional rigid body [26] the 119899-dimensional gyrostat wasintroduced [8 9] as the 119899-dimensional analog of the Volterraequations (3) the latter recovered at 119899 = 3 It has turned outthat the 6-mode extension of the Lorenz model (1) recentlysuggested as a more appropriate minimal model of the two-dimensional RBC [12] could also be treated in terms of G-models namely as a 4-dimensional gyrostat [27]

G-models are all of the above gyrostatic LOMs Volterragyrostats coupled Volterra gyrostats and 119899-dimensionalgyrostats Any G-model has a quadratic integral of motion(interpreted as some form of energy) which eliminatesunphysical behaviors that have often plagued LOMs obtainedthrough ad hoc Galerkin truncations This integral is often agood candidate for a Hamiltonian function thus paving wayfor developing Hamiltonian LOMs [9] which is importantsince the conservative part of various atmospheric models(the primitive equations shallow water equations and quasi-geostrophic equations) is Hamiltonian (eg [28])

3 Results and Discussion

31 G-Model Form of the VallisModel After the linear changeof variables

119906 997888rarr radic119861119879 (1198833minus 1)

119879119908997888rarr 119879119883

2

119879119890997888rarr 119879119883

2

119905 997888rarr 2Δ119909radic119861119879

(6)

(2) take the form of gyrostat (3) in a forced regime

1= minus119883

21198833

+1198832minus1198833minus12057211198831+ 1198651

2= 119883

31198831minus 1198833minus1198831

minus12057221198832+ 1198652

3= 119883

2+1198831minus12057231198833+ 1198653

(7)

Complexity 3

where

1198651= 1 + 2Δ119909119860119879lowastradic1198611198793

1198652= 1 minus 2Δ119909119860119879lowastradic1198611198793

1198653= 119862(radic119861119879 + 119906lowast)

1205721= 1205722= 2Δ119909119860radic119861119879

1205723= 2Δ119909119862radic119861119879

(8)

In comparison to the general form (3) the gyrostat in (7)has only twononlinear terms but it has all three pairs of lineargyrostatic terms unlike the Lorenz gyrostat in (4) that hasonly one such pair In addition the external force in (7) hasthree components versus one in (4)

These differences between the Lorenz model (see (1) and(4)) and the Vallis model (see (2) and (7)) could perhapsbe clarified using a mechanical interpretation of the originalversion of (3) [20 21]

11986811= (1198682minus 1198683) 12059621205963+ ℎ21205963minus ℎ31205962

11986822= (1198683minus 1198681) 12059631205961+ ℎ31205961minus ℎ11205963

11986833= (1198681minus 1198682) 12059611205962+ ℎ11205962minus ℎ21205961

(9)

as a rigid body containing an axisymmetric rotor that rotateswith a constant angular velocity about an axis fixed in thecarrier body In (9) then 119868

119894are the principal moments of

inertia of the gyrostat 120596 is the angular velocity of the carrierand h is the fixed angular momentum caused by the relativemotion of the rotor (the gyrostatic motion) For the Lorenzgyrostat (4) this means [22] that the rotor is rotating aroundits principal axis 1 (ℎ

2= ℎ3= 0) with a constant external

force directed along this axis while its ellipsoid of inertia isthe ellipsoid of rotation around another principal axis axis3 (1198681= 1198682) The latter is also true for the gyrostat in G-

model (7) but in this case both h and F have three nonzerocomponents

Of particular importance for using the Vallis model (see(2) and (7)) in ENSO studies is that similar to the Lorenzmodel (see (1) and (4)) it exhibits both stable and chaoticbehaviors [18 19 29]

32 G-Models as Effective LOMs for Atmospheric Studies TheVallis model augments the list of phenomena described by G-models Among them are shell models of turbulence [22 24]models of convection in rotating fluid [30 31] of a barotropicatmosphere with topography and of the thermal convectionwith shear [32] and Hamiltonian LOMs [9] In fact we havefound [33] that all physically sound LOMs of 2D RBC that

have appeared in recent publications are equivalent to G-models while the LOMs lacking this form exhibit violationsof energy conservation

A new promising application of G-models is their use asnovel atmospheric time series models [33] thereby utilizingboth their deterministic and probabilistic facets It was moti-vated by current problems with handling atmospheric dataand by recent progress in statistical properties of dynamicalsystems In particular it has been proved that the flow ofthe Lorenz model (1) possesses a physical ergodic invariantprobability measure [34] and satisfies the central limit theo-rem [35 36] that is series of observations on this model mayexhibit statistics of sequences of randomvariables In contrastto common models (borrowed from traditional time seriesanalysis and having little to do with the atmosphere per se)G-models are derived from the underlying equations and sotheir statistical behavior is in better agreement with reality

LOMs reveal basic mechanisms and their interplaythrough the focus on key elements and retaining onlyminimal number of degrees of freedom Any G-modelas mentioned above has a quadratic integral of motionwhich eliminates certain unphysical behaviors which oftenplague other LOMs Another attractive feature of G-modelsis that increasing the order of approximation in the Galerkinprocedure results in adding to the system new gyrostats (orblocks of gyrostats) whose linear terms represent variouseffects important in atmospheric dynamics such as stratifi-cation rotation or topography In this way larger G-modelsbecome even more useful as they provide increasingly betterapproximations to the original system This is because thedynamics generated by fundamental mathematical models offluid flows are in a sense asymptotically finite-dimensional[37]

For all these reasons G-models may probably offer a gen-eral framework for deriving effective LOMs of atmosphericdynamics and atmospheric time series analysis

4 Conclusions

We have shown that the Vallis model of ENSO is a G-modeland argued advantages of this type of LOMs Even with ever-increasing computer power LOMs remain important sinceas noted by Smith [38] ldquoalthough it is unreasonable to expectsolutions to low-dimensional problems to generalize to amillion dimensional spaces so too it is unlikely that problemsidentified in the simplified models will vanish in operationalmodelsrdquo

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by NSF Grant AGS-1050588

4 Complexity

References

[1] I M Held ldquoThe gap between simulation and understandingin climate modelingrdquo Bulletin of the American MeteorologicalSociety vol 86 no 11 pp 1609ndash1614 2005

[2] A Pasini V Pelino and S Potesta ldquoTorsion and attractors in theKolmogorov hydrodynamical systemrdquo Physics Letters A vol241 no 1-2 pp 77ndash83 1998

[3] A Pasini and V Pelino ldquoA unified view of Kolmogorov andLorenz systemsrdquo Physics Letters A vol 275 no 5-6 pp 435ndash446 2000

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] E N Lorenz ldquoLow-order models of atmospheric circulationsrdquoJournal of the Meteorological Society of Japan vol 60 pp 255ndash267 1982

[6] A M Obukhov ldquoOn integral characteristics in hydrodynamictype systemsrdquo Soviet PhysicsmdashDoklady vol 14 pp 32ndash35 1969

[7] A M Obukhov ldquoOn the problem of nonlinear interactions influid dynamicsrdquo Gerlands Beitrage zur Geophysik vol 14 pp282ndash290 1973

[8] A Gluhovsky ldquoNonlinear systems that are superpositions ofgyrostatsrdquo Soviet PhysicsmdashDoklady vol 27 pp 823ndash825 1982

[9] A Gluhovsky ldquoEnergy-conserving and Hamiltonian low-ordermodels in geophysical fluid dynamicsrdquo Nonlinear Processes inGeophysics vol 13 no 2 pp 125ndash133 2006

[10] P Nevir and R Blender ldquoHamiltonian and Nambu represen-tation of the non-dissipative Lorenz equationsrdquo Beitraege zurPhysik der Atmosphaere vol 67 Article ID 133140 pp 133ndash1401994

[11] A Bihlo ldquoRayleighndashBenard convection as a Nambu-metriplectic problemrdquo Journal of Physics A Mathematicaland Theoretical vol 41 no 29 Article ID 292001 2008

[12] A Bihlo and J Staufer ldquoMinimal atmospheric finite-modemodels preserving symmetry and generalized Hamiltonianstructuresrdquo Physica D Nonlinear Phenomena vol 240 no 7 pp599ndash606 2011

[13] R Blender and V Lucarini ldquoNambu representation of anextended Lorenz model with viscous heatingrdquo Physica DNonlinear Phenomena vol 243 pp 86ndash91 2013

[14] V Pelino andA Pasini ldquoDissipation in Lie-Poisson systems andthe Lorenz-84 modelrdquo Physics Letters A vol 291 no 6 pp 389ndash396 2001

[15] V Pelino and F Maimone ldquoEnergetics skeletal dynamicsand long-term predictions on Kolmogorov-Lorenz systemsrdquoPhysical Review E vol 76 no 4 Article ID 046214 2007

[16] V Pelino F Maimone and A Pasini ldquoOscillating forcingsand new regimes in the Lorenz system a four-lobe attractorrdquoNonlinear Processes in Geophysics vol 19 no 3 pp 315ndash3222012

[17] V Pelino F Maimone and A Pasini ldquoEnergy cycle for theLorenz attractorrdquo Chaos Solitons amp Fractals vol 64 pp 67ndash772014

[18] G K Vallis ldquoEl Nino a chaotic dynamical systemrdquo Science vol232 no 4747 pp 243ndash245 1986

[19] G K Vallis ldquoConceptual models of El Nino and the SouthernOscillationrdquo Journal of Geophysical Research vol 93 no 11 pp13979ndash13991 1988

[20] V Volterra ldquoSur la theorie des variations des latitudesrdquo ActaMathematica vol 22 no 1 pp 201ndash357 1899

[21] J Wittenburg Dynamics of Multibody Systems Springer 2008

[22] A Gluhovsky and C Tong ldquoThe structure of energy conservinglow-order modelsrdquo Physics of Fluids vol 11 no 2 pp 334ndash3431999

[23] A Gluhovsky ldquoOn systems of coupled gyrostats in problemsof geophysical hydrodynamicsrdquo Izvestiya Atmospheric andOceanic Physics vol 22 pp 543ndash549 1986

[24] A Gluhovsky ldquoCascade system of coupled gyrostats for mod-eling fully developed turbulencerdquo Izvestiya Atmospheric andOceanic Physics vol 23 pp 952ndash958 1987

[25] C Tong and A Gluhovsky ldquoEnergy-conserving low-ordermodels for three-dimensional Rayleigh-Benard convectionrdquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 65 no 4 Article ID 046306 2002

[26] V Arnold ldquoSur la geometrie differentielle des groupes de lie dedimension infinie et ses applications a lrsquohydrodynamique desfluides parfaitsrdquo Annales de lrsquoInstitut Fourier vol 16 pp 316ndash361 1966

[27] A Gluhovsky ldquoComment on ldquominimal atmospheric finite-mode models preserving symmetry and generalized Hamilto-nian structures Physica D 240 (2011) 599ndash606rdquordquo Physica DNonlinear Phenomena vol 268 pp 118ndash120 2014

[28] T G Shepherd ldquoSymmetries Conservation laws and Hamil-tonian structure in geophysical fluid dynamicsrdquo Advances inGeophysics vol 32 pp 287ndash338 1990

[29] K K TungTopics inMathematicalModeling PrincetonUniver-sity Press Princeton NJ USA 2007

[30] A Gluhovsky and F V Dolzhansky ldquoThree-componentgeostrophic models of convection in a rotating fluidrdquo IzvestiyaAtmospheric and Oceanic Physics vol 16 pp 311ndash318 1980

[31] G A Leonov ldquoExistence criterion of homoclinic trajectories inthe Glukhovsky-Dolzhansky systemrdquo Physics Letters A vol 379no 6 pp 524ndash528 2015

[32] A Gluhovsky C Tong and E Agee ldquoSelection of modesin convective low-order modelsrdquo Journal of the AtmosphericSciences vol 59 no 8 pp 1383ndash1393 2002

[33] A Gluhovsky and K Grady ldquoEffective low-order models foratmospheric dynamics and time series analysisrdquo Chaos vol 26no 2 Article ID 023119 2016

[34] V Araujo M J Pacifico E R Pujals and M Viana ldquoSingular-hyperbolic attractors are chaoticrdquo Transactions of the AmericanMathematical Society vol 361 no 5 pp 2431ndash2485 2009

[35] M Holland and I Melbourne ldquoCentral limit theorems andinvariance principles for Lorenz attractorsrdquo Journal of theLondon Mathematical Society Second Series vol 76 no 2 pp345ndash364 2007

[36] V Araujo and P Varandas ldquoRobust exponential decay of cor-relations for singular-flowsrdquo Communications in MathematicalPhysics vol 311 no 1 pp 215ndash246 2012

[37] J C Robinson ldquoAttractors and finite-dimensional behaviour inthe 2D Navier-Stokes equationsrdquo ISRN Mathematical Analysisvol 2013 Article ID 291823 29 pages 2013

[38] L A Smith ldquoWhat might we learn from climate forecastsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 1 pp 2487ndash2492 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Complexity 3

where

1198651= 1 + 2Δ119909119860119879lowastradic1198611198793

1198652= 1 minus 2Δ119909119860119879lowastradic1198611198793

1198653= 119862(radic119861119879 + 119906lowast)

1205721= 1205722= 2Δ119909119860radic119861119879

1205723= 2Δ119909119862radic119861119879

(8)

In comparison to the general form (3) the gyrostat in (7)has only twononlinear terms but it has all three pairs of lineargyrostatic terms unlike the Lorenz gyrostat in (4) that hasonly one such pair In addition the external force in (7) hasthree components versus one in (4)

These differences between the Lorenz model (see (1) and(4)) and the Vallis model (see (2) and (7)) could perhapsbe clarified using a mechanical interpretation of the originalversion of (3) [20 21]

11986811= (1198682minus 1198683) 12059621205963+ ℎ21205963minus ℎ31205962

11986822= (1198683minus 1198681) 12059631205961+ ℎ31205961minus ℎ11205963

11986833= (1198681minus 1198682) 12059611205962+ ℎ11205962minus ℎ21205961

(9)

as a rigid body containing an axisymmetric rotor that rotateswith a constant angular velocity about an axis fixed in thecarrier body In (9) then 119868

119894are the principal moments of

inertia of the gyrostat 120596 is the angular velocity of the carrierand h is the fixed angular momentum caused by the relativemotion of the rotor (the gyrostatic motion) For the Lorenzgyrostat (4) this means [22] that the rotor is rotating aroundits principal axis 1 (ℎ

2= ℎ3= 0) with a constant external

force directed along this axis while its ellipsoid of inertia isthe ellipsoid of rotation around another principal axis axis3 (1198681= 1198682) The latter is also true for the gyrostat in G-

model (7) but in this case both h and F have three nonzerocomponents

Of particular importance for using the Vallis model (see(2) and (7)) in ENSO studies is that similar to the Lorenzmodel (see (1) and (4)) it exhibits both stable and chaoticbehaviors [18 19 29]

32 G-Models as Effective LOMs for Atmospheric Studies TheVallis model augments the list of phenomena described by G-models Among them are shell models of turbulence [22 24]models of convection in rotating fluid [30 31] of a barotropicatmosphere with topography and of the thermal convectionwith shear [32] and Hamiltonian LOMs [9] In fact we havefound [33] that all physically sound LOMs of 2D RBC that

have appeared in recent publications are equivalent to G-models while the LOMs lacking this form exhibit violationsof energy conservation

A new promising application of G-models is their use asnovel atmospheric time series models [33] thereby utilizingboth their deterministic and probabilistic facets It was moti-vated by current problems with handling atmospheric dataand by recent progress in statistical properties of dynamicalsystems In particular it has been proved that the flow ofthe Lorenz model (1) possesses a physical ergodic invariantprobability measure [34] and satisfies the central limit theo-rem [35 36] that is series of observations on this model mayexhibit statistics of sequences of randomvariables In contrastto common models (borrowed from traditional time seriesanalysis and having little to do with the atmosphere per se)G-models are derived from the underlying equations and sotheir statistical behavior is in better agreement with reality

LOMs reveal basic mechanisms and their interplaythrough the focus on key elements and retaining onlyminimal number of degrees of freedom Any G-modelas mentioned above has a quadratic integral of motionwhich eliminates certain unphysical behaviors which oftenplague other LOMs Another attractive feature of G-modelsis that increasing the order of approximation in the Galerkinprocedure results in adding to the system new gyrostats (orblocks of gyrostats) whose linear terms represent variouseffects important in atmospheric dynamics such as stratifi-cation rotation or topography In this way larger G-modelsbecome even more useful as they provide increasingly betterapproximations to the original system This is because thedynamics generated by fundamental mathematical models offluid flows are in a sense asymptotically finite-dimensional[37]

For all these reasons G-models may probably offer a gen-eral framework for deriving effective LOMs of atmosphericdynamics and atmospheric time series analysis

4 Conclusions

We have shown that the Vallis model of ENSO is a G-modeland argued advantages of this type of LOMs Even with ever-increasing computer power LOMs remain important sinceas noted by Smith [38] ldquoalthough it is unreasonable to expectsolutions to low-dimensional problems to generalize to amillion dimensional spaces so too it is unlikely that problemsidentified in the simplified models will vanish in operationalmodelsrdquo

Competing Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by NSF Grant AGS-1050588

4 Complexity

References

[1] I M Held ldquoThe gap between simulation and understandingin climate modelingrdquo Bulletin of the American MeteorologicalSociety vol 86 no 11 pp 1609ndash1614 2005

[2] A Pasini V Pelino and S Potesta ldquoTorsion and attractors in theKolmogorov hydrodynamical systemrdquo Physics Letters A vol241 no 1-2 pp 77ndash83 1998

[3] A Pasini and V Pelino ldquoA unified view of Kolmogorov andLorenz systemsrdquo Physics Letters A vol 275 no 5-6 pp 435ndash446 2000

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] E N Lorenz ldquoLow-order models of atmospheric circulationsrdquoJournal of the Meteorological Society of Japan vol 60 pp 255ndash267 1982

[6] A M Obukhov ldquoOn integral characteristics in hydrodynamictype systemsrdquo Soviet PhysicsmdashDoklady vol 14 pp 32ndash35 1969

[7] A M Obukhov ldquoOn the problem of nonlinear interactions influid dynamicsrdquo Gerlands Beitrage zur Geophysik vol 14 pp282ndash290 1973

[8] A Gluhovsky ldquoNonlinear systems that are superpositions ofgyrostatsrdquo Soviet PhysicsmdashDoklady vol 27 pp 823ndash825 1982

[9] A Gluhovsky ldquoEnergy-conserving and Hamiltonian low-ordermodels in geophysical fluid dynamicsrdquo Nonlinear Processes inGeophysics vol 13 no 2 pp 125ndash133 2006

[10] P Nevir and R Blender ldquoHamiltonian and Nambu represen-tation of the non-dissipative Lorenz equationsrdquo Beitraege zurPhysik der Atmosphaere vol 67 Article ID 133140 pp 133ndash1401994

[11] A Bihlo ldquoRayleighndashBenard convection as a Nambu-metriplectic problemrdquo Journal of Physics A Mathematicaland Theoretical vol 41 no 29 Article ID 292001 2008

[12] A Bihlo and J Staufer ldquoMinimal atmospheric finite-modemodels preserving symmetry and generalized Hamiltonianstructuresrdquo Physica D Nonlinear Phenomena vol 240 no 7 pp599ndash606 2011

[13] R Blender and V Lucarini ldquoNambu representation of anextended Lorenz model with viscous heatingrdquo Physica DNonlinear Phenomena vol 243 pp 86ndash91 2013

[14] V Pelino andA Pasini ldquoDissipation in Lie-Poisson systems andthe Lorenz-84 modelrdquo Physics Letters A vol 291 no 6 pp 389ndash396 2001

[15] V Pelino and F Maimone ldquoEnergetics skeletal dynamicsand long-term predictions on Kolmogorov-Lorenz systemsrdquoPhysical Review E vol 76 no 4 Article ID 046214 2007

[16] V Pelino F Maimone and A Pasini ldquoOscillating forcingsand new regimes in the Lorenz system a four-lobe attractorrdquoNonlinear Processes in Geophysics vol 19 no 3 pp 315ndash3222012

[17] V Pelino F Maimone and A Pasini ldquoEnergy cycle for theLorenz attractorrdquo Chaos Solitons amp Fractals vol 64 pp 67ndash772014

[18] G K Vallis ldquoEl Nino a chaotic dynamical systemrdquo Science vol232 no 4747 pp 243ndash245 1986

[19] G K Vallis ldquoConceptual models of El Nino and the SouthernOscillationrdquo Journal of Geophysical Research vol 93 no 11 pp13979ndash13991 1988

[20] V Volterra ldquoSur la theorie des variations des latitudesrdquo ActaMathematica vol 22 no 1 pp 201ndash357 1899

[21] J Wittenburg Dynamics of Multibody Systems Springer 2008

[22] A Gluhovsky and C Tong ldquoThe structure of energy conservinglow-order modelsrdquo Physics of Fluids vol 11 no 2 pp 334ndash3431999

[23] A Gluhovsky ldquoOn systems of coupled gyrostats in problemsof geophysical hydrodynamicsrdquo Izvestiya Atmospheric andOceanic Physics vol 22 pp 543ndash549 1986

[24] A Gluhovsky ldquoCascade system of coupled gyrostats for mod-eling fully developed turbulencerdquo Izvestiya Atmospheric andOceanic Physics vol 23 pp 952ndash958 1987

[25] C Tong and A Gluhovsky ldquoEnergy-conserving low-ordermodels for three-dimensional Rayleigh-Benard convectionrdquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 65 no 4 Article ID 046306 2002

[26] V Arnold ldquoSur la geometrie differentielle des groupes de lie dedimension infinie et ses applications a lrsquohydrodynamique desfluides parfaitsrdquo Annales de lrsquoInstitut Fourier vol 16 pp 316ndash361 1966

[27] A Gluhovsky ldquoComment on ldquominimal atmospheric finite-mode models preserving symmetry and generalized Hamilto-nian structures Physica D 240 (2011) 599ndash606rdquordquo Physica DNonlinear Phenomena vol 268 pp 118ndash120 2014

[28] T G Shepherd ldquoSymmetries Conservation laws and Hamil-tonian structure in geophysical fluid dynamicsrdquo Advances inGeophysics vol 32 pp 287ndash338 1990

[29] K K TungTopics inMathematicalModeling PrincetonUniver-sity Press Princeton NJ USA 2007

[30] A Gluhovsky and F V Dolzhansky ldquoThree-componentgeostrophic models of convection in a rotating fluidrdquo IzvestiyaAtmospheric and Oceanic Physics vol 16 pp 311ndash318 1980

[31] G A Leonov ldquoExistence criterion of homoclinic trajectories inthe Glukhovsky-Dolzhansky systemrdquo Physics Letters A vol 379no 6 pp 524ndash528 2015

[32] A Gluhovsky C Tong and E Agee ldquoSelection of modesin convective low-order modelsrdquo Journal of the AtmosphericSciences vol 59 no 8 pp 1383ndash1393 2002

[33] A Gluhovsky and K Grady ldquoEffective low-order models foratmospheric dynamics and time series analysisrdquo Chaos vol 26no 2 Article ID 023119 2016

[34] V Araujo M J Pacifico E R Pujals and M Viana ldquoSingular-hyperbolic attractors are chaoticrdquo Transactions of the AmericanMathematical Society vol 361 no 5 pp 2431ndash2485 2009

[35] M Holland and I Melbourne ldquoCentral limit theorems andinvariance principles for Lorenz attractorsrdquo Journal of theLondon Mathematical Society Second Series vol 76 no 2 pp345ndash364 2007

[36] V Araujo and P Varandas ldquoRobust exponential decay of cor-relations for singular-flowsrdquo Communications in MathematicalPhysics vol 311 no 1 pp 215ndash246 2012

[37] J C Robinson ldquoAttractors and finite-dimensional behaviour inthe 2D Navier-Stokes equationsrdquo ISRN Mathematical Analysisvol 2013 Article ID 291823 29 pages 2013

[38] L A Smith ldquoWhat might we learn from climate forecastsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 1 pp 2487ndash2492 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Complexity

References

[1] I M Held ldquoThe gap between simulation and understandingin climate modelingrdquo Bulletin of the American MeteorologicalSociety vol 86 no 11 pp 1609ndash1614 2005

[2] A Pasini V Pelino and S Potesta ldquoTorsion and attractors in theKolmogorov hydrodynamical systemrdquo Physics Letters A vol241 no 1-2 pp 77ndash83 1998

[3] A Pasini and V Pelino ldquoA unified view of Kolmogorov andLorenz systemsrdquo Physics Letters A vol 275 no 5-6 pp 435ndash446 2000

[4] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[5] E N Lorenz ldquoLow-order models of atmospheric circulationsrdquoJournal of the Meteorological Society of Japan vol 60 pp 255ndash267 1982

[6] A M Obukhov ldquoOn integral characteristics in hydrodynamictype systemsrdquo Soviet PhysicsmdashDoklady vol 14 pp 32ndash35 1969

[7] A M Obukhov ldquoOn the problem of nonlinear interactions influid dynamicsrdquo Gerlands Beitrage zur Geophysik vol 14 pp282ndash290 1973

[8] A Gluhovsky ldquoNonlinear systems that are superpositions ofgyrostatsrdquo Soviet PhysicsmdashDoklady vol 27 pp 823ndash825 1982

[9] A Gluhovsky ldquoEnergy-conserving and Hamiltonian low-ordermodels in geophysical fluid dynamicsrdquo Nonlinear Processes inGeophysics vol 13 no 2 pp 125ndash133 2006

[10] P Nevir and R Blender ldquoHamiltonian and Nambu represen-tation of the non-dissipative Lorenz equationsrdquo Beitraege zurPhysik der Atmosphaere vol 67 Article ID 133140 pp 133ndash1401994

[11] A Bihlo ldquoRayleighndashBenard convection as a Nambu-metriplectic problemrdquo Journal of Physics A Mathematicaland Theoretical vol 41 no 29 Article ID 292001 2008

[12] A Bihlo and J Staufer ldquoMinimal atmospheric finite-modemodels preserving symmetry and generalized Hamiltonianstructuresrdquo Physica D Nonlinear Phenomena vol 240 no 7 pp599ndash606 2011

[13] R Blender and V Lucarini ldquoNambu representation of anextended Lorenz model with viscous heatingrdquo Physica DNonlinear Phenomena vol 243 pp 86ndash91 2013

[14] V Pelino andA Pasini ldquoDissipation in Lie-Poisson systems andthe Lorenz-84 modelrdquo Physics Letters A vol 291 no 6 pp 389ndash396 2001

[15] V Pelino and F Maimone ldquoEnergetics skeletal dynamicsand long-term predictions on Kolmogorov-Lorenz systemsrdquoPhysical Review E vol 76 no 4 Article ID 046214 2007

[16] V Pelino F Maimone and A Pasini ldquoOscillating forcingsand new regimes in the Lorenz system a four-lobe attractorrdquoNonlinear Processes in Geophysics vol 19 no 3 pp 315ndash3222012

[17] V Pelino F Maimone and A Pasini ldquoEnergy cycle for theLorenz attractorrdquo Chaos Solitons amp Fractals vol 64 pp 67ndash772014

[18] G K Vallis ldquoEl Nino a chaotic dynamical systemrdquo Science vol232 no 4747 pp 243ndash245 1986

[19] G K Vallis ldquoConceptual models of El Nino and the SouthernOscillationrdquo Journal of Geophysical Research vol 93 no 11 pp13979ndash13991 1988

[20] V Volterra ldquoSur la theorie des variations des latitudesrdquo ActaMathematica vol 22 no 1 pp 201ndash357 1899

[21] J Wittenburg Dynamics of Multibody Systems Springer 2008

[22] A Gluhovsky and C Tong ldquoThe structure of energy conservinglow-order modelsrdquo Physics of Fluids vol 11 no 2 pp 334ndash3431999

[23] A Gluhovsky ldquoOn systems of coupled gyrostats in problemsof geophysical hydrodynamicsrdquo Izvestiya Atmospheric andOceanic Physics vol 22 pp 543ndash549 1986

[24] A Gluhovsky ldquoCascade system of coupled gyrostats for mod-eling fully developed turbulencerdquo Izvestiya Atmospheric andOceanic Physics vol 23 pp 952ndash958 1987

[25] C Tong and A Gluhovsky ldquoEnergy-conserving low-ordermodels for three-dimensional Rayleigh-Benard convectionrdquoPhysical Review E Statistical Nonlinear and SoftMatter Physicsvol 65 no 4 Article ID 046306 2002

[26] V Arnold ldquoSur la geometrie differentielle des groupes de lie dedimension infinie et ses applications a lrsquohydrodynamique desfluides parfaitsrdquo Annales de lrsquoInstitut Fourier vol 16 pp 316ndash361 1966

[27] A Gluhovsky ldquoComment on ldquominimal atmospheric finite-mode models preserving symmetry and generalized Hamilto-nian structures Physica D 240 (2011) 599ndash606rdquordquo Physica DNonlinear Phenomena vol 268 pp 118ndash120 2014

[28] T G Shepherd ldquoSymmetries Conservation laws and Hamil-tonian structure in geophysical fluid dynamicsrdquo Advances inGeophysics vol 32 pp 287ndash338 1990

[29] K K TungTopics inMathematicalModeling PrincetonUniver-sity Press Princeton NJ USA 2007

[30] A Gluhovsky and F V Dolzhansky ldquoThree-componentgeostrophic models of convection in a rotating fluidrdquo IzvestiyaAtmospheric and Oceanic Physics vol 16 pp 311ndash318 1980

[31] G A Leonov ldquoExistence criterion of homoclinic trajectories inthe Glukhovsky-Dolzhansky systemrdquo Physics Letters A vol 379no 6 pp 524ndash528 2015

[32] A Gluhovsky C Tong and E Agee ldquoSelection of modesin convective low-order modelsrdquo Journal of the AtmosphericSciences vol 59 no 8 pp 1383ndash1393 2002

[33] A Gluhovsky and K Grady ldquoEffective low-order models foratmospheric dynamics and time series analysisrdquo Chaos vol 26no 2 Article ID 023119 2016

[34] V Araujo M J Pacifico E R Pujals and M Viana ldquoSingular-hyperbolic attractors are chaoticrdquo Transactions of the AmericanMathematical Society vol 361 no 5 pp 2431ndash2485 2009

[35] M Holland and I Melbourne ldquoCentral limit theorems andinvariance principles for Lorenz attractorsrdquo Journal of theLondon Mathematical Society Second Series vol 76 no 2 pp345ndash364 2007

[36] V Araujo and P Varandas ldquoRobust exponential decay of cor-relations for singular-flowsrdquo Communications in MathematicalPhysics vol 311 no 1 pp 215ndash246 2012

[37] J C Robinson ldquoAttractors and finite-dimensional behaviour inthe 2D Navier-Stokes equationsrdquo ISRN Mathematical Analysisvol 2013 Article ID 291823 29 pages 2013

[38] L A Smith ldquoWhat might we learn from climate forecastsrdquoProceedings of the National Academy of Sciences of the UnitedStates of America vol 99 no 1 pp 2487ndash2492 2002

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of