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Singular spectrum analysis (SSA) or extended empirical orthogonal function methods are powerful, commonly used data‐driven techniques to identify modes of variability in time series and space‐time data sets. Due to the time‐lagged embedding, these methods can provide inaccurate reconstructions of leading modes near the endpoints, which can hinder the use of these methods in real time. A modified version of the traditional SSA algorithm, referred to as SSA with conditional predictions (SSA‐CP), is presented to address these issues. It is tested on low‐dimensional, approximately Gaussian data, high‐dimensional non‐Gaussian data, and partially observed data from a multiscale model. In each case, SSA‐CP provides a more accurate real‐time estimate of the leading modes of variability than the traditional reconstruction. SSA‐CP also provides predictions of the leading modes and is easy to implement. SSA‐CP is optimal in the case of Gaussian data, and the uncertainty in real‐time estimates of leading modes is easily quantified. Singular spectrum analysis (SSA) is a powerful, commonly used technique to identify prominent patterns in observed data. However, SSA has some difficulty in providing accurate estimates near the endpoints of the time series, which can hinder its use in real time. A modified version of the SSA algorithm, referred to as SSA with conditional predictions, is presented to address these issues. SSA with conditional predictions provides a more accurate real‐time estimate of the leading modes of variability than the traditional method in a variety of tests. It can also be used to predict these patterns, and it is easy to implement. The uncertainty in the real‐time estimates of leading patterns is easily quantified as well. Singular spectrum analysis (SSA) and extended empirical orthogonal function (EEOF) methods suffer from endpoint issuesSSA with conditional predictions (SSA‐CP) is presented as a simple modification to improve real‐time estimates near endpointsForecasts are also possible, including error estimates, and are optimal for Gaussian data and shown to also be skillful for non‐Gaussian data

The skill of the global climate models (GCMs) to realistically simulate the monsoon intraseasonal oscillations (MISOs) is related to the sensitivity of their convective parameterization schemes. Here we show that by coupling a simple multicloud parameterization to a coarse‐resolution aquaplanet GCM, realistic MISOs can be simulated. We conduct three different simulations with a fixed nonhomogeneous sea surface temperature mimicking the Indian Ocean/western Pacific warm pool (WP) centered at the three latitudes 5°N, 10°N, and 15°N, respectively, to replicate the seasonal migration of the Tropical Convergence Zone (TCZ). This results in the generation of mean circulation resembling the monsoonal flow pattern in boreal summer. Succession of eastward propagating Madden‐Julian Oscillation (MJO) disturbances with phase speed, amplitude, and structure similar to summer MJOs are simulated when the WP is at 5°N. When the WP is located over 10°N, northward and eastward propagating MISOs are simulated. This case captures the meridional seesaw of convection between continental and oceanic TCZ observed during boreal summer over South Asia. Westward propagating Rossby wave‐like disturbances are simulated when the WP is over 15°N congruous with the synoptic disturbances seen over the monsoon trough. The initiation of intraseasonal oscillations in the model can occur internally through organization of convective events above the WP associated with internal dynamics. Monsoon ISO simulated using multicloud modelThree‐cloud structure, key for Monsoon ISOsA more realistic parameterization scheme for monsoon ISOs

The main mechanisms for the initiation and propagation of the Madden‐Julian Oscillation (MJO) are still widely debated. The capacity of operational global climate models (GCMs) to correctly simulate the MJO is hindered by the inadequacy of the underlying cumulus parameterizations. Here we show that a coarse resolution GCM, coupled to a simple multicloud model parameterization mimicking the observed dynamics and physical structure of organized tropical convection, simulates the MJO in an idealized setting of an aquaplanet without ocean dynamics. We impose a fixed nonhomogeneous sea‐surface temperature replicating the Indian Ocean/Western Pacific warm pool. This results in a succession of MJOs with realistic phase speed, amplitude, and physical structure. Each MJO event is initiated at a somewhat random location over the warm pool and dies sometimes near the eastern boundary of the warm pool and sometimes at a random location way beyond the warm pool. Also occasionally the MJO events stall at the center of maximum heating. This is reminiscent of the fact that in nature some MJOs stall over the maritime continent while others reach the central Pacific Ocean and beyond. The initiation mechanism in the model is believed to be a combination of persistent intermittent convective events interacting with observed large‐scale flow patterns and internal tropical dynamics. The large‐scale flow patterns are associated with planetary‐scale dry Kelvin waves that are triggered by preceding MJO events and circle the globe, while congestus cloud decks on the flanks of the warm pool are believed to force Rossby gyres which then funnel moisture toward the equatorial region. Realistic MJO initiation and dynamics in a GCMUsed a simple three layer multicloud convection scheme in the simulationA different initiation mechanism for the MJO as opposed to the existing theory

In dynamical systems with intrinsic chaos, many degrees of freedom, and many conserved quantities, a fundamental issue is the statistical relevance of suitable subsets of these conserved quantities in appropriate regimes. The Galerkin truncation of the Burgers-Hopf equation has been introduced recently as a prototype model with solutions exhibiting intrinsic stochasticity and a wide range of correlation scaling behavior that can be predicted successfully by simple scaling arguments. Here it is established that the truncated Burgers-Hopf model is a Hamiltonian system with Hamiltonian given by the integral of the third power. This additional conserved quantity, beyond the energy, has been ignored in previous statistical mechanics studies of this equation. Thus, the question arises of the statistical significance of the Hamiltonian beyond that of the energy. First, an appropriate statistical theory is developed that includes both the energy and Hamiltonian. Then a convergent Monte Carlo algorithm is developed for computing equilibrium statistical distributions. The probability distribution of the Hamiltonian on a microcanonical energy surface is studied through the Monte-Carlo algorithm and leads to the concept of statistically relevant and irrelevant values for the Hamiltonian. Empirical numerical estimates and simple analysis are combined to demonstrate that the statistically relevant values of the Hamiltonian have vanishingly small measure as the number of degrees of freedom increases with fixed mean energy. The predictions of the theory for relevant and irrelevant values for the Hamiltonian are confirmed through systematic numerical simulations. For statistically relevant values of the Hamiltonian, these simulations show a surprising spectral tilt rather than equipartition of energy. This spectral tilt is predicted and confirmed independently by Monte Carlo simulations based on equilibrium statistical mechanics together with a heuristic formula for the tilt. On the other hand, the theoretically predicted correlation scaling law is satisfied both for statistically relevant and irrelevant values of the Hamiltonian with excellent accuracy. The results established here for the Burgers-Hopf model are a prototype for similar issues with significant practical importance in much more complex geophysical applications. Several interesting mathematical problems suggested by this study are mentioned in the final section. © 2002 Wiley Periodicals, Inc.

There has been a recent burst of activity in the atmosphere-ocean sciences community in utilizing stable linear Langevin stochastic models for the unresolved degrees of freedom in stochastic climate prediction. Here a systematic mathematical strategy for stochastic climate modeling is developed, and some of the new phenomena in the resulting equations for the climate variables alone are explored. The new phenomena include the emergence of both unstable linear Langevin stochastic models for the climate mean variables and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances. All of these phenomena are derived from a systematic self-consistent mathematical framework for eliminating the unresolved stochastic modes that is mathematically rigorous in a suitable asymptotic limit. The theory is illustrated for general quadratically nonlinear equations where the explicit nature of the stochastic climate modeling procedure can be elucidated. The feasibility of the approach is demonstrated for the truncated equations for barotropic flow with topography. Explicit concrete examples with the new phenomena are presented for the stochastically forced three-mode interaction equations. The conjecture of Smith and Waleffe [Phys. Fluids 11 (1999), 1608–1622] for stochastically forced three-wave resonant equations in a suitable regime of damping and forcing is solved as a byproduct of the approach. Examples of idealized climate models arising from the highly inhomogeneous equilibrium statistical mechanics for geophysical flows are also utilized to demonstrate self-consistency of the mathematical approach with the predictions of equilibrium statistical mechanics. In particular, for these examples, the reduced stochastic modeling procedure for the climate variables alone is designed to reproduce both the climate mean and the energy spectrum of the climate variables. © 2001 John Wiley & Sons, Inc.

Abstract.: We develop a point-vortex equilibrium statistical model for baroclinic quasigeostrophic vortices within the context of a two-layer quasigeostrophic fluid that evolves in all of space. Angular momentum, which follows from the rotational symmetry of the unbounded domain, is the key conserved quantity, introducing a length scale that confines the most probable states of the statistical theory. We apply the theory as a model of localized convection in a preconditioned gyre. To illustrate this application, the preconditioned cyclonic, largely barotropic gyres are modeled as „zero inverse temperature” states, which are explicit solutions to the mean-field equations with a Gaussian probability distribution of vortices. Convection is modeled by a cloud of point-vortex hetons – purely baroclinic arrangements of point vortices, cyclonic above and anticyclonic below – which capture the short-term, geostrophically balanced response to strong surface cooling. Numerical heton studies (Legg and Marshall, 1993, 1998) have shown that a preexisting barotropic rim current can suppress baroclinic instability and confine anomalies of potential vorticity and temperature introduced by the cold-air outbreak. Here, we demonstrate that the lateral extent of the most probable states of the statistical theory are constrained by the angular momentum. Without resolution of the detailed dynamics, the equilibrium statistical theory predicts that baroclinic instability is suppressed for preconditioned flows with potential vorticity of the same sign in each layer provided that the strength of convective overturning does not change the sign of potential vorticity in one of the layers. This result agrees with detailed simulations (Legg and Marshall, 1998) and supports the potential use of these statistical theories as parametrizations for crude closure.

A “most probable state” equilibrium statistical theory for random distributions of hetons in a closed basin is developed here in the context of two-layer quasigeostrophic models for the spreading phase of open-ocean convection. The theory depends only on bulk conserved quantities such as energy, circulation, and the range of values of potential vorticity in each layer. For a small Rossby deformation radius typical for open-ocean convection sites, the most probable states that arise from this theory strongly resemble the saturated baroclinic states of the spreading phase of convection, with a rim current and localized temperature anomaly. Furthermore, rigorous explicit nonlinear stability analysis guarantees the stability of these steady states for a suitable range of parameters. Both random heton distributions in a basin with quiescent flow as well as heton addition to an ambient barotropic flow in the basin are studied here. Also, systematic results are presented on the influence of the Rossby deformation radius compared to the basin scale on the structure of the predictions of the statistical theory.

A new Monte Carlo algorithm for constructing and sampling stationary isotropic Gaussian random fields with power-law energy spectrum, infrared divergence, and fractal self-similar scaling is developed here. The theoretical basis for this algorithm involves the fact that such a random field is well approximated by a superposition of random one-dimensional plane waves involving a fixed finite number of directions. In general each one-dimensional plane wave is the sum of a random shear layer and a random acoustical wave. These one-dimensional random plane waves are then simulated by a wavelet Monte Carlo method for a single space variable developed recently by the authors. The computational results reported in this paper demonstrate remarkable low variance and economical representation of such Gaussian random fields through this new algorithm. In particular, the velocity structure function for an imcorepressible isotropic Gaussian random field in two space dimensions with the Kolmogoroff spectrum can be simulated accurately over 12 decades with only 100 realizations of the algorithm with the scaling exponent accurate to 1.1% and the constant prefactor accurate to 6%; in fact, the exponent of the velocity structure function can be computed over 12 decades within 3.3% with only 10 realizations. Furthermore, only 46,592 active computational elements are utilized in each realization to achieve these results for 12 decades of scaling behavior. Copyright 1995, 1999 Academic Press

A new algorithm is developed here for Monte Carlo simulation of turbulent diffusion with random velocity fields having a power law spatial spectrum, arbitrarily many statistical scales, long range correlations, and infrared divergence. This algorithm uses an expansion of the moving average representation of a Gaussian random field via a specific orthonormal basis which exploits the scaling behavior of the random field and yields a compact representation of the field despite infrared divergence. The orthonormal basis involves a family of wavelets due to Alpert and Rokhlin and the vanishing of a large number of moments for this basis (order 3 or 4) is crucial for compact representation of the statistical velocity field. The authors also develop a rigorous practical a priori energy criterion for truncation of the wavelet expansion; furthermore, for moment cancellations with orders 3 or 4 and a fixed energy tolerance, the number of Gaussian random basis elements needed by the method is sublinear in the exponent m, where 1 < |x| < 2mdenotes the range of active velocity scales. The algorithm developed here has remarkably low variance as regards sampling errors. For example, for the infrared divergent velocity spectrum corresponding to the Kolmogoroff spectrum in an exactly solvable model, the velocity structure function is simulated accurately for separation distances over more than five decades with only 792 random basis elements and 100 ensemble realizations; with this same singular spectrum in the model problem, the pair dispersion statistic for the passive scalar can be determined within errors of 3% throughout over five decades of pair separation distance with only 792 random elements, 1000 realizations, and a few hours calculation on a workstation. All of the computational algorithms and results are presented for a nontrivial exactly solvable model and are described within a mathematical framework of error analysis where analytic, sampling, and discretization errors are treated separately.

Avellaneda and one of the authors ([1], [3]) have recently established that an upper bound for the enhanced diffusivity in the large scale, long time advection‐diffusion with periodic steady incompressible velocity fields has the form where Peis the Peclet number and is the reciprocal of the Prandtl number. In this paper, flow fields with maximal and minimal enhanced diffusion are studied. Maximal enhanced diffusion requires that in some directions the enhanced diffusion tensor also has the lower bound . For minimal enhanced diffusion, the effect of the velocity field is to boost the enhanced diffusivity by a negligible amount that is bounded by a fixed constant times the bare diffusivity regardless of Peclet number. Stieltjes measure formulas are used to develop a simple, necessary, and sufficient condition for maximal enhanced diffusion and also to characterize minimal enhanced diffusion. It is established here that constant mean flows can have a dramatic effect on maximal and minimal enhanced diffusion. In particular, for flows in two space dimensions, an explicit criterion is developed that guarantees the surprising fact that mean flows with rational ratios typically generate maximal enhanced diffusion through interaction with an arbitrary steady periodic incompressible flow with zero mean. In contrast, a simple criterion for flows without stagnation points is developed here that guarantees that the effect of mean flows with irrational ratios on advection‐diffusion in two dimensions creates minimal enhanced diffusion. The theory for the phenomena mentioned above is elementary yet mathematically rigorous. Examples are emphasized throughout this work including a discussion of enhanced diffusivity for a class of flows recently introduced by Childress and Soward [8]. The theory developed here is also supplemented by a series of numerical experiments that both verify the theoretical predictions and display interesting crossover phenomena at rather large but finite Peclet numbers.

Various facets of the equilibrium statistical theories for large-scale coherent structures for two-dimensional flows with and without topography are studied here. The classical few-constraint statistical theories involving energy-enstrophy principles or point vortices are shown to be statistically sharp in the more recent statistical theories with an infinite number of constraints; in other words, at the macrostates of the few-constraint theories, the many-constraint theory provides no additional statistical information. These results are established through a general link between these statistical theories, generalized “selective decay” principles, and statistical sharpness. Through an asymptotic procedure, the many-constraint statistical theories for flows with topography and small-potential vorticity are shown to yield the simpler energy-enstrophy macroscopic states at leading order with systematic higher-order corrections involving a renormalized topography that includes higher moments of the microscopic potential vorticity distribution. For nonequilibrium flows with and without topography, the utility of crude approximate dynamics based on “adiabatic approximation” through the macrostates of few-constraint statistical theory is developed here. It is established that for nonequilibrium decaying flows with viscous dissipation, the crude dynamics based on macrostates involving statistical point vortices yields an excellent approximation; the role of “selective decay” principles is also clarified and compared quantitatively in this context through both mathematical theory and numerical experiments. Surprisingly, these approximate dynamics yield a much poorer approximation with moderate Ekman drag as the dissipative mechanism, and a simple analytical explanation is provided here. Finally, all of these issues are pursued more briefly for damped and driven flows with topography. © 1997 John Wiley & Sons, Inc.

Here the propagation of flame fronts for turbulent reaction diffusion equations is established for model velocity fields that are random and fractal unidirectional shear flows. It is established here that the flame fronts that exhibit anomalous scaling and are likely to be fractal themselves can be described through solutions of a suitable variational inequality. Despite the special nature of the random shear flows, these are the first rigorous results on front propagation with fractal fields. © 1998 John Wiley & Sons, Inc.

In classical work, Mathéron and the Marsilly showed that superdiffusive scaling of mean-square displacements occurs in transport diffusion for stratified flows with steady simple shear layers and long-range spatial correlations. More recently the authors have calculated a formula for the non-Gaussian large-scale long-time renormalized Green function for these problems. Here the scaling laws and renormalized Green functions for diffusion in “nearly stratified” flows are studied; in such flows the simple shear layer with long-range correlations is perturbed by incompressible flows with short-range correlations. Here it is established that these flows belong to the same universality class as the simple shear layers, with a renormalized Green function with a similar structure but reflecting homogenization by the transverse displacements. The tools in the analysis involve a modification of homogenization theory and also rigorous diagrammatic perturbation theory.

The inertial range for a statistical turbulent velocity field consists of those scales that are larger than the dissipation scale but smaller than the integral scale. Here the complete scale-invariant explicit inertial range renormalization theory for all the higher-order statistics of a diffusing passive scalar is developed in a model which, despite its simplicity, involves turbulent diffusion by statistical velocity fields with arbitrarily many scales, infrared divergence, long-range spatial correlations, and rapid fluctuations in time-such velocity fields retain several characteristic features of those in fully developed turbulence. The main tool in the development of this explicit renormalization theory for the model is an exact quantum mechanical analogy which relates higher-order statistics of the diffusing scalar to the properties of solutions of a family ofN- body parabolic quantum problems. The canonical inertial range renormalized statistical fixed point is developed explicitly here as a function of the velocity spectral parameter?, which measures the strength of the infrared divergence: for?<2, mean-field behavior in the inertial range occurs with Gaussian statistical behavior for the scalar and standard diffusive scaling laws; for?>2 a phase transition occurs to a fixed point with anomalous inertial range scaling laws and a non-Gaussian renormalized statistical fixed point. Several explicit connections between the renormalization theory in the model and intermediate asymptotics are developed explicitly as well as links between anomalous turbulent decay and explicit spectral properties of Schrödinger operators. The differences between this inertial range renormalization theory and the earlier theories for large-scale eddy diffusivity developed by Avellaneda and the author in such models are also discussed here.

A new hierarchical method for the Monte Carlo simulation of random fields called the Fourier–wavelet method is developed and applied to isotropic Gaussian random fields with power law spectral density functions. This technique is based upon the orthogonal decomposition of the Fourier stochastic integral representation of the field using wavelets. The Meyer wavelet is used here because its rapid decay properties allow for a very compact representation of the field. The Fourier–wavelet method is shown to be straightforward to implement, given the nature of the necessary precomputations and the run-time calculations, and yields comparable results with scaling behavior over as many decades as the physical space multiwavelet methods developed recently by two of the authors. However, the Fourier–wavelet method developed here is more flexible and, in particular, applies to anisotropic spectra generated through solutions of differential equations. Simulation results using this new technique and the well-known nonhierarchical simulation technique, the randomization method, are given and compared for both a simple shear layer model problem as well as a two-dimensional isotropic Gaussian random field. The Fourier–wavelet method results are more accurate for statistical quantities depending on moments higher than order 2, in addition to showing a quite smooth decay to zero on the scales smaller than the scaling regime when compared with the randomization method results. The only situation in which the nonhierarchical randomization method is more computationally efficient occurs when no more than four decades of scaling behavior are needed and the statistical quantities of interest depend only on second moments.

Precise necessary and sufficient conditions on the velocity statistics for mean field behavior in advection-diffusion by a steady incompressible velocity field are developed here. Under these conditions, a rigorous Stieltjes integral representation for effective diffusivity in turbulent transport is derived. This representation is valid for all Péclet numbers and provides a rigorous resummation of the divergent perturbation expansion in powers of the Péclet number. One consequence of this representation is that convergent upper and lower bounds on effective diffusivity for all Peclet numbers can be obtained utilizing a prescribed finite number of terms in the perturbation series. Explicit rigorous examples of steady incompressible velocity fields are constructed which have effective diffusivities realizing the simplest upper or lower bounds for all Péclet numbers. A nonlocal variational principle for effective diffusivity is developed along with applications to advection-diffusion by random arrays of vortices. A new class of rigorous examples is introduced. These examples have an explicit Stieltjes measure for the effective diffusivity; furthermore, the effective diffusivity behaves likek0(Pe)1/2 in the limit of large Péclet numbers wherek0 is the molecular diffusivity. Formal analogies with the theory of composite materials are exploited systematically.

In contrast to the Miles–Howard theorem for inviscid steady shear flow in stably stratified fluids, explicit elementary time-periodic solutions of the Boussinesq equations are developed here which are unstable for arbitrarily large Richardson numbers. These elementary flows are parameterized through solutions of a nonlinear pendulum equation and involve spatially constant but temporally varying vorticity and density gradients which interact through advection and baroclinic vorticity production. Exact nonlinear solutions for arbitrary wave-like disturbances for these flows are developed here and Floquet theory combined with elementary numerical calculations is utilized to demonstrate instability at all large Richardson numbers. The dominant inviscid instability for these non-parallel flows is a purely two-dimensional parametric instability with twice the period of the elementary flow and persists for all Reynolds numbers and the wide range of Prandtl numbers, 1Pr200, investigated here. Similar elementary time-periodic solutions of the Boussinesq equations in a constant external strain field are developed here which reduce to uniform shear flows in one extreme limit and the time-periodic vortical flows in the other extreme limit. These flows are stable in a strict sense for large Richardson numbers; however there is transient large-amplitude non-normal behaviour which yields effective instability for a wide range of Richardson numbers. For example, suitable initial perturbations can amplify by at least a factor of fifty with exponential growth for short times for ℛi=1, ∣σ∣0.5 and ℛi=5, ∣σ∣0.1, with σ the amplitude of the external strain.

The authors introduce a new concept of measure-valued solution for the 3-D incompressible Euler equations in order to incorporate the complex phenomena present in limits of approximate solutions of these equations. One application of the concepts developed here is the following important result: a sequence of Leray-Hopf weak solutions of the Navier-Stokes equations converges in the high Reynolds number limit to a measure-valued solution of 3-D Euler defined for all positive times. The authors present several explicit examples of solution sequences for 3-D incompressible Euler with uniformly bounded local kinetic energy which exhibit complex phenomena involving both persistence of oscillations and development of concentrations. An extensions of the concept of Young measure is developed to incorporate these complex phenomena in the measure-valued solutions constructed here.

Abstract:: Various facets of recent mathematical theories for averaging over fast gravity waves on advective time scales for geophysical flows with unbalanced initial data are presented here including nonlinear Rossby adjustment and simplified reduced dynamics. This work is presented within the context of simplified geophysical models involving the rotating shallow-water equations and the rotating stably stratified Boussinesq equations. Novel mechanisms for enhanced gravity wave dissipation through the catalytic interaction with potential vortical modes are also developed here within the context of the rotating shallow-water equations.

Nonlinear geometric optics is developed to construct translating self-similar high-frequency nonlinear wave patterns as perturbed solutions behind reacting shock waves. The geometric conditions involving these wave patterns that yield the maximum acoustic amplification lead to criteria for both the formation and regular spacing dimension for Mach stems in reacting gases.