Stochastic and Statistical Methods in Climate, Atmosphere, and Ocean Science

Introduction The behavior of the atmosphere, oceans, and climate is intrinsically uncertain. The basic physical principles that govern atmospheric and oceanic flows are well known, for example, the Navier-Stokes equations for fluid flow, thermodynamic properties of moist air, and the effects of density stratification and Coriolis force. Notwithstanding, there are major sources of randomness and uncertainty that prevent perfect prediction and complete understanding of these flows. The climate system involves a wide spectrum of space and time scales due to processes occurring on the order of microns and milliseconds such as the formation of cloud and rain droplets to global phenomena involving annual and decadal oscillations such as the EL Nio-Southern Oscillation (ENSO) and the Pacific Decadal Oscillation (PDO) [5]. Moreover, climate records display a spectral variability ranging from 1 cycle per month to 1 cycle per 100, 000 years [23]. The complexity of the climate system stems in large part from the inherent nonlinearities of fluid mechanics and the phase changes of water substances. The atmosphere and oceans are turbulent, nonlinear systems that display chaotic behavior (e.g., [39]). The time evolutions of the same chaotic system starting from two slightly different initial states diverge exponentially fast, so that chaotic systems are marked by limited predictability. Beyond the so-called predictability horizon (on the order of 10 days for the atmosphere), initial state uncertainties (e.g., due to imperfect observations) have grown to the point that straightforward forecasts are no longer useful. Another major source of uncertainty stems from the fact that numerical models for atmospheric and oceanic flows cannot describe all relevant physical processes at once. These models are in essence discretized partial differential equations (PDEs), and the derivation of suitable PDEs (e.g., the so-called primitive equations) from more general ones that are less convenient for computation (e.g., the full Navier-Stokes equations) involves approximations and simplifications that introduce errors in the equations. Furthermore, as a result of spatial discretization of the PDEs, numerical models have finite resolution so that small-scale processes with length scales below the model grid scale are not resolved. These limitations are unavoidable, leading to model error and uncertainty. The uncertainties due to chaotic behavior and unresolved processes motivate the use of stochastic and statistical methods for modeling and understanding climate, atmosphere, and oceans. Models can be augmented with random elements in order to represent time-evolving uncertainties, leading to stochastic models. Weather forecasts and climate predictions are increasingly expressed in probabilistic terms, making explicit the margins of uncertainty inherent to any prediction

Recent Advances Concerning Certain Class of Geophysical Flows

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.Comment: arXiv admin note: text overlap with arXiv:1507.0523

Analysis of a General Family of Regularized Navier-Stokes and MHD Models

We consider a general family of regularized Navier-Stokes and Magnetohydrodynamics (MHD) models on n-dimensional smooth compact Riemannian manifolds with or without boundary, with n greater than or equal to 2. This family captures most of the specific regularized models that have been proposed and analyzed in the literature, including the Navier-Stokes equations, the Navier-Stokes-alpha model, the Leray-alpha model, the Modified Leray-alpha model, the Simplified Bardina model, the Navier-Stokes-Voight model, the Navier-Stokes-alpha-like models, and certain MHD models, in addition to representing a larger 3-parameter family of models not previously analyzed. We give a unified analysis of the entire three-parameter family using only abstract mapping properties of the principle dissipation and smoothing operators, and then use specific parameterizations to obtain the sharpest results. We first establish existence and regularity results, and under appropriate assumptions show uniqueness and stability. We then establish results for singular perturbations, including the inviscid and alpha limits. Next we show existence of a global attractor for the general model, and give estimates for its dimension. We finish by establishing some results on determining operators for subfamilies of dissipative and non-dissipative models. In addition to establishing a number of results for all models in this general family, the framework recovers most of the previous results on existence, regularity, uniqueness, stability, attractor existence and dimension, and determining operators for well-known members of this family.Comment: 37 pages; references added, minor typos corrected, minor changes to revise for publicatio

Coarse-grained stochastic models for tropical convection and climate

Prototype coarse-grained stochastic parametrizations for the interaction with unresolved features of tropical convection are developed here. These coarse-grained stochastic parametrizations involve systematically derived birth/death processes with low computational overhead that allow for direct interaction of the coarse-grained dynamical variables with the smaller-scale unresolved fluctuations. It is established here for an idealized prototype climate scenario that, in suitable regimes, these coarse-grained stochastic parametrizations can significantly impact the climatology as well as strongly increase the wave fluctuations about an idealized climatology. The current practical models for prediction of both weather and climate involve general circulation models (GCMs) where the physical equations for these extremely complex flows are discretized in space and time and the effects of unresolved processes are parametrized according to various recipes. With the current generation of supercomputers, the smallest possible mesh spacings are ≈50–100 km for short-term weather simulations and of order 200–300 km for short-term climate simulations. There are many important physical processes that are unresolved in such simulations such as the mesoscale sea-ice cover, the cloud cover in subtropical boundary layers, and deep convective clouds in the tropics. An appealing way to represent these unresolved features is through a suitable coarse-grained stochastic model that simultaneously retains crucial physical features of the interaction between the unresolved and resolved scales in a GCM. In recent work in two different contexts, the authors have developed both a systematic stochastic strategy (1) to parametrize key features of deep convection in the tropics involving suitable stochastic spin-flip models and also a systematic mathematical strategy to coarse-grain such microscopic stochastic models (2) to practical mesoscopic meshes in a computationally efficient manner while retaining crucial physical properties of the interaction. This last work (2) is general with potential applications in material sciences, sea-ice modeling, etc. Crucial new scientific issues involve the fashion in which a stochastic model effects the climate mean state and the strength and nature of fluctuations about the climate mean. The main topic of this article is to discuss development of a family of coarse-grained stochastic models for tropical deep convection by combining the systematic strategies from refs. 1 and 2 and to explore their effect on both the climate mean and fluctuations for an idealized prototype model parametrization in the simplest scenario for tropical climate involving the Walker circulation, the east–west climatological state that arises from local region of enhanced surface heat flux, mimicking the Indonesian marine continent

Coarse-grained stochastic models for tropical convection and climate

Prototype coarse-grained stochastic parametrizations for the interaction with unresolved features of tropical convection are developed here. These coarse-grained stochastic parametrizations involve systematically derived birth/death processes with low computational overhead that allow for direct interaction of the coarse-grained dynamical variables with the smaller-scale unresolved fluctuations. It is established here for an idealized prototype climate scenario that, in suitable regimes, these coarse-grained stochastic parametrizations can significantly impact the climatology as well as strongly increase the wave fluctuations about an idealized climatology. The current practical models for prediction of both weather and climate involve general circulation models (GCMs) where the physical equations for these extremely complex flows are discretized in space and time and the effects of unresolved processes are parametrized according to various recipes. With the current generation of supercomputers, the smallest possible mesh spacings are ≈50–100 km for short-term weather simulations and of order 200–300 km for short-term climate simulations. There are many important physical processes that are unresolved in such simulations such as the mesoscale sea-ice cover, the cloud cover in subtropical boundary layers, and deep convective clouds in the tropics. An appealing way to represent these unresolved features is through a suitable coarse-grained stochastic model that simultaneously retains crucial physical features of the interaction between the unresolved and resolved scales in a GCM. In recent work in two different contexts, the authors have developed both a systematic stochastic strategy (1) to parametrize key features of deep convection in the tropics involving suitable stochastic spin-flip models and also a systematic mathematical strategy to coarse-grain such microscopic stochastic models (2) to practical mesoscopic meshes in a computationally efficient manner while retaining crucial physical properties of the interaction. This last work (2) is general with potential applications in material sciences, sea-ice modeling, etc. Crucial new scientific issues involve the fashion in which a stochastic model effects the climate mean state and the strength and nature of fluctuations about the climate mean. The main topic of this article is to discuss development of a family of coarse-grained stochastic models for tropical deep convection by combining the systematic strategies from refs. 1 and 2 and to explore their effect on both the climate mean and fluctuations for an idealized prototype model parametrization in the simplest scenario for tropical climate involving the Walker circulation, the east–west climatological state that arises from local region of enhanced surface heat flux, mimicking the Indonesian marine continent

Stochastic and mesoscopic models for tropical convection

A new way to parametrize certain aspects of tropical convection through stochastic and mesoscopic models is developed here. The technical idea is to adapt tools from statistical physics and materials science to model important unresolved features of tropical convection. This new strategy consists of modeling the unresolved effects of convective inhibition in a coarse mesh mesoscopic parametrization through a “heat bath” model involving a stochastic spin flip model with very natural interaction rules for convective inhibition combined with a suitable external potential defined by the coarse mesh values. In turn, the values of the order parameter from this heat bath alter the vertical mass flux in regions of deep convection. Both stochastic and systematic deterministic mesoscopic parametrizations are developed here. The deterministic mesoscopic models derived in this fashion exhibit new phenomena such as multiple radiative equilibria in suitable parameter regimes. The simplest first numerical experiments reported here with the mesoscopic deterministic parametrization qualitatively reproduce several key features of the observational record regarding convectively coupled tropical waves. The systematic stochastic modeling strategy proposed here could also be very useful for capturing other features of tropical convection such as those involving cloud radiation feedbacks

Four Theories of the Madden-Julian Oscillation.

Studies of the Madden-Julian Oscillation (MJO) have progressed considerably during the past decades in observations, numerical modeling, and theoretical understanding. Many theoretical attempts have been made to identify the most essential processes responsible for the existence of the MJO. Criteria are proposed to separate a hypothesis from a theory (based on the first principles with quantitative and testable assumptions, able to predict quantitatively the fundamental scales and eastward propagation of the MJO). Four MJO theories are selected to be summarized and compared in this article: the skeleton theory, moisture-mode theory, gravity-wave theory, and trio-interaction theory of the MJO. These four MJO theories are distinct from each other in their key assumptions, parameterized processes, and, particularly, selection mechanisms for the zonal spatial scale, time scale, and eastward propagation of the MJO. The comparison of the four theories and more recent development in MJO dynamical approaches lead to a realization that theoretical thinking of the MJO is diverse and understanding of MJO dynamics needs to be further advanced