Abstracts of the invited talks

Tamás Bódai (IBS Center for Climate Physics, Busan)

Multistability in climate and weather: The case of fractal basin boundaries

Persistence or regime behaviour is a common effect, which might emerge from the multistability of an underlying low-dimensional dynamics with a perturbation superimposed. Transition paths might be actually quite restricted, in association with the stationary measure of the dynamics. I will consider the case when the underlying low-dimensional dynamics features a fractal basin boundary, which embeds a chaotic saddle. The fractality of the basin boundary might be filamentary or "rough", or a mixture of these. The perturbed stationary measure can be predicted by large deviation theory, in terms of a so-called nonequilibrium potential based on the unperturbed dynamics and a correction to it, which selects specific instantons as most probable transition paths. The potential difference between the potential well and a saddle will also determine the expected sojourn or exit time. I will present an efficient algorithm to determine this potential difference from exit time data when the perturbation strength is under the control of the experimenter.

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Roberto Buizza (Scuola Superiore Sant’Anna, Pisa, Italy)

Numerical models of the Earth-system are essential tools to understand how the system evolved and will evolve in the future. This is true whether we are aiming to understand and predict the time evolution of weather patterns, or to estimate how the past climate changed and which way the future climate could evolve.

Advances in the models’ formulation in the past decades, among which the adoption of more sophisticated data-assimilation methods and the paradigm shift from single to ensemble forecasts, have made it possible to extend the forecast skill horizon for weather prediction. Together with model improvements, the introduction of stochastic terms in the numerical model equations has led to more accurate and reliable simulations. In climate studies, the application of data-assimilation methods to past data has allowed us to reconstruct the evolution of the past climate, and ensembles of projects allow us to estimate how the future Earth climate will look like, under a range of emission scenarii.

Weather and climate needs have led, in the past decade, to an increasing convergence between the models used for weather prediction and for climate studies, and this has generated benefits for both communities. Indeed, the fact that the same ‘model’ is used over the different time scales, have forced developers to apply more thorough quality controls, with the end result being more accurate and reliable simulations. Improvements developed by one community have brought benefits to both.

During my seminar, I will briefly illustrate what is a numerical model of the Earth-system and how they are initialized to generate a weather forecast, and I will illustrate open challenges on which people are working on. I will also discuss how the same model can be used to reconstruct the past climate, and generate future climate scenario.

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Paola Cessi

The global overturning circulation of the ocean

Revolutionary observational arrays, together with a new generation of ocean and climate models, have provided new and intriguing insight into the Atlantic Meridional Overturning Circulation (AMOC) over the last two decades. Theoretical models have also contributed to changing our view of the AMOC, providing a dynamical framework for understanding the new observations and the results of complex models. Recent theoretical advances on the processes maintaining the AMOC, the mid-depth and abyssal stratification are discussed, together with the conceptual understanding that has resulted. Recent theoretical models are discussed that address issues such as the interplay between surface buoyancy and wind forcing, the extent to which the AMOC is adiabatic, the interaction between the mid-depth North Atlantic Deep Water cell and the abyssal Antarctic Bottom Water cell, the role of basin geometry and bathymetry, and the importance of a three-dimensional multiple-basin perspective. These theories show that dynamics in the Antarctic circumpolar region are essential in determining the deep and abyssal stratification. In addition, they show that a mid-depth cell consistent with observational estimates is powered by the wind stress in the Antarctic circumpolar region, while the abyssal cell relies on interior diapycnal mixing, which is bottom intensified. Simple theoretical models remain a vital and powerful tool for articulating our understanding of the AMOC, the global overturning circulation, the deep stratification and identifying the processes that are most critical to represent in the next generation of numerical ocean and climate models.

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Michel Crucifix (Earth and Life Institute, UCLouvain)

Mathematical games around Quaternary ice ages

After the glaciation of Antarctica (which became definitive around 15 Ma (million years) ago), the glaciation of the Northern Hemisphere started around 3 Myr ago. It defines the entry into the Pleistocene. The phenomenon started with modest glaciations, interrupted by quasi-periodic increases in incoming summer solar radiation caused by the movements of obliquity and the phenomenon of precession. However, unlike Antarctica, the Northern glaciation never became definitive. Glacial maxima are invariably --- and somewhat predictably --- followed by large deglaciations, suggesting that the presence of continental-wide ice sheets in the Northern Hemisphere is unstable. Merely postulating and encoding this instability in a low-order dynamical system model suffices to reproduce the sequence of glacial-interglacial cycles observed throughout the Pleistocene, once the astronomical forcing is accounted for. In this sense, there is not much mystery about the period and amplitude of Pleistocene glaciations.

More challenging is developing a theory for the full range of climatic fluctuations over the frequency range 1 ka -- 1Ma, accounting for millennial variability and the spectral fluctuation background which seemingly connects all time scales. 
Our somewhat inaccurate knowledge of the physical and biological constraints on ice age dynamics gives us some freedom for toying with mathematical concepts: dynamical systems, stochastic dynamical systems, self-similar stochastic processes,  spectral analysis and Bayesian inference. In this brief presentation, we will present two case studies involving non-intuitive concepts, and open the discussion as whether we should allow mathematical fun to take over physical insight.

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Anne-Laure Dalibard 

Theoretical analysis of boundary layer problems in oceanographic models

The purpose of this talk is to introduce some mathematical techniques to study boundary layer systems and to apply these techniques to classical oceanographic models (Ekman and Munk boundary layers, internal layers in stratified fluids…) Note that we will tackle both the construction of the boundary layer and the proof of its validity.

I will also insist on the limitations of these techniques, and on some future challenges and open problems.

Gábor Drótos

What the snapshot/pullback attractor of an Earth system model can tell about climate change: a case study about the ENSO--Indian monsoon teleconnection 

The climate of a time instant under explicitly time-dependent forcing seems to be most plausibly defined by the natural probability measure of the corresponding snapshot attractor (the instantaneous section of the full pullback attractor), and climate change is described by the time evolution of this probability measure. This probability measure is numerically represented by an ensemble of trajectories (realizations) emanating from different initial conditions, but only after memory loss takes place. Recently, communities in Earth system modeling have started to develop `initial-condition large ensembles' in state-of-the-art models. I will present a statistical analysis about the change of a correlation coefficient relating the El Ni\~no--Southern Oscillation and the Indian monsoon in two of these `large ensembles', and illustrate that the proper analysis may lead to conclusions qualitatively different from those relying on the time evolution of a single realization, questioning conclusions drawn about observations in the literature. After pointing out practical implications of the attractor framework (as compared to arbitrary analyses utilizing `initial-condition large ensembles'), I will discuss open questions about long time scales and temporal characteristics.

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Thomas Dubos (Ecole Polytechnique)

Are geophysical hydrodynamics Newtonian?

Geophysical hydrodynamics rely on a variety of models which approximate classical, Newtonian fluid motion in rotating frames. Even accurate approximations, especially pertaining to the Coriolis pseudo-force, can have profound consequences, such as the loss of inertial frames. This questions the relationship between geophysical hydrodynamics and Newtonian hydrodynamics. I will address this matter by also investigating the relationship of geophysical hydrodynamics to relativistic hydrodynamics, which does not postulate the existence of inertial frames. 

Bérengère Dubrulle (Service de Physique de l’Etat Condensé, CNRS, CEA Saclay, Université Paris-Saclay)

Dissipation and singularities

Turbulent flows are characterized by a self-similar energy spectrum, signature of fluid movements at all scales. This organization has been described for more than 70 years by the phenomenology of "Kolmogorov cascade": the energy injected on a large scale by the work of the force that moves the fluid (e. g. a turbine) is transferred to smaller and smaller scales with a constant dissipation rate, up to the Kolmogorov scale, where it is transformed into heat and dissipated by viscosity.

I will explain why this image, which Landau questioned in the 1950s, is false. I will use recent velocity measurements obtained by very high resolution laser velocimetry to show that the energy "cascade" is in fact driven by extreme events on a very small scale, which are the signature of quasi-singularities of the Navier-Stokes equations existing under the Kolmogorov scale.

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Gary Froyland

Spectral methods for geophysical fluid dynamics

I will survey recent transfer operator theory and methods for analysing geophysical fluid flows.  These include a selection from: tracking coherent features from the meso-scale to the global scale using models and observations, computing optimal stochastic and deterministic perturbations to enhance fluid mixing, rigorous computation of statistical limit laws, and theory for quenched random limit laws.

Isabelle Gallagher

Some mathematical methods in the analysis of geophysical flows

The aim of this talk is to present some techniques that can be used to analyze mathematically geophysical fluid equations, in terms of well-posedness, and of the asymptotic behaviour of the solutions. In order to be the most accessible possible, this overview will be made in simplified settings (in particular in terms of the number of parameters involved and of boundary conditions) and no attempt will be made to present the most up-to-date results in the field, which involve many more technicalities.

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Michael Ghil (ENS, Paris, and UCLA)

The Irruption of Nonautonomous and Random Dynamical Systems Into the Climate Sciences  

H. Poincaré already raised doubts about the predictability of weather due to the divergence of orbits of dynamical systems associated more recently with chaos. Progress in the theory of nonlinear, deterministic dynamical systems (DDS theory), on the one hand, and the highly ingenious and somewhat independent work of E.N. Lorenz, on the other, justified Poincaré’s early doubts. The theory of autonomous DDSs, with time-independent forcing and coefficients, provided a solid mathematical basis for much of the work on weather predictability over several decades.

More recently, an interesting and highly stimulating convergence occurred between studies of climate predictability, on the one hand, and the development of the theory of nonautonomous and random dynamical systems (NDS and RDS), on the other. While the diurnal and the seasonal cycle of insolation played a more limited role in weather predictability for 10–15 days, it was impossible to ignore the role of the seasonal cycle and of anthropogenic effects in climate predictability for years to decades.

At the same time, the theory of purely deterministic, skew product flows, as well as that of RDSs, incorporated time-dependent forcing and coefficients and took huge mathematical strides, including the rigorous formulation and application of pullback attractors. A parallel development in the physical literature formulated and applied in a more intuitive fashion the closely related concept of snapshot attractors.

These mathematical and physical advances were seized upon by several groups of researchers interested in climate modeling and predictability. In this talk, I will try to present some of the mathematical background, as well as some of the applications to the climate sciences. These will include, as time permits, (i) the use of pullback and snapshot attractors for the proper understanding of the effects of time-dependent forcing, both deterministic and stochastic, natural as well as anthropogenic, upon intrinsic climate variability; (ii) the use of Wasserstein distance between time-dependent invariant measures to estimate these effects; (iii) the topological aspects of nonautonomous effects upon the intrinsic variability; and (iv) a “grand unification” between the nonlinear, deterministic and autonomous point of view espoused by E.N. Lorenz and the linear, stochastically driven one of K. Hasselmann.

This talk reflects joint work with G. Charó, M.D. Chekroun, A. DiGarbo, Y. Feliks, S. Galatolo, F.-F. Jin, D. Kondrashov, V. Lucarini, L. Marangio, J.D. Neelin, S. Pierini, D. Sciamarella, E. Simonnet, Y. Sato, J. Sedro, L. Sushama, and I. Zaliapin.

Anna S. von der Heydt 

Uncertainty quantification of climate sensitivity: State-dependence, extreme values and the probability of tipping

The Equilibrium Climate Sensitivity (ECS) remains not very well constrained, either by climate models, observational, historical or palaeoclimate data. Next to the classical (measurement) uncertainty, the spread in ECS values is due to dynamical aspects: (i) The climate system has strong internal variability on many timescales such that the equilibrium will only be relative to fixing slow processes. This implies the assumption that time scale separation exists and ECS values from palaeoclimate time series can be compared to short model simulations. Palaeoclimate records often determine the Earth System Sensitivity, which includes the integrated effect of slow processes and boundary conditions (e.g. geography, vegetation and land ice). (ii) The background state dependence of fast feedback processes: Information from the late Pleistocene ice age cycles indicates that ECS varies considerably between regimes because of fast feedback processes changing their relative strength over one cycle. (iii) Tipping elements in the climate system: Extreme values of palaeo-derived ECS suggest that the climate response is in a region where the assumption of linear response to perturbations breaks down. Here we show for climate system models with more than one regime and occasional switches between these regimes, we can empirically determine the probability of change in regime and confirm that extremes of climate sensitivity are associated with very high probabilities of tipping.

Joint work with Peter Ashwin

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Darryl D. Holm (Mathematics, Imperial College, London)

Variational principles for Stochastic Fluid Dynamics

The lecture will be based on Hamilton’s principle for ideal fluids and its infinite-dimensional symmetries for temporally stochastic evolution of fluids under spatially smooth invertible maps. This means that the Lagrangian particle trajectories exist as stochastic paths generated by Eulerian stochastic vector fields. Thus, Hamilton’s principle retains its original gauge symmetry under the relabelling of Lagrangian particles. Because this gauge symmetry persists, Noether’s theorem delivers Kelvin’s theorem for circulation dynamics of the stochastic flow in the same form as for the deterministic case, but now Kelvin’s circulation loop moves with the fluid along a stochastic path. The corresponding nonlinear stochastic PDEs for the fluid motion provide a probabilistic estimation of model error, based on the observed spatial correlations of uncertainties in the fluid transport. Thus, this variational approach yields a data-driven model of stochastic flow for probabilistic estimates of uncertainty in geophysical fluid dynamics (GFD). Details of the derivation of the equations from Hamilton’s principle and examples for GFD may be found in: D.D. Holm, Proc Roy Soc A, 471: 20140963 (2015). Derivation of the stochastic path for Lagrangian trajectories as a diffusive limit of deterministic Lagrangian multi-time dynamics via temporal homogenisation may be found in C. J. Cotter, G. A. Gottwald, D. D. Holm, Proc Roy Soc A, Vol 473 page 20170388 (2017). Analytical properties of solutions of the 3D stochastic Euler fluid equation may be found in D. Crisan, F. Flandoli, D. D. Holm, J Nonlinear Sci 29(3): 813-870 (2019).

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David Lannes

Nonlinear dispersive decomposition of internal waves in a continuously stratified fluid

We consider here a continuously stratified fluid and consider the propagation of internal waves. At first order, perturbations of the hydrostatic equilibrium decompose into several normal modes travelling at different speeds provided by the eigenvalues of a Sturm-Liouville problem associated to the underlying stratification. For larger times, dispersive and nonlinear effects have to be considered and complicate the analysis since the evolutions of the different modes are then coupled. We propose an asymptotic description of this coupling. 

This is a joint work with B. Desjardins and J.-C. Saut. 

Juergen Kurths 

Predictability of Extreme Climate Events via a Complex Network Approach

Complex networks are a powerful tool in many engineering and science fields. Here, we analyse climate dynamics from a complex network approach. This leads to an inverse problem: Is there a backbone-like structure underlying the climate system? For this, we propose a method to reconstruct and analyze a complex network from data generated by a spatio-temporal dynamical system. This approach enables us to uncover relations to global circulation patterns in oceans and the atmosphere.

This concept is then applied to Monsoon data; in particular, we develop a general framework to predict extreme events by combining a non-linear synchronization technique with complex networks. This way we analyze the Indian Summer Monsoon (ISM) and identify two regions of high importance. By estimating an underlying critical point, this leads to a substantially improved prediction of the onset of the ISM by two weeks compared to the available method.

References

Boers, N., B. Bookhagen, N. Marwan, J. Kurths, and J. Marengo, Geophys. Res. Lett. 40, 4386 (2013)

N. Boers, B. Bookhagen, H.M.J. Barbosa, N. Marwan, J. Kurths, and J.A. Marengo, Nature Communications 5, 5199 (2014) 

N. Boers, R. Donner, B. Bookhagen, and J. Kurths, Climate Dynamics 45, 619 (2015)

J. Runge et al., Nature Communications 6, 8502 (2015)

V. Stolbova, E. Surovyatkina, B. Bookhagen, and J. Kurths, Geophys. Res. Lett. (2016)

D. Eroglu, F. McRobies, I. Ozken, T. Stemler, K. Wyrwoll, S. Breitenbach, N. Marwan, J. Kurths, Nature Communications 7, 12929 (2016)

B. Goswami, N. Boers, A. Rheinwalt, N. Marwan, J. Heitzig, S. Breitenbach, J. Kurths, Nature Communications 9, 48(2018)

N. Boers, B. Goswami, A. Rheinwalt, B. Bookhagen, B. Hoskins, J. Kurths, Nature 566, 373 (2019)

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Honghu Liu

A variational approach to closure of nonlinear dynamical systems 

In this talk, we discuss the closure problem of nonlinear evolution systems, with a focus on systems subject to autonomous forcings and placed in parameter regimes for which no slaving principle holds. Adopting a variational framework, we will show that efficient parameterizations can be explicitly determined as parametric deformations of invariant manifolds; such deformations themselves are optimized by minimization of cost functionals naturally associated with the dynamics. The minimizers are objects, called the optimal parameterizing manifolds, that are intimately tied to the conditional expectation of the original system, i.e. the best vector field of the reduced state space resulting from averaging of the unresolved variables with respect to a probability measure conditioned on the resolved variables. Applications to the closure of low-order models of Atmospheric Primitive Equations will then be discussed. The approach will be finally illustrated---in the context of the Kuramoto-Sivashinsky turbulence with many unstable modes---to provide efficient closures without slaving for a cutoff scale placed within the inertial range and the reduced state space just spanned by the unstable modes. This talk is based on joint work with Mickael D. Chekroun (UCLA) and James McWilliams (UCLA).

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Valerio Lucarini

Global Stability Properties of the Climate: Melancholia States, Invariant Measures, and Phase Transitions

For a wide range of values of the incoming solar radiation, the Earth features at least two attracting states, which correspond to competing climates. The warm climate is analogous to the present one; the snowball climate features global glaciation and conditions that can hardly support life forms. Paleoclimatic evidences suggest that in past our planet flipped between these two states. The main physical mechanism responsible for such instability is the ice-albedo feedback. Following an idea developed by Eckhardt and co. for the investigation of multistable turbulent flows, we study the global instability giving rise to the snowball/warm multistability in the climate system by identifying the climatic edge state, a saddle embedded in the boundary between the two basins of attraction of the stable climates. We refer to these states as Melancholia States. We then introduce random perturbations as modulations to the intensity of the incoming solar radiation. We observe noise-induced transitions between the competing basins of attractions. In the weak noise limit, large deviation laws define the invariant measure and the statistics of escape times. By empirically constructing the instantons, we show that the Melancholia states are the gateways for the noise-induced transitions. In the region of multistability, in the zero-noise limit, the measure is supported only on one of the competing attractors. For low (high) values of the solar irradiance, the limit measure is the snowball (warm) climate. The changeover between the two regimes corresponds to a first order phase transition in the system. The framework we propose seems of general relevance for the study of complex multistable systems. Finally, we propose a new method for constructing Melancholia states from direct numerical simulations, thus bypassing the need to use the edge-tracking algorithm.
Refs.
V. Lucarini, T. Bodai, Transitions across Melancholia States in a Climate Model: Reconciling the Deterministic and Stochastic Points of View, Phys. Rev. Lett. 122,158701 (2019)
V. Lucarini, T. Bodai, Edge States in the Climate System: Exploring Global Instabilities and Critical Transitions, Nonlinearity 30, R32 (2017)

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David Neelin (Department of Atmospheric and Oceanic Sciences, UCLA)

Stochastic process models for precipitation processes in dialogue with observational and climate model diagnostics

Stochastic process models based on simplifications of climate model equations suggest that economical assumptions can yield simple connections between underlying physics and important aspects of observed precipitation statistics and their relationship to the water vapor-temperature environment. These include characteristic shapes of probability distributions of precipitation accumulations, time-averaged intensities and spatial clusters, and factors controlling changes in extremes. The memory associated with prognostic water vapor permits understanding of relationships between wet and dry regimes, and probability distributions across these.  A dialogue between such theoretical underpinnings, observational analysis and pragmatic diagnostics for climate models aims to help understand biases in the large numerical models and aspects of the stochastic models that should be expanded.

Joint work with: Yi-Hung Kuo, Cristian Martinez-Villalobos, Fiaz Ahmed 

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Yuzuru Sato  

Anomalous Diffusion in Random Dynamical Systems

Consider a chaotic dynamical system generating diffusion-like Brownian motion. Consider a second, nonchaotic system in which all particles localize. Let a particle experience a random combination of both systems by sampling between them in time. What type of diffusion is exhibited by this random dynamical system? We show that the resulting dynamics can generate anomalous diffusion, where in contrast to Brownian normal diffusion the mean square displacement of an ensemble of particles increases nonlinearly in time. Randomly mixing simple deterministic walks on the line, we find anomalous dynamics characterized by aging, weak ergodicity breaking, breaking of self-averaging, and infinite invariant densities. This result holds for general types of noise and for perturbing nonlinear dynamics in bifurcation Scenarios.

Reference: https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.122.174101

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Tamas Tel

Climate changes in mechanical systems

We investigate paradigmatic examples of low-dimensional chaotic systems subjected to different scenarios of parameter drifts of non-negligible rates. To characterize such dynamics, trajectory ensembles should be used since any single trajectory description is nonrepresentative.

In dissipative cases, we show that
- starting on a chaotic attractor, the complexity of the dynamics, characterized by means of snapshot/pullback attractors, remains very pronounced even when the driving amplitude has decayed to rather small values. When after the death of chaos the strength of the forcing is increased, chaos is found to revive but with a different history;
- the underlying  structures can be made visible, for arbitrary parameter drift scenarios, by prescribing a certain type of history for an ensemble of trajectories in phase space and by analyzing the trajectories fulfilling this constraint;
- a number of novel types of tippings can be observed due to the topological complexity underlying general systems. We argue for a probabilistic approach and propose the use of tipping probabilities as a measure of tipping.

We briefly discuss Hamiltonian cases and show that trajectory ensembles initiated on tori of the initial phase space provide the basis for an appropriate statistical characterization of the time-dependent dynamics.

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Shouhong Wang

Dynamic and Topological Phase Transitions in Geophysical Fluid Dynamics 

A dynamic phase transition refers to transitions of the underlying physical system from one state to another, as the control parameter crosses certain critical threshold. The notion of dynamic phase transitions is applicable to all dissipative systems, including nonlinear dissipative systems in statistical physics, quantum physics, fluid dynamics, atmospheric and oceanic sciences, biological and chemical systems etc. A topological phase transition (TPT) refers to the change of the topological structure in the physical space as certain system control parameter crosses a critical threshold. In this talk, we present the basic theories for these transitions and applications to statistical physics, and to geophysical fluid dynamics.

This is joint work with Tian Ma.

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Jeroen Wouters 

Stochastic model reduction for slow-fast systems with moderate time-scale separation

We propose a stochastic model reduction strategy for deterministic and stochastic slow-fast systems with a moderate time- scale separation. The stochastic model reduction strategy improves the approximation of systems with finite time-scale separation when compared to classical homogenization theory, by incorporating deviations from the infinite time-scale limit considered in homogenization, as described by an Edgeworth expansion in the time-scale separation parameter. To approximate these deviations from the limiting homogenized system in the reduced model, a surrogate system is constructed the parameters of which are matched to produce the same Edgeworth expansion as in the original multi-scale system, up to any desired order. We corroborate our analytical findings by numerical examples, showing significant improvements to classical homogenized model reduction.

Joint with Georg A. Gottwald

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William Young

Horizontal convection: ocean energetics, heat flux, upper-bounds and scaling

I’ll begin by surveying the current understanding of the mechanical energy balance of the ocean i.e., the identification of the main sources of the energy that power ocean circulation, turbulence and mixing. This motivates the study of convection driven by differential heating along one horizontal boundary (e.g., the sea surface) of a fluid-filled volume. This problem of horizontal convection is the main fluid-mechanical model used to understand the role of surface buoyancy forcing in oceanography. Horizontal convection also serves as an interesting counterpoint to Rayleigh-Benard convection. In contrast to Rayleigh-Benard convection, it is easy to show that horizontal convection violates the so-called "zeroth law of turbulence”. I’ll conclude with a discussion of rigorous bounds on the heat flux on the horizontal convection, and the large gap between those bounds and recent numerical results.