Zonal flow driven by convection and convection driven by internal heating

In the first part, Rayleigh-Benard convection is studied in a two-dimensional, horizontally periodic domain with free-slip top and bottom boundaries. This configuration encourages mean horizontal flows of zero horizontal wavenumber, which we study as an idealization of zonal flows in tokamaks, planetary atmospheres, and annular cylindrical convection experiments. These systems often satisfy free-slip conditions on at least one boundary and are approximately two-dimensional. Stable steady states with zonal flow are found for Prandtl numbers up to 0.3. Stable and unstable steady states with horizontal periods up to six times the layer height are computed for a Prandtl number of 0.1 and Rayleigh numbers, Ra, up to 2*10⁵. Concurrently stable states with and without zonal flow are found where the state without zonal flow convects heat over 10 times faster. Steady zonal flow arises subcritically whenever the horizontal period is not forced to be narrow, contrary to most prior predictions by truncated models. Steady states and their bifurcations are studied in a truncated model that does predict subcriticality. Direct numerical simulations are performed with a horizontal period twice the layer height, Prandtl numbers between 1 and 10, and Ra between 5*10⁵ and 2*10⁸. Zonal flow arises subcritically as Ra is raised but is seen in all quasi-steady states at large Ra. The fraction of the total kinetic energy comprised by zonal flow approaches unity as Ra grows. At a Prandtl number of 1, vertical convective heat transport occurs in temporal bursts, nearly vanishing in between, and is non-monotonic in Ra. At Prandtl numbers of 3 and 10, convective transport at no time nearly vanishes, and time-averaged Nusselt numbers scale as Ra⁰·⁰⁷⁷ and Ra⁰·¹⁹, respectively. Both growth rates are below the range accepted for Rayleigh-Benard convection without zonal flow. In the second part, two-dimensional direct numerical simulations are conducted for convection sustained by uniform internal heating in a horizontal fluid layer. Top and bottom boundary temperatures are fixed and equal. Prandtl numbers range from 0.01 to 100. A control parameter, R, that is similar to the usual Rayleigh number is varied up to 5*10^5 times its critical value at the onset of convection. The asymmetry between upward and downward heat fluxes is non-monotonic in R. In a broad high-R regime, dimensionless mean temperature scales as R^-1/5. We discuss the scaling of mean temperature and heat-flux-asymmetry, which we find to be better diagnostic quantities than the conventionally used top and bottom Nusselt numbers

Bounding extreme events in nonlinear dynamics using convex optimization

We study a convex optimization framework for bounding extreme events in nonlinear dynamical systems governed by ordinary or partial differential equations (ODEs or PDEs). This framework bounds from above the largest value of an observable along trajectories that start from a chosen set and evolve over a finite or infinite time interval. The approach needs no explicit trajectories. Instead, it requires constructing suitably constrained auxiliary functions that depend on the state variables and possibly on time. Minimizing bounds over auxiliary functions is a convex problem dual to the non-convex maximization of the observable along trajectories. This duality is strong, meaning that auxiliary functions give arbitrarily sharp bounds, for sufficiently regular ODEs evolving over a finite time on a compact domain. When these conditions fail, strong duality may or may not hold; both situations are illustrated by examples. We also show that near-optimal auxiliary functions can be used to construct spacetime sets that localize trajectories leading to extreme events. Finally, in the case of polynomial ODEs and observables, we describe how polynomial auxiliary functions of fixed degree can be optimized numerically using polynomial optimization. The corresponding bounds become sharp as the polynomial degree is raised if strong duality and mild compactness assumptions hold. Analytical and computational ODE examples illustrate the construction of bounds and the identification of extreme trajectories, along with some limitations. As an analytical PDE example, we bound the maximum fractional enstrophy of solutions to the Burgers equation with fractional diffusion.Comment: Revised according to comments by reviewers. Added references and rearranged introduction, conclusions, and proofs. 38 pages, 7 figures, 4 tables, 4 appendices, 87 reference

Convex computation of maximal Lyapunov exponents

We describe an approach for finding upper bounds on an ODE dynamical system's maximal Lyapunov exponent among all trajectories in a specified set. A minimization problem is formulated whose infimum is equal to the maximal Lyapunov exponent, provided that trajectories of interest remain in a compact set. The minimization is over auxiliary functions that are defined on the state space and subject to a pointwise inequality. In the polynomial case -- i.e., when the ODE's right-hand side is polynomial, the set of interest can be specified by polynomial inequalities or equalities, and auxiliary functions are sought among polynomials -- the minimization can be relaxed into a computationally tractable polynomial optimization problem subject to sum-of-squares constraints. Enlarging the spaces of polynomials over which auxiliary functions are sought yields optimization problems of increasing computational cost whose infima converge from above to the maximal Lyapunov exponent, at least when the set of interest is compact. For illustration, we carry out such polynomial optimization computations for two chaotic examples: the Lorenz system and the H\'enon-Heiles system. The computed upper bounds converge as polynomial degrees are raised, and in each example we obtain a bound that is sharp to at least five digits. This sharpness is confirmed by finding trajectories whose leading Lyapunov exponents approximately equal the upper bounds.Comment: 29 page

Bounds for deterministic and stochastic dynamical systems using sum-of-squares optimization

We describe methods for proving upper and lower bounds on infinite-time averages in deterministic dynamical systems and on stationary expectations in stochastic systems. The dynamics and the quantities to be bounded are assumed to be polynomial functions of the state variables. The methods are computer-assisted, using sum-of-squares polynomials to formulate sufficient conditions that can be checked by semidefinite programming. In the deterministic case, we seek tight bounds that apply to particular local attractors. An obstacle to proving such bounds is that they do not hold globally; they are generally violated by trajectories starting outside the local basin of attraction. We describe two closely related ways past this obstacle: one that requires knowing a subset of the basin of attraction, and another that considers the zero-noise limit of the corresponding stochastic system. The bounding methods are illustrated using the van der Pol oscillator. We bound deterministic averages on the attracting limit cycle above and below to within 1%, which requires a lower bound that does not hold for the unstable fixed point at the origin. We obtain similarly tight upper and lower bounds on stochastic expectations for a range of noise amplitudes. Limitations of our methods for certain types of deterministic systems are discussed, along with prospects for improvement.Comment: 25 pages; Added new Section 7.2; Added references; Corrected typos; Submitted to SIAD