Direct statistical simulation of out-of-equilibrium jets

Direct statistical simulation of the Lorenz96 system in model reduction approaches

Direct statistical simulation (DSS) of nonlinear dynamical systems bypasses the traditional route of accumulating statistics by lengthy direct numerical simulations by solving the equations that govern the statistics themselves. DSS suffers, however, from the curse of dimensionality as the statistics (such as correlations) generally have higher dimensions than the underlying dynamical variables. Here we investigate two approaches to reduce the dimensionality of DSS, illustrating each method with numerical experiments with the Lorenz96 dynamical system. The forms of DSS chosen here involve approximate closures at second and third order in the equal-time cumulants. We demonstrate significant reduction in computational effort that can be achieved without sacrificing the accuracy of DSS. The methods developed here can be applied to turbulent fluid and magnetohydrodynamical systems.

Statistics of inhomogeneous turbulence in large-scale quasigeostrophic dynamics

A remarkable feature of two-dimensional turbulence is the transfer of energy from small to large scales. This process can result in the self-organization of the flow into large, coherent structures due to energy condensation at the largest scales. We investigate the formation of this condensate in a quasigeostropic flow in the limit of small Rossby deformation radius, namely the large-scale quasigeostrophic model. In this model potential energy is transferred up-scale while kinetic energy is transferred down-scale in a direct cascade. We focus on a jet mean flow and carry out a thorough investigation of the second-order statistics for this flow, combining a quasilinear analytical approach with direct numerical simulations. We show that the quasilinear approach applies in regions where jets are strong and is able to capture all second-order correlators in that region, including those related to the kinetic energy. This is a consequence of the blocking of the direct cascade by the mean flow in jet regions, suppressing fluctuation-fluctuation interactions. The suppression of the direct cascade is demonstrated using a local coarse-graining approach allowing us to measure space dependent interscale kinetic energy fluxes, which we show are concentrated in between jets in our simulations. We comment on the possibility of a similar direct cascade arrest in other two-dimensional flows, arguing that it is a special feature of flows in which the fluid element interactions are local in space.

Two-Dimensional Turbulence with Local Interactions: Statistics of the Condensate

Two-dimensional turbulence self-organizes through a process of energy accumulation at large scales, forming a coherent flow termed a condensate. We study the condensate in a model with local dynamics, the large-scale quasigeostrophic equation, observed here for the first time. We obtain analytical results for the mean flow and the two-point, second-order correlation functions, and validate them numerically. The condensate state requires partiy+time-reversal symmetry breaking. We demonstrate distinct universal mechanisms for the even and odd correlators under this symmetry. We find that the model locality is imprinted in the small scale dynamics, which the condensate spatially confines.

Direct statistical simulation of low-order dynamosystems

In this paper, we investigate the effectiveness of direct statistical simulation (DSS) for two low-order models of dynamo action. The first model, which is a simple model of solar and stellar dynamo action, is third order and has cubic nonlinearities while the second has only quadratic nonlinearities and describes the interaction of convection and an aperiodically reversing magnetic field. We show how DSS can be used to solve for the statistics of these systems of equations both in the presence and the absence of stochastic terms, by truncating the cumulant hierarchy at either second or third order. We compare two different techniques for solving for the statistics: timestepping, which is able to locate only stable solutions of the equations for the statistics, and direct detection of the fixed points. We develop a complete methodology and symbolic package in Python for deriving the statistical equations governing the low-order dynamic systems in cumulant expansions. We demonstrate that although direct detection of the fixed points is efficient and accurate for DSS truncated at second order, the addition of higher order terms leads to the inclusion of many unstable fixed points that may be found by direct detection of the fixed point by iterative methods. In those cases, timestepping is a more robust protocol for finding meaningful solutions to DSS.

The interplay between cancer type, panel size and tumor mutational burden threshold in patient selection for cancer immunotherapy

The tumor mutational burden (TMB) is increasingly recognized as an emerging biomarker that predicts improved outcomes or response to immune checkpoint inhibitors in cancer. A multitude of technical and biological factors make it difficult to compare TMB values across platforms, histologies, and treatments. Here, we present a mechanistic model that explains the association between panel size, histology, and TMB threshold with panel performance and survival outcome and demonstrate the limitations of existing methods utilized to harmonize TMB across platforms.

An increasing number of studies have demonstrated the benefit of tumor mutation burden (TMB), the number of non-silent mutations in the genome, as a predictive biomarker in a clinical setting. Most clinical trials utilize a smaller panel, instead of whole exome sequencing (WES), to estimate the exome-wide mutational load. However, the use of panels introduces panel size dependent sampling noise that could affect the performance of the TMB biomarker. In this work we create a mathematical model of the cancer histology, treatment response, and TMB device system to assess the interplay between cancer type, panel size and tumor mutational burden threshold in patient selection for cancer immunotherapy.

Multiple scales analysis of slow-fast quasi-linear systems

This article illustrates the application of multiple scales analysis to two archetypal quasi-linear systems; i.e. to systems involving fast dynamical modes, called fluctuations, that are not directly influenced by fluctuation-fluctuation nonlinearities but nevertheless are strongly coupled to a slow variable whose evolution may be fully nonlinear. In the first case, fast waves drive a slow, spatially inhomogeneous evolution of their celerity field. Multiple scales analysis confirms that, although the energy E, the angular frequency ω and the modal structure of the waves evolve, the wave action E/ω is conserved in the absence of forcing and dissipation. In the second system, the fast modes undergo an instability that is saturated through a feedback on the slow variable. A new multi-scale analysis is developed to treat this case. The key technical point, confirmed by the analysis, is that the fluctuation energy and mode structure evolve slowly to ensure that the slow field remains in a state of near marginal stability. These two model systems appear to be generic, being representative of many if not all quasi-linear systems. In each case, numerical simulations of both the full and reduced dynamical systems are performed to highlight the accuracy and efficiency of the multiple scales approach. Python codes are provided as electronic supplementary material.

Rare Event Algorithm Links Transitions in Turbulent Flows with Activated Nucleations

Many turbulent flows undergo drastic and abrupt configuration changes with huge impacts. As a paradigmatic example we study the multistability of jet dynamics in a barotropic beta plane model of atmosphere dynamics. It is considered as the Ising model for Jupiter troposphere dynamics. Using the adaptive multilevel splitting, a rare event algorithm, we are able to get a very large statistics of transition paths, the extremely rare transitions from one state of the system to another. This new approach opens the way for addressing a set of questions that are out of reach through direct numerical simulations. We demonstrate for the first time the concentration of transition paths close to instantons, in a numerical simulation of genuine turbulent flows. We show that the transition is a noise-activated nucleation of vorticity bands. We address for the first time the existence of Arrhenius laws in turbulent flows. The methodology we developed shall prove useful to study many other transitions related to drastic changes for the turbulent dynamics of climate, geophysical, astrophysical, and engineering applications. This opens a new range of studies impossible so far, and bring turbulent phenomena in the realm of nonequilibrium statistical mechanics.

Generalized quasilinear approximation of the interaction of convection and mean flows in a thermal annulus

In this paper, we examine the interaction of convection, rotation and mean flows in a thermal annulus. In this system, mean flows are driven by correlations induced by rotation leading to non-trivial Reynolds stresses. The mean flows act back on the convective turbulence acting as a barrier to transport. For this system, we demonstrate that the generalized quasilinear approximation (Marston et al . 2016 Phys. Rev. Lett. 116 , 214501. ( doi:10.1103/PhysRevLett.116.214501 )) may provide a much better approximation to the complicated full nonlinear dynamics than the widely used quasilinear approximation. This result will enable the construction of more accurate statistical theories for the description of geophysical and astrophysical flows.

Turbulence Statistics in a Two-Dimensional Vortex Condensate

Disentangling the evolution of a coherent mean-flow and turbulent fluctuations, interacting through the nonlinearity of the Navier-Stokes equations, is a central issue in fluid mechanics. It affects a wide range of flows, such as planetary atmospheres, plasmas, or wall-bounded flows, and hampers turbulence models. We consider the special case of a two-dimensional flow in a periodic box, for which the mean flow, a pair of box-size vortices called "condensate," emerges from turbulence. As was recently shown, a perturbative closure describes correctly the condensate when turbulence is excited at small scales. In this context, we obtain explicit results for the statistics of turbulence, encoded in the Reynolds stress tensor. We demonstrate that the two components of the Reynolds stress, the momentum flux and the turbulent energy, are determined by different mechanisms. It was suggested previously that the momentum flux is fixed by a balance between forcing and mean-flow advection: using unprecedently long numerical simulations, we provide the first direct evidence supporting this prediction. By contrast, combining analytical computations with numerical simulations, we show that the turbulent energy is determined only by mean-flow advection and obtain for the first time a formula describing its profile in the vortex.

Statistics of the stochastically forced Lorenz attractor by the Fokker-Planck equation and cumulant expansions

We investigate the Fokker-Planck description of the equal-time statistics of the three-dimensional Lorenz attractor with additive white noise. The invariant measure is found by computing the zero (or null) mode of the linear Fokker-Planck operator as a problem of sparse linear algebra. Two variants are studied: a self-adjoint construction of the linear operator and the replacement of diffusion with hyperdiffusion. We also access the low-order statistics of the system by a perturbative expansion in equal-time cumulants. A comparison is made to statistics obtained by the standard approach of accumulation via direct numerical simulation. Theoretical and computational aspects of the Fokker-Planck and cumulant expansion methods are discussed.

Geophysical fluid dynamics: whence, whither and why?

This article discusses the role of geophysical fluid dynamics (GFD) in understanding the natural environment, and in particular the dynamics of atmospheres and oceans on Earth and elsewhere. GFD, as usually understood, is a branch of the geosciences that deals with fluid dynamics and that, by tradition, seeks to extract the bare essence of a phenomenon, omitting detail where possible. The geosciences in general deal with complex interacting systems and in some ways resemble condensed matter physics or aspects of biology, where we seek explanations of phenomena at a higher level than simply directly calculating the interactions of all the constituent parts. That is, we try to develop theories or make simple models of the behaviour of the system as a whole. However, these days in many geophysical systems of interest, we can also obtain information for how the system behaves by almost direct numerical simulation from the governing equations. The numerical model itself then explicitly predicts the emergent phenomena-the Gulf Stream, for example-something that is still usually impossible in biology or condensed matter physics. Such simulations, as manifested, for example, in complicated general circulation models, have in some ways been extremely successful and one may reasonably now ask whether understanding a complex geophysical system is necessary for predicting it. In what follows we discuss such issues and the roles that GFD has played in the past and will play in the future.

Interaction between mean flow and turbulence in two dimensions

This short note is written to call attention to an analytic approach to the interaction of developed turbulence with mean flows of simple geometry (jets and vortices). It is instructive to compare cases in two and three dimensions and see why the former are solvable and the latter are not (yet). We present the analytical solutions for two-dimensional mean flows generated by an inverse turbulent cascade on a sphere and in planar domains of different aspect ratios. These solutions are obtained in the limit of small friction when the flow is strong while turbulence can be considered weak and treated perturbatively. I then discuss when these simple solutions can be realized and when more complicated flows may appear instead. The next step of describing turbulence statistics inside a flow and directions of possible future progress are briefly discussed at the end.

Generalized Quasilinear Approximation: Application to Zonal Jets

Quasilinear theory is often utilized to approximate the dynamics of fluids exhibiting significant interactions between mean flows and eddies. We present a generalization of quasilinear theory to include dynamic mode interactions on the large scales. This generalized quasilinear (GQL) approximation is achieved by separating the state variables into large and small zonal scales via a spectral filter rather than by a decomposition into a formal mean and fluctuations. Nonlinear interactions involving only small zonal scales are then removed. The approximation is conservative and allows for scattering of energy between small-scale modes via the large scale (through nonlocal spectral interactions). We evaluate GQL for the paradigmatic problems of the driving of large-scale jets on a spherical surface and on the beta plane and show that it is accurate even for a small number of large-scale modes. As GQL is formally linear in the small zonal scales, it allows for the closure of the system and can be utilized in direct statistical simulation schemes that have proved an attractive alternative to direct numerical simulation for many geophysical and astrophysical problems.

Nonperturbative mean-field theory for minimum enstrophy relaxation

The dual cascade of enstrophy and energy in quasi-two-dimensional turbulence strongly suggests that a viscous but otherwise potential vorticity (PV) conserving system decays selectively toward a state of minimum potential enstrophy. We derive a nonperturbative mean field theory for the dynamics of minimum enstrophy relaxation by constructing an expression for PV flux during the relaxation process. The theory is used to elucidate the structure of anisotropic flows emerging from the selective decay process. This structural analysis of PV flux is based on the requirements that the mean flux of PV dissipates total potential enstrophy but conserves total fluid kinetic energy. Our results show that the structure of PV flux has the form of a sum of a positive definite hyperviscous and a negative or positive viscous transport of PV. Transport parameters depend on zonal flow and turbulence intensity. Turbulence spreading is shown to be related to PV mixing via the link of turbulence energy flux to PV flux. In the relaxed state, the ratio of the PV gradient to zonal flow velocity is homogenized. This homogenized quantity sets a constraint on the amplitudes of PV and zonal flow in the relaxed state. A characteristic scale is defined by the homogenized quantity and is related to a variant of the Rhines scale. This relaxation model predicts a relaxed state with a structure which is consistent with PV staircases, namely, the proportionality between mean PV gradient and zonal flow strength.

Emergence of large scale structure in barotropic β-plane turbulence

In this Letter, we use a nonequilibrium statistical theory, the stochastic structural stability theory (SSST), to show that an extended version of this theory can make predictions for the formation of nonzonal as well as zonal structures (lattice and stripe patterns) in forced homogeneous turbulence on a barotropic β plane. Comparison of the theory with nonlinear simulations demonstrates that SSST predicts the parameter values for the emergence of coherent structures and their characteristics (scale, amplitude, phase speed) as they emerge and at finite amplitude. It is shown that nonzonal structures (lattice states or zonons) emerge at lower energy input rates of the stirring compared to zonal flows (stripe states) and their emergence affects the dynamics of jet formation.