arXiv subjects

James C. McWilliams

Publications and source records attributed to James C. McWilliams.

17 recordsLinked to original sources

arXiv2026

A Symplectic Theory of Turbulence Closure: Hidden Reservoir Dynamics, Endogenous Stochastic Transport, and Kraichnan Dual Cascades

The equations of fluid motion are known; retaining their physics after removing most degrees of freedom remains a fundamental problem. Closure must preserve the geometry of the dynamics it replaces. We introduce Symplectic Geometric Closure (SGC), a prognostic stochastic field theory carrying Euler's Hamiltonian, area-preserving transport into two-dimensional and $β$-plane turbulence. SGC couples resolved vorticity to a hidden stochastic reservoir. The coupling conserves augmented enstrophy while permitting bidirectional energy transfer. This yields a compact random attractor in the hyperviscous Navier--Stokes--$β$ realization. Numerically, SGC sustains jets, vortices and filaments in high-Reynolds-number turbulence with high fidelity to filtered DNS at an inertial-range cutoff, preserving geometric identities to machine precision. The induced transport folds, stretches and rearranges vorticity: cross-gradient coupling selects interactions through resolved--reservoir gradient misalignment, retaining geometric selectivity and memory. The same geometry addresses spurious sweeping decorrelation, a longstanding obstacle to Eulerian closure. The interaction vertex excludes uniform translation and is Random Galilean compatible. Eliminating the reservoir generates finite memory, stochastic backscatter and a Dyson--Volterra equation for the dressed propagator. Its one-loop, line-renormalized self-energy reproduces Kraichnan-type Direct-Interaction Approximation architecture with fourth-order infrared suppression of sweeping. Deformation-controlled memory and conservative triad constraints yield the $k^{-5/3}$ inverse-energy and $k^{-3}$ forward-enstrophy cascades under standard assumptions. SGC thus realizes Kraichnan's program through Eulerian field dynamics, opening a route to data-driven closures that learn admissible geometric interactions rather than unconstrained forces.

physics.flu-dyn↗

arXiv2026

Diapycnal material transport driven by submesoscale frontogenesis

Submesoscale fronts, occurring at intermediate scales between mesoscale eddies and boundary layer turbulence, play a crucial role in driving vertical transport from the ocean surface into the interior. Their dynamics involve complex interactions between submesoscale currents and turbulence. However, the mechanisms by which these multiscale processes combine to transport tracers such as pollutants or nutrients remain less well understood. This study uses large-eddy simulation to investigate passive tracer transport associated with submesoscale fronts. Intense turbulence develops during frontogenesis, leading to strong diapycnal tracer transport into the ocean interior. While part of this transport arises from the direct turbulent flux, represented by the covariance between turbulent velocity and tracer concentration fluctuations, a substantial portion is due to an advective diapycnal flux driven by the mean diapycnal velocity. The mean diapycnal velocity results from the evolving secondary circulation in the presence of turbulent density mixing. These findings reveal an underexplored diapycnal transport pathway in submesoscale frontal zones, with implications for improved representation of vertical exchange in ocean models.

physics.flu-dyn↗

arXiv2025

Abyssal Slope Currents

Realistic computational simulations in different oceanic basins reveal prevalent prograde mean flows (i.e. in the direction of topographic Rossby wave propagation along isobaths; a.k.a. topostrophy) on topographic slopes in the deep ocean, consistent with the barotropic theory of eddy-driven mean flows. Attention is focused on the Western Mediterranean Sea with strong currents and steep topography. These prograde mean currents induce an opposing bottom drag stress and thus a turbulent boundary-layer mean flow in the downhill direction, evidenced by a near-bottom negative mean vertical velocity. The slope-normal profile of diapycnal buoyancy mixing results in down-slope mean advection near the bottom (a tendency to locally increase the mean buoyancy) and up-slope buoyancy mixing (a tendency to decrease buoyancy) with associated buoyancy fluxes across the mean isopycnal surfaces (diapycnal downwelling). In the upper part of the boundary layer and nearby interior, the diapycnal turbulent buoyancy flux divergence reverses sign (diapycnal upwelling), with upward Eulerian mean buoyancy advection across isopycnal surfaces. These near-slope tendencies abate with further distance from the boundary. An along-isobath mean momentum balance shows an advective acceleration and a bottom-drag retardation of the prograde flow. The eddy buoyancy advection is significant near the slope, and the associated eddy potential energy conversion is negative, consistent with mean vertical shear flow generation for the eddies. This cross-isobath flow structure differs from previous proposals, and a new one-dimensional model is constructed for a topostrophic, stratified, slope bottom boundary layer. The broader issue of the return pathways of the global thermohaline circulation remains open, but the abyssal slope region is likely to play a dominant role.

physics.ao-ph↗

arXiv2024

Non-Markovian Reduced Models to Unravel Transitions in Non-equilibrium Systems

This work proposes a general framework for capturing noise-driven transitions in spatially extended non-equilibrium systems and explains the emergence of coherent patterns beyond the instability onset. The framework relies on stochastic parameterizations to reduce the original equations' complexity while capturing the key effects of unresolved scales. It works for both Gaussian and Levy-type noise. Our parameterizations offer two key advantages. First, they approximate stochastic invariant manifolds when the latter exist. Second, even when such manifolds break down, our formulas can be adapted by a simple optimization of its constitutive parameters. This allows us to handle scenarios with weak time-scale separation where the system has undergone multiple transitions, resulting in large-amplitude solutions not captured by invariant manifold or other time-scale separation methods. The optimized stochastic parameterizations capture how small-scale noise impacts larger scales through the system's nonlinear interactions. This effect is achieved by the very fabric of our parameterizations incorporating non-Markovian coefficients into the reduced equation. Such coefficients account for the noise's past influence using a finite memory length, selected for optimal performance. The specific "memory" function, which determines how this past influence is weighted, depends on the noise's strength and how it interacts with the system's nonlinearities. Remarkably, training our theory-guided reduced models on a single noise path effectively learns the optimal memory length for out-of-sample predictions, including rare events. This success stems from our "hybrid" approach, which combines analytical understanding with data-driven learning. This combination avoids a key limitation of purely data-driven methods: their struggle to generalize to unseen scenarios, also known as the "extrapolation problem."

math.DS↗

arXiv2024

Spontaneous emission of internal waves by a radiative instability

The spontaneous emission of internal waves (IWs) from balanced mesoscale eddies has been previously proposed to provide a source of oceanic IW kinetic energy (KE). This study examines the mechanisms leading to the spontaneous emission of spiral-shaped IWs from an anticyclonic eddy with an order-one Rossby number, using a high-resolution numerical simulation of a flat-bottomed, wind-forced, reentrant channel flow configured to resemble the Antarctic Circumpolar Current. It is demonstrated that IWs are spontaneously generated as a result of a loss of balance process that is concentrated at the eddy edge, and then radiate radially outward. A 2D linear stability analysis of the eddy shows that the spontaneous emission arises from a radiative instability which involves an interaction between a vortex Rossby wave supported by the radial gradient of potential vorticity and an outgoing IWs. This particular instability occurs when the perturbation frequency is superinertial. This finding is supported by a KE analysis of the unstable modes and the numerical solution, where it is shown that the horizontal shear production provides the source of perturbation KE. Furthermore, the horizontal length scale and frequency of the most unstable mode from the stability analysis agree well with those of the spontaneously emitted IWs in the numerical solution.

physics.flu-dyn↗

arXiv2024

The High-Frequency and Rare Events Barriers to Neural Closures of Atmospheric Dynamics

Recent years have seen a surge in interest for leveraging neural networks to parameterize small-scale or fast processes in climate and turbulence models. In this short paper, we point out two fundamental issues in this endeavor. The first concerns the difficulties neural networks may experience in capturing rare events due to limitations in how data is sampled. The second arises from the inherent multiscale nature of these systems. They combine high-frequency components (like inertia-gravity waves) with slower, evolving processes (geostrophic motion). This multiscale nature creates a significant hurdle for neural network closures. To illustrate these challenges, we focus on the atmospheric 1980 Lorenz model, a simplified version of the Primitive Equations that drive climate models. This model serves as a compelling example because it captures the essence of these difficulties.

math.DS↗

arXiv2024

Langmuir turbulence in suspended kelp farms

This study investigates the influence of suspended kelp farms on ocean mixed layer hydrodynamics in the presence of currents and waves. We use the large eddy simulation method, where the wave effect is incorporated by solving the wave-averaged equations. Distinct Langmuir circulation patterns are generated within various suspended farm configurations, including horizontally uniform kelp blocks and spaced kelp rows. Intensified turbulence arises from the farm-generated Langmuir circulation, as opposed to the standard Langmuir turbulence observed without a farm. The creation of Langmuir circulation within the farm is attributed to two primary factors depending on farm configuration: (1) enhanced vertical shear due to kelp frond area density variability, and (2) enhanced lateral shear due to canopy discontinuity at lateral edges of spaced rows. Both enhanced vertical and lateral shear of streamwise velocity, representing the lateral and vertical vorticity components respectively, can be tilted into downstream vorticity to create Langmuir circulation. This vorticity tilting is driven by the Craik- Leibovich vortex force associated with the Stokes drift of surface gravity waves. In addition to the farm-generated Langmuir turbulence, canopy shear layer turbulence is created at the farm bottom edge due to drag discontinuity. The intensity of different types of turbulence depends on both kelp frond area density and the geometric configuration of the farm. The farm-generated turbulence has substantial consequences for nutrient supply and kelp growth. These findings also underscore the significance of the presence of obstacle structures in modifying ocean mixed layer characteristics.

physics.flu-dyn↗

arXiv2023

Optimal Parameterizing Manifolds for Anticipating Tipping Points and Higher-order Critical Transitions

A general, variational approach to derive low-order reduced systems is presented. The approach is based on the concept of optimal parameterizing manifold (OPM) that substitutes the more classical notions of invariant or slow manifold when breakdown of "slaving" occurs, i.e. when the unresolved variables cannot be expressed as an exact functional of the resolved ones anymore. The OPM provides, within a given class of parameterizations of the unresolved variables, the manifold that averages out optimally these variables as conditioned on the resolved ones. The class of parameterizations retained here is that of continuous deformations of parameterizations rigorously valid near the onset of instability. These deformations are produced through integration of auxiliary backward-forward (BF) systems built from the model's equations and lead to analytic formulas for parameterizations. In this modus operandi, the backward integration time is the key parameter to select per scale/variable to parameterize in order to derive the relevant parameterizations which are doomed to be no longer exact, away from instability onset, due to breakdown of slaving typically encountered e.g. for chaotic regimes. The selection criterion is then made through data-informed minimization of a least-square parameterization defect. It is thus shown, through optimization of the backward integration time per scale/variable to parameterize, that skilled OPM reduced systems can be derived for predicting with accuracy higher-order critical transitions or catastrophic tipping phenomena, while training our parameterization formulas for regimes prior to these transitions take place.

math.DS↗

arXiv2023

Turbulence closure with small, local neural networks: Forced two-dimensional and $β$-plane flows

We parameterize sub-grid scale (SGS) fluxes in sinusoidally forced two-dimensional turbulence on the $β$-plane at high Reynolds numbers (Re$\sim$25000) using simple 2-layer Convolutional Neural Networks (CNN) having only O(1000)parameters, two orders of magnitude smaller than recent studies employing deeper CNNs with 8-10 layers; we obtain stable, accurate, and long-term online or a posteriori solutions at 16X downscaling factors. Our methodology significantly improves training efficiency and speed of online Large Eddy Simulations (LES) runs, while offering insights into the physics of closure in such turbulent flows. Our approach benefits from extensive hyperparameter searching in learning rate and weight decay coefficient space, as well as the use of cyclical learning rate annealing, which leads to more robust and accurate online solutions compared to fixed learning rates. Our CNNs use either the coarse velocity or the vorticity and strain fields as inputs, and output the two components of the deviatoric stress tensor. We minimize a loss between the SGS vorticity flux divergence (computed from the high-resolution solver) and that obtained from the CNN-modeled deviatoric stress tensor, without requiring energy or enstrophy preserving constraints. The success of shallow CNNs in accurately parameterizing this class of turbulent flows implies that the SGS stresses have a weak non-local dependence on coarse fields; it also aligns with our physical conception that small-scales are locally controlled by larger scales such as vortices and their strained filaments. Furthermore, 2-layer CNN-parameterizations are more likely to be interpretable and generalizable because of their intrinsic low dimensionality.

physics.flu-dyn↗

arXiv2022

Transitions in Stochastic Non-equilibrium Systems: Efficient Reduction and Analysis

A central challenge in physics is to describe non-equilibrium systems driven by randomness, such as a randomly growing interface, or fluids subject to random fluctuations that account e.g. for local stresses and heat fluxes not related to the velocity and temperature gradients. For deterministic systems with infinitely many degrees of freedom, normal form and center manifold theory have shown a prodigious efficiency to often completely characterize how the onset of linear instability translates into the emergence of nonlinear patterns. However, in presence of random fluctuations, this reduction procedure is seriously challenged due to large excursions caused by the noise, and the approach needs to be revisited. We present an alternative framework to cope with these difficulties exploiting the approximation theory of stochastic invariant manifolds and energy estimates measuring the defect of parameterization of the high-modes. To operate for fluid problems, these error estimates are derived under assumptions regarding dissipation effects brought by the high-modes that suitably counterbalance the loss of regularity due to the nonlinear terms. The approach enables us to predict, from the reduced equations, the occurrence in large probability of a stochastic analogue to the pitchfork bifurcation, as long as the noise's intensity and the eigenvalue's magnitude of the mildly unstable mode scale accordingly. Our parameterization formulas involve non-Markovian coefficients, which depend explicitly on the history of the noise path that drives the SPDE dynamics, and their memory content is self-consistently determined by the intensity of the random force and its interaction through the SPDE's nonlinear terms. Applications to a stochastic Rayleigh-Benard problem are detailed, for which conditions for a stochastic pitchfork bifurcation (in large probability) to occur, are clarified.

math.AP↗

arXiv2021

Overlapping boundary layers in coastal oceans

Boundary layer turbulence in coastal regions differs from that in deep ocean because of bottom interactions. In this paper, we focus on the merging of surface and bottom boundary layers in a finite-depth coastal ocean by numerically solving the wave-averaged equations using a large eddy simulation method. The ocean fluid is driven by combined effects of wind stress, surface wave, and a steady current in the presence of stable vertical stratification. The resulting flow consists of two overlapping boundary layers, i.e. surface and bottom boundary layers, separated by an interior stratification. The overlapping boundary layers evolve through three phases, i.e. a rapid deepening, an oscillatory equilibrium and a prompt merger, separated by two transitions. Before the merger, internal waves are observed in the stratified layer, and they are excited mainly by Langmuir turbulence in the surface boundary layer. These waves induce a clear modulation on the bottom-generated turbulence, facilitating the interaction between the surface and bottom boundary layers. After the merger, the Langmuir circulations originally confined to the surface layer are found to grow in size and extend down to the sea bottom (even though the surface waves do not feel the bottom), reminiscent of the well-organized Langmuir supercells. These full-depth Langmuir circulations promote the vertical mixing and enhance the bottom shear, leading to a significant enhancement of turbulence levels in the vertical column.

physics.ao-ph↗

arXiv2021

Generation of attached Langmuir circulations by a suspended macroalgal farm

In this study, we focus on Langmuir turbulence in the deep ocean with the presence of a large macroalgal farm using a Large Eddy Simulation method. The wave-current interactions are modelled by solving the wave-averaged equations. The hydrodynamic process over the farm is found to drive a persistent flow pattern similar to Langmuir circulations but is locked in space across the farm. These secondary circulations are generated because the cross-stream shear produced by the rows of canopy elements leads to a steady vertical vorticity field, which is then rotated to the downstream direction under the effect of vortex force. Since the driving mechanism is similar to the CraikLeibovich type 2 instability theory, these secondary circulations are also termed as attached Langmuir circulations. We then apply a triple decomposition on the flow field to unveil the underlying kinematics and energy transfer between the mean flow, the secondary flow resulting from the farm drag, and the transient eddies. Flow visualizations and statistics suggest that the attached Langmuir circulations result from the adjustment of the upper ocean mixed layer to the macroalgal farm, and they will weaken (if not disappear) when the flow reaches an equilibrium state within the farm. The tripledecomposed energy budgets reveal that the energy of the secondary flow is transferred from the mean flow under the action of canopy drag, while the transient eddies feed on wave energy transferred by the Stokes drift and energy conversion from the secondary flow.

physics.flu-dyn↗

arXiv2019

Variational approach to closure of nonlinear dynamical systems: Autonomous case

A general, variational approach to derive low-order reduced systems for nonlinear systems subject to an autonomous forcing, is introduced. The approach is based on the concept of optimal parameterizing manifold (PM) that substitutes the more classical notion of slow manifold or invariant manifold when breakdown of slaving occurs. An optimal PM provides the manifold that describes the average motion of the neglected scales as a function of the resolved scales and allows, in principle, for determining the best vector field of the reduced state space that describes e.g. the dynamics' slow motion. The underlying optimal parameterizations are approximated by dynamically-based formulas derived analytically from the original equations. These formulas are contingent upon the determination of only a few (scalar) parameters obtained from minimization of cost functionals, depending on training dataset collected from direct numerical simulation. In practice, a training period of length comparable to a characteristic recurrence or decorrelation time of the dynamics, is sufficient for the efficient derivation of optimized parameterizations. Applications to the closure of low-order models of Atmospheric Primitive Equations and Rayleigh-Bénard convection are then discussed. The approach is finally illustrated --- in the context of the Kuramoto-Sivashinsky turbulence --- as providing efficient closures without slaving for a cutoff scale $k_c$ placed within the inertial range and the reduced state space is just spanned by the unstable modes, without inclusion of any stable modes whatsoever. The underlying optimal PMs obtained by our variational approach are far from slaving and allow for remedying the excessive backscatter transfer of energy to the low modes encountered by classical invariant manifold approximations in their standard forms when the latter are used at this cutoff wavelength.

math.DS↗

arXiv2011

The Elemental Shear Dynamo

A quasi-linear theory is presented for how randomly forced, barotropic velocity fluctuations cause an exponentially-growing, large-scale (mean) magnetic dynamo in the presence of a uniform shear flow, $\vec{U} = S x \vec{e}_y$. It is a "kinematic" theory for the growth of the mean magnetic energy from a small initial seed, neglecting the saturation effects of the Lorentz force. The quasi-linear approximation is most broadly justifiable by its correspondence with computational solutions of nonlinear magneto-hydrodynamics, and it is rigorously derived in the limit of large resistivity, $η\rightarrow \infty$. Dynamo action occurs even without mean helicity in the forcing or flow, but random helicity variance is then essential. In a sufficiently large domain and with small wavenumber $k_z$ in the direction perpendicular to the mean shearing plane, a positive exponential growth rate $γ$ can occur for arbitrary values of $η$, the viscosity $ν$, and the random-forcing correlation time $t_f$ and phase angle $θ_f$ in the shearing plane. The value of $γ$ is independent of the domain size. The shear dynamo is "fast", with finite $γ> 0$ in the limit of $η\rightarrow 0$. Averaged over the random forcing ensemble, the mean magnetic field grows more slowly, if at all, compared to the r.m.s. field (or magnetic energy). In the limit of small Reynolds numbers ($η, \ ν\rightarrow \infty$), the dynamo behavior is related to the well-known alpha--omega {\it ansatz} when the forcing is steady ($t_f \rightarrow \infty$) and to the "incoherent" alpha--omega {\it ansatz} when the forcing is purely fluctuating.

astro-ph.GA↗

arXiv2003

Forced Stratified Turbulence: Successive Transitions with Reynolds Number

Numerical simulations are made for forced turbulence at a sequence of increasing values of Reynolds number, R, keeping fixed a strongly stable, volume-mean density stratification. At smaller values of R, the turbulent velocity is mainly horizontal, and the momentum balance is approximately cyclostrophic and hydrostatic. This is a regime dominated by so-called pancake vortices, with only a weak excitation of internal gravity waves and large values of the local Richardson number, Ri, everywhere. At higher values of R there are successive transitions to (a) overturning motions with local reversals in the density stratification and small or negative values of Ri; (b) growth of a horizontally uniform vertical shear flow component; and (c) growth of a large-scale vertical flow component. Throughout these transitions, pancake vortices continue to dominate the large-scale part of the turbulence, and the gravity wave component remains weak except at small scales.

physics.flu-dyn↗

arXiv2001

Local Gravitational Instability of Magnetized Shear Flows

The effect of magnetic shear and shear flow on local gravitationally induced instabilities is investigated. A simple model is constructed allowing for an arbitrary entropy gradient and a shear plasma flow in the Boussinesq approximation. A transformation to shearing magnetic coordinates achieves a model with plasma flow along the magnetic field lines where the coordinate lines are coincident with the field lines. The solution for the normal modes of the system depends on two parameters: the Alfven Mach number of the plasma flow and the entropy gradient. The behavior of the unstable normal modes of this system is summarized by a stability diagram. Important characteristics of this stability diagram are the following: magnetic shear is stabilizing and the entropy gradient must exceed a threshold value for unstable mode growth to occur; flow acts to suppress mode growth in a substantially unstable regime as expected, yet near marginal stability it can lessen the stabilizing effect of magnetic shear and enhance the growth rates of the instability; and, as the Alfven Mach number approaches one, the instability is completely stabilized. Analytical work is presented supporting the characteristics of the stability diagram and illuminating the physical mechanisms controlling the behavior of the model. The implications of this work for astrophysical and fusion applications and the potential for future research extending the results to include compressibility are discussed.

astro-ph↗

arXiv1998

Lagrangian Dynamics in High-Dimensional Point-Vortex Systems

We study the Lagrangian dynamics of systems of N point vortices and passive particles in a two-dimensional, doubly periodic domain. The probability distribution function of vortex velocity, p_N, has a slow-velocity Gaussian component and a significant high-velocity tail caused by close vortex pairs. In the limit for N -> oo, p_N tends to a Gaussian. However, the form of the single-vortex velocity causes very slow convergence with N; for N ~ 10^6 the non-Gaussian high-velocity tails still play a significant role. At finite N, the Gaussian component is well modeled by an Ornstein-Uhlenbeck (OU) stochastic process with variance sigma_N = sqrt{N ln N /2 pi}. Considering in detail the case N=100, we show that at short times the velocity autocorrelation is dominated by the Gaussian component and displays an exponential decay with a short Lagrangian decorrelation time. The close pairs have a long correlation time and cause nonergodicity over at least the time of the integration. Due to close vortex dipoles the absolute dispersion differs significantly from the OU prediction, and shows evidence of long-time anomalous dispersion. We discuss the mathematical form of a new stochastic model for the Lagrangian dynamics, consisting of an OU model combined with long-lived close same-sign vortices engaged in rapid rotation and long-lived close dipoles engaged in ballistic motion. From a dynamical-systems perspective this work indicates that systems of dimension O(100) can have behavior which is a combination of both low-dimensional behavior, i.e. close pairs, and extremely high-dimensional behavior described by traditional stochastic processes.

chao-dyn↗