## Publications

331 results found

Holm DD, Hu R, Street OD, 2023, Lagrangian reduction and wave mean flow interaction, *Physica D: Nonlinear Phenomena*, Vol: 454, ISSN: 0167-2789

How does one derive models of dynamical feedback effects in multiscale, multiphysics systems such as wave mean flow interaction (WMFI)? We shall address this question for hybrid dynamical systems, defined as systems whose motion can be expressed as the composition of two or more Lie-group actions. Hybrid systems abound in fluid dynamics. Examples include: the dynamics of complex fluids such as liquid crystals; wind-driven waves propagating with the currents moving on the sea surface; turbulence modelling in fluids and plasmas; and classical–quantum hydrodynamic models in molecular chemistry. From among these examples, the motivating question here is: How do wind-driven waves produce ocean surface currents? The paper first summarises the geometric mechanics approach for deriving hybrid models of multiscale, multiphysics motions in ideal fluid dynamics. It then illustrates this approach for WMFI in the examples of 3D WKB waves and 2D wave amplitudes governed by the nonlinear Schrödinger (NLS) equation propagating in the frame of motion of an ideal incompressible inhomogeneous Euler fluid flow. The results for these examples tell us that the mean flow in WMFI does not create waves, although it does transport the waves. However, feedback in the opposite direction is possible, since the 3D WKB and 2D NLS wave dynamics discussed here do in fact create circulatory mean flow.

Crisan D, Holm DD, Korn P, 2023, An implementation of Hasselmann’s paradigm for stochastic climate modelling based on stochastic Lie transport, *Nonlinearity*, Vol: 36, Pages: 4862-4903, ISSN: 0951-7715

A generic approach to stochastic climate modelling is developed for the example of an idealised Atmosphere-Ocean model that rests upon Hasselmann’s paradigm for stochastic climate models. Namely, stochasticity is incorporated into the fast moving atmospheric component of an idealised coupled model by means of stochastic Lie transport, while the slow moving ocean model remains deterministic. More specifically the stochastic model stochastic advection by Lie transport (SALT) is constructed by introducing stochastic transport into the material loop in Kelvin’s circulation theorem. The resulting stochastic model preserves circulation, as does the underlying deterministic climate model. A variant of SALT called Lagrangian-averaged (LA)-SALT is introduced in this paper. In LA-SALT, we replace the drift velocity of the stochastic vector field by its expected value. The remarkable property of LA-SALT is that the evolution of its higher moments are governed by deterministic equations. Our modelling approach is substantiated by establishing local existence results, first, for the deterministic climate model that couples compressible atmospheric equations to incompressible ocean equation, and second, for the two stochastic SALT and LA-SALT models.

Diamantakis T, Holm DD, Pavliotis GA, 2023, Variational principles on geometric rough paths and the Lévy area correction, *SIAM Journal on Applied Dynamical Systems*, Vol: 22, Pages: 1182-1218, ISSN: 1536-0040

In this paper, we describe two effects of the Lévy area correction on the invariant measure of stochastic rigid body dynamics on geometric rough paths. From the viewpoint of dynamics, the Lévy area correction introduces an additional deterministic torque into the rigid body motion equation on geometric rough paths. When the rigid body dynamics is driven by colored noise, and damped by double-bracket dissipation, our theoretical and numerical results show that the additional deterministic torque due to the the Lévy area correction shifts the center of the probability distribution function by shifting the Hamiltonian function in the exponent of the Gibbsian invariant measure.

Holm DD, Pan W, 2023, Deterministic and stochastic Euler-Boussinesq convection, *PHYSICA D-NONLINEAR PHENOMENA*, Vol: 444, ISSN: 0167-2789

Arnaudon A, Holm D, Sommer S, 2023, Stochastic Shape Analysis, Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging: Mathematical Imaging and Vision, Pages: 1325-1348, ISBN: 9783030986605

The chapter describes stochastic models of shapes from a Hamiltonian viewpoint, including Langevin models, Riemannian Brownian motions and stochastic variational systems. Starting from the deterministic setting of outer metrics on shape spaces and transformation groups, we discuss recent approaches to introducing noise in shape analysis from a physical or Hamiltonian point of view. We furthermore outline important applications and statistical uses of stochastic shape models, and we discuss perspectives and current research efforts in stochastic shape analysis.

Crisan D, Holm DD, Lang O,
et al., 2023, Theoretical analysis and numerical approximation for the stochastic thermal quasi-geostrophic model, *Stochastics and Dynamics*, ISSN: 0219-4937

This paper investigates the mathematical properties of a stochastic version of the balanced 2D thermal quasigeostrophic (TQG) model of potential vorticity dynamics. This stochastic TQG model is intended as a basis for parametrization of the dynamical creation of unresolved degrees of freedom in computational simulations of upper ocean dynamics when horizontal buoyancy gradients and bathymetry affect the dynamics, particularly at the submesoscale (250m-10km). Specifically, we have chosen the Stochastic Advection by Lie Transport (SALT) algorithm introduced in [D. D. Holm, Variational principles for stochastic fluid dynamics, Proc. Roy. Soc. A: Math. Phys. Eng. Sci. 471 (2015) 20140963, http://dx.doi.org/10.1098/rspa.2014.0963] and applied in [C. Cotter, D. Crisan, D. Holm, W. Pan and I. Shevchenko, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Found. Data Sci. 2 (2020) 173, https://doi.org/10.3934/fods.2020010; Numerically modeling stochastic lie transport in fluid dynamics, SIAM Multiscale Model. Simul. 17 (2019) 192-232, https://doi.org/10.1137/18M1167929] as our modeling approach. The SALT approach preserves the Kelvin circulation theorem and an infinite family of integral conservation laws for TQG. The goal of the SALT algorithm is to quantify the uncertainty in the process of up-scaling, or coarse-graining of either observed or synthetic data at fine scales, for use in computational simulations at coarser scales. The present work provides a rigorous mathematical analysis of the solution properties of the thermal quasigeostrophic (TQG) equations with SALT [D. D. Holm and E. Luesink, Stochastic wave-current interaction in thermal shallow water dynamics, J. Nonlinear Sci. 31 (2021), https://doi.org/10.1007/s00332-021-09682-9; D. D. Holm, E. Luesink and W. Pan, Stochastic mesoscale circulation dynamics in the thermal ocean, Phys. Fluids 33 (2021) 046603, https://doi.org/10.1063/5.0040026].

Crisan D, Holm DD, Luesink E,
et al., 2023, Theoretical and Computational Analysis of the Thermal Quasi-Geostrophic Model., *J Nonlinear Sci*, Vol: 33, ISSN: 0938-8974

This work involves theoretical and numerical analysis of the thermal quasi-geostrophic (TQG) model of submesoscale geophysical fluid dynamics (GFD). Physically, the TQG model involves thermal geostrophic balance, in which the Rossby number, the Froude number and the stratification parameter are all of the same asymptotic order. The main analytical contribution of this paper is to construct local-in-time unique strong solutions for the TQG model. For this, we show that solutions of its regularised version α-TQG converge to solutions of TQG as its smoothing parameter α→0 and we obtain blow-up criteria for the α-TQG model. The main contribution of the computational analysis is to verify the rate of convergence of α-TQG solutions to TQG solutions as α→0, for example, simulations in appropriate GFD regimes.

Crisan D, Holm DD, Leahy J-M,
et al., 2022, Variational principles for fluid dynamics on rough paths, *arXiv*

In this paper, we introduce a new framework for parametrization schemes (PS)in GFD. Using the theory of controlled rough paths, we derive a class of roughgeophysical fluid dynamics (RGFD) models as critical points of rough actionfunctionals. These RGFD models characterize Lagrangian trajectories in fluiddynamics as geometric rough paths (GRP) on the manifold of diffeomorphic maps.Three constrained variational approaches are formulated for the derivation ofthese models. The first is the Clebsch formulation, in which the constraintsare imposed as rough advection laws. The second is the Hamilton-Pontryaginformulation, in which the constraints are imposed as right-invariant roughvector fields. The third is the Euler--Poincar\'e formulation in which thevariations are constrained. These variational principles lead directly to theLie--Poisson Hamiltonian formulation of fluid dynamics on geometric roughpaths. The GRP framework preserves the geometric structure of fluid dynamicsobtained by using Lie group reduction to pass from Lagrangian to Eulerianvariational principles, thereby yielding a rough formulation of the Kelvincirculation theorem. The rough-path variational approach includes non-Markovianperturbations of the Lagrangian fluid trajectories. In particular, memoryeffects can be introduced through this formulation through a judicious choiceof the rough path (e.g. a realization of a fractional Brownian motion). In thespecial case when the rough path is a realization of a semimartingale, werecover the SGFD models in Holm (2015). However, by eliminating the need forstochastic variational tools, we retain a pathwise interpretation of theLagrangian trajectories. In contrast, the Lagrangian trajectories in thestochastic framework are described by stochastic integrals which do not have apathwise interpretation. Thus, the rough path formulation restores thisproperty.

Crisan D, Holm DD, Leahy J-M,
et al., 2022, Solution properties of the incompressible Euler system with rough path advection, *arXiv*

We consider the Euler equations for the incompressible flow of an ideal fluidwith an additional rough-in-time, divergence-free, Lie-advecting vector field.In recent work, we have demonstrated that this system arises from Clebsch andHamilton-Pontryagin variational principles with a perturbative geometric roughpath Lie-advection constraint. In this paper, we prove local well-posedness ofthe system in $L^2$-Sobolev spaces $H^m$ with integer regularity $m\ge \lfloord/2\rfloor+2$ and establish a Beale-Kato-Majda (BKM) blow-up criterion in termsof the $L^1_tL^\infty_x$-norm of the vorticity. In dimension two, we show thatthe $L^p$-norms of the vorticity are conserved, which yields globalwell-posedness and a Wong-Zakai approximation theorem for the stochasticversion of the equation.

Holm DD, Hu R, 2022, Nonlinear dispersion in wave-current interactions, *Journal of Geometric Mechanics*, Vol: 14, ISSN: 1941-4889

Via a sequence of approximations of the Lagrangian in Hamilton's principlefor dispersive nonlinear gravity waves we derive a hierarchy of Hamiltonianmodels for describing wave-current interaction (WCI) in nonlinear dispersivewave dynamics on free surfaces. A subclass of these WCI Hamiltonians admits\emph{emergent singular solutions} for certain initial conditions. Thesesingular solutions are identified with a singular momentum map for left actionof the diffeomorphisms on a semidirect-product Lie algebra. Thissemidirect-product Lie algebra comprises vector fields representing horizontalcurrent velocity acting on scalar functions representing wave elevation. We usecomputational simulations to demonstrate the dynamical interactions of theemergent wavefront trains which are admitted by this special subclass ofHamiltonians for a variety of initial conditions. In particular, weinvestigate: (1) A variety of localised initial current configurations in stillwater whose subsequent propagation generates surface-elevation dynamics on aninitially flat surface; and (2) The release of initially confinedconfigurations of surface elevation in still water that generate dynamicallyinteracting fronts of localised currents and wave trains. The results of thesesimulations show intricate wave-current interaction patterns whose structuresare similar to those seen, for example, in Synthetic Aperture Radar (SAR)images taken from the space shuttle.

Holm DD, Rawlinson J, Tronci C, 2021, The bohmion method in nonadiabatic quantum hydrodynamics, *JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL*, Vol: 54, ISSN: 1751-8113

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- Citations: 2

Crisan D, Holm DD, Street OD, 2021, Wave-current interaction on a free surface, *STUDIES IN APPLIED MATHEMATICS*, Vol: 147, Pages: 1277-1338, ISSN: 0022-2526

Holm DD, Hu R, 2021, Stochastic effects of waves on currents in the ocean mixed layer, *Journal of Mathematical Physics*, Vol: 62, ISSN: 0022-2488

This paper introduces an energy-preserving stochastic model for studying wave effects on currents in the ocean mixing layer. The model is called stochastic forcing by Lie transport (SFLT). The SFLT model is derived here from a stochastic constrained variational principle, so it has a Kelvin circulation theorem. The examples of SFLT given here treat 3D Euler fluid flow, rotating shallow water dynamics and the Euler-Boussinesq equations. In each example, one sees the effect of stochastic Stokes drift and material entrainment in the generation of fluid circulation. We also present an Eulerian-averaged SFLT model (EA SFLT), based on decomposing the Eulerian solutions of the energy-conserving SFLT model into sums of their expectations and fluctuations.

Bendall TM, Cotter CJ, Holm DD, 2021, Perspectives on the formation of peakons in the stochastic Camassa-Holm equation, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 477, ISSN: 1364-5021

A famous feature of the Camassa–Holm equation is its admission of peaked soliton solutions known as peakons. We investigate this equation under the influence of stochastic transport. Noting that peakons are weak solutions of the equation, we present a finite-element discretization for it, which we use to explore the formation of peakons. Our simulations using this discretization reveal that peakons can still form in the presence of stochastic perturbations. Peakons can emerge both through wave breaking, as the slope turns vertical, and without wave breaking as the inflection points of the velocity profile rise to reach the summit.

Holm D, Luesink E, Pan W, 2021, Stochastic mesoscale circulation dynamics in the thermal ocean, *Physics of Fluids*, Vol: 33, Pages: 1-22, ISSN: 1070-6631

In analogy with similar effects in adiabatic compressible fluid dynamics, the effects of buoyancy gradients on incompressible stratified flows are said to be “thermal.” The thermal rotating shallow water (TRSW) model equations contain three small nondimensional parameters. These are the Rossby number, the Froude number, and the buoyancy parameter. Asymptotic expansion of the TRSW model equations in these three small parameters leads to the deterministic thermal versions of the Salmon’s L1 (TL1) model and the thermal quasi-geostrophic (TQG) model, upon expanding in the neighborhood of thermal quasi-geostrophic balance among the flow velocity and the gradients of free surface elevation and buoyancy. The linear instability of TQG at high wavenumber tends to create circulation at small scales. Such a high- wavenumber instability could be unresolvable in many computational simulations, but its presence at small scales may contribute signifi- cantly to fluid transport at resolvable scales. Sometimes, such effects are modeled via “stochastic backscatter of kinetic energy.” Here, we try another approach. Namely, we model “stochastic transport” in the hierarchy of models TRSW/TL1/TQG. The models are derived via the approach of stochastic advection by Lie transport (SALT) as obtained from a recently introduced stochastic version of the Euler–Poincare var- iational principle. We also indicate the potential next steps for applying these models in uncertainty quantification and data assimilation of the rapid, high-wavenumber effects of buoyancy fronts at these three levels of description by using the data-driven stochastic parametrization algorithms derived previously using the SALT approach.

Holm DD, Luesink E, 2021, Stochastic Wave-Current Interaction in Thermal Shallow Water Dynamics, *JOURNAL OF NONLINEAR SCIENCE*, Vol: 31, ISSN: 0938-8974

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- Citations: 3

Holm DD, 2021, Stochastic Variational Formulations of Fluid Wave-Current Interaction, *JOURNAL OF NONLINEAR SCIENCE*, Vol: 31, ISSN: 0938-8974

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- Citations: 1

Holm DD, Luesink E, 2021, Stochastic Geometric Mechanics with Diffeomorphisms, *Springer Proceedings in Mathematics and Statistics*, Vol: 378, Pages: 169-185, ISSN: 2194-1009

Noether’s celebrated theorem associating symmetry and conservation laws in classical field theory is adapted to allow for broken symmetry in geometric mechanics and is shown to play a central role in deriving and understanding the generation of fluid circulation via the Kelvin-Noether theorem for ideal fluids with stochastic advection by Lie transport (SALT).

Drivas TD, Holm D, 2020, Circulation and energy theorem preserving stochastic fluids, *Proceedings of the Royal Society of Edinburgh: Section A Mathematics*, Vol: 150, Pages: 2776-2814, ISSN: 0308-2105

Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem [1]. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin’s theorem, introduced by Constantin–Iyer (2008) [3]. In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler–Poincare´ and stochastic Navier-Stokes–Poincare´ equations respectively. The stochastic Euler–Poincare equations were previously derived from a stochastic ´ variational principle by Holm (2015) [20], which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.

Cotter C, Crisan D, Holm DD, et al., 2020, Modelling uncertainty using stochastic transport noise in a 2-layer quasi-geostrophic model, Publisher: arXiv

The stochastic variational approach for geophysical fluid dynamics wasintroduced by Holm (Proc Roy Soc A, 2015) as a framework for derivingstochastic parameterisations for unresolved scales. This paper applies thevariational stochastic parameterisation in a two-layer quasi-geostrophic modelfor a beta-plane channel flow configuration. We present a new method forestimating the stochastic forcing (used in the parameterisation) to approximateunresolved components using data from the high resolution deterministicsimulation, and describe a procedure for computing physically-consistentinitial conditions for the stochastic model. We also quantify uncertainty ofcoarse grid simulations relative to the fine grid ones in homogeneous (teamedwith small-scale vortices) and heterogeneous (featuring horizontally elongatedlarge-scale jets) flows, and analyse how the spread of stochastic solutionsdepends on different parameters of the model. The parameterisation is tested bycomparing it with the true eddy-resolving solution that has reached somestatistical equilibrium and the deterministic solution modelled on alow-resolution grid. The results show that the proposed parameterisationsignificantly depends on the resolution of the stochastic model and gives goodensemble performance for both homogeneous and heterogeneous flows, and theparameterisation lays solid foundations for data assimilation.

Geurts BJ, Holm DD, Luesink E, 2020, Lyapunov Exponents of Two Stochastic Lorenz 63 Systems, *JOURNAL OF STATISTICAL PHYSICS*, Vol: 179, Pages: 1343-1365, ISSN: 0022-4715

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- Citations: 5

Alonso-Oran D, de Leon AB, Holm DD,
et al., 2020, Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler-Boussinesq Equation with Transport Noise, *JOURNAL OF STATISTICAL PHYSICS*, Vol: 179, Pages: 1267-1303, ISSN: 0022-4715

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- Citations: 6

Holm D, 2020, Stochastic modelling in fluid dynamics: It\^o vs stratonovich, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 476, Pages: 1-12, ISSN: 1364-5021

Suppose the observations of Lagrangian trajectories for fluid flow in some physical situation can be modelled sufficiently accurately by a spatially correlated Itô stochastic process (with zero mean) obtained from data which is taken in fixed Eulerian space. Suppose we also want to apply Hamilton’s principle to derive the stochastic fluid equations for this situation. Now, the variational calculus for applying Hamilton’s principle requires the Stratonovich process, so we must transform from Itô noise in the data frame to the equivalent Stratonovich noise. However, the transformation from the Itô process in the data frame to the corresponding Stratonovich process shifts the drift velocity of the transformed Lagrangian fluid trajectory out of the data frame into a non-inertial frame obtained from the Itô correction. The issue is, ‘Will non-inertial forces arising from this transformation of reference frames make a difference in the interpretation of the solution behaviour of the resulting stochastic equations?’ This issue will be resolved by elementary considerations.

Holm DD, 2020, Stochastic modelling in fluid dynamics: Ito versus Stratonovich, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 476, ISSN: 1364-5021

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- Citations: 2

Gay-Balmaz F, Holm DD, 2020, PREDICTING UNCERTAINTY IN GEOMETRIC FLUID MECHANICS, *DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S*, Vol: 13, Pages: 1229-1242, ISSN: 1937-1632

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- Citations: 3

Cotter C, Crisan D, Holm D, et al., 2020, Data Assimilation for a Quasi-Geostrophic Model with Circulation-Preserving Stochastic Transport Noise, Publisher: SPRINGER

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- Citations: 14

de Leon AB, Holm DD, Luesink E, et al., 2020, Implications of Kunita-Ito-Wentzell Formula for k-Forms in Stochastic Fluid Dynamics, Publisher: SPRINGER

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- Citations: 8

Drivas TD, Holm DD, Leahy J-M, 2020, Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids, *JOURNAL OF STATISTICAL PHYSICS*, Vol: 179, Pages: 1304-1342, ISSN: 0022-4715

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- Citations: 8

Holm DD, Naraigh LO, Tronci C, 2020, A geometric diffuse-interface method for droplet spreading, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 476, ISSN: 1364-5021

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- Citations: 1

Holm DD, 2019, Stochastic Closures for Wave-Current Interaction Dynamics, Publisher: SPRINGER

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- Citations: 8

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