## Publications

340 results found

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: 4

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: 4

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: 1

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

Foskett MS, Holm DD, Tronci C, 2019, Geometry of nonadiabatic quantum hydrodynamics, *Acta Applicandae Mathematicae*, Vol: 162, Pages: 63-103, ISSN: 0167-8019

The Hamiltonian action of a Lie group on a symplectic manifold induces a momentum map generalizing Noether’s conserved quantity occurring in the case of a symmetry group. Then, when a Hamiltonian function can be written in terms of this momentum map, the Hamiltonian is called ‘collective’. Here, we derive collective Hamiltonians for a series of models in quantum molecular dynamics for which the Lie group is the composition of smooth invertible maps and unitary transformations. In this process, different fluid descriptions emerge from different factorization schemes for either the wavefunction or the density operator. After deriving this series of quantum fluid models, we regularize their Hamiltonians for finite ℏ by introducing local spatial smoothing. In the case of standard quantum hydrodynamics, the ℏ≠0 dynamics of the Lagrangian path can be derived as a finite-dimensional canonical Hamiltonian system for the evolution of singular solutions called ‘Bohmions’, which follow Bohmian trajectories in configuration space. For molecular dynamics models, application of the smoothing process to a new factorization of the density operator leads to a finite-dimensional Hamiltonian system for the interaction of multiple (nuclear) Bohmions and a sequence of electronic quantum states.

Cotter C, Crisan D, Holm DD, et al., 2019, A Particle Filter for Stochastic Advection by Lie Transport (SALT): A case study for the damped and forced incompressible 2D Euler equation, Publisher: arXiv

In this work, we apply a particle filter with three additional procedures(model reduction, tempering and jittering) to a damped and forcedincompressible 2D Euler dynamics defined on a simply connected bounded domain.We show that using the combined algorithm, we are able to successfullyassimilate data from a reference system state (the ``truth") modelled by ahighly resolved numerical solution of the flow that has roughly $3.1\times10^6$degrees of freedom for $10$ eddy turnover times, using modest computationalhardware. The model reduction is performed through the introduction of a stochasticadvection by Lie transport (SALT) model as the signal on a coarser resolution.The SALT approach was introduced as a general theory using a geometricmechanics framework from Holm, Proc. Roy. Soc. A (2015). This work follows onthe numerical implementation for SALT presented by Cotter et al, SIAMMultiscale Model. Sim. (2019) for the flow in consideration. The modelreduction is substantial: The reduced SALT model has $4.9\times 10^4$ degreesof freedom. Forecast reliability and estimated asymptotic behaviour of the particlefilter are also presented.

Crisan D, Flandoli F, Holm DD, 2019, Solution properties of a 3D stochastic euler fluid equation, *Journal of Nonlinear Science*, Vol: 29, Pages: 813-870, ISSN: 0938-8974

We prove local well-posedness in regular spaces and a Beale–Kato–Majda blow-up criterion for a recently derived stochastic model of the 3D Euler fluid equation for incompressible flow. This model describes incompressible fluid motions whose Lagrangian particle paths follow a stochastic process with cylindrical noise and also satisfy Newton’s second law in every Lagrangian domain.

Holm D, 2019, Stochastic evolution of augmented Born–Infeld equations, *Journal of Nonlinear Science*, Vol: 29, Pages: 115-138, ISSN: 0938-8974

This paper compares the results of applying a recently developed method of stochastic uncertainty quantification designed for fluid dynamics to the Born–Infeld model of nonlinear electromagnetism. The similarities in the results are striking. Namely, the introduction of Stratonovich cylindrical noise into each of their Hamiltonian formulations introduces stochastic Lie transport into their dynamics in the same form for both theories. Moreover, the resulting stochastic partial differential equations retain their unperturbed form, except for an additional term representing induced Lie transport by the set of divergence-free vector fields associated with the spatial correlations of the cylindrical noise. The explanation for this remarkable similarity lies in the method of construction of the Hamiltonian for the Stratonovich stochastic contribution to the motion in both cases, which is done via pairing spatial correlation eigenvectors for cylindrical noise with the momentum map for the deterministic motion. This momentum map is responsible for the well-known analogy between hydrodynamics and electromagnetism. The momentum map for the Maxwell and Born–Infeld theories of electromagnetism treated here is the 1-form density known as the Poynting vector. Two appendices treat the Hamiltonian structures underlying these results.

Holm DD, 2019, Stochastic parametrization of the Richardson triple, *Journal of Nonlinear Science*, Vol: 29, Pages: 89-113, ISSN: 0938-8974

A Richardson triple is an ideal fluid flow map (Formula presented.) composed of three smooth maps with separated time scales: slow, intermediate and fast, corresponding to the big, little and lesser whorls in Richardson’s well-known metaphor for turbulence. Under homogenization, as (Formula presented.), the composition (Formula presented.) of the fast flow and the intermediate flow is known to be describable as a single stochastic flow (Formula presented.). The interaction of the homogenized stochastic flow (Formula presented.) with the slow flow of the big whorl is obtained by going into its non-inertial moving reference frame, via the composition of maps (Formula presented.). This procedure parameterizes the interactions of the three flow components of the Richardson triple as a single stochastic fluid flow in a moving reference frame. The Kelvin circulation theorem for the stochastic dynamics of the Richardson triple reveals the interactions among its three components. Namely, (1) the velocity in the circulation integrand is kinematically swept by the large scales and (2) the velocity of the material circulation loop acquires additional stochastic Lie transport by the small scales. The stochastic dynamics of the composite homogenized flow is derived from a stochastic Hamilton’s principle and then recast into Lie–Poisson bracket form with a stochastic Hamiltonian. Several examples are given, including fluid flow with stochastically advected quantities and rigid body motion under gravity, i.e. the stochastic heavy top in a rotating frame.

Cotter CJ, Crisan D, Holm DD,
et al., 2019, Numerically modelling stochastic lie transport in fluid dynamics, *SIAM Journal on Scientific Computing*, Vol: 17, Pages: 192-232, ISSN: 1064-8275

We present a numerical investigation of stochastic transport in ideal fluids.According to Holm (Proc Roy Soc, 2015) and Cotter et al. (2017), the principlesof transformation theory and multi-time homogenisation, respectively, imply aphysically meaningful, data-driven approach for decomposing the fluid transportvelocity into its drift and stochastic parts, for a certain class of fluidflows. In the current paper, we develop new methodology to implement thisvelocity decomposition and then numerically integrate the resulting stochasticpartial differential equation using a finite element discretisation forincompressible 2D Euler fluid flows. The new methodology tested here is foundto be suitable for coarse graining in this case. Specifically, we performuncertainty quantification tests of the velocity decomposition of Cotter et al.(2017), by comparing ensembles of coarse-grid realisations of solutions of theresulting stochastic partial differential equation with the "true solutions" ofthe deterministic fluid partial differential equation, computed on a refinedgrid. The time discretization used for approximating the solution of thestochastic partial differential equation is shown to be consistent. We includecomprehensive numerical tests that confirm the non-Gaussianity of the streamfunction, velocity and vorticity fields in the case of incompressible 2D Eulerfluid flows.

Arnaudon A, Holm DD, Sommer S, 2019, Stochastic Metamorphosis with Template Uncertainties

In this paper, we investigate two stochastic perturbations of the metamorphosis equations of image analysis, in the geometrical context of the Euler-Poincaré theory. In the metamorphosis of images, the Lie group of diffeomorphisms deforms a template image that is undergoing its own internal dynamics as it deforms. This type of deformation allows more freedom for image matching and has analogies with complex uids when the template properties are regarded as order parameters. The first stochastic perturbation we consider corresponds to uncertainty due to random errors in the reconstruction of the deformation map from its vector field. We also consider a second stochastic perturbation, which compounds the uncertainty of the deformation map with the uncertainty in the reconstruction of the template position from its velocity field. We apply this general geometric theory to several classical examples, including landmarks, images, and closed curves, and we discuss its use for functional data analysis.

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