Theoretical and Computational Analysis of the Thermal Quasi-Geostrophic Model.

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.

Stochastic Variational Formulations of Fluid Wave-Current Interaction.

We are modelling multiscale, multi-physics uncertainty in wave-current interaction (WCI). To model uncertainty in WCI, we introduce stochasticity into the wave dynamics of two classic models of WCI, namely the generalised Lagrangian mean (GLM) model and the Craik-Leibovich (CL) model. The key idea for the GLM approach is the separation of the Lagrangian (fluid) and Eulerian (wave) degrees of freedom in Hamilton's principle.

Stochastic modelling in fluid dynamics: Itô versus Stratonovich.

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 to the equivalent Stratonovich noise.

A geometric diffuse-interface method for droplet spreading.

This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation valid in the case of large-scale droplet spreading-the geometric diffuse-interface method. The method possesses some advantages when compared with the existing models of droplet spreading, namely the slip model, the precursor-film method and the diffuse-interface model. These advantages are discussed and a case is made for using the geometric diffuse-interface method for the purpose of numerical simulations.

New variational and multisymplectic formulations of the Euler-Poincaré equation on the Virasoro-Bott group using the inverse map.

We derive a new variational principle, leading to a new momentum map and a new multisymplectic formulation for a family of Euler-Poincaré equations defined on the Virasoro-Bott group, by using the inverse map (also called 'back-to-labels' map). This family contains as special cases the well-known Korteweg-de Vries, Camassa-Holm and Hunter-Saxton soliton equations. In the conclusion section, we sketch opportunities for future work that would apply the new Clebsch momentum map with 2-cocycles derived here to investigate a new type of interplay among nonlinearity, dispersion and noise.

Stochastic Geometric Models with Non-stationary Spatial Correlations in Lagrangian Fluid Flows.

Inspired by spatiotemporal observations from satellites of the trajectories of objects drifting near the surface of the ocean in the National Oceanic and Atmospheric Administration's "Global Drifter Program", this paper develops data-driven stochastic models of geophysical fluid dynamics (GFD) with non-stationary spatial correlations representing the dynamical behaviour of oceanic currents. Three models are considered. Model 1 from Holm (Proc R Soc A 471:20140963, 2015) is reviewed, in which the spatial correlations are time independent.

Dynamics of non-holonomic systems with stochastic transport.

This paper formulates a variational approach for treating observational uncertainty and/or computational model errors as stochastic transport in dynamical systems governed by action principles under non-holonomic constraints. For this purpose, we derive, analyse and numerically study the example of an unbalanced spherical ball rolling under gravity along a stochastic path. Our approach uses the Hamilton-Pontryagin variational principle, constrained by a stochastic rolling condition, which we show is equivalent to the corresponding stochastic Lagrange-d'Alembert principle.

Noise and Dissipation on Coadjoint Orbits.

We derive and study stochastic dissipative dynamics on coadjoint orbits by incorporating noise and dissipation into mechanical systems arising from the theory of reduction by symmetry, including a semidirect product extension. Random attractors are found for this general class of systems when the Lie algebra is semi-simple, provided the top Lyapunov exponent is positive. We study in details two canonical examples, the free rigid body and the heavy top, whose stochastic integrable reductions are found and numerical simulations of their random attractors are shown.

Un-reduction in field theory.

The un-reduction procedure introduced previously in the context of classical mechanics is extended to covariant field theory. The new covariant un-reduction procedure is applied to the problem of shape matching of images which depend on more than one independent variable (for instance, time and an additional labelling parameter). Other possibilities are also explored: nonlinear [Formula: see text]-models and the hyperbolic flows of curves.

Stochastic discrete Hamiltonian variational integrators.

Variational integrators are derived for structure-preserving simulation of stochastic Hamiltonian systems with a certain type of multiplicative noise arising in geometric mechanics. The derivation is based on a stochastic discrete Hamiltonian which approximates a type-II stochastic generating function for the stochastic flow of the Hamiltonian system. The generating function is obtained by introducing an appropriate stochastic action functional and its corresponding variational principle.

Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics.

In Holm (Holm 2015 , 20140963. (doi:10.1098/rspa.

Multipole Vortex Blobs (MVB): Symplectic Geometry and Dynamics.

Vortex blob methods are typically characterized by a regularization length scale, below which the dynamics are trivial for isolated blobs. In this article, we observe that the dynamics need not be trivial if one is willing to consider distributional derivatives of Dirac delta functionals as valid vorticity distributions. More specifically, a new singular vortex theory is presented for regularized Euler fluid equations of ideal incompressible flow in the plane.

-Strands on symmetric spaces.

We study the -strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group , and we treat in more detail examples with symmetric space (2)/ and (4)/(3).

Variational principles for stochastic soliton dynamics.

We develop a variational method of deriving stochastic partial differential equations whose solutions follow the flow of a stochastic vector field. As an example in one spatial dimension, we numerically simulate singular solutions (peakons) of the stochastically perturbed Camassa-Holm (CH) equation derived using this method. These numerical simulations show that peakon soliton solutions of the stochastically perturbed CH equation persist and provide an interesting laboratory for investigating the sensitivity and accuracy of adding stochasticity to finite dimensional solutions of stochastic partial differential equations.

Variational principles for stochastic fluid dynamics.

This paper derives stochastic partial differential equations (SPDEs) for fluid dynamics from a stochastic variational principle (SVP). The paper proceeds by taking variations in the SVP to derive stochastic Stratonovich fluid equations; writing their Itô representation; and then investigating the properties of these stochastic fluid models in comparison with each other, and with the corresponding deterministic fluid models. The circulation properties of the stochastic Stratonovich fluid equations are found to closely mimic those of the deterministic ideal fluid models.

Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework.

We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry.

Quasiconservation laws for compressible three-dimensional Navier-Stokes flow.

We formulate the quasi-Lagrangian fluid transport dynamics of mass density ρ and the projection q=ω·∇ρ of the vorticity ω onto the density gradient, as determined by the three-dimensional compressible Navier-Stokes equations for an ideal gas, although the results apply for an arbitrary equation of state. It turns out that the quasi-Lagrangian transport of q cannot cross a level set of ρ. That is, in this formulation, level sets of ρ (isopycnals) are impermeable to the transport of the projection q.

Quantum splines.

A quantum spline is a smooth curve parametrized by time in the space of unitary transformations, whose associated orbit on the space of pure states traverses a designated set of quantum states at designated times, such that the trace norm of the time rate of change of the associated Hamiltonian is minimized. The solution to the quantum spline problem is obtained, and is applied in an example that illustrates quantum control of coherent states. An efficient numerical scheme for computing quantum splines is discussed and implemented in the examples.

Simultaneous multi-scale registration using large deformation diffeomorphic metric mapping.

In the framework of large deformation diffeomorphic metric mapping (LDDMM), we present a practical methodology to integrate prior knowledge about the registered shapes in the regularizing metric. Our goal is to perform rich anatomical shape comparisons from volumetric images with the mathematical properties offered by the LDDMM framework. We first present the notion of characteristic scale at which image features are deformed.

Extreme events in solutions of hydrostatic and non-hydrostatic climate models.

Initially, this paper reviews the mathematical issues surrounding hydrostatic primitive equations (HPEs) and non-hydrostatic primitive equations (NPEs) that have been used extensively in numerical weather prediction and climate modelling. A new impetus has been provided by a recent proof of the existence and uniqueness of solutions of viscous HPEs on a cylinder with Neumann-like boundary conditions on the top and bottom. In contrast, the regularity of solutions of NPEs remains an open question.

Simultaneous fine and coarse diffeomorphic registration: application to atrophy measurement in Alzheimer's disease.

In this paper, we present a fine and coarse approach for the multiscale registration of 3D medical images using Large Deformation Diffeomorphic Metric Mapping (LDDMM). This approach has particularly interesting properties since it estimates large, smooth and invertible optimal deformations having a rich descriptive power for the quantification of temporal changes in the images. First, we show the importance of the smoothing kernel and its influence on the final solution.

Singular solutions of a modified two-component Camassa-Holm equation.

The Camassa-Holm (CH) equation is a well-known integrable equation describing the velocity dynamics of shallow water waves. This equation exhibits spontaneous emergence of singular solutions (peakons) from smooth initial conditions. The CH equation has been recently extended to a two-component integrable system (CH2), which includes both velocity and density variables in the dynamics.

Emergent singular solutions of nonlocal density-magnetization equations in one dimension.

We investigate the emergence of singular solutions in a nonlocal model for a magnetic system. We study a modified Gilbert-type equation for the magnetization vector and find that the evolution depends strongly on the length scales of the nonlocal effects. We pass to a coupled density-magnetization model and perform a linear stability analysis, noting the effect of the length scales of nonlocality on the system's stability properties.

Highly turbulent solutions of the Lagrangian-averaged Navier-Stokes alpha model and their large-eddy-simulation potential.

We compute solutions of the Lagrangian-averaged Navier-Stokes alpha - (LANS alpha ) model for significantly higher Reynolds numbers (up to Re approximately 8300 ) than have previously been accomplished. This allows sufficient separation of scales to observe a Navier-Stokes inertial range followed by a second inertial range specific to the LANS alpha model. Both fully helical and nonhelical flows are examined, up to Reynolds numbers of approximately 1300.

Aggregation of finite-size particles with variable mobility.

New model equations are derived for dynamics of aggregation of finite-size particles. The differences from standard Debye-Hückel and Keller-Segel models are that the mobility of particles depends on the configuration of their neighbors and linear diffusion acts on locally averaged particle density. The evolution of collapsed states in these models reduces exactly to finite-dimensional dynamics of interacting particle clumps.