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. 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.

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. In the example of the rolling ball, the stochasticity represents uncertainty in the observation and/or error in the computational simulation of the angular velocity of rolling. The influence of the stochasticity on the deterministically conserved quantities is investigated both analytically and numerically. Our approach applies to a wide variety of stochastic, non-holonomically constrained systems, because it preserves the mathematical properties inherited from the variational principle.

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

In Holm (Holm 2015 Proc. R. Soc. A471, 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.

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.

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. With this HPE regularity result in mind, the second issue examined in this paper is whether extreme events are allowed to arise spontaneously in their solutions. Such events could include, for example, the sudden appearance and disappearance of locally intense fronts that do not involve deep convection. Analytical methods are used to show that for viscous HPEs, the creation of small-scale structures is allowed locally in space and time at sizes that scale inversely with the Reynolds number.

CO2 molecule as a quantum realization of the 1:1:2 resonant swing-spring with monodromy.

We consider the wide class of systems modeled by an integrable approximation to the 3 degrees of freedom elastic pendulum with 1:1:2 resonance, or the swing-spring. This approximation has monodromy which prohibits the existence of global action-angle variables and complicates the dynamics. We study the quantum swing-spring formed by bending and symmetric stretching vibrations of the CO2 molecule. We uncover quantum monodromy of CO2 as a nontrivial codimension 2 defect of the three dimensional energy-momentum lattice of its quantum states.

Transient vortex events in the initial value problem for turbulence.

A vorticity surge event that could be a paradigm for a wide class of bursting events in turbulence is studied. The coherent mechanism is characterized by locally transverse vortex configurations that are intrinsically helical in both physical and Fourier space when there is a peak of the maximum vorticity parallel omega parallel(infinity)(t). At no time are nonhelical, antiparallel vorticity elements observed. This event precedes the appearance of the traditional signatures of an energy cascade such as strong growth of the dissipation, spectra approaching -5/3, and strongly Beltramized vortex tubes. Comparing how different large-eddy simulations reproduce these properties demonstrates the importance of properly modeling nonlinear transport of both energy and helicity.

An integrable shallow water equation with linear and nonlinear dispersion.

We use asymptotic analysis and a near-identity normal form transformation from water wave theory to derive a 1+1 unidirectional nonlinear wave equation that combines the linear dispersion of the Korteweg-deVries (KdV) equation with the nonlinear/nonlocal dispersion of the Camassa-Holm (CH) equation. This equation is one order more accurate in asymptotic approximation beyond KdV, yet it still preserves complete integrability via the inverse scattering transform method. Its traveling wave solutions contain both the KdV solitons and the CH peakons as limiting cases.