Rapidly rotating plane layer convection with zonal flow

The onset of convection in a rapidly rotating layer in which a thermal wind is present is studied. Diffusive effects are included. The main motivation is from convection in planetary interiors, where thermal winds are expected due to temperature variations on the core-mantle boundary. The system admits both convective instability and baroclinic instability. We find a smooth transition between the two types of modes, and investigate where the transition region between the two types of instability occurs in parameter space. The thermal wind helps to destabilise the convective modes. Baroclinic instability can occur when the applied vertical temperature gradient is stable, and the critical Rayleigh number is then negative. Long wavelength modes are the first to become unstable. Asymptotic analysis is possible for the transition region and also for long wavelength instabilities, and the results agree well with our numerical solutions. We also investigate how the instabilities in this system relate to the classical baroclinic instability in the Eady problem. We conclude by noting that baroclinic instabilities in the Earth's core arising from heterogeneity in the lower mantle could possibly drive a dynamo even if the Earth's core were stably stratified and so not convecting.Comment: 20 pages, 7 figure

Grassmannian flows and applications to nonlinear partial differential equations

We show how solutions to a large class of partial differential equations with nonlocal Riccati-type nonlinearities can be generated from the corresponding linearized equations, from arbitrary initial data. It is well known that evolutionary matrix Riccati equations can be generated by projecting linear evolutionary flows on a Stiefel manifold onto a coordinate chart of the underlying Grassmann manifold. Our method relies on extending this idea to the infinite dimensional case. The key is an integral equation analogous to the Marchenko equation in integrable systems, that represents the coodinate chart map. We show explicitly how to generate such solutions to scalar partial differential equations of arbitrary order with nonlocal quadratic nonlinearities using our approach. We provide numerical simulations that demonstrate the generation of solutions to Fisher--Kolmogorov--Petrovskii--Piskunov equations with nonlocal nonlinearities. We also indicate how the method might extend to more general classes of nonlinear partial differential systems.Comment: 26 pages, 2 figure

Lax pair and first integrals for two of nonlinear coupled oscillators

The system of two nonlinear coupled oscillators is studied. As partial case this system of equation is reduced to the Duffing oscillator which has many applications for describing physical processes. It is well known that the inverse scattering transform is one of the most powerful methods for solving the Cauchy problems of partial differential equations. To solve the Cauchy problem for nonlinear differential equations we can use the Lax pair corresponding to this equation. The Lax pair for ordinary differential or systems or for system ordinary differential equations allows us to find the first integrals, which also allow us to solve the question of integrability for differential equations. In this report we present the Lax pair for the system of coupled oscillators. Using the Lax pair we get two first integrals for the system of equations. The considered system of equations can be also reduced to the fourth-order ordinary differential equation and the Lax pair can be used for the ordinary differential equation of fourth order. Some special cases of the system of equations are considered.Comment: 9 page

Trace Formulas for Schroedinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds

We investigate trace formulas for one-dimensional Schroedinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg-de Vries hierarchy in this class and relate them to the reflection coefficients via Abelian integrals on the underlying hyperelliptic Riemann surface.Comment: 14 page

Memory-encoding vibrations in a disconnecting air bubble

Many nonlinear processes, such as the propagation of waves over an ocean or the transmission of light pulses down an optical fibre1, are integrable in the sense that the dynamics has as many conserved quantities as there are independent variables. The result is a time evolution that retains a complete memory of the initial state. In contrast, the nonlinear dynamics near a finite-time singularity, in which physical quantities such as pressure or velocity diverge at a point in time, is believed to evolve towards a universal form, one independent of the initial state2. The break-up of a water drop in air3 or a viscous liquid inside an immiscible oil4,5 are processes that conform to this second scenario. These opposing scenarios collide in the nonlinearity produced by the formation of a finite-time singularity that is also integrable. We demonstrate here that the result is a novel dynamics with a dual character

From bore-soliton-splash to a new wave-to-wire wave-energy model

We explore extreme nonlinear water-wave amplification in a contraction or, analogously, wave amplification in crossing seas. The latter case can lead to extreme or rogue-wave formation at sea. First, amplification of a solitary-water-wave compound running into a contraction is disseminated experimentally in a wave tank. Maximum amplification in our bore–soliton–splash observed is circa tenfold. Subsequently, we summarise some nonlinear and numerical modelling approaches, validated for amplifying, contracting waves. These amplification phenomena observed have led us to develop a novel wave-energy device with wave amplification in a contraction used to enhance wave-activated buoy motion and magnetically induced energy generation. An experimental proof-of-principle shows that our wave-energy device works. Most importantly, we develop a novel wave-to-wire mathematical model of the combined wave hydrodynamics, wave-activated buoy motion and electric power generation by magnetic induction, from first principles, satisfying one grand variational principle in its conservative limit. Wave and buoy dynamics are coupled via a Lagrange multiplier, which boundary value at the waterline is in a subtle way solved explicitly by imposing incompressibility in a weak sense. Dissipative features, such as electrical wire resistance and nonlinear LED loads, are added a posteriori. New is also the intricate and compatible finite-element space–time discretisation of the linearised dynamics, guaranteeing numerical stability and the correct energy transfer between the three subsystems. Preliminary simulations of our simplified and linearised wave-energy model are encouraging and involve a first study of the resonant behaviour and parameter dependence of the device

Kelvin-Helmholtz instability in the presence of variable viscosity for mudflow resuspension in estuaries

The temporal stability of a parallel shear flow of miscible fluid layers of dif- ferent density and viscosity is investigated through a linear stability analysis and direct numerical simulations. The geometry and rheology of this Newto- nian fluid mixing can be viewed as a simplified model of the behavior of mud- flow at the bottom of estuaries for suspension studies. In this study, focus is on the stability and transition to turbulence of an initially laminar configuration. A parametric analysis is performed by varying the values of three control pa- rameters, namely the viscosity ratio, the Richardson and Reynolds numbers, in the case of initially identical thickness of the velocity, density and viscosity profiles. The range of parameters has been chosen so as to mimic a wide variety of real configurations. This study shows that the Kelvin-Helmholtz instability is controlled by the local Reynolds and Richardson numbers of the inflection point. In addition, at moderate Reynolds number, viscosity strat- ification has a strong influence on the onset of instability, the latter being enhanced at high viscosity ratio, while at high Reynolds number, the influ- ence is less pronounced. In all cases, we show that the thickness of the mixing layer (and thus resuspension) is increased by high viscosity stratification, in particular during the non-linear development of the instability and especially pairing processes. This study suggests that mud viscosity has to be taken into account for resuspension parameterizations because of its impact on the inflec- tion point Reynolds number and the viscosity ratio, which are key parameters for shear instabilities