Nonlinear stability and the saturation of instabilities to axisymmetric vortices

Nonlinear stability theorems are presented for axisymmetric vortices under the restriction that the disturbance is independent of either the azimuthal or the axial coordinate. These stability theorems are then used, in both cases, to derive rigorous upper bounds on the saturation amplitudes of instabilities. Explicit examples of such bounds are worked out for some canonical profiles. The results establish a minimum order for the dependence of saturation amplitude on supercriticality, and are thereby suggestive as to the nature of the bifurcation at the stability threshold.

Lower-stratospheric radiative damping and polar-night jet oscillation events

The effect of stratospheric radiative damping time scales on stratospheric variability and on stratosphere–troposphere coupling is investigated in a simplified global circulation model by modifying the vertical profile of radiative damping in the stratosphere while holding it fixed in the troposphere. Perpetual-January conditions are imposed, with sinusoidal topography of zonal wavenumber 1 or 2. The depth and duration of the simulated sudden stratospheric warmings closely track the lower-stratospheric radiative time scales. Simulations with the most realistic profiles of radiative damping exhibit extended time-scale recoveries analogous to polar-night jet oscillation (PJO) events, which are observed to follow sufficiently deep stratospheric warmings. These events are characterized by weak lower-stratospheric winds and enhanced stability near the tropopause, which persist for up to 3 months following the initial warming. They are obtained with both wave-1 and wave-2 topography. Planetary-scale Eliassen–Palm (EP) fluxes entering the vortex are also suppressed, which is in agreement with observed PJO events. Consistent with previous studies, the tropospheric jets shift equatorward in response to the warmings. The duration of the shift is closely correlated with the period of enhanced stability. The magnitude of the shift in these runs, however, is sensitive only to the zonal wavenumber of the topography. Although the shift is sustained primarily by synoptic-scale eddies, the net effect of the topographic form drag and the planetary-scale fluxes is not negligible; they damp the surface wind response but enhance the vertical shear. The tropospheric response may also reduce the generation of planetary waves, further extending the stratospheric dynamical time scales.

The effect of non-uniform radiative damping on the zonal-mean dynamics of the extratropical middle atmosphere

The effect of spatial and temporal variations in the radiative damping rate on the response to an imposed forcing or diabatic heating is examined in a zonal-mean model of the middle atmosphere. Attention is restricted to the extratropics, where a linear approach is viable. It is found that regions with weak radiative damping rates are more sensitive in terms of temperature to the remote influence of the diabatic circulation. The delay in the response in such regions can mean that ‘downward’ control is not achieved on seasonal time-scales. A seasonal variation in the radiative damping rate modulates the evolution of the response and leaves a transient-like signature in the annual mean temperature field. Several idealized examples are considered, motivated by topical questions. It is found that wave drag outside the polar vortex can significantly affect the temperatures in its interior, so that high-latitude, high-altitude gravity-wave drag is not the only mechanism for warming the southern hemisphere polar vortex. Diabatic mass transport through the 100 hPa surface is found to lag the seasonal evolution of the wave drag that drives the transport, and thus cannot be considered to be in the downward control regime. On the other hand, the seasonal variation of the radiative damping rate is found to make only a weak contribution to the annual mean temperature increase that has been observed above the ozone hole. Copyright © 2002 Royal Meteorological Society.

First order k-th moment finite element analysis of nonlinear operator equations with stochastic data

We develop and analyze a class of efficient Galerkin approximation methods for uncertainty quantification of nonlinear operator equations. The algorithms are based on sparse Galerkin discretizations of tensorized linearizations at nominal parameters. Specifically, we consider abstract, nonlinear, parametric operator equations J(\alpha ,u)=0 for random input \alpha (\omega ) with almost sure realizations in a neighborhood of a nominal input parameter \alpha _0. Under some structural assumptions on the parameter dependence, we prove existence and uniqueness of a random solution, u(\omega ) = S(\alpha (\omega )). We derive a multilinear, tensorized operator equation for the deterministic computation of k-th order statistical moments of the random solution's fluctuations u(\omega ) - S(\alpha _0). We introduce and analyse sparse tensor Galerkin discretization schemes for the efficient, deterministic computation of the k-th statistical moment equation. We prove a shift theorem for the k-point correlation equation in anisotropic smoothness scales and deduce that sparse tensor Galerkin discretizations of this equation converge in accuracy vs. complexity which equals, up to logarithmic terms, that of the Galerkin discretization of a single instance of the mean field problem. We illustrate the abstract theory for nonstationary diffusion problems in random domains.

A finite element method for nonlinear elliptic problems

We present a Galerkin method with piecewise polynomial continuous elements for fully nonlinear elliptic equations. A key tool is the discretization proposed in Lakkis and Pryer, 2011, allowing us to work directly on the strong form of a linear PDE. An added benefit to making use of this discretization method is that a recovered (finite element) Hessian is a byproduct of the solution process. We build on the linear method and ultimately construct two different methodologies for the solution of second order fully nonlinear PDEs. Benchmark numerical results illustrate the convergence properties of the scheme for some test problems as well as the Monge–Amp`ere equation and the Pucci equation.

Available potential energy density for a multicomponent Boussinesq fluid with arbitrary nonlinear equation of state

In this paper, the concept of available potential energy (APE) density is extended to a multicomponent Boussinesq fluid with a nonlinear equation of state. As shown by previous studies, the APE density is naturally interpreted as the work against buoyancy forces that a parcel needs to perform to move from a notional reference position at which its buoyancy vanishes to its actual position; because buoyancy can be defined relative to an arbitrary reference state, so can APE density. The concept of APE density is therefore best viewed as defining a class of locally defined energy quantities, each tied to a different reference state, rather than as a single energy variable. An important result, for which a new proof is given, is that the volume integrated APE density always exceeds Lorenz’s globally defined APE, except when the reference state coincides with Lorenz’s adiabatically re-arranged reference state of minimum potential energy. A parcel reference position is systematically defined as a level of neutral buoyancy (LNB): depending on the nature of the fluid and on how the reference state is defined, a parcel may have one, none, or multiple LNB within the fluid. Multiple LNB are only possible for a multicomponent fluid whose density depends on pressure. When no LNB exists within the fluid, a parcel reference position is assigned at the minimum or maximum geopotential height. The class of APE densities thus defined admits local and global balance equations, which all exhibit a conversion with kinetic energy, a production term by boundary buoyancy fluxes, and a dissipation term by internal diffusive effects. Different reference states alter the partition between APE production and dissipation, but neither affect the net conversion between kinetic energy and APE, nor the difference between APE production and dissipation. We argue that the possibility of constructing APE-like budgets based on reference states other than Lorenz’s reference state is more important than has been previously assumed, and we illustrate the feasibility of doing so in the context of an idealised and realistic oceanic example, using as reference states one with constant density and another one defined as the horizontal mean density field; in the latter case, the resulting APE density is found to be a reasonable approximation of the APE density constructed from Lorenz’s reference state, while being computationally cheaper.

Efficient fully nonlinear data assimilation for geophysical fluid dynamics

A potential problem with Ensemble Kalman Filter is the implicit Gaussian assumption at analysis times. Here we explore the performance of a recently proposed fully nonlinear particle filter on a high-dimensional but simplified ocean model, in which the Gaussian assumption is not made. The model simulates the evolution of the vorticity field in time, described by the barotropic vorticity equation, in a highly nonlinear flow regime. While common knowledge is that particle filters are inefficient and need large numbers of model runs to avoid degeneracy, the newly developed particle filter needs only of the order of 10-100 particles on large scale problems. The crucial new ingredient is that the proposal density cannot only be used to ensure all particles end up in high-probability regions of state space as defined by the observations, but also to ensure that most of the particles have similar weights. Using identical twin experiments we found that the ensemble mean follows the truth reliably, and the difference from the truth is captured by the ensemble spread. A rank histogram is used to show that the truth run is indistinguishable from any of the particles, showing statistical consistency of the method.

Chaotic destruction of Anderson localization in a nonlinear lattice

We consider a scattering problem for a nonlinear disordered lattice layer governed by the discrete nonlinear Schrodinger equation. The linear state with exponentially small transparency, due to the Anderson localization, is followed for an increasing nonlinearity, until it is destroyed via a bifurcation. The critical nonlinearity is shown to decay with the lattice length as a power law. We demonstrate that in the chaotic regimes beyond the bifurcation the field is delocalized and this leads to a drastic increase of transparency. Copyright (C) EPLA, 2008

A nonlinear oscillator describing storm track variability

We construct a two-variable model which describes the interaction between local baroclinicity and eddy heat flux in order to understand aspects of the variance in storm tracks. It is a heuristic model for diabatically forced baroclinic instability close to baroclinic neutrality. The two-variable model has the structure of a nonlinear oscillator. It exhibits some realistic properties of observed storm track variability, most notably the intermittent nature of eddy activity. This suggests that apparent threshold behaviour can be more accurately and succinctly described by a simple nonlinearity. An analogy is drawn with triggering of convective events.

Adaptive nonlinear equalizer using a mixture of gaussians based on-line density estimator

This paper introduces a new adaptive nonlinear equalizer relying on a radial basis function (RBF) model, which is designed based on the minimum bit error rate (MBER) criterion, in the system setting of the intersymbol interference channel plus a co-channel interference. Our proposed algorithm is referred to as the on-line mixture of Gaussians estimator aided MBER (OMG-MBER) equalizer. Specifically, a mixture of Gaussians based probability density function (PDF) estimator is used to model the PDF of the decision variable, for which a novel on-line PDF update algorithm is derived to track the incoming data. With the aid of this novel on-line mixture of Gaussians based sample-by-sample updated PDF estimator, our adaptive nonlinear equalizer is capable of updating its equalizer’s parameters sample by sample to aim directly at minimizing the RBF nonlinear equalizer’s achievable bit error rate (BER). The proposed OMG-MBER equalizer significantly outperforms the existing on-line nonlinear MBER equalizer, known as the least bit error rate equalizer, in terms of both the convergence speed and the achievable BER, as is confirmed in our simulation study

Representation errors and retrievals in linear and nonlinear data assimilation

This article shows how one can formulate the representation problem starting from Bayes’ theorem. The purpose of this article is to raise awareness of the formal solutions,so that approximations can be placed in a proper context. The representation errors appear in the likelihood, and the different possibilities for the representation of reality in model and observations are discussed, including nonlinear representation probability density functions. Specifically, the assumptions needed in the usual procedure to add a representation error covariance to the error covariance of the observations are discussed,and it is shown that, when several sub-grid observations are present, their mean still has a representation error ; socalled ‘superobbing’ does not resolve the issue. Connection is made to the off-line or on-line retrieval problem, providing a new simple proof of the equivalence of assimilating linear retrievals and original observations. Furthermore, it is shown how nonlinear retrievals can be assimilated without loss of information. Finally we discuss how errors in the observation operator model can be treated consistently in the Bayesian framework, connecting to previous work in this area.

Advances in the study of boundary value problems for nonlinear integrable PDEs

In this review I summarise some of the most significant advances of the last decade in the analysis and solution of boundary value problems for integrable partial differential equations in two independent variables. These equations arise widely in mathematical physics, and in order to model realistic applications, it is essential to consider bounded domain and inhomogeneous boundary conditions. I focus specifically on a general and widely applicable approach, usually referred to as the Unified Transform or Fokas Transform, that provides a substantial generalisation of the classical Inverse Scattering Transform. This approach preserves the conceptual efficiency and aesthetic appeal of the more classical transform approaches, but presents a distinctive and important difference. While the Inverse Scattering Transform follows the "separation of variables" philosophy, albeit in a nonlinear setting, the Unified Transform is a based on the idea of synthesis, rather than separation, of variables. I will outline the main ideas in the case of linear evolution equations, and then illustrate their generalisation to certain nonlinear cases of particular significance.

The elliptic sine-Gordon equation: a nonlinear elliptic integrable PDE

In this paper, we summarise this recent progress to underline the features specific to this nonlinear elliptic case, and we give a new classification of boundary conditions on the semistrip that satisfy a necessary condition for yielding a boundary value problem can be effectively linearised. This classification is based on formulation the equation in terms of an alternative Lax pair.