Steady periodic water waves with constant vorticity: regularity and local bifurcation

This paper studies periodic traveling gravity waves at the free surface of water in a flow of constant vorticity over a flat bed. Using conformal mappings the free-boundary problem is transformed into a quasilinear pseudodifferential equation for a periodic function of one variable. The new formulation leads to a regularity result and, by use of bifurcation theory, to the existence of waves of small amplitude even in the presence of stagnation points in the flow.

A geometric approach to generalized Stokes conjectures

We consider the Stokes conjecture concerning the shape of extreme two-dimensional water waves. By new geometric methods including a nonlinear frequency formula, we prove the Stokes conjecture in the original variables. Our results do not rely on structural assumptions needed in previous results such as isolated singularities, symmetry and monotonicity. Part of our results extends to the mathematical problem in higher dimensions.

Condition number estimates for combined potential integral operators in acoustics and their boundary element discretisation

We consider the classical coupled, combined-field integral equation formulations for time-harmonic acoustic scattering by a sound soft bounded obstacle. In recent work, we have proved lower and upper bounds on the $L^2$ condition numbers for these formulations, and also on the norms of the classical acoustic single- and double-layer potential operators. These bounds to some extent make explicit the dependence of condition numbers on the wave number $k$, the geometry of the scatterer, and the coupling parameter. For example, with the usual choice of coupling parameter they show that, while the condition number grows like $k^{1/3}$ as $k\to\infty$, when the scatterer is a circle or sphere, it can grow as fast as $k^{7/5}$ for a class of `trapping' obstacles. In this paper we prove further bounds, sharpening and extending our previous results. In particular we show that there exist trapping obstacles for which the condition numbers grow as fast as $\exp(\gamma k)$, for some $\gamma>0$, as $k\to\infty$ through some sequence. This result depends on exponential localisation bounds on Laplace eigenfunctions in an ellipse that we prove in the appendix. We also clarify the correct choice of coupling parameter in 2D for low $k$. In the second part of the paper we focus on the boundary element discretisation of these operators. We discuss the extent to which the bounds on the continuous operators are also satisfied by their discrete counterparts and, via numerical experiments, we provide supporting evidence for some of the theoretical results, both quantitative and asymptotic, indicating further which of the upper and lower bounds may be sharper.

Stability of solutions of linear difference equations with periodic coefficients

This paper represents the last technical contribution of Professor Patrick Parks before his untimely death in February 1995. The remaining authors of the paper, which was subsequently completed, wish to dedicate the article to Patrick. A frequency criterion for the stability of solutions of linear difference equations with periodic coefficients is established. The stability criterion is based on a consideration of the behaviour of a frequency hodograph with respect to the origin of coordinates in the complex plane. The formulation of this criterion does not depend on the order of the difference equation.

Characterising the variability and extremes of the stratospheric polar vortices using 2D moment analysis

The mean state, variability and extreme variability of the stratospheric polar vortices, with an emphasis on the Northern Hemisphere vortex, are examined using 2-dimensional moment analysis and Extreme Value Theory (EVT). The use of moments as an analysis to ol gives rise to information about the vortex area, centroid latitude, aspect ratio and kurtosis. The application of EVT to these moment derived quantaties allows the extreme variability of the vortex to be assessed. The data used for this study is ECMWF ERA-40 potential vorticity fields on interpolated isentropic surfaces that range from 450K-1450K. Analyses show that the most extreme vortex variability occurs most commonly in late January and early February, consistent with when most planetary wave driving from the troposphere is observed. Composites around sudden stratospheric warming (SSW) events reveal that the moment diagnostics evolve in statistically different ways between vortex splitting events and vortex displacement events, in contrast to the traditional diagnostics. Histograms of the vortex diagnostics on the 850K (∼10hPa) surface over the 1958-2001 period are fitted with parametric distributions, and show that SSW events comprise the majority of data in the tails of the distributions. The distribution of each diagnostic is computed on various surfaces throughout the depth of the stratosphere, and shows that in general the vortex becomes more circular with higher filamentation at the upper levels. The Northern Hemisphere (NH) and Southern Hemisphere (SH) vortices are also compared through the analysis of their respective vortex diagnostics, and confirm that the SH vortex is less variable and lacks extreme events compared to the NH vortex. Finally extreme value theory is used to statistically mo del the vortex diagnostics and make inferences about the underlying dynamics of the polar vortices.

Cumulant-based deconvolution and identification: several new families of linear equations

This paper presents several new families of cumulant-based linear equations with respect to the inverse filter coefficients for deconvolution (equalisation) and identification of nonminimum phase systems. Based on noncausal autoregressive (AR) modeling of the output signals and three theorems, these equations are derived for the cases of 2nd-, 3rd and 4th-order cumulants, respectively, and can be expressed as identical or similar forms. The algorithms constructed from these equations are simpler in form, but can offer more accurate results than the existing methods. Since the inverse filter coefficients are simply the solution of a set of linear equations, their uniqueness can normally be guaranteed. Simulations are presented for the cases of skewed series, unskewed continuous series and unskewed discrete series. The results of these simulations confirm the feasibility and efficiency of the algorithms.

Applying the extended Kalman filter to systems described by nonlinear differential-algebraic equations

This paper describes a method for the state estimation of nonlinear systems described by a class of differential-algebraic equation models using the extended Kalman filter. The method involves the use of a time-varying linearisation of a semi-explicit index one differential-algebraic equation. The estimation technique consists of a simplified extended Kalman filter that is integrated with the differential-algebraic equation model. The paper describes a simulation study using a model of a batch chemical reactor. It also reports a study based on experimental data obtained from a mixing process, where the model of the system is solved using the sequential modular method and the estimation involves a bank of extended Kalman filters.

Optimal control of nonlinear systems represented by differential algebraic equations

An iterative procedure is described for solving nonlinear optimal control problems subject to differential algebraic equations. The procedure iterates on an integrated modified simplified model based problem with parameter updating in such a manner that the correct solution of the original nonlinear problem is achieved.

Development and application of a novel algorithm for steady-state optimising control using dynamic information

A novel optimising controller is designed that leads a slow process from a sub-optimal operational condition to the steady-state optimum in a continuous way based on dynamic information. Using standard results from optimisation theory and discrete optimal control, the solution of a steady-state optimisation problem is achieved by solving a receding-horizon optimal control problem which uses derivative and state information from the plant via a shadow model and a state-space identifier. The paper analyzes the steady-state optimality of the procedure, develops algorithms with and without control rate constraints and applies the procedure to a high fidelity simulation study of a distillation column optimisation.

Velocity-based moving mesh methods for nonlinear partial differential equations

This article describes a number of velocity-based moving mesh numerical methods formultidimensional nonlinear time-dependent partial differential equations (PDEs). It consists of a short historical review followed by a detailed description of a recently developed multidimensional moving mesh finite element method based on conservation. Finite element algorithms are derived for both mass-conserving and non mass-conserving problems, and results shown for a number of multidimensional nonlinear test problems, including the second order porous medium equation and the fourth order thin film equation as well as a two-phase problem. Further applications and extensions are referenced.