Water wave scattering by finite arrays of circular structures

The scattering of small amplitude water waves by a finite array of locally axisymmetric structures is considered. Regions of varying quiescent depth are included and their axisymmetric nature, together with a mild-slope approximation, permits an adaptation of well-known interaction theory which ultimately reduces the problem to a simple numerical calculation. Numerical results are given and effects due to regions of varying depth on wave loading and free-surface elevation are presented.

On the convergence of the no response test

The no response test is a new scheme in inverse problems for partial differential equations which was recently proposed in [D. R. Luke and R. Potthast, SIAM J. Appl. Math., 63 (2003), pp. 1292–1312] in the framework of inverse acoustic scattering problems. The main idea of the scheme is to construct special probing waves which are small on some test domain. Then the response for these waves is constructed. If the response is small, the unknown object is assumed to be a subset of the test domain. The response is constructed from one, several, or many particular solutions of the problem under consideration. In this paper, we investigate the convergence of the no response test for the reconstruction information about inclusions D from the Cauchy values of solutions to the Helmholtz equation on an outer surface $\partial\Omega$ with $\overline{D} \subset \Omega$. We show that the one‐wave no response test provides a criterion to test the analytic extensibility of a field. In particular, we investigate the construction of approximations for the set of singular points $N(u)$ of the total fields u from one given pair of Cauchy data. Thus, the no response test solves a particular version of the classical Cauchy problem. Also, if an infinite number of fields is given, we prove that a multifield version of the no response test reconstructs the unknown inclusion D. This is the first convergence analysis which could be achieved for the no response test.

A multiwave range test for obstacle reconstructions with unknown physical properties

We develop a new multiwave version of the range test for shape reconstruction in inverse scattering theory. The range test [R. Potthast, et al., A ‘range test’ for determining scatterers with unknown physical properties, Inverse Problems 19(3) (2003) 533–547] has originally been proposed to obtain knowledge about an unknown scatterer when the far field pattern for only one plane wave is given. Here, we extend the method to the case of multiple waves and show that the full shape of the unknown scatterer can be reconstructed. We further will clarify the relation between the range test methods, the potential method [A. Kirsch, R. Kress, On an integral equation of the first kind in inverse acoustic scattering, in: Inverse Problems (Oberwolfach, 1986), Internationale Schriftenreihe zur Numerischen Mathematik, vol. 77, Birkhäuser, Basel, 1986, pp. 93–102] and the singular sources method [R. Potthast, Point sources and multipoles in inverse scattering theory, Habilitation Thesis, Göttingen, 1999]. In particular, we propose a new version of the Kirsch–Kress method using the range test and a new approach to the singular sources method based on the range test and potential method. Numerical examples of reconstructions for all four methods are provided.

A new Rossby Wave-breaking interpretation of the North Atlantic Oscillation

This paper proposes the hypothesis that the low-frequency variability of the North Atlantic Oscillation (NAO) arises as a result of variations in the occurrence of upper-level Rossby wave–breaking events over the North Atlantic. These events lead to synoptic situations similar to midlatitude blocking that are referred to as high-latitude blocking episodes. A positive NAO is envisaged as being a description of periods in which these episodes are infrequent and can be considered as a basic, unblocked situation. A negative NAO is a description of periods in which episodes occur frequently. A similar, but weaker, relationship exists between wave breaking over the Pacific and the west Pacific pattern. Evidence is given to support this hypothesis by using a two-dimensional potential-vorticity-based index to identify wave breaking at various latitudes. This is applied to Northern Hemisphere winter data from the 40-yr ECMWF Re-Analysis (ERA-40), and the events identified are then related to the NAO. Certain dynamical precursors are identified that appear to increase the likelihood of wave breaking. These suggest mechanisms by which variability in the tropical Pacific, and in the stratosphere, could affect the NAO.

Sudden stratospheric warmings as noise-induced transitions

Sudden stratospheric warmings (SSWs) are usually considered to be initiated by planetary wave activity. Here it is asked whether small-scale variability (e.g., related to gravity waves) can lead to SSWs given a certain amount of planetary wave activity that is by itself not sufficient to cause a SSW. A highly vertically truncated version of the Holton–Mass model of stratospheric wave–mean flow interaction, recently proposed by Ruzmaikin et al., is extended to include stochastic forcing. In the deterministic setting, this low-order model exhibits multiple stable equilibria corresponding to the undisturbed vortex and SSW state, respectively. Momentum forcing due to quasi-random gravity wave activity is introduced as an additive noise term in the zonal momentum equation. Two distinct approaches are pursued to study the stochastic system. First, the system, initialized at the undisturbed state, is numerically integrated many times to derive statistics of first passage times of the system undergoing a transition to the SSW state. Second, the Fokker–Planck equation corresponding to the stochastic system is solved numerically to derive the stationary probability density function of the system. Both approaches show that even small to moderate strengths of the stochastic gravity wave forcing can be sufficient to cause a SSW for cases for which the deterministic system would not have predicted a SSW.

The influence of bottom topography on the decay of modeled Agulhas rings

The influence of a large meridional submarine ridge on the decay of Agulhas rings is investigated with a 1 and 2-layer setup of the isopycnic primitive-equation ocean model MICOM. In the single-layer case we show that the SSH decay of the ring is primarily governed by bottom friction and secondly by the radiation of Rossby waves. When a topographic ridge is present, the effect of the ridge on SSH decay and loss of tracer from the ring is negligible. However, the barotropic ring cannot pass the ridge due to energy and vorticity constraints. In the case of a two-layer ring the initial SSH decay is governed by a mixed barotropic–baroclinic instability of the ring. Again, radiation of barotropic Rossby waves is present. When the ring passes the topographic ridge, it shows a small but significant stagnation of SSH decay, agreeing with satellite altimetry observations. This is found to be due to a reduction of the growth rate of the m = 2 instability, to conversions of kinetic energy to the upper layer, and to a decrease in Rossby-wave radiation. The energy transfer is related to the fact that coherent structures in the lower layer cannot pass the steep ridge due to energy constraints. Furthermore, the loss of tracer from the ring through filamentation is less than for a ring moving over a flat bottom, related to a decrease in propagation speed of the ring. We conclude that ridges like the Walvis Ridge tend to stabilize a multi-layer ring and reduce its decay.

Stochastic resonance in a nonlinear model of a rotating, stratified shear flow, with a simple stochastic inertia-gravity wave parameterization

We report on a numerical study of the impact of short, fast inertia-gravity waves on the large-scale, slowly-evolving flow with which they co-exist. A nonlinear quasi-geostrophic numerical model of a stratified shear flow is used to simulate, at reasonably high resolution, the evolution of a large-scale mode which grows due to baroclinic instability and equilibrates at finite amplitude. Ageostrophic inertia-gravity modes are filtered out of the model by construction, but their effects on the balanced flow are incorporated using a simple stochastic parameterization of the potential vorticity anomalies which they induce. The model simulates a rotating, two-layer annulus laboratory experiment, in which we recently observed systematic inertia-gravity wave generation by an evolving, large-scale flow. We find that the impact of the small-amplitude stochastic contribution to the potential vorticity tendency, on the model balanced flow, is generally small, as expected. In certain circumstances, however, the parameterized fast waves can exert a dominant influence. In a flow which is baroclinically-unstable to a range of zonal wavenumbers, and in which there is a close match between the growth rates of the multiple modes, the stochastic waves can strongly affect wavenumber selection. This is illustrated by a flow in which the parameterized fast modes dramatically re-partition the probability-density function for equilibrated large-scale zonal wavenumber. In a second case study, the stochastic perturbations are shown to force spontaneous wavenumber transitions in the large-scale flow, which do not occur in their absence. These phenomena are due to a stochastic resonance effect. They add to the evidence that deterministic parameterizations in general circulation models, of subgrid-scale processes such as gravity wave drag, cannot always adequately capture the full details of the nonlinear interaction.