Planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin waves on the f-plane and β-plane in the presence of a uniform zonal flow

<p>A linear wave theory of the Rotating Shallow Water Equations (RSWE) is developed in a channel on either the mid-latitude f-plane/β-plane or on the equatorial β-plane in the presence of a uniform mean zonal flow that is balanced geostrophically by a meridional gradient of the fluid surface height. We show that this surface height gradient is a potential vorticity (PV) source that generates Rossby waves even on the f-plane similar to the generation of these waves by PV sources such as the β–effect, shear of the mean flow and bottom topography. Numerical solutions of the RSWE show that the resulting planetary (Rossby), Inertia-Gravity (Poincaré) and Kelvin-like waves differ from their counterparts without mean flow in both their phase speeds and meridional structures. Doppler shifting of the “no mean-flow” phase speeds does not account for the difference in phase speeds, and the meridional structure does not often oscillate across the channel but is trapped near one the channel's boundaries in mid latitudes or behaves as Hermite function in the case of an equatorial channel. The phase speed of Kelvin-like waves is modified by the presence of a mean flow compared to the classical gravity wave speed but their meridional velocity does not vanish. The gaps between the dispersion curves of adjacent Poincaré modes are not uniform but change with the zonal wavenumber, and the convexity of the dispersion curves also changes with the zonal wavenumber. In some cases, the Kelvin-like dispersion curve crosses those of Poincaré modes, but it is not an evidence for the existence of instability since the Kelvin waves are not part of the solutions of an eigenvalue problem. </p>

Generalized Wehrl entropies and Euclidean Landau levels

We are concerned with a class of generalized coherent states (GCS) attached to Euclidean Landau levels (or to [Formula: see text]-true-polyanalytic spaces), which can be obtained by displacing a Gaussian–Hermite function as an admissible (or window) function. Precisely, we evaluate the Wehrl entropy for a density operator representing a projector on a Fock state (pure states) and we give an upper bound for this entropy. We also establish an exact formula of this entropy for the heat operator (mixed states) associated with the harmonic oscillator. In this case, the behavior of the entropy with respect to the temperature parameter shows the dependence of its minimum to the Landau level or equivalently to the window function by means of which the GCS involved in the Wehrl entropy were constructed.

Comparative Study of Pulse Shaping Filters in FBMC

Filter Bank Multicarrier (FBMC) is a multitone modulation technique that is expected to replace the Orthogonal Frequency Division Multiplexing (OFDM) due to its inherent characteristics that makes it immune to channel dispersive effect on the transmitted signal in both time and frequency. The most effective ingredient in the FBMC is the pulse shaping that the OFDM symbol lacks. In this paper, a comparative study is presented between different pulse shapes used in the FBMC like the RRC, PHYDIAS, IOTA and Hermite function alongside the conventional OFDM.

Quantitative evaluation of numerical integration schemes for Lagrangian particle dispersion models

A rigorous methodology for the evaluation of integration schemes for Lagrangian particle dispersion models (LPDMs) is presented. A series of one-dimensional test problems are introduced, for which the Fokker–Planck equation is solved numerically using a finite-difference discretisation in physical space and a Hermite function expansion in velocity space. Numerical convergence errors in the Fokker–Planck equation solutions are shown to be much less than the statistical error associated with a practical-sized ensemble (N = 106) of LPDM solutions; hence, the former can be used to validate the latter. The test problems are then used to evaluate commonly used LPDM integration schemes. The results allow for optimal time-step selection for each scheme, given a required level of accuracy. The following recommendations are made for use in operational models. First, if computational constraints require the use of moderate to long time steps, it is more accurate to solve the random displacement model approximation to the LPDM rather than use existing schemes designed for long time steps. Second, useful gains in numerical accuracy can be obtained, at moderate additional computational cost, by using the relatively simple “small-noise” scheme of Honeycutt.

Higher powers of analytical operators and associated ∗-Lie algebras

We introduce a new product of two test functions denoted by [Formula: see text] (where [Formula: see text] and [Formula: see text] in the Schwartz space [Formula: see text]). Based on the space of entire functions with [Formula: see text]-exponential growth of minimal type, we define a new family of infinite dimensional analytical operators using the holomorphic derivative and its adjoint. Using this new product [Formula: see text], such operators give us a new representation of the centerless Virasoro–Zamolodchikov-[Formula: see text]∗-Lie algebras (in particular the Witt algebra) by using analytical renormalization conditions and by taking the test function [Formula: see text] as any Hermite function. Replacing the classical pointwise product [Formula: see text] of two test functions [Formula: see text] and [Formula: see text] by [Formula: see text], we prove the existence of new ∗-Lie algebras as counterpart of the classical powers of white noise ∗-Lie algebra, the renormalized higher powers of white noise (RHPWN) ∗-Lie algebra and the second quantized centerless Virasoro–Zamolodchikov-[Formula: see text]∗-Lie algebra.

Quantitative evaluation of numerical integration schemes for Lagrangian particle dispersion models

A rigorous methodology for the evaluation of integration schemes for Lagrangian particle dispersion models (LPDMs) is presented. A series of one-dimensional test problems are introduced, for which the Fokker-Planck equation is solved numerically using a finite-difference discretisation in physical space, and a Hermite function expansion in velocity space. Numerical convergence errors in the Fokker-Planck equation solutions are shown to be much less than the statistical error associated with a practical-sized ensemble (N = 106) of LPDM solutions, hence the former can be used to validate the latter. The test problems are then used to evaluate commonly used LPDM integration schemes. The results allow for optimal time-step selection for each scheme, given a required level of accuracy. The following recommendations are made for use in operational models. First, if computational constraints require the use of moderate to long time steps it is more accurate to solve the random displacement model approximation to the LPDM, rather than use existing schemes designed for long time-steps. Second, useful gains in numerical accuracy can be obtained, at moderate additional computational cost, by using the relatively simple "small-noise" scheme of Honeycutt.