We introduce a new mathematical tool for quantifying the symmetry contents of molecular structures: the Symmetry Operation Measures. In this approach, we measure the minimal distance between a given structure and the structure which is obtained after applying a selected symmetry operation on it. If the given operation is a true symmetry operation for the structure, this distance is zero; otherwise it gives an indication of how different the transformed structure is from the original one. Specifically, we provide analytical solutions for measures of all the improper rotations, S n p, including mirror symmetry and inversion, as well as for all pure rotations, C n p. These measures provide information complementary to the Continuous Symmetry Measures (CSM) that evaluate the distance between a given structure and the nearest structure which belongs to a selected symmetry point-group.

No abstract

Conditions determining the existence of localized steadily translating two-layer vortices (modons) of arbitrary symmetric form on the $\beta$-plane are considered. A numerical method for direct construction of modon solutions is suggested and its accuracy is analysed in relation to the parameters of the computational procedure and the geometrical and physical parameters of the modon sought. Using this method, several non-circular baroclinic solutions are constructed marked by nonlinearity of the dependence of the potential vorticity (PV) on the streamfunction in the trapped-fluid area of the modon, i.e. where the streamlines are closed. The linearity of this dependence and the circularity of the trapped-fluid area are shown to be equivalent properties of a modon. Special attention is given to elliptical modons – extended both in the direction of the modon propagation and in the orthogonal direction, the baroclinic PV component being assumed continuous. The differences between the two types of elliptical modons are discussed. The simplest vortical couples and shielded modons are considered. In the context of the continuity of the baroclinic PV field, the stability of modons is discussed based on numerical simulations.

The objective of this work is a numerical study of the stability properties and the evolution of the eastward-travelling baroclinic modons – coherent vortex structures specific to stratified geophysical fluids where differential rotation (the β-effect) is of the essence. In the vortices under study, the initial dependence of the potential vorticity (PV) upon the streamfunction is piecewise-linear, the barotropic component is dipolar, the baroclinic component is circularly symmetric about the vertical axis, and the boundary of the trapped-fluid region (in which the vorticity contours are closed) is a circular cylinder. These modons are shown to be stable for a wide range of parameters. In two- and three-layer fluids, modons of this type are shown to be able to transit to even more durable states, in which the trapped-fluid area is oval in shape and the PV versus streamfunction dependence in this domain is nonlinear. Possible transition mechanisms and linkage between the circular and oval modons are discussed.

A family of semianalytical solutions is presented describing multipolar vortical structures with zero total circulation in a variety of two-dimensional models. Analytics are used to determine the form of a multipole edge, or separatrix, and the solution outside this separatrix. The interior is solved using a Newton-Kantorovich (successive linearization) procedure combined with a collocation method. The models considered are the quasigeostrophic f- and γ-planes, with either the rigid-lid or free-surface conditions. A multipole, termed also an (m+1)-pole, is a vortical system that possesses an m-fold symmetry (m≥2) and is comprised of a central core vortex and m satellite vortices surrounding the core. Fluid parcels in the core and the satellites revolve oppositely, and the multipole as a whole rotates steadily. The characteristics of the multipoles are examined as functions of m and a parameter that incorporates the Rossby deformation radius, γ-effect, and the vortex’s angular velocity. The analogy between the β-plane modons and γ-plane multipoles is tracked.