On the Validity of the β-Plane Approximation in the Dynamics and the Chaotic Advection of a Point Vortex Pair Model on a Rotating Sphere

Abstract

The dynamics of modulated point vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). In this setting the authors point out a qualitative difference between the full spherical dynamics and the one obtained in a β-plane approximation. In particular, dipole trajectories starting at the same location evolve to completely different directions under these two treatments, despite the fact that the deviations from the initial latitude remain small. This is a strong indication for the mathematical inconsistency of the traditional β-plane approximation. At the same time, a consistently linearized set of equations of motion leads to trajectories agreeing with those obtained under the full spherical treatment. The β-plane advection patterns due to chaotic advection in the velocity field of finite-sized vortex pairs are also found to considerably deviate from those of the full spherical treatment, and quantities characterizing transport properties (e.g., the escape rate from a given region) strongly differ.

Point vortices dynamics on a rotating sphere and modeling of global atmospheric vortices interaction

“…Study of the point vortices (PV) on a sphere is important for understanding the processes of the atmospheric and oceanic vortices evolution and modeling of them [1][2][3][4][5][6][7][8][9][10] . Conceptual models of point vortices are useful in fluid dynamics for identification and evaluation of physical mechanisms affecting the structure and interaction of atmospheric and oceanic vortices 11 .…”

Global coarse-grained mesoscale eddy statistics based on integrated kinetic energy and enstrophy correlations

“…The picture is further complicated by the weak nonlinear effects present in a quasi-geostrophic approach, resulting in an amplitude (thus traveling distance) dependence of propagating speeds (see, e.g., Figs. 9 and 10 in Early et al, 2011).…”

“…A well-known and widely analyzed feature of eddy trajectories is the general tendency for westward propagation in the absences of strong countercurrents (Cushman- Roisin et al, 1990;Chelton et al, 2007Chelton et al, , 2011Early et al, 2011;Kurian et al, 2011;Drótos and Tél, 2015;Brach et al, 2018;Cetina-Heredia et al, 2019;van Sebille et al, 2020;Wichmann et al, 2021, just to mention of a few references from the existing vast literature). The usual parsimonious explanation is based on the beta-plane effect: the Coriolis parameter is slightly different on the two opposite sides of an eddy (in the meridional direction).…”

Mathematical Modeling of Vortex Interaction Using a Three-Layer Quasigeostrophic Model. Part 1: Point-Vortex Approach

“…Further development of the theory of point vortices is reflected in monographs and reviews [6][7][8][9][10][11][12][13][14][15][16]. The classical concept of point vortices was used in problems of meteorology [17][18][19][20][21][22][23] and oceanography in [24][25][26][27][28][29][30]. Gryanik first generalized the theory of two-dimensional vortices to the case of a two-layer [31] and then to an N-layer rotating fluid [32].…”