Interaction between global-scale atmospheric vortices: Modeling with Hamiltonian dynamic system of antipodal point vortices on a rotating sphere

TL;DR

This paper models the interaction of global atmospheric vortices on a rotating sphere using Hamiltonian dynamics of antipodal point vortices, providing exact solutions that account for sphere rotation and stability conditions.

Contribution

It introduces an exact Hamiltonian model for antipodal point vortices on a rotating sphere, capturing global vortex interactions and stability in atmospheric flows.

Findings

Derived exact equations for antipodal vortex dynamics on a rotating sphere.

Identified conditions for stability of vortex pairs and blocks.

Modeled interaction of atmospheric centers of action with sphere rotation effects.

Abstract

We get point vortices dynamics equations on a rotating sphere surface directly from the hydrodynamic equations as representing their weak exact solution contrary to the conventional case of the use of a kinematic relationship between a given singular vortex field and velocity field. It is first time that the effect of a sphere rotation on the vortices interaction is accounted for in exact form. We show that only the stream function of a vortex pair of antipodal vortices (APV), and only it satisfies the original three-dimensional hydrodynamics equations on a sphere. We prove that only APV pair with two point vortices in the diameter-conjugated points of a sphere with equal by quantity but different sign circulations may be correctly considered as an elementary (stationary, not self-affecting) singular point object on a sphere. We suggest using the axis connecting the two point vortices in an APV for describing of an axis of rotation of the global vortices introduced in (Barrett, 1958) to reflect the observed global rotation of atmospheric masses with the rotation axes not coinciding with the planet rotation axis and precessing about it. Up to now, the question about interaction of global vortices corresponding to such solutions with rotations about different axis was not even posed. This is the first model describing interaction of the Barrett-type global vortices corresponding to atmospheric centers of action (ACA). The new steady-state and its stability conditions for N=2 are obtained and used for the analysis of coupled cyclone-anticyclone ACAs over oceans in the Northern Hemisphere. On the base of corresponding exact solutions, we show acceptability of modelling of the stable blocks of splitting flow type when in line with the exact accounting for the sphere rotation, we define also conditions of the polar vortices affecting on the stability boundaries of the block modes.