The N -vortex problem on a rotating sphere. III. Ring configurations coupled to a background field

Abstract

We study the evolution of N -point vortices in ring formation embedded in a background flowfield that initially corresponds to solid-body rotation on a sphere. The evolution of the point vortices is tracked numerically as an embedded dynamical system along with the M contours which separate strips of constant vorticity. The full system is a discretization of the Euler equations for incompressible flow on a rotating spherical shell, hence a ‘barotropic’ model of the one-layer atmosphere. We describe how the coupling creates a mechanism by which energy is exchanged between the ring and the background, which ultimately serves to break up the structure. When the centre-of-vorticity vector associated with the ring is initially misaligned with the axis of rotation of the background field, it sets up the propagation of Rossby waves around the sphere which move retrograde to the solid-body rotation. These waves pass energy to the ring (in the case when the solid-body field and the ring initially co-rotate) or extract energy from the ring (when the solid-body field and the ring initially counter-rotate), hence the Hamiltonian and the centre-of-vorticity vector associated with the N -point vortices are no longer conserved as they are for the one-way coupled model described by Newton & Shokraneh. In the first case, energy is transferred to the ring, the length of the centre-of-vorticity vector increases, while its tip spirals in a clockwise manner towards the North Pole. The ring stays relatively intact for short times, but ultimately breaks-up on a longer time-scale. In the latter case, energy is extracted from the ring, the length of the centre-of-vorticity vector decreases while its tip spirals towards the North Pole and the ring loses its coherence more quickly than in the co-rotating case. The special case where the ring is initially oriented so that its centre-of-vorticity vector is perpendicular to the axis of rotation is also examined as it shows how the coupling to the background field breaks this symmetry. In this case, both the length of the centre-of-vorticity vector and the Hamiltonian energy of the ring achieve a local maximum at roughly the same time.