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

• Main Authors: NEWTON, Paul K., SAKAJO, Takashi
• Format: Article in Journal/Newspaper
• Language: English
• Published: Department of Mathematics, Hokkaido University
• Subjects: N-vortex problem; rotating sphere; Background vorticity gradients; Embedded dynamical system; 410; North Pole
• Online Access: http://hdl.handle.net/2115/69605; https://doi.org/10.14943/83947

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 are 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 center-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-Haurwitz 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 center-of-vorticity vector associated with the N-point vortices are no longer conserved as they are for the one-way coupled model described in Newton & Shokraneh (2006a). In the first case, energy is transferred to the ring, the length of the center-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 timescale. In the later case, energy is extracted from the ring, the length of the center-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 center-of-vorticity vector is perpendicular to the axis of rotation is also examined as it shows how the coupling to the background field breaks ...