Geodesic interpretation of the global quasi-geostrophic equations

Abstract We give an interpretation of the global shallow water quasi-geostrophic equations on the sphere $$\mathbb {S}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:math> as a geodesic equation on the central extension of the quantomorphism group on $$\mathbb {S}^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:math> . The study includes deriving the model as a geodesic equation for a weak Riemannian metric, demonstrating smooth dependence on the initial data, and establishing global-in-time existence and uniqueness of solutions. We also prove that the Lamb parameter in the model has a stabilizing effect on the dynamics: if it is large enough, the sectional curvature along the trade-wind current is positive, implying conjugate points.