Hamiltonian Structure and a Variational Principle for Grounded Abyssal Flow on a Sloping Bottom in a Mid‐Latitude β‐Plane

Abstract

Observations, numerical simulations, and theoretical scaling arguments suggest that in mid‐latitudes, away from the high‐latitude source regions and the equator, the meridional transport of abyssal water masses along a continental slope correspond to geostrophic flows that are gravity or density driven and topographically steered. These dynamics are examined using a nonlinear reduced‐gravity geostrophic model that describes grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. It is shown that this model possesses a noncanonical Hamiltonian formulation. General nonlinear steady solutions to the model can be obtained for arbitrary bottom topography. These solutions correspond to nonparallel shear flows that flow across the planetary vorticity gradient. If the in‐flow current along the poleward boundary is strictly equatorward, then no shock can form in the solution in the mid‐latitude domain. It is also shown that the steady solutions satisfy the first‐order necessary conditions for an extremal to a suitably constrained potential energy functional. Sufficient conditions for the definiteness of the second variation of the constrained energy functional are examined. The theory is illustrated with a nonlinear steady solution corresponding to an abyssal flow with upslope and down slope groundings in the height field.

Meridional dynamics of grounded abyssal water masses on a sloping bottom in a mid-latitude -plane

“…Equation (2.9) may be interpreted as a variant of the so-called planetary geostrophic wave equation introduced by Anderson & Killworth (1979), Dewar (1987), Wright & Willmott (1992) and Edwards, Willmott & Killworth (1998), generalized to allow for meridional flow on a mid-latitude β-plane with variable bottom topography. The model equation (2.8) possesses a non-canonical Hamiltonian formulation (Swaters 2018). One useful property of the model (2.9) is that it ensures that the appropriate kinematic condition associated with a grounding (i.e.…”

“…where τ = x on y = y 0 . General steady solutions to (3.1) satisfy the first-order necessary conditions for an extremum to a suitably constrained energy functional (Swaters 2018). The characteristics, which are the isolines in the (x, y) plane for constant τ , are also the geostrophic streamlines since (3.2) and (3.3) can be combined to yield…”