Bounds on dissipation in stress-driven flow in a rotating frame

Abstract

We calculate a rigorous dual bound on the long-time-averaged mechanical energy dissipation rate e within a channel of an incompressible viscous fluid of constant kinematic viscosity ν, depth h and rotation rate f, driven by a constant surface stress τ = ρu 2 * i, where u * is the friction velocity. It is well known that e ≤ e Stokes = u 4 * /v, i.e. the dissipation is bounded above by the dissipation associated with the Stokes flow. Using an approach similar to the variational 'background method' (due to Constantin, Doering & Hopf), we generate a rigorous dual bound, subject to the constraints of total power balance and mean horizontal momentum balance, in the inviscid limit ν→0 for fixed values of the friction Rossby number Ro * = u * /(fh) = GE, where G=τh 2 /(ρν 2 ) is the Grashof number, and E=v/fh 2 is the Ekman number. By assuming that the horizontal dimensions are much larger than the vertical dimension of the channel, and restricting our attention to particular, analytically tractable, classes of Lagrange multipliers imposing mean horizontal momentum balance analogous to the ones used in Tang, Caulfield & Young, we show that e ≤ e max = u 4 * /v-2.93u 2 * f, an improved upper bound from the Stokes dissipation, and e ≥ e min = 2.795u 3 */h, a lower bound which is independent of the kinematic viscosity ν