Gyre turbulence: Anomalous dissipation in a two-dimensional ocean model

The exploration of a two-dimensional wind-driven ocean model with no-slip boundaries reveals the existence of a turbulent asymptotic regime where energy dissipation becomes independent of fluid viscosity. This asymptotic flow represents an out-of-equilibrium state, characterized by a vigorous two-dimensional vortex gas superimposed onto a western-intensified gyre. The properties of the vortex gas are elucidated through scaling analysis for detached Prandtl boundary layers, providing a rationalization for the observed anomalous dissipation. The asymptotic regime demonstrates that boundary instabilities alone can be strong enough to evacuate wind-injected energy from the large-scale oceanic circulation.

Figure 1

(a) The $\beta$-plane model with an inlet for the streamlines of the time-average sea surface velocity field in the North Atlantic. Colors show the corresponding relative vorticity field. (b) Temporal average of the streamlines in the gyre turbulence regime (${\nu }^{*}=2.5\times {10}^{-5}$), superimposed on the corresponding relative vorticity field (colors). (c) Snapshot of relative vorticity in the gyre turbulence regime (${\nu }^{*}=2\times {10}^{-6}$). Blue: anticyclonic vortices; red: cyclonic vortices (see Supplemental Material [33] movie S2 for an animation).

Figure 2

(a) Parameter space (no-slip boundary conditions). Insets show typical instantaneous vorticity fields (color) and stream functions (lines). (b) Energy dissipation rate, rescaled by the energy injection through a Sverdrup interior. (c) Energy rescaled by the energy of the inertial western boundary layer. Error bars show the standard deviation.

Figure 3

(a) Contour plot of the mean stream function $\langle \psi \rangle$ at ${\nu }^{*}=2.5\times {10}^{-5}$, with colored areas identified as inertial boundary layers. (b) $q-\psi$ relations in the inertial boundary layers.

Figure 4

(a) Probability distribution functions $\mathrm{\Pi }$ of $log\left({\omega }^2\right)$ for different values of ${\nu }^{*}$. Also shown are the values responsible for dissipation, ${\omega }^2\mathrm{\Pi }$. (b) Values responsible for dissipation after rescaling by ${\omega }_{\text{max}}=1/\sqrt{{\nu }^{*}}$. The area under the curves is equal to the total dissipation $\langle \varepsilon \rangle$.