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Abstract Current eddy‐permitting and eddy‐resolving ocean models require dissipation to prevent a spurious accumulation of enstrophy at the grid scale. We introduce a new numerical scheme for momentum advection in large‐scale ocean models that involves upwinding through a weighted essentially non‐oscillatory (WENO) reconstruction. The new scheme provides implicit dissipation and thereby avoids the need for an additional explicit dissipation that may require calibration of unknown parameters. This approach uses the rotational, “vector invariant” formulation of the momentum advection operator that is widely employed by global general circulation models. A novel formulation of the WENO “smoothness indicators” is key for avoiding excessive numerical dissipation of kinetic energy and enstrophy at grid‐resolved scales. We test the new advection scheme against a standard approach that combines explicit dissipation with a dispersive discretization of the rotational advection operator in two scenarios: (a) two‐dimensional turbulence and (b) three‐dimensional baroclinic equilibration. In both cases, the solutions are stable, free from dispersive artifacts, and achieve increased “effective” resolution compared to other approaches commonly used in ocean models. 

Gradient ascent methods are developed to compute incompressible flows that maximize heat transport between two isothermal no-slip parallel walls. Parameterizing the magnitude of the velocity fields by a Péclet number $Pe$ proportional to their root-mean-square rate of strain, the schemes are applied to compute two-dimensional flows optimizing convective enhancement of diffusive heat transfer, i.e. the Nusselt number $Nu$ up to $$Pe\approx 10^{5}$$ . The resulting transport exhibits a change of scaling from $$Nu-1\sim Pe^{2}$$ for $Pe<10$ in the linear regime to $$Nu\sim Pe^{0.54}$$ for $$Pe>10^{3}$$ . Optimal fields are observed to be approximately separable, i.e. products of functions of the wall-parallel and wall-normal coordinates. Analysis employing a separable ansatz yields a conditional upper bound $${\lesssim}Pe^{6/11}=Pe^{0.\overline{54}}$$ as $$Pe\rightarrow \infty$$ similar to the computationally achieved scaling. Implications for heat transfer in buoyancy-driven Rayleigh–Bénard convection are discussed.