Evaluating and improving wave and non-wave stress parametrisations for oceanic flows

TL;DR

The paper addresses the challenge of representing topographic-stress (internal-wave drag) in global ocean models where fine-scale bathymetry is unresolved. It systematically evaluates prevailing tidal, steady, and mixed-flow parametrisations (notably JSL2001, SAH2020, Bell, KLP) against hundreds of high-resolution 2D/3D simulations around an isolated Gaussian hill, identifying regimes where linear theory succeeds and where it requires refinement. The authors introduce practical refinements, such as velocity-scaling corrections (e.g., $U_m$ and $\tilde U_m$) and Fr/ Ro- dependent adjustments, to extend the applicability of these parametrisations to larger hills and near resonant latitudes. They propose a set of updated, piecewise expressions to implement in ocean models, while acknowledging the need to test against more complex topography, variable stratification, and vertical-stress placement, ultimately advancing toward a robust, generalisable framework for topographic-stress parametrisation.

Abstract

Whenever oceanic currents flow over rough topography, there is an associated stress that acts to modify the flow. In the deep ocean, this stress is predominantly a form drag due to pressure differentials across topography, caused by the formation of internal waves and other baroclinic motions: processes that act on such small scales most global ocean models cannot resolve. Despite the need to incorporate this stress into ocean models, existing parametrisations are limited in their applicability. For instance, most parametrisations are only suitable for small-scale topography and are either for periodic or steady flows, but rarely a combination thereof. Here we summarise some of the most widely used parametrisations and evaluate the accuracy of a carefully selected subset using hundreds of idealised two-dimensional and three-dimensional simulations spanning a wide parameter space. We focus on the case of an isolated Gaussian hill as an idealised representation of a seamount. In cases where the parametrisations prove to be inaccurate, we use our data to suggest improved formulations. Our results thus provide a starting point for a comprehensive parameterisation of topographic stresses in ocean models where fine scale topography is unresolved.

Figures (9)

• Figure 1: A schematic of the model domain. A spatially mean flow $U(t)$ interacts with the topography $h(x,y)$ to generate baroclinic motions and an associated stress.
• Figure 2: The amplitude of the oscillatory stress given by $F_{\text{SAH2d}}$ and the simulation data for two-dimensional tidal flow. In Figures (e)--(g) a refinement of the $F_{\text{SAH2d}}$ parametrisation with a scaled velocity (see Equation \ref{['tildeum']}) is also shown in red. The default parameters for each test are as in Table \ref{['paramtable']} with $\omega=10^{-4}\ s^{-1}$ ($M_2$ tide) at a latitude of $20^\circ S$ ($f = 5 \times 10^{-5}\ s^{-1}$). The only exception is the bottom-trapped tests in Figure (h) which use $\omega=7.3\times 10^{-5}\ s^{-1}$ ($K_1$ tide) at a latitude of $90^\circ S$ ($f=1.46\times 10^{-4}\ s^{-1}$). In all tests, the uncertainty in the computed stress was less than 3% so the small error bars have been omitted.
• Figure 3: The fitted phase of the stress for our two-dimensional tidal flow simulations varying latitude. Here, the phase $\phi'=\phi\times(\omega/2\pi)$ is given in hours as opposed to radians (cf. \ref{['sinesteq']}), and the critical latitude ($\approx 74.5^{\circ}S$) is displayed in red. In each test, the uncertainty was less than $0.05$ hours so the small error bars have been omitted.
• Figure 4: The amplitude of the oscillatory stress given by $F_{\text{SAH3d}}$ and the simulation data for three-dimensional tidal flow (panels (a)--(f)), and the phase of the stress as the latitude is varied (panel (g)). The default values for each test are the same as the two-dimensional case (see Figure \ref{['fig:2dtidaltests']} and Table \ref{['paramtable']}). In panels (a)--(f), the uncertainty in the computed values was less than 10% so the small error bars have been omitted. In Figure (g), error bars have only been included in the single case where the uncertainty in the phase was greater than 0.1 hours.
• Figure 5: A comparison of the stress predicted by the bell1975topographically and KLP2010 parametrisations with our simulation data for two-dimensional steady flow. Here, $N$ denotes the buoyancy frequency, $h_0$ the hill height, $U$ the flow speed, and $f$ the Coriolis parameter. Note that for Figures (a) and (b), the ratio between the stress as predicted the bell1975topographically theory, $F_{\text{Bell2d}}$, and the stress output from the simulations, $F_{\text{2d}}$, is plotted. On the other hand, in Figures (c) to (e) the numerical values of the stress predicted by KLP2010 parametrisation, $F_{\text{KLP}}$, is shown. The default parameters are as in Table \ref{['paramtable']} and error bars are shown whenever the uncertainty in the stress was greater than 10%.