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

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.

💡 Research Summary

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This paper tackles a long‑standing problem in ocean modelling: how to represent the stress exerted by unresolved rough seafloor topography on the large‑scale flow. When a deep‑ocean current encounters a seamount or ridge, internal waves are generated, creating a pressure differential across the feature that manifests as a form stress (often called “topographic drag”). Because global climate models cannot resolve the sub‑kilometre scales at which these waves are produced, the stress must be parameterised. Existing parameterisations, however, are typically derived under restrictive assumptions – small topographic height, linear wave dynamics, steady or purely tidal flow, isotropic topography, and often neglect the out‑of‑phase “spring” forces that arise in bottom‑trapped tides.

The authors focus on four widely used formulations: Bell (1975), Jayne & St Laurent (2001; hereafter JSL2001), Klymak et al. (2010), and Shakespeare, Arbic & Hogg (2020; hereafter SAH2020). Bell’s linear theory provides a time‑averaged internal‑tidal energy flux, which JSL2001 scales to a stress using a tunable wavenumber κ and the product N h² U. Klymak et al. add a correction for flow blocking, while SAH2020 derives a fully analytic stress expression that incorporates the full Fourier spectrum of the topography, the Coriolis frequency f, and the tidal frequency ω, thereby capturing the “spring” forces that are invisible to energy‑flux‑based schemes.

To assess the validity of these schemes, the authors conduct an extensive numerical experiment campaign. They simulate flow over an isolated Gaussian hill (characterised by height h₀ and half‑width a) in both two‑dimensional (x‑z) and three‑dimensional (x‑y‑z) domains using a high‑resolution, non‑hydrostatic Boussinesq model. Hundreds of simulations span a wide parameter space: background flow speed U, buoyancy frequency N, Coriolis parameter f (covering equatorial to polar latitudes), tidal frequency ω, and a range of h₀/a ratios from 0.02 to 0.3. Bottom pressure fields p_h(x,y) are extracted, and the form stress is computed directly from the definitions (1) and (2) in the manuscript.

Key findings are:

1. Small‑height regime (h₀/H ≪ 0.05) – Both JSL2001 and SAH2020 reproduce the simulated stress within ~20 % error, confirming that the linear‑wave assumption is adequate when the hill is shallow relative to the water depth.

2. Intermediate to large hills (h₀/H ≥ 0.1) – JSL2001 systematically underestimates the stress, sometimes by a factor of two, because its single‑wavenumber representation cannot capture the broadband topographic spectrum. SAH2020 performs better but still deviates by up to 30 % when ω approaches f (sub‑critical latitudes).

3. Latitude dependence – Near the equator (|f| ≈ ω) the stress changes sign, a phenomenon captured only by the (ω² − f²) term in SAH2020. JSL2001, being latitude‑independent, fails dramatically in this regime.

4. Blocking effects – The correction proposed by Klymak et al. becomes noticeable only for Rossby numbers of order unity; for most of the parameter space the stress is dominated by wave drag rather than flow blockage.

5. Spectral representation – When the authors replace the single κ in JSL2001 with the actual integral ∫|ĥ(K)|²K² dK derived from the Gaussian hill’s analytic Fourier transform, the prediction improves markedly, highlighting the importance of retaining the full topographic spectrum.

Based on these diagnostics, the authors propose three concrete improvements to existing parameterisations:

• Non‑linear height correction – Multiply the stress by (1 + β (h₀/H)²) to account for finite‑amplitude effects; β is calibrated from the simulation suite.

• Explicit spectral integration – Use the analytically known (or observationally derived) Fourier spectrum of the hill, inserting the integral of |ĥ(K)|²K² directly into the stress formula, thereby eliminating the need for an ad‑hoc κ.

• Generalised frequency‑latitude factor – Replace the simple (ω² − f²) term with (ω² − f² + γ f²), where γ is a tunable coefficient that restores the correct stress magnitude across the full range of f/ω ratios, including the equatorial limit.

When these three modifications are combined into a single “enhanced SAH” formulation, the root‑mean‑square error against the high‑resolution simulations drops by roughly 35 % relative to the original SAH2020, and by more than 50 % for large hills (h₀ ≈ 2 km) at high latitudes. The authors argue that implementing this composite scheme in Earth‑system models will improve the representation of internal‑wave‑driven energy dissipation, bottom‑drag feedback on currents, and ultimately the large‑scale overturning circulation (e.g., the Atlantic Meridional Overturning Circulation and the Antarctic Circumpolar Current).

In summary, the paper provides a rigorous, data‑driven evaluation of the most common wave‑and‑non‑wave stress parameterisations, identifies their systematic biases, and offers a physically motivated, empirically calibrated improvement that is ready for incorporation into next‑generation global ocean models.