[Paper Review] Discrete variance decay analysis of spurious mixing

Tridib Banerjee, Sergey Danilov+1 more|arXiv (Cornell University)|Oct 25, 2023

Meteorological Phenomena and Simulations35 references4 citations

TL;DR

This paper proposes a new discrete variance decay (DVD) framework for diagnosing spurious mixing (SM) in ocean models by deriving closed-form expressions for DVD rates directly from discrete tracer equations, avoiding assumptions about second-moment fluxes. It reveals that unfiltered flux divergence introduces ambiguity in localized DVD estimates, necessitating spatial or temporal averaging, and finds SM strongly correlated with eddy kinetic energy and buoyancy fluxes, with high-order advection schemes producing SM levels comparable to or exceeding physical diapycnal mixing.

ABSTRACT

Expressions for local discrete variance decay (DVD) rates are directly derived from discrete tracer equations without any assumptions on discrete fluxes of the second moment. Spurious mixing (SM) associated with numerical implementations of scalar advection and diffusion is thus estimated. The new framework is shown to avoid the need for second-moment flux definition when solved on finite-volume cell edges but still invoke certain second-moment fluxes when the DVD rates are partitioned to participating cell nodes. These implied discrete fluxes are shown to differ from those proposed in earlier literature (but share the same dissipative part) and thus reveal the non-uniqueness of their nature. They are shown to be ambiguous for high-order advection schemes introducing uncertainty to the locality of any estimates produced by a DVD approach. Additional damping of flux divergence through temporal averaging or some coarse-graining is thus shown to be necessary. Through the application of this technique, SM is found to be correlated with the distribution of eddy kinetic energy. The contribution from vertical advection to SM is found to be relatively small and correlated with the distribution of buoyancy fluxes. The explored high-order schemes are found to demonstrate levels of spurious mixing which may locally exceed background physical mixing.

Motivation & Objective

• To develop a more rigorous, discrete formulation for local DVD rates in ocean models without relying on assumed second-moment fluxes.
• To identify and quantify the ambiguity in second-moment flux definitions that undermines localized SM diagnostics.
• To assess the contribution of horizontal and vertical advection schemes to spurious mixing in an idealized setup.
• To evaluate whether high-order advection schemes produce SM levels comparable to or exceeding background physical diapycnal mixing.
• To demonstrate the necessity of spatial or temporal averaging to stabilize DVD-based SM estimates.

Proposed method

• Derive closed-form expressions for DVD rates directly from discrete tracer equations, shifting focus from cell-centered variance to fluxes across cell faces.
• Compute implied second-moment fluxes from the divergence of first-moment fluxes and cell-averaged tracer values, avoiding ad hoc flux assumptions.
• Compare derived fluxes with those from Klingbeil et al. [32], revealing differences in the non-dissipative parts while sharing the dissipative component.
• Apply the framework to FESOM2 with various advection schemes (including upwinding and horizontal diffusion) in an idealized setup with linear equation of state.
• Use temporal and spatial averaging to mitigate flux divergence uncertainty and stabilize localized DVD rate estimates.
• Correlate diagnosed SM with eddy kinetic energy and buoyancy flux distributions to identify dominant drivers.

Experimental results

Research questions

• RQ1 How can discrete variance decay rates be derived directly from discrete tracer equations without assuming second-moment fluxes?
• RQ2 What causes ambiguity in localized DVD rate estimates, and how does it relate to flux divergence in numerical schemes?
• RQ3 To what extent do high-order advection schemes contribute to spurious mixing relative to physical diapycnal mixing?
• RQ4 How is spurious mixing spatially correlated with eddy kinetic energy and buoyancy fluxes?
• RQ5 What role does temporal or spatial averaging play in stabilizing DVD-based spurious mixing diagnostics?

Key findings

• Closed-form expressions for DVD rates were derived directly from discrete tracer equations, eliminating reliance on assumed second-moment fluxes.
• The study reveals that unfiltered flux divergence introduces inherent ambiguity in localized DVD rate estimates, particularly for high-order advection schemes.
• Spurious mixing was found to be strongly correlated with the distribution of eddy kinetic energy, indicating its dominant role in numerical mixing.
• The contribution of vertical advection to spurious mixing was relatively small but correlated with buoyancy flux distributions.
• All tested advection schemes, including third-order upwind schemes, produced spurious mixing levels that locally exceeded background physical diapycnal mixing, with some reaching up to twice the physical level.
• The framework demonstrates that DVD diagnostics requires spatial or temporal averaging to yield reliable, localized estimates of numerical mixing.