On the stability of tracer simulations with opposite-signed diffusivities

Abstract

Many recent studies have diagnosed opposite-signed diffusion eigenvalues to be a prevalent feature of the transfer tensor for diffusive tracer transport by oceanic mesoscale eddies. This diagnosed tensor, which we refer to as the diffusion tensor, therefore accounts for tracer filamentation effects. The preferential orientation of this filamentation is quantified by the principal axis of the diffusion tensor, namely the diffusion axis. Parameterisations of eddy diffusion commonly invoke a diffusion tensor, typically one with non-negative eigenvalues to avoid numerical issues. Motivated by the need to parameterise tracer filamentation, in this study we examine diffusion of a Gaussian tracer patch with imposed opposite-signed diffusion eigenvalues, and in particular we focus on the time scale for the onset of instability. For a fixed diffusion axis, numerical instability is an inevitable consequence of persistent up-gradient fluxes associated with the negative eigenvalue. For typical oceanic scales and diffusion magnitudes, this time scale is of the order of $100$ days, but is shorter for larger negative eigenvalues or for finer grid resolutions. We show that imposing a time-dependent diffusion axis can lead to simulations with no onset of instability after 100 000 days of tracer evolution. Although motivated by oceanographic fluid dynamics, our results have much broader applications since diffusive processes are present in a wide range of fluid flows.

Impact of Parameterized Isopycnal Diffusivity on Shelf‐Ocean Exchanges Under Upwelling‐Favorable Winds: Offline Tracer Simulations Augmented by Artificial Neural Network

“…For computational efficiency and to disentangle the isopycnal mixing from other eddy effects, we have opted for offline calculations, in which the mean flow quantities are prescribed using the solutions of MITgcm simulations. Similar offline calculations have recently been conducted to quantify the stability of tracer simulations with negative‐signed diffusivities but no background flow (Haigh & Berloff, 2022) and to assess the validity of the flux‐gradient relation in parameterizing the eddy‐driven fluxes of reactive tracers using a kinematic flow model (Prend et al., 2021). However, to convert our proposed Redi variants into actual parameterizations in predictive ocean models, fully online prognostic tests of these Redi variants are imperative.…”

“…As such, these calculations rely on fully stationary mean flows. Haigh and Berloff (2022) conducted analogous simulations that solve the tracer diffusion equation without prescribing any background flow (see their Equation 3.1). Our rationale for using stationary mean flows is that an eddy parameterization scheme must accurately represent the corresponding eddy effect in the simplest possible flow scenario if it is to be applied for more complex oceanic environments (Bachman & Fox-Kemper, 2013).…”

“…Parameterizing these eddy stresses together with isopycnal eddy mixing is essential for developing a complete closure for mesoscale turbulence over continental slopes, which, however, requires parallel prognostic tests for other eddy parameterization schemes (e.g., the GM scheme) and is therefore beyond the scope of this work. Thus, to isolate the influence of isopycnal mixing from other eddy effects (e.g., eddy restratification and lateral momentum transfer), we opt for an offline simulation approach (e.g., Haigh & Berloff, 2022). Specifically, the governing equation for a passive tracer that incorporates the Redi parameterization is solved by a parameterized ocean model based on the mean flow properties produced by an eddy-resolving model with almost identical configurations.…”

On transport tensor of dynamically unresolved oceanic mesoscale eddies

“…The eigenvalues of the -tensor (figure 3) are predominantly polar, which is an important result obtained here for the first time for active tracers and from the brute-force estimation of the full transport tensor. Note that it is unusual to deal simultaneously with both positive and negative eddy diffusion (of density) and viscosity (of momentum); more on that side can be found in Haigh & Berloff (2021, 2022) analyses of passive tracers. Firstly, by construction, our transport tensor is the same for the involved dynamical properties – PV, relative-vorticity (i.e.…”

The Maxwell Effect and the Material Transport by Transient Eddies

“…Recent studies (Haigh et al., 2020; Kamenkovich et al., 2021; Sun et al., 2021) raised significant concerns regarding our ability to fully represent the eddy‐induced transport using parameterizations that are based exclusively on Fick's model. For instance, eddy‐resolving simulations revealed ubiquitous negative Fick‐based diffusivities, which can severely undermine attempts to incorporate eddy parameterizations in global ocean models (Bachman et al., 2020; Haigh & Berloff, 2022). At the same time, neglecting the regional variability in the intensity of turbulent mixing is bound to produce large biases in the numerical and observational estimates of eddy transfer.…”