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\begin{document}

\title{Dynamically-connected tracer and buoyancy mixing coefficients in eddy
parameterization schemes}



\author[1]{Dhruv Balwada}%
\author[1]{K Shafer Smith}%
\author[2]{Ryan Abernathey}%
\affil[1]{New York University}%
\affil[2]{Columbia University}%


\vspace{-1em}



  
  \date{June 17, 2024}


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\begin{abstract}
The Gent-McWilliams (GM) and Redi eddy parameterizations are essential
features to ocean climate models. GM helps to maintain stratification,
balancing the steepening of isopycnals by Ekman forcing and convection
with a relaxation that dissipates potential energy adiabatically. The
Redi parametrization represents unresolved isopycnal mixing of tracers,
while keeping diabatic mixing small. Due to its direct impact on the
simulated circulation, research has focused more on theories for the GM
than Redi coefficient, the latter typically being set equal to the
former without justification. Theories for the GM coefficient invariably
rely on an assumption of down-gradient eddy buoyancy fluxes, despite
that estimates of the latter in eddy-resolving models and nature often
show up-gradient tendencies. When tuned to values of O(500)
\$m\^{}2s\^{}\{-1\}\$, GM-based simulations are able to reproduce
observed ocean stratification. By contrast, observational estimates of
along-isopycnal mesoscale diffusivity (the Redi part) are typically an
order of magnitude larger. Setting the Redi coefficient to the too-small
GM value results in serious errors in biogeochemical tracers like
oxygen. Here I will describe an alternate approach that requires only
small changes to the existing infrastructure and resolves the
discrepancy in values. The idea relies on three results: (1)
materially-conserved tracers are mixed down-gradient, with diffusivities
well-estimated by mixing-length theory; (2) the mixing rate of tracers
and potential vorticity (PV) are very similar; (3) eddy PV and buoyancy
fluxes are related through an integral relationship derived from
quasigeostrophic theory. Therefore, I argue that PV flux theories should
be applied to setting the Redi coefficient, with the GM coefficient
determined diagnostically. This idea is explored using a high-resolution
MITgcm simulation of an idealized Southern Ocean channel, run with 10
independent tracers, each driven by different mean gradients. The
tracers are used to extract an estimate of the mixing tensor, and hence
of the mixing coefficients in question.%
\end{abstract}%



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