GEOMETRIC outline¶

TL;DR [08 Jun 2023]: My versions of the GEOMETRIC codes for NEMO and/or MITgcm can be found in this repository. The official NEMO one may be found here, with thanks to Andrew Coward at NOC-Southampton. Current version does not support the newer RK3 time-step (still needs the leap-frog), but that is on a to-do list.

GEOMETRIC (Geometry and Energetics of Ocean Mesoscale Eddies and Their Rectified Impact on Climate) is an approach to representing the unresolved turbulent eddies in ocean climate models, first derived in [MMB12]. David Marshall’s page has an excellent outline and summary of GEOMETRIC, so this page will focus on outlining the details relating to the NEMO implementation.

The implementation of GEOMETRIC was done in NEMO by providing a new module ldfeke.f90 and adding appropriate calls and variables to ldftra.f90, step.f90 step_oce.f90 and nemogcm.f90. This was initially done in SVN version 8666, which is somewhere between the 3.6 stable and 4.0 beta, by myself and Gurvan Madec back in November 2017. The current implementation of GEOMETRIC is what may be considered GM-based [GM90] and follows the prescription described in [MMMM22]. The GEOMETRIC scaling gives \(\kappa_{\rm gm} = \alpha E (N / M^2)\) (see below for symbol definitions). While \(\alpha\) is prescribed and \(M\) and \(N\) are given by the coarse resolution ocean model, information relating to \(E\) is provided by a parameterised eddy energy budget. The recipe for GEOMETRIC then is as follows:

time-step the parameterised eddy energy budget to get \(E\) with info provided by the GCM
calculate the new \(\kappa_{\rm gm}\)
use the existing GM routines with new \(\kappa_{\rm gm}\) and time-step the GCM. Cycle as appropriate.

The current NEMO implementation considers an eddy energy field that varies in longitude, latitude and time (and so \(\kappa_{\rm gm}\) inherits this spatio-temporal dependence), given by

\[\frac{\mathrm{d}}{\mathrm{d} t} \int E\; \mathrm{d}z + \nabla \cdot \left( (\tilde{\boldsymbol{u}} - |c|\boldsymbol{e}_1 ) \int E\; \mathrm{d}z \right) = \int \kappa_{\rm gm} \frac{M^4}{N^2}\; \mathrm{d}z - \lambda \int E\; \mathrm{d}z + \nu_E \nabla^2 \int E\; \mathrm{d}z,\]

(respecively, the time-evolution, advection, source, dissipation and diffusion of eddy energy), with \(\kappa_{\rm gm}\) calculated as

\[\kappa_{\rm gm} = \alpha \frac{\int E\; \mathrm{d}z}{\int \Gamma (M^2 / N)\; \mathrm{d}z} \Gamma(z).\]

The symbols are as follows:

symbol	definition	units
\(\alpha\)	eddy efficiency parameter non-dimensional, \(\|\alpha\| \leq 1\)	\(--\)
\(E\)	total eddy energy	\(m^2\ s^{-2}\)
\(M, N\)	mean horizontal and vertical buoyancy gradient	\(s^{-1}\)
\(\tilde{\boldsymbol{u}}\)	depth-mean flow	\(m^2\ s^{-1}\)
\(\|c\|\)	magnitude of long Rossby phase speed of 1st baroclinic mode	\(m^2\ s^{-1}\)
\(\kappa_{\rm gm}\)	Gent–McWilliams coefficient	\(m^2\ s^{-1}\)
\(\lambda\)	linear damping rate of eddy energy	\(s^{-1}\)
\(\nu_{E}\)	Laplacian diffusion of eddy energy	\(m^2\ s^{-1}\)

Advection¶

The advection of eddy energy is given in flux form and has a contribution from the depth-mean flow as well as a contribution associated with the westward propagation of eddies at the long Rossby phase speed (motivated by e.g. [CSS11] and [KM14]). The advection is by the barotropic mean flow already computed in NEMO, with a first order upwind scheme. The baroclinic Rossby wave speed is obtained by computing the eigenvalue associated with the first baroclinic mode (see e.g. eq. 6.11.8 of [Gil82]) and uses two subroutines (eke_rossby and eke_thomas) via the WKB expression given in [CdeSzoekeS+98] (their equation 2.2):

\[c_n \approx \frac{1}{n\pi} \int^0_{-H} N(z)\; \mathrm{d}z\]

and the long-phase speed that the total eddy energy is to be advected at is computed as (e.g. eq. 12.3.13 of [Gil82])

\[|c_p| \approx \frac{\beta}{f_0^2}c_1^2 = c_1^2 \frac{\cos\phi_0}{2\Omega R \sin^2 \phi_0}\]

In practice the expression diverges at the equator and the actual wave contribution to eddy energy advection as implemented in GEOMETRIC is bounded above by the magnitude tropical planetary wave phase speed (e.g. eq. 12.3.14 of [Gil82]), i.e.,

\[|c| = \min\left(|c_p|, \left|c_1/3\right|\right)\]

See here for usage and implementation details.

Note

As of Feb 2019 the removal of the routines to solve the tri-diagonal eigenvalue problem means the nn_wav_cal variable in namelist_cfg has been removed.

Source¶

The source of mesoscale eddy energy here is only from the slumping of neutral surfaces through the eddy induced velocity as parameterised by the GM scheme (note that it is positive-definite). These are straight-forwardly computed as is (rather than using the quasi-Stokes streamfunction) using the already limited slopes computed in NEMO. See here for implementation details.

Dissipation¶

The damping of eddy energy is linearly damped and the coefficient is specified in namelist_cfg as a time-scale in days (which is subsequently converted to per seconds in ldf_eke_init). There is an option to read in an externally prepared NetCDF file geom_diss_2D.nc that varies in longitude and latitude in anticipation of further investigation. See here for usage details, here for a sample Python Notebook to generate the file, and [MAD+22] and the associated Zenodo repository for some scripts to sample an estimate onto a grid onto a global grid (obtained from a finite element calculation, requires the vtk package in Python to probe the spherical immersed mesh).

Diffusion¶

The diffusion of eddy energy is through a Laplacian (cf. [EG08]), done through relevant copy and pasting of code that are in other NEMO modules. The GEOMETRIC scheme is actually stable (most likely because of the upwinding scheme). The diffusion may be switched off by setting rn_eke_lap = 0. in namelist_cfg which will bypass the relevant loop in ldf_eke.

CSS11: D. B. Chelton, M. G. Schlax, and R. M. Samelson. Global observations of nonlinear mesoscale eddies. Prog. Oceanog., 91:167–216, 2011. doi:10.1016/j.pocean.2011.01.002.
CdeSzoekeS+98: D. B. Chelton, R. A. de Szoeke, M. G. Schlax, K. El Naggar, and N. Siwertz. Geographical variability of the first baroclinic Rossby radius of deformation. J. Phys. Oceanogr., 28:433–459, 1998. doi:10.1175/1520-0485(1998)028<0433:GVOTFB>2.0.CO;2.
EG08: C. Eden and R. J. Greatbatch. Towards a mesoscale eddy closure. Ocean Modell., 20:223–239, 2008. doi:10.1016/j.ocemod.2007.09.002.
GM90: P. R. Gent and J. C. McWilliams. Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20:150–155, 1990. doi:10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2.
Gil82(1,2,3): A. E. Gill. Atmospheric-Ocean Dynamics. Academic Press, 1982.
KM14: A. Klocker and D. P. Marshall. Advection of baroclinic eddies by depth mean flow. Geophys. Res. Lett., 41:L060001, 2014. doi:10.1002/2014GL060001.
MAD+22: J. Mak, A. Avdis, T. W. David, H. S. Lee, Y. Na, and F. E. Yan. On constraining the mesoscale eddy energy dissipation time-scale. J. Adv. Model. Earth Syst., 14:e2022MS003223, 2022. doi:10.1029/2022MS003223.
MMMM22: J. Mak, D. P. Marshall, G. Madec, and J. R. Maddison. Acute sensitivity of global ocean circulation and heat content to eddy energy dissipation time-scale. Geophys. Res. Lett., 49(8):e2021GL097259, 2022. doi:10.1029/2021GL097259.
MMB12: D. P. Marshall, J. R. Maddison, and P. S. Berloff. A framework for parameterizing eddy potential vorticity fluxes. J. Phys. Oceanogr., 42:539–557, 2012. doi:10.1175/JPO-D-11-048.1.