Loose, Nora, Gustavo Marques, Alistair Adcroft, Scott D Bachman, Stephen M Griffies, Ian Grooms, Robert Hallberg, and Malte Jansen, December 2023: Comparing two parameterizations for the restratification effect of mesoscale eddies in an isopycnal ocean model. Journal of Advances in Modeling Earth Systems, 15(12), DOI:10.1029/2022MS003518. Abstract
There are two distinct parameterizations for the restratification effect of mesoscale eddies: the Greatbatch and Lamb (1990, GL90, https://journals.ametsoc.org/view/journals/phoc/20/10/1520-0485_1990_020_1634_opvmom_2_0_co_2.xml?tab_body=abstract-display) parameterization, which mixes horizontal momentum in the vertical, and the Gent and McWilliams (1990, GM90, https://journals.ametsoc.org/view/journals/phoc/20/1/1520-0485_1990_020_0150_imiocm_2_0_co_2.xml) parameterization, which flattens isopycnals adiabatically. Even though these two parameterizations are effectively equivalent under the assumption of quasi-geostrophy, GL90 has been used much less than GM90, and exclusively in z-coordinate models. In this paper, we compare the GL90 and GM90 parameterizations in an idealized isopycnal coordinate model, both from a theoretical and practical perspective. From a theoretical perspective, GL90 is more attractive than GM90 for isopycnal coordinate models because GL90 provides an interpretation that is fully consistent with thickness-weighted isopycnal averaging, while GM90 cannot be entirely reconciled with any fully isopycnal averaging framework. From a practical perspective, the GL90 and GM90 parameterizations lead to extremely similar energy levels, flow and vertical structure, even though their energetic pathways are very different. The striking resemblance between the GL90 and GM90 simulations persists from non-eddying through eddy-permitting resolution. We conclude that GL90 is a promising alternative to GM90 for isopycnal coordinate models, where it is more consistent with theory, computationally more efficient, easier to implement, and numerically more stable. Assessing the applicability of GL90 in realistic global ocean simulations with hybrid coordinate schemes should be a priority for future work.
We describe an idealized primitive-equation model for studying mesoscale turbulence and leverage a hierarchy of grid resolutions to make eddy-resolving calculations on the finest grids more affordable. The model has intermediate complexity, incorporating basin-scale geometry with idealized Atlantic and Southern oceans and with non-uniform ocean depth to allow for mesoscale eddy interactions with topography. The model is perfectly adiabatic and spans the Equator and thus fills a gap between quasi-geostrophic models, which cannot span two hemispheres, and idealized general circulation models, which generally include diabatic processes and buoyancy forcing. We show that the model solution is approaching convergence in mean kinetic energy for the ocean mesoscale processes of interest and has a rich range of dynamics with circulation features that emerge only due to resolving mesoscale turbulence.
We document the configuration and emergent simulation features from the Geophysical Fluid Dynamics Laboratory (GFDL) OM4.0 ocean/sea‐ice model. OM4 serves as the ocean/sea‐ice component for the GFDL climate and Earth system models. It is also used for climate science research and is contributing to the Coupled Model Intercomparison Project version 6 Ocean Model Intercomparison Project (CMIP6/OMIP). The ocean component of OM4 uses version 6 of the Modular Ocean Model (MOM6) and the sea‐ice component uses version 2 of the Sea Ice Simulator (SIS2), which have identical horizontal grid layouts (Arakawa C‐grid). We follow the Coordinated Ocean‐sea ice Reference Experiments (CORE) protocol to assess simulation quality across a broad suite of climate relevant features. We present results from two versions differing by horizontal grid spacing and physical parameterizations: OM4p5 has nominal 0.5° spacing and includes mesoscale eddy parameterizations and OM4p25 has nominal 0.25° spacing with no mesoscale eddy parameterization.
MOM6 makes use of a vertical Lagrangian‐remap algorithm that enables general vertical coordinates. We show that use of a hybrid depth‐isopycnal coordinate reduces the mid‐depth ocean warming drift commonly found in pure z* vertical coordinate ocean models. To test the need for the mesoscale eddy parameterization used in OM4p5, we examine the results from a simulation that removes the eddy parameterization. The water mass structure and model drift are physically degraded relative to OM4p5, thus supporting the key role for a mesoscale closure at this resolution.
Jansen, Malte, Alistair Adcroft, and Sina Khani, et al., August 2019: Towards an energetically consistent, resolution aware parameterization of ocean mesoscale eddies. Journal of Advances in Modeling Earth Systems, 11(8), DOI:10.1029/2019MS001750. Abstract
A subgrid‐scale eddy parameterization is developed which makes use of an explicit eddy kinetic energy (EKE) budget and can be applied at both “non‐eddying” and “eddy permitting” resolutions. The subgrid‐scale eddies exchange energy with the resolved flow in both directions via a parameterization of baroclinic instability (based on the established formulation of Gent and McWilliams) and bi‐harmonic and negative Laplacian viscosity terms. This formulation represents the turbulent cascades of energy and enstrophy consistent with our current understanding of the turbulent eddy energy cycle. At the same time, the approach is simple and general enough to be readily implemented in ocean climate models, without adding significant computational cost.
The closure has been implemented in the Modular Ocean Model (MOM6) and tested in the “Neverworld” configuration, which employs an idealized analytically defined topography designed as a testbed for mesoscale eddy parameterizations. The parameterization performs well over a range of resolutions, seamlessly connecting non‐eddying and eddy resolving regimes.
General circulation models use subgrid‐scale (SGS) parameterizations to represent the effects of unresolved mesoscale eddies on large‐scale motions. Most of the current SGS parameterizations are based on a theoretical understanding of transient eddies, where the mean flow is a temporal average. Here, we use a spatial filtering analysis to better understand the scale‐dependent characteristics of the SGS fluxes. Specifically, we apply the filtering approach to diagnose SGS eddy volume fluxes and eddy velocity scales in a hierarchy of model configurations from a flat‐bottom channel to an idealized Southern Hemisphere. Importantly, SGS volume fluxes include significant contributions from standing meanders; unlike for transient eddies, the vertically integrated SGS volume flux does not necessarily integrate to zero. To accommodate net vertically integrated eddy fluxes, we define a SGS eddy diffusivity based on planetary potential vorticity (PV) diffusion. We diagnose the transient and standing contributions to SGS fluxes and associated effective diffusivities. In the presence of bottom topography or continental barriers the standing component of the PV diffusivity becomes dominant at large filter scales in the westerly wind region, while the transient component remains dominant in the easterly wind region. Our results suggest that the diagnosed PV diffusivity can be parameterized using mixing length theory based on a priori estimates of SGS velocity and length scales.
Chai, Junyi, Malte Jansen, and Geoffrey K Vallis, August 2016: Equilibration of a baroclinic planetary atmosphere toward the limit of vanishing bottom friction. Journal of the Atmospheric Sciences, 73(8), DOI:10.1175/JAS-D-15-0329.1. Abstract
This paper discusses whether and how a baroclinic atmosphere can equilibrate with very small bottom friction in a dry, primitive equation, general circulation model. The model is forced by a Newtonian relaxation of temperature to a prescribed temperature profile, and it is damped by a linear friction near the lower boundary. When friction is decreased by four orders of magnitude, kinetic energy dissipation by friction gradually becomes negligible, while “energy recycling” becomes dominant. In this limit kinetic energy is converted back into potential energy at the largest scales, thus closing the energy cycle without significant frictional dissipation. The momentum fluxes are of opposite sign in the upper and lower atmosphere: in the upper atmosphere, eddies converge momentum into the westerly jets, however, in the lower atmosphere, the eddies diverge momentum out of the westerly jets. The secondary circulation driven by the meridional eddy momentum fluxes thus acts to increase the baroclinicity of the westerly jet. This regime may be relevant for the Jovian atmosphere, where the frictional time scale may be much larger than the radiative damping time scale.
Jansen, Malte, and R Ferrari, January 2015: Diagnosing the Vertical Structure of the Eddy Diffusivity in Real and Idealized Atmospheres. Quarterly Journal of the Royal Meteorological Society, 141(687), DOI:10.1002/qj.2387. Abstract
Earth's extra-tropical troposphere is equilibrated by turbulent eddy fluxes of potential temperature and momentum. The equilibrated state has the remarkable characteristic that isentropic slopes leaving the surface in the sub-tropics reach the tropopause near the poles. It has been speculated that turbulent eddy fluxes maintain this state for a wide range of radiative forcing and planetary parameters. In a previous study the authors showed that this state needs to be associated with an eddy diffusivity of Ertel potential vorticity that is largest at the surface and decays through the troposphere to approximately zero at the tropopause. This result is confirmed in this study using atmospheric reanalysis and idealized numerical simulations. However, it is also shown that the vertical profile of the eddy diffusivity can change, resulting in different isentropic slopes and climates. This is illustrated with a series of idealized numerical simulations with varying planetary scales and rotation rates.
It has recently been proposed to formulate eddy diffusivities in ocean models based on a mesoscale eddy kinetic energy (EKE) budget. Given an appropriate length scale, the mesoscale EKE can be used to estimate an eddy diffusivity based on mixing length theory. This paper discusses some of the open questions associated with the formulation of an EKE budget and mixing length, and proposes an improved energy budget-based parameterization for the mesoscale eddy diffusivity. A series of numerical simulations is performed, using an idealized flat-bottomed β-plane channel configuration with quadratic bottom drag. The results stress the importance of the mixing length formulation, as well as the formulation for the bottom signature of the mesoscale EKE, which is important in determining the rate of EKE dissipation. In the limit of vanishing planetary vorticity gradient, the mixing length is ultimately controlled by bottom drag, though the frictional arrest scale predicted by barotropic turbulence theory needs to be modified to account for the effects of baroclinicity. Any significant planetary vorticity gradient, β, is shown to suppress mixing, and limit the effective mixing length to the Rhines scale. While the EKE remains moderated by bottom friction, the bottom signature of EKE is shown to decrease as the appropriately non-dimensionalized friction increases, which considerably weakens the impact of changes in the bottom friction compared to barotropic turbulence. For moderate changes in the bottom-friction, eddy fluxes are thus reasonably well approximated by the scaling relation proposed by Held, I.M., Larichev, V.D., 1996. A scaling theory for horizontally homogeneous baroclinically unstable ow on a beta plane. J. Atmos. Sci. 53, 946–952., which ignores the effect of bottom friction.
It has recently been proposed to formulate eddy diffusivities in ocean models based on a mesoscale eddy kinetic energy (EKE) budget. Given an appropriate length scale, the mesoscale EKE can be used to estimate an eddy diffusivity based on mixing length theory. This paper discusses some of the open questions associated with the formulation of an EKE budget and mixing length, and proposes an improved energy budget-based parameterization for the mesoscale eddy diffusivity. A series of numerical simulations is performed, using an idealized flat-bottomed β-plane channel configuration with quadratic bottom drag. The results stress the importance of the mixing length formulation, as well as the formulation for the bottom signature of the mesoscale EKE, which is important in determining the rate of EKE dissipation. In the limit of vanishing planetary vorticity gradient, the mixing length is ultimately controlled by bottom drag, though the frictional arrest scale predicted by barotropic turbulence theory needs to be modified to account for the effects of baroclinicity. Any significant planetary vorticity gradient, β, is shown to suppress mixing, and limit the effective mixing length to the Rhines scale. While the EKE remains moderated by bottom friction, the bottom signature of EKE is shown to decrease as the appropriately non-dimensionalized friction increases, which considerably weakens the impact of changes in the bottom friction compared to barotropic turbulence. For moderate changes in the bottom-friction, eddy fluxes are thus reasonably well approximated by the scaling relation proposed by Held and Larichev (1996), which ignores the effect of bottom friction.
Zurita-Gotor, Pablo, Isaac M Held, and Malte Jansen, September 2015: Kinetic energy-conserving hyperdiffusion can improve low-resolution atmospheric models. Journal of Advances in Modeling Earth Systems, 7(3), DOI:10.1002/2015MS000480. Abstract
Motivated by findings that energetically-consistent subgrid dissipation schemes can improve eddy-permitting ocean simulations, this work investigates the impact of the subgrid dissipation scheme on low-resolution atmospheric dynamical cores. A kinetic energy-conserving dissipation scheme is implemented in the model adding a negative viscosity term that injects back into the eddy field the kinetic energy dissipated by horizontal hyperdiffusion. The kinetic energy-conserving scheme enhances numerical convergence when horizontal resolution is changed with fixed vertical resolution and gives superior low-resolution results. Improvements are most obvious for eddy kinetic energy but also found in other fields, particularly with strong or little scale-selective horizontal hyperdiffusion. One advantage of the kinetic energy-conserving scheme is that it reduces the sensitivity of the model to changes in the subgrid dissipation rate, providing more robust results.
Ferrari, R, and Malte Jansen, et al., June 2014: Antarctic sea ice control on ocean circulation in present and glacial climates. Proceedings of the National Academy of Sciences, 111(24), DOI:10.1073/pnas.1323922111. Abstract
In the modern climate, the ocean below 2 km is mainly filled by waters sinking into the abyss around Antarctica and in the North Atlantic. Paleoproxies indicate that waters of North Atlantic origin were instead absent below 2 km at the Last Glacial Maximum, resulting in an expansion of the volume occupied by Antarctic origin waters. In this study we show that this rearrangement of deep water masses is dynamically linked to the expansion of summer sea ice around Antarctica. A simple theory further suggests that these deep waters only came to the surface under sea ice, which insulated them from atmospheric forcing, and were weakly mixed with overlying waters, thus being able to store carbon for long times. This unappreciated link between the expansion of sea ice and the appearance of a voluminous and insulated water mass may help quantify the ocean’s role in regulating atmospheric carbon dioxide on glacial–interglacial timescales. Previous studies pointed to many independent changes in ocean physics to account for the observed swings in atmospheric carbon dioxide. Here it is shown that many of these changes are dynamically linked and therefore must co-occur.
In the near future we expect the resolution of many IPCC-class ocean models to enter the “eddy-permitting” regime. At this resolution models can produce reasonable eddy-like disturbances, but can still not properly resolve geostrophic eddies at all relevant scales. Adequate parameterizations representing sub-grid eddy effects are thus necessary. Most eddy-permitting models presently employ some kind of hyper-viscosity, which is shown to cause a significant amount of energy dissipation. However, comparison to higher resolution simulations shows that only enstrophy, but almost no energy, should be dissipated below the grid-scale. As a result of the artificial energy sink associated with viscous parameterizations, the eddy fields in eddy permitting models are generally not energetic enough.
To overcome this problem, we propose a class of sub-grid parameterizations which dissipate enstrophy but little or no energy. The idea is to combine a standard hyperviscous closure with some mechanism to return dissipated energy to the resolved flow. Enstrophy dissipation remains ensured because the energy is returned at larger scales. Two simple ways to return the energy are proposed: one using a stochastic excitation and one using a negative Laplacian viscosity. Both approaches are tested in an idealized two-layer quasi-geostrophic model. Either approach is shown to greatly improve the solutions in simulations with typical eddy-permitting resolutions. The adaptation of the proposed parameterization for use in realistic ocean models is discussed.