Implementation and optimisation of geostrophic eddy parameterisations in ocean circulation models

Lead Research Organisation: University of Edinburgh
Department Name: Sch of Mathematics


The global ocean is populated by a vigorous and dynamic eddy field. This eddy field has a significant impact on the wider structure of the ocean circulation, but it is computationally extremely expensive to resolve directly. While recent simulations have allowed for direct eddy resolving calculations to be performed, equilibrated eddy resolving calculations are unlikely to become routine at any point in the near future. There therefore remains an urgent need for eddy parameterisation schemes, which aim to capture the aggregated effects of the eddy field in coarse resolution numerical models.

The development of ocean eddy closures remains a formidable challenge. However, one can apply fundamental principles in order to rule out eddy closures which lead to unphysical behaviour. For example, the eddies require energy in order to mix ocean properties, and must observe the principle of conservation of momentum. A somewhat more subtle property is that ocean eddies must, on average, mix out gradients in the fluid potential vorticity. Eddy parameterisations which fail to observe these constraints can be ruled out as a possible solution for the underlying eddy dynamics.

A particularly important challenge for ocean eddy parameterisations is the accurate reproduction of ocean sensitivities to forcing changes. These responses have implications for the long term ocean response to climate change. Recent studies indicate that the Southern Ocean may exhibit reduced responses to wind forcing changes when the influence of the eddies is well sufficiently resolved. Coarse resolution models, which parameterise the eddies and do not resolve them directly, often give wildly incorrect predictions for the ocean responses.

This study proposes the implementation of a suite of new and existing ocean eddy parameterisations. The parameterisations are to be implemented in a simplified context (in a "quasi-geostrophic" model), and then in a full ocean circulation model. This research has three key novel aspects. First, optimisation techniques will be employed in order to assess the performance of the closures in the simplified quasi-geostrophic case. These techniques can be used to identify the best possible configuration of a given eddy parameterisation scheme, enabling the best-case performance of the scheme to be rigorously identified. This will provide a robust comparison of a broad range of possible approaches for eddy parameterisation. Secondly, the research aims to impose the three physical principles, imposed by energetic constraints, momentum conservation, and the need to mix potential vorticity, simultaneously. Thirdly, the research will investigate the performance of a broad range of parameterisations in determining the ocean sensitivity to wind forcing changes.

Particularly important questions to be addressed are: How important are the fundamental physical constraints in controlling the ocean eddy dynamics? Can the restoration of these constraints in ocean eddy parameterisation schemes lead to improved coarse resolution simulations?

Planned Impact

This project primarily involves the practical application of abstract theory in order to implement and study new geostrophic ocean eddy parameterisations. The primary impacts will be within the academic community.

However, ocean circulation models form an essential component of complex models used to study and predict the impact of future climate change. The representation of ocean eddies in these models can have a significant influence upon the resulting model predictions. A successful conclusion of the research has the potential to generate significant impacts on the ways in which ocean eddies are modelled. Such advancements would be of benefit to climate modellers in general, and to modellers at the Hadley centre in particular.

While the theoretical background of the research is somewhat esoteric in nature, the wider implications of the research itself are not. The response of the ocean circulation to forcing changes forms an important ingredient in our understanding of the wider climate system. Outreach activities will be undertaken to communicate the links between the new research and its wider implications.

This research also involves the application of recent developments in automated code generation, specifically including the application of automated adjoining and optimisation technologies. This research provides an opportunity for interdisciplinary communication, enabling novel applications of the new automated code generation technologies to the fields of physical oceanography and ocean modelling. The research additionally has the potential to raise awareness of the technical and performance demands of ocean models within the automated code generation community, thereby ensuring its continued relevance for future applications in oceanographic research.

The impact of the research will be increased via:
- Publication of articles in high quality international journals.
- Presentation of key results at international conferences.
- Attendance at smaller specialised meetings within the United Kingdom.
- Active engagement with developers of automated code generation software.
- The regular maintenance of an online blog, and active participation in social media.
- The online distribution of model figures and videos.
- The attraction of mathematics students to NERC related science activities in general, and to oceanographic research in particular.


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Tamarin T (2016) A Geometric Interpretation of Eddy Reynolds Stresses in Barotropic Ocean Jets in Journal of Physical Oceanography

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Marshall D (2017) Eddy saturation and frictional control of the Antarctic Circumpolar Current in Geophysical Research Letters

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Mak J (2017) Vortex disruption by magnetohydrodynamic feedback in Physical Review Fluids

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Maddison J (2017) Optimal Constrained Interpolation in Mesh-Adaptive Finite Element Modeling in SIAM Journal on Scientific Computing

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De Ridder S (2018) Full wavefield inversion of ambient seismic noise in Geophysical Journal International

Description The project studied the development of new closures for ocean mesoscale turbulence, and their implementation in numerical ocean models. This was achieved through a combination of studies of ocean turbulence using simplified models and novel diagnostic approaches, and the direct implementation of a new closure in numerical models. A key aim of the project was to develop and test parameterisations in a configuration in which they were previously known to perform poorly - in the representation of the response of Southern Ocean transports with respect to changes in surface wind forcing.

Initially the properties of ocean turbulence were studied in a simplified setting, using a quasi-geostrophic ocean model. This model represents key features of ocean turbulence on the ocean mesoscale (scales of approximately 10km and above), while removing many complicating features. Key parameters appearing in standard forms of ocean turbulence parameterisations were diagnosed using a novel diagnostic approach. An optimisation problem for the parameters was formed, seeking parameter values which matched diagnosed properties of the turbulence in an optimal sense. This allowed the structure of these parameters to be diagnosed in detail, and further allowed the study of the influence of including information about turbulent energetics.

It has been found in numerical studies which more accurately resolve ocean mesoscale turbulence that the transport of the Southern Ocean is relatively insensitive to changes in the strength of surface wind forcing - a principle known as eddy saturation. Existing ocean turbulence parameterisation schemes struggle to represent this process, and may for example suggest significant increases in Southern Ocean transport with increases in surface wind strength. Recent developments in ocean turbulence theory have suggested a new form of turbulence parameterisation which allows both the inclusion of information about the influence of turbulent energetics, and also observes key physical conservation principles. A new turbulence closure of this form was developed and implemented in an idealised numerical ocean model. Crucially the new parameterisation allows for a representation of a mechanism behind eddy saturation: an increased surface wind forcing increases the turbulent energy, which in turn strengthens the dynamical influence of this turbulence, and then allows it to more easily dissipate the additional momentum input by the wind. This led to a notable project success - a new parameterisation which was both able to represent eddy saturation effects, and also provided a representation of a plausible physical mechanism behind this process.

The new parameterisation was implemented initially in a highly idealised numerical model, where its performance was assessed against existing parameterisation schemes. A form of the parameterisation was then implemented in an ocean circulation model, MITgcm. In work that has been submitted for publication it has been found that many of the the key properties of the parameterisation, developed initially in a theoretical setting, and then transferred to an idealised numerical model, do indeed transfer to the much more complicated setting of a primitive equation ocean model.
Exploitation Route Through the research developed in this project it has been demonstrated that a new form of ocean mesoscale turbulence parameterisation is able to improve the representation of key large-scale ocean processes, and in particular to capture the principle of eddy saturation. This has important potential applications in climate modelling.

While it has been found that a representation of ocean eddy energetics allows for a more accurate representation of the eddy saturation effect, a key area of investigation left open is the detailed representation of eddy energetics. In particular, a large scale surface wind stress leads indirectly to generation of turbulent eddy energy, which then feeds back to oppose the wind stress which generated it. However an essential additional element of this process is the dissipation of eddy energy. For example strong dissipation of turbulent energy weakens the ability of the turbulence to oppose the wind, leading (counter-intuitively) to increased transport. A fully developed form of this parameterisation must therefore consider further aspects of the ocean turbulent energetics, and in particular consider mechanisms for dissipation of this energy.

The outputs of this project are being further developed in the NERC funded GEOMETRIC project, which is studying the development of the parameterised representation of the turbulent energetics in the ocean. This project further implements the new parameterisation in NEMO. This provides a natural pathway for this research to be applied to the study of long-term ocean responses to forcing changes, and to the study of future climate.
Sectors Environment

Description GEOMETRIC: Geometry and Energetics of Ocean Mesoscale Eddies and Their Representation in Climate models
Amount £461,687 (GBP)
Funding ID NE/R000999/1 
Organisation Natural Environment Research Council 
Sector Public
Country United Kingdom
Start 09/2017 
End 03/2022
Title Eddy diffusivity diagnostics data 
Description Coarse resolution numerical ocean models must typically include a parameterisation for mesoscale turbulence. A common recipe for such parameterisations is to invoke down-gradient mixing, or diffusion, of some tracer quantity, such as potential vorticity or buoyancy. However, it is well known that eddy fluxes include large rotational components which necessarily do not lead to any mixing; eddy diffusivities diagnosed from unfiltered fluxes are thus contaminated by the presence of these rotational components. 
Type Of Material Database/Collection of data 
Year Produced 2015 
Provided To Others? Yes  
Title Emergent eddy saturation diagnostic data 
Description Dataset for the article "Emergent eddy saturation from an energy constrained eddy parameterisation". 
Type Of Material Database/Collection of data 
Year Produced 2017 
Provided To Others? Yes  
Impact Associated with the publication "Emergent eddy saturation from an energy constrained eddy parameterisation", Mak, Marshall, Maddison, and Bachman, Ocean Modelling, 2017 
Title GEOMETRIC (MITgcm) diagnostic data 
Description Dataset for the article "Implementation of a geometrically and energetically constrained mesoscale eddy parameterisation in an ocean circulation model". 
Type Of Material Database/Collection of data 
Year Produced 2018 
Provided To Others? Yes  
Impact Associated with the submitted article "Implementation of a geometrically and energetically constrained mesoscale eddy parameterization in an ocean circulation model", Mak, Maddison, Marshall, Munday, submitted to Journal of Physical Oceanography 
Title Quasi-geostrophic double gyre force function data 
Type Of Material Database/Collection of data 
Year Produced 2015 
Provided To Others? Yes  
Title SUPERSEDED - Emergent eddy saturation diagnostic data 
Description Dataset for the article "Emergent eddy saturation from an energy constrained eddy parameterisation". Superceded by data set with doi 10.7488/ds/1715. 
Type Of Material Database/Collection of data 
Year Produced 2016 
Provided To Others? Yes  
Impact Associated with the publication "Emergent eddy saturation from an energy constrained eddy parameterisation", Mak, Marshall, Maddison, and Bachman, Ocean Modelling, 2017