A geostrophic eddy energy dissipation rate due to the interaction of the large-scale wind field and mesoscale ocean currents, or

Satellite altimetry data have revealed an ocean surface scattered with geostrophic eddies

The representation of mesoscale eddies in coarse resolution ocean models is usually carried out using the Gent–McWilliams (GM) parameterisation

With all this in mind, a new fleet of GM style eddy parameterisations have been developed that aim to be more energetically consistent

Whilst energy budget-based eddy parameterisations offer improvements, there are current uncertainties surrounding the dissipation rate of eddy energy, which will feed back into uncertainties in the GM coefficient. It was revealed by

One important dissipation mechanism of eddy energy is relative wind stress, a process that can directly spin down mesoscale eddies by applying surface friction

In this paper we will derive a constrained eddy energy dissipation rate due to relative wind stress damping, validating this approach against a numerical model. In Sect.

The first objective of the theoretical framework is to derive an analytical expression that approximates the damping of eddy energy due to relative wind stress. This can be done by making some assumptions on eddy shape and wind profile.

A comprehensive study by

Idealised Gaussian eddy with anticyclonic rotation:

Recall the bulk formula for relative wind stress in Eq. (

The next step in deriving the analytical expression for relative wind stress damping is to find the work done by winds on the surface geostrophic motion. This is done by taking the dot product of relative wind stress and surface geostrophic velocities, and making use of Eq. (

To find the analytical expression, we put analytical equations for geostrophic velocities Eq. (

Horizontal views showing differences between relative and absolute wind stress calculated over an idealised Gaussian anticyclonic eddy:

Mesoscale ocean eddies take on a complex vertical structure, making them hard to accurately model. However, studies such as the one by

Two-layer shallow water equations are used to describe the baroclinic eddy:

Before progressing with the derivation of the baroclinic eddy energy equation, some points are discussed first. The two-layer shallow water equations in the form shown in Eq. (

The derivation of the two-layer energy equation is done as follows: Equation (

We now want to acquire an analytical equation for Eq. (

After integrating Eq. (

The numerical experiments were performed using the hydrostatic MIT general circulation model

The numerical model is set up on an

The baroclinic eddy is initialised using analytical equations. The stratification is given by a 3D temperature field of the form

So that an adequate comparison of the two-layer baroclinic eddy in Sect.

The wind field used in this setup is uniform in one direction and is designed to represent a large-scale background wind. (See

When the model is first initialised it is allowed to run for 10 d with zero wind forcing. This allows any inertial waves to die down and also lets the equations of motion form a balance that could be slightly different from that of geostrophy. After this adjustment phase, the wind forcing is turned on and the model is run for 400 d.

Key experimental parameters.

Time series of total eddy energy,

To validate the evolution of baroclinic eddy energy in the analytical model (Sect.

Horizontal views of MITgcm surface geostrophic relative vorticity normalised by Coriolis frequency in an anticyclonic eddy for absolute (top) and relative (bottom) wind stress at days

The time evolution of analytical eddy energy is achieved by time-stepping Eq. (

In this section we present our first set of results, comparing the time evolution of the analytical and numerical eddy energy budgets. Figure

We first focus on the first 150 d of the ACE (Fig.

Beyond day 150, Pred and MIT time series begin to diverge, with MIT undergoing a sudden reduction in total energy of around

Similar results are also observed for the CE (Fig.

Time series of total wind power input in relative wind stress simulation,

In this section we have compared the evolution of total eddy energy between an analytical and a numerical model. The results tell us that a two-layer analytical model can reasonably explain the evolution of total eddy energy in the MIT simulation. However, the agreement between both models diminishes due to an instability process in the MIT simulation. Nevertheless, we find the timescale of around

Global maps between the latitudes of 70

An eddy energy dissipation rate due to relative wind stress takes the form

Global maps between the latitudes of 70

We approach the computation of the dissipation rate

A global dissipation rate for relative wind stress damping,

Following

Figure

Figure

A global dissipation rate is now presented, culminating from the variable climatology data calculated in Sect.

Beginning with the Rossby radius of deformation

Kernel density estimation of

Figure

Contrasting the two choices of eddy length scale is summarised using a density plot of

In this work we have presented a constrained eddy energy dissipation rate for a well-known and important mesoscale dissipation pathway, relative wind stress. Deriving this dissipation rate draws on our fundamental understanding of relative wind stress damping, vertical eddy structure, and eddy energy. The intention with this dissipation rate is for it to fit into an existing eddy energy budget-based eddy parameterisation (e.g. GEOMETRIC) and offer improvements to the relatively unconstrained and spatially homogenous dissipation rate currently employed.

Before the proposition of a dissipation rate, an approximate expression for relative wind stress damping, termed

The key component of this work lies in the proposed dissipation rate for eddy energy due to relative wind stress, outlined in Sect.

A global map of the dissipation rate is presented in Sect.

The dissipation rate is based on a simple energy budget derived from a two-layer analytical model, which by design neglects many phenomena that take place in the ocean, such as instabilities and wave dynamics. In the time evolution of total eddy energy (Fig.

An alternative approach to the one taken in this paper might be to employ a different representation for the first baroclinic mode, and therefore analytical and numerical setups. In this work we followed the ideas of

This study presents a constrained eddy energy dissipation rate due to relative wind stress damping. Although relative wind stress is not the only mechanism associated with eddy energy dissipation, its focus in this study is grounded in the effects it has on ocean dynamics and ocean processes

NCEP–NCAR Reanalysis 1 wind speed data are provided by the NOAA PSL (NOAA PSL,

All authors contributed to the conception and design of this work. TW worked on the analytical derivations and their numerical solutions, optimised model design, carried out formal analysis and figure production, and contributed to the writing (original and review). XZ provided supervision of the work, administered the project, assisted in solving the eigenvalue problem, and contributed to the writing (review and editing). DM provided supervision of the work, assisted in the analytical work and MITgcm setup, and contributed to the writing (review and editing). MJ provided supervision of the work and contributed to the writing (review and editing).

The authors declare that they have no conflict of interest.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors.

TW thanks XZ, DM, and MJ for their guidance and mentorship throughout this work. Further thanks goes to Julian Mak and an anonymous reviewer who provided constructive feedback and thought-provoking suggestions. The research presented in this paper was carried out on the High Performance Computing Cluster supported by the Research and Specialist Computing Support service at the University of East Anglia (UK). The authors thank the Open Access Team at the University of Reading (UK) for their assistance in organising the funding of this paper.

This work was funded by the Natural Environment Research Council through the EnvEast Doctoral Training Partnership (grant no. NE/L002582/1) and the European Union's Horizon 2020 research and innovation programme under grant agreement no. 101003536 (ESM2025 – Earth System Models for the Future).

This paper was edited by Bernadette Sloyan and reviewed by two anonymous referees.