Due to the massive disparity between the largest and smallest eddies in the atmosphere and ocean, it is not possible to simulate these flows by explicitly resolving all scales on a computational grid. Instead the large scales are explicitly resolved, and the interactions between the unresolved subgrid turbulence and large resolved scales are parameterised. If these interactions are not properly represented then an increase in resolution will not necessarily improve the accuracy of the large scales. This has been a significant and long-standing problem since the earliest climate simulations. Historically subgrid models for the atmosphere and ocean have been developed in isolation, with the structure of each motivated by different physical phenomena. Here we solve the turbulence closure problem by determining the parameterisation coefficients (eddy viscosities) from the subgrid statistics of high-resolution quasi-geostrophic atmospheric and oceanic simulations. These subgrid coefficients are characterised into a set of simple unifying scaling laws, for truncations made within the enstrophy-cascading inertial range. The ocean additionally has an inverse energy cascading range, within which the subgrid model coefficients have different scaling properties. Simulations adopting these scaling laws are shown to reproduce the statistics of the reference benchmark simulations across resolved scales, with orders of magnitude improvement in computational efficiency. This reduction in both resolution dependence and computational effort will improve the efficiency and accuracy of geophysical research and operational activities that require data generated by general circulation models, including weather, seasonal, and climate prediction; transport studies; and understanding natural variability and extreme events.

Eddies in the atmosphere and ocean range in size from thousands of kilometres
down to the millimetre scale, with energy and enstrophy transferred over
these scales via complex non-linear inter-eddy interactions

The effect that the small unresolved subgrid scales have on the large
resolved scales is typically parameterised by defining a form of eddy
viscosity. In most subgrid models, including the most widely celebrated and
adopted ones

As in general it is only possible to parameterise the statistical effects of
the subgrid eddies

We use the method of

Historically subgrid models for the atmosphere and ocean have been developed
in isolation, with the derivation of the functional forms of the subgrid
models often motivated by very different physical phenomena. Here we provide
evidence that the effects of subgrid turbulence in the atmosphere and ocean
actually have much in common. When non-dimensionalised appropriately, subgrid
coefficients calculated from atmospheric

Here we present a first systematic comparison of subgrid models of QG
turbulence in the atmosphere and ocean, and develop simple unifying scaling
laws that represent both fluid media within their enstrophy cascading
inertial ranges. A large set of simulations is analysed, which covers a broad
range of flow parameters, including an order of magnitude change in the
Rossby radius of deformation and the energy-containing scale. By focussing on
the enstrophy cascading inertial range in both media, the large number of
simulations and wide parameter range has enabled the establishment of robust
scaling laws. In Sect.

The atmospheric and oceanic flows are generated by solving the two-level QG
potential vorticity equation (QGPVE). The numerical integration of the QGPVE
is a computationally efficient means of simulating geophysical flows. It
captures the essential dynamics of baroclinic and barotropic instabilities,
and the interaction of coherent structures with inhomogeneous Rossby wave
turbulence

The two-level QG, equations of motion in physical space are

In our study we solve Eq. (

In Eq. (

All simulations are driven toward a mean state

By definition the bare dissipation,

In the benchmark atmospheric simulations, the Rossby radius of deformation

In the initial benchmark oceanic simulation the Rossby radius is

The strength of the flow field on each level is quantified by the potential
enstrophy flux, and is required for scaling the magnitude of the eventual
subgrid coefficients. The enstrophy flux,

Instantaneous fields and climate states of the benchmark
simulations. Contours of instantaneous eddy (non-zonal) stream function, and
vectors of instantaneous velocity (wind/current) on the upper level of the

Spectral properties of the benchmark simulations. Potential
enstrophy flux spectra on the upper vertical level (level

The wavenumber extent of the large energy-containing scales is required for
scaling the spectral slope of the subgrid coefficients. Within the inertial
ranges the external forcing and dissipation are negligible, and the transfer
of energy is dominated by non-linear triadic interactions

Using a series of the above-discussed simulations, we study the inter-eddy
interactions by removing vortices smaller than a certain cut-off size, or
equivalently larger than a specified truncation wavenumber (

The resolution of a LES is lower than the associated
benchmark simulation, and confined to the resolved scale wavenumber set

The QDIA closure provides the theoretical justification for modelling the
subgrid tendency for a particular wavenumber pair as a function of the
resolved fields at only that same wavenumber pair

Backscatter is the physical process by which kinetic energy is transferred
from small to large scales. The subgrid model in
Eq. (

Real component of the upper diagonal subgrid eddy viscosities.
Anisotropic drain eddy viscosity at

For the atmosphere the subgrid model coefficients are presented at a
truncation of

We now consider the drain eddy viscosity in the ocean at the same resolution
of

The self-similarity of the eddy viscosities is most clearly illustrated by
the isotropised (averaged over zonal wavenumber

We have calculated the subgrid parameterisation coefficients (eddy
viscosities) for the atmosphere and ocean at various resolutions (

First, we present the power exponents of the drain eddy viscosities
(

Scaling of the isotropic eddy viscosities. Slope of the

Equivalent powers of the Laplacian for the subgrid net eddy
viscosity of atmospheric and oceanic simulations at various angular grid
spacings (

Scaling laws for the maximum values are again non-dimensionalised using the
energy-containing wavenumber, and additionally a timescale based on the
potential enstrophy flux

These scaling laws allow us to determine the drain and backscatter terms at
the desired resolution (

We now determine if LES with subgrid models defined by the eddy viscosities
presented above, can replicate the statistics of the higher-resolution
benchmark simulations. The equation governing the LES is equivalent to that
of the benchmark simulation in Eq. (

Scale by scale comparison of the benchmark simulation (dashed line)
to the LES variants (red solid line). Kinetic energy spectra at the upper
level of the

We compare the benchmark simulation results to LES comprising of both stochastic and
deterministic subgrid models, with the model coefficients in their original
anisotropic form (as in Fig.

The atmospheric benchmark simulation of maximum wavenumber

A general stochastic modelling approach

The scaling laws developed here can be implemented directly into spectral
simulations, and are expected to improve the efficiency and accuracy of
numerical weather and climate simulations

Finally, the stochastic modelling approach adopted here is not confined
to fluid mechanics but can also be used to represent non-linear interactions in
any classical multi-scale dynamical system

We acknowledge the Commonwealth Scientific Industrial Research Organisation for funding this research. Edited by: H. J. Fernando Reviewed by: three anonymous referees