This paper describes a reduced-order quasi-geostrophic coupled
ocean–atmosphere model that allows for an arbitrary number of atmospheric
and oceanic modes to be retained in the spectral decomposition. The
modularity of this new model allows one to easily modify the model physics.
Using this new model, coined the “Modular Arbitrary-Order Ocean-Atmosphere
Model” (

The atmosphere at mid-latitudes displays a variability on a wide range of
space scales and timescales, and in particular a low-frequency variability at
interannual and decadal timescales as suggested by the analyses of different
time series developed in the past years

Recently the impact of the coupling between the ocean and the atmosphere at
mid-latitudes on the atmospheric predictability

In the

In this article, we present a model that generalizes the

The model is composed of a two-layer quasi-geostrophic (QG) atmosphere,
coupled both thermally and mechanically to a QG shallow-water ocean layer, in
the

The Coriolis parameter

The equation of motion for the streamfunction

The time evolution of the atmosphere and ocean temperatures

The hydrostatic relation in pressure coordinates

The prognostic equations for these four fields are then non-dimensionalized
by dividing time by

All the parameters of the model equations used in the present work are listed
in Table

Values of the parameters of the model that are used in the analyses of Sect.

In non-dimensionalized coordinates

Analogously, the oceanic basis functions must be of the form

For example, the spectral truncation used by

For the given ranges of

Ordering the basis functions as in Eqs. (

Furthermore, the short-wave radiation or insolation is determined by

Substituting the fields in Eqs. (

To construct the dynamical equations of these variables, one has to compute
the various projections or inner products with the basis functions, for which
the following shorthand notation will be used:

As described by

Substituting the fields by Eqs. (

Two implementations of

The form of Eq. (

This section details some key results obtained with the model for various
levels of spectral truncation, with the set of parameter values given in
Table

For the atmospheric part of the model, a previous study

Number of variables, transient time, and effective runtime of the runs (in years).

Cross section of the attractors for various model resolutions. The
atmospheric and oceanic resolutions are both indicated above each panel. The
parameters are given in Table

Cross section of the attractors for various model resolutions
(continued from Fig.

This question is now addressed for the

Variance distributions of the

The first panel of Fig.

Regarding the question of convergence, the variability of the atmospheric
variables becomes quite stable as the resolution increases beyond

The impact of the resolution on the solutions can also be examined by
computing the variance of each variable of the barotropic and baroclinic
streamfunctions, since these are associated with the kinetic and potential
energy of the system

Variance distributions of the

Variance distributions of the

Let us now focus on the development of the LFV in these different model
configurations, and let us define the geopotential height difference

Variance distributions of the

Time series of the geopotential height difference (m) between
locations (

Time series of the geopotential height difference (continued from
Fig.

The climatologies of the atmospheric barotropic streamfunction expressed in
geopotential height further highlight the changes in the statistical
properties of the model as a function of resolution. As shown in
Figs.

The previous results point toward the important question of the optimal
resolution of the oceanic component needed to get a sufficiently
low-resolution model while keeping a dynamics with strong similarities to a
very high-resolution model. To answer this question, we have performed some
higher-resolution integrations, but on shorter time spans. The time span for
each integration is given in Table

The variance distributions of the oceanic streamfunction variables (see
Fig.

Climatologies for the geopotential height field

Climatologies for the geopotential height field

However, the comparison between the atm.

Climatologies for the oceanic streamfunction field

Climatologies for the oceanic streamfunction field

Finally, the dynamics of the model for the various resolutions are also
illustrated in the videos provided as supplementary material. These videos
depict the time evolution of the streamfunction and temperature fields, as
well as the geopotential height difference and the three-dimensional
phase-space projection shown in Figs.

A new reduced-order coupled ocean–atmosphere model is presented, extending
the low-resolution versions previously published

Variance distributions of the

In the present work, we have studied the impact of the resolution on the
model solution's dynamics, by investigating the properties of the attractors
and the variance distributions in both the oceanic and atmospheric
components. The conclusion that can be drawn is that the convergence of the
atmospheric component of the system is quite fast (as noted in

Climatologies for the oceanic streamfunction

The robustness of the LFV pattern, one of the most interesting features of
the model, has also been explored. As it turns out, a LFV is still present in
a large portion of the model configurations explored (not in

Another interesting finding is the change of structure of the climatologies
of the ocean gyres when choosing even or odd wave numbers (

Finally, the aim of the model is to study the effects of specific physical interaction mechanisms between the ocean and the atmosphere on the mid-latitude climate, both at large and intermediate scales. The modular design of the code of the model is adapted to such purposes, with the possibility of implementing new components, such as oceanic active transport, time-dependent forcings, or salinity fields.

In the formulae of the inner products of the atmospheric modes,

In the following, we consider the ordering of the basis function used in
Eqs. (

These coefficients correspond to the eigenvalues of the Laplacian operator
acting on the spectral expansion basis functions:

These coefficients are needed to evaluate the contribution of the

These coefficients are given by

These coefficients are given by

These coefficients encode the inner products between the atmospheric and
oceanic basis functions:

These coefficients are related to the forcing of the ocean on the atmosphere.
They are given by the formula

These coefficients are related to the forcing of the atmosphere on the ocean.
They are given by

These coefficients identify with the eigenvalues of the Laplacian acting on
the oceanic basis functions:

These coefficients are needed to evaluate the contribution of the

These coefficients are given by

These coefficients are given by

These coefficients are related to the short-wave radiative forcing of the
ocean and are given by

The system of non-dimensionalized ODEs for the model variables is encoded in
the model tensor

The components of the tensor for the atmosphere streamfunction are given by

The atmospheric temperature equations are determined by the tensor elements

The components of the tensor for the ocean streamfunction are

Finally, the equations for the ocean temperature are determined by

The time evolution of the model dynamics is illustrated by a set of videos,
which are available online at

This work is partly supported by the Belgian Federal Science Policy Office
under contract BR/121/A2/STOCHCLIM. The figures and videos have been prepared
with the Matplotlib software