A new framework is proposed for the evaluation of stochastic subgrid-scale parameterizations in the context of the Modular Arbitrary-Order Ocean-Atmosphere Model (MAOOAM), a coupled ocean–atmosphere model of intermediate complexity. Two physically based parameterizations are investigated – the first one based on the singular perturbation of Markov operators, also known as homogenization. The second one is a recently proposed parameterization based on Ruelle's response theory. The two parameterizations are implemented in a rigorous way, assuming however that the unresolved-scale relevant statistics are Gaussian. They are extensively tested for a low-order version known to exhibit low-frequency variability (LFV), and some preliminary results are obtained for an intermediate-order version. Several different configurations of the resolved–unresolved-scale separations are then considered. Both parameterizations show remarkable performances in correcting the impact of model errors, being even able to change the modality of the probability distributions. Their respective limitations are also discussed.

Climate models are not perfect, as they cannot encompass the whole world in
their description. Model inaccuracies, also called model errors, are
therefore always present

Recently, a revival of interest in stochastic parameterization methods for
climate systems has occurred, due to the availability of new mathematical
methods to perform the reduction of ordinary differential equations (ODEs)
systems: either based on the conditional
averaging

The usual way to test the effectiveness of a parameterization method is to
consider a well-known climate low-resolution model on which other methods
have already been tested. For instance, several methods cited above have been
tested on the Lorenz 96 model

Timescale separation, or the existence of a spectral gap, is a crucial ingredient on which numerous parameterization methods rely.

, for instance the evaluation of the MTV parameterization on barotropic and baroclinic modelsIn this work, we investigate two parameterizations in the context of the
MAOOAM (Modular Arbitrary-Order
Ocean-Atmosphere Model) ocean–atmosphere coupled model

The particular problem of the atmospheric impact on the ocean could be
addressed in this context as in

This paper is organized as follows: in Sect.

The Modular Arbitrary-Order Ocean-Atmosphere Model is a coupled
ocean–atmosphere model for midlatitudes. It is composed of a two-layer
atmosphere over a shallow-water ocean layer on a

MAOOAM model schematic representation.

The dynamical fields of the model include the atmospheric barotropic
streamfunction

By recasting these expansions into the partial differential equations of the
model, one obtains a set of ODEs for its coefficients

Now let us consider a more general system of ordinary differential equations:

More precisely a parameterization of the subsystem

The first one is based on Ruelle's response theory

As said above, the parameterization is based on Ruelle's response theory,
which quantifies the contribution of the perturbations

This approach is based on the singular perturbation methods that were
developed for the analysis of the linear Boltzmann equation in an asymptotic
limit

Note that in
homogenization theory

As MAOOAM is a model whose nonlinearities consist solely of
quadratic terms, the decomposition of Eq. (

For
two matrices

This decomposition can be chosen arbitrarily since the only requirement is
that

The definition of

In the framework of the MTV method, the measure of the

In both cases, the other terms

In

It is interesting to note that there is no a priori justification for one or
the other assumption. For both parameterization methods, the decomposition of
the unresolved dynamics could be based on Eq. (

The MTV and WL parameterizations described above are presented in more
detail in the Appendices

To take into account model errors or the impact of smaller scales, the
present implementation of MAOOAM allows for the addition of Gaussian white
noise in each components, resolved and unresolved, for both the ocean and the
atmosphere. It also includes the timescale separation parameter

The relative performance and the interesting features of the
parameterizations described in the previous section require us to consider
multiple versions and resolutions of the model. We thus shall consider in the
following two different resolutions. The first one is the 36-variable model
version considered in

We shall also consider different sets of parameters for the model
configuration. To control the results obtained with the code implementation
provided as a Supplement, we will compare our results with those obtained
in

Finally, for a given resolution and a given parameter set, multiple different
parameterization experiments can be designed, by using for example the
unresolved dynamics (

Unless otherwise specified, the subsequent results were obtained by
considering long trajectories lasting

The main parameters used in the parameterization experiments. For a description of the parameters,
see

Probability density functions (PDFs) of the dominant variables of
the system dynamics with the invariant manifold decomposition for the

Component averaged Kullback–Leibler divergence with respect to the distributions of the full coupled system, in the case of the wavenumber 2 atmospheric variables parameterization. The abbreviations “atm.” and “oc.” refer to atmospheric and oceanic variables.

For this model version, the atmospheric Fourier modes are denoted as

Probability density functions (PDFs) of the dominant variables of
the system dynamics as in Fig.

Correlation function of various variables for the wavenumber 2
parameterization and with the parameter set DDV2016. The correlation of
the full coupled system (

Probability density functions (PDFs) of the dominant variables of
the system dynamics as in Fig.

Correlation function of various variables for the wavenumber 2
parameterization and with the parameter set noLFV. The correlation of the
full coupled system (

Probability density functions (PDFs) of the dominant variables of
the system dynamics as in Fig.

We first consider a parameterization based on the presence in the VDDG model
of a genuine invariant manifold. As stated in

The code used in

In fact, for the present parameterization based on the invariant manifold, the expression of both methods is very close and coincides for an infinite timescale separation.

Both the MTV and the WL methods correct the oceanic variables better, whereas
the atmospheric variables seem to display very different dynamical
behaviors between the coupled and
uncoupled systems which are difficult to correct. As stated
in

As the present implementation allows for an arbitrary selection of the resolved–unresolved components, we shall now consider cases of the VDDG model version with different unresolved components.

We consider now a smaller set of unresolved variables

The unresolved variables

A first comment on the results obtained with that particular configuration is
that the WL method seems to be unstable for all the parameter sets
investigated. As stated in the Appendix Sect.

Probability density functions (PDFs) of the dominant variables of
the system dynamics as in Fig.

Correlation function of various variables for the wavenumber 2
parameterization and with the parameter set DV2017. The correlation of the
full coupled system (

Probability density functions (PDFs) of the dominant variables of
the system dynamics as in Fig.

Component averaged Kullback–Leibler divergence with respect to the distributions of the full coupled system, in the case of the wavenumber 2 atmospheric baroclinic variables' parameterization.

The quality of the solutions obtained with the MTV method alone, applied to
the system with the three parameter sets of Table

As depicted in Figs.

Probability density functions (PDFs) of the dominant variables of
the system dynamics as in Fig.

Component averaged Kullback–Leibler divergence with respect to the
distributions of the full coupled system, in the case of the wavenumber 2 and

The impact of the MTV parameterization on the correlation is also
particularly important, as seen in Figs.

Additional marginal PDF and autocorrelation function figures are available
for each variable of the system in the Supplement. The Kullback–Leibler
divergences (

Since the WL parameterization does not work in the current case, we cannot properly compare both methods. To do so, we shall consider two other parameterization configurations in the following sections.

Parameterization of the

Let us now consider that only the two baroclinic variables

The parameterizations correct quite well the ocean components, except for the MTV parameterization of the baroclinic component in the noLFV case. The MTV parameterization is better in the DDV2016 case, and the WL one is better in the two other cases.

The barotropic component of the atmosphere is only well corrected in the DV2017 case.
Additionally, the MTV fails to correct also the baroclinic PDFs in the two other cases. In fact,
the MTV method seems to only perform well in the DV2017 case. Looking to the divergence for
every variable (see the Supplement), we note that those underperformances are
due to the incorrect representation of the small-scale wavenumber 2 barotropic variables, namely
the

Interestingly, the PDFs of the dominant variables

Regarding the correlation functions, a first general comment is that, in this
configuration, the decorrelation time of the large scales (mode

Another possibility to test the WL parameterization is to remove the
aforementioned cubic interactions by considering that the wavenumber 2 modes
and the meridional mode

The global averaged Kullback–Leibler divergences are given in
Table

The PDFs of the three main variables (see
Fig.

A higher-resolution test with the MTV method has also been performed by
considering the

In the present work, we have introduced a new framework to test different
stochastic parameterization methods in the context of the ocean–atmosphere
coupled model MAOOAM. We have implemented two methods: (i) a homogenization
method based on the singular perturbation of Markovian operators and known as
MTV

Within this framework, we have considered two different model resolutions and
performed a model reduction. We have performed several reductions in the case
of a model version with 36 dimensions. We first parameterized atmospheric
modes related to the existence of an invariant manifold present in the
dynamics and the results previously obtained in

Additionally, we have found that these methods are able to change correctly
the modality of the distributions in some cases. However, in some other
cases, they can also trigger a LFV that is absent from the full system. This
leads us to underline the profound impact that a stochastic parameterization,
and noise in general, can have on models. For instance,

The MTV parameterization has also been tested in an intermediate-order version of the model, showing that this parameterization reduces the anomaly of the PDF of the two dominant oceanic modes. The atmospheric modes are however less well corrected. In this case the number of modes that are removed is large and one can wonder whether reducing this number or increasing the resolution will help. More work needs to be done to assess the impact of the parameterizations on a higher-order version of the models in the future.

The MTV method is simpler, less involved than the WL one, with no memory term
estimations needed, and thus no integrals are being computed at every step.
The memory term could however be Markovianized as in

Finally, it would be interesting to consider that the unresolved dynamics
used to perform the averaging may have non-Gaussian statistics. In the
present work, as stated in Appendix

The source code for MAOOAM v1.3 is available on GitHub at

We now consider Eqs. (

It is due to the fact that we use
directly the measure

The measure

As stated in Sect.

We will now derive explicitly the MTV parameterization for the
system (

We consider first the case defined by Eq. (

Thus, for vectors and matrices it is the standard product.

. The productWith the notable exception of

This solves the case when the singular perturbation

The Eq. (

The
notation

The Wouters–Lucarini method is considered with the
decomposition (

This is the averaging term, which is defined as

This is the correlation term, which here can be written as follows:

Now, the process

This is the memory term which is defined as

The tendencies appearing in Eq. (

The
notation

Both authors have developed the experiments presented in this paper. JD implemented the concepts and ran the experiments. Both authors have contributed to the redaction of the manuscript.

Stéphane Vannitsem is a member of the editorial board of the journal. Jonathan Demaeyer declares that he has no conflict of interest.

This article is part of the special issue “Numerical modeling, predictability and data assimilation in weather, ocean and climate: A special issue honoring the legacy of Anna Trevisan (1946–2016)”. It is not associated with a conference.

The authors thank Lesley De Cruz for her authorization to use the
Fig.