Constructing accurate, flexible, and efficient parametrizations is one of the great challenges in the numerical modeling of geophysical fluids. We consider here the simple yet paradigmatic case of a Lorenz 84 model forced by a Lorenz 63 model and derive a parametrization using a recently developed statistical mechanical methodology based on the Ruelle response theory. We derive an expression for the deterministic and the stochastic component of the parametrization and we show that the approach allows for dealing seamlessly with the case of the Lorenz 63 being a fast as well as a slow forcing compared to the characteristic timescales of the Lorenz 84 model. We test our results using both standard metrics based on the moments of the variables of interest as well as Wasserstein distance between the projected measure of the original system on the Lorenz 84 model variables and the measure of the parametrized one. By testing our methods on reduced-phase spaces obtained by projection, we find support for the idea that comparisons based on the Wasserstein distance might be of relevance in many applications despite the curse of dimensionality.

The climate is a forced and dissipative system featuring variability on a
large range of spatial and temporal scales, as a result of many complex and
coupled dynamical processes inside it

The triad of terms – deterministic, stochastic, and non-Markovian – was also
found by

Conceptually similar results have been found through bottom up, data driven
approaches, by

Even when a parametrization is efficient enough to represent unresolved
phenomena with the desired precision, problems arise when it comes to dealing
with scale adaptivity. Re-tuning the parametrization to a new set of
parameters of the model usually means running again long simulations, adding
further computational costs. For this reason the development of a scale-adaptive
parametrization is considered to be a central task in geosciences

In this paper, we wish to apply the WL parametrization to a simple dynamical
system introduced by

In Sect.

The Lorenz 84 model

The Lorenz 63 model is probably the most iconic chaotic dynamical system

The full model used in this paper, proposed by

It is important to underline that the coupling between the Lorenz 84 and the Lorenz 63 is unidirectional: the latter model affects the former and, acts as an external forcing, with no feedback acting the other way around.

In what follows, we choose fairly classical values for the parameters:

Henceforth, we will refer to the standard Lorenz 84 model as the uncoupled model, whilst the Lorenz 84 subject to the coupling with the Lorenz 63 will be called the coupled model.

As discussed in

The coupling strength

The deterministic term

Since the coupling shown in Eq. (

The last term in Eq. (

After the implementation of the Wouters and Lucarini procedure, Eq. (

We wish to assess how well a parametrization allows us to reproduce the
statistical properties of the full coupled system. In this regard, it seems
relevant to quantify to what extent the projected invariant measure of the
full coupled model on the variables of interest differs from the invariant
measures of the surrogate models containing the parametrization. In order to
evaluate how much such measures differ, we resort to considering their
Wasserstein distance

Starting from two distinct spatial distributions of points, described by the
measures

We can also define the Wasserstein distance also in the case of two discrete
distributions

This latter definition of Wasserstein distance has already been proven
effective

Poincaré
section in

Hereby we propose to further assess the reliability of the WL stochastic
parametrization by studying the Wasserstein distance between the projected
invariant measure of the original system on the first three variables

The coordinates of the cubes will then be equal to the location

In this section we show the results corresponding to the case

Poincaré section in

We first provide a qualitative overview of the performance of the
parametrization by investigating a few Poincaré sections, which provide a
convenient and widely used method to visualize the dynamics of a system in a
two-dimensional plot

A 3-D view of the attractor of

Metaphorically, our parametrization aims at describing as accurately as
possible the impact of “convection” on the “westerlies”. It is insightful to
look at how it affects the properties of the two variables –

Probability density of the

Probability density of the

Probability density of the

Wasserstein distances from the coupled model with respect to number of
cubes per side:

In order to provide further qualitative evidence of our results, in Fig.

Expectation values for the ensemble average of the first two moments of the variables

Poincaré section in

Wasserstein distances from the coupled model with respect to number of
cubes per side:

Further to the qualitative inspection, we provide here quantitative comparisons to support our study. All the remaining simulations in this section are run for 100 years (7300 time units) with a time step of 0.005; thus, each attractor is constructed with 1 460 000 points. We have tested that the results presented below are virtually unchanged when considering a smaller time step of 0.001.

We first look into the probability density functions (PDFs)
of the variables

Aside from the analysis of the PDF, a further statistical investigation can
be provided by looking into the numerical results provided by first moments
of the variables and their uncertainty, which is computed as the standard
deviation derived from the analysis of an ensemble of runs. We have performed
just 10 runs, but our results are very robust. The results for the
statistics of the first two moments are reported in Table

If the considered PDFs depart strongly from unimodality, the analysis of the
first moments can be of little use, and it becomes hard to have a
thorough understanding of the statistics by adopting this point of view. As
discussed above, we wish to supplement this simple analysis with a more
robust evaluation of the performance of the parametrizations by taking into
account the Wasserstein distance. A first issue to deal with in order to compute
the Wasserstein distance consists of carefully choosing the number of cubes
used for the Ulam projection. Figure

A well-known problem of Ulam's method is the fact that it can hardly be
adapted to high-dimensional spaces – this is the so-called curse of
dimensionality. Additionally, evaluating the Wasserstein distance in high
dimensions itself becomes extremely computationally challenging. In order to
partially address these problems, we repeat the analysis shown in
Fig.

In order to extend the scope of our study, we have repeated the analysis
described above for the case

Figure

The analysis performed considering the Wasserstein distance between the
measures is shown in Fig.

Developing parametrizations able to surrogate efficiently and accurately the dynamics of unresolved degrees of freedom is a central task in many areas of science, and especially in geosciences. There is no obvious protocol in testing parametrizations for complex systems, because one is bound to look only at specific observables of interest. This procedure is not error-free, because optimizing a parametrization against one or more observables might lead to unfortunate effects on other aspects of the system and worsen, in some other aspects, its performance.

In this paper we have addressed the problem of constructing a parametrization for a simple yet meaningful two-scale system, and then testing its performance in a possibly comprehensive way. We have considered a simple six-dimensional system constructed by coupling a Lorenz 84 system and a Lorenz 63 system, with the latter acting as forcing to the former, and the former being the subsystem of interest. We have included a parameter controlling the timescale separation of the two systems and a parameter controlling the intensity of the coupling. We have built a first order and a second order parametrization able to surrogate the effects of the coupling using the scale-adaptive WL method. The second order scheme includes a stochastic term, which has proved to be essential for radically improving the quality of the parametrization with respect to the purely deterministic case (first order parametrization), as already visually shown by looking at suitable Poincaré sections.

We show here that, in agreement with what was discussed in previous papers, the WL approach provides an accurate and flexible framework for constructing parametrizations. Nonetheless, the main novelty of this paper lies in our use of the Wasserstein distance as a comprehensive tool for measuring how different the invariant measures (“the climates”) of the uncoupled Lorenz 84 model, and of its two versions with deterministic and stochastic parametrizations are from the projection of the measure of the coupled model on the variables of the Lorenz 84 model. We discover that the Wasserstein distance provides a robust tool for assessing the quality of the parametrization, and, quite encouragingly, meaningful results can be obtained when considering a very coarse-grained representation of the phase space. A well-known issue with using a methodology like the Wasserstein distance is the so-called curse of dimensionality: the procedure itself becomes unfeasible when the system has a number of degree of freedom above a few units. We have addressed (partially) this issue by looking at the Wasserstein distance of the projected measures on the three two-dimensional spaces spanned by two of the three variables of the Lorenz 84 model. We find that the properties of the Wasserstein distance in the reduced spaces follow closely those found in the full space. We maintain that diagnostics based on the Wasserstein distance in suitably defined reduced-phase spaces should become standard in the analysis of the performance of parametrizations and in intercomparing models of any level of complexity.

The data used for plotting the figures contained in the paper were generated using codes available from Gabriele Vissio upon request.

The authors declare that they have 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 wish to thank the reviewers and the editor for providing
constructive criticism for the paper, which has stimulated an improvement of
the quality of the paper. The authors wish to thank Gabriel Peyré for making the
Matlab software related to Wasserstein distance publicly available. Gabriele Vissio was
supported by the Hans Ertel Center for Weather Research (HErZ), a
collaborative project involving universities across Germany, the Deutscher
Wetterdienst, and funded by the BMVI (Federal Ministry of Transport and
Digital Infrastructure, Germany, grant agreement number U4603BMV1501). Valerio Lucarini
acknowledges financial support provided by the DFG SFB/Transregio project
TRR181 (grant agreement number U4603SFB160110) and by the Horizon2020
projects Blue-Action (grant agreement number 727852) and CRESCENDO (grant
agreement number 641816). Valerio Lucarini wishes to thank Michael Ghil for having suggested the
relevance of the Wasserstein distance, and