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 . Numerical models are able to explicitly resolve only a relatively short range of such scales. In particular, it is crucial to derive efficient and accurate ways to surrogate the effect of dynamical processes occurring on the small spatial and temporal scales that are not explicitly resolved (e.g., because of excessive computational or storage costs) by the model. The operation of constructing so-called parametrizations is key to the development of geophysical fluid dynamical models and stimulates the investigation of the fundamental laws defining the multiscale properties of the coupled atmosphere–ocean dynamics . Traditionally, the development of parametrizations boiled down to deriving deterministic empirical laws able to describe the effect of the small-scale dynamical processes. More recently, it has become apparent that it is important to include stochastic terms in the parametrization that are able to provide a theoretically more coherent representation of such effects and that lead, on a practical level, to an improved skill . A first way to derive or at least justify the need for stochastic parametrizations comes from homogenization theory , which leads to constructing an approximate representation of the impact of the fast scales on the slow variables as the sum of two terms, a mean field term and a white noise term. Such an approach suffers from the fact that one has to take the rather nonphysical hypothesis that an infinite timescale separation exists between the fast and the slow scale. As the climate is a multiscale system, such a methodology is a bit problematic to adopt. Yet, this point of view has been crucial in the development of methods aimed at deriving reduced order models for a system of geophysical interest (see, e.g., ).

and analyzed, in the context of statistical mechanics, the related problem of studying how one can project the effect of a group of variables, with the goal of constructing effective evolution equations for a subset of variables of interest. They reformulated the dynamics of such variables expressing them as a sum of three terms: a deterministic term, a stochastic forcing, and a memory term. The memory term defines a non-Markovian contribution where the past states of the variables of interest enter the evolution equation. In the limit of infinite timescale separation, the last term tends to zero, whilst the random forcing approaches the form of (in general, multiplicative) white noise.

The triad of terms – deterministic, stochastic, and non-Markovian – was also found by , who proposed a method (we refer to it in what follows as WL parametrization) for constructing parametrizations based on the Ruelle response theory . They interpreted the coupling between the variables of interest and those one wants to parametrize as a weak perturbation of the otherwise unperturbed dynamics of the two groups of variables. A useful feature of this approach is that it can be applied on a wide variety systems that do not feature a clear-cut separation of scales. The parametrizations obtained along these lines match the result of the perturbative expansion of the projection operator introduced by Mori and Zwanzig for describing the effective dynamics of the variables of interest . Another quality of the WL parametrization is that it is not tailored to optimize the representation of the statistics of some specific statistical property, but rather approximates coherently well all observables of the system of interest. This method has already been successfully tested in simple to intermediate-complexity multiscale models by , and .

Conceptually similar results have been found through bottom up, data driven approaches, by , and . Specifically, constructed effective models from climatic time series through an extension of the nonlinear case of the multilevel linear regressive method, while showed how non-Markovian data-driven parametrizations emerge naturally when we consider partial observations from a large-dimensional system.

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 a previous paper, the authors demonstrated the scale adaptivity of the WL approach by testing it in a mildly modified version of the Lorenz 96 model . A further degree of flexibility of this approach has been explored in another recent publication , where the authors provided explicit formulas for modifying the parametrization when the parameters controlling the dynamics of the full system are altered.

In this paper, we wish to apply the WL parametrization to a simple dynamical system introduced by and constructed by coupling the Lorenz 84 model with the Lorenz 63 model. In what follows, we want to parametrize the dynamical effect of the variables corresponding to the Lorenz 63 system on the variables corresponding to the Lorenz 84 system. We analyze two different scenarios, where the Lorenz 63 model acts first as a fast and then as a slow forcing, taking into account that the WL parametrization is adaptive and able to seamlessly treat both of them. Compared to what was studied in , the models investigated here have simpler dynamics, as they are not spatially extended and their coupling is simpler, since it is only one-way. Nonetheless, we propose a significant advance with respect to our previous work in terms of methodology for evaluating the performance of the parametrization. We wish to extend what was studied in by focusing on a systematic comparison of the properties of the projected measure of the original coupled system on the subspace spanned by the variables of the Lorenz 84 model with the actual measure of the parametrized model. In particular, we will study the Wasserstein distance between the coarse-grained estimates of the two 3-dimensional invariant measures. Additionally, we will look at the Wasserstein distance of the measures obtained by projecting onto two of the three variables of interest, which allows for a comprehensive evaluation of how different the one-time statistical properties of the two systems are. The Wasserstein distance has been proposed by as a tool for studying the climate variability and response to forcings, and applied by in a simplified setting.

In Sect.  we thoroughly describe the individual models and the full coupled model, while in Sect.  we briefly review Wouters and Lucarini parametrization and its application to the Lorenz 84–Lorenz 63 coupled model. Section  is dedicated to discussing the Wasserstein distance and in particular (a) whether it is efficient in summarizing the quality of the parametrization, (b) how sensitive our analysis is to the coarse graining of the phase space, and (c) whether useful conclusions can be drawn by looking at the problem in a projected space of two variables only. Section  provides the main results of our analysis. In the last section we draw our conclusions and propose future investigations.

presented a top-down method suitable for constructing parametrizations for chaotic dynamical systems in the form dKdt=FK(K)+ϵΨK(K,J),dJdt=FJ(J)+ϵΨJ(K,J), where K=(X,Y,Z) is the vector of the variables we are interested in and the J=(x̃,ỹ,z̃) is the vector of the variables we want to parametrize. The coefficient ϵ controls the strength of the couplings, i.e., ΨK(K,J) and ΨJ(K,J). The parametrization is obtained assuming the chaotic reference and applying Ruelle response theory ; the effect of the coupling in Eq. () is approximated, up to the second order in ϵ, by three terms: the first order consists of a deterministic term, while the second order includes a stochastic forcing and a non-Markovian term. The general form of the parametrization (e.g., ) is dKdt=FK(K)+ϵD(K)+ϵS(K)+ϵ2M(K), where D, S, and M indicate, respectively, the deterministic, stochastic, and memory terms and are defined below in Eqs. ()–(). Note that the projection onto the variables of interest of invariant measure of the full system given in Eqs. ()–() and the invariant measure of the system give in Eq. () are the same up to second order in the coupling parameter ϵ, as discussed in . Since the couplings are seen as a perturbation applied to an otherwise uncoupled system, the three terms in Eq. () can be calculated considering the statistical properties of the unperturbed equations dKdt=FK(K),dJdt=FJ(J). The numerical integration of Eqs. ()–() may allow to use less computational resources with respect to Eqs. ()–(), particularly in the case of multiscale systems.

As discussed in , the WL parametrization has the remarkable feature of having a good degree of adaptivity in terms of changes to the timescale separation between the K and J variables, to be performed by rescaling, e.g., t→τ in Eq. (). In this scale, the term D(K) in Eq. () is unchanged, while the timescale of the autocorrelation of the noise term S(K) and of the memory term M(K) are reduced by a factor τ/t. In the specific case of the Lorenz 96 system studied in , the adaptivity is more general than the one related to changes in the timescale separation only, and points to the possibility of developing general adaptive parametrization schemes beyond such specific model. It is not yet clear whether this might lead to constructing spatial scale-adaptive parametrizations.

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 . Such a distance quantifies the minimum “effort” in morphing one measure into the other, and was originally introduced by , somewhat unsurprisingly, to study problems of military relevance, and later improved by .

Starting from two distinct spatial distributions of points, described by the measures μ and ν, we can define the optimal transport cost as the minimum cost to move the set of points corresponding to μ into the set of points corresponding to ν: C(μ,ν)=inf⁡π∈Π(μ,ν)∫c(x,y)dπ(x,y), where c(x,y) is the cost for transporting one unit of mass from x to y and Π(μ,ν) is the set of all joint probability measures whose marginals are μ and ν. The function C(μ,ν) in Eq. () is called the Kantorovich–Rubinstein distance. In the rest of the paper, we will consider the Wasserstein distance of order 2: W2(μ,ν)=inf⁡π∈Π(μ,ν)∫[d(x,y)]2dπ(x,y)12, where d(x,y) is the Euclidean distance between x and y. Euclidean distance is given by d(μ,ν)=∑i=1n(xi-yi)212.

We can also define the Wasserstein distance also in the case of two discrete distributions μ=∑i=1nμiδxi,ν=∑i=1nνiδyi, where xi and yi represent the location of the different points, with mass given, respectively, by μi and νi. We can construct the order 2 Wasserstein distance for discrete distributions as follows: W2(μ,ν)=inf⁡γij∑i,jγij[d(xi,yj)]212, where γij is the fraction of mass transported from xi to xj.

This latter definition of Wasserstein distance has already been proven effective for providing a quantitative measurement of the difference between the snapshot attractors of the Lorenz 84 system in the instance of summer and winter forcings.

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 (X,Y,Z) and the invariant measures obtained using the surrogate dynamics corresponding to the first and second order parametrization. Nevertheless, since the numerical computations for optimal transport through linear programming theory are not cheap, a new approach is required. In order to accomplish it, we perform a standard Ulam discretization of the measure supported on the attractor, by coarse graining on a set of cubes with constant sides across the phase space. We will discuss below the impact of changing the sides of such cubes.

The coordinates of the cubes will then be equal to the location xi, while the corresponding densities of the points are used to define γij; finally, we exclude from the subsequent calculation all the grid boxes containing no points at all. Our calculations are performed using a modified version of the software for Matlab written by Gabriel Peyré and made available at http://www.numerical-tours.com/matlab/optimaltransp_1_linprog/ (last access: 1 March 2018), conveniently modified to include the subdivision of the phase space in cubes and the assignment of corresponding density to those cubes.

In this section we show the results corresponding to the case τ=5. Therefore, Lorenz 84 and Lorenz 63 are seen as the slow and the fast dynamical systems, respectively.

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.