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Some issues which are relevant for the recent state in climate modeling have been considered. A detailed overview of literature related to this subject is given. The concept in modeling of climate, as a complex system, seen through Gödel’s theorem and Rosen’s definition of complexity and predictability is discussed. Occurrence of chaos in computing the environmental interface temperature from the energy balance equation given in a difference form is pointed out. A coupled system of equations, often used in climate models, was analyzed. It is shown that the Lyapunov exponent mostly has positive values allowing presence of chaos in this system. The horizontal energy exchange between environmental interfaces, which is described by the dynamics of driven coupled oscillators, was analyzed. Their behavior and synchronization, when a perturbation is introduced in the system, as a function of the coupling parameter, the logistic parameter, and the parameter of exchange, were studied calculating the Lyapunov exponent under simulations with the closed contour of

Among the most interesting and fascinating phenomena that are predicted/predictable is the chaotic ocean/atmosphere/land system called weather and its longtime average, climate. While weather is not predictable beyond a few days, aspects of the climate may be predictable for years, decades, and perhaps longer [

Earth’s atmosphere has evolved into a complex system in which life and climate are intricately interwoven. The interface between Earth and atmosphere as a “pulsating biophysical organism” is a complex system itself. We use the term complex system in Rosen’s sense (Rosen, 1991) as it was explicated in the comment by Collier (2003) as follows: “In Rosen’s sense a complex system cannot be decomposed non-trivially into a set of part [sic] for which it is the logical sum. Rosen’s modeling relation requires this. Other notions of modeling would allow complete models of Rosen style complex systems, but the models would have to be what Rosen calls

Generally, predictability refers to the degree that a correct forecast of a system’s state can be made either qualitatively or quantitatively. For example, while the second law of thermodynamics can tell us about the equilibrium that a system will evolve to and steady states in dissipative systems can sometimes be predicted, there exists no general rule to predict the time evolution of systems far from equilibrium, that is, chaotic systems, if they do not approach some kind of equilibrium. Their predictability usually deteriorates with time. To quantify predictability, the rate of divergence of system trajectories in phase space can be measured (Kolmogorov-Sinai entropy, Lyapunov exponents).

Lorenz (1984) discussed several issues in the predictability of weather systems [

The above insight of the predictability is underlined in the context of the “environmental predictability” (primarily linked to the climate change issues); we finish with the following question:

Let us return to the question we were asking ourselves after we had shortly considered climate modeling (i.e., predictability) beyond the complexity. To our mind there is a significant space for “improvement” of models and their capabilities to provide good forecasts. It can be done only if the modeling attempts are directed towards the following steps: from structures and states to processes and functions; from self-correcting to self-organizing systems; from hierarchical steering to participation; from conditions of equilibrium to dynamic balances of nonequilibrium; from single trajectories to bundles of trajectories; from linear causality to circular causality; from predictability to relative chance; from order and stability to instability, chaos, and dynamics; from certainty and determination to a larger degree of risk, ambiguity, and uncertainty; from reductionism to emergentism; from being to becoming.

In this paper we address two issues that, to our mind, are important for further improvements in designing the climate models. (1) The phenomenon of chaos: (i) in behavior of the environmental interface temperature computed from the energy balance equation, (ii) in coupling of processes of vertical and horizontal energy transfers in climate models which can result in something that is much more complex than the deterministic chaos of those models (Section

The target of global climate models is the Earth’s climate system, consisting of the physical and chemical components of the atmosphere, ocean, land surface, and cryosphere. In climate simulations, the objective is to correctly simulate the spatial variation of climate conditions in some average sense. There exists a hierarchy of different climate models, ranging from simple energy balance models to the very complex global circulation models. These models attempt to account for as many processes as possible to simulate the detailed evolution of the atmosphere, ocean, cryosphere, and land system, at a horizontal resolution that is typically 100 s of km. Climate model complexity is result of the nonlinearity of the equations, high dimensionality, and the linking of multiple subsystems. However, the secret of understanding of climate model complexity lies in the nonlinear dynamics of the atmosphere and oceans, which is described by the Navier-Stokes equations whose solution is one of the most vexing problems in all of mathematics.

We shortly enhance the main current issues related to the modeling of the global climate system [

In Section

In Section

Traditional mathematical analysis of physical systems tacitly assumes that integers and all real numbers, no matter how large or how small, are physically possible and all mathematically possible trajectories are physically possible [

According to van der Vaart many models for environmental problems have been and will be built in the form of differential equations or systems of such equations [

In this subsection we analyze the energy balance equation in procedure of computing the environmental interface temperature and the deeper soil layer temperature commonly used in climate models. The environmental interface is defined as

There are a lot of examples of environmental interfaces in the nature, but here we deal with the ground surface, where there exist all three mechanisms of energy transfer: incoming and outgoing radiation, convection of heat and moisture into the atmosphere, and conduction of heat into deeper soil layers of ground (Figure

Energy balance equation terms.

One of the most important conditions for functioning of any complex system is a proper supply of the system with energy. Dynamics of energy flow is based on the energy balance equation [

Finally, system (

Now we analyse the stability of physical solutions of coupled maps (

Lyapunov exponent of the coupled system (

Irregularities in solution of the system of difference equation (

There are three major sets of processes that must be considered when constructing a climate model: (i) radiative (the transfer of radiation through the climate system, e.g., absorption and reflection); (ii) dynamic (the horizontal and vertical transfer of energy, e.g., advection, convection, and diffusion); and (iii) surface process (inclusion of processes involving land/ocean/ice and the effects of albedo, emissivity, and surface-atmosphere energy exchanges). If the nonlinearities in these processes are treated improperly, then while designing the model, the complexity and thus its reliability will not be retained in the highest degree. In Section

As noted above, the horizontal exchange of energy between environmental interfaces is considered as diffusion-like process. The dynamics of energy exchange behavior on environmental interface are typically expressed as a logistic map

Schematic diagram of horizontal energy exchange between two environmental interfaces. Parameters

Since these and many other processes on environmental interface are defined as diffusion-like, we will explore (i) how these processes can be better represented in climate models by introducing parameter of exchange

In considering these problems we have to include observational heterarchy, a challenging topic when dealing with complex systems. Essentially, observational heterarchy reveals that it is impossible to unambiguously determine to which subsystems an element belongs [

Observational heterarchy consists of two sets of intralayer maps, called Intent and Extent perspectives, and interlayer operations satisfying the following conditions. (1) The interlayer operations inherit the mixture of intra- and interlayer operations and (2) there is a procedure by which the interlayer operation can be regarded as an adjoint functor. If the interlayer operation satisfies the conditions (1) and (2), it is called a prefunctor [

The time development of the environmental surface dynamics

We perform our analysis following the procedure described in [

Normalized frequency of synchronization,

Here, we address the behavior of active coupling [

In order to see how perturbation enhances robust behavior in the framework of observational heterarchy in a multienvironmental interface system represented by closed contour of coupled environmental interfaces exchanging the energy horizontally, then the system of coupled difference equations for

Simulations with the active coupling, defined by (

Diagram of Lyapunov exponent,

In calculating

In the introduction we have considered the complex ocean/atmosphere/land dynamical system, called weather and its long time average climate, as a complex one. This system is modeled by climate models having different levels of sophistication. An important concept in climate system modeling is that of a spectrum of models of differing levels of complexity, each being optimum for answering specific questions. It is not meaningful to judge one level as being better or worse than another independently of the context of analysis. What is important is that each model be asked questions appropriate for its level of complexity and quality of its simulation [

Clearly, an answer to the above question requires (i) a definition and a measure of complexity and (ii) that this measure is equally applicable to the model and to the data, because some sort of comparison is necessary. It is a hard task to find that measure even approximately. However, intuitively we can put a cadence on a view of complexity which is more related to a model’s dynamical properties rather than its architecture. Thus, we can say that, in developing tools, an advantage will be given to a tool which gives answers to the following questions: (i) what is the maximal dynamical complexity a given model can generate? and (ii) what kind of different dynamical behaviors can a given model generate? as it is underlined by Boschetti (2008). For our consideration we will rely on Boschetti (2008) who defined the complexity of an ecological model as the statistical complexity of the output it produces that allows a direct comparison between data and model complexity [

In this subsection we will illustrate an example of comparison between complexities of global and regional model. Here, we do not deal with statistical complexity of the global and regional models. Our intention is just to show possible differences in complexities of time series of precipitation as well as air temperature for both models, applying the algorithm for calculating the Kolmogorov complexity.

We have calculated the Kolmogorov complexity following Lempel and Ziv [

Encode the time series by constructing a sequence

Calculate the complexity counter

Calculate the normalized complexity measure

In order to calculate complexities of model time series we have used (i) air temperature and (ii) precipitation time series which are outputs from climate simulations for Belgrade and Novi Sad in Serbia [

We have calculated the Kolmogorov complexity for each time series obtained when each sample, in the original time series, is used as a threshold (

Kolmogorov complexity (KL and its maximum: KLM) values for the precipitation and air temperature time series for Belgrade and Novi Sad, in Serbia, obtained from climate simulations using different models.

Quantity | Measure | Model | |||
---|---|---|---|---|---|

Global | Regional | ||||

SINTEX-G | ECHAM5 | Eta-POM | RegCM | ||

Temperature | KL | 0.176 | 0.207 | ||

(Belgrade) | KLM | 0.326 | 0.331 | ||

Temperature | KL | 0.241 | 0.251 | ||

(Novi Sad) | KLM | 0.318 | 0.354 | ||

Precipitation | KL | 0.705 | 0.671 | ||

(Belgrade) | KLM | 0.834 | 0.793 | ||

Precipitation | KL | 0.265 | 0.871 | ||

(Novi Sad) | KLM | 0.289 | 0.935 |

Kolmogorov complexity for the (a) precipitation and (b) air temperature time series for Belgrade and (c) precipitation and (d) temperature for Novi Sad, in Serbia, obtained from climate simulations using different models. On

From Figure

Now, we analyze the air temperature and precipitation time series for Novi Sad obtained by the global ECHAM5 and regional RegCM models. From Figure

We have considered climate predictions through two issues: (i) occurrence of chaos and (ii) complexity in climate models. We have given a detailed overview of literature related to this subject. Then, we considered the climate modeling through the light of Gödel’s theorem that says that Number Theory is more

First, we have pointed out occurrence of chaos in computing the environmental interface temperature from the energy balance equation when the given differential equation is replaced by a difference equation. For that purpose we have analyzed a coupled system of equations, often used in climate models. It is shown that the Lyapunov exponent mostly has positive values approving presence of chaos in this system, but there are still some strait regions where the Lyapunov exponent is negative, that is, where there exist physically meaningful solutions. This analysis, set into context of the climate modeling, points out the fact that there exists set of domains where the environmental interface temperature cannot be calculated by the physics of currently designed climate models.

Second, we have analyzed the coupling of processes of vertical and horizontal energy exchange between environmental interfaces which is described by the dynamics of driven coupled oscillators. To study that coupling, when a perturbation is introduced in the system, as a function of the coupling parameter, the logistic parameter, and the parameter of exchange, we have considered dynamics of two maps serving the diffusive coupling. Then, we have performed simulations, calculating the Lyapunov exponent, with the closed contour of

Finally, we have explored possible differences in complexities of two global and two regional climate models using their output time series for the precipitation and air temperature. We have applied the algorithm for calculating the Kolmogorov complexity on those time series. We have found differences in the level of complexity among models. However, for more reliable conclusion that could be given we need to test outputs of many different GCM and RegCM models.

The authors declare that there is no conflict of interests regarding the publication of this paper.

This paper was realized as a part of the Project “Studying Climate Change and Its Influence on the Environment: Impacts, Adaptation and Mitigation” (III43007) financed by the Ministry of Education and Science of the Republic of Serbia within the framework of integrated and interdisciplinary research for the period 2011–2014. The authors are grateful to the Provincial Secretariat for Science and Technological Development of Vojvodina for the support under the Project “Climate Projections for the Vojvodina Region up to 2030 Using a Regional Climate Model” funded by the Contract no. 114-451-2151/2011-01. The authors are grateful to Professor D. Kapor for the invested effort and useful suggestions.