Second differential of the entropy as a criterion for the stability in low-dimensional climate models

The second differential of the entropy is used for analysing the stability of a thermodynamic climatic model. A delay time for the heat flux is introduced whereby it becomes an independent variable. Two different expressions for the second differential of the entropy are used: one follows classical irreversible thermodynamics theory; the second is related to the introduction of response time and is due to the extended irreversible thermodynamics theory. the second differential of the classical entropy leads to unstable solutions for high values of delay times. the extended expression always implies stable states for an ice-free earth. When the ice-albedo feedback is included, a discontinuous distribution of stable states is found for high response times. Following the thermodynamic analysis of the model, the maximum rates of entropy production at the steady state are obtained. A latitudinally isothermal earth produces the extremum in global entropy production. the material contribution to entropy production (by which we mean the production of entropy by material transport of heat) is a maximum when the latitudinal distribution of temperatures becomes less homogeneous than present values

Extremal climatic states simulated by a 2-dimensional model Part I: Sensitivity of the model and present state

The criterion, based on the thermodynamics theory, that the climatic system tends to extremizesome function has suggested several studies. In particular, special attention has been devoted to the possibility that the climate reaches an extremal rate of planetary entropy production.Due to both radiative and material effects contribute to total planetary entropy production,climatic simulations obtained at the extremal rates of total, radiative or material entropy production appear to be of interest in order to elucidate which of the three extremal assumptions behaves more similar to current data. In the present paper, these results have been obtainedby applying a 2-dimensional (2-Dim) horizontal energy balance box-model, with a few independent variables (surface temperature, cloud-cover and material heat fluxes). In addition, climatic simulations for current conditions by assuming a fixed cloud-cover have been obtained. Finally,sensitivity analyses for both variable and fixed cloud models have been carried out

Extremal climatic states simulated by a 2-dimensional model Part II: Different climatic scenarios

Different climatic simulations have been obtained by using a 2-Dim horizontal energy balancemodel (EBM), which has been constrained to satisfy several extremal principles on dissipationand convection. Moreover, 2 different versions of the model with fixed and variable cloud-coverhave been used. The assumption of an extremal type of behaviour for the climatic system canacquire additional support depending on the similarities found with measured data for pastconditions as well as with usual projections for possible future scenarios

Using a Parameterization of a Radiative Transfer Model to Build High-Resolution Maps of Typical Clear-Sky UV Index in Catalonia, Spain

To perform a climatic analysis of the annual UV index (UVI) variations in Catalonia, Spain (northeast of the Iberian Peninsula), a new simple parameterization scheme is presented based on a multilayer radiative transfer model. The parameterization performs fast UVI calculations for a wide range of cloudless and snow-free situations and can be applied anywhere. The following parameters are considered: solar zenith angle, total ozone column, altitude, aerosol optical depth, and single-scattering albedo. A sensitivity analysis is presented to justify this choice with special attention to aerosol information. Comparisons with the base model show good agreement, most of all for the most common cases, giving an absolute error within 0.2 in the UVI for a wide range of cases considered. Two tests are done to show the performance of the parameterization against UVI measurements. One uses data from a high-quality spectroradiometer from Lauder, New Zealand [45.04°S, 169.684°E, 370 m above mean sea level (MSL)], where there is a low presence of aerosols. The other uses data from a Robertson–Berger-type meter from Girona, Spain (41.97°N, 2.82°E, 100 m MSL), where there is more aerosol load and where it has been possible to study the effect of aerosol information on the model versus measurement comparison. The parameterization is applied to a climatic analysis of the annual UVI variation in Catalonia, showing the contributions of solar zenith angle, ozone, and aerosols. High-resolution seasonal maps of typical UV index values in Catalonia are presented

Greenhouse gases and climatic states of minimum entropy production

The hypothesis of minimum entropy production is applied to a simple one-dimensional energy balance model and is analysed for different values of the radiative forcing due to greenhouse gases. The extremum principle is used to determine the planetary “conductivity” and to avoid the “diffusive” approximation, which is commonly assumed in this type of model. For present conditions the result at minimum radiative entropy production is similar to that obtained by applying the classical model. Other climatic scenarios show visible differences, with better behaviour for the extremal case

Periodic Solutions in Low-Dimensional Climatic Models

Classic climatic models use constitutive laws without any response time. A more realistic approach to the natural processes governing climate dynamics must introduce response time for heat and radiation fluxes. Extended irreversible thermodynamics (EIT) is a good thermodynamical framework for introducing nonclassical constitutive laws. In the present study EIT has been used to analyze a Budyko–Sellers one-dimensional energybalance model developed by G. R. North. The results present self-sustained periodic oscillations when the response time is greater than a critical value. The high-frequency (few kiloyears) damped and nondamped oscillations obtained can be related to abrupt climatic changes without any variation in the external forcing of the system

States of maximum entropy production in a onedimensional vertical model with convective adjustment

We investigate the hypothesis that the atmosphere is constrained to maximize its entropy production by using a one-dimensional (1-D) vertical model. We prescribe the lapse rate in the convective layer as that of the standard troposphere. The assumption that convection sustains a critical lapse rate was absent in previous studies, which focused on the vertical distribution of climatic variables, since such a convective adjustment reduces the degrees of freedom of the system and may prevent the application of the maximum entropy production (MEP) principle. This is not the case in the radiative–convective model (RCM) developed here, since we accept a discontinuity of temperatures at the surface similar to that adopted in many RCMs. For current conditions, the MEP state gives a difference between the ground temperature and the air temperature at the surface ≈10 K. In comparison, conventional RCMs obtain a discontinuity ≈2 K only. However, the surface boundary layer velocity in the MEP state appears reasonable (≈3 m s-¹). Moreover, although the convective flux at the surface in MEP states is almost uniform in optically thick atmospheres, it reaches a maximum value for an optical thickness similar to current conditions. This additional result may support the maximum convection hypothesis suggested by Paltridge (1978)