Two-layer Energy Balance Model Research Articles

Analysis of a two-layer energy balance model: Long time behavior and greenhouse effect.

We study a two-layer energy balance model that allows for vertical exchanges between a surface layer and the atmosphere. The evolution equations of the surface temperature and the atmospheric temperature are coupled by the emission of infrared radiation by one level, that emission being partly captured by the other layer, and the effect of all non-radiative vertical exchanges of energy. Therefore, an essential parameter is the absorptivity of the atmosphere, denoted εa. The value of εa depends critically on greenhouse gases: increasing concentrations of CO2 and CH4 lead to a more opaque atmosphere with higher values of ϵa. First, we prove that global existence of solutions of the system holds if and only if εa∈(0,2) and blow up in finite time occurs if εa>2. (Note that the physical range of values for εa is (0,1].) Next, we explain the long time dynamics for εa∈(0,2), and we prove that all solutions converge to some equilibrium point. Finally, motivated by the physical context, we study the dependence of the equilibrium points with respect to the involved parameters, and we prove, in particular, that the surface temperature increases monotonically with respect to εa. This is the key mathematical manifestation of the greenhouse effect.

Polar jet stream fluctuations in an energy balance model

We investigate the effect of increased longwave radiative forcing (a proxy for increased greenhouse gas concentration) on the zonally averaged location of the eddy-driven jet stream in a latitude dependent, two-layer Energy Balance Model. The model includes separate terms for atmospheric and surface albedos, and takes into account reflections of shortwave radiation between the surface and atmospheric layers. We introduce the notion of a cloud factor function, which depends on temperature gradients, to simulate the eddy-driven jet. An increase in longwave radiative forcing initially results in a poleward movement of the jet stream’s mean latitude, but as the forcing increases, the location of the jet stream becomes quasi-periodic and its mean location moves equatorward.

An energy balance model of heterogeneous extensive green roofs

We study a two-layer energy balance model of an extensive green roof. The model represents the evolution of the temperature in both the vegetation and the substrate layers. We focus on the modeling and the numerical approximation of the energy balance on the vegetation cover and the substrate. One characteristic of the model is that the vegetation layer is heterogeneous, that is, some parameters describing the vegetation are time–space-dependent. An approximate solution of the model is obtained by means of a numerical scheme based on finite volume method. This method has made possible to solve the model without any linearization. This work also includes the validation of the model with experimental data. Different scenarios have been simulated to verify how changes in some parameters affect the energy balance. Chosen parameters are: LAI, vegetation height and soil humidity. As expected, looking at the numerical results of the simulations, we can conclude that if any of these parameters increases green roof temperature is reduced.

Understanding the links between climate feedbacks, variability and change using a two-layer energy balance model

A simple, two-layer energy balance model (EBM) is used to investigate climate variability in Coupled Model Intercomparison Project Phase 5 (CMIP5) models and examine possible links between variability and climate sensitivity, and the roles of stochastic variability, radiative feedbacks and ocean mixing. The EBM represents global variability that, while somewhat stronger than the CMIP5 models, simulates reasonable ratios between shorter and longer timescales. Variability in the EBM to the range of parameters from the Global Climate Models is found to be particularly sensitive to stochastic variability, especially on interannual time-scales. Radiative feedbacks and ocean mixing parameters are also important, particularly for decadal timescales. A modest amount of the model-to-model spread in the variance of global temperature in the CMIP5 models is explained by the EBM. The EBM exhibits a stronger link between equilibrium climate sensitivity and the magnitude of the variability than it does for transient climate response, especially on decadal and longer timescales. The EBM results suggests that spread in stochastic forcing across the CMIP5 models is the single greatest factor degrading the correlation between variability and climate sensitivity, although model to model differences in radiative forcing and mixing into the deep ocean are also important. The findings suggest that from a theoretical point of view investigating constraints from variability may be a fruitful exercise. However, they also suggest that normalizing variability in general circulation models by stochastic forcing, uptake into the deep ocean and radiative forcing are all important first steps to reduce factors that will otherwise confound the correlations.

Equilibrium- and Transient-State Dependencies of Climate Sensitivity: Are They Important for Climate Projections?

AbstractThe effective equilibrium climate sensitivity is generally assumed to be constant in climate change studies, whereas it may vary due to different mechanisms. This study assesses the importance of the different types of state dependencies of the radiative feedbacks for constraining climate projections from the historical record. In transition, the radiative feedbacks may vary with the changes in the warming pattern due to inhomogeneous ocean heat uptake. They may also vary in equilibrium due to their dependence on both temperature and CO2 concentration. A two-layer energy balance model (EBM) that accounts for these effects is shown to improve the representation of any CO2 pathway for the CMIP5 ensemble. Neglecting the nonlinear effects in constraint studies of climate projections from the historical record may induce errors in the estimated future warming. The EBM framework is used to study these errors for three characteristic CO2 pathways. The results show that the pattern effect of ocean heat uptake is not of major importance by inducing a median error of roughly −2% for a high-emission scenario. In contrast, assuming a log-linear CO2–ERF relationship and neglecting the equilibrium-state dependencies induce a larger median error of roughly −10%. This median error is likely due to the non-log-linear dependency of the instantaneous (nonadjusted) forcing, suggesting that the equilibrium-state dependencies do not induce any systematic error. However, they contribute to increasing uncertainties in future warming estimation.

Extending CMIP5 projections of global mean temperature change and sea level rise due to thermal expansion using a physically-based emulator

We present a physically-based emulator approach to extending 21st century CMIP5 model simulations of global mean surface temperature (GMST) and global thermal expansion (TE) to 2300. A two-layer energy balance model that has been tuned to emulate the CO2 response of individual CMIP5 models is combined with model-specific radiative forcings to generate an emulated ensemble to 2300 for RCP2.6, RCP4.5 and RCP8.5. Errors in the emulated time series are quantified using a subset of CMIP5 models with data available to 2300 and factored into the ensemble uncertainty. The resulting projections show good agreement with 21st century ensemble projections reported in IPCC AR5 and also compare favourably with individual CMIP5 model simulations post-2100. There is a tendency for the two-layer model simulations to overestimate both GMST rise and TE under RCP2.6, which is suggestive of a systematic error in the applied radiative forcings. Overall, the framework shows promise as a basis for extending process-based projections of global sea level rise beyond the 21st century time horizon that typifies CMIP5 simulations. The results also serve to illustrate the differing responses of GMST and Earth’s energy imbalance (EEI) to reductions in greenhouse gas emissions. GMST responds relatively quickly to changes in emissions, leading to a negative trend post-2100 for RCP2.6, although temperature remains substantially elevated compared to present day at 2300. In contrast, EEI remains positive under all RCPs, and results in ongoing sea level rise from TE.

The impact of non-isothermal soil moisture transport on evaporation fluxes in a maize cropland

The process of evaporation interacts with the soil, which has various comprehensive mechanisms. Multiphase flow models solve air, vapour, water, and heat transport equations to simulate non-isothermal soil moisture transport of both liquid water and vapor flow, but are only applied in non-vegetated soils. For (sparsely) vegetated soils often energy balance models are used, however these lack the detailed information on non-isothermal soil moisture transport. In this study we coupled a multiphase flow model with a two-layer energy balance model to study the impact of non-isothermal soil moisture transport on evaporation fluxes (i.e., interception, transpiration, and soil evaporation) for vegetated soils. The proposed model was implemented at an experimental agricultural site in Florida, US, covering an entire maize-growing season (67 days). As the crops grew, transpiration and interception became gradually dominated, while the fraction of soil evaporation dropped from 100% to less than 20%. The mechanisms of soil evaporation vary depending on the soil moisture content. After precipitation the soil moisture content increased, exfiltration of the liquid water flow could transport sufficient water to sustain evaporation from soil, and the soil vapor transport was not significant. However, after a sufficient dry-down period, the soil moisture content significantly reduced, and the soil vapour flow significantly contributed to the upward moisture transport in topmost soil. A sensitivity analysis found that the simulations of moisture content and temperature at the soil surface varied substantially when including the advective (i.e., advection and mechanical dispersion) vapour transport in simulation, including the mechanism of advective vapour transport decreased soil evaporation rate under wet condition, while vice versa under dry condition. The results showed that the formulation of advective soil vapor transport in a soil-vegetation-atmosphere transfer continuum can affect the simulated evaporation fluxes, especially under dry condition.

Simulations of coupled non-isothermal soil moisture transport and evaporation fluxes in a forest area

Abstract This study focuses on the quantification of non-isothermal soil moisture transport and evaporation fluxes in vegetated area. A one-dimensional numerical model is developed by integrating a multi-phase flow model with a twolayer energy-balance model. The non-isothermal multi-phase flow model solves four governing equations for coupled air, vapour, moisture, and heat transport in soil porous medium. The two-layer energy balance model estimates evaporation fluxes from transpiration, interception, and soil surface. The model was implemented to an oak forest area in Missouri, USA. For model calibration and validation, measurements of energy fluxes, soil moisture, and soil temperature were used. The proposed model is compared with a simple model that couples the Penman-Monteith equation with the Richards’ equation. The results indicate that the simple model underestimate the total evaporation rate. On the contrary, the proposed model includes a more detailed description of energy transfer, which could improve the accuracy in estimating evaporation rates. The proposed model could be a promising tool to quantify the energy and moisture fluxes in a soil-vegetation-atmosphere continuum in vegetated area.

Meridional Energy Transport in the Coupled Atmosphere–Ocean System: Compensation and Partitioning

Abstract The variability and compensation of the meridional energy transport in the atmosphere and ocean are examined with the state-of-the-art GFDL Climate Model, version 2.1 (CM2.1), and the GFDL Intermediate Complexity Coupled Model (ICCM). On decadal time scales, a high degree of compensation between the energy transport in the atmosphere (AHT) and ocean (OHT) is found in the North Atlantic. The variability of the total or planetary heat transport (PHT) is much smaller than the variability in either AHT or OHT alone, a feature referred to as “Bjerknes compensation.” Natural decadal variability stems from the Atlantic meridional overturning circulation (AMOC), which leads OHT variability. The PHT is positively correlated with the OHT, implying that the atmosphere is compensating, but imperfectly, for variations in ocean transport. Because of the fundamental role of the AMOC in generating the decadal OHT anomalies, Bjerknes compensation is expected to be active only in coupled models with a low-frequency AMOC spectral peak. The AHT and the transport in the oceanic gyres are positively correlated because the gyre transport responds to the atmospheric winds, thereby militating against long-term variability involving the wind-driven flow. Moisture and sensible heat transports in the atmosphere are also positively correlated at decadal time scales. The authors further explore the mechanisms and degree of compensation with a simple, diffusive, two-layer energy balance model. Taken together, these results suggest that compensation can be interpreted as arising from the highly efficient nature of the meridional energy transport in the atmosphere responding to ocean variability rather than any a priori need for the top-of-atmosphere radiation budget to be fixed.

Transient Climate Response in a Two-Layer Energy-Balance Model. Part II: Representation of the Efficacy of Deep-Ocean Heat Uptake and Validation for CMIP5 AOGCMs

AbstractIn this second part of a series of two articles analyzing the global thermal properties of atmosphere–ocean coupled general circulation models (AOGCMs) within the framework of a two-layer energy-balance model (EBM), the role of the efficacy of deep-ocean heat uptake is investigated. Taking into account such an efficacy factor is shown to amount to representing the effect of deep-ocean heat uptake on the local strength of the radiative feedback in the transient regime. It involves an additional term in the formulation of the radiative imbalance at the top of the atmosphere (TOA), which explains the nonlinearity between radiative imbalance and the mean surface temperature observed in some AOGCMs. An analytical solution of this system is given and this simple linear EBM is calibrated for the set of 16 AOGCMs of phase 5 of the Coupled Model Intercomparison Project (CMIP5) studied in Part I. It is shown that both the net radiative fluxes at TOA and the global surface temperature transient response are well represented by the simple EBM over the available period of simulations. Differences between this two-layer EBM and the previous version without an efficacy factor are analyzed and relationships between parameters are discussed. The simple model calibration applied to AOGCMs constitutes a new method for estimating their respective equilibrium climate sensitivity and adjusted radiative forcing amplitude from short-term step-forcing simulations and more generally a method to compute their global thermal properties.

Transient Climate Response in a Two-Layer Energy-Balance Model. Part I: Analytical Solution and Parameter Calibration Using CMIP5 AOGCM Experiments

AbstractThis is the first part of a series of two articles analyzing the global thermal properties of atmosphere–ocean coupled general circulation models (AOGCMs) within the framework of a two-layer energy-balance model (EBM). In this part, the general analytical solution of the system is given and two idealized climate change scenarios, one with a step forcing and one with a linear forcing, are discussed. These solutions give a didactic description of the contributions from the equilibrium response and of the fast and slow transient responses during a climate transition. Based on these analytical solutions, a simple and physically based procedure to calibrate the two-layer model parameters using an AOGCM step-forcing experiment is introduced. Using this procedure, the global thermal properties of 16 AOGCMs participating in phase 5 of the Coupled Model Intercomparison Project (CMIP5) are determined. It is shown that, for a given AOGCM, the EBM tuned with only the abrupt 4×CO2 experiment is able to reproduce with a very good accuracy the temperature evolution in both a step-forcing and a linear-forcing experiment. The role of the upper-ocean and deep-ocean heat uptakes in the fast and slow responses is also discussed. One of the main weaknesses of the simple EBM discussed in this part is its ability to represent the evolution of the top-of-the-atmosphere radiative imbalance in the transient regime. This issue is addressed in Part II by taking into account the efficacy factor of deep-ocean heat uptake.

On the Meaning of Feedback Parameter, Transient Climate Response, and the Greenhouse Effect: Basic Considerations and the Discussion of Uncertainties

In this paper we discuss the meaning of feedback parameter, greenhouse effect and transient climate response usually related to the globally averaged energy balance model of Schneider and Mass. After scrutinizing this model and the corresponding planetary radiation balance we state that (a) this globally averaged energy balance model is flawed by unsuitable physical considerations, (b) the planetary radiation balance for the Earth in the absence of an atmosphere is fraught by the inappropriate assumption of a uniform surface temperature, the so-called radiative equilibrium temperature of about 255 K, and (c) the effect of the radiative anthropogenic forcing, considered as a perturbation to the natural system, is much smaller than the uncertainty involved in the solution of the model of Schneider and Mass. This uncertainty is mainly related to the empirical constants suggested by various authors and used for predicting the emission of infrared radiation by the Earth's skin. Furthermore, after inserting the absorption of solar radiation by atmospheric constituents and the exchange of sensible and latent heat between the Earth and the atmosphere into the model of Schneider and Mass the surface temperatures become appreciably lesser than the radiative equilibrium temperature. Moreover, both the model of Schneider and Mass and the Dines-type two-layer energy balance model for the Earthatmosphere system, containing the planetary radiation balance for the Earth in the absence of an atmosphere as an asymptotic solution, do not provide evidence for the existence of the so-called atmospheric greenhouse effect if realistic empirical data are used.

The Two-layer Surface Energy Balance Parameterization Scheme (TSEBPS) for estimation of land surface heat fluxes

Abstract. A Two-layer Surface Energy Balance Parameterization Scheme (TSEBPS) is proposed for the estimation of surface heat fluxes using Thermal Infrared (TIR) data over sparsely vegetated surfaces. TSEBPS is based on the theory of the classical two-layer energy balance model, as well as a set of new formulations derived from assumption of the energy balance at limiting cases. Two experimental data sets are used to assess the reliabilities of TSEBPS. Based on these case studies, TSEBPS has proven to be capable of estimating heat fluxes at vegetation surfaces with acceptable accuracy. The uncertainties in the estimated heat fluxes are comparable to in-situ measurement uncertainties.

Temperature response of Earth to the annual solar irradiance cycle

We directly determine the sensitivity and time delay of Earth's surface temperature response to annual solar irradiance variations from 60 years of data. A two-layer energy balance model is developed to interpret the results. Explaining both the resulting low sensitivity and time delay of 1–2 months requires negative feedback.

Forward and inverse modeling of land surface energy balance using surface temperature measurements

Land surface temperature measurements have been widely used to estimate surface energy balance. However, because land surface temperature and energy balance both depend on a complex suite of factors, precise estimation of surface energy exchanges using thermal remote sensing is difficult. In recent years, a variety of methods have been developed that overcome previous limitations and show substantial promise for robust estimation of surface fluxes from remote sensing. This paper reviews recent progress in this domain and describes a two-layer energy balance model designed for use with thermal remote sensing. An important aspect of the model is that it is specifically designed to account for the complex micrometeorology and thermal properties of land surfaces possessing a range of density in vegetation. Further, the physics underlying this model are complementary to the physics of land surface thermal remote sensing. Comparisons between field measurements and modeled fluxes show good agreement, which suggests that the model describes land surface energy balance processes with good realism. More importantly, these results reinforce the conclusions of other recent studies that have demonstrated the compatibility of two-layer energy balance models with remote sensing observations and, by extension, the viability of using thermal remote sensing to model surface energy balance.