Analytical solution of the energy balance equation for thermal induction plasmas in argon

We present the Fractional Energy Balance Equation (FEBE): a generalization of the standard EBE.&#160; The key FEBE novelty is the assumption of a hierarchy of energy storage mechanisms: scaling energy storage.&#160; Mathematically the storage term is of fractional rather than integer order.&#160; The special half-order case (HEBE) can be classically derived from the continuum mechanics heat equation used by Budyko and Sellers simply by introducing a vertical coordinate and using the correct conductive-radiative surface boundary conditions (the FEBE is a mild extension).<div> We use the FEBE to determine the temperature response to both historical forcings and to future scenarios.&#160; Using historical data, we estimate the 2 FEBE parameters: its scaling exponent (H = 0.38&#177;0.05; H = 1 is the standard EBE) and relaxation time (4.7&#177;2.3 years, comparable to box model relaxation times). We also introduce two forcing parameters: an aerosol re-calibration factor, to account for their large uncertainty, and a volcanic intermittency exponent so that the intermittency volcanic signal can be linearly related to the temperature. The high frequency FEBE regime not only allows for modelling responses to volcanic forcings but also the response to internal white noise forcings: a theoretically motivated error model (approximated by a fractional Gaussian noise). The low frequency part uses historical data and long memory for climate projections, constraining both equilibrium climate sensitivity and historical aerosol forcings.&#160;Parameters are estimated in a Bayesian framework using 5 global observational temperature series, and an error model which is a theoretical consequence of the FEBE forced by a Gaussian white noise. Using the CMIP5 Representative Concentration Pathways (RCPs) and CMIP6 Shared Socioeconomic Pathways (SSPs) scenario, the FEBE projections to 2100 are shown alongside the CMIP5 MME. The Equilibrium Climate Sensitivity = 2.0&#177;0.4 oC/CO2 doubling implies slightly lower temperature increases.&#160; &#160;However, the FEBE&#8217;s 90% confidence intervals are about half the CMIP5 size so that the new projections lie within the corresponding CMIP5 MME uncertainties so that both approaches fully agree.&#160; &#160;The mutually agreement of qualitatively different approaches, gives strong support to both. &#160;We also compare both generations of General Circulation Models (GCMs) outputs from CMIP5/6 alongside with the projections produced by the FEBE model which are entirely independent from GCMs, contributing to our understanding of forced climate variability in the past, present and future. Following the same methodology, we apply the FEBE to regional scales: estimating model and forcing parameters to produce climate projections at 2.5ox2.5o resolutions. We compare the spatial patterns of climate sensitivity and projected warming between the FEBE and CMIP5/6 GCMs.&#160; </div>

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