Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model

Bermejo, Rodolfo; Carpio, Jaime; Díaz, Jesús Ildefonso Díaz; Tello, J. Ignacio

doi:10.1016/j.mcm.2008.04.010

Cited by 18 publications

(22 citation statements)

References 17 publications

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“…These numerical results were rigorously justified and generalized to the case of simplistic with respect to the natural variability of climate [3,14,15]. For instance, such EBMs do not support the spatio-temporal chaos characteristics of climate dynamics (see [3,14,16] and references therein).…”

Section: Introductionmentioning

confidence: 99%

Parameter estimation for energy balance models with memory

et al. 2014

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We study parameter estimation for one-dimensional energy balance models with memory (EBMMs) given localized and noisy temperature measurements. Our results apply to a wide range of nonlinear, parabolic partial differential equations with integral memory terms. First, we show that a space-dependent parameter can be determined uniquely everywhere in the PDE's domain of definition D, using only temperature information in a small subdomain E ⊂ D. This result is valid only when the data correspond to exact measurements of the temperature. We propose a method for estimating a model parameter of the EBMM using more realistic, error-contaminated temperature data derived, for example, from ice cores or marine-sediment cores. Our approach is based on a so-called mechanistic-statistical model that combines a deterministic EBMM with a statistical model of the observation process. Estimating a parameter in this setting is especially challenging, because the observation process induces a strong loss of information. Aside from the noise contained in past temperature measurements, an additional error is induced by the age-dating method, whose accuracy tends to decrease with a sample's remoteness in time. Using a Bayesian approach, we show that obtaining an accurate parameter estimate is still possible in certain cases.

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Section: Introductionmentioning

confidence: 99%

Parameter estimation for energy balance models with memory

et al. 2014

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“…, there exists a function h u satisfying conditions (2), and for every test-function η ∈ C 1 (C) ∩ C 0 (C), vanishing as t = T , the following identity holds:…”

Section: The Change Of Independent Variables: a Mechanical Approachmentioning

confidence: 99%

“…where h u is the function associated with u by conditions (2). The function p depends on t as a parameter.…”

Section: The Function P(x T) Is Thus Defined As Followsmentioning

confidence: 99%

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

Díaz

Shmarev

2008

Arch Rational Mech Anal

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We study the dynamics and regularity of level sets in solutions of the semilinear parabolic equationwhere Ω ⊂ R n is a ring-shaped domain, a and µ are given positive constants, H(·) is the Heaviside maximal monotone graph:Such equations arise in climatology (the so-called Budyko energy balance model), as well as in other contexts such as combustion. We show that under certain conditions on the initial data the level sets Γ µ = {(x, t) : u(x, t) = µ} are n-dimensional hypersurfaces in the (x, t)-space and show that the dynamics of Γ µ is governed by a differential equation which generalizes the classical Darcy law in filtration theory. This differential equation expresses the velocity of advancement of the level surface Γ µ through spatial derivatives of the solution u. Our approach is based on the introduction of a local set of Lagrangian coordinates: the equation is formally considered as the mass balance law in the motion of a fluid and the passage to Lagrangian coordinates allows us to watch the trajectory of each of the fluid particles.

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“…Many interesting results on the subject have been proved by various authors, among them Diaz, Hetzer, Tello. For an overview of these studies, we may refer the reader to [16][17][18]21,5] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Determination of the insolation function in the nonlinear Sellers climate model

Tort

Vancostenoble

2012

Ann. Inst. H. Poincaré Anal. Non Linéaire

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We are interested in the climate model introduced by Sellers in 1969 which takes the form of some nonlinear parabolic equation with a degenerate diffusion coefficient. We investigate here some inverse problem issue that consists in recovering the so-called insolation function. We not only solve the uniqueness question but also provide some strong stability result, more precisely unconditional Lipschitz stability in the spirit of the well-known result by Imanuvilov and Yamamoto (1998) [22]. The main novelties rely in the fact that the considered model is degenerate and above all nonlinear. Indeed we provide here one of the first result of Lipschitz stability in a nonlinear case.

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Mathematical and numerical analysis of a nonlinear diffusive climate energy balance model

Cited by 18 publications

References 17 publications

Parameter estimation for energy balance models with memory

Parameter estimation for energy balance models with memory

Lagrangian Approach to the Study of Level Sets: Application to a Free Boundary Problem in Climatology

Determination of the insolation function in the nonlinear Sellers climate model

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