Long-Time Behavior of State Functions for Badyko Models
In this note we examine the long-time behavior of state functions for a climate energy balance model (Budyko Model) in the strongest topologies of the phase and the extended phase spaces. Strongest convergence results for all weak solutions are obtained.
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Long-Time Behavior of State Functions for Badyko Models Nataliia V. Gorban, Mark O. Gluzman, Pavlo O. Kasyanov and Alla M. Tkachuk
Abstract In this note we examine the long-time behavior of state functions for a climate energy balance model (Budyko Model) in the strongest topologies of the phase and the extended phase spaces. Strongest convergence results for all weak solutions are obtained. New structure and regularity properties for global and trajectory attractors are justified.
18.1 Introduction and Setting of the Problem Let (M , g) be a C ∞ compact connected oriented two-dimensional Riemannian manifold without boundary (e.g., M = S 2 the unit sphere of R3 ). Consider the problem: ∂u ∂t
− u + Re (x, u) ∈ QS(x)β(u), (x, t) ∈ R+ × M ,
(18.1)
where u = divM (∇M u); ∇M is understood in the sense of the Riemannian metric g. Note that (18.1) is the so-called climate energy balance model. It was proposed in Budyko [4] and Sellers [38] and examined also in D´iaz et al. [10–13]. The unknown N.V. Gorban · P.O. Kasyanov Institute for Applied System Analysis, National Technical University of Ukraine “Kyiv Polytechnic Institute”, Peremogy ave., 37, build, 35, Kyiv 03056, Ukraine e-mail: [email protected] P.O. Kasyanov e-mail: [email protected] M.O. Gluzman (B) Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA e-mail: [email protected] A.M. Tkachuk Faculty of Automation and Computer Systems, National University of Food Technologies, Volodymyrska st., 68, Kyiv 01601, Ukraine e-mail: [email protected] © Springer International Publishing Switzerland 2016 V.A. Sadovnichiy and M.Z. Zgurovsky (eds.), Advances in Dynamical Systems and Control, Studies in Systems, Decision and Control 69, DOI 10.1007/978-3-319-40673-2_18
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u(x, t) represents the average temperature of the Earth’s surface. In Budyko [4] the energy balance is expressed as heat variation = Ra − Re + D. Here Ra = QS(x)β(u). It represents the solar energy absorbed by the Earth, Q > 0 is a solar constant, S(x) is an insolation function (the distribution of solar radiation falling on upper atmosphere), β represents the ratio between absorbed and incident solar energy at the point x of the Earth’s surface (so-called the co-albedo function). The term Re represents the energy emitted by the Earth into space, and as usual, it is assumed to be an increasing function on u. The term D is the heat diffusion, and we assume (for simplicity) that it is constant. As usual, the term Re may be chosen according to the Newton cooling law as linear function on u, Re = Bu + C (here B and C are some positive constants) [4], or according to the Stefan–Boltzmann law, Re = σ u4 [38]. In this note we consider Re = Bu as in Budyko [4]. Let S : M → R be a function such that S ∈ L ∞ (M ), and there exist S0 , S1 > 0 such that 0 < S0 ≤ S(x) ≤ S1 . Suppose also that β is a bounded maximal monotone graph of R2 ; that is, there exist m, M ∈ R, such that for all s ∈ R and z ∈ β(s) m ≤ z ≤ M. Through the note we co
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