Abstract
A new stochastic model of heat transfer in the atmospheric surface layer is proposed. The model is based on the experimentally confirmed fact that the horizontal wind velocity can be treated as a random process. Accordingly, the model is formalized using a differential equation with random coefficients. Explicit formulas for the expectation and the second moment function of the solution to the heat transfer equation with random coefficients are given. The error induced by replacing the random coefficient of the equation with its expectation is estimated. An example is given demonstrating the efficiency of the proposed approach in the case of a Gaussian distribution of the horizontal wind velocity, when the expectation and the second moment function can be determined within the framework of model representations.
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REFERENCES
A. M. Denisov, “Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation,” Comput. Math. Math. Phys. 56 (10), 1737–1742 (2016).
V. E. Troshchiev and Yu. V. Troshchiev, “Monotonic difference schemes with weight for the transfer equation in a plane layer,” Mat. Model. 15 (1), 3–13 (2003).
V. S. Nozhkin et al., “Stochastic model of moisture motion in atmosphere,” J. Phys.: Conf. Ser. 1096, 012167 (2018). https://doi.org/10.1088/1742-6596/1096/1/012167
A. A. Abramov and L. F. Yukhno, “Solving some problems for systems of linear ordinary differential equations with redundant conditions,” Comput. Math. Math. Phys. 57 (8), 1277–1284 (2017).
S. V. Bogomolov and L. V. Dorodnitsyn, “Equations of stochastic quasi-gas dynamics: Viscous gas case,” Math. Model. Comput. Simul. 3 (4), 457–467 (2011).
A. Zh. Bayev and S. V. Bogomolov, “On the stability of the discontinuous particle method for the transfer equation,” Math. Model. Comput. Simul. 10 (2), 186–197 (2018).
A. M. Denisov, “Inverse problem for a quasilinear system of partial differential equations with a nonlocal boundary condition,” Comput. Math. Math. Phys. 54 (10), 1513–1521 (2014).
O. V. Germider, V. N. Popov, and A. A. Yushkanov, “Heat transfer process in an elliptical channel,” Math. Model. Comput. Simul. 9 (4), 521–528 (2017).
L. Dzierzbicka-Glowacka, J. Jakacki, M. Janecki, and A. Nowicki, “Activation of the operational ecohydrodynamic model (3D CEMBS): The hydrodynamic part,” Oceanologia 55 (3), 519–541 (2013).
L. Gimeno, F. Dominguez, and R. Nieto, “Major mechanisms of atmospheric moisture transport and their role in extreme precipitation events,” Rev. Adv. 41, 25 (2016).
M. G. Hadfield, G. J. Rickard, and M. J. Uddstrom, “A hydrodynamic model of Chatham Rise, New Zealand,” N. Z. J. Mar. Freshwater Res. 41, 239–264 (2007).
K. C. Mo, M. Chelliah, and M. L. Carrera, “Atmospheric moisture transport over the United States and Mexico as evaluated in the NCEP regional reanalysis,” J. Hydrometeorol. 6, 710–728 (2005).
L. R. Dmitrieva-Arrago, “Methods of short-term forecasting of nonconvective clouds and precipitation using a moisture transformation model, with microphysics parametrization: 1. Moisture transformation model and nonconvective cloud forecasting,” Russ. Meteorol. Hydrol. 29, 1–18 (2004).
Ya. N. Belov, E. P. Borisenkov, and B. D. Panin, Numerical Methods of Weather Forecasting (Gidrometeoizdat, Leningrad, 1989) [in Russian].
V. G. Zadorozhniy, “Linear chaotic resonance in vortex motion,” Comput. Math. Math. Phys. 53 (4), 486–502 (2013).
V. G. Zadorozhniy, “Stabilization of linear systems by a multiplicative random noise,” Differ. Equations 54 (6), 728–747 (2018).
V. G. Zadorozhniy, Methods of Variational Analysis (RKhD, Moscow, 2006) [in Russian].
B. Oksendal, Stochastic Differential Equations (Springer, Berlin, 2003).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1989; Dover, New York, 1999).
G. E. Shilov, Mathematical Analysis: Second Special Course (Fizmatlit, Moscow, 1965) [in Russian].
E. J. Allen and C. Huff, “Derivation of stochastic differential equations for sunspot activity,” Astron. Astrophys. 516 (2010). https://doi.org/10.1051/0004-6361/200913978
R. Kozlov, “Random Lie symmetries of Ito stochastic differential equations,” J. Phys. A.: Math. Theor. 51 (30), 305203 (2018).
W. Mao, L. Hu, and X. Mao, “Approximate solutions for a class of doubly perturbed stochastic differential equations,” Adv. Differ. Equations (2018). https://doi.org/10.1186/s13662-018-1490-5
Funding
The work by M.E. Semenov (see Sections 2, 3) was supported by the Russian Science Foundation, grant no. 19-11-00197.
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Translated by I. Ruzanova
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Zadorozhniy, V.G., Nozhkin, V.S., Semenov, M.E. et al. Stochastic Model of Heat Transfer in the Atmospheric Surface Layer. Comput. Math. and Math. Phys. 60, 459–471 (2020). https://doi.org/10.1134/S0965542520030173
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DOI: https://doi.org/10.1134/S0965542520030173