A coupled thermodynamic system of sea ice and its parameter identification. (English) Zbl 1187.80024

In this paper, an inverse coefficient identification thermal problem in a triple layer heat conductor is investigated. The layers represent the snow, sea ice and ocean mixed layer, whilst the unknowns are the thermal properties, i.e. density \(\rho_i\), specific heat \(c_i\), and thermal conductivity \(\kappa_i\), of each layer \(i= 1,2,3\), totalling a number of 9 positive unknown constant parameters. A certain physical constraint is imposed, namely that for the snow \(\kappa_1= a\rho^2_1\), where \(a\) is a given positive constant, such that the number of unknown positive constants is reduced to 9. However, it is impossible to determine \(\rho_2\), \(c_2\), \(\rho_3\) and \(c_3\) separately, but only their products, namely \(\rho_2\cdot c_2\) and \(\rho_3\cdot c_3\), i.e. the heat capacity of the ice and ocean mixed layers, respectively.
The thermodynamic system measures the temperature throughout the triple layer system which is probably too much and impossible in practice. The existence of an optimal control for the least-squares minimization problem is discussed and necessary conditions for optimality are derived. No regularization and no numerical results are presented. There are a few English mistakes and it seems that the paper has not been proof checked properly.


80A23 Inverse problems in thermodynamics and heat transfer
86A05 Hydrology, hydrography, oceanography
86A22 Inverse problems in geophysics
Full Text: DOI


[1] Kang, J. C.; Tang, S. L.; Liu, L. B., Antarctic sea ice and climate, Adv. Earth Sci., 20, 7, 86-793 (2005), (in Chinese)
[2] Maykut, G. A.; Untersteiner, N., Some results from a time-dependent thermodynamic model of sea ice, J. Geophys. Res., 76, 1550-1575 (1971)
[3] Parkinson, C. L.; Washington, W. M., A large-scale numerical model of sea ice, J. Geophys. Res., 84, C1, 311-337 (1979)
[4] Omstedt, A., A coupled one-dimensional sea-ice-ocean model applied to a semienclosed basin, Tellus, 42A, 568-582 (1990)
[5] Lemke, P., A coupled one-dimensional sea-ice-ocean model, J. Geophys. Res., 92, C12, 13164-13172 (1987)
[6] Leppäranta, M., A growth model for black ice, snow-ice and snow thickness in subarctic basins, Nordic Hyreol., 14, 59-70 (1983)
[7] Cox, G. F.N.; Weeks, W. F., Numerical simulations of the profile properties of undeformed first-year sea-ice during the growth season, J. Geophys. Res., 93, C10, 12449-12460 (1988)
[8] Gabison, R., A thermodynamic model of the formation, growth and decay of first-year sea-ice, J. Glaciol., 33, 105-109 (1987)
[9] Ebert, E. E.; Curry, J. A., An intermediate one-dimensional thermodynamic sea-ice model for investigating ice-atmosphere interaction, J. Geophys. Res., 98, C6, 10085-10109 (1993)
[11] Shidfar, A.; Karamali, G. R., Numerical solution of inverse heat conduction problem with nonstationary measurements, Appl. Math. Comput., 168, 1, 540-548 (2005) · Zbl 1079.65097
[12] Li, G. S., Data compatibility and conditional stability for an inverse source problem in the heat equation, Appl. Math. Comput., 173, 566-581 (2006) · Zbl 1105.35144
[13] Shidfar, A.; Karamali, G. R.; Damirchi, J., An inverse heat conduction problem with a nonlinear source term, Nonlinear Anal., 65, 615-621 (2006) · Zbl 1106.35141
[14] Ikehata, M., An inverse source problem for the heat equation and the enclosure method, Inverse Probl., 23, 183-202 (2007) · Zbl 1111.35116
[15] Christov, C. I.; Marinov, T., Identification of heat-conduction coefficient via method of variational imbedding, Math. Comput. Modell., 27, 3, 109-116 (1998) · Zbl 1185.65170
[16] Yang, C. Y., Estimation of the temperature-dependent thermal conductivity in inverse heat condition problems, Appl. Math. Modell., 23, 469-478 (1999) · Zbl 0934.35209
[17] Engl, H. W.; Zou, J., A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Probl., 16, 1907-1923 (2000) · Zbl 0968.35124
[18] Telejko, T.; Malinowski, A., Application of an inverse solution to the thermal conductivity identification using the finite element method, J. Mater. Process. Technol., 146, 145-155 (2004)
[19] Pierce, A., Unique identification of eigenvalues and coefficients in a parabolic problem, SIAM J. Control Optim., 17, 494-499 (1979) · Zbl 0415.35035
[20] Belov, Yu. Ya., On an inverse problem for a semilinear parabolic equation, Sov. Math. Dokl., 43, 216-220 (1991) · Zbl 0761.35110
[21] Li, C. F.; Feng, E. M.; Liu, J. W., Optimal control of systems of parabolic PDEs in exploitation of oil, J. Appl. Math. Comput., 13, 247-259 (2003) · Zbl 1039.49025
[22] Kurylev, Y. U.; Mandache, N.; Peat, K. S., Hausdorff moments in an inverse problem for the heat equation: numerical experiment, Inverse Probl., 19, 253-264 (2003) · Zbl 1221.35450
[23] Wang, Q. F.; Feng, D. X.; Cheng, D. Z., Parameter identification for a class of abstract nonlinear parabolic distributed parameter systems, Comput. Math. Appl., 48, 1847-1861 (2004) · Zbl 1071.65131
[24] Wu, Z. Q.; Yin, J. X.; Wang, C. P., Elliptic and Parabolic Equations (2003), Science Press: Science Press Beijing, (in Chinese)
[25] Ahmed, N. U., Optimization and Identification of Systems Governed by Evolution Equations on Banach Space (1989), Longman Scientific and Technical: Longman Scientific and Technical England · Zbl 0674.93018
[26] Wang, Y. D., \(L^2\) Theories on Partial Differential Equation (1989), Beijing University Press: Beijing University Press Beijing, (in Chinese)
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