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2015, 5(1): 47-57. doi: 10.3934/naco.2015.5.47

Optimality of piecewise thermal conductivity in a snow-ice thermodynamic system

Wei Lv 1, and  Ruirui Sui 1,

1. 

Department of Mathematics, Shanghai University, Shanghai 200444, China, China

Received  December 2014 Revised  March 2015 Published  March 2015

This article is intended to provide the optimality of piecewise thermal conductivity in a snow-ice thermodynamic system. Based on the temperature distribution characteristics of snow and sea ice, we construct a piecewise smooth thermodynamic system coupled by snow and sea ice. Taking the piecewise thermal conductivities of snow and sea ice as control variables and the temperature deviations obtained from the system and the observations as the performance criterion, an identification model with state constraints is given. The dependency relationship between state and control variables is proven, and the existence of the optimal control is discussed. The work can provide a theoretical foundation for simulating temperature distributions of snow and sea ice.
Keywords: Optimality, thermodynamic system, snow, piecewise thermal conductivity, sea ice..
Mathematics Subject Classification: Primary: 49J20; Secondary: 35K0.
Citation: Wei Lv, Ruirui Sui. Optimality of piecewise thermal conductivity in a snow-ice thermodynamic system. Numerical Algebra, Control & Optimization, 2015, 5 (1) : 47-57. doi: 10.3934/naco.2015.5.47
References:
[1]

N. Calonne, F. Flin, S. Morin, B. Lesaffre, S. du Roscoat, C. Geindreau, F. St Martin dHeres, F. Grenoble and I. Grenoble, Numerical and experimental investigations of the effective thermal conductivity of snow, J. Geophys. Res., 38 (2010), L23501. doi: 10.1029/2011GL049234.  Google Scholar

[2]

C. I. Christov and T. Marinov, Identification of heat-conduction coefficient via method of variational imbedding, Math. Comput. Modell., 27 (1998), 109-116. doi: 10.1016/S0895-7177(97)00269-0.  Google Scholar

[3]

S. Dutman and J. Ha, Identifiability of piecewise constant conductivity in a heat conduction process, SIAM J.Control.Optim., 46 (2007), 694-713. doi: 10.1137/060657364.  Google Scholar

[4]

H. W. Engl and J. Zou, A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Probl., 16 (2000), 1907-1923. doi: 10.1088/0266-5611/16/6/319.  Google Scholar

[5]

L. C Evans, Partial differential equations, 2nd edition, Graduate studies in mathemastic 19. Province, RI: American Mathemastic Society, 2010, 350-358.  Google Scholar

[6]

H. Fang, J. Wang, E. Feng and Z. Li, Parameter identification and application of a distributed parameter coupled system with a movable inner boundary, Computers and Mathematics with Applications, 62 (2011), 4015-4020. doi: 10.1016/j.camwa.2011.09.035.  Google Scholar

[7]

T. Fichefet, B. Tartinville and H. Goosse, Sensitivity of the Antarctic sea ice to the thermal conductivity of snow, Geophys. Res. Lett., 27 (2000), 401-404. Google Scholar

[8]

S. Gutman, Identification of discontinuous parameter in flow equation, SIAM J.Control Optim., 28 (1990), 1049-1060. doi: 10.1137/0328057.  Google Scholar

[9]

S. Larrsson and V. Themée, Partial differential equations with numerical methods, Springer-Verlag, New York, 1971. Google Scholar

[10]

RO. Lecomte, T. Fichefet, M. Vancoppenolle, F. Domine, F. Massonnet, P. Mathiot, S. Morin and P. Y. Barriat, On the formulation of snow thermal conductivity in large-scale sea ice models, J. Adv. Model. Earth Syst., 5 (2013). doi: 10.1002/jame.20039.  Google Scholar

[11]

R. Lei, Z. Li, B. Cheng, Z. Zhang and P. Heil, Annual cycle of landfast sea ice in Prydz Bay, east Antarctica, Geophys. Res. Lett., 115 (2010), C02006. doi: 10.1029/2008JC005223.  Google Scholar

[12]

P. Lemke, W. Owens and W. D. Hibler III, A coupled sea ice-mixed layer-pycnocline model for the Weddell Sea, J. Geophys. Res., 95 (1990), 9513-9525. Google Scholar

[13]

W. Lv, E. Feng and Z. Li, A coupled thermodynamic system of sea ice and its parameter identification, Appl. Math. Model., 32 (2008), 1198-1207. doi: 10.1016/j.apm.2007.03.006.  Google Scholar

[14]

W. Lv, E. Feng and Z. Li, Properties and optimality conditions of a three-dimension non-smooth thermodynamic system of sea ice, Appl. Math. Model., 33 (2009), 2324-2333. doi: 10.1016/j.apm.2008.07.002.  Google Scholar

[15]

G. A. Maykut and W. M. Washington, Some results from a time-dependent thermodynamic model of sea ice, J. Geophys. Res., 276 (1971), 1550-1575. Google Scholar

[16]

M. J. McGuinness, K. Collins, H. J. Trodahl and T. G. Haskell, Nonlinear thermal transport and brine convection in first year sea ice, Ann. Glaciol., 72 (1998), 471-476. Google Scholar

[17]

S. Omatu and J. H. Seinfeld, Distributed parameter system, Cla.Press, 274 (1989), Oxford.  Google Scholar

[18]

C. L. Parkinson and W. M. Washington, A large-scale numerical model of sea ice, J. Geophys. Res., 84 (1979), 311-337. Google Scholar

[19]

D. J. Pringle, H. Eicken, H. J. Trodahl and L. G. E. Backstrom, Thermal conductivity of landfast Antarctic and Arctic sea ice, J. Geophys. Res., 112 (2007), C04017. doi: 10.1029/2006JC003641.  Google Scholar

[20]

M. C. Serreze, M. M. Holland and J. Stroeve, Perspectives on the Arctic's shrinking sea-ice cover, Science, 315 (2007), 1533-1536. Google Scholar

[21]

H. J. Trodahl, S. Wilkinson, M. McGuinness and T. Haskell, Thermal conductivity of sea ice: Dependence on temperature and depth, Geophys. Res. Lett., 28 (2001), 1279-1282. Google Scholar

[22]

X. Wu, I. Simmonds and W. Budd, Modeling of Antarctic sea ice in a general circulation model, J. Clim., 10 (1997), 593-609. Google Scholar

[23]

C. Y. Yang, Estimation of the temperature-dependent thermal conductivity in inverse heat condition problems, Appl. Math. Modell., 23 (1999), 469-478. Google Scholar

show all references

References:
[1]

N. Calonne, F. Flin, S. Morin, B. Lesaffre, S. du Roscoat, C. Geindreau, F. St Martin dHeres, F. Grenoble and I. Grenoble, Numerical and experimental investigations of the effective thermal conductivity of snow, J. Geophys. Res., 38 (2010), L23501. doi: 10.1029/2011GL049234.  Google Scholar

[2]

C. I. Christov and T. Marinov, Identification of heat-conduction coefficient via method of variational imbedding, Math. Comput. Modell., 27 (1998), 109-116. doi: 10.1016/S0895-7177(97)00269-0.  Google Scholar

[3]

S. Dutman and J. Ha, Identifiability of piecewise constant conductivity in a heat conduction process, SIAM J.Control.Optim., 46 (2007), 694-713. doi: 10.1137/060657364.  Google Scholar

[4]

H. W. Engl and J. Zou, A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Probl., 16 (2000), 1907-1923. doi: 10.1088/0266-5611/16/6/319.  Google Scholar

[5]

L. C Evans, Partial differential equations, 2nd edition, Graduate studies in mathemastic 19. Province, RI: American Mathemastic Society, 2010, 350-358.  Google Scholar

[6]

H. Fang, J. Wang, E. Feng and Z. Li, Parameter identification and application of a distributed parameter coupled system with a movable inner boundary, Computers and Mathematics with Applications, 62 (2011), 4015-4020. doi: 10.1016/j.camwa.2011.09.035.  Google Scholar

[7]

T. Fichefet, B. Tartinville and H. Goosse, Sensitivity of the Antarctic sea ice to the thermal conductivity of snow, Geophys. Res. Lett., 27 (2000), 401-404. Google Scholar

[8]

S. Gutman, Identification of discontinuous parameter in flow equation, SIAM J.Control Optim., 28 (1990), 1049-1060. doi: 10.1137/0328057.  Google Scholar

[9]

S. Larrsson and V. Themée, Partial differential equations with numerical methods, Springer-Verlag, New York, 1971. Google Scholar

[10]

RO. Lecomte, T. Fichefet, M. Vancoppenolle, F. Domine, F. Massonnet, P. Mathiot, S. Morin and P. Y. Barriat, On the formulation of snow thermal conductivity in large-scale sea ice models, J. Adv. Model. Earth Syst., 5 (2013). doi: 10.1002/jame.20039.  Google Scholar

[11]

R. Lei, Z. Li, B. Cheng, Z. Zhang and P. Heil, Annual cycle of landfast sea ice in Prydz Bay, east Antarctica, Geophys. Res. Lett., 115 (2010), C02006. doi: 10.1029/2008JC005223.  Google Scholar

[12]

P. Lemke, W. Owens and W. D. Hibler III, A coupled sea ice-mixed layer-pycnocline model for the Weddell Sea, J. Geophys. Res., 95 (1990), 9513-9525. Google Scholar

[13]

W. Lv, E. Feng and Z. Li, A coupled thermodynamic system of sea ice and its parameter identification, Appl. Math. Model., 32 (2008), 1198-1207. doi: 10.1016/j.apm.2007.03.006.  Google Scholar

[14]

W. Lv, E. Feng and Z. Li, Properties and optimality conditions of a three-dimension non-smooth thermodynamic system of sea ice, Appl. Math. Model., 33 (2009), 2324-2333. doi: 10.1016/j.apm.2008.07.002.  Google Scholar

[15]

G. A. Maykut and W. M. Washington, Some results from a time-dependent thermodynamic model of sea ice, J. Geophys. Res., 276 (1971), 1550-1575. Google Scholar

[16]

M. J. McGuinness, K. Collins, H. J. Trodahl and T. G. Haskell, Nonlinear thermal transport and brine convection in first year sea ice, Ann. Glaciol., 72 (1998), 471-476. Google Scholar

[17]

S. Omatu and J. H. Seinfeld, Distributed parameter system, Cla.Press, 274 (1989), Oxford.  Google Scholar

[18]

C. L. Parkinson and W. M. Washington, A large-scale numerical model of sea ice, J. Geophys. Res., 84 (1979), 311-337. Google Scholar

[19]

D. J. Pringle, H. Eicken, H. J. Trodahl and L. G. E. Backstrom, Thermal conductivity of landfast Antarctic and Arctic sea ice, J. Geophys. Res., 112 (2007), C04017. doi: 10.1029/2006JC003641.  Google Scholar

[20]

M. C. Serreze, M. M. Holland and J. Stroeve, Perspectives on the Arctic's shrinking sea-ice cover, Science, 315 (2007), 1533-1536. Google Scholar

[21]

H. J. Trodahl, S. Wilkinson, M. McGuinness and T. Haskell, Thermal conductivity of sea ice: Dependence on temperature and depth, Geophys. Res. Lett., 28 (2001), 1279-1282. Google Scholar

[22]

X. Wu, I. Simmonds and W. Budd, Modeling of Antarctic sea ice in a general circulation model, J. Clim., 10 (1997), 593-609. Google Scholar

[23]

C. Y. Yang, Estimation of the temperature-dependent thermal conductivity in inverse heat condition problems, Appl. Math. Modell., 23 (1999), 469-478. Google Scholar

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