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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 and 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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

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

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.

[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.

[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.

[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.

[5]

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

[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.

[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.

[8]

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

[9]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[18]

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

[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.

[20]

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

[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.

[22]

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

[23]

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

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