• Previous Article
    Optimal dilution strategy for a microbial continuous culture based on the biological robustness
  • NACO Home
  • This Issue
  • Next Article
    Analysis of complexity of primal-dual interior-point algorithms based on a new kernel function for linear optimization
2015, 5(1): 47-57. doi: 10.3934/naco.2015.5.47

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

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.
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).  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.  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.  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.  doi: 10.1088/0266-5611/16/6/319.  Google Scholar

[5]

L. C Evans, Partial differential equations,, 2nd edition, (2010), 350.   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.  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.   Google Scholar

[8]

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

[9]

S. Larrsson and V. Themée, Partial differential equations with numerical methods,, Springer-Verlag, (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).  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.   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.  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.  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.   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.   Google Scholar

[17]

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

[18]

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

[23]

C. Y. Yang, Estimation of the temperature-dependent thermal conductivity in inverse heat condition problems,, Appl. Math. Modell., 23 (1999), 469.   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).  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.  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.  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.  doi: 10.1088/0266-5611/16/6/319.  Google Scholar

[5]

L. C Evans, Partial differential equations,, 2nd edition, (2010), 350.   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.  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.   Google Scholar

[8]

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

[9]

S. Larrsson and V. Themée, Partial differential equations with numerical methods,, Springer-Verlag, (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).  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.   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.  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.  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.   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.   Google Scholar

[17]

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

[18]

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

[23]

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

[1]

Victor Isakov, Shingyu Leung, Jianliang Qian. A three-dimensional inverse gravimetry problem for ice with snow caps. Inverse Problems & Imaging, 2013, 7 (2) : 523-544. doi: 10.3934/ipi.2013.7.523

[2]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[3]

María Teresa González Montesinos, Francisco Ortegón Gallego. The evolution thermistor problem with degenerate thermal conductivity. Communications on Pure & Applied Analysis, 2002, 1 (3) : 313-325. doi: 10.3934/cpaa.2002.1.313

[4]

María Teresa González Montesinos, Francisco Ortegón Gallego. The thermistor problem with degenerate thermal conductivity and metallic conduction. Conference Publications, 2007, 2007 (Special) : 446-455. doi: 10.3934/proc.2007.2007.446

[5]

Xiaoping Fang, Youjun Deng. Uniqueness on recovery of piecewise constant conductivity and inner core with one measurement. Inverse Problems & Imaging, 2018, 12 (3) : 733-743. doi: 10.3934/ipi.2018031

[6]

J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485

[7]

Masahiro Yamaguchi, Yasuhiro Takeuchi, Wanbiao Ma. Population dynamics of sea bass and young sea bass. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 833-840. doi: 10.3934/dcdsb.2004.4.833

[8]

Adriana C. Briozzo, María F. Natale, Domingo A. Tarzia. The Stefan problem with temperature-dependent thermal conductivity and a convective term with a convective condition at the fixed face. Communications on Pure & Applied Analysis, 2010, 9 (5) : 1209-1220. doi: 10.3934/cpaa.2010.9.1209

[9]

Elena Bonetti, Pierluigi Colli, Mauro Fabrizio, Gianni Gilardi. Modelling and long-time behaviour for phase transitions with entropy balance and thermal memory conductivity. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1001-1026. doi: 10.3934/dcdsb.2006.6.1001

[10]

Tao Wang. One dimensional $p$-th power Newtonian fluid with temperature-dependent thermal conductivity. Communications on Pure & Applied Analysis, 2016, 15 (2) : 477-494. doi: 10.3934/cpaa.2016.15.477

[11]

Asim Aziz, Wasim Jamshed. Unsteady MHD slip flow of non Newtonian power-law nanofluid over a moving surface with temperature dependent thermal conductivity. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 617-630. doi: 10.3934/dcdss.2018036

[12]

Kari Eloranta. Archimedean ice. Discrete & Continuous Dynamical Systems - A, 2013, 33 (9) : 4291-4303. doi: 10.3934/dcds.2013.33.4291

[13]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. On the one-dimensional version of the dynamical Marguerre-Vlasov system with thermal effects. Conference Publications, 2009, 2009 (Special) : 536-547. doi: 10.3934/proc.2009.2009.536

[14]

Irena Lasiecka, To Fu Ma, Rodrigo Nunes Monteiro. Long-time dynamics of vectorial von Karman system with nonlinear thermal effects and free boundary conditions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1037-1072. doi: 10.3934/dcdsb.2018141

[15]

Jishan Fan, Fucai Li, Gen Nakamura. Regularity criteria for the Boussinesq system with temperature-dependent viscosity and thermal diffusivity in a bounded domain. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 4915-4923. doi: 10.3934/dcds.2016012

[16]

Ciro D'Apice, Olha P. Kupenko, Rosanna Manzo. On boundary optimal control problem for an arterial system: First-order optimality conditions. Networks & Heterogeneous Media, 2018, 13 (4) : 585-607. doi: 10.3934/nhm.2018027

[17]

Vaughn Climenhaga. A note on two approaches to the thermodynamic formalism. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 995-1005. doi: 10.3934/dcds.2010.27.995

[18]

Yong Fang. Thermodynamic invariants of Anosov flows and rigidity. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1185-1204. doi: 10.3934/dcds.2009.24.1185

[19]

Lili Du, Mingshu Fan. Thermal runaway for a nonlinear diffusion model in thermal electricity. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2349-2368. doi: 10.3934/dcds.2013.33.2349

[20]

Jihua Yang, Erli Zhang, Mei Liu. Limit cycle bifurcations of a piecewise smooth Hamiltonian system with a generalized heteroclinic loop through a cusp. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2321-2336. doi: 10.3934/cpaa.2017114

 Impact Factor: 

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]