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Mathematical Problems in Engineering
Volume 2015 (2015), Article ID 915793, 14 pages
http://dx.doi.org/10.1155/2015/915793
Research Article

Inversion Study of Vertical Eddy Viscosity Coefficient Based on an Internal Tidal Model with the Adjoint Method

1Laboratory of Physical Oceanography, Ocean University of China, Qingdao 266100, China
2College of Engineering, Ocean University of China, Qingdao 266100, China

Received 28 March 2014; Revised 17 August 2014; Accepted 18 August 2014

Academic Editor: Fatih Yaman

Copyright © 2015 Guangzhen Jin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. M. Rattray, “On the coastal generation of internal tides,” Tellus, vol. 12, no. 1, pp. 54–62, 1960. View at Google Scholar
  2. P. G. Baines, “The generation of internal tides by flat-bump topography,” Deep-Sea Research and Oceanographic Abstracts, vol. 20, no. 2, pp. 179–205, 1973. View at Publisher · View at Google Scholar · View at Scopus
  3. T. H. Bell, “Topographically generated internal waves in the open ocean,” Journal of Geophysical Research, vol. 80, pp. 320–327, 1975. View at Google Scholar
  4. P. G. Baines, “On internal tide generation models,” Deep Sea Research A: Oceanographic Research Papers, vol. 29, no. 3, pp. 307–338, 1982. View at Publisher · View at Google Scholar · View at Scopus
  5. P. D. Craig, “Solutions for internal tidal generation over coastal topography,” Journal of Marine Research, vol. 45, pp. 83–105, 1987. View at Publisher · View at Google Scholar
  6. T. Gerkema, “A unified model for the generation and fission of internal tides in a rotating ocean,” Journal of Marine Research, vol. 54, no. 3, pp. 421–450, 1996. View at Publisher · View at Google Scholar · View at Scopus
  7. S. G. Llewellyn Smith and W. R. Young, “Conversion of the barotropic tide,” Journal of Physical Oceanography, vol. 32, no. 5, pp. 1554–1566, 2002. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. S. K. Kang, M. G. G. Foreman, W. R. Crawford, and J. Y. Cherniawsky, “Numerical modeling of internal tide generation along the Hawaiian Ridge,” Journal of Physical Oceanography, vol. 30, no. 5, pp. 1083–1098, 2000. View at Publisher · View at Google Scholar · View at Scopus
  9. Y. Niwa and T. Hibiya, “Numerical study of the spatial distribution of the M2 internal tide in the Pacific Ocean,” Journal of Geophysical Research C: Oceans, vol. 106, no. 10, pp. 22441–22449, 2001. View at Publisher · View at Google Scholar · View at Scopus
  10. P. F. Cummins, J. Y. Cherniawsky, and M. G. G. Foreman, “North Pacific internal tides from the Aleutian ridge: altimeter observations and modeling,” Journal of Marine Research, vol. 59, no. 2, pp. 167–191, 2001. View at Publisher · View at Google Scholar · View at Scopus
  11. Y. Niwa and T. Hibiya, “Three-dimensional numerical simulation of M2 internal tides in the East China Sea,” Journal of Geophysical Research C: Oceans, vol. 109, no. 4, Article ID C04027, 2004. View at Publisher · View at Google Scholar · View at Scopus
  12. S. Jan, C.-S. Chern, J. Wang, and S.-Y. Chao, “Generation of diurnal K1 internal tide in the Luzon Strait and its influence on surface tide in the South China Sea,” Journal of Geophysical Research C: Oceans, vol. 112, no. 6, Article ID C06019, 2007. View at Publisher · View at Google Scholar · View at Scopus
  13. R. C. Pacanowski and S. G. H. Philander, “Parameterization of vertical mixing in numerical models of tropical oceans,” Journal of Physical Oceanography, vol. 11, no. 11, pp. 1443–1451, 1981. View at Google Scholar · View at Scopus
  14. B. Henderson-Sellers, “A simple formula for vertical eddy diffusion coefficients under conditions of nonneutral stability,” Journal of Geophysical Research, vol. 87, pp. 5860–5864, 1982. View at Google Scholar
  15. N. S. Heaps, “Three-dimensional model for tides and surges with vertical eddy viscosity prescribed in two layers—I. Mathematical formulation,” Geophysical Journal of the Royal Astronomical Society, vol. 64, pp. 291–302, 1981. View at Google Scholar
  16. N. S. Heapsand and J. E. Jones, “Three-dimensional model for tides and surges with vertical eddy viscosity prescribed in two layers—II. Irish Sea with bed friction layer,” Geophysical Journal of the Royal Astronomical Society, vol. 64, pp. 303–320, 1981. View at Google Scholar
  17. V. P. Kochergin, “Three-dimensional prognostic models. In: three-dimensional coastal ocean models,” Coastal and Estuarine Science, vol. 4, pp. 201–208, 1987. View at Publisher · View at Google Scholar
  18. T. Pohlmann, “Calculating the annual cycle of the vertical eddy viscosity in the North Sea with a three-dimensional baroclinic shelf sea circulation model,” Continental Shelf Research, vol. 16, no. 2, pp. 147–161, 1996. View at Publisher · View at Google Scholar · View at Scopus
  19. A. F. Bennett and P. C. McIntosh, “Open ocean modeling as an inverse problem: tidal theory,” Journal of Physical Oceanography, vol. 12, no. 10, pp. 1004–1018, 1982. View at Google Scholar
  20. L. Yu and J. J. O'Brien, “Variational estimation of the wind stress drag coefficient and the oceanic eddy viscosity profile,” Journal of Physical Oceanography, vol. 21, pp. 709–719, 1991. View at Google Scholar
  21. R. W. Lardner, “Optimal control of open boundary conditions for a numerical tidal model,” Computer Methods in Applied Mechanics and Engineering, vol. 102, no. 3, pp. 367–387, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. U. Seiler, “Estimation of open boundary conditions with the adjoint method,” Journal of Geophysical Research, vol. 98, no. 12, pp. 22855–22870, 1993. View at Publisher · View at Google Scholar · View at Scopus
  23. I. M. Navon, “Practical and theoretical aspects of adjoint parameter estimation and identifiability in meteorology and oceanography,” Dynamics of Atmospheres and Oceans, vol. 27, no. 1–4, pp. 55–79, 1998. View at Publisher · View at Google Scholar · View at Scopus
  24. N. Ayoub, “Estimation of boundary values in a North Atlantic circulation model using an adjoint method,” Ocean Modelling, vol. 12, no. 3-4, pp. 319–347, 2006. View at Publisher · View at Google Scholar · View at Scopus
  25. J. Zhang and X. Lu, “Inversion of three-dimensional tidal currents in marginal seas by assimilating satellite altimetry,” Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 49–52, pp. 3125–3136, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. J. C. Zhang and X. Q. Lu, “Parameter estimation for a three-dimensional numerical barotropic tidal model with adjoint method,” International Journal for Numerical Methods in Fluids, vol. 57, no. 1, pp. 47–92, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. H. Chen, C. Miao, and X. Lv, “A three-dimensional numerical internal tidal model involving adjoint method,” International Journal for Numerical Methods in Fluids, vol. 69, no. 10, pp. 1584–1613, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  28. J. Zhang and H. Chen, “Semi-idealized study on estimation of partly and fully space varying open boundary conditions for tidal models,” Abstract and Applied Analysis, vol. 2013, Article ID 282593, 14 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  29. A. Cao, H. Chen, J. Zhang, and X. Lv, “Optimization of open boundary conditions in a 3D internal tidal model with the adjoint method around Hawaii,” Abstract and Applied Analysis, vol. 2013, Article ID 950926, 11 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus
  30. H. Chen, C. Miao, and X. Lv, “Estimation of open boundary conditions for an internal tidal model with adjoint method: a comparative study on optimization methods,” Mathematical Problems in Engineering, vol. 2013, Article ID 802136, 12 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  31. X. Lu and J. Zhang, “Numerical study on spatially varying bottom friction coefficient of a 2D tidal model with adjoint method,” Continental Shelf Research, vol. 26, no. 16, pp. 1905–1923, 2006. View at Publisher · View at Google Scholar · View at Scopus
  32. J. Zhang, X. Lu, P. Wang, and Y. P. Wang, “Study on linear and nonlinear bottom friction parameterizations for regional tidal models using data assimilation,” Continental Shelf Research, vol. 31, no. 6, pp. 555–573, 2011. View at Publisher · View at Google Scholar · View at Scopus
  33. R. W. Lardner and S. K. Das, “Optimal estimation of eddy viscosity for a quasi-three-dimensional numerical tidal and storm surge model,” International Journal for Numerical Methods in Fluids, vol. 18, no. 3, pp. 295–312, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  34. J. E. Richardson and V. G. Panchang, “A modified adjoint method for inverse eddy viscosity estimation for use in coastal circulation models,” in Proceedings of the 2nd International Conference on Estuarine and Coastal Modeling, pp. 733–745, November 1992. View at Scopus
  35. G. P. Cressman, “An operational objective analysis system,” Monthly Weather Review, vol. 87, no. 10, pp. 367–374, 1959. View at Google Scholar
  36. S. J. Wright and J. Nocedal, Numerical optimization, Springer, New York, NY, USA, 1999. View at Publisher · View at Google Scholar · View at MathSciNet
  37. W. F. Mascarenhas, “The BFGS method with exact line searches fails for non-convex objective functions,” Mathematical Programming, vol. 99, no. 1, pp. 49–61, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. A. K. Alekseev, I. M. Navon, and J. L. Steward, “Comparison of advanced large-scale minimization algorithms for the solution of inverse ill-posed problems,” Optimization Methods and Software, vol. 24, no. 1, pp. 63–87, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  39. X. Zou, I. M. Navon, M. Berger, K. H. Phua, T. Schlick, and F. Le Dimet, “Numerical experience with limited-memory quasi-Newton and truncated Newton methods,” SIAM Journal on Optimization, vol. 3, no. 3, pp. 582–608, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  40. J. Nocedal, “Updating quasi-Newton matrices with limited storage,” Mathematics of Computation, vol. 35, no. 151, pp. 773–782, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  41. R. Malouf, “A comparison of algorithms for maximum entropy parameter estimation,” in Proceedings of the 6th Conference on Natural Language Learning (CoNLL '02), pp. 49–55, 2002.
  42. G. Andrew and J. Gao, “Scalable training of L1-regularized log-linear models,” in Proceedings of the 24th International Conference on Machine Learning (ICML '07), pp. 33–40, June 2007. View at Publisher · View at Google Scholar · View at Scopus
  43. A. K. Alekseev, I. M. Navon, and J. L. Steward, “Comparison of advanced large-scale minimization algorithms for the solution of inverse ill-posed problems,” Optimization Methods & Software, vol. 24, no. 1, pp. 63–87, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  44. D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Mathematical Programming, vol. 45, no. 1–3, pp. 503–528, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  45. L. J. Nocedal, “BFGS subroutine, software for large-scale unconstrained optimization,” http://www.ece.northwestern.edu/~nocedal/lbfgs.html.