Some Computational Problems of Oceanography
Three computational problems will be discussed. The first is numerical forecasting of hurricane-induced storm surges. Because population densities are highest around bays and estuaries, forecast models must be capable of representing the irregular shorelines of these natural basins. The computational domain mist extend from the shallow inland waters out into the deeper waters of the continental shelf. The distance between adjacent computational points should vary inversely with the square root of the depth, if the solutions are to be uniformly accurate throughout the entire basin. And a storm which takes twelve hours in crossing the basin must be simulated in only a few minutes due to operational constraints involved with issuing warnings to evacuate vulnerable coastal areas. Finite element methods are too slow to be used for this problem, and coordinate transformation methods are incompatible with the required distribution of spatial points. However, a finite difference method can represent the irregular coastline while providing the proper interior resolution and computational efficiency.
The other two problems are encountered in numerically forecasting wave conditions in the ocean. Problem two is a new twist to the well-known problem of solving the advection equation. For this problem there are a family of distributions, one for each frequency and direction of the waves making up the spectrum of the chaotic sea. Thus, a family of advection equations must be solved simultaneously, each with a different advecting velocity. As a result, after long times numerical solutions must be characterized by anomalous dispersion. Because the finite-element method provides a means of interpolating the advecting velocity between discrete wave components of the numerical model, it provides a direction in which to look for a solution to this problem.
The third problem concerns the nonlinear interaction among the components of the wave spectrum. This involves the evaluation of a six-dimensional integral at each time-step of a wave forecast model. The finite element method can be used to replace this integral equation by a discrete analog, thus allowing the nonlinear interactions to be explicitly included in wave models.
KeywordsFinite Difference Method Storm Surge Wave Spectrum Anomalous Dispersion Advecting Velocity
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- Arakawa, A. (1966) Computational Design for Long Term Numerical Integration of the Equations of Motion: Two-Dimensional ncompressible Flow. Part I. J. Comp. Phys., 1: 119–143.Google Scholar
- Birchfield, G. E. and T. S. Marty (1974) A Numerical Model for Wind-Driven Circulation in Lakes Michigan and Huron. Mon. Wea. Rev., 102: 157–165.Google Scholar
- Chu, Wen-Hwa (1971) Development of a General Finite Difference Approximation for a General Domain. Part I: Machine Transformation. J. Comp. Phys., 8: 392–408.Google Scholar
- Fritts, M. J. and J. P. Boris (1979) The Lagrangian Solution of Transient Problems in Hydrodynamics using a Triangular Mesh. J. Comp. Phys., 31: 173–215.Google Scholar
- Gray, W. G. and G. F. Pinder (1976) On the Relationship between the Finite Element and Finite Difference Methods. Int. J. Num. Math. Engng., 10: 893–923.Google Scholar
- Haussling, H. J. (1979) Boundary-Fitted Coordinates for Accurate Numerical Solution of Multi-Body Flow Problems. J. Comp. Phys., 30: 107–124.Google Scholar
- Jespersen, D. C. (1974) Arakawa’s Method is a Finite-Element Method. J. Comp. Phys., 16: 383–390.Google Scholar
- MacNeal, R. J. (1953) An Asymmetrical Finite-Difference Network. Q. Appl. Math., 11: 285–310.Google Scholar
- Meyder, R. (1975) Solving the Conservation Equations in Fuel Rod Bundles Exposed to Parallel Flow by Means of Curvilinear-Orthogonal Coordinates. J. Comp. Phys., 17: 53–67.Google Scholar
- Noh, W. F. (1964) CEL: A Time-Dependent Two-Space-Dimensional Coupled Eulerian-Lagrange Code. Meth. Comp. Phys., 3: 117–179.Google Scholar
- Potter, D. E. and G. H. Tuttle (1973) The Construction of Discrete Orthogonal Coordinates. J. Comp. Phys., 13: 483–501.Google Scholar
- Reid, R. D. and A. C. Vastano (1966) Orthogonal Coordinates for the Analysis of Long Gravity Waves Near Islands. Coastal Engineering, Proc., Santa Barbara Specialty Conf., American Soc. Civil Eng., 1–20.Google Scholar
- Sadourny, R., A. Arakawa and Y. Mintz (1968) Integration of the Nondivergent Barotropic Vorticity Equation with an Icosahedral-Hexagonal Grid for the Sphere. Mon. Wea. Rev., 96: 351–356.Google Scholar
- Shen, Shan-fu (1977) Finite-Element Methods in Fluid Mechanics. Ann. Rev. Fluid Mach., 9: 421–445.Google Scholar
- Thacker, W. C. (1977) Irregular Grid Finite-Difference Techniques: Simulations of Oscillations in Shallow Circular Basins. J. Phys. Oceanogr., 7: 284–292.Google Scholar
- Thacker, W. C. (1978) Comparison of Finite-Element and Finite Difference Schemes. Part II: Iwo-Dimensional Gravity Wave Motion. J. Phys. Oceanogr., 8: 680–689.Google Scholar
- Thacker, W. C. (1980) A Geodesic Finite-Difference Method for Curved Domains: Simulation of Tidal Motion on a Sphere. J. Comp. Phys., 37: 355–369.Google Scholar
- Thacker, W. C., A. Gonzalez and G. E. Putland (1980) A Method for Automating the Construction of Irregular Computational Grids for Storm Surge Forecast Models. J. Comp. Phys., 37: 371–387.Google Scholar
- Thompson, J. F., F. C. Thames and C. W. Mastin (1977) TOMCAT: A Code for Numerical Generation of Boundary-Fitted Curvilinear Coordinate Systems on Fields Containng any Number of Arbitrary Two-Dimensional Bodies. J. Comp. Phys., 24: 274–302.Google Scholar
- Wanstrath, J. J., R. E. Whitaker, R. O. Reid and A. C. Vastano (1976) Storm Surge Simulation in Transformed Coordinates. Vol I: Theory and Application. U.S. Army Corps of Engineers Tech. Rept. No. 76–3.Google Scholar