Abstract
Area-time averaged Navier-Stokes equations of turbulent fluid motions are simplified by single-layer approximations to model tidal elevations and currents in the real world oceans. The resulting ocean tidal equations (OTE’s) are solved by a finite difference scheme, which allows for a realistic integration of empirical tide data into the computed tidal field. The unique hydrodynamical-interpolation technique takes advantage of the massive empirical values, to search by systematic variation for the a-priori unknown turbulent momentum exchange or mixing and friction parameters occurring in the OTE’s. The characteristic features of the new interpolation scheme are pointwise demonstrated for the constructed M2-tide model by tidal charts displaying side-by-side empirical and computed data for direct evaluation. The global quality of the model is shown by the computed realistically balanced budgets of angular momentum and energy. The latter results reveal the true nature of eddy dissipation as a momentum exchange: while the former is completely negligible, the latter is of major significance.
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References
Accad, Y., and C.L. Pekeris, Solution of the Tidal Equations for the M2 and S2 Tides in the World Oceans from a Knowledge of the Tidal Potential Alone Phil, Trans,, Roy, Soc„ London,, A, 220, p. 235, 1978.
Bagriantsev, NV., A.I. Danilov, and V.O. Ivehenko, Orientational Effects in Geophysical Fluid Dynamics, Int J. Eng,Séi„ 21, p. 725, 1983.
Bartels, J., Gezeitenkraefte,In Handbuch der Physik XLVIII, Geophysik II, edited by S. Fluegge and J. Bartels, Springer, Berlin, 1957.
Boussinesq, J., Expression du Frottement Extérieur dans L’écoulement Tumultueux d’une Fluide, Comptes Rend. Acad. Séi„ 122, p. 1445, 1896a.
Boussinesq, J., Formules du Coefficient des Frottements Intérieurs dans L’écoulement Tumultueux Graduellement Varie des Liquides Comptes Rend, Acad, Sei„ 122, p. 1517, 1896b.
British Admiralty, Tide Tables Vols 1,2, and 3, 1984.
Busse, F.H., and J.A. Whitehead, Instabilities of Convection
Rolls in a High Prandtl Number Fluid, J, Fluid Meeh„ 42, p. 305, 1971.
Cartwright, D.E., Ocean Tides, Rep Progr Phys. 40, p. 665, 1977.
Cartwright, D.E., and A.C. Edden, Corrected Tables of Tidal Harmonics, Geophys. Js Roy. Astrs BOé,.33, p. 253, 1973.
Cartwright, D.E., A.C. Edden, R. Spencer, and J.M. Vassie, The Tides of the Northern Atlantic Ocean, Phil, Trans, Roy Soé 29, London, p. 87, 1980.
Cartwright, D.E., and R.J. Tayler, New Computations of the Tide-Generating Potential, Geophyss Js Roy, Astr, Socs, 23, p. 45, 1971.
Cartwright, D.E., B.D. Zetler, and H.V. Hamon, Pelagic. Tidal
Constants, IAPSO Publication Scientifique No 30, 1979. Cox, M.D., A Mathematical Model of the Indian Ocean, DeepSea Res1, p. 45, 1970.
Crowley, W.P., A Global Numerical Model: Part 1, J,Çomp, Phys:.,., p. 1 1 1, 1968.
Crowley, W.P., A Numerical Model for Viscous, Free-Surface, Barotropic Wind-Driven Ocean Circulations, J=Comp,:Phys,,, a, p. 139, 1970.
Darwin, G. H., Report on the Harmonic Analysis of Tidal Observations, Brit. Ass. for adySeiRep,,, 1883, see also Sci,Papi, Cambridge, 1907.
Davies, A.M., The Numerical Solution of the Three-Dimensional Hydrodynamic Equations Using a B-Spline Representation of the Vertical Current Profile; Three-Dimensional Model with Depth-Varying Eddy Viscosity, In Bottom Tgrby= lenee,, Proceedings of the 8th Liege Colloquium op Ocean Hy drodynamies,12, edited by J.C.J. Nihoul, Elsevier Oceanography Series, p.1, 1977.
Davies, A.M., and G. K. Furness, Observed and Computed M2 Tidal Currents in the North Sea, J Phys,. Oceanogr„10, p. 237, 1980.
Dietrich, G., Die Gezeiten des Weltmeeres als Geographische Erscheinung, Zeitschr. d i Gcsselhsch,., Erdkynde,,3L, 1944a.
Dietrich, G., Die Schwingungssyteme der Halb-und Eintaegigen Tiden in den Ozeanen, Veroeff,, Inst MeereskgnJe„Oniy, Berlin, N = F„1.No = 41, 1944b.
Doodson, A.T., The harmonic Development of the Tide-Generating Potential, Prpg, Roy, Sgg„ hogggp, A. 100, p. 305, 1921.
Eliassen, A., and E. Kleinschmidt, Dynamic Meteorology, In Handhueh der Physik XLVII,geophysik II, edited by S. Fluegge and J. Bartels, Springer, Berlin, 1957.
Estes, R.H., A Computer Software System for the Generation of Global Ocean Tides Including Self–Gravitation and Crustal Loading Effects, NASA, TR–X–920–77–82, Goddard Space Flight Center, 1977.
Estes, R.H., A Simulation of Global Ocean Tide Recovery Using Altimeter Data with Systematic Orbit Error, Marine Geodesy,3., p. 75, 1980.
Eyris, M., Maregraphs de Grandes Profoundeurs, Cahiers 0ceanographigues, 20, p. 355, 1968.
Farrell, W.E., Deformation of the Earth by Surface Loads, Rev s Geophys, Space Phys„10, p. 261, 1972.
Filloux, J.H., Bourdon Tube Deep Sea Tide Gauges, In Tsunamis in the Pacific Ocean, edited by W.M. Adams, East-West Center Press, Honolulu, 1969.
Friedrich, H.J., Preliminary Results from a Numerical Multi-layer Model for the Circulation in the North Atlantic, Deutsche Hydr, Zeitsch s 2a, p. 145, 1970.
Gill, S.K., and D.L. Porter, Theoretical Offshore Tide Range Derived from a Simple Defant Tidal Model Compared with Observed Offshore Tides, Int. Hydrogr. Reyie,H LVII, Monaco, p. 155, 1980.
Grace, S.F., The Semidiurnal Lunar Tidal Motion of the Red Sea, Mon. Not. Roy. Astr. Soc„ Geophys. Supp,I,2, p. 273, 1930a.
Grace, S.F., The Influence of the Friction on the Tidal Motion of the Gulf of Suez, Mon, Not. Roy, Astr. Soc„ Oeophys. Suppl.,2, p. 316, 1930b.
Greenberg, D.A., Modeling of the Mean Barotropic, Circulation in the Bay of Fundy and Gulf of Maine, JPhys, Oceanogr„ 11, 1983.
Hansen, W., Die Ermittlung der Gezeiten Beliebig Gestalteter Meeresgebiete mit Hilfe des Randwertverfahrens, Deutsche Hydr. Zeitsch. 4 1, p. 157, 1948.
Hansen, W., Die Reproduktion der Bewegungsvorgaenge im Meere mit Hilfe Hydrodynamisch-Numerischer Verfahren, Mitt,.des Inst. f. Meereskunde der Univ. Hamburg, V, 1966.
Harris, R.A., Manual of Tides Parts I and II, U.S. Coast and Geodetic Survey, 1897.
Heaps, N.S., On the Numerical Solution of the Three-Dimensional Hydrodynamical Equations for Tides and Storm Surges, Mem. Soo Roy, Sei, Liege, Sep, 6, 2, p. 143, 1972.
Heiskanen, W., Ueber den Einfluss der Gezeiten auf die Saekulare Acceleration des Mondes, Ann, Acad, Sci„ Fennieae A,.. 18, p. 1, 1921.
Hendershott, M.C., The Effects of Solid-Earth Deformation on Global Ocean Tides, Geophys. J. Roy, Astr, Soc„a2, p. 389, 1972.
Holland, W. R., and A. D. Hirschman, A Numerical Calculation in the North Atlantic Ocean, J, Phys. Oceanogr,. 1 2, p. 336, 1972.
International Hydrographie Bureau, Tides, Harmonic Constants, Computer Tape Monaco, 1978.
Irish, J.D., W.H. Munk, and F.E. Snodgrass, M2 Amphidrome in the Northeast Pacific, Geophys, Fluid Dyn„ 2, p. 355, 1971.
Jachens, R.C., and J.T. Kuo, The 01 Tide in the North Atlantic Ocean as Derived from Land-based Tidal Gravity Measurements, Proceedings of the Seventh Symposium on Earth Tides, Sopron, Hungary Akad. Kiado Budapest, 1973.
Jeffreys, H., Tidal Friction in Shallow Seas, Phil, Trans„ Roy, Soc. A.,, 221, p. 239, 1920.
Johns, B., Vertical Structure of Tidal Flows in River Estu- aries, Geophys. J. Res„ Astr. Soc.,. 12, p. 103, 1966.
Joseph, D.D., Stability of Fluid Motions I, Springer, Berlin, 1976.
Kagan, B.A., Resistance Law of Tidal Flow, Izy,, Acad, Sci, USSR,. Atm. and Oce. Phys., 5, p. 302, 1972.
Kraav, V.R., Computation of the Semidiurnal Tide and Turbulence Parameters in the North Sea, Oceanology,., p. 332, 1969.
Ladyzhenskaya, 0.A., The Mathematical Theory of Viscous Incompressible Flog, Gordon and Breach, New York, 1969.
Lamb, H., Hydrodynamics, Dover Publications, New York, 1932. Laplace, P.S., Recherches sur Quelques Points de Systeme du Monde, Memo Acad,,,, Roy,, Sci,,, 88, 1775.
Leith, C.E., Two-Dimensional Eddy Viscosity Coefficients, Proc, WMOLIUGG Symp,, on Numerical Weather Prediction, Tokyo, p. 140, 1968.
Luther, D.S., and C. Wunsch, Tidal Charts of the Central Pacific Ocean, J. Phys. Oce.,, 2, p. 227, 1975.
Marchuk, G.I., and B.A. Kagan, OCEAN Tides, Mathematical Models and Numerical Experiments, Pergamon Press, Oxford, 1984.
McGregor, R.C., The Influence of Eddy Viscosity on the Vertical Distribution of Velocity in the Tidal Estuary, Geophys J, Roy. Astr. Soc„ 21, p. 103, 1972.
Miller, G.R., The Flux of Tidal Energy out of the Deep Oceans, J, Geophys. Res., 21, p. 2485, 1966.
Mofjeld, H.O., Empirical Model for Tides in the Western North Atlantic Ocean, Nat. Oceanic and Atmos, Admin, Rep. TR ERL 240-AOML 19., 1975.
Mofjeld, H.O., and J. W. Lavelle, Bottom Boundary Layer Studies in Tidally Dominated Regimes, paper presented at the XVIII General Assembly of the International Union of Geodesy and Geophysics Symposium on Coastal and Near Shore Zone Processes, H.mburg, August 15–27, 1983.
Munk, W.H., Abyssal Recipes, Deep-Sea Res„ 12, p. 707, 1966.
Munk, W.H., F. Snodgrass, and M. Wimbush, Tides Offshore: Transition from California Coastal to Deep-Sea Waters, Geophys Fluid Dyn., 1, p. 161, 1970.
Neumann, G., and W.J. Pierson, Jr., Principles of Physical Oceanography, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1966.
Newton, I., Philosophiae Naturali. Principia Mathematica, London, 1687.
Nihoul, J.C.J., Three-Dimensional Model of Tides and Storm Surges in a Shallow Well-Mixed Continental Sea, Dyn Atm. Oceans,,2, p. 29, 1977.
OBrien, J.J., A Two-Dimensional Model of the Wind-Driven North Pacific, Investig. Pesguera. 3. 5, p. 331, 1971.
Parke, M.E., 01, P1, N2 Models of the Global Ocean Tide on an Elastic Earth Plus Surface Potential and Spherical Harmonic Decompositions for M2, S2 and K1, Marine Geodesy, 6, P. 35, 1982.
Parke, M.E., and M.C. Hendershott, M2,,82, K1 Models of the Global Ocean Tide on an Elastic Earth, Marine Geodesy„ 3, P. 379, 1980.
Pekeris, C.L., and Y. Accad, Solution of Laplaces Equations for the M2 Tide in the World Oceans, Phil. Trans,, Roy„, Soé t London, A = 265, p. 413, 1969.
Pearson., C.A., Deep-Sea Tide Observations off the South-Eastern United States, Tech. Memo., NOS 17, Nat. Oceanic and Atmos. Admin., Rockville, Md., 1975.
Prandtl, L., Heber die Ausgebildete Turbulenz, ZAMM,.a, p. 136, 1925.
Pratt, J.G.D., Tides at Shackleton, Weddel Sea, Trans-Ant,, Expts 1225=51, Soit Repu 4, London, 1960.
Proudman, J., Deformation of Earth-Tides by Means of Water-Tides in Narrow Seas, Bull No 11, Sect,, Oceanogr,. s Cons, de Recherches yenedig, 1928.
Proudman, J., Dynamical Oceanography, Dover Publications, New York, 1952.
Reynolds, O., On the Dynamical Theory of Incompressible Viscous Fluids and the Determination of the Criterion, Phil, Trans„ Roy, Soc„ 186, London A, p. 123, 1894.
Richardson, L.F., Weather Prediction.y Numerical Methods, Cambridge University Press, New York, 1922.
Schlichting, H., Boundary = Layer Theory, McGraw-Hill Book Co., New York, 1968.
Schwiderski, E.W., Bifurcation of Convection in Internally Heated Fluid Layers, Phys. Fluids, 15, p. 1882, 1972.
Schwiderski, E.W., Global Ocean Tides, Part I: A Detailed Hydrodynamical Interpolation Model, NSWC/DL-TR 3866, 1978a. Schwiderski, E.W., Hydrodynamically Defined Ocean Bathymetry, NSWCLDL-TR3868, 1978b.
Schwiderski, E.W., Global Ocean Tides, Part II: The Semidiurnal Principal Lunar Tide (M2), Atlas of Tidal Charts and Maps, NSWCTR/2–414, 1979.
Schwiderski, E.W., On Charting Global Ocean Tides, Reviews of Geophys. and SptPhys:,. 1, 243, 1980a.
Schwiderski, E.W., Ocean Tides, Part I: Global Tidal Equations, Marine geodesy, 3, p. 161, 1980b.
Schwiderski, E.W., Ocean Tides, Part II: A Hydrodynamical Interpolation Model, Marine Geodesys3, p. 219, 1980c.
Schwiderski, E.W., Global Ocean Tides, Parts III-I%: S2, K1, 01, N2, P1, K2, Q1, NSWC TRs 81–122, -121=144„ 218„=220„=222„=224, 1981a.
Schwiderski, E.W., Exact Expansions of Arctic Ocean Tides, NSWC TR 11–4111, 1981b.
Sohwiderski, E.W., Global Ocean Tides, Parts X-XII: Mf, Mm, Ssa, NSWC TRs 12–15J -14Z, ß 141, 1982.
Schwiderski, E.W., Atlas of Ocean Tidal Charts and Maps, Part I: Semidiurnal Principal Lunar Tide M2, Marine Geodesy, 6, p. 219, 1983.
Sohwiderski, E.W., Combined Hydrodynamical and Empirical Modeling of Ocean Tides, Mar,Geophys,Res„/, p. 215, 1984.
Schwiderski, E.W., On Tidal Friction and the Decelerations of the Earths Rotation and Moons Revolution, Marine Geodesy,,Q, p. 417, 1985.
Smagorinsky, J., General Circulation Experiments With the Primitive Equations. I. The Basic Experiment, Mon, Weather Rey~.4.11, P. 99, 1963.
Smith, S.M., H.M. Menard, and G. Sharman, Worldwide Ocean Depths and Continental Elevations Averaged for Areas Approximating One-Degree Squares of Latitude and Longitude, Scripps Inst. of Oceanography, Ref,61$, 1966.
Snodgrass, F.E., Deep-Sea Instrument Capsule, Science, 162, p. 78, 1968.
Suendermann, J., and P. Brosche, Numerical Computation of Tidal Friction for Present and Ancient Oceans, In Tidal Friction and the Earths Rotation, edited by P. Brosche and J. Suendermann, Springer, Berlin, 1978.
Takahasi, R., Tilting Motion of the Earth Crust Caused by
Tidal Loading, Bull, Earthquake Res Inst„ 6, p. 85, 1929. Taylor, G.I., Tidal Friction in the Irish Sea, Phil,Trans, Roy, Soc,.,. London, A, 220, p. 1, 1919.
Thiel, E., A.P. Crary, R.A. Haubrieh, and J.C. Behrendt, Gravimetric Determination of Ocean Tide, Weddel and Ross Seas, Antartica, J,Geophys,Res„ 6Q, p. 629, 1960.
Thomson, W. (Lord Kelvin), Report of Committee for the Purpose of Harmonic Analysis of Tidal Observations, Brit,=Ass, Ady. Sci. Rep„ London, 1868. O.S. National Ocean Service, Tide Tables, 1985.
Whewell, W., Essay Towards a First Approximation to a Map of Co-tidal Lines, Phil, Trans„Roy,Soc„ London, 1, 147, 1833.
Whitaker, S., Introdyction to Fluid Mechanics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968.
Wunsch, C., Internal Tides in the Ocean, Rev,ofGeophys, and Sp. Phys =,, 13, p. 167, 1975.
Young, T., Tides, in Encyclopedia Britanica, 8th Ed. Vol. 21, Little and Brown, Boston, 1823.
Zahel, W., Die Reproduktion Gezeitenbedingter Bewegungsvorgaenge im Weltozean Mittels des Hydrodynamisch-Numerischen Verfahrens, Mitt, des Inst, f, Meereskgnde der Oniy,, Hamburg, X VII, 1970.
Zahel, W., A Global Hydrodynamical-Numerical 10-Model of the Ocean Tides; the Oscillation System of the M2-Tide and its Distribution of Energy Dissipation, Ann s ßophys11s 433, p. 31, 1977.
Zahel, W., The Influence of Solid Earth Deformations on Semi-diurnal and Diurnal Oceanic Tides, In Tidal Friction and the Earths Rotation, edited by P. Brosche and J. Suender-mann, Springer, Berlin, 1978.
Zetler, B.W., H. Munk, H. Mofjeld, W. Brown, and F. Dormer, MODE Tides,. Phys s Oceanogr„, p. 430, 1975.
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Schwiderski, E.W. (1986). Worldwide Ocean Tide Modelling. In: O’Brien, J.J. (eds) Advanced Physical Oceanographic Numerical Modelling. NATO ASI Series, vol 186. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-0627-8_20
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