Skip to main content

Part of the book series: NATO ASI Series ((ASIC,volume 186))

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 259.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 329.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 329.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

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.

    Article  Google Scholar 

  • Bagriantsev, NV., A.I. Danilov, and V.O. Ivehenko, Orientational Effects in Geophysical Fluid Dynamics, Int J. Eng,Séi„ 21, p. 725, 1983.

    Article  Google Scholar 

  • Bartels, J., Gezeitenkraefte,In Handbuch der Physik XLVIII, Geophysik II, edited by S. Fluegge and J. Bartels, Springer, Berlin, 1957.

    Google Scholar 

  • Boussinesq, J., Expression du Frottement Extérieur dans L’écoulement Tumultueux d’une Fluide, Comptes Rend. Acad. Séi„ 122, p. 1445, 1896a.

    Google Scholar 

  • Boussinesq, J., Formules du Coefficient des Frottements Intérieurs dans L’écoulement Tumultueux Graduellement Varie des Liquides Comptes Rend, Acad, Sei„ 122, p. 1517, 1896b.

    Google Scholar 

  • British Admiralty, Tide Tables Vols 1,2, and 3, 1984.

    Google Scholar 

  • Busse, F.H., and J.A. Whitehead, Instabilities of Convection

    Google Scholar 

  • Rolls in a High Prandtl Number Fluid, J, Fluid Meeh„ 42, p. 305, 1971.

    Google Scholar 

  • Cartwright, D.E., Ocean Tides, Rep Progr Phys. 40, p. 665, 1977.

    Article  Google Scholar 

  • Cartwright, D.E., and A.C. Edden, Corrected Tables of Tidal Harmonics, Geophys. Js Roy. Astrs BOé,.33, p. 253, 1973.

    Google Scholar 

  • 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.

    Google Scholar 

  • Cartwright, D.E., and R.J. Tayler, New Computations of the Tide-Generating Potential, Geophyss Js Roy, Astr, Socs, 23, p. 45, 1971.

    Google Scholar 

  • Cartwright, D.E., B.D. Zetler, and H.V. Hamon, Pelagic. Tidal

    Google Scholar 

  • Constants, IAPSO Publication Scientifique No 30, 1979. Cox, M.D., A Mathematical Model of the Indian Ocean, DeepSea Res1, p. 45, 1970.

    Google Scholar 

  • Crowley, W.P., A Global Numerical Model: Part 1, J,Çomp, Phys:.,., p. 1 1 1, 1968.

    Google Scholar 

  • Crowley, W.P., A Numerical Model for Viscous, Free-Surface, Barotropic Wind-Driven Ocean Circulations, J=Comp,:Phys,,, a, p. 139, 1970.

    Google Scholar 

  • Darwin, G. H., Report on the Harmonic Analysis of Tidal Observations, Brit. Ass. for adySeiRep,,, 1883, see also Sci,Papi, Cambridge, 1907.

    Google Scholar 

  • 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.

    Google Scholar 

  • Davies, A.M., and G. K. Furness, Observed and Computed M2 Tidal Currents in the North Sea, J Phys,. Oceanogr„10, p. 237, 1980.

    Google Scholar 

  • Dietrich, G., Die Gezeiten des Weltmeeres als Geographische Erscheinung, Zeitschr. d i Gcsselhsch,., Erdkynde,,3L, 1944a.

    Google Scholar 

  • Dietrich, G., Die Schwingungssyteme der Halb-und Eintaegigen Tiden in den Ozeanen, Veroeff,, Inst MeereskgnJe„Oniy, Berlin, N = F„1.No = 41, 1944b.

    Google Scholar 

  • Doodson, A.T., The harmonic Development of the Tide-Generating Potential, Prpg, Roy, Sgg„ hogggp, A. 100, p. 305, 1921.

    Article  Google Scholar 

  • Eliassen, A., and E. Kleinschmidt, Dynamic Meteorology, In Handhueh der Physik XLVII,geophysik II, edited by S. Fluegge and J. Bartels, Springer, Berlin, 1957.

    Google Scholar 

  • 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.

    Google Scholar 

  • Estes, R.H., A Simulation of Global Ocean Tide Recovery Using Altimeter Data with Systematic Orbit Error, Marine Geodesy,3., p. 75, 1980.

    Google Scholar 

  • Eyris, M., Maregraphs de Grandes Profoundeurs, Cahiers 0ceanographigues, 20, p. 355, 1968.

    Google Scholar 

  • Farrell, W.E., Deformation of the Earth by Surface Loads, Rev s Geophys, Space Phys„10, p. 261, 1972.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Grace, S.F., The Semidiurnal Lunar Tidal Motion of the Red Sea, Mon. Not. Roy. Astr. Soc„ Geophys. Supp,I,2, p. 273, 1930a.

    Google Scholar 

  • 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.

    Google Scholar 

  • Greenberg, D.A., Modeling of the Mean Barotropic, Circulation in the Bay of Fundy and Gulf of Maine, JPhys, Oceanogr„ 11, 1983.

    Google Scholar 

  • Hansen, W., Die Ermittlung der Gezeiten Beliebig Gestalteter Meeresgebiete mit Hilfe des Randwertverfahrens, Deutsche Hydr. Zeitsch. 4 1, p. 157, 1948.

    Google Scholar 

  • Hansen, W., Die Reproduktion der Bewegungsvorgaenge im Meere mit Hilfe Hydrodynamisch-Numerischer Verfahren, Mitt,.des Inst. f. Meereskunde der Univ. Hamburg, V, 1966.

    Google Scholar 

  • Harris, R.A., Manual of Tides Parts I and II, U.S. Coast and Geodetic Survey, 1897.

    Google Scholar 

  • 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.

    Google Scholar 

  • Heiskanen, W., Ueber den Einfluss der Gezeiten auf die Saekulare Acceleration des Mondes, Ann, Acad, Sci„ Fennieae A,.. 18, p. 1, 1921.

    Google Scholar 

  • Hendershott, M.C., The Effects of Solid-Earth Deformation on Global Ocean Tides, Geophys. J. Roy, Astr, Soc„a2, p. 389, 1972.

    Google Scholar 

  • Holland, W. R., and A. D. Hirschman, A Numerical Calculation in the North Atlantic Ocean, J, Phys. Oceanogr,. 1 2, p. 336, 1972.

    Google Scholar 

  • International Hydrographie Bureau, Tides, Harmonic Constants, Computer Tape Monaco, 1978.

    Google Scholar 

  • Irish, J.D., W.H. Munk, and F.E. Snodgrass, M2 Amphidrome in the Northeast Pacific, Geophys, Fluid Dyn„ 2, p. 355, 1971.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Jeffreys, H., Tidal Friction in Shallow Seas, Phil, Trans„ Roy, Soc. A.,, 221, p. 239, 1920.

    Article  Google Scholar 

  • Johns, B., Vertical Structure of Tidal Flows in River Estu- aries, Geophys. J. Res„ Astr. Soc.,. 12, p. 103, 1966.

    Google Scholar 

  • Joseph, D.D., Stability of Fluid Motions I, Springer, Berlin, 1976.

    Google Scholar 

  • Kagan, B.A., Resistance Law of Tidal Flow, Izy,, Acad, Sci, USSR,. Atm. and Oce. Phys., 5, p. 302, 1972.

    Google Scholar 

  • Kraav, V.R., Computation of the Semidiurnal Tide and Turbulence Parameters in the North Sea, Oceanology,., p. 332, 1969.

    Google Scholar 

  • Ladyzhenskaya, 0.A., The Mathematical Theory of Viscous Incompressible Flog, Gordon and Breach, New York, 1969.

    Google Scholar 

  • 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.

    Google Scholar 

  • Leith, C.E., Two-Dimensional Eddy Viscosity Coefficients, Proc, WMOLIUGG Symp,, on Numerical Weather Prediction, Tokyo, p. 140, 1968.

    Google Scholar 

  • Luther, D.S., and C. Wunsch, Tidal Charts of the Central Pacific Ocean, J. Phys. Oce.,, 2, p. 227, 1975.

    Google Scholar 

  • Marchuk, G.I., and B.A. Kagan, OCEAN Tides, Mathematical Models and Numerical Experiments, Pergamon Press, Oxford, 1984.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Miller, G.R., The Flux of Tidal Energy out of the Deep Oceans, J, Geophys. Res., 21, p. 2485, 1966.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Munk, W.H., Abyssal Recipes, Deep-Sea Res„ 12, p. 707, 1966.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • Neumann, G., and W.J. Pierson, Jr., Principles of Physical Oceanography, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1966.

    Google Scholar 

  • Newton, I., Philosophiae Naturali. Principia Mathematica, London, 1687.

    Google Scholar 

  • 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.

    Google Scholar 

  • OBrien, J.J., A Two-Dimensional Model of the Wind-Driven North Pacific, Investig. Pesguera. 3. 5, p. 331, 1971.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Prandtl, L., Heber die Ausgebildete Turbulenz, ZAMM,.a, p. 136, 1925.

    Google Scholar 

  • Pratt, J.G.D., Tides at Shackleton, Weddel Sea, Trans-Ant,, Expts 1225=51, Soit Repu 4, London, 1960.

    Google Scholar 

  • 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.

    Google Scholar 

  • Proudman, J., Dynamical Oceanography, Dover Publications, New York, 1952.

    Google Scholar 

  • 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.

    Google Scholar 

  • Richardson, L.F., Weather Prediction.y Numerical Methods, Cambridge University Press, New York, 1922.

    Google Scholar 

  • Schlichting, H., Boundary = Layer Theory, McGraw-Hill Book Co., New York, 1968.

    Google Scholar 

  • Schwiderski, E.W., Bifurcation of Convection in Internally Heated Fluid Layers, Phys. Fluids, 15, p. 1882, 1972.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Schwiderski, E.W., Global Ocean Tides, Part II: The Semidiurnal Principal Lunar Tide (M2), Atlas of Tidal Charts and Maps, NSWCTR/2–414, 1979.

    Google Scholar 

  • Schwiderski, E.W., On Charting Global Ocean Tides, Reviews of Geophys. and SptPhys:,. 1, 243, 1980a.

    Article  Google Scholar 

  • Schwiderski, E.W., Ocean Tides, Part I: Global Tidal Equations, Marine geodesy, 3, p. 161, 1980b.

    Article  Google Scholar 

  • Schwiderski, E.W., Ocean Tides, Part II: A Hydrodynamical Interpolation Model, Marine Geodesys3, p. 219, 1980c.

    Google Scholar 

  • 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.

    Google Scholar 

  • Schwiderski, E.W., Exact Expansions of Arctic Ocean Tides, NSWC TR 11–4111, 1981b.

    Google Scholar 

  • Sohwiderski, E.W., Global Ocean Tides, Parts X-XII: Mf, Mm, Ssa, NSWC TRs 12–15J -14Z, ß 141, 1982.

    Google Scholar 

  • Schwiderski, E.W., Atlas of Ocean Tidal Charts and Maps, Part I: Semidiurnal Principal Lunar Tide M2, Marine Geodesy, 6, p. 219, 1983.

    Article  Google Scholar 

  • Sohwiderski, E.W., Combined Hydrodynamical and Empirical Modeling of Ocean Tides, Mar,Geophys,Res„/, p. 215, 1984.

    Google Scholar 

  • Schwiderski, E.W., On Tidal Friction and the Decelerations of the Earths Rotation and Moons Revolution, Marine Geodesy,,Q, p. 417, 1985.

    Google Scholar 

  • Smagorinsky, J., General Circulation Experiments With the Primitive Equations. I. The Basic Experiment, Mon, Weather Rey~.4.11, P. 99, 1963.

    Google Scholar 

  • 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.

    Google Scholar 

  • Snodgrass, F.E., Deep-Sea Instrument Capsule, Science, 162, p. 78, 1968.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Takahasi, R., Tilting Motion of the Earth Crust Caused by

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Article  Google Scholar 

  • 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.

    Google Scholar 

  • Whewell, W., Essay Towards a First Approximation to a Map of Co-tidal Lines, Phil, Trans„Roy,Soc„ London, 1, 147, 1833.

    Article  Google Scholar 

  • Whitaker, S., Introdyction to Fluid Mechanics, Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1968.

    Google Scholar 

  • Wunsch, C., Internal Tides in the Ocean, Rev,ofGeophys, and Sp. Phys =,, 13, p. 167, 1975.

    Article  Google Scholar 

  • Young, T., Tides, in Encyclopedia Britanica, 8th Ed. Vol. 21, Little and Brown, Boston, 1823.

    Google Scholar 

  • Zahel, W., Die Reproduktion Gezeitenbedingter Bewegungsvorgaenge im Weltozean Mittels des Hydrodynamisch-Numerischen Verfahrens, Mitt, des Inst, f, Meereskgnde der Oniy,, Hamburg, X VII, 1970.

    Google Scholar 

  • 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.

    Google Scholar 

  • 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.

    Google Scholar 

  • Zetler, B.W., H. Munk, H. Mofjeld, W. Brown, and F. Dormer, MODE Tides,. Phys s Oceanogr„, p. 430, 1975.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1986 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

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

Download citation

  • DOI: https://doi.org/10.1007/978-94-017-0627-8_20

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-8428-6

  • Online ISBN: 978-94-017-0627-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics