Ocean Dynamics

, Volume 69, Issue 4, pp 427–441 | Cite as

Viscous waves on a beta-plane and its zonal asymmetry

  • Valery N. Zyryanov
  • Marianna K. ChebanovaEmail author


Effects of the viscosity, Earth rotation, and sphericity (beta-effect) on the long-wave dynamics are investigated based on the linear model. The basic equation for the complex amplitudes of gravitational long waves is obtained. It is shown that the viscosity plays a significant role in the long-wave dynamics. Stokes’ layer thickness is the criterion which separates two regimes of long-wave evolution: low viscosity and viscous flows. Two Stokes’ layers occur in the rotating fluid. The thickness of the first approaches to infinity as the frequency tends to inertial frequency. Considering the role of the Stokes’ layer as a criterion of viscosity influence, we can conclude that for the waves of the near-inertial frequency, viscosity always plays a significant role irrespective of ocean depths. The beta-effect leads to the planetary drift velocity occurrence in the equation. The planetary drift velocity can have either eastward or westward direction depending on the wave frequency. Thus, Earth sphericity causing the planetary drift plays a major role in the asymmetry of the eastward and westward directions in wave dynamics. Friction is another reason for the asymmetry of the eastward and westward directions in wave dynamics. Damping decrements of the westward and eastward waves differ with the biggest difference for waves with the near-inertial frequencies. Group velocities of eastward and westward waves are nonsymmetrical too. Moreover, in a certain range of the near-inertial frequencies, group velocities of both westward and eastward waves are directed exceptionally eastward. Thus, the beta-effect and fluid viscosity can be the reasons for the asymmetry of western and eastern bays in the tidal wave dynamics.


Stokes layer Beta-plane Viscous fluid Friction Tidal wave 



The authors are grateful to the anonymous reviewer for the detailed analysis of our paper and very useful comments and discussions.


This work was supported by the Russian Foundation for Basic Research (RFBR project 16-05-00209 А).


  1. Bowden KF (1988) Physical oceanography of coastal waters. Ellis Horwood Ltd, New YorkGoogle Scholar
  2. Defant A (1961) Physical oceanography. Pergamon, New YorkGoogle Scholar
  3. Djordjevic VD (1980) On the dissipation of interfacial and internal long gravity waves. J of Appl Math Phys(ZAMP) 31:318–331Google Scholar
  4. Djordjevic VD (1983) On the effect of viscosity on some nonwave motions of liquids with the free surface. Acta Mech 48:219–226CrossRefGoogle Scholar
  5. Dronkers J J (1964) Tidal computations in rivers and coastal waters. AmsterdamGoogle Scholar
  6. Eckart C (1960) Hydrodynamics of oceans and atmospheres. PergamonGoogle Scholar
  7. Ekman VW (1905) On the influence of the Earth’s rotation on ocean currents. Aroh math astron fhs 2:1–53Google Scholar
  8. Friedrichs CT, Aubrey DG (1994) Tidal propagation in strongly convergent channels. J Geophys Res 99:3321–3336CrossRefGoogle Scholar
  9. Gerkema T, Shrira VI (2005) Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J Fluid Mech 529:195–219CrossRefGoogle Scholar
  10. Gerkema T, Zimmerman JTF, Maas LRM, Van Haren H (2008) Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev Geophys 46:RG2004CrossRefGoogle Scholar
  11. Gill AE (1982) Atmosphere-Ocean Dyn. Academic, NewYorkGoogle Scholar
  12. Grimshaw RHJ, Ostrovsky LA, Shrira VI, Stepanyants YA (1998) Long nonlinear surface and internal gravity waves in a rotating ocean. Surv Geophys 19:289–338CrossRefGoogle Scholar
  13. Harleman DRF (1966) Tidal dynamics in estuaries, part II: Real estuaries. In: Estuary and Coastline Hydrodynamics, edited by A.T. Ippen. McGraw-Hill, New YorkGoogle Scholar
  14. Hunt JN (1964) Tidal oscillations in estuaries. Geophys J R Astron Soc 8:440–455CrossRefGoogle Scholar
  15. Hydrodynamics of Lakes (1984), Kolumban Hutter, Ed., Wien: Springer Verlag. 341 p. ISBN: 978–2–211-81812-1Google Scholar
  16. Ippen AT (1966) Estuary and coastline hydrodynamics. McGraw Hill, New YorkGoogle Scholar
  17. Jay DA (1991) Green’s law revisited: tidal long-wave propagation in channels with strong topography. J Geophys Res 96:20585–20598CrossRefGoogle Scholar
  18. Jelesnianski CP (1970) Bottom stress time-history in linearized equations of motion for storm surges. Mon Weather Rev 98:462–478CrossRefGoogle Scholar
  19. Jordan TP, Baker JR (1980) Vertical structure of time-dependent flow dominated by friction in a well-mixed fluid. J Phys Oceanogr 10:1091–1103CrossRefGoogle Scholar
  20. Kakutani T, Matsuuchi K (1975) Effect of viscosity on long gravity waves. J Phys Soc Jpn 39:237–246CrossRefGoogle Scholar
  21. Lanzoni S, Seminara G (1998) On tide propagation in convergent estuaries. J Geophys Res 103:30793–30812CrossRefGoogle Scholar
  22. Lanzoni S, Seminara G (2002) Long-term evolution and morphodynamic equilibrium of tidal channels. J Geophys Res 107:1–13CrossRefGoogle Scholar
  23. Le Blond PH (1978) On tidal propagation in shallow rivers. J Geophys Res 83:4717–4721CrossRefGoogle Scholar
  24. Le Blond PH, Mysek LA (1978) Waves in the ocean. Elsevier, AmsterdamGoogle Scholar
  25. Lighthill J (1978) Waves in fluids. Univ. Press, CambridgeGoogle Scholar
  26. Maz’ya VG (1972) On a degenerating problem with directional derivative. Math USSR-Sb 16(3):429–469CrossRefGoogle Scholar
  27. Mofjeld HO (1980) Effects of vertical viscosity on Kelvin waves. J Phys Oceanogr 10:1039–1050CrossRefGoogle Scholar
  28. Nakaya C (1974) Spread of fluid drops over a horizontal plane. J Phys Soc Jpn 37:539–543CrossRefGoogle Scholar
  29. Pedlosky J (1982) Geophysical fluid dynamics. Springer-Verlag, New YorkCrossRefGoogle Scholar
  30. Perroud P (1959) The Propagation of tidal waves into channels of gradually varying cross section. Technical memorandum. Beach Erosion Board. Washington, D. C. 112Google Scholar
  31. Platzman GW (1963) The dynamical prediction of wind tides on Lake Erie. The dynamical prediction of wind tides on Lake Erie. American Meteorological Society, Boston, MA, pp 1–44CrossRefGoogle Scholar
  32. Poincare H. (1910) Lecons de Mecanique celeste. III. ParisGoogle Scholar
  33. Prandle D (1985) Classification of tidal response in estuaries from channel geometry. Geophys J R Astron Soc 80:209–221CrossRefGoogle Scholar
  34. Prandle D (2009) Estuaries. Dynamics, mixing, sedimentation and morphology. Univ. Press, CambridgeCrossRefGoogle Scholar
  35. Prandle D, Rahman M (1980) Tidal response in estuaries. J Phys Oceanogr 10:1552–1573CrossRefGoogle Scholar
  36. Proudman J (1925) Tides in a channel. Philos Mag (49)6:465Google Scholar
  37. Proudman J (1953) Dynamical oceanography. LondonGoogle Scholar
  38. Savenije HHG (1992) Lagrangian solution of St. Venant's equations for an alluvial estuary. J Hydraul Eng 118:1153–1163CrossRefGoogle Scholar
  39. Savenije HHG (2005) Salinity and tides in alluvial estuaries. Elsevier, AmsterdamGoogle Scholar
  40. Savenije HHG, Veling EJM (2005) The relation between tidal damping and wave celerity in estuaries. J Geophys Res 110:1–10CrossRefGoogle Scholar
  41. Taylor GI (1922) Tidal oscillations in gulfs and rectangular basins. Proc Lond Math Soc 20:148–181CrossRefGoogle Scholar
  42. Van Rijn LC (2011) Analytical and numerical analysis of tides and salinity in estuaries. Pt I. Tidal wave propagation in convergent estuaries. Ocean Dyn 61:1719–1741CrossRefGoogle Scholar
  43. Welander P (1957) Wind action on a shallow sea: some generalizations of Ekman’s theory. Tellus 9:45–52Google Scholar
  44. Zyryanov V N (1995) Topographic eddies in sea current dynamics. Moscow (in Russian) Google Scholar
  45. Zyryanov VN (2014) Nonlinear pumping in oscillatory diffusive processes: the impact on the oceanic deep layers and lakes. Commun Nonlinear Sci Numer Simul 19:2131–2139CrossRefGoogle Scholar
  46. Zyryanov VN, Chebanova MK (2016) Hydrodynamic effects at the entry of tidal waves into estuaries. Water Res 43:621–628CrossRefGoogle Scholar
  47. Zyryanov VN, Chebanova MK (2017a) Dissipative-convergent intermittency in dynamics of tidal waves in estuaries. Fluid Dynamics 52:722–732CrossRefGoogle Scholar
  48. Zyryanov VN, Chebanova MK (2017b) Experimental studies of the right and left bays asymmetry in the tidal waves dynamics. Process Geo-media 1(10):410–418Google Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of Water Problems of RASMoscowRussia

Personalised recommendations