Advertisement

On Internal Waves in Partially Mixed and Stratified Tidal Flows

  • C. Kranenburg

Abstract

The various mechanisms producing internal waves in estuaries are reviewed and a short discussion of lee waves is given. Elementary analytical methods are applied to demonstrate the interaction between large-scale internal waves and turbulence in shallow estuaries. The production of turbulent kinetic energy owing to long internal waves is calculated for horizontal and undulating beds. The relevance to the numerical modelling of estuarine flows is briefly outlined. Two-layer models are discussed in some detail, and it is shown that including friction introduces an instability beyond a certain internal Froude number.

Keywords

Turbulent Kinetic Energy Internal Wave Stratify Flow Stratify Fluid Internal Solitary Wave 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abbott MB, Torbe I (1963) On flows and fronts in a stratified fluid. Proc R Soc 273A:12–40Google Scholar
  2. Abraham G, Karelse M, Os AG van (1979) On the magnitude of interfacial shear of subcritical stratified flows in relation with interfacial stability. J Hydraul Res 17:273–287CrossRefGoogle Scholar
  3. Alavian V (1986) Behavior of density currents on an incline. J Hydraul Eng 112:27–42CrossRefGoogle Scholar
  4. Armi L (1986) The hydraulics of two flowing layers with different densities. J Fluid Mech 163:27–58CrossRefGoogle Scholar
  5. Baines PG (1977) Upstream influence and Long’s model in stratified flows. J Fluid Mech 82:147–159CrossRefGoogle Scholar
  6. Baines PG (1979) Observations of stratified flow over two-dimensional obstacles in fluid of finite depth. Tellus 31:351–371CrossRefGoogle Scholar
  7. Baines PG (1984) A unified description of two-layer flow over topography. J Fluid Mech 146:127–167 Chereskin TK (1983) Generation of internal waves in Massachusetts Bay. J Geophys Res C 88:2649–2661CrossRefGoogle Scholar
  8. Chereskin TK (1983) Generation of internal waves in Massachusetts Bay. J Geophys Res C 88:2649–2661CrossRefGoogle Scholar
  9. Davies PA, Peltier WR (1976) Resonant parallel shear instability in the stably stratified planetary boundary layer. J Atmos Sci 33:1287–1300CrossRefGoogle Scholar
  10. Davis PA, Peltier WR (1979) Some characteristics of the Kelvin-Helmholtz and resonant overreflec-tion modes of shear flow instability and of their interaction through vortex pairing. J Atmos Sci 36:2394–2412CrossRefGoogle Scholar
  11. Dermissis V, Partheniades E (1984) Interfaeial resistance in stratified flows. J Waterway Port Coastal Ocean Eng 110:231–250CrossRefGoogle Scholar
  12. Dermissis V, Partheniades E (1985) Dominant shear stresses in arrested saline wedges. J Waterway Port Coatal Ocean Eng 111:733–752 Dyer KR (1982) Mixing caused by lateral internal seiching within a partially mixed estuary. Estuarine Coastal Shelf Sci 15:443–457 Dyer KR (1985) Tidally generated estuarine mixing processes. In: Kjerfve BJ (ed) Hydrodynamics of estuaries. CRC Press, Cleveland, Boca Raton On Internal Waves in Partially Mixed and Stratified Tidal FlowsCrossRefGoogle Scholar
  13. Dyer KR (1982) Mixing caused by lateral internal seiching within a partially mixed estuary. Estuarine Coastal Shelf Sci 15:443–457CrossRefGoogle Scholar
  14. Dyer KR (1985) Tidally generated estuarine mixing processes. In: Kjerfve BJ (ed) Hydrodynamics of estuaries. CRC Press, Cleveland, Boca Raton On Internal Waves in Partially Mixed and Stratified Tidal FlowsGoogle Scholar
  15. Dyer KR, New AL (1985) Intermittency in estuarine mixing. Proc 8th Estuarine Research Federation conference on estuarine variability, July 1985. Durham, New HampshireGoogle Scholar
  16. Farmer DM, Freeland HJ (1983) The physical oceanography of fjords. Prog Oceanogr 12:147–220CrossRefGoogle Scholar
  17. Farmer DM, Smith JD (1978) Nonlinear internal waves in a fjord. In: Nihoul JCJ (ed) Hydrodynamics of estuaries and fjords. Elsevier, AmsterdamGoogle Scholar
  18. Freeland HJ, Farmer DM, Levings CD (1980) Fjord oceanography. Plenum, New YorkGoogle Scholar
  19. Gardner GB, Smith JD (1978) Turbulent mixing in a salt-wedge estuary. In: Nihoul JCJ (ed) Hydrodynamics of estuaries and fjords. Elsevier, AmsterdamGoogle Scholar
  20. Gardner GB, Smith JD (1980) Observations of time-dependent, stratified flow in a small salt-wedge estuary. In: Carstens T, McClimans T (eds) Stratified flows, vol 2. Tapir, TrondheimGoogle Scholar
  21. Gargett A (1980) Turbulence measurements through a train of breaking internal waves in Knight Inlet, B.C. In: Freeland HJ, Farmer DM, Levings CD (eds) Fjord oceanography. Plenum, New YorkGoogle Scholar
  22. Geyer WR, Cannon GA (1982) Sill processes related to deep-water renewal in a fjord. J Geophys Res C87:7985–7996CrossRefGoogle Scholar
  23. Helfrich KR, Melville WK (1986) On long nonlinear internal waves over slope-shelf topography. J Fluid Mech 167:285–308CrossRefGoogle Scholar
  24. Howard LN (1961) Note on a paper of John W. Miles. J Fluid Mech 10:509–512CrossRefGoogle Scholar
  25. Huppert HE (1980) Topographic effects in stratified fluids. In: Freeland HJ, Farmer DM, Levings CD (eds) Fjord oceanography. Plenum, New YorkGoogle Scholar
  26. Janowitz GS (1981) Stratified flow over a bounded obstacle in a channel of finite height. J Fluid Mech 110:161–170CrossRefGoogle Scholar
  27. Kao TW, Pan FS, Renouard D (1985) Internal solitons on the pycnocline: generation, propagation, and shoaling and breaking over a slope. J Fluid Mech 159:19–53CrossRefGoogle Scholar
  28. Koop CG (1981) A preliminary investigation of the interaction of internal gravity waves with a steady shearing motion. J Fluid Mech 113:347–386CrossRefGoogle Scholar
  29. Kranenburg C (1983) Internal kinematic waves and the large-scale stability of two-layer stratified flow. J Hydraul Res 21:259–275CrossRefGoogle Scholar
  30. Kranenburg C (1987a) Boundary-induced entrainment in two-layer stratified flow. J Geophys Res C 92:5417–5425CrossRefGoogle Scholar
  31. Kranenburg C (1987b) A model of long internal waves in salt-wedge estuaries with tidal motion. Presented at Third int symp on stratified flows, February 1987, PasadenaGoogle Scholar
  32. Krauss W (1966) Methoden und Ergebnisse der theoretischen Ozeanographie, Bd II: Interne Wellen. Bornträger, BerlinGoogle Scholar
  33. Lamb VR, Britter RE (1984) Shallow flow over an isolated obstacle. J Fluid Mech 147:291–313CrossRefGoogle Scholar
  34. Lawrence GA (1985) The hydraulics and mixing of two-layer flow over an obstacle. Report UCB/HL-85/02, University of California, BerkeleyGoogle Scholar
  35. LeBlond PH, Mysak LA (1987) Waves in the ocean. Elsevier, AmsterdamGoogle Scholar
  36. Lee JD, Su CH (1977) A numerical method for stratified shear flows over a long obstacle. J Geophys Res 64:420–426CrossRefGoogle Scholar
  37. Lewis RE (1985) Intense mixing periods in an estuary. In: Hunt JCR (ed) Turbulence and diffusion in stable environments. Clarendon, OxfordGoogle Scholar
  38. Lighthill J (1978) Waves in fluids. Cambridge University Press, CambridgeGoogle Scholar
  39. Lindzen RS, Barker JW (1985) Instability and wave over-reflection in stably stratified shear flow. J Fluid Mech 151:189–217CrossRefGoogle Scholar
  40. Liu AK, Benney DJ (1981) The evolution of nonlinear wave trains in stratified shear flows. Stud. Appl. Math. 64:247–269Google Scholar
  41. Long RR (1953) Some aspects of the flow of stratified fluids. I. A theoretical investigation. Tellus 5:42–57CrossRefGoogle Scholar
  42. Long RR (1955) Some aspects of the flow of stratified fluids. III. Continuous density gradients. Tellus 7:342–357CrossRefGoogle Scholar
  43. Long RR (1974) Some experimental observations of upstream disturbances in a two-fluid system. TeUus 26:313–317Google Scholar
  44. Maxworthy T (1979) A note on the internal solitary waves produced by tidal flow over a three-dimensional ridge. J Geophys Res C 84:338–346CrossRefGoogle Scholar
  45. Miles JW (1961) On the stabiUty of heterogeneous shear flows. J Fluid Mech 10:496–508CrossRefGoogle Scholar
  46. Miles JW, Huppert HE (1969) Lee waves in a stratified flow. Part 4. Perturbation approximations. J Fluid Mech 35:497–525CrossRefGoogle Scholar
  47. Muir LR (1980) Internal tides in a partially-mixed estuary. In: Carstens T, McClimans T (eds) Stratified flows, vol. 1. Tapir, TrondheimGoogle Scholar
  48. Nakano W, Yakuwa I (1985) Evolution of interfacial waves along an unsteady salt wedge. In: Edge BL (ed) Proc 19th coastal engineering conference, Houston, Texas, vol III. ASCE, New YorkGoogle Scholar
  49. Needham DJ, Merkin JH (1984) On roll waves down an open inclined channel. Proc R Soc Lond A 394:259–278CrossRefGoogle Scholar
  50. Nezu I (1977) Turbulent structure in open-channel flows. Ph D thesis, Kyoto University, KyotoGoogle Scholar
  51. Partch EN, Smith JD (1978) Time dependent mixing in a salt wedge estuary. Estuarine Coastal Mar Sci 6:3–19CrossRefGoogle Scholar
  52. Phillips OM (1977) The dynamics of the upper ocean. Cambridge University Press, CambridgeGoogle Scholar
  53. Rajaratnam N, Subramanyan S (1985) Plane turbulent buoyant surface jets and jumps. J Hydraul Res 23:131–146CrossRefGoogle Scholar
  54. Rodi W (1980) Turbulence models and their application in hydraulics — a state of the art review. IAHR, DelftGoogle Scholar
  55. Schijf JB, Schönfeld JC (1953) Theoretical considerations on the motion of salt and fresh water. In: Proc Minnesota int hydraulics convention. IAHR, Delft. ASCE, New YorkGoogle Scholar
  56. Stillinger DC, Heiland KN, Van Atta CW (1983) Experiments on the transition of homogeneous turbulence to internal waves in a stratified fluid. J Fluid Mech 131:91–122CrossRefGoogle Scholar
  57. Su CH (1976) Hydraulic jumps in an incompressible stratified fluid. J Fluid Mech 73:3–47CrossRefGoogle Scholar
  58. Thorpe SA (1986) Transitional phenomena and the development of turbulence in stratified fluids; a review. J Geophys Res C 92:5231–5248CrossRefGoogle Scholar
  59. Turner JS (1973) Buoyancy effects in fluids. Cambridge University Press, CambridgeGoogle Scholar
  60. Weinstock J (1984) Effect of gravity waves on turbulence decay in stratified fluids. J Fluid Mech 140:11–26CrossRefGoogle Scholar
  61. West JR, Shiono K (1985) A note on turbulent perturbations of salinity in a partially mixed estuary. Estuarine Coastal Shelf Sci 20:55–78CrossRefGoogle Scholar
  62. Wood IR, Simpson JE (1984) Jumps in layered miscible fluids. J Fluid Mech 140:329–342CrossRefGoogle Scholar
  63. Yih CS (1980) Stratified flows. Academic Press, New YorkGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • C. Kranenburg
    • 1
  1. 1.Department of Civil EngineeringDelft University of TechnologyDelftThe Netherlands

Personalised recommendations