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Are cascading flows stable?

Published online by Cambridge University Press:  08 October 2007

S. A. THORPE  and
B. OZEN
S. A. THORPE*
Affiliation:
School of Ocean Sciences, Marine Science Laboratories, Menai Bridge, Anglesey LL59 5AB, UK
B. OZEN
Affiliation:
Laboratoire de Recherches Hydrauliques, Ecole Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
*
†Correspondence to ‘Bodfryn’, Glanrafon, Llangoed, Anglesey LL58 8PH, UK. oss413@sos.bangor.ac.uk.
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Abstract

The stability of flows cascading down slopes as dense inclined plumes is examined, with particular reference to flows observed in Lake Geneva during winter periods of severe cooling. A previous conjecture by Turner that the flow may be in a state of marginal stability is confirmed: the observed mean velocity and density profiles are unstable to Kelvin–Helmholtz instability, but only marginally so; the growth rates of the most unstable small disturbances to the cascading flow in Lake Geneva are small, with e-folding periods of about 2 h. A reduction in the maximum velocity by about 20% is required to stabilize the flow.

The possibility that stationary hydraulic jumps may occur in the observed flow is also considered. Several plausible flow states downstream of transitions are examined, allowing for mixing and density changes to occur, ranging from one that preserves the shape of the density and velocity profiles to one in which, as a consequence of mixing, the velocity and density become uniform in depth within the cascading flow. Neither of these extreme states is found to conserve the fluxes of volume, mass and momentum through a transition in which the energy flux does not increase, and to be unique or ‘stable’ in the sense that no further transition is possible to a similar flow state without more entrainment. Stable transitions to intermediate downstream flows that conserve flow properties and reduce energy flux are, however, found, although the smallest value of the flow parameter, Fr≡ Umax2/gΔ h (where Umax is the maximum flow speed, g is the acceleration due to gravity, Δ is a fractional density difference within the flow and h is the flow thickness) at which transitions may occur is only slightly less than that of the cascading flow in Lake Geneva. In this sense, the observed flow is marginally unstable to a finite-amplitude transition or hydraulic jump. Velocity and density profiles of possible flows downstream of a transition are found. The amplitudes of possible transitions and the flux of water entrained from the ambient overlying water mass are limited to narrow ranges.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 589 , 25 October 2007 , pp. 411 - 432
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Baines, P. G. 1995 Topographic Effects in Stratified Flows. Cambridge University Press.Google Scholar
Bell, T. H. 1974 Effects of shear on the properties of internal gravity waves. Deutsche Hydrograph. Z. 27, 57–62.CrossRefGoogle Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448–460.Google Scholar
Cenedese, C., Whitehead, J. A., Ascarelli, T. A. & Ohiwa, M. 2004 A dense current flowing down a sloping bottom in a rotating fluid. J. Phys. Oceanogr. 34, 188–203.2.0.CO;2>CrossRefGoogle Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Ellison, T. H. & Turner, J. S. 1959 Turbulent entrainment in stratified fluids. J. Fluid Mech. 6, 423–448.CrossRefGoogle Scholar
Fer, I., Lemmin, U. & Thorpe, S. A. 2002 a Winter cascading of cold water in Lake Geneva. J. Geophys. Res. 107, C6, 10.1029/2001JC000828, 13-1–13-16.CrossRefGoogle Scholar
Fer, I., Lemmin, U. & Thorpe, S. A. 2002 b Contribution of entrainment and vertical plumes to the winter cascading of cold waters in a deep lake. Limnol. Oceanogr. 47, 576–580.CrossRefGoogle Scholar
Gaster, M. 1962 A note on the relation between temporally-increasing and spatially-increasing disturbances in an infinitely deep, continuously stratified fluid. J. Fluid Mech. 14, 222–224.CrossRefGoogle Scholar
Gray, T. E., Alexander, J. & Leeder, M. R. 2006 Longitudinal flow evolution and turbulence structure of dynamically similar, saline density and turbidity currents. J. Geophys. Res. 111, C08015, doi:10.1029/2005JC003089.CrossRefGoogle Scholar
Hazel, P. 1972 Numerical studies of the stability of inviscid stratified shear flows. J. Fluid Mech. 51, 39–61.CrossRefGoogle Scholar
Holland, D. M., Rosales, R. R., Stefanica, D. & Tabak, E. G. 2002. Internal hydraulic jumps and mixing in two-layer flows. J. Fluid Mech. 470, 63–83.CrossRefGoogle Scholar
Lighthill, J. 1978 Waves in Fluids. Cambridge University Press.Google Scholar
Miles, J. W. 1967 Internal waves in a continuously stratified atmosphere or ocean. J. Fluid Mech. 28, 305–310.CrossRefGoogle Scholar
Mittendorf, G. H. 1961 The instability of stratified flow. MSc thesis. State University of Iowa.Google Scholar
Neiman, P. J., Hardesty, R. M., Shapiro, M. A. & Cupp, R. E. 1988 Doppler lidar observations of a downslope windstorm. Mon. Weather Rev. 116, 2265–2275.2.0.CO;2>CrossRefGoogle Scholar
Pawlak, G. & Armi, L. 2000 Mixing and entrainment in developing stratified currents. J. Fluid Mech. 424, 45–73.CrossRefGoogle Scholar
Peltier, W. R. & Scinocca, J. F. 1990 The origin of severe downslope windstorm pulsations. J. Atmos. Sci. 47, 2853–2870.2.0.CO;2>CrossRefGoogle Scholar
Pratt, L., Deese, H., Murray, S. & Johns, W. 2000 Continuous dynamical modes in straits having arbitrary cross sections, with applications to the Bab el Mandab. J. Phys. Oceanogr. 30, 2515–2534.2.0.CO;2>CrossRefGoogle Scholar
Press, W. H., Teukolski, S. A., Vetterling, W. T. & Flannery, B.P. 1992 Numerical Recipes in C: the Art of Scientific Computing, 2nd edn. CambridgeUniversity Press.Google Scholar
Scorer, R. 1972 Clouds of the World. Lothian, Melbourne.Google Scholar
Simpson, J. E. 1997 Gravity currents. 2nd edn. Cambridge University Press.Google Scholar
Thorpe, S. A. 1978 On the shape and breaking of finite amplitude internal gravity waves in a shear flow. J. Fluid Mech. 85, 7–31.CrossRefGoogle Scholar
Thorpe, S. A. 1999 A note on the breaking of internal waves in the ocean. J. Phys. Oceanogr. 29, 2433–2441.2.0.CO;2>CrossRefGoogle Scholar
Thorpe, S. A. 2007 Dissipation in hydraulic transitions in flows through abyssal channels. J. Mar. Res.CrossRefGoogle Scholar
Thorpe, S. A. & Hall, A. J. 1977 Mixing in upper layer of lake during heating cycle. Nature, 265, 719–722.CrossRefGoogle Scholar
Turner, J. S. 1973 Buoyancy Effects in Fluids. Cambridge University Press.CrossRefGoogle Scholar