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Mechanics of thermohaline interleaving: beyond the empirical flux laws

Published online by Cambridge University Press:  10 March 2011

TIMOUR RADKO*
Affiliation:
Department of Oceanography, Naval Postgraduate School, Monterey, CA 93943, USA
*
Email address for correspondence: tradko@nps.edu

Abstract

An analytical theory is developed which illustrates the dynamics of the spontaneous generation of thermohaline intrusions in the stratified ocean with density compensated lateral temperature and salinity gradients. Intrusions in the model are driven by the interaction with the initially homogeneous field of salt fingers, whose amplitude and spatial orientation is weakly modulated by the long wavelength perturbations introduced into the system. The asymptotic multiscale analysis makes it possible to identify intrusive instabilities resulting from the positive feedback of salt fingers on large-scale perturbations and analyse the resulting patterns. The novelty of the proposed analysis is related to our ability to avoid using empirical double-diffusive flux laws – an approach taken by earlier models. Instead, we base our analytical explorations directly on the governing (Navier–Stokes) equations of motion. The model predictions of the growth rates and preferred slopes of intrusions are in general agreement with the laboratory and field measurements.

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Papers
Creative Commons
This is a work of the U.S. Government and is not subject to copyright protection in the United States.
Copyright
Copyright © Cambridge University Press 2011. This is a work of the U.S. Government and is not subject to copyright protection in the United States.

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References

REFERENCES

Baines, P. G. & Gill, A. E. 1969 on thermohaline convection with linear gradients. J. Fluid Mech. 37, 289306.CrossRefGoogle Scholar
Balmforth, N. J., Ghadge, S. A., Kettapun, A. & Mandre, S. D. 2006 Bounds on double-diffusive convection. J. Fluid Mech. 569, 2950.CrossRefGoogle Scholar
Balmforth, N. J. & Young, Y.-N. 2002 Stratified Kolmogorov flow. J. Fluid Mech. 450, 131167.CrossRefGoogle Scholar
Balmforth, N. J. & Young, Y.-N. 2005 Stratified Kolmogorov flow. Part 2. J. Fluid Mech. 528, 2342.CrossRefGoogle Scholar
Canuto, V., Howard, A., Cheng, Y. & Dubikov, M. 2002 Ocean turbulence. Part II: Vertical diffusivities of momentum, heat, mass, salt and passive scalars. J. Phys. Oceanogr. 32, 240264.2.0.CO;2>CrossRefGoogle Scholar
Gama, S., Vergassola, M. & Frisch, U. 1994 Negative eddy viscosity in isotropically forced 2-dimensional flow – linear and nonlinear dynamics. J. Fluid Mech. 260, 95126.CrossRefGoogle Scholar
Gargett, A. E. & Schmitt, R. W. 1982 Observations of salt fingers in the central waters of the eastern North Pacific. J. Geophys. Res. 87, 80178092.CrossRefGoogle Scholar
Garrett, C. 1982. On the parameterization of diapycnal fluxes due to double-diffusive intrusions. J. Phys. Oceanogr. 12, 952959.2.0.CO;2>CrossRefGoogle Scholar
Hart, J. E. 1973 Finite amplitude sideways diffusive convection. J. Fluid Mech. 59, 4764.CrossRefGoogle Scholar
Holyer, J. Y. 1981 On the collective instability of salt fingers. J. Fluid Mech. 110, 195207.CrossRefGoogle Scholar
Holyer, J. Y. 1983 Double-diffusive interleaving due to horizontal gradients. J. Fluid Mech. 137, 347362.CrossRefGoogle Scholar
Holyer, J. Y. 1984 The stability of long, steady, two-dimensional salt fingers. J. Fluid Mech. 147, 169185.CrossRefGoogle Scholar
Holyer, J. Y. 1985 The stability of long steady three-dimensional salt fingers to long wavelength perturbations. J. Fluid Mech. 156, 495503.CrossRefGoogle Scholar
Holyer, J., Jones, T., Priestly, M. & Williams, N. 1987 The effect of vertical temperature and salinity gradients on double diffusive interleaving. Deep-Sea Res. 34, 517530.CrossRefGoogle Scholar
Joyce, T. M., Zenk, W. & Toole, J. M. 1978 The anatomy of the Antarctic Polar Front in the Drake Passage. J. Geophys. Res. 83, 60936113.CrossRefGoogle Scholar
Kerr, O. S. 1990 Heating a salinity gradient from a vertical sidewall: nonlinear theory. J. Fluid Mech. 217, 529546.CrossRefGoogle Scholar
Kevorkian, J. & Cole, J. D. 1996 Multiple Scale and Singular Perturbation Methods. Springer, 632 pp.CrossRefGoogle Scholar
Krishnamurti, R. 2006 Double-diffusive interleaving on horizontal gradients. J. Fluid Mech. 558, 113131.CrossRefGoogle Scholar
Landau, L. D. 1944 On the problem of turbulence. Dokl. Akad. Nauk SSSR 44, 311314.Google Scholar
Manfroi, A. & Young, W. 1999 Slow evolution of zonal jets on the beta plane. J. Atmos. Sci. 56, 784800.2.0.CO;2>CrossRefGoogle Scholar
Manfroi, A. & Young, W. 2002 Stability of beta-plane Kolmogorov flow. Physica D 162, 208232.CrossRefGoogle Scholar
May, B. & Kelley, D. 1997 Effect of baroclinicity on double-diffusive interleaving. J. Phys. Oceanogr. 27, 19972008.2.0.CO;2>CrossRefGoogle Scholar
McDougall, T. J. 1985 a Double-diffusive interleaving. Part I: Linear stability analysis. J. Phys. Oceanogr. 15, 15321541.2.0.CO;2>CrossRefGoogle Scholar
McDougall, T. J. 1985 b Double-diffusive interleaving. Part II: Finite amplitude steady state interleaving. J. Phys. Oceanogr. 15, 15421556.2.0.CO;2>CrossRefGoogle Scholar
McDougall, T. J. 1986 Oceanic intrusions: some limitations of the Ruddick & Turner (1979) mechanism. Deep-Sea Res. 33, 16531664.CrossRefGoogle Scholar
Mei, C. C. & Vernescu, M. 2010 Homogenization Methods for Multiscale Mechanics. World Scientific Publishing, 330 pp.CrossRefGoogle Scholar
Meshalkin, L. & Sinai, Y. 1961 Investigation of the stability of a stationary solution of a system of equations for the plane movement of an incompressible viscous fluid. J. Appl. Math. Mech. 25, 17001705.CrossRefGoogle Scholar
Novikov, A. & Papanicolau, G. 2001 Eddy viscosity of cellular flows. J. Fluid Mech. 446, 173198.CrossRefGoogle Scholar
Radko, T. 2008 The double-diffusive modon. J. Fluid Mech. 609, 5985.CrossRefGoogle Scholar
Radko, T. 2010 Equilibration of weakly nonlinear salt fingers. J. Fluid Mech. 645, 121143.CrossRefGoogle Scholar
Radko, T. 2011 On the generation of large-scale structures in a homogeneous eddy field. J. Fluid Mech. (in press).CrossRefGoogle Scholar
Radko, T. & Stern, M. E. 1999 Salt fingers in three dimensions. J. Mar. Res. 57, 471502.CrossRefGoogle Scholar
Radko, T. & Stern, M. E. 2000 Finite amplitude salt fingers in a vertically bounded layer. J. Fluid Mech. 425, 133160.CrossRefGoogle Scholar
Ruddick, B. 2003 Laboratory studies of interleaving. Progr. Oceanogr. 56, 549–547.CrossRefGoogle Scholar
Ruddick, B. & Hebert, D. 1988 The mixing of Meddy ‘Sharon’. In Small-scale Turbulence and Mixing in the Ocean (ed. Nihoul, J. & Jamart, B.), pp. 249261, Elsevier.Google Scholar
Ruddick, B. & Kerr, O. 2003 Oceanic thermohaline intrusions: theory. Prog. Oceanogr. 56, 483497.CrossRefGoogle Scholar
Ruddick, B. & Richards, K. 2003 Oceanic thermohaline intrusions: observations. Prog. Oceanogr. 56, 499527.CrossRefGoogle Scholar
Ruddick, B. R., Phillips, O. M. & Turner, J. S. 1999 A laboratory and quantitative model of finite-amplitude intrusions. Dyn. Atmos. Oceans 30, 7199.CrossRefGoogle Scholar
Ruddick, B. R. & Turner, J. S. 1979 The vertical length scale of double-diffusive intrusions. Deep-Sea Res. 26A, 903913.CrossRefGoogle Scholar
Rudels, B., Bjork, G., Muench, R. D. & Schauer, U. 1998 Double-diffusive layering in the Eurasian Basin of the Arctic Ocean. J. Mar. Syst. 21, 327.CrossRefGoogle Scholar
Schmitt, R. W. 1979 The growth rate of supercritical salt fingers. Deep-Sea Res. 26A, 2344.CrossRefGoogle Scholar
Schmitt, R. W. 1983 The characteristics of salt fingers in a variety of fluid systems, including stellar interiors, liquid metals, oceans, and magmas. Phys. Fluids 26, 23732377.CrossRefGoogle Scholar
Schmitt, R. W. & Georgi, D. T. 1982 Finestructure and microstructure in the North Atlantic Current. J. Mar. Res. 40 Suppl., 659705.Google Scholar
Shen, C. Y. & Schmitt, R. W. 1996 The wavenumber spectrum of salt fingers. In Double-Diffusive Convection (ed. Brandt, S. A. & Fernando, H.), vol. 94, pp. 305312. AGU Geophysical Monograph.Google Scholar
Simeonov, J. & Stern, M. E. 2004 Double-diffusive intrusions on a finite-width thermohaline front. J. Phys. Oceanogr. 34, 17231740.2.0.CO;2>CrossRefGoogle Scholar
Simeonov, J. & Stern, M. E. 2007 Equilibration of two-dimensional double-diffusive intrusions. J. Phys. Oceanogr. 37, 625643.CrossRefGoogle Scholar
Simeonov, J. & Stern, M. E. 2008 Double-diffusive intrusions in a stable salinity gradient ‘heated from below’. J. Phys. Oceanogr. 38, 22712282.CrossRefGoogle Scholar
Sivashinsky, G. 1985 Weak turbulence in periodic flows. Physica D 17, 243255.CrossRefGoogle Scholar
Smyth, W. 2007 Instabilities of a baroclinic, double diffusive frontal zone. J. Phys. Oceanogr. 38, 840861.CrossRefGoogle Scholar
Smyth, W. D. & Ruddick, B. 2010 Effects of ambient turbulence on interleaving at a baroclinic front. J. Phys. Oceanogr. 40, 685712.CrossRefGoogle Scholar
Stern, M. E. 1960 The “salt-fountain” and thermohaline convection. Tellus 12, 172175.CrossRefGoogle Scholar
Stern, M. E. 1967 Lateral mixing of water masses. Deep-Sea Res. 14, 747753.Google Scholar
Stern, M. E. 1969 Collective instability of salt fingers. J. Fluid Mech. 35, 209218.CrossRefGoogle Scholar
Stern, M. E., Radko, T. & Simeonov, J. 2001 3D salt fingers in an unbounded thermocline with application to the Central Ocean. J. Mar. Res. 59, 355390.CrossRefGoogle Scholar
Stern, M. E. & Simeonov, J. 2004 Amplitude equilibration of sugar–salt fingers. J. Fluid Mech. 508, 265286.CrossRefGoogle Scholar
Stern, M. E. & Simeonov, J. 2005 The secondary instability of salt fingers. J. Fluid Mech. 533, 361380.CrossRefGoogle Scholar
St. Laurent, L. & Schmitt, R. W. 1999 The contribution of salt fingers to vertical mixing in the North Atlantic tracer release experiment. J. Phys. Oceanogr. 29, 14041424.2.0.CO;2>CrossRefGoogle Scholar
Taylor, J. 1993 Anisotropy of salt fingers. J. Phys. Oceanogr. 23, 554565.2.0.CO;2>CrossRefGoogle Scholar
Toole, J. & Georgi, D. 1981 On the dynamics of double diffusively driven intrusions. Prog. Oceanogr. 10, 123145.CrossRefGoogle Scholar
Walsh, D. & Ruddick, B. 2000 Double diffusive interleaving in the presence of turbulence: the effect of a nonconstant flux ratio. J. Phys. Oceanogr. 30, 22312245.2.0.CO;2>CrossRefGoogle Scholar
Young, Y. & Rosner, R. 1998 Linear and weakly nonlinear analysis of doubly-diffusive vertical slot convection. Phys. Rev. E 57, 55545563.CrossRefGoogle Scholar