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Mechanics of merging events for a series of layers in a stratified turbulent fluid

Published online by Cambridge University Press:  19 April 2007

TIMOUR RADKO
TIMOUR RADKO*
Affiliation:
Department of Oceanography, Naval Postgraduate School, Monterey, CA 93943, [email protected]
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Abstract

This study attempts to explain the evolutionary pattern of a series of well-mixed layers separated by thin high-gradient interfaces frequently observed in stratified fluids. Such layered structures form as a result of the instability of the equilibrium with uniform stratification, and their subsequent evolution is characterized by a sequence of merging events which systematically increase the average layer thickness. The coarsening of layers can take one of two forms, depending on the realized vertical buoyancy flux law. Layers merge either when the high-gradient interfaces drift and collide, or when some interfaces gradually erode without moving vertically. The selection of a preferred pattern of coarsening is rationalized by the analytical theory – the merging theorem – which is based on linear stability analysis for a series of identical layers and strongly stratified interfaces. The merging theorem suggests that the merger by erosion of weak interfaces occurs when the vertical buoyancy flux decreases with the buoyancy variation across the step. If the buoyancy flux increases with step height, then coarsening of a staircase may result from drift and collision of the adjacent interfaces. Our model also quantifies the time scale of merging events and makes it possible to predict whether the layer merging continues indefinitely or whether the coarsening is ultimately arrested. The merging theorem is applied to extant one-dimensional models of turbulent mixing and successfully tested against the corresponding fully nonlinear numerical simulations. It is hypothesized that the upscale cascade of buoyancy variance associated with merging events may be one of the significant sources of the fine-scale (∼ 10m) variability in the ocean.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 577 , 25 April 2007 , pp. 251 - 273
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Balmforth, N. J. & Young, Y.-N. 2002 Stratified Kolmogorov flow. J. Fluid Mech. 450, 131–167.CrossRefGoogle Scholar
Balmforth, N. J. & Young, Y.-N. 2005 Stratified Kolmogorov flow. Part 2. J. Fluid Mech. 528, 23–42.CrossRefGoogle Scholar
Balmforth, N. J., LlewellynSmith, S. G. Smith, S. G. & Young, W. R. 1998 Dynamics of interfaces and layers in a stratified turbulent fluid. J. Fluid Mech. 355, 329–358.CrossRefGoogle Scholar
Bates, P. & Xun, J. 1995 Metastable patterns for the Cahn Hilliard equation, Part II. J. Diffl Eqns 117, 165–216.CrossRefGoogle Scholar
Cahn, J. W. & Hilliard, J. E. 1958 Free energy of a nonuniform system. I. Interfacial free energy. J. Chem. Phys. 28, 258–267.CrossRefGoogle Scholar
Chapman, C. & Proctor, M. 1980 Nonlinear Rayleigh–Bénard convection with poorly conducting boundaries. J. Fluid Mech. 101, 759–782.CrossRefGoogle Scholar
Holford, J. M. & Linden, P. F. 1999 Turbulent mixing in a stratified fluid. Dyn. Atmos. Oceans 30, 173–198.CrossRefGoogle Scholar
Huppert, H. E. 1971 On the stability of a series of double-diffusive layers. Deep-Sea Res. 18, 1005–1021.Google Scholar
Kelley, D. E., Fernando, H. J. S., Gargett, A. E., Tanny, J. & Ozsoy, E. 2003 The diffusive regime of double-diffusive convection. Prog. Oceanogr. 56, 461–481.CrossRefGoogle Scholar
Legras, B., Frisch, U. & Villone, B. 1999 Dispersive stabilization of the inverse cascade for the Kolmogorov flow. Phys. Rev. Lett. 82, 4440–4443.CrossRefGoogle Scholar
Manfroi, A. & Young, W. 1999 Slow evolution of zonal jets on the beta-plane, J. Atmos. Sci. 56, 784–800.2.0.CO;2>CrossRefGoogle Scholar
Merryfield, W. J. 2000 Origin of thermohaline staircases. J. Phys. Oceanogr. 30, 1046–1068.2.0.CO;2>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. Z. Angew. Math. Mech. 25, 1700–1705.CrossRefGoogle Scholar
Panetta, R. L. 1993 Zonal jets in wide baroclinically unstable regions: persistence and scale selection. J. Atmos. Sci. 50, 2073–2106.2.0.CO;2>CrossRefGoogle Scholar
Park, Y.-G., Whitehead, J. A. & Gnanadeskian, A. 1994 Turbulent mixing in stratified fluids: layer formation and energetics. J. Fluid Mech. 279, 279–311.CrossRefGoogle Scholar
Phillips, O. M. 1972 Turbulence in a strongly stratified fluid: is it unstable? Deep-Sea Res. 19, 79–81.Google Scholar
Posmentier, E. S. 1977 The generation of salinity finestructure by vertical diffusion. J. Phys. Oceanogr. 7, 298–300.2.0.CO;2>CrossRefGoogle Scholar
Radko, T. 2003 A mechanism for layer formation in a double-diffusive fluid. J. Fluid Mech. 497, 365–380.CrossRefGoogle Scholar
Radko, T. 2005 What determines the thickness of layers in a thermohaline staircase? J. Fluid Mech. 523, 79–98.CrossRefGoogle Scholar
Rhines, P. 1975 Waves and turbulence on a beta-plane. J. Fluid Mech. 69, 417–443.CrossRefGoogle Scholar
Ruddick, B. R., McDougall, T. J. & Turner, J. S. 1989 The formation of layers in a uniformly stirred density gradient. Deep-Sea Res. 36, 597–609.CrossRefGoogle Scholar
Schmitt, R. W. 1994 Double diffusion in oceanography. Annu. Rev. Fluid Mech. 26, 255–285.CrossRefGoogle Scholar
Schmitt, R. W. 2003 Observational and laboratory insights into salt finger convection. Prog. Oceanogr. 56, 419–433.CrossRefGoogle Scholar
Simeonov, J. & Stern, M. E. 2007 Equilibration of two-dimensional double diffusive intrusions. J. Phys. Oceanogr. (in press).Google Scholar
Simpson, J. H. & Woods, J. D. 1970 Temperature microstructure in a freshwater thermocline. Nature 226, 832–834.CrossRefGoogle Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. John Wiley.Google Scholar
Zodiatis, G. & Gasparini, G. P. 1996 Thermohaline staircase formations in the Tyrrhenian Sea. Deep-Sea Res. 43, 665–678.CrossRefGoogle Scholar