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Multilayer shallow water equations with complete Coriolis force. Part 3. Hyperbolicity and stability under shear

Published online by Cambridge University Press:  16 April 2013

Andrew L. Stewart  and
Paul J. Dellar
Andrew L. Stewart
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, UK
Paul J. Dellar*
Affiliation:
OCIAM, Mathematical Institute, University of Oxford, 24-29 St Giles’, Oxford OX1 3LB, UK
*
†Email address for correspondence: [email protected]
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Abstract

We analyse the hyperbolicity of our multilayer shallow water equations that include the complete Coriolis force due to the Earth’s rotation. Shallow water theory represents flows in which the vertical shear is concentrated into vortex sheets between layers of uniform velocity. Such configurations are subject to Kelvin–Helmholtz instabilities, with arbitrarily large growth rates for sufficiently short-wavelength disturbances. These instabilities manifest themselves through a loss of hyperbolicity in the shallow water equations, rendering them ill-posed for the solution of initial value problems. We show that, in the limit of vanishingly small density difference between the two layers, our two-layer shallow water equations remain hyperbolic when the velocity difference remains below the same threshold that also ensures the hyperbolicity of the standard shallow water equations. Direct calculation of the domain of hyperbolicity becomes much less tractable for three or more layers, so we demonstrate numerically that the threshold for the velocity differences, below which the three-layer equations remain hyperbolic, is also unchanged by the inclusion of the complete Coriolis force. In all cases, the shape of the domain of hyperbolicity, which extends outside the threshold, changes considerably. The standard shallow water equations only lose hyperbolicity due to shear parallel to the direction of wave propagation, but the complete Coriolis force introduces another mechanism for loss of hyperbolicity due to shear in the perpendicular direction. We demonstrate that this additional mechanism corresponds to the onset of a transverse shear instability driven by the non-traditional components of the Coriolis force in a three-dimensional continuously stratified fluid.

JFM classification

Geophysical and Geological Flows: Waves in rotating fluids Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Ocean processes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 723 , 25 May 2013 , pp. 289 - 317
Copyright
©2013 Cambridge University Press 

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Footnotes

‡

Now at: Environmental Science and Engineering, California Institute of Technology, Pasadena, CA 91125, USA.

References

Abramowitz, M. & Stegun, I. A. 1965 Handbook of Mathematical Functions. Dover.Google Scholar
Adcroft, A. & Hallberg, R. 2006 On methods for solving the oceanic equations of motion in generalized vertical coordinates. Ocean Model. 11, 224–233.CrossRefGoogle Scholar
Benzoni-Gavage, S. & Serre, D. 2007 Multi-Dimensional Hyperbolic Partial Differential Equations: First Order Systems and Applications. Oxford University Press.Google Scholar
Bleck, R. & Chassignet, E. 1994 Simulating the oceanic circulation with isopycnic-coordinate models. In The Oceans: Physical–Chemical Dynamics and Human Impact (ed. Majumdar, S. K., Miller, E. W., Forbe, G. S., Schmalz, R. F. & Panah, A. A.), pp. 17–39. Pennsylvania Academy of Science.Google Scholar
Bleck, R., Rooth, C., Hu, D. & Smith, L. T. 1992 Salinity-driven thermocline transients in a wind- and thermohaline-forced isopycnic coordinate model of the North Atlantic. J. Phys. Oceanogr. 22, 1486–1505.Google Scholar
Boss, E., Paldor, N. & Thompson, L. 1996 Stability of a potential vorticity front: from quasi-geostrophy to shallow water. J. Fluid Mech. 315, 65–84.Google Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Charney, J. G. 1947 The dynamics of long waves in a baroclinic westerly current. J. Meteorol. 4, 136–162.Google Scholar
Choboter, P. F. & Swaters, G. E. 2004 Shallow water modeling of Antarctic bottom water crossing the Equator. J. Geophys. Res. C109, 03038.Google Scholar
Chumakova, L., Menzaque, F. E., Milewski, P. A., Rosales, R. R., Tabak, E. G. & Turner, C. V. 2009a Shear instability for stratified hydrostatic flows. Commun. Pure Appl. Maths 62, 183–197.CrossRefGoogle Scholar
Chumakova, L., Menzaque, F. E., Milewski, P. A., Rosales, R. R., Tabak, E. G. & Turner, C. V. 2009b Stability properties and nonlinear mappings of two and three-layer stratified flows. Stud. Appl. Maths 122, 123–137.CrossRefGoogle Scholar
Colin de Verdière, A. 2012 The stability of short symmetric internal waves on sloping fronts: beyond the traditional approximation. J. Phys. Oceanogr. 42, 459–475.Google Scholar
Dellar, P. J. & Salmon, R. 2005 Shallow water equations with a complete Coriolis force and topography. Phys. Fluids 17, 106601.CrossRefGoogle Scholar
Eady, E. T. 1949 Long waves and cyclone waves. Tellus 1, 33–52.Google Scholar
Eckart, C. 1960 Hydrodynamics of Oceans and Atmospheres. Pergamon.Google Scholar
Eliassen, A. 1951 Slow thermally or frictionally controlled meridional circulation in a circular vortex. Astrophys. Norvegica 5, 19–60.Google Scholar
Fjørtoft, R. 1950 Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geofys. Publ. 17, 1–52.Google Scholar
Friedrichs, K. O. 1954 Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Maths 7, 345–392.Google Scholar
Garver, R. 1933 On the nature of the roots of a quartic equation. Math. News Lett. 7, 6–8.Google Scholar
Gerkema, T. & Shrira, V. I. 2005a Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195–219.Google Scholar
Gerkema, T. & Shrira, V. I. 2005b Near-inertial waves on the non-traditional $\beta $ -plane. J. Geophys. Res. 110, C10003.Google Scholar
Gerkema, T., Zimmerman, J. T. F., Maas, L. R. M. & van Haren, H. 2008 Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev. Geophys. 46, RG2004.Google Scholar
Gill, A. E. 1982 Atmosphere Ocean Dynamics. Academic.Google Scholar
Godlewski, E. & Raviart, P.-A. 1996 Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer.Google Scholar
Goldstein, S. 1931 On the stability of superposed streams of fluids of different densities. Proc. R. Soc. Lond. A 132, 524–548.Google Scholar
Green, A. E. & Naghdi, P. M. 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237–246.Google Scholar
Hadamard, J. 1923 Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press.Google Scholar
van Haren, H. & Millot, C. 2005 Gyroscopic waves in the Mediterranean Sea. Geophys. Res. Lett. 32, L24614.Google Scholar
Hathaway, D. H., Gilman, P. A. & Toomre, J. 1979 Convective instability when the temperature gradient and rotation vector are oblique to gravity. I. Fluids without diffusion. Geophys. Astrophys. Fluid Dyn. 13, 289–316.Google Scholar
Heywood, K. J., Naveira Garabato, A. C. & Stevens, D. P. 2002 High mixing rates in the abyssal Southern Ocean. Nature 415, 1011–1014.Google Scholar
Hoskins, J. B. 1974 The role of potential vorticity in symmetric stability and instability. Q. J. R. Meteorol. Soc. 100, 480–482.Google Scholar
Houghton, D. & Isaacson, E. 1970 Mountain winds. In Numerical Solutions of Nonlinear Problems (ed. Ortega, J. M. & Rheinboldt, W. C.), Studies in Numerical Analysis, vol. 2, pp. 21–52. Society for Industrial and Applied Mathematics.Google Scholar
Howard, L. N. 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512.Google Scholar
Hua, B. L., Moore, D. W. & Le Gentil, S. 1997 Inertial nonlinear equilibration of equatorial flows. J. Fluid Mech. 331, 345–371.Google Scholar
Irving, R. S. 2004 Integers, Polynomials, and Rings. Springer.Google Scholar
Jeffery, N. & Wingate, B. 2009 The effect of tilted rotation on shear instabilities at low stratifications. J. Phys. Oceanogr. 39, 3147–3161.Google Scholar
Joseph, D. D. & Saut, J. C. 1990 Short-wave instabilities and ill-posed initial-value problems. Theor. Comput. Fluid Dyn. 1, 191–227.Google Scholar
Kelder, H. & Teitelbaum, H. 1991 A note on instabilities of a Kelvin–Helmholtz velocity profile in different approximations. Il Nuovo Cimento C 14, 107–118.Google Scholar
Ku, W. 1965 Explicit criterion for the positive definiteness of a general quartic form. IEEE Trans. Autom. Control 10, 372–373.Google Scholar
Kuo, H.-L. 1954 Symmetrical disturbances in a thin layer of fluid subject to a horizontal temperature gradient and rotation. J. Met. 11, 399–411.Google Scholar
Lamb, H. 1932 Hydrodynamics, 6th edn. Cambridge University Press.Google Scholar
Lawrence, G. A. 1990 On the hydraulics of Boussinesq and non-Boussinesq two-layer flows. J. Fluid Mech. 215, 457–480.Google Scholar
Lazier, J. R. N. 1980 Oceanographic conditions at Ocean Weather Ship Bravo, 1964–1974. Atmos.-Ocean 18, 227–238.Google Scholar
LeBlond, P. H. & Mysak, L. A. 1978 Waves in the Ocean. Elsevier.Google Scholar
Leibovich, S. & Lele, S. K. 1985 The influence of the horizontal component of Earth’s angular velocity on the instability of the Ekman layer. J. Fluid Mech. 150, 41–87.Google Scholar
Liska, R., Margolin, L. & Wendroff, B. 1995 Nonhydrostatic two-layer models of incompressible flow. Comput. Maths Applics 29, 25–37.Google Scholar
Liska, R. & Wendroff, B. 1997 Analysis and computation with stratified fluid models. J. Comput. Phys. 137, 212–244.Google Scholar
Long, R. R. 1956 Long waves in a two-fluid system. J. Met. 13, 70–74.2.0.CO;2>CrossRefGoogle Scholar
Marchal, O. & Nycander, J. 2004 Nonuniform upwelling in a shallow-water model of the Antarctic Bottom Water in the Brazil Basin. J. Phys. Oceanogr. 34, 2492–2513.CrossRefGoogle Scholar
Marshall, J. & Schott, F. 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 1–64.Google Scholar
Miles, J. W. 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496–508.Google Scholar
Mooers, C. N. K. 1975 Several effects of a baroclinic current on the crossstream propagation of inertial-internal waves. Geophys. Fluid Dyn. 6, 245–275.Google Scholar
Ooyama, K. 1966 On the stability of the baroclinic circular vortex: a sufficient criterion for instability. J. Atmos. Sci. 23, 43–53.Google Scholar
Pedlosky, J. 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.Google Scholar
Persson, A. 2005 The Coriolis effect: four centuries of conflict between common sense and mathematics. Part I. A history to 1885. Hist. Meteorol. 2, 1–24.Google Scholar
Phillips, N. A. 1954 Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6, 273–286.Google Scholar
Phillips, N. A. 1968 Reply to comment by Veronis. J. Atmos. Sci. 25, 1155–1157.2.0.CO;2>CrossRefGoogle Scholar
Phillips, N. A. 1973 Principles of large scale numerical weather prediction. In Dynamic Meteorology: Lectures delivered at the Summer School of Space Physics of the centre National d’Études Spatiales (ed. Morel, P.), pp. 3–96. D. Reidel Pub. Co.Google Scholar
Queney, P. 1950 Adiabatic perturbation equations for a zonal atmospheric current. Tellus 2, 35–51.Google Scholar
Rayleigh, Lord. 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. Ser. A 93, 148–154.Google Scholar
Renardy, M. & Rogers, R. C. 2004 An Introduction to Partial Differential Equations, 2nd edn. Springer.Google Scholar
Salmon, R. 1982 The shape of the main thermocline. J. Phys. Oceanogr. 12, 1458–1479.Google Scholar
Salmon, R. 1998 Lectures on Geophysical Fluid Dynamics. Oxford University Press.Google Scholar
Sawyer, J. S. 1949 The significance of dynamic instability in atmospheric motions. Q. J. R. Meteorol. Soc. 75, 364–374.Google Scholar
Stephens, J. C. & Marshall, D. P. 2000 Dynamical pathways of Antarctic bottom water in the Atlantic. J. Phys. Oceanogr. 30, 622–640.Google Scholar
Stewart, A. L. & Dellar, P. J. 2010 Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech. 651, 387–413.Google Scholar
Stewart, A. L. & Dellar, P. J. 2011a Cross-equatorial flow through an abyssal channel under the complete Coriolis force: two-dimensional solutions. Ocean Model. 40, 87–104.Google Scholar
Stewart, A. L. & Dellar, P. J. 2011b The rôle of the complete Coriolis force in cross-equatorial flow of abyssal ocean currents. Ocean Model. 38, 187–202.Google Scholar
Stewart, A. L. & Dellar, P. J. 2012a Cross-equatorial channel flow with zero potential vorticity under the complete Coriolis force. IMA J. Appl. Maths 77, 626–651.Google Scholar
Stewart, A. L. & Dellar, P. J. 2012b Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. J. Fluid Mech. 690, 16–50.Google Scholar
Stone, P. H. 1966 On non-geostrophic baroclinic stability. J. Atmos. Sci. 23, 390–400.Google Scholar
Stone, P. H. 1971 Baroclinic stability under non-hydrostatic conditions. J. Fluid Mech. 45, 659–671.Google Scholar
Strang, G. 1966 Necessary and insufficient conditions for well-posed Cauchy problems. J. Differ. Equ. 2, 107–114.Google Scholar
Sun, W. Y. 1995 Unsymmetrical symmetrical instability. Q. J. R. Meteorol. Soc. 121, 419–431.CrossRefGoogle Scholar
Taylor, G. I. 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499–523.Google Scholar
Thuburn, J., Wood, N. & Staniforth, A. 2002 Normal modes of deep atmospheres. I: spherical geometry. Q. J. R. Meteorol. Soc. 128, 1771–1792.Google Scholar
Vallis, G. K. 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.Google Scholar
Wang, F. & Qi, L. 2005 Comments on ‘explicit criterion for the positive definiteness of a general quartic form’. IEEE Trans. Autom. Control 50, 416–418.Google Scholar
White, A. A. 2002 A view of the equations of meteorological dynamics and various approximations. In Large-Scale Atmosphere-Ocean Dynamics 1: Analytical Methods and Numerical Models (ed. Norbury, J. & Roulstone, I.), pp. 1–100. Cambridge University Press.Google Scholar
White, A. A. & Bromley, R. A. 1995 Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Q. J. R. Meteorol. Soc. 121, 399–418.Google Scholar
Whitham, G. B. 1974 Linear and Nonlinear Waves. Wiley Interscience.Google Scholar
Xui, Q. & Clark, J. H. E. 1985 The nature of symmetric instability and its similarity to convective and inertial instability. J. Atmos. Sci. 42, 2880–2883.Google Scholar