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

  • Andrew L. Stewart (a1) and Paul J. Dellar (a1)

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.

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

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A. Adcroft & R. Hallberg 2006 On methods for solving the oceanic equations of motion in generalized vertical coordinates. Ocean Model. 11, 224233.

R. Bleck , C. Rooth , D. Hu & L. T. Smith 1992 Salinity-driven thermocline transients in a wind- and thermohaline-forced isopycnic coordinate model of the North Atlantic. J. Phys. Oceanogr. 22, 14861505.

E. Boss , N. Paldor & L. Thompson 1996 Stability of a potential vorticity front: from quasi-geostrophy to shallow water. J. Fluid Mech. 315, 6584.

J. G. Charney 1947 The dynamics of long waves in a baroclinic westerly current. J. Meteorol. 4, 136162.

L. Chumakova , F. E. Menzaque , P. A. Milewski , R. R. Rosales , E. G. Tabak & C. V. Turner 2009a Shear instability for stratified hydrostatic flows. Commun. Pure Appl. Maths 62, 183197.

L. Chumakova , F. E. Menzaque , P. A. Milewski , R. R. Rosales , E. G. Tabak & C. V. Turner 2009b Stability properties and nonlinear mappings of two and three-layer stratified flows. Stud. Appl. Maths 122, 123137.

A. Colin de Verdière 2012 The stability of short symmetric internal waves on sloping fronts: beyond the traditional approximation. J. Phys. Oceanogr. 42, 459475.

P. J. Dellar & R. Salmon 2005 Shallow water equations with a complete Coriolis force and topography. Phys. Fluids 17, 106601.

E. T. Eady 1949 Long waves and cyclone waves. Tellus 1, 3352.

C. Eckart 1960 Hydrodynamics of Oceans and Atmospheres. Pergamon.

K. O. Friedrichs 1954 Symmetric hyperbolic linear differential equations. Commun. Pure Appl. Maths 7, 345392.

R. Garver 1933 On the nature of the roots of a quartic equation. Math. News Lett. 7, 68.

T. Gerkema & V. I. Shrira 2005a Near-inertial waves in the ocean: beyond the ‘traditional approximation’. J. Fluid Mech. 529, 195219.

T. Gerkema & V. I. Shrira 2005b Near-inertial waves on the non-traditional $\beta $-plane. J. Geophys. Res. 110, C10003.

T. Gerkema , J. T. F. Zimmerman , L. R. M. Maas & H. van Haren 2008 Geophysical and astrophysical fluid dynamics beyond the traditional approximation. Rev. Geophys. 46, RG2004.

A. E. Gill 1982 Atmosphere Ocean Dynamics. Academic.

E. Godlewski & P.-A. Raviart 1996 Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer.

S. Goldstein 1931 On the stability of superposed streams of fluids of different densities. Proc. R. Soc. Lond. A 132, 524548.

A. E. Green & P. M. Naghdi 1976 A derivation of equations for wave propagation in water of variable depth. J. Fluid Mech. 78, 237246.

H. van Haren & C. Millot 2005 Gyroscopic waves in the Mediterranean Sea. Geophys. Res. Lett. 32, L24614.

D. H. Hathaway , P. A. Gilman & J. Toomre 1979 Convective instability when the temperature gradient and rotation vector are oblique to gravity. I. Fluids without diffusion. Geophys. Astrophys. Fluid Dyn. 13, 289316.

K. J. Heywood , A. C. Naveira Garabato & D. P. Stevens 2002 High mixing rates in the abyssal Southern Ocean. Nature 415, 10111014.

J. B. Hoskins 1974 The role of potential vorticity in symmetric stability and instability. Q. J. R. Meteorol. Soc. 100, 480482.

L. N. Howard 1961 Note on a paper of John W. Miles. J. Fluid Mech. 10, 509512.

B. L. Hua , D. W. Moore & S. Le Gentil 1997 Inertial nonlinear equilibration of equatorial flows. J. Fluid Mech. 331, 345371.

N. Jeffery & B. Wingate 2009 The effect of tilted rotation on shear instabilities at low stratifications. J. Phys. Oceanogr. 39, 31473161.

D. D. Joseph & J. C. Saut 1990 Short-wave instabilities and ill-posed initial-value problems. Theor. Comput. Fluid Dyn. 1, 191227.

H. Kelder & H. Teitelbaum 1991 A note on instabilities of a Kelvin–Helmholtz velocity profile in different approximations. Il Nuovo Cimento C 14, 107118.

W. Ku 1965 Explicit criterion for the positive definiteness of a general quartic form. IEEE Trans. Autom. Control 10, 372373.

H.-L. Kuo 1954 Symmetrical disturbances in a thin layer of fluid subject to a horizontal temperature gradient and rotation. J. Met. 11, 399411.

G. A. Lawrence 1990 On the hydraulics of Boussinesq and non-Boussinesq two-layer flows. J. Fluid Mech. 215, 457480.

J. R. N. Lazier 1980 Oceanographic conditions at Ocean Weather Ship Bravo, 1964–1974. Atmos.-Ocean 18, 227238.

P. H. LeBlond & L. A. Mysak 1978 Waves in the Ocean. Elsevier.

S. Leibovich & S. K. Lele 1985 The influence of the horizontal component of Earth’s angular velocity on the instability of the Ekman layer. J. Fluid Mech. 150, 4187.

R. Liska , L. Margolin & B. Wendroff 1995 Nonhydrostatic two-layer models of incompressible flow. Comput. Maths Applics 29, 2537.

R. Liska & B. Wendroff 1997 Analysis and computation with stratified fluid models. J. Comput. Phys. 137, 212244.

R. R. Long 1956 Long waves in a two-fluid system. J. Met. 13, 7074.

O. Marchal & J. Nycander 2004 Nonuniform upwelling in a shallow-water model of the Antarctic Bottom Water in the Brazil Basin. J. Phys. Oceanogr. 34, 24922513.

J. Marshall & F. Schott 1999 Open-ocean convection: observations, theory, and models. Rev. Geophys. 37, 164.

J. W. Miles 1961 On the stability of heterogeneous shear flows. J. Fluid Mech. 10, 496508.

C. N. K. Mooers 1975 Several effects of a baroclinic current on the crossstream propagation of inertial-internal waves. Geophys. Fluid Dyn. 6, 245275.

K. Ooyama 1966 On the stability of the baroclinic circular vortex: a sufficient criterion for instability. J. Atmos. Sci. 23, 4353.

J. Pedlosky 1987 Geophysical Fluid Dynamics, 2nd edn. Springer.

N. A. Phillips 1954 Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level, quasi-geostrophic model. Tellus 6, 273286.

N. A. Phillips 1968 Reply to comment by Veronis. J. Atmos. Sci. 25, 11551157.

P. Queney 1950 Adiabatic perturbation equations for a zonal atmospheric current. Tellus 2, 3551.

Lord. Rayleigh 1917 On the dynamics of revolving fluids. Proc. R. Soc. Lond. Ser. A 93, 148154.

R. Salmon 1982 The shape of the main thermocline. J. Phys. Oceanogr. 12, 14581479.

J. S. Sawyer 1949 The significance of dynamic instability in atmospheric motions. Q. J. R. Meteorol. Soc. 75, 364374.

J. C. Stephens & D. P. Marshall 2000 Dynamical pathways of Antarctic bottom water in the Atlantic. J. Phys. Oceanogr. 30, 622640.

A. L. Stewart & P. J. Dellar 2010 Multilayer shallow water equations with complete Coriolis force. Part 1. Derivation on a non-traditional beta-plane. J. Fluid Mech. 651, 387413.

A. L. Stewart & P. J. Dellar 2011a Cross-equatorial flow through an abyssal channel under the complete Coriolis force: two-dimensional solutions. Ocean Model. 40, 87104.

A. L. Stewart & P. J. Dellar 2011b The rôle of the complete Coriolis force in cross-equatorial flow of abyssal ocean currents. Ocean Model. 38, 187202.

A. L. Stewart & P. J. Dellar 2012a Cross-equatorial channel flow with zero potential vorticity under the complete Coriolis force. IMA J. Appl. Maths 77, 626651.

A. L. Stewart & P. J. Dellar 2012b Multilayer shallow water equations with complete Coriolis force. Part 2. Linear plane waves. J. Fluid Mech. 690, 1650.

P. H. Stone 1966 On non-geostrophic baroclinic stability. J. Atmos. Sci. 23, 390400.

P. H. Stone 1971 Baroclinic stability under non-hydrostatic conditions. J. Fluid Mech. 45, 659671.

G. Strang 1966 Necessary and insufficient conditions for well-posed Cauchy problems. J. Differ. Equ. 2, 107114.

W. Y. Sun 1995 Unsymmetrical symmetrical instability. Q. J. R. Meteorol. Soc. 121, 419431.

G. I. Taylor 1931 Effect of variation in density on the stability of superposed streams of fluid. Proc. R. Soc. Lond. A 132, 499523.

J. Thuburn , N. Wood & A. Staniforth 2002 Normal modes of deep atmospheres. I: spherical geometry. Q. J. R. Meteorol. Soc. 128, 17711792.

G. K. Vallis 2006 Atmospheric and Oceanic Fluid Dynamics. Cambridge University Press.

F. Wang & L. Qi 2005 Comments on ‘explicit criterion for the positive definiteness of a general quartic form’. IEEE Trans. Autom. Control 50, 416418.

A. A. White 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. J. Norbury & I. Roulstone ), pp. 1100. Cambridge University Press.

A. A. White & R. A. Bromley 1995 Dynamically consistent, quasi-hydrostatic equations for global models with a complete representation of the Coriolis force. Q. J. R. Meteorol. Soc. 121, 399418.

Q. Xui & J. H. E. Clark 1985 The nature of symmetric instability and its similarity to convective and inertial instability. J. Atmos. Sci. 42, 28802883.

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