Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-04T18:26:09.917Z Has data issue: false hasContentIssue false
  • English
  • Français

On the internal shear layers spawned by the critical regions in oscillatory Ekman boundary layers

Published online by Cambridge University Press:  26 April 2006

R. R. Kerswell
R. R. Kerswell
Affiliation:
Department of Mathematics and Statistics, University of Newcastle upon Tyne, Newcastle NE1 7RU, UK
Get access Rights & Permissions [Opens in a new window]

Abstract

The Ekman boundary layer scalings associated with small oscillatory flows in a rapidly rotating system are well known to break down at the critical latitudes. The effect of these anomalous boundary regions on the interior flow has long been an issue of some concern. We argue, using analytical results derived in a model problem based on Stewartson's (1957) split-disk configuration, that these boundary regions spawn internal shear layers. These shear layers have their natural length scale of E1/3 and carry velocities only O(E1/6) smaller than the Rossby number – where E is the Ekman number – if the boundary is concave away from the fluid at the critical latitude. Otherwise, only a weaker shear layer is produced of length scale E1/5 containing velocities of O(E3/10) smaller than the Rossby number rather than the usual O(E1/5) value for viscous flows in the interior. We show how these layers can carry angular momentum by themselves or more significantly in combination with the interior inviscid flow, so as to influence the mean flow at leading order. These conclusions are then discussed in the context of a precessing oblate spheroid of fluid for which detailed experimental observations are available.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 298 , 10 September 1995 , pp. 311 - 325
Copyright
© 1995 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barrett, K. E. 1969 Resonant torsional oscillations of a pair of discs in a rotating fluid. Z. Angew. Math. Phys. 20, 721–729.Google Scholar
Bnney, D. J. 1965 The flow induced by a disk oscillating about a state of steady rotation. Q. J. Mech. Appl. Maths 18, 333–345.Google Scholar
Bondi, H. & Lyttleton, R. A. 1953 On the dynamical theory of the rotation of the Earth: the effect of precession on the motion of the liquid core. Proc. Camb. Phil. Soc. 49, 498.Google Scholar
Busse, F. H. 1968 Steady fluid flow in a precessing spheroidal shell. J. Fluid Mech. 33, 739–751.Google Scholar
Gans, R. F. 1983 Boundary layers on characteristic surfaces for time-dependent rotating flows. Trans. ASMEE: J. Appl. Mech. 50, 251–254.Google Scholar
Görtler, H. 1944 Einige Bemerkungen über Strömungen in rotierenden Flüssigkeiten. Z. Angew. Math. Mech. 24, 210–214.Google Scholar
Görtler, H. 1957 On forced oscillations in rotating fluids. 5th Midwestern Conf. on Fluid Mech., pp. 1–10.
Greenspan, H. P. 1964 On the transient motion of a contained rotating fluid. J. Fluid Mech. 20, 673–696.Google Scholar
Greenspan, H. P. 1965 On the general theory of contained rotating fluid motions. J. Fluid Mech. 22, 449–462.Google Scholar
Greenspan, H. P. 1968 The Theory of Rotating Fluids. Cambridge University Press; reprinted Breukelen Press, Brookline (1990).
Greenspan, H. P. & Howard, L. N. 1963 On a time dependent motion of a rotating fluid. J. Fluid Mech. 17, 385–404.Google Scholar
Hollerbach, R. & Kerswell, R. R. 1995 Oscillatory internal shear layers in rotating and precessing flows. J. Fluid Mech. 298, 327–339.Google Scholar
Jacobs, C. 1971 Transient motions produced by disks oscillating torsionally about a state of rigid rotation. Q. J. Mech. Appl. Maths 24, 221–236.Google Scholar
Jacobs, S. J. 1964 The Taylor column problem. J. Fluid Mech. 20, 581–591.Google Scholar
Kerswell, R. R. 1994 Tidal excitation of hydromagnetic waves and their damping in the Earth. J. Fluid Mech. 274, 219–241.Google Scholar
Kerswell, R. R. & Barenghi, C. F. 1995 On the viscous decay rates of inertial waves in a rotating circular cylinder. J. Fluid Mech. 285, 203–214.Google Scholar
Malkus, W. V. R. 1968 Precession of the Earth as the cause of geomagnetism. Science 169, 259–264.Google Scholar
Mcewan, A. D. 1970 Inertial oscillations in a rotating fluid cylinder. J. Fluid Mech. 40, 603–640.Google Scholar
Moore, D. W. & Saffman, P. G. 1969 The structure of free vertical shear layers in a rotating fluid and the motion produced by a slowly rising body. Phil. Trans. R. Soc. Lond. A 264, 597–634.Google Scholar
Morrison, J. A. & Morgan, G. W. 1956 The slow motion of a disc along the axis if a viscous rotating fluid. Tech. Rep. Div. Appl. Math., Brown University, vol. 8.
Oser, H. 1957 Erzwungene Schwingungen in rotierenden Flussigkeiten. Arch. Rat. Mech. Anal. 1, 81–96.Google Scholar
Oser, H. 1958 Experimentelle Untersuchung uber harmonische Schwingungen in rotierenden Flussigkeiten. Z. angew. Math. Mech. 38, 386–391.Google Scholar
Phillips, O. M. 1963 Energy transfer in rotating fluids by reflection of inertial waves. Phys. Fluids 6, 513–520.Google Scholar
Poincaré, H. 1910 Sur la précession des corps deformable. Bull. Astron. 27, 321.Google Scholar
Proudman, I. 1956 The almost-rigid rotation of viscous fluid between concentric spheres. J. Fluid Mech. 1, 505–516.Google Scholar
Reynolds, A. 1962a Forced oscillations in a rotating liquid (I). Z. angew. Math. Phys. 13, 460–468.Google Scholar
Reynolds, A. 1962b Forced oscillations in a rotating liquid (II). Z. angew. Math. Phys. 13, 561–572.Google Scholar
Roberts, P. H. & Stewartson, K. 1963 On the stability of a Maclaurin spheroid of small viscosity. Astrophys. J. 137 777–790.Google Scholar
Sneddon, I. N. 1951 Fourier Transforms. McGraw-Hill.
Stewartson, K. 1957 On almost rigid rotations. J. Fluid Mech. 3, 17–26.Google Scholar
Stewartson, K. 1966 On almost rigid rotations. Part 2. J. Fluid Mech. 26 131–144.Google Scholar
Thornley, C. 1968 On Stokes and Rayleigh layers in a rotating system. Q. J. Mech. Appl. Maths 21, 451–461.Google Scholar
Vanyo, J. P., Wilde, P., Cardin, P. & Olson, P. 1992 Flow structures induced in the liquid core by precession. Abstract AGU EOS 73 63 (Supplement).Google Scholar
Vanyo, J. P., Wilde, P., Cardin, P. & Olson, P. 1995 Experiments on precessing flows in the Earth's liquid core. Geophys. J. Intl 121, 136–142.Google Scholar
Walton, I. C. 1975a Viscous shear layers in an oscillating rotating fluid. Proc. R. Soc. Lond. A 344, 101–110.Google Scholar
Walton, I. C. 1975b On waves in a thin rotating spherical shell of slightly viscous fluid. Mathematika 22, 46–59.Google Scholar
Wood, W. W. 1965 Properties of inviscid, recirculating flows. J. Fluid Mech. 22, 337–346.Google Scholar
Wood, W. W. 1966 An oscillatory disturbance of rigidly rotating fluid. Proc. R. Soc. Lond. A 298, 181–212.Google Scholar
Wood, W. W. 1977 Inertial oscillations in a rigid axisymmetric container. Proc. R. Soc. Lond. A 358, 17–30.Google Scholar
Wood, W. W. 1981 Inertial modes with large azimuthal wavenumbers in an axisymmetric container. J. Fluid Mech. 105, 427–449.Google Scholar