Hostname: page-component-77c89778f8-gvh9x Total loading time: 0 Render date: 2024-07-17T07:46:50.415Z Has data issue: false hasContentIssue false
  • English
  • Français

The piecewise constant symmetric potential vorticity vortex in geophysical flows

Published online by Cambridge University Press:  16 October 2008

ÁLVARO VIÚDEZ
ÁLVARO VIÚDEZ*
Affiliation:
Institut de Ciències del Mar, CSIC, 08003 Barcelona, Spain
Get access Rights & Permissions [Opens in a new window]

Abstract

The concept of piecewise constant symmetric vortex in the context of three-dimensional baroclinic balanced geophysical flows is explored. The pressure gradients generated by horizontal cylinders and spherical balls of uniform potential vorticity (PV), or uniform material invariants, are obtained either analytically or numerically, in the general case of Boussinesq and f-plane dynamics as well as under the quasi-geostrophic and semigeostrophic dynamical approximations. Based on the order of magnitude of the different terms in the PV inversion equation, approximated PV equations are deduced. In some of these cases, radial solutions are possible and the interior and exterior solutions are found analytically. In the case of non-radial dependence, exterior solutions can be found numerically. Linear, and upper and lower bound approximations to the full PV inversion equations, and their respective solutions, are also included. However, the general solution for the pressure gradient in the vortex exterior does not have spherical symmetry and remains as an important theoretical challenge. It is suggested that, in order to maintain everywhere the inertial and static stability of the balanced geophysical flows, small balls of finite radius, rather than PV singularities, could become, specially in numerical applications, useful mathematical objects.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 614 , 10 November 2008 , pp. 145 - 172
Copyright
Copyright © Cambridge University Press 2008

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

REFERENCES

Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Beltrami, E. 1871 Sui principii fondamentali dell'idrodinamica razionali. Memorie della Accademia delle Scienze dell'Instituto di Bologna 1, 431–476.Google Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 1087–1095.2.0.CO;2>CrossRefGoogle Scholar
Dritschel, D. G. & Viúdez, A. 2003 A balanced approach to modelling rotating stablystratified geophysical flows. J. Fluid Mech. 488, 123–150.CrossRefGoogle Scholar
Eliassen, A. 1949 The quasi-static equations of motion with pressure as independent variable. Geofys. Publ. 17 (3), 1–44.Google Scholar
Ertel, H. 1942 Ein neuer hydrodynamischer erhaltungssatz. Naturwissenschaften 30, 543–544.CrossRefGoogle Scholar
Fjortoft, R. 1962 On the integration of a system of geostrophically balanced prognostic equations. In Proc. Intl Symp. Numerical Weather Prediction. Met. Soc. Japan, Tokyo.Google Scholar
Head, A. K. 1979 The monodromic Galois groups of the sextic equation of anisotropic elasticity. J. Elasticity 9, 321–324.CrossRefGoogle Scholar
Holton, J. R. 2004 An Introduction to Dynamic Meteorology. Elsevier.Google Scholar
Hoskins, B. J. 1975 The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci. 32, 233–242.2.0.CO;2>CrossRefGoogle Scholar
Hoskins, B. J. & Draghici, I. 1977 The forcing of ageostrophic motion according to the semi-geostrophic equations and in an isentropic coordinate model. J. Atmos. Sci. 34, 1859–1867.2.0.CO;2>CrossRefGoogle Scholar
Kurgansky, M. V. & Tatarskaya, M. S. 1987 The potential vorticity concept in meteorology: a review. Izv. Atmos. Ocean. Phys. 23, 587–606.Google Scholar
Lamb, H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Müller, P. 1995 Ertel's potential vorticity theorem in physical oceanography. Rev. Geophys. 33, 67–97.CrossRefGoogle Scholar
Nolan, D. S. & Montgomery, M. T. 2002 Nonhydrostatic, three-dimensional perturbations to balanced, hurricane-like vortices. Part I: Linearized formulation, stability, and evolution. J. Atmos. Sci. 59, 2989–3020.2.0.CO;2>CrossRefGoogle Scholar
Rossby, C.-G. 1940 Planetary flow patterns in the atmosphere. Q. J. R. Met. Soc. 66 (suppl.), 68–87.CrossRefGoogle Scholar
Saffman, P. G. 1992 Vortex Dynamics. Cambridge University Press.Google Scholar
Shutts, G. J. 1991 Some exact solutions to the semi-geostrophic equations for uniform potential vorticity flows. Geophys. Astrophys. Fluid Dyn. 57, 99–114.CrossRefGoogle Scholar
Thorpe, A. J. & Bishop, C. H. 1994 Potential vorticity and the electrostatics analogy – quasi-geostrophic theory. Quart. J. Roy. Met. Soc. 120, 713–731.Google Scholar
Thorpe, A. J. & Bishop, C. H. 1995 Potential vorticity and the electrostatics analogy – Ertel–Rossby formulation. Q. J. R. Met. Soc. 121, 1477–1495.Google Scholar
Viúdez, A. 2001 The relation between Beltrami's material vorticity and Rossby–Ertel's potential vorticity. J. Atmos. Sci. 58, 2509–2517.2.0.CO;2>CrossRefGoogle Scholar
Viúdez, A. 2005 The vorticity–velocity gradient cofactor tensor and the material invariant of the semigeostrophic theory. J. Atmos. Sci. 62, 2294–2301.CrossRefGoogle Scholar
Viúdez, A. & Dritschel, D. G. 2003 Vertical velocity in mesoscale geophysical flows. J. Fluid Mech. 483, 199–223.CrossRefGoogle Scholar
Voropayev, S. I. & Afanasyev, Y. D. 1994 Vortex Structures in a Stratified Fluid. Order from Chaos. Chapman & Hall.CrossRefGoogle Scholar
White, A. A. 2002 A view of the equations of meteorological dynamics and various approximations. In Large-Scale Atmosphere-Ocean Dynamics I (ed. Norbury, J. & Roulstone, I.), pp. 1–100. Cambridge University Press.Google Scholar