Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-28T09:32:37.520Z Has data issue: false hasContentIssue false
  • English
  • Français

Elliptic and hyperbolic interior solutions of piecewise-constant potential vorticity geophysical vortices

Published online by Cambridge University Press:  28 February 2012

Álvaro Viúdez
Álvaro Viúdez*
Affiliation:
Institut de Ciències del Mar, CSIC, 08003 Barcelona, Spain
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Elliptic and hyperbolic geopotential solutions, for a homogeneous distribution of potential vorticity (PV), are obtained via PV inversion in geophysical vortices. The flow in the axisymmetrical three-dimensional vortices is steady and horizontal, where the centripetal acceleration plus the Coriolis acceleration equals the pressure anomaly gradient term (gradient wind or cyclo-geostrophic balance). It is found that the family of geopotential solutions in the vortex interior is completely parameterized by the PV density in the vortex and the squared aspect ratio between the horizontal and vertical semi-axes of the ellipsoidal or hyperbolic geopotential surfaces. Thus, the PV inversion task consists of obtaining, via solution of algebraic cubic equations, the absolute vertical vorticity and vertical stratification as a function of PV and aspect ratio. It is found that there is always a critical aspect ratio, which depends on PV, beyond which the PV inversion solutions are multi-valued. The complete vorticity and stratification solutions for the different regions in the PV and aspect ratio space are obtained and analysed with emphasis on the inertial and static instability of the vortex flow.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Rotating flows Vortex Flows: Vortex dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 696 , 10 April 2012 , pp. 301 - 318
Copyright
Copyright © Cambridge University Press 2012

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

1. Beltrami, E. 1871 Sui principii fondamentali dell’idrodinamica razionali. Mem. Accad. Sci. Inst. Bologna 1, 431–476.Google Scholar
2. Bishop, C. H. & Thorpe, A. J. 1994 Potential vorticity and the electrostatics analogy: Quasi-geostrophic theory. Q. J. R. Meteorol. Soc. 120, 713–731.Google Scholar
3. Casey, J. & Naghdi, P. M. 1991 On the Lagrangian description of vorticity. Arch. Rat. Mech. Anal. 115, 1–14.CrossRefGoogle Scholar
4. Ertel, H. 1942 Ein neuer hydrodynamischer erhaltungssatz. Naturwissenschaften 30, 543–544.CrossRefGoogle Scholar
5. Fultz, D. 1991 Quantitative nondimensional properties of the gradient wind. J. Atmos. Sci. 48, 869–875.2.0.CO;2>CrossRefGoogle Scholar
6. Gustafson, A. F. 1953 On anomalous winds in the free atmosphere. Bull. Am. Meteorol. Soc. 34, 196–201.CrossRefGoogle Scholar
7. Holton, J. R. 2004 An Introduction to Dynamic Meteorology. Elsevier.Google Scholar
8. Hoskins, B. J., McIntyre, M. E. & Robertson, A. W. 1985 On the use and significance of isentropic potential-vorticity maps. Q. J. R. Meteorol. Soc. 111, 877–946.CrossRefGoogle Scholar
9. Kurganskiy, M. V. & Tatarskaya, M. S. 1987 The potential vorticity concept in meteorology: A review. Izv. Atmos. Ocean. Phys. 23, 587–606.Google Scholar
10. McWilliams, J. C., Weiss, J. B. & Yavneh, I. 1999 The vortices of homogeneous geostrophic turbulence. J. Fluid Mech. 410, 1–26.CrossRefGoogle Scholar
11. Müller, P. 1995 Ertel’s potential vorticity theorem in physical oceanography. Rev. Geophys. 33, 67–97.CrossRefGoogle Scholar
12. Reinaud, J. N., Dritschel, D. G. & Koudella, C. R. 2003 The shape of vortices in quasi-geostrophic turbulence. J. Fluid Mech. 474, 175–192.CrossRefGoogle Scholar
13. Rossby, C.-G. 1936 Dynamics of steady ocean currents in the light of experimental fluid mechanics. Pap. Phys. Oceanogr. Meteor. 5, 1–43.Google Scholar
14. Rossby, C.-G. 1940 Planetary flow patterns in the atmosphere. Q. J. R. Meteorol. Soc. 66 (suppl), 68–87.CrossRefGoogle Scholar
15. Santurette, P. & Georgiev, C. G. 2005 Weather Analysis and Forecasting. Applying Satellite Water Vapor Imagery and Potential Vorticity Analysis. Elsevier Academic.Google Scholar
16. Thorpe, A. J. & Bishop, C. H. 1994 Potential vorticity and the electrostatics analogy – quasi-geostrophic theory. Q. J. R. Meteorol. Soc. 120, 713–731.Google Scholar
17. Thorpe, A. J. & Bishop, C. H. 1995 Potential vorticity and the electrostatics analogy – Ertel–Rossby formulation. Q. J. R. Meteorol. Soc. 121, 1477–1495.Google Scholar
18. 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
19. Viúdez, A. 2008 The piecewise constant symmetric potential vorticity vortex in geophysical flows. J. Fluid Mech. 614, 145–172.CrossRefGoogle Scholar
20. Viúdez, A. 2012 Potential vorticity and inertia–gravity waves. Geophys. Astrophys. Fluid Dyn. 106, 67–88.CrossRefGoogle Scholar
21. Viúdez, A. & Dritschel, D. G. 2003 Vertical velocity in mesoscale geophysical flows. J. Fluid Mech. 483, 199–223.CrossRefGoogle Scholar
22. Viúdez, A. & Dritschel, D. G. 2004 Optimal potential vorticity balance of geophysical flows. J. Fluid Mech. 521, 343–352.CrossRefGoogle Scholar
23. Woan, G. 2003 The Cambridge Handbook of Physics Formulas. Cambridge University Press.Google Scholar