• Bishop, G. H., and A. J. Thorpe, 1994: Potential vorticity and the electrostatics analogy—Quasi-geostrophic theory. Quart. J. Roy. Meteor. Soc.,120, 713–731.

  • Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc.,92, 335–345.

  • Buizza, R., and T. N. Palmer, 1995: The singular-vector structure of the atmospheric global circulation. J. Atmos. Sci.,52, 1434–1456.

  • Cho, H.-R., and J. N. Koshyk, 1989: Dynamics of frontal discontinuities in the semigeostrophic theory. J. Atmos. Sci.,46, 2166–2177.

  • Courant, R., and D. Hilbert, 1962: Methods of Mathematical Physics. Vol. II. Interscience Publishing, 830 pp.

  • Davis, C., and K. A. Emanuel, 1991: Potential vorticity diagnostics of cyclogenesis. Mon. Wea. Rev.,119, 1929–1953.

  • Gelaro, R., R. Buizza, T. N. Palmer, and E. Klinker, 1998: Sensitivity analysis of forecast errors and the construction of optimal perturbations using singular vectors. J. Atmos. Sci.,55, 1012–1037.

  • Hoskins, B. J., 1975: The geostrophic momentum approximation and the semi-geostrophic equations. J. Atmos. Sci.,32, 233–242.

  • ——, and F. P. Bretherton, 1972: Atmospheric frontogenesis models:Mathematical formulation and solution. J. Atmos. Sci.,29, 11–37.

  • ——, and I. Draghici, 1977: The forcing of ageostrophic motion according to the semi-geostrophic equations and in an isentropic coordinate model. J. Atmos. Sci.,34, 1859–1867.

  • ——, M. E. McIntyre, and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc.,111, 877–946.

  • Koshyk, J. N., and H.-R. Cho, 1992: Dynamics of mature front in a uniform potential vorticity semigeostrophic model. J. Atmos. Sci.,49, 497–510.

  • Kushner, P. J., and T. G. Shepherd, 1995a: Wave-activity conservation-laws and stability theorems for semi-geostrophic dynamics. Part 1: Pseudomomentum-based theory. J. Fluid Mech.,290, 67–104.

  • ——, and ——, 1995b: Wave-activity conservation-laws and stability theorems for semi-geostrophic dynamics. Part 2: Pseudoenergy-based theory. J. Fluid Mech.,290, 105–129.

  • Palmer, T. N., R. Gelaro, J. Barkmeijer, and R. Buizza, 1998: Singular vectors, metrics, and adaptive observations. J. Atmos. Sci.,55, 633–653.

  • Purser, R. J., and M. J. P. Cullen, 1987: A duality principle in semigeostrophic theory. J. Atmos. Sci.,44, 3449–3468.

  • Ren, S. Z., 1998: Linear stability of the three-dimensional semigeostrophic model in geometric coordinates. J. Atmos. Sci.,55, 3392–3402.

  • Thorpe, A. J., and G. H. Bishop, 1995: Potential vorticity and the electrostatics analogy—Ertel–Rossby formulation. Quart. J. Roy. Meteor. Soc.,121, 1477–1495.

  • Xu, Q., 1988: Baroclinic waves and frontogenesis with an embedded zone of small moist symmetric stability. Quart. J. Roy. Meteor. Soc.,114, 1221–1251.

  • ——, 1990: Cold and warm frontal circulations in an idealized moist semigeostrophic baroclinic wave. J. Atmos. Sci.,47, 2337–2352.

  • ——, 1994: Semibalance model—Connection between geostrophic-type and balanced-type intermediate models. J. Atmos. Sci.,51, 953–970.

  • ——, W. Gu, and J. Gao, 1998: Baroclinic Eady wave and fronts. Part I: Viscous semigeostrophy and the impact of boundary condition. J. Atmos. Sci.,55, 3598–3615.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 111 14 3
PDF Downloads 9 8 2

Baroclinic Eady Wave and Fronts. Part II: Geostrophic Potential Vorticity Dynamics in Semigeostrophic Space

Qin XuNaval Research Laboratory, Monterey, California

Search for other papers by Qin Xu in
Current site
Google Scholar
PubMed
Close
and
Wei GuCIMMS, University of Oklahoma, Norman, Oklahoma

Search for other papers by Wei Gu in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The geostrophic coordinate transformation is applied to the viscous semigeostrophic (SG) Eady wave. In the transformed space, called the SG space, the potential temperature anomaly can be treated as a delta-function anomaly of geostrophic potential vorticity (GPV) at the physical boundary or imaginary boundary (along the top of the boundary layer). Since the delta-function anomaly is analogous to the surface charge of a problem in electrostatics with the induced geopotential playing the role of the electric potential, the development of the Eady wave and fronts can be interpreted in terms of the interaction between the “surface charges” at two imaginary boundaries. It is shown that this interpretation and related GPV thinking for the viscous SG Eady wave can be made nearly as concise as its inviscid paradigm during the boundary stage (until the inviscid surface front collapses in physical space).

When the viscous SG Eady wave develops into the interior stage, strong interior GPV anomalies, analogous to “body charges,” are generated by the diffusive GPV flux. These body charges form two domes in the SG space. The geostrophic flow field induced by the body charge in each dome produces diffusive GPV fluxes that converge at the upstream edge of each dome and thus keep the body charge in step against the horizontal advection. The growth of the geometric area of each dome of body charge (or the penetration of the front into the interior in physical space), however, is caused mainly by the ageostrophic circulation forced by the geostrophic flow. It is also shown that the body charge in each dome can be represented by the surface charge (potential temperature anomaly) on an imaginary boundary that covers the dome (above the boundary layer). The growth of these surface charges can be explained by a simplified GPV thinking applied only to the nearly inviscid interior region, similar to its inviscid counterpart for the interior stage (beyond the time that the inviscid surface front collapses in physical space).

* Current affiliation: National Severe Storms Laboratory, Norman, Oklahoma.

Corresponding author address: Dr. Qin Xu, National Severe Storm Laboratory, 1313 Halley Circle, Norman, Oklahoma 73069.

Email: qin.xu@nssl.noaa.gov

Abstract

The geostrophic coordinate transformation is applied to the viscous semigeostrophic (SG) Eady wave. In the transformed space, called the SG space, the potential temperature anomaly can be treated as a delta-function anomaly of geostrophic potential vorticity (GPV) at the physical boundary or imaginary boundary (along the top of the boundary layer). Since the delta-function anomaly is analogous to the surface charge of a problem in electrostatics with the induced geopotential playing the role of the electric potential, the development of the Eady wave and fronts can be interpreted in terms of the interaction between the “surface charges” at two imaginary boundaries. It is shown that this interpretation and related GPV thinking for the viscous SG Eady wave can be made nearly as concise as its inviscid paradigm during the boundary stage (until the inviscid surface front collapses in physical space).

When the viscous SG Eady wave develops into the interior stage, strong interior GPV anomalies, analogous to “body charges,” are generated by the diffusive GPV flux. These body charges form two domes in the SG space. The geostrophic flow field induced by the body charge in each dome produces diffusive GPV fluxes that converge at the upstream edge of each dome and thus keep the body charge in step against the horizontal advection. The growth of the geometric area of each dome of body charge (or the penetration of the front into the interior in physical space), however, is caused mainly by the ageostrophic circulation forced by the geostrophic flow. It is also shown that the body charge in each dome can be represented by the surface charge (potential temperature anomaly) on an imaginary boundary that covers the dome (above the boundary layer). The growth of these surface charges can be explained by a simplified GPV thinking applied only to the nearly inviscid interior region, similar to its inviscid counterpart for the interior stage (beyond the time that the inviscid surface front collapses in physical space).

* Current affiliation: National Severe Storms Laboratory, Norman, Oklahoma.

Corresponding author address: Dr. Qin Xu, National Severe Storm Laboratory, 1313 Halley Circle, Norman, Oklahoma 73069.

Email: qin.xu@nssl.noaa.gov

Save