(Academic Press, 2003, vol 2, pp. 680-685 and 685-694)
Copyright © Michael E. McIntyre
by agreement with Academic.
Final proof corrections have been incorporated into the files for download, links below. The published version suffers from occasional though mostly harmless copy-editing damage, not reproduced in the files below. Note in particular that, in the published version, the phrase `the submanifold in phase space' became `the in phase space submanifold' --- could the copy editor have been thinking in German? --- and the phrase `the stratospheric transport or Brewer-Dobson circulation' became `the stratospheric or transport Brewer-Dobson circulation'.
Notes added subsequently: In June 2004 I discovered to my embarrassment that the statement about non-Hamiltonian `velocity splitting' in the Balanced Flow article is wrong. (Splitting means here that the model has two velocity fields: one to advect mass, and another to advect and evaluate the exact potential vorticity.) The statement in the Balanced Flow article is true, to be sure, of all the most accurate balanced models known at the time of writing. They all exhibit velocity splitting. And it seemed to be supported by a strong heuristic argument, given in the concluding section of my paper with W. A. Norton published in 2000. In brief the argument says that, because any balanced model has to suppress the local mass rearrangement involved in the spontaneous-adjustment emission of inertia-gravity waves, there should be a tradeoff between accuracy and splitting. That is, non-Hamiltonian balanced models constrained to be free of velocity splitting should, by that fact, be unable to attain as high an accuracy as non-Hamiltonian balanced models not so constrained.
However, subsequent work with Dr A. R. Mohebalhojeh resulted in the discovery of a novel class of `hyperbalance equations', free of velocity splitting yet capable of arbitrarily high formal accuracy. The equations involve functional derivatives as well as ordinary partial derivatives. It is possible that this technicality explains why the hyperbalance equations remained undiscovered for so long.
Very careful numerical work by Dr Mohebalhojeh
then showed that the hyperbalance equations are capable of
extraordinarily high numerical accuracy as well.
High formal accuracy does not, of course,
necessarily imply high numerial accuracy.
But from extensive numerical tests it appeared that
there is, after all, no systematic tradeoff between accuracy and splitting,
for non-Hamiltonian balanced models.
Though surprising and embarrassing to me personally,
this was excellent news theoretically.
It seems that in the hyperbalance equations
we now have in our possession what had long seemed
an unattainable prize
-- a class of non-Hamiltonian potential-vorticity-conserving balanced models
competitive with all known such models in terms of accuracy, yet also
(because the splitting is healed)
having an exactly conserved potential enstrophy, and exact
conservation of all the other non-Hamiltonian Casimirs.
This work was published in June 2007
as a pair of papers
in J. Atmos. Sci 64, 1782-1793 and 1794-1810.
Pdf reprints are available here
(copyright © American Meteorological Society):
Hyperbalance Equations Part I (0.2 Mbyte),
Hyperbalance Equations Part II (2.6 Mbyte).
The results are for the shallow-water equations only.
It remains possible that a tradeoff will be found for
2-layer and multi-layer models.
A
short paper to appear in ADGEO
(Advances in Geosciences)
gives
a brief summary of the hyperbalance equations
(.pdf, 0.16 Mbyte).
Related
issues of balance and imbalance are discussed in a paper in the
J. Atmos. Sci. Special Collection on `Spontaneous Imbalance', entitled
Spontaneous imbalance and hybrid vortex-gravity structures
(.pdf, 0.7 Mbyte).
As for the Encyclopedia articles, that on Balanced Flow (vol 2, pp. 680-685) can be downloaded as an uncompressed postscript file (144 kbyte) from here or as a gzipped postscript file (56 kbyte) from here, or as a .pdf (acrobat) file (61 kbyte) from here.
The text of the article on Potential Vorticity (vol 2, pp. 685-694) -- which includes some hard-to-find history -- can be downloaded as an uncompressed postscript file (196 kbyte) from here or as a gzipped postscript file (76 kbyte) from here, or as a .pdf (acrobat) file (87 kbyte) from here.
The Potential Vorticity article has two accompanying figures, and also mentions figures from Professor Holton's introduction/overview (not available here), referring to page numbers in the Encyclopedia supplied by the copy editors. Here are the two accompanying figures:
Fig. 1. Estimated isentropic distribution of Rossby-Ertel potential vorticity (PV) on the 320K isentropic surface on 14 May 1992 at 1200 UT (Greenwich mean time), derived from observations as explained in the text. Over Europe the 320 K surface lies near jetliner cruising altitudes z ~ 10 km. The estimate used data from the operational weather-prediction analyses of the European Centre for Medium Range Weather Forecasts (ECMWF). Values from 1 PVU upwards are colored rainbow-wise from dark blue to red, with contour interval 1 PVU; 1 PVU = 10-6m2s-1K kg-1. From the paper by Appenzeller et al., J. Geophys. Res. 101, 1435-1456 (1996), on ``Fragmentation of stratospheric intrusions''.
Fig. 2. Latitude-altitude cross-section for January 1993 showing the monthly and zonally averaged temperature as dashed contours, and potential temperature as light solid contours. The heavy solid contour shows the nominal extratropical tropopause as defined by the 2 PVU contour of the PV field calculated from Rossby's formula (3), using zonally averaged potential temperature, horizontal wind and pressure/height fields obtained from UK Met Office analyses. From the review by Holton et al., Revs. Geophys., 33, 403-439 (1995) on ``Stratosphere-troposphere exchange'', to which the reader is referred for further detail.
