Acta Geod. Geoph. Hung., Vol. 41(1), pp. 143–146 (2006)

DOI: 10.1556/AGeod.41.2006.1.11

HANS ERTEL, RELATIVISTIC POTENTIALVORTICITY AND GEOPHYSICAL APPLICATIONS

W Schroder and H-J Treder

Geophysical Commission, Hechelstrasse 8, D-28777 Bremen, Germany

[Manuscript received August 3, 2004; accepted September 7, 2004]

Short review on the geophysical application of Ertel’s potential vorticity theorem.

Keywords: Ertel’s potential vorticity

Introduction

In 1942 Hans Ertel derived his well-known hydrodynamical theorem which isnow general referred to in the international literature as “Ertel’s potential vortic-ity”. A short time later F Moran (1942) gave an interesting discussion of Ertel’sformula but in the following years, due to a lack of international scientific communi-cation during and immediately after the Second World War, few studies are knownrelated to Ertel’s meteorological and hydrodynamical results. More generalisationand application of Ertel’s work was presented by Truesdell in his earlier papers(Truesdell 1951) in the 1950 and in his “Handbuch-article” in 1960. More recentlyin the 1980’s the Cambridge and Reading research groups (Hoskins et al. 1985)started programmes which used and analysed the results of Ertel’s potential vor-ticity. Moreover, the monographs by Gill and Pedlosky (Gill 1982) have developedthe topic further and their descriptions of Ertel’s result in English have made thiswork more widely known to the scientific. Recently, some applications in relativisticphysics and astrophysics has been presented by Friedman (1978), by Friedman andSchutz (1978), by Meiss and Horton (1983), and by Schroder and Treder (1993).

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144 W SCHRODER and H J TREDER

1. Ertel’s vorticity theorems

Hans Ertel the great theoretic meteorologist, whose lifework is above all con-nected with vorticity theory for real fluids. Internationally, he has first of all becomeknown by his remarks on the cosmological constances and relations between micro-and macrocosmos. All his live he has been especially interested in cosmology and asan expert in theory of relativity as well in vorticity dynamics of relativistic cosmicdimensions.

Therefore, professor Ertel invited one of us (HJT) to take part in the curatorsof the “Gerlands Beitrage zur Geophysik” edited by him. He was so kind as toaccept an article as a presentation copy to his 65th birthday about the subject“Geophysik und Kosmologie”, he is especially interested in. In our explanations, wereferred to the hodogetic meaning of the opinions represented by A S Eddington,H Ertel, R Furth and A Haas that there should exist apriori-relations between el-ementary particle physics and cosmology. At the same time, we remarked in thesame sense of Hans Ertel: the earth and its environments register several billionyears of “self-experienced history”. The interpretation of the particular observa-tions can be very complex. Therefore, these observations are much more informa-tive and diract in regard to the theory of cognition than astrophysical informationwith its space-time retardation, and the necessary realisation of the modulation ofelectromagnetic waves. Therefore cosmological theories and hypotheses principallyshould be rather decisive with the help of geophysical and similar tests (physics ofmeteorits) than with pure astrophysical observationes only. After having severaland long discussions with Ertel, we was now asked by him the question about thegeneral-relativistic formulation of the vorticity theoremes of Helmholtz. My answer,that he published in the “Gerlands Beitrage” was “Die allgemein-kovariante, rela-tivistische Verallgemeinerung des 1. Helmholtzschen Wirbelsatzes” (1970). In thisconnection, Ertels interpretation that the covariant general-relativistic derivation ofthese vorticity theoremes would be more understandable than the non-relativisticone has been confirmed. There has been much critisism of Helmholtz’s derivationof his vorticity theoremes (even in A Sommerfield’s textbook of continuum mechan-ics). But, instead of the imprimitive Galilei-group in general relativistic theory theprimitive Lorentz-Poincare-group is substituted and the general convariance of theequations at the same time. From identities (all indices run from 1 to 4)

Duα

Dτ= uα

βUβ = 0

follows forUα;jβ − uβ;α = ωαβ

the relativistic generalization of Helmholtz’s vorticity theoreme (the semicolon standsfor the covariant derivative). We have

D

Dτωαβ = ωλ

βUα;λ − ωλαuβ;λ .

Evidently, this is the general fourdimensional formulation of the Helmholtz’ vor-ticity theoreme. One can see that the change in time of the rotation vanishes, if

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HANS ERTEL 145

ωαβ vanishes itself. After that in Ertels opinion all obscurities of Helmholtz’ deriva-tion were cleared up. That is why, we were demanded by Hans Ertel to give somemore: “Die allgemein-relativistische und kovariants Integralform der Wirbeltheo-reme” (1970) and he was much delighted that he had known all the things before.The Cartan-Stokes integral-theoremes supply as well the relativistic form of theHelmholtz-Kelvin theoremes:

D

Dτ

∫ωαβdxα

∧dxβ = 0 .

Ertel gave a third task: to set up relativisticly “Die Boltzmannsche Form derHelmholtzschen Wirbelsatze”. We was successful in doing it because of one identitygiven by Ertel (1962). But in this work we did a howler. When we told Ertel aboutthis regrettably after having published my paper, he comforted us making the fol-lowing remark: An analogue mistake was even contained in Einstein’s first versionof his “Grundlagen der Allgemeinen Relativitatstheorie” (Ann. der Phys., 1916)with respect to the relativistic formulation of Euler’s equation. (In his later is-suses “Grundzuge der Relativitatstheorie” Einstein redefined a variable and, wefollowed this pattern.) Encouraged so much by Ertel we wrote a further paper “ZurAllgemein-relativistischen Wirbeldynamik” (1970) wich satisfied now even Hans Er-tel fully.

The co-operation between professor Ertel and us, that means in this case thequestions raised by Ertel and our mathematictechnical abilities supposed by him,arose his favourite cosmological problem, taken over from his early days, whichmeans the question about the “Quellen und Senken des universellen Schwere-Feldes”.“Heuristische Betrachtungen zur Kosmologie” (Ann. der Phys., 1971) asked by thefamous Munic mechanic A Foppl to ask one more again. Unfortunately there wasno continuation of the discussion because of H Ertels death. But it seems to us thatthe ideas which Ertel brought into this article are higly interestingly even today(Ertel 1962, 1963).

2. Conclusions

From the earliest stages of his scientific career Hans Ertel worked on hydrody-namical problems, and on geophysical and meteorological hydrodynamics. In May1971 Hans Ertel visited the university of Uppsala. He delivered lectures dealing withthe problems of inertia in ocean and atmosphere, the hydro-dynamical problems ofEl Nino, and some special problems of theoretical geomorphology. Therefore, dur-ing the General Assembly of the International Union of Geophysics and Geodesy, aspecial commemorative session of the History commission of the International As-sociation of Geomagnetism and Aeronomy was organised. The paper of this sessionhas been published in 1993 by Schroder and Treder (1993). It may be noted thatin 2004 we celebrate Hans Ertel’s 100th birthday.

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146 W SCHRODER and H J TREDER

References

Ertel H 1962: Monatsberichte dtsch. Acad. Wiss. Berlin, 4, 292–296.Ertel H 1963: Monatsberichte dtsch. Acad. Wiss. Berlin, 5, 362–365.Friedman J L 1978: Comm. Math. Phys., 62, 247–278.Friedman J L, Schutz B F 1978: Astrophysics J., 221, 937–957.Gill A E 1982: Atmosphere-ocean dynamics. Academic Press, LondonHoskins B J, McIntyre M E, Robertson W A 1985: Quart. J. Royal Met. Soc., 111, 887–

946.Meiss J D, Horton W 1983: Phys. Fluids, 26, 990–997.Schroder W, Treder H J 1993: Theoretical concepts and observational implications in

meteorology and geophysics. Science Edition, Bremen-RonnebeckTruesdell C 1951: Z. Angew. Math. Mech., 2, 109–114.

Acta Geod. Geoph. Hung. 41, 2006