Ceophys. J. zyxwvutsr R. astr. zyxwvutsr SOC. zyxwvuts (1976) 44, 725-728. zyxwvut Letter to the Editors Comments on Tore Undertones with Rotation' by D. J. Crossley Martin L. Smith (Received 1975 August 11) 1. Introdaction zyxwvuts Seismologists have, until recently, tended to neglect the existence of suites of ve~--loow-fiequency elastic-gravitational normal modes which are principally confined to the Earth's fluid outer core. The existence of such modes (which are well known to fluid dynamicists as zyxwvu ' internal gravity waves ') for certain classes of non-rotating Earth models is theoretically certain, and much is known about their properties (Cowling 1941; Eckart 1960; Dahlen 1974). One important property is that the eigenfrequenciesof these oscillations are uniquely sensitive to the density stratification in the fluid outer core and thus, should any such motions ever be observed, we might hope to use them to extend greatly our knowledge of the Earth's deep interior. In the spherically symmetric, non-rotating case (hereafter SNREI) the eigenfrequencies of these modes become dense near o = 0, where w denotes angular frequency. A geophysically useful study of these modes will require a theoretical treatment that takes proper account of the Earth's diurnal rotation since for at least a part of every overtone sequence, the effects of rotation wilI be too great to treat with small parameter perturbation theory. Crossley (1975) has reported the results of a study designed to, at least partially, satisfy this requirement. One of Dr Crossley's principal conclusions is that the internal gravity modes in the fluid core of a rotating Earth all must have angular eigenfrequencies not less than twice the angular frequency of steady rotation. The basis for this conclusion is a rather unusual interpretation of what Crossley (1975) calls ' transverse inertia ', an interpretation which he attributes to D. E. Smiley. I believe that this interpretation is misleading and that, as a result, the numerical results of Crossley's calculations cannot be usefully interpreted in terms of the normal-mode eigenspectrum of the rotating Earth. Section 2 of this note is devoted to a justification of this assertion. 2. The coefficient of transverse inertia Let zyxwvut s(x) denote the infinitesimal Lagrangian particle displacement associated with some elastic-gravitational normal mode of a rotating, axisymmetric planet which is also symmetric under inversion through its centre-of-mass. (The rotating spherical Earth studied by Crossley (1975) is a special case of such a planet.) In a co-ordinate system aligned with the rotation axis, s(x) must vary with longitude (p as exp(im(p) for 725 N Downloaded from https://academic.oup.com/gji/article-abstract/44/3/725/583884 by guest on 01 June 2020