## Spin and angular momentum in gravity waves

M. S.  Longuet-Higgins a1
a1 Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge, England, and Institute of Oceanographic Sciences, Wormley, Surrey

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## Abstract

The angular momentum A per unit horizontal distance of a train of periodic, progressive surface waves is a well-defined quantity, independent of the horizontal position of the origin of moment.

The Lagrangian-mean angular momentum $\overline{A}_L$ consists of two parts, arising from the orbital motion and from the Stokes drift respectively. Together these contribute a positive sum, nearly proportional to the energy density (when the origin is taken in the mean surface level). If moments are taken about some point P not at the mean surface level, the angular momentum will differ by an amount proportional to the elevation of P. There is just one elevation for which the Lagrangian-mean angular momentum about P vanishes. This elevation is called the level of action. For infinitesimal waves in deep water the level of action is at a height above the mean surface equal to ½k, that is ¼π times the wavelength.

Just as for ordinary fluid velocities, the Lagrangian-mean angular momentum $\overline{A}_L$ differs from the Eulerian-mean $\overline{A}_L$, the latter being zero to second order. The difference between $\overline{A}_L$ and $\overline{A}_E$ is associated with the displacement of the lateral boundaries of any given mass of fluid.

For waves of finite amplitude, an initially rectangular mass of fluid becomes ultimately quite distorted by the Stokes drift. Nevertheless it is possible to define a long-time average l.t.$\overline{A}$ and to calculate its numerical value accurately in waves of finite amplitude. In low waves, l.t.$\overline{A}$ is equal to $\overline{A}_L$. Defining the level of action ya in the general case as l.t.$\overline{A}/I$, where I is the linear momentum, we find that ya rises from 0·5k−1 for infinitesimal waves to about 0·6k−1 for steep waves. Thus ya is about the same as the height yx of the wave crests above the mean level in limiting waves, a fact which may account for why steep irrotational waves can support whitecaps in a quasi-steady state. The same argument suggests that Gerstner waves (in which the particle orbits are theoretically circular) could not support whitecaps.