2 results
On inertial waves and oscillations in a rapidly rotating spheroid
- KEKE ZHANG, XINHAO LIAO, PAUL EARNSHAW
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- Journal:
- Journal of Fluid Mechanics / Volume 504 / 10 April 2004
- Published online by Cambridge University Press:
- 16 April 2004, pp. 1-40
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The problem of fluid motions in the form of inertial waves or inertial oscillations in an incompressible viscous fluid contained in a rotating spheroidal cavity was first formulated and studied by Poincaré (1885) and Bryan (1889). Upon realizing the limitation of Bryan's implicit solution using complicated modified spheroidal coordinates, Kudlick (1966) proposed a procedure that may be used to compute an explicit solution in spheroidal coordinates. However, the procedure requires an analytical expression for the $N$ real and distinct roots of a polynomial of degree $N$, where $N$ is a key parameter in the problem. When $0\,{\le}\,N\,{\le}\,2$, an explicit solution can be derived by using Kudlick's procedure. When $3\,{\le}\,N\,{\le}\,4$, the procedure cannot be practically used because the analytical expression for the $N$ distinct roots becomes too complicated. When $N\,{>}\,4$, Kudlick's procedure cannot be used because of the non-existence of an analytical expression for the $N$ distinct roots. For the inertial wave problem, Kudlick thus restricted his analysis to several modes for $1\,{\le}\,N\,{\le}\,2$ with the azimuthal wavenumbers $1\,{\le}\,m\,{\le}\,2$. We have found the first explicit general analytical solution of this classical problem valid for $0\,{\le}\,N\,{<}\,\infty$ and $0\,{\le}\,m\,{<}\,\infty$. The explicit general solution in spheroidal polar coordinates represents a possibly complete set of the inertial modes in an oblate spheroid of arbitrary eccentricity. The problem is solved by a perturbation analysis. In the first approximation, the effect of viscosity on inertial waves or oscillations is neglected and the corresponding inviscid solution, the pressure and the three velocity components in explicit spheroidal coordinates, is obtained. In the next approximation, the effect of viscous dissipation on the inviscid solution is examined. We have derived the first explicit general solution for the viscous spheroidal boundary layer valid for all inertial modes. The boundary-layer flux provides the solvability condition that is required to solve the higher-order interior problem, leading to an explicit general expression for the viscous correction of all inertial modes in a rapidly rotating, general spheroidal cavity. On the basis of the general explicit solution, some unusual and intriguing properties of the spheroidal inertial waves or oscillation are discovered. In particular, we are able to show that $\int_V ( {\bm u} \,{\bm \cdot}\, \nabla^2 {\bm u} )\,{\rm d}V \equiv 0$, where ${\bm u}$ is the velocity of any three-dimensional inviscid inertial waves or oscillations in an oblate spheroid of arbitrary eccentricity and $\int_V$ denotes three-dimensional integration over the volume of the spheroidal cavity.
On inertial waves in a rotating fluid sphere
- KEKE ZHANG, PAUL EARNSHAW, XINHAO LIAO, F. H. BUSSE
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- Journal:
- Journal of Fluid Mechanics / Volume 437 / 25 June 2001
- Published online by Cambridge University Press:
- 22 June 2001, pp. 103-119
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Several new results are obtained for the classical problem of inertial waves in a rotating fluid sphere which was formulated by Poincaré more than a century ago. Explicit general analytical expressions for solutions of the problem are found in a rotating sphere for the first time. It is also discovered that there exists a special class of three-dimensional inertial waves that are nearly geostrophic and always travel slowly in the prograde direction. On the basis of the explicit general expression we are able to show that the internal viscous dissipation of all the inertial waves vanishes identically for a rotating fluid sphere. The result contrasts with the finite values obtained for the internal viscous dissipation for all other cases in which inertial waves have been studied.