On vortex waves in compressible fluids. I. The hollow–core vortex

Paul H. Roberts


In 1880, Kelvin published his analysis of small–amplitude waves carried by a straightline vortex in an incompressible, inviscid fluid. The most significant of these waves are the ‘bending modes’, in which the axis of the vortex becomes helical. The corresponding angular wavenumber, m, is 1, but Kelvin found solutions for all m and all axial wavenumbers k. For the hollow–core vortex, the model also studied here, he found two modes for each k and m. For ka ≪ 1, where a is the core radius, the waves are of two types. The majority are ‘fast’, with frequencies ω of the order of κ/a 2, where κ is the vortex circulation. The axisymmetric (m = 0) modes and one bending mode are ‘slow’; for these, ω is of the order of κk 2, apart from a logarithmic factor.

The effects of compressibility on Kelvin7apos;s results are studied here. It is found that ω for the fast waves may be complex. Such ‘free modes’ radiate sound away from the vortex. For waves of sufficiently short wavelength, ω is, as in Kelvin's case, real; these may be considered to be ‘bound states’. The borderline between the free and bound modes is analysed and additional compressive modes are located. It is shown that the slow free modes are weakly unstable.

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