The radial temperature differences at which the transitions from one wave number to the next occur have been measured with either increasing or decreasing positive radial temperature gradients, at five different rotation rates, with the fluid being always in thermal equilibrium and being in contact with an upper rigid lid. Hysteresis has been observed in all wave number transitions, and also in the transition to upper symmetry. There are, nevertheless, regions in the stability diagram where the wave number is unique. There is an excluded region where the wave number four cannot be obtained through quasi-steady procedure. There is a reversal of the sense of the hysteresis of the transitions. At low #x394;T, a wave number transition with increasing radial temperature difference occurs at a higher #x394;T, than the same transition with decreasing temperature difference. On the other hand, at large values of #x394;T, a wave number transition with increasing radial temperature difference occurs at a lower #x394;T, than the same transition with decreasing temperature difference. Wave number transitions with increasing #x394;T, occur spontaneously out of amplitude oscillations. Wave number transitions with decreasing #x394;T, occur via slow wave splitting in association with phase modulations of the waves. The uniqueness of the wave number in the unique areas of the stability diagram has been confirmed by sudden start experiments.
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