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The principles of restricted superposition of circularly polarizedarbitrary-amplitude waves for several hydrodynamic type models are illustratedsystematically with helical representation in a unified sense. It is shown thatthe only general modes satisfying arbitrary-amplitude superposition to kill thegeneric nonlinearity are the mono-wavelength homochiral Beltrami mode and theone-dimensional-two-component stratified vorticity mode, which we call the XYzflow/wave; while, there are other special superposition principles for somespecific cases. We try to remark on the possible connections with the geo-and/or astro-physical fluid and magnetohydrodynamic turbulence issues, such asthe rotating turbulence, dynamo and solar atmosphere turbulence, especiallywith the introduction of disorder locally frozen in some (randomly distributed)space-time regions. Recent disagreements about exact solutions of Hall andfully two-fluid magnetohydrodynamics are also settled down by such a treatment.This work complements, by studying the modes which completely kill the triadicinteractions or the nonlinearities, previous studies on the thermalizationpurely from the triadic interactions, and in turn offers alternativeperspectives of the nonlinearities.

On the exact solutions of (magneto)hydrodynamic systems and the superposition principles of nonlinear helical waves