The Craik–Criminale class of exact solutions is examined for a nonlinear-reactive fluids theory that includes a family of turbulence closure models. These may be formally regarded as either large eddy simulation or Reynolds-averaged Navier–Stokes models of turbulence. All of the turbulence closure models in the class under investigation preserve the existence of elliptic instability, although they shift its angle of critical stability as a function of the rotation rate Ω of the coordinate system, the wave number β of the Kelvin wave, and the model parameter α, the turbulence correlation length. Elliptic instability allows a comparison among the properties of these models. It is emphasized that the physical mechanism for this instability is not wave–wave interaction, but rather wave, mean-flow interaction as governed by the choice of a model’s nonlinearity.