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Counterflow instability and turbulence in a spin-1 spinor Bose--Einstein condensate

We theoretically study counterflow instability and turbulence in a spin-1 spinor Bose--Einstein condensate by the Gross--Pitaevskii equation and the Bogoliubov--de Gennes equation. Our study considers (i) the dynamics induced by the counterflow of two components with different magnetic quantum numbers, which leads to turbulence with spin degrees of freedom, and (ii) the properties of the turbulence. For (i), the behavior of the condensate induced by the counterflow strongly depends on whether the spin-dependent interaction is ferromagnetic or antiferromagnetic, leading to different behaviors for the dispersion relation and the spin density vector, $etc$. For (ii), we numerically calculate the spectrum of the spin-dependent interaction energy, which also depends on the spin-dependent interaction. The spectrum of the spin-dependent interaction energy in the ferromagnetic case exhibits a -7/3 power law, whereas that in the antiferromagnetic case does not. The -7/3 power law can be explained by scaling analysis.