A numerical study of superfluid turbulence in the self-induction approximation.

*(English)*Zbl 0639.76136Two stable numerical methods are presented to solve the self-induction equations of vortex theory. These numerical methods are validated by comparison with known exact solutions. A new self-similar solution of the self-induction equation is presented and the approximate solutions are shown to converge to the exact solution for the self-similar solution. The numerical method is then generalized to solve the equations of motion of a superfluid vortex in the self-induction approximation where reconnection is allowed. A careful numerical study shows that the mesh spacing of the method must be restricted so that the approximate solutions are accurate. The line length density of a system of superfluid vortices is calculated. Contrary to earlier results it is found that the line length density produced does not scale as the velocity squared and therefore is not characteristic of homogeneous turbulence. It is concluded that the model equation used is inadequate to describe superfluid turbulence.

##### MSC:

76Y05 | Quantum hydrodynamics and relativistic hydrodynamics |

81V99 | Applications of quantum theory to specific physical systems |

76M99 | Basic methods in fluid mechanics |

76F99 | Turbulence |

##### Keywords:

stable numerical methods; self-induction equations of vortex theory; exact solutions; self-similar solution; approximate solutions; superfluid vortex; self-induction approximation; superfluid turbulence
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\textit{T. F. Buttke}, J. Comput. Phys. 76, No. 2, 301--326 (1988; Zbl 0639.76136)

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