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We numerically simulated quantum turbulence in superfluid $^4$He toinvestigate the emission of vortex rings from a localized vortex tangle.Turbulence is characterized by some universal statistical laws. Although thereare a lot of studies on statistical laws in bulk quantum turbulence, studies ininhomogeneous or localized turbulence is scarce. We first investigate thestatistical laws of localized quantum turbulence, referring to two statisticallaws deduced from the vibrating wire experiments in [Yano $et$ $al.$, J. LowTemp. Phys. $\bf{196}$, $184\ (2019)$]. The first law is the Poisson processfor the detection of vortex rings; the vortex tangle emits vortex rings withfrequencies depending on their sizes. The second law is the power law betweenthe frequency and the size of the emitted vortex rings, showing theself-similarity of the tangle. To study these statistical laws numerically, wedeveloped a system similar to experiments. First, we generate a localizedstatistically steady vortex tangle by injecting vortex rings from two oppositesides and causing collisions. We investigated the conditions that aid theformation of the tangles and the anisotropy of the emission of vortex ringsfrom the tangle. Second, from the data on emitted rings, we reconstruct the twostatistical laws. Results from our numerical investigations are consistent withthe known self-similarity of emitted vortex rings and localized tangles.

Statistical laws and self-similarity of vortex rings emitted from a localized vortex tangle in superfluid ${}^4$He