Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, United Kingdom.
Phys Rev E. 2016 Mar;93(3):032218. doi: 10.1103/PhysRevE.93.032218. Epub 2016 Mar 22.
We demonstrate the existence of localized structures along quantized vortex filaments in superfluid helium under the quantum form of the local induction approximation (LIA), which includes mutual friction and normal fluid effects. For small magnitude normal fluid velocities, the dynamics are dissipative under mutual friction. On the other hand, when normal fluid velocities are sufficiently large, we observe parametric amplification of the localized disturbances along quantized vortex filaments, akin to the Donnelly-Glaberson instability for regular Kelvin waves. As the waves amplify they will eventually cause breakdown of the LIA assumption (and perhaps the vortex filament itself), and we derive a characteristic time for which this breakdown occurs under our model. More complicated localized waves are shown to occur, and we study these asymptotically and through numerical simulations. Such solutions still exhibit parametric amplification for large enough normal fluid velocities, although this amplification may be less uniform than would be seen for more regular filaments such as those corresponding to helical curves. We find that large rotational velocities or large wave speeds of nonlinear waves along the filaments will result in more regular and stable structures, while small rotational velocities and wave speeds will permit far less regular dynamics.
我们证明了在超流氦中，在量子局部感应近似（LIA）的形式下，存在沿着量子涡旋丝的局部结构，其中包括了相互摩擦和正常流体效应。对于小幅度的正常流体速度，相互摩擦下的动力学是耗散的。另一方面，当正常流体速度足够大时，我们观察到沿着量子涡旋丝的局部扰动的参数放大，类似于正则开尔文波的唐纳利-格拉伯森不稳定性。随着波的放大，它们最终会导致 LIA 假设的破坏（也许还有涡旋丝本身），我们根据我们的模型推导出了这种破坏发生的特征时间。我们还展示了更复杂的局部波的发生，并通过数值模拟对其进行了研究。对于足够大的正常流体速度，这些解仍然表现出参数放大，尽管这种放大可能不如更规则的纤维（例如对应于螺旋曲线的纤维）那么均匀。我们发现，纤维上的大旋转速度或非线性波的大波速将导致更规则和稳定的结构，而小的旋转速度和波速将允许极不规则的动力学。
