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Steady State of Counterflow Quantum Turbulence: Vortex filament Simulation with the Full Biot-Savart Law

We perform a numerical simulation of quantum turbulence produced by thermal counterflow in superfluid $^4$He by using the vortex filament model with the full Biot--Savart law. The pioneering work of Schwarz has two shortcomings: it neglects the non-local terms of the Biot--Savart integral (known as the localized induction approximation, LIA) and it employs an unphysical mixing procedure to sustain the statistically steady state of turbulence. For the first time we have succeeded in generating the statistically steady state under periodic boundary conditions without using the LIA or the mixing procedure. This state exhibits the characteristic relation $L=γ^2 v_{ns}^2$ between the line-length density $L$ and the counterflow relative velocity $v_{ns}$ and there is quantitative agreement between the coefficient $γ$ and some measured values. The parameter $γ$ and some anisotropy parameters are calculated as functions of temperature and the counterflow relative velocity. The numerical results obtained using the full Biot--Savart law are compared with those obtained using the LIA. The LIA calculation constructs a layered structure of vortices and does not proceed to a turbulent state but rather to another anisotropic vortex state; thus, the LIA is not suitable for simulations of turbulence.