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We present an analytical study of fluctuations of the Vortex Line Density(VLD) $<\delta \mathcal{L}(\omega) \delta \mathcal{L}(-\omega)>$ in turbulentflows of quantum fluids. Two cases are considered. The first one is thecounterflowing (Vinen) turbulence, where the vortex lines are disordered, andthe evolution of quantity $\mathcal{L}(t)$ obeys the Vinen equation. The secondcase is the quasi-classic turbulence, where vortex lines are believed to formthe so called vortex bundles, and their dynamics is described by the HVBKequations. The latter case, is of a special interest, since a number of recentexperiments demonstrate the $\omega ^{-5/3}$ dependence for spectrum VLD,instead of $\omega ^{1/3}$ law, typical for spectrum of vorticity. Innonstationary situation, in particular, in the fluctuating turbulent flow thereis a retardation between the instantaneous value of the normal velocity and thequantity $\mathcal{L}$. This retardation tends to decrease in the accordancewith the inner dynamics, which has a relaxation character. In both cases therelaxation dynamics of VLD is related to fluctuations of the relative velocity,however if for the Vinen case the rate of temporal change for $\mathcal{L}(t)$is directly depends on $\delta \mathbf{v}_{ns}$, for the HVBK dynamics itdepends on $\nabla \times \delta \mathbf{v}_{ns}$. As a result, for thedisordered case the spectrum $<\delta \mathcal{L}(\omega) \delta\mathcal{L}(-\omega)>$ coincides with the spectrum $\omega ^{-5/3} $. In thecase of the bundle arrangement, the spectrum of the VLD varies (at differenttemperatures) from $\omega ^{1/3}$ to $\omega ^{-5/3}$ dependencies. Thisconclusion may serve as a basis for the experimental determination of what kindof the turbulence is implemented in different types of generation.

Fluctuations of the vortex line density in turbulent flows of quantum fluids