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We discuss the energy and vorticity spectra of turbulent superfluid $^4$He inall the temperature range from $T=0$ up to the phase transition "$\lambda$point", $T_\lambda\simeq 2.17\,$K. Contrary to classical developed turbulencein which there are only two typical scales, i.e. the energy injection $L$ andthe dissipation scales $\eta$, here the quantization of vorticity introducestwo additional scales, i.e the vortex core radius $a_0$ and the mean vortexspacing $\ell$. We present these spectra for the super- and normal-fluidcomponents in the entire range of scales from $L$ to $a_0$ including thecross-over scale $\ell$ where the hydrodynamic eddy-cascade is replaced by thecascade of Kelvin waves on individual vortices. At this scale a bottleneckaccumulation of the energy was found earlier at $T=0$.  We show that even very small mutual friction dramatically suppresses thebottleneck effect due to the dissipation of the Kelvin waves. Using our resultsfor the spectra we estimate the Vinen "effective viscosity" $\nu'$ in theentire temperature range and show agreement with numerous experimentalobservation for $\nu'(T)$.

Energy and Vorticity Spectra in Turbulent Superfluid $^4$He from $T=0$ to $T_\lambda$