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We investigate the interplay of phase mixing and the nonlinear turbulentcascade in the evolution and dissipation of Alfv\'en waves using compressiblemagnetohydrodynamics numerical simulations. We consider perturbations in theform of torsional waves, both propagating and standing, or turbulentfluctuations, or a combination of the two. The main purpose is to study howphase mixing and nonlinear couplings jointly work to produce small scales indifferent regimes.  We conduct a numerical campaign to explore the typical parameters as the looplength, the amplitude and spatial profile of the perturbations, and thedissipative coefficients. A pseudo-spectral code is employed to solve thethree-dimensional compressible magnetohydrodynamic equations, modeling theevolution of perturbations propagating in a flux tube corresponding to anequilibrium configuration with cylindrical symmetry.  We find that phase mixing takes place for moderate amplitudes of theturbulent component even in a distorted, non-axisymmetric configuration,building small scales that are locally transverse to the density gradient. Thedissipative time decreases with increasing the percentage of the turbulentcomponent. This behavior is verified both for propagating and standing waves.Even in the fully turbulent case, a mechanism qualitatively similar to phasemixing occurs: it actively generates small scales together with the nonlinearcascade, thus providing the shortest dissipative time. General considerationsare given to identify this regime in the parameter space. The turbulentperturbation also distorts the background density, locally increasing theAlfv\'en velocity gradient and further contributing to accelerating theformation of small scales.

Joint action of phase mixing and nonlinear effects in MHD waves propagating in coronal loops