Studies of the Jet in BL Lacertae. II. Superluminal Alfv\'en Waves

We study the kinematics of ridge lines on the pc-scale jet of the active galactic nucleus BL Lac. We show that the ridge lines display transverse patterns that move superluminally downstream, and that the moving patterns are analogous to waves on a whip. Their apparent speeds $\beta_\mathrm{app}$ (units of $c$) range from 3.9 to 13.5, corresponding to $\beta_\mathrm{wave}^\mathrm{gal}= 0.981 - 0.998$ in the galaxy frame. We show that the magnetic field in the jet is well-ordered with a strong transverse component, and assume that it is helical and that the transverse patterns are Alfv\'en waves propagating downstream on the longitudinal component of the magnetic field. The wave-induced transverse speed of the jet is non-relativistic ($\beta_\mathrm{tr}^\mathrm{gal} \lesssim 0.09$). In 2010 the wave activity subsided and the jet then displayed a mild wiggle that had a complex oscillatory behaviour. The Alfv\'en waves appear to be excited by changes in the position angle of the recollimation shock, in analogy to exciting a wave on a whip by shaking the handle. A simple model of the system with plasma sound speed $\beta_\mathrm{s}=0.3$ and apparent speed of a slow MHD wave $\beta_\mathrm{app,S}=4$ yields Lorentz factor of the beam $\Gamma_\mathrm{beam} \sim 4.5$, pitch angle of the helix (in the beam frame) $\alpha\sim 67^\circ$, Alfv\'en speed $\beta_\mathrm{A}\sim 0.64$, and magnetosonic Mach number $M_\mathrm{ms}\sim 4.7$. This describes a plasma in which the magnetic field is dominant and in a rather tight helix, and Alfv\'en waves are responsible for the moving transverse patterns.