Varve-like parallel laminated and homogeneous muds on natural levees of submarine channels result from the spill-over of the dilute upper parts of channelized turbidity currents. On the levees of the Northwest Atlantic Mid-Ocean Channel, Labrador Sea, laminated turbidite muds show a systematic upward thickness variation of the granular silt laminae and the clay laminae. In many depositional units the former decrease logarithmically upwards in thickness while the latter increase logarithmically. The origin of parallel lamination in current-deposited, fine-grained sediments is interpreted in terms of the ‘burst-and-sweep model’ for the viscous sublayer of turbulent flows. Individual silt laminae of laminated turbidite muds form due to shear sorting during burst-and-sweep events which have a winnowing effect on the clay fraction. In contrast to sand-transporting flows the viscous sublayer of silt- and clay-carrying dilute turbidity currents cannot be permanently in a turbulent state. The presence of clay laminae suggests that the boundary layer relaminarizes periodically. It is speculated that this is caused by pressure waves with negative pressure gradients favouring the suppression of bursts and sweeps. The observed vertical thickness variations suggest that the periods of laminar boundary-layer flow increase in duration as turbulence of the main current decays. The upward thickness decrease of the silt laminae is explained in terms of decreasing supply rates to the boundary layer from the main flow. A multiple bursting-cycle mechanism is favored over a single bursting cycle mechanism for the formation of individual silt laminae by shear sorting.