Thin layers of plankton: Formation by shear and death by diffusion

We show that a steady vertically-sheared current can produce a thin layer of plankton by differentially advecting an initial patch whose vertical and horizontal dimensions are H 0 and L 0 , respectively. Our model treats the plankton as an inert passive tracer with vertical diffusivity κ v and subject to a vertically-sheared horizontal current with shear α . After a transient of duration L 0 / α H 0 the vertical thickness H of the patch decreases with H ( t ) ≈ L 0 / α t . This shear-driven thinning is halted by diffusion at a time of order α - 2 / 3 κ v - 1 / 3 L 0 2 / 3 , and at this time the layer achieves a minimum layer thickness of order α - 1 / 3 κ v 1 / 3 L 0 1 / 3 . For typical oceanic parameters, such as κ v ∼ 10 - 5 m 2 s - 1 , α ∼ 10 - 2 s - 1 , and L 0 ∼ 1000 m the initial transient is about 3 h and the layer achieves a minimum thickness of order 1 m in a time of order 1 day. During the shear thinning the intensity of the layer decreases by a factor of 3 - 1 / 2 ≈ 0.58 , which means that the intensity of the thin layer is comparable to the intensity of the patch from which it was formed. Subsequently the layer thickens and its intensity decreases; the coup de grace is delivered by shear dispersion at a time of order H 0 2 / κ v . The lifetime of the thin layer, defined by the condition that the maximum concentration is comparable to the initial maximum concentration, is the same order as the time to achieve minimum thickness. Additionally, analysis of a nutrient–phytoplankton model shows that phytoplankton growing in a sheared patch of nutrients can result in a layer of phytoplankton that develops as an initially thin feature.