Multiple Jets as PV Staircases: The Phillips Effect and the Resilience of Eddy-Transport Barriers
Authors: Dritschel D. G. , McIntyre M. E. 
Published: March 2008

Abstract
A review is given that focuses on why the sideways mixing of potential vorticity (PV) across its background gradient tends to be inhomogeneous, arguably a reason why persistent jets are commonplace in planetary atmospheres and oceans, and why such jets tend to sharpen themselves when disturbed. PV mixing often produces a sideways layering or banding of the PV distribution and therefore a corresponding number of jets, as dictated by PV inversion. There is a positive feedback in which mixing weakens the “Rossby wave elasticity” associated with the sideways PV gradients, facilitating further mixing. A partial analogy is drawn with the Phillips effect, the spontaneous layering of a stably stratified fluid, in which vertically homogeneous stirring produces vertically inhomogeneous mixing of the background buoyancy gradient. The Phillips effect has been extensively studied and has been clearly demonstrated in laboratory experiments. However, the “eddy-transport barriers” and sharp jets characteristic of extreme PV inhomogeneity, associated with strong PV mixing and strong sideways layering into Jupiter-like “PV staircases,” with sharp PV contrasts Δqbarrier, say, involve two additional factors besides the Rossby wave elasticity concentrated at the barriers. The first is shear straining by the colocated eastward jets. PV inversion implies that the jets are an essential, not an incidental, part of the barrier structure. The shear straining increases the barriers’ resilience and amplifies the positive feedback. The second is the role of the accompanying radiation-stress field, which mediates the angular-momentum changes associated with PV mixing and points to a new paradigm for Jupiter, in which the radiation stress is excited not by baroclinic instability but by internal convective eddies nudging the Taylor–Proudman roots of the jets.
Some examples of the shear-straining effects for strongly nonlinear disturbances are presented, helping to explain the observed resilience of eddy-transport barriers in the Jovian and terrestrial atmospheres. The main focus is on the important case where the nonlinear disturbances are vortices with core sizes ∼LD, the Rossby (deformation) length. Then a nonlinear shear-straining mechanism that seems significant for barrier resilience is the shear-induced disruption of vortex pairs. A sufficiently strong vortex pair, with PV anomalies ±Δqvortex, such that Δqvortex ≫ Δqbarrier, can of course punch through the barrier. There is a threshold for substantial penetration through the barrier, related to thresholds for vortex merging. Substantial penetration requires Δqvortex ≳ Δqbarrier, with an accuracy or fuzziness of order 10% when core size ∼LD, in a shallow-water quasigeostrophic model. It is speculated that, radiation stress permitting, the barrier-penetration threshold regulates jet spacing in a staircase situation. For instance, if a staircase is already established by stirring and if the stirring is increased to produce Δqvortex values well above threshold, then the staircase steps will be widened (for given background PV gradient β) until the barriers hold firm again, with Δqbarrier increased to match the new threshold. With the strongest-vortex core size ∼LD this argument predicts a jet spacing 2b = Δqbarrier/β ∼ L2Rh (Uvortex)/LD in order of magnitude, where LRh(Uvortex) = (Uvortex/β)1/2, the Rhines scale based on the peak vortex velocity Uvortex, when 2b ≳ LD. The resulting jet speeds Ujet are of the same order as Uvortex; thus also 2b ∼ L2Rh(Ujet)/LD. Weakly inhomogeneous turbulence theory is inapplicable here because there is no scale separation between jets and vortices, both having scales ∼LD in this situation.