Effective degrees of nonlinearity in a family of generalized models of two-dimensional turbulence

We study the small-scale behavior of generalized two-dimensional turbulence governed by a family of model equations, in which the active scalar $\theta=(-\Delta)^{\alpha/2}\psi$ is advected by the incompressible flow $\u=(-\psi_y,\psi_x)$. The dynamics of this family are characterized by the material conservation of $\theta$, whose variance is preferentially transferred to high wave numbers. As this transfer proceeds to ever-smaller scales, the gradient $\nabla\theta$ grows without bound. This growth is due to the stretching term $(\nabla\theta\cdot\nabla)\u$ whose ``effective degree of nonlinearity'' differs from one member of the family to another. This degree depends on the relation between the advecting flow $\u$ and the active scalar $\theta$ and is wide ranging, from approximately linear to highly superlinear. Linear dynamics are realized when $\nabla\u$ is a quantity of no smaller scales than $\theta$, so that it is insensitive to the direct transfer of the variance of $\theta$, which is nearly passively advected. This case corresponds to $\alpha\ge2$, for which the growth of $\nabla\theta$ is approximately exponential in time and non-accelerated. For $\alpha<2$, superlinear dynamics are realized as the direct transfer of entails a growth in \nabla\u, thereby enhancing the production of $\nabla\theta$. This superlinearity reaches the familiar quadratic nonlinearity of three-dimensional turbulence at $\alpha=1$ and surpasses that for $\alpha<1$. The usual vorticity equation ($\alpha=2$) is the border line, where \nabla\u and $\theta$ are of the same scale, separating the linear and nonlinear regimes of the small-scale dynamics. We discuss these regimes in detail, with an emphasis on the locality of the direct transfer.Comment: 6 journal pages, to appear in Physical Review

The limits of β-plane turbulence

The quasigeostrophic shallow-water system on the mid-latitude β plane with weak, small-scale turbulent forcing is explored in the limit of large energy. Forcing is weak in the sense that the energy input rate relative to the energy of the flow is very small, of the order of 10−5–10−10, and the potential vorticity assumes an approximate staircase structure. The flow has large energy in the sense that the jet spacing is equal to the domain width so that no further jet mergers can occur. Quasi-stationary numerical experiments, in which the energy grows linearly, reveal late-time quasi-steady, translating solutions comprising a single jet and vortex dipole, with details of the jet-vortex configuration depending on the deformation radius. At a smaller deformation radius the jet may traverse the entire domain in the y direction one or more times, giving a jet orientation that is predominantly north–south, rather than the usual east–west orientation characteristic of β-plane jets at lower energy. In these meandering cases, a mode number is proposed that quantifies the degree of meandering relative to the vortices. Besides the steadily translating solutions, topological changes in the jet-vortex structure are identified that occur via a transient interaction of a meandering jet with a vortex. At high energy, these give rise to apparently periodic solutions of the system; at low energy, before a single, domain-wide jet is established, they indicate that jet merger may occur through more complicated processes than the simple merging of neighbouring jets.Publisher PDFPeer reviewe

Vortical control of forced two-dimensional turbulence

A new numerical technique for the simulation of forced two-dimensional turbulence (Dritschel and Fontane, 2010) is used to examine the validity of Kraichnan-Batchelor scaling laws at higher Reynolds number than previously accessible with classical pseudo-spectral methods,making use of large simulation ensembles to allow a detailed consideration of the inverse cascade in a quasi-steady state. Our results support the recent finding of Scott (2007), namely that when a direct enstrophy cascading range is well-represented numerically, a steeper energy spectrum proportional to k^(−2) is obtained in place of the classical k^(−5/3) prediction. It is further shown that this steep spectrum is associated with a faster growth of energy at large scales, scaling like t^(−1) rather than Kraichnan’s prediction of t^(−3/2). The deviation from Kraichnan’s theory is related to the emergence of a population of vortices that dominate the distribution of energy across scales, and whose number density and vorticity distribution with respect to vortex area are related to the shape of the enstrophy spectrum. An analytical model is proposed which closely matches the numerical spectra between the large scales and the forcing scale

The role of planetary waves in the tropospheric jet response to stratospheric cooling

K.L.S. is funded in part by a Natural Sciences and Engineering Council of Canada Postdoctoral Fellowship. R.K.S. acknowledges support from the National Science Foundation.An idealized general circulation model is used to assess the importance of planetary-scale waves in determining the position of the tropospheric jet, specifically its tendency to shift poleward as winter stratospheric cooling is increased. Full model integrations are compared against integrations in which planetary waves are truncated in the zonal direction, and only synoptic-scale waves are retained. Two series of truncated integrations are considered, using (i) a modified radiative equilibrium temperature or (ii) a nudged-bias correction technique. Both produce tropospheric climatologies that are similar to the full model when stratospheric cooling is weak. When stratospheric cooling is increased, the results indicate that the interaction between planetary- and synoptic-scale waves plays an important role in determining the structure of the tropospheric mean flow and rule out the possibility that the jet shift occurs purely as a response to changes in the planetary- or synoptic-scale wave fields alone.Publisher PDFPeer reviewe

Internal interannual variability of the winter polar vortex in a simple model of the seasonally evolving stratosphere

We investigate persistent low-frequency variability of the stratospheric winter polar vortex in a rotating spherical shallow-water model under the action of topographic wave-forcing and radiative cooling to a simple time-varying equilibrium state representative of the seasonal cycle in solar heating. A range of modes of variability is obtained, dependent on wave forcing amplitude and characterized by the distribution of quiescent and disturbed winters, defined as winters in which the vortex is either close to radiative equilibrium, with low planetary wave amplitude, or else strongly disturbed from equilibrium by the wave forcing. At low forcing amplitude the vortex is typically quiescent every year, while at higher amplitude it is typically disturbed; in both cases there is little year-to-year variation of the vortex state. For a range of intermediate forcing amplitudes, however, the vortex transitions between quiescent and disturbed states from one winter to the next with a persistent and well-defined pattern of variability. To investigate the extent to which the low-frequency variability found here may be explained in terms of a low-latitude flywheel mechanism, we conduct additional experiments varying a linear drag on the zonal mean flow in the tropics and find that sufficiently strong drag can completely suppress the variability. The robustness of the variability is demonstrated by further experiments using a modified radiative equilibrium profile, associated with a tropical westerly flow: similar variability is obtained but the modified profile is less effective at constraining the tropical flow from a persistent easterly acceleration.PostprintPeer reviewe

A moist-thermal quasigeostrophic model for monsoon depressions

Funding: AKC is supported by a St Leonard’s College Interdisciplinary Doctoral Scholarship awarded by the University of St Andrews.Monsoon depressions (MDs) are synoptic-scale storms that occur during the summer phase of the global monsoon cycle and whose dynamical mechanisms remain incompletely understood. To gain insight into the dynamics governing the large-scale structure of MDs, we formulate an idealised moist-thermal quasi-geostrophic model that includes distinct thermal and moisture fields in simple forms. A linear-stability analysis of the model, with basic states corresponding to typical monsoon conditions, shows three distinct mode classifications: thermal-Rossby modes, heavy precipitating modes, and a moist-thermal mode. In the linearised model, the presence of a background precipitation gradient strengthens thermal-Rossby modes by coupling the dynamics to latent heating. The separation of heavy precipitating modes from fast-propagating thermal-Rossby modes is further examined with numerical experiments of large-amplitude MDs. Wind-induced evaporation is found to amplify large-amplitude MDs in conditions analogous to those over the northern Bay of Bengal. An energetic analysis shows the pathways by which the MDs derive energy from the background state. A further series of experiments through a continuum of meridional temperature gradients demonstrates the sensitivity of large-scale MD dynamics to the background state and suggests a possible mechanism to explain variations in the propagation direction of MDs.Peer reviewe

Gang members are entangled in a web of violence that leads the gunman of today to become the victim of tomorrow

While the media often portrays a stark line between the victims of crime and offenders the reality is much more blurred. New research from David Pyrooz, Richard K. Moule, and Scott H. Decker find that this is especially the case for gang members who find that they are twice as likely to be both victims and offenders as non-gang members. They argue that gang membership is a large risk factor in this victim-offender overlap, as single acts of violence between gang members often lead to acts of retribution between gangs as a whole