Irrational Conformal Field Theory

This is a review of irrational conformal field theory, which includes rational conformal field theory as a small subspace. Central topics of the review include the Virasoro master equation, its solutions and the dynamics of irrational conformal field theory. Discussion of the dynamics includes the generalized Knizhnik-Zamolodchikov equations on the sphere, the corresponding heat-like systems on the torus and the generic world- sheet action of irrational conformal field theory.Comment: 195 pages, Latex, 12 figures, to appear in Physics Reports. Typos corrected in Sections 13 and 14, and a footnote added in Section 1

Mechanisms underlying transient growth of planar perturbations in unbounded compressible shear flow

Non-modal mechanisms underlying transient growth of propagating acoustic waves and non-propagating vorticity perturbations in an unbounded compressible shear flow are investigated, making use of closed form solutions. Propagating acoustic waves amplify mainly due to two mechanisms: growth due to advection of streamwise velocity that is typically termed as the lift-up mechanism leading for large Mach numbers to an almost linear increase in streamwise velocity with time and growth due to the downgradient irrotational component of the Reynolds stress leading to linear growth of acoustic wave energy for large times. Synergy between these mechanisms along with the downgradient solenoidal component of the Reynolds stress produces large and robust energy amplification. On the other hand, non-propagating vorticity perturbations amplify due to kinematic deformation of vorticity by the mean flow. For weakly compressible flows, an initial vorticity perturbation abruptly excites acoustic waves with exponentially small amplitude, and the energy gained by vorticity perturbations is transferred back to the mean flow. For moderate Mach numbers, a strong coupling between vorticity perturbations and acoustic waves is found with the energy gained by vorticity perturbations being transferred to acoustic waves that are abruptly excited by the vortex. Calculation of the optimal perturbations for a viscous flow shows that for low Mach numbers, acoustic wave excitation by vorticity perturbations and the subsequent growth of acoustic waves leads to robust energy growth of the order of Reynolds number, while for large Mach numbers, synergy between the lift-up mechanism and the downgradient solenoidal component of the Reynolds stress dominates the growth and leads to a comparable large amplification of streamwise velocity. © 2009 Cambridge University Press

Is spontaneous generation of coherent baroclinic flows possible?

Geophysical turbulence is observed to self-organize into large-scale flows such as zonal jets and coherent vortices. Previous studies of barotropic-plane turbulence have shown that coherent flows emerge from a background of homogeneous turbulence as a bifurcation when the turbulence intensity increases. The emergence of large-scale flows has been attributed to a new type of collective, symmetry-breaking instability of the statistical state dynamics of the turbulent flow. In this work, we extend the analysis to stratified flows and investigate turbulent self-organization in a two-layer fluid without any imposed mean north-south thermal gradient and with turbulence supported by an external random stirring. We use a second-order closure of the statistical state dynamics, that is termed S3T, with an appropriate averaging ansatz that allows the identification of statistical turbulent equilibria and their structural stability. The bifurcation of the statistically homogeneous equilibrium state to inhomogeneous equilibrium states comprising zonal jets and/or large-scale waves when the energy input rate of the excitation passes a critical threshold is analytically studied. Our theory predicts that there is a large bias towards the emergence of barotropic flows. If the scale of excitation is of the order of (or larger than) the deformation radius, the large-scale structures are barotropic. Mixed barotropic-baroclinic states with jets and/or waves arise when the excitation is at scales shorter than the deformation radius with the baroclinic component of the flow being subdominant for low energy input rates and insignificant for higher energy input rates. The predictions of the S3T theory are compared with nonlinear simulations. The theory is found to accurately predict both the critical transition parameters and the scales of the emergent structures but underestimates their amplitude. © 2019 Cambridge University Press

Structural stability theory of two-dimensional fluid flow under stochastic forcing

Large-scale mean flows often emerge in turbulent fluids. In this work, we formulate a stability theory, the stochastic structural stability theory (SSST), for the emergence of jets under external random excitation. We analytically investigate the structural stability of a two-dimensional homogeneous fluid enclosed in a channel and subjected to homogeneous random forcing. We show that two generic competing mechanisms control the instability that gives rise to the emergence of an infinitesimal jet: advection of the eddy vorticity by the mean flow that is shown to be jet forming and advection of the vorticity gradient of the jet by the eddies that is shown to hinder the formation of the mean flow. We show that stochastic forcing with small streamwise coherence and an amplitude larger than a certain threshold leads to the emergence of jets in the channel through a bifurcation of the non-linear SSST system. © Cambridge University Press 2011

Modal and nonmodal growths of inviscid planar perturbations in shear flows with a free surface

Shear flows with a free surface possess diverse branches of modal instabilities. By approximating the mean flow with a piecewise linear profile, an understanding and classification of the instabilities can be achieved by studying the interaction of the edge waves that arise at the density discontinuity at the surface and the vorticity waves that are supported at the mean vorticity gradient discontinuities in the interior. The various branches of instability are identified and their physical origin is clarified. The edge waves giving rise to the modal instabilities can also lead to a modest transient growth that extends into the regions of neutrality of the flow. However, when the continuous spectrum is excited substantial transient growth can arise and the optimal perturbations attain greater energy when compared with the energy of the fastest modal growing perturbation. These optimal perturbations utilize the continuous spectrum to excite at large amplitude the neutral or amplifying modes of the system. © 2009 American Institute of Physics

A theory for the emergence of coherent structures in beta-plane turbulence

Planetary turbulent flows are observed to self-organize into large-scale structures such as zonal jets and coherent vortices. One of the simplest models of planetary turbulence is obtained by considering a barotropic flow on a beta-plane channel with turbulence sustained by random stirring. Nonlinear integrations of this model show that as the energy input rate of the forcing is increased, the homogeneity of the flow is broken with the emergence of non-zonal, coherent, westward propagating structures and at larger energy input rates by the emergence of zonal jets. We study the emergence of non-zonal coherent structures using a non-equilibrium statistical theory, stochastic structural stability theory (S3T, previously referred to as SSST). S3T directly models a second-order approximation to the statistical mean turbulent state and allows the identification of statistical turbulent equilibria and study of their stability. Using S3T, the bifurcation properties of the homogeneous state in barotropic beta-plane turbulence are determined. Analytic expressions for the zonal and non-zonal large-scale coherent flows that emerge as a result of structural instability are obtained. Through numerical integrations of the S3T dynamical system, it is found that the unstable structures equilibrate at finite amplitude. Numerical simulations of the nonlinear equations confirm the characteristics (scale, amplitude and phase speed) of the structures predicted by S3T. © © 2014 Cambridge University Press

Gravity waves in a horizontal shear flow. Part I: Growth mechanisms in the absence of potential vorticity perturbations

Interaction of internal gravity waves with a horizontal shear flow in the absence of potential vorticity perturbations is investigated making use of closed-form solutions. Localized wave packet trajectories are obtained, the energy growth mechanisms occurring are identified, and the potential role of perturbation growth in wave breaking is assessed. Regarding meridional propagation, the wave packet motion is limited by turning levels where the waves are reflected and trapping levels where the waves stagnate. Regarding perturbation energy amplification, two growth mechanisms can be distinguished: growth due to advection of zonal velocity and growth due to downgradient Reynolds stresses. The three-dimensional perturbations producing optimal energy growth reveal that these two mechanisms produce large and robust amplification of zonal velocity and/or density and vertical velocity, potentially leading to shear or convective instability. For large static stability, amplification of density perturbations in conjunction with vertical orientation of the constant phase lines close to the trapping level potentially leads to a convective collapse of the wave packet near the trapping level, in agreement with existing direct numerical simulation studies. For lower static stability and for waves with phase lines oriented horizontally, growth due to advection of zonal velocity dominates, leading to rapid growth of streamwise streaks within the localized wave packet and potentially to shear instability. © 2009 American Meteorological Society

Gravity waves in a horizontal shear flow. Part II: Interaction between gravity waves and potential vorticity perturbations

Interaction among potential vorticity perturbations and propagating internal gravity waves in a horizontally sheared zonal flow is investigated. In the strong stratification limit, an initial vorticity perturbation weakly excites two propagating gravity waves while the density component of the potential vorticity perturbation is significantly amplified, potentially leading to convective collapse. If stratification is sufficiently weak, a strong coupling between vorticity perturbations and gravity waves is found and spontaneous gravity wave generation occurs. This coupling can be traced to the nonnormal interaction between the potential vorticity and gravity wave manifolds in the weak stratification limit. Vorticity perturbations amplify in energy due to downgradient Reynolds stress when their phase lines tilt against the shear and the large growth attained is transferred to propagating gravity waves. When the flow geometry is such that the excited gravity waves are confined in the vicinity of the vorticity perturbation by their trapping levels, an overall convective collapse of this region can be anticipated. On the other hand, when the flow geometry permits wave propagation, significant gravity wave emission occurs. © 2009 American Meteorological Society

Momentum and energy transport by gravity waves in stochastically driven stratified flows. Part I: Radiation of gravity waves from a shear layer

In this paper, the emission of internal gravity waves from a local westerly shear layer is studied. Thermal and/or vorticity forcing of the shear layer with a wide range of frequencies and scales can lead to strong emission of gravity waves in the region exterior to the shear layer. The shear flow not only passively filters and refracts the emitted wave spectrum, but also actively participates in the gravity wave emission in conjunction with the distributed forcing. This interaction leads to enhanced radiated momentum fluxes but more importantly to enhanced gravity wave energy fluxes. This enhanced emission power can be traced to the nonnormal growth of the perturbations in the shear region, that is, to the transfer of the kinetic energy of the mean shear flow to the emitted gravity waves. The emitted wave energy flux increases with shear and can become as large as 30 times greater than the corresponding flux emitted in the absence of a localized shear region. Waves that have horizontal wavelengths larger than the depth of the shear layer radiate easterly momentum away, whereas the shorter waves are trapped in the shear region and deposit their momentum at their critical levels. The observed spectrum, as well as the physical mechanisms influencing the spectrum such as wave interference and Doppler shifting effects, is discussed. While for large Richardson numbers there is equipartition of momentum among a wide range of frequencies, most of the energy is found to be carried by waves having vertical wavelengths in a narrow band around the value of twice the depth of the region. It is shown that the waves that are emitted from the shear region have vertical wavelengths of the size of the shear region. © 2007 American Meteorological Society

On the mechanism underlying the spontaneous emergence of barotropic zonal jets

Zonal jets are commonly observed to spontaneously emerge in a b-plane channel from a background of turbulence that is sustained in a statistical steady state by homogeneous stochastic excitation and dissipation of vorticity. The mechanism for jet formation is examined in this work within the statistical wave-mean flow interaction framework of stochastic structural stability theory (SSST) that makes predictions for the emergence of zonal jets in b-plane turbulence. Using the coupled dynamical SSST system that governs the joint evolution of the second-order statistics and the mean flow, the structural stability of the spatially homogeneous statistical equilibrium with no mean zonal jets is studied. It is shown that close to the structural stability boundary, the eddy-mean flow dynamics can be split into two competing processes. The first, which is shearing of the eddies by the local shear described by Orr dynamics in a b plane, is shown in the limit of infinitesimal shear to lead to the formation of jets. The second, which is momentum flux divergence resulting from lateral wave propagation on the nonuniform local mean vorticity gradient, is shown to oppose jet formation. The upgradient momentum fluxes due to shearing of the eddies are shown to act exactly as negative viscosity for an anisotropic forcing and as negative hyperviscosity for isotropic forcing. The downgradient fluxes due to wave flux divergence are shown to act hyperdiffusively. © 2013 American Meteorological Society