A method for estimating the extent of denitrification of Arctic polar vortex air from tracer-tracer scatter plots

A method for estimating the extent of denitrification of Arctic polar vortex air is proposed. Previous estimates of denitrification using tracer-tracer scatter plots have not allowed for mixing-induced changes in tracer-tracer relationships in a sufficiently general way. This difficulty is overcome by constructing an artificial "reference tracer'' from a linear combination of other long-lived tracers. The reference tracer is designed so that, as far as possible, it has a linear canonical relationship with NOy in midlatitudes. A linear relationship is unaffected by mixing, so denitrification is apparent as deviations of vortex measurements from the linear midlatitude relationship. The method is first demonstrated using data from a chemical transport model in which no denitrification processes are present. It is then applied to balloon, aircraft and shuttle-borne measurements made before and during the breakdown of the Arctic vortex in 1992-1993 and 1996-1997. In each case the method indicates that little or no denitrification had occurred in any of the vortex air encountered. When the method is applied to the southern hemisphere vortex in 1994, by contrast, denitrified air is clearly seen to be present around 19-23 km in the vortex

Adaptive stochastic trajectory modelling in the chaotic advection regime

Motivated by the goal of improving and augmenting stochastic Lagrangian models of particle dispersion in turbulent flows, techniques from the theory of stochastic processes are applied to a model transport problem. The aim is to find an efficient and accurate method to calculate the total tracer transport between a source and a receptor when the flow between the two locations is weak, rendering direct stochastic Lagrangian simulation prohibitively expensive. Importance sampling methods that combine information from stochastic forward and back trajectory calculations are proposed. The unifying feature of the new methods is that they are developed using the observation that a perfect strategy should distribute trajectories in proportion to the product of the forward and adjoint solutions of the transport problem, a quantity here termed the ‘density of trajectories’ D(x,t). Two such methods are applied to a ‘hard’ model problem, in which the prescribed kinematic flow is in the large-Péclet-number chaotic advection regime, and the transport problem requires simulation of a complex distribution of well-separated trajectories. The first, Milstein’s measure transformation method, involves adding an artificial velocity to the trajectory equation and simultaneously correcting for the weighting given to each particle under the new flow. It is found that, although a ‘perfect’ artificial velocity v∗ exists, which is shown to distribute the trajectories according to D, small errors in numerical estimates of v∗ cumulatively lead to difficulties with the method. A second method is Grassberger’s ‘go-with-the-winners’ branching process, where trajectories found unlikely to contribute to the net transport (losers) are periodically removed, while those expected to contribute significantly (winners) are split. The main challenge of implementation, which is finding an algorithm to select the winners and losers, is solved by a choice that explicitly forces the distribution towards a numerical estimate of D generated from a previous back trajectory calculation. The result is a robust and easily implemented algorithm with typical variance up to three orders of magnitude lower than the direct approach

Non-dispersive and weakly dispersive single-layer flow over an axisymmetric obstacle: the equivalent aerofoil formulation

Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Gamma)M-5/3, where Gamma = (F - 1)M-2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the 'equivalent aerofoil' specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M less than or similar to 0.4 and vertical bar Gamma vertical bar less than or similar to 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev-Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.For Gamma greater than or similar to 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be c(d)M(2)/(F root F-2 - 1), where c(d) is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Kortewegde Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of A = 3M/(2 delta(2) root F-2 - 1), where delta is the ratio of the undisturbed layer depth to the radial scale of the obstacle

Equilibrium energy spectrum of point vortex motion with remarks on ensemble choice and ergodicity

The dynamics and statistical mechanics of N chaotically evolving point vortices in the doubly periodic domain are revisited. The selection of the correct microcanonical ensemble for the system is first investigated. The numerical results of Weiss and McWilliams [Phys. Fluids A 3, 835 (1991)], who argued that the point vortex system with N=6 is nonergodic because of an apparent discrepancy between ensemble averages and dynamical time averages, are shown to be due to an incorrect ensemble definition. When the correct microcanonical ensemble is sampled, accounting for the vortex momentum constraint, time averages obtained from direct numerical simulation agree with ensemble averages within the sampling error of each calculation, i.e., there is no numerical evidence for nonergodicity. Further, in the N→∞ limit it is shown that the vortex momentum no longer constrains the long-time dynamics and therefore that the correct microcanonical ensemble for statistical mechanics is that associated with the entire constant energy hypersurface in phase space. Next, a recently developed technique is used to generate an explicit formula for the density of states function for the system, including for arbitrary distributions of vortex circulations. Exact formulas for the equilibrium energy spectrum, and for the probability density function of the energy in each Fourier mode, are then obtained. Results are compared with a series of direct numerical simulations with N=50 and excellent agreement is found, confirming the relevance of the results for interpretation of quantum and classical two-dimensional turbulence

Unique stochastic Lagrangian particle dispersion models for inhomogeneous turbulent flows

It is well established that Lagrangian particle dispersion models, for inhomogeneous turbulent flows, must satisfy the ‘well-mixed condition’ of Thomson (J. Fluid Mech., vol. 180, 1987, pp. 529–556) in order to produce physically reasonable results. In more than one dimension, however, the well-mixed condition is not sufficient to define the dispersion model uniquely. The non-uniqueness, which is related to the rotational degrees of freedom of particle trajectories, permits models with trajectory curvatures and velocity autocorrelation functions which are clearly unphysical. A spin condition is therefore introduced to constrain the models. It requires an ensemble of particles with fixed initial position and velocity to have, at short times, expected angular momentum, measured relative to the mean position and velocity of an ensemble of fluid particles with initially random velocity, equal to the relative angular momentum of the mean flow at the ensemble mean location. The resulting unique model is found explicitly for the canonical example of inhomogeneous Gaussian turbulence and is characterised by accelerations which are exponential in the particle velocity. A simpler unique model with a quadratic acceleration is obtained using a weaker version of the spin condition. Unlike previous models, the unique models defined by the spin condition lead to particles having the correct (ensemble mean) angular speed in a turbulent flow in solid-body rotation. The properties of the new models are discussed in the settings of a turbulent channel flow and an idealised turbulent atmospheric boundary-layer flow

Transcritical rotating flow over topography

The flow of a one-and-a-half layer fluid over a three-dimensional obstacle of non-dimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a non-dimensional parameter B (inverse Burger number). The transcritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is close to unity is considered in the shallow-water (small-aspect-ratio) limit. For weakly rotating flow over a small isolated obstacle (M -> 0) a similarity theory is developed in which the behaviour is shown to depend on the parameters Gamma = (F - 1)M-2/3 and nu = (BM-1/3)-M-1/2. The flow pattern in this regime is determined by a nonlinear equation in which Gamma and nu appear explicitly, termed here the 'rotating transcritical small-disturbance equation' (rTSD equation, following the analogy with compressible gasdynamics). The rTSD equation is forced by 'equivalent aerofoil' boundary conditions specific to each obstacle. Several qualitatively new flow behaviours are exhibited, and the parameter reduction afforded by the theory allows a (Gamma, nu) regime diagram describing these behaviours to be constructed numerically. One important result is that, in a supercritical oncoming flow in the presence of sufficient rotation (nu greater than or similar to 2), hydraulic jumps can appear downstream of the obstacle even in the absence of an upstream jump. Rotation is found to have the general effect of increasing the amplitude of any existing downstream hydraulic jumps and reducing the lateral extent and amplitude of upstream jumps. Numerical results are compared with results from a shock-capturing shallow-water model, and the (Gamma, nu) regime diagram is found to give good qualitative and quantitative predictions of flow patterns at finite obstacle height (at least for M less than or similar to 0.4). Results are compared and contrasted with those for a two-dimensional obstacle or ridge, for which rotation also causes hydraulic jumps to form downstream of the obstacle and acts to attenuate upstream jumps

Universal statistics of point vortex turbulence

A new methodology, based on the central limit theorem, is applied to describe the statistical mechanics of two-dimensional point vortex motion in a bounded container D, as the number of vortices N tends to infinity. The key to the approach is the identification of the normal modes of the system with the eigenfunction solutions of the so-called hydrodynamic eigenvalue problem of the Laplacian in D. The statistics of the projection of the vorticity distribution onto these eigenfunctions (‘vorticity projections’) are then investigated. The statistics are used first to obtain the density-of-states function and caloric curve for the system, generalising previous results to arbitrary (neutral) distributions of vortex circulations. Explicit expressions are then obtained for the microcanonical (i.e. fixed energy) probability density functions of the vorticity projections in a form that can be compared directly with direct numerical simulations of the dynamics. The energy spectra of the resulting flows are predicted analytically. Ensembles of simulations with N=100, in several conformal domains, are used to make a comprehensive validation of the theory, with good agreement found across a broad range of energies. The probability density function of the leading vorticity projection is of particular interest because it has a unimodal distribution at low energy and a bimodal distribution at high energy. This behaviour is indicative of a phase transition, known as Onsager–Kraichnan condensation in the literature, between low-energy states with no mean flow in the domain and high-energy states with a coherent mean flow. The critical temperature for the phase transition, which depends on the shape but not the size of D, and the associated critical energy are found. Finally the accuracy and the extent of the validity of the theory, at finite N, are explored using a Markov chain phase-space sampling method

Quantitative evaluation of numerical integration schemes for Lagrangian particle dispersion models

A rigorous methodology for the evaluation of integration schemes for Lagrangian particle dispersion models (LPDMs) is presented. A series of one-dimensional test problems are introduced, for which the Fokker–Planck equation is solved numerically using a finite-difference discretisation in physical space and a Hermite function expansion in velocity space. Numerical convergence errors in the Fokker–Planck equation solutions are shown to be much less than the statistical error associated with a practical-sized ensemble (N = 106) of LPDM solutions; hence, the former can be used to validate the latter. The test problems are then used to evaluate commonly used LPDM integration schemes. The results allow for optimal time-step selection for each scheme, given a required level of accuracy. The following recommendations are made for use in operational models. First, if computational constraints require the use of moderate to long time steps, it is more accurate to solve the random displacement model approximation to the LPDM rather than use existing schemes designed for long time steps. Second, useful gains in numerical accuracy can be obtained, at moderate additional computational cost, by using the relatively simple “small-noise” scheme of Honeycutt

Dynamical Elliptical Diagnostics of the Antarctic Polar Vortex

Elliptical diagnostics provide dynamical and climatological information about the behavior of the Arctic and Antarctic stratospheric polar vortices. Here Kida’s model, describing the evolution of a uniform vortex in a linear, but possibly unsteady, background flow, is used to interpret the observed evolution of the Antarctic vortex in late winter during 1999–2018. Kida’s model has oscillatory solutions that can undergo an amplitude bifurcation, which serves as a simple model for the onset of vortex-splitting stratospheric sudden warmings (SSWs). A data assimilation method is used to find solutions of Kida’s equations consistent with the observations. A phase-plane analysis reveals large interannual variability in the amplitude of oscillations of the vortex. In 2002, the year of the only observed vortex-splitting Antarctic SSW, the system is found to cross a separatrix in phase space, associated with the SSW amplitude bifurcation, in late September. An output of the data assimilation is the linear background flow experienced by the vortex. The rotational component of this linear flow is consistent with the vortex being embedded in an anticyclonic background. The time-mean strain flow is weak but has a clear orientation, consistent with the presence of stationary forcing due to planetary-scale topography and land–sea contrast. The time-varying strain flow is comparatively large in magnitude, illustrating the relative importance of the planetary-scale component of the turbulent dynamics occurring at tropopause level. Unlike in the Northern Hemisphere, therefore, the direction of future Antarctic vortex splits will not necessarily align with the direction of the 2002 split

Wave propagation in rotating shallow water in the presence of small-scale topography

The question of how finite-amplitude, small-scale topography affects small-amplitude motions in the ocean is addressed in the framework of the rotating shallow water equations. The extent to which the dispersion relations of Poincaré, Kelvin and Rossby waves are modified in the presence of topography is illuminated, using a range of numerical and analytical techniques based on the method of homogenisation. Both random and regular periodic arrays of topography are considered, with the special case of regular cylinders studied in detail, because this case allows for highly accurate analytical results. The results show that, for waves in a β-channel bounded by sidewalls, and for steep topographies outside of the quasi-geostrophic regime, topography acts to slow Poincaré waves slightly, Rossby waves are slowed significantly and Kelvin waves are accelerated for long waves and slowed for short waves, with the two regimes being separated by a narrow band of resonant wavelengths. The resonant band, which is due to the excitation of trapped topographic Rossby waves on each seamount, may affect any of the three wave types under the right conditions, and for physically reasonable results requires regularisation by Ekman friction. At larger topographic amplitudes, for cylindrical topography, a simple and accurate formula is given for the correction to the Rossby wave dispersion relation, which extends previous results for the quasi-geostrophic regime