An integrated approach to mixing sensitivities in tropospheric chemistry: a basis for the parameterization of subgrid-scale emissions for chemistry transport models

The net effect on the global atmosphere of a continuous isolated chemical source is considered under idealized conditions. A general framework is described that allows M i , the steady state global perturbation to the ith species due to the source, to be calculated. This is achieved by exploiting the fact that once the emissions are sufficiently dilute, far from the source, they decay with the timescales of the chemical eigenstates of the background atmosphere. Both M i and the level of excitation of the longer-lived eigenstates are shown to depend on the details of the mixing of emissions near the source. If the details of the dilution of the emissions plume are known, it is also shown that “equivalent emissions” can be calculated. Equivalent emissions are designed so that when diluted instantaneously into the background atmosphere they result in the same global perturbation to each species as the original slowly diluted plume. The framework is then applied to test the sensitivity to mixing of a simple O3-NO x -CO-HO x tropospheric chemistry system. M i is calculated for a NO-CO source of constant strength as the mixing scenario undergone by the emissions is varied. The global increase in O3 due to the source is found to increase with the rate at which emissions are mixed, whereas the global increase in CO is reduced. The equivalent emissions for each plume dilution mechanism are then calculated. In a simple plume box model it is shown that the equilibrium state obtained when the model is forced by emissions that are first diluted in entraining plumes can be reproduced in a standard box model (i.e., with instantaneous mixing of emissions) by the corresponding equivalent emissions. It is argued that the concept of equivalent emissions can be exploited straightforwardly to derive a parameterization of unresolved subgrid plumes in order to reduce systematic error in global models

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

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

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

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

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

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

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

Backflow Correlations for the Electron Gas and Metallic Hydrogen

We justify and evaluate backflow-threebody wavefunctions for a two component system of electrons and protons. Based on the generalized Feynman-Kacs formula, many-body perturbation theory, and band structure calculations, we analyze the use and the analytical form of the backflow function from different points of view. The resulting wavefunctions are used in Variational and Diffusion Monte Carlo calculations of the electron gas and of solid and liquid metallic hydrogen. For the electron gas, the purely analytic backflow and three-body form gives lower energies than those of previous calculations. For bcc hydrogen, analytical and optimized backflow-threebody wavefunctions lead to energies nearly as low as those from using LDA orbitals in the trial wavefunction. However, compared to wavefunctions constructed from density functional solutions, backflow wavefunctions have the advantage of only few parameters to estimate, the ability to include easily and accurately electron-electron correlations, and that they can be directly generalized from the crystal to a disordered liquid of protons.Comment: 16 pages, 6 figure