EAS471 -- Lecture Log -- 2014

• Tues 8 April. Finish windbreak presentation. Inverse dispersion to deduce ground-air emission rate, implemented as MO-bLS in WindTrax, or as 3D-bLS (bLS: the atmospheric dispersion model invoked for inverse dispersion is a backward Lagrangian stochastic model).

• Thurs 3 April. Continue lecture of previous class (windbreak flows).

• Tues 1 April. Modelling disturbed micrometeorological flows (local advection; windbreak flows). Continue exercise: tuning an eddy viscosity closure to match a "reference flow".

• Thurs 27 March. Course evaluation. Downscaling a reanalysis using the WRF-EMS model: a period of extreme cold in New Zealand.

• Tues 25 March. An alternative K closure for the ABL. Overview of NCEP's Numerical Weather Prediction (NWP) models NAM/WRF & GFS

• Thurs 20 March. Numerical schemes for the heat/diffusion equation: Crank-Nicholson, Adams-Bashfort, Dufort-Frankel, Richardson. Delage's classic model of the nocturnal boundary layer (NBL or SBL), including a discussion of turbulence energetics from the perspective of the TKE equation and variance budget equations.

• Tues 18 March. Parameterization of clouds & precip, including some background on the dynamics/physics separation in NWP and GCM models. Not covered: Fritsch & Chappell scheme.

• 11 & 13 March. Simulating dispersion in the CBL: introduction to the convective boundary layer (CBL); a well-mixed 1D LS model for skew turbulence; the turbulent kinetic energy k equation; energetics of turbulence; Thomson's well-mixed 2D model for horizontally-homogeneous Gaussian turbulence; depostion velocity (its definition, and how it can be determined) and its relationship to particle reflection probability at the ground; simulating heavy particle trajectories.

• 6 March. Continuing on the theory underpinning modern LS models (if interested in a history: Thomson & Wilson, 2012). Equivalence of the Lagrangian stochastic (or Monte-Carlo, or ensemble of stochastic realizations) description of particle dispersion, and the deterministic statistical description in terms of the evolution of joint probability density function p(z,w,t). Representation of particle state in the z-w phase space, diagrammatically and in terms of a formula, e.g. in case position and velocity are both exactly known (z=z₁, w=w₁) at time t₁: a point in phase space corresponding to

  p(z,w,t₁) = δ(z, z₁) δ(w, w₁)

  (the Dirac δ-function having the properties of being infinitely narrow with unit area, i.e. non-zero only in an infinitesimal region around z=z₁, for instance). The transition density fuction for the 1D stochastic model dZ = W dt, dW = a(Z,W) dt + b dξ, a product of a delta-function for position change and a Gaussian for the velocity change:

  p(z,w,t+Δt | z₀, w₀,t) = δ(z-z₀, w₀ dt) * [(2 π)^1/2 b² dt]^-1 * exp [- {w - w₀ - a(z₀, w₀) dt}² / {2 b² dt}].

  Fokker-Planck equation (which was derived from the Chapman-Kolmogorov equation by physicists of the early 1900s) for the time evolution of the joint PDF for velocity and position p(z,w,t), and its interpretation in terms of (the divergences of) advective and diffusive fluxes of probability in the z-w space.

  Principle of non-decreasing entropy (once in a state of maximum disorder - well mixed - a natural system does not become more ordered). Thomson's (1987) "well-mixed constraint" for Lagrangian stochastic models. Meaning of "Gaussian turbulence." The unique well-mixed 1D LS model for vertically-inhomogeneous Gaussian turbulence. Its simplication for homogeneous turbulence (applicable to a stationary, horizontally-homogeneous, neutrally-stratified ASL), viz. the Langevin equation.

• 4 March. Theory of "first-order" Lagrangian stochastic (LS) trajectory models. Justifying dc/dt=0 (high Peclet number). Link between trajectories and mean concentration in a sampling volume (mean residence time). Markovian state variable (Z,W) where W=dZ/dt. Most general algorithm dW = a(Z,W) dt + b(Z,W) dξ. Consistency with the Kolmogorov conjecture determines b=(C₀ ε)^1/2, where ε is the turbulent kinetic energy dissipation rate. Phase space view of the process (z-w space) and the joint PDF for velocity and position p(z,w,t). Chapman-Kolmogorov equation for evolution of p(z,w,t) over discrete steps in time. The transition density function p(z,w,t+Δt | z₀, w₀,t)

• 27 Feb. Back to the subject of turbulent mixing, but henceforth with searching treatments that admit (vertical) inhomogeneity -- and (thus) apply more naturally (if not necessarily rigorously) to the turbulent atmospheric boundary layer. The drunkard's walk (also known as the Random Displacement Model and as the 0th-order Lagrangian stochastic model) can be adaped to vertically-inhomogeneous turbulence, viz.

  dZ = (∂K/∂z) dt + (2 K dt)^0.5 r

  where r ∈ N[0,1]. This is a useful model for many purposes. Before proceeding to the 1st-order LS model, we covered two (related) simple Eulerian dispersion models that are (still) commonly applied for air pollution modelling: we systematically derived the Gaussian puff and Gaussian plume models from the Reynolds-averaged mass conservation equation (here is a recent article on performance of several regulatory models, including CALPUFF).

• 25 Feb. Galerkin methods. Spectral method for solution of the 1D, non-linear advection equation.

• 13 Feb. Midterm exam.

• 11 Feb. Buckingham Pi theorem. Method of indices. Applications of dimensional analysis: period of a pendulum; terminal velocity of a falling sphere; profile of mean potential temperature in the horizontally homogeneous atmospheric surface layer (hh_ASL). Scaling parameters for the hh_ASL (u*, θ*, L).

• 6 Feb. Analytic solution of the heat equation using Fourier transform. Fourier transform pair. The Laplacian as a wavelength-selective filter. Elliptic problems, and their solution using relaxation method (illustrated for 2D heat eqn., setup for scored assignment 2).

• 4 Feb. Overview of the micrometeorological theory (and a numerical model) of winds over gently-sloping hills, and comparison with measurements at Lacombe, Alberta (a current project of JDW).

• 30 Jan. Introduction to spectral decomposition/representation: a field or curve θ(x) is represented by a (possibly infinite) sum (superposition) of "modes" (if the modes are sines and cosines, we have a Fourier series representation). Aliasing (example) and Shannon's sampling theorem. Non-linear Computational Instability, illustrated by discretizing the 1D non-linear advection equation and examining the evolution of a single mode over a single time step. Role of the curvature operator (Laplacian), which typically appears naturally in the governing equations and may be artifically boosted by cranking up the viscosity, as a scale-selective filter -- damping the spurious high wavenumber energy generated by the non-linear (wave-wave) interactions.

• 28 Jan. Continuing quick sketch of the micrometeorology of the hh_ASL. Turbulent (eddy) momentum fluxes and the friction velocity. Eddy heat flux (or to use all the helpful adjectives: the time-averaged turbulent convective vertical flux density of sensible heat); eddy vapour flux. "Kinematic" fluxes. Eddy flux convergence across the atmospheric boundary layer (ABL). The ASL as (roughly) a "constant flux layer." Monin-Obukhov theory of the hh_ASL. Wind and diffusivity profiles in the ASL. A simplified advection-diffusion equation for dispersion in the ASL. Control volume discretization of the latter.

• 23 Jan. (Tridiagonal) coefficient matrix C for an implicit treatment of the heat eqn. with only four gridpoints (i.e. two boundary points and two internal points). Inversion of the matrix "by hand" (please complete the steps yourelves; then compare with the solution given by the following Matlab TDMA function). Note added 7 Feb: the instructor may have wrongly placed a 0 in the 4th row of the matrix on the rhs, which ought to have been [0 1 2 2]. The solution matrix should be [0 -26/29 -40/29 18/29]. The linear 1D advection equation (for constant velocity U; its exact (sybolic) solution in terms of the initial state; a conditionally stable "leapfrog scheme" (3 time-level discretization) of it (deduced by way of another "intuitive" stability analysis: we shall not cover the formal Von Neuman method). The CFL (Courant-Friedrichs-Lewy) condition. Began some background material on the micrometeorology of the horizontally-homogeneous atmospheric surface layer ("hh_ASL"), intended to provide context for the first scored computing assignment (to be posted very soon). Focused on the prevailing idealizations or approximations, the statistics of most interest (mean velocities, velocity standard deviations, velocity covariances...), and the fact that mean vertical fluxes (e.g. the mean heat flux Q_H) are in essence "covariances" between fluctuating vertical velocity and the flucuation in the transported quantity.

• 21 Jan. Classification of errors arising in numerical methods: Truncation error in relation to consistency (compatibility) of a difference equation with the differential eqn. it represents. Discretization error. Stability error. Lax equivalence theorem. "Explicit" vs. "implicit" algorithms, illustrated w.r.t. 1D heat equation. Informal stability analysis of the algorithm for computational lab 2: a limitation on the magnitude of the "diffusion number" (conditionally unstable scheme). Richardson's (unconditionally unstable) leapfrog (3-time level) scheme. Crank-Nicholson scheme for the heat/diffusion eqn. as an example of an implicit scheme (this algorithm, as it turns out, being unconditionally stable). Matrix formulation of the latter, CT=D, where C is a square matrix of coefficients, T is the solution vector for the unknown temperature at time "n+1" and D, which is "known", involves the solution at the earlier time "n"... then formally, T=C^-1D.

• 16 Jan. Begin material on "formalities of numerical methods in fluid mechanics." Setup for Eulerian derivation (in 3D) of conservation eqn for arbitrary property whose quantity per unit volume is θ (detailed steps same as earlier derivation in 1D, so omitted). Coordinate-independent representation of this conservation eqn. using the grad operator (∇). Classification of terms (storage, transport & source terms). Specification of the flux vector (diffusive, convective and radiative transport). Lagrangian time derivative. Continuity eqn (for air density ρ) expressed alternatively in flux (transport) form, advection form and Lagrangian form. Non-divergent ("incompressible") flows (∇ ⋅ u=0). Heat equation (θ → ρ c T) treating density and specific heat as constant. Laplacian operator (∇²=∇ ⋅ ∇), known also as the "diffusion" operator and as "curvature operator" (slope of slope). Discretization using finite differences, illustrated for 1D heat equation (truncated Taylor series). Forward, central and backward "computational molecules" for the time derivative. "Explicit" algorithm for the 1D heat equation, using forward difference in time (order Δt) and central difference in space (order Δx²).

• 14 Jan. G.I. Taylor's (1921) analytical Lagrangian theory for dispersion in stationary, homogeneous turbulence (p81, Turbulence in the Atmosphere, Wyngaard). Transition density function for particle position p(z,t|0,0) and its relation to mean square displacement. [Time permitting, we'll return to Lagrangian modelling of particle dispersion later in the term, to look at the modern numerical generalization of Taylor's approach].

• 9 Jan. Continuation on modelling of dispersion in homogeneous turbulence. Analytic solution for the mean square displacement of the drunkard as function of time (or number of steps), i.e. outcome of the Drunkard's Walk. Derivation of a mass conservation equation in one space dimension; convective and diffusive mass fluxes; simplication to conditions paralleling the drunkard's walk, i.e. diffusion only; resulting (1D) diffusion equation. Mean square displacement as function of time according to this diffusion equation.

• 7 Jan. Course organization. Familiarization with computing environment. Began developing "drunkard's walk" algorithm for spread of particles released independently at the origin into a field of stationary, homogeneous, unbounded, two-dimensional turbulence. We covered some terminology and notation, and touched on Reynolds averaging.

Last Modified: 8 Apr., 2014