EAS471 -- Lecture Log -- 2016

• Thurs 7 April. Varied applications of the Lagrangian stochastic trajectory model on the micrometeorological scale: examples covering both horizontally-homogeneous flow and disturbed flows.

• Tues 5 April. Lecture on wind and turbulence in the plant canopy layer.

• Thurs 31 Mar. Micrometeorological methods to determine ground-air exchange, focusing on an "inverse dispersion" method that combines measurements with a Lagrangian stochastic atmospheric transport model.

• Tues 29 Mar. Parameterization of clouds and precip in NWP models. Q&A for Assignment 3. Exercise: surface layer TKE budget.

• Thurs 24 Mar. Continuation of Tuesday's lecture. Specific ASL flows covered included "local advection" due to change in surface properties; and "windbreak flows". (Several slides that were not specifically covered during the lectures have been cut.)

• Tues 22 Mar. Lecture on disturbed surface layer flows, and approaches to their modelling. (Continued 24 March.)

• Thurs 17 Mar. Delage's model of the development of the nocturnal boundary layer (NBL), typifying a classical approach in NWP.

• Tues 15 Mar. Lecture conveyed an overview of the horizontally-homogeneous ABL and ASL; further elements of MO Similarity Theory (MOST); and introduced the turbulent kinetic energy (TKE) budget.

• Thurs 10 Mar. Lecture continued on subject of modern LS dispersion models, culminating with the well-mixed model for vertically-inhomogeneous Gaussian turbulence. Second part of lecture began a refresher on the horiz.-homogeneous ASL (background for final assignment), including an exercise illustrating dimensional analysis and leading to a length scale that, to within a factor -1/k_v, is the Obukhov length.

• Tues 8 Mar. Lecture: exercise on dimensional analysis; implications of G.I. Taylor's Lagrangian theory of turbulent dispersion; background on modern Lagrangian stochastic (LS) particle trajectory models. Exercises:

  • Take-home: represent the daily mean temperatures at YEG for February 2016 by way of a histogram (of relative frequencies, F_i=N_i/29, where N_i is the count in the i^th bin). Use eight "bins", each with a width of 2^oC, to cover the range -12 to +4^oC. Divide your F_i by the bin width, to obtain the empirical PDF, p_i. Plot this alongside the Gaussian distribution having the same mean and standard deviation as the (YEG February 2016) observations.

  • Done in class (instructor's result above): dimensional analysis. Assuming a unidirectional, laminar flow (of a fluid with density ρ and kinematic viscosity ν) in a pipe of radius R such that the velocity vector u(r) has only a single component u=u(r), find the form of the law u=u(r). Assume the flow is "driven" by a uniform (constant) pressure gradient ∂p/∂x.

• Thurs 3 Mar. Lecture developed Sir G.I. Taylor's Lagrangian theory of dispersion in stationary homogeneous turbulence, with an aside on the subject of probability density functions. The "far field eddy diffusivity" is defined by statistics of the Lagrangian velocity, i.e. the product σ_w² τ of velocity variance with the Lagrangian integral timescale.

• Tues 1 Mar. Lecture slot spent working on assignment 2.

• Thurs 25 Feb. Lecture gave an overview of the Eulerian approach to turbulent dispersion, illustrating a typical sequence of simplifications to the mass conservation equation, leading to the closure problem, and after the briefest allusion to higher-order closure pursuing first order (or eddy diffusion, or K-theory, or flux-gradient) closure which (in the case that the eddy K's are constant) leads to the Gaussian plume model. Then a switch back to the Lagrangian approach. Link between statistics of displacement from release position, and the mean concentration field.

• Tues 23 Feb. Lecture included a further exercise on dimensional analysis (drag on a porous windbreak), followed by topics related to the second scored lab (steady-state conduction in the plane): character of elliptic PDEs (Laplace's eqn, Poisson's eqn), and the "relaxation method" for their solution.

• Tues 9 Feb. Lecture on applications of "superposition-of-waves representation." (i) Action of the Laplacian as a lowpass filter. (ii) governing equation for a fluid thermometer (with Ohm's Law model for heat exchange), and: radiation error; device time constant and step response; transfer function, sinusoidal response, power gain. (iii) Aliasing, in relation to (iv) Non-Linear Computational Instability.

  Testing the stability of explicit schemes for the 1D linear advection equation, one finds:

  • Euler scheme, i.e. forward in time and central in space - conditionally stable
  • Central in time (leapfrog) and central in space - conditionally stable [100 time steps with C=0.5; and 38 time steps with C=1.5].
  • Forward in time and downwind in space - unconditionally unstable

• Thurs 4 Feb. Lecture on the generalities of representing a curve (or field) as a superposition of waves. Laplace transform. Fourier series. Fourier transform. Wave-wave interaction, and its operation in the non-linear 1D advection equation.

• Tues 2 Feb. Lecture: Algorithms for the 1D linear advection equation: CFL condition deduced intuitively. Students' confirmation, using Matlab prgrm linearadvection.m, that the explicit forward-in-time, upwind-in-space scheme is indeed subject to the CFL condition, e.g. here an unstable solution for the advection of a property θ=θ(x,t) on -1≤x≤1, after n=46 timesteps with C≡UΔt/Δx=1.6 [gridlength Δx=2/99; periodic boundary condition; initial condition a unit sine wave, θ(x,0)=sin(πx) ].

• Thurs 28 Jan. Lecture: Matrix formulation for "marching" solution of the advection-diffusion equation (continuation from previous class). "Hand" solution of a simpler marching problem (3 x 3 matrix inversion). Tri-diagonal matrix algorithm (TDMA).

• Tues 26 Jan. Lecture: (A) Reconciliation of the drunkard's walk (RDM) with the diffusion equation; (B) Reynolds-averaged mass conservation equation, and its manipulation to a form suitable for dispersion in the ASL; (C) An implicit discretization of the Reynolds-averaged, steady-state advection-diffusion equation**, leading to a tri-diagonal matrix inversion problem (this derivation is a precursor for the first scored lab, an Eulerian treatment of surface layer dispersion with a comparison to observations).

  **In two space dimensions: horizontal advection by the mean wind and vertical (eddy) diffusion.

• Thurs 21 Jan. Lecture covered explicit and implicit discretizations for the 1D heat equation. Stability condition on the "diffusion number" for the explicit scheme. (Balance of lecture time spent on the unscored computing tasks.)

• Tues 19 Jan. Lecture followed on from the generalized conservation eqn of previous class, letting φ ⇒ ρ c_pT to express conservation of sensible heat. Heat equation. Discretization using finite differences. Terminology for errors in numerical methods. Lax equivalence theorem. Continued work on the first computing task.

• Thurs 14 Jan. Derivation of an equation expressing the conservation principle. Begin (unscored) computing assignment 1, a drunkard's walk simulation (see flowchart). Here is JDW's solution, based on 1000 paths, with Δt=0.1).

• Tues 12 Jan. No class.

• Thurs. 7 Jan. Finalise course structure - balance of formal lectures vs. computing work? [We decided to partition lecture periods between "formal" lectures and time spent working on assignments]. Lecture covering dimensional analysis; and the random walk in position, as possible model for the dispersion problem introduced in Tuesday's class, and a lead in to our first computational exercise.

• Tues. 5 Jan. 2016. Orientation to the course. A thought experiment to promote discussion and thinking: how, starting from scratch, might one go about constructing a quantitative theory for the dispersion of a gas released into the atmosphere? We acknowledged the necessity to take a statistical approach; identified elements of the meteorology that would come into play; made the compromise of idealizing the problem, including the adoption of symmetries; saw that we can go a long way just on the basis of dimensional analysis; then embarked on the question of a rigorous approach based on conservation principles. Many elements of our course were introduced: symbols & terminology; the mass conservation eqn (proven in the handout); dimensional analysis (see handout); resolved versus unresolved processes or mechanisms.

Last Modified: 8 Apr., 2016