PHY1460H Course Project: Mixing and Chaotic Advection in Atmospheric Dynamics Andre R. Erler April 20, 2009 1 Andre R. Erler PHY1460H April 20, 2009 Contents 1 Introduction 3 2 The Theory of Kinematic Mixing 3 2.1 Kinematic Foundations . .3 2.1.1 Measures of Mixing . .6 2.2 Chaotic Advection and Hamiltonian Chaos . .7 2.3 Mixing in Discrete Maps . 11 3 Chaotic Advection in the Atmosphere 11 3.1 Idealized Examples of Kinematic Mixing . 12 3.1.1 The Barotropic Rossby Wave . 12 3.1.2 A Simple Shear{Flow Map . 16 3.2 Realistic Mixing in the Atmosphere . 19 3.3 Dynamically Consistent Tracer Advection . 21 3.3.1 Dynamically Active and Passive Tracers . 25 4 Summary and Conclusion 27 2 Andre R. Erler PHY1460H April 20, 2009 1 Introduction Like many problems in fluid dynamics mixing is both, theoretically challenging and math- ematically involved, and at the same time of very high practical importance. Applications range from the chemical industry over life{sciences to geophysical fluid dynamics and as- trophysics. Most often mixing is associated with diffusion, however, in most fluids advection is a much faster process, so that kinematic mixing becomes dominant. Furthermore kinematic mixing is often understood to be synonymous with turbulence, which in the proper sense only exists in three dimensions. Yet, kinematic mixing does occur in two{dimensional flow. In this essay I will review our current understanding of kinematic mixing in two{ dimensional flow and its application to atmospheric dynamics. I will further discuss the mixing of dynamically active tracers and the formation of mixing barriers in the form of jets ant large vortices in the atmosphere. 2 The Theory of Kinematic Mixing Mixing as discussed in this paper is a purely kinematic phenomenon. The Eulerian flow field v(x; t) is assumed to be know; the theory of mixing is then concerned with the stretching and folding of material fluid elements along their trajectories (streaklines) as seen in Fig. 1. In stationary flow streaklines and streamlines generally coincide, but in time{dependent flow trajectories (streaklines) typically exhibit a much more complicated structure. 2.1 Kinematic Foundations Most of the discussion in this section follows Ottino (1989). The treatment here will be focused on two{dimensional flows, in three dimensions \line elements" will typically be 3 Andre R. Erler PHY1460H April 20, 2009 Figure 1: Mixing in a journal{bearing flow (a)-(d) and a time{periodic cavity flow (e)-(f). Panels (b), (e), and (f) show stretching and folding of line elements; this process is what effects kinematic mixing. (a) shows typical initial conditions, (c) illustrates the initial stretching of a line element, and (d) gives an idea of the final mixed state. In panels (a), (c), and (d) line elements break up due to interfacial tension between the fluids. Note that the area around the upper blob does not participate in the mixing at all. (Ottino, 1990) 4 Andre R. Erler PHY1460H April 20, 2009 replaces by \surface elements" and the relations below change accordingly. The infinitesimal deformation of a line{element dx0 by the flow is given by dx = F · dx0 ; (1) where F = rv denotes the deformation tensor. Since we are not primarily interested in absolute length or orientation, we introduce the stretch parameter λ as a local measure of line stretching: jdxj 1 λ = lim = eT C e 2 : (2) jdxj!0 jdx0j The tensor C = FT F is called the Cauchy{Green strain tensor and is a measure for local (and directional) stretching;1 e is a unit vector which defines the direction of strain under consideration. The rate of deformation of a line element is given by D dx = dx · rv ; (3) Dt so that he rate of stretching can be written as D ln λ = eT D e ; (4) Dt 1 T with the symmetric deformation tensor D = 2 (rv) + rv . Given the measure of stretching λ, an approach to quantify mixing would be to estimate the magnitude of the deformation D along a typical trajectory in the Eulerian flow field (the stretching function), and integrate the rate of deformation (4) forward in time. If fluid trajectories are sufficiently chaotic, the deformation a line element is subject to is essentially random and can be approximated by a mean value (cf. random strain theory). A constant 1Deformation can be decomposed into rotation and stretching: F = RU; with RT R = 1, we obtain C = FT F = UT U, a measure for stretching. 5 Andre R. Erler PHY1460H April 20, 2009 stretch rate will result in exponential stretching of line elements but can usually not be achieved in a finite volume; for the same reason, non{vanishing stretch rates in a finite volume also imply folding. Only stretching and folding together achieve efficient mixing. Note that stationary area preserving 2D–flow can not produce efficient mixing: streamline must close inside the domain (and can not cross), so that the amount of folding is bounded and absolute stretching due to shear is at most linear in time. 2.1.1 Measures of Mixing There are numerous ways to quantify mixing, and which one is used largely depends on the specific purpose; here I will give some examples to illustrate some common approaches. Inspired by the concept of stretching of line elements, a measure of mixing efficiency often used in atmospheric science is contour stretching: the stretching of iso{lines of some materially conserved tracer (Haynes et al., 2001, and references therein). Contour lines are either obtained from the Eulerian grid point values, or from Lagrangian contour advection schemes (Dritschel, 1989). The latter usually resolve finer scales of mixing. Another way of quantifying mixing is via Lagrangian decorrelation times: essentially flow characteristics of a material fluid element are correlated with the characteristics of the same fluid element at a previous time{step (lag{auto{correlation). The time{scale at which the auto{correlation becomes small, is the decorrelation time (Ngan and Shepherd, 1999, and references therein). Pierrehumbert (1990) takes a slightly different approach, where he considers the separa- tion of fluid parcels which are initially close together. The evolution of the distribution (as a function of separation distance) permits inferences upon the characteristics of the mixing process. Pierrehumbert and Yang (1993) also give estimates for striation spacing and fold- ing length and uses the characteristic length{ and time{scales to characterize kinematic mixing (as opposed to diffusive mixing for instance). 6 Andre R. Erler PHY1460H April 20, 2009 2.2 Chaotic Advection and Hamiltonian Chaos Intuitively it is obvious that chaotic motion will likely lead to good mixing. Furthermore, we might expect that chaotic dynamics will thermalize a system and equilibrate gradients. While the former is genrally true the latter is only partially true: we will see that chaotic systems can maintain a degree of order which can lead to the steepening of gradients. A straight forward connection between mixing and chaotic dynamics arises, when Lya- punov exponents are related to the rate deformation: Lyapunov exponents are the eigenval- ues of the Lagrangian long{time average of the deformation tensor D(x; t), and are thus an estimate for the stretching rate: if dx is taken to be a line element between two infinitely close end points, the divergence of these two end points is proportional to the stretch of the line element (sensitive dependence on initial conditions). But in non{divergent 2D{flow the relevance of chaos theory for kinematic mixing goes much beyond that: if one exploits the formal analogy between the stream{function formu- lation and Hamiltonian dynamics, results from Hamiltonian chaos theory can be directly transferred to vortical (i.e. non{divergent) flow in two dimensions. Identifying the (area{preserving) stream{function with the Hamiltonian of a dynamical system Ψ () H ; (5) the components of the flow vector v = (u; v) correspond exactly to the generalized co- ordinate q and momentum p of a one{dimensional Hamiltonian system (two degrees of freedom, excluding time): @ dq @ u = − Ψ () = − H (6) @y dt @p @ dp @ v = Ψ () = H : (7) @x dt @q 7 Andre R. Erler PHY1460H April 20, 2009 Critical points correspond to extrema of the stream{function (i.e. v = 0). They are clas- sified into hyperbolic, elliptic, and parabolic points. Elliptic points correspond to true extrema of the stream{function and are associated with vortices. Hyperbolic points are actually saddle points of the stream{function and are called stagnation points in the fluid dynamics literature; hyperbolic points have stable and unstable manifolds (directions of contraction and stretching, respectively). Depending on the orientation, orbits (or stream- lines) crossing a hyperbolic point can either belong to the stable or the unstable manifold. Orbits which belong to both, the stable and the unstable manifold, are called homoclinic; if the stable and unstable manifolds do not belong to the same hyperbolic point, they are called heteroclinic (cf. Fig. 2a). The most important result of chaos theory concerns the behaviour of orbits in the vicin- ity of critical points under perturbation: if G(x; t) is a time{dependent perturbation the Hamiltonian/stream{function of the system H0(x; t) = H(x) + G(x; t)( being a small parameter), attains more than two degrees of freedom and looses its integrability;2 it can exhibit chaotic behaviour. Closed orbits are structurally unstable and break up under perturbation; this means that homoclinic and heteroclinic connections disconnect, as illustrated in Fig. 2b. Orbits on the stable manifold are slightly perturbed and begin to fold up and stretch under increasingly divergent transverse flow as they approach the hyperbolic point. Because the (Hamiltonian) flow preserves area, the striation of the folds becomes increasingly thinner.