16th US National Congress of Theoretical and Applied Mechanics June 27-July 2, 2010, State College, Pennsylvania, USA

USNCTAM2010-1025

IDENTIFYING DYNAMICAL BOUNDARIES AND PHASE SPACE TRANSPORT USING LAGRANGIAN COHERENT STRUCTURES

Phanindra Tallapragada Virginia Tech Blacksburg, VA, USA [email protected]

ABSTRACT In many problems in dynamical systems one is interested in the identification of dynamical barriers which organize transport in phase space. The method of Lagrangian coherent structures (LCS), the analogues of invariant manifolds for non autonomous systems, utilizes the spatially non uniform rates of stretching to identify dynamical barriers. LCS have been shown to act as dynamical boundaries in phase space over finite times. The method of LCS is applied to two problems. The first is the non trivial motion of inertial spherical particles in a fluid whose motion is governed by the Maxey Riley equation.We explain the separation of inertial particles by partitioning the phase space with dynamical boundaries using the method of LCS. The second problem is the meso scale transport of passive tracers in the lower atmosphere, specifically plant pathogens that use the atmosphere to move from one habitat to another. We seek to explain spatial and temporal fluctuations of the concentrations of atmospheric pathogens by identifying atmospheric transport barriers using the method of LCS.

INTRODUCTION In dynamical systems the theory of invariant manifolds plays an important role in identifying dynamical barriers, distinct sets of solutions to the system and phase space transport. But many scientific and engineering problems are non linear, time dependent and often non smooth. Moreover many problems are defined only by spatially limited discrete data sets. These preclude the direct application of the theory of invariant manifolds. The method of LCS is an intuitive way to generalize the concept of dynamical barriers by identifying them with regions of high (exponential) rates of stretch in the phase space. Points on either side of the repelling (attracting) barrier separate (converge) exponentially. LCS are structures through which the flux is minimal and thus they partition the phase space into sets of distinct fates.

Shane D Ross Virginia Tech Blacksburg, VA, USA [email protected]

MATHEMATICAL FORMULATION The basic definitions and results of LCS are reviewed. Let φ : M × R 7→ M be a smooth time dependent flow over a smooth n dimensional manifold M, with the associated vector field x˙ = f(x,t). Let δ x(t0 , x) be a perturbed trajectory around an arbitrary reference trajectory at time t0 . The growth of the perturbation obtained by linearizing around

2 the reference trajectory

δ x (t0 ) ) The maximal finite (t ,t, x)δ x(t ) + O( is δ x(t) = dφ 0 0 dx time Lyapunov exponent is defined by [1] 1 σ (x,t0 ,t) = ln t − t0

s

dφ

(t0 ,t, x)

dx

(1)

2

where k·k2 denotes the matrix norm. A topographical ridge in the scalar field, σ , is a codimension one structure that is formally called the LCS [2]. An advantage of this formulation is that for high dimensional systems one can choose a subset of interest and study the rates of stretching restricted to the subset caused by perturbations in any subspace of the system. We call the LCS so obtained as the partial LCS.

Segregation of Inertial Particles Inertial particles in a fluid can have non zero velocity relative to the fluid (W). A simplified form of the Maxey Riley equation is used to describe the dynamics of neutrally buoyant spherical particles having small Stokes number, St, [3], [4]. The evolution of the position and relative velocity of the inertial particles W = −(J + 2 St −1 I) · W respecis given by ddtx = W + u and ddt 3 tively, where J is Jacobian of the underlying fluid velocity. For a simple cellular flow with stream function ψ(x, y,t) = a cos x cos y, we studied the sensitivity of final positions of particles with initial perturbations restricted to initial relative velocity (W ) subspace to obtain a partial LCS. The location of the partial LCS is 1

2010 USNCTAM

(a)partial LCS

(b)T = 0.005

(c) T = 0.03

(d)T = 0.185

FIGURE 1.

(a) Ridges in the sensitivity field for particles with St = 0.2 (red) and St = 0.1(blue). The initial position of all particles is (x0 , y0 ) = (3π/8, 3π/8) and the initial relative velocities belong to the region sandwiched between the ridges of the two Stokes numbers. (b-d) The particles are segregated by size.

a function of Stokes number. From a plot of partial LCS for two different sized particles, one can choose initial W which leads to segregation of particles by Stokes number, [5], as shown in Fig.1

ATMOSPHERIC TRANSPORT BARRIERS (ATBs) Plant pathogens are often transported hundreds of kilometers from one habitat to another in the lower atmosphere [6], [7]. Individual trajectories even on a dense set of initial conditions do not tell us about the transport of sets or how sensitive the trajectories are to initial conditions. Transport barriers defined by invariant manifolds and LCS organize the Lagrangian information of individual trajectories in a geometric framework that highlights the main features of geophysical flows and have been used extensively such as [8], [9]. Additionally individual trajectories may have uncertainties because of errors in velocity data, numerical interpolation and integration, but the LCS are more robust and reliable despite large data errors [10]. Our problem of interest is the transport of Fusarium in the eastern USA. The atmospheric flow field is given by the NAM12 velocity data generated by NOAA. As a first step pathogens are considered to be small enough to ignore inertial effects. A dense grid of air particles are seeded at fixed points at regular intervals of time. The final positions of particles for integration times of 24 hours are obtained by integrating the interpolated velocity data. Once again we used the partial LCS by considering the evolution of perturbations in the ‘horizontal’ plane. This is justified because vertical velocities are very small compared to horizontal velocities. By stacking the partial LCS at different heights, we obtained a 3D transport barrier. Fig.2 shows the movement of dynamical barriers over North America and specifically Blacksburg, Virginia. ATBs partition the atmosphere into sets of distinct composition for finite time scales. Preliminary spore collection results suggest that our hypothesis is true: the concentration of Fusarium spores are correlated to the passage of dynamical barriers.

FIGURE 2.

Top left picture shows ATBs on a continental scale. The next

three plots show the movement of a 3D LCS over the experimental spore sampling point near Blacksburg.

REFERENCES [1] Haller, G., 2001, Distinguished material surfaces and coherent structures in 3D fluid flows. Physica D(149) pp. 248-277. [2] Shadden, S.C., Lekien, F. and Marsden, J., 2005, Definition and properties of Lagrangian coherent structures from finitetime Lyapunov exponents in two-dimensional aperiodic flows. Physica D(212), pp. 271-304. [3] Maxey, M.R., Riley, J.J., 1983, Equation of motion of a small rigid sphere in a non uniform flow. Phys. Fluid(26), pp. 883889. [4] Babiano, A., Cartwright, J.H.E., Piro, O. and Provenzale, A., 2000, Dynamics of a small neutrally buoyant sphere in a fluid and targeting in Hamiltonian systems. Physical Review Letters(84), pp. 5764-5767. [5] Tallapragada, P. and Ross, S. D., 2008, Particle segregation by Stokes number for small neutrally buoyant spheres in a fluid. Physical Review E(78). [6] Isard, S.A., Gage. S.H., Comtois, P. and Russo. J.H., 2005, Principles of the Atmospheric Pathway for Invasive Species Applied to Soybean Rust. BioScience(55), pp. 851-861. [7] Schmale, D. G., Shields, E. J., and Bergstrom, G. C., 2006, Night-time spore deposition of the Fusarium head blight pathogen, Gibberella zeae. Can. J. Plant Pathology(28), pp. 100-108 [8] Lekien, F., Coulliette, C., Mariano, A.J., Ryan, E.H., Shay, L.K., Haller, G., Marsden, J.E., [2005], Pollution Release Tied to Invariant Manifolds: A case Study for the Coast of Florida. Physica D(210), pp. 1-20. [9] Lekien, F. and Ross, S.D., 2010, The computation of finitetime Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds. Chaos(20), 017505. [10] Haller, G., 2001, Lagrangian coherent structures from approximate velocity data. Phys. Fluids(A14), pp. 1851-1861.

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2010 USNCTAM