1. Adnan Khan Department of Mathematics Lahore University of Management Sciences

2.  Introduction  Theory of Periodic Homogenization  The Advection Diffusion Equation – Eulerian and Lagrangian Pictures  Non Standard Homogenization Theory  Summary International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 20102

3.  Many physical systems involve more than one time/space scales  Usually interested in studying the system at the large scale  Multiscale techniques have been developed for this purpose  We would like to capture the information at the fast/small scales in some statistical sense International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 20103

4.  Heterogeneous Porous Media  Bhattacharya et.al, Asymptotics of solute dispersion in periodic porous media, SIAM J. APPL. MATH 49(1):86-98, 1989  Plasma Physics  Soward et.al, Large Magnetic Reynold number dynmo action in spatially periodic flow with mean motion, Proc. Royal Soc. Lond. A 33:649-733  Ocean Atmospheric Science  Cushman-Roisin et.al, Interactions between mean flow and finite amplitude mesoscale eddies in a baratropic ocean Geophys. Astrpophys. Fkuid Dynamics 29:333-353, 1984  Astrophysics  Knobloch et.al, Enhancement of diffusive transport in Oscillatory Flows, Astroph. J., 401:196-205, 1992  Fully Developed Turbulence  Lesieur. M., Turbulence in Fluids, Fluid Mechanics and its Applications 1, Kluwer, Dordrecht, 1990 International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 20104

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6.  To smooth out small scale heterogeneities  Assume periodicity at small scales for mathematical simplification  Capture the behavior of the small scales in some ‘effective parameter’  Obtain course grained ‘homogenized’ equation at large scale International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 20106

7.  As a ‘toy’ problem consider the following Dirichlet Problem  D is periodic in the second ‘fast’ argument International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 20107

8.  Using the ‘ansatz’  Where are periodic functions  We obtain International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 20108

9.  Collecting terms with like powers of ε we obtain the following asymptotic hierarchy  O(1):  O( ε ):  O( ε 2 ): International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 20109

10.  Applying periodicity and zero mean conditions  O(1)  O( ε ) where → The ‘Cell Problem’  O( ε 2 ) on Homogenized on Equation International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201010

11.  We have obtained an ‘homogenized’ equation  The effective diffusivity is given by  Where the average over a period is  a is obtained by solving the ‘cell problem’ International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201011

12.  Transport is governed by the following non dimensionalized Advection Diffusion Equation  There are different distinguished limits Weak Mean Flow Equal Strength Mean Flow Strong Mean Flow International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010 12

13.  We study the simplest case of two scales with periodic fluctuations and a mean flow  The case of weak and equal strength mean flows has been well studied  For the strong mean flow case standard homogenization theory seems to break down International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201013

14.  For the first two cases we obtain a coarse grained effective equation  is the effective diffusivity given by  is the solution to the ‘cell problem’  The goal is to try an obtain a similar effective equation for the strong mean flow case International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010 14

15.  We study the transport using Monte Carlo Simulations for tracer trajectories  We compare our MC results to numerics obtained by extrapolating homogenization code  We develop a non standard homogenization theory to explain our results International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010 15

16.  We use Monte Carlo Simulations for the particle paths to study the problem  The equations of motion are given by  The enhanced diffusivity is given by International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 2010 16

17.  Some MC runs with Constant Mean Flow & CS fluctuations  MC and homogenization results agree  Need a modified Homogenization theory International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201017

18.  We consider one distinguished limit where we take  We develop a Multiple Scales calculation for the strong mean flow case in this limit  We get a hierarchy of equations (as in standard Multiple Scales Expansion) of the form  is the advection operator, is a smooth function with mean zero over a cell International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201018

19.  We develop the correct solvability condition for this case  We want to see if becomes large on time scales  This is equivalent to estimating the following integral  The magnitude of this integral will determine the solvability condition International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201019

20.  has mean zero over a ‘cell’  Two cases  Low order rational ratio  High Order rational ratio  Magnitude of Integral in both these cases International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201020

21. International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201021  Change to coordinates ‘s’ & ‘t’ along and perpendicular to the characteristics  Estimate magnitude of the integral in traversing the cell over the characteristics

22.  Analysis of the integral gives the following  Hence the magnitude of the integral depends on the ratio of and  For low order rational ratio the integral gets in time  For higher order rational ratio the integral stays small over time International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201022

23.  We develop the asymptotic expansion in both the cases  We have the following multiple scales hierarchy  We derive the effective equation for the quantity International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201023

24.  For the low order rational case we get  For the high order rational ratio case we get International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201024

25.  Brief exposition of Periodic Homogenization  Toy Problem to illustrate the process  Advection Diffusion Equation  Eulerian Approach – Homogenization  Lagrangian Approach – Monte Carlo Simulation  Non Standard Homogenization Theory International Symposium on Frontiers of Computational Sciences (ISFCS), Islamabad, June 7-8 201025