Adnan KhanLahore University of Management Sciences

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES OCTOBER 8, 2012

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

To study turbulent transport of passive scalars

Transport occurs via two mechanisms Advection (at Macro and Meso Scales) Diffusion (Eddy Diffusivity)

The advective field being turbulent has waves at a continuum of wavelengths

We study simplified models with widely separated scales to understand these issues

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

Want to study the problem at scales of observation (Macro Scales – Mean Flow)

Use homogenization theory to capture effects of the meso scales where we assume periodic fluctuations

The idea is to capture the effects of the flow as a lumped parameter, in this case as an enhancement in diffusivity

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

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

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

As a ‘toy’ problem consider the following Dirichlet Problem

D is periodic in the second ‘fast’ variable

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

)()()xD(x, xfxu =

∇∇

εΩ∈x

Ω∂∈= xxgxu )()(

Using the ‘ansatz’

We obtain

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

εε∇+∇→∇

1x

)(),(),( 210 εε Oyxuyxuu ++≅

( ) ( )( ) )(2 xfuD xyxy εεε =∇+∇∇+∇

Collecting terms with like powers of ε we obtain the following asymptotic hierarchy

O(1):

O(ε):

O(ε2):

( ) 0),(. 0 =∇∇ uyxD yy

( ) 01 .. uDuD xyyy ∇−∇=∇∇

( ) )()().().(. 0112 xfuDuDuDuD xxyxxyyy +∇∇−∇∇−∇−∇=∇∇

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

Applying periodicity and no secularity conditions

O(1)

O(ε)where

O(ε2) onon

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

)(),( 00 xuyxu ≡⇒

)().(),( 01 xcuyayxu x +∇=⇒

( ) )()()( 0 xfxuxD xx =∇∇⇒ Ω)()(0 xgxu = Ω∂

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

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’

( ) )()( 0 xfuxD xx =∇∇

pjyijpij aDDDi

>∂<+>=< δ

∫ΩΩ

=><0

),(1),(0

dAyxvyxv p

( ) DaDiyiyy −∂=∇∇

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

TPeaTtxavtxVtT

l ∆=∇⋅

++

∂∂ −1),(),( δ

ηδ

1>>a)1(Oa ≈

1<<a

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

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

)),((),(_

*_

_

txTKTtxVtT

∇⋅∇=∇⋅+∂∂

*K><−= −

jiijlij vPeK χδ1χ

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

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

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

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

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

11

111 2),(),( dWaPedttxavtxVdX l−+

+=

ηδ

21

222 2),(),( dWPeadttxavtxVdX l−+

+= δ

ηδ

>−−<= ))0()())(0()((21

jjiiij XTXXTXT

K

Some MC runs with Constant Mean Flow & CS fluctuations

MC and homogenization results agree

Need a modified Homogenization theory

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

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

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

δ=a

fgL =0

0L f

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

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

g

δ1O

<<δ10 Ot

∫δt

dssYsXf0

))(),((

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

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

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

∫ ++

≤δ

δδ

t

OVVVVCdssf

02

22

1

22

21 )()(

1V 2V

)1(δ

O)1(

δO

)1(δ

O

We develop the asymptotic expansion in both the cases

We have the following multiple scales hierarchy

We derive the effective equation for the quantity

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

22

10 TTTT δδ ++=

000 =TL00110 =+ TLTL

0021120 =++ TLTLTL003122130 =+++ TLTLTLTL

For the low order rational case we get

Where the operators are given by

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

0)( 32

21 =++ TRRR δδ

αα α ∇⋅+∇−∇⋅+∂∂

= − )(~~211 vPeV

tR lx

jijijiijvPePevR lxxxlx ααααααααα χαχχρρα ∂∂∂+∇⋅∇+∇⋅∇+∂∂−∇⋅−∇⋅∇−∇⋅= −−

22221

211

2 2)(~~2)(~~

))((~~22

211

213 xxlxxl vPePeR ∇+∇∇⋅+∇∇+∇⋅+∇−= −−

ααα αχχρ

iR

For the high order rational ratio case we get the following homogenized equation

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

021221 =∇+

∇−∇⋅+∂∂ −− TPeTPeVt xllx δξ

Random Homogenization – No restriction of periodicity on fluctuations

Lagrangian Inference Problem – In the Lagrangian formulation estimate the ‘drift’ and ‘diffusivity’ from large scale data

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012

FIRST NATIONAL CONFERENCE ON SPACE SCIENCES ISPA, KU, October 2012