The Time Varying Shear Layer: Rigorous Joint

Asymptotics for an Enhanced Di!usion Surface.

James Bonn, Department of Mathematics, University of North Carolina, Chapel Hill, NC, 27599,

bonn@amath.unc.edu

Roberto Camassa, Department of Mathematics, University of North Carolina, Chapel Hill, NC,

27599, camassa@amath.unc.edu

Kenneth D T R McLaughlin, Department of Mathematics, University of North Carolina, Chapel

Hill, NC, 27599, mcl@amath.unc.edu

Richard M. McLaughlin, Department of Mathematics, University of North Carolina, Chapel

Hill, NC, 27599, rmm@amath.unc.edu

Abstract

We derive new, rigorous asymptotic formulae for the enhanced di!usion surface induced

by a temporally varying shear flow in the limit of vanishing Strouhal number, and infinite

Peclet number. These formulae document rich structure in the enhanced di!usion surface

characterized by regions in the Peclet-Strouhal plane with distinct scaling behavior, and regions

with strong non-monotonic dependence in the Strouhal number. The asymptotics additionally

rigorize within this class of models the quasi-steady approximation employed by Bonn and

McLaughlin [1]. High accuracy numerical calculations supporting the analysis are presented.

1 Introduction

The evolution of a passive tracer has received considerable attention in the context of fluid dynamics,

specifically for aiding in the understanding of many environmentally relevant transport problems

[2], in the study of turbulent closures [3, 4, 5, 6], in the study of subsurface transport in porous

media [7, 8, 9], in understanding evolving probability measures [10, 11, 12, 13, 14, 15, 16, 17, 18, 19,

20, 34, 21], and in helping to elucidate sub-grid scale phenomena [25, 32, 26, 27, 4, 28, 29, 33, 30, 1].

Focussing upon this latter issue, the fundamental problem is, in the simplified case of a passive

scalar di!using in the presence of a prescribed fluid flow, to deduce the e!ect which a small (sub-

grid) scale fluid flow has upon the large scale, bulk tracer transport. The full evolution is governed

1

To Appear: Discrete and Continuous Dynamical Systems

by the passive scalar equation:

!T

!t+ v(x, t) ·!T = ""T

T |t=0 = T0(x) (1.1)

Here, v is the here periodic incompressible fluid velocity, T is the tracer concentration, and "

denotes the tracer molecular di!usivity. We non-dimensionalize the problem using the Peclet and

the Strouhal numbers, defined as Pe = V L! , and S = L

TpV, with characteristic velocities, V , spatial

velocity period, L, and temporal velocity period, Tp. We will use the notation for the reciprocal of

the Peclet number, # = 1/Pe , and Strouhal number $ = S. Letting

x! =x

L, t! =

t

Tp, v! =

v

V, (1.2)

and dropping the primes, the non-dimensionalized evolution is:

$!T

!t+ v(x, t) ·!T = #"T (1.3)

Under suitable control of the excited scales in the prescribed fluid velocity field, (see Avellaneda

Majda for details [4]), methods of homogenized averaging may be applied to explicitly deduce the

e!ective long time, limiting dynamics of the tracer field. In this case, the limiting dynamics are

always di!usive, with an explicit enhanced di!usion tensor [25, 4], and the main problem is to

study the enhanced di!usion tensor, and its dependence upon the various fluid flow parameters.

This problem is well studied in the case of periodic, steady fluid flows [28, 29, 8, 26, 33, 27, 32]

with a variety of interesting predictions specifically regarding the high Peclet number (parameter

defined below measuring relative importance of advection compared with di!usion) asymptotic

scalings. Recent work of Bonn and McLaughlin [1] extends some of these predictions [28, 8, 29]

to temporally varying flows. In such a context, the addition of time variation adds considerable

complexity, and an additional nondimensional parameter, the Strouhal number, a ratio of flow

timescales, emerges. Computational and formal asymptotic studies of the Peclet-Strouhal plane by

Bonn and McLaughlin [1] have exhibited distinct regions which distinguish the steady flows from

unsteady flows, and additionally isolated regions in which the enhanced di!usion coe#cients exhibit

non-monotonic behavior in the various flow parameters. Here, our purpose is to analyze rigorously

an exactly solvable time varying shear model introduced by Majda and Kramer [6], and studied

computationally by Bonn and McLaughlin [1]. This analysis will successfully rigororize many of the

formal observations made by Bonn and McLaughlin, but more importantly, presents a complete two-

parameter asymptotic expansion of the enhanced di!usion surface in the limit of vanishing Strouhal

2

number, and infinite Peclet number. These asymptotics show a very rich and complex behavior

in the enhanced di!usion. We organize the paper as follows. In Section 2, we derive the solution

formula for the special geometry, reviewing the homogenized averaging employed at deriving the

solution formula. We additionally derive a new, convolution formula which will be especially useful

in deducing several asymptotic limits. In Section 3, we present mathematically rigorous asymptotics

for several di!erent regimes in the Peclet-Strouhal plane. New regimes which are distinct from those

calculated by Majda and Kramer whose analyses were restricted to the diagonal with $ = # [6, 35]

are developed here. These asymptotics show regions where strong oscillations in the enhanced

di!usion surface exist, and elucidates the role which multiple, same order zeros of the large scale

fluctuating wind play in creating these oscillations. A universally valid formula (joint in Peclet and

Strouhal numbers) is presented for the special case of fluctuating winds possessing simple zeros

through detailed Bessel function identities available in this case. In Section 4, we present a highly

e#cient FFT based scheme for numerically computing the necessary quadratues to calculate the

enhanced di!usion at finite Peclet and finite Strouhal numbers. Numerically constructed enhanced

di!usivities for very small Strouhal numbers (O(10"8), and very large Peclet numbers (O(108)) are

attainable. Lastly, in Section 5, we present formal asymptotic formulae which connect the steepest

descents regions with the quasi steady integration by parts regimes, and discuss the mathematical

details necessary to rigorize these formal approximations. Using the numerical method developed

in Section 4, direct comparisons to the rigorous and formal asymptotic formulas are presented, and

detailed enhanced di!usion surfaces are calculated showing rich structure in the Peclet-Strouhal

plane, and showing strong success of the asymptotic program in fully describing the enhanced

di!usion.

2 Homogenization and the time varying shear formula

We begin with a brief review of the method of homogenized averaging. Here, we will assume that

the velocity field is a fixed, deterministic, incompressible flow. In that case, the long time e!ective

dynamics for a passive scalar governed by the partial di!erential equation given in (1.1) are always

purely di!usive governed by a constant coe#cient e!ective di!usion equation with coe#cients given

by homogenized averaging [4, 25, 31, 6, 35]. These e!ective large scale dynamics equivalently may

be observed at finite times in the limit of a rapidly oscillating (in space and/or time) fluid flow

3

(yielding strong scale separation between the passive scalar and fluid velocity field), and are given

by the following:

!T

!t=

d!

i,j=1

(#ij + "ij)!2T

!xi!xj, (2.1)

where T is leading order asymptotic expansion of the original T field in the homogenization limit [31].

The e!ective heat coe#cients are given through a laborious tabulation proceedure which requires

the inner products of solution gradients of an associated set of elliptic (parabolic) boundary value

problems referred to as the cell problems:

"ij =1

2%

" 2"

0

"

T d!&i ·!&jdyd' (2.2)

where the auxiliary functions, &i satisfy the periodic boundary value problem

$!&i

!t+ v(y, ') ·!&i " #"&i = "(v(y, ')i " #v(y, ')i$) (2.3)

Here #·$ %# 10

#T d ·dyd' , the space-time average over the fast scales in the problem with T d denoting

the standard unit torus. The functions, &i, satisfy unit periodic boundary conditions in space and 2%

periodic in time, &i(y+ej, ') = &i(y, '+1) = &i(y, '), reflecting the particular nondimensionalization

employed, here ej denoting any unit vector. This nondimensionalization assumes the fluid flow

field have identical spatial period in all directions, and scales are nondimensionalized with respect

to the Peclet-Strouhal group [1], the Peclet number measuring the relative importance of e!ects of

advection as compared to molecular di!usion, and the Strouhal number being a ratio of intrinsic

flow timescales, the sweep time as compared to intrinsic fluctuation time. The usual molecular

di!usion is denoted in formula (2.1) above via the Kronecker delta, #ij, and the enhanced di!usion

is given by equation (2.2) in terms of an inner product of gradients of the auxiliary functions &i’s.

2.1 Time varying shear layer

The cell problem above in equation (2.3) is generally not exactly solvable. Further, the solution

dependence upon the Peclet and Strouhal numbers is highly-nontrivial, and generally requires nu-

merical tabulation. A full discussion of the cell problem in the case of periodic flow geometries may

be found in the literature [28, 33, 1]. One important class of examples for which the cell problem

is explicitly solvable in closed form is the case of a shear geometry:

v(x, t) =

$

%&g(t)

v(x1)

'

() (2.4)

4

where g(t) and v(x1) are unit periodic functions of time and space. In the special case in which the

function g(t) is identically constant was examined by Majda and McLaughlin [28]. The behavior

di!ers greatly between cases in which this constant is zero, or non-zero, the form of the ensuing

enhanced di!usion in this simple case being:

"22 =1

#2 + A(2.5)

where the constant, A, is proportional to the (for present discussion) constant function, g. A

few remarks regarding this case are in order. First, the enhanced di!usion is independent of the

Strouhal number in this case. Second, the Peclet scaling of the enhanced di!usion switches from

being constant at large Peclet numbers if g &= 0, to scaling quadratically with the Peclet number if

g = 0. More generally, if the function g(t) is not identically constant, a more complicated integral

expression for the enhanced di!usion was derived by Majda and Kramer, [6, 35]. We present

this integral here, and defer its derivation to Appendix A. Here for brevity in exposition, we have

assumed that the shear profile is v(x1) = 2'

% cos 2%x and take the time period to be 2%.

"22 = #"2 Re" #

0dy e"4"2y

" 2"

0d' exp

*

iG(', #

$y)

$

+

, (2.6)

where Re denotes real part and

G(', y) =" %"y

%g(s)ds .

and g(s) = 2%g(s) The remainder of this paper will concern the asymptotic characterization of the

enhanced di!usion surface as a function of Strouhal number ($), and reciprocal Peclet number #

given by this integral in equation (2.6). We will establish that this surface possesses rich structure

involving multiple regions of distinct behavior.

2.2 Convolution formula

Rescale ' and y in (2.6) as

y ( #

$y , G(', y) = f(' " y)" f(') , (2.7)

where f(t) is the primitive of the function g(t) = f !(t). We define

( % 4%2

PeS% 4%2#

$(2.8)

to rewrite (2.6) as

"22 =1

$ #Re

" #

0dy e"&y

" 2"

0d' exp

,i

$(f(' " y)" f('))

-. (2.9)

5

Henceforth, we will study the integral

I %" #

0dy e"&y

" 2"

0d' exp

,i

$(f(' " y)" f('))

-, (2.10)

where the function f is the primitive of the periodic function g(') in [0, 2%). From expression (2.9),

the integral I determines the di!usivity through

"22 =1

# $Re I . (2.11)

Introduce the function

h(' ; $) % e"i! f(%) ;

integral can be written as

I =" #

0dy e"&y

" 2"

0d' l(y " ' ; $) h(' ; $) , (2.12)

where

l(' ; $) % ei! f("%) = h("' ; $) .

(Complex conjugation denoted by (·) .) Second integral is a convolution. If we define

hn($) % 1

2%

" 2"

0h(' ; $)e"in% d' , ln($) % 1

2%

" 2"

0l(' ; $)e"in% d' , (2.13)

we have " 2"

0d' l(y " ')h(') = 2%

+#!

n="#ln($)hn($)einy .

The series sign can be taken outside of the remaining integral (dominated convergence of the partial

sum sequence assures that this is possible) and performing the integration in y

" #

0e"&y+iny dy =

1

"( + ine"&y+iny

...#

0=

1

(" in

yields

I = 2%+#!

n="#

ln($)hn($)

(" in

or

I = 2%

*l0($)h0($)

(+

#!

n=1

*ln($)hn($)

(" in+

l"n($)h"n($)

( + in

++

.

Observe that

ln($) % 1

2%

" 2"

0l(' ; $)e"in% d' =

1

2%

" 2"

0h("' ; $) e"in% d' =

1

2%

" 0

"2"h(' ; $) ein% d' ,

6

and, by periodicity of h,

1

2%

" 0

"2"h(' ; $) ein% d' =

1

2%

" 2"

0h(' ; $) ein% d' = hn($) ,

so that

ln($) = hn($) .

Hence

I = 2%

*|h0($)|2

(+

#!

n=1

*|hn($)|2

(" in+|h"n($)|2

( + in

++

,

which shows explicitly that ReI ) 0, so that the di!usivity is positive. By relation (2.11) between

"22 and I, the di!usivity can be written as

"22 =2%

$#

*|h0($)|2

(+ (

#!

n=1

|hn($)|2 + |h"n($)|2

(2 + n2

+

,

and, taking into account the definition of ( (2.8), finally as

"22(#, $) =2%

$ #(

*

|h0($)|2 +#!

n=1

|hn($)|2 + |h"n($)|2

1 + n2/(2

+

=1

2%#2

*

|h0($)|2 +#!

n=1

|hn($)|2 + |h"n($)|2

1 + n2$2/(4%#)2

+

. (2.14)

Moreover, if the function f is even, i.e., f("') = f(') then h("' ; $) = h(' ; $) and

h"n($) =1

2%

" 2"

0h(' ; $) ein% d' =

1

2%

" 0

"2"h("' ; $) e"in% d' =

1

2%

" 2"

0h(' ; $) e"in% d' = hn($) ,

(2.15)

so that

I = 2%

*|h0($)|2

(+ 2(

#!

n=1

|hn($)|2

(2 + n2

+

or

I =2%

(

*

|h0($)|2 + 2#!

n=1

|hn($)|2

1 + n2/(2

+

. (2.16)

With this expression, the e!ective di!usivity (2.9) becomes

"22 =2%

($#

*

|h0($)|2 + 2#!

n=1

|hn($)|2

1 + n2/(2

+

. (2.17)

7

3 Summary of results

In Sections 3-5 below we present a detailed study of the behavior of the enhanced di!usion as a

function of the Strouhal and Peclet numbers. We briefly summarize the main results:

• In Section 4.1 we present rigorous asymptotics in the limit of vanishing Strouhal at fixed Peclet

number. This yields formulas for the enhanced di!usion which are equivalent to those obtained

through the quasi steady approximation of the cell problem. The high Peclet asymptotics of

these quasi-steady formulas are immediate, and are derived in Appendix D. In Section 4.2,

by estimating the remainder of the previous asymptotics, we find the critical algebraic scaling

region between Peclet and Strouhal number where the quasi-steady asymptotics breaks down.

This region is seen to depend upon the order of the zero of the transverse time varying wind,

g(t). Conversely, in Section 4.3, we rigorously compute the opposite limit, namely, we send

Peclet to infinity at fixed Strouhal number. These asymptotic formulas may have an infinite

number of zeros with accumulation point at S = 0. In Appendix C, we elucidate on the origin

of these zeros using asymptotic expansions in the vanishing Strouhal number using stationary

phase methods. In Section 4.4, we provide an explicit example for the simple sinusoidal

transverse wind in which the complete joint asymptotics can be derived.

• In Section 5, we develop a highly e#cient numerical method for computing the enhanced

di!usion which is based upon an fast Fourier transform applied to the convolution formula

presented in Section 2. This scheme allows the enhanced di!usion to be accurately computed

for the entire Strouhal-Peclet plane, and in particular allows for Peclet to be as large as 107,

and Strouhal as small as 10"7.

• in Section 6, we present formal asymptotic calculations joint in both the Peclet and Strouhal

number. In Section 6.1, we derive a general connection formula bridging the quasi-steady

asymptotics of Section 4.1 across the critical scaling identified in Section 4.2 to the envelope

asymptotics S = O(1/Pe ) using iterated stationary phase methods. In Section 6.2, we derive

the general connection formula bridging the envelope asymptotics to the ordered asymptotic

results presented in Section 4.3.

8

4 Rigorous asymptotics

We are mainly interested in studying the behavior of "22 in the limiting cases, e.g., $ ( 0 and

# ( 0, following the observations of Bonn and McLaughlin [1].

To this end, we first observe the formal quasi-steady limit recognized by Bonn and McLaughlin

(see also similar work in combustion by Bourlioux, Khouider, and Majda [36, 37]), which we may

now rigorize in the shear geometry:

4.1 Quasi steady regime: integration by parts estimates

We are interested in the complete joint asymptotic description of the enhanced di!usion at small

Strouhal number, and large Peclet number. Here we explore the ordered asymptotic regime in which

the Strouhal number is first sent to zero, followed by the limit of large Peclet number. We further

show that this regime extends into the Peclet-Strouhal plane, and give a precise description of the

region in which these asymptotics are valid.

We note that the integral expression given in (2.6) is absolutely convergent, and we may inter-

change orders integration. In this ordering of limits, first letting the Strouhal number vanish, Bonn

and McLaughlin [1] used formal asymptotic arguments on the cell problem to generally yield the

quasi-steady (adiabatic) approximation. In the present context of the time varying shear geometry,

we may now rigorize this formal argument. First, we switch order of integration between ' and y

which is possible on account of the absolute convergence and Fubini’s theorem. Next, we perform

a Taylor expansion of the exponent in the integrand in the y variable about y = 0 in the integral

(2.6): An optimal way to document the success of this approach is to first add and subtract the

leading order term in the exponent:

"22 = #"2 Re" 2"

0d'

" #

0dy e"(4"2+i g(")

# )yei/

G(",!y#

)

! + g(")#

0

If we next integrate by parts, integrating the first exponential, we obtain the following:

= #"2 Re

$

%%&

" 2"

0

d'

4%2 + ig(%)$

+i

#

" 2"

0d'

" #

0dy

e"/

4"2y+iG(", !

#y)

!

0

4%2 + ig(%)$

!F (y, ', $, #)

!y

'

(() (4.1)

where the function, F , is given exactly as:

F (y, ', $, #) =

*G(', #y

$ )

$+ y

g(')

#

+

9

Recalling that G is defined as an integral through

G(',$y

#) =

" %" !y#

%g(s)ds ,

the partial derivative with respect to y may be explicitly calculated to yield

!F

!y=

1

#(g(')" g(' " $y

#)) ,

which, by using Taylor’s mean value theorem, yields:

= #"2 Re

$

%%&

" 2"

0

d'

4%2 + ig(%)$

+i

#

" 2"

0d'

" #

0dy

e"/

4"2y+iG(", !

#y)

!

0

4%2 + ig(%)$

(g(')" g(' " $y

#)

'

(() (4.2)

= #"2 Re

$

&" 2"

0

d'

4%2 + ig(%)$

+i

$

" 2"

0d'

" #

0udu

e"(4"2 #! u+iG(",u)

! )

4%2 + ig(%)$

dg()(u))

du

'

) (4.3)

= #"2 Re

$

%%&

" 2"

0

d'

4%2 + ig(%)$

+i$

#2

" 2"

0d'

" #

0ydy

e"/

4"2y+iG(", !

#y)

!

0

4%2 + ig(%)$

dg()( #$y))

dy

'

(() (4.4)

= #"2 Re

$

%%&

" 2"

0

d'

4%2 + ig(%)$

+i$

#2

" 2"

0d'

" #

0ydy

e"/

4"2y+iG(", !

#y)

!

0

4%2 + ig(%)$

dg()$(y))

dy

'

(() . (4.5)

Here ) and )$ are numbers on the line, as guaranteed by Taylor’s theorem with remainder.

The first observation is that the limit of vanishing Strouhal number $ ( 0, is now explicitly

given by the first integral on account of the boundedness of the second integral with respect to the

parameter $:

lim#%0

"22 = #"2 Re

$

&" 2"

0

d'

4%2 + ig(%)$

'

) (4.6)

= 4%2" 2"

0

d'

16%4#2 + (g('))2. (4.7)

We note that the leading order asymptotic behavior of this integral in (4.6) as the Peclet number

diverges (# ( 0) is elementary. We defer this calculation to Appendix D, citing only the formula

here: For small #, the leading order behavior for this integral is rigorously given by the following

formula:

"22 = A#"2+ 1M + O

/#"2+ 2

M

0,

where M is the order of the largest order zero of the function g('), and A is an explicit constant

given in the appendix. Considering the analogous enhanced di!usion formula for steady shears given

10

above in (2.5), the enhanced di!usion formula in (4.6) amounts to precisely the temporal average

of the enhanced di!usion for the steady shear layer with cross wind, thereby giving a rigorous

derivation of the formal asymptotics employed by Bonn and McLaughlin [1].

It should be noted that Majda and Kramer [6] explored the asymptotic expansion of the enhanced

di!usion given by the double integral above in equation (2.6) for the very particular curve in the

Peclet- Strouhal plane in which $ = #. They correctly obtained the envelope for the enhanced

di!usion in this case, however substantial oscillation in the enhanced di!usion along this trajectory

at leading order was missed in their calculation, we discuss this further below in Section 4.3.

Interestingly, the envelope predicted by Majda and Kramer di!ers substantially from the quasi

steady asymptotics just derived, with their calculation prediction "22 * #"2M

M+1 . These formulae

only agree for simple zeros with M = 1, and suggests that a fairly rich enhanced di!usion surface

may be lurking in the Peclet-Strouhal plane, as we will soon find. First we show that the quasi-

steady asymptotic scalings extend to an M dependent algebraic curve in the Peclet-Strouhal plane.

4.2 Quasi-steady asymptotics in Peclet-Strouhal plane

In this subsection, we compute the precise region in the Peclet-Strouhal plane over which the quasi-

steady asymptotics presented in the prior section extend uniformly. This will seen to be a region

in which $ + #1+ 1M , a slightly larger region than immediately inferred from the formulae of the

prior section. We obtain this extension from carrying the Taylor expansion to higher order. This

estimate is sharp, as we will see from results of Section 6.

To deduce over what region in the Peclet-Strouhal plane the quasi steady regime holds, we first

observe that the formula in (4.3) shows immediately that

"22 = 4%2" 2"

0

d'

16%4#2 + (g('))2+ O

,$

#4

-. (4.8)

Using the leading order asymptotics available for the first integral in the limit of small #, we have

"22 = A#"2+ 1M + O

/#"2+ 2

M

0+ O

,$

#4

-, (4.9)

which shows that if $ = #2+ 1M +' for * > 0, the corrections will be sub-dominant as compared to the

leading order term. This defines a curve in the Peclet-Strouhal plane below which the quasi-steady

asymptotics holds, and shows only Peclet dependence, with the enhanced di!usion in this regime

being independent of the Strouhal number. It is also worth remarking here that the leading order

contributions from lower order zeros of the function g(') are sub-dominant to the first term above

11

in equation (4.9). Moreover, it is a routine calculation to further show that corrections arising from

expansion about lower order zeros are asymptotically smaller in general than corrections retained

from expansion about the higher order (M) zero, and hence do not alter the critical regime over

which the quasi steady regime holds.

The quasi-steady asymptotics do not hold in the entire plane as expected, and documented in

simulations by Bonn and McLaughlin [1]. We will next see that this curve is not the critical curve,

but one can move slightly higher in the Peclet-Strouhal plane with the critical relation being:

$crit = #1+ 1M (4.10)

To see that this is the case, we work on the correction term given in the second term in equation

(4.2):

II = #"2 Re

$

& i

$

" 1

0d'

" #

0du

(g(')" g(' " u))/4%2 + ig(%)

$

0 e"(4"2 #! u+iG(",u)

! )

'

) (4.11)

Assuming that the function, g(') has an Mth order zero at ' = 0, we expand g(' " y) about y = 0

through order M + 1 using Taylor’s theorem with remainder:

g(')" g(' " u) =M!

j=1

("1)j+1g(j)(')uj

j!+ R(u, ')

R(u, ') =uM+1g(M+1)()1)

(M + 1)!.

Inserting into (4.11), we obtain

II = #"2 Re

$

%&i

$

" 1

0d'

" #

0du

1Mj=1

("1)j+1g(j)(%)uj

j! + R(u, ')/4%2 + ig(%)

$

0 e"(4"2 #! u+iG(",u)

! )

'

() .

Taking a modulus, gives

|II| , #"2

$

%&1

$

" 1

0d'

" #

0du

1Mj=1

|g(j)(%)|uj

j! + |R(u, ')|2

16%4 + (g(%))2

$2

e"4"2 #! y

'

() .

Performing the u integrations, noting that##0 uje"audu = a"j+1j!, we obtain

|II| , 1

$#2

M+1!

j=1

Ij

Ij = Aj$j+1

#j+1

" 1

0d'

...g(j)(')...

216%4 + (g(%))2

$2

, j = 1, · · · , M

IM+1 = AM+1$M+1

#M+1

" 1

0d'

...g(M+1)()1)...

216%4 + (g(%))2

$2

,

12

where the coe#cients, Aj depend on the shape details of the function g(t), but are independent of

the parameters $ and #. For each j = 2, · · · , M + 1, it is straightforward to show that each of these

integrals satisfy the following bound for all M > 1:

Ij < Cj$j+1

#j(1+1/M)"1/M(4.12)

For some constants Cj independent of $ and #. We note that for j=1, the argument will involve a

logarithm, but the argument may be easily extended:

I1 < C1$2 log #

#. (4.13)

We further note that the case M=1 is handled directly by the uniform joint asymptotics developed

below, and, consequently, we need not consider that case here. Putting this together, we find that

the size of the correction, II, is:

|II| < C1$ log #

#3+

M+1!

j=2

Cj$j

#j(1+1/M)+2"1/M(4.14)

Observing that the leading order term is of size, #"2+ 1M , we establish the asymptotic sub-dominance

of the term II by showing that the ratio vanishes in the limit of vanishing # over appropriate values

of $:

|II|#"2+1/M

< C1$ log #

#1+1/M+

M+1!

j=2

Cj

,$

#1+1/M

-j

, (4.15)

and the right hand side will vanish for those values lying below the critical curve in the Peclet-

Strouhal plane, $crit = #1+ 1M . We can show this by considering di!erent curves in the plane,

indexed by a positive parameter, * > 0. Let

$ = #1+ 1M +' (4.16)

Then the asymptotic sub-dominance follows from:

|II|#"2+ 1

M

< C1#' log # +

M+1!

j=2

Cj#'j (4.17)

( 0 . (4.18)

This expression tends to zero with vanishing # for all positive values of *, and for j = 1, · · ·M+1,

thereby showing that for all $ < #1+ 1M the corrections are (rigorously) sub-dominant.

13

4.3 Fixed Strouhal, large Peclet

Here we consider the counterpart to that in the previous Section, in the limit large Peclet number

at fixed Strouhal number, and discover a remarkable oscillatory phenomena in which leading order

asymptotics breaks over a measure zero set of special Strouhal values accumulating at S = 0.

Recall the definition of the Fourier coe#cients (2.13) in the series expression for I (2.16),

I =2%

(

*

|h0($)|2 + 2#!

n=1

|hn($)|2

1 + n2/(2

+

.

Properties of Fourier series [39] assure that

|h0($)|2 +#!

n=1

|hn($)|2 +#!

n=1

|h"n($)|2 =1

2%

" 2"

0|h(' ; $)|2 d' = 1 , (4.19)

and in particular for g even

|h0($)|2 + 2#!

n=1

|hn($)|2 = 1 , (4.20)

while

|hn($)|2 , V 2($)

4%2n2

since h(' ; $) is well behaved over [0, 2%), with the constant V ($) being the total variation of h(' ; $).

The limit of large Peclet at fixed Strouhal corresponds to # ( 0, fixed $. Then (cf. (2.8)),

( = O(#), and from (2.16) we see that the second term (the series) is vanishingly small. In fact,

from (4.20)#!

n=1

|hn($)|2

1 + n2/(2, 1

2

#!

n=1

1

1 + n2/(2, %2(2

12.

Thus,

I =2%

(|h0($)|2 + O(() =

$

2%#|h0($)|2 + O(#) (4.21)

as ( ( 0 for # ( 0, fixed $. Notice that estimate (4.20) and the definition of ( = 4%2#/$ show

explicitly the 1/$ dependence of the constant(s) in the order relation O(#) in (4.21).

This evaluation of the integral I and the definition of the di!usivity (2.11) yields

"22 =1

2%#2|h0($)|2 + O(1) . (4.22)

Notice that depending on Strouhal ($ = S) the Fourier coe#cient h0($) may be zero. For instance,

for the simple sinusoidal case, h0($) = J0(1/$) (see below), and hence the di!usivity does not follow

the Pe2-large Peclet asymptotic at values of S that coincide with the inverse of the zeros of the

Bessel function J0. Notice that remainder in (4.22) is bounded (independently of #) by a coe#cient

14

of order O(1/$2). Notice further that existence of such values of Strouhal that lead to vanishing

coe#cient in the leading order term (4.22) is related to the symmetric occurrence of same-order

zeros in the time dependency of the wind g(t).

We remark that vanishing of the first Fourier coe#cients has a simple interpretation from the

viewpoint of particle trajectories defined by the velocity field (2.4). These can be computed by

quadratures,

x(t) = x0 +1

2%$f(t), y(t) = y0 +

1

$

" t

0cos

*f(s)

$+ 2%x0

+

ds , (4.23)

where we have used the notation f(t) (with f(0) = 0) for the primitive of the wind g(t), and we

have taken into account the rescaling (1.2), and its e!ect (1.3), for the fluid particle velocity (2.4).

Imposing that the vertical component of trajectories be periodic, so that the shear does not lead to

unbounded stretching of the fluid parcels along its direction, yields

y(2%)" y0 = 0 =" 2"

0cos

*f(s)

$+ 2%x0

+

ds

for all x0, which is precisely the condition h0($) = 0 (cf. definition (2.13) that kills the leading order

term of the enhanced di!usion (4.22).

The asymptotic expansion by stationary phase as $ ( 0 for fixed n of the Fourier coe#-

cients hn($) gives an approximate criterion for when suppression of enhancement due to vanishing

of the first Fourier coe#cients occurs. In fact, the absolute value in (4.22) takes out oscillatory

factors of unit modulus, and only symmetry of stationary points can contribute to non-unit moduli

(see example for M = 3 in Appendix C). Thus, for M > 1, for cases involving a unique zero of

order M , the leading asymptotics does not vanish at any value of Strouhal $, and in this case it

may be shown that the asymptotic formulae presented in this section are rigorously dominant over

all sweeps $ = #( with 0 < + < 1 which corresponds to the upper triangular region in the $-# plane

bounded from below by the curve $ = #. For cases with multiple equal order zeros, the analysis

requires going to the next term in the expansion, i.e., replacing the bound on the remainder in (4.22)

with a precise order estimate, and the regimes of validity in the plane near the values where the

leading order coe#cient |h0($)|2 vanishes are yet to be determined. For the case with M = 1 with

a non-unique zero, such as for g(t) = sin(t), the following section provides the answer through a

leading order asymptotic evaluation that is uniformly valid over the whole (#, $)-plane, for # and $

small.

15

4.4 The simple sinusoidal case

With choice of the function f(t) = cos t the Fourier coe#cients hn($) can be evaluated explicitly

with Bessel functions of integer order,

hn($) = Jn(1/$) .

Then the double integral

I %" #

0dy e"&y

" 2"

0d' e

i! (cos %"cos(%"y)) (4.24)

coincides with the following expansion in series of Bessel functions,

I = 2%

*1

(J2

0 (1/$) + 2(#!

n=1

1

(2 + n2J2

n(1/$)

+

, (4.25)

which behaves asymptotically for $ ( 0 as

I - 2%$ coth(%()[1 + sech(%() sin(2/$)] . (4.26)

Appendix B carries out the necessay estimates to establish this asymptotic result; these are

established for the case ( = O(1) with respect to either # or $. This constraint is then relaxed by

looking at the asymptotics for “sweeps” in the parameter plane along one-parameter curves $ = ,(#)

that fill the plane. The asymptotic results along these curves together with the results from the

general analysis above allow us to the establish the global behavior of the e!ective di!usivity in the

(#, $)-plane for this simple zero case.

We demonstrate the agreement of this approximation in Figure 4.1, where a plot obtained with

a direct numerical integration of (4.24) is compared with its asymptotic expression (4.25) for the

range of (moderate) values Pe = 1/$ . [30, 40], and # = $. Notice the oscillations of the integral I,

which implies that the e!ective di!usivity

"22 -2%

#coth(4%3#/$)[1 + sech(4%3#/$) sin(2/$)] (4.27)

does not behave monotonically as a function of the Pe along the line $ = # (or Pe = 1/S).

4.4.1 Sweeps through the (Pe, S)-plane in the sinusoidal case

In taking the limits of large Peclet and small Strouhal it is useful to consider “sweeps” through the

(Pe, S)-plane along the curves

Pe S = S$

$+1 , (4.28)

16

32 34 36 38 40

0.15

0.17

0.19

0.21Pe

/Peκ

22

Figure 4.1: Comparison between (4.24) and the asymptotic expression (4.26)

or

S = $ = #(+1

so that along these sweeps,

( = 4%2/#( ,

and the di!usivity rewrites as

"22 =1

#2Re

" #

0dy e"4"2y

" 1

0d' exp

*

iG(', #( y)

#(+1

+

.

The case ! > 0

From expression (2.10), it is clear the sinusoidal case above quickly generalizes to the case + > 0.

Simply substitute that ( = 4%2/#( , and $ is $ = #(+1 with # ( 0, everywhere in the formulae of

Appendix B, keeping track in particular of overall factor 2( for the sum.

The final expression for the di!usivity (2.9), for the sinusoidal case with + > 0 is

"22 -2%#(+1

$#=

2%

#= 2% Pe

as Pe ( /. The only di!erence with respect to + = 0 is the absence of oscillations, i.e., the

asymptotic behavior is monotonic in Pe.

The case with !1 < ! < 0

Inserting ( = 4%2/#( , and $ = #(+1 in the asymptotic approximation of I (4.26) for # ( 0

yields

"22 -2%

#coth

*4%3

#(

+ 3

1 + sech

*4%3

#(

+

sin,

2

#(+1

-4

. (4.29)

17

5 10 15 20 25 30

2

4

6

8

! 22

103

105

X

X

Pe

Figure 4.2: Oscillations of e!ective di!usivity from the asymptotic expression (4.29), with + = "1/2.

The dashed curves track the extrema "e22 according to (4.30) .

In the range "1 < + < 0, the hyperbolic functions can be simply approximated by

coth(x) - 1/x + x/3 sech(x) - 1" x2/2 as x ( 0

thereby producing at leading order for (4.29)

"22 -1

2%2#1"(

51 + sin

,2

#(+1

-6.

Thus, "22(Pe) oscillates as Pe (/ with a period that tends to / as + ( "1.

The minima and maxima of these oscillations therefore occur along the curves

"e22 =

2%

#coth

*4%3

#(

+ 3

1 ± sech

*4%3

#(

+4

=

788888889

8888888:

2%

#coth

*4%3

#(

+

- 1

2%2#1"(

2%

#tanh

*4%3

#(

+

- 8 %3

#1+(

. (4.30)

so that the location of maxima grows as the (convex) scaling law Pe1"( and the minima as the

(concave) scaling Pe1+(. Notice that as + ( "1 the growth of minima becomes slower, while

the maxima approach the scaling Pe2, in agreement with the special limit (4.22). Figure 4.2

demonstrates these properties of "22 for the case + = "1/2.

18

5 Numerical computation of the enhanced di!usivity

The series expression of the enhanced di!usivity (2.17) o!ers a starting point for a highly optimized

numerical algorithm for computing "22 via a Fast Fourier Transform (FFT). Recall that coe#cients

hn’s are the Fourier coe#cients

hn($) =1

2%

" 2"

0ein%+ i

! f(%)d' =1

2%

" 2"

0ein%e

i! f(%)d'

and f(t) is the antiderivative of g(t). In all the examples we study we will take g as a trigonometric

polynomial, e.g.,

g(t) = sin(t), g(t) = sin(t) + 1/2 sin(2t), etc. .

Once the infinite sum is cut o! at a finite number, all of the hn’s needed can be generated by one

FFT. The cut o! number will naturally depend on $, since the 1/$ in the exponent determines the

oscillatory nature of the function to be used as input for the FFT. Morever, the optimal number of

terms for convergence may depend on Peclet.

In the computations we have performed, the criteria we use to determine the size of the

FFT, or how many terms we keep in our sum, is as follows: let j = next integer greater than

log2 max(Pe, 1/S) (remember that S = $); then Nmax = 2(2j) which is therefore the size of the FFT

required. This generates modes "Nmax/2+1, . . . , Nmax. Hence, we actually only have Nmax/2 = 2j

modes in the sum since we only need the positive modes due to the symetry (2.15). In all com-

putations, to check for convergence we have doubled the number of modes being used and have

verified that no significant di!erence can be oobserved, with the largest relative error kept of the

order of 10"7.

With this algorithm, the largest parameter space that we have explored is of max(Pe, 1/S) = 108.

By using Nmax = 2j with j defined as before, the FFT leads to a data structure of size 227. This

takes approximately 4.8 Gigabytes of memory and requires about 3 minutes to find the FFT and

compute the sum for the enhanced di!usion coe#cient on a 525 Mhz dec alphaserver.

We show results from our numerical computation in Figure 5.1 to 6.2, where we change the order

of the zero(s) of the function g(t) from M = 1 to M = 5 and plot the level curves of the di!usivity

k22(#, $) in the quarter plane # > 0, $ > 0 spanning the range 10"5 to 10"6.

19

1 2 3 4 5 6 7 8 9 10x 10-5

1

2

3

4

5

6

7

8

9

10x 10-5

1/Pe

Stro

uhal

g(t)=sin(t)

Figure 5.1: Contour plot of the e!ective di!usivity "22 from the numerical evaluation of the series

expression (2.17), with a simple zero (M = 1).

1 2 3 4 5 6 7 8 9 10x 105

1

2

3

4

5

6

7

8

9

10x 105

1/Pe

S

g(t)=sin(t)+0.5sin(2t)

Figure 5.2: Contour plot of the e!ective di!usivity "22 from the numerical evaluation of the series

expression (2.17) with a single triple zero (M = 3) .

20

6 5 4 3 2 1 0 1 26

5

4

3

2

1

0

1

2

log10 (1/Pe)

log 10

S

log10 !22 for g(t) = sin(t)+0.5sin(2t)

Figure 5.3: Large scale log-log contour plot of the e!ective di!usivity "22 for that shown in Figure 5.2

1 2 3 4 5 6 7 8 9 10x 105

1

2

3

4

5

6

7

8

9

10x 105

1/Pe

Stro

uhal

sin3(t)g(t)=

Figure 5.4: Contour plot of the e!ective di!usivity "22 from the numerical evaluation of the series

expression(2.17) with a two triple zeros (M = 3).

21

1 2 3 4 5 6 7 8 9 10x 105

1

2

3

4

5

6

7

8

9

10x 105

1/Pe

Stro

uhal

g(t)=sin(t)+5/8sin(2t)1/16sin(4t)

Figure 5.5: Contour plot of the e!ective di!usivity "22 from the numerical evaluation of the series

expression(2.17) with a single quintic zero (M = 5).

6 Formal asymptotic calculations

We now turn to using more formal, and less rigorous methods to connect these di!erent asymp-

totic regimes of small Strouhal number with fixed Peclet number, and large Peclet number with

fixed Strouhal number. Our purpose here is to calculate connection formulae between the results

computed in Sections 4.1 and 4.2 with those computed in Sections 4.3 and 4.4. This provides a

connection between the di!erent ordered asymptotic limits between Strouhal to zero and Peclet to

infinity. Where possible, we discuss which of these calculations are capable of being made math-

ematically rigorous. In Section 6.1, iterated stationary phase methods are employed to compute

the quasi-steady regime the envelope asymptotics computed by Kramer and Majda, who explored

the distinguished limit with $ = O(#). In Section 6.2, the connection between the envelope asymp-

totics and the regime of Section 4.3 is computed through an asymptotic expansion of the Fourier

sums given in Section 2. This requires a careful study of the asymptotics of the Fourier coe#cients

themselves in the limit of vanishing Strouhal number, and how this feeds into the leading order

contributions to the infinite series. Lastly, in Section 6.3, we document strong agreement of these

formal asymptotic calculations with the e#cient numerical simulations developed in Section 6.

22

6.1 A connection identity for M &= 1

To connect to the envelope asymptotics valid along $ = # in the Peclet-Strouhal plane, where Majda

and Kramer originally predicted that the enhanced di!usion should follow the M dependent scaling

law Pe2M

M+1 , to the quasi-steady regime we employ a formal iterated stationary phase calculation.

We will see that above the critical curve (recall that in the present notation, the Strouhal number,

S = $, and the Peclet number is Pe = #"1), defined by $ = #1+1/M , a di!erent asymptotic formula

holds which connects the quasi-steady asymptotics to the Majda-Kramer scaling. This formula is

"22 = O$

&$1!M1+M

#

'

) (6.1)

Observe that along the curve # = $ this expression reduces to

"22 = O$

&#1!M1+M

#

'

) (6.2)

= O/#"

2MM+1

0(6.3)

which is the Majda-Kramer prediction. Along the critical curve, $ = #1+ 1M , we recover the quasi-

steady formula:

"22 = O$

&#(1+1/M) 1!M1+M

#

'

) (6.4)

= O/#"2+ 1

M

0(6.5)

We now give the iterated stationary phase argument. First, recall the general expression for the

enhanced di!usion:

"22 = #"2 Re" #

0dy e"4"2y

" 1

0d' exp

*

iG(', #

$y)

$

+

, (6.6)

where Re denotes real part and

G(', y) =" %"y

%g(s)ds . (6.7)

Through periodic shifting we may reduce the above integral to a finite area, double integral:

"22 =1

#$(1" e"4%2#

! )

" 1

0d'

" 1

0dtei i&(",t)

! (6.8)

where the complex phase is defined to be:

-(t, ') = G(' " t) + i4%2#t .

23

For the purposes of the present discussion, we take the primitive function g(t) = tM , which

localizes the periodic function g(t) around its single zero of order M at the origin t = 0. Computing

points of stationary phase yields

!-

!t= "g(' " t) + i4%2# = 0 (6.9)

= (' " t)M + i4%2# = 0 (6.10)

which gives

tc = ' " (i4%2#)1M (6.11)

We discuss below the multiple zeros, and first evaluate - and its derivatives at one of the saddle

points:

-t(', tc) = 0 (6.12)

-(j)(', tc) = Aj(')#M!j+1

M (6.13)

Expanding, we find:

"22 =1

#$(1" e"4%2#

! )

" 1

0d'e

i&(",tc)!

" 1

0dte

i!

;)(%,tc)+

1M+1

j=2&(j)(",tc)

j (t"tc)j+O((t"tc)M+1)<

(6.14)

Changing scales through $!1

M+1 (t" tc) = w though which

"22 =1

#$M

M+1 (1" e"4%2#

! )

" 1

0d'e

i&(",tc)!

" (1"tc)#!1

M+1

"tc#!1

M+1dwei)(w,$,#) (6.15)

where

- =M!

j=2

Aj(')#M!j+1

M $j!M!1

M+1 wj + AM+1(')wM+1 + o(wM+1) (6.16)

and the integrals are taken along suitably selected steepest descent contours in the complex w plane.

At first glance, it may appear hard to justify dropping terms in the summand, but this is possible

when exploring the asymptotic region

#1+ 1M + $ + 1

$ + #

24

and so, consequently, the factors of # in the summand conspire to justify dropping the sum as we

now establish, while the higher order terms in w can be formally dropped on the basis of localization

near the saddle point. Let $ = #', the #-$ product in the sum above then reduces to

#M!j+1

M $j!M!1

M+1 % #Q

Q =(M + 1)(M " j + 1) + M(j "M " 1)*

M(M + 1)

=(j " (M + 1))

,(M + 1)

1+ 1!jM

j"M"1 + *-

M + 1

= "(M + 1" j)

,* " 1+(M+1) 1!j

MM+1"j

-

M + 1

= "(M + 1" j)

/* " (1 + 1

M )0

M + 1

To work within the sought asymptotic regime, we must select * < 1 + 1M . We need to show a

positive Q to guarantee asymptotic sub-dominance. To this end, first note that the first bracket in

the numerator of the above expression is strictly positive for j = 2, · · · , M , and consequently, we

are finished provided the second term in brackets is strictly negative, i.e. * , 1 + 1M , which is true.

So we need only retain the term of order wM+1.

Continuing, we have that in this asymptotic regime:

"22 * 1

#$M

M+1 (1" e"4%2#

! )

" 1

0d'e

i&(",tc)!

" 1

0dweiAM+1wM+1

(6.17)

* 1

#$M

M+1

CM

" 1

0d'

1

(AM+1)1

M+1

ei&(",tc)

! (6.18)

-(', tc) =" (i4"2$)

1M

%g(s)ds + i4%2(' " (i4%2#)

1M ) (6.19)

Now we must repeat (iterate) the stationary phase argument. Seeking the stationary phase

points in ' , we have

-(', tc)% = "g(') + i4%2# (6.20)

= "'M + i4%2# (6.21)

so that,

'c = O/#

1M

0(6.22)

25

Expanding the argument of the exponential as before,

"22 * 1

#$M

M+1

CM

" 1

0d'e

i!

1M+1

j=2$

M!j+1M Bj(%"%c)j+O(%"%c)M+1

(6.23)

this argument is now identical to the previous case. Accounting for a new overall integration

constant, and $ factor, we have

"22 * CM$

1!MM+1

#, (6.24)

which concludes the argument.

We note that several things must be carefully explored to rigorize this analysis. Firstly, upon

seeking the first saddle point for the t integration, an M th root must be computed, which will

generically yield M solutions in the complex t plane. For the particular singular limit we study,

namely # ( 0, these saddle points will collapse to the origin. Nonetheless, the appropriate contour

deformation through these saddle regions must be computed correctly to properly localize the

integral. One clear complication is that the location of these saddles additionally are dependent

upon the other variable of integration, ' . Anticipating a second stationary phase in the complex

' plane makes it necessary to argue for localization through the appropriate saddle for complex

values of ' . We remark that we have successfully carried out this program for the special case

involving g(t) = t3. This calculation is quite lengthy, and we plan a subsequent article to further

explore this regime. It is worth noting that preliminary calculations using the numerical approach

developed in Section 4 show scaling agreement with this intermediate regime when sweeps $ = #(

with + > 1 + 1/M are considered. The theory of several complex variables combined with steepest

descents asymptotics could provide the appropriate language to discuss this problem generally.

Next, we present a more detailed asymptotic calculation for exploring this same intermediate

regime which is based upon the series representation in terms of Fourier coe#cients. This approach

is successful in calculating much more than just the scaling relations identified here.

6.2 Uniform asymptotics for small Strouhal " and large Peclet

We study the series (2.16) in the limit of small $ and #. We use the notation

4%2# % #", so that ( = #" /$ ,

and focus on the series

$$($) %1

2|h0($)|2 +

#!

n=1

|hn($)|2

1 + n2$2/#"2, (6.25)

26

so that

I =2%

($$($) .

Notice that this expression has the natural structure of the trapezoidal rule of integration for the

continuous function

F$(m, $) % |h(m, $)|2

1 + m2/#"2

evaluated at the discrete points m = n$. Thus, the leading order contribution as $ ( 0 to the series

is (formally) given by its integral approximation,

1

$

" #

0F$(m, $) dm + O($) ,

provided the function F$(m, $) is integrable on the positive real axis m > 0.

From the definition of the Fourier coe#cients hn($), one can see that they decay exponentially

fast to zero as n > N($) = O(1/$), and so does the function F$(m, ·) for m > N($)$ = O(1).

To handle this type of asymptotic regime, we follow the techniques used in the long time limit of

dispersive wave trains [40].

In fact, suppose

m > maxt&[0,2")

f !(t) 0 N($) % f !max/$ .

The steepest descent method applied to the Fourier coe#cient hn($)

hn($) % h(m, $) =1

2%

" 2"

0exp

,i

$

/m' " f(')

0-d' (6.26)

with m = n$ then shows that saddle points ' = 's(m), i.e., the solutions of

f !('s) = m , (6.27)

are complex (assume that g admits an analytic extension o! the real axis), which contributes a

factor

exp("c1N($)) = exp("c2/$)

to the leading order asymptotic evaluation of the integral (6.26),

hn($) - 1

2%

!

saddles

=2%$

|f !!('s)|exp

,i

$

/m 's " f('s)

0" %i

4sgn(f !!('s))

-, (6.28)

with the appropriate choice of the saddle t = 's and some positive constants c1 and c2. Hence, for

n > N($),

#!

n=N(#)+1

|hn($)|2

1 + n2/(2<

#!

n=N(#)+1

|hn($)|2 = O(e"c1N(#))#!

n=1

O(e"c1n) = O(e"c1N(#)) .

27

We now proceed to compute the leading order asymptotics of the Fourier coe#cients in the

range 1 < n < N($), or, equivalently, $ < m < mmax. In order to get real solutions to the saddle

point equation (6.27) it must be 0 , m , f !max. The asymptotics of the Fourier coe#cients for small

$, m fixed, is dominated by the contribution of the saddle points, and one can already anticipate

that the asymptotics will not be uniform in m. In fact, expression (6.28) shows explicitly that the

!

1

g(")

"

m

" " (2) (1)s s

" max

Figure 6.1: Saddle points 's(m) for a case with a single zero of order M = 3. Of the two saddle

points corresponding to 0 < m < fmax, the one labeled ' (2)s (m) approaches the third order zero at

' = % as m ( 0.

asymptotic expansion has to break down for the saddle points 's such that f !!('s) ( 0, that is,

the asymptotics is not uniform in an open interval m > 0. Around the points where f !! vanishes

the dominant asymptotic decay rate in $ changes from that of a square root to a slower one. This

corrected behavior is defined by a local expansion of f in the neighborhood of the higher order

saddles. Figure 6.1 shows that for a smooth function f !(t) this must happen at least at one point 's

corresponding to the maximum of f !('). This saddle will contribute an overall factor with a decay

rate slower than'

$ for n - N($). A variety of other possibilities can be envisioned depending on

the graph of the function g(t).

Let us focus on the simplest case where there exists one other value t = & such that simulta-

neously f !(&) = 0 = f !!(&) = . . . f (M)(&) = 0, see Figure 6.1. Then as m ( 0 one of the saddle

branches 's ( &. We need to determine the range of m in which the asymptotic expression (6.28)

is uniform. The lower boundary of this range, m > µ($) is determined by the requirement that the

28

leading order term in (6.28) becomes comparable to that obtained by the local asymptotic analysis

at m = 0 0 's = &. This (see, e.g., Appendix C) is proportional to

h0($) = O($1

M+1 ) ,

if the (smooth) function f(') is such that f !(') possesses a single isolated zero at ' = & of order

M in (0, 2%) (that is, f !(&) = 0 together with M " 1 derivatives). The boundary of the range such

that the asymptotic expansion (6.28) remains valid is therefore defined by the condition that

$

|f !!('s(µ))| = O($2

M+1 ) 0 |f !!('s(µ))| = O($M!1M+1 ) .

This functional dependence µ($) can then be extracted by expanding f ! around &. We have

m = f !(') =f (M+1)(&)

M !(' " &)M + O((' " &)M+1)

for ' ( &, so that the zero '(m), for m small, is

'(m)" & -*

M ! m

f (M+1)(&)

+ 1M

% m1M /CM , (6.29)

where by 's % ' we denote the saddle’s branch which approaches & as m ( 0. Then

f !!(') - f (M+1)(&)

(M " 1)!(' " &)M"1

and so

f !!(') = M

*f (M+1)(&)

M !

+ 1M

m1" 1M %MCM m1" 1

M (6.30)

which then implies that the boundary layer for the change in asymptotic behavior occurs at

µ($) = const. $M

M+1 .

In the spirit of matched asymptotics, we now use this result for the dominant contribution in the

asymptotic expansion of |h(m, $)|2 to split the sums (and the corresponding integral approximation)

around the transition value m = µ($). If we denote by . the corresponding value of n,

.($)$ = µ($) 0 .($) = O($"1

M+1 )

we split the sums in the three ranges n < .1($), .1($) < n < .2($), and n > .2($), where

1 + .1($) + .($) + .2($) + N($)

orN(#)!

n=1

=*1(#)!

1

+*2(#)!

*1(#)+1

+N(#)!

*2(#)+1

.

We can compute the last sum with the integral approximation, and evaluate the first sum directly.

We work separately on each of the sums.

29

6.2.1 The “inner” approximation 1 < n < #1(")

When 1 , n , .1($) the leading order term in the asymptotic expansion as $ ( 0 of hn($) is (see

Appendix C) independent of n (except for a phase factor when multiple zeros of the same order are

present), or

|hn($)|2 - AM$2

M+1 , 0 , n < .1($) , (6.31)

where the constant AM is

AM %$

& 2

M + 1%

/ 1

M + 1

0 *f (M+1)(&)

(M + 1)!

+" 1M+1

'

)2

.

Notice that n < .($) is precisely the condition for the remainder in (6.31) to be bounded by the

leading order term. Thus, we can write

*1(#)!

1

|hn($)|2

1 + n2$2/#"2- AM$

2M+1

*(#)!

1

1

1 + n2$2/#"2 = AM$2

M+1

$

&#!

1

"#!

*!(#)+1

'

) 1

1 + n2$2/#"2 .

The last splitting of the sum is convenient because an exact summation formula is known for both

terms,#!

1

1

1 + n2$2/#"2 =%

2

#"$

coth/#"

$

0" 1

2,

and#!

*1(#)+1

1

1 + n2$2/#"2 =#"$1(/(. + 1 + i#"/$)) ,

respectively.

Since .1($) (/ as $ ( 0, the asymptotics of the /-function (the logarithmic derivative of the

gamma function)

/(z) = log z " 1

2z+ O

/ 1

z2

0

shows that the leading order contribution from the last sum is

#"$1(/(. + 1 + i#"/$)) - #"

$arctan

/ #"$(.1($) + 1)

0,

which of course coincides with the result from the integral approximation to the sum with m = n$,#!

*"

1

1 + n2$2/#"2 =1

$

" #

#*"

1

1 + m2/#"2 dm - #"$

*%

2" arctan

/$. !($)

#"0+

=#"$

arctan/ #"$. !($)

0,

with . !($) % .1($) + 1. The exact expression for the sum also yields explicitly the error of the

integral approximation through the asymptotics of /(z) when z = . ! + i#"/$,

"1

21

,1

z

-=

1

2

1(z)

|z|2 =1

2

$#"$2. !2 + #"2

30

which would be useful in collecting the estimates of all the remainders for determining the influence

of the non-leading order terms.

Thus, collecting all leading order contributions for the range 0 , n < .1($) in the series (6.25)

we have

1

2|h0($)|2 +

*(#)!

n=1

|hn($)|2

1 + n2$2/#"2- AM #" $

1!M1+M

*%

2coth

/#"$

0" arctan

/ #"$.1($)

0+

. (6.32)

The error coming from the approximation of the sums of this expression is

1

2#" $

1!M1+M

$#"µ2 + #"2

= O(#2$2

1+M ) .

Notice that if we extend .1($) all the way to .($) and let $ ( 0 in such a way that $.($) = µ($) + #,

that is

$M

M+1 2 #

the coee#cient in parentheses of the leading order term (6.32) vanishes. This vanishing takes place

when crossing the balance relation between the two small parameters expressed by

# = O(µ($)) = O($M

M+1 ) or $ = O(#1+ 1M ) ,

which therefore identifies a curve in the (small) (#, $)-plane where a change in asymptotic behavior

can be detected.

6.2.2 The “outer” approximation #2(") < n < N(")

We now work on the range .2($) < n < N($). We focus on the contribution from the leading order

terms, as any cancellation occurring at this level as the #, $ parameter plane is sampled is enough to

identify a change in di!usivity behavior. We remark that the error terms can be estimated by the

higher order terms in the asymptotics of hn($) and the use of the Euler-Mclaurin formula for sums.

This does not seem to present any conceptual obstacles, and it amounts to a laborious bookkeeping

job. It is expected that no fundamental surprises would come from the remainder terms, much as

in the case of the integral approximation above, however rigorous analysis would obviously require

this step.

We will evaluate the sumN(#)!

n=*2(#)+1

|hn($)|2

1 + n2$2/#"2

31

by its integral appoximation with m = n$. While in principle we should take into account the change

in dominant asymptotic behavior brought about by the maximum of f ! where f !! = 0 (i.e., when

m = maxf !), this change is in general dwarfed by a factor O(#2) coming from the denominator when

n$ = O(1) and 0 < # + 1. Thus, these contributions will enter at higher order in the asymptotics,

beyond the formal considerations carried out here. With this in mind, we simply replace the sum

by its integral approximation and use the asymptotics (6.28) for h(m, $). We can anticipate that

the dominant contribution will come from the saddle that limits onto the higher order zero (because

of the singular behavior in m (6.30) of the integrand) so by taking into account just this term we

have

|h(m, $)|2 =1

4%2

2%$

|f !!('(m))|so that

N(#)!

n=*2(#)+1

|hn($)|2

1 + n2$2/#"2- 1

2%

" g"max

µ2(#)

1

1 + m2/#"2

dm

|f !!('(m))| ,

with µ2($) % $(.2($) + 1). It is again convenient to rewrite the integral as the sum of two terms in

order to isolate the contribution from 0 < µ2($) + 1 as $ ( 0," f "max

µ2(#)=

" f "max

0"

" µ2(#)

0

The first integral is independent of $ and assumes a compact form when the definition of the saddle

is taken into account as a change of variables,

m = f !(') 0 dm

|f !!('(m))| = d'

and so " g"max

0

1

1 + m2/#"2

dm

|f !!('(m))| =" +

%max

d'

1 + (f !(')/#")2.

The asymptotics of this term as # ( 0 is governed by the behavior of f ! near &. By the rescaling

(f !(')/#")1M = x

it is easy to show (following the arguments in Appendix D),

" +

%max

d'

1 + (f !(')/#")2- #" 1

M

CM

" #

0

dx

1 + x2M=

#" 1M

CM

%

2Mcsc

/ %

2M

0% #" 1

M BM . (6.33)

From the scaling (6.29) and from this expression, it is possible to see the role played by the remaining

integral explicitly even before evaluating it," µ2(#)

0

1

1 + m2/#"2

dm

|g!!('(m))| -1

CM#" 1

M

" ,(#,$)

0

dx

1 + x2M

32

where the upper limit of integration is

0($, #) = (µ2($)/#")1M .

If we extend µ2($) all the way to µ($) and let # ( 0 in such a way that µ($) 2 #", the integration

in the rescaled variable x e!ectively takes place on a large interval, and this integral will go into

canceling the term contributed by (6.33). Once again, this vanishing of the leading order term takes

place when crossing the balance relation between the two small parameters expressed by

# = O(µ($)) = O($M

M+1 ) or $ = O(#1+ 1M ) .

Below this curve in the (#, $) plane, or # + µ($) the second integral becomes subdominant, and the

dominant term is given by the first integral’s contribution (6.33).

The second integral (6.33) has a compact expression in terms of the Hypergeometric function

F (a, b; c; z),1

MCM

" µ2(#)

0

dm

m1" 1M (1 + m2/#"2)

=1

CM0F

/1,

1

2M; 1 +

1

2M;"µ2

#"2

0.

From (2.16), the terms entering the dominant behavior of the enhanced di!usivity "22 collected so

far are

"22 * 1

2%AM

$1!M1+M

#

*%

2coth

/#"$

0" arctan

/ #"µ1($)

0+

+4%2

#"2" 1M

$

&BM "µ

1M2

CMF

/1,

1

2M; 1 +

1

2M;"02M

0'

) ,

which already contains the mechanism by which, across the curve µ1($) - µ2($) - #, the contribution

to "22 switches from that dictated by the first term for $ 2 #1+ 1M ,

"22 * 1

4AM

$1!M1+M

#coth

/#"$

0,

to that dictated by the second term for $ + #1+ 1M ,

"22 * 4%2

#"2" 1M

BM

=1

4%2#2" 1M

(4%2)1M %

2MCM sin ( "2M )

which coincides with the rigorous evaluation of the stationary integral.

33

6.2.3 Intermediate sums #1(") < n < #2(")

Proceeding formally, we again use the Taylor approximation to the function g(t) around the multiple

zero t = &, which with no loss of generality we can place at t = 0 by periodicity. Rescaling t by

t = 0($)s the Fourier integral becomes

hn($) 3 1

2%

" 2"

0exp

,i (n t" c

$tM+1)

-dt =

0

2%

" 2"/,

0exp

,i (n 0s" c

$0M+1sM+1)

-.

Choosing

0 =

*$

c(M + 1)

+ 1M+1

turns the integral into

hn($) 3 0

2%

" 2"/,

0exp

*

i/xs" sM+1

M + 1

0+

ds

where z % n 0. Since 0 ( 0 as $ ( 0, the upper extremum of integration can be replaced by /,

and so the Fourier coe#cient is eventually determined by the integral

AiM(z) %" #

0

*

i/zs" sM+1

M + 1

0+

ds

which can be viewed as an integral representation of the solution to the M -th order (complex) ODE

dMw

dzM= (i)Mz w ,

a generalization of the Airy equation for M > 2. The detailed study of this equation and its solution

will be postponed to future publications, where a rigorous uniformly valid asymptotic approximation

to the enhanced di!usivity will be provided. This in turn will allow to close the asymptotic gap left

by the intermediate sums .1($) < n , .2($).

Iterated stationary phase analysis addressed directly the asymptotics for the double integral

form (2.6) of the e!ective di!usivity. Its alternative representation via the series of Fourier coe#-

cients (2.17) o!ers a di!erent route, which however has at its heart the same multiple saddle point

analysis for highly oscillatory integrals in the case of winds with higher order zeros. Ultimately, of

course, both approaches must lead to the same estimates, as already evidenced here by the (formal)

identification of the switch-over scaling $ = O(#1+ 1M ) in either of these analyses.

34

1 2 3 4 5 6 7 8 9 10x 105

1

2

3

4

5

6

7

8

9

10x 106

1/Pe

S

g(t)=sin(t)+0.5sin(2t)

Figure 6.2: Contour plot of the e!ective di!usivity "22 from the numerical evaluation of the series

expression(2.17) as in Figure 5.2, only zooming in on the lower tenth of the figure. Also shown is

the curve connecting the inflection points of each curve which separates quasi-steady asymptotic

regime (integration by parts) from steepest descents regime.

6.3 Comparison between asymptotic formulae and numerical compu-

tations.

Here, we document the success in the asymptotic program in capturing many of the features and

behavior of "22. To begin, to highlight the precise transition identified between the rigorous quasi-

steady regime and the formal stationary phase regimes, we redraw Figure 5.2 in Figure 6.2, focusing

upon the lower tenth of that figure. We draw a solid curve connecting the inflection points of each

contour.

In Figure 6.3, we draw this curve connecting the inflection points from Figure 6.2 as a function

of the Strouhal number in log-log coordinates. Also shown are various linear curves for providing

a basis of comparison between the di!erent transitional scalings for several M values. Observe, for

the value of M = 3, the correct transitional curve clearly matches the numerics.

Next, in the first panel of Figure 6.4, we show a comparison between the asymptotic formula with

fixed small # = 5410"8, and varying $ in the range (5410"8, 5410"7) and the numerical computations.

35

5.5 5 4.5 4 3.5 37

6.5

6

5.5

5

4.5

4

3.5

log10 (1/Pe)

log 10

S

g(t) = sin(t) + 0.5sin(2t)

Min(abs(2nd derivative of contour))S = (1/Pe)4/3

slope = 1

slope = 6/5

Figure 6.3: Location in the Peclet-Strouhal plane of inflection points in contour plots from Fig-

ure 6.2, depicted with log-log plot of the coordinates. The critical transition curve for this case has

slope 4/3 and is drawn as a solid straight line. Also shown for comparison, a slope one curve and

the critical transition curve for a di!erent case with a 5th order zero (slope 6/5) demonstrate a

clear distinction.

For the higher values of $, the agreement is exceptional, while for smaller values, the agreement

deteriorates, as expected, as the quasi-steady regime is entered. On the right panel is shown for

this case, the uniform joint asymptotic formula given by equation (4.27). The discrepancies in the

left panel should be expected to get larger as # and $ become comparable, as the validity of the

asymptotics is broken. However, in contrast, the right panel documents the utility of having the

complete, joint, uniformly valid asymptotic expansion, and documents the excellent agreement with

the highly resolved numerical evaluation.

Lastly, in Figure 6.5, a similar regime crossing between the quasi-steady asymptotics and the

stationary phase asymptotics as shown in Figure 6.4 is shown. Here, we show only the numerical

simulation, with the left panel involving a fluctuating wind possessing a two third zeros, while the

right panel has a unique third order zero. Clearly, the transition is evident as the Strouhal number

is decreased. The e!ective di!usivity switches from decreasing to increasing as the quasi-steady

regime is entered with decreasing Strouhal number. Finally, the more prominent of the oscillations

36

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 10-7

5

5.5

6

6.5

7

7.5

8

8.5

9

Strouhal

log 10

κ22

Computed π Pe2 J0

2(1/S)

g(t)=sin(t)

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5x 107

7

7.2

7.4

7.6

7.8

8

8.2

8.4

8.6

Strouhal

log 10

κ22

g(t)=sin(t)

Computed π/δ coth(π α)(1 + sech(π α)sin(2/ε))

Figure 6.4: Comparison between several asymptotic formulae with the numerical computations for

small Strouhal behavior of the enhanced di!usivity for simple zero case in the high Peclet asymptotic

limit. On the left: the small Strouhal asymptotics compared to the numerical computations. On

the right, the uniformly valid asymptotic expansion compared to the numerical computations. In

both cases, Pe = 5 4 107.

for the case with two distinct third order zeros.

Acknowledgements

R. Camassa is partially supported by NSF grant DMS-0104329, NSF DMS-0509423, and gratefully

acknowledges hospitality by the Theoretical Division of Los Alamos Natl. Lab. during some of this

work, K. McLaughlin is partially supported by NSF grants DMS-0451495 and DMS-0200749, and

RMM is supported by NSF grants DMS-0308687, DMS-97019242, RTG NSF DMS-0502266. RC

and RM further thank NSF for their support to the completion of this work through grant CMG-

ATM-0327906. All authors thanks Air Force Grant F496209810501 for assistance in the purchase

of the UNC Applied Mathematics Alphaserver. The authors thank Norberto Kerzman for useful

discussions regarding steepest descent methods for functions of several complex variables, and also

thank Lou Howard for a helpful remark concerning the connection between zeros of the Fourier

coe#cients, and fluid particle trajectories. Additionally, the authors wish to thank the referees for

providing insightful comments and suggestions.

37

7 6 5 4 3 2 1 05

6

7

8

9

10

11

12

13

log10 S

log 10

κ22

g(t) = sin3(t), Pe = 10 6

7 6 5 4 3 2 1 09

9.5

10

10.5

11

11.5

12

12.5

log10 S

log 10

κ22

g(t) = sin(t)+0.5sin(2t), Pe = 10 6

Figure 6.5: Numerically computed enhanced di!usions plotted for Pe = 106, varying the Strouhal

number. On the left, a case with a fluctuating wind possessing two third order zeros, while on the

right, the analogous case involving only one third order zero. Observe the larger oscillations for the

case with multiple high order zeros. Further observe in both cases the vanishing of oscillation and

the increase of enhanced di!usion with decreasing Strouhal number as the quasi steady (integration

by parts) regime is entered, which is the canonical behavior expected when leaving a stationary

phase region.

Appendix A. The integral formula for the e!ective di!usivity

In this Appendix we give the detals of the derivation of expression (2.6) for the e!ective di!usivity.

We simplify notation by using & in place of &2 and x in place of x1. The general formula (2.2) for

the case of cross-wind shear velocity field (2.4) relies on the space-time periodic solutions of the cell

problem

$&t + g(t)&x " #&xx = "v(x) .

This is e#ciently solved by Fourier series

&(x, t) =+#!

k="#e"2"ikx&k(t), v(x) =

+#!

k="#e"2"ikxVk,

which yields a first-order inhomogeneous ODE for the Fourier coe#cients

$&k " 2%ikg(t)&k + 4%2k2#2&k = "Vk . (A.1)

38

The general solution of this ODE is given by the sum of the homogeneous solution

&(h)k = C exp

*

"4%2k2#

$t +

2%ik

$

" t

0g(s) ds

+

and the particular solution

&(p)k = "

" t

0

Vk

$exp

*

"4%2k2#

$(t" ') +

2%ik

$

" t

%g(s) ds

+

d' .

Seeking bounded solutions as t( "/ constrains the constant C to be

C =" "#

0

Vk

$exp

*4%2k2#

$' " 2%ik

$

" %

0g(s) ds

+

d' ,

so that the bounded solutions of (A.1) have the form

&k(t) = "" t

"#

Vk

$exp

*

"4%2k2#

$(t" ') +

2%ik

$

" t

%g(s) ds

+

d'

= "" #

0

Vk

$exp

*

"4%2k2#

$' ! +

2%ik

$

" t

t"% "g(s) ds

+

d' ! . (A.2)

It is easy to see that this solution is Tp-periodic in time if so is the time varying wind g(t), e.g.,

from the first form of &k(t) above

&k(t + Tp) =" t+Tp

"#exp

*

A(t + Tp " ') + iB" t+Tp

%g(s) ds

+

d'

=" t

"#exp

*

A(t" ' !) + iB" t+Tp

% "+Tp

g(s) ds

+

d' !

=" t

"#exp

,A(t" ' !) + iB

" t

% "g(s! + Tp) ds!

-d' !

= &k(t) , (A.3)

for any two constants A and B, which in our case are

A = "4%2k2#

$, B =

2%k

$.

(Here we have omitted the multiplicative constant Vk/$; it is worth mentioning that for time-

dependent velocity v(x, t), whereby its Fourier coe#cients become time dependent functions Vk(t),

the same argument leads to a periodic &k(t) if v(·, t + Tp) = v(·, t).)To compute the e!ective di!usivity according to (2.2), the space-time average of &2

x is now needed.

By Parseval’s equality, the space average can be computed directly in terms of the modulus of the

Fourier coe#cients &k,

|&k(t)|2 =|Vk|2

$2

" #

0

" #

0exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

d' d' ! , (A.4)

39

and this must then be time-averaged.

The resulting triple integral can be compressed into a double one. First, Fubini’s theorem

for interchanging integration order and splitting the integration domain in two along the bisectrix

' ! = ' yields

" #

0

" #

0exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

d' d' ! =

=" #

0

" #

%exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

d' d' ! +" #

0

" %

0exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

d' d' !

=" #

0

" #

%exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

d' d' ! +" #

0

" #

% "exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

d' ! d'

=" #

0

" #

%exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

d' d' ! + complex conjugate , (A.5)

by relabelling the dummy integration variables in the second integral. Hence, the final result is

determined simply by (twice) the real part of the first integral. Next, introduce the “diagonal”

coordinate transformation

' ! + ' = w ' ! " ' = y ,

whose Jacobian determinant is 1/2, whereby the integration in the upper triangular region ' ! > '

becomes " #

0

" #

%d' d' ! =

1

2

" #

0

" w

0dw dy =

1

2

" #

0

" #

ydy dw ,

with the integrand

exp

*

A(' + ' !) + iB" t"% "

t"%g(s) ds

+

= exp

*

Aw + iB" t"w/2"y/2

t"w/2+y/2g(s) ds

+

.

The full space time-averaging for e!ective di!usivity then requires the evaluation of the triple

integral

I3 =" #

0

" #

y

" Tp

0exp

*

Aw + iB" t"w/2"y/2

t"w/2+y/2g(s) ds

+

dy dw dt ,

and by shifting the time integration by t! = t" w/2 + y/2 this reads

I3 =" #

0

" #

yeAw

" Tp+r

reiB

# t"!y

t" g(s) dsdy dw dt! = " 1

A

" #

0eAy

" Tp

0eiB

# t"!y

t" g(s) dsdy dt! , (A.6)

where we have used the fact that for any periodic integrand f(t + Tp) = f(t)

" Tp+r

rf(t)dt =

" Tp

0f(t)dt ,

40

so that the imaginary integral in the argument of the exponential is in fact independent of r %"w/2 + y/2.

Expression (A.6) leads to the final form for the e!ective di!usivity average,

1

Tp

" Tp

0

" 1

0|&x(x, t)|2dt dx =

2

Tp#$

#!

k=1

|Vk|2 Re" #

0e"

4%2k2#! y

" Tp

0e

2%ik!

# t"!y

t" g(s) dsdy dt!

=2

Tp#2

#!

k=1

|Vk|2 Re" #

0e"4"2k2y"

" Tp

0e

2%ik!

# t"! !#

y"

t" g(s) dsdy! dt! , (A.7)

which, once specialized to monochromatic velocity, k = 1 say, yields expression (2.6).

Appendix B: Estimates for the sinusoidal case

In this appendix, we consider the special case of pure sinusoidal winds, for which we can take advan-

tage of the body of knowledge existing for Bessel functions, their identities, and their asymptotics.

We first explore the case with $/# fixed, while $ vanishes. Secondly, we explore the case where $ is

an algebraic power of # (sweeps).

The double integral

I %" #

0dy e"&y

" 2"

0d' e

i! (cos %"cos(%"y)) (B.1)

through the identity

cos a" cos b = "2 sin

*a + b

2

+

sin

*a" b

2

+

and the definition of Bessel function of order zero

J0(x) % 1

2%

" 2"

0ei x sin )d-

is equivalent to

I = 4%" #

0dze"2&zJ0

,2

$sin z

-. (B.2)

Now, J0(x) is an entire function of x with power series expression

J0(x) =#!

n=0

("1)n

/x2

02n

(n!)2.

Replace J0 with its power series in I, take the integral inside the series and notice that

" #

0dze"2&z sin2n z =

$

%&2n

n

'

()1

22n+1(+

1

22n(

n!

m=1

("1)m

$

%&2n

n"m

'

()1

(2 + m2,

41

so that

I =2%

(

#!

n=0

("1)n 2n!

(n!)4

1

(2$)2n+ 4%(

#!

n=1

("1)n 1

(n!)2

1

(2$)2n

#!

m=1

("1)m 1

(2 + m2

$

%&2n

n"m

'

()

=2%

(

#!

n=0

("1)n 2n!

(n!)4

1

(2$)2n+ 4%(

#!

m=1

1

(2 + m2

1

(2$)2m

#!

n=1

("1)n

$

%&2(n + m)

n

'

()1

((n + m)!)2

1

(2$)2n

(B.3)

Recall the identity for products of Bessel functions

Jµ(x) J*(x) =#!

k=0

("1)k

/x2

0µ+*+2k%(µ + . + 2k + 1)

k!%(µ + . + k + 1)%(µ + k + 1)%(. + k + 1),

and take µ = . = 0 for the first term in (B.3), µ = . = m for the second term in (B.3), respectively,

so that the square of the Bessel functions J20 and J2

m can be recognized. We obtain

I =2%

(J2

0 (1/$) + 4%(#!

m=1

1

(2 + m2J2

m(1/$) , (B.4)

i.e., an alternative derivation of (2.16) for this particular (sinusoidal) case.

Next, take Hankel’s form of Bessel J-function

Jn(x) =,

2

%x

- 12

[P (x, n) cos 1"Q(x, n) sin 1]

where 1 = x + 12n% " 1

4% and the functions P (n, x) and Q(n, x) are defined by, respectively,

P (x, n) =1

2%/n + 1

2

0" #

0du e"u un" 1

2

79

:

,1 +

iu

2x

-n" 12

+,1" iu

2x

-n" 12

>?

@

Q(x, n) =1

2%/n + 1

2

0" #

0du e"u un" 1

2

79

:

,1 +

iu

2x

-n" 12

",1" iu

2x

-n" 12

>?

@ .

We have

J2n(x) =

,2

%x

- ;P 2(x, n) cos2 1 + Q2(x, n) sin2 1" 2P (x, n) Q(x, n) sin 1 cos 1

<

=,

1

%x

- ;P 2(x, n) + Q2(x, n) + (P 2(x, n)"Q2(x, n)) cos 21" 2P (x, n) Q(x, n) sin 21

<

=,

1

%x

- AP 2(x, n) + Q2(x, n) + ("1)n

;(P 2(x, n)"Q2(x, n)) sin 2x" 2P (x, n) Q(x, n) cos 2x

<B.

(B.5)

42

Consider the asymptotic expansions of P (x, n) and Q(x, n) for large x, n fixed. It can be shown

that

P (x, n) = 1 +p"1!

k=1

("1)k K(n, 2k)

(2k)!(8x)2k+ RP (n, p, x) (B.6)

Q(x, n) =q"1!

k=0

("1)k K(n, 2k + 1)

(2k + 1)!(8x)2k+1+ RQ(n, q, x) (B.7)

where the coe#cients K(n, m) are

K(n, 2k) = (4n2 " 1)(4n2 " 32) · · · (4n2 " (2k " 1)2)

and

K(n, 2k + 1) = (4n2 " 1)(4n2 " 32) · · · (4n2 " (2k + 1)2) .

and the remainder terms RP and RQ can be estimated for if the integers p and q are su#ciently

large. Specifically, if 2p > n and 2q > n" 1 respectively for RP and RQ in (B.6) and (B.7), then

|RP (n, p, x)| ,.....(4n2 " 1)(4n2 " 32) · · · (4n2 " (2p)2)

(n)!(8x)n

.....

|RQ(n, q, x)| ,.....(4n2 " 1)(4n2 " 32) · · · (4n2 " (2q + 1)2)

(n" 1)!(8x)n

..... .

By factoring out 4n2, and provided that p = [n/2], q = [n/2]" 1, where [n/2] indicates the smallest

integer larger than n/2, each term in the asymptotic sums can be bounded as follows:

.....(4n2 " 1)(4n2 " 32) · · · (4n2 " (2k " 1)2)

(2k)!(8x)2k

..... ,1

(2k)!

,n

4x

-2k

,

and .....(4n2 " 1)(4n2 " 32) · · · (4n2 " (2k + 1)2)

(2k + 1)!(8x)2k+1

..... ,1

(2k + 1)!

,n

4x

-2k+1

,

respectively for P (x, n) and Q(x, n). Therefore, by choosing the minimum of p and q for the estimate

of the remainders RP and RQ

|P (x, n)| , 1 +[n/2]!

k=1

1

(2k)!

,n

4x

-2k

, cosh,

n

4x

-(B.8)

and similarly

|Q(x, n)| ,[n/2]"1!

k=0

1

(2k + 1)!

,n

4x

-2k+1

, sinh,

n

4x

-. (B.9)

43

The series in expression (B.4) for the integral I can be split into a finite and an infinite component

and the second term bounded from above,

N!

n=1

1

(2 + n2J2

n(1/$) +#!

n=N+1

1

(2 + n2J2

n(1/$) ,N!

n=1

1

(2 + n2J2

n(1/$) +1

N2

#!

n=N+1

J2n(1/$) . (B.10)

The identity

1 = J20 (x) + 2

#!

n=1

J2n(x)

shows that the second sum in (B.10) is such that

#!

n=N+1

J2n(1/$) , 1

2, (B.11)

uniformly in $. For the first sum in (B.10), we have from (B.5), (B.8) and (B.9)

N!

n=1

J2n(1/$)

(2 + n2=

$

%

N!

n=1

1

(2 + n2

AP 2

n(1/$, n) [1 + ("1)n sin(2/$)]+

+Q(1/$, n) [Qn(1/$, n) (1" ("1)n sin(2/$))" 2 ("1)nPn(1/$, n) cos(2/$)]}

, $

%

CN!

n=1

1 + ("1)n sin(2/$)

(2 + n2+

N!

n=1

cosh2(n$/4)" 1 + 2 sinh(n$/4) exp(n$/4)

(2 + n2

D

.

(B.12)

If we now choose

N = N($) % [1/$,] with1

2< 0 < 1

write cosh2(n$/4)" 1 = sinh2(n$/4), and use sinh(n$/4) , const. n$/4 for n$/4 , 1, with const. =

e" 1/e, etc., the second sum in (B.12) is bounded by

N!

n=1

cosh2(n$/4)" 1 + 2 sinh(n$/4) exp(n$/4)

(2 + n2, C(()N($)$ , O($1",) (B.13)

(the constant C(() takes into account the factor contributed by the (convergent) series1#

n=1((2 +

n2)"1), so that expression (B.10) for I can be bounded by

I , 2$

E

F 1

((1 + sin(2/$)) + 2(

N!

n=1

1 + ("1)n sin(2/$)

(2 + n2+ O($1",)

G

H + O($2,)

, 2$

31

((1 + sin(2/$)) + 2(

#!

n=1

1 + ("1)n sin(2/$)

(2 + n2+ O($,) + O($1",)

4

+ O($2,) (B.14)

44

The two series in the square bracket evaluate to

#!

n=1

1

(2 + n2=

%

2(coth(%()" 1

2(2,

#!

n=1

("1)n

(2 + n2=

%

2( sinh(%()" 1

2(2,

so that the asymptotic term in (4.26) in Section 4.4 finally emerges,

I , 2%$ coth(%()[1 + sech(%() sin(2/$)] + O($,+1) + O($2",) + O($2,) . (B.15)

A bound for I from below is established by dropping the infinite series in (B.10) (which amounts

to neglecting the last O-term above). The second sum in (B.12) is subtracted o! in modulus, and

bounded away by the same choice of 0. The asymptotic relation (4.26) hence follows.

We have so far treated ( as independent of # and $, that is, from (2.8), we have considered the

ratio #/$ as fixed. This corresponds to + = 0 in the sweeping curves (4.28). It remains to be seen

if the general case (2.8) for ( leads to the same asympotic formula. In any case, in terms of the

di!usivity, from (2.11)

"22 -%

#coth(%()[1 + sech(%() sin(2/$)] . (B.16)

Along the line S = 1/Pe this corresponds to

"22 - %Pe coth(4%2)[1 + sech(4%2) sin(2Pe)] . (B.17)

From this, one can see that the asymptotic dependence on Pe may not be monotonic along S =

1/Pe, since the derivative of the right-hand side

% coth(4%2)[1 + sech(4%2)( sin(2Pe) + 2Pe cos(2Pe))]

changes sign if Pe is su#ciently large (although this means Pe 3 exp(4%2) 3 1017 !)

Sweeps through the (Pe, S)-plane in the sinusoidal case with + > 0

From expression (2.10), it is clear that the above estimates generalizes to the case + > 0. Simply

notice that ( is

( = 4%/#( ,

and $ is

$ = #(+1

45

with # ( 0, everywhere in the formulae above. In particular, one needs to be keep track of the

overall factor 2(. Crucial estimate (B.13) is now

2(N!

n=1

cosh2(n$/4)" 1 + 2 sinh(n$/4) exp(n$/4)

(2 + n2, 2(C(()N($)$ , O(#(1+()(1",))

where we have used N = [$,] again, and we have noticed that C(() is bounded by

C(() ,#!

n=1

1

(2 + n2=

%

2(coth(%()" 1

2(2,

so that 2(C(() ( %/2 as ( ( /. The factor 2( in front of the tail estimates (B.10) and (B.11)

modify the tail estimate to

2(

N2

#!

n=N+1

J2n(1/$) , O

,(

N2

-= O

*$2,

#(

+

= O(#2,(1+()"() .

Hence, if we take

20(1 + +)" + > 1 + +

and

(1" 0)(1 + +) > 0

that is,1 + 2+

2(1 + +)< 0 < 1 (B.18)

which is always possible for + > 0, we have, in place of the estimate (B.14),

I , 2$

3

2(#!

n=1

1 + ("1)n sin(2/$)

(2 + n2+ O(($,) + O(#(1+()(1",))

4

+ O(#2,(1+()"()

, 2%#1+(;1 + O(#,(1+()"()) + O(#(1+()(1",))

<+ O(#2,(1+()"() , (B.19)

that is

I , 2%#1+( + o(#1+()

since when 0 is in the range (B.18), the second order term in (B.19) is subdominant, i.e.,

0(1 + +)" + > 0 .

Notice that unlike the case + = 0 when ( is constant, ( = 4%2, the contribution from the oscillating

term in the sum (B.19) proportional to 1/ sinh(%() has to be dropped because of its exponential

46

smallness as ( ( /. Lower bound estimates proceed similarly. Putting this into the expression

for the di!usivity (2.9), we get for the sinusoidal case with + > 0

"22 -2%#(+1

2$#=

%

#= % Pe

as Pe ( /. The only di!erence with respect to + = 0 is the absence of oscillations, i.e., the

asymptotic behavior is monotonic in Pe.

Appendix C: Asymptotics of hn(") as " " 0, n fixed, for

M = 3

We take

f(') = cos(') +1

4cos(2') ,

so that

f !(%) = 0 , f !!(%) = 0 , f !!!(%) = 0 , f (IV)(%) = 3 ,

that is, ' = % is a zero of order three for the derivative f !('). As in the sinusoidal case, a simple

zero occurs at ' = 0, 2%. Following [38], the first three terms in an asymptotic expansion for small

$ of the integral

hn($) =1

2%

" 2"

0ein%+ i

! f(%)d' (C.1)

are

hn -1

2%

*

("1)n %(14)

(2x)14

e"3i4 x+ i

8" +%(1

2)

x12

e5i4 x" i

4" + ("1)n %(34)(1" 4n2)

2(2x)34

e"3i4 x+ 3i

8 "

+

. (C.2)

Figure 3.6 shows a comparison between the absolute values of h1 vs. x = 1/$ as computed through

the integration in (C.1) and the asymptotic expression (C.2).

We remark that expression (C.2) for n = 0 used in the limit of fixed Strouhal, with Peclet going

to infinity (4.22) leads to a small $ approximation

"22 =$

12

16%2#2

(%(14))

2

'2

which, unlike the simple sinusoidal case, shows that the di!usivity does behave monotonically in $

(or S ) with the same power law 1/#2 = Pe2 for this higher-order-zero case. This is in contrast with

the double higher-order-zero (cubic) case

g(t) = sin3(t) ,

47

101 102 103 104

0.13

0.14

0.15

0.16

0.17

h ( )ε

ε

n

1/

Figure 3.6: Comparison between a numerical evaluation of (C.1) (solid) and the asymptotic expres-

sion (C.2) (dash)

where at the leading order we have

h0 -1

%

%(14)

(2x)14

cos,"3

4x +

%

8

-.

Thus, for this cubic cubic case,

"22 =$

12

4%2#2

(%(14))

2

'2

cos2," 3

4$+

%

8

-.

Appendix D: Estimates for the quasi steady formula

We first derive the leading order, large Peclet (# ( 0) asymptotics for the quasi steady integral

(in the limit vanishing Strouhal number, $ ( 0 ) given in (4.6). We consider only the cases with

M > 1, and note that for M = 1, for the case with g(t) = sin(t), a general formula is available

from the analysis of the series of Fourier coe#cients which rigorously applies in this region. Also

note that the general case for functions g(t) having only isolated first order zeros may be handled

directly via explicit change of variables.

High Peclet Asymptotics for Integral (4.6)

Consider the integral given above in equation (4.6):

"22 = 4%2" 2"

0

d'

16%4#2 + g2(')

48

The asymptotics for small # will be dominated near the zeroes of the the periodic function g(').

For ease in exposition, we assume that there will be a single zero of order M , located at ' = 0.

The contribution to the asymptotics from the other lower order zeros (which are necessary in this

case for the continuous function g(t) to have zero mean) is easily shown to be sub-dominant. The

final answer will be doubled from the symmetric contribution from ' = 1, the presence of which is

excluded from the following to simplify the discussion. To this end, let the function, g(') admit the

following convergent Taylor expansion:

g(') = 'M (1 + r('))

r(') = a' + · · ·

|r(')| , c ' on (0, 22)

|r(')" a' | , d ' 2 on (0, 22)

Let 2 = #1

2M , and write

A = B + C, (D.1)

B =" -

0

d'

16%2#2 + g(')2, (D.2)

C =" 1

-

d'

16%2#2 + g(')2. (D.3)

Estimate for C:

g(') = 'M (1 + a' + · · ·)

which implies g(') ) c2M for some c > 0, provided ' . (2, 1).

Thus1

16%2#2 + g(')2, 1

#2 + c 22M

, c12"2M , c1#

"1 for ' . (2, 1).

So we have the following bound which will show that contributions from C will be sub-dominant:

C , c1#"1

Asymptotics for B:

B = #"2" -

0

d'

16%2 +,

%

$1M

-2M

(1 + r('))2.

Now setting u = %

$1M

, we have B = #"2+ 1M

" -$!1M

0

du

16%2 + u2M/1 + r(#

1M u)

02 .

49

Now we may expand the integrand as follows using the identity

1

1 + x= 1" x +

x2

1 + x

applying to the integrand, we have

B =

#"2+ 1M

" -$!1M

0

du

16%4 + u2M

$

%&1" u2M

16%4 + u2Mr(#

1M u)(2 + r(#

1M u)) +

/u2M

16"4+u2M r(#1M u)(2 + r(#

1M u))

02

16%4 + u2M

16"4+u2M r(#1M u)(2 + r(#

1M u))

'

()

= #"2+ 1M (I " II + III)

The upper limit of integration approaches / as # ( 0, and as such it is worth noting that each

of these three integrals are integrable for all M > 1. We first derive the leading order asymptotics

from the first integral, and then show the remaining integrals to give sub-dominant contributions.

Now we may approximate the first integral:

(4%2)1M

16%4

" -$!1M

0

1

1 + u2Mdu =

(4%2)1M

16%4

," #

0

1

1 + u2Mdu"

" #

-$!1M

1

1 + u2Mdu

-

=(4%2)

1M

16%4

" #

0

1

1 + u2Mdu + O

/#2" 1

M

0

=(4%2)

1M

16%4

%

2M sin (%/2M)+ O

/#2" 1

M

0(D.4)

And similarly we may approximate the second integral:

|II| ," -$!

1M

0

u2M

(1 + u2M)2

;2c#

1M u + c2#

2M u2

<

= O/#

1M

0

Likewise, the third integral, III, is seen to be subdominant via:

1

16%4 + u2M

/u2M

16"4+u2M f(#1M u)(2 + f(#

1M u))

02

16%4 + u2M

16"4+u2M f(#1M u)(2 + f(#

1M u))

,

/Acu2M+1#

1M

02

(16%4 + u2M)3

= O/#

2M

0

so that III = O/#

2M

0. Consequently, we find that

"22 = #"2+ 1M

(4%2)1M

4%2

%

2M sin (%/2M)+ O

/#"2+ 2

M

0(D.5)

provided M > 1.

50

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