Turbulent Dispersion

Benoit Cushman-Roisin

Thayer S hool of Engineering

Dartmouth College

Hanover, New Hampshire 03755, USA

Contribution to Mathemati s of Marine Modeling

Handbook edited by Henk M. S huttelaars, Arnold W. Heemink & Eri Deleersnijder

to be published by Springer

September 29, 2018

ABSTRACT

This hapter proposes a novel method for the modeling of turbulent dispersion in the absen e of buoyan y

e�e ts. Starting from a few salient observations, properties that an e�e tive model should possess are

identi�ed, and a model is subsequently developed to in orporate these ne essary properties. The model

turns out to make use of fra tional al ulus and leads to a non-lo al operator, whi h is hallenging from a

omputational perspe tive. Appli ations to dispersion by turbulent jets (round and planar) and the marine

Ekman layer (surfa e and bottom) demonstrate the usefulness of the model.

1

Turbulent Dispersion 2

1 Introdu tion

The purpose of this hapter is not to review the �eld observations, laboratory experiments, or turbulen e-

resolving numeri al simulations that have ontributed to our present knowledge of turbulent dispersion.

Su h reviews an be found in Csanady (1973), Fis her et al. (1979), Roberts and Webster (2002), Garrett

(2006), and Cushman-Roisin (2013), and referen es therein. Rather, our purpose is to take the salient

properties thus dis overed and to formulate a modeling method that reprodu es those properties without

the need for resolution of turbulent �u tuations. Su h non-eddy-resolving methods are ne essary in the

pursuit of large-s ale o ean modeling, espe ially at the basin s ale.

The vast majority of non-eddy-resolving models handle unresolved, sub-grid pro esses by means of an

eddy di�usivity DE (eddy vis osity νE for momentum), whi h is then made to vary heuristi ally with spa e

and time a ording to the propensity of the broader �ow to generate eddies via instabilities, su h as shear

instability. A prime example is the parameterization proposed by Joseph Smagorinsky (1963),

DE ∼ νE = ∆x∆y

√

(

∂u

∂x

)2

+

(

∂v

∂y

)2

+1

2

(

∂u

∂y+

∂v

∂x

)2

, (1)

whi h in reases with the divergen e and shear of the lo al �ow �eld under the reasoning that the greater

the gradient of the velo ity omponents, the greater the propensity of the �ow to develop eddy-generating

instabilities. Somewhat more sophisti ated models, su h as the family of k turbulen e losure models (k,

k− ǫ, k−ω; for a summary, see Cushman-Roisin & Be kers, 2011, Se tions 14.3 & 14.4), solve an evolution

equation for the turbulent kineti energy k and make the eddy di�usivity (vis osity) dependent of the level

of turbulent kineti energy. For example (Umlauf & Bur hard, 2005), the eddy di�usivity may be expressed

as:

DE = Ck2

ǫ, (2)

in whi h the fa tor C depends on the velo ity shear, and ǫ is the energy dissipation rate, whi h like k needs

its own evolution equation. The fun tion C and several terms in the evolution equations for k and ǫ areformulated based more on inferred phenomenology and empiri al eviden e than basi physi s.

Likewise, in the presen e of density strati� ation (Umlauf & Bur hard, 2005), the eddy di�usivity

(vis osity) is made to depend on the lo al Ri hardson number in su h a way that when buoyan y for es are

stabilizing (destabilizing), the di�usivity is redu ed (augmented). Again, the formulations are heuristi and

a epted based more on post-model validation than pre-model physi s. Some values have also been inferred

from measurements of dispersion (e.g., Yanagi et al., 1982), by what might be alled reverse engineering.

Aside from the fa t that these parameterizations involve mu h heuristi s and therefore stand on shaky

grounds, the eddy di�usivity approa h su�ers from a major de� ien y, namely that it predi ts a growth

proportional to the square root of time for the size of a tra er pat h, with L ≃√2DEt in whi h L is the

size of the pat h undergoing dispersion and t is time. A multipli ity of observations (Cushman-Roisin,

2013) point out beyond a doubt that pat h size in a turbulent �ow environment grows rather like the �rst

power of time or even faster (like t32), and a better model ought to predi t L ≃ u∗t in whi h u∗ =

√2k

is a turbulent velo ity or L ≃ ǫ12 t

32in whi h ǫ is the energy dissipation rate. A lear example (albeit

from the atmosphere, not the o ean) is shown in Figure 1. The unmistakable triangular shape of the plume

indi ates linear growth over distan e, whi h in a larger-s ale and therefore lo ally uniform wind orresponds

to spreading proportional to the �rst power of time.

†It should be noted that, for a ase like this, spatially

dependent eddy di�usivities as those proposed in (1) and (2) are of no use. The straight path of the plume

†Be ause buoyan y a�e ts the verti al dispersion of hot plumes, it is more instru tive when onsidering me hani al dis-

persion alone to use the horizontal view provided by aerial or satellite imagery than ground-level visualization that inevitably

looks sideways.

Turbulent Dispersion 3

Figure 1: Aerial view of an ash and smoke

plume emanating from the Mt. Etna vol-

ano in Si ily, aptured by the MODIS sen-

sor onboard NASA's Terra satellite, on 27

O tober 2002. Note the unmistakable tri-

angular shape of the plume, whi h indi-

ates linear growth over time in the hor-

izontal plane. [Photo redit : NASA℄

indi ates a uniform wind �eld from whi h we an presume homogeneous turbulen e onditions and thus a

uniform eddy di�usivity. The fa t that the plume does not widen like the square root of distan e points

to a fundamental �aw with the eddy di�usivity approa h. Put another way, the mathemati s of turbulent

dispersion, at least in this ase, should be governed by an operator other than a se ond derivative as in

mole ular di�usion. This argument was already voi ed by Ri hardson and Stommel (1948) who remarked

in the ontext of marine dispersion that �The variation of K depends on a geometri al quantity σ, andFi k's equation is also geometri al in so far as it ontains ∂2/∂x2. For this reason it is di� ult to regard

the variation of K as an outer ir umstan e deta hed from Fi k's equation. There appears to be a fault

in the equation itself.� (Note: In this quote the quantities K and σ stand respe tively for the di�usivity

and pat h size, noted DE and L above.) Notwithstanding their own statement, Ri hardson and Stommel

(1948) retained the eddy di�usivity on ept and Fi k's equation, preferring to rely on a di�usivity that

grows as a power of pat h size and proposing (in the present notation) DE ∼ σαwith α varying from 1.0

to 1.4. In a follow-up study of turbulent dispersion in the sea, Stommel (1949) reiterated the on lusion

that the Fi kian model fails to des ribe horizontal di�usion in the sea.

Ri hardson and Stommel (1948) were not the only ones to suggest a pat h-size-dependent di�usivity; oth-

ers followed, in luding Okubo (1971) and Clark et al. (1996), among others. But this is wholly inadequate

in modeling. Indeed, how ould a model be so onstru ted when the system in ludes multiple overlapping

pat hes at various stages of development? What should be the lo al value of DE where an older, wider

pat h overtakes a more re ent and smaller pat h? To be e�e tive, a model of turbulent dispersion ought to

in lude a manner by whi h the vigor of dispersion automati ally adapts to the spatial distribution of the

tra er's on entration. The answer lies in the formulation of a di�erent operator than the se ond derivative

pre eded by an eddy di�usivity. Put another way, turbulent dispersion in the environment pro eeds in a

qualitatively di�erent manner than mole ular di�usion with an enhan ed and varying di�usivity, and one

ought to pursue other mathemati al formulations.

2 Model Requirements

Here, we are not interested in developing arguments about the statisti s of dispersion su h as onsidering

the evolution of two-parti le separation or moments of displa ements (e.g. Taylor, 1921; Ri hardson &

Stommel, 1948; Hunt, 1985; Ferrari, 2007). Instead, we are strongly dire ted toward the development of a

mathemati al tool that an readily be implemented in existing models, su h as basin-wide o eanographi

Turbulent Dispersion 4

models, in whi h unresolved sub-grid s ale pro esses an be represented on the resolving grid by means of

al ulations that provide the spatio-temporal evolution of on entration �elds.

The onsiderations outlined in the pre eding se tion lead us to formulate a model of turbulent dispersion

that possesses the following three properties:

1. It must be based on physi al prin iples rather than hie�y be validated after trials;

2. It should lead to pat h growth proportional to the �rst or higher power of time; for example, in

wall turbulen e hara terized by a eddy velo ity u∗ =√

τwall/ρ, it should lead to pat h growth

proportional to the �rst power of time (L ∼ u∗t); and

3. It must be su h that the vigor of dispersion automati ally adapts to the spatial distribution of the

tra er's on entration.

Figure 2: Pat h of a passive tra er

of size L in a �eld of eddies of vari-

ous s ales. Eddies of diameter mu h

shorter than the pat h size (d1 <<L) stir the inside of the pat h while

marginally in reasing its size along its

edges. Eddies of size omparable to

that of the pat h (d2 ∼ L) greatly dis-tort the pat h and augment its size by

means of stret hing. Finally, eddies

mu h larger than the pat h (d3 >>L) merely translate the pat h around

their orbits, ausing large displa e-

ments but little dispersion. The on-

lusion is that the eddies that most

e�e tively ontribute to the enlarge-

ment of the pat h are those of size

omparable to the pat h size. As the

pat h grows in time, eddies take their

turn at being those of the most e�e -

tive kind.

To establish a working model, it is helpful to re all the essen e of turbulent dispersion. The ause

of spreading is the eddy �eld inside whi h the tra er pat h resides, with eddies moving, distorting and

stirring the pat h (Figure 2). The eddies that are signi� antly larger than the pat h sweep the latter

around their orbits, merely hanging its lo ation without a�e ting its shape or size signi� antly. At the

opposite end of the spe trum, the eddies mu h smaller than the pat h size only stir the inside of the pat h,

while marginally in reasing its extent along the edges. In ontrast, the eddies of diameter omparable

to the pat h size greatly distort it, enlarge its overall extent by means of stret hing, and ause the most

dispersion. While the eddy population may remain statisti ally un hanged over time, the eddies take their

turn in e�e ting dispersion, with the smaller eddies a ting �rst when the pat h is small and in reasingly

larger eddies a ting sequentially as the pat h widens. In terms of an eddy di�usivity, this means that the

value of the di�usivity must somehow vary with the dominant s ale of the pat h. Thus, if the pat h is

hara terized by a dominant wavenumber

† k ∼ 1L , the di�usivity must be a fun tion of k, and as k evolves,

so does the eddy di�usivity, lo ally and instantaneously without re ording the time elapsed or distan e

overed sin e the start.

†Note the subtlety in notation: straight k for the turbulent kineti energy, now itali k for the wavenumber magnitude,

and, a bit later, bold k for the wavenumber ve tor.

Turbulent Dispersion 5

The rate of growth of the pat h depends on how the orbital velo ity hanges with eddy size. Growth

rate is onstant in time when eddies all share the same orbital velo ity s ale regardless of diameter (as in

wall turbulen e hara terized by a single orbital velo ity u∗) or is in reasing with time when larger eddies

have larger orbital velo ities (as in inertial turbulen e as ade). Growth proportional to the square root

of time, as a onstant di�usivity would have it, would imply growth that slows down in time ( rate of

growth proportional to t−12) and would orrespond to a situation where eddy orbital velo ity de reases

with in reasing size, a situation never en ountered in geophysi al turbulen e.

To put these elements in mathemati al terms, let us assume that the eddy �eld is a olle tion of vorti es

of various sizes d with orbital velo ity u∗(d), i.e. with all eddies of omparable size having similar orbital

velo ities. The orresponding half-turnaround time is ∆t = πd/u∗(d), and over this time the displa ement

aused by the eddies of size d is one diameter. In terms of an eddy di�usivity, we would have

DE =1

2

dL2

dt= L

dL

dt= L

d

∆t=

1

πLu∗(d) . (3)

As remarked earlier, the eddies ontributing most to the dispersion are those with diameters omparable

to the pat h size. Thus, the dominant ontribution to DE is due to d = O(L) and

DE = O(Lu∗(L)), (4)

or, in terms of the dominant wavenumber k ∼ L−1,

DE = O(

u∗(k−1)

k

)

. (5)

In Fourier spa e, the di�usion term DE∇2c of a on entration �eld c(x, t) is −k2DE c(k, t), and the

pre eding onsiderations suggest that we repla e this by

−k2DE c → −C ku∗(k−1) c, (6)

with a onstant dimensionless oe� ient C in front and in whi h k stands for the magnitude of the three-

dimensional wavenumber k. The inverse Fourier transform provides the substitution expression in spa e:

DE ∇2c → −C1

(2π)32

∫∫∫ ∞

−∞ku∗(k

−1) c(k, t) eik·x dk . (7)

For wall turbulen e with uniform orbital velo ity u∗ a ross s ales, the substituted dispersion operator

ought to be

2Cu∗π2

∫∫∫

c(x′, t)− c(x, t)

|x′ − x|4 dx , (8)

while for the Kolmogorov inertial as ade with u∗(k−1) ≃ (ǫ/k)

13, it is

(...) Cǫ13

∫∫∫

c(x′, t)− c(x, t)

|x′ − x| 113dx , (9)

in whi h the triple integral overs the 3D in�nite spa e.

†We re ognize here expressions of the fra tional

Lapla ian (Kwa±ni ki, 2017).

†The treatment of boundaries is deli ate and will be addressed later in the ontext of one-dimensional modeling.

Turbulent Dispersion 6

3 Model Development

We now pro eed to re-derive the pre eding alternative expressions from basi physi s. For this, we begin

with the elementary adve tion equation for a passive tra er �eld c(x, t) in three dimensions:

∂c

∂t+ u · ∇c = 0, (10)

in whi h the velo ity �eld u is a three-dimensional random �ow �eld with known probability density fun tion

(pdf) f(u). Over a su� iently short time interval ∆t, the pat h is merely adve ted from x−u∆t to x, i.e.c(x, t+ ∆t) = c(x − u∆t, t), but sin e the value of u is one of many, the a tual on entration will be the

expe ted value over all possible values of u:

c(x, t+∆t) =

∫∫∫

c(x− u∆t, t) f(u) du. (11)

Note in passing how we pro eeded by solving the equation �rst and then averaging its solution, in omplete

reversal from Osborne Reynolds' original de omposition with an averaging of the equations �rst followed

by (an attempt at) their solutions. Sin e the pdf of u must be normalized (

∫∫∫

f(u)du = 1), this an be

rewritten as

c(x, t+∆t)− c(x, t)

∆t=

∫∫∫

c(x− u∆t, t)− c(x, t)

∆tf(u) du. (12)

Epps and Cushman-Roisin (2018) have shown that su h an expression may be taken to the limit ∆t → 0as long as the pdf f(u) is a stable α-Lévy distribution (a family of pdf's with parameter 0 < α ≤ 2), whi hwe hen eforth denote fα(u). The general expression for this type of distribution is best given in terms of

its inverse Fourier transform:

†

fα(u) =1

(2π)3

∫∫∫

e−|γku|α eiku·u dku (13)

in whi h the fa tor γ > 0 is a s aling fa tor (also alled elasti ity) with the same dimensions as u, whi h is

length per time here. For large argument u and for α < 2, the tail of the distribution is given by (Nolan,

2006; Epps & Cushman-Roisin, 2018):

fα(u) ≃Cα γα

|u|α+3, (14)

in whi h the oe� ient Cα is given by

Cα =2αΓ

(

α+32

)

π32 |Γ(−α

2)|

. (15)

Parti ular values are C2 = 0 (the pdf falls o� exponentially rather than algebrai ally), C1 = 1π2 and

C2/3 = 0.066011.

For α = 2, the distribution is Gaussian (the only member of the family with non-algebrai tails):

f2(u) =1

(4π)32γ3

e−

|u|2

4γ2 . (16)

The non-singular limit ∆t → 0 is obtained by taking γ =√

D/∆t so that Equation (12) be omes

c(x, t+∆t)− c(x, t)

∆t=

(

1

4πD

)32

∆t12

∫∫∫

[c(x− u∆t, t)− c(x, t)] e−|u|2∆t

4D du. (17)

†We assume here velo ity distributions with zero mean. Adding a mean omponent u to u is relatively straightforward and

yields the expe ted adve tion term u · ∇c added to the time rate of hange ∂c/∂t.

Turbulent Dispersion 7

De�ning the anterior position x

′ = x − u∆t, swit hing from u to x

′as the integration variable (with

du = −dx/∆t3), and taking the limit of a vanishing time interval ∆t, we obtain

∂c

∂t=

(

1

4πD

)32

lim∆t→0

∆t−52

∫∫∫

[c(x′, t)− c(x, t)] e−|x′−x|2

4D∆t dx′

=

(

1

4πD

)32

lim∆t→0

∆t−52

∫∫∫[

1

2(x′ − x)2

∂2c

∂x2+

1

2(y′ − y)2

∂2c

∂y2+

1

2(z′ − z)2

∂2c

∂z2

]

× e−(x′−x)2+(y′−y)2+(z′−z)2

4D∆t dxdydz

=2D

π32

∫∫∫[

ξ2∂2c

∂x2+ η2

∂2c

∂y2+ ζ2

∂2c

∂z2

]

e−ξ2−η2−ζ2 dξdηdζ = D ∇2c . (18)

Thus, we re over Fi kian di�usion when taking α = 2. Sin e this predi ts pat h growth proportional to

the square root of time, it is deemed unsuitable to model turbulent dispersion, and we reje t the possibility

α = 2.

Put another way, the probability density fun tion of velo ity �u tuations must have a so- alled �fat tail�

(that is, an algebrai ally de aying tail instead of an exponential tail) in order to be appli able to turbulen e.

This stands to reason sin e turbulen e is hara terized by a population of eddies, some of whi h may be as

large as the entire domain, leading to the likelihood of displa ements always as large as the pat h size.

For α = 1, the pdf is the Cau hy distribution,

f1(u) =1

π2

au∗(|u|2 + a2u2∗)

2, (19)

with γ = au∗ (a being a free multipli ative onstant), and Equation (12) be omes

∂c

∂t=

au∗π2

lim∆t→0

∫∫∫

c(x− u∆t, t)− c(x, t)

∆t

1

(|u|2 + a2u2∗)2du.

=au∗π2

lim∆t→0

∫∫∫

c(x′, t)− c(x, t)

(|x′ − x|2 + a2u2∗∆t2)2dx′

= C u∗

∫∫∫

c(x′, t)− c(x, t)

|x′ − x|4 dx′, (20)

with C = a/π2, in whi h we re ognize the integral operator anti ipated in (8). So, we are assured that

this model will produ e tra er pat hes that grow with the �rst power of time. In other words, the hoi e

α = 1 is suitable for the modeling of dispersion in shear turbulen e hara terized by a single turbulent

velo ity u∗. This is the ase, for example, near the surfa e of the o ean under the a tion of a wind stress

τwind = ρwateru2∗, in whi h the u∗ value is unequivo ally set by the surfa e stress.

The pdf for α = 23, whi h we anti ipate to orrespond to inertial turbulen e (Kolmogorov as ade),

annot be expressed in simpler terms than expression (13), but its asymptoti behavior for large values of

u is given in (14)-(15):

f2/3 ≃C2/3 γ

23

|u| 113with C2/3 = 0.066011 . (21)

Sin e inertial turbulen e is governed by the energy dissipation rate ǫ of dimensions L2/T 3, we now take

γ = a√ǫ∆t, and Equation (12) in the limit of ∆t → 0 be omes

∂c

∂t= C2/3 a

23 lim∆t→0

(ǫ∆t)13

∫∫∫

c(x− u∆t, t)− c(x, t)

∆t|u|− 11

3 du

= C ǫ13 lim∆t→0

∆t13

∆t

∫∫∫

[c(x′, t)− c(x, t)]

( |x′ − x|∆t

)− 113 dx′

∆t3

= C ǫ13

∫∫∫

c(x′, t)− c(x, t)

|x′ − x| 113dx′ , (22)

Turbulent Dispersion 8

in whi h the two multipli ative onstants were lumped into one, C3/2 a23 = C.

The general ase α < 2 is treated brie�y as follows. The asymptoti expression of fα(u) generates a

fa tor |u|−α−3, whi h after use of u = (x−x′)/∆t provides a fa tor ∆tα+3

. This is in addition to the ∆t−1

fa tor in [c(x′, t) − c(x, t)]/∆t and another fa tor ∆t−3arising from the hange of variable du = dx/∆t3.

The net is a fa tor ∆tα−1. The limit ∆t → 0 will be meaningful only if this power of ∆t is negated by a

fa tor of opposing power in γα in the numerator of the asymptoti expression (14) of fα(u). Thus, γ must

be made proportional to ∆t1−αα, and we write

γ = q1α ∆t

1−αα , (23)

with the quantity q having the dimensions Lα/T so that γ has the required dimensions of a velo ity.

This sets the rule to as ribe the value of α: Identify the physi al quantity that governs the nature of the

turbulent �ow �eld, determine its dimensions, and then raise this quantity to the power that transforms its

dimensions to the form Lα/T 1; the exponent of L then sets the value to be adopted for α. For example,

in wall turbulen e, the quantity governing the turbulent �eld is the fri tion velo ity u∗, with dimensions

L/T ; thus, α = 1 in this ase. For inertial turbulen e, the pertinent quantity is the energy dissipation rate

ǫ with dimensions L2/T 3, and the orresponding quantity is q = ǫ

13of dimensions L

23/T , setting α = 2

3.

This rule even applies to the non-turbulent regime of mole ular di�usion, whi h is governed by a di�usivity

D of dimensions L2/T , thus requiring α = 2.

In summary, we have developed an alternative to Fi kian di�usion that should be preferable for modeling

turbulent dispersion. It onsists of a family of fra tional Lapla ian operators with parameter 0 < α < 2.This is not entirely, for fra tional al ulus has already been proposed for the modeling of dispersion in

porous media (S humer et al., 2001; S humer et al., 2009). The �rst ase applies to the ase of boundary

turbulen e (shear turbulen e) where the turbulent velo ity �u tuations are all, regardless of eddy size, on

the order of the fri tion velo ity imposed by the stress at the boundary, u∗ =√

τwall/ρ, giving α = 1. Theequation governing the on entration c(x, t) of a passive tra er is (20), whi h with adve tion by the mean

�ow in orporated is:

∂c

∂t+ u · ∇c = C u∗

∫∫∫

c(x′, t)− c(x, t)

|x′ − x|4 dx′ . (24)

The se ond ase applies in the ase of inertial turbulen e (Kolmogorov as ade) where the velo ity �u tua-

tions u∗ vary with eddy size d a ording to u∗ = (ǫd)13. The equation for on entration is (22), whi h with

adve tion terms in luded is:

∂c

∂t+ u · ∇c = C ǫ

13

∫∫∫

c(x′, t)− c(x, t)

|x′ − x| 113dx′ . (25)

The general formulation for 0 < α < 2 is:

∂c

∂t+ u · ∇c = C q

∫∫∫

c(x′, t)− c(x, t)

|x′ − x|α+3dx′ , (26)

in whi h the fa tor q with dimensions Lα/T is the physi al quantity that governs the turbulent �eld.

The multipli ative onstant C should depend on detailed statisti s of the eddy �eld

†, but pra ti ality

suggests to use them as tunable fa tors. Sour es and sinks an be easily in luded as additional terms on the

right-hand side. The triple integrals in (24) and (25) are two parti ular ases of the fra tional Lapla ian

(Kwa±ni ki, 2017) as it an be shown that their Fourier transforms are −kαc (with α = 1, 23, respe tively).

†For shear turbulen e ase, the value of C ould perhaps be related to the ratio of the most probable velo ity magnitude

|u| to u∗.

Turbulent Dispersion 9

The pre eding formulations were developed without spe ifying boundaries and thus apply stri tly to an

in�nite domain in all three dimensions of spa e. The nature of boundary onditions a ompanying the

fra tional Lapla ian is a subje t of urrent debate (Lis hke et al., 2018). In pra ti al situations, a �rst

approximation is simply to restri t the triple integration to the �nite physi al domain. The weighing fa tor

1|x′−x|α+3 de ays with distan e, and the trun ation to �nite distan es should not be a problem in the interior

of the domain. Closer to the boundaries, a re�e tion term should preferentially be added, as dis ussed in

the following se tion.

4 Redu tion to One Dimension with Boundaries

The pre eding 3D formulations may be redu ed to a single dimension in the ase where the tra er on en-

tration depends on a single spatial variable, say z in the verti al. In marine situations, this would be the

ase when verti al mixing o urs in the presen e of mu h weaker gradients in the two horizontal dimensions.

In su h a ase, the on entration spatial di�eren e inside the integrals is c(z′, t) − c(z, t), and integration

over x′ and y′ an be performed analyti ally prior to numeri al implementation. The result is:

∂c

∂t+ w

∂c

∂z= C1 u∗

∫

c(z′, t)− c(z, t)

(z′ − z)2dz′ (27a)

∂c

∂t+ w

∂c

∂z= C1 ǫ

13

∫

c(z′, t)− c(z, t)

|z′ − z| 53dz′ , (27b)

for shear and inertial turbulen e, respe tively. The multipli ative onstant C1 in front of the dispersion

operator is now designated with a subs ript 1 to indi ate that it orresponds to one dimension. The relation

between the onstants at 1 and 3 dimensions is:

C1 = πΓ(

α+12

)

Γ(

α+32

) C, (28)

yielding C1 = πC for α = 1 (wall turbulen e) and C1 = 0.248855 C for α = 23(inertial turbulen e).

Stri tly, these expressions apply for the in�nite interval −∞ < z < ∞. When the domain is semi-in�nite,

say with a boundary at z = 0 so that 0 ≤ z < ∞, the integral is restri ted to this physi al interval in return

for one additional term that may be interpreted as a re�e tion term (Epps & Cushman-Roisin, 2018)

†.

Sin e turbulen e near a wall is most often of the shear turbulen e type, only the u∗ formulation needs to

be onsidered:

∂c

∂t+ w

∂c

∂z= C1 u∗

∫ ∞

0

[c(z′, t)− c(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ . (29)

The ase of two boundaries (say 0 ≤ z ≤ H) is more ompli ated and ne essitates the addition of an in�nite

series of re�e tion terms (Epps & Cushman-Roisin, 2018):

∂c

∂t+w

∂c

∂z= C1 u∗

∫ H

0

[c(z′, t)− c(z, t)]

{

1

(z′ − z)2+

1

(z′ + z)2

+

∞∑

n=1

[

1

(2nH + z′ + z)2+

1

(2nH + z′ − z)2+

1

(2nH − z′ + z)2+

1

(2nH − z′ − z)2

]

}

dz′ .

(30)

†The origin of these additional terms is found in the �boun e-ba k� ondition of the probability density fun tion in a

Boltzmann kineti s framework.

Turbulent Dispersion 10

5 Appli ation to Dispersion in Turbulent Jets

An interesting appli ation is to dispersion aused by a turbulent jet be ause oftentimes ontaminants are

released by means of a smokesta k or dis harge pipe into the atmosphere or water body. We distinguish

here the round jet from the planar jet.

Figure 3: A turbulent round jet

of dyed water being dis harged into

lean water. The half-angle value of

11.8◦ tends to be universal at high

Reynolds numbers. [Photo taken in

the Thayer S hool's Fluids Lab at

Dartmouth College℄

5.1 Turbulent Round Jet

The turbulent round jet (Figure 3 is reated by a �ow of velo ity U and on entration c0 exiting from

a ir ular ori� e of radius R and penetrating in an otherwise quies ent and pure �uid (no �ow, no on-

entration). The quantity governing the �strength� of the jet is its momentum inje tion m = ρ(πR2)U2,

whi h is onserved all along the jet for la k of downstream pressure for e.

†Thus the quantity governing

the turbulen e is m, whi h leads us to adopting q = RU with dimensions L2/T , and we set α = 2, whi hbrings us ex eptionally to using the lassi al di�usion model with D proportional to RU .

The governing equations in ylindri al oordinates with x dire ted along the enterline of the jet and rradially a ross it, and with the assumptions of axisymmetry, steady state and negligible radial momentum

are:

∂u

∂x+

1

r

∂

∂r(rv) = 0 (31a)

u∂u

∂x+ v

∂u

∂r= C RU

(

∂2u

∂x2+

∂2u

∂r2+

1

r

∂u

∂r

)

(31b)

u∂c

∂x+ v

∂c

∂r= C RU

(

∂2c

∂x2+

∂2c

∂r2+

1

r

∂c

∂r

)

, (31 )

in whi h u(x, r) is the downstream velo ity, v(x, r) is the radial velo ity (positive outward and mu h

weaker than u), and c(x, r) is the tra er's on entration �eld. In both momentum and tra er equations,

the di�usivity is taken as CRU , based on the assumption that in a turbulent environment all quantities

†It an be shown rather easily that there annot be any pressure for e a ross and down the jet as long as the pressure in

the quies ent �uid away from the jet is uniform and radial a eleration is weak, thus ensuring onservation of momentum.

Radial a eleration is weak be ause the jet is mu h longer than it is wide (so- alled �thin jet approximation�). Note that in

ontrast mass is not onserved in the jet as it entrains �uid form the quies ent surroundings. The amount of tra er in the jet

is onserved as long as the ambient �uid is tra er-free. The tra er on entration de reases by dilution with tra er-free �uid.

Turbulent Dispersion 11

an disperse at an equal rate be ause the stirring is aused by the shared turbulent velo ity �u tuations.

†

Be ause the jet is nearly unidire tional (|v| << u), mass onservation (31a) requires that the downstream

variations be mu h weaker than the radial variations (

∂∂x << ∂

∂r ), and we may negle t downstream di�usion

ompared to ross-jet di�usion, redu ing the momentum and on entration equations to

u∂u

∂x+ v

∂u

∂r= C RU

(

∂2u

∂r2+

1

r

∂u

∂r

)

(32a)

u∂c

∂x+ v

∂c

∂r= C RU

(

∂2c

∂r2+

1

r

∂c

∂r

)

, (32b)

There are two upstream onstraints that serve to set the velo ity and on entration amplitudes, namely

the in�ux of momentum and tra er at the jet's origin, whi h are both onserved along the jet:

∫ ∞

0

u2(r, x) 2πr dr = πR2 U2(33a)

∫ ∞

0

u(r, x) c(r, x) 2πr dr = πR2 Uc0. (33b)

The above set of equations and onstraints possesses a similarity solution of the form:

u(x, r) =RU

xu(η), v(x, r) =

RU

xv(η), c(x, r) =

Rc0x

c(η) (34)

with the similarity variable η de�ned as the ratio of spatial oordinates:

η =r

x. (35)

We note that the equal powers of r and x in the numerator and denominator of η imply that the solution

will represent a jet that widens linearly with distan e, in agreement with observations (Pope, 2000, page

100). The value η = 0 orresponds to the enterline.

The redu ed equations governing the ross-jet pro�le fun tions u(η), v(η) and c(η) are:

−u− ηdu

dη+

1

η

d(ηv)

dη= 0 (36a)

−u2 − ηudu

dη+ v

du

dη= C

(

d2u

dη2+

1

η

du

dη

)

(36b)

−uc− ηudc

dη+ v

dc

dη= C

(

d2c

dη2+

1

η

dc

dη

)

. (36 )

The a ompanying onstraints (33a)-(33b) be ome:

∫ ∞

0

u2(η) η dη =1

2(37a)

∫ ∞

0

u(η) c(η) η dη =1

2. (37b)

Constraints (37a)-(37b) serve as substitutes for boundary onditions on u(η) and c(η), to whi h we add the

obvious u → 0 and c → 0 for η → ∞ be ause the jet is on�ned. Clearly, the solution for c(η) will be thesame as u(η) be ause substitution of u for c in (36 ) and (37b) reprodu es (36b) and (37a), respe tively.

†There is eviden e (Jis ha & Rieke, 1979) that the turbulent Prandtl (heat vs. momentum) and S hmidt (tra er vs.

momentum) numbers depart from unity only when the mole ular Prandtl and S hmidt numbers are mu h less than one, as in

liquid metals, and the dependen e of the departure depends on the Reynolds number of the �ow, with smaller departures at

high Reynolds numbers.

Turbulent Dispersion 12

Figure 4: Pro�les of the downstream

velo ity u(η) (in blue) and ross-jet

radial velo ity v(η) (in red, multipliedby 10 to highlight its features) that

are solutions to (36a)-(36b) and given

by (38a)-(38b), for C = 0.033. While

the jet velo ity is fairly on�ned, the

ross-jet velo ity is not. Away from

the jet, the latter ontributes to �ow

toward the jet and, therefore, to en-

trainment into the jet. Around the

enter of the jet, the ross-�ow re-

verses, ontributing to jet divergen e

to ompensate for the downstream

onvergen e of the slowing jet velo ity

along the enterline.

Sin e these equations with a onstant vis osity apply equally well to laminar �ow, they are not new, and

S hli hting (1933) has already found the solution, whi h an be expressed analyti ally:

u(η) = c(η) =3

8C

1

(1 + aη2)2(38a)

v(η) =3

16C

η (1− aη2)

(1 + aη2)2, (38b)

in whi h the onstant a is related to the dispersion oe� ient by

a =3

64 C2. (39)

The jet velo ity pro�le u(η) is symmetri a ross the enterline, whereas the ross-jet velo ity v(η) is anti-symmetri a ross the enterline and reverses sign at η = ±1/

√a. The lower the value of C, the faster the

enterline velo ity (3/8C) and the narrower the jet (η50% = 64(√2 − 1)C2/3). The pro�les of u(η) and

v(η) are plotted in Figure 4 for C = 0.033. The jet velo ity pro�le u(η) (Figure 5) is not Gaussian, with

algebrai rather than exponential tails, but ould be mistaken for Gaussian around the peak.

Ba k with dimensions in the (x,r) plane, the velo ity and on entration �elds are:

u(x, r) =3RU

8C x

1(

1 + a r2

x2

)2, v(x, r) =

3RU r

16C x21− a r2

x2

(

1 + a r2

x2

)2, c(x, r) =

3Rc08C x

1(

1 + a r2

x2

)2. (40)

An important hara teristi of the round jet is the rate at whi h it widens with downstream distan e, alled

the spreading rate and de�ned from the distan e r50% at whi h the jet velo ity drops to half of its enterline

value:

S =r50%x

=

√√2− 1

a= 8C

√√2− 1

3= 2.973 C . (41)

Laboratory observations (Hussein et al., 1994) for Re = 95, 500 give S = 0.098 ± 0.004, yielding C =0.033± 0.013. This orresponds to a oni al jet with half-angle of 11.8◦ when measured where velo ity has

dropped to 12% of its enterline value (Figure 3).

Turbulent Dispersion 13

Figure 5: Close-up on the downstream velo ity pro�le (solid bla k line), re-s aled by enterline value versus

η/η50% in omparison with the best-�tting Gaussian distribution (red dashed line) and data fromWygnanski

and Fiedler (1969) for Re ≈ 105. The ore of the jet has a very nearly Gaussian stru ture but has thi ker

tails. In the tail, the data seem to onform better with the Gaussian pro�le, but in this region at the

edge of the jet, intermitten y is very pronoun ed, and velo ity values are highly variable, as sket hed by

Wygnanski and Fiedler (1969) by means of envelope lines and hat hings.

While the on entration pro�le c(η) is identi al to the velo ity pro�le u(η) in the similarity solution, the

measurement of on entration values in the laboratory is an independent endeavor. The experiments by

Be ker et al. (1967), who used oil smoke as a passive tra er in a turbulent air jet (with S hmidt number of

about 38, 000), on�rm that the on entration pro�le, like the velo ity pro�le, exhibits a similarity behavior,

furthermore with a radius of half value in reasing with downstream distan e 0.106 x, whi h is nearly the

same as that of velo ity (Hussein et al., 1994). Figure 6 ompares the similarity solution c(η) given by

(38a) to the laboratory data of Be ker et al. (1967).

5.2 Turbulent Planar Jet

The turbulent planar jet is reated by a �ow of velo ity U and on entration c0 exiting from an elongated

slit of width W and penetrating in an otherwise quies ent and pure �uid (no �ow, no on entration).

The quantity governing the �strength� of the jet is its momentum inje tion per unit length along the slit

m = ρWU2, whi h is onserved along the jet for la k of downstream pressure for e (for the same reason as

for the round jet). Thus the quantity governing the turbulen e is m, whi h leads us to adopting q =√WU2

with dimensions L32/T , with α = 3

2in this ase.

Turbulent Dispersion 14

Figure 6: Comparison of the on en-

tration pro�le in a turbulent round jet

as predi ted by the similarity solution

(38a) with C = 0.033 (normalized by

enterline value) and laboratory data

reported by Be ker et al. (1967). The

s atter in the data is primarily aused

by turbulent �u tuations and turbu-

len e intermitten y at the edges, not

experimental ina ura ies. The �t is

ex ellent in both enter of jet and at

the edge.

The governing equations in Cartesian oordinates with x dire ted downstream along the enterline of

the jet, y transversely in the dire tion along the slit, and z a ross the jet, and with the assumptions of

steady state, negligible ross-jet momentum, no variation in the y−dire tion are:

∂u

∂x+

∂w

∂z= 0 (42a)

u∂u

∂x+ w

∂u

∂z= C

√WU2

∫∫∫

u(x′)− u(x)

|x′ − x| 92dx′dy′dz′ (42b)

u∂c

∂x+ w

∂c

∂z= C

√WU2

∫∫∫

c(x′)− c(x)

|x′ − x| 92dx′dy′dz′ , (42 )

in whi h u(x, z) is the downstream velo ity, w(x, z) is the ross-jet velo ity (mu h weaker than u), andc(x, z) is the tra er's on entration �eld.

Applying now the assumption of mu h weaker dispersion in the downstream dire tion x than in the

ross-jet dire tion z, as we did for the round jet, allows us to retain only the z variations in [u(x′)− u(x)]and [c(x′)− c(x)] inside the integrals and thus to integrate over x′ and y′ on e and for all:

∫∫ ∞

−∞

1

[(x′ − x)2 + (y′ − y)2 + (z′ − z)2]94

dx′dy′ = πΓ(

54

)

Γ(

94

)

1

|z′ − z| 52=

2.513274

|z′ − z| 52. (43)

The redu ed equations are

u∂u

∂x+ w

∂u

∂z= C1

√WU2

∫ ∞

−∞

u(x, z′)− u(x, z)

|z′ − z| 52dz′ (44a)

u∂c

∂x+ w

∂c

∂z= C1

√WU2

∫ ∞

−∞

c(x, z′)− c(x, z)

|z′ − z| 52dz′ , (44b)

with C1 = 2.513274 C. The inje tion of momentum and tra er, whi h are both onserved downstream,

provide the following two onditions:

∫ ∞

−∞u2 dz = WU2

(45a)

∫ ∞

−∞uc dz = WUc0 . (45b)

Turbulent Dispersion 15

As for the round jet, a similarity solution exists, this time with

u(x, z) = U

√

W

xu(η), w(x, z) = U

√

W

xw(η), c(x, z) = c0

√

W

xc(η) (46)

with similarity variable

η =z

x, (47)

whi h means that the jet widens linearly with downstream distan e like the round jet, but the strength of

the jet now de ays with distan e x like 1/√x instead of 1/x. This is exa tly what is observed (Pope, 2000,

page 135).

The equations governing the similarity pro�les are:

−1

2u− η

du

dη+

dw

dη= 0 (48a)

−1

2u2 − ηu

du

dη+ w

du

dη= C1

∫ ∞

−∞

u(η′)− u(η)

|η′ − η| 52dη′ (48b)

−1

2uc− ηu

dc

dη+ w

dc

dη= C1

∫ ∞

−∞

c(η′)− c(η)

|η′ − η| 52dη′ . (48 )

There is a single dimensionless parameter, C1, whi h measures the strength of both momentum and tra er

dispersion. Integration of (48a) from w = 0 along the enterline by symmetry yields:

w = ηu− 1

2

∫ η

0

u(η′) dη′. (49)

The inje tion onditions (50a)-(50b) that set the amplitudes of u and c redu e to:

∫ ∞

−∞u2(η) dη = 1 (50a)

∫ ∞

−∞u(η)c(η) dη = 1 . (50b)

As for the round jet, the solution for c(η) will be the same as u(η) be ause substitution of u for c in (48 )

and (50b) reprodu es (48b) and (50a), respe tively.

In the absen e of a known analyti al solution, we pro eed numeri ally as follows. An initial, bell-shape

guess is made for the fun tion u(η) with amplitude set by (50a). From it, w(η) is al ulated using (49).

The two fun tions are then inserted into the x−momentum equation (48b) in whi h a fake-time derivative

du/dt is introdu ed on the left. This permits to update u(η), on whi h ondition (50a) is imposed again,

and the steps are repeated until onvergen e. Figure 7 displays the two velo ity pro�les for C1 = 0.011(C = 0.00438), the value that yields a jet spreading rate S = dz50%/dx = 0.1 to mat h observations (Pope,

2000, page 138). Figure 8 ompares the half-jet pro�le obtained numeri ally with the solution obtained

with onstant eddy vis osity and laboratory data, both from Heskestad (1965).

6 Turbulent Flow along a Wall � The Logarithmi Velo ity Pro�le

It was shown in Se tion 4 that in the presen e of a wall boundary at z = 0 and with q = u∗ as the governingquantity ( ase α = 1), the dispersion operator is that given in (29):

∂c

∂t+ w

∂c

∂z= C1 u∗

∫ ∞

0

[c(z′, t)− c(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ . (51)

Turbulent Dispersion 16

Figure 7: Pro�les of the downstream

velo ity u(η) (in blue) and transverse

velo ity w(η) (in red) that are solu-

tions to (48a)-(48b), for C1 = 0.011.While the jet velo ity is fairly on-

�ned, the transverse velo ity is not.

Away from the jet, the latter on-

tributes to �ow toward the jet and,

therefore, to entrainment into the

jet. Around the enter of the jet,

the ross-�ow reverses, ontributing

to jet divergen e to ompensate for

the downstream onvergen e of the

slowing jet velo ity along the enter-

line.

We now apply this to unidire tional turbulent �ow along a wall by repla ing the on entration c(z, t) bythe velo ity u(z, t). In steady state and absen e of a pressure gradient, the x−momentum equation redu es

to a single term:

0 = C1 u∗

∫ ∞

0

[u(z′)− u(z)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ . (52)

It an be shown (Cushman-Roisin & Jenkins, 2006) that the exa t solution to this integral equation with

two degrees of freedom is:

u(z) =u∗κ

lnz

z0, (53)

in whi h the fa tors κ and z0 serve, respe tively, has a multipli ative onstant and an additive onstant.

We re ognize here the well known logarithmi pro�le of velo ity for turbulent �ow along a wall with κ being

the von Kármán onstant and z0 the roughness height.

Epps and Cushman-Roisin (2018) showed that the shear stress asso iated with the integral in (52) is:

τxz = C1ρu∗

∫ ∞

0

[u(z′)− u(z)]

[

1

z′ − z− 1

z′ + z

]

dz′. (54)

The logarithmi velo ity pro�le renders this stress onstant, as it should be in a steady �ow with no

nonlinear adve tion and no pressure gradient. After substituting (53) in (54), we obtain:

τxz = C1ρu∗

∫ ∞

0

u∗κ

ln

(

z′

z

) [

1

z′ − z− 1

z′ + z

]

dz′ = C1ρu2∗κ

π2

2=

π2C1

2κρu2∗ . (55)

Sin e the turbulent velo ity u∗ is de�ned from the stress as τxz = τwall = ρu2∗, it follows that:

π2C1

2κ= 1 → C1 =

2κ

π2= 0.0811 . (56)

for κ = 0.40. This suggests whi h value should be used for the onstant C1 when the quantity governing

the turbulen e is the turbulent velo ity u∗. The orresponding onstant in three dimensions is C = C1/π =2κ/π3 = 0.0258. We note that this value is fairly lose to the ones determined for turbulent jets from �t

with observations, namely C = 0.033 for the round jet, C1 = 0.011 for the planar jet.

Turbulent Dispersion 17

Figure 8: Close-up on the downstream velo ity pro�le (solid bla k line), re-s aled by enterline value versus

η/η50% in omparison with the analyti al solution obtained for onstant eddy vis osity (red dashed line)

and data from Heskestad (1965) for Re = 3.4 × 104. The data reveal a tail thi ker than that predi ted

by the onstant eddy di�usivity solution but thinner than that of the present solution. At the far end

(η/η50% ≥ 3), the laboratory data ontinue to show non-zero values in agreement with the heavy tail of the

present solution and in ontrast to the solution with onstant eddy di�usivity.

7 Appli ation to the Marine Ekman Layer

7.1 Surfa e Ekman Layer

The surfa e Ekman layer in the o ean is a manifestation of downward momentum dispersion from a wind

stress at the surfa e

†. The governing equations are those of Ekman dynami s(Cushman-Roisin & Be kers,

2011, Chapter 8) ex ept that we now substitute the pre eding 1D turbulent dispersion operator to represent

the fri tional terms. In the absen e of horizontal gradients (and thus of horizontal adve tion and a horizontal

pressure for e) and over an in�nitely deep bottom, the equations to be solved are:

∂u

∂t− fv = C1 u∗

∫ 0

−∞[u(z′, t)− u(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ (57a)

∂v

∂t+ fu = C1 u∗

∫ 0

−∞[v(z′, t)− v(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ , (57b)

in whi h u∗ is set by the stress exerted on the surfa e by the wind, u∗ =√

τwind/ρ, whi h we take in the

x−dire tion and onstant over time suddenly started at t = 0 over an o ean initially at rest. The value of

C1 is given by (56) under the premise that the presen e of rotation does not appre iably a�e t the nature of

the shear turbulen e near the surfa e. The se ond term inside the bra kets in the integrand of (57a)-(57b)

†Re all that a shear stress is a momentum �ux.

Turbulent Dispersion 18

is in luded to take into a ount the presen e of the surfa e boundary at z = 0. Boundary onditions are:

τx = ρu2∗, τy = 0 at z = 0 (58a)

u → 0, v → 0 as z → −∞. (58b)

After s aling the velo ity omponents by u∗, depth by u∗/f and time by 1/f , we obtain the following

dimensionless equations

∂u

∂t− v = C1

∫ 0

−∞[u(z′, t)− u(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ (59a)

∂v

∂t+ u = C1

∫ 0

−∞[v(z′, t)− v(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ . (59b)

The solution pro eeds numeri ally from rest after a suddenly imposed stress in the x−dire tion at z = 0(enfor ed as a body for e over the top grid ell) and until steady state is rea hed. Figure 9 shows that,

after a several inertial os illations, the �ow settles into a steady state. Turbulent fri tion has a damping

e�e t, as expe ted.

Figure 9: Evolution of the surfa e ve-

lo ity following the sudden imposition

of a surfa e stress in the x−dire tion.Inertial os illations are damped, and

a steady �ow is a hieved. The surfa e

velo ity makes an angle of 27◦ to the

right of the wind stress.

Steady-state velo ity pro�les (Figure 10) show an interplay between the two velo ity omponents hinting

at veering with depth. If the depth of the Ekman layer is de�ned as the depth where the velo ity magnitude

has dropped to 5% of its surfa e value, the result is 0.63u∗/f . This agrees with observations (Cushman-

Roisin & Be kers, 2011, page 255), both in its analyti al expression as well as in its numeri al value. The

veering of the velo ity with depth is prominently displayed in Figure 11. The angle between surfa e stress

and surfa e velo ity is 27◦ (see also Figure 9), noti eably lower than the 45◦ of the lassi al Ekman veering

obtained with a onstant eddy di�usivity. We also note from the time evolution shown in Figure 9 that the

angle between wind stress and surfa e velo ity an be very mu h smaller than the ultimate angle of 27

◦in

the early part of the �rst inertial os illation. Thus, in a situation with variable winds, the surfa e urrent

an be expe ted to be more losely aligned with the wind than in steady state (Sta ey et al., 1986).

7.2 Bottom Ekman Layer

The bottom Ekman layer di�ers from the surfa e Ekman layer by its for ing: Instead of being driven by

an applied boundary stress and having a vanishing velo ity at great distan e, the bottom Ekman layer is

driven by a non-zero geostrophi �ow above and has a vanishing velo ity at the boundary. The velo ity

equations are the same as (57a)-(57b) ex ept for the bounds of integration now running upward from 0 to

Turbulent Dispersion 19

Figure 10: Velo ity pro�les in the tur-

bulent surfa e Ekman layer a ording

to equations (59a)-(59b). The hori-

zontal bar indi ates the level at whi h

the velo ity magnitude has dropped

to 5% of its surfa e value, serving as

a measure of the Ekman layer depth.

Figure 11: Velo ity hodograph for the

surfa e Ekman layer. The angle be-

tween the surfa e velo ity and the ap-

plied stress is 27

◦. A ve tor velo ity

is indi ated every 0.15 dimensionless

depth units (every 10 grid points with

∆z = 0.015).

in�nity. The boundary onditions are:

u = v = 0 at z = 0 (60a)

u → ug, v → vg as z → ∞. (60b)

By aligning the x−axis with the dire tion of the geostrophi �ow, we take vg = 0. After s aling the velo ity omponents by ug, height by ug/f , and time by 1/f , the equations redu e to

∂u

∂t− v = C1 λ

∫ ∞

0

[u(z′, t)− u(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ (61a)

∂v

∂t+ u = C1 λ

∫ ∞

0

[v(z′, t)− v(z, t)]

[

1

(z′ − z)2+

1

(z′ + z)2

]

dz′ , (61b)

in whi h λ = u∗/ug. The value of C1 remains that given in (56).

The numeri al solution pro eeds as follows: First, a guess value is made for the parameter λ sin e the

bottom stress is not known a priori. The velo ity �eld is initially set as the geostrophi �ow throughout

the water olumn (u = 1, v = 0), and the bottom value u(z = 0) is suddenly set to zero to enfor e no slip

on the bottom boundary in a ordan e with (60a). The equations are then mar hed numeri ally in time

until steady state is rea hed. On e the velo ity pro�le is known, the two stress omponents are determined

with use of (54), on e with velo ity u(z) and on e again with v(z). The magnitude of the stress ve tor

(nondimensionalized by ρu2g) yields a value for λ2, and the resulting λ is ompared to the initial guess.

Iterations are performed until the starting and ending values of λ oin ide. That value is found to be

λ = 0.0905 for C1 = 0.0811.

Turbulent Dispersion 20

Figure 12 displays the velo ity pro�les. If the thi kness of the bottom Ekman layer is de�ned as the

height at whi h the magnitude of the velo ity �rst rea hes the magnitude of the geostrophi velo ity aloft

(Garratt, 1992, page 287), as indi ated by the horizontal bars a ross the velo ity pro�les in Figure 12, the

expression is

h = 0.34u∗f

. (62)

Not only is the expression proportional to the ratio u∗/f as observed, but the numeri al oe� ient is also

within the range 0.2 − 0.4 derived from multiple observations of the neutral atmospheri boundary layer

(Garratt, 1992, page 288, and referen es therein) and 0.3− 0.4 for bottom marine Ekman layers at various

lo ations (Kundu, 1976; Mer ado & Van Leer, 1976). Figure 13 displays the orresponding hodograph,

revealing a small angle of 18◦ between near-bottom velo ity and geostrophi �ow aloft, on�rming that in a

turbulent situation the angle is mu h less than the 45◦ predi ted by a theory with a onstant di�usivity, as

seen in observations. Perlin et al. (2007) quote a range of values between 10◦ and 23◦ for bottom Ekman

layers in various oastal regions.

Figure 12: Velo ity pro�les in the

turbulent bottom Ekman layer a -

ording to equations (61a)-(61b) us-

ing 1501 grid points overing the �-

nite domain 0 ≤ z ≤ 2 (only the bot-

tom 10% are shown here) and run un-

til steady state is a hieved. The hor-

izontal bars a ross the velo ity pro-

�les indi ates the level at whi h the

velo ity magnitude �rst equals the

geostrophi value.

Figure 13: Velo ity hodograph for the

bottom Ekman layer. The angle be-

tween the bottom velo ity and the

geostrophi �ow aloft is 18

◦.

8 Con lusions

Notwithstanding the wide use of the eddy vis osity and di�usivity formulation in urrent marine models, it

is argued that a fundamentally di�erent approa h is required for the proper modeling of turbulent dispersion

in the o ean. The ne essary ingredient is a mathemati al operator that represents stirring by eddies of size

omparable to the length s ale of the instantaneous on entration �eld, and thus by eddies of in reasingly

Turbulent Dispersion 21

longer s ales as the on entration �eld widens over time. Put another way, the formulation needs to

re ognize in some automated way that, as a non-uniform distribution of a substan e

†evolves over time, the

a tion of turbulen e is to disperse this substan e by the sele tive a tion of those eddies that o ur on the

same length s ale as the distribution in that region and at that time. These arguments lead to an integral

operator that repla es the lassi al di�usion operator:

DE∇2c −→ C q

∫∫∫

c(x′, t)− c(x, t)

|x′ − x|α+3dx′, (63)

in whi h C is a dimensionless onstant, q is a dimensional variable hara teristi s of the lo al turbulent �eld,

with dimensions Lα/T , and α is the power of L in the dimensions of q, whi h must be 0 and 2. Examples

are q = u∗ with α = 1 in wall turbulen e and q = ǫ13with α = 2

3for inertial turbulen e. The possible ase

of multiple turbulent hara terizations superimposed onto one another remains to be elu idated.

Formulation (63) was applied to ases of round and planar turbulent jets, showing the ability of the

formulation to produ e solutions in good agreement with laboratory data. Before a value was as ribed

to the onstant C, it was shown that the model possesses a similarity solution that mat hes the observed

behavior, namely a enterline velo ity de reasing with downstream distan e at the orre t power (1 for the

round jet and

12for the planar jet) and a jet width in reasing proportionally to downstream distan e. After

the value of the parameter C was adjusted to reprodu e one feature of the solution, hosen here to be the

angle at whi h the jet widens with downstream distan e, the predi tion of other information pertaining to

the jet, su h as its ross-stream velo ity pro�le, followed without further tuning. What sets the value of Cis not yet lear, but it appears that the various adjusted values are not too disparate (0.0044 ≤ C ≤ 0.033).Assignment of a value for C remains an area of inquiry.

The presen e of a boundary auses the appearan e a se ond, re�e tion-like term in the integrand of (63)

[see (29) in the 1D ase of wall turbulen e℄. Appli ation of this operator to the ase of a turbulent �ow

along a wall reprodu es the well known logarithmi pro�le. A onne tion is then possible between the von

Kármán onstant and the new onstant C, whi h leads to C = 0.0258 (C1 = 0.0811 in one dimension) for

wall turbulen e. There is then no parameter left to tune! Using this value, the formulation was applied to

the surfa e and bottom Ekman layers, reprodu ing a urately the key aspe ts of observed turbulent Ekman

layers in both atmosphere and o ean, namely angle of veering and Ekman layer thi kness.

The integral operator in (63) demands that, at every grid point at every time step, di�eren es be

omputed with values at all other grid points. This is omputationally burdensome. The appli ations

presented herein did not take any short ut, but it is hoped that numeri al modelers an �nd ways to

redu e the number of al ulations that are ne essary to obtain approximate but a urate evaluations of the

integrals. A point of departure is to re ognize that di�eren es be ome less important at larger distan es by

virtue of the in reasing distan e |x′ − x| raised to a power α + 3 > 3 in the denominator. The questions

are: How far or near an the integral be trun ated to guarantee an a eptable approximation? Or, better,

is there a way to approximate analyti ally the ontributions of tails in order to avoid al ulating di�eren es

a ross distant points at any time? Kämpf and Cox (2016) have begun this investigation.

ACKNOWLEDGEMENT

The author thanks Professor Brenden Epps of Dartmouth College for his assistan e with the Lévy distri-

butions and their asymptoti behavior, and the editors of this volume for their invitation, en ouragements,

and support.

†The word �substan e� should be understood broadly here. It may be a passive s alar or an a tive s alar, su h as temper-

ature, or even momentum.

Turbulent Dispersion 22

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