SLIDE 18 Scale-Dependent Eddy-Diffusivity
,-
i22
I>. Y. Richardson.
r31* ,B
.L;,&,, data are s~l~nmariset'i
in the following t:ible :---
I
I
Xcfc rc~iice.
k
I
I
..-
.
.-- I.-_
~ C C - I
Clli.
!
ii
from nlolecalar diffasicn of o:;ygcn into nilrcr:_;t>n (Kayo and i.rhy9s ' P!iysi~.ni nnri c
t ) . 1.
7 x 0 1
5 x L
O '
J+or P; s o preceding discnssioii.
1
K ai-9 nietxcs above ground from sncmometsrs at, lleigllts ~d 8, iii nnri 32 mrtn,s (12.8chmidi. ' LVlra. Aliid. 1
13.2 Y 1
"
1.5 s 10" Fiizb.,' I'la, rol. 126, p.
_
773 (1917)). 1 i
1
__ 1.4 :
: 10' . . .
I< from p;lot balloti~sat fieight~
helwcon 100 and 800 metres (Taylor, '
- Phil. Trans.,' A, rol. 215, p. 21 (1914),
6 x 10' :r,lsoHc~selberp
ant1 iiverdrnp, ' 1,eipzig Geophps. Inst.,' ,%r, 8, Heft 10 (19?5)). Meteom!ogical Soc,iety n'lomoirs,' No. I ).
VO~CZII;,
a h , same referenr.~ as last
5 x 10"
1
5 % loG ....................
. . . . . . . ._____--....~-l.I.-___._-_---_.~. I
_. _^__l-_l_l__/_
1 !
!
Dii?'u%io?i due to cvcloriea ~ n a r d e d
as devint,io~,sfrom
" t1,r rwan circulation of tiic latitltde (Ueiant, Xeo'.
Ant,.,' K. :I, also (1921), '
- Wicn. Aliad. \Vls..;. firtzb.7
Iln, u(bl 130, p. 401 (192l)).
Bijice, when nob obstrtlcted by the gron~icl,smoke spreads a,bout as rrillcll iiorizorrtally as it does vertically,':. the obfiervations a t t,ho s~~lallec valncs of I, f;hon,gnmaole in -the vertical, c8n be treated its applicable to the horizonta'l.
Tli~ii the wllole collection is coherent.
l'ilr: logarihhms 01 K and I when p!otted on a graph (fig. 8
)are seen .i:o lie
eIx,si: l
- a line of siigEiL cur-wture. I%
is harcily ~rorth while to tliscuss cictails tmi-ii ohserwtioiis linw b9en rnntic ixl n manner appopriate for the cltti;i.~:minatIo~l
' (1) rat11.e~
i;h:ru of K. .How such observatioizz: could be
- bta,iued xvill be discusscil in 7.
r1.1;~ straight line on the logarithnlk diagram wh-icil eorrei;poilils to K 2
;
IT
0.2 k"'"
also fitx the ~bser-rra~tions almost 8s :is .the curve in the limited
i7,-c!l
i : 3 1 1 g B between i = =
~rietr:: ~+,i:d1 -
1 10 kiiometreu. For ~-milierna.ticd
qinipiieity tAis orrn-uiil i ~ l l lix: riscd ia t'rle .illus%rielions whiclt follow. T!:rri3 in this range F ( I ) = : 0.4 lUi" a~proxin~a-i-ely, wheri t'lre amit,.; :hr.e cen iirrarlres a,nd seconds.
$ : c. 1. vro,ylt3r,
Richardson’s table of raw data Richardson’s approach was semi-empirical. By estimating “effective diffusivity” K = |∆x|2/t as a function of ℓ =
data that K(ℓ) ∼ K0ℓ4/3. He proposed that the probability density func- tion of the separation vector ℓ = x1−x2 would satisfy a diffusion equation ∂tP(ℓ, t) = ∂ ∂ℓi
∂ℓi (ℓ, t)
- with scale-dependent 2-particle eddy-diffusivity.
This equation predicts at long times that |x1(t) − x2(t)|2 ∼ t3, averaging over velocity realizations.