follow\: }.'or non-
the first two terms of the series (continuing to higher order terms is in
pcncritl coniputation;illy impractical). However. its time goes on (in partictilar for larger Reynolds numbers) the processes requirc progressively more
terms lor adcquate representation : the rcprcscntation bccomes less riipidly
sonvcrp&t. The difticulty was discussed by Orsnig and Rissonette ( 1967). I f
onc employs ;I white noise process i n the representation. which process is
fixcd for all time. then its time progresscs the physical process becomes less
and less well rclatcd to the initial white noise process. The representation has
hccn shown by Wicncr to he complete for processes of physical interest. In
order to represent the later process i n tcrnis of the enrl) whitc noise it turns
o u t that more and more terms in thc \c'ricb arc required.
To remedy this dilficitlty v;tricw investigators have pIopos~tui altering the
white noise proccss with time s o ;is to reduce the effect of this out-of-date
characteristic: see Bodner (1969). ('anavan and Leith ( 1968). and C'levcr and
Mecchiim (1972). Tlic result ofthis work shows a considerahlc improvcnicnt
over ciirlier results. and examples from Burgers' model are presented below.
One lnily deduce from this discussion t h a t the W- H representation is not
unique. Thus we may represent ; I process using the white noise process
nppropriatc Ibr it itt the initial time. Or alternatively we may use a white
noise process ;ippropriate iit ;I later timc. in order to improve the coiivcrgeiicc of the representation. O f course, one often says that usually ;I process
at later time heconics substantially statistically independent of its earlier
wlucs. t o r most purposes. that statement may he taken to be accurate.
Howcvcr. t he late process cannot literally be statistically independent of the
c;irIy process: the lutc process must i n some way grow out of the early
process. I n lilct. OIIC c;in plausibly argitc (see below) that certain high order
iiioiiiciiIs o f lhe cilrly process will he well correlated with the later process
( 1 1 0 practic;il iisc lor this interesting result is oflered here).
I t is s e w thiit tlicrc iIrc in;iny diflercnt ways to represent a given random
proccss using thc W I I expansion. Thcsc diflerent modes o f representation
arc' rcliltcci to one another by what :ire known as measure-preserving transformatiotis. In tho present pitper we propose using some of the manifest
arhitrwrincss or the representation to iniprove convergence. It is proposed
that this he done by calculating for the process using one of the presently
available (time-dcpendcnt) representations for the white noise process. Chit.
citlcukitch for some time, then stops the calculntion, readjusts the terms in
the series. prcscrving certain necessarj physical chiiritcteristics of thu
process. l h e rc;idjustment is done in such ii way as to improve convergence.
Then thc calculation is starlcti with tliesc renormiili/.ed values. continlied
the first two terms of the series (continuing to higher order terms is in
pcncritl coniputation;illy impractical). However. its time goes on (in partictilar for larger Reynolds numbers) the processes requirc progressively more
terms lor adcquate representation : the rcprcscntation bccomes less riipidly
sonvcrp&t. The difticulty was discussed by Orsnig and Rissonette ( 1967). I f
onc employs ;I white noise process i n the representation. which process is
fixcd for all time. then its time progresscs the physical process becomes less
and less well rclatcd to the initial white noise process. The representation has
hccn shown by Wicncr to he complete for processes of physical interest. In
order to represent the later process i n tcrnis of the enrl) whitc noise it turns
o u t that more and more terms in thc \c'ricb arc required.
To remedy this dilficitlty v;tricw investigators have pIopos~tui altering the
white noise proccss with time s o ;is to reduce the effect of this out-of-date
characteristic: see Bodner (1969). ('anavan and Leith ( 1968). and C'levcr and
Mecchiim (1972). Tlic result ofthis work shows a considerahlc improvcnicnt
over ciirlier results. and examples from Burgers' model are presented below.
One lnily deduce from this discussion t h a t the W- H representation is not
unique. Thus we may represent ; I process using the white noise process
nppropriatc Ibr it itt the initial time. Or alternatively we may use a white
noise process ;ippropriate iit ;I later timc. in order to improve the coiivcrgeiicc of the representation. O f course, one often says that usually ;I process
at later time heconics substantially statistically independent of its earlier
wlucs. t o r most purposes. that statement may he taken to be accurate.
Howcvcr. t he late process cannot literally be statistically independent of the
c;irIy process: the lutc process must i n some way grow out of the early
process. I n lilct. OIIC c;in plausibly argitc (see below) that certain high order
iiioiiiciiIs o f lhe cilrly process will he well correlated with the later process
( 1 1 0 practic;il iisc lor this interesting result is oflered here).
I t is s e w thiit tlicrc iIrc in;iny diflercnt ways to represent a given random
proccss using thc W I I expansion. Thcsc diflerent modes o f representation
arc' rcliltcci to one another by what :ire known as measure-preserving transformatiotis. In tho present pitper we propose using some of the manifest
arhitrwrincss or the representation to iniprove convergence. It is proposed
that this he done by calculating for the process using one of the presently
available (time-dcpendcnt) representations for the white noise process. Chit.
citlcukitch for some time, then stops the calculntion, readjusts the terms in
the series. prcscrving certain necessarj physical chiiritcteristics of thu
process. l h e rc;idjustment is done in such ii way as to improve convergence.
Then thc calculation is starlcti with tliesc renormiili/.ed values. continlied