450
w. ( ' . 511~1"111\L1
in I.'ig. 2. I'Iicr-c the rclwsentation pivc\ \ irttlitlly perfect results for moderate
iimcs through Kc = 10. Marhed lie\ iations occur for this niodilic:ition iII
Re = 70. The W t { results ;ire Cotllpitrcd with numerical cxprinicnts pcrIbriiicd on thc model [.Eel. ( I )I.
This work hits shown, ;is discussed Libove, thiit the second term in thc
expiinsion beconics comparablc to tlic tirst. and ultiniatcly exceeds it i n
viilut': the expansion n o longer converges sufficiently rapidly so that the first
two terms ;ire adequate to represent the process. We are faced then with the
problem of altering the expansion in such il way that the higher orcicr lcriiis
rcniiiin smilll compared with the first term. Of course. if the process develops
largc non-Gaussian parts. this is not possible even in principle. Hurgcrs'
model (see ('lever itlid Meccham. 1972) may not be ideally suited in this way
;is ii testing ground since it shows stlhstantiiil non-Gaussian portions as the
process tlcvclops. Ncvcrthelcss we propose to test the renornialization suggestion otrercd here against this niodcl equation. ( I t is worth noting t h a t
driven random Ilurgcrs' processes arc probably niorc easily treated using the
W t1 expansion tliiin i\rc decaying processes. This may be SO becallst! O W
a r n use the white noisc process associated with the currcnt vnlue of the
forcing term i n the reprcsentation. We are thus not dependent tipon it white
noise process determined at an initial time. One might expect for this re3soii
bcttcr convcrgcncc properties for such 11 forced riindom process. Many diffusion processes niay be treated in this u;iy.)
3. KI~NOHMAI.I/AI'ION FOR Ht:HC;tRS' M0I)E'I.
Starting from givan initiiit values for the kernels [using Eq. (3)] we CLIICUtitle thc Viilttcs of the kcrncls up to ;I later time. Then the calculation is
stopped: the tr;inspired time is chosen i n such a way that the higher order
Icrni K'" rcl1iiiilis stii;ill comp;ircd with K'". Thus one may reasonably hope
~ l i i t ~
tlic expiinsion is still riipidly convergent. Once the calculation is
vtoppcd. onc will in gencriil lind that the higher ortier term in the rcprcscntation of the cncrgy spectrum !i2 [see Eq. ( 6 ) ] has increased in size conipared
with its value a t the initial time. We now propose renormiilizing.. changing
the kcrncls K"' and K'" i n such ;I way ;IS to minimize E 2 , at the same time
niinimiring the crror in the transfer term T ( k ) given by Eq. (7) and preserving I.:(k). To simplify the prohlem we shall adjust K f 2 ' choosing new valucs
A"' and using the calculated values for K"'. 'The problem then is a standard
quadratic variational problem for the determination of I?". Once this miniiniziition has been accomplished. we use Eq. (0) to determine
using rhc
VUIUC of E ( k ) foiiiid froni tlic unmodified kernels. One could itcriitc this
process using tiow P I ' in place of K"'; we wilt not pursue tliis furiher.
mcxiiliciition here. Ldct represent the viilue of 7 when K'" is replaced hy
w. ( ' . 511~1"111\L1
in I.'ig. 2. I'Iicr-c the rclwsentation pivc\ \ irttlitlly perfect results for moderate
iimcs through Kc = 10. Marhed lie\ iations occur for this niodilic:ition iII
Re = 70. The W t { results ;ire Cotllpitrcd with numerical cxprinicnts pcrIbriiicd on thc model [.Eel. ( I )I.
This work hits shown, ;is discussed Libove, thiit the second term in thc
expiinsion beconics comparablc to tlic tirst. and ultiniatcly exceeds it i n
viilut': the expansion n o longer converges sufficiently rapidly so that the first
two terms ;ire adequate to represent the process. We are faced then with the
problem of altering the expansion in such il way that the higher orcicr lcriiis
rcniiiin smilll compared with the first term. Of course. if the process develops
largc non-Gaussian parts. this is not possible even in principle. Hurgcrs'
model (see ('lever itlid Meccham. 1972) may not be ideally suited in this way
;is ii testing ground since it shows stlhstantiiil non-Gaussian portions as the
process tlcvclops. Ncvcrthelcss we propose to test the renornialization suggestion otrercd here against this niodcl equation. ( I t is worth noting t h a t
driven random Ilurgcrs' processes arc probably niorc easily treated using the
W t1 expansion tliiin i\rc decaying processes. This may be SO becallst! O W
a r n use the white noisc process associated with the currcnt vnlue of the
forcing term i n the reprcsentation. We are thus not dependent tipon it white
noise process determined at an initial time. One might expect for this re3soii
bcttcr convcrgcncc properties for such 11 forced riindom process. Many diffusion processes niay be treated in this u;iy.)
3. KI~NOHMAI.I/AI'ION FOR Ht:HC;tRS' M0I)E'I.
Starting from givan initiiit values for the kernels [using Eq. (3)] we CLIICUtitle thc Viilttcs of the kcrncls up to ;I later time. Then the calculation is
stopped: the tr;inspired time is chosen i n such a way that the higher order
Icrni K'" rcl1iiiilis stii;ill comp;ircd with K'". Thus one may reasonably hope
~ l i i t ~
tlic expiinsion is still riipidly convergent. Once the calculation is
vtoppcd. onc will in gencriil lind that the higher ortier term in the rcprcscntation of the cncrgy spectrum !i2 [see Eq. ( 6 ) ] has increased in size conipared
with its value a t the initial time. We now propose renormiilizing.. changing
the kcrncls K"' and K'" i n such ;I way ;IS to minimize E 2 , at the same time
niinimiring the crror in the transfer term T ( k ) given by Eq. (7) and preserving I.:(k). To simplify the prohlem we shall adjust K f 2 ' choosing new valucs
A"' and using the calculated values for K"'. 'The problem then is a standard
quadratic variational problem for the determination of I?". Once this miniiniziition has been accomplished. we use Eq. (0) to determine
using rhc
VUIUC of E ( k ) foiiiid froni tlic unmodified kernels. One could itcriitc this
process using tiow P I ' in place of K"'; we wilt not pursue tliis furiher.
mcxiiliciition here. Ldct represent the viilue of 7 when K'" is replaced hy