f?'): E , ( k ) represent E , ( k ) with A"2' replaced hy k'". We see that we have
great arbitrariness in the description of thc functions E and T because the
kcrnel function k"2) is a function of t w o independent arguments.
U;c clioosc a system ol' I'unctions to rr'present the renornializd kernel
. u hcrc thc rcprc'sentiitioii will include ;I set of constitiits lo be
under the interchange of its two wave number arguments and should be a n
odd function of the two arguments taken together. We have
( 8 )
K12)(kl. k , ) = ( k , 4- k,)trm,~.L,.(kl)L,,(k2)
with ( J ~ , , ~ ,
= t i ,,., and lit'. 11' = -- 2. 0. 7,
The L are even functions of their
iirguments and we assume summation on repeated indices over the whole set
of fiinctions to he used. It is convenient to define subsidiary function3
(9)
M,,,.( k ) = K' ' '( k )Lwl, ( k ).
A'"' i? thc function cnlculatcd by integration of Eqs. (4). prior t o the stop for
renormalization. We represent the modified kernels by Hermite p(>lynoniiills
with thcir (Gaiissian) weighting function
( 10)
k/,ll(k) Z= ~ ~ C V l ( ~ '
' k ) CXp( - k 2 ! 2 ) ;
171. / I = 0, 2. . . .
The lirsr lilllction. A / ? . is described below. The functions M,, are mutually
orthogonal. We sh;ill adhere to the convention on range indicated for
primed and unprimcd indices, Eqs. ( 8 ) and (10).
I n using thc representation given hy Eq. (8) it will be desirable to preserve
tlic asymptotic form o f the transfer for large wave numbers in order to
prcscrve the inertial subrange. Considcr the first term in the expansion of
Ilq. (8) ;ind suppose that the coefficient u . z = I ; this single could represent the tr;inslcr 7' exactly for all wave nunihers as we shall see [from
l ~ i l .
(7)). Suhstitutc that single term in Eq. (7). use the definition given in
Ilcl. (9). t;ikc thc fourier trillisform of Eq. (7) after substitution to find for the
function M
the following rclution
( 1 1 )
A! ' >(.\.) -- 2,W ~ ( . Y ) , W JO) - '~'.F(.Y) =T 0
with
.F(.\.) = (. ~'"'k ~ " l ' ( k ) dk and hf Jv) = (. F ' M - ? ( k ) tlk
( i t is casily shown Ihiit 7' hcgins like k Z . for small k, so thiit the Fourier
Irii1lsli)rtli dcsc-l-ilmi lierc is convergent.) Equation ( 1 I ) can be solved.
(12)
hf #I) is h i n d froin I;q. ( 1 I). From Eq. (9) we inay reasonably expect that
A I ?(.\.) = : M . JO) - p.Iz 2(o) + z?r~.Y-(.Y)]'
great arbitrariness in the description of thc functions E and T because the
kcrnel function k"2) is a function of t w o independent arguments.
U;c clioosc a system ol' I'unctions to rr'present the renornializd kernel
. u hcrc thc rcprc'sentiitioii will include ;I set of constitiits lo be
under the interchange of its two wave number arguments and should be a n
odd function of the two arguments taken together. We have
( 8 )
K12)(kl. k , ) = ( k , 4- k,)trm,~.L,.(kl)L,,(k2)
with ( J ~ , , ~ ,
= t i ,,., and lit'. 11' = -- 2. 0. 7,
The L are even functions of their
iirguments and we assume summation on repeated indices over the whole set
of fiinctions to he used. It is convenient to define subsidiary function3
(9)
M,,,.( k ) = K' ' '( k )Lwl, ( k ).
A'"' i? thc function cnlculatcd by integration of Eqs. (4). prior t o the stop for
renormalization. We represent the modified kernels by Hermite p(>lynoniiills
with thcir (Gaiissian) weighting function
( 10)
k/,ll(k) Z= ~ ~ C V l ( ~ '
' k ) CXp( - k 2 ! 2 ) ;
171. / I = 0, 2. . . .
The lirsr lilllction. A / ? . is described below. The functions M,, are mutually
orthogonal. We sh;ill adhere to the convention on range indicated for
primed and unprimcd indices, Eqs. ( 8 ) and (10).
I n using thc representation given hy Eq. (8) it will be desirable to preserve
tlic asymptotic form o f the transfer for large wave numbers in order to
prcscrve the inertial subrange. Considcr the first term in the expansion of
Ilq. (8) ;ind suppose that the coefficient u . z = I ; this single could represent the tr;inslcr 7' exactly for all wave nunihers as we shall see [from
l ~ i l .
(7)). Suhstitutc that single term in Eq. (7). use the definition given in
Ilcl. (9). t;ikc thc fourier trillisform of Eq. (7) after substitution to find for the
function M
the following rclution
( 1 1 )
A! ' >(.\.) -- 2,W ~ ( . Y ) , W JO) - '~'.F(.Y) =T 0
with
.F(.\.) = (. ~'"'k ~ " l ' ( k ) dk and hf Jv) = (. F ' M - ? ( k ) tlk
( i t is casily shown Ihiit 7' hcgins like k Z . for small k, so thiit the Fourier
Irii1lsli)rtli dcsc-l-ilmi lierc is convergent.) Equation ( 1 I ) can be solved.
(12)
hf #I) is h i n d froin I;q. ( 1 I). From Eq. (9) we inay reasonably expect that
A I ?(.\.) = : M . JO) - p.Iz 2(o) + z?r~.Y-(.Y)]'