Study of the Mellin transform of the velocity field with sharp cut-off and of the first order interaction vertex for the passive scalar

The Mellin transform defined as

Mellin[f][x, z]          = ∫_0^∞w/ ... 4;u e^-u f[e^(-u/(η - z))] = -(x^zL^(η - z))/(z - η) f[0] + O (η - z)

is inverted by a Cauchy integral closed in the positive-ordinate z semiplane. The MINUS SIGN is compensated by the clockwise orientation of the contour

Packages and substitution rules

Velocity field

Mellin[D_ (α β)] (x)        = D_0ξ∫_0^ͩ ... 3308;q/(2 π)^d   e^(I qx - u q^2) (q^2δ_αβ - q_αq_β)

Gaussian integral

Evaluation of the Fourier anti-transform

∫q/(2 π)^d   e^(I qx - u q^2) (q_αq_β)/q^(2 + d + ... ma[(d + z + 2)/2] ∫q/(2 π)^d   e^(I qx - u q^2) q_αq_β

Computation of the term proportional to q_αq_β

Scalar term

Reconstruction of the Mellin transform of the velocity correlation

Small scale asymptotics of the velocity field

First order function

Mellin[V_ (α β)] (x, z + 2)       = -m^(z - ξ)/(z - &# ... #960;)^d (1 - e^(I q x))    e^(-u q^2) (q^2δ_αβ - q_αq_β)

where D is the EDDY DIFFUSIVITY. the result is FORMALLY equivalent to

Mellin[V_ (α β)] (x, z + 2)        = m^(z - ξ)/(z ... ;q/(2 π)^de^(I q x)    e^(-u q^2) (q^2δ_αβ - q_αq_β)

if only poles occurring for z larger than MINUS two are taken into account. Note that the overall minus sign cancels out

Mellin transform of the term proportional to q_αq_β

Mellin transform of the scalar term

Mellin transform of the leading order correction to the structure function

structure[1] = scalar[1] - tensor[2]/.gammasubsrule[1] ;

An overall MINUS sign is introduced in order to compute the Mellin transform of the structure function. An overall factor TWO is due to the structure
function operation.

structure[2] = ( m^(z - ξ) ξ)/(z - ξ) (2 Dzero[1])/diffusivity[2] Simplify[Coef ... ucture[1], x[α] x[β]]/Coefficient[structure[1], K[α, β]]] x[α] x[β])

(2^(1 - z) m^z (1 + d + z) ξ Abs[x]^(2 + z) Gamma[1 + d/2] Gamma[1 - z/2] (K[α, ^ ... 6;])/((1 + d + z) Abs[x]^2)))/((-1 + d) z (2 + z) (2 + d + z) (z - ξ) Gamma[1/2 (2 + d + z)])

double pole at ξ equal zero

double[1] = structure[2]/ξ/.ξ0

(2^(1 - z) m^z (1 + d + z) Abs[x]^(2 + z) Gamma[1 + d/2] Gamma[1 - z/2] (K[α, β] - ( ... 45;] x[β])/((1 + d + z) Abs[x]^2)))/((-1 + d) z^2 (2 + z) (2 + d + z) Gamma[1/2 (2 + d + z)])

single pole at ξ equal zero

single[1] = SeriesCoefficient[Series[double[1], {z, 0, 1}], -1]/.{Log[a_] Log[2] FullSimplify[Log[a]/Log[2]], PolyGamma[0, 1 + d/2] Simplify[PolyGamma[0, 1 + d/2]]} ;

Coefficients

Small scale asymptotics of the structure function at ξ equal zero

the residues must be multiplied by MINUS one in order to take into account the clockwise orientation of the contour

smallscale[1] = - Residue[double[1], {z, 0}]/.{Log[a_] Log[2] FullSimplify[Log[a]/Log[2]], PolyGamma[0, 1 + d/2] Simplify[PolyGamma[0, 1 + d/2]]} ;

smallscale[1]//Simplify

-1/(2 (-1 + d) (2 + d)^2) (Abs[x]^2 K[α, β] (-3 d - d^2 + 2 EulerGamma + 3 d EulerGa ... ) Log[m] + 4 Log[Abs[x]] + 2 d Log[Abs[x]] - (2 + d) PolyGamma[0, (2 + d)/2]) x[α] x[β])

Coefficients

Expression of the interaction vertex

ivec[a_] = K[α, β] - a/(d + a - 1) (x[α] x[β])/Abs[x]^2 ;

ivecratio[1] = Coefficient[smallscale[1], Log[Abs[x]]]/ivec[2]//Simplify

-((1 + d) Abs[x]^2)/(-2 + d + d^2)

vertex[1] = ivecratio[1] ivec[2] (Log[m Abs[x]/2] + 1/2EulerGamma - 1/2PolyGamma[0, (4 + d)/2]) ;

FullSimplify[vertex[1] - smallscale[1]]/.  Conjugate[x] Abs[x]^2/( x)

(Abs[x]^2 K[α, β])/(-4 - 2 d)

vertex[2] = vertex[1] + Abs[x]^2/(2 (2 + d)) K[α, β] ;

FullSimplify[vertex[2] - smallscale[1]]

0

Correction to the scaling dimension

Taking into account that

x^2∂^2 (x^(2 n) Y_jl) =[2n (2n + d - 2) + j (j + d - 2)] x^(2 n) Y_jl

and

x^αx^β∂_α ∂_β (x^(2 n) Y_jl) =[(x^α∂_α )^2 - (x^α∂_α )] (x^(2 n) Y_jl) = 2 n (2n - 1) x^(2 n) Y_jl

one obtains

scalexp[1] = Coefficient[smallscalecoeff[1][[2]], K[α, β]] (2n (2n + d - 2) - j (j + d - 2))/.Abs[x]^21

-((1 + d) (-j (-2 + d + j) + 2 n (-2 + d + 2 n)))/(-2 + d + d^2)

scalexp[2] = Coefficient[smallscalecoeff[1][[2]], x[α] x[β]] 2 n (2 n - 1)/.Abs[x]^21

(4 n (-1 + 2 n))/(-2 + d + d^2)

scalexp[3] = Collect[Sum[scalexp[i], {i, 1, 2}], j, FullSimplify]

((-2 + d) (1 + d) j)/(-2 + d + d^2) + ((1 + d) j^2)/(-2 + d + d^2) - (2 n (d + 2 n))/(2 + d)

Created by Mathematica  (March 14, 2006)