  For a shell model of the fully developed turbulence and the incompressible
Navier-Stokes equations in the Fourier space, when a Gaussian white noise is
artificially added to the equation of each mode, an expression of the mean
linear response function in terms of the velocity correlation functions is
derived by applying the method developed for nonequilibrium Langevin systems
[Harada and Sasa, Phys. Rev. Lett. 95, 130602 (2005)]. We verify numerically
for the shell model case that the derived expression of the response function,
as the noise tends to zero, converges to the response function of the noiseless
shell model.
