  We consider a modification of the three-dimensional Navier--Stokes equations
and other hydrodynamical evolution equations with space-periodic initial
conditions in which the usual Laplacian of the dissipation operator is replaced
by an operator whose Fourier symbol grows exponentially as $\ue ^{|k|/\kd}$ at
high wavenumbers $|k|$. Using estimates in suitable classes of analytic
functions, we show that the solutions with initially finite energy become
immediately entire in the space variables and that the Fourier coefficients
decay faster than $\ue ^{-C(k/\kd) \ln (|k|/\kd)}$ for any $C<1/(2\ln 2)$. The
same result holds for the one-dimensional Burgers equation with exponential
dissipation but can be improved: heuristic arguments and very precise
simulations, analyzed by the method of asymptotic extrapolation of van der
Hoeven, indicate that the leading-order asymptotics is precisely of the above
form with $C= C_\star =1/\ln2$. The same behavior with a universal constant
$C_\star$ is conjectured for the Navier--Stokes equations with exponential
dissipation in any space dimension. This universality prevents the strong
growth of intermittency in the far dissipation range which is obtained for
ordinary Navier--Stokes turbulence. Possible applications to improved spectral
simulations are briefly discussed.
