For the inverse energy cascade, the assumption of a constant
flux of energy
toward large scales reproduces
3d-like scaling laws for velocities and characteristic times:
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The hypothesis of locality of triadic interactions in the inverse cascade
is consistent with the spectrum. The transfer
is associated with the distortion of the velocity field by its own shear.
The effective shear at given wavenumber
is expected to be negligibly
affected by wave-numbers
because the integral
, which measure the mean-square shear,
converges at
. Also the contribution by high wavenumber
is negligible because vorticity associated with those wave-numbers
fluctuates rapidly in space and times and gives a small effective shear
across distances of order
.
In absence of a large-scale sink of energy the inverse cascade
can only be quasi-steady because the peak of the energy spectrum
keeps moving down to ever-lower wave-numbers as
The presence of friction stops the energy cascade at wavenumber
In two-dimensional turbulence it is possible to demonstrate
the analogous of the Kolmogorov's four-fifths law.
In the limit of infinite Reynolds number the third order
(longitudinal) structure function of two-dimensional
homogeneous isotropic turbulence, evaluated for increments
small compared to the integral scale, and larger than the forcing
correlation length, is given in terms of the mean energy flux
per unit mass
by
At variance with the three-dimensional case, where the
dimensional prediction for the scaling exponents is modified
by the presence of small scale intermittency, the statistics of velocity
fluctuations at different scale in the scaling range of the
inverse energy cascade are found to be roughly self-similar [21],
with small deviations from gaussianity.