  We look at various correlation functions, which include those that involve
both the velocity and the vorticity fields, in two-dimensional (2D) isotropic
homogeneous unforced turbulence. We adopt the more intuitive approach due to
Kolmogorov (and subsequently, Landau in his text on fluid dynamics) and show
that how the 2D turbulence's results, obtainable using other methods, may be
established in a simpler way. Same method is used to calculate some third-order
structure functions for quasi-geostrophic (QG) turbulence for the forward
cascade of pseudo-potential enstrophy and the inverse energy cascade in
quasi-geostrophic turbulence. These results motivate us to study the two-point
third order structure function in the context of the two-dimensionalisation
effect. Consequent studies enable us to give a reason for the inverse energy
cascade in the two-dimensionalised rapidly rotating three dimensional (3D)
incompressible turbulence. For such a system, literature shows a possibility of
the exponent of wavenumber in the energy spectrum's relation to lie between -2
and -3. We argue the existence of a stricter range of -2 to -7/3 for the
exponent in the case of rapidly rotating turbulence which is in accordance with
the recent experiments. Also, a derivation for the two point third order
structure function has been provided helping one to argue that even with slow
rotation one gets, although dominated, a spectrum with the exponent -2.87,
thereby hinting at the initiation of the two-dimensionalisation effect with
rotation. Moreover, using the Gledzer-Ohkitani-Yamada (GOY) shell model,
modified for rotation, these signatures of two-dimensionalisation effect have
been verified.
