  At high angular frequencies the thermal Sunyaev-Zel'dovich (tSZ) effect
constitutes the dominant signal in the CMB sky. The tSZ effect is caused by
large scale pressure fluctuations in the baryonic distribution in the Universe
so its statistical properties provide estimates of corresponding properties of
the projected 3D pressure fluctuations. It's power spectrum is a sensitive
probe of the density fluctuations, and the bispectrum can be used to separate
the bias associated with pressure. The bispectrum is often probed with a
one-point real-space analogue, the skewness. In addition to the skewness the
morphological properties, as probed by the well known Minkowski Functionals
(MFs), also require the generalized one-point statistics, which at the lowest
order are identical to the skewness parameters. The concept of generalized
skewness parameters can be extended to define a set of three associated
generalized skew-spectra. We use these skew-spectra to probe the morphology of
the tSZ sky or the y-sky. We show how these power spectra can be recovered from
the data in the presence of arbitrary mask and noise templates using the well
known Pseudo-Cl (PCL) approach for arbitrary beam shape. We also employ an
approach based on the halo model to compute the tSZ bispectrum. The bispectrum
from each of these models is then used to construct the generalized
skew-spectra. We consider the performance of an all-sky survey with Planck-type
noise and compare the results against a noise-free ideal experiment using a
range of smoothing angles. We find that the skew-spectra can be estimated with
very high signal-to-noise ratio from future frequency cleaned tSZ maps that
will be available from experiments such as Planck. This will allow their mode
by mode estimation for a wide range of angular frequencies and will help us to
differentiate them from various other sources of non-Gaussianity.
