  We study quasi-geostrophic turbulence and plasma drift turbulence within the
Charney-Hasegawa-Mima (CHM) model. We focus, theoretically and using numerical
simulations, on conservation of {\em zonostrophy} and on its role in the
formation of the zonal jets. The zonostrophy invariant was first predicted in
\cite{perm,BNZ_invariant} in two special cases -- large-scale turbulence and
anisotropic turbulence. Papers \cite{perm,BNZ_invariant} also predicted that
the three invariants, energy, enstrophy and zonostrophy, will cascade
anisotropically into non-intersecting sectors in the $k$-space, so that the
energy cascade is "pushed" into the large-scale zonal scales. In the present
paper, we consider the scales much less than the Rossby deformation radius and
generalise the Fj{\o}rtoft argument of \cite{perm,BNZ_invariant} to find the
directions of the three cascades in this case. For the first time, we
demonstrate numerically that zonostrophy is well conserved by the CHM model,
and that the energy, enstrophy and zonostrophy cascade as prescribed by the
Fj{\o}rtoft argument if the nonlinearity is sufficiently weak. Moreover,
numerically we observe that zonostrophy is conserved surprisingly well at late
times and the triple-cascade picture is rather accurate even if the initial
nonlinearity is strong.
