  We conduct an in-depth analysis of statistical flow properties calculated
from the reference high-resolution Saturn simulation obtained by global climate
modelling in Part II. In the steady state of this reference simulation,
strongly energetic, zonally dominated, large-scale structures emerge, which
scale with the Rhines scale. Spectral analysis reveals a strong anisotropy in
the kinetic energy spectra, consistent with the zonostrophic turbulent flow
regime. By computing spectral energy and enstrophy fluxes we confirm the
existence of a double cascade scenario related to 2D-turbulent theory. To
diagnose the relevant 3D dynamical mechanisms in Saturn's turbulent atmosphere,
we run a set of four simulations using an idealized version of our Global
Climate Model devoid of radiative transfer, with a well-defined Taylor-Green
forcing and over several rotation rates (4, 1, 0.5, and 0.25 times Saturn's
rotation rate). This allows us to identify dynamics in three distinctive
inertial ranges: (1) a ``residual-dominated'' range, in which non-axisymmetric
structures dominate with a -5/3 spectral slope; (2) a ``zonostrophic inertial''
range, dominated by axisymmetric jets and characterized by the pile-up of
strong zonal modes with a steeper, nearly -3, spectral slope; and (3) a
``large-scale'' range, beyond Rhines' typical length scale, in which the
reference Saturn simulation and our idealized simulations differ. In the latter
range, the dynamics is dominated by long-lived zonal modes 2 and 3 when a
Saturn-like seasonal forcing is considered (reference simulation), and a steep
energetic decrease with the idealized Taylor-Green forcing. Finally,
instantaneous spectral fluxes show the coexistence of upscale and downscale
enstrophy/energy transfers at large scales, specific to the regime of
zonostrophic turbulence in a 3D atmosphere.
