Extended MHD is a one-fluid model that incorporates two-fluid effects such as
electron inertia and the Hall drift. This model is used to construct fully
nonlinear Alfv\'enic wave solutions, and thereby derive the kinetic and
magnetic spectra by resorting to a Kolmogorov-like hypothesis based on the
constant cascading rates of the energy and generalized helicities of this
model. The magnetic and kinetic spectra are derived in the ideal $\left(k <
1/\lambda_i\right)
,Hall\left(1/\lambda_i < k < 1/\lambda_e \right)$, and
electron inertia
\left(k > 1/\lambda_e\right) regimes;
k is the wavenumber
and
λs=c/ωps is the skin depth of species `
s'. In the
Hall regime, it is shown that the emergent results are fully consistent with
previous numerical and analytical studies, especially in the context of the
solar wind. The focus is primarily on the electron inertia regime, where
magnetic energy spectra with power-law indexes of
−11/3 and
−13/3 are
always recovered. The latter, in particular, is quite close to recent
observational evidence from the solar wind with a potential slope of
approximately
−4 in this regime. It is thus plausible that these spectra may
constitute a part of the (extended) inertial range, as opposed to the standard
`dissipation' range paradigm.