  We study numerically a Constantin-Lax-Majda-De Gregorio model generalized by
Okamoto, Sakajo and Wunsch, which is a model of fluid turbulence in one
dimension with an inviscid conservation law. In the presence of the viscosity
and two types of the large-scale forcings, we show that turbulent cascade of
the inviscid invariant, which is not limited to quadratic quantity, occurs and
that properties of this model's turbulent state are related to singularity of
the inviscid case by adopting standard tools of analyzing fluid turbulence.
