  We examine the relation between the absolute equilibrium state of the
spectrally truncated Euler equations (TEE) predicted by Kraichnan (1973) to the
forced and dissipated flows of the spectrally truncated Navier-Stokes (TNS)
equations. In both of these idealized systems a finite number of Fourier modes
is kept contained inside a sphere of radius $k_{max}$ but while the first
conserves energy in the second energy is injected by a body-force $\bf{f}$ and
dissipated by the viscosity $\nu$. For the TNS system stochastically forced
with energy injection rate $\mathcal{I}_\mathcal{E}$ we show, using an
asymptotic expansion of the Fokker-Planck equation, that in the limit of small
$k_{max}\eta$ (where $\eta=(\nu^3/\mathcal{I}_\mathcal{E})^{1/4}$ the
Kolmogorov lengthscale) the flow approaches the absolute equilibrium solution
of Kraichnan with such an effective "temperature" so that there is a balance
between the energy injection and the energy dissipation rate. Numerical
simulations verify the predictions of the model for small values of
$k_{max}\eta$. For intermediate values of $k_{max}\eta$ a transition from the
quasi-equilibrium "thermal" state to Kolmogorov turbulence is observed. If the
forcing is applied at small scales and the dissipation acts only at large
scales then the equipartition spectrum appears at all scales for all values of
$\nu$. In both cases a finite forward or inverse flux is present even for the
cases where the flow is close to the equilibrium state solutions. However,
unlike the classical turbulence where an energy cascade develops with a mean
energy flux that is large compared to its fluctuations, the quasi-equilibrium
state has a mean flux of energy that is subdominant to the large flux
fluctuations observed.
