  A Hamiltonian six-field gyrofluid model is constructed, based on closure
relations derived from the so-called "quasi-static" gyrokinetic linear theory
where the fields are assumed to propagate with a parallel phase velocity much
smaller than the parallel particle thermal velocities. The main properties of
this model, primarily aimed at exploring basic phenomena of interest for space
plasmas such as the solar wind, are its ability to provide a reasonable
agreement with kinetic theory for linear low-frequency modes, and at the same
time to ensure a Hamiltonian structure in the absence of explicit dissipation.
The model accounts for equilibrium temperature anisotropy, ion and electron
finite Larmor radius corrections, electron inertia, magnetic fluctuations along
the direction of a strong guide field, and parallel Landau damping. Remarkably,
the quasi-static closure leads to exact expressions for the nonlinear terms
involving gyroaveraged electromagnetic perturbations. One of the consequences
is that a rather natural identification of the Hamiltonian structure of the
model becomes possible when Landau damping is neglected. A slight variant of
the model consists of a four-field Hamiltonian reduction of the original
six-field model, which we use for the subsequent linear analysis. In the
latter, the dispersion relations of kinetic Alfv\'en waves and the firehose
instability are shown to be correctly reproduced, relatively far in the sub-ion
range (depending on the plasma parameters), while the spectral range where the
slow-wave dispersion relation and the field-swelling instabilities are
precisely described is less extended. This loss of accuracy originates from the
breaking of the condition of small phase velocity, relative to the parallel
thermal velocity of the electrons (for kinetic Alfv\'en waves and firehose
instability) or of the ions (in the case of the field-swelling instabilities).
