  Alfv\'en waves may be generated via mode conversion from fast
magneto-acoustic waves near their reflection level in the solar atmosphere,
with implications both for coronal oscillations and for active region
helio-seismology. In active regions this reflection typically occurs high
enough that the Alfv\'en speed $a$ greatly exceeds the sound speed $c$, well
above the $a=c$ level where the fast and slow modes interact. In order to focus
on the fundamental characteristics of fast/Alfv\'en conversion, stripped of
unnecessary detail, it is therefore useful to freeze out the slow mode by
adopting the gravitationally stratified cold MHD model $c\to0$. This provides a
benchmark for fast-to-Alfv\'en mode conversion in more complex atmospheres.
Assuming a uniform inclined magnetic field and an exponential Alfv\'en speed
profile with density scale height $h$, the Alfv\'en conversion coefficient
depends on three variables only; the dimensionless
transverse-to-the-stratification wavenumber $\kappa=kh$, the magnetic field
inclination from the stratification direction $\theta$, and the polarization
angle $\phi$ of the wavevector relative to the plane containing the
stratification and magnetic field directions. We present an extensive
exploration of mode conversion in this parameter space and conclude that
near-total conversion to outward-propagating Alfv\'en waves typically occurs
for small $\theta$ and large $\phi$ ($80^\circ$--$90^\circ$), though it is
absent entirely when $\theta$ is exactly zero (vertical field). For wavenumbers
of helioseismic interest, the conversion region is broad enough to encompass
the whole chromosphere.
