  Kinetic Alfv\'{e}n waves (KAWs) are the short-wavelength extension of the MHD
Alfv\'{e}n-wave branch in the case of highly-oblique propagation with respect
to the background magnetic field. Observations of space plasma show that
small-scale turbulence is mainly KAW-like. We apply two theoretical approaches,
collisional two-fluid theory and collisionless kinetic theory, to obtain
predictions for the KAW polarizations depending on $\beta_\mathrm{p}$ (the
ratio of the proton thermal pressure to the magnetic pressure) at the ion
gyroscale in terms of fluctuations in density, bulk velocity, and pressure. We
perform a wavelet analysis of MMS magnetosheath measurements and compare the
observations with both theories. We find that the two-fluid theory predicts the
observations better than kinetic theory, suggesting that the small-scale
KAW-like fluctuations exhibit a fluid-like behavior in the magnetosheath
although the plasma is weakly collisional. We also present predictions for the
KAW polarizations in the inner heliosphere that are testable with Parker Solar
Probe and Solar Orbiter.
