  The properties of filamentary interstellar clouds observed at sub-millimetre
wavelengths, especially by the Herschel Space Observatory, are analysed with
polytropic models in cylindrical symmetry. The observed radial density profiles
are well reproduced by negative-index cylindrical polytropes with polytropic
exponent $1/3\lesssim \gamma_{\rm p} \lesssim 2/3$ (polytropic index
$-3\lesssim n \lesssim -3/2$), indicating either external heating or
non-thermal pressure components. However, the former possibility requires
unrealistically high gas temperatures at the filament's surface and is
therefore very unlikely. Non-thermal support, perhaps resulting from a
superposition of small-amplitude Alfv\'en waves (corresponding to $\gamma_{\rm
p}=1/2$), is a more realistic possibility, at least for the most massive
filaments. If the velocity dispersion scales as the square root of the density
(or column density) on the filament's axis, as suggested by observations, then
polytropic models are characterised by a uniform width. The mass per unit
length of pressure-bounded cylindrical polytropes depends on the conditions at
the boundary and is not limited as in the isothermal case. However, polytropic
filaments can remain stable to radial collapse for values of the
axis-to-surface density contrast as large as the values observed only if they
are supported by a non-isentropic pressure component.
