  Motivated by the recent numerical results of Khalid et al., Phys. Rev. Lett.,
127, 134502 (2021), we consider the large-Weissenberg-number ($W$) asymptotics
of the centre mode instability in inertialess viscoelastic channel flow. The
instability is of the critical layer type in the distinguished ultra-dilute
limit where $W(1-\beta)=O(1)$ as $W \rightarrow \infty$ ($\beta$ is the ratio
of solvent-to-total viscosity). In contrast to centre modes in the
Orr-Sommerfeld equation, $1-c=O(1)$ as $W \rightarrow \infty$ where $c$ is the
phase speed normalised by the centreline speed as a central `outer' region is
always needed to adjust the non-zero cross-stream velocity at the critical
layer down to zero at the centreline. The critical layer acts as a pair of
intense `bellows' which blows the flow streamlines apart locally and then sucks
them back together again. This compression/rarefaction amplifies the
streamwise-normal polymer stress which in turn drives the streamwise flow
through local polymer stresses at the critical layer. The streamwise flow
energises the cross-stream flow via continuity which in turn intensifies the
critical layer to close the cycle. We also treat the large-Reynolds-number
($Re$) asymptotic structure of the upper (where $1-c=O(Re^{-2/3})$) and lower
branches of the $Re$-$W$ neutral curve confirming the inferred scalings from
previous numerical computations. Finally, we argue that the viscoelastic centre
mode instability was actually first found in viscoelastic Kolmogorov flow by
Boffetta et al., J. Fluid Mech., 523, 161-170 (2005).
