  This work explains a scaling law of the first Landau coefficient of the
derived Ginzburg-Landau equation (GLE) in the weakly nonlinear analysis of
axisymmetric viscoelastic pipe flows in the large-Weissenberg-number ($Wi$)
limit, recently reported in Wan et al. J. Fluid Mech. (2021), vol. 929, A16.
Using an asymptotic method, we derive a reduced system, which captures the
characteristics of the linear centre-mode instability near the critical
condition in the large-$Wi$ limit. Based on the reduced system we then conduct
a weakly nonlinear analysis using a multiple-scale expansion method, which
readily explains the aforementioned scaling law of the Landau coefficient and
some other scaling laws. Particularly, the equilibrium amplitude of disturbance
near linear critical conditions is found to scale as $Wi^{-1/2}$, which may be
of interest to experimentalists. The current analysis reduces the numbers of
parameters and unknowns and exemplifies an approach to studying the
viscoelastic flow at large $Wi$, which could shed new light on the
understanding of its nonlinear dynamics.
