  We perform an extensive study of numerical convergence for hot-Jupiter
atmospheric flow solutions in simulations employing a setup commonly-used in
extrasolar planet studies, a resting state thermally forced to a prescribed
temperature distribution on a short time-scale at high altitudes. Convergence
is assessed rigorously with: (i) a highly-accurate pseudospectral model which
has been explicitly verified to perform well under hot-Jupiter flow conditions
and (ii) comparisons of the kinetic energy spectra, instantaneous (unaveraged)
vorticity fields and temporal evolutions of the vorticity field from
simulations which are numerically equatable. In the simulations, the
(horizontal and vertical) resolutions, dissipation operator order and viscosity
coefficient are varied with identical physical and initial setups. All of the
simulations are compared against a fiducial, reference simulation at high
horizontal resolution and dissipation order (T682 and $\nabla^{16}$,
respectively) -- as well as against each other. Broadly, the reference solution
features a dynamic, zonally (east-west) asymmetric jet with a copious amount of
small-scale vortices and gravity waves. Here we show that simulations converge
to the reference simulation only at T341 resolution and with $\nabla^{16}$
dissipation order. Below this resolution and order, simulations either do not
converge or converge to unphysical solutions. The general convergence behaviour
is independent of the vertical range of the atmosphere modelled, from $\sim\!
2\!\times\! 10^{-3}$ MPa to $\sim\! 2\!\times\! 10^1$ MPa. Ramifications for
current extrasolar planet atmosphere modelling and observations are discussed.
