  In the region close to the thermodynamic critical point and in the proximity
of the pseudo-boiling (Widom) line, strong property variations substantially
alter the growth of modal instabilities, as revealed in Ren et al. (J. Fluid
Mech., vol. 871, 2019, pp. 831-864). Here, we study non-modal disturbances in
the spatial framework using an eigenvector decomposition of the linearized
Navier-Stokes equations under the assumption of locally parallel flow. The
boundary layers with the fluid at supercritical pressure are heated or cooled
by prescribing the wall and free-stream temperatures so that the temperature
profile is either entirely subcritical (liquid-like), supercritical (gas-like),
or transcritical (across the Widom line). The free-stream Mach number is set to
$10^{-3}$. In the non-transcritical regimes, the resulting
streamwise-independent streaks originate from the lift-up effect. Wall cooling
enhances the energy amplification for both subcritical and supercritical
regimes. When the temperature profile is increased beyond the Widom line, a
strong sub-optimal growth is observed over very short streamwise distances due
to the Orr mechanism. The non-modal growth is put in perspective with modal
growth by means of an $N$-factor comparison. In the non-transcritical regimes,
modal stability dominates regardless of a wall-temperature variation. In
contrast, in the transcritical regime, non-modal $N$-factors are found to
resemble the imposition of an adverse pressure gradient in the ideal-gas
regime. When cooling beyond the Widom line, optimal growth is greatly enhanced,
yet strong inviscid instability prevails. When heating beyond the Widom line,
optimal growth could be sufficiently large to favor transition, particularly
with a high free-stream turbulence level.
