  As a first step towards the asymptotic description of coherent structures in
compressible shear flows, we present a description of nonlinear equilibrium
solutions of the Navier--Stokes equations in the compressible asymptotic
suction boundary layer (ASBL). The free-stream Mach number is assumed to be $<
0.8$ so that the flow is in the subsonic regime and we assume a perfect gas. We
extend the large-Reynolds number free-stream coherent structure theory of
\cite{deguchi_hall_2014a} for incompressible ASBL flow to describe a nonlinear
interaction in a thin layer situated just below the free-stream which produces
streaky disturbances to both the velocity and temperature fields, which can
grow exponentially towards the wall. We complete the description of the growth
of the velocity and thermal streaks throughout the flow by solving the
compressible boundary-region equations numerically. We show that the velocity
and thermal streaks obtain their maximum amplitude in the unperturbed boundary
layer. Increasing the free-stream Mach number enhances the thermal streaks,
whereas varying the Prandtl number changes the location of the maximum
amplitude of the thermal streak relative to the velocity streak. Such nonlinear
equilibrium states have been implicated in shear transition in incompressible
flows; therefore, our results indicate that a similar mechanism may also be
present in compressible flows.
