  We study the flow field induced by a sphere translating in a viscous
density-stratified ambient, specifically, in the limit of small Reynolds $(Re =
\rho U a/\mu \ll 1)$, and viscous Richardson numbers $(Ri_v = \gamma a^3 g/\mu
U\ll 1)$, and large Peclet number $(Pe = Ua/D\gg 1)$. Here, $a$ is the sphere
radius, $U$ its translational velocity, $\rho$ an appropriate reference density
within the Boussinesq framework, $\mu$ the ambient viscosity, $\gamma$ the
absolute value of the background density gradient, and $D$ the diffusivity of
the stratifying agent. For the scenario where buoyancy forces first become
comparable to viscous forces at large distances, corresponding to the
Stokes-stratification regime defined by $Re \ll Ri_v^{1/3} \ll 1$ for $Pe \gg
1$, important flow features such as a vertical reverse jet and a horizontal
wake, on scales larger than the primary screening length of
$\mathcal{O}(aRi_v^{-1/3})$, have been identified by Varanasi and Subramanian
(2022). Here, we show that the reverse jet is only the central portion of a
columnar structure with multiple annular cells. In the absence of diffusion
this columnar structure extends to downstream infinity with the number of
annular cells diverging in this limit. We provide expressions for the boundary
of the structure, and the number of cells within, as a function of the
downstream distance. For small but finite diffusion, two additional length
scales emerge - a secondary screening length of $O(aRi_v^{-1/2}Pe^{1/2})$,
where diffusion starts to smear out density variations across cells, leading to
exponentially decaying flow field; and a tertiary screening length, of
$O(aRi_v^{-1/2}Pe^{1/2}\ln(Ri_v^{-1}Pe^3))$, beyond which the columnar
structure ceases to exist and the downstream disturbance field reverts from an
exponential to eventual algebraic decay, analogous to that prevalent at large
distances upstream.
