= Solution
The global mean absorbed solar flux is $L_\infty=0.8\times584/4=116.8\,\mathrm{W\,m^{-2}}$. Using $\sigma=5.6704\times10^{-8}\,\mathrm{W\,m^{-2}\,K^{-4}}$ gives
$$
\boxed{T_g\simeq240.5\,\mathrm K,\qquad T(0)\simeq219.4\,\mathrm K,\qquad T_\infty\simeq179.1\,\mathrm K.}
$$
For the requested graphical parcel estimate, start a <adiabatic process>[dry adiabat] at the warm ground, $T_{\rm ad}(z)=240.5-4.4z$ with $z$ in km, and intersect it with the radiative atmospheric profile $179.1[1+1.25e^{-z/10}]^{1/4}$. Solving gives
$$
\boxed{z_{\rm mix}\simeq9.9\,\mathrm{km},\qquad T(z_{\rm mix})\simeq197.1\,\mathrm K.}
$$
The original diagram below shows the intersection and the initial ground–air <temperature> discontinuity. A parcel heated towards the ground <temperature> remains buoyant until nearly one <scale height> in this model.
The atmospheric radiative lapse rate itself is only about $3.05\,\mathrm{K\,km^{-1}}$ at the bottom and decreases with height, smaller than the stated adiabatic $4.4\,\mathrm{K\,km^{-1}}$. Thus the interior radiative atmosphere is statically stable; the driver of this particular estimate is the hotter ground and discontinuity at the lower boundary, not a radiative lapse rate everywhere steeper than adiabatic. <Convection> would smooth that discontinuity and redistribute <energy>.
This is a diagnostic depth using the \i[unadjusted] radiative ground and atmospheric <temperatures>, not a fully <energy>-conserving radiative–convective equilibrium after adjustment. A self-consistent adjustment changes the surface <temperature>, infrared fluxes and boundary height. <Mars> also has spectral CO$_2$ absorption, dust, day–night variation, seasonal changes and geographically uneven heating; a <grey atmosphere>[gray], horizontally averaged, constant-scale-height column with fixed lapse rate captures none of those details. The calculation indicates the possible importance of a deep surface-driven mixed layer under the stated assumptions, not a universal Martian boundary-layer measurement.
\Image[/past-exam-of-the-physics-of-atmospheres-and-oceans-course-of-the-university-of-oxford/2018-radiating-level-and-mars.png]
{title=Emission weighting and a diagnostic surface-driven mixed-layer depth in the gray Mars model}
{description=Original calculations. The emission pressure is normalized by surface pressure; the Mars panel compares the unadjusted radiative atmosphere with the adiabat launched at its radiative ground temperature.}
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