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\begin{document}

\title{Scaling laws for mixed-heated stagnant-lid convection and application to
Europa}



\author[1]{Frederic Deschamps}%
\author[2]{Kenny vilella}%
\affil[1]{Academia Sinica}%
\affil[2]{Hokkaido University}%


\vspace{-1em}



  
  \date{May 26, 2024}


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\begin{abstract}
Because rocks and ices viscosities strongly depend on temperature,
planetary mantles and ice shells are often thought to be animated by
stagnant-lid convection. Their dynamics is further impacted by the
release of internal heat either through radioactive isotopes decay or
tidal dissipation. Here, we quantify the impact of internal heating on
stagnant-lid convection. We performed numerical simulations of
convection combining strongly temperature-dependent viscosity and mixed
(basal and internal) heating in 3D-Cartesian and spherical geometries,
and used these simulations to build scaling laws relating surface heat
flux, \selectlanguage{greek}Φ\selectlanguage{english}\textsubscript{surf}, interior temperature,
\emph{T}\textsubscript{m}, and stagnant lid thickness,
\emph{d}\textsubscript{lid}, to the system Rayleigh number, heating
rate, \emph{H}, and top-to-bottom viscosity ratio, \selectlanguage{greek}Δη. \selectlanguage{english}These
relationships show that \emph{T}\textsubscript{m} increases with
\emph{H} but decreases with \selectlanguage{greek}Δη, \selectlanguage{english}while \selectlanguage{greek}Φ\selectlanguage{english}\textsubscript{surf} increases
with \emph{H} and \selectlanguage{greek}Δη. \selectlanguage{english}Importantly, they also describe heterogeneously
heated systems well, provided that the maximum dissipation occurs in
hottest regions. For \emph{H} larger than a critical value
\emph{H}\textsubscript{crit}, the bottom heat flux turns negative and
the system cools down both at its top and bottom. Two additional
interesting results are that 1) while the rigid lid stiffens with
increasing \emph{H}, it also thins; and 2) \emph{H}\textsubscript{crit}
increases with increasing \selectlanguage{greek}Δη. \selectlanguage{english}We then use our scaling laws to assess the
impact of tidal heating on Europa's ice shell properties and evolution.
Our calculations suggest a shell thickness in the range 20-80 km,
depending on viscosity and dissipated power, and show that internal
heating has a stronger influence than the presence of impurities in the
sub-surface ocean.%
\end{abstract}%



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