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\begin{document}

\title{Long-wave isentropic ocean-atmosphere dynamics:~ providing
faster-than-real-time, predictive modelling of the 2022 Hunga Tonga
event}



\author[1]{Stephen D. Winn}%
\author[2,3]{Emile Touber}%
\author[2]{Adel F. Sarmiento}%
\affil[1]{Affiliation not available}%
\affil[2]{Okinawa Institute of Science and Technology}%
\affil[3]{Imperial College London}%


\vspace{-1em}



  
  \date{June 14, 2024}


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\begin{abstract}
Starting from the fully-compressible Euler equations, a two-way-coupled
system governing the long-wave behaviour of thin layers (with respect to
the radius of Earth) representing the ocean and atmosphere, under an
isentropic constraint, was derived. This approach incorporates
bathymetry and topographic features as well as three-dimensional
atmospheric non-uniformities through their depth-average over a
spherical shell. Linear analysis of the obtained system yields two pairs
of gravito-acoustic waves which are found to be representative of the
fast-travelling atmospheric wave (with a propagation speed mainly
governed by the atmospheric-layer-averaged speed of sound) and the
slower-travelling gravity waves in the ocean (with a propagation speed
mainly governed by local water depth). Remarkably, the `Proudman
resonance', observed in the forced shallow-water equation framework and
invoked to justify, in part, observed large wave-heights, vanishes in
favour of a continuous transition past the critical water depth,
occurring when the two wave propagation speeds are closest.
Two-dimensional non-linear global simulations were performed, using
atmospheric conditions on the day, showcasing the predictive ability of
the model. Local maxima of water-height disturbance in the farfield from
the volcano, linked to the atmospheric wave deformation over time, are
observed, emphasising the importance of the atmospheric-layer modelling
and two-way coupling for any daylong predictions. An efficient
implementation of the modelling strategy was carried out in the open
source computational framework
\href{https://github.com/dNamiLab/dNami}{dNami} to demonstrate the
ability to perform faster-than-real-time simulations despite the
additional equations in the governing system. Future work would see the
strategy extended to incorporate additional layers and physics e.g.
ocean and atmosphere stratification, interaction with the upper
atmosphere.%
\end{abstract}%



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