Abstract

Thermally driven flows in fractures play a key role in enhancing the heat transfer and fluid mixing across the Earth's lithosphere. Yet the energy pathways in such confined environments have not been characterised. Building on \cite{2019_Letelier_JFM}, we introduce novel expressions for energy transfer rates--energetics--of geometrically constrained Rayleigh-B\'enard convection in Hele-Shaw cells, HS-RBC, based on two different conceptual frameworks. First, we derived the energetics following the well-established framework introduced by \cite{1995_Winters_JFM}, in which the gravitational potential energy, $E_{\sf p}$, is decomposed into its available, $E_{\sf ap}$, and background, $E_{\sf bp}$, components. Secondly, we derived the energetics considering a new decomposition for $E_{\sf p}$, named dynamic, $E_{\sf dp}$, and reference, $E_{\sf rp}$, potential energies. $E_{\textsf{dp}}$ is defined as the departure of the system's potential energy from the reference state $E_{\sf rp}$, determined by the `energy' of the scalar fluctuations. For HS-RBC, both frameworks lead to the same energy transfer rates at a steady state, satisfying the relationship $\langle E_{\textsf{ap}} \rangle_{\uptau} = \langle E_{\textsf{dp}} \rangle_{\uptau} + 1/6$. Consistent with the work by \cite{2013_Hughes_JFM} on 3D-RBC, we report analytical expressions for the energetics and efficiencies of HS-RBC in terms of the Rayleigh number and the global Nusselt number. Additionally, we performed numerical experiments to illustrate the application of the energetics for the analysis of HS-RBC. Finally, we discuss the impact of the thermal forcing and the geometrical control exerted by Hele-Shaw cells on the development of boundary layers, protoplumes and the self-organisation of large-scale flows.