Flow reorganization and transport enhancement in two-dimensional horizontal convection near a density extremum

Horizontal convection serves as a canonical model for geophysical and industrial flows. While the Oberbeck-Boussinesq approximation is well established, the impact of a nonlinear equation of state, specifically the density extremum of water near $4^\circ\mathrm{C}$, remains underexplored. Here we investigate this effect using two-dimensional direct numerical simulations over the Rayleigh number range $10^6 \le Ra \le 5\times 10^{10}$. We examine four configurations, contrasting extremum (EXT) and monotonic (MON) buoyancy boundary conditions against linear (LENT) and nonlinear (NELT) equations of state. Our results reveal that the EXT-NELT case undergoes a pronounced reorganization of the large-scale flow, evolving from a bicellular structure to a single-roll circulation driven by central `mixing plumes'. This reorganization manifests as transitional anomalies in the $Re$ scaling, while the emergence of full-depth plumes alters the heat transport mechanism. Consequently, distinct from the Rossby scaling ($Nu \sim Ra^{1/5}$) observed in the reference cases, the EXT-NELT case exhibits an enhanced heat transport scaling ranging from $Nu \sim Ra^{1/4}$ to $Nu \sim Ra^{1/3}$. To interpret this behaviour, we examine the total energy budget and identify an additional potential-energy transfer term, \(\Phi_{i2}\), arising from the nonlinear equation of state. The scaling argument suggests that the magnitude of this contribution is controlled by the characteristic plume height ($\hat{z}$). Specifically, when plumes penetrate the entire cavity depth ($\hat{z} \sim H$), as observed in the EXT-NELT case, the global kinetic energy dissipation is no longer described by the standard OB HC energy closure alone. The resulting model captures the main trends of the numerical data and provides a possible energy budget interpretation of the enhanced transport observed in this two-dimensional configuration.