Abrupt transition from slow to fast melting of ice

How fast ice melts in turbulent flows is key to many natural and industrial processes, most notably the melting of ice in the polar regions. To get a better quantitative understanding of the physical mechanics at play, as a model system we pick vertical convection, consisting of ice and fresh water, and examine the lateral melting behavior through numerical simulations and theory. We find that the melting rate of ice as a function of an increasing heating temperature undergoes an abrupt transition from a slow- to a fast-melting state, contrary to the intuition of a gradual transition. The abrupt transition of the ice melting rate is due to the emergence of a reversed buoyant flow, due to the density anomaly of water near the melting point. A theoretical model based on energy conservation gives rise to a universal expression to relate the global heat fluxes and the ice melting rate which is consistent with our data. Besides their fundamental significance, our findings improve our understanding of how phase transitions couple to adjacent turbulent flow.

Figure 1

The normalized equilibrium ice-water interface height as a function of the heating temperature $T_h$: comparison of our numerical results with the prior experiments and numerical simulations by Wang et al. [31].

Figure 2

(a) The schematic sketch of the setup of melting in Rayleigh-Bénard convection from Ulvrova et al. [6]. (b) Comparison of the time evolution of the mean Nusselt number over the hot bottom boundary between the results of Ulvrova et al. [6] and our results.

Figure 3

Vertical profiles of the temperature at $x=0.25$ (in the horizontal direction) for four different times (a) $t=200$, (b) $t=2000$, (c) $t=10000$, and (d) $t=20000$ in free-fall time units. Solid red lines represent our simulation results. Blue points represent the results from Ulvrova et al. [6].

Figure 4

(a) Schematic sketch of the setup of melting in vertical convection from Ulvrova et al. [6]. (b) The comparison of the interface position at time $t=2236$ free-fall time units ($t=0.1$ diffusive time unit). (c) Comparison of the time evolution of the mean Nusselt number over the hot bottom boundary between the results of Ulvrova et al. [6] and our results.

Figure 5

3D visualization of (a) the simulation setup and the instantaneous temperature field at $T_h=10{}^{\circ }\mathrm{C}$ (b) without and (c) with density anomaly. The 2D instantaneous temperature field at $T_h=10{}^{\circ }\mathrm{C}$ (d) without and (e) with a density anomaly, where the white dashed line shows the contour of $T_x=4{}^{\circ }\mathrm{C}$. The solid-liquid interface evolution (f) without and (g) with density anomaly at $T_h=6.7,10,20{}^{\circ }\mathrm{C}$, correspondingly to ${\text{Ra}}^{*}=1.25\times {10}^8,2.7\times {10}^8,1\times {10}^9$.

Figure 6

The overall melting rate $\overline{f}$ (calculated from the characteristic length scale and time from completely ice to completely water), normalized by the overall melting rate at $4{}^{\circ }\mathrm{C}$, as a function of $T_h$ (a) without density anomaly and (b) with density anomaly. (c), (d) Overall averaged ${\text{Nu}}_h$ and ${\text{Nu}}_c$, as a function of $T_h$ (c) without and (d) with density anomaly. (e), (f) Net heat flux $\mathrm{\Delta }\text{Nu}$, normalized by the net heat flux at $4{}^{\circ }\mathrm{C}$, as a function of $T_h$ (e) without and (f) with density anomaly. The darker shaded part shows the region of high-melting state.

Figure 7

The flow and temperature field for $T_h=20,13.3,10,8$, and $6.7{}^{\circ }\mathrm{C}$ for water with the density anomaly, where the white dashed line shows the contour line of $T=4{}^{\circ }\mathrm{C}$ and the solid white line the solid-liquid interface. The black and white curved arrows indicate the clockwise large-scale circulation and the reversed buoyant flow. The red dashed boxes shown in (b) and (c) capture the region near the cooled plate where the shielding of buoyant reversal flow takes effect, which clearly explains the $\mathrm{\Delta }\text{Nu}$ change and the melting rate.

Figure 8

(a) Illustration of the heat flux into and out of the domain during the melting process. (b) Plot of the relation between the effective net heat flux $\mathrm{\Delta }\mathrm{Nu}/\overline{h}$ and the renormalized melting rate $\sqrt{\text{Ra}Pr}(\overline{\theta }+{\mathrm{St}}^{-1})\overline{f}$. The dashed line represents the linear fitting curve according to Eq. (6) with slope 1 and intercept 3.0.