3.1. Convection regime

Let us begin by discussing the convection regime, taking the case $(Ra, {\varLambda }) = (10^6, 10^{-2})$ as an example. Figure 3(a) provides snapshots of the temperature field (left half), the concentration field (right half) and the solid–liquid interfaces (white solid lines). Clearly, a ‘cell merging’ phenomenon, similar to that observed in the melting RBC system driven solely by temperature (Rabbanipour Esfahani et al. Reference Rabbanipour Esfahani, Hirata, Berti and Calzavarini2018; Favier et al. Reference Favier, Purseed and Duchemin2019), is evident. The number of convective cells decreases as the interface rises, as convection spreads horizontally with the increasing liquid layer. In this case, ${\varLambda } = 10^{-2}$ is too small to induce any observable stabilizing effect on the flow. It is important to note that although a thin stably stratified layer (SSL) is initially generated beneath the interface due to the locally huge concentration gradient, it is quite vulnerable. The lower convection flow can easily penetrate this layer, leading to the formation of convexities and cusps on the interface features that are qualitatively similar to those observed at ${\varLambda } = 0$. This penetration behaviour and interface deformation are reminiscent of the findings reported by Wang et al. (Reference Wang, Calzavarini, Sun and Toschi2021b) for the icing problem in pure water, where density anomalies also lead to the coexistence of stable and unstable stratifications. Given that the flow structure in the convection regime has been investigated thoroughly by Favier et al. (Reference Favier, Purseed and Duchemin2019), and additional numerical results for quantitative illustration are presented in Appendix A, we will not delve into further details on the flow structure in this regime.

Figure 3.

Characteristics of the convection regime at $(Ra, {\varLambda }) = (10^6, 10^{-2})$. (a) Snapshots of the temperature field (left-hand image), the concentration field (right-hand image), and the solid–liquid interfaces (white solid lines) at different times: $t = 5\times 10^{-3}, 0.1, 0.3, 0.6$. (b–d) Time evolution of the horizontally averaged (b) temperature $\tilde {\theta }$, (c) concentration $\tilde {\phi }$, and (d) effective density $\tilde {\rho }$ along the vertical direction $z$. Here, $\tilde {\theta }$, $\tilde {\phi }$ and $\tilde {\rho }$ denote the averaged values at a given $z$. The red portion of a single line represents the liquid phase, while the purple portion indicates the solid phase, with the gradient depicting the presence of a wavy interface at this height. The insets of (b) and (c) present sharp temperature and concentration profiles within the slice $x=2$ at $t=0.48$, revealing interfacial values $\theta _\varGamma \approx 0$ and $\phi _\varGamma \approx 0.2$, discussed further in § 4.2.

In figure 3(b), the temporal distribution of temperature along the vertical direction is illustrated, where $\tilde {\theta }(z,t) = ({1}/{L})\int _0^L\theta (x,z,t)\,\mathrm {d}\kern0.07em x$ represents the averaged temperature at a given dimensionless height $z$ and time moment $t$. The red portion of a single line denotes the liquid phase, and the purple portion represents the solid phase, with the gradient indicating the presence of a wavy interface at this height. The dotted line represents the initial temperature distribution. Observations reveal that as RBC initiates during the melting process, the initially linearly distributed temperature transforms into a stair-like profile within the liquid. This pattern is characterized by the well-known ‘BL-shortcut-BL’ structure seen in classic RBC (Grossmann & Lohse Reference Grossmann and Lohse2000), with BL abbreviating the boundary layer. Specifically, $\tilde {\theta }$ experiences a sharp decrease in the lower and upper boundary layers, levelling off in a plateau between the two boundary layers. Notably, the value of this plateau remains nearly constant over time, estimated to be $1/2$ (indicated by the dashed horizontal line) from the numerical results. This observation aligns with findings reported by Favier et al. (Reference Favier, Purseed and Duchemin2019), and we will utilize this conclusion in § 4 for quantitative analysis. Additionally, Chong et al. (Reference Chong, Yang, Yang, Verzicco and Lohse2020b) observed a similar temperature profile in DDC at ${\varLambda } = 0.1$. In figure 3(c), the corresponding curves for the distribution of concentration $\tilde {\phi }$ are depicted, representing the averaged value at $z$. Similar BL-shortcut-BL profiles are observed, showing a sharp decrease at the lower and upper boundary layers in the liquid, but stabilizing at a plateau $\tilde {\phi } = 1/2$ at the developed stage. Notably, the thickness of the concentration boundary layer is smaller than that of the temperature, attributable to the solute concentration having a smaller diffusion coefficient compared to temperature, characterized by ${Le} = 100$. It is crucial to acknowledge that the $\tilde {\cdot }$ notation is employed to smooth out the discontinuity in the gradient of $\theta$ and $\phi$ at the interface. Consequently, $\tilde {\theta }$ or $\tilde {\phi }$ may not precisely capture their interfacial values at a specific $x$. To illustrate this point, we examine the vertical distribution of $\theta$ and $\phi$ at $x=2$ and time $t=0.48$, as depicted in the insets of figures 3(b) and 3(c). Here, we observe that the local interfacial temperature is approximately $\theta _\varGamma \approx 0$, while the local concentration is approximately $\phi _\varGamma \approx 0.2$. Importantly, these interfacial values remain consistent across various $x$ locations, a topic we will delve into further in § 4.2. Additionally, we estimate the averaged effective density at height $z$ using $\tilde {\rho }(z,t)={\varLambda }\,\tilde {\phi }(z,t) - \tilde {\theta }(z,t)$, and present it in figure 3(d). Here, we observe a stair-like profile that generally increases with $z$, indicating that the lighter fluid at the bottom of the liquid tends to rise upwards, triggering RBC.

However, the question arises: does the concentration field act as a passive tracer at small ${\varLambda }$? To elucidate the role that the solute plays in this scenario, figure 4(a) presents the temporal evolution of the temperature distribution located at $(x, \bar {h}(x,t)/2)$, with constant $Ra=10^6$ but ${\varLambda }$ varying between $0.1, 0.5, 1$. The white dotted line in the image indicates the first moment of observing cell merging. Clearly, a larger ${\varLambda }$ results in earlier cell merging, suggesting that the solute helps to destabilize the flow. This observation appears to be inconsistent with the previously mentioned solute-stabilizing effect. However, this apparent paradox can be understood by considering the significant difference in diffusivities between the two scalars, with ${Le} = \alpha /D = 100$. In simpler terms, when the descending flow reaches the bottom boundary layer, the melted fresh fluid carried along by the thermal plume cannot immediately adapt its concentration with the surrounding fluid due to the small diffusivity of the solute concentration. These diluted fluids are then swept horizontally, generating a fluctuating upward buoyancy force because of the lower concentration. This fluctuated motion can penetrate and agitate the thermal boundary layer at the bottom, eventually creating additional thermal plumes that further destabilize the flow. Figures 4(b) and 4(c) support this argument by focusing on the flow structures at $t = 0.228$ for ${\varLambda } = 1$. We observe thermal plumes (figure 4c) launching from positions where lower concentrations of solute are located (figure 4b), as highlighted by the closed dotted circles.

Figure 4.

Influences of ${\varLambda }$ in the convection regime while maintaining the Rayleigh number at $Ra = 10^6$. (a) Temporal evolution of the temperature distribution located at $(x,\bar {h}(t)/2)$, with ${\varLambda }$ varied between $0.1$, $0.5$ and $1$. The white dotted line indicates the occurrence of the first cell merging in each case. (a) Images reveal that a larger ${\varLambda }$ leads to earlier cell merging. Contour maps of (b) solute concentration and (c) temperature are shown at $t = 0.228$ for ${\varLambda } = 1$. Comparing (b,c) indicates that thermal plumes are launched from positions where fresher plumes are located.

3.2. Diffusion regime

Now let us examine the diffusion regime. Figure 5 presents snapshots of the temperature field (left-hand images), the concentration field (right-hand images) and the solid–liquid interfaces (white solid lines), with figures 5(a,b) corresponding to the cases $(Ra, {\varLambda }) =(10^6, 5)$ and $(Ra, {\varLambda }) =(10^6, 10^3)$, respectively. In the case $(Ra, {\varLambda }) =(10^6, 5)$, it is evident that although convection rolls are still observable at $t = 10^{-2}$, their strength diminishes by $t = 5\times 10^{-2}$ and is fully suppressed by $t = 1$. As the solid melts, the fresh fluid forms an SSL that develops downwards, compressing the convection structure. However, the convection rolls are unable to penetrate the SSL. Consequently, the solute-stabilizing effect gradually dominates over the temperature-destabilizing effect, and ultimately, the fluid becomes almost static, with heat and mass transported only in a diffusive manner. In the case $(Ra, {\varLambda }) =(10^6, 10^3)$, as shown in figure 5(b) at the same time moments, we observe the convection flow transitioning to a diffusion structure as soon as the melt begins. This transition happens more rapidly than in the former case. Additionally, figure 5 reveals that the temperature exhibits stratification only in the liquid phase for ${\varLambda } = 5$, while such stratification spreads to the solid phase for ${\varLambda } = 10^3$. These distinct scenarios imply that the underlying physics of melting for ${\varLambda } = 5$ and $10^3$ are not identical, even though both belong to the diffusion regime.

Figure 5.

Snapshots depicting the temperature field (left-hand images), the concentration field (right-hand images), and the positions of the solid–liquid interfaces (illustrated by white solid lines) within the diffusion regime. These snapshots correspond to scenarios at (a) $(Ra, {\varLambda }) = (10^6, 5)$ and (b) $(Ra, {\varLambda }) = (10^6, 10^3)$. The progression of time is displayed from bottom to top: $t = 10^{-2}$, $5\times 10^{-2}$ and $1$.

To elucidate this issue, let us revisit the Gibbs–Thomson correlation that couples the temperature and concentration at the interface: $\theta _\varGamma (x,t) = \theta _M + m_L\,\phi _\varGamma (x,t)$ (see (2.5)), where $m_L\,\phi _\varGamma (x,t)$ represents the concentration-induced undercooling effect. It is important to note that in our study, we have $m_L = -3\times 10^{-3}{\varLambda }$. For a small ${\varLambda }$, this implies $\theta _\varGamma (x,t) \approx \theta _M$, resulting in an adiabatic interface and an almost homogeneous temperature in the solid phase during melting. This observation aligns with what we see in figure 5(a) for ${\varLambda } = 5$. In contrast, for a much larger value of ${\varLambda }$, the undercooling effect becomes more pronounced, causing $\theta _\varGamma (x,t)$ to increase with time. This variation in $\theta _\varGamma (x,t)$ leads to a more continuous heat flux across the interface, as depicted in figure 5(b). To provide a more quantitative understanding, figure 6 illustrates the time evolution of the temperature and concentration distribution across the entire domain for different values of ${\varLambda }$ (specifically, $5$, $10^2$ and $10^3$). As we increase ${\varLambda }$, we observe the final temperature distribution aligning with our expectations: ${\varLambda } = 5$ results in $\tilde {\theta }$ distributing quasi-linearly only in the liquid, but remaining almost $0$ in the solid, while ${\varLambda } > 10^2$ corresponds to a quasi-linear $\tilde {\theta }$ in both liquid and solid. Additionally, a small ${\varLambda }$ corresponds to a more concave and nonlinear curve for $\tilde {\phi }$, while a larger ${\varLambda }$ leads to a more pronounced jump in $\tilde {\phi }$ at the interface.

Figure 6.

Temporal evolution of the horizontally averaged (a,c,e) temperature and (b,d, f) concentration in the vertical direction, maintaining $Ra = 10^6$ and varying ${\varLambda }$ between $5, 10^2, 10^3$. The descriptions are consistent with those of figure 3. It is evident that at ${\varLambda } = 5$, the temperature remains nearly constant at $\tilde {\theta } = 0$ within the solid, while becoming linearly distributed across the entire domain at ${\varLambda } > 10^2$. Concerning the concentration, a higher ${\varLambda }$ leads to a more pronounced discontinuity of $\tilde {\phi }$ at the interface.

Having understood the impact of ${\varLambda }$ on temperature and concentration distributions in the diffusion regime, we revisit the Stefan condition (2.6), which governs the propagation velocity of the interface. At relatively small ${\varLambda }$ in this regime, such as ${\varLambda } = 5$, the discontinuity of the thermal flux at the interface still drives the phase change, thus it can be considered a ‘melting’ phenomenon (Woods Reference Woods1992). However, in the case of a very large ${\varLambda }$, e.g. ${\varLambda } = 10^3$, the negligible jump in the thermal flux at the interface means that the phase change is no longer driven primarily by temperature but strongly depends on the solute concentration. In this case, this diffusion regime should be identified as ‘dissolution’ rather than ‘melting’, as distinguished by Woods (Reference Woods1992). We will distinguish these two diffusion regimes, namely melting and dissolution, in the following paragraphs because they exhibit different behaviours in mass and heat transfer.

3.3. Layering regime

Finally, let us discuss the observed layering structure at moderate values of ${\varLambda }$. The term ‘layering’ is used here to describe the stratified structures similar to those documented in the DDC literature (Rosevear et al. Reference Rosevear, Gayen and Galton-Fenzi2021, Reference Rosevear, Gayen and Galton-Fenzi2022; Yang et al. Reference Yang, Verzicco, Lohse and Caulfield2022, Reference Yang, Howland, Liu, Verzicco and Lohse2023a). This structure shares similarities with the flow characteristics observed in DDC scenarios. In this regime, we anticipate that the solute-stabilizing effect will prevail in the upper liquid layer, as convection is unable to fully penetrate the stably stratified layer (SSL). Conversely, the temperature-destabilizing effect will dominate in the lower liquid layer, since the SSL does not extend downwards sufficiently to completely inhibit the convection flow. As a result, an equilibrated double-layer structure may emerge, reflecting the interplay between solute and temperature effects in this intermediate regime.

Figure 7(a) illustrates a case within the layering regime at $(Ra, {\varLambda }) = (10^7, 3)$, where the coexistence of upper and lower liquid layers is evident as the solid melts. Additionally, the final distributions of temperature and concentration, as shown in figures 7(b) and 7(c), reveal a clear double-layer structure (Wang et al. Reference Wang, Reiter, Lohse and Shishkina2021a; Ding & Wu Reference Ding and Wu2021) within the liquid, akin to the findings reported by Rosevear et al. (Reference Rosevear, Gayen and Galton-Fenzi2021, Reference Rosevear, Gayen and Galton-Fenzi2022) in their study of ocean-driven melting of horizontal ice. This structure transitions from a BL-shortcut-BL to linear diffusion from the bottom to the top of the liquid. The former represents the convective flow, while the latter characterizes the diffusive behaviour. To quantify this layering structure, we define a new dimensionless parameter, the local density ratio, given by ${\varLambda }^*(\boldsymbol {x},t) = {\varLambda }\,\partial _z \phi (\boldsymbol {x},t) / \partial _z \theta (\boldsymbol {x},t)$. We then estimate the averaged value of ${\varLambda }^*(\boldsymbol {x},t)$ at a given height, and figure 7(e) presents the spatio-temporal evolution of $\widetilde {\varLambda ^{*}}$. The figure shows that at any given time $t$, the convection and diffusion layers are distinctly separated along the height $z$, with a narrow band of high $\widetilde {\varLambda ^{*}}$ (indicated by the dark red colour) situated between them. Furthermore, the solid–liquid interface (denoted by the blue solid line) rises as the diffusion layer thickens. This ‘narrow band region’ was also identified by Rosevear et al. (Reference Rosevear, Gayen and Galton-Fenzi2022) in their investigation of DDC in ice–ocean interactions; however, the development of the diffusion layer was not as apparent in their study. The larger melting time scale in their work resulted in a nearly stationary solid–liquid interface during the investigated period. Additionally, within the diffusion layer, $\widetilde {\varLambda ^{*}}$ is observed to decrease with height, consistent with the diminishing density gradient with height (i.e. a more flattened curve with $z$) as shown in figure 7(d). Specifically, $\widetilde {\varLambda ^{*}}$ decreases to a value less than 1 in the fluid layer adjacent to the solid–liquid interface, where $\widetilde {\varLambda ^{*}} < 1$ indicates an unstably stratified flow. This observation raises the question: can this unstable stratification in the diffusion layer trigger convection?

Figure 7.

Characteristics of the layering regime with penetrative convection at $(Ra, {\varLambda }) = (10^7, 3)$. (a) Snapshots of the temperature field (left-hand image), concentration field (right-hand image) and solid–liquid interfaces (indicated by white solid lines) at times $t = 10^{-2}$, $0.1$, $0.5$ and $1.5$, from bottom to top. (b–d) The time evolutions of the vertical distribution of the averaged temperature, concentration and density at a given height, respectively. Descriptions are consistent with those provided in figure 3. (e) The time evolution of the averaged local density ratio along the height, $\widetilde {{\varLambda }^*}(z,t)$, in the $(t,z)$ plot, with the blue solid line denoting the position of the solid–liquid interface.

We investigate this by increasing ${Ra}$ to a higher value, such as $({Ra}, {\varLambda }) = (10^8, 3)$, as shown in figure 8. Figure 8(a) illustrates a convection–convection double-layer structure in the liquid. Here, the unstable stratification below the interface destabilizes the diffusion layer, leading to its transition into the upper convection layer. This double-layer structure resembles the well-known ‘thermohaline staircase’ observed in the DDC literature (Rosevear et al. Reference Rosevear, Gayen and Galton-Fenzi2021; Yang et al. Reference Yang, Chen, Verzicco and Lohse2020, Reference Yang, Verzicco, Lohse and Caulfield2022). It is characterized by the double staircases in the profiles of $\tilde {\theta }$ and $\tilde {\phi }$, as depicted in figures 8(b) and 8(c). As presented in figure 8(e), the two layers are separated by a narrow band of high $\widetilde {{\varLambda }^*}$, whose strength decreases along the melting process. This implies a further destabilization of this double-layer structure. Although this transition is not captured in this case due to the top boundary, we have further simulated a case $({Ra}, {\varLambda }) = (10^7, 2.5)$ with an enlarged domain in Appendix B, which allows us to capture the entire transition from a convection–diffusion layering regime to a convection regime. Additionally, figure 2(a) demonstrates that for $Ra < 3 \times 10^6$, the layering regime does not manifest in the simulations. These observations strongly suggest that the layering regime is transient, ultimately evolving into either a convection-dominant regime or a diffusion-dominant regime as melting progresses.

Figure 8.

Characteristics of the layering regime with convection–convection double-layer structure at $(Ra, {\varLambda }) = (10^8, 3)$, with descriptions identical to those in figure 7, except that the time moments of the snapshots in (a) are $t=10^{-2}$, 0.1, 0.3 and $0.5$ from bottom to top.

We now revisit figure 2(a), focusing on the region of relatively low Rayleigh number ($Ra$), specifically below the green dotted line that denotes the threshold for the onset of RBC at the beginning of the simulation. In this regime, both the temperature and concentration fields initially exhibit purely diffusive transport. The onset of convection, as the solid melts, is influenced by local stratification. According to Radko (Reference Radko2013), a critical value ${\varLambda }_{{cr}} = ({Pr}+1)/({Pr}+{Le}^{-1}) \approx 1.1$ is predicted for the onset of thermohaline convection through linear stability analysis (LSA). This critical value is slightly higher than our numerical result ${\varLambda }_{{cr}} \approx 1.0$. This discrepancy is attributed to the fact that LSA assumes a linear concentration profile, whereas our study observes a concave profile due to its lower diffusivity. This nonlinearity leads to a larger local density ratio, making convection more difficult to initiate. In contrast, examining results in the higher ${Ra}$ region, above the green dotted line, reveals that an increased ${Ra}$ promotes the convection regime. This observation indicates that at higher Rayleigh numbers, the threshold ${\varLambda }_{{cr}}$ for transitioning from the convection regime to the layering regime increases. The observed increase in ${Lambda}_{{cr}}$ can be attributed to the dynamics of penetrative convection, where a convection-dominated fluid layer extends into an initially stably stratified layer (an SSL). As demonstrated by Wang et al. (Reference Wang, Reiter, Lohse and Shishkina2021a,Reference Wang, Calzavarini, Sun and Toschib), a higher Rayleigh number enhances the heat transfer within the convection layer, necessitating a reduction in the thickness of the SSL to balance the increased heat flux from the convection layer. Consequently, with higher ${Ra}$, the SSL becomes thinner and more susceptible to penetration. Under these conditions, a higher ${\varLambda }$ is required to provide a stronger stabilizing effect, which is essential for maintaining the equilibrium structure of the layering regime at elevated ${Ra}$ values.

It is also noteworthy to discuss the unusual distribution of $\tilde {\rho }$ within the convection layers of the layering regime. Contrary to the ascending density profile observed in figure 3(d) for $\varLambda = 10^{-2}$, the density profiles presented in figures 7(d) and 8(d) exhibit a descending, stair-like pattern. This suggests that lighter fluid is situated above heavier fluid, which would typically indicate a stably stratified flow rather than an unstable convective flow. To resolve this inconsistency, the insets in figures 7(d) and 8(d) display magnified views of the density distribution within the bulk liquid, revealing a gravitationally unstable stratified region characterized by increasing density with height. At $t=0$, the convective motions are initiated by this initially unstable stratification, as indicated by the dotted line in figures 7(d) and 8(d). Over time, the solutal boundary layer evolves and forms a thick, stably stratified boundary layer, while the bulk liquid retains its unstable stratification. This phenomenon is consistent with observations reported by Chong et al. (Reference Chong, Yang, Yang, Verzicco and Lohse2020b), who describe it as a subcritical behaviour in DDC.

4.1. Heat and mass transfer

To assess heat transfer in a melting system, a crucial dimensionless parameter is the thermal Nusselt number, denoted as ${Nu}_T$, which characterizes the ratio of heat transfer caused by convection to conduction. It is important to note a distinction in ${Nu}_T$ between the RBC system with and without melting. In classical RBC, the entire system is in energetic equilibrium, allowing ${Nu}_T$ to be calculated from any horizontal plane. In the presence of melting, however, the injected energy must include latent heat due to melting, resulting in spatial variations in ${Nu}_T$. Here, we define ${Nu}_T$ using the injected flux, expressed as

(4.1)\begin{equation} {Nu}_T(t) = \frac{Q_C(t)}{Q_D(t)} = \frac{\displaystyle\int\nolimits_{0}^L \left.-\dfrac{\partial\theta}{\partial z}\right|_{z=0}(x, t)\,\mathrm{d}\kern0.07em x}{L\,\dfrac{1-\bar{\theta}_\varGamma(t)}{\bar{h}(t)} \left(1+\dfrac{2\,{\Delta A}(t)}{\rm \pi}\right)}, \end{equation}

where $Q_C$ and $Q_D$ are the injected thermal flux realized by the convection and by the pure diffusion, respectively. We recall that $x=L=4$ is the position of the right-hand wall where periodic boundary condition is imposed. Besides, the term $2\,{\Delta A(t)}/{\rm \pi}$ in (4.1) represents a second-order correction considering topography, with $\Delta A = A - 1$, and $A$ the area of the solid–liquid interface per unit length (Favier et al. Reference Favier, Purseed and Duchemin2019). Figure 9(a) illustrates the evolution of ${Nu}_T(t)$ with respect to $Ra_e(t)$, showing a growth with melting, as explained earlier. Notably, for a melting system driven solely by temperature (${\varLambda } = 0$), the thermal Nusselt number follows a scaling ${Nu}_T \sim Ra_e^{1/4}$, as confirmed by plotting the result of $(Pr, St) = (10, 0.1)$ from Rabbanipour Esfahani et al. (Reference Rabbanipour Esfahani, Hirata, Berti and Calzavarini2018). In our study, we find that for cases with ${\varLambda } \leqslant 1$ in the convection regime, all curves ultimately collapse onto the correlation ${{Nu}_T} \sim {Ra}_e^{1/4}$ after initial jumps due to the onset of convection. Furthermore, they share identical prefactors $0.25$, indicating that the solute concentration, despite varying with ${\varLambda }$, has minimal influence on the heat transfer of the system. However, when ${\varLambda }$ approaches the borderline between convection and layering regimes (e.g. ${\varLambda }=2$ at ${Ra}=10^7$ indicating weak convection), the stabilizing solutal effect becomes non-negligible for heat transfer. Specifically, we observe ${Nu}_T$ initially dropping but eventually converging to ${Ra}_e^{1/4}$ after a sharp rise. This initial drop is attributed to the formation of the stabilizing effect from the SSL temporarily separating the convective liquid layer from the solid–liquid interface. However, as the interface height rises during melting, $Ra_e$ increases, allowing convection to penetrate the SSL, deform the interface, and sharply enhance heat transfer. Similarly, the ${Nu}_T$ profile corresponding to the layering regime (e.g. ${\varLambda } = 3$ at $Ra= 10^8$) exhibits a variation trend similar to that of the weak convection regime, but with exponent values smaller than $1/4$. Moreover, for the diffusion regime with large ${\varLambda }$, the convection flow is entirely eliminated by the growth of SSL. Consequently, ${Nu}_T$ initially jumps to a large value due to the onset of convection, but ultimately converges to 1 in a short time period.

Figure 9.

The temporal evolution of (a) the thermal Nusselt number ${Nu}_T(t)$ and (b) the solutal Nusselt number ${Nu}_C(t)$, plotted against ${Ra}_e(t)$ for varying $Ra$ (${Ra}=10^6,10^7,10^8$) and ${\varLambda }$ ($10^{-2} \leqslant {\varLambda } \leqslant 10^3$). In (b), the dash-dotted line represents the predicted value of ${Nu}_C\approx 2.49$ for the ‘melting’-dominated diffusion regime (${\varLambda } \leqslant 10$) obtained through scaling analysis. For better readability, the different groups of profiles with ${Ra}=10^6$, $10^7$ and $10^8$ are distinguished by different transparency. Also, the range of ${Ra}_e$ for different ${Ra}$ is highlighted in both plots.

To evaluate mass transfer in this system, we introduce the solutal Nusselt number, denoted as ${Nu}_C$. This parameter is defined in a manner analogous to the thermal Nusselt number ${Nu}_T$ and is expressed as

(4.2)\begin{equation} {Nu}_C(t) = \frac{\displaystyle\int\nolimits_{0}^L \left.-\dfrac{\partial\phi}{\partial z}\right|_{z=0}(x, t)\, \mathrm{d}\kern0.07em x}{L\,\dfrac{1-\bar{\phi}_\varGamma(t)}{\bar{h}(t)}\left(1+\dfrac{2\,{\Delta A}(t)}{\rm \pi}\right)}, \end{equation}

which represents the ratio between the mass flux induced by convective processes and that resulting from pure diffusion. Figure 9(b) presents the results, noting that since the binary fluid has an initially homogeneous concentration $\phi = 1$, independent of ${\varLambda }$, all ${Nu}_C$ values start from 0. In the convection and layering regimes, the evolution of ${Nu}_C$ closely mirrors that of ${Nu}_T$ because both the injected thermal and mass fluxes are influenced by the same convective structures. In particular, the solutal Nusselt number converges to ${Nu}_C = 1.4\,{Ra}_e^{1/4}$ at ${\varLambda } \leqslant 1$, while it is ${Nu}_T = 0.25\,{Ra}_e^{1/4}$ for the thermal Nusselt number. It is crucial to understand the ratio of the two prefactors, i.e. why ${Nu}_C(t)/{Nu}_T(t) \approx 1.4/0.25 = 5.6$. A scaling analysis based on the energy/mass budget helps to clarify this ratio.

Regarding heat transfer, the injected thermal flux serves two primary functions. First, it maintains the liquid at an average temperature, denoted by $\langle \theta \rangle _\ell \approx 1/2$, as illustrated in figure 3(b), where $\langle \cdot \rangle _\ell$ represents the average over the liquid phase. Second, it provides the necessary heat flux to facilitate the melting of the solid. These contributions are quantified by $({L}/{2}) ({\mathrm {d}\bar {h}(t)}/{\mathrm {d}t})$ and $\int _{s\unicode{x2013}l\ interface} [(- \boldsymbol {\nabla }\theta ^\ell )\boldsymbol {\cdot }\boldsymbol {n}](s,t)\,\mathrm {d}s$, respectively, with $\int _{s\unicode{x2013}l\ interface}$ indicating the line integral of the heat flux along the solid–liquid interface. Consequently, the energy budget can be expressed as

(4.3) \begin{equation} \int_{0}^L \left.-\frac{\partial\theta}{\partial z}\right|_{z=0}(x, t)\,\mathrm{d}\kern0.07em x \approx \frac{L}2\,\frac{\mathrm{d}\bar{h}(t)}{\mathrm{d}t} + \int_{s\unicode{x2013}l\ interface} [(- \boldsymbol{\nabla}\theta^\ell)\boldsymbol{\cdot}\boldsymbol{n}](s,t)\,\mathrm{d}s. \end{equation}

Substituting (4.1) and (2.6) into this equation further reformulates it to

(4.4)\begin{equation} \frac{{Nu}_T(t)\,L}{\bar{h}(t)} \approx \frac{L}2\,\frac{\mathrm{d}\bar{h}(t)}{\mathrm{d}t} + {St}^{{-}1}\int_{s\unicode{x2013}l\ interface} v_\varGamma(s,t) \,\mathrm{d}s, \end{equation}

where several approximations are adopted: (1) the change in the interface area is negligible, i.e. $\Delta A \ll 1$; (2) the averaged interfacial temperature in the convection regime is nearly zero, i.e. $\bar {\theta }_\varGamma \approx 0$, as discussed in § 4.2; (3) the heat flux contribution from the solid phase is insignificant, i.e. $(\boldsymbol {\nabla }\theta ^s)\boldsymbol {\cdot }\boldsymbol {n} \approx 0$, as identified in figure 3(b). Additionally, the integral of the interface propagation rate, which appears as the second term on the right-hand side of (4.4), can be estimated based on the average height as

(4.5)\begin{equation} \int_{s\unicode{x2013}l\ interface}v_\varGamma(s,t)\,\mathrm{d}s = L\,\frac{\mathrm{d}\bar{h}(t)}{\mathrm{d}t}. \end{equation}

Finally, (4.3) becomes

(4.6)\begin{equation} \frac{{{Nu}_T}(t)}{\bar{h}(t)} \approx \frac{\mathrm{d} \bar{h}(t)}{\mathrm{d}t}\left(\frac12 + {St}^{{-}1}\right), \end{equation}

which describes the dynamic coupling between the heat transfer and the evolution of the solid–liquid interface.

Regarding the mass budget, the injected mass flux serves to maintain the liquid at an average concentration $\langle \phi \rangle _\ell \approx 1/2$, as shown in figure 3(c). The mass budget can be expressed as

(4.7)\begin{equation} {Le}^{{-}1}\int_{0}^L \left.-\frac{\partial\phi}{\partial z}\right|_{z=0}(x, t)\,\mathrm{d}x\approx \frac{L}2\,\frac{\mathrm{d}\bar{h}(t)}{\mathrm{d}t}. \end{equation}

Similarly, by substituting (4.2) and applying the approximation $\Delta A \ll 1$, the injected mass flux can be reformulated as

(4.8)\begin{equation} {Le}^{{-}1}\,\frac{{Nu}_C(t)\,(1-\phi_\varGamma(t))}{\bar{h}(t)} \approx \frac12\, \frac{\mathrm{d}\bar{h}(t)}{\mathrm{d}t}, \end{equation}

which reflects the interplay between mass transfer and the evolving interface. Combining the energy and mass conservation laws given by (4.6) and (4.8), we derive the expression

(4.9)\begin{equation} \frac{{Nu}_C(t)}{{{Nu}_T}(t)} \approx \frac{{Le}}{(1+2\,{St}^{{-}1})(1- \bar{\phi}_\varGamma(t))} \approx 6, \end{equation}

where $\bar {\phi }_\varGamma (t)$ is known to be approximately 0.2 when ${\varLambda } \leqslant 1$ (see figure 3 and the corresponding text), and this relationship will be discussed further in § 4.2. The predicted value ${Nu}_C(t)/{Nu}_T(t) \approx 6$ closely matches the numerical observation $5.6$. This analysis provides a quantitative understanding of the ratio between solutal and thermal Nusselt numbers in the context of mass and energy conservation.

In the diffusion regime, which is categorized into ‘melting’ or ‘dissolution’ patterns, we can provide further insights. For the dissolution cases (${\varLambda } \geqslant 10^3$), where the $\phi$ profile is almost linearly distributed in the liquid (see figures 6b,d, f), we expect the solutal Nusselt number to approach 1, similar to the thermal Nusselt number in the diffusion regime. However, for the melting cases, the situation becomes more complex as the concave $\phi$ profile in the liquid exhibits more nonlinearity (see figures 6b,d, f). In this scenario, we establish a theoretical argument by approximating the entire process as a fully melting one-dimensional Stefan problem, enabling us to obtain self-similar solutions. We assume that $\theta _\varGamma$ and $\phi _\varGamma$ are maintained at 0 during melting, as illustrated in figure 6. The temperature and concentration fields in the liquid can then be formulated as $\theta (z,t)=1-\mathrm {erf}(z/2\sqrt {t})/\mathrm {erf}(\xi )$ and $\phi (z,t)=1-\mathrm {erf}(z\sqrt {{Le}}/2\sqrt {t})/\mathrm {erf}(\xi \sqrt {{Le}})$. The interface evolves as $h(t)=2\xi \sqrt {t}$, with $\xi$ being obtained from the heat and mass balance at the interface, given that ${St} = \xi \,\mathrm {erf}(\xi )\exp (\xi ^2)\sqrt {{\rm \pi} }$. Note that these solutions have been discussed in our previous study (Xue et al. Reference Xue, Zhao, Ni and Zhang2023). Based on the presented arguments, the solutal Nusselt number can be predicted as ${Nu}_C = -\partial _z\phi |_{z=0}\,h(t)\approx 2.49$. Consequently, for all cases in the diffusion regime, we would expect the stabilized ${Nu}_C$ to fall in a region $1 < {Nu}_C < 2.49$, with the two extremes being labelled by the dashed lines and dash-dotted lines in figure 9(b). This prediction aligns with our numerical results at ${\varLambda } \geqslant 5$ in the same plot.

4.2. Interfacial temperature and concentration

In this subsection, we investigate the temporal evolution of the averaged temperature and concentration at the interface, denoted by $\bar {\theta }_\varGamma (t)$ and $\bar {\phi }_\varGamma (t)$, respectively. We emphasize their strong coupling through the Gibbs–Thomson correlation (2.5) that $\bar {\theta }_\varGamma (t)=\theta _M + m_L\,\bar {\phi }_\varGamma (t)$, where $m_L = -3\times 10^{-3}{\varLambda }$.

For small to moderate ${\varLambda }$, where ${\varLambda } \leqslant 10$, the average interfacial temperature $\bar {\theta }_\varGamma (t)$ remains at $0$, largely due to the negligible solutal undercooling effect. However, as is evident from figure 3, the interfacial concentration appears to stabilize around $\phi _\varGamma \approx 0.2$ for low ${\varLambda }$, warranting further investigation. Figures 10(a)–10(c) present the time histories of $\bar {\phi }_\varGamma (t)$ at different ${\varLambda }$ (${\varLambda } \leqslant 10$), with ${Ra}=10^6$, $10^7$ and $10^8$ depicted in separate plots. Additionally, figure 10(d) showcases their $\bar {\phi }_\varGamma$ values at the final moment. As melting commences, $\bar {\phi }_\varGamma (t)$ decreases from $1$, and notably, for cases where ${\varLambda } < 1$, $\bar {\phi }_\varGamma (t)$ almost stabilizes at a constant value $0.2$, independent of ${\varLambda }$. Even at ${Ra} = 10^7$ and $10^8$, which exhibit secondary convection in the upper liquid layer, $\bar {\phi }_\varGamma (t)$ converges to approximately $0.2$. This consistency suggests that the ultimate $\bar {\phi }_\varGamma$ significantly depends on the convective structure near the solutal boundary layer beneath the interface, irrespective of whether the liquid comprises one layer or double layers.

Figure 10.

Time evolution of $\bar {\phi }_\varGamma (t)$ within the range of small to moderate ${\varLambda }$ (${\varLambda } \leqslant 10$) depicted at (a) ${Ra=10^6}$, (b) ${Ra}=10^7$, and (c) ${Ra}=10^8$, respectively. The corresponding ${\varLambda }$ values are represented by different line colours, consistent with those in figure 9. (d) The ultimate value of $\bar {\phi }_\varGamma$ as a function of ${\varLambda }$ for different ${Ra}$. (e) A schematic representation of the one-dimensional BL-shortcut-BL structure for the concentration $\phi$ and temperature $\theta$ profiles. The upper and lower solutal (thermal) boundary layers are symbolized by $\delta _{\phi,u}$ and $\delta _{\phi,\ell }$ ($\delta _{\theta,u}$ and $\delta _{\theta,\ell }$), respectively.

To elucidate why $\bar {\phi }_\varGamma (t)$ approaches $0.2$, we establish a theoretical model based on two assumptions. First, inspired by the profiles within the slice $x=2$ (insets of figures 3b,c), we approximate the concentration and temperature profiles in the liquid using a one-dimensional BL-shortcut-BL structure (figure 10e) observed in the convection regime. We assume $\phi$ to be linearly distributed in the boundary layer, with value $1/2$ in the bulk flow. The thicknesses of the upper and lower solutal boundary layers are given by

(4.10a,b)\begin{align} \delta_{\phi,u}(t) = \left.\left(\frac{1}{2} - \phi_\varGamma(t)\right)\right/\left(\left.-\frac{\partial\phi}{\partial z}\right|_{z=\bar{h}}(t)\right) \quad \mathrm{and} \quad \delta_{\phi,\ell}(t) = \left.\frac{1}{2}\right/\left(\left.-\frac{\partial\phi}{\partial z}\right|_{z=0}(t)\right). \end{align}

The ratio between the thicknesses of the lower and upper solutal boundary layers is given by

(4.11)\begin{equation} \frac{\delta_{\phi,\ell}(t)}{\delta_{\phi,u}(t) } = \frac{\left.\dfrac{\partial\phi}{\partial z}\right|_{z=\bar{h}}(t)}{\left(1-2\phi_\varGamma(t)\right)\left.\dfrac{\partial\phi}{\partial z}\right|_{z=0}(t)}, \end{equation}

whereas the upper mass flux can be estimated using the solute-rejection relation (2.6), with the approximation ${\partial \phi }/{\partial z}|_{z=\bar {h}}(t) \approx -{Le}\,\phi _\varGamma (t)\,({\mathrm {d}\bar {h}(t)}/{\mathrm {d}t})$, and the lower mass flux is estimated from the mass budget (4.7), with ${\partial \phi }/{\partial z}|_{z=0}(t) \approx -({{Le}}/{2})({\mathrm {d}\bar {h}(t)}/{\mathrm {d}t})$. Substituting these approximations into (4.11) yields

(4.12)\begin{equation} \frac{\delta_{\phi,\ell}(t)}{\delta_{\phi,u}(t)} \approx \frac{2\phi_\varGamma(t)}{1-2\phi_\varGamma(t)}. \end{equation}

Following a method analogous to (4.10a,b), the thicknesses of the upper and lower temperature boundary layers are estimated as

(4.13a,b)\begin{equation} \delta_{\theta,u}(t) = \left.\frac{1}{2}\right/\left(\left.-\frac{\partial\theta}{\partial z}\right|_{z=\bar{h}}(t)\right) \quad\mathrm{and}\quad \delta_{\theta,\ell}(t) = \left.\frac{1}{2}\right/\left(\left.-\frac{\partial\theta}{\partial z}\right|_{z=0}(t)\right), \end{equation}

thus the thickness ratio is

(4.14)\begin{equation} \frac{\delta_{\theta,\ell}(t)}{\delta_{\theta,u}(t)} = \left.\left(\left.-\frac{\partial{\theta}}{\partial z}\right|_{z=\bar{h}}(t)\right)\right/ \left(\left.-\frac{\partial{\theta}}{\partial z}\right|_{z=0}(t)\right). \end{equation}

The upper heat flux is related to the Stefan condition (2.6), so we have $({\partial {\theta }}/{\partial z})|_{z=\bar {h}}(t) \approx {St}^{-1}\, ({\mathrm {d}\bar {h}(t)}/{\mathrm {d}t})$, by neglecting the heat flux contribution from the solid side. Additionally, the lower heat flux is determined from the energy budget (4.3) and (4.6), which gives $-({\partial {\theta }}/{\partial z})|_{z=0}(t) \approx ({\mathrm {d} \bar {h}(t)}/{\mathrm {d}t})(\frac 12 + {St}^{-1})$. Combining these two correlations with (4.14) leads to

(4.15)\begin{equation} \frac{\delta_{\theta,\ell}(t)}{\delta_{\theta,u}(t)} \approx \frac{2}{{St}+2}. \end{equation}

Additionally, it is reasonable to assume that the thickness ratio between the lower and upper boundary layers is the same for both the concentration and temperature fields. Thus we approximate that

(4.16)\begin{equation} \frac{\delta_{\phi,\ell}(t)}{\delta_{\phi,u}(t)} \approx \frac{\delta_{\theta,\ell}(t)}{\delta_{\theta,u}(t)}, \end{equation}

which can be further simplified by considering (4.12) and (4.15), leading to the correlation

(4.17)\begin{equation} \phi_\varGamma \approx \frac{1}{{St}+4} \approx 0.24. \end{equation}

This value closely matches our numerical solution. For other cases with ${\varLambda } \leqslant 10$, including the ‘melting’-dominated diffusion regime and the layering regime without secondary convection, the concentration gradually drops and converges to $0$ due to the slower diffusion of concentration compared to temperature in the diffusion layer beneath the melting interface.

For cases where ${\varLambda } \geqslant 10^2$, dominated by the ‘dissolution’ process, an intriguing interplay between $\bar {\theta }_\varGamma (t)$ and $\bar {\phi }_\varGamma (t)$ emerges. The solid lines in figure 11 illustrate the evolution of $\bar {\phi }_\varGamma$ and $\bar {\theta }_\varGamma$ with respect to the average interface height $\bar {h}$ for ${\varLambda } = 10^2$ and $10^3$. Due to a significant solutal undercooling effect, the decrease in $\bar {\phi }_\varGamma$ leads to a noticeable increase in $\bar {\theta }_\varGamma$, ultimately converging to the guide lines in the figure during the fully developed stage. Understanding this convergence is linked to the characteristics discussed in figure 6. These characteristics reveal a quasi-linear temperature profile across the entire domain in the fully developed stage, allowing the interfacial temperature $\bar {\theta }_\varGamma$ to be approximated as

(4.18)\begin{equation} \bar{\theta}_\varGamma(t) \approx 1 - \bar{h}(t). \end{equation}

Considering the solutal undercooling effect (2.5), the interfacial concentration $\bar {\phi }_\varGamma$ can be approximated as

(4.19)\begin{equation} \bar{\phi}_\varGamma(t) \approx \frac{1-\bar{h}(t)}{m_L} + 1. \end{equation}

Expressions (4.18) and (4.19) are presented as dotted lines in figure 11 for both cases, and demonstrate excellent agreement with our numerical solution in the developed stage.

Figure 11.

Evolution of the averaged interfacial values of temperature $\bar {\theta }_\varGamma$ and concentration $\bar {\phi }_\varGamma$ in the ‘dissolution’-dominated regime, with the Rayleigh number maintained at ${Ra}=10^6$, for (a) ${\varLambda } = 10^2$, and (b) ${\varLambda } = 10^3$. Solid lines represent numerical results, while the dotted and dash-dotted lines depict the analytical solutions of (4.18) and (4.19), respectively.

4.3. Interface evolution

We then shift our focus to the time evolution of the interface height $\bar {h}(t)$, analysing it through an energy argument, a method employed previously by others (Rabbanipour Esfahani et al. Reference Rabbanipour Esfahani, Hirata, Berti and Calzavarini2018; Favier et al. Reference Favier, Purseed and Duchemin2019). In both the convection regime and the ‘melting’-dominated diffusion regime, we make two key approximations. First, we assume that the average temperature in the liquid is $\langle \theta \rangle _\ell = 1/2$. Second, we assume that $\theta _\varGamma (x,t)$ remains at 0. We define a universal correlation ${Nu}_T(t)=\gamma _1\,{Ra}_e^{\gamma _2}(t)$, with $(\gamma _1,\gamma _2) = (0.25,1/4)$ for the convection regime, and $(\gamma _1, \gamma _2) = (1, 0)$ for the ‘melting’-dominated diffusion regime, from the observation in figure 9(a). Substituting this correlation into (4.6) yields an ordinary differential equation (ODE)

(4.20)\begin{equation} \gamma_1\,{Ra}^{\gamma_2}\,\bar{h}^{3\gamma_2-1}(t)\approx\frac{\mathrm{d}\bar{h}(t) }{\mathrm{d}t}\left(\frac12+{St}^{{-}1}\right), \end{equation}

with the relation ${Ra}_e(t) = {Ra}\,\bar {h}^3(t)$. We then obtain the solution of (4.20), which reads

(4.21)\begin{equation} \bar{h}(t) \approx \left[ \frac{(2-3\gamma_2)\gamma_1\,{Ra}^{\gamma_2}}{\dfrac12+{St}^{{-}1}}\,t + h_0^{2-3\gamma_2}\right]^{1/({2-3\gamma_2})}. \end{equation}

For $Ra = 10^6$, the predicted behaviours for $\bar {h}(t)$ based on (4.21) are

(4.22)\begin{equation} \bar{h}(t) \approx \left\{\begin{array}{@{}ll@{}} (0.941t+0.134)^{4/5} & \text{for convection regime}, \\ (0.190t+0.040)^{1/2} & \text{for `melting'-dominated diffusion regime}. \end{array} \right. \end{equation}

Consequently, the averaged interface height propagates exponentially, with scaling exponents $4/5$ and $1/2$ corresponding to the convection regime and the ‘melting’-dominated diffusion regime, respectively. Accordingly, the melting rate of the solid follows the scaling exponents $-1/5$ and $-1/2$ in these two regimes. In figure 12(a), a comparison is presented between the theoretical predictions (solid black line and dashed black line for the two options of (4.22)) and the numerical solution in these two regimes, showing excellent agreement. Additionally, the scaling exponents $1/2$ and $4/5$ are plotted as grey dash-dotted lines, providing an accurate prediction of the development trend of the corresponding numerical solutions in the two regimes.

Figure 12.

Characteristics of the interface height during melting. (a) Averaged interface height at ${Ra}=10^6$. (b) Roughness of the interface height at ${Ra}=10^6$, $10^7$ and $10^8$. The colours of the lines correspond to different values of ${\varLambda }$ and are consistent with those described in figure 9. In (a), the black solid line represents the theoretical predictions from (4.22) for the convection regime, the black dashed line from (4.22) for the ‘melting’-dominated diffusion regime, and the black dotted line from (4.25) for the diffusion-dominated regime. The inset in (a) presents the numerical solution for the time evolution of the average height $\bar {h}$ over an extended period for ${\varLambda } = 10^3$, illustrating the very slow dissolution process, which aligns well with the prediction from (4.25) for the diffusion-dominated regime.

To estimate $\bar {h}(t)$ in the ‘dissolution’-dominated diffusion regime (e.g. at ${\varLambda } = 10^3$), the assumption $\theta _\varGamma = 0$ is no longer valid, as seen in figures 6(a,c,e). However, it is still possible to approximate it with the asymptotic correlation between the average interfacial concentration $\bar {\phi }_\varGamma$ and the average interface height $\bar {h}$ of (4.19):

(4.23)\begin{equation} \bar{\phi}_\varGamma(t) \approx \frac{1-\bar{h}(t)}{m_L} + 1 = \frac{1}{3}\,\bar{h}(t) + \frac{2}{3}. \end{equation}

Recalling that $m_L = -3 \times 10^{-3}{\varLambda } = -3$, the ‘dissolution’ dominated by the solute $\phi$ allows us to approximate the $\phi$ profile in the liquid with the one-dimensional linear function

(4.24)\begin{equation} \phi(z,t) \approx \frac{\bar{\phi}_\varGamma(t) - 1}{\bar{h}(t)}\,z + 1. \end{equation}

Combining (4.23), the vertical gradient of $\phi$ from (4.24), and the solute-rejection relation (2.6), we obtain the ODE

(4.25)\begin{equation} {Le}^{{-}1}\,\frac{1 - \left(\dfrac{1}{3}\bar{h}(t) + \dfrac{2}{3}\right)}{\bar{h}(t)} \approx \left(\frac{1}{3}\,\bar{h}(t) + \frac{2}{3}\right)\frac{\mathrm{d}\bar{h}(t)}{\mathrm{d}t}. \end{equation}

By solving (4.25), the analytical solution for the interface height $\bar {h}(t)$ at ${\varLambda } = 10^3$ is plotted in figure 12(a), showing good agreement with the numerical solution, except for a slight underestimation at the initial stage. The deviation arises from the faster melting by the initial triggered convection, which is not considered by the theoretical estimation.

Another relevant characteristic that requires investigation is the roughness of the interface, which is characterized as

(4.26)\begin{equation} h_{{rms}}(t)=\sqrt{\overline{(h(x,t)-\bar{h}(t))^2}}\,{Ra}^{1/3} = \sqrt{\overline{h'(x,t)^2}}\,{Ra}^{1/3}, \end{equation}

where ${Ra}^{1/3}$ is used for normalization at different ${Ra}$ (Yang et al. Reference Yang, Howland, Liu, Verzicco and Lohse2023b). The numerical results are presented in figure 12(b). We observe that in all cases, $h_{{rms}}$ remains at $0$ until the onset of convection deforms the melting interface, as described previously. In the convection regime corresponding to very small ${\varLambda }$, the developing trend of the curves follows that of $h_{{rms}} \sim {Ra}_e^{1/3}$, consistent with what was derived by Yang et al. (Reference Yang, Howland, Liu, Verzicco and Lohse2023b) for the pure temperature-driven melting problem. For the diffusion and layering regimes characterized by promoting ${\varLambda }$, the interface roughness is observed to reduce in the picture. This reduction is attributed to the weakening or disappearance of convection rolls in these two regimes. Additionally, even if there are cusps and convexities along the interface, the cusp regions melt more quickly since the local temperature has a negative gradient there, resulting in a decreasing $h_{{rms}}$.