3.1. Overview of the flow development

In this section, the key scaling results are summarized to provide an overview of the flow development. When the water in the present reservoir model is exposed to a constant heat flux from the water surface for a sufficiently long time, the flow in the reservoir eventually reaches a quasi-steady state, in which temperature gradient and flow velocity become steady everywhere in the water body. The possible pathways through which the flow develops towards the final quasi-steady state, which depend on the Rayleigh number, are summarized in table 1. A schematic of the development pathways is shown in figure 2, and a schematic of the movements of the interfaces between the different subregions shown in figure 2 is given in figure 3.

Table 1.

Possible development pathways of the flow.

Figure 2.

Schematic of possible flow development pathways towards the final quasi-steady state. The horizontal axis denotes the distance from the tip for the slope region. Here $t_{d}$ is the time scale for the thermal boundary layer to reach the sloping bottom; $t_{h}$ is the time scale to reach the quasi-steady state in the conduction-dominated subregion; $t_{cL}$ and $t_{cR}$ are the time scales at which the convection balance is achieved; and $t_{m}$ is the time scale for the discharge flow.

Figure 3.

Schematic of the movement of the interfaces between different flow regions: (a)  $t<t_{B}$ ; (b)  $t_{B}<t<t_{F}$ ; (c)  $t>t_{F}$ . Here $t_{B}$ denotes the time scale when the convection balance initially occurs, and $t_{F}$ denotes the time scale when the right boundary of the conduction-dominated subregion meets the left boundary of the convection-dominated subregion. The dashed lines denotes the interfaces: $x_{d}$ denotes the interface between the distinct and indistinct thermal boundary layer regions; $x_{h}$ denotes the right boundary of the conduction-dominated subregion; $x_{cL}$ and $x_{dR}$ denote the left and right boundaries of the convection-dominated subregion, respectively; $x_{B}$ denotes the position where the convection balance initially occurs; $x_{F}$ denotes the position where the left boundary of the convection-dominated subregion and the right boundary of the conduction-dominated subregion meet each other; $x_{D}$ denotes the position of the right boundary of the convection-dominated subregion at the quasi-steady state. The dash-dotted line denotes the boundary layer. The arrow indicates the direction of the movement.

Upon the initiation of the heat flux at the water surface, the thermal boundary layer initially grows downwards. Once the thermal boundary layer reaches the bottom, it becomes indistinct over the local water depth. When that happens, the thermal boundary layer is referred to as an indistinct thermal boundary layer. In contrast, a thermal boundary layer with thickness less than the full local water depth is referred to as a distinct thermal boundary layer. A horizontal temperature gradient develops in the indistinct thermal boundary layer region due to the variation of the water depth, which results in unequal heat gain over the local depth. A horizontal pressure gradient is then induced by buoyancy, which drives a horizontal flow in the offshore direction along the surface.

The flow is weak initially and conduction is dominant in the whole reservoir. The indistinct thermal boundary layer region expands in the offshore region (the dashed line $x_{d}$ in figure 3 a). Within the indistinct thermal boundary layer region, a conduction-dominated subregion develops and also expands in the offshore direction (the dashed line $x_{h}$ in figure 3 a). Within the conduction-dominated subregion, the flow reaches the quasi-steady state. Convection in the rest of the domain enhances with time. The expansion of the indistinct thermal boundary layer region stops when the interface $x_{d}$ reaches the position $x_{B}$ ; at the time ( $t\sim t_{B}$ ) convection initially balances horizontal conduction at $x_{B}$ . The conduction-dominated subregion continues to extend offshore and the horizontal temperature gradient in the region between $x_{h}$ and $x_{B}$ keeps increasing. The convection-dominated subregion begins to expand in both the onshore and offshore directions (the dashed lines $x_{cL}$ and $x_{cR}$ in figure 3 b). The thermal boundary layer within the convection-dominated subregion (between $x_{cL}$ and $x_{cR}$ ) shrinks with time. The thermal boundary layer in the region on the right side of $x_{cR}$ still grows. When the interfaces $x_{h}$ and $x_{cL}$ eventually meet at the position $x_{F}$ (at the corresponding time $t\sim t_{F}$ ), both the conduction-dominated and convection-dominated subregions stop developing. At that time ( $t\sim t_{F}$ ), the right-hand interface of the convection-dominated subregion stops at the position $x_{D}$ (figure 3 c). For $t>t_{F}$ , the flow discharges into the region on the offshore side of $x_{D}$ , which resembles an intrusion flow from the shore side of $x_{D}$ . A detailed discussion on the flow intrusion can be found in Patterson & Imberger (Reference Patterson and Imberger1980). In this study, we mainly focus on the flow development at $t\leqslant t_{F}$ .

In what follows, the detailed transient flow developments at different stages and in various subregions are described through scaling analysis.

3.2. Initiation of the flow

At time $t=0$ , there is no flow and no heat transfer in the reservoir. At the early stage, both horizontal conduction and convection are negligible so that the heat input from the water surface is diffused vertically downwards to raise the water temperature. As a consequence, a thermal boundary layer starts to grow downwards underneath the water surface. A balance between the unsteady term and the diffusion term in the energy equation (2.4) yields a scale for the thickness of the thermal boundary layer (Mao et al. Reference Mao, Lei and Patterson2010)

(3.1) $$\begin{eqnarray}{\it\delta}_{T}\sim ({\it\kappa}t)^{1/2}.\end{eqnarray}$$

The temperature in this surface layer increases due to the heat input through the surface specified in (2.8). The temperature scale thus can be obtained as

(3.2) $$\begin{eqnarray}T_{s}\sim H_{0}{\it\kappa}^{-1/2}t^{1/2}.\end{eqnarray}$$

Note that the scale $T_{s}$ represents the temperature rise relative to the initial temperature $T_{0}$ . In this very early stage, the fluid is still at rest. This situation holds until the thermal boundary layer reaches the sloping bottom and becomes indistinct.

A comparison between the thickness of the thermal boundary layer (3.1) and the local water depth $Ax$ results in the time scale for the thermal boundary layer to reach the sloping bottom,

(3.3) $$\begin{eqnarray}t_{d}\sim A^{2}x^{2}{\it\kappa}^{-1}.\end{eqnarray}$$

Clearly, with increasing offshore distance, it takes a longer time for the thermal boundary layer to reach the sloping bottom. This indicates that the indistinct thermal boundary layer region extends in the offshore direction. Within the indistinct thermal boundary layer region, the scale for the boundary layer thickness is the local water depth, i.e. 

(3.4) $$\begin{eqnarray}{\it\delta}_{T}\sim Ax.\end{eqnarray}$$

At an arbitrary position $x$ within the indistinct thermal boundary layer region, considering an infinitesimal water column of width $\text{d}x$ , the energy balance in this water column gives

(3.5) $$\begin{eqnarray}I_{0}\,\text{d}x\sim {\it\rho}_{0}C_{p}{\it\delta}_{T}\,\text{d}x\,\frac{\text{d}T_{i}}{\text{d}t},\end{eqnarray}$$

where $T_{i}$ is the temperature of the water column. Substituting (3.4) into (3.5) yields the temperature scale within the indistinct thermal boundary layer region,

(3.6) $$\begin{eqnarray}T_{i}\sim H_{0}A^{-1}x^{-1}t.\end{eqnarray}$$

Again, the scale $T_{i}$ is not the temperature itself but the temperature rise relative to the initial temperature $T_{0}$ .

Equation (3.6) indicates that a horizontal temperature gradient establishes within the indistinct thermal boundary layer region, which results in a horizontal pressure gradient that drives a flow away from the tip. As the bottom slope is assumed to be very small (usually $\ll 1$ in field situations), the vertical velocity component is negligible compared to the horizontal velocity component. A balance between the pressure term and the buoyancy term in the vertical momentum equation (2.3) yields a scale for the pressure $p$ , by applying (3.4) and (3.6),

(3.7) $$\begin{eqnarray}p\sim g{\it\beta}{\it\rho}_{0}T_{i}{\it\delta}_{T}\sim g{\it\beta}{\it\rho}_{0}H_{0}t.\end{eqnarray}$$

Mao et al. (Reference Mao, Lei and Patterson2010) have shown that the viscous term in the horizontal momentum equation (2.2) is dominant in the indistinct thermal boundary layer region at the initial stage. This indistinct thermal boundary layer is shallow so that $\partial /\partial y\gg \partial /\partial x$ . The balance in the horizontal momentum equation is thus between the viscous term and the buoyancy-induced pressure gradient, which yields a velocity scale of

(3.8) $$\begin{eqnarray}u_{n}\sim g{\it\beta}H_{0}A^{2}{\it\nu}^{-1}xt\sim \mathit{Ra}\,A^{2}{\it\kappa}^{2}h^{-4}xt,\end{eqnarray}$$

where $\mathit{Ra}$ is the global Rayleigh number defined as

(3.9) $$\begin{eqnarray}\mathit{Ra}=g{\it\beta}H_{0}h^{4}/{\it\kappa}^{2}{\it\nu}.\end{eqnarray}$$

Previous numerical and experimental studies (see e.g. Horsch et al. Reference Horsch, Stefan and Gavali1994; Bednarz, Lei & Patterson Reference Bednarz, Lei and Patterson2009b ; Mao et al. Reference Mao, Lei and Patterson2010) have shown that the properties of the convective flow in a reservoir model such as the flow velocity vary along the horizontal direction. Adopting a fixed length scale in the scaling analysis would obscure the position-dependent flow features to be revealed. Thus, a variable horizontal length scale $x$ is adopted in the present study. The same approach has been adopted in the previous scaling analyses of Mao et al. (Reference Mao, Lei and Patterson2009, Reference Mao, Lei and Patterson2010) and has been proved to be appropriate by corresponding numerical simulations.

In the distinct thermal boundary layer region, the flow is passively driven by the pressure gradient induced in the indistinct thermal boundary layer region. Note that, in the distinct thermal boundary layer region, the boundary layer thickness and temperature scales are still governed by (3.1) and (3.2), respectively. The balance between the viscous term and the pressure gradient yields a velocity scale

(3.10) $$\begin{eqnarray}u_{f}\sim g{\it\beta}H_{0}{\it\kappa}{\it\nu}^{-1}x^{-1}t^{2}\sim \mathit{Ra}\,{\it\kappa}^{3}h^{-4}x^{-1}t^{2}.\end{eqnarray}$$

It is worth noting that the velocity scale (3.10) is the same as that reported in Mao et al. (Reference Mao, Lei and Patterson2010). Although the heat flux boundary conditions at the surface in these two studies have opposite signs (one for heating and the other for cooling), the same mechanism drives the flow in both cases, and thus the same velocity scale is expected. It is also worth noting that the horizontal flow considered here is in the opposite direction to that considered in Mao et al. (Reference Mao, Lei and Patterson2010).

3.3. Extension of the indistinct thermal boundary layer region

The velocity scales given by (3.8) and (3.10) indicate that the velocity initially increases and then decreases along the $x$ direction at an instantaneous time $t$ (see figure 4). The peak velocity $u_{m}$ occurs at the interface between the indistinct and distinct thermal boundary layer regions. By substituting (3.3) into (3.8) and (3.10), we found that the two velocity scales are identical at the interface, which represents the peak velocity at any given location $x$ (achieved at time $t_{d}$ ),

(3.11) $$\begin{eqnarray}u_{m}\sim \mathit{Ra}\,A^{4}{\it\kappa}h^{-4}x^{3}.\end{eqnarray}$$

With the passage of time, the indistinct thermal boundary layer region extends in the offshore direction at a speed of $u_{i}$ . The position where the peak velocity occurs ( $x_{u}\sim ({\it\kappa}t)^{1/2}/A$ ) also moves at the velocity $u_{i}$ in the offshore direction, which is given by

(3.12) $$\begin{eqnarray}u_{i}\sim {\it\delta}_{T}A^{-1}t^{-1}\sim A^{-1}{\it\kappa}^{1/2}t^{-1/2}.\end{eqnarray}$$

While heat is transferred into the thermal boundary layer through the surface heat flux, a horizontal temperature gradient is established after the thermal boundary layer reaches the sloping bottom. The heat entering the thermal boundary layer is thus conducted away by the horizontal temperature gradient. On the other hand, the horizontal temperature gradient induces a horizontal pressure gradient, which drives a flow in the offshore direction. Therefore, heat is also convected away by the flow induced by the horizontal pressure gradient.

Figure 4.

Sketch of the velocity profile along the $x$ direction. Here $u_{m}$ denotes the peak velocity and $x_{u}$ denotes the corresponding location of $u_{m}$ . Initially, with the passage of time, the peak velocity increases and its location moves in the offshore direction. The solid line denotes data at the water surface ( $y=0$ ) from numerical simulation, whereas the dash-dotted line denotes data from scaling analysis.

3.4. Development of a conduction-dominated subregion

It is expected that the flow will achieve a steady or quasi-steady state after a certain time period, which is characterized by the steady velocity distribution in the whole reservoir. At the steady or quasi-steady state, the temperature gradient becomes steady in the whole reservoir; otherwise, the variation of temperature gradient would induce additional forces to accelerate or decelerate the flow. To maintain a steady temperature gradient across the entire reservoir, the temperature has to be constant or increase at the same rate everywhere in the reservoir. Since the reservoir model considered here is an enclosed domain with heat entering from the surface and no heat escaping from anywhere, it is expected that the temperature increases at the same rate but the temperature gradient remains steady in the entire reservoir at the quasi-steady state. Based on the above argument, the average temperature increase rate in the whole reservoir (considering that the flat-bottom region is much longer than the sloping-bottom region) can be calculated as

(3.13) $$\begin{eqnarray}\frac{\text{d}T_{avg}}{\text{d}t}\sim H_{0}h^{-1}\quad \text{or}\quad T_{avg}\sim H_{0}th^{-1}.\end{eqnarray}$$

It is worth noting that the heat advected out of the domain is neglected in (3.13), which is appropriate for a sufficiently long domain. At a given location, when the convective flow reaches the quasi-steady state, part of the energy entering from the top surface is absorbed by the water and the rest is either conducted or convected away. In the indistinct thermal boundary layer region, the comparison between convection and horizontal conduction in (2.4) gives

(3.14) $$\begin{eqnarray}\frac{\text{convection}}{\text{horizontal conduction}}\sim \frac{uTx^{-1}}{{\it\kappa}Tx^{-2}}\sim ux{\it\kappa}^{-1}\sim \mathit{Ra}\,A^{2}{\it\kappa}h^{-4}x^{2}t.\end{eqnarray}$$

Note that the velocity $u$ in (3.14) is governed by scale (3.8).

From (3.14), it is found that conduction is dominant when $t_{com}<\mathit{Ra}^{-1}A^{-2}{\it\kappa}^{-1}h^{4}x^{-2}$ , where $t_{com}$ represents the time scale for heat transfer to switch from conduction-dominated to convection-dominated at a given $x$ value. Within the indistinct thermal boundary layer region, for a particular location $x_{0}$ , the thermal boundary layer reaches the bottom at $t_{dx0}\sim A^{2}x_{0}^{2}{\it\kappa}^{-1}$ . If $t_{dx0}<t_{com}$ , the flow is still conduction-dominant, which gives $x_{0}<\mathit{Ra}^{-1/4}A^{-1}h$ . Thus, we can always find an $x_{0}\sim \mathit{Ra}^{-1/4}A^{-1}h$ so that within the region $x<x_{0}$ the horizontal conduction term is larger than the convection term. In this region, the velocity stops increasing when horizontal conduction is strong enough to remove the excessive heat input from the surface that is not absorbed by the water. The energy balance in this region is thus written as

(3.15) $$\begin{eqnarray}I_{0}x\sim {\it\rho}_{0}C_{p}{\it\kappa}\left(-\frac{\partial T}{\partial x}\right)Ax+\frac{1}{2}{\it\rho}_{0}C_{p}xAx\frac{\text{d}T_{avg}}{\text{d}t}.\end{eqnarray}$$

Substituting (3.13) into (3.15) gives

(3.16) $$\begin{eqnarray}-\frac{\partial T}{\partial x}\sim H_{0}{\it\kappa}^{-1}A^{-1}\left(1-\frac{Ax}{2h}\right).\end{eqnarray}$$

Substituting (3.6) into (3.16) yields the time scale to reach the quasi-steady state,

(3.17) $$\begin{eqnarray}t_{h}\sim \left(1-\frac{Ax}{2h}\right)x^{2}{\it\kappa}^{-1}.\end{eqnarray}$$

Equation (3.17) implies that the region where the quasi-steady state is achieved extends in the $x$ direction with time. Horizontal conduction dominates in this region, which carries away the excessive heat that is not absorbed by the water. We shall refer to this region as the conduction-dominated subregion, which is a subregion in the indistinct thermal boundary layer region. It is worth noting that the internal energy change and the horizontal conduction were not accounted for in Mao et al. (Reference Mao, Lei and Patterson2010), and the conduction-dominated and the indistinct thermal boundary layer regions were not distinguished in their study. As implied by (3.17), the conduction-dominated subregion expands in the offshore direction with time. Comparing (3.3) and (3.17) gives

(3.18) $$\begin{eqnarray}\frac{t_{h}}{t_{d}}\sim \frac{1-Ax/(2h)}{A^{2}}\sim A^{-2}-A^{-1}xh^{-1}.\end{eqnarray}$$

We can then confirm that $t_{h}/t_{d}\gg 1$ because $A\ll 1$ and the maximum $x$ value is $x\sim h/A$ . Thus, the conduction-dominated subregion is always contained by the indistinct thermal boundary layer region.

3.5. Initiation of balance between convection and horizontal conduction

Equations (3.8) and (3.10) indicate that the velocity in the reservoir increases with time, resulting in an increase in convection. The convection term may eventually become comparable with the horizontal conduction term over a certain region. In such a region, the unabsorbed energy from the surface is removed from the layer by convection. It is shown above that the peak velocity occurs at the interface between the indistinct and distinct thermal boundary layer regions. At that point, local convection is maximum, which indicates that the energy balance should be initially achieved there, which gives

(3.19) $$\begin{eqnarray}I_{0}x\sim {\it\rho}_{0}C_{p}u_{m}{\it\delta}_{T}T_{c}+\frac{1}{2}{\it\rho}_{0}C_{p}xAx\frac{\text{d}T_{avg}}{\text{d}t}.\end{eqnarray}$$

Note that in (3.19) the thickness of the thermal boundary layer and the velocity are governed by (3.4) and (3.11) respectively. The temperature scale $T_{c}$ can be calculated from (2.8), giving

(3.20) $$\begin{eqnarray}T_{c}\sim H_{0}Ax{\it\kappa}^{-1}.\end{eqnarray}$$

Substituting (3.4), (3.11), (3.13) and (3.20) into (3.19) gives

(3.21) $$\begin{eqnarray}x^{4}+0.5\mathit{Ra}^{-1}h^{3}A^{-5}x-\mathit{Ra}^{-1}h^{4}A^{-6}\sim 0.\end{eqnarray}$$

From (3.21) we can obtain the position where the convection balance initially occurs ( $x_{B}$ ). Although (3.21) can be solved analytically, it would be more convenient to yield the solution by asymptotic analysis or a numerical method. The asymptotic solution of $x_{B}$ can be written as $x_{B}\sim [\mathit{Ra}^{-1/4}A^{-3/2}-(1/8)\mathit{Ra}^{-1/2}A^{-2}-(1/128)\mathit{Ra}^{-3/4}A^{-5/2}]h$ . If the Rayleigh number is big enough so that $Ax_{B}/(2h)\ll 1$ , we can obtain

(3.22) $$\begin{eqnarray}x_{B}\sim \mathit{Ra}^{-1/4}A^{-3/2}h.\end{eqnarray}$$

The corresponding time scale $t_{B}$ is obtained by substituting (3.22) into (3.3), which gives

(3.23) $$\begin{eqnarray}t_{B}\sim \mathit{Ra}^{-1/2}{\it\kappa}^{-1}A^{-1}h^{2}.\end{eqnarray}$$

The variation of $x_{B}$ with $\mathit{Ra}$ is shown in figure 5. With increasing $\mathit{Ra}$ , these solutions converge.

Figure 5.

Variation of $x_{B}$ , $x_{F}$ and $x_{D}$ with Ra (see the caption of figure 3 for the definitions of $x_{B}$ , $x_{F}$ and $x_{D}$ ). For $x_{B}$ , the solid, dotted and dash-dotted-dotted lines denote the numerical solution of (3.21), the asymptotic solution and the approximate solution given by (3.22), respectively.

The thermal boundary layer stops growing once the energy balance is reached. This indicates that the indistinct thermal boundary layer region stops extending at the position $x_{B}$ and the corresponding time $t_{B}$ . Although the convection balance is achieved at $x\sim x_{B}$ and $t\sim t_{B}$ , this does not mean that the development of flow within the indistinct thermal boundary layer region ( $x<x_{B}$ ) stops. In fact, the region where the convection balance is achieved extends in both onshore and offshore directions from the initial position $x\sim x_{B}$ as demonstrated in § 3.6. The indistinct thermal boundary layer region thus shrinks towards the tip region.

3.6. Expansion of the convection-dominated subregion

As shown in § 3.4, the conduction-dominated subregion, which is within the indistinct thermal boundary layer region, extends from the tip region in the offshore direction. At the time $t\sim t_{B}$ , the right boundary of the conduction-dominated subregion $x_{h}$ satisfies $x_{h}<x_{B}$ . Although the local energy balance is achieved at $x<x_{h}$ (conduction) and $x\sim x_{B}$ (convection), the horizontal temperature gradient in the region $x_{h}<x<x_{B}$ still increases, as neither convection nor horizontal conduction is strong enough to remove the excessive heat from that region. The horizontal pressure gradient also increases in this region, and so does the flow velocity. Thus, with the passage of time, convection outside the conduction-dominated subregion (i.e. for the region $x>x_{h}$ ) continuously increases until energy balance is achieved in this region.

For $t>t_{B}$ , the conduction-dominated subregion continuously extends in the offshore direction with time, that is, $x_{h}$ increases with time. As convection increases, convection in the region $x_{h}<x<x_{B}$ may eventually become strong enough to remove the unabsorbed heat from the region, which gives

(3.24) $$\begin{eqnarray}I_{0}x\sim {\it\rho}_{0}C_{p}u{\it\delta}_{T}T_{c}+\frac{1}{2}{\it\rho}_{0}C_{p}xAx\frac{\text{d}T_{avg}}{\text{d}t}.\end{eqnarray}$$

In the region $x_{h}<x<x_{B}$ , the thermal boundary layer is indistinct. Thus the velocity in (3.24) is governed by (3.8). The thickness of the thermal boundary layer and the temperature scale $T_{c}$ are governed by (3.4) and (3.20), respectively. The time scale at which the convection balance is achieved can then be obtained as

(3.25) $$\begin{eqnarray}t_{cL}\sim \mathit{Ra}^{-1}h^{4}A^{-4}{\it\kappa}^{-1}x^{-2}\left(1-\frac{Ax}{2h}\right).\end{eqnarray}$$

The corresponding position $x_{cL}$ where the convection balance is achieved can be calculated from (3.25), which implies that $x_{cL}$ moves backwards towards the tip region with time. Note that, at $t\sim t_{B}$ , $x_{cL}$ coincides with $x_{B}$ . Thus, for $t>t_{B}$ , the region where the convection balance is achieved extends from $x_{B}$ towards the onshore region. We shall refer to the region as the convection-dominated subregion.

For the distinct thermal boundary layer region ( $x>x_{B}$ ), convection also increases with time. Convection in this region may eventually become strong enough to remove the unabsorbed heat, which is also governed by (3.24). However, the thickness of the thermal boundary layer, the temperature and the velocity are now governed by (3.1), (3.2) and (3.10), respectively. This energy balance yields the following time scale at which the convection balance is achieved:

(3.26) $$\begin{eqnarray}t_{cR}\sim \mathit{Ra}^{-1/3}h^{4/3}{\it\kappa}^{-1}x^{2/3}\left(1-\frac{Ax}{2h}\right)^{1/3}.\end{eqnarray}$$

Once the convection balance is achieved, the thermal boundary layer stops growing. The corresponding position at which the convection balance is achieved, $x_{cR}$ , can be calculated from (3.26), which implies that $x_{cR}$ moves in the offshore direction with time. Note that, at $t\sim t_{B}$ , $x_{c}$ coincides with $x_{B}$ . Thus, for $t>t_{B}$ , the convection-dominated subregion also extends from $x_{B}$ towards the offshore region.

The above analysis shows that for $t>t_{B}$ the convection-dominated subregion extends from $x_{B}$ in both the onshore and offshore directions. The left and right boundaries of the convection-dominated subregion are $x_{cL}$ and $x_{cR}$ , respectively. For the region between $x_{h}$ and $x_{cL}$ , the horizontal temperature gradient still develops because no energy balance has been achieved. Thus, for the regions outside the conduction-dominated subregion ( $x>x_{h}$ ), the horizontal pressure gradient still increases, and so does the velocity. This indicates that, even in the convection-dominated subregion, convection still keeps increasing after the convection balance is initially achieved. The thermal boundary layer in the convection-dominated subregion has to adjust in order to maintain the energy balance. This dynamic energy balance can still be described by (3.24). However, the velocity, temperature and thermal boundary layer thickness scales in (3.24) have to be re-evaluated.

In the convection-dominated subregion, the balance in the horizontal momentum equation is between the viscous term and the buoyancy-induced pressure gradient, which gives

(3.27) $$\begin{eqnarray}\frac{p}{{\it\rho}_{0}x}\sim {\it\nu}\frac{u}{{\it\delta}_{T}^{2}}.\end{eqnarray}$$

It is worth noting that the pressure $p$ in the above relation is still determined by the balance between the pressure gradient term and the buoyancy term in the vertical momentum equation (2.3), which leads to a scale for pressure similar to (3.7), that is, $p\sim g{\it\beta}{\it\rho}_{0}T_{c}{\it\delta}_{T}$ . However, in this case the thermal boundary layer thickness is unknown and is determined by neither (3.1) nor (3.4). Equation (3.27) can be rewritten as

(3.28) $$\begin{eqnarray}u\sim g{\it\beta}H_{0}{\it\nu}^{-1}t{\it\delta}_{T}^{2}x^{-1}\sim \mathit{Ra}\,h^{-4}{\it\kappa}^{2}t{\it\delta}_{T}^{2}x^{-1}.\end{eqnarray}$$

The temperature scale can be derived from the boundary condition (2.8), which gives

(3.29) $$\begin{eqnarray}T_{c}\sim H_{0}{\it\delta}_{T}{\it\kappa}^{-1}.\end{eqnarray}$$

Substituting (3.28) and (3.29) into (3.24) gives

(3.30) $$\begin{eqnarray}{\it\delta}_{T}\sim \mathit{Ra}^{-1/4}{\it\kappa}^{-1/4}t^{-1/4}x^{1/2}h\left(1-\frac{Ax}{2h}\right)^{1/4}.\end{eqnarray}$$

The velocity scale can then be obtained by substituting (3.30) into (3.28), yielding

(3.31) $$\begin{eqnarray}u\sim \mathit{Ra}^{1/2}{\it\kappa}^{3/2}t^{1/2}h^{-2}\left(1-\frac{Ax}{2h}\right)^{1/2}.\end{eqnarray}$$

Equation (3.30) indicates that the thermal boundary layer thickness decreases with time. This means that the thickness of thermal boundary layer shrinks in the convection-dominated subregion while this region extends in both the onshore and offshore directions.

3.7. Quasi-steady state of convection- and conduction-dominated subregions

Equation (3.17) implies that the right boundary of the conduction-dominated subregion moves in the offshore direction with time. The analysis in § 3.6 shows that the convection-dominated subregion extends from $x_{B}$ in both the onshore and offshore directions. In the region between the conduction-dominated and the convection-dominated subregions, the horizontal temperature gradient is still growing because neither horizontal conduction nor convection is sufficiently strong to carry away the unabsorbed heat. The horizontal temperature gradient stops growing only when the right boundary of the conduction-dominated subregion meets the left boundary of the convection-dominated subregion, i.e.  $t_{h}\sim t_{cL}$ , which gives

(3.32) $$\begin{eqnarray}x_{F}\sim \mathit{Ra}^{-1/4}A^{-1}h,\end{eqnarray}$$

and the corresponding time is

(3.33) $$\begin{eqnarray}t_{F}\sim (1-{\textstyle \frac{1}{2}}\mathit{Ra}^{-1/4})\mathit{Ra}^{-1/2}{\it\kappa}^{-1}A^{-2}h^{2}.\end{eqnarray}$$

The variation of $x_{F}$ with $\mathit{Ra}$ is shown in figure 5. As $A\ll 1$ , $t_{F}$ is always larger than  $t_{B}$ .

At $t\sim t_{F}$ , the conduction-dominated subregion stops extending in the offshore direction. At the quasi-steady state, the velocity scale for the conduction-dominated subregion can be obtained by substituting (3.17) into (3.8) as follows:

(3.34) $$\begin{eqnarray}u\sim g{\it\beta}H_{0}A^{2}{\it\nu}^{-1}xt_{h}\sim \mathit{Ra}\,A^{2}{\it\kappa}h^{-4}x^{3}\left(1-\frac{Ax}{2h}\right)\quad \text{for}~x\lesssim x_{F}.\end{eqnarray}$$

Once the horizontal temperature gradient stops increasing, the horizontal pressure gradient remains constant, and so does the velocity. Thus, the velocity in the convection-dominated subregion stops increasing, the region stops expanding, and the thickness of the thermal boundary layer stops shrinking. The left boundary of the convection-dominated subregion and the right boundary of the conduction-dominated subregion meet at $x\sim x_{F}$ . The right boundary of the convection-dominated subregion stops at $t\sim t_{F}$ , which can be obtained by substituting (3.33) into (3.26) as follows:

(3.35) $$\begin{eqnarray}x^{2/3}\left(1-\frac{Ax}{2h}\right)^{1/3}\sim \left(1-\frac{1}{2}\mathit{Ra}^{-1/4}\right)\mathit{Ra}^{-1/6}A^{-2}h^{2/3}.\end{eqnarray}$$

From (3.35) we can obtain the position of the right boundary of the convection-dominated subregion $x_{D}$ . Equation (3.35) can be solved either analytically or numerically. For sufficiently high Rayleigh numbers $Ax_{D}/2h\ll 1$ , we can obtain

(3.36) $$\begin{eqnarray}\displaystyle x_{D} & {\sim} & \displaystyle \mathit{Ra}^{-1/4}A^{-3}h,\end{eqnarray}$$

(3.37) $$\begin{eqnarray}\displaystyle t_{D} & {\sim} & \displaystyle \mathit{Ra}^{-1/2}{\it\kappa}^{-1}A^{-2}h^{2}.\end{eqnarray}$$

The variation of $x_{D}$ with $\mathit{Ra}$ is also shown in figure 5. Note that the solid line denotes the numerical solution of (3.35) whereas the dashed line denotes the approximate solution given by (3.36). At $t\sim t_{F}$ , the thermal boundary layer in the convection-dominated subregion stops shrinking. The final scales for the thickness of the thermal boundary layer and the velocity can be obtained by substituting (3.33) into (3.30) and (3.31), respectively,

(3.38) $$\begin{eqnarray}\displaystyle {\it\delta}_{T} & {\sim} & \displaystyle \displaystyle \mathit{Ra}^{-1/8}x^{1/2}A^{1/2}h^{1/2}\left(1-\frac{1}{2}\mathit{Ra}^{-1/4}\right)^{-1/4}\left(1-\frac{Ax}{2h}\right)^{1/4},\end{eqnarray}$$

(3.39) $$\begin{eqnarray}\displaystyle u & {\sim} & \displaystyle \displaystyle \mathit{Ra}^{1/4}{\it\kappa}A^{-1}h^{-1}\left(1-\frac{1}{2}\mathit{Ra}^{-1/4}\right)^{1/2}\left(1-\frac{Ax}{2h}\right)^{1/2}.\end{eqnarray}$$

The flow on the right side of the convection-dominated subregion may still develop after $t>t_{F}$ . We shall refer to this region as the free-developing region. In this region, the flow is driven by the pressure gradient induced by the temperature gradient in the indistinct boundary layer region, which remains constant after $t>t_{F}$ ,

(3.40) $$\begin{eqnarray}p_{F}\sim g{\it\beta}{\it\rho}_{0}H_{0}t_{F}.\end{eqnarray}$$

The balance in the horizontal momentum equation is between the viscous term and the buoyancy-induced pressure gradient, which gives

(3.41) $$\begin{eqnarray}\frac{p_{F}}{{\it\rho}_{0}x}\sim {\it\nu}\frac{u}{{\it\delta}_{T}^{2}}.\end{eqnarray}$$

The scale of the thermal boundary layer thickness in (3.41) is governed by (3.1). The velocity scale can be obtained by substituting (3.1) and (3.40) into (3.41), which yields

(3.42) $$\begin{eqnarray}u\sim \mathit{Ra}^{1/2}(1-{\textstyle \frac{1}{2}}\mathit{Ra}^{-1/4})A^{-2}h^{-2}{\it\kappa}^{2}x^{-1}t.\end{eqnarray}$$

The flow stops developing when the energy balance described by (3.24) is reached, which yields the time scale

(3.43) $$\begin{eqnarray}t_{s}\sim \mathit{Ra}^{-1/4}\left(1-\frac{1}{2}\mathit{Ra}^{-1/4}\right)^{-1/2}\left(1-\frac{Ax}{2h}\right)^{1/2}Axh{\it\kappa}^{-1}.\end{eqnarray}$$

The final thermal boundary layer thickness and velocity scales at the quasi-steady state can then be obtained by substituting (3.43) into (3.1) and (3.42), respectively, which are exactly the same as (3.38) and (3.39).

3.8. Possible flow regimes

The time-dependent behaviour of the flow varies with the Rayleigh number. The possible developmental pathways of the flow at different Rayleigh numbers are summarized in table 1. If the Rayleigh number is sufficiently small ( $\mathit{Ra}<1$ ), the condition $h/A<x_{F}$ holds. In this case, horizontal conduction is always greater than convection. After sufficient time has elapsed, the thermal boundary layer attaches to the bottom in the whole reservoir. Subsequently, the horizontal temperature gradient continues to increase until it becomes significant enough to conduct away the heat input from the water surface and the quasi-steady state is achieved.

For medium Rayleigh numbers ( $1<\mathit{Ra}<\mathit{Ra}_{i}$ ), the condition $x_{F}<h/A<x_{B}$ holds. Note that $\mathit{Ra}_{i}$ can be obtained by letting $x_{B}\sim h/A$ , where $x_{B}$ is determined by (3.21). The value of $\mathit{Ra}_{i}$ is $O(A^{-2})$ , which is ${\sim}50$ if $A=0.1$ . After sufficient time has elapsed, the thermal boundary layer eventually attaches to the bottom in the whole reservoir, but the flow velocity keeps increasing. Horizontal conduction is dominant in the whole reservoir. Convection first becomes significant in the region with the maximum depth and the thermal boundary layer becomes distinct there again, which occurs at $t\sim t_{B}$ . The indistinct thermal boundary layer region shrinks until $t\sim t_{F}$ . At the quasi-steady state, the whole reservoir can be divided into two subregions, i.e. the conduction- and convection-dominated subregions. In the conduction-dominated subregion, the velocity scale at the quasi-steady state is given by (3.34). In the convection-dominated subregion, the flow becomes steady at $t\sim t_{F}$ ; the corresponding thermal boundary layer thickness and velocity scales are described by (3.38) and (3.39), respectively.

For sufficiently high Rayleigh numbers ( $\mathit{Ra}>\mathit{Ra}_{i}$ ), the condition $h/A>x_{B}$ holds. In this Rayleigh-number regime, the indistinct thermal boundary layer region does not occupy the whole reservoir during the developing stage and stops expanding when $t\sim t_{B}$ at $x\sim x_{B}$ . The indistinct thermal boundary layer region then shrinks until $t\sim t_{F}$ . At the quasi-steady state, the whole reservoir can be divided into the conduction-dominated and convection-dominated subregions.

5.1. Unsteady scales for the indistinct thermal boundary layer region

At the initial stage, the scaling analysis indicates that the instantaneous horizontal velocity increases and then decreases in the $x$ direction ((3.8) and (3.10)). The peak velocity is located at the interface between the indistinct and distinct thermal boundary layer regions, which moves in the $x$ direction with time. If the Rayleigh number is sufficiently low, the interface eventually reaches the end of the slope region at ${\it\tau}\sim 1$ . At the same time, the thermal boundary layer in the whole flat region also reaches the bottom and becomes indistinct. As a consequence, the thermal boundary layer becomes indistinct over the entire flow domain, and the interface between the distinct and indistinct thermal boundary layer regions no longer exists. Since the peak velocity initially moves with the interface as shown in § 3.3, it remains at the end of the slope region after the disappearance of the distinct thermal boundary layer (i.e. for ${\it\tau}\gtrsim 1$ ).

Figure 6 shows the horizontal velocity profile along the water surface at different time for $\mathit{Ra}=0.14$ . For ${\it\tau}\lesssim 1$ , the velocity first increases and then decreases in the offshore direction. The velocity is almost zero for $X$ bigger than a certain value. The position of the peak velocity moves towards the offshore direction when ${\it\tau}\lesssim 1$ , corresponding to the expansion of the indistinct thermal boundary layer region (3.12). The thermal boundary layer attaches to the bottom in the entire region at the time scale ${\it\tau}\sim 1$ (3.3). The position of the peak velocity then remains fixed with $X$ of order $h/A$ for ${\it\tau}\gtrsim 1$ . These behaviours are consistent with the scaling analysis.

Figure 6.

Horizontal velocity profiles along the water surface at different times for $\mathit{Ra}=0.14$ . The symbols on the plots are used to distinguish the curves at different times instead of representing the $x$ locations where the $U$ velocity is sampled. The arrows indicate the locations of the peak velocity.

Mao et al. (Reference Mao, Lei and Patterson2010) also proposed a velocity scale for the early stage of the flow development, which is the same as the present velocity scale (3.10). They considered this velocity scale as the sole velocity scale governing the flow in the early stage and reported that the velocity at any given time decreases monotonically with the offshore distance. However, both the present scaling analysis and numerical simulation indicate that the velocity initially increases and then decreases with the offshore distance. Therefore, the present study has provided a more comprehensive picture of the transient flow behaviour.

In the flow regime $\mathit{Ra}<1$ , the interface between the distinct and indistinct thermal boundary layer regions initially moves in the offshore direction with the extension of the indistinct thermal boundary layer region $(x_{u}\sim ({\it\kappa}t)^{1/2}/A)$ until it reaches the end of the sloping region at ${\it\tau}\sim 1$ . The interface disappears at ${\it\tau}\sim 1$ because the thermal boundary layer in the whole flat region also reaches the bottom and becomes indistinct as mentioned above. The location of the peak velocity then remains at the end of the sloping region for ${\it\tau}\gtrsim 1$ . In the flow regime $\mathit{Ra}>1$ , the corresponding interface initially also moves in the offshore direction with the extension of the indistinct thermal boundary layer region $(x_{u}\sim ({\it\kappa}t)^{1/2}/A)$ and then moves back towards the tip region with the shrinkage of the indistinct thermal boundary layer region (which can be derived from (3.25)). We record the position $X_{u}$ of the peak $U$ velocity along the water surface for the numerical simulation at different Rayleigh numbers and plot the data in figure 7(a). The curves for $\mathit{Ra}<1$ indicate that the location of the peak velocity moves rapidly in the offshore direction when ${\it\tau}\lesssim 1$ and reaches around $X\sim 8.5$ at ${\it\tau}\sim 1$ . For ${\it\tau}\gtrsim 1$ , the location of the peak velocity stays around $X\sim O(10)$ . The curves for $\mathit{Ra}>1$ shows that $X_{u}$ initially moves in the offshore direction and then retracts. All these results agree well with the scaling prediction. However, the velocity scale provided by Mao et al. (Reference Mao, Lei and Patterson2010) indicated that the position of the peak $U$ velocity only moves in the offshore direction, which cannot explain the retracting behaviour of the location of the peak velocity.

Figure 7.

The transient behaviour of the position of the peak $U$ velocity along the surface: (a) time history of the position; (b) validation of the scale for $t<t_{B}$ ; (c) validation of the scale for $t_{B}<t<t_{F}$ ; (d) validation of the scale for the quasi-steady state ( $\mathit{Ra}>1$ ). The Rayleigh numbers are indicated in the panels. The symbols denote the numerical results and the solid line is a linear fit in (b–d).

Note that there is a coarse stagger on the curves for $\mathit{Ra}<1$ in figure 7(a). As mentioned in § 4, the computational meshes are non-uniform, with higher mesh density near all the boundaries. The distance between two neighbouring points around $X\sim O(10)$ is relatively large. The location of the peak velocity may drift between two neighbouring points due to numerical errors, which may in turn cause the obvious staggers on the curves. These staggers may be avoided by using a finer mesh in the horizontal direction. However, this would require a significant increase in computational time. Given that the present numerical results are sufficiently accurate to capture the major flow features, the present mesh is adopted, although its resolution in the region around $X\sim O(10)$ is relatively low.

The time scale for the thermal boundary layer to reach the sloping bottom is governed by the scale (3.3). The dimensionless form of the scale can be rewritten as ${\it\tau}_{d}\sim A^{2}X^{2}$ . As mentioned previously, the peak flow velocity occurs at the interface between the indistinct and distinct thermal boundary layer regions. Accordingly, $X_{u}^{2}$ linearly increases with time at the initial stage for all the Rayleigh numbers as shown in figure 7(b). The curves deviate from the linear relationship for $t>t_{B}$ (curves for $\mathit{Ra}>1$ ) or $t>h^{2}/{\it\kappa}$ (curves for $\mathit{Ra}<1$ ).

The time scale for the shrinkage of the indistinct thermal boundary layer is governed by the scale (3.25). The dimensionless form of the scale can be rewritten as ${\it\tau}_{cL}\sim (1-2AX/3)/(\mathit{Ra}\,A^{4}X^{2})$ if the new prefactor $(1-2Ax/(3h))$ determined by (5.1) is applied. Note that the position of the peak velocity along the surface $X_{u}$ is located at the interface between the indistinct and distinct thermal boundary layer regions, which moves in the $-x$ direction with the shrinkage of the indistinct thermal boundary layer region as shown in figure 7(a). Figure 7(c) shows the time history of $(1-2AX_{u}/3)/(\mathit{Ra}\,A^{4}X_{u}^{2})$ . The time is rescaled by $t_{F}$ (refer to the scale (3.33)). As shown in figure 7(c), the function $(1-2AX_{u}/3)/(\mathit{Ra}\,A^{4}X_{u}^{2})$ increases linearly with time over the time period of $0.4<t/t_{F}<1.5$ for $\mathit{Ra}=2.1\times 10^{3}$ and $2.1\times 10^{4}$ . This is consistent with the behaviour of the scale (3.25) in the retracting stage ( $t_{B}<t<t_{F}$ ). It is also seen in figure 7(c) that the curves for $\mathit{Ra}=2.1\times 10^{3}$ and $2.1\times 10^{4}$ do not collapse onto each other. This is because the magnitude of ${\it\tau}_{cL}$ in the scale (3.25) is both Ra- and position-dependent, and the scale (3.25) is only valid when $t_{B}<t<t_{F}$ . Therefore, it is difficult to rescale the plots to make the curves for the different Rayleigh numbers collapse onto each other. Nevertheless, figure 7(c) shows a linear relationship around $t/t_{F}\sim O(1)$ for the different Rayleigh numbers, which is consistent with the prediction of the scale (3.25).

At the final quasi-steady state, the location of the interface $X_{u}$ remains at $x_{F}$ (3.32) for $\mathit{Ra}>1$ . Figure 7(d) shows the locations of $X_{u}$ at the quasi-steady state obtained from the simulation for different Rayleigh numbers against the scaling of $x_{F}\sim \mathit{Ra}^{-1/4}A^{-1}h$ . It demonstrates an approximately linear correlation between the numerical data and the scaling prediction, which further validates the present scaling analysis. However, it may be argued that the numerical data shown in figure 7(d) exhibit a power-law relationship with the scaling prediction. As described in the above scaling analysis, at the quasi-steady state, the peak velocity is located at the interface between the conduction-dominated and convection-dominated subregions. The scaling analysis is based on the assumption that there is a sharp transition between the conduction- and convection-dominated subregions. However, a transitional region, in which both conduction and convection are important, exists in reality. This modifies the energy balances ((3.19) and (3.24)) and causes the deviation between the numerical simulation and scaling prediction.

5.2. Unsteady scales for the conduction-dominated subregion

The scaling analysis suggests that the conduction-dominated subregion extends in the offshore direction with time. As described in the preceding scaling analysis, the conduction-dominated subregion is always within the indistinct thermal boundary layer region. For very low Rayleigh numbers ( $\mathit{Ra}<1$ ), the conduction-dominated subregion keeps expanding even after the thermal boundary layer has encompassed the whole sloping region. The final quasi-steady state in the entire reservoir occurs at the time scale ${\it\tau}\sim A^{-2}/2=50$ (estimated from (3.17)).

It is worth mentioning that Mao et al. (Reference Mao, Lei and Patterson2010) obtained the same time scale $t_{d}\sim A^{2}x^{2}/{\it\kappa}$ for the thermal boundary layer to reach the sloping bottom. They also considered $t_{d}$ as the time scale for the flow in the indistinct thermal boundary layer region to reach the quasi-steady state, i.e. the flow reaches the quasi-steady state once the thermal boundary layer reaches the sloping bottom. Therefore, there is no distinction between the indistinct thermal boundary layer region and the conduction-dominated subregion in their analysis. The present scaling analysis reveals that the developments of the flow and the horizontal temperature gradient in the indistinct thermal boundary layer region continue after the thermal boundary layer has reached the sloping bottom. The conduction-dominated subregion develops within the indistinct thermal boundary layer region until $t\sim t_{h}$ . The developing time for the flow to reach the quasi-steady state is thus much longer than that predicted by Mao et al. (Reference Mao, Lei and Patterson2010). The numerical results indeed support our scaling analysis, as will be shown later.

Figure 6 shows that the horizontal velocity keeps increasing when ${\it\tau}\gg \sim 1$ , which suggests that the horizontal temperature gradient is still increasing after the thermal boundary layer reaches the bottom in the entire reservoir. The development of the flow stops only if horizontal conduction is significantly strong to remove the excessive heat entering from the surface.

We also record the time histories of the peak horizontal velocity along the water surface for $\mathit{Ra}=0.014$ and 0.14 in figure 8(a). Here the peak velocity presented in figure 8(a) is further normalized by Ra. The scaling analysis indicates that the peak velocity is governed by (3.11). When ${\it\tau}\gtrsim 1$ , the thermal boundary layer has reached the bottom in the entire reservoir. The velocity is governed by scale (3.8). As the peak velocity is located at $x\sim h/A$ for ${\it\tau}\gtrsim 1$ , the peak velocity scale becomes $u_{m}\sim \mathit{Ra}\,A{\it\kappa}^{2}h^{-3}t$ . Thus, the peak velocity is a linear function of time for ${\it\tau}\gtrsim 1$ . This situation holds until ${\it\tau}\sim A^{-2}/2=50$ (see figure 8 a), i.e. the time scale for the quasi-steady state in the sloping region as given by the scale (3.17). As expected, the calculated velocities for the two different Ra values collapse onto one curve, as shown in figure 8(a).

Figure 8.

Time histories of (a) the peak $U$ velocity along the water surface and (b) the $u$ velocity at two fixed locations on the water surface. The Rayleigh numbers are indicated in the panels. The velocity and time are normalized as shown.

The above scaling analysis shows that, in the indistinct thermal boundary layer region, the velocity is a linear function of both the position and time. The flow within this region reaches the quasi-steady state at $t\sim t_{h}$ , which is position-dependent, as indicated by (3.17). At the quasi-steady state, the velocity in the conduction-dominated subregion is governed by (3.34). To validate the velocity scale, the time histories of the horizontal surface velocity at different Rayleigh numbers from the simulations are plotted with the position as a parameter in figure 8(b). The velocity and time in the figure are normalized by the scales $u\sim \mathit{Ra}\,A^{2}{\it\kappa}h^{-4}x^{3}(1-2Ax/(3h))$ (scale (3.34) with the new prefactor) and $t_{h}\sim (1-2Ax/(3h))x^{2}{\it\kappa}^{-1}$ (scale (3.17) with the new prefactor), respectively. The scaling analysis shows that the conduction-dominated subregion is confined in the tip region $x<x_{F}$ . Thus, we choose relatively small values of $x$ to ensure that the positions where the velocities are recorded are well within this region.

As shown in figure 8(b), all the curves collapse after the velocity and the time from the simulations are normalized by the corresponding scales. Figure 8(b) also shows that, at the initial stage, the velocity increases approximately linearly with time. The development of the velocity stops at $t/[x^{2}{\it\kappa}^{-1}(1-2Ax/(3h))]\sim O(1)$ . These flow behaviours agree well with the prediction of the scaling analysis.

Figure 9 shows the normalized velocity profiles along the $x$ direction by using the modified scale (5.2), i.e. with the new prefactor $(1-2Ax/(3h))$ . The curves for $\mathit{Ra}=0.014$ and 0.14 collapse onto each other, indicating that conduction is dominant in the entire slope region at these low Rayleigh numbers. It is noted in figure 9 that the curves for $\mathit{Ra}=21$ and 70 deviate from those for $\mathit{Ra}<1$ when $X$ is beyond a certain value. As indicated by the scaling analysis, the region far away from the tip region becomes convection-dominated for $\mathit{Ra}>1$ . The velocity in the convection-dominated subregion does not follow the velocity scale of (5.2) and thus deviates from those for $\mathit{Ra}<1$ . The position of the deviating point moves towards the tip region with increasing Ra because the conduction-dominated subregion becomes smaller at larger Ra (refer to the variation of $x_{F}$ with Ra in figure 5). It is worth noting that using the prefactor $(1-Ax/(2h))$ , which is appropriate for a large and effectively infinite rectangular domain, does not match the expected asymptote, but the new prefactor $(1-2Ax/(3h))$ derived for the finite domain does. This observation further confirms that the change in the internal energy of the water body has to be taken into account when considering the energy balance. All these results further verify the present scaling analysis.

Figure 9.

Profiles of the $U$ velocity along the water surface from the simulation.

5.3. Unsteady scales for the convection-dominated subregion

For high Rayleigh numbers ( $\mathit{Ra}>1$ ), convection becomes dominant in the region far from the tip region. The scaling analysis shows that the horizontal pressure gradient still increases after the balance between convection and horizontal conduction is initially achieved at $x\sim x_{B}$ . Thus, the horizontal velocity also increases and convection enhances. The convection-dominated subregion expands in both the onshore and offshore directions, while the indistinct thermal boundary layer region shrinks. The shrinkage of the indistinct thermal boundary layer region has been demonstrated by the time history of the location of the peak surface horizontal velocity $X_{u}$ , as shown in figure 7(a). To further show the expansion of the convection-dominated subregion in the onshore direction, the time evolution of the isotherms for $\mathit{Ra}=2.1\times 10^{5}$ are shown in figure 10. At the very early time $t=0.005$ , the isotherms are horizontal and parallel in most regions of the reservoir except for the tip region. The isotherms are curled in the tip region in order to satisfy the no-flux conduction there, indicating that the thermal boundary layer attaches to the bottom and an indistinct thermal boundary layer region forms. The contour lines with a very small non-zero value of $10^{-4}$ are marked in figure 10(b–f). It is shown that the $10^{-4}$ contour line first moves in the offshore direction and then retracts in the onshore direction. If this contour line is regarded as an indicator of the interface between the distinct and indistinct thermal boundary layer regions, the expansion and retraction of the indistinct thermal boundary layer region is then indicated by the positions of the $10^{-4}$ contour lines. The retraction of the $10^{-4}$ contour line also indicates a decrease in temperature there.

Figure 10.

Isotherms in the reservoir at different time instants for $\mathit{Ra}=2.1\times 10^{5}$ . Isotherms at: (a)  $t=0.005$ with an interval of 0.02; (b)  $t=0.01$ with an interval of 0.05; (c)  $t=0.05$ with an interval of 0.1; (d)  $t=0.25$ with an interval of 0.2; (e)  $t=0.5$ with an interval of 0.5; (f)  $t=1$ with an interval of 0.5.

The above scaling analysis reveals that the dominant mode of heat transfer varies with time and position. To validate this, the horizontal heat transfer rates by conduction and convection are calculated from the simulation results. The total horizontal heat transfer rate, including the contributions of both conduction and convection, is averaged over the local water depth and defined in a dimensionless form as

(5.3) $$\begin{eqnarray}H(X)=\frac{1}{|Y_{b}|}\int _{Y_{b}}^{0}(U{\it\theta}^{\ast }-{\it\theta}_{x}^{\ast })\,\text{d}Y,\end{eqnarray}$$

where ${\it\theta}^{\ast }$ is the spatial variation of the temperature (spatial deviation from the average temperature in the reservoir) and $Y_{b}$ is the corresponding $Y$ coordinate of the bottom at a given horizontal position $X$ .

It is worth mentioning that (5.3) has been successfully used in the previous studies (Lei & Patterson Reference Lei and Patterson2005; Mao et al. Reference Mao, Lei and Patterson2009, Reference Mao, Lei and Patterson2010) to characterize the quasi-steady flow behaviours. The present study will further show the transient behaviours of flow properties. Figure 11 shows the profiles of the horizontal heat transfer rates averaged over the local water depth calculated at different times for $\mathit{Ra}=2.1\times 10^{3}$ and $2.1\times 10^{4}$ , respectively, which also compares the relative intensities of convection and conduction along the $x$ direction. The intersection of the convection and conduction curves at a particular time indicates that convection and conduction are equally important there. Clearly, this figure shows that the intersection moves towards the tip region with time, which indicates the expansion of the convection-dominated subregion towards the onshore direction. All the transient flow results presented above provide strong evidence to support the present scaling analysis.

Figure 11.

Profiles of the horizontal convection (solid lines) and conduction (dashed lines) heat transfer rates averaged over the local water depth at different time instants for (a) $\mathit{Ra}=2.1\times 10^{3}$ and (b) $\mathit{Ra}=2.1\times 10^{4}$ .

Further, the time history of the $x$ coordinate of the intersection $X_{i}$ obtained from the numerical simulation for $\mathit{Ra}=2.1\times 10^{3}$ and $2.1\times 10^{4}$ is recorded. The scaling analysis shows that the time scale for the balance between convection and vertical conduction is governed by (3.25). It is expected that there is a linear relationship between $(1-2AX_{i}/3)/(\mathit{Ra}\,A^{4}X_{i}^{2})$ and $t/t_{F}$ during the developing stage of the convection-dominated subregion, i.e. for $t_{B}<t<t_{F}$ , which is confirmed in figure 12(a). The two curves for $\mathit{Ra}=2.1\times 10^{3}$ and $2.1\times 10^{4}$ do not collapse onto each other. The reason for this is the same as that for $X_{u}$ , as explained in figure 7(c).

Figure 12.

(a) Time history of the $x$ coordinate of the intersection of the horizontal convection and conduction heat transfer curves $X_{i}$ from the numerical simulation for $\mathit{Ra}=2.1\times 10^{3}$ and $2.1\times 10^{4}$ . (b) The locations of $X_{i}$ at the quasi-steady state from the simulation versus the scaling predictions at various Rayleigh numbers ( $\mathit{Ra}>1$ ). The symbols denote the numerical results whereas the solid lines are linear fits.

At the quasi-steady state, the location of the interface $X_{i}$ stays at $x_{F}$ (3.32) if $\mathit{Ra}>1$ . The locations of $X_{i}$ at the quasi-steady state obtained from the simulation for different Rayleigh numbers are plotted against the scaling of $x_{F}\sim \mathit{Ra}^{-1/4}A^{-1}h$ in figure 12(b). The excellent linear correlation between the numerical data and the scaling prediction demonstrates that the location of $X_{i}$ at the quasi-steady state is well predicted by the scaling analysis.

The scaling analysis indicates that the flow in the convection-dominated subregion reaches the quasi-steady state at the same time. To validate this time scale, the surface velocities at different locations for $\mathit{Ra}=2.1\times 10^{5}$ are shown in figure 13(a). All the locations selected are within the convection-dominated subregion and also within the sloping-bottom region to ensure the validity of the scale. Indeed, all the curves in this figure indicate that the flow in the whole convection-dominated subregion reaches the quasi-steady state at the same time. Simulations for other Rayleigh numbers also show the same result. The previous numerical simulations reported in Yu et al. (Reference Yu, Lei and Patterson2012b ) also support this result.

Figure 13.

Time history of (a) the $U$ velocity and (b) the renormalized $u$ velocity at fixed locations on the water surface. The locations and the Rayleigh numbers are indicated in the panels. The velocity and time are normalized as shown.

It is worth noting that Mao et al. (Reference Mao, Lei and Patterson2010) reported a time scale $t_{c}\sim x^{2/3}\mathit{Ra}^{-1/3}h^{4/3}{\it\kappa}^{-1}$ for the flow in the convection-dominated subregion to reach the quasi-steady state, which suggests that the steady-state time depends on the location. Clearly the present scaling analysis and numerical simulation do not support their prediction (refer to figure 13 a).

To validate the velocity scale for the convection-dominated subregion, the time histories of the horizontal surface velocity at different Rayleigh numbers from the simulations are plotted with the position as a parameter in figure 13(b). The velocity and time in the figure are normalized by the scales $u\sim \mathit{Ra}^{1/4}{\it\kappa}A^{-1}h^{-1}$ $\times (1-2\mathit{Ra}^{-1/4}/3)^{1/2}(1-2Ax/(3h))^{1/2}$ ((3.39) with the new prefactor) and $t_{F}\sim$ $(1-2\mathit{Ra}^{-1/4}/3)\mathit{Ra}^{-1/2}{\it\kappa}^{-1}A^{-2}h^{2}$ ((3.33) with the new prefactor), respectively. The values of $x$ are chosen carefully to ensure that they are within the convection-dominated subregion. As shown in figure 13(b), with the normalization, all the curves become fairly close to each other despite some scattering, suggesting that the scaling gives a reasonable prediction for the transient flow velocity. It is also seen in figure 13(b) that the development of the flow velocity at all the Rayleigh numbers and locations stops at $t/t_{F}\sim 5$ , again confirming the validity of the scale for the steady-state time.

Figure 14 shows the profiles of the $U$ velocity along the water surface at the quasi-steady state. The velocity is normalized by the velocity scale for the convection-dominated subregion at the quasi-steady state (3.39). It is expected that the normalized velocity in the convection-dominated subregion at different Ra collapses onto a horizontal straight line because this velocity scale embodies the Rayleigh number and position dependences. As can be seen in figure 14, the normalized velocity profiles approximately converge to a horizontal line for $X$ beyond a certain value (this value is Ra-dependent and corresponds to the convection-dominated subregion). Although the curve for $\mathit{Ra}=2.1\times 10^{5}$ deviates slightly from the other curves for $X>9$ , the deviation is considered negligibly small for the purpose of demonstrating the scaling. It is also worth emphasizing that the unscaled values of $U$ in figures 13(b) and 14 vary by a factor of 4, but the rescaling reduces the variability to less than 30 % of the mean value. Thus, the simulation results generally support the scaling analysis.

Figure 14.

Profiles of the $U$ velocity along the water surface from the numerical simulation normalized by the relevant scale for the convection-dominated subregion.