A comparative study of linear and step forcing temperature profiles in horizontal convection^(a)

The meridional overturning circulation (MOC) plays a crucial role in the Earth's climate and energy system [1–4]. Potential candidates for the MOC energy sources include wind and tide [5], geothermal heating [6], and surface thermal forcing. An important conceptual model in the study of the effect of surface thermal forcing is horizontal convection (HC), in which the flow is driven by the surface temperature difference applied horizontally on a fluid layer [6–12]. There are two dimensionless control parameters in this system: $Ra\equiv\alpha g\Delta L^3/(\kappa\nu)$ is the Rayleigh number representing the strength of thermal driving, and $Pr\equiv\nu/\kappa$ is the ratio between momentum and thermal diffusivity of the working fluid. Here α is the thermal expansion coefficient, g the gravitational acceleration, $\Delta=T_{hot}-T_{cold}$ the temperature difference between hot and cold plates, L the length of the convective cell, ν refers to the kinetic viscosity, and κ the thermal diffusivity. Nusselt number Nu and Reynolds number Re are two of the main responding parameters of this system. The Nusselt number is defined as $Nu\equiv\langle|\partial T/\partial z|\rangle_{S^*,t}/(\Delta/L)$, where $\langle\cdot\rangle_{S^*,t}$ represents the averaging over time and the surface with non-zero heat flux. The Reynolds number is defined by $Re\equiv UL/\nu$, where U is the characteristic velocity of the system. A recent study [11,13] extended the Grossmann and Lohse (GL) theory for the Rayleigh-Bénard convection [14–17] to HC and revealed various scaling laws of Nu and Re with Ra based on the step forcing profile, in which the hot and cold plates take up a fraction of the top (or bottom) surface and each with uniform temperature are located at the two ends of the cell. For $Pr\gtrsim 1$, a scaling relationship $Nu\sim Ra^{1/4}$ is predicted when Ra is low (denoted by the I^* _lregime). In this regime, the boundary layer dominates the global heat transport and the viscous boundary layer thickness is a constant of the order of the cell height [11,15]. As Ra increases, the Rossby scaling [7] can be obtained (also denoted by I_lin refs. [11,13]), in which the heat transport is dominated by the boundary layer. For sufficiently high Ra such that the heat transport is controlled by the bulk flow, a scaling relationship $Nu\sim Ra^{1/4}$ was predicted (corresponding to the IV_uregime in ref. [11]). In ref. [18], the authors proposed a similar 1/4-scaling when the bulk flow is significantly turbulent. In that study, a HC system with spatially periodic thermally forcing was used. The transition to $Nu\sim Ra^{1/4}$ was observed when the scales of turbulent motion exceed the forcing scale and become comparable to the width of the box.

There is no standard choice for the type of forcing profile for horizontal convection, per se. Thus, various forcing profiles have been used in studies of HC, which include step, linear [19], sinusoidal [18,20–22], and mixed profile of fixed heat flux and fixed temperature [9]. Because of experimental and theoretical convenience, the step forcing profile has become the most frequently used in the studies of the HC system [6,11–13,23–27].

However, step forcing is obviously less relevant to the real ocean when compared with other types of forcing profiles with continuously varying temperature along the surface. To apply the results obtained based on step forcing profiles to the other types, it is important to know how the key quantities of the flow will be affected. Tsai et al. [28] compared the influence that different forcing profiles have on the global heat transfer and on the transitional behaviour of the system to the turbulent state. They proposed that the linear forcing profile gives a power law scaling closer to $Nu\sim Ra^{1/4}$ than that given by step forcing, and forcing profiles can induce different transitional behaviour from laminar to turbulent flow. However, how some of the important quantities, such as the Reynolds number, the boundary layer thickness, and the generation rate of available potential energy, behave under the different profiles remains to be explored. To find out such differences, we have conducted a comparative study of the step and continuous forcing profiles. Among different continuous forcing profiles, we use linear forcing in this study, which can already illustrate the main differences with the step forcing.

In this study a series of three-dimensional direct numerical simulations (DNS) are conducted. We consider a body of fluid (water in the present case) with non-slip boundary conditions at all boundaries. All lateral and bottom boundaries are adiabatic. As illustrated in fig. 1, the temperature difference driving the system is applied on the top surface. We use two kinds of thermal forcing profiles: a) Step forcing profile. In this configuration, the top surface is divided into three regions, a hot plate at one end and a cold plate at the other end. Each plate takes up 1/10 of the length of the top surface, with the remaining part of the surface being adiabatic. The hot and cold plates have uniform temperature T_hot and T_cold , respectively. b) Linear forcing profile, in which the temperature is linearly distributed over the length of the entire top surface, i.e., $T_{top}(x,y)=T_{cold}+(T_{hot}-T_{cold})\cdot(x/L)$. The geometrical properties of the convective cell is characterized by the aspect ratio $\Gamma=L/H$, where H is the height of the convection cell. The system is governed by Navier-Stokes equations in the Oberbeck-Boussinesq approximation

$$\begin{align} &\nabla\cdot \mathbf{u}=0 ,\\ &\frac{\partial \mathbf{u}}{\partial t}+\mathbf{u}\cdot\nabla \mathbf{u}=-\frac{1}{\rho_0}\nabla p+\nu\nabla^2\mathbf{u}+\alpha g\theta \mathbf{e_z},\\ &\frac{\partial \theta}{\partial t}+\mathbf{u}\cdot\nabla \theta=\kappa\nabla^2\theta , \end{align} \tag{ 1 }$$

where $\rho_0$, u, p and $\theta=[T-(T_{hot}+T_{cold})/2]$ are the mean density, velocity, pressure and reduced temperature, respectively. The governing equations (1) are solved in a dimensionless form. We non-dimensionalized the equations using $x_{ref}=L$, $u_{ref}=\left(\alpha gL\Delta\right)^{1/2}$, $T_{ref}=\Delta$ and $t_{ref}=x_{ref}/u_{ref}$.

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We conduct the simulations using a DNS code called CUPS [29–31], which is based on the finite-volume method and achieves 4th-order spatial precision. Temperature and velocity fields are discreted in staggered grids. The characteristic lengths of temperature and velocity are the Batchelor and the Kolmogorov length scale, respectively. For Pr > 1, the Batchelor length scale is smaller than the Kolmogorov length scale. In the traditional single resolution scheme, only one set of meshes is used. To ensure that both momentum and temperature fields are resolved, the meshes to be used are required to resolve the smallest length scale in the system, i.e., the Batchelor length scale. This scheme will inevitably bring some unnecessary computational costs in the momemtum solver. In the multiple resolution scheme, the momemtum and temperature fields are solved in their respective set of meshes, so that only the meshes for the temperature field needs to resolve the Batchelor length scale and the momentum field can be solved in coarser meshes. Thus, computational costs can be efficiently saved without any sacrifices in precision [31]. For simulations with $Ra\geq1\times10^{10}$, the multiple resolution scheme is used; while for $Ra<1\times10^{10}$ we use the single resolution method. Lists of the meshes used in our simulations and other statistical quantities are provided in the Supplementary Material Supplementarymaterial.pdf . Tests have been done comparing global quantities of the system, e.g., Nu, and we find that the differences between two numerical methods are within 1%. To ensure that small local length scales are resolved, grids are refined near boundaries. For the step forcing profile, the normalized total area of hot and cold plates, denoted as $\lambda=L_{hot}/L=L_{cold}/L$, is fixed to be 0.1. We fixed $Pr=8.0$ and $\Gamma=10$ in this study, and varied Ra from $1\times10^8$ to $1\times10^{12}$.

We first discuss and compare our results of Nu with other studies. In fig. 2(a), we present Nu compensated by Ra^0.23, which is the scaling relationship we observed for both forcing profiles after plume emissions occur. This scaling relationship agrees with the results for high Pr observed in refs. [13,23,24]. For comparison, we also plot the results from refs. [23,24], which use step forcing and the same configuration and geometry as ours. Here we only use those with Pr = 10 from their results, which is the closest Pr to ours. The results of both step and linear forcing in ref. [28] for $Pr=6.14$, $\Gamma=6.25$ and $\lambda=0.5$ are also plotted in fig. 2(a). As shown in fig. 2(a), our results for the step forcing profile agree well with those of refs. [23,24]. Comparing our results with those of ref. [28], we find that for $Ra\gtrsim 1\times10^{10}$ both data sets yield similar scaling behavior for both step and linear forcing. However, there exist significant differences in the magnitudes of Nu for linear forcing and scaling relationships at $Ra\lesssim1\times10^{10}$ for both forcing types. Such differences may be attributed to the different Γ used in the two studies.

For linear forcing, heat transfer is less efficient in the middle part than in the two ends, since the temperature contrast between the the top plate and the bulk flow is small in the middle. When evaluating Nu, the whole top plate is taken into account for linear forcing, which makes the magnitudes for linear forcing overall smaller than the step forcing, as shown in fig. 2(a). For $Ra<1\times10^9$ a steep scaling is observed. As Ra increases, we observe the Rossby scaling $Nu\sim Ra^{0.2}$ [7,11] for about a decade $(1\times10^9\lesssim Ra\lesssim1\times10^{10})$ for both forcing profiles. For $Ra\gtrsim 1\times10^{10}$, plume emissions occur and a scaling relationship $Nu\sim Ra^{0.23}$ is observed. However, the transitional behaviour between linear and step forcing appears to be different, i.e., for step forcing the system directly changes from the Rossby scaling $Nu\sim Ra^{0.2}$ to $Nu\sim Ra^{0.23}$ at $Ra\approx1\times10^{10}$, but for linear forcing a gradual transition is observed. Nevertheless, for both profiles the system shows a clear $Nu\sim Ra^{0.23}$ scaling relationship when Ra reaches $1\times10^{11}$. Moreover, though Tsai et al. [28] argued that the linear forcing profile gives a power law scaling relationship closer to $Nu\sim Ra^{1/4}$ than the step one, we find that the difference in scaling between step and linear forcings is actually insignificant for their results with both being close to $Nu\sim Ra^{0.23}$ for high Ra.

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The thermal boundary layer thickness $\delta_\theta$ has a deep connection with Nu. The relationship between Nu and $\delta_\theta$ can be written as

$$\begin{equation} \frac{\delta_\theta}{L}\approx\frac{T_{hot}-T_{bulk}}{\Delta}\frac{1}{Nu}, \end{equation} \tag{ 2 }$$

where T_bulk is the bulk temperature. For Rayleigh-Bénard convection (RBC), since $T_{bulk}\approx(T_{hot}+T_{cold})/2$, the thermal boundary layer thickness is usually given by $\delta_\theta/L\approx0.5/Nu$. This approximation has also been applied to the HC system [11–13]. To examine this approximation, we measure the thermal boundary layer thickness $\delta_\theta/L$ for both forcing profiles. The thermal boundary layer thickness is first measured locally using the slope method [32] and then averaged over $x/L\geq0.9$ for both forcing profiles. The results of $\delta_\theta/L$ compensated by $Ra^{-0.23}$ are shown in fig. 2(b). One can observe that for $Ra\geq1\times10^{10}$, $\delta_\theta/L$ follows a less steep scaling relationship than $Ra^{-0.23}$ (we observe $\delta_\theta/L\sim Ra^{-0.21}$), suggesting that $\delta_\theta/L$ does not follow the assumption $\delta_\theta/L\sim1/Nu$. In the HC system, plume emissions occur only in the vicinity of the cold plate and the cold downwelling flow is dominant. For this reason, T_bulk is closer to T_cold than T_hot . As Ra increases, the cold plumes emit more frequently, which further decreases T_bulk . Thus, the prefactor $(T_{hot}-T_{bulk})/\Delta$ in (2) increases as Ra increases, which makes the Ra dependence of $\delta_\theta/L$ being less steep than $1/Nu$. Nevertheless, as Ra further increases and the bulk temperature becomes sufficiently close to the cold plate, the dependence of the bulk temperature on Ra becomes much weaker than in low Ra, and it would then be valid to assume $\delta_\theta/L\sim1/Nu$.

We next discuss the results of the Reynolds number for the two forcing profiles. Two kinds of characteristic velocities U_max and U_ke are used to evaluate Re. U_max is obtained by taking the average value of the local maximum u_x over the region $x/L\in[0.2,0.8]$. Figure 3(b) shows examples of the u_x profiles at three representative positions. We use U_max here because it may be used to characterize the velocity of the thermocline in the context of oceanic circulations [33]. We remark here that from fig. 3(b) it is clear that the convective flow is mostly confined to the upper half of the convection layer, i.e., the circulation is shallow. This is consistent with a previous experimental observation that convective flows in HC do not penetrate deep into the convection cell [27]. U_ke is defined according to the mean kinetic energy of the system $U_{ke}=\sqrt{\langle u^2\rangle_{t,V}}$. Corresponding to these two velocities, two Reynolds numbers can be defined: $Re_{max}=U_{max}L/\nu$ represents the flow stength of the thermocline, and $Re_{ke}=U_{ke}L/\nu$ indicates the mean flow strength of the system. We present the obtained Re_ke and Re_max in fig. 3(a). The results for Re_ke for Pr = 10 in ref. [23] are also shown in fig. 3(a) for comparison, which show good agreement with the present ones. Multiple transitions can be seen for both Re_ke and Re_max . These transitions are consistent with those in Nu –Ra shown in fig. 2(a). For $Ra\lesssim1\times10^{10}$, one can observe very similar behaviour in both Re_max and Re_ke for the two forcing profiles. After a steep scaling corresponding to the $I_l^*$ regime in refs. [11,13] for small Ra, a scaling relationship $Re\sim Ra^{0.4}$ can be observed for both Re_max and Re_ke when Ra increases. This scaling relationship agrees with the predictions in ref. [9] and the I_l (boundary layer dominating) regime in refs. [11,13]. When Ra further increases, the transition in the Re –Ra scaling can be observed for both Reynolds numbers. In this regime, Re_ke and Re_max give different scaling exponents. For the former, we observe a scaling relation close to $Re_{ke}\sim Ra^{1/3}$, which agrees with ref. [23]. As for the latter, the scaling exponent is approximately 0.44. This exponent is larger than any previous predictions [9,11], suggesting that one may have underestimated the velocity of the thermocline when applying the current theories in HC to the oceanic problems. The difference in the scaling relationship between Re_max and Re_ke also suggests the high degree of inhomogeneity in this system. Moreover, the linear forcing produces larger magnitude for both Re_max and Re_ke than step forcing, although the difference for Re_ke is significantly smaller than that for Re_max . These results also suggest that the type of forcing profiles has stronger influence on the thermocline than it has on the bulk flow.

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From the above discussions on Nu and Re, one sees that the influence of the forcing profile is mainly on the magnitudes of these quantities. The scaling relations in Nu and Re observed in step forcing profiles are similar to those of linear forcing. In the rest of this paper, we focus on the differences in other quantities for these two forcing profiles.

The available potential energy (APE) is the portion of potential energy that can be converted into kinetic energy [34,35]. The generation rate of APE can be expressed as [34]

$$\begin{equation} G_{APE}=\alpha g\kappa\oint z^*\frac{\partial\theta}{\partial z} \textrm{d}S, \end{equation} \tag{ 3 }$$

where $z^*$ is the reference position for the state of minimum potential energy of the corresponding fluid parcel. For the step forcing profile, $z^*$ equals H at the hot plate and 0 at the cold plate, while for the rest of the cell surface $\nabla\theta=0$. Thus, from (3) we can derive an exact relationship between G_APE and Nu [12]:

$$\begin{equation} G_{APE}=\frac{VRaPr\kappa^3}{L^4}\lambda Nu. \end{equation} \tag{ 4 }$$

This exact relation provides a direct connection between heat transfer efficiency and the generation rate of APE. However, for the linear forcing profile, $\partial\theta/\partial z\neq0$ at the whole top surface, and the local $z^*$ depends on the bulk temperature distribution. Thus, one cannot derive an exact relationship between Nu and G_APE . Since $Nu\sim Ra^{0.23}$ for large Ra, which is the same for both types of forcing profiles, one can find that $G_{APE}\sim Ra^{1.23}$ according to (4). To examine the relationship between Nu and G_APE , we evaluate G_APE using (3) and show the normalized results in fig. 4. Interestingly, we find that the G_APE shown in fig. 4 behaves similarly as the compensated Nu plotted in fig. 2(a), although in fig. 4 the normalized G_APE for linear forcing is larger than that of step forcing. This result suggests that for the linear forcing profile there could exist a relationship that is formally the same as (4). To check this statement, we first define an effective forcing area ratio $\lambda^*$ for linear forcing profile according to

$$\begin{equation} G_{APE}=\frac{VRaPr\kappa^3}{L^4}\lambda^* Nu. \end{equation} \tag{ 5 }$$

The results of $\lambda^*$ for linear forcing are shown in the inset of fig. 4, and one finds $\lambda^*$ approximately equal to a constant 0.38. With the effective forcing area ratio $\lambda^*$ determined, we can estimate G_APE with Nu according to (5). More importantly, this result suggests a general connection between heat transfer and the generation rate of APE. This connection does not rely on a specific forcing profile. It would be interesting to examine whether $\lambda^*$ is still a constant for other forcing profiles.

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In the HC system, the rate of change in potential energy $E_p\equiv\int_V\alpha g\theta z\textrm{d}V$ and kinetic energy $E_k\equiv\frac{1}{2}\int_V\mathbf{u}^2 \textrm{d}V$ are given, respectively, by [34,35]

$$\begin{eqnarray} \frac{\textrm{d}E_p}{\textrm{d}t}&= -\phi_z+\phi_i, \end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray} \frac{\textrm{d}E_k}{\textrm{d}t}&= \phi_z-\mathcal{E}. \end{eqnarray} \tag{ 7 }$$

Here $\phi_z\equiv\int_V \alpha gu_z\theta \textrm{d}V$ is the conversion rate from E_p to E_k , $\phi_i\equiv\alpha g\kappa\oint\theta\mathbf{e}_z \cdot\mathbf{n}\textrm{d}A$ is the rate of production of E_p , and $\mathcal{E}\equiv\int_V\epsilon_u \textrm{d}V$ is the global viscous dissipation, where $\epsilon_u\equiv\nu\sum_{i,j}(\partial u_i/\partial x_j)^2$. Taking the temporal averaging of (6) and (7), one can obtain

$$\begin{equation} \frac{L^4Pr^2}{\nu^3Ra}\langle\epsilon_u\rangle_{t,V}=\frac{\langle\theta\rangle_{t,z=H}-\langle\theta\rangle_{t,z=0}}{\Delta}\Gamma. \end{equation} \tag{ 8 }$$

The non-dimensionalized visous dissipation (8) is found to be constant for the step forcing profile [13]. In fig. 5(a) we show the normalized dissipation rate for the two types of forcing profiles. It is seen that the normalized viscous dissipation for step forcing is approximately a constant $(\sim2.25)$ in our study, which is close to the value of 2 observed in ref. [13]. For the linear forcing, it is clear that the normalized viscous dissipation rate increases with increasing Ra. This result can be understood as follows. From (8) one sees that the normalized viscous dissipation is directly related to the normalized temperature difference between the top and the bottom. For linear forcing, the mean top temperature $\langle\theta\rangle_{t,z=H}/\Delta$ is always zero. On the other hand, the bottom is cooled by the downwelling plumes detached from the cold plate, which becomes colder as Ra increases. Thus, the bottom temperature $\langle\theta\rangle_{t,z=0}/\Delta$ decreases and $(\langle\theta\rangle_{t,z=H}-\langle\theta\rangle_{t,z=0})/\Delta$ increases as Ra increases for the linear forcing profile, as shown in fig. 5(a). However, since $\langle\theta\rangle_{t,z=0}/\Delta>-0.5$, one expects that for sufficiently high Ra, $(\langle\theta\rangle_{t,z=H}-\langle\theta\rangle_{t,z=0})/\Delta$ approaches to its upper limit 0.5 and becomes insensitive to the change of Ra. In such limiting case, the normalized viscous dissipation will be approximately a constant $0.5\Gamma$. In the present study, $0.33\Gamma$ is observed for the highest Ra.

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Beside the global viscous dissipation, the dissipation rate of temperature variance also shows significant difference for the step and linear forcing profiles. The dissipation rate of temperature variance is given by $\epsilon_\theta\equiv\kappa\sum_i(\partial\theta/\partial x_i)^2$. Multiplying (1c) by θ and integrating in time and volume gives

$$\begin{equation} \langle\epsilon_\theta\rangle_{t,V}=\frac{\kappa}{H}\left\langle\theta\frac{\partial\theta}{\partial z}\right\rangle_{t,z=H}. \end{equation} \tag{ 9 }$$

For the step forcing profile, the regions with non-zero $\partial\theta/\partial z$ have constant temperature θ, so (9) leads to an exact relation between $\langle\epsilon_\theta\rangle_{t,V}$ and the mean surface heat flux Nu:

$$\begin{equation} \langle\epsilon_\theta\rangle_{t,V}=\lambda\Gamma\frac{\kappa\Delta^2}{L^2}Nu. \end{equation} \tag{ 10 }$$

Thus, one can define a Nusselt number using the mean dissipation rate of thermal variance:

$$\begin{equation} Nu_{\epsilon_\theta}\equiv\frac{\langle\epsilon_\theta\rangle_{t,V}}{\lambda\Gamma}\frac{L^2}{\kappa\Delta^2}. \end{equation} \tag{ 11 }$$

As for linear forcing (or other continuous forcings), the coupling between θ and $\partial\theta/\partial z$ forbids an exact relation between Nu and $\langle\epsilon_\theta\rangle_{t,V}$. To solve this problem, the key is to decouple θ and $\partial\theta/\partial z$. We assume that the local heat flux is positively correlated to the forcing temperature. For simplicity, we further assume that such correlation is linear, i.e., $\partial\theta/\partial z\sim\theta$. Thus, the surface heat flux can be expressed as

$$\begin{equation} \frac{\partial\theta}{\partial z}\big/\frac{\Delta}{L}\approx 4Nu\frac{\theta}{\Delta}=4Nu\Big(\frac{x}{L}-\frac{1}{2}\Big). \end{equation} \tag{ 12 }$$

By substituting (12) into (9), one obtains the following relation for the linear forcing profile:

$$\begin{equation} \langle\epsilon_\theta\rangle_{t,V}\approx\frac{\Gamma}{3}\frac{\kappa\Delta^2}{L^2}Nu. \end{equation} \tag{ 13 }$$

From (13) we can find an approximate expression for Nu

$$\begin{equation} Nu_{\epsilon_\theta}^*\equiv\frac{3\langle\epsilon_\theta\rangle_{t,V}}{\Gamma}\frac{L^2}{\kappa\Delta^2}\approx Nu. \end{equation} \tag{ 14 }$$

The results of $Nu_{\epsilon_\theta}^*/Nu$ for the linear forcing profile are shown in fig. 5(b). For comparison and verification of our simulations, we also plot $Nu_{\epsilon_\theta}/Nu$ for step forcing. One sees that for $Ra<1\times10^{10}\,Nu_{\epsilon_\theta}^*$ agrees well with Nu, suggesting that the linear approximation can properly describe the relationship between θ and $\partial\theta/\partial z$. As for $Ra>1\times10^{10}$, $Nu_{\epsilon_\theta}^*/Nu$ increases as Ra increases. To explain this deviation, we first examine the linear approximation (12). We show the longitudinal distribution of the normalized heat flux on the top surface for the linear forcing profile with $Ra=1\times10^{11}$ in the inset of fig. 5(b). For the hotter half $(x/L>0.5)$, one can find that the linear approximation (red dashed line) agrees with the actual surface heat flux. However, clear disagreement is seen at the cold end, where the linear approximation significantly underestimates the local heat flux. This is because abounded plumes are emitted at the cold end and enhance the local heat flux there. Since the cold end contributes a large fraction of the global heat transfer, the linear approximation in the model leads to an underestimation in (13). In other words, when plume emissions occur, (13) becomes

$$\begin{equation} \langle\epsilon_\theta\rangle_{t,V}\gtrsim\frac{\Gamma}{3}\frac{\kappa\Delta^2}{L^2}Nu. \end{equation} \tag{ 15 }$$

Therefore, $Nu_{\epsilon_\theta}^*$, which is proportional to $\langle\epsilon_\theta\rangle_{t,V}$ according to its definition (14), should be larger than Nu. Since plume emissions become more frequent as Ra increases, the underestimation of local heat flux in the linear approximation of (12) becomes more significant. Thus, $Nu_{\epsilon_\theta}^*/Nu$ increases with Ra for $Ra>1\times10^{10}$, as observed in fig. 5(b). This result also implies that for the linear forcing profile the dissipation rate of temperature variance $\langle\epsilon_\theta\rangle_{t,V}$ should have a stronger Ra dependence than Nu.

In this study we have investigated horizontal convection with step and linear forcing profiles, respectively. Our study shows how forcing profiles can affect the global heat transfer, flow strength, the generation rate of APE, and the viscous and temperature variance dissipation rates in HC. We show that although the magnitudes of Nu depend on forcing profiles, scaling-wise, both forcing profiles have the same Nu –Ra power law relationships. A transition from $Nu\sim Ra^{0.2}$ to $Nu\sim Ra^{0.23}$ can be observed at $Ra\approx1\times10^{10}$ for both forcing profiles. The exponent 0.23 is close to but less than 1/4, which corresponds to the regime where heat transport is governed by the bulk. One may expect to see a transition to the 1/4-scaling at higher Ra when the bulk turbulence is sufficiently strong. We evaluate two Reynolds numbers Re_max and Re_ke , corresponding to the strength of the thermocline and the global flow strength, respectively. We find a scaling relationship $Re\sim Ra^{0.4}$ for both Reynolds numbers when $Ra\approx1\times10^9$. As Ra further increases, Re_max and Re_ke show different power law relationships, and the one for Re_max gives a stronger Ra dependence. This suggests that horizontal convection is more efficient in driving the thermocline than the whole system. It also demonstrates the presence of strong inhomogeneity in this system. Similar to the influence on Nu, forcing profiles only affect the magnitudes of Re and their influences on the scaling exponent are negligible.

For the step forcing profile, one can derive an exact relationship between G_APE and Nu. As for linear forcing, however, the coupling between $z^*$ and $\partial\theta/\partial z$ at the top surface makes it impossible to derive such an exact relationship. We define an effecitive forcing area ratio $\lambda^*$ according to (5), which affords the G_APE for linear forcing a relation formally similar to the exact relation (4) for step forcing. Therefore, we can examine the connection between G_APE and Nu using $\lambda^*$, even though the exact relation between these two quantities does not exist. Our results show that for linear forcing $\lambda^*\approx0.38$ and is approximately constant with respect to varying Ra. This suggests that the connection between the heat transfer and the generation rate of APE is probably generic and does not rely on a specific forcing profile. It would be interesting to examine whether $\lambda^*$ is still constant for other forcing profiles.

The influences of the forcing profile on viscous and thermal dissipation rates are investigated. For the step forcing profile, the normalized global viscous dissipation in (8) is approximately independent of Ra. For linear forcing, $(\langle\theta\rangle_{t,z=H}-\langle\theta\rangle_{t,z=0})/\Delta$ increases as Ra increases, due to the rigorous cooling of the bottom surface by the cold downwelling plumes. According to the global balance given by (8), the increasing temperature difference also means the increase of normalized global viscous dissipation. Since the bottom temperature cannot be colder than the temperature of the cold plate, $(\langle\theta\rangle_{z=H}-\langle\theta\rangle_{z=0})/\Delta$ will gradually become less sensitive to Ra and approaches to 0.5 for very large values of Ra.

As for the dissipation rate of temperature variance $\epsilon_\theta$, one can obtain an exact relation connecting Nu and $\langle\epsilon_\theta\rangle_{t,V}$ using the step forcing profile. But for linear forcing, as a result of the coupling between θ and $\partial\theta/\partial z$ at the surface, such relationship does not exist. We use a linear approximation and obtain a relation (14) connecting Nu and $\langle\epsilon_\theta\rangle_{t,V}$. We find that for small Ra, (14) agrees well with our measurements. But for $Ra\gtrsim 1\times10^{10}$ (when plume emissions become pronounced), the linear approximation cannot properly describe the ensuing local heat transfer enhancement. The linear approximation can lead the right-hand side of (9) to a term related to Nu, but it also underestimates (9) if plume emission occurs, as indicated by (15). This suggests that the dissipation rate of temperature variance $\langle\epsilon_\theta\rangle_{t,V}$ should have a stronger Ra dependence than Nu for the linear forcing profile.

In practice, the step forcing configuration is experimentally easier to implement and easier to tackle analytically. Thus, although the step forcing profile is less realistic for the real oceans when compared with the linear forcing, it is still a very useful configuration for most problems. As for the quantities such as global thermal dissipation, one should be aware of the differences between the step and other types of forcing profiles, and take caution when applying the results of step forcing to more general situations.

The authors would like to thank Lu Zhang and Yu-Hao He for stimulating discussions. We gratefully acknowledge support of this work by the National Natural Science Foundation of China (Grant No. 12072144); the Research Grants Council of HKSAR (Grant No. CUHK14302317); and the Department of Science and Technology of Guangdong Province (Grant No. 2019B21203001). Numerical resources for this study are supported by the Center for Computational Science and Engineering at Southern University of Science and Technology.

Data availability statement: The data that support the findings of this study are available upon reasonable request from authors.