4.1 Flow structure and small-scale convection

All of the solutions are highly unsteady. We therefore begin by describing the solutions in terms of instantaneous snapshots of the flow structure, followed by an examination of the time-averaged flow. Emphasis is given to the location of vertical convective transport.

Snapshots of vertical velocity and temperature on a vertical plane are shown for strong rotation ( $Ra=7.4\times 10^{12}$ , $Ro_{\unicode[STIX]{x0394}T}=1.1\times 10^{-3}$ ) in figure 1, where velocity is normalised by the simplest scale ( $\sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}$ ) for buoyancy-driven inertial flow and the temperature difference from the cold boundary is normalised by the imposed $\unicode[STIX]{x0394}T$ . The snapshots illustrate the shallow, very small-scale convection within the thermal boundary layer close to $\hat{x}\approx 0.5$ , where the boundary layer flow first encounters a destabilising boundary buoyancy flux. These snapshots also show deep, vertically coherent warm plumes in the interior at $\hat{x}<0.3$ (figure 1 b), having horizontal scale much larger than the boundary layer convection and strong upward velocities throughout the depth of the box (figure 1 a). In contrast, where the boundary buoyancy flux is stabilising at $\hat{x}>0.5$ , there is weak downwelling into the thermal boundary layer and the flow is laminar.

The thermal boundary layer is clearly shown by vertical profiles of temperature above the cooling region of the base (figure 2). Temperature profiles at other locations are similar, and like those in the non-rotating case (Gayen et al. Reference Gayen, Griffiths and Hughes2014), apart from the presence of a mixed layer in the lower portion of the thermal boundary layer over the heated region of the base. The temperature gradient is stable everywhere outside the mixed layer and the very thin sublayer on the heated region of the base. The thermal boundary layer thickness, defined here as the height encompassing 90 % of the top-to-bottom temperature difference (zonally and time-averaged at $\hat{x}=0.75$ ), is listed in table 1 and was previously shown to be consistent with the geostrophic scaling (2.4) (Vreugdenhil et al. Reference Vreugdenhil, Gayen and Griffiths2016). The boundary layer is thinner for smaller Rossby numbers. Outside the boundary layer, the stratification is larger for smaller Rossby numbers.

Figure 2.

Vertical profiles of temperature in the cooled half at $\hat{x}=0.92$ averaged in the ‘zonal’ direction for $Ra=7.4\times 10^{11}$ (no-slip sidewalls, solid curves) and $Ra=7.4\times 10^{12}$ (broken curve). Line colour indicates $Ro_{\unicode[STIX]{x0394}T}$ .

Figure 3.

Snapshots of flow structure represented by the shape of two surfaces having fixed vertical velocity for $Ra=7.4\times 10^{11}$ and three values of the Rossby number: (a) $Ro_{\unicode[STIX]{x0394}T}=1.6\times 10^{-1}$ , (b) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ , (c) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ and (d)  $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ (all with no-slip sidewalls). Isosurfaces show upward (grey) and downward (green) normalised velocities $w/\sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}$ at values chosen to most effectively illustrate the flow: (a) $\pm 3.2\times 10^{-3}$ , (b) $\pm 1.6\times 10^{-3}$ , (c) $\pm 3.2\times 10^{-4}$ and (d)  $\pm 3.2\times 10^{-4}$ . Only the region above the heated half of the base is shown; the cooled half extends to the right.

The dependence of flow structure on rotation rate is further illustrated in figure 3, where vertical velocity isosurfaces are shown for several values of the ratio $Ro_{\unicode[STIX]{x0394}T}$ of buoyancy forcing to rotation rate. Velocities are again normalised by the buoyancy–inertia scale involving the least assumptions and independent of rotation rate. At $Ro_{\unicode[STIX]{x0394}T}>10^{-1}$ (weak rotation, figure 3 a) the flow shows coherent two-dimensional convection rolls aligned with the local flow direction near the leading edge of the heated region. The depth of the boundary layer convection increases with distance downstream and the rolls transition after a short distance to three-dimensional turbulence. The turbulence remains capped by boundary layer stratification except at the end of the basin, where it is carried into a large, full-depth endwall plume. All of these features are similar to those previously reported for the non-rotating case (Mullarney et al. Reference Mullarney, Griffiths and Hughes2004; Gayen et al. Reference Gayen, Griffiths and Hughes2014). For strong rotation (shown for $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ in figure 3 b), the onset of boundary layer instability can be seen closer to the leading edge of the heated region of the base, and streamwise rolls are again parallel to the (now deflected and slower) mean flow of the boundary layer and become turbulent within a shorter distance. Full-depth convection, which remains essential in order to maintain the temperature near the top boundary, now involves many vortical plumes, which are broadly distributed over an area at $\hat{x}<0.3$ . Thus there is no longer a single endwall plume, although weak upwelling at the endwall persists. With still stronger rotation ( $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ and $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ ; figure 3 c,d), the vertical convection is confined to the boundary layer in only a very small region and the vortical plumes become more numerous, more widespread and also more coherent between the interior and boundary layer. Under these conditions, there is no longer a scale separation between the small-scale convection in the boundary layer and the vortical coherent plumes. There is broader, slower and less organised downwelling around the upward vortical plumes.

Snapshots of the vertical component of relative vorticity, shown for strong rotation in figure 4, again reveal the small-scale boundary layer convection, its deepening as it is carried along (and across) the basin by the mean flow between $\hat{x}\approx 0.5$ and $\hat{x}\approx 0.3$ (figure 4 a), and its evolution further ‘downstream’ into deep vortical plumes. The base of each plume is hot compared to the surroundings (figure 1 b and schematic in figure 4 c) due to convergence of warm water (in the heated sublayer below the bulk of the cold, dense boundary layer) into the base of the plume. Vortex stretching in the plumes leads to cyclonic relative vorticity (figure 4 a–c). Higher in the water column, beyond the thermal boundary layer, there is radial divergence in each plume, vortex compression and anticyclonic relative vorticity (Helfrich Reference Helfrich1994; Maxworthy & Narimousa Reference Maxworthy and Narimousa1994; Marshall & Schott Reference Marshall and Schott1999). The horizontal scale of the plumes decreases with increasing rotation rate or decreasing Rossby number.

Figure 4.

Vertical component of relative vorticity, normalised by $f$ , $\unicode[STIX]{x1D714}/f=f^{-1}\unicode[STIX]{x1D735}\times (u,v)$ , for $Ra=7.4\times 10^{11}$ and $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ (no-slip) on (a) a vertical plane ${\hat{y}}=0.2$ and (b) a horizontal plane $\hat{z}=0.01$ (within the thermal boundary layer, outside the Ekman layer). Contour lines in (a) show isosurfaces of upward velocity $w/\sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}=3.2\times 10^{-4}$ (white) and downward velocity $w/\sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}=-3.2\times 10^{-4}$ (black). Black arrows in (b) show the direction of the strongest mean current from the cooled to the heated region. White arrows in (a,b) show the general direction of migration of deep plumes (the area $\hat{z}<0.6$ is shown). (c) Zoom showing the plume in the white rectangle in (a), with colours showing temperature $\hat{T}$ , white contour showing isosurface of upward velocity $w/\sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}=3.2\times 10^{-4}$ and blue arrows indicating direction of local circulation. (d) Zoom on the sidewall boundary current in the white rectangle in (b) and a vertical plane ${\hat{y}}=0.96$ , with colour showing temperature $\hat{T}$ . Large black arrows show the direction of the boundary current near the base, and thin arrows show the current in the eddy.

The columnar vortices at rapid rotation shown by the vertical velocity isosurfaces in figure 3 may bear some resemblance to the structures of strongly rotating Rayleigh–Bénard convection (King, Stellmach & Aurnou Reference King, Stellmach and Aurnou2012; Julien et al. Reference Julien, Rubio, Grooms and Knobloch2012, Reference Julien, Aurnou, Calkins, Knobloch, Marti, Stellmach and Vasil2016; Plumley et al. Reference Plumley, Julien, Marti and Stellmach2016). However, it is clear from figure 4 that the columnar vortices are not the same, even locally, because the columns penetrate through or break down a stably stratified boundary layer and a weakly stable stratification in the interior. In addition, the cooling is supplied to the convecting region by horizontal flow (from the cold base) in the boundary layer, setting the present case apart from rotating Rayleigh–Bénard convection.

Relative vorticity (in figure 4 a,b) also highlights a highly unsteady, cold current along the boundary ${\hat{y}}=1$ (which we refer to as the ‘eastern’ boundary despite the uniform Coriolis parameter) over the heated region and continuing towards the ‘northern’ endwall ( $\hat{x}=0$ ). This current is in the thermal boundary layer against the base and links to a flow along the opposite sidewall ${\hat{y}}=0$ in the cooled region, which crosses the basin near the mid-section ( $\hat{x}\approx 0.6$ ). The mean velocity component $v$ in the current across the basin is in thermal wind balance with the temperature gradient in the $x$ direction. For free-slip sidewalls the boundary current is narrower and faster, but otherwise there is little difference between the solutions with free- or no-slip conditions. The dense current along the ‘eastern’ boundary undergoes wave and eddy instability that has horizontal scales comparable to the boundary layer Rossby deformation radius and which appear to be a result of baroclinic instability of a local gravitationally stable density gradient (rather than of convection driven by a density inversion). This instability produces cold, anticyclonic eddies surrounded by filaments of cyclonic relative vorticity (figure 4 d). The baroclinic eddies propagate slowly towards the mid-section $\hat{x}=0.5$ (in the opposite direction to the mean current), where they disappear in the co-flowing mean motion of a large anticyclonic gyre. The enhanced heat input from the base to the cold eddies enhances small-scale convection in the most weakly stratified, anticyclonic regions of the baroclinic waves. The size of the eddies decreases with increasing rotation rate, approximately proportional to the Rossby radius. Moving from $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ to $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ at fixed Rayleigh number sees a 20-fold decrease in the inertial radius $U/f$ ( ${\sim}Ro_{\unicode[STIX]{x0394}T}$ ), a 6.3-fold decrease in the deformation radius $\unicode[STIX]{x1D706}\sim \sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}/f$ ( ${\sim}(RaPr)^{1/10}Ro^{3/5}$ ) and a 4.6-fold decrease in an alternative deformation radius based on the boundary layer depth ( $\unicode[STIX]{x1D706}_{bl}\sim \sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T\unicode[STIX]{x1D6FF}}/f\sim Ro_{\unicode[STIX]{x0394}T}^{1/2}$ ). An approximate four-fold decrease in wavelength is observed. Baroclinic eddies are also generated from the current crossing the basin near the mid-section, contributing to lateral transport of water and heat through this section and a mix of geostrophic turbulence and vertical convection in this energetic site.

4.2 Mean circulation

The mean flow is estimated by averaging the velocities over a period of 600 (for $Ra=7.4\times 10^{11}$ ) or 65 (for $Ra=7.4\times 10^{12}$ ) buoyancy time scales $\unicode[STIX]{x1D70F}_{b}$ , a period that is as long as the solutions could practically be run after reaching the thermally equilibrated state. However, it is long enough to average over all but very low-frequency fluctuations.

The variation with rotation rate (at a fixed Rayleigh number) of the horizontal mean circulation within the thermal boundary layer is represented in figure 5 by the $x$ component of the mean velocity on the plane $z=\unicode[STIX]{x1D6FF}/2$ (which is outside the Ekman layer in all cases) and the horizontal streamfunction $\unicode[STIX]{x1D713}_{H}$ , defined by $(u,v)=(\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{H}/\unicode[STIX]{x2202}y,-\unicode[STIX]{x2202}\unicode[STIX]{x1D713}_{H}/\unicode[STIX]{x2202}x)$ , calculated on the same plane. The solutions show a mostly uniform flow from the cooled to the heated region at weak rotation and two or three quasi-steady gyres at strong rotation. In particular, for $10^{-2}>Ro_{\unicode[STIX]{x0394}T}>10^{-4}$ , an anticyclonic gyre dominates the region of stabilising boundary buoyancy flux. There is a smaller but stronger anticyclonic gyre in the middle region above the destabilising boundary flux, and a strong current across the width of the basin (in the $y$ direction) at $0.3<\hat{x}<0.4$ . There is weak cyclonic circulation near the endwall of the heated region. At the strongest rotation rate ( $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ ), there is a general anticyclonic (clockwise) circulation but with a more complicated current near the middle of the basin. Comparison of the solutions at the two Rayleigh numbers used here, as well as viscous solutions at $Ra=7.4\times 10^{8}$ given by Vreugdenhil et al. (Reference Vreugdenhil, Gayen and Griffiths2016), show that the mean boundary layer flow is stronger at higher Rayleigh numbers (for a given rotation rate), consistent with (2.4). The mean flow is weaker at larger rotation rate (at a given Rayleigh number) because the geostrophic constraints increasingly inhibit large-scale circulation. Note that the small-scale quasi-linear features across the heated region at rapid rotation are tracks of the columnar vortices. They appear in figure 5(c,d) because the horizontal mean flow is increasingly inhibited at larger rotation rates and the vortices are advected more slowly. All four rotation rates have the same time averaging period of 600 $\unicode[STIX]{x1D70F}_{b}$ (as this is based on the buoyancy time scale $\unicode[STIX]{x1D70F}_{b}=(L/g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T)^{1/2}$ ) and so in the strongly rotating cases there is a remnant signal of the vortex movement even in our time average.

Figure 5.

Time-averaged horizontal flow in the boundary layer over 600 buoyancy time scales for different Rossby numbers: colour scale shows normalised velocity in the $x$ direction, $u/\sqrt{g^{\prime }L}$ , on the plane $z=\unicode[STIX]{x1D6FF}/2$ ; line contours show the normalised streamfunction $\unicode[STIX]{x1D713}_{H}/(L\sqrt{g^{\prime }L})$ computed at this height. The solutions are for $Ra=7.4\times 10^{11}$ and Rossby number (a) $Ro_{\unicode[STIX]{x0394}T}=1.6\times 10^{-1}$ (streamfunction increment $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=2.1\times 10^{-4}$ ), (b) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=1.0\times 10^{-4}$ ), (c) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ , no-slip ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=3.9\times 10^{-5}$ ), and (d) $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=7.8\times 10^{-6}$ ). Black lines show clockwise (anticyclonic) circulation, thick black line is the zero contour, thin broken lines are anticlockwise and $g^{\prime }=g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T$ .

The dependence of the corresponding mid-depth horizontal circulation on rotation rate is shown in figure 6 by the $x$ component of the mean velocity and the corresponding horizontal transport $\unicode[STIX]{x1D713}_{H}$ at $z=H/2$ . The weakly rotating case is dominated by a domain-sized cyclonic gyre, with a small anticyclonic gyre near the endwall plume over the destabilising heated region. As rotation increases, a cyclonic gyre remains present over the stabilising half of the domain, while one or two anticyclonic gyres form over the heated region. At $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ the anticyclonic gyre centred at $\hat{x}=0.4$ is strong, but with increasing rotation the anticyclonic gyre over the destabilising region becomes relatively weak compared with the cyclonic gyre over the stabilising region.

Figure 6.

Time-averaged horizontal flow at mid-depth over 600 buoyancy time scales for different Rossby numbers: colour scale shows normalised velocity in the $x$ direction, $u/\sqrt{g^{\prime }L}$ , on the plane $z/H=0.5$ ; line contours show the normalised streamfunction $\unicode[STIX]{x1D713}_{H}/(L\sqrt{g^{\prime }L})$ computed at this height. The solutions are for $Ra=7.4\times 10^{11}$ and Rossby number (a) $Ro_{\unicode[STIX]{x0394}T}=1.6\times 10^{-1}$ (streamfunction increment $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=3.1\times 10^{-5}$ ), (b) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=5.2\times 10^{-5}$ ), (c) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ , no-slip ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=3.4\times 10^{-5}$ ), and (d) $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=1.0\times 10^{-5}$ ). Black lines show clockwise (anticyclonic) circulation, thick black line is the zero contour, thin broken lines are anticlockwise and $g^{\prime }=g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T$ .

The boundary layer and mid-depth horizontal velocity and streamfunctions for the larger $Ra$ are shown in figure 7. The dynamics are similar to those at the smaller $Ra$ at similar $Ro_{\unicode[STIX]{x0394}T}$ ( $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ and $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ ). At mid-depth there is a strong cyclonic gyre over the stabilising region and anticyclonic flow over the destabilising region (figure 7 a). In the boundary layer an anticyclonic gyre dominates much of the flow, with remnants of a weak cyclonic gyre over the heated end of the domain near the endwall (figure 7 b). The structure of the mean flow is thus primarily dependent on Rossby number.

Figure 7.

A comparison of time-averaged horizontal flow (over 65 buoyancy time scales) in the interior and boundary layer: colour scale shows normalised velocity in the $x$ direction, $u/\sqrt{g^{\prime }L}$ ; line contours show the normalised streamfunction $\unicode[STIX]{x1D713}_{H}/(L\sqrt{g^{\prime }L})$ computed at this height. The solutions are for $Ra=7.4\times 10^{12}$ and Rossby number $Ro_{\unicode[STIX]{x0394}T}=1.1\times 10^{-3}$ on the plane (a) at mid-depth $z/H=0.5$ (streamfunction increment $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=3.2\times 10^{-5}$ ), and (b) in the boundary layer $z=\unicode[STIX]{x1D6FF}/2$  ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}_{H}/L\sqrt{g^{\prime }L}=6.6\times 10^{-5}$ ). Black lines show clockwise (anticyclonic) circulation, thick black line is the zero contour, thin broken lines are anticlockwise and $g^{\prime }=g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T$ .

The mean overturning (vertical) circulation, shown in figure 8, is illustrated by zonal ( $y$ ) averages of the time-averaged $x$ component of velocity along with contours of the zonally integrated transport $\unicode[STIX]{x1D709}_{vert}$ (2.5). For weak rotation, the main upwelling leg of the circulation occurs near the endwall, whereas for strong rotation relative to weak buoyancy forcing ( $Ro_{\unicode[STIX]{x0394}T}<10^{-1}$ ) it shifts to a relatively wide area above the heated base, separated from the endwall. For the strongest rotation ( $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ ; figure 8 d) there is a relatively weak counter-rotating cell at $\hat{x}<0.25$ . Although this may appear to be a dramatic change in the mean overturning circulation, it is important to note that the flow is three-dimensional (including the upwelling near the walls) and the counter-rotating cell is likely to be an artefact of the zonal averaging. Rotation has less affect on the pattern of the overturning transport above the cooled region of the base. The solution for the larger value of the Rayleigh number ( $Ra=7.4\times 10^{12}$ ) shows overturning similar in pattern and magnitude to that in figure 8(b) over the heated region and a similar pattern but smaller strength compared to figure 8(c) over the cooled region. In addition to the mean overturning transport (2.5) shown in figure 8 we also calculate the ‘residual mean circulation’ in appendix A, but find that, in the present case for flow in a closed box and driven solely by buoyancy, it does not reveal any new information about the flow.

Figure 8.

Time-averaged and zonally averaged velocity in the $x$ direction, $u/\sqrt{g^{\prime }L}$ , projected on a vertical plane (colour contours) and zonally integrated mean overturning transport $\unicode[STIX]{x1D709}_{vert}/L^{2}\sqrt{g^{\prime }L}$ (line contours) for $Ra=7.4\times 10^{11}$ and Rossby number (a)  $Ro_{\unicode[STIX]{x0394}T}=1.6\times 10^{-1}$ (transport increment $\unicode[STIX]{x0394}\unicode[STIX]{x1D709}_{vert}/L^{2}\sqrt{g^{\prime }L}=5.0\times 10^{-6}$ ), (b)  $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D709}_{vert}/L^{2}\sqrt{g^{\prime }L}=2.5\times 10^{-6}$ ), (c) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ , no-slip ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D709}_{vert}/L^{2}\sqrt{g^{\prime }L}=1.2\times 10^{-6}$ ), and (d) $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ ( $\unicode[STIX]{x0394}\unicode[STIX]{x1D709}_{vert}/L^{2}\sqrt{g^{\prime }L}=5.0\times 10^{-7}$ ). Time averaging is over 600 buoyancy time scales. Thin black lines show clockwise overturning, thick black line is the zero contour, thin broken lines are anticlockwise and $g^{\prime }=g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T$ .

The dependence of the mean circulation on governing parameters can be summarised by the maximum values of the integrated transports $\unicode[STIX]{x1D709}_{H}$ and $\unicode[STIX]{x1D709}_{vert}$ defined in (2.6) and (2.5), respectively, and plotted in figure 9. Also shown is the maximum value of the baroclinic component of the horizontal transport in the boundary layer $\unicode[STIX]{x1D709}_{B}$ defined in (2.7). All transport values in figure 9 are normalised by the established scaling (2.8) for the overturning transport in the inertial boundary layer regime of the non-rotating, large-Rayleigh-number case. The results show that all measures of the transport decrease with decreasing Rossby number. The barotropic contribution to the total horizontal transport in the boundary layer is small and insignificant in all cases except at the smallest Rossby number, where the barotropic component reduces the total boundary layer transport by ${\sim}30\,\%$ of the baroclinic transport. Hence this convectively driven geostrophic circulation can develop a significant barotropic component despite zero net momentum input. The total horizontal boundary layer transport (at least as measured by the transport maximum) is less than the maximum in the baroclinic contribution due to differences in the sign of the horizontal streamfunctions in the interior and boundary layer over significant areas of the basin.

Figure 9.

Maximum values of (a) the overturning transport and (b) the total horizontal transport in the boundary layer $\unicode[STIX]{x1D709}_{H}$ (circles, no-slip; diamond, free-slip) and its baroclinic component $\unicode[STIX]{x1D709}_{B}$ (triangles, no-slip; square, free-slip). All transports are normalised by the non-rotating scaling for the overturning (2.8) (with constant $c_{0}=1$ ) for $Ra=7.4\times 10^{11}$ (filled symbols). Points at $Ro_{\unicode[STIX]{x0394}T}=1.1\times 10^{-3}$ are for large Rayleigh number $Ra=7.4\times 10^{12}$ (open symbols). The solid line is the geostrophic–diffusive boundary layer scaling prediction (2.10)  $\unicode[STIX]{x1D709}/\unicode[STIX]{x1D709}_{0}=c_{1}Ro_{\unicode[STIX]{x0394}T}^{1/5}$ fitted to no-slip $Ra=7.4\times 10^{11}$ results (filled circles) for $Ro_{\unicode[STIX]{x0394}T}\leqslant 0.05$ with prefactor value (a) $c_{1}=1.1$ for $\unicode[STIX]{x1D709}_{vert}$ and (b) $c_{1}=1.3$ for $\unicode[STIX]{x1D709}_{B}$ . Broken lines are best-fitting power laws to no-slip $Ra=7.4\times 10^{11}$ results (filled circles) for (a)  $\unicode[STIX]{x1D709}_{vert}/\unicode[STIX]{x1D709}_{0}=1.7Ro_{\unicode[STIX]{x0394}T}^{0.28}$ and (b)  $\unicode[STIX]{x1D709}_{B}/\unicode[STIX]{x1D709}_{0}=1.1Ro_{\unicode[STIX]{x0394}T}^{0.18}$ . The overturning transport at $Ro_{\unicode[STIX]{x0394}T}=0.16$ is greater than that for the non-rotating case due to Ekman transport.

The computed transport in figure 9 can be compared with the geostrophic scaling predictions. The scaling for the geostrophic regime (2.9) predicts that the two streamfunctions will have different dependence on rotation rate, whereas the integrated overturning transport (2.10) (hence the net boundary layer transport in the $x$ direction) will be in fixed proportional to the depth-integrated baroclinic horizontal transport in the recirculating gyres. The results in figure 9 indicate instead that the total boundary layer transport $\unicode[STIX]{x1D709}_{H}$ trends like the overturning transport $\unicode[STIX]{x1D709}_{vert}$ , whereas only the baroclinic contribution to the boundary layer transport trends close to the predicted thermal wind scaling $Ro_{\unicode[STIX]{x0394}T}^{1/5}$ . The baroclinic boundary layer transport is larger than the overturning at $Ro_{\unicode[STIX]{x0394}T}<10^{-2}$ . We conclude that both the vertical overturning and total horizontal transports are more strongly inhibited at strong rotation than is the horizontal baroclinic component of the flow in the boundary layer. Since the time-averaged flow is approximately geostrophic, the baroclinic component is essentially the thermal wind. The results also show little difference in the computed transports between the cases of free-slip sidewall and no-slip sidewall boundary conditions. The success of the geostrophic scaling in predicting the horizontal transport provides evidence that the circulation is governed by a vertical advection–diffusion balance (above the region of stabilising boundary heat flux) together with a geostrophic balance in the basin-scale flow of the boundary layer, and that the interior dynamics play a lesser role.

5.1 Heat transport diagnostics

Given that all scales of flow and diffusion are resolved in the present simulations and the surface boundary conditions are imposed temperatures, the heat flux at the base is diffusive and the total heat transported through the system is the integrated flux through the area of positive boundary flux. The use of an almost piecewise-uniform distribution of the imposed boundary temperatures leads to no significant differences between the area of positive boundary flux from the area of imposed higher temperature $T_{h}$ . The total heat throughput is therefore $(\unicode[STIX]{x1D70C}_{r}c_{p})^{-1}FA_{hot}=\unicode[STIX]{x1D705}\int _{A_{hot}}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}\,\text{d}a$ , where the area of the heated base is $A_{hot}\approx WL/2$ and the overbar indicates a time average. The Nusselt number is evaluated as $Nu=2\int _{A_{hot}}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}\,\text{d}a/(\unicode[STIX]{x0394}TW)$ .

For the values of $Ra$ considered here, the boundary layer on the cool region of the base is laminar. Downwelling flow of warmer water from the interior into the boundary layer leads to a balance of advection and vertical diffusion of heat. Above the heated region of the base, the average temperature gradient at the boundary must be equal to but of opposite sign from that at the cold boundary, so that there is no net heat input to the box. However, this causes much of the thermal boundary layer on the heated region to be unstable, and a mixed layer of small-scale turbulent convection forms between a very thin diffusive sublayer on the boundary and an overlying stably stratified portion of the boundary layer flow (Mullarney et al. Reference Mullarney, Griffiths and Hughes2004). The convective circulation also includes plumes in the interior and near the side boundaries above the heated half of the base, carrying water and heat from the boundary layer into the interior.

The mechanisms responsible for heat transport can be examined further by distinguishing the contributions to heat transport by the time mean and fluctuating components of flow in local regions within the basin. Focusing particularly on the vertical transport, the vertical velocity $w=\overline{w}+w^{\prime }$ and temperature field $T=\overline{T}+T^{\prime }$ imply heat transfer $Q_{h}$ across a horizontal plane of horizontal area $a$ at a rate

(5.1) $$\begin{eqnarray}(\unicode[STIX]{x1D70C}_{r}c_{p})^{-1}Q_{h}=\int \overline{wT}\,\text{d}a-\unicode[STIX]{x1D705}\int \overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}\,\text{d}a,\end{eqnarray}$$

and this can be expressed in terms of the mean and fluctuating contributions as

(5.2) $$\begin{eqnarray}(\unicode[STIX]{x1D70C}_{r}c_{p})^{-1}Q_{h}-\overline{T}_{ref}\int \overline{w}\,\text{d}a=\int \overline{w}(\overline{T}-\overline{T}_{ref})\,\text{d}a+\int \overline{w^{\prime }T^{\prime }}\,\text{d}a-\unicode[STIX]{x1D705}\int \overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}\,\text{d}a,\end{eqnarray}$$

where the reference temperature $\overline{T}_{ref}$ is introduced in order to remove the large and irrelevant transport of heat by mean flow having a non-zero absolute temperature. The mean and fluctuation contributions of advective transport on the right of (5.2) will be examined for the full area of the basin, but also in subregions chosen to encompass the side boundary processes, the interior vertical convection, or the heated and cooled halves. The reference temperature $\overline{T}_{ref}={\mathcal{A}}^{-1}\int _{{\mathcal{A}}}T\,\text{d}a$ was chosen as the horizontally averaged temperature over the full area ${\mathcal{A}}=LW$ of the basin on the horizontal plane of interest. Hence the temperature anomaly $T^{\ast }=T-T_{ref}$ is such that positive heat transport is either by upwelling of water that is relatively warm compared to the average temperature over the whole area of the basin at the same level, or by downwelling of water that is relatively cool for that level. When integrating over the full area of the basin, the net advection at the reference temperature on the left of (5.2) vanishes (by continuity $\int _{{\mathcal{A}}}\overline{w}\,\text{d}a=0$ ). When considering a time average of the thermal equilibrium state and integrating over the full area, there must also be zero net heat transport across any horizontal level in the fluid ( $\int _{{\mathcal{A}}}\overline{wT}\,\text{d}a=\unicode[STIX]{x1D705}\int _{{\mathcal{A}}}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}\,\text{d}a$ ).

5.2 Contributions to heat transport

The overall heat transport through the basin was reported for the lower-Rayleigh-number cases in Vreugdenhil et al. (Reference Vreugdenhil, Gayen and Griffiths2016) and the Nusselt-number values are repeated in table 1. Those values, along with that for the new solution at the higher Rayleigh number, are consistent with both the $Ra$ and $Ro_{\unicode[STIX]{x0394}T}$ dependence predicted by the geostrophic scaling (2.4). Here we examine the vertical transport of heat into the interior and the mechanisms responsible.

Snapshots of vertical velocity, temperature and the vertical advective heat flux at mid-depth ( $\hat{z}=0.5$ ) are shown in figure 10 for the higher Rayleigh number $Ra=7.4\times 10^{12}$ and $Ro_{\unicode[STIX]{x0394}T}=1.1\times 10^{-3}$ . The mid-depth plane is chosen here in order to remain clear of the spatially and temporally varying thermal boundary layer thickness, particularly at the strongest rotation rate. Results for the smaller $Ra$ at $Ro_{\unicode[STIX]{x0394}T}<10^{-1}$ are similar, including the case having free-slip sidewalls. Upwelling velocities are largest in the warm vortical plumes, which are distributed through the region $\hat{x}<0.3$ (figure 10 a,b). These plumes are generally surrounded by significantly slower downward motion of water that is cooler than the plumes but warmer than the basin-wide average $\overline{T}_{ref}$ on this plane, hence contributing downward heat flux.

Figure 10.

Snapshots at mid-depth ( $\hat{z}=0.5$ ) for $Ra=7.4\times 10^{12}$ , $Ro_{\unicode[STIX]{x0394}T}=1.1\times 10^{-3}$ : (a) normalised vertical velocity $w/\sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}$ ; (b) normalised temperature $\hat{T}$ ; and (c) normalised advective heat flux (per unit area) $wT^{\ast }/\unicode[STIX]{x1D705}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}_{hot}$ through the horizontal plane, where $\unicode[STIX]{x1D705}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}_{hot}$ is the average input heat flux (per unit area) from the warm half of the base.

The corresponding time-averaged quantities (vertical velocity, temperature and advective vertical heat flux) on the mid-depth plane (figure 11) reveal a substantial role of the side boundaries. There is mean upwelling of warm water at the three walls bounding the destabilised half of the basin. There is only a very small mean vertical volume flux in the interior region of deep columnar convection, and it is in the opposite direction. For the case shown in figure 11(a), the average vertical velocity over the area outside the boundary plumes and at $\hat{x}<0.3$ , at mid-depth, is just $-0.3\,\%$ (i.e. downwards) of the maximum time-averaged velocity in the side boundary plumes. The width of this sidewall upwelling, plotted in figure 12, was estimated by measuring the distance from the wall at which the vertical velocity is 10 % of the maximum mean vertical velocity, at three chosen locations (shown in figure 11) and averaging these. For weak rotation ( $Ro_{\unicode[STIX]{x0394}T}=0.16$ ) there was no distinct upwelling on the ‘eastern’ and ‘western’ boundaries, and in place of this the width of the endwall plume is plotted for this case. The width of boundary upwelling decreases with increasing rotation, with a best-fitting power law $Ro_{\unicode[STIX]{x0394}T}^{1/3}\sim f^{-5/9}$ : this dependence is significantly stronger than that predicted for viscous Stewartson layers ( $f^{-1/4}$ ) but significantly weaker than in a scaling with the Rossby deformation radius, as has previously been suggested (Spall Reference Spall2011; Cenedese Reference Cenedese2012). An important result is that similar side boundary upwelling zones also occur in the case of free-slip sidewalls, where they are only 20 % narrower (filled triangle in figure 12). Hence the upwelling is not a viscous boundary layer phenomenon and is instead buoyancy-driven, stationary line plumes at the boundaries. As they are driven by gravitationally unstable temperature differences, the width of the plumes is not directly related to the deformation radius based on the overall density difference and that characterises the stable density stratification elsewhere in the circulation.

Figure 11.

Time-averaged quantities at mid-depth ( $\hat{z}=0.5$ ) corresponding to figure 10, for an averaging period 65 $\unicode[STIX]{x1D70F}_{b}$ : (a) mean vertical velocity $\overline{w}/\sqrt{g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}TL}$ ; (b) mean temperature $\overline{\hat{T}}$ ; and (c,d) time-averaged advective heat flux (per unit area) through the plane by, respectively, the mean $\overline{w}\overline{T^{\ast }}/\unicode[STIX]{x1D705}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}_{hot}$ and fluctuating $\overline{{w^{\prime }T}^{\prime }}/\unicode[STIX]{x1D705}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}_{hot}$ components of the flow. Heat fluxes are normalised by $\unicode[STIX]{x1D705}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}_{hot}$ , the time- and area-averaged heat flux (per unit area) into the basin from the heated area of the base. In (a) the arrows show the locations where the width of the boundary upwelling zone is measured, and the broken line is the average of the three measured widths.

Figure 12.

Width of the side boundary upwelling zones relative to basin length $L$ as a function of Rossby number for $Ra=7.4\times 10^{11}$ (closed symbols, triangle for no-slip sidewalls) and $Ra=7.4\times 10^{12}$ (open square). The broken line indicates theoretical Stewartson layer thickness $\unicode[STIX]{x1D6FF}_{S}/L=(2E)^{1/4}\sim Ro_{\unicode[STIX]{x0394}T}^{3/20}$ ; solid line is the best fit ( $0.22Ro_{\unicode[STIX]{x0394}T}^{1/3}$ ) for $Ra=7.4\times 10^{11}$ , $Ro_{\unicode[STIX]{x0394}T}<10^{-1}$ (no-slip sidewalls).

Advective heat flux due to the mean flow, $\overline{w}\overline{T^{\ast }}$ (figure 11 c), reflects the mean vertical velocities and is largest in the side boundary upwelling zones. It is positive also through the bulk of the cooled half of the basin (corresponding to a slow downward motion of water cooler than $\overline{T}_{ref}$ ), where it approximately balances the mean downward diffusion of heat in the mean temperature gradient. In contrast, the contribution of the fluctuating (plume) component of flow, $\overline{w^{\prime }T^{\prime }}$ (figure 11 d), carries heat upwards in much of the interior at $\hat{x}<0.3$ . The averaging period of $65\unicode[STIX]{x1D70F}_{b}$ (which for the case in figure 11 equates to 1600 rotation periods) does not completely remove the signature of the interior vortical plumes, as these are long-lived, slowly migrating structures. The solution for $Ra=7.4\times 10^{11},~Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ (not shown) has an additional weak downward transport of heat by fluctuations inside a large anticyclonic gyre near $\hat{x}=0.6$ .

Figure 13.

Time-averaged heat transport contributions through the horizontal mid-depth plane $z=H/2$ , plotted as a function of Rossby number, carried by the fluctuating component of flow $\int _{{\mathcal{A}}}\overline{w^{\prime }T^{\prime }}\,\text{d}a$ (red triangles; as shown in figure 11 d) and by the mean flow $\int \overline{w}\overline{T^{\ast }}\,\text{d}a$ (as shown in figure 11 c), where the latter is separated into contributions from horizontal sub-areas $a$ in the interior (blue circles) and sidewall upwelling zones (cyan circles): $Ra=7.4\times 10^{11}$ (solid symbols, no-slip sidewall results linked by the broken lines; free-slip wall results are those lying slightly off the lines at $Ro_{\unicode[STIX]{x0394}T}=4\times 10^{-4}$ );  $Ra=7.4\times 10^{12}$ (open symbol). The inset shows the total mean diffusive component of vertical heat transport $\unicode[STIX]{x1D705}\int _{{\mathcal{A}}}\unicode[STIX]{x2202}\overline{T}/\unicode[STIX]{x2202}z\,\text{d}a$ at mid-depth. Heat transports are normalised by the total heat throughput $\unicode[STIX]{x1D705}\int _{A_{hot}}\overline{(\unicode[STIX]{x2202}T/\unicode[STIX]{x2202}z)}|_{z=0}\,\text{d}a$ and are averaged over 600 $\unicode[STIX]{x1D70F}_{b}$ for $Ra=7.4\times 10^{11}$ and 65 $\unicode[STIX]{x1D70F}_{b}$ for $Ra=7.4\times 10^{12}$ .

Area-integrated contributions to mean vertical heat transport through the mid-depth plane in subregions of the basin are compared in figure 13. Shown here are the mean flow contributions in the subregions encompassing either the side boundary upwelling zones or the interior above the heated half of the base, along with the heat transport by fluctuations over the whole basin area (which is not spatially decomposed, as the fluctuations are overwhelmingly the plumes in the interior region of chimney convection). Similarly the heat transport by diffusion (inset in figure 13) is shown only for the whole basin area. The diffusive transport in all cases transports heat downwards as a result of the stable mean temperature gradient: it is very small for $Ro_{\unicode[STIX]{x0394}T}>10^{-1}$ (as in the non-rotating case), increases with increasing rotation (reflecting increased stratification in the interior; figure 2), and is notably larger for the smallest Rossby number. The total of all contributions sum to zero net heat transport through the plane. All contributions at this level are small compared to the total heat throughput (which is used to normalise each contribution in figure 13), as most of the throughput is carried horizontally by flow in the thermal boundary layer.

The vertical heat transport by the mean flow does not trend systematically at moderate Rossby numbers. However, it is positive in the sidewall upwelling zones and negative in the interior. At $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ these two contributions are comparable, implying that the vertical heat transport is largely by a coherent zonal mean overturning, which we associate with a strong anticyclonic gyre near the centre of the basin, present only in this run. At stronger rotation ( $Ro_{\unicode[STIX]{x0394}T}<10^{-3}$ ) interior transport by mean flow becomes negligible, the mean sidewall upwelling and interior fluctuating flow give comparable (and positive) contributions, and the total mean heat transport is balanced by the mean downward diffusion of heat, the latter becoming the largest term at extreme rotation. The heat transport by fluctuating flow is downward at weak rotation, where it is confined to the endwall plume in which $\text{d}\overline{T}/\text{d}z<0$ , and changes sign at $Ro_{\unicode[STIX]{x0394}T}\approx 10^{-2}$ . At Rossby numbers less than this, the chimney convection in the interior takes place within the positive mean temperature gradient. The fluctuating flow, representing the vortical plumes, gives an increasingly large upward heat transport with increasing rotation, becoming the second largest contribution (after mean diffusion) at extreme rotation due to a larger number of vortical plumes.

6.1 Energy budget diagnostics

The energy source for the flow is the generation of available potential energy (APE) by the boundary buoyancy forcing. The APE is converted (by the reversible buoyancy flux, integrated over the full volume ${\mathcal{V}}$ of the basin) to kinetic energy at a matching rate $\unicode[STIX]{x1D6F7}_{z}=g\int _{{\mathcal{V}}}w\unicode[STIX]{x1D70C}\,\text{d}V$ (Winters et al. Reference Winters, Lombard, Riley and D’Asaro1995; Hughes, Hogg & Griffiths Reference Hughes, Hogg and Griffiths2009). This energy transfer can be decomposed into contributions from the time-averaged and fluctuating components of the flow, $\overline{\unicode[STIX]{x1D6F7}_{z}}=g\int _{{\mathcal{V}}}\overline{w}\bar{\unicode[STIX]{x1D70C}}\,\text{d}V$ and ${\unicode[STIX]{x1D6F7}_{z}}^{\prime }=g\int _{{\mathcal{V}}}\overline{w^{\prime }\unicode[STIX]{x1D70C}^{\prime }}\,\text{d}V$ , respectively, corresponding to large-scale circulation and small-scale convection (Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013, Reference Gayen, Griffiths and Hughes2014). The roles of these contributions have not previously been examined for geostrophic circulation with resolved convection. The roles of these contributions in the case of non-rotating convection have been discussed in an alternative framework of a locally defined APE density (Scotti & White Reference Scotti and White2014). In the present study we choose to use the APE framework of Winters et al. (Reference Winters, Lombard, Riley and D’Asaro1995) and Hughes et al. (Reference Hughes, Hogg and Griffiths2009), with energy conversions computed from the mean and fluctuating parts of the velocity and buoyancy fluxes, an approach that is widely familiar in fluid dynamics.

There are two sinks in the mechanical energy budget: viscous dissipation and mixing of density (Winters et al. Reference Winters, Lombard, Riley and D’Asaro1995; Peltier & Caulfield Reference Peltier and Caulfield2003; Hughes et al. Reference Hughes, Hogg and Griffiths2009). Viscous dissipation converts kinetic energy into internal energy. The rate of viscous dissipation $\unicode[STIX]{x1D700}$ in a volume of interest $V$ is $\unicode[STIX]{x1D700}=\unicode[STIX]{x1D70C}_{r}\unicode[STIX]{x1D708}\int (\unicode[STIX]{x2202}u_{i}/\unicode[STIX]{x2202}x_{j})^{2}\,\text{d}V$ , where $u_{i}$ are components of the velocity, $x_{i}$ are displacements, the subscripts $i$ and $j$ are orthogonal directions and summation is implied. In the time-averaged thermally equilibrated state, the present simulations confirm that the total rates of generation and dissipation of kinetic energy ( $\unicode[STIX]{x1D700}_{T}$ when integrated over the whole volume of the basin) are accurately matched. They also match the net production of potential energy,

(6.1) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{i}=-g\unicode[STIX]{x1D705}WL(\langle \unicode[STIX]{x1D70C}\rangle _{H}-\langle \unicode[STIX]{x1D70C}\rangle _{0}),\end{eqnarray}$$

which is due only to diffusion of heat down the small mean top-to-bottom vertical temperature gradient ( $\langle \unicode[STIX]{x1D70C}\rangle _{H}$ and $\langle \unicode[STIX]{x1D70C}\rangle _{0}$ being the horizontally averaged densities at the top and bottom boundaries, respectively (Paparella & Young Reference Paparella and Young2002; Hughes et al. Reference Hughes, Hogg and Griffiths2009; Vreugdenhil et al. Reference Vreugdenhil, Gayen and Griffiths2016)). That is, $\unicode[STIX]{x1D700}_{T}=\unicode[STIX]{x1D6F7}_{i}$ . The results further imply that the rate of viscous dissipation, in the turbulent boundary layer regime, is approximately independent of rotation, and that any weak dependence is a consequence of a rotation-dependent top-to-bottom density difference.

The second energy sink is conversion of APE due to the mixing of the density field. The rate of mixing in any subvolume $V$ is given by $\unicode[STIX]{x1D6F7}_{d}=-g\unicode[STIX]{x1D705}\int (\text{d}z^{\ast }/\text{d}\unicode[STIX]{x1D70C})(\unicode[STIX]{x2202}\unicode[STIX]{x1D70C}/\unicode[STIX]{x2202}x_{i})^{2}\,\text{d}V$ , where $z^{\ast }$ is the height of the water parcel (above the base) if all the water parcels in the basin were adiabatically reordered into the rest state of gravitational equilibrium (Hughes et al. Reference Hughes, Hogg and Griffiths2009). Energetics frameworks for incompressible fluids under the Boussinesq approximation and with a linear equation of state describe the role of this mixing as an irreversible transfer of APE to background potential energy (Winters et al. Reference Winters, Lombard, Riley and D’Asaro1995; Peltier & Caulfield Reference Peltier and Caulfield2003; Hughes et al. Reference Hughes, Hogg and Griffiths2009). (For a nonlinear equation of state or a compressible fluid, a more complete description is required, in which the changes in APE and background potential energy are not equal and the APE is irreversibly converted to internal energy, hence a contribution to Joule heating (Tailleux Reference Tailleux2009, Reference Tailleux2013, Reference Tailleux2018).) For the Boussinesq fluid with a linear equation of state considered here, in a time-averaged thermally equilibrated state, the mixing rate $\unicode[STIX]{x1D6F7}_{d}$ integrated over the whole volume of the basin must equal the rate of generation of APE by buoyancy forcing at the boundary (Hughes et al. Reference Hughes, Hogg and Griffiths2009; Tailleux Reference Tailleux2009; Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013, Reference Gayen, Griffiths and Hughes2014). A previous theoretical result (Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013), confirmed by DNS (Gayen et al. Reference Gayen, Griffiths and Hughes2014), is that the mixing rate is related simply to the heat transport in a manner that is independent of rotation ( $\unicode[STIX]{x1D6F7}_{d}/\unicode[STIX]{x1D700}_{T}\sim NuA_{x}$ ) and this can be re-expressed in terms of the Rossby number defined here:

(6.2) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{d}/\unicode[STIX]{x1D700}_{T}\sim (RaPrRo_{\unicode[STIX]{x0394}T})^{1/5}.\end{eqnarray}$$

The energy conversions outlined above can be used to define a global mixing efficiency

(6.3) $$\begin{eqnarray}\unicode[STIX]{x1D702}=\frac{\unicode[STIX]{x1D6F7}_{d}-\unicode[STIX]{x1D6F7}_{i}}{\unicode[STIX]{x1D6F7}_{d}-\unicode[STIX]{x1D6F7}_{i}+\unicode[STIX]{x1D700}_{T}}\end{eqnarray}$$

(Peltier & Caulfield Reference Peltier and Caulfield2003; Hughes et al. Reference Hughes, Hogg and Griffiths2009). Applying the steady-state result that $\unicode[STIX]{x1D700}_{T}=\unicode[STIX]{x1D6F7}_{i}$ , the mixing efficiency becomes

(6.4) $$\begin{eqnarray}\unicode[STIX]{x1D702}=1-\frac{\unicode[STIX]{x1D700}_{T}}{\unicode[STIX]{x1D6F7}_{d}}.\end{eqnarray}$$

The scaling above ( $\unicode[STIX]{x1D6F7}_{d}/\unicode[STIX]{x1D700}_{T}\sim NuA_{x}$ ) coupled with (2.10) then gives (Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013)

(6.5) $$\begin{eqnarray}\unicode[STIX]{x1D702}\approx 1-(NuA_{x})^{-1}.\end{eqnarray}$$

The efficiency will therefore be smaller for larger rotation rates.

Whereas the global mixing efficiency helps to reveal the overall energy balances of the system, further insights can be obtained from the distribution of energy sinks in the flow and a local mixing efficiency. In order to define a local mixing efficiency, we recognise that $\unicode[STIX]{x1D6F7}_{i}$ is only defined as a global value, and use the result that at large Rayleigh numbers $\unicode[STIX]{x1D6F7}_{d}\gg \unicode[STIX]{x1D6F7}_{i}$ . A useful definition for a local mixing efficiency is then $\unicode[STIX]{x1D702}_{l}=\unicode[STIX]{x1D6F7}_{dl}/(\unicode[STIX]{x1D6F7}_{dl}+\unicode[STIX]{x1D700})$ , where $\unicode[STIX]{x1D6F7}_{dl}$ and $\unicode[STIX]{x1D700}$ are local (grid-scale) values of the mixing rate and viscous dissipation (Sohail, Gayen & Hogg Reference Sohail, Gayen and Hogg2018).

Figure 14.

Snapshots of (a,b) viscous dissipation, (c,d) mixing rates (per unit volume), and (e,f) local mixing efficiency for $Ra=7.4\times 10^{12}$ , $Ro_{\unicode[STIX]{x0394}T}=1.1\times 10^{-3}$ . Panels (a,c,e) show zonally averaged quantities and (b,d,e) show quantities at a single depth of $\hat{z}=0.0055$ (within the thermal boundary layer, just outside of the Ekman layer). The dissipation and mixing are normalised by $\unicode[STIX]{x1D6F7}_{i}/HWL$ , where $\unicode[STIX]{x1D6F7}_{i}~(=\unicode[STIX]{x1D700}_{T})$ is from (6.1). The colour scale is logarithmic for dissipation and mixing.

The DNS for the smaller of the two Rayleigh-number values used here (Vreugdenhil et al. Reference Vreugdenhil, Gayen and Griffiths2016) showed the turbulent boundary layer regime and confirmed the behaviour predicted by (6.1)–(6.5). The basin-integrated mixing of density is smaller for larger rotation rates (as a result of increased boundary layer thickness and decreased buoyancy throughput), while the turbulent dissipation is unaffected by rotation. Thus the global mixing efficiencies are large (listed in table 1). In this paper we revisit the two energy sinks and discuss their spatial distribution.

6.2 Buoyancy flux, dissipation and mixing

The solutions can be interrogated to find the distributions of dissipation $\unicode[STIX]{x1D700}$ , mixing of density $\unicode[STIX]{x1D6F7}_{d}$ and the local mixing efficiency $\unicode[STIX]{x1D702}_{l}$ . The zonally averaged values and those on a plane at $\hat{z}=0.0055$ , just outside the Ekman layer, are shown in figure 14 for the case $Ra=7.4\times 10^{12}$ and $Ro_{\unicode[STIX]{x0394}T}=1.1\times 10^{-3}$ , where the rates of energy conversion are normalised by $\unicode[STIX]{x1D6F7}_{i}/HWL$ , the volume average rate of production of potential energy. The viscous dissipation is elevated at the base of the domain over the heated region (figure 14 a) and is especially large near the mid-section of the basin, where it is associated with combined small-scale convection and shear in the strong mean flow of the boundary layer current across the basin width and along the ‘eastern’ boundary (figure 14 b). Modest dissipation rates also appear in isolated patches beneath the vortical plumes. The mixing rate is strongest at the base, especially over the cooled region where the vertical temperature gradient is largest (figure 14 c). In the boundary layer, the largest mixing rates are around the edge of the convection chimney (the dark region in figure 14 d). Otherwise the mixing is distributed across the heated base and in the very small scales of convection. There is little mixing in the core of the chimney region at or above the height of the plane shown. At this height the two sinks are of comparable magnitude (figure 14 b,d). However, when summed over the volume of the basin, the mixing rate is an order of magnitude greater than the total dissipation rate, with the main site of mixing closer to the boundary than the plane chosen for figure 14(d). The local mixing efficiency shows values close to unity above the cooled region (figure 14 e) and at the edge of the convective chimney region (figure 14 f). When zonally averaged, most of the interior is at small to moderate local mixing efficiency. The columnar plumes through the depth appear as local highs in the interior dissipation, mixing and mixing efficiency, even in the zonal averages.

Figure 15.

Distribution of dissipation $\unicode[STIX]{x1D700}$ for $Ra=7.4\times 10^{11}$ runs with (a)  $Ro_{\unicode[STIX]{x0394}T}=1.6\times 10^{-1}$ , (b) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ , (c) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ and (d) $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ . Horizontal planes are taken within the boundary layer at (a) $\unicode[STIX]{x1D6FF}/H=0.011$ , (b) $\unicode[STIX]{x1D6FF}/H=0.011$ , (c) $\unicode[STIX]{x1D6FF}/H=0.006$ and (d) $\unicode[STIX]{x1D6FF}/H=0.0027$ . Excepting the weak rotation case (a), each plane is taken just outside the Ekman layer. The dissipation is normalised by $\unicode[STIX]{x1D6F7}_{i}/HWL$ .

Figure 16.

Distribution of mixing $\unicode[STIX]{x1D6F7}_{d}$ for $Ra=7.4\times 10^{11}$ runs with (a)  $Ro_{\unicode[STIX]{x0394}T}=1.6\times 10^{-1}$ , (b) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-3}$ , (c) $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ and (d) $Ro_{\unicode[STIX]{x0394}T}=1.7\times 10^{-5}$ . Horizontal planes are taken within the boundary layer at (a) $\unicode[STIX]{x1D6FF}/H=0.011$ , (b) $\unicode[STIX]{x1D6FF}/H=0.011$ , (c) $\unicode[STIX]{x1D6FF}/H=0.006$ and (d) $\unicode[STIX]{x1D6FF}/H=0.0027$ . Excepting the weak rotation case (a), each plane is taken just outside the Ekman layer. The dissipation is normalised by $\unicode[STIX]{x1D6F7}_{i}/HWL$ .

The effect of rotation rate on distributions of dissipation (figure 15) and mixing (figure 16) rates in the thermal boundary layer are shown for $Ra=7.4\times 10^{11}$ . Dissipation and mixing both appear to be associated with small-scale convection and the intermediate eddy scale (where eddies are at the mid-section and along the boundary currents). The cold front results in the largest vertical temperature gradients that drive direct convection, hence large values of the mixing rate.

The distribution of energy sinks between key subregions is summarised in figure 17(a). The total dissipation in the basin is largely confined (79–82 %) to the average thermal boundary layer of thickness $\unicode[STIX]{x1D6FF}$ (given in table 1). For strong rotation, this estimate leaves some of the boundary layer dissipation attributed to the interior region because the true boundary layer thickness varies in time and space. Dissipation in the Ekman layer on the base (a subset of the thermal boundary layer volume) accounts for most of the dissipation at weak rotation but only a small fraction of the total at strong rotation. There is no significant difference between results for the two values of $Ra$ . The free-slip side boundary condition leads to slightly greater dissipation in the Ekman layer due to faster flow in the boundary layer and hence greater Ekman transport.

Figure 17.

(a) Volume-integrated contributions to viscous dissipation by subvolumes of the flow, expressed as a fraction of the total basin-integrated dissipation rate $\unicode[STIX]{x1D700}_{T}$ and plotted as a function of Rossby number. Subregion contributions shown are $\unicode[STIX]{x1D700}_{B}$ in the thermal boundary layer (green triangles), $\unicode[STIX]{x1D700}_{E}$ in the Ekman layer (blue circles) and $\unicode[STIX]{x1D700}_{I}$ in the interior (red squares). The thermal boundary layer includes the Ekman layer volume. (b) Basin-integrated buoyancy flux by the mean ( $\overline{\unicode[STIX]{x1D6F7}_{z}}$ , black inverted triangles) and fluctuating ( $\unicode[STIX]{x1D6F7}_{z}^{\prime }$ , pink diamonds) flow, shown as a fraction of the total basin-integrated buoyancy flux $\unicode[STIX]{x1D6F7}_{z}$ and plotted as a function of Rossby number, for $Ra=7.4\times 10^{11}$ (closed symbols) and $Ra=7.4\times 10^{12}$ (open symbols). Results for no-slip sidewalls are linked by the broken curve, and the free-slip case is shown by the same symbol lying off the lines at $Ro_{\unicode[STIX]{x0394}T}=3.6\times 10^{-4}$ . All results are averaged over 600 $\unicode[STIX]{x1D70F}_{b}$ for $Ra=7.4\times 10^{11}$ and 65 $\unicode[STIX]{x1D70F}_{b}$ for $Ra=7.4\times 10^{12}$ . Total dissipation $\unicode[STIX]{x1D700}_{T}$ used for normalisation in (a) is equal to the net buoyancy flux $\unicode[STIX]{x1D6F7}_{z}$ used in (b).

The rate of APE conversion by mixing, when similarly integrated over the volume of subregions (not shown here), is in all cases overwhelmingly confined $({\geqslant}98\,\%)$ to the thermal boundary layer, where it is associated with large mean temperature gradients at the boundary. This is similar in the non-rotating case, in which an even larger fraction of mixing ( $99.6\,\%$ at $Ra\approx 10^{12}$ ) is in the thermal boundary layer (Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013). Both the mixing and dissipation distributions are insensitive to rotation. As previously reported (Vreugdenhil et al. Reference Vreugdenhil, Gayen and Griffiths2016) and listed in table 1, the mixing efficiency $\unicode[STIX]{x1D702}$ given by (6.4), at $Ra\approx 10^{12}$ , decreases with increasing rotation rate in a manner consistent with the predicted Nusselt-number dependence (6.5). The mixing efficiency is also predicted to be insensitive to the magnitude of buoyancy forcing at very large Rayleigh and Nusselt numbers. Thus the solution at $Ra\approx 10^{13}$ yields a mixing efficiency $\unicode[STIX]{x1D702}=0.93$ , although this value is slightly smaller than predicted by the scaling analysis (Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013; Vreugdenhil et al. Reference Vreugdenhil, Gayen and Griffiths2016) because the solution could not be run long enough to obtain a perfect equilibrium dissipation rate. However, the quantity $\unicode[STIX]{x1D6F7}_{i}$ was fully equilibrated, and if this is used as an estimate for the final dissipation, the mixing efficiency is $\unicode[STIX]{x1D702}=0.94$ in line with the scaling prediction.

The contributions of the mean and fluctuating reversible buoyancy fluxes to the mechanical energy budget ( $\overline{\unicode[STIX]{x1D6F7}_{z}}$ and $\unicode[STIX]{x1D6F7}_{z}^{\prime }$ , respectively, shown in figure 17(b) as net values integrated over the full volume of the basin) reflect the change in role of the mean flow and small-scale convection between moderate and extreme rotation, consistent with the time-averaged heat transport contributions by the mean and fluctuating flow in figure 13. At the weakest rotation, the relative contributions of mean and fluctuating flow are similar to results for the non-rotating case (Gayen et al. Reference Gayen, Griffiths and Hughes2014). There is an increase in the role of mean flow buoyancy flux with increasing rotation at weak rotation, which can be attributed to Ekman transport. However, the mean flow plays its maximum role at $Ro_{\unicode[STIX]{x0394}T}\approx 3\times 10^{-3}$ , and at stronger rotation than this the buoyancy flux due to fluctuating flow (small-scale deep vortical convection) increases with rotation while the role of mean flow decreases. The conversion of potential to kinetic energy by the mean flow becomes less than the conversion by fluctuations for $Ro_{\unicode[STIX]{x0394}T}<10^{-4}$ , as a result of geostrophic constraints on the mean flow, which cause the small scales to play a larger role in releasing the APE. At the most extreme rotation examined, the small scales of convection contribute 70 % of the total buoyancy flux. Free-slip sidewalls cause a slightly smaller fraction of the buoyancy flux to be carried by the mean flow. Corresponding results for the higher Rayleigh number indicate that the partition of energy conversion is not solely dependent on $Ro_{\unicode[STIX]{x0394}T}$ , the fluctuations playing a slightly greater role at the larger $Ra$ (interpolated to the same  $Ro_{\unicode[STIX]{x0394}T}$ ).