2.1 Non-rotating scaling

The non-rotating scaling for very large Rayleigh numbers is briefly reviewed here for the purpose of comparison with rotating cases. After neglecting rotation and eliminating the pressure in $x$ and $z$ components of the momentum equation (2.3) an inertial–buoyancy balance gives

Taking vertical gradients in the boundary layer much larger than horizontal gradients and scaling terms in (2.5) with $U$ for horizontal velocity, $\unicode[STIX]{x1D6FF}_{m}$ for the turbulent momentum boundary layer thickness, $L$ for horizontal length and $\unicode[STIX]{x0394}T$ for temperature, this becomes

The viscous scaling of Rossby (Reference Rossby1965, Reference Rossby1998) was based on momentum and thermal boundary layers of similar thickness, and we argue that the same result holds for the turbulent scaling. The stable thermal boundary layer is sustained by diffusion over the cooled region, which sets up a horizontal temperature gradient and leads to advection in the boundary layer towards the heated region of the boundary. In the turbulent case the drag is dominated by Reynolds stress produced by the small-scale convective turbulence (Gayen et al. Reference Gayen, Griffiths, Hughes and Saenz2013a , Reference Gayen, Griffiths and Hughes2014), and this is confined to a layer within, or beneath, the stably stratified thermal boundary layer, growing to the full thermal boundary layer thickness toward the end of the box. Hence the momentum boundary layer is within the thermal boundary layer and follows the same scaling, $\unicode[STIX]{x1D6FF}\sim \unicode[STIX]{x1D6FF}_{m}$ . This is strongly supported for the inertial-buoyancy regime by DNS and large eddy simulations (LES) in the ranges [ $Ra\approx 10^{12}$ , $0.05\leqslant Pr\leqslant 5$ ] and [ $10^{12}<Ra<10^{15}$ , $Pr=5$ ] respectively (Gayen et al. Reference Gayen, Griffiths and Hughes2014). Matching thermal diffusion in the vertical within the thermal boundary layer against vertical advection of heat (again as in the previous viscous scaling), and matching the vertical mass transport to the horizontal transport by continuity, the heat equation (2.4) gives

where $\overline{w}$ is the mean vertical velocity into the stable regions of the boundary layer. The flux boundary condition implies

Solving (2.6), (2.7) and (2.8) gives

where the zero subscript is added to denote the non-rotating case. The Nusselt number can also be written as $Nu_{0}\sim L/\unicode[STIX]{x1D6FF}_{0}$ . As $L/\unicode[STIX]{x1D6FF}_{0}\gg L/H$ , the Nusselt number is assured to be large when $L/H>1$ . The scaling (2.9) can alternatively be expressed in terms of the external parameters defined in (2.1):

where $c_{i}$ are constant prefactors of $O(1)$ . There is no dependence on viscosity. The volume transport in the boundary layer, $\unicode[STIX]{x1D6F9}_{0}\sim U_{0}\unicode[STIX]{x1D6FF}_{0}W$ , becomes $\unicode[STIX]{x1D6F9}_{0}\sim (\unicode[STIX]{x1D705}L)^{1/2}W(BL)^{1/6}$ , or

Previous non-rotating laboratory experiments with imposed flux for $10^{12}<Ra_{F}<10^{14}$ and $Pr\approx 4$ (Mullarney et al. Reference Mullarney, Griffiths and Hughes2004) are consistent with the inertial scaling (2.10) and give $c_{1}=0.29$ , $c_{2}=3.3$ and $c_{3}=0.65$ . The solution (2.10), including a successful theoretical prediction of the prefactors, was alternatively derived from a heuristic inviscid model coupling a turbulent plume to the interior through turbulent entrainment, and assuming a vertical advection–diffusion balance in the interior throughout the depth (Hughes et al. Reference Hughes, Griffiths, Mullarney and Peterson2007). Entrainment into the plume increases the overturning transport by adding to that passing through the boundary layer (2.11). The inertial–buoyancy scaling (when extended to the case of an applied temperature difference) is also supported by DNS (Gayen et al. Reference Gayen, Griffiths and Hughes2014).

2.2 Geostrophic boundary layer scaling

With background rotation the Coriolis term in (2.3) can be large, causing the boundary stress to be confined to an Ekman layer of thickness $\unicode[STIX]{x1D6FF}_{E}$ . For strong rotation the Ekman thickness is much less than the thickness of the thermal boundary layer, leaving the bulk of the thermal boundary layer free of boundary stress and in geostrophic balance (Hignett et al. Reference Hignett, Ibbetson and Killworth1981). Following derivations by Robinson & Stommel (Reference Robinson and Stommel1959), Robinson (Reference Robinson1960), Bryan (Reference Bryan1987) and Winton (Reference Winton1996) we assume a regime in which the mean horizontal flow (having length scales comparable to the basin length $L$ ) within the thin boundary layer is characterised by a geostrophic, thermal wind balance:

where $u$ and $v$ are velocity components in $x$ and $y$ , respectively. In the boundary layer the strongly stratified flow is then quasi-geostrophic. In the interior the net mass transport through any vertical plane at a fixed $x$ must be equal and opposite to that in the thin thermal boundary layer and the interior flow is therefore characterised by relatively small velocities, for at least the large length scales. We therefore follow the early geostrophic ocean modelling and assume that the dynamics of the faster boundary layer flow governs the overall transport. The presence of side walls may play a significant role in the circulation and heat transport through formation of boundary currents that allow geostrophic flow along the box (in the $x$ -direction) that would otherwise be prevented by geostrophy, and temperature gradients to be established in the $y$ -direction (these would be, respectively, the ‘meridional’ and ‘zonal’ directions in a planetary context). From (2.12) $V_{g}/U_{g}\sim W/L$ and

where $U_{g}$ , $V_{g}$ characterise the geostrophic velocities in the thermal boundary layer in the $x$ and $y$ directions, respectively. Previous derivations have proceeded assuming $V_{g}\sim U_{g}$ , but the effect of the horizontal aspect ratio $A_{y}=W/L$ is retained in the following derivation.

Assuming the scalings (2.7) and (2.8) for the vertical advection–diffusion balance in the stable region of the boundary layer and the boundary flux condition are unchanged by rotation effects, they are solved with (2.13) to give

Rewriting (2.14) in terms of the dimensionless parameters defined in (2.1) yields a familiar form (Park & Whitehead Reference Park and Whitehead1999):

where a dependence on horizontal aspect ratio is retained and the prefactor $c_{4}$ will be evaluated from experiments.

Previous work identified dynamical regimes in terms of $Q=(\unicode[STIX]{x1D6FF}_{0}/\unicode[STIX]{x1D6FF}_{E})^{2}$ . Transition from the viscous non-rotating regime to the (laminar) geostrophic boundary layer regime was predicted at $Q\gg 1$ (Hignett et al. Reference Hignett, Ibbetson and Killworth1981). A similar boundary layer control dependent on $Q$ has been shown for rotating Rayleigh–Bénard convection (King et al. Reference King, Stellmach, Noir, Hansen and Aurnou2009; King, Stellmach & Aurnou Reference King, Stellmach and Aurnou2012). DNS of horizontal convection in a rectangular basin with very large Rayleigh number and turbulent thermal boundary layer (under an applied temperature difference, Vreugdenhil et al. Reference Vreugdenhil, Gayen and Griffiths2016) showed the transition at $QPr\approx 10$ , where inclusion of the Prandtl number removes dependence on viscosity. For the imposed flux case considered here $QPr\sim (Ra_{F}Pr)^{-1/3}(PrE^{-1})$ . However, the decoupling of the thermal boundary layer from boundary friction when the Ekman layer thickness is very much smaller than the thermal boundary layer thickness implies that the dynamics in the geostrophic flow is more appropriately considered in terms of a comparison of the local horizontal advection accelerations in (2.3) to the Coriolis acceleration, hence in terms of a convective Rossby number $Ro=U/fL$ , where $U$ is a characteristic convection velocity produced by the buoyancy forcing. By dimensional analysis the Rossby number can be defined in terms of the velocity scale $(B/f)^{1/2}$ (Maxworthy & Narimousa Reference Maxworthy and Narimousa1994; Klinger & Marshall Reference Klinger and Marshall1995). This was also shown to be the case for rotating Rayleigh–Bénard convection in the geostrophic regime (Boubnov Reference Boubnov1984; Boubnov & Golitsyn Reference Boubnov and Golitsyn1986, Reference Boubnov and Golitsyn1990), where the characteristic velocity can be expressed in terms of the rate of dissipation of kinetic energy, $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D6FC}gF/\unicode[STIX]{x1D70C}c_{p}$ , with $U\approx 2(\unicode[STIX]{x1D716}/f)^{1/2}$ . Thus

In the geostrophic boundary layer regime of horizontal convection this is equivalent to $Ro=(Ra_{F}E^{3})^{1/2}/Pr$ or $Ro=(QPr)^{-3/2}$ . As the motion is driven by buoyancy, the Rossby number (2.16) is a measure of the relative importance of buoyancy forcing and rotation.

The geostrophic boundary layer scaling (2.14) can be rewritten in the form

Focusing on the effect of rotation relative to buoyancy in (2.17), it is useful to normalise by the corresponding non-rotating solution (2.10):

The boundary layer transport $\unicode[STIX]{x1D6F9}_{g}\sim U_{g}\unicode[STIX]{x1D6FF}_{g}W$ becomes,

where $c_{i}$ are constants that will be evaluated from experimental results.

The transition to the geostrophic boundary layer regime is predicted at

or equivalently, at $Ra_{F}=c_{7}^{-12}A_{y}^{3}Pr^{2}E^{-3}$ .

At $Ro\ll Ro_{crit}$ and $\unicode[STIX]{x1D6FF}\ll H$ the Ekman layer is much thinner than the thermal boundary layer and the thermal boundary layer is thin compared to the depth of the basin. Geostrophic balance then begins to govern the largest scale $L$ (the mean flow) and for smaller $Ro$ geostrophic balance extends to smaller scales. However, for extremely strong rotation the boundary layer is thick ( $\unicode[STIX]{x1D6FF}_{g}>H$ at $Ro\leqslant (c_{2}c_{6}/A)^{6}A_{y}^{3/2}(Ra_{F}Pr)^{-1}$ ) and the boundary layer analysis does not apply. For these extreme conditions, DNS for an applied temperature difference has shown that advection is greatly reduced and heat transport is primarily by conduction (Sheard et al. Reference Sheard, Hussam and Tsai2016; Vreugdenhil et al. Reference Vreugdenhil, Gayen and Griffiths2016). In the following we discuss the potential for additional regimes, in which transport is controlled by vertical convection rather than the boundary layer.

2.3 Chimney regimes

The laboratory observations reported in § 5 will show that at sufficiently rapid rotation rates vertical convection in the area of destabilising boundary buoyancy flux forms cyclonic vortical plumes that penetrate through any remaining stratification of the thermal boundary layer. The vortical plumes become more numerous with increasing $f$ and become columnar structures extending through the full depth of the laboratory box. We adopt the ocean modelling term ‘chimney’ for this region of plumes (Marshall & Schott Reference Marshall and Schott1999) and aim to describe the way in which the chimney convection is maintained within, and coupled to, a larger basin-scale circulation.

3.1 Apparatus

The laboratory experiments were carried out with a rectangular acrylic box of dimensions $L\times W\times H=1.25\times 0.3\times 0.2$ m (figure 1) with a rigid lid. The same box was used in previous studies (Stewart et al. Reference Stewart, Hughes and Griffiths2011; Griffiths et al. Reference Griffiths, Hughes and Gayen2013). Except for the base, all sides were triple-glazed with argon gas in the gaps minimising heat loss to the room. When temperatures alone were to be measured, $0.1$ m of expanded polystyrene foam was placed around the box. The base was a 10 mm thick copper plate, the upper surface of which was levelled, in both the $x$ and $y$ directions, to within $0.5~\text{mm}~\text{m}^{-1}$ . One half of the base was heated by an electrical resistance heater ( $0.600\times 0.305~\text{m}^{2}$ ), which was supplied with an electrical power held constant by a controller to within $\pm 0.1~\text{W}$ . The heater was designed to give a uniform heat flux. Experiments used three values of the power input (530, 155, and 27 W) giving the heat fluxes and three Rayleigh numbers $2\times 10^{13}-6\times 10^{14}$ listed in table 1. The other half of the base was cooled using a heat exchanger coupled to a water bath held at a fixed temperature $T_{c}$ between 9 and $22\,^{\circ }\text{C}$ . Insulation 50 mm wide was placed between the heat source and cooler to ensure that the heat transfer by conduction along the base was negligible. Room temperature was held at $26\,^{\circ }\text{C}$ ( $\pm 2\,^{\circ }\text{C}$ ). Heat loss from the tank, estimated from calorimetry tests (Stewart et al. Reference Stewart, Hughes and Griffiths2011; Griffiths et al. Reference Griffiths, Hughes and Gayen2013), was 1–4 % of total heat input depending on the bulk temperature in the box, which in turn was dependent on the applied heat flux. The convecting fluid was de-aerated water with a small amount of dissolved salt. The Prandtl number listed in table 1 was based on the molecular values for water at the bulk temperature in the thermal equilibrium states, with diffusivity $\unicode[STIX]{x1D705}$ weakly dependent on water temperature and viscosity assumed constant at $\unicode[STIX]{x1D708}=6.11\times 10^{-7}~\text{m}^{2}~\text{s}^{-1}$ .

Table 1.

Summary of experimental conditions. The applied heat fluxes $F$ give buoyancy fluxes $B=g\unicode[STIX]{x1D6FC}F/\unicode[STIX]{x1D70C}_{0}c_{p}$ (for $g=9.8~\text{ms}^{-2}$ , $\unicode[STIX]{x1D70C}_{0}\approx 1025~\text{kg m}^{-3}$ and $c_{p}=4179~\text{Jkg}^{-1}~\text{K}^{-1}$ ). The properties are from Ruddick & Shirtcliffe (Reference Ruddick and Shirtcliffe1979) and derived from the polynomial interpolation of values given in appendix 1 of Batchelor (Reference Batchelor1967).

The apparatus was set on a rotating table, with the exception of the constant temperature water bath, which was connected to the experiment via rotating fluid connections. Rotation was anticlockwise and rotation rates covered the range $\unicode[STIX]{x1D6FA}=0{-}0.8~\text{rad}~\text{s}^{-1}$ and the Coriolis parameter $f=2\unicode[STIX]{x1D6FA}=0{-}1.6~\text{rad}~\text{s}^{-1}$ (table 1). This range was limited by the paraboloidal curvature of surfaces of constant potential resulting from centrifugal acceleration. Isopotential surfaces are of the form $\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}_{0}=f^{2}r^{2}/8g$ , where $\unicode[STIX]{x1D702}_{0}$ is the height of a surface at the axis of rotation and $r$ is the radius about the axis. For illumination purposes described below, the position of the tank was offset from the axis of rotation by 0.195 m. The across-tank height difference was in all cases negligible. The difference across the largest radius ( $r=L/2$ ) can become significant near the end walls at the largest rotation rates used and a small correction will be applied (discussed in § 6). However, rotation rates were kept within the range for which this maximum isopotential surface height difference was less than the thermal boundary layer thickness, $\unicode[STIX]{x1D702}(x=0,L)-\unicode[STIX]{x1D702}_{0}(x=L/2)<\unicode[STIX]{x1D6FF}$ and $f^{2}<(32g/L)(Ra_{F}PrRo)^{-1/6}$ using (2.17), ensuring that the planar base of the tank did not act effectively as a topographic barrier to the stably stratified, cold boundary layer.

The temperature of the hot and cold boundary regions was monitored by four thermistors set into the copper plate, two near the hot end ( $x/L=0.08$ , $y/W=0.25,0.75$ ; of type Thermometrics P60DB163M) and two near the cold end ( $x/L=0.92$ , $y/W=0.25,0.75$ ; Thermometrics P60DB472M). In some experiments the hot plate thermistor at $y/W=0.25$ malfunctioned.

Vertical profiles of temperature in the water were obtained from an array of 8–12 fast response thermistors (Thermometrics Fastip FP07DA103N) that were mounted on 2 mm diameter rods passing through holes in the lid. The thermistor array was traversed downward by a SmartMotor through the box at a speed $4~\text{mm}~\text{s}^{-1}$ (a 50 s transit time through the depth) with a sampling time interval $2\times 10^{-3}~\text{s}$ . Readings were averaged over 20 consecutive samples and values recorded at a time interval 0.04 s. The vertical resolution, limited by the resolution of the SmartMotor output, was 0.2 mm. In order to protect the thermistors from damage, the temperature profiles stopped 2 mm from the base, except for one case in which the closest approach to the base was set at 1.5 mm. Vertical temperature profiles were generally taken every 2 min for 24 h after the flow had reached thermal equilibrium, and time averaged to obtain the final profile.

Experiments generally began with a new fill of water at uniform temperature, spun up to the desired rotation rate and brought to thermal equilibrium. A series of additional runs (not included in table 1) were designed to examine the equilibration adjustment in more detail. The convection was first brought to thermal equilibrium with heat input $F=2623~\text{W}~\text{m}^{-2}$ (effectively a 10 % decrease in each $Ra_{F}$ for Experiments 1–6 in table 1) for a certain rotation rate. The heat input was then increased to $F=2896~\text{W}~\text{m}^{-2}$ (corresponding to the stated values of $Ra_{F}$ in table 1) and the adjustment of the interior temperature over more than 12 h was recorded at $x/L=0.08$ and $x/L=0.92$ , with $y/W=0.5$ and $z/H=0.5$ . Thus the adjustment was to a modest perturbation of the thermal forcing conditions while the choice of the largest heat flux afforded maximum temperature differences in the flow.

3.2 Dye visualisation experiments

In separate runs red and blue dyes were released slowly into the tank at selected locations using 1 mm metal tubes. The locations, shown in figure 3 below, were chosen such that the tracer advection most effectively revealed the flow patterns. The releases were generally within 1 mm of the base, in the stably stratified boundary layer, where the tracer was first carried in the boundary layer flow before entering the interior in vertical convection above the heated region of the base and subsequently advected back toward the cooled end in the interior flow. The density of the dye was carefully matched to that of the boundary layer water by adding salt to the dye. For these runs the polystyrene insulation around the tank was removed to allow illumination and visibility. Two Sony HDR-HC7E video cameras ( $1920\times 1080$ pixels) mounted in the rotating reference frame 2 m above the convection box recorded the tracer motions in planform. Another two video cameras mounted 1 m from the side wall recorded the flow from the side. Each camera recorded one half the length of the domain. Given the complex three-dimensional flow the dyes provided qualitative but invaluable insight into the structure of the convection.

3.3 Particle tracking velocimetry

Horizontal flow velocities relative to the rotating frame were measured in three horizontal planes using particle tracking velocimetry (PTV). A light emitting diode (LED) source (1 m long by 0.1 m wide) was placed 1 m away from the side wall and the light passed through horizontal slits to create a horizontal sheet of light (10 mm thick in the vertical). The light sheet was adjusted to three different heights, $z/H=0.15,0.5$ and 0.85. The cameras mounted above the tank were focused on the height of the light sheet. PTV could not be utilised in the bottom thermal boundary layer due to uncertainty in the vertical position of illuminated particles, resulting from the large vertical temperature gradient near the base and strong refraction of light as rays crossed the relatively large width of the box, as well as from the thickness of the light sheet. PTV was not possible in the region closest to the walls (5 mm at $y/W=0$ , $1$ and 12.5 mm at $x/L=0$ , $1$ ) due to refraction of light near the edges of the triple-glazed lid. Visibility constraints also prohibited simultaneous use of temperature profiling and PTV.

The PTV used Pliolite resin particles suspended in the water. In order to make the particles neutrally buoyant, NaCl was added to achieve a density $\unicode[STIX]{x1D70C}\approx 1025~\text{kg}~\text{m}^{3}$ at  $21\,^{\circ }\text{C}$ . Once thermal equilibrium was reached 10 ml of water containing the particles was slowly added to the tank at several locations. The system was then left for 2 h while the convection stirred the particles throughout the box, after which the flow was recorded for 3 h at one image per second. The population of particles in suspension decreased over time, limiting the useful measurement period to 3 h. The particles settled on the top and bottom surfaces decreasing the visibility of suspended particles and meaning that a fresh fill of water was required for each of the three light sheet heights. As it took a significant amount of time to refill the tank and bring the system to a thermally equilibrated state, PTV was reserved for the largest $Ra_{F}$ cases only (Experiments 1–6 in table 1).

The video was processed (using Streams 2.01 software; Nokes (Reference Nokes2014)) to obtain horizontal fields of the horizontal velocity. Instantaneous horizontal velocity fields at each level were obtained as averages over 10 frames (10 s) with a vector calculated in each of 4000 windows, each window representing $12.25~\text{mm}\times 7.25~\text{mm}$ of the area of the box. This combination ensured an accurate representation of the fluctuating velocity field for length scales greater than or equal to the PTV window. The 3 h measurement period proved sufficient for computation of reliable time-averaged velocity fields that converged with increasing averaging times and captured the frequencies of significant fluctuations.

6.1 Boundary layer thickness

The boundary layer thickness is defined here as the height from the base containing 90 % of the top-to-bottom buoyancy difference. The thickness varies with location as a result of geostrophic horizontal flow, including the circulation gyres on the scale of the box and side wall boundary currents. It also varies with distance from the axis of rotation due to parabolic curvature of isopotential surfaces (appendix A). Over the cooled region the boundary layer is thinner for larger Rayleigh numbers and thicker for more rapid rotation (figure 7 a). Rotation begins to influence the boundary layer at $E<10^{-5}$ given the other experimental conditions. The effect of rotation is seen more clearly in figure 7(b), where all thickness measurements are normalised by the thickness given by the scaling law (2.10) for the non-rotating case. When comparing the measured thickness at this location near the ends of the box with theoretical predictions, a correction is added to the predicted thickness scaling in order to account for the parabolic curvature of isopotential surfaces (appendix A). In figure 7(b) the geostrophic boundary layer scaling (2.18) with the correction for isopotential curvature matches the measured boundary layer thicknesses in the range $Ro<10^{-1}$ . The data are consistent with past experimental results at smaller Rayleigh number (Park & Whitehead Reference Park and Whitehead1999). The transition to the geostrophic boundary layer regime starts at $Ro\approx 10^{-1}$ .

Figure 7.

(a) Boundary layer thickness near the cooled end (at $x/L=0.92$ , $y/W=0.5$ ) against Ekman number and (b) the boundary layer thickness normalised by the non-rotating scaling against Rossby number for three different $Ra_{F}$ . The three data points after the axis break show the measured boundary layer thicknesses in the non-rotating cases. In (a) the broken lines are the geostrophic boundary layer scaling (2.15) and the solid curves are that scaling with a correction for isopotential curvature (A 1 in appendix A), both with $c_{4}=1.96$ . The horizontal dotted line indicates the box height. In (b) the normalisation uses the non-rotating scaling (2.10) for $\unicode[STIX]{x1D6FF}_{0}$ fitted to the three non-rotating results, which give $c_{2}=2.0$ . The broken line is the normalised geostrophic boundary layer scaling (2.18) and the solid curves are that scaling with the isopotential curvature correction (A 2), all with prefactor $c_{6}=0.97$ . In (a) the non-rotating $Ra_{F}\approx 6\times 10^{14}$ (blue circle) case lies underneath the $Ra_{F}\approx 1\times 10^{14}$ (red triangle) case.

The boundary layer becomes more variable with increasing rotation rate. Figure 8 shows the normalised boundary layer thicknesses at three locations across the mid-section ( $x/L=0.5$ ) of the box. The boundary layer is asymmetric in the presence of rotation and the thickness trends depend on the cross-stream location. Near the ‘western’ wall (left-hand side when looking in the direction of net boundary layer flow; $y/W=0.027$ ) the boundary layer thickness remains unchanged by rotation until $Ro<10^{-3}$ , and at stronger rotation it becomes thinner. At the centre point of the box ( $y/W=0.5$ ) the boundary layer is thicker for stronger rotation if $Ro>10^{-3}$ but becomes thinner at $Ro<10^{-3}$ . Near the ‘eastern’ wall ( $y/W=0.973$ ) the thickness monotonically increases for stronger rotation, in a manner consistent with the geostrophic boundary layer scaling (2.18). Thus the boundary layer thickness is again largely consistent with the geostrophic boundary layer scaling at $10^{-3}<Ro<10^{-1}$ , but there is more complex behaviour at $Ro<10^{-3}$ .

Figure 8.

Boundary layer thickness on the mid-section of the domain ( $x/L=0.5$ ) at (a) $y/W=0.027$ , (b) $y/W=0.5$ and (c) $y/W=0.973$ . Boundary layer thickness is normalised by the non-rotating scaling (2.10) fitted to the non-rotating results (the points after the axis break), which give $c_{2}=1.7$ for all cases. The solid line is the geostrophic boundary layer scaling (2.18) with prefactors (b) $c_{6}=1.26$ and (c) $c_{6}=1.09$ . The averaged thickness across the width at $x/L=0.5$ is shown in figure 11(c).

6.2 Nusselt number

In the case of imposed heat flux the (inverse) Nusselt number (2.2) serves as the dimensionless measure of temperature differences in the flow. The Nusselt number is measured as the time-averaged temperature difference $\unicode[STIX]{x0394}T$ between the heated (where $T_{h}$ was from the single functioning thermistor) and cooled (where $T_{c}$ was the mean of two thermistors) regions of the plate. The Nusselt number, shown normalised by non-rotating values for each Rayleigh number in figure 9, is smaller for smaller heat flux or more rapid rotation, and hence for smaller Rossby number. However, the dependence is weaker than predicted by the geostrophic boundary layer scaling (2.18). Across the full range of conditions achieved, the dimensionless temperature difference required to drive the imposed heat flux increased by only 20 %. A similar trend was found by Park & Whitehead (Reference Park and Whitehead1999), over a small range of Coriolis parameter, and it was argued that the simple geostrophic boundary layer scaling requires a correction to account for the volume and heat transport in the Ekman layer (which we extend to the present Rossby number expressions in appendix B). The Ekman transport correction is negligible at very small $Ro$ and becomes large when the Ekman layer approaches the thickness of the thermal boundary layer (at $Q\approx 1$ or $Ro\approx 10^{-1}$ ) and there is no longer a significant geostrophic part of the boundary layer. Thus in figure 9 the Ekman corrected geostrophic boundary layer scaling (B 1) is shown up to $Ro\approx 5\times 10^{-2}$ beyond which its formulation is invalid. The correction provides an estimate of the expected flow in transitional conditions (the weak rotation regime of Hignett et al. (Reference Hignett, Ibbetson and Killworth1981)) between the non-rotating regime and the strong rotation (geostrophic) regime. The present results place the transitional regime at $10^{-2}<Ro<10^{-1}$ , although the magnitude of the Ekman correction to the scaling remains significant (compared with the data uncertainties) for $Ro<10^{-3}$ . A correction for thickening of the thermal boundary layer toward the end of the box (figure 7) resulting from the curvature of isopotential surfaces decreases the predicted Nusselt number (as $Nu\sim \unicode[STIX]{x1D6FF}^{-1}$ averaged over the area of the cooling boundary). The results show that the dynamics in the laboratory basin follows the geostrophic boundary layer scaling over only a small range of Rossby numbers, within $10^{-3}<Ro<10^{-2}$ .

Figure 9.

Nusselt number as a function of Rossby number, where the Nusselt number is based on $\unicode[STIX]{x0394}T$ and normalised by the non-rotating scaling (2.10) fitted to the three non-rotating results (the points after the axis break), which gives $c_{3}=0.47$ . Both the broken curve and the dotted curve are the geostrophic boundary layer scaling (2.18) with different prefactors; the unbroken black curve is the geostrophic boundary layer scaling corrected for Ekman transport ((B 1); Park & Whitehead Reference Park and Whitehead1999); the coloured solid curves for three different $Ra_{F}$ are the stress-free geostrophic boundary layer scaling corrected for effects of curvature of isopotential surfaces (A 4). The prefactor $c_{7}=1.6$ is used for all rotating scaling, excepting the dotted curve geostrophic boundary layer scaling (2.18) which has $c_{7}=1.45$ . Error bars are dominated by the heat flux uncertainty, which is shown as the fraction (at most 4 %) of heat input lost from the box to the room and is estimated from calibrations of the heat loss (Stewart et al. Reference Stewart, Hughes and Griffiths2011). Since the error is always a heat loss, the bars only extend downward. The vertical and horizontal axes are both logarithmic.

The Nusselt number results become largely independent of Rossby number at $Ro<10^{-3}$ , meaning that the temperature difference required to drive the imposed heat flux was approximately constant at larger rotation rates. For the largest heat flux the interior temperature was even observed to be smaller at the largest rotation rate (hence $Nu/Nu_{0}$ is larger) compared with the second largest $f$ , as is also seen from the profiles in figure 6. This strong deviation of the Nusselt number from the geostrophic boundary layer scaling is in the opposite direction from Ekman transport or isopotential curvature effects and inhibits the net transport of heat. These data are, on the other hand, consistent with the behaviour predicted for the inertial chimney regime (2.24), (2.25), specifically the absence of a dependence of Nusselt number on rotation rate. An estimate of the prefactor in (2.24) from the data is $c_{9}\approx 2\times 10^{-2}$ . The deviation from the geostrophic boundary layer scaling at $Ro\approx 10^{-3}$ is consistent with the sudden change in behaviour of the boundary layer thickness (figure 8), which decreased rapidly with further increases in $Ro$ across at least the ‘western’ half of the box width at the mid-section, leaving only the ‘eastern’ portion of the boundary layer continuing to increase in the manner predicted by the geostrophic boundary layer scaling.

6.3 Thermal wind and boundary layer transport

For the rotating cases with $Ro<10^{-1}$ a geostrophic balance is assumed to dominate much of the flow field, especially in regions with a strong vertical buoyancy gradient that inhibits vertical motion and sustains quasi-horizontal motion. Geostrophy may break down where convection drives strong vertical motion, in strong vortices having large relative vorticity, and at the small scales of turbulence, for which a local Rossby number is large. Thus the larger scales of the flow in the thermal boundary layer through the mid-section of the box ( $x/L=0.5$ ) are expected to be closely approximated by geostrophic balance and we use the thermal wind equation to find the geostrophic velocities and the net boundary layer transport through this section. This transport serves as a measure of net overturning transport given that all of the boundary heat input to the box is on one side of this section, and all of the heat withdrawal is on the other side.

There were five thermistors at $x/L=0.5$ spanning the cross-stream $y$ -direction. The thermal wind equation (2.12) between two adjacent temperature profiles a distance $\unicode[STIX]{x0394}y$ apart gives

and profiles of the velocity component $u$ in the $x$ -direction were calculated from

where $z_{0}$ is a reference level. A suitable reference level is one for which we have the best approximation of the velocity $u(z_{0})$ . The no-slip condition at the base was not useful because temperature profiles stopped at a distance $0.01H$ (2 mm) from the base, which was greater than the Ekman layer thickness ( $\unicode[STIX]{x1D6FF}_{E}<0.01H$ for $Ro<10^{-1}$ ). Another option could be to assume a velocity reversal at the top edge of the thermal boundary layer, which may apply on average, but a reversal is not required at each location in the presence of horizontal recirculation and the resulting thermal wind velocities were not consistent with velocity fields obtained from particle tracking velocimetry (see § 7.1). Hence the PTV measurements of horizontal velocities were used to reference the thermal wind profiles, choosing the PTV velocity field closest to the base (in the plane $z/H=0.15$ ; see § 7.1), and to obtain the best estimate of the boundary layer transport. The calculated geostrophic velocities at $x/L=0.5$ , shown in figure 10 for the largest Rayleigh number and three rotation rates, thus include a reference velocity $u(z_{0}=0.15H)$ . The time-averaged temperature profiles were used. The three-dimensional results are complex. For example, the time-averaged geostrophic velocities close to the base near the ‘eastern’ side wall ( $y/W=0.903$ ) change from negative (toward to heated end) for weak rotation (figure 10 a), to positive for strong rotation (figure 10 b), and then back to negative for extreme rotation (figure 10 a). The velocities near the ‘eastern’ and ‘western’ walls tend to be of opposite sign through most of the depth of the box, but on average across the transect the boundary layer velocities are negative and the interior velocities are positive. There is generally no indication of a decrease in velocity towards zero at the base, consistent with the Ekman layer lying entirely within the region for which there are no data.

Figure 10.

Vertical profiles of horizontal geostrophic velocity $u$ through the mid-section of the domain ( $x/L=0.5$ ) assuming thermal wind balance for $Ra_{F}\approx 6\times 10^{14}$ and three different rotation rates: (a) $f=0.04~\text{s}^{-1}$ , $Ro=1.7\times 10^{-1}$ , (b) $f=0.4~\text{s}^{-1}$ , $Ro=5.6\times 10^{-3}$ and (c) $f=1.6~\text{s}^{-1}$ , $Ro=7.1\times 10^{-4}$ . Locations in the key are half-way between the thermistor profiles. In (c) only four thermistors worked, giving only three velocity profiles. Positive $u$ indicates movement from the heated end to the cooled end. Uncertainty bars are propagated from measurement errors $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x0394}T}=0.02\,^{\circ }\text{C}$ , $\unicode[STIX]{x1D70E}_{\unicode[STIX]{x0394}y}=0.005$  m and $\unicode[STIX]{x1D70E}_{z}=0.001$  m. It is uncertain whether thermal wind approximates the horizontal velocity field in the interior above the cooled boundary, as vertical and horizontal velocities may be comparable there; nonetheless the computed profiles are shown through the full depth.

The geostrophic volume transport through each segment, $\unicode[STIX]{x1D6F9}_{i}$ , was found by integrating the thermal wind velocity over the (local) boundary layer depth and multiplying by the separation distance $\unicode[STIX]{x0394}y_{i}$ between each thermistor pair,

where $i=1,n$ is the segment number and $\unicode[STIX]{x1D6FF}_{i}$ is the average boundary layer thickness in each segment. The net boundary layer transport through $x/L=0.5$ is the sum of the segment transports $\unicode[STIX]{x1D6F9}=\sum _{i=1,n}\unicode[STIX]{x1D6F9}_{i}$ . The thermistors closest to the side walls were at $y=0.0027W$ and $y=(1{-}0.0027)W$ (ie. 0.8 mm from the walls) and the transport in this region was not included. However, this neglects only a very small fraction of the total width and much of this is expected to be within a viscous side wall boundary flow with zero velocity at the walls.

The net transport can be written as $\unicode[STIX]{x1D6F9}=U_{av}\unicode[STIX]{x1D6FF}_{av}W$ , where the average boundary layer thickness $\unicode[STIX]{x1D6FF}_{av}$ across the box width is given by

The transport in this problem is expressed in terms of the volume flux in preference over a volume flux per unit width of the box (as commonly used for the non-rotating horizontal convection problem) in view of both the three-dimensionality of the circulation and a possible dependence on the aspect ratio $W/L$ (which is not investigated in the experiments). The average velocity $U_{av}$ across the section is $U_{av}=\unicode[STIX]{x1D6F9}/\unicode[STIX]{x1D6FF}_{av}W$ , after taking $W\approx \sum _{i=1,n}\unicode[STIX]{x0394}y_{i}$ .

Figure 11.

(a) The net volume transport (6.3) and (b) average geostrophic velocity in the $x$ -direction calculated from the thermal wind equation for the mid-section of the box for $Ra_{F}\approx 6\times 10^{14}$ , and (c) average boundary layer thickness (6.4) across the box at the mid-section ( $x/L=0.5$ ) for three Rayleigh numbers. The non-rotating transport used to normalise $\unicode[STIX]{x1D6F9}$ in (a) is given by (2.11) with numerical coefficient set to one given that the actual coefficient has not been evaluated. The velocity in (b) is normalised by the non-rotating scaling (2.10), with $c_{1}=0.29$ taken from past results (Mullarney et al. Reference Mullarney, Griffiths and Hughes2004), as the thermal wind balance cannot be used for the non-rotating case. The boundary layer thickness in (c) is normalised by the non-rotating scaling (2.10) fitted to the three non-rotating experiments (the data points after the axis break), which give $c_{2}=1.7$ . The solid lines are the geostrophic boundary layer scaling ((2.19) and (2.18)) with (a) $c_{8}=0.44$ , (b) $c_{5}=0.80$ and (c) $c_{6}=1.09$ .

The calculated thermal wind transport and average velocities in the boundary layer, and the average boundary layer thickness at the mid-section of the domain are shown in figure 11, where the uncertainties include propagation of the uncertainties from the thermal wind profiles (coloured error bounds on figure 10) and allowances for the missing regions at the base and at the side walls. The transports include a correction using a simple linear interpolation of the velocity profiles to the base from its value at $z/H=0.01$ to zero at the boundary. The uncertainty estimate includes the difference in transport between this assumption and an alternative simple linear extrapolation of the velocity to a maximum at the edge of the Ekman layer (at $\unicode[STIX]{x1D6FF}_{E}=LE^{1/2}$ ) and a linear decrease through the Ekman layer to zero velocity at the base. In most cases these extrapolations add to the net transport and therefore the uncertainty is asymmetric, the upper error bars in figure 11(a,b) being larger than the lower bars. The result at the smallest Rossby number has a larger uncertainty because this case had the thinnest Ekman layer and the temperature profiles do not extend as close to the expected height of the maximum velocity. All of the results are reasonably described by the geostrophic boundary layer scaling in the range $10^{-3}<Ro<10^{-1}$ . As foreseen from figure 11, the average boundary layer thickness at this mid-section deviates strongly from the scaling at $Ro<10^{-3}$ . However, the data for geostrophic transport and velocity were obtained only for a sequence of runs with the largest heat flux and are not sufficient to determine the behaviour of the average boundary layer velocity and transport at very small $Ro$ .

7.1 Interior horizontal velocities

The PTV velocity fields obtained at 1 s intervals during the thermally equilibrated state indicated large-amplitude, low frequency fluctuations consistent with those inferred qualitatively from the dye tracer observations. Time-averaged velocity fields were obtained by averaging the instantaneous fields over periods of 20, 60 and 180 min after tracer particles were introduced and the results were approximately independent of the averaging period. As an additional test of whether the length of the averaging period was sufficient, one experiment (with $Ra_{F}=6.5\times 10^{14}$ and $Ro=5.6\times 10^{-3}$ ) was repeated with a new fill of the tank and the two time-averaged velocity fields (from 3 h periods) taken at mid-depth were compared. To quantify the standard deviation between the two time-averaged velocity fields at mid-depth, we calculate the spatial average of the difference between the two velocity fields $\overline{u_{1}}(x,y)$ and $\overline{u_{2}}(x,y)$ as a fraction of the maximum speed in the $u_{2}$ field,

where $i,j$ are the PTV windows in the field. The data give $\unicode[STIX]{x1D70E}_{m}=0.1$ , meaning that the two fields match to within 10 %. Thus we confidently use averages over 3 h (and 10800 frames) as faithful representations of the time-averaged horizontal flow.

For ‘weak rotation’ ( $Ro=1.7\times 10^{-1}$ ) a weak and basin-scale cyclonic gyre (having circulation in the same sense of the anticlockwise background rotation) occupied much of the domain, as shown in figure 12(a) (where the $u$ component at mid-depth is plotted along with contours of the two-dimensional horizontal streamfunction derived from the horizontal velocity field). At mid-depth the mean velocities on the ‘western’ side of the basin ( $y/W<0.5$ ) in this gyre were largely in the opposite direction to those in the underlying thermal boundary layer (where they are largely negative, figure 10 a), whereas on the ‘eastern’ side of the gyre the velocities were in the same direction as the boundary layer. However, when averaged across the box width at the mid-section of the box, the net flow was towards the heated end (in the same direction as the boundary layer transport). This became a more uniform motion toward the end wall in the region $x/L<0.2$ . A nearly uniform velocity $u$ tended to be maintained up to very close to the end wall, consistent with a significant vertical convection in a narrow region against the end wall. There was a second but much smaller region of cyclonic circulation above the ‘western’ corner of the heated region. The magnitude of the velocities at this Rossby number was everywhere less than $0.1fL$ , and not much smaller than the calculated geostrophic boundary layer speeds (figure 10 a). These observations are largely consistent with the (substantially less accurate) thermal wind calculations for the interior flow in figure 10(a). They are also similar to the circulation at mid-depth in the non-rotating experiments and in corresponding direct numerical simulations of the non-rotating case (Gayen et al. Reference Gayen, Griffiths and Hughes2014). At this moderately large value of the Rossby number, the internal radius of deformation characterising the boundary layer flow, $\unicode[STIX]{x1D706}\approx (g\unicode[STIX]{x1D6FC}\unicode[STIX]{x0394}T\unicode[STIX]{x1D6FF})^{1/2}/f$ , or the inertial radius $\unicode[STIX]{x1D706}=U/f$ , is of the same order as the domain width ( $\unicode[STIX]{x1D706}/W=RoL/W\approx 0.7$ ). Hence this natural length scale is not available to determine the size of geostrophically balanced flow structures.

Figure 12.

Velocity component $u$ in the $x$ -direction measured by PTV at mid-depth ( $z/H=0.5$ ) and contours of calculated streamfunction for $Ra_{F}=6.2\times 10^{14}$ and $Ro=1.7\times 10^{-1}$ ( $f=0.04~\text{s}^{-1}$ ): (a) time averaged ( $\overline{u}$ ) over 3 h and (b) an instantaneous field. The velocities are normalised by $fL$ (scale shown by colour bar); positive velocities (red) indicate flow to the right toward the cooled region. The streamfunction contours have equal spacing $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=4\times 10^{-5}~\text{m}^{3}~\text{s}^{-1}$ per unit depth: white contours indicate cyclonic (anticlockwise) circulation and grey is the zero contour.

In contrast to the flow at mid-depth, the time average near the top of the domain ( $z/H=$ 0.85) shows a net flow through the mid-section directed towards the cooled end. This is similar to the return flow that closes the overturning circulation through the upper half of the depth in the non-rotating case. Adapting the measure defined in (7.1) to compare the time-averaged velocity fields at different depths we set $\overline{u_{1}}=\overline{u}(z/H=0.85)$ and $\overline{u_{2}}=\overline{u}(z/H=0.5)$ and replace $\unicode[STIX]{x1D70E}_{m}$ by $\unicode[STIX]{x1D70E}_{z}$ . We find $\unicode[STIX]{x1D70E}_{z}=0.7$ for $Ro=1.7\times 10^{-1}$ . Hence the velocities in the upper and mid-depth planes show that the flow varied substantially with height, although less so than that in the non-rotating case (in which we find $\unicode[STIX]{x1D70E}_{z}=1.5$ ). The variability in the velocity field under these conditions is a significant fraction of the time average, but the general characteristics of the circulation described above reflect the instantaneous flow fields at nearly all times, as in the instantaneous field shown in figure 12(b). In particular there was no evidence of vortical plumes extending through the height of the box.

At a small value of the Rossby number ( $Ro=5.6\times 10^{-3}$ ), for which the boundary layer data in § 6 place the flow in the geostrophic boundary layer regime, the interior circulation was more substantially influenced by rotation: there were five basin-wide gyres along the length of the domain (figure 13 a). The time-averaged flow field, shown here for the PTV plane at mid-depth, in this ‘strong rotation’ case was much the same at all three PTV levels ( $\unicode[STIX]{x1D70E}_{z}=0.13$ ). The mean velocity fields were also mostly consistent with the time-averaged velocities calculated from thermal wind balance in the interior (figure 10 b) (although there were several discrepancies that may relate to uncertainties in the thermal wind calculation with very small horizontal temperature differences). Above the cooled region of the base there was a very weak anticyclonic gyre (indicated here only by the grey $\unicode[STIX]{x1D6F9}=0$ contour) that was largely overshadowed by two stronger cyclonic gyres. Above the heated region of the base there was a weak cyclonic region at the end of the box and a strong anticyclonic gyre, the latter having circulation comparable to the adjacent cyclonic gyre above the cooled base. Thus the mean horizontal circulation was concentrated in the central half of the basin, nearest the imposed gradient of thermal boundary conditions.

Figure 13.

Velocity component $u$ from PTV at mid-depth and horizontal streamfunction contours for $Ra_{F}=6.5\times 10^{14}$ and $Ro=5.6\times 10^{-3}$ ( $f=0.4~\text{s}^{-1}$ ): (a) time-averaged over 3 h and (b–d) instantaneous fields (time interval from (b) to (c) is 3.7 min; (c) to (d) is $16.1$  min). Velocities are normalised by $fL$ ; note that velocity colour scale is an order of magnitude smaller than in figure 12. Streamfunction contour spacing is $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=4\times 10^{-5}~\text{m}^{3}~\text{s}^{-1}$ per unit depth: white contours indicate cyclonic (anticlockwise) circulation, grey is the zero contour and black is anticyclonic circulation.

Instantaneous interior velocity fields at this small Rossby number (figure 13 b–d) show that the flow above the heated base experienced larger fluctuations compared to the ‘weak rotation’ case of figure 12. The strong anticyclonic gyre varied in shape and maximum transport, but was always present. The weak cyclonic gyre near the heated end of the basin was more variable, shifting its location and often splitting into two (cyclonic and anticyclonic) parts. Vertical convection (as seen with tracer advection in figure 4 and with the advection of fluorescein dye tracer in other runs with the PTV sheet illumination) can be linked to many of the smaller structures, which were also relatively short lived. In this experiment the Rossby radius of deformation was much smaller than the domain width ( $\unicode[STIX]{x1D706}/W\approx 0.02$ ) and could influence the size of flow structures, whether they were eddies formed by baroclinic instability or vortical plumes generated by vertical convection events. Above the cooled base the flow was less variable, although there was some fluctuations of the gyre structures.

At very strong rotation ( $Ro=7.1\times 10^{-4}$ ) the time-averaged velocity fields from PTV (shown for mid-depth in figure 14 a) show only two dominant gyres. The flow was again approximately independent of depth ( $\unicode[STIX]{x1D70E}_{z}=0.07$ ) and mostly consistent with the interior thermal wind results of figure 10(c). Circulation above the cooled half of the base was again predominantly cyclonic, but in this case formed a single gyre having larger lateral extent and smaller transport. The largest time-averaged interior velocities were associated with a large, strong anticyclonic gyre centred at $x/L\approx 0.4$ above the heated base but extending across the mid-section of the box. This anticyclonic gyre is therefore expected to provide an important contribution to the measured change in Nusselt number trend, as this mean flow will tend to transport heat more directly than the time-averaged flow pattern in figure 13. The mean flow was relatively weak near the end of the box above the heated base, consistent with the hypothesis that chimney convection above a large area of the heated base shifts the site of vertical heat transport away from an end wall plume and into an ‘open ocean’ chimney, despite the uniform heat flux boundary condition.

Figure 14.

Velocity component $u$ from PTV at mid-depth and horizontal streamfunction contours for $Ra_{F}=6.8\times 10^{14}$ and $Ro=7.1\times 10^{-4}$ ( $f=1.6~\text{s}^{-1}$ ): (a) time-averaged velocity over 3 h and (b–d) instantaneous fields (time interval from (b) to (c) is 5.2 min; (c) to (d) is 9.3 min). Velocities are normalised by $fL$ ; note the velocity colour scale is $1/4$ of that used in figure 13; streamfunction contour spacing is $\unicode[STIX]{x0394}\unicode[STIX]{x1D713}=4\times 10^{-5}~\text{m}^{3}~\text{s}^{-1}$ per unit depth.

Instantaneous PTV velocity fields illustrated in figure 14(b–d) show that the strong anticyclonic gyre at such rapid rotation was present at all times, although it fluctuated very substantially in position, size and maximum transport (the transport varying in time by almost 50 %). Associated with these fluctuations were large changes in a transient, even more variable, cyclonic gyre close to the end wall. The fluctuations were of relatively high frequency, with time scales of a few minutes or 40–60 inertial periods. Smaller structures in the velocity field, such as the two or three small cyclonic eddies and two small anticyclonic eddies in figure 14(b), are consistent with a snap shot of vortical plumes penetrating through the depth of the box and the very small deformation radius ( $\unicode[STIX]{x1D706}/W\sim 2\times 10^{-3}$ ). The greater variability and smaller eddy scales at smaller values of the Rossby number are expected to contribute to lateral heat transfer in the chimney regime.

7.2 Overturning transport from PTV

An overturning transport was defined in § 6.3 as the net volume transport through the mid-section $x/L=0.5$ in the boundary layer and was estimated from temperature profiles assuming thermal wind balance. An independent estimate of the overturning is obtained from the interior velocity measurements, specifically from the horizontal divergence of velocity on any of the three PTV measurement planes. From the continuity equation (2.4) the vertical velocity $w$ and horizontal divergence satisfy

and invoking the Taylor–Proudman theorem in the interior for strong rotation, giving ( $u$ , $v$ ) independent of depth, equation (7.2) is integrated to find the velocity

where $z_{0}$ is a reference height. At the edge of the Ekman layer on the upper boundary, $z_{E}=H-\unicode[STIX]{x1D6FF}_{E}=H-\sqrt{2\unicode[STIX]{x1D708}/f}$ and $w(z_{E})$ is the Ekman pumping velocity

Hence we take $z_{0}=z_{E}$ at the upper boundary (which avoids the stratified bottom boundary layer) and (7.3) becomes

Figure 15.

Overturning volume transport calculated from the horizontal divergence of the horizontal velocity fields from PTV at three vertical levels, normalised by the non-rotating scaling (2.11) with prefactor of one. The line shows the geostrophic boundary layer scaling (2.19) with prefactor $c_{8}=0.44$ and is the same line as in figure 11(a).

The vertical velocity was computed using (7.5) and the time-averaged horizontal velocity field on each of the three PTV measurement planes ( $z/H=0.15$ , 0.5 and 0.85) for each experiment at the largest heat flux ( $Ra_{F}\approx 6\times 10^{14}$ ). The positive values of $w$ were integrated over the area of the plane to give a total upward transport $\unicode[STIX]{x1D6F9}_{pos}=\int _{0}^{W}\!\!\int _{0}^{L}w_{pos}(z)\,\text{d}x\,\text{d}y$ and separately the negative values of $w$ were integrated to give a total downward transport $\unicode[STIX]{x1D6F9}_{neg}=\int _{0}^{W}\!\!\!\int _{0}^{L}w_{neg}(z)\,\text{d}x\,\text{d}y$ . On figure 15 are plotted the average of the two $\unicode[STIX]{x1D6F9}_{vert}$ , and any difference between them is shown as an estimate of the uncertainty. In order to allow comparison with the boundary layer transport from thermal wind calculations (figure 11 a), $\unicode[STIX]{x1D6F9}_{vert}$ is again normalised by the scaling for the boundary layer transport in the non-rotating case (2.11). The values obtained from PTV on the upper plane are within 10 % of the boundary layer estimates, while those at mid-depth are approximately twice as large, consistent with a significant fraction of the vertical transport passing laterally from one end to the other at levels below $z/H=0.85$ , or an increased amount of local vertical motion both upwards and downwards. Much larger vertical transports are estimated from PTV on the lower plane and this is attributed to some of the boundary layer being included in that plane of illumination, which can cause the Taylor–Proudman approximation assumed in the method to break down. The depth dependence of horizontal velocities in the interior as indicated by the thermal wind calculations in figure 10, suggests that the approximation may already be inaccurate in the interior. However, the vertical overturning results are again reasonably described by the geostrophic boundary layer scaling (2.19) at all vertical levels. At mid-depth the results give a prefactor for (2.19) of $c_{8}\approx 0.75$ .

A further independent estimate of the overturning transport, which would measure the same quantity as the net boundary layer transport, could in principle be obtained by integrating the $u$ component of velocity from PTV over the area of the vertical mid-section $x/L=0.5$ . However, this required an extrapolation of the time-averaged $u$ fields measured at $z/H=0.85$ and $z/H=0.5$ , through the depth of the fluid outside the boundary layer, given a contamination of the velocities at $z/H=0.15$ by boundary layer flow. The uncertainties in the method proved too large for it to be of use.