2.1 Apparatus, set-up and flow measurement

The spin-up experiments were performed in a sliced circular cylinder mounted centrally on a variable-speed turntable. The transparent cylinder was of acrylic construction with an internal radius $L=17~\text{cm}$ , a mean depth $H=23~\text{cm}$ and a base slope $\unicode[STIX]{x1D6FC}=0.175~\text{rad}$ (i.e.  $\unicode[STIX]{x1D6FC}=10^{\circ }$ ). A schematic view of the set-up is shown in figure 1. The sliced bottom of the cylinder was sealed by a rigid base plane and its horizontal top by a removable transparent lid. The angular frequency of the turntable could be adjusted continuously between 0 and $1.2~\text{rad}~\text{s}^{-1}$ . In all experiments reported here the table’s rotation was in the anticlockwise direction.

Each experiment was set up by first filling the cylinder with a linearly stratified salt-water solution using the standard double-bucket technique (Economidou & Hunt Reference Economidou and Hunt2009). The salt used was NaCl and the buoyancy frequency defined as $N=[g(\unicode[STIX]{x1D70C}_{0}-\unicode[STIX]{x1D70C}_{H})/\unicode[STIX]{x1D70C}_{H}H]^{1/2}$ , where $\unicode[STIX]{x1D70C}_{H}$ and $\unicode[STIX]{x1D70C}_{0}$ denote the fluid’s density at the free surface and at the centre of the base plane, respectively. The standard double-bucket technique can be used to set-up a linear density stratification in a container of uniform depth and uniform cross-section. Hence, to apply this method to a sliced cylinder the following approach was used. The inclined base was mounted inside an ‘outer’ rectangular container (of $35~\text{cm}\times 39~\text{cm}$ cross-section and of uniform depth 31 cm) with its lower edge 2 cm above the base of the container, and the sliced-cylinder section initially suspended inside the outer container, 1 cm above the inclined base. The double-bucket technique was used to gradually fill the outer container from below (so that the density of the fluid entering the container gradually increased). As the outer container filled, the fluid was free to flow over the upper and lower edges of the inclined base and to enter the suspended cylinder. This produced a salt-water solution with a linear density gradient. The cylinder was then carefully lowered onto the inclined base, and fixed into position with its vertical axis aligned with the table’s rotation axis. Once filled, the transparent lid was fitted, which was designed to completely displace the fluid’s free surface to prevent the entrapment of air pockets on the lid’s underside. The table was then brought slowly, from rest, into rotation and its angular frequency incrementally increased over a period of at least 5 hours to the required initial value $\unicode[STIX]{x1D6FA}-\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}$ , and then left for at least 16 hours to allow the fluid to reach a state of near-solid-body rotation. The experiment was started (at time $t^{\ast }=0$ ) by instantaneously increasing the table’s rotation rate to the final angular frequency $\unicode[STIX]{x1D6FA}$ . We note that buoyancy frequency $N$ was measured prior to spin-up and after use (following a very gradual spin-down of the container) by using a syringe fitted with a long pipetting needle (as outlined in § 3.3 of Economidou & Hunt Reference Economidou and Hunt2009) to sample the fluid at ten equally spaced incremental heights between the centre of the base slope and the free surface. There was little difference in the measured ‘before’ and ‘after’ values of $N$ .

The key dimensionless parameters for each experiment are the Burger ( $S$ ), Ekman ( $E$ ) and Rossby ( $\unicode[STIX]{x1D716}$ ) numbers, defined here as

(2.1a-c ) $$\begin{eqnarray}\displaystyle S=\left(\frac{N}{\unicode[STIX]{x1D6FA}}\right)^{2},\quad E=\frac{\unicode[STIX]{x1D708}}{\unicode[STIX]{x1D6FA}L^{2}},\quad \unicode[STIX]{x1D716}=\frac{\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}}{\unicode[STIX]{x1D6FA}}. & & \displaystyle\end{eqnarray}$$

For the range of salinities used, the corresponding Schmidt number was $Sc\approx 670$ (Munro, Foster & Davies Reference Munro, Foster and Davies2010) and so molecular diffusion of salinity was negligible. The aspect ratio $h=H/L$ was throughout fixed at $h=1.35$ . Here we present results from five experiments. In all cases, the Ekman number was small with $E=O(10^{-4})$ , or less. Under these conditions, the structure and decay rate of the Rossby waves was determined by the magnitude of the Burger number, which was varied between $S=0.21$ and $S=20$ , by changing $\unicode[STIX]{x1D6FA}$ (as outlined in table 1) but also by changing $N$ . The Rossby number was kept fixed at $\unicode[STIX]{x1D716}=0.02$ ; this is as small as the system would allow for the larger values of $S$ considered here. Table 1 lists the key parameters for the five experiments reported here, which are henceforth labelled A–E. We note that the values of angular frequency given in table 1 are those set via the digital drive used to control the turntable’s motor. The set angular frequency (both initial and final) was always verified by taking several independent measurements of the rotation period; the difference between the measured and set values of angular frequency was small and always less than 0.1 %.

Table 1.

The key parameters for the experiments, where $S=(N/\unicode[STIX]{x1D6FA})^{2}$ , $\unicode[STIX]{x1D716}=\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}/\unicode[STIX]{x1D6FA}$ and $E=\unicode[STIX]{x1D708}/\unicode[STIX]{x1D6FA}L^{2}$ . Also given are values of the vertical length scale $S^{-1/2}L$ , normalized by the container’s mean depth $H$ (namely, $S^{-1/2}h^{-1}$ ).

van de Konijnenberg et al. (Reference van de Konijnenberg, Naulin, Rasmussen, Stenum and van Heijst2000) reported a series of experiments on the spin-up of a homogenous fluid in a sliced circular cylinder, and highlighted how the observed flow was dependent on the comparative magnitudes of the Rossby number ( $\unicode[STIX]{x1D716}$ ) and the base slope ( $\unicode[STIX]{x1D6FC}$ ). That is, in experiments performed with $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6FC}\geqslant O(1)$ the flow was dominated by the nonlinear separation of the sidewall boundary layer and subsequent shedding of cyclonic vorticity into the interior region. However, an experiment performed with $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6FC}\approx 0.04$ – corresponding to when the time scale associated with the formation of Rossby waves is small compared to the time scale associated with vortex shedding – showed clear evidence of Rossby-wave formation, with measurements in good agreement with the analytical results of Pedlosky & Greenspan (Reference Pedlosky and Greenspan1967). Therefore, in an effort to limit the effects associated with sidewall separation, we have $\unicode[STIX]{x1D716}/\unicode[STIX]{x1D6FC}\approx 0.1$ in all experiments reported here.

The flow was visualized and measured using particle image velocimetry (PIV). To facilitate the use of this technique, small tracer particles were suspended within the fluid column, which were added as the cylinder was filled and allowed to settle freely into suspension. A table-mounted light box was used to project a narrow light sheet through the cylinder’s sidewall to illuminate the tracer particles within a horizontal plane at a height $z_{0}^{\ast }$ above the centre of the base slope (see figure 1). The experiments listed in table 1 were first performed with $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ . So that the key height-dependent features of the flow could be identified, each experiment was then repeated, under near-identical conditions, with $z_{0}^{\ast }=5~\text{cm}\approx 0.2H$ (note that the top level of the base slope corresponds to height $0.14H$ ). We were unable to take measurements at heights below $0.2H$ due to optical effects caused by the light sheet illuminating tracer particles that had settled onto the slope’s surface.

A co-rotating digital camera was mounted on the turntable and positioned to point vertically down into the cylinder’s interior (see figure 1), and used to record the particle motion within the illuminated horizontal plane, in the frame of reference that rotates with the turntable. The images were recorded at a frame rate of 10 Hz (at $1280\times 1024$ pixel resolution) and processed at the end of the experiment in terms of the co-rotating coordinates $(x^{\ast },y^{\ast },z^{\ast })$ , which are defined relative to the centre of the base plane, as shown in figure 1. The surface of the base plane is defined by $z^{\ast }=x^{\ast }\tan \unicode[STIX]{x1D6FC}$ . To relate these coordinates to the standard $\unicode[STIX]{x1D6FD}$ -plane convention (for anticlockwise rotation), we shall also refer to the up-slope $x$ -direction as north, and the long-slope $y$ -direction as west (see figure 1). Measurements of the horizontal velocity components $(u^{\ast },v^{\ast })$ and vertical vorticity field $\unicode[STIX]{x1D714}^{\ast }=v_{x^{\ast }}^{\ast }-u_{y^{\ast }}^{\ast }$ , in the illuminated plane at $z^{\ast }=z_{0}^{\ast }$ , were obtained by applying a standard PIV algorithm to the images (Dalziel Reference Dalziel2006). The PIV calculations were performed on consecutive images using square interrogation windows of $13\times 13$ pixels, with an 8-pixel spacing overlap, which corresponds to a physical spacing between velocity vectors of approximately 0.04 cm and a temporal resolution of 0.1 s. The corresponding streamfunction was calculated from the measured vorticity field using a Poisson solver. We used a SLOR (successive line over-relaxation) method for the computations. In the remainder of this section we will use contours of the measured streamfunction to describe the key features of the observed flows; typical examples of these streamlines are shown in figure 2 (see caption for details).

Figure 2.

Streamlines from experiment A ( $\unicode[STIX]{x1D6FA}=1.04~\text{rad}~\text{s}^{-1}$ , $S=0.21$ , $\unicode[STIX]{x1D716}=0.020$ ), measured at the near mid-height $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ . The dimensionless times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }$ at which the data were taken are (a) 0.18, (b) 3.2, (c) 6.8, (d) 12, (e) 24 and (f) 45. The top (north) and bottom (south) of the base slope have been indicated in (a). The streamfunction values shown on each contour are dimensionless, and have been scaled by $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}L^{2}$ .

2.2 Observations

We have already noted that homogeneous spin-up in a sliced cylinder differs greatly from spin-up in a regular cylinder ( $\unicode[STIX]{x1D6FC}=0$ ), which is a gradual decay of the axisymmetric initial condition on the time scale $E^{-1/2}\unicode[STIX]{x1D6FA}^{-1}$ (Greenspan & Howard Reference Greenspan and Howard1963). In the sliced-cylinder configuration, the vortex lines of the initial condition when advected down the slope are stretched, producing vorticity relative to the rotating reference frame. The result is the formation, on the $\unicode[STIX]{x1D6FC}^{-1}\unicode[STIX]{x1D6FA}^{-1}$ time scale, of alternating cyclonic and anticyclonic vorticity waves that propagate across the slope; these two-dimensional vorticity waves gradually fill the fluid’s interior and are the primary mechanism resulting in the fluid’s spin-up. Pedlosky & Greenspan (Reference Pedlosky and Greenspan1967) described these vorticity waves theoretically for the case $\unicode[STIX]{x1D716}=0$ – showing they are a combination of all excited Rossby modes – and van de Konijnenberg et al. (Reference van de Konijnenberg, Naulin, Rasmussen, Stenum and van Heijst2000) reported observations and measurements from an experiment with $\unicode[STIX]{x1D716}=0.01$ , $\unicode[STIX]{x1D6FC}=0.244~\text{rad}$ ( $14^{\circ }$ ) which were in good agreement with the theory.

Here we describe how the observed flow differs when the fluid’s density field is linearly stratified, over the range of Burger number ( $S$ ) listed in table 1 and for $\unicode[STIX]{x1D716}=0.02$ , $\unicode[STIX]{x1D6FC}=0.175~\text{rad}$ ( $10^{\circ }$ ). Under these conditions the key experimental observations can be summarised as follows. At small $S$ the flow is very similar to that observed in the homogeneous case, with spin-up accomplished as two-dimensional, alternating vorticity waves fill the fluid’s interior. For $S\approx 1$ or greater, the strength of the fluid’s vertical density gradient is now sufficient to inhibit vortex stretching. As a result, the vorticity waves that form have a three-dimensional structure and are confined to a layer above the slope, the height of which decreases with increasing $S$ . The vorticity waves are also damped more rapidly as $S$ is increased. Therefore, at increasingly large $S$ the slope-induced effects become confined to an increasingly shallower layer, with the bulk-interior flow above remaining largely unaffected by the presence of the slope. We now illustrate this description using the experimental data.

To begin we consider experiment A, corresponding to the case of a weak stratification ( $S=0.21$ ). Figure 2 shows streamline data from experiment A, taken at the near mid-height $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ and at times ranging from the start of the experiment through to when the fluid is practically spun-up. The streamlines show the structure of the alternating vorticity waves as they propagate across the slope. To aid the reader in differentiating between the cyclonic and anticyclonic vorticity waves, we have shown negative streamlines as broken lines (anticyclonic flow) and positive streamlines as solid lines (cyclonic flow). For the configuration used here – with anticlockwise background rotation and north corresponding to the top of the slope – the vorticity waves appear first in the east side of the container and propagate westwards across the slope. We also note that the data obtained under these conditions at the near-slope height $z_{0}^{\ast }=5~\text{cm}\approx 0.2H$ were little different from the data shown in figure 2 (and so are not shown here). The key point here being that at small- $S$ the flow remained primarily two-dimensional throughout, in agreement with observations previously reported for the homogeneous case (van de Konijnenberg et al. Reference van de Konijnenberg, Naulin, Rasmussen, Stenum and van Heijst2000).

Immediately after the start of the experiment, the flow was an axisymmetric, anticyclonic vortex of the form shown in figure 2(a). The first vorticity wave (which has cyclonic circulation) can be seen forming adjacent to the eastern sidewall in figure 2(b), as the axis of the initial anticyclone moved across the slope into the western half of the domain. The first vorticity wave is shown at a later time in figure 2(c) as it proceeds across the container’s slope towards the western sidewall. In figure 2(b,c) there is a noticeable north–south asymmetry to the flow, evident by the structure of cyclonic vorticity wave being not fully extended north–south across the container. This asymmetry was also evident, to a similar degree, in the data obtained at the near-slope height $z^{\ast }=0.2H$ . The vorticity waves continue to form at subsequent times, entering from the east and propagating westwards, each with opposite-sense circulation to the previous one. For example, figures 2(d) and 2(e) show, respectively, the ‘second’ and ‘fourth’ vorticity waves shortly after formation, occupying the eastern half of the container, both with anticyclonic circulation and both preceded by a cyclonic vorticity wave (i.e. the ‘first’ and ‘third’ waves). As time increases, more and more vorticity waves fill the domain and the flow approaches a near spun-up state. It is known for the homogeneous case that viscous corrections due to the top and bottom Ekman layers lead to decay of the form $\exp [-2(E^{1/2}/h)\unicode[STIX]{x1D6FA}t^{\ast }]$ (Pedlosky & Greenspan Reference Pedlosky and Greenspan1967). The corresponding spin-up time scale for experiment A is equivalent to $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }\approx 14$ , and so we expect the effects of Ekman suction to start becoming significant for times of the order or greater than this. There was no evidence of vorticity waves forming at times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }\gtrsim 34$ , and thereafter the residual relative flow took the form of a weak, slowly decaying, asymmetric anticyclone, as shown in figure 2(f).

We also note the presence of the small cyclonic vortex in the northwest perimeter region in figure 2(d–f). This is not a vorticity wave but is the result of vortex shedding from the western sidewall boundary layer. The time scale for shedding is of the order $\unicode[STIX]{x1D716}^{-1}\unicode[STIX]{x1D6FA}^{-1}$ , which was chosen here to be large in comparison to the time scale on which the Rossby waves form, which is of the order $\unicode[STIX]{x1D6FC}^{-1}\unicode[STIX]{x1D6FA}^{-1}$ : specifically, $\unicode[STIX]{x1D6FC}/\unicode[STIX]{x1D716}\approx 9$ in all experiments, and so the onset time of shedding, if it occurs, corresponds to $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }\approx 12$ (as is the case figure 2). Under these conditions the shed vortex has a limited effect on the flow, and little if any effect on the initial Rossby waves, which consist of the largest-scale modes. The key effect – evident in figure 2(d,e) – is that the formation and shedding of this vortex resulted in a slight increase in the degree of north–south asymmetry evident in the flow. We will discuss in more detail the shedding process in § 7.

Figure 3.

Streamlines from experiment B ( $\unicode[STIX]{x1D6FA}=0.48~\text{rad}~\text{s}^{-1}$ , $S=1.0$ , $\unicode[STIX]{x1D716}=0.020$ ), measured at the near-slope height $z_{0}^{\ast }=5~\text{cm}\approx 0.2H$ . The dimensionless times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }$ at which the data were taken are (a) 0.16, (b) 3.4, (c) 5.4, (d) 11, (e) 19 and (f) 37. The top (north) and bottom (south) of the base slope have been indicated in (a). The streamfunction values shown on each contour are dimensionless, and have been scaled by $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}L^{2}$ .

Figure 4.

Streamlines from experiment B ( $\unicode[STIX]{x1D6FA}=0.48~\text{rad}~\text{s}^{-1}$ , $S=1.0$ , $\unicode[STIX]{x1D716}=0.020$ ), measured at the near mid-height $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ . The dimensionless times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }$ at which the data were taken are (a) 0.16, (b) 3.4, (c) 5.4, (d) 11, (e) 19 and (f) 37. The top (north) and bottom (south) of the base slope have been indicated in (a). The streamfunction values shown on each contour are dimensionless, and have been scaled by $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}L^{2}$ .

For comparison, figures 3 and 4 show streamline data for $S=1.0$ (experiment B). Under these conditions – unlike the data just described – the flow had a clear height-dependent structure and so for this case we have shown data at both heights: $z_{0}^{\ast }=5~\text{cm}\approx 0.2H$ (figure 3) and $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ (figure 4). At early times, the starting flow was a two-dimensional, axisymmetric anticyclone, as shown in figures 3(a) and 4(a). At subsequent times, however, clear differences in the flow’s structure were evident at the two heights: consider first the near-slope data in figure 3 (taken at $z_{0}^{\ast }=0.2H$ ), which clearly show the presence of alternating vorticity waves at this height. The first vorticity wave – which has cyclonic circulation (solid streamlines) – is shown shortly after forming in figure 3(b) and at a later time in figure 3(c), having moved westwards across the slope; again, this cyclonic wave is proceeded by the initial anticyclone and followed by the second vorticity wave, which has anticyclonic circulation. When compared with the small- $S$ data in figure 2, there are a number of key differences to highlight: (i) there is a greater degree of north–south asymmetry evident in the streamlines; in particular, the first vorticity wave, as it moved across the slope, remained confined to the northern half of the plane, while the second wave (anticyclonic) that followed remained positioned in the southern half of the plane, as shown in figure 3(b,c). (ii) Relative to the scaled time $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }$ , the vorticity waves formed earlier and travelled more rapidly across the slope; this can be seen by comparing figure 3(b) with figure 2(b), which both correspond to $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }=3.4$ . (iii) Figure 3(c) shows the first vorticity wave (cyclonic) approaching the north-west region of the sidewall; upon reaching the sidewall, this vortex remained positioned within the western-sidewall region, which appeared to preclude the development of large adverse pressure gradients in this region. As a result, there was no vortex shedding and subsequent formation of a persistent cyclonic vortex adjacent to the north-west sidewall region (as was the case in figure 2). (iv) Vorticity waves were seen to form at times subsequent to this, however, only waves with anticyclonic circulation. For example, figure 3(d) shows the second vorticity wave (anticyclonic) as it approaches the western sidewall, followed by another anticyclonic wave located near the central axis of the cylinder. (v) There was no formation of vorticity waves for times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }\gtrsim 14$ , indicating that at this larger value of $S$ the Rossby modes were damped more rapidly. At times subsequent to this the measured streamlines took the form of the quasi-steady, asymmetric anticyclone shown in figure 3(e,f), which thereafter decayed gradually.

Streamline data at corresponding times, but taken at the near mid-depth $z_{0}^{\ast }=0.4H$ , are shown in figure 4. Again, the starting-flow anticyclone, once formed, moved westwards across the slope (see figure 4 b), although, there was no following cyclonic vortex wave present within the bulk interior at this height. However, we draw attention to the thin region of cyclonic vorticity in figure 4(b), which at this time is positioned adjacent to the north-north-east region of the sidewall, which subsequently moved westwards across the slope, but throughout remained confined within a narrow region close to the sidewall – for example, figure 4(c) shows this cyclonic vortex at a later time positioned adjacent to the sidewall, at north-west. Moreover, the westward trajectory of this near-wall vortex coincides with the trajectory of the first vorticity wave evident at height $z^{\ast }=0.2H$ ; this can be seen by comparing figure 4(b,c) with figure 3(b,c). (We note that this vortex is not a result of shedding at the north-west sidewall, as shedding would occur at a later time, $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }\approx 12$ .) It appears, therefore, that under these conditions the cyclonic vortex has a three-dimensional structure, becoming ‘thinner’ and increasingly more confined closer to the sidewall as height above the slope is increased. On reaching the western sidewall region, the cyclonic vortex remained there. There was again evidence of vorticity waves forming at this height at later times, but as was the case at the near-slope height (figure 3), only waves with anticyclonic circulation – for example, see figure 4(c,d). No waves were seen to form for times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }\gtrsim 14$ , and thereafter the flow took the form of a quasi-steady, asymmetric anticyclone which decayed slowly, as shown in figure 4(e,f).

Figure 5.

Streamlines from experiment D ( $\unicode[STIX]{x1D6FA}=0.42~\text{rad}~\text{s}^{-1}$ , $S=10$ , $\unicode[STIX]{x1D716}=0.020$ ). The data in (a–c), were measured at the near-slope height $z_{0}^{\ast }=5~\text{cm}\approx 0.2H$ ; the data in (d–f), were measured at the near-mid-height $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ . The dimensionless times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }$ at which the data were taken are (a,d) 1.1, (b,e) 2.0 and (c,f) 18. The top (north) and bottom (south) of the base slope are shown in (d). The streamfunction values shown on each contour are dimensionless, and have been scaled by $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}L^{2}$ .

The streamline data from experiment C ( $S=2.5$ ) are qualitatively very similar to the data shown in figures 3 and 4 for $S=1$ , and so are not shown here. However, at $S=2.5$ we did note a comparative increase in the degree of north–south asymmetry evident in the first vorticity wave (cyclonic) at the height $z_{0}^{\ast }\approx 0.2H$ – it being confined even further north in the plane – and that no vorticity waves were seen to form for times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }\gtrsim 9$ .

The effects of further increasing $S$ can be seen in figure 5, which shows data for $S=10$ (experiment D). Comparatively fewer features were evident in these data and so the figure is formatted with the left-hand column showing streamlines measured at the near-slope height ( $z_{0}^{\ast }=5~\text{cm}\approx 0.2H$ ), and the right-hand column showing data measured at the near mid-height ( $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ ). See figure captions for details. Consider first the data at the near-slope height: unlike the data already presented, the initial anticyclone at this height was north–south asymmetric from the outset, forming with its centre located in the southern half of the plane. Once formed, the anticyclone quickly moved westwards, as shown in figure 5(a), and the first cyclonic vortex wave formed adjacent to the north-east region of the sidewall, which subsequently moved westwards around the northern sidewall of the container (as shown in figure 5 b). Comparing these streamlines with those in figure 3 (also taken at $z_{0}^{\ast }\approx 0.2H$ , but for $S=1$ ) confirms that further increasing the relative strength of the stratification resulted in a more rapid formation (and propagation) of the cyclonic vorticity wave, relative to the scaled time $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }$ , and with the wave at this height being further confined within a region close to the northern sidewall (i.e. an increased north–south asymmetry). In addition, at this larger value of $S$ the Rossby modes were damped more rapidly; as a result, following the formation of the first wave, no vorticity waves were seen to form at this height, at subsequent times. Instead, the interior core region throughout remained occupied by the initial anticyclone, whose core moved from its displaced western position back into the southern half of the domain as the cyclonic vortex wave moved around the northern sidewall, as shown by comparing figure 5(a,b). Shortly after this, the cyclonic vortex entered the western boundary region where it remained, and thereafter the flow took the form of a quasi-steady, asymmetric anticyclone which decayed slowly. Figure 5(c) shows streamlines at a much later time, which have not changed significantly in comparison with those shown in figure 5(b). We also stress that the cyclonic vortex evident in the north-west sidewall region of figure 5(c) is the remnant of the first vorticity wave and is not a result of shedding from the western-sidewall region.

In contrast, the streamline data in figure 5(d–f), taken at corresponding times but at $z_{0}^{\ast }\approx 0.4H$ , show that from the outset, and throughout the period shown, the flow at the near mid-height took the form of a slowly decaying, near-axisymmetric anticyclone. Slight irregularities were evident in the streamlines about the cylinder’s axis, but the outer streamlines remained near circular throughout. Notably, none of the slope-induced flow features described above were observed in the data, and so the flow at this height appears to have been little affected by the presence of the slope. The slope-induced features were certainly confined within a layer above the slope of height less than $0.4H$ .

Further evidence to confirm this bottom trapping of slope-induced features is shown in figure 6, which shows streamline data from our extreme case $S=20$ (experiment E). Again, given the comparative featureless nature of these data, the figure is formatted with the data in the left-hand and right-hand columns taken, respectively, at $z_{0}^{\ast }\approx 0.2H$ and $z_{0}^{\ast }\approx 0.4H$ . In this case, at both heights, the flow had the form of slowly decaying, axisymmetric anticyclone, although at the near-slope height the streamlines did exhibit slight but noticeable irregularities (similar to those observed at the near mid-height in figure 5). These data clearly show that any flow features induced by the presence of the slope must have been confined to heights $z^{\ast }<0.2H$ . There are certainly no Rossby waves evident in these data. (We have already noted that we were unable to take measurements at heights much below $z^{\ast }=0.2H$ , due to optical effects caused by the light sheet illuminating tracer particles that had settled onto the slope’s surface. Recall, the top level of the base slope corresponds to $z^{\ast }=0.14H$ .)

The theory presented in § 6 below shows that as $S$ gets large the stratified Rossby waves become increasingly confined to a layer of height of the order $S^{-1/2}L$ above the mid-slope. We have therefore included in table 1 the corresponding height scales for each experiment, normalized by the container’s mean depth $H$ (namely, $S^{-1/2}h^{-1}$ ). Comparing these scales with the experimental observations shows a good qualitative confirmation for the presence of the layer. That is, for experiments A, B and C we have $S^{-1/2}h^{-1}\geqslant 0.47$ and in each of these cases the Rossby waves were observed at both of the sample heights $z^{\ast }/H=0.2$ and 0.4. In contrast, $S^{-1/2}h^{-1}=0.23$ in experiment D and the Rossby waves were observed only at $z^{\ast }/H=0.2$ , but not at $z^{\ast }/H=0.4$ . Under the more extreme conditions of experiment E, with $S^{-1/2}h^{-1}=0.17$ , no Rossby waves were observed at either of the sample heights.

Figure 6.

Streamlines from experiment E ( $\unicode[STIX]{x1D6FA}=0.24~\text{rad}~\text{s}^{-1}$ , $S=20$ , $\unicode[STIX]{x1D716}=0.020$ ). The data in (a–c), were measured at the near-slope height $z_{0}^{\ast }=5~\text{cm}\approx 0.2H$ ; the data in (d–f), were measured at the near-mid-height $z_{0}^{\ast }=10~\text{cm}\approx 0.4H$ . The dimensionless times $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }$ at which the data were taken are (a,d) 0.27, (b,e) 10 and (c,f) 30. The top (north) and bottom (south) of the base slope are shown in (d). The streamfunction values shown on each contour are dimensionless, and have been scaled by $\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FA}L^{2}$ .

5.1 The steady state

There are stratified Rossby waves in the container, and frictional decay. For $\unicode[STIX]{x1D702}\unicode[STIX]{x1D70F}\rightarrow \infty$ , the solution to the equations developed in appendix A takes a steady structure, and this structure erodes slowly on a viscous time scale. It is shown in appendix A that the steady solution is

(5.8a ) $$\begin{eqnarray}\displaystyle p_{0} & = & \displaystyle K_{3}\left[\frac{2}{S}(1-r^{2})+z^{2}-zh+\mathop{\sum }_{k=1}^{\infty }\frac{c_{k}}{2\unicode[STIX]{x1D707}_{k0}^{2}}\frac{\sinh [\unicode[STIX]{x1D707}_{k0}(z-h)]-\sinh (\unicode[STIX]{x1D707}_{k0}z)}{\sinh (\unicode[STIX]{x1D707}_{k0}h)}\;J_{0}(\unicode[STIX]{x1D6FC}_{k0}r)\right]\nonumber\\ \displaystyle & & \displaystyle -\,\mathop{\sum }_{k=1}^{\infty }\frac{c_{k}}{\unicode[STIX]{x1D6FC}_{k0}^{2}}\;\frac{\sinh (\unicode[STIX]{x1D707}_{k0}z)-\sinh [\unicode[STIX]{x1D707}_{k0}(z-h)]-\sinh (\unicode[STIX]{x1D707}_{k0}h)}{\sinh (\unicode[STIX]{x1D707}_{k0}h)}\;J_{0}(\unicode[STIX]{x1D6FC}_{k0}r),\end{eqnarray}$$

(5.8b ) $$\begin{eqnarray}\displaystyle K_{3}=-\frac{8\unicode[STIX]{x1D70E}\;S}{hS^{1/2}-16\unicode[STIX]{x1D70E}}, & & \displaystyle\end{eqnarray}$$

(5.8c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}\equiv \mathop{\sum }_{k=1}^{\infty }\frac{\cosh (\unicode[STIX]{x1D707}_{k0}h)-1}{\unicode[STIX]{x1D6FC}_{k0}^{3}\sinh (\unicode[STIX]{x1D707}_{k0}h)}. & & \displaystyle\end{eqnarray}$$

We now examine two extreme cases. First of all, for large $S$ , the $\unicode[STIX]{x1D70E}$ sum can be seen to be of order one, so from (5.8b ), we can see that $K_{3}=O(S^{1/2})$ . In (5.8a ), the square bracket terms multiplying $K_{3}$ are order $S^{-1}$ , so that term takes a limiting form that is of order $S^{-1/2}$ , and is hence small for large $S$ . The final term in (5.8a ) gives the dominant behaviour

(5.9) $$\begin{eqnarray}\displaystyle p_{0}\sim \mathop{\sum }_{k=1}^{\infty }\frac{c_{k}}{\unicode[STIX]{x1D6FC}_{k0}^{2}}\;J_{0}(\unicode[STIX]{x1D6FC}_{k0}r)=1-r^{2}+O(S^{1/2}),\quad \text{for }S\text{ large}. & & \displaystyle\end{eqnarray}$$

The same result is given in § 6.3.1 below for the large- $S$ theory. The order- $S^{1/2}$ terms noted are all functions of $z$ alone, and so involve shifts in the isopycnals, but do not affect the velocity field. So, the interior does not spin up at all on this scale – the spin-up seems to be confined to regions of height $S^{-1/2}$ near the upper and lower boundaries.

For $S$ small, the $\unicode[STIX]{x1D70E}$ sum is of order $S^{1/2}$ . Therefore, $K_{3}$ is of order $S$ . The square brackets in the summations are of $O(1)$ , and hence each term in $p_{0}$ is at most of $O(S)$ , and hence $p_{0}\rightarrow 0$ as $S\rightarrow 0$ , so in the homogeneous limit, we recover the fact that the entire interior is spun-up on this time scale.

Suppose we are interested in the rate of rotation at the mid-plane. Then, with $v_{0}=p_{0r}/2$ we get

(5.10) $$\begin{eqnarray}\displaystyle v_{0}|_{z=h/2}=-\frac{S/4}{1-\displaystyle \frac{16\unicode[STIX]{x1D70E}}{hS^{1/2}}}\mathop{\sum }_{k=1}^{\infty }\unicode[STIX]{x1D6FC}_{k0}\unicode[STIX]{x1D719}_{k}\left[1-\frac{1}{\cosh (\unicode[STIX]{x1D707}_{k0}h/2)}\right]J_{1}(\unicode[STIX]{x1D6FC}_{k0}r). & & \displaystyle\end{eqnarray}$$

In this expression, the quantity $\unicode[STIX]{x1D719}_{k}$ is used for convenience, defined as $\unicode[STIX]{x1D719}_{k}=c_{k}/2\unicode[STIX]{x1D707}_{k0}^{2}$ .

Observing that

(5.11) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{k=1}^{\infty }\unicode[STIX]{x1D6FC}_{k0}\unicode[STIX]{x1D719}_{k}J_{1}(\unicode[STIX]{x1D6FC}_{k0}r)=\frac{4r}{S}, & & \displaystyle\end{eqnarray}$$

we can see from (5.10) that, for large $S$ , since $\unicode[STIX]{x1D70E}=O(1)$ , $v_{0}\rightarrow -r$ . Further, for small $S$ , $v_{0}=O(S)$ . Note that the motion is solid-body rotation if the hyperbolic secant in the numerator of (5.10) is negligible – at a value of $S$ of about $4$ for $h=1$ . However, for the amplitude of the rigid rotation to be one, (5.10) indicates that the neglect of the $\unicode[STIX]{x1D70E}$ term in the denominator requires that $S$ be much larger than $4$ . So, at moderately large $S$ , there is rigid rotation, but at an amplitude that decays toward one very slowly for increasing $S$ .

It is also interesting to note that the motion is solid-body rotation if the hyperbolic secant term in (5.10) is negligible, which occurs at a much smaller value of $S$ – approximately $4$ for $h=1$ – than the value of $S$ where the $\unicode[STIX]{x1D70E}$ term in the denominator of the term multiplying the summation in (5.10) is negligible. The swirl velocity $v_{0}(h/2)$ versus $r$ , obtained using equation (5.10), is shown in figure 7(a), at values of $S$ corresponding to experiments A, B, C and E.

Figure 7.

(a) Azimuthal velocity component, $v_{0}$ , obtained using equation (5.10) at the mid-height $z=0.5h$ , and plotted against $r$ for a range of values of $S$ (labelled). Note, as $v_{0}$ is relative to the new rotation rate, a fully spun-up state would correspond to $v_{0}=0$ . The dashed line shows the initial solid-body rotation. (b) A comparison of the composite solution in equation (5.16) with measurements of $v_{0}$ obtained from the experiments. Here the comparison is made at the height $z=0.4h$ , with the experimental data taken from the azimuth $\unicode[STIX]{x1D703}=3\unicode[STIX]{x03C0}/2$ (i.e. the east coordinate). Again, the corresponding values of $S$ for each case are shown on the figure and the dashed line shows the initial solid-body rotation. Note that the experimental data for $S=20$ are essentially axisymmetric, whereas the data for $S=0.21$ , 1.0 and 2.5 are not, but instead exhibit a degree of (primarily) north–south asymmetry. In (b), the error bars show an estimate of the uncertainty in each set of data, obtained by calculating the peak root-mean-square deviation from 30 velocity profiles about each profile shown (corresponding to a period of 3 s).

Finally, notice that

(5.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70E}=\mathop{\sum }_{n=1}^{\infty }\frac{\tanh (\unicode[STIX]{x1D707}_{n0}h/2)}{\unicode[STIX]{x1D6FC}_{n0}^{3}}<\frac{S^{1/2}}{4}h\mathop{\sum }_{n=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{n0}^{2}}, & & \displaystyle\end{eqnarray}$$

and the sum $\sum 1/\unicode[STIX]{x1D6FC}_{n0}^{2}=1/4$ (Sneddon Reference Sneddon1960), so we have

(5.13) $$\begin{eqnarray}\displaystyle \frac{16\unicode[STIX]{x1D70E}}{hS^{1/2}}<1, & & \displaystyle\end{eqnarray}$$

so the denominator in the $v_{0}$ expression is always positive and never zero.

5.2 Sidewall boundary layers and the composite motion

In Munro & Foster (Reference Munro and Foster2014) – but in more detail in Foster & Munro (Reference Foster and Munro2012) and Munro et al. (Reference Munro, Hewitt and Foster2015) – we have shown that even at small Rossby numbers, the sidewall boundary layers are fully nonlinear Prandtl boundary layers, and as such are subject to spatio-temporal eruption at finite times – the mechanism for the formation of vertical vortices in these flows. Indeed, Munro et al. (Reference Munro, Hewitt and Foster2015) provided a strong affirmation of that mechanism in experimental results, boundary-layer computations and Navier–Stokes calculations. However, for times not too long, the sidewall boundary layers begin as Rayleigh layers, well known to have physical thickness $(\unicode[STIX]{x1D708}t^{\ast })^{1/2}$ , or $(Et)^{1/2}$ in our dimensionless notation. On the time scales investigated in the analysis in this paper, where $t\sim E^{-1/2}$ , such layers obviously scale like $E^{1/4}$ . Neglect of the nonlinear terms on this short time scale requires that $t\ll \unicode[STIX]{x1D716}^{-1}$ , guaranteed under the proviso that $\unicode[STIX]{x1D716}\ll E^{1/2}$ , as we have already noted.

Hence, the Rayleigh-layer equation for the azimuthal velocity component near the sidewall is

(5.14) $$\begin{eqnarray}\displaystyle \bar{v}_{\bar{t}}=\bar{v}_{\bar{r}\bar{r}}, & & \displaystyle\end{eqnarray}$$

where $r=1-E^{1/4}\bar{r}$ and $t=E^{-1/2}\bar{t}$ . The solution is well known, namely,

(5.15) $$\begin{eqnarray}\displaystyle \bar{v}=v_{0}(1,\unicode[STIX]{x1D703},z,\unicode[STIX]{x1D702}\unicode[STIX]{x1D70F})\text{erf}\left(\frac{\bar{r}}{2\sqrt{\bar{t}}}\right). & & \displaystyle\end{eqnarray}$$

Note that $\unicode[STIX]{x1D703}$ and $z$ variations in the flow at the edge of the layer are parametric in the solution.

Then, for $\unicode[STIX]{x1D702}\unicode[STIX]{x1D70F}\rightarrow \infty$ , the ‘steady state’ discussed above is achieved for $v_{0}$ , but the Rayleigh layer is still thickening in time. In this narrow segment of time, we may form a quasi-steady velocity profile for the horizontal motion outside the upper and lower bounding surfaces as an additive composite, that is

(5.16) $$\begin{eqnarray}\displaystyle v=v_{0}(r,\unicode[STIX]{x1D703},z,\infty )\text{erfc}\left(\frac{1-r}{2\sqrt{Et}}\right). & & \displaystyle\end{eqnarray}$$

We plot in figure 7(b) this composite azimuthal velocity profile compared with measured velocity data from four of the experiments listed in table 1. Recall, the ‘steady state’ solution corresponds not to when the fluid is fully spun up, but to when the slope-induced flow features have effectively been damped by the stratification and friction. Hence, we have attempted to make the comparisons at times that satisfy this condition. For $S=20$ , the comparison time was chosen based on the large- $S$ analysis presented in § 6 below. There we show that the Rossby modes are damped as $\exp [\unicode[STIX]{x1D706}(\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}S^{1/2}t^{\ast })]$ , and so the comparison time was chosen to be $t^{\ast }=1/|\unicode[STIX]{x1D706}|\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}S^{1/2}$ , where $\unicode[STIX]{x1D706}=-0.2$ is the computed eigenvalue corresponding to the slowest damped mode (see § 6.3.2 below). The corresponding time for $S=20$ (Exp E) was $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }=3.2$ . For $S=0.21$ , 1.0 and 2.5, the comparison times were chosen based on visual inspection of the streamline data, when the formation of Rossby waves had ceased: these were $\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6FA}t^{\ast }=34$ , 14 and 9.1 for $S=0.21$ (Exp A), $S=1.0$ (Exp B) and $S=2.5$ (Exp C) respectively. We note that the data for $S=10$ (Exp D) were little different from the $S=20$ data, and so have not be included in figure 7(b), to avoid over saturation of data points. Unsurprisingly, the data for $S=20$ , which are essentially axisymmetric throughout, are in very good agreement with the theoretical prediction. At the smaller values of $S$ shown the agreement is generally good, although is clearly affected by a degree of asymmetry that is evident in the experimental data at the times at which the comparison is made.

5.3 The final, viscous decay

Friction at top and bottom surfaces causes the decay of the Rossby waves, the process ending with the steady state given in the above section. In a linearly stratified fluid the final ‘spun-up’ state is achieved on the diffusive time scale $E^{-1}\unicode[STIX]{x1D6FA}^{-1}$ , unlike the case for a homogeneous fluid where the spun-up state is reached more rapidly on the much shorter $E^{-1/2}\unicode[STIX]{x1D6FA}^{-1}$ time scale (Walin Reference Walin1969). The final stage of the decay has been explored by Duck & Foster (Reference Duck and Foster2001), on a time scale $\tilde{t}\equiv Et$ . That discussion is for an axisymmetric situation only, but we have seen above that this flow is in fact axisymmetric beyond the spin-up times. We do not repeat the analysis here, but the solution may be written as

(5.17) $$\begin{eqnarray}\displaystyle v=\mathop{\sum }_{n=1}^{\infty }H_{n}(z,\tilde{t})J_{1}(\unicode[STIX]{x1D6FC}_{1n}r), & & \displaystyle\end{eqnarray}$$

and the coefficient function is itself a Fourier sine series in $z$ ,

(5.18) $$\begin{eqnarray}\displaystyle H_{n}=\mathop{\sum }_{m=1}^{\infty }h_{nm}(\tilde{t})\sin (m\unicode[STIX]{x03C0}z/h), & & \displaystyle\end{eqnarray}$$

where the sine series coefficient is

(5.19) $$\begin{eqnarray}\displaystyle h_{nm}=h_{nm}(0)\text{e}^{-\unicode[STIX]{x1D714}_{nm}\tilde{t}}. & & \displaystyle\end{eqnarray}$$

Substitution into the appropriate equations in this time range, as given in Duck & Foster (Reference Duck and Foster2001), leads to a formula for the time decay coefficients

(5.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}_{nm}=\frac{S}{4}\unicode[STIX]{x1D6FC}_{1n}^{2}\frac{m^{2}\unicode[STIX]{x03C0}^{2}+\unicode[STIX]{x1D6FC}_{1n}^{2}h^{2}}{m^{2}\unicode[STIX]{x03C0}^{2}+(S/4)\unicode[STIX]{x1D6FC}_{1n}^{2}h^{2}}. & & \displaystyle\end{eqnarray}$$

The value of $h_{nm}(0)$ is found from the initial condition for this motion – the steady state from (5.8) – by means of the orthogonality of the sine series and the Fourier–Bessel series. Doing that results in

(5.21a ) $$\begin{eqnarray}\displaystyle & \displaystyle h_{nm}(0)=\frac{2K_{3}}{S}\left[\mathop{\sum }_{k=1}^{\infty }\frac{c_{k}\unicode[STIX]{x1D6E4}_{kn}m\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D6FC}_{0k}}\frac{1-(-1)^{m}}{m^{2}\unicode[STIX]{x03C0}^{2}+\unicode[STIX]{x1D707}_{0k}^{2}h^{2}}\right]-\mathop{\sum }_{k=1}^{\infty }\frac{1-(-1)^{m}}{m\unicode[STIX]{x03C0}\unicode[STIX]{x1D6FC}_{0k}}\frac{c_{k}\unicode[STIX]{x1D707}_{0k}^{2}h^{2}\unicode[STIX]{x1D6E4}_{kn}}{m^{2}\unicode[STIX]{x03C0}^{2}+\unicode[STIX]{x1D707}_{0k}^{2}h^{2}}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.21b ) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6E4}_{kn}=\frac{2J_{1}(\unicode[STIX]{x1D6FC}_{0k})}{(\unicode[STIX]{x1D6FC}_{0k}^{2}-\unicode[STIX]{x1D6FC}_{1n}^{2})J_{0}(\unicode[STIX]{x1D6FC}_{1n})}. & \displaystyle\end{eqnarray}$$

$\unicode[STIX]{x1D714}_{nm}$

This viscous decay was examined using experiment E ( $S=20$ ), which was run over a period of several hours. In § 2 we showed that, under these conditions, the flow within the bulk interior remained largely axisymmetric throughout (see figure 6). Figure 8 shows measurements taken from experiment E of the azimuthal speed, $|v|$ , at the interior point $r=0.5$ , $\unicode[STIX]{x1D703}=3\unicode[STIX]{x03C0}/2$ , $z=0.4h$ . The data are shown (circles) on semi-log-scale and are consistent with the decay rate computed from $\unicode[STIX]{x1D714}_{11}$ , which is shown as the broken line.

Figure 8.

The circles show measurements from Experiment E ( $S=20$ ) of the azimuthal speed $|v|$ at the point $r=0.5$ , $\unicode[STIX]{x1D703}=3\unicode[STIX]{x03C0}/2$ , $z=0.4h$ . The broken line shows decay proportional to $\exp (-\unicode[STIX]{x1D714}_{11}Et)$ , with $\unicode[STIX]{x1D714}_{11}=1.271$ . Note that, $Et=1$ corresponds to the viscous time scale $E^{-1}\unicode[STIX]{x1D6FA}^{-1}$ , which in this case is ${\approx}8~\text{h}$ . The error bars show an estimate of the uncertainty in each measurement, and are based on the time-averaged root-mean-square deviation in the data (over durations of 3 s at the early times and up to 1 min at the much later times).

6.1 Overview of the motion

The mathematical details of this fully three-dimensional flow are algebraically complicated, as we shall see, so it is important to outline the overall flow structure and interactions between regions before proceeding. Figure 9 shows a basic sketch of the flow structure. From (6.1), the motion in the interior ‘core’ of the cylinder is axisymmetric, and the azimuthal velocity component is $z$ -independent. All of the vertical structure and non-axisymmetry is confined to thin layers of height $S^{-1/2}$ below the top and above the bottom boundary, i.e. the ‘upper’ and ‘lower’ layers shown in figure 9. This large- $S$ structure of these waves is well known, probably first noted by Walin (Reference Walin1969), but also discussed by Rhines (Reference Rhines1970), who pointed out that the large stratification – that is, large $S$ – inhibits vertical motion in the core; only near the lower wall does the geometric forcing stretch the vortex lines to generate waves; further, the decay to the interior is exponential.

Requiring that $S\ll 1/\unicode[STIX]{x1D6FC}^{2}$ assures that the sloping surface is wholly contained within this $S^{-1/2}$ lower layer. The temporal evolution of $C$ and $X$ in (6.1) is determined by matching to the upper and lower layers. Because of the linearity of the problem, we can most easily decompose $C$ and $X$ into various components.

Because of the three zones (core, upper layer and lower layer), and the various motions that occur within those zones, we need to supply some decoding help for the reader. Once the general formulation is given below, we consider first the steady-state values for $C$ and $X$ , corresponding to the limit $\unicode[STIX]{x1D70F}^{\prime }\rightarrow \infty$ . We denote those values simply by $C$ and $X$ , with no other notation. The amplitudes of the unsteady components, with a time dependence $\exp (\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}^{\prime })$ , are denoted by $c$ and $\unicode[STIX]{x1D712}$ respectively. However, there are two differing sorts of motion in the upper and lower layers: there are modes that decay because of top and bottom friction, and the amplitudes $c$ and $\unicode[STIX]{x1D712}$ that correspond to those unsteady motions are denoted by $c_{f}$ and $\unicode[STIX]{x1D712}_{f}$ . In the discussion of the Rossby-wave modes, no added notation is utilized, since there is no ambiguity there. Within the upper and lower layers, there are other quantities denoted by, for example, $k_{\ell }$ and $k_{u}$ , for upper- and lower-layer properties. Such quantities associated with frictional modes are similarly denoted by $k_{\ell f}$ and $k_{uf}$ . In the end, then, $C$ is a superposition of the steady-state value, a sum over all frictional modes, and a sum over all Rossby-wave modes.

Finally, a word about the Rossby waves themselves. One can relatively easily work out the spectrum of the waves over a sloping wall in a circular container. However, as is evident in the analysis of § 5, and will also emerge below, the spectrum for waves involved in spin-up is more complicated than a näive approach suggests. First, the eruption of Ekman fluid in the lower corners of the tank alters the spectrum; the presence of the $\unicode[STIX]{x1D70E}$ term in (5.8) is evidence of that in the $S=O(1)$ case. Secondly, the lower-layer dynamics is coupled to the interior core and upper-layer motion in a fashion that also alters the Rossby-wave spectrum.

Below, we begin with the upper-layer motion, then examine the lower-layer dynamics which includes the Rossby waves, and, throughout, match the regions together in a way the ultimately gives the evolution of $C$ and $X$ in time.

Figure 9.

Schematic of the flow structure for large $S$ .

6.2 Upper layer

Near the upper surface, we write $z=h+S^{-1/2}\unicode[STIX]{x1D701}$ , in which case (5.2) becomes

(6.3) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FB}_{1}^{2}P+4P_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}=-4, & & \displaystyle\end{eqnarray}$$

where $P$ denotes the solution for $p_{0}$ in this layer. The boundary condition (3.7b ) becomes,

(6.4) $$\begin{eqnarray}\displaystyle P_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D70F}^{\prime }}=-\unicode[STIX]{x1D702}-\unicode[STIX]{x1D702}P_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\mathscr{E}_{h}. & & \displaystyle\end{eqnarray}$$

The general form of the solution in this axisymmetric layer, based on the matching requirements to (6.1), is

(6.5) $$\begin{eqnarray}\displaystyle P=\mathop{\sum }_{m=1}^{\infty }A_{m}(\unicode[STIX]{x1D701},\unicode[STIX]{x1D70F}^{\prime })J_{0}(\unicode[STIX]{x1D6FC}_{m0}r)-C(r^{2}-1)+\frac{1}{2}(C-1)\unicode[STIX]{x1D701}^{2}+K_{u}\unicode[STIX]{x1D701}. & & \displaystyle\end{eqnarray}$$

Matching this solution to the solution in the core, (6.1), leads to the connection

(6.6) $$\begin{eqnarray}\displaystyle K_{u}=-\frac{hS^{1/2}}{2}(1-C)+\frac{X}{S^{1/2}}. & & \displaystyle\end{eqnarray}$$

Using the facts,

(6.7a,b ) $$\begin{eqnarray}\displaystyle \int _{0}^{1}rJ_{0}(\unicode[STIX]{x1D6FC}_{m0}r)\,\text{d}r=\frac{J_{1}(\unicode[STIX]{x1D6FC}_{m0})}{\unicode[STIX]{x1D6FC}_{m0}},\quad \int _{0}^{1}r[J_{0}(\unicode[STIX]{x1D6FC}_{m0}r)]^{2}\,\text{d}r=\frac{J_{1}^{2}(\unicode[STIX]{x1D6FC}_{m0})}{2}, & & \displaystyle\end{eqnarray}$$

and requiring that the averaged velocity be zero over $z=h$ leads to

(6.8) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x03C0}\mathop{\sum }_{m=1}^{\infty }A_{m,\unicode[STIX]{x1D701}\unicode[STIX]{x1D70F}}\int _{0}^{1}rJ_{0}(\unicode[STIX]{x1D6FC}_{m0}r)\,\text{d}r+\unicode[STIX]{x03C0}K_{u,\unicode[STIX]{x1D70F}^{\prime }}=0,\quad \unicode[STIX]{x1D701}=0, & & \displaystyle\end{eqnarray}$$

or

(6.9) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{m=1}^{\infty }A_{m,\unicode[STIX]{x1D701}\unicode[STIX]{x1D70F}^{\prime }}\frac{J_{1}(\unicode[STIX]{x1D6FC}_{m0})}{\unicode[STIX]{x1D6FC}_{m0}}+\frac{1}{2}K_{u,\unicode[STIX]{x1D70F}^{\prime }}=0,\quad \unicode[STIX]{x1D701}=0, & & \displaystyle\end{eqnarray}$$

and, on integrating and using the homogeneous initial conditions as required,

(6.10) $$\begin{eqnarray}\displaystyle K_{u}=-2\mathop{\sum }_{m=1}^{\infty }A_{m,\unicode[STIX]{x1D701}}\frac{J_{1}(\unicode[STIX]{x1D6FC}_{m0})}{\unicode[STIX]{x1D6FC}_{m0}}. & & \displaystyle\end{eqnarray}$$

Using the orthogonality of the members of the Bessel function series, the wall boundary condition (6.4) becomes

(6.11) $$\begin{eqnarray}\displaystyle (A_{m,\unicode[STIX]{x1D701}\unicode[STIX]{x1D70F}^{\prime }}+\unicode[STIX]{x1D702}A_{m,\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}})\frac{1}{2}J_{1}^{2}(\unicode[STIX]{x1D6FC}_{m0})=-(\unicode[STIX]{x1D702}C+K_{u,\unicode[STIX]{x1D70F}^{\prime }})\frac{J_{1}(\unicode[STIX]{x1D6FC}_{m0})}{\unicode[STIX]{x1D6FC}_{m0}}\quad \text{on}~\unicode[STIX]{x1D701}=0. & & \displaystyle\end{eqnarray}$$

From (6.3), the equation satisfied by $A_{m}$ is

(6.12) $$\begin{eqnarray}\displaystyle A_{m,\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6FC}_{m0}^{2}}{4}A_{m}=0, & & \displaystyle\end{eqnarray}$$

so the matchable solution is

(6.13) $$\begin{eqnarray}\displaystyle A_{m}=a_{m}\text{e}^{\unicode[STIX]{x1D6FC}_{m0}\unicode[STIX]{x1D701}/2}. & & \displaystyle\end{eqnarray}$$

Substituting into (6.11) leads to

(6.14) $$\begin{eqnarray}\displaystyle a_{m,\unicode[STIX]{x1D70F}^{\prime }}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}\unicode[STIX]{x1D702}a_{m}=-(\unicode[STIX]{x1D702}C+K_{u,\unicode[STIX]{x1D70F}^{\prime }})\frac{4}{J_{1}(\unicode[STIX]{x1D6FC}_{m0})\unicode[STIX]{x1D6FC}_{m0}^{2}}. & & \displaystyle\end{eqnarray}$$

Putting in the evaluation of $K_{u}$ from (6.10), we have

(6.15) $$\begin{eqnarray}\displaystyle a_{m,\unicode[STIX]{x1D70F}^{\prime }}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}\unicode[STIX]{x1D702}a_{m}=\frac{4}{J_{1}(\unicode[STIX]{x1D6FC}_{m0})\unicode[STIX]{x1D6FC}_{m0}^{2}}\left[-\unicode[STIX]{x1D702}C+\mathop{\sum }_{r=1}^{\infty }a_{r,\unicode[STIX]{x1D70F}^{\prime }}J_{1}(\unicode[STIX]{x1D6FC}_{r0})\right]. & & \displaystyle\end{eqnarray}$$

6.2.1 Steady state

The particular solution to (6.15) corresponds to the long-time behaviour ( $\unicode[STIX]{x1D702}\unicode[STIX]{x1D70F}^{\prime }\rightarrow \infty$ ), after the transient, and is clearly given simply by

(6.16) $$\begin{eqnarray}\displaystyle \tilde{a}_{m}=-\frac{8C}{\unicode[STIX]{x1D6FC}_{m0}^{3}J_{1}(\unicode[STIX]{x1D6FC}_{m0})}, & & \displaystyle\end{eqnarray}$$

letting $\tilde{a}_{m}$ denote the particular solution for $a_{m}$ .

Multiplying by $J_{1}(\unicode[STIX]{x1D6FC}_{m0})$ and summing over $m$ gives the steady-state relation between $K_{u}$ and $C$ as

(6.17) $$\begin{eqnarray}\displaystyle K_{u}=8C\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}. & & \displaystyle\end{eqnarray}$$

Combining with (6.6), we find a relationship between $C$ and $X$ , namely

(6.18) $$\begin{eqnarray}\displaystyle -\frac{S^{1/2}}{2}h(1-C)+\frac{X}{S^{1/2}}=8C\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}. & & \displaystyle\end{eqnarray}$$

Thus, we have a steady-state relationship between $C$ and $X$ ,

(6.19) $$\begin{eqnarray}\displaystyle C=\left(1-\frac{16}{S^{1/2}h}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}\right)^{-1}\left(1-\frac{2X}{Sh}\right). & & \displaystyle\end{eqnarray}$$

The completion of the steady result is in § 6.3.1, since it depends on the effects of the lower layer.

Combining gives the long-time behaviour of the pressure perturbation as

(6.20) $$\begin{eqnarray}\displaystyle P=-C\left\{8\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}\left[\text{e}^{\unicode[STIX]{x1D6FC}_{m0}\unicode[STIX]{x1D701}/2}\frac{J_{0}(\unicode[STIX]{x1D6FC}_{m0}r)}{J_{1}(\unicode[STIX]{x1D6FC}_{m0})}-\unicode[STIX]{x1D701}\right]-\frac{hS^{1/2}}{2}\unicode[STIX]{x1D701}+r^{2}-1\right\}+\frac{1}{2}(C-1)\unicode[STIX]{x1D701}^{2}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Evaluating the azimuthal component in the layer,

(6.21) $$\begin{eqnarray}\displaystyle v=\frac{1}{2}\frac{\unicode[STIX]{x2202}P}{\unicode[STIX]{x2202}r}=-C\left[r-4\mathop{\sum }_{m=1}^{\infty }\frac{J_{1}(\unicode[STIX]{x1D6FC}_{m0}r)}{\unicode[STIX]{x1D6FC}_{m0}^{2}J_{1}(\unicode[STIX]{x1D6FC}_{m0})}\,\text{e}^{\unicode[STIX]{x1D6FC}_{m0}\unicode[STIX]{x1D701}/2}\right]. & & \displaystyle\end{eqnarray}$$

From the properties of Bessel functions,

(6.22) $$\begin{eqnarray}\displaystyle r=\mathop{\sum }_{m=1}^{\infty }\frac{4}{\unicode[STIX]{x1D6FC}_{m0}^{2}}\,\frac{J_{1}(\unicode[STIX]{x1D6FC}_{m0}r)}{J_{1}(\unicode[STIX]{x1D6FC}_{m0})}. & & \displaystyle\end{eqnarray}$$

Thus, we see that

(6.23) $$\begin{eqnarray}\displaystyle v=0\quad \text{on}~\unicode[STIX]{x1D701}=0\;(\text{or}~z=h), & & \displaystyle\end{eqnarray}$$

so that the fluid is totally spun-up at the upper wall, but the velocity grows rapidly away from $z=h$ , so that in the interior, the velocity is $Cr$ .

6.2.2 Unsteadiness

The time dependence of the motion is determined by putting $a_{m}=\hat{a}_{m}\exp (\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}^{\prime })$ into the homogeneous version of (6.14), leading to the eigenvalue problem

(6.24) $$\begin{eqnarray}\displaystyle \left(\unicode[STIX]{x1D706}+\frac{1}{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{m0}\right)\hat{a}_{m}=\frac{4}{\unicode[STIX]{x1D6FC}_{m0}^{2}J_{1}(\unicode[STIX]{x1D6FC}_{m0})}\left(-\unicode[STIX]{x1D702}c-\unicode[STIX]{x1D706}k_{u}\right), & & \displaystyle\end{eqnarray}$$

where we have also written $K_{u}=k_{u}\exp (\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}^{\prime })$ and $C=c\exp (\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}^{\prime })$ . Solving for $\hat{a}_{m}J_{1}(\unicode[STIX]{x1D6FC}_{m0})$ , and then summing,

(6.25) $$\begin{eqnarray}\displaystyle k_{u}=4\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D706}+\frac{1}{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{m0}\right)}(\unicode[STIX]{x1D702}c+\unicode[STIX]{x1D706}k_{u}). & & \displaystyle\end{eqnarray}$$

Putting $X=\unicode[STIX]{x1D712}\exp (\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}^{\prime })$ , and using (6.6) to relate $c$ and $k_{u}$ to $\unicode[STIX]{x1D712}$ leads eventually to

(6.26) $$\begin{eqnarray}\displaystyle k_{u}\left[\left(\unicode[STIX]{x1D706}+\frac{2\unicode[STIX]{x1D702}}{hS^{1/2}}\right)\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D706}+\frac{1}{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{m0}\right)}-\frac{1}{4}\right]=\unicode[STIX]{x1D702}\frac{2\unicode[STIX]{x1D712}}{hS}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D706}+\frac{1}{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{m0}\right)}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Two possible scalings arise of these equations, delineated below.

1. (i) $\unicode[STIX]{x1D706}=O(1)$ : Rossby waves

   We know that there are Rossby waves in the container, so $\unicode[STIX]{x1D706}=O(1)$ as $\unicode[STIX]{x1D702}\rightarrow 0$ for the waves. The $1/4$ in the square bracket in (6.26) may be replaced by $\sum _{m=1}^{\infty }1/\unicode[STIX]{x1D6FC}_{m0}^{2}$ , as previously quoted from Sneddon (Reference Sneddon1960), and the two terms in the square bracket in equation become, without approximation,

   (6.27) $$\begin{eqnarray}\displaystyle \frac{2\unicode[STIX]{x1D702}}{hS^{1/2}}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D706}+\frac{1}{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{m0}\right)}-\frac{\unicode[STIX]{x1D702}}{2}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}\left(\unicode[STIX]{x1D706}+\frac{1}{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{m0}\right)}. & & \displaystyle\end{eqnarray}$$
   As $\unicode[STIX]{x1D702}\rightarrow 0$ , the second of these series is divergent, and in fact behaves like $\log \unicode[STIX]{x1D702}$ – see Munro & Foster (Reference Munro and Foster2014). Note that, therefore, (6.26) determines $k_{u}$ to be of order $\unicode[STIX]{x1D712}/\log \unicode[STIX]{x1D702}$ , so the $k_{u}$ term is negligible in (6.6), leading to
   (6.28) $$\begin{eqnarray}\displaystyle \frac{hS^{1/2}}{2}c+\frac{\unicode[STIX]{x1D712}}{S^{1/2}}=0. & & \displaystyle\end{eqnarray}$$
   Note that this relationship between $c$ and $\unicode[STIX]{x1D712}$ is valid only on the Rossby-wave scale.

2. (ii) $\unicode[STIX]{x1D706}=O(\unicode[STIX]{x1D702})$ : frictional decay

   In this case, we write $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D702}\unicode[STIX]{x1D6EC}$ , in which case, (6.26) becomes

   (6.29) $$\begin{eqnarray}\displaystyle k_{uf}\left[\left(\unicode[STIX]{x1D6EC}+\frac{2}{hS^{1/2}}\right)\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D6EC}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}\right)}-\frac{1}{4}\right]=\frac{2\unicode[STIX]{x1D712}_{f}}{hS}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D6EC}+\frac{1}{2}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{m0}\right)}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
   Recall that we have appended an $f$ -subscript to distinguish those elements of $c$ , $\unicode[STIX]{x1D712}$ and $k_{u}$ that are associated with frictional decay only, with unsubscripted values of $c,\unicode[STIX]{x1D712}$ and $k_{u}$ corresponding to Rossby waves. In addition, (6.25) is
   (6.30) $$\begin{eqnarray}\displaystyle k_{uf}=4\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D6EC}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}\right)}(c_{f}+\unicode[STIX]{x1D6EC}k_{uf}). & & \displaystyle\end{eqnarray}$$

As we shall determine in § 6.3.3, in the lower $S^{-1/2}$ layer, on this time scale, the relationship between $c$ and $\unicode[STIX]{x1D712}$ is given as (6.60), namely

(6.31) $$\begin{eqnarray}\displaystyle \left(8\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}-\frac{hS^{1/2}}{2}\right)c=-\frac{\unicode[STIX]{x1D712}}{S^{1/2}}. & & \displaystyle\end{eqnarray}$$

Combining with (6.30) and (6.29), and after some algebra, gives the eigenvalue problem

(6.32) $$\begin{eqnarray}\displaystyle \left[\unicode[STIX]{x1D6EC}-\left(8\mathop{\sum }_{m=1}^{\infty }1/\unicode[STIX]{x1D6FC}_{m0}^{3}-hS^{1/2}\right)^{-1}\right]\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D6EC}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}\right)}-\frac{1}{4}=0. & & \displaystyle\end{eqnarray}$$

Neglecting for now the $S^{-1/2}$ corrections, this becomes

(6.33) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{m=1}^{\infty }\frac{\unicode[STIX]{x1D6EC}}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D6EC}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}\right)}=\frac{1}{4}. & & \displaystyle\end{eqnarray}$$

Assume that $\unicode[STIX]{x1D6EC}$ has non-zero imaginary part. Multiply (6.33) by the conjugate of $\unicode[STIX]{x1D6EC}$ . Then multiplying numerator and denominator of the term in the sum by the conjugate of $\unicode[STIX]{x1D6EC}+\unicode[STIX]{x1D6FC}_{m0}/2$ , and taking the imaginary part of the equation leads to the result, for non-zero imaginary part of $\unicode[STIX]{x1D6EC}$ ,

(6.34) $$\begin{eqnarray}\displaystyle \frac{1}{4|\unicode[STIX]{x1D6EC}|^{2}}=\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}|\unicode[STIX]{x1D6EC}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}|^{2}}. & & \displaystyle\end{eqnarray}$$

Clearly, $|\unicode[STIX]{x1D6EC}+(1/2)\unicode[STIX]{x1D6FC}_{m0}|^{2}\geqslant |\unicode[STIX]{x1D6EC}|^{2}$ by use of the triangle inequality. However, these two quantities can be equal only if $\unicode[STIX]{x1D6FC}_{m0}$ is zero, which it cannot be. Hence, $|\unicode[STIX]{x1D6EC}+(1/2)\unicode[STIX]{x1D6FC}_{m0}|^{2}>|\unicode[STIX]{x1D6EC}|^{2}$ , so we have

(6.35) $$\begin{eqnarray}\displaystyle \frac{1}{4}<\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}}, & & \displaystyle\end{eqnarray}$$

which is a contradiction, since $\sum _{m=1}^{\infty }1/\unicode[STIX]{x1D6FC}_{m0}^{2}=1/4$ . Therefore, $\unicode[STIX]{x1D6EC}$ can have no imaginary part.

So, if $\unicode[STIX]{x1D6EC}$ is purely real, assume that it is positive. In that case,

(6.36) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}\left(\unicode[STIX]{x1D6EC}+\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}\right)}<\frac{1}{\unicode[STIX]{x1D6EC}}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{2}}=\frac{1}{4\unicode[STIX]{x1D6EC}}, & & \displaystyle\end{eqnarray}$$

which contradicts (6.33), so we conclude that

(6.37) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EC}\;\text{is real and negative}, & & \displaystyle\end{eqnarray}$$

for all solutions of (6.33).

The eigenvalue equation (6.33) may alternatively be written in the simpler form

(6.38) $$\begin{eqnarray}\displaystyle \mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}\left(\frac{1}{2}\unicode[STIX]{x1D6FC}_{m0}-\unicode[STIX]{x1D708}\right)}=0, & & \displaystyle\end{eqnarray}$$

where we have written $\unicode[STIX]{x1D6EC}=-\unicode[STIX]{x1D708}$ , and the result from Sneddon (Reference Sneddon1960) has been used again. A geometric argument from this or (6.25) leads to the fact that the values of $\unicode[STIX]{x1D708}$ interlace with $\{\unicode[STIX]{x1D6FD}_{i}\}/2$ , that is, for the $j$ th eigenvalue,

(6.39) $$\begin{eqnarray}\displaystyle {\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{j}<\unicode[STIX]{x1D708}_{j}<{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FD}_{j+1},\quad j=1,2,3,\ldots . & & \displaystyle\end{eqnarray}$$

For every solution to (6.38) $\unicode[STIX]{x1D708}_{j}$ , there corresponds a solution to (6.24), so the general solution for $\hat{a}_{n}$ is constituted by a linear combination of all possible solutions, that is

(6.40) $$\begin{eqnarray}\displaystyle \hat{a}_{n}=\mathop{\sum }_{j=1}^{\infty }c_{nj}\text{e}^{-\unicode[STIX]{x1D702}\unicode[STIX]{x1D708}_{j}\unicode[STIX]{x1D70F}^{\prime }}. & & \displaystyle\end{eqnarray}$$

That being the case, for every eigenvalue, there is a value for $\tilde{C}$ , hence

(6.41) $$\begin{eqnarray}\displaystyle c=\mathop{\sum }_{j=1}^{\infty }c_{j}\text{e}^{-\unicode[STIX]{x1D702}\unicode[STIX]{x1D708}_{j}\unicode[STIX]{x1D70F}^{\prime }}. & & \displaystyle\end{eqnarray}$$

An important conclusion is that the core flow decays by friction – $C$ to a constant, and $X$ to zero – for $\unicode[STIX]{x1D702}\unicode[STIX]{x1D70F}^{\prime }\rightarrow \infty$ .

6.3 Lower layer

Proceeding much as in the layer near the upper surface, we write $z=S^{-1/2}\unicode[STIX]{x1D701}$ . Because the lower wall is sloped at angle $\unicode[STIX]{x1D6FC}$ , in order to be able to ignore slope in this layer, we require that

(6.42) $$\begin{eqnarray}\displaystyle S^{-1/2}\gg \unicode[STIX]{x1D6FC}. & & \displaystyle\end{eqnarray}$$

When $S^{1/2}\unicode[STIX]{x1D6FC}=O(1)$ , the slope of the lower boundary must come into the boundary condition in this layer.

We write the solution in this region in a form that guarantees matching to (6.1), namely,

(6.43) $$\begin{eqnarray}\displaystyle P & = & \displaystyle C(1-r^{2})-\frac{1}{2}(1-C)\unicode[STIX]{x1D701}^{2}+K_{\ell }(\unicode[STIX]{x1D70F}^{\prime })\unicode[STIX]{x1D701}\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=0}^{\infty }a_{mn}(\unicode[STIX]{x1D70F}^{\prime })\text{e}^{-\unicode[STIX]{x1D6FC}_{mn}\unicode[STIX]{x1D701}/2}\cos (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r)\nonumber\\ \displaystyle & & \displaystyle +\,\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }b_{mn}(\unicode[STIX]{x1D70F}^{\prime })\text{e}^{-\unicode[STIX]{x1D6FC}_{mn}\unicode[STIX]{x1D701}/2}\sin (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r).\end{eqnarray}$$

On matching,

(6.44) $$\begin{eqnarray}\displaystyle K_{\ell }=\frac{hS^{1/2}}{2}(1-C)+\frac{X}{S^{1/2}}. & & \displaystyle\end{eqnarray}$$

The boundary condition is

(6.45) $$\begin{eqnarray}\displaystyle P_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D70F}^{\prime }}=\mathscr{E}_{0}+\unicode[STIX]{x1D702}P_{\unicode[STIX]{x1D701}\unicode[STIX]{x1D701}}+\frac{1}{2}\left(\sin \unicode[STIX]{x1D703}P_{r}+\frac{\cos \unicode[STIX]{x1D703}}{r}P_{\unicode[STIX]{x1D703}}\right)+\unicode[STIX]{x1D702}\quad \text{at}~\unicode[STIX]{x1D701}=0. & & \displaystyle\end{eqnarray}$$

Substitution gives

(6.46) $$\begin{eqnarray}\displaystyle & & \displaystyle K_{\ell }^{\prime }-\frac{1}{2}\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D6FC}_{mn}a_{mn}^{\prime }\cos (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r)-\frac{1}{2}\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{mn}b_{mn}^{\prime }\sin (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r)\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{1}{2}\sin \unicode[STIX]{x1D703}\left[\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D6FC}_{mn}a_{mn}\cos (n\unicode[STIX]{x1D703})J_{n}^{\prime }(\unicode[STIX]{x1D6FC}_{mn}r)+\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{mn}b_{mn}\sin (n\unicode[STIX]{x1D703})J_{n}^{\prime }(\unicode[STIX]{x1D6FC}_{mn}r)\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\cos \unicode[STIX]{x1D703}}{2r}\left[-\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=0}^{\infty }na_{mn}\sin (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r)+\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }nb_{mn}\cos (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r)\right]\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x1D702}}{4}\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=0}^{\infty }\unicode[STIX]{x1D6FC}_{mn}^{2}a_{mn}\cos (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{\unicode[STIX]{x1D702}}{4}\mathop{\sum }_{m=1}^{\infty }\mathop{\sum }_{n=1}^{\infty }\unicode[STIX]{x1D6FC}_{mn}^{2}b_{mn}\sin (n\unicode[STIX]{x1D703})J_{n}(\unicode[STIX]{x1D6FC}_{mn}r)+C\unicode[STIX]{x1D702}+rC\sin \unicode[STIX]{x1D703}+\mathscr{E}_{0}.\end{eqnarray}$$

The prime here denotes differentiation with respect to $\unicode[STIX]{x1D70F}^{\prime }$ . Averaging over the circle for the left-hand side terms – as above – leads to the formula for $K_{\ell }^{\prime }$ , which may then be integrated to give

(6.47) $$\begin{eqnarray}\displaystyle K_{\ell }=\mathop{\sum }_{m=1}^{\infty }a_{m0}(\unicode[STIX]{x1D70F}^{\prime })J_{1}(\unicode[STIX]{x1D6FC}_{m0}). & & \displaystyle\end{eqnarray}$$

Then, on using the orthogonality of the sines and cosines and also the Bessel functions gives, as in the general case in § 5 and discussed in detail in appendix A,

(6.48a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\unicode[STIX]{x1D6FC}_{k0}a_{k0}^{\prime }+\frac{1}{4}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{k0}^{2}a_{k0}=\frac{2}{\unicode[STIX]{x1D6FC}_{k0}J_{1}(\unicode[STIX]{x1D6FC}_{k0})}(K_{\ell }^{\prime }-C\unicode[STIX]{x1D702})-\frac{1}{4}\mathop{\sum }_{m=1}^{\infty }b_{m,1}U_{mk}^{(1)}, & \displaystyle\end{eqnarray}$$

(6.48b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\unicode[STIX]{x1D6FC}_{kn}a_{kn}^{\prime }+\frac{1}{4}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{kn}^{2}a_{kn}=-\frac{1}{4}\mathop{\sum }_{m=1}^{\infty }b_{m,n+1}U_{mk}^{(n+1)}-\frac{1}{4}\mathop{\sum }_{m=1}^{\infty }b_{m,n-1}V_{mk}^{(n-1)},\quad n=1,2,3,\ldots & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(6.48c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\unicode[STIX]{x1D6FC}_{k1}b_{k1}^{\prime }+\frac{1}{4}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{k1}^{2}b_{k1}=-\frac{1}{4}\mathop{\sum }_{m=1}^{\infty }[2a_{m0}r_{mk}^{(0)}-a_{m,2}U_{mk}^{(2)}]-\frac{2C}{\unicode[STIX]{x1D6FC}_{k1}J_{0}(\unicode[STIX]{x1D6FC}_{k1})}, & \displaystyle\end{eqnarray}$$

(6.48d ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\unicode[STIX]{x1D6FC}_{kn}b_{kn}^{\prime }+\frac{1}{4}\unicode[STIX]{x1D702}\unicode[STIX]{x1D6FC}_{kn}^{2}b_{kn}=\frac{1}{4}\mathop{\sum }_{m=1}^{\infty }a_{m,n+1}U_{mk}^{(n+1)}+\frac{1}{4}\mathop{\sum }_{m=1}^{\infty }a_{m,n-1}V_{mk}^{(n-1)},\quad n=2,3,4,\ldots . & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

6.3.1 Steady state

As $\unicode[STIX]{x1D702}\unicode[STIX]{x1D70F}^{\prime }\rightarrow \infty$ , we recover a solution quite similar to the one in the $z=h$ layer – so the non-axisymmetry has completely disappeared by this time. That large- $\unicode[STIX]{x1D70F}^{\prime }$ solution is seen to be

(6.49) $$\begin{eqnarray}\displaystyle \tilde{a}_{k0}\rightarrow -\frac{8C}{\unicode[STIX]{x1D6FC}_{k0}^{3}J_{1}(\unicode[STIX]{x1D6FC}_{k0})}. & & \displaystyle\end{eqnarray}$$

Then, from (6.44), we have

(6.50) $$\begin{eqnarray}\displaystyle \frac{hS^{1/2}}{2}(1-C)+\frac{X}{S^{1/2}}=-8C\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}. & & \displaystyle\end{eqnarray}$$

This equation and equation (6.18) constitute simultaneous equations for $X$ and $C$ , and they lead immediately to the conclusion that $X=0$ in this steady state, and hence, from (6.19), we have the steady value of $C$ to two orders in $S$ to be

(6.51) $$\begin{eqnarray}\displaystyle C=\left[1-\frac{16}{S^{1/2}h}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}\right]^{-1}+O(S^{-1}). & & \displaystyle\end{eqnarray}$$

This result agrees precisely with the large- $S$ limit of the exact solution, discussed previously in § 5.1.

6.3.2 Unsteadiness for $\unicode[STIX]{x1D706}=O(1)$ : Rossby waves

We found a relationship between the transient portions of $X$ and $C$ – denoted by $\unicode[STIX]{x1D712}$ and $c$ respectively – from the upper layer, given in (6.28) for $\unicode[STIX]{x1D706}=O(1)$ . Equation (6.44) becomes

(6.52) $$\begin{eqnarray}\displaystyle k_{\ell }=-\frac{hS^{1/2}}{2}c+\frac{\unicode[STIX]{x1D712}}{S^{1/2}}. & & \displaystyle\end{eqnarray}$$

Solving these simultaneously gives formulas for $c$ and $\unicode[STIX]{x1D712}$ in terms of $k_{\ell }$ ,

(6.53a,b ) $$\begin{eqnarray}\displaystyle k_{\ell }=-hS^{1/2}c,\quad \unicode[STIX]{x1D712}=\frac{S^{1/2}}{2}k_{\ell }. & & \displaystyle\end{eqnarray}$$

The eigenvalue problem in this lower layer is then formulated by putting in a time dependence $\exp (\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}^{\prime })$ into (6.48) above with $\unicode[STIX]{x1D702}$ set to zero. Insofar as the determination of the eigenvalues – the Rossby-wave frequencies – we note from (6.48) that all values of $a_{kn}$ are zero for $n$ odd and all values of $b_{kn}$ are zero for $n$ even. We use that fact to formulate a single eigenvalue problem, in a way detailed in appendix B.

Before showing the results of the Rossby-wave computations, a few comments are in order. In appendix B, we show that the eigenvalue problem, when truncated to a finite number of terms $N$ in the Fourier–Bessel series, reduces to a single, non-standard, matrix eigenvalue problem, given in detail in (B 3), which it turns out may be written as

(6.54) $$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D645}-\unicode[STIX]{x1D706}\unicode[STIX]{x1D644})\boldsymbol{P}=-\unicode[STIX]{x1D706}k_{\ell }\boldsymbol{V}+c\boldsymbol{V}^{\prime }, & & \displaystyle\end{eqnarray}$$

where eigenvector $\boldsymbol{P}$ and constant vectors $\boldsymbol{V}$ and $\boldsymbol{V}^{\prime }$ have dimension $(N+1)^{2}$ .

1. (i) Putting $c=k_{\ell }=0$ in (6.54) – the homogeneous problem – physically corresponds to Rossby waves that might exist in this lower layer in the absence of Ekman eruptions and interior motion.

2. (ii) Ekman-layer eruptions come into the solution of (6.54) through the $k_{\ell }$ term, thus altering the frequencies from the homogeneous values.

3. (iii) The coupling of the lower-layer Rossby waves to the motion in the core modifies the spectrum through the $c$ -term in (6.54). Note however that the effect is weaker, since $c=O(k_{\ell }/S^{1/2})$ .

4. (iv) The final solution for the unsteady portion of each $\hat{a}_{kn}$ is, of course, a summation over all eigenvalues and eigenvectors. Similarly, $c$ itself includes a summation over all modes, leading therefore to oscillations in the interior swirl motion. However, since members of $\{a_{kn}\}$ must necessarily be $O(1)$ to satisfy initial conditions, $k_{\ell }$ is also $O(1)$ , and so $c$ is $O(S^{-1/2})$ , hence the fraction of energy in the Rossby-wave oscillations in the core scales with $S^{-1}$ , and so for the large- $S$ case considered here is therefore very small and essentially unobservable.

5. (v) We find in this large- $S$ case that all Rossby-wave frequencies lie between $0$ and $1$ , the distribution of which is quite complicated – owing to the complications of some of the items listed above.

Finally, there are corrections to these $\unicode[STIX]{x1D706}=O(1)$ Rossby-wave modes due to small but non-zero values of $\unicode[STIX]{x1D702}$ . In an asymptotic context, we would write $\unicode[STIX]{x1D706}=\unicode[STIX]{x1D706}_{0}+g(\unicode[STIX]{x1D702})\unicode[STIX]{x1D706}_{1}+\cdots \,$ ; the quantity $g(\unicode[STIX]{x1D702})$ is a gauge function that has the property $g(0)=0$ . Obviously, $\unicode[STIX]{x1D706}_{0}$ comes from the Rossby-wave analysis, and $\unicode[STIX]{x1D706}_{1}$ arises from the upper- and lower-wall frictional damping. The value of $\unicode[STIX]{x1D706}_{1}$ and the determination of $g(\unicode[STIX]{x1D702})$ may in principle be found by using the Fredholm alternative, as in Munro & Foster (Reference Munro and Foster2014) for the square cylinder.

Here, instead, we retain all $\unicode[STIX]{x1D702}$ terms in the differential equations (6.48), which alters the matrix $\unicode[STIX]{x1D645}$ in (6.54) in a way described in detail in appendix B. Doing that computation for $\unicode[STIX]{x1D702}=0.1$ – the value appropriate to these experiments – gives results shown in figure 10. Recall from § 4 that the real part of all values of $\unicode[STIX]{x1D706}$ for $S=0$ is $-2\unicode[STIX]{x1D702}/h$ , so that the $S=0$ spectrum, which is not shown, would lie along a vertical line in figure 10, instead of the much more complicated array of modes shown for $S=20$ . Our approach for finding $\unicode[STIX]{x1D706}_{1}$ is strictly numerical, and as such leads to no determination of the gauge function $g(\unicode[STIX]{x1D702})$ . We expect that $g(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D702}\log \unicode[STIX]{x1D702}$ as for the square cylinder, but we have not demonstrated that.

At first blush, there appears to be little order or structure to the large array of damped waves shown in figure 10. There is no simple or obvious algebraic relation between the location of a particular eigenvalue in this plane and the wavenumbers of its associated eigenvector.

For the eigenvalue labelled ▴ in figure 10, corresponding streamlines are shown in figure 11(a–c), at three different heights $z=0$ , 0.2 and 0.4. At $z=0.4$ , one is effectively out of the boundary layer for $S=20$ . This mode is of course the last one to survive in the spin-up, because it has the smallest damping rate. Unlike the sorts of Rossby modes found for $S=0$ by Pedlosky & Greenspan (Reference Pedlosky and Greenspan1967), this particular mode is very nearly axisymmetric – which is precisely what we see in the experiments toward the end of the spin-up time scale (see, for example, figure 6). The truncation error for this eigenvalue at $N=40$ is quite small, since

(6.55a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=-0.20636+0.77240\text{i},\quad N=40, & & \displaystyle\end{eqnarray}$$

(6.55b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=-0.20635+0.77241\text{i},\quad N=30. & & \displaystyle\end{eqnarray}$$

Another very slowly damped mode is marked ▪ in figure 10, located at

(6.56) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=-0.52521+0.91936\,i,\quad N=40. & & \displaystyle\end{eqnarray}$$

again with negligible truncation error. The corresponding streamlines again at heights $z=0$ , 0.2 and 0.4 are shown in figure 12(a–c). Note that strong east–west asymmetry does exist in this mode, indicating that close to the end of the spin-up, some significant asymmetry might remain.

By contrast, the eigenvalue marked ♦ in figure 10, obviously very strongly damped,

(6.57) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D706}=-2.28265+0.33442\,i,\quad N=40. & & \displaystyle\end{eqnarray}$$

turns out also to be a strongly asymmetric, short-wavelength mode, which is discussed further in appendix B.

There seems no a priori way to know, for a given mode in figure 10, what a sufficiently large value of $N$ is required to satisfy a certain accuracy criterion. Typically, though certainly not all modes have been checked, it appears that eigenvectors associated with the eigenvalues toward the left of the display in figure 10 tend to have larger wavenumber content, and hence require larger values of $N$ for good resolution. Trial and error suggests that $N=40$ , for modes with moderate decay rates, gives reliable $5$ -digit accurate numbers. Figures 13–15 in appendix B confirm the validity of the results displayed here, and further discussion of this point may be found there.

Figure 10.

Solutions to (B 3) with $\mathscr{A}_{n}$ matrix on the main diagonal. $S=20$ and $\unicode[STIX]{x1D702}=0.1$ . There are $961$ roots, with the conjugates not shown, for the case $N=40$ . The three particular cases labelled with ▴, ▪ and ♦ symbols are discussed in the text.

Figure 11.

Horizontal plane streamlines on (a) $z=0$ , (b) $z=0.2$ and, (c), $z=0.4$ for the eigenfunction at a particular instant of time, that corresponds to the eigenvalue in (6.55), labelled ▴ in figure 10. For this computation, $S=20$ , $\unicode[STIX]{x1D702}=0.1$ and $N=40$ . (d) The dashed curve is the azimuthal velocity at $r=1$ for this eigenfunction, and the solid line is the negative of the azimuthal pressure gradient, $v_{0}\,\text{d}v_{0}/\text{d}\unicode[STIX]{x1D703}$ , also at $r=1$ . The top (north) and bottom (south) of the base slope have been shown in (b). Note that $\unicode[STIX]{x1D703}=0$ corresponds to north.

Figure 12.

Horizontal plane streamlines on (a) $z=0$ , (b) $z=0.2$ and, (c), $z=0.4$ for the eigenfunction at a particular instant of time, that corresponds to the eigenvalue in equation (6.56), labelled ▪ in figure 10. For this computation, $S=20$ , $\unicode[STIX]{x1D702}=0.1$ and $N=40$ . (d) The dashed curve is the azimuthal velocity at $r=1$ for this eigenfunction, and the solid line is the negative of the azimuthal pressure gradient, $v_{0}\,\text{d}v_{0}/\text{d}\unicode[STIX]{x1D703}$ , also at $r=1$ . The top (north) and bottom (south) of the base slope have been shown in (b). Note that $\unicode[STIX]{x1D703}=0$ corresponds to north.

The eigenfunctions computed herein for $S=20$ using this asymptotic theory correspond to experiment E in table 1, whose streamlines at $z=0.2h$ and $0.4h$ are shown in figure 6. Clearly the near axisymmetry of the data shown in figure 6 corresponds to what is evident in figures 11 and 12 above. It would have been desirable to obtain experimental data much closer to the lower slope, where the Rossby-wave amplitudes are larger, but as noted in § 2.2, optical problems with light reflections off the slope made that impossible. At $S=20$ there is certainly very weak evidence in the $z=0.4$ data for any boundary-layer eruption, and consequent vortex formation. Eigenfunctions corresponding to eigenvalues toward the left of the array shown in figure 10 tend to exhibit smaller scales, with a high degree of asymmetry – for example the mode shown as ♦ – see appendix B.

6.3.3 Unsteadiness for $\unicode[STIX]{x1D706}=O(\unicode[STIX]{x1D702})$ : frictional decay

We know from the upper layer that there are modes there with $\unicode[STIX]{x1D706}=O(\unicode[STIX]{x1D702})$ , and those modes are present as a part of $c$ in the interior. Careful examination of the third of equations (6.48) shows that for $\unicode[STIX]{x1D706}=O(\unicode[STIX]{x1D702})$ , the first Fourier–Bessel coefficient is simply related to $C$ – very like what happens in the steady state. Then,

(6.58) $$\begin{eqnarray}\displaystyle \hat{a}_{k0}=-\frac{8c_{f}}{\unicode[STIX]{x1D6FC}_{k0}^{3}J_{1}(\unicode[STIX]{x1D6FC}_{k0})}, & & \displaystyle\end{eqnarray}$$

and since $k_{\ell }$ is simply related to a sum over $a_{k0}$ as in (6.47), we have

(6.59) $$\begin{eqnarray}\displaystyle k_{\ell f}=-8c_{f}\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}, & & \displaystyle\end{eqnarray}$$

from which, using (6.44), we have the relationship between $c_{f}$ and $\unicode[STIX]{x1D712}_{f}$ as

(6.60) $$\begin{eqnarray}\displaystyle \left[8\mathop{\sum }_{m=1}^{\infty }\frac{1}{\unicode[STIX]{x1D6FC}_{m0}^{3}}-\frac{hS^{1/2}}{2}\right]c_{f}=-\frac{\unicode[STIX]{x1D712}_{f}}{S^{1/2}}. & & \displaystyle\end{eqnarray}$$

Using this result gives the complete solution for $\unicode[STIX]{x1D6EC}$ values on this long time scale, in § 6.2.2.

However, things are actually more complicated than is indicated above in (6.58): it turns out that, in the notation of equations (B 1) rather than (6.48), on this time scale all of the vectors, $\boldsymbol{A}_{n}$ with $n$ even are identically zero (actually, they are of order $\unicode[STIX]{x1D702}^{2}$ ), apart from $\boldsymbol{A}_{0}$ . For $n$ odd, we write $\boldsymbol{A}_{n}=\unicode[STIX]{x1D702}\tilde{\boldsymbol{A}}_{n}$ . In the same notation as in (B 1), we have

(6.61) $$\begin{eqnarray}\displaystyle \mathscr{D}_{0}\tilde{\boldsymbol{A}}_{1}=-(c_{f}+\unicode[STIX]{x1D6EC}k_{\ell f})\boldsymbol{v}+(\unicode[STIX]{x1D6EC}+\unicode[STIX]{x1D64B}_{0})\boldsymbol{A}_{0},\quad \boldsymbol{A}_{0}=c_{f}\;\mathscr{C}_{0}^{-1}\boldsymbol{v}^{\prime }, & & \displaystyle\end{eqnarray}$$

where the matrix $\unicode[STIX]{x1D64B}_{0}$ is diagonal, with its $i$ th element given by $\unicode[STIX]{x1D6FD}_{i}/2$ . For the other odd terms,

(6.62) $$\begin{eqnarray}\displaystyle \mathscr{D}_{n}\tilde{\boldsymbol{A}}_{n+1}=-\mathscr{C}_{n}\tilde{\boldsymbol{A}}_{n-1},n=2,4,6,\ldots . & & \displaystyle\end{eqnarray}$$