3.1. The dimensionless angular velocity $\omega$

It is convenient to express the reduced volume and momentum balances in dimensionless form. We scale lengths with $a$, and velocity with $W$. We introduce the scaled angular velocity

(3.7)\begin{equation} \omega(r) = T^{-{1/2}}\,v(r)/r \end{equation}

and the dimensionless $\varepsilon = (H/2)^{1/2} T^{-1/4}$. After some algebra, the above-mentioned equation for $v(r)$ (obtained by the combination of volume continuity with the azimuthal momentum equation) can be expressed as

(3.8)\begin{equation} 2 \varepsilon ^2 (1 - f(r)/H)\left(\frac{{\rm d}^2 \omega}{{\rm d} r^2} + \frac{3}{r}\,\frac{{\rm d} \omega}{{\rm d} r}\right ) - [1+ (1 + {f'} ^2)^{1/4}] \omega = 1 \quad (0 \le r \le 1) \end{equation}

with the boundary conditions $\omega (1) =0,\ ({\rm d} \omega /{\rm d} r) (0) = 0$. The drag prediction (3.6) is then expressed as

(3.9)\begin{equation} D = 4 {\rm \pi}T^{3/2} \int_0^1 (|\omega(r) |)\,r^3 \,{\rm d} r \times (W \nu \rho a). \end{equation}

The shear term on the left-hand side of (3.8) represents the contribution of the $1/4$ layer to the outward radial transport of the fluid; the next term represents the contribution of the Ekman layers. The right-hand side of (3.8) represents the upward motion that generates the radial fluxes. Evidently, a negative $\omega$ is needed in the upper core. (For the lower core, not solved here, (3.8) with $-1$ on the right-hand side applies. Therefore $-\omega$ of the upper core is the solution for the lower core.)

For $\varepsilon = 0$, we obtain analytically the geostrophic solution

(3.10a,b)\begin{align} \omega_0(r) =- \frac{(1 - r^2)^{1/4}}{1 + (1 - r^2)^{1/4} }, \quad D_0 = \left(\frac{43}{105}\,{\rm \pi}\right) T^{3/2} \times (W \nu \rho a) = 1.29T^{3/2} (W \nu \rho a). \end{align}

For a finite $\varepsilon$, we use a finite-difference solution for (3.8) and (3.9). In this case, we must specify $H$ and $T$ (or $\varepsilon$ and one of these parameters). We obtain, numerically,

(3.11a,b)\begin{equation} \omega_{qg}(r; \varepsilon,H),\quad D_{qg} = \mathcal{D}(\varepsilon,H)\,T^{3/2} \times (W \nu\rho a). \end{equation}

We expect $\omega _{qg}(r)/\omega _0(r) <1$ and $\mathcal {D} < 1.29$. Note the difference between (3.10a,b) and (3.11a,b): the first depends only on $T$; the second predicts a more complex behaviour, because for a given $T$, various values of $\varepsilon$ and $H$ are possible practically.

By setting $f =0$, the rising disk problem is recovered; in this case, for $\varepsilon \ll 1$, (3.8) reduces to $\varepsilon ^2 \omega '' - \omega = 1/2$, hence

(3.12)\begin{equation} \omega_{qg\ disk} =- \tfrac{1}{2} [ 1 - \exp ((r-1)/ \varepsilon) ], \end{equation}

which illustrates the classical Stewartson $1/4$ layer in this context. When $\varepsilon$ is not very small, curvature terms distort the exponential decay, and the $\omega (1) =0$ boundary condition affects the entire domain, including a reduction of $|\omega (0)|$. For simplicity of discussion, we include this behaviour in the concept of the $1/4$ Stewartson layer. Quantitatively, the numerical solution of (3.8) that is used in our comparisons contains the curvature terms and the effect of non-small $\varepsilon$.

We note that the reduced formulation (3.8) and (3.9) is based on reliable physical balances (Ekman layer transport, volume conservation, momentum equations in radial, azimuthal and axial directions). The simplicity of the result is a consequence of simplifications that can be justified for asymptotic limits of the parameters. The physical relevance of the results is for finite values of $Ro$, $\varepsilon$ and $\varepsilon _{1/3}$. The accuracy of the predictions for realistic values of the parameters must be tested by comparison with realistic data.

4.1. Difficulties

The assessment of the theory by the available experiments encounters various difficulties. Formally, experiments for small $Ro$ are expected to provide $D/D_0$ and also $\omega (r)$, whose dimensionless values are of the order of unity, that can be compared straightforwardly with the theory. A close inspection of the major available data reveals that (1) they do not cover the parameter range needed for a conclusive comparison, and (2) there are uncertainties that cast some doubts on the data.

4.2. The angular velocity $\omega (r)$

It is convenient to start the comparisons with the angular velocity in the cores. There is consensus in the literature that the flow is dominated by the Ekman layers, and this requires that (1) the azimuthal velocity $v$ is of the order of magnitude $T^{1/2} W$, and (2) $v_u<0$ while $v_l >0$. (Again, $u, l$ denote the upper and lower cores, respectively.) The behaviour of the scaled $v/r$, denoted $\omega$, as a function of the dimensionless $r$, is of interest; see § 3.1.

We note that the comparison concerning the data of $\omega (r)$ of the fluid in the cores is inconclusive. Few data have been reported, and there is no overlap between Max68 (using die tests) and K23 (using a particle image velocimetry technique).

Figure 10 of Max68 displays data of $|\omega |$ close to the centre versus $Ro\,T$ obtained with a towed sphere, for $4000< T<16\,000$. The figure provides the following conclusions. (1) As expected, $\omega _u(0)<0$ while $\omega _l(0)>0$. (2) The value $|\omega _u(0)|$ is slightly smaller than $\omega _l(0)$. This observation is surprising, because it contradicts the prediction of the theory and is in contrast with the more detailed measurements of K23. (To understand this effect, the extension of the quasi-geostrophic analysis for finite $Ro$ must be used. Qualitatively, as suggested by figure 1, the advection terms compress $\varepsilon _u$ and enlarge $\varepsilon _l$, therefore $|\omega _u|/\omega _l >1$ is expected.) (3) The typical value of $\omega (0)$ decreases from 0.45 to 0.35 as $Ro\,T$ increases from 7 to 28. The interpretation of this variation is problematic because apparently both $Ro$ and $T$ were varied between the points (no details are provided). Max68 remarks that there was big scatter in these data.

Figure 7 of K23 provides more detailed $\omega (r)$ information for $T = 2890$ and $Ro = 0.0015$ (approximately). We consider the measurements for a sphere in mid-position (the points with open circle and full diamond in that figure). The digitized and rescaled data of $\omega (r)$ are plotted in figure 2, together with the theoretical predictions (geostrophic, quasi-geostrophic and corrected quasi-geostrophic).

Figure 2.

Plots of $|\omega |$ versus $r$ for $T = 2890$. The lines with symbols are data for the upper and lower cores from K23 figure 7. Also shown are the theoretical results: geostrophic (dotted line), quasi-geostrophic (solid line), and corrected quasi-geostrophic (dash-dotted line).

We note that a quantitative comparison of $\omega$ data between Max68 and K23 is not possible because of significant incompatibly of $T$ and $r$ position of the reported points. The agreement concerning the signs of $\omega _l$ and $\omega _u$ is a trivial confirmation of the Ekman-layer control of the flow. However, K23 report $|\omega _u| > \omega _l|$, in agreement with the theory (in contrast with Max68). On the other hand, we note that the difference between $\omega _l$ and $|\omega _u|$ reported by K23 is much larger than in Max68. We have no theoretical explanation for this observation. Such a significant asymmetry is expected to produce a torque and thus rotation of the sphere; but figure 4 of K23 indicates no such rotation for a sphere in middle position. We speculate that the $\omega _l$ of the K23 data was contaminated by some instability or measurement problem. We conjecture that the data for $\omega _u$ are correct, and, given the very small $Ro = 0.0015$ of the experiment, a repeated experiment will find a close match in the lower core.

4.2.1. Semi-empirical corrections of $\omega$ and $D$

The $\omega (r)$ data of K23 throw some light on the theory. Figure 2 indicates a significant discrepancy between the predicted and measured $\omega (r)$ about the cylinder $r=1$. The theory uses the boundary condition $\omega (r = 1) = 0$. The data indicate that this condition is fulfilled at $r \approx 1.3$. We note that a similar shift of the $\omega = 0$ point has been detected theoretically for a disk; see figure 5 of Minkov et al. (Reference Minkov, Ungarish and Israeli2000). We attribute this shift of the $\omega = 0$ condition to the presence of a thick $1/3$ Stewartson layer: here, $\varepsilon _{1/3} = 0.15$. The exact mechanism that connects the Ekman layers to this $1/3$ layer is obscure presently. The theoretical analysis of this flow is expected to be complicated and the results impractical, because small powers of $E=(1/T)$ are involved, and an asymptotic separation of terms can be attained only for extremely large values of $T$ (say $10^{12}$). For progress, we adopt a semi-empirical approach.

Figure 2 suggests that the quasi-geostrophic $|\omega (r)|$ decreases too sharply to $0$ at $r=1$. A more realistic profile is obtained by a shift of the $\omega = 0$ point to $r = 1 + \varDelta$, where $\varDelta =c (2 H /T)^{1/3}$, and $c$ is a coefficient of the order of 1. (This $1/3$ layer is on the outer side of the sphere, hence the relevant dimensionless height is $2H$, and the thickness is $2^{1/3} \varepsilon _{1/3}$.) Since the shift is due to the Stewartson layer, it should not affect the centre domain. We postulate the corrected profile as

(4.1)\begin{equation} \omega_c(r) = \left\{\begin{array}{@{}ll@{}} \omega_{qg}(r=0) & (0 \le r \le \varDelta), \\[3pt] \omega_{qg}(r-\varDelta) & ( \varDelta < r \le 1+\varDelta), \end{array}\right. \end{equation}

where $\omega _{qg}(r)$ is the quasi-geostrophic result defined in § 3. Numerical tests indicate that $c=1.2$ is a good choice for the range of parameters considered in this paper. The corrected $|\omega (r)|$ is shown in figure 2. The agreement with the data for the upper core is improved significantly, but no reliable conclusions can be drawn from one value of $T$. The importance of this correction is that it suggest a significant contribution to the drag force, which can be tested later for the points of tables 1 and 2.

Table 1.

Max68 data, where $\varepsilon$ and $\varepsilon _{1/3}$ are calculated with $H = 10.5$. A $*$ in the last column indicates that the point is discarded because of too large $Ro\,T^{1/2}$ or some big scatter from neighbouring points. Here, $x_3= Ro\,T^{2/3}$, $y_3 = c_D\,Ro\,T^{-{1/2}}$ are the coordinates in figure 3 of Max68.

Table 2.

Data points of K23. Set I corresponds to $j_p = 1\unicode{x2013}9$, set II corresponds to $j_p=10\unicode{x2013}24$, and the rest of the points are for set III. Here, $x_3,y_3$ are defined as in the caption of table 1.

Recall the connection (3.9) between the drag and the angular velocity. The integral is based on the balance between the pressure and Coriolis over/below the sphere. The shift of the boundary condition $\omega = 0$ to a larger radius does not affect this balance. Consequently, the corrected $\omega _c(r)$ (4.1) yields the corrected drag

(4.2) \begin{align} D_c &= 4 {\rm \pi}T^{3/2} \int_0^{1} |\omega_c(r)|\,r^3 \,{\rm d} r \nonumber\\ &=4 {\rm \pi} T^{3/2} \large \left[\frac{1}{4}\,\varDelta^4\omega_{qg}(0) + (1 + \varDelta)^3 \int_0^{1-\varDelta} |\omega_{qg}(r)|\,r^3 \,{\rm d} r\large\right]\!, \end{align}

where $\varDelta = 1.2 (2 H /T)^{1/3}$. The implementation of this correction is straightforward. The accuracy and insights of this correction will be discussed later.

4.3. The drag

We recall (2.6)–(2.8a–c). We realize that the experiments of Max68 and K23 provide (upon some simple reprocessing) about 70 data points of $D/D_0$ for various combinations of large $T$ and small $Ro$. However, the strategies of variation of these parameters were different between these studies. The full methodology of Max68 has not been reported in the paper, and the original data are no longer available, so we must fill the gaps by some plausible assumptions. The first assumption is that all experiments of Max68 share the same $\nu$ (of water; this hypothesis cannot be verified). Max68 used a cylinder of height 40 cm and outer radius $r_O = 14$ cm.

Max68 presents clusters of $D/D_0$ for a fixed $T$ and various $Ro$. We infer that the same size of sphere and the same $\varOmega$ were used for a certain $T$; the spheres differed in $g'$, and this produced various buoyant forces $B$, and hence various $W$ and various $Ro$. This explains the clusters. However, a difficulty arises because Max68 has spheres of two sizes, $a = 1.905$ and $3.885$ cm. Therefore, the unspecified parameter $H$ was either $10.5$ or $5.2$. We infer that the first value is relevant to the drag data. The justification is that the agreement with the data of K23 (which have a known $H=9.4$) suggests that the value of $H$ in the Max68 drag measurements was close to 9.4. For use in this paper, the point data of Max68 were digitized from figures 2 and 3 of that paper; see table 1.

In the experiments of K23, only one sphere was used, of radius $a=1.20$ cm and density $\rho _p = 0.90\ {\rm g}\ {\rm cm}^{-3}$. (K23 uses the notation $\rho _f$ and $\rho _S$ for the densities of the fluid and sphere.) The embedding cylinder was of height 22.5 cm and radius $r_O = 2.65$ cm. Therefore only one value of the parameter $H =9.4$ applies to all the experiments. K23 has three sets of fluid density $\rho$ and kinematic viscosity $\nu$, as follows (in cgs units):

(4.3)\begin{equation} \left.\begin{gathered} {\rm I}\quad \rho =1.170, \nu = 0.107;\\ {\rm II}\quad \rho =1.127, \nu = 0.047;\\ {\rm III}\quad \rho =1.000, \nu = 0.010. \end{gathered}\right\} \end{equation}

Each set has a fixed $g'$ (and hence a fixed buoyant force $B$). For each set, the variation of $\varOmega \in (35, 90)$ s$^{-1}$ produced various $T$ (determined by the definition (2.2a,b)) and $Ro$ (calculated from the measured $W$). This methodology does not produce clusters of data points with the same $T$ (as in Max68). Since $T \propto 1/\nu$, the typical $T$ increases from set I to II, and finally III. The point data used in our work were obtained from the first author of K23 in private communication as $(W, \varOmega )$ for the sets I–III (with the correction that in set II, the values are $\rho = 1.127$, $\nu =0.047$, cgs units, not as printed in the journal); see table 2.

Max68 and K23 released the buoyant particle near the bottom and recorded the upward motion, from which the value of $W$ was calculated. The initiations of the motion were different: Max68 released the particle from a cage, while K23 changed rapidly the orientation of the spinning cylinder from horizontal to vertical. In our opinion, this difference has negligible influence on the subsequent propagation. There is, however, an important point that apparently has been missed by both papers. The theory assumes that the flow field is quasi-steady. This implies that the angular velocity of the cores between the Ekman layers has been spun-up (at release, the fluid is in solid-body rotation $\omega =0$). The estimate (verified by Ungarish Reference Ungarish1997) of the spin-up time is $2 H T^{{1/2}}/ \varOmega$, which must be smaller than $H a / W$. The corresponding restriction is $Ro\,T^{1/2} \ll 1$. It turns out that not all the data points satisfy this condition, and this may create considerable confusion (we will discard points with $Ro\,T^{1/2} >0.4$).

Another important difference is the ratio $r_O/a$, which in Max68 was 7.3 (small particle) and 3.6 (large particle), while in K23 it was only 2.2. Thus, as explained later, the influence of the outer wall can be safely discarded for Max68, but could play some role in the sets I and II of K23 (which display thick $1/4$ layers).

Consider the data points for the drag analysis listed in tables 1 and 2.

Inspection of the tables reveals that in general, Max68 used larger values of $\psi = g' a^3/\nu ^2$ than K23. We attribute this to the fact that K23 used more viscous fluids (larger $\nu$) for sets I and II. This parameter was large in all cases, hence this difference between Max68 and K23 is insignificant.

The main free parameter in these tables is $T$. The $T$ overlap between the tables is the interval 2500–11 000; since the data of Max68 for $T= 2500$ violated the $Ro\,T^{1/2} < 0.4$ restriction, the practical overlap is only from $T_1 = 4300$ to $T_2= 11\,000$. The large interval (6700) is an illusion. The theory indicates that $D/D_0$ depends on $\varepsilon$ (the thickness of the Stewartson $1/4$ layer), i.e. the relative effective range variation is $(T_2/T_1)^{1/4} -1 = 0.26$. In other words, agreement of $D/D_0$ between Max68 and K23 will be encouraging, but cannot serve as a criterion for assessing the accuracy of the combined data. (We reiterate that the value of $H$ was not the same, and this is also a source for disagreement between Max68 and K23.)

K23 made a brief attempt at data comparison with Max68, using a plot of $\log W$ (similar to $- \log D$) versus $T$; see figure 8 of K23. The scatter and differences are obscured by the log-log plot, and were not discussed. The suggested reconciliation between theory and experiment (of both Max68 and K23) was based on a curve-fit formula for $W$ (scaled with $4 g' a^2/\nu$) as a function of $T$. More details on this issue are presented in Appendix A.

The only comparison with theory in the papers of Max68 and K23 concerns the geostrophic prediction (2.6). We attempt a more detailed analysis. In particular, we argue that the dependent variable $D/D_0$ (or $W/W_0$) provides a much more clearer and reliable accuracy test than $W$ or $D$ (in some scaled form) used in the previous studies.

Figure 3 displays the available $D/D_0$ data versus $T$, where squares and circles correspond to Max68 and K23. Figure 3(a) shows all the data points of tables 1 and 2. We observe (1) a big scatter in the data of Max68, and (2) a change of slope, from negative to positive, for the data of K23 for $T< 800$. (Points $j_p = 1,2$ of K23 (table 2) have large $\varepsilon$ and were prone to the influence of the outer wall of the cylinder. This may explain the unexpected slope.) An inspection of these (apparently problematic) points reveals that most of them belong to non-small values of $Ro\,T^{{1/2}}$, i.e. violate the quasi-steady-state assumption. We eliminate the problematic points (marked with asterisks in the tables) from our analysis. The remaining data points are displayed in figure 3(b). The subsequent analysis of $D/D_0$ uses only these points, which satisfy the restriction $Ro\,T^{1/2} < 0.4$.

Figure 3.

The data points $D/D_0$ versus $T$: squares for Max68, circles for K23. (a) All the points of tables 1 and 2. (b) The data points after excluding those marked by asterisks in the tables, and hence satisfying $Ro\,T^{1/2} < 0.4$.

The quasi-geostrophic theory predicts that $D/D_0$ is a function of $\varepsilon = (H/2)^{1/2} T^{-1/4}$ and $H$. Since $H \approx 10$ for all the data, we expect that $D/D_0$ collapses on a line of $\varepsilon$. This has motivated the plot in figure 4. We see that the data of Max68 and K23 collapse fairly well on the trend line $1-0.9 \varepsilon$. There is some scatter in both directions, which can be attributed to measurement errors. The scatter is smaller for the data of K23, which reflects the significant progress of measurement and recording methods. (It is possible that the novel techniques of particle release used by K23 have also contributed in this direction.) The figure confirms that there is only a small overlap between the data of Max68 and K23: sets I and II of K23 have significantly larger $\varepsilon$ than the points of Max68. (We note that the assessment of the theory could benefit greatly from smaller values of $\varepsilon$. Therefore, additional experiments are still needed.)

Figure 4.

Drag as a function of $\varepsilon$; data of Max68 (squares) and K23 (circles). Also shown are the trend line, the quasi-geostrophic prediction and the corrected quasi-geostrophic prediction. The predictions were obtained by finite-difference solution of (3.8), (3.9) and (4.2) for $H=10$.

The predictions of the geostrophic and quasi-geostrophic theories are also shown in figure 4. The data correspond to $\varepsilon >0.2$. The more than $20\,\%$ discrepancy with the geostrophic approximation, derived for $\varepsilon =0$, could be anticipated. However, the weakness of the geostrophic theory is also on the qualitative aspect: it misses completely the effect of the drag decrease when $\varepsilon$ increases. The quasi-geostrophic line shows clearly this qualitative effect. However, the quasi-geostrophic drag reduction with increasing $\varepsilon$ is exaggerated as compared to the data. We recall that the quasi-geostrophic approximation assumes $\varepsilon _{1/3} = 0$, or rather $\varepsilon _{1/3}/\varepsilon \ll 1$, i.e. a very thin $1/3$ Stewartson layer. An inspection of the data in the tables reveals that this requirement is fulfilled neither by Max68 nor by K23. Since the $1/3$ and $1/4$ layers are inseparable, we infer that low accuracy of the quasi-geostrophic prediction for the experimental data should be attributed to the presence of a thick $1/3$ layer.

This hypothesis is consistent with the measured profile of $\omega (r)$ discussed in § 4.2.1, where a semi-empirical correction to $D/D_0$ has been suggested. The effect of the semi-empirical correction (4.2) is shown in figure 4. Evidently, the line for the corrected drag $D_c/D_0$ is much closer to the data than the lines of geostrophic and quasi-geostrophic prediction. Since now the agreement is over a range of $T$, we can infer that this correction captures correctly the physical mechanism that was missing in the original theories. The geostrophic theory discards the influence of the Stewartson layers, and hence predicts a constant $D = D_0$ for all $T$. The quasi-geostrophic theory takes into account the $1/4$ Stewartson layer, and hence predicts correctly that $D/D_0$ decreases with $\varepsilon \propto T^{-1/4}$, but overpredicts the rate of decrease (as compared with the available data). The present investigation suggests that the $1/3$ Stewartson layer reduces the influence of the $1/4$ layers. The details of this reduction are complicated, but there is evidence that the net effect is a shift of the $\omega = 0$ condition to a larger radius. This shift increases significantly the quasi-geostrophic drag.

The agreement of the corrected drag $D_c$ with the data is encouraging, but no clear-cut conclusion can be drawn yet. We must keep in mind that the tests were performed for only one value of $H$, while the correction contains an adjustable constant, $c$. The robustness of such a correction needs additional verification.

4.3.1. The effect of $Ro$

Max68 has suggested that the difference between theory and data is due to inertial effects, i.e. the non-sufficiently small $Ro$. Here, we reconsider this interpretation. Historically, Max 68 was of course unaware of the quasi-geostrophic theory (Ungarish Reference Ungarish1996) that attributes the drag discrepancy to the Stewartson layers (mostly). Max68 detected correctly that some inertial terms become important in some momentum balances when $Ro$ is not very small, and naturally attributed to these terms the discrepancy with the linear ($Ro =0$) theory. However, the geostrophic theory is just a branch of the linear theory for $\varepsilon \to 0$. The flows of Max68 display non-small $\varepsilon$. We argue that the drag (which is an integral result) can be affected more by the finite $\varepsilon$ than by the finite $Ro$. Indeed, a careful inspection of the inertial terms reveals that in steady state, they make opposing contributions to the pressure in the upper and lower cores; therefore, the net contribution to the drag cancels out in favour of the linear drag result. A significant inertial influence on the drag reduction (compared to $D_0$) is during spin-up, expected for $Ro\,T^{1/2} > 0.5$, because both the upper and lower cores are close to the initial condition of zero drag. Max68 included such points in the analysis. K23 did not extend the discussion of the inertial terms. We have deliberately excluded from the analysis data points corresponding to $Ro\,T^{1/2}>0.4$.

An inspection of the inertial terms discarded in the geostrophic and quasi-geostrophic linear theories indicates that the relative local contributions are of the orders of magnitude $Ro$, $Ro\,T^{1/2}$ and $Ro\,T^{2/3}$. The amplification of $Ro$ appears because initial accelerations and local flow-field gradients enhance the effect of the advection terms. We can now check the influence of these parameters on the drag data.

Figure 5 displays the data points satisfying $Ro\,T^{1/2} < 0.4$ as a function of $Ro$, $Ro\,T^{1/2}$ and $Ro\,T^{3/2}$. It is evident that there is no tendency of $D/D_0$ to approach 1 as $Ro \to 0$. While the data of Max68 show a slight decrease of $D/D_0$ as $Ro$ increases, the data sets of K23 show a remarkable invariance with $Ro$. Note that the lines for sets I and II of K23 are clearly below the lines of set III and Max68 for all values of $Ro$, $Ro\,T^{1/2}$ and $Ro\,T^{2/3}$. The points of the lower lines have larger $\varepsilon$ (about 0.35) than these in the upper line (about 0.2); see figure 4. The change of $D/D_0$ between the various sets is certainly not a result of different values of $Ro$.

Figure 5.

Drag data as a function of (a) $Ro$, (b) $Ro\,T^{1/2}$ and (c) $Ro\,T^{2/3}$. Data of Max68 (squares) and K23 (circles).

Note that figure 5(c) for $(D/D_0)$ versus $Ro\,T^{2/3}$ is closely related to figure 3 of Max68. With the aid of (2.8a–c), we find that the vertical axis of this figure is $0.82 (D/D_0)$, while the horizontal axis is also $Ro\,T^{2/3}$. We argue, however, that the representation in the figure of Max68 is confusing because: (1) it uses data of accelerating (during the spin-up process) flows, which creates a bias toward small drag at larger $Ro$; (2) the horizontal axis is logarithmic, which precludes extrapolation to $Ro =0$. Our figure 5(c) is more insightful, and, moreover, it has the benefit of the additional data of K23. We can derive, with confidence, a conclusion concerning the effect of $Ro$.

The conclusion from figure 5 is that $Ro$ had a negligible effect on $D/D_0$ for the data used in our analysis, i.e. for $Ro\,T^{1/2} < 0.4$. The effect of $\varepsilon$ shown in figure 4 is sharper, and supported by a clear-cut theory.