3.1. The generated wave beam in the absence of topographies

To begin with, we check the vertical density profile measured in the system, which is shown in figure 3(a). Here, the inset shows the measured temperature profile, which is used to calculate the density profile by (2.1). The buoyancy frequency $N$ (calculated by (2.2)) is shown in figure 3(b), where there are four data sets that were measured over a period of one month. The overlap of these four data sets shows excellent stability of the established stratification. The buoyancy frequency is not uniform in the vertical direction. However, within the measuring region, indicated by the two dashed lines in figure 3(b), the buoyancy frequency is nearly constant, with mean value 0.42. As indicated by (1.1), the value of $N$ determines the upper limit of the frequency of the generated wave.

Figure 3.

(a) The density profile from data set 1, which is calculated by the measured temperature profile (inset). (b) The profiles of buoyancy frequency from four data sets measured over a period of one month. The dashed lines indicate the vertical range of the measuring window.

We now examine the structure of the generated wave in our system in the absence of topographies. An example of the vector maps of the measured internal wave field is plotted in figure 4(a). Here, $A/D=0.03$ and $\omega /N=0.21$. Also, $T$ is the length of the wave period, and $t=T/4$ is the time when the cylinder reaches the equilibrium position and moves downwards.

Figure 4.

Vector maps of internal wave velocity with magnitude $\sqrt {u^{2}+v^{2}}$ ${\rm cm}\ {\rm s}^{-1}$ coded in both colour and arrow length at $A/D=0.03$, $\omega /N=0.21$ and $t=T/4$: (a) experimental measurement; (b) theoretical prediction. The black solid line and red dashed line indicate, respectively, the perpendicular and parallel directions of the energy propagation of the waves.

Assuming that the boundary condition at the surface of the wave generator is free-slip, and that the variation of the excited internal wave in the across-beam direction is much larger than that in the along-beam direction, Hurley & Keady (Reference Hurley and Keady1997) proposed a viscous linear theory to describe the internal waves generated by a vibrating elliptic cylinder in stratified fluid of constant buoyancy frequency. Such a theory (referred to as HK theory hereafter) has been used to compare with previous experiments (Sutherland et al. Reference Sutherland, Dalziel, Hughes and Linden1999; Zhang, King & Swinney Reference Zhang, King and Swinney2007). In figure 4(b), we plot the wave field predicted by the HK theory with the same values of the parameters as in the experiment. One sees that the experimentally measured wave field shows good correspondence with that predicted by theory.

To make a more quantitative comparison, we plot the profiles of along-beam velocity (velocity in the direction of wave propagation) $V_a$ in figure 5, which are taken across the black lines shown in figure 4. Here, $b$ represents the relative distance of the points on the black line to the centre of the line, and the minus or plus sign of $b$ means the point is located below or above the midpoint on the black line. From figures 5(a) and 5(c), we see that the experimental results can be described very well by the HK theory. In figures 5(b) and 5(d), where the wave frequency becomes larger, it can be seen that HK theory under-predicts the wave amplitude when $b$ is positive. The increase of the deviations between the experiments and the HK theory when $\omega$ increases was also found in Sutherland et al. (Reference Sutherland, Dalziel, Hughes and Linden1999). This will not be discussed further as it is not the main focus in the present study.

Figure 5.

The profiles of along-beam velocity when $t=T/4$: (a) $A/D=0.03$, $\omega /N=0.21$; (b) $A/D=0.03$, $\omega /N=0.30$; (c) $A/D=0.09$, $\omega /N=0.21$; (d) $A/D=0.09$, $\omega /N=0.30$.

Figure 6 shows the variation of the wave amplitude in the direction of the group velocity from our measurement and the prediction from the linear HK theory in a semi-logarithmic plot. One can see clearly that neither the experimental data nor the model predication follows the exponential decay described by the classical wave attenuation equation for a monochromatic wave (Lighthill Reference Lighthill1978). The figure also shows some differences between the experimental data and linear model prediction; the difference is about 10 % in the largest case. Such a deviation is also found in some previous works. For example, in Zhang et al. (Reference Zhang, King and Swinney2007), the difference between the theory and experimental data can be as large as 50 % (figure 6(d) in their paper).

Figure 6.

The variation of wave amplitude in the direction of the group velocity. The data are taken at the positions indicated by the red dashed lines in figure 4. $q$ is the coordinate along the direction of the group velocity of the waves. The origin of $q$ is the cross point of black solid lines and red dashed lines in figure 4. Note that the vertical scale in this figure is logarithmic.

3.2. Wave field near the rough slope

We now place the topography into the box and study the wave fields near the slopes with different surface roughness. The scattering of the incident internal waves can be seen clearly in figure 7, where we show the contour maps for the fields of time-averaged kinetic energy:

(3.1)\begin{equation} E =\frac{1}{t_2-t_1}\int_{t_1}^{t_2}(u^{2}/2+w^{2}/2)\,\mathrm{d} t. \end{equation}

Here, $u$ and $w$ are the horizontal and vertical velocities, respectively, $t_1$ is a time when the flow reaches steady state, and $t_2$ is chosen such that $t_2-t_1$ is larger than at least 30 oscillating periods. For simplicity, we show only the results when $A/D=0.03$ and $\gamma =0.50$. Figures 7(a) to 7(e) show that the internal waves are modulated by the surface roughness of the slope: The wave patterns exhibit light and dark stripes in the whole flow field. This is because of the constructive/destructive interference between the incident waves and the reflected waves. This effect becomes increasingly prominent as the roughness parameter $\lambda$ is increased from 0 to 0.5. As $\lambda$ becomes larger than 0.5, and the wavelength of the roughness becomes small, this kind of modulation is weakened. When $\lambda =1.5$, the wave field is very close to that in smooth surface case. One may also find that high-intensity regions of the wave are around the peak positions of the roughness elements, which is due to the energy concentration during the wave reflection. Low-intensity regions correspond to the valley positions of the roughness elements, because part of the wave is trapped near the valley and hence becomes stagnant. Moreover, figure 7 shows that the kinetic energy of an internal wave is the largest in the region very close to the surface of the topography. Surprisingly, this maximum value shows no significant dependence on $\lambda$.

Figure 7.

Contour maps for the fields of time-averaged kinetic energy at different $\lambda$ when $A/D=0.03$ and $\gamma =0.50$: (a) $\lambda =0$, (b) $\lambda =0.25$, (c) $\lambda =0.5$, (d) $\lambda =0.75$, (e) $\lambda =1.5$. The units are ${\rm cm}^{2}\ {\rm s}^{-2}$.

In figure 8, we show the fields of time-averaged dissipation rate:

(3.2)\begin{equation} \epsilon =\frac{1}{t_2-t_1}\int_{t_1}^{t_2}\varepsilon\,\mathrm{d} t, \end{equation}

where $\varepsilon$ is the instantaneous dissipation rate of kinetic energy in two-dimensional flow (Pope Reference Pope2000):

(3.3)\begin{equation} \varepsilon=\nu\left[2\left(\frac{\partial u}{\partial x}\right)^{2}+2 \left(\frac{\partial w}{\partial z}\right)^{2}+\left(\frac{\partial u}{\partial z}+\frac{\partial w}{\partial x}\right)^{2}\right]. \end{equation}

Here, $\nu$ is the kinematic viscosity of water. To measure the dissipation rate accurately, the Kolmogorov length scale should be resolved in the PIV measurement. However, the spatial resolution should not be too high, otherwise the measurement error increases (Tokgoz et al. Reference Tokgoz, Elsinga, Delfos and Westerweel2012). In this experiment, it is difficult to realize fully the above requirements at each grid point because of the spatial inhomogeneity, i.e. the resolution is a bit too low for some high-dissipation region, but may be a little too high for some low-dissipation region. Hence a direct comparison between the measured dissipation field in the rough surface cases and that in the smooth surface case can be only qualitative.

Figure 8.

Contour maps for energy dissipation at different $\lambda$ when $A/D=0.03$ and $\gamma =0.50$: (a) $\lambda =0$, (b) $\lambda =0.25$, (c) $\lambda =0.5$, (d) $\lambda =0.75$, (e) $\lambda =1.5$. The value of $\log _{10}\langle \epsilon \rangle _t$ is coded in colour. The units are ${\rm cm}^{2}\ {\rm s}^{-3}$.

Figures 8(a) to 8(e) show the modulation of the surface roughness on the energy dissipation of internal waves. For all cases, the energy dissipation is the largest near the slope, and the maximum energy dissipation in the smooth surface case is even larger than that in the rough surface case. On the other hand, in the roughness case, the region of intense dissipation seems to be less localized and is extended beyond the vicinity of the surface. This results in the distribution of dissipation rate in the rough cases being more uniform in space. Again, the dissipation field at $\lambda =1.5$ is found to be close to that in the smooth surface case.

3.3. Profiles of dissipation rate and kinetic energy

To analyse quantitatively the effect of roughness on internal wave field, we plot and then examine the profiles of the averaged dissipation rate along the direction normal to the slope. We first introduce a new coordinate system shown in figure 8(b): the origin is set at halfway along the slope surface. Two variables, $s$ and $\sigma$, represent the coordinates parallel and perpendicular to the slope surface, respectively. We then take the average of $\epsilon$ in the $s$ direction:

(3.4)\begin{equation} \langle \epsilon \rangle(\sigma)=\frac{1}{\varDelta}\int_{s_1}^{s_2} \epsilon (\sigma,s)\,\mathrm{d} s, \end{equation}

where $s_1$ and $s_2$ are equal or close to $-l/2$ and $l/2$, respectively, in the region near the slope, $l$ being the length of the rough surface of the slope ($l=16$ cm). Also, $\varDelta =(s_2-s_1)\xi$, where $\xi$ is the ratio of the number of the grid points outside the roughness elements to the total number of grid points at a certain $\sigma$. In the region far away from the slope, the length of the average region, i.e. $s_2-s_1$, would be smaller than $l$ due to the finite size of the measuring window, which may partly reduce the accuracy of $\langle \epsilon \rangle$. However, as most of the energy is dissipated near the slope, such a finite size effect is negligibly small in the final results. Although the measured energy dissipation may be less than accurate at some grid points, as mentioned before, we have verified that the obtained profiles of $\langle \epsilon \rangle$ are accurate and reliable (for details about the verification, please refer to Appendix A). For simplicity, we will show only the dissipation profiles that were measured with $A/D=0.03$.

In figures 9(a,b), we plot the normalized average dissipation rate $\langle \epsilon \rangle /\epsilon _0$ versus the normalized distance normal to the slope, $\sigma /h$. Here, $\epsilon _0=A^{2}\omega ^{3}$, and $h$ is the height of the roughness element, as we said before. These figures show that the surface roughness reduces the maximum value in the dissipation profile. It is also seen that the peak width of the profiles in the rough cases is larger than that in the smooth surface case. This is consistent with the observation from figure 8 that the energy dissipation becomes less localized and is spatially more uniform when the slope surface is rough. Another feature shown in these figures is that when $\lambda \geqslant 0.50$, the energy dissipation is very weak when $\sigma$ is very close to the valley of the roughness ($\sigma \approx 0$). A possible reason is that part of the incident wave is trapped near the valley of the roughness elements and hence become stagnant, as we said before. The results are the same for other values of $\gamma$. In figures 9(c,d), we plot the profiles of the normalized accumulated dissipation rate, $\epsilon _{T}/\epsilon _{T0}$. Here, $\epsilon _{T}$ is defined as

(3.5)\begin{equation} \epsilon_T(\sigma)=\frac{\rho W_0l}{s_2-s_1}\int_{0}^{\sigma}\int_{s_1}^{s_2} \epsilon (\sigma,s)\,\mathrm{d} s\,\mathrm{d\sigma}, \end{equation}

where $\rho _0$ is the density of water, and $W_0$ is the width of the slope. Also, $l/(s_2-s_1)$ is the normalization factor, which is used to make the integral length in the $s$ direction remain the same at different $\sigma$, and thus can reduce the impact from the finite size of the measuring window. As (3.5) shows, $\epsilon _{T}(\sigma )$ reflects the total amount of energy dissipation in the region below a certain $\sigma$. We have $\epsilon _{T0}=\rho _0 {l_0}^{3}A^{2}\omega ^{3}$, where $l_0$ is the unit length (1 cm). Both figures 9(c) and 9(d) show that the surface roughness can also reduce the total amount of the energy dissipation, and this roughness-induced reduction shows a clear dependence on $\lambda$. When comparing figures 9(c) with 9(d), we see that the roughness-induced reduction is much more prominent at $\gamma =0.17$, suggesting that such a reduction is also highly related to the off-criticality.

Figure 9.

Profiles of (a,b) the along-slope and time-averaged dissipation rates, and (c,d) the accumulated dissipation rate. For (a,c) $\gamma =0.5$, and for (b,d) $\gamma =0.17$. Different symbols represent different roughnesses as shown in (a).

In figure 10, we examine the influence of the off-criticality on the energy dissipation, where the profiles of the averaged and accumulated dissipation rates are both shown for various values of $\gamma$. It is found that there exists a non-zero optimal off-criticality ($\gamma _{{optimal}}=0.17$) for energy dissipation in both the smooth and rough surface cases. Surprisingly, unlike surface roughness, the off-criticality is found to apparently have no effect on the peak width of the dissipation profile, but affects its shape only far from the slope. This suggests a weak dependence of the spatial distribution of the dissipation on off-criticality, which is confirmed further in figure 11.

Figure 10.

Profiles of (a,b) the normalized average dissipation rate, and (c,d) the accumulated dissipation rate. For (a,c) $\lambda =0$, and for (b,d) $\lambda =0.5$. Different symbols represent different off-criticality.

Figure 11.

Full width at half-maximum (FWHM) of the profiles of the averaged dissipation rate versus $\lambda$ at different $\gamma$.

In figure 11, we plot the normalized full width at half-maximum (FWHM) of the peaks in the profiles of the averaged dissipation rate versus $\lambda$ at different $\gamma$. It is seen that the peak width of the dissipation profile indeed depends weakly on off-criticality, as the values of FWHM/$h$ at different $\gamma$ are close. Rather, FWHM/$h$ of the dissipation profiles is determined mainly by the surface roughness. The peak width in rough surface cases can be much larger than that in the smooth surface case, as figure 9 shows. Moreover, the peak width increases monotonically when $\lambda$ decreases from 1.50 to 0.25. Figure 11 demonstrates quantitatively that the region of intense dissipation is broadened due to the effect of surface roughness.

From both figures 9 and 10, we see that the energy dissipation accumulates rapidly in the region near the slope (except for the regions where $\sigma$ is very close to 0 in some rough surface cases, i.e. near the valley of the roughness elements), and then accumulates very slowly in the region far from the slope. We introduce a quantity called the displacement boundary layer (DBL), which is used to separate the rapid accumulation and slow accumulation regions. The method to determine the thickness of the DBL is shown in figure 12(a). We extrapolate the linear parts of the accumulated dissipation profile both near the slope and far from the slope, and the distance from the slope at which these two extrapolating lines cross corresponds to the thickness of the DBL. We plot the thickness of the DBL ($\delta$) versus the off-criticality $\gamma$ at different surface roughnesses $\lambda$ in figure 12(b), which shows that $\delta$ has a weak dependence on $\gamma$, and the data scatter may be due to the errors in determining the DBL thickness. The figures also show that the DBL thickness for the smooth surface is much smaller than for the rough cases. The differences between the DBL thicknesses in the smooth and rough surface cases are close to the height of the roughness element. This suggests that such differences are determined mainly by the height of the roughness element.

Figure 12.

(a) Illustration of determining the thickness of the displacement boundary layer (DBL) by the slope method. (b) The thickness of the DBL versus off-criticality at different surface roughnesses.

To examine the effects of roughness on energy dissipation more completely, we plot in figure 13(a) the maximum value of the averaged dissipation $\langle \epsilon \rangle _{max}$ in the profiles measured at all $\gamma$, versus the surface roughness $\lambda$. It is seen that for each $\gamma$, $\langle \epsilon \rangle _{max}$ first decreases and then increases with increasing $\lambda$. Except for the case with $\gamma =0$, $\langle \epsilon \rangle _{max}$ reaches a minimum at $\lambda =0.5$, where the modulation effect is most pronounced, as shown in figure 7. The ratio of $\langle \epsilon \rangle _{max}$ at $\lambda =0$ (smooth surface case) to that at $\lambda =0.5$ can be as large as $\sim$10. We also plot the total dissipation rate at $\sigma =2\delta$, namely $\epsilon _T|_{\sigma =2\delta }$, versus $\lambda$ in figure 13(c). When $\sigma$ reaches $2\delta$, the dissipation has decayed significantly and the accumulated dissipation $\epsilon _T$ increases very slowly with the distance $\sigma$, hence $\epsilon _T|_{\sigma =2\delta }$ can be used as a measure of the total energy dissipation. Although the relation between $\epsilon _T|_{\sigma =2\delta }$ and $\lambda$ is somewhat complex, a broad feature is that the surface roughness reduces $\epsilon _T|_{\sigma =2\delta }$ under most circumstances. Figure 13(c) also shows that the total energy dissipation has less dependence on $\lambda$ when compared with the maximum value in the dissipation profile. The reason is that although the surface roughness reduces the maximum value of the dissipation rate, it broadens the region of intense dissipation. The combination of these two effects reduces the dependence of the total accumulated dissipation on the surface roughness. In figures 13(b) and 13(d), we plot $\langle \epsilon \rangle _{max}$ and $\epsilon _T|_{\sigma =2\delta }$ versus off-criticality $\gamma$, respectively. It is clear that there exist an optimal off-criticality at which both $\langle \epsilon \rangle _{max}$ and $\epsilon _T|_{\sigma =2\delta }$ reach their respective maximums. This optimum off-criticality remains invariant for different surface roughnesses, and equals 0.17. It should be noted that because of the sparsity of the data, this number serves to provide only a rough measure of the true optimal value. Another feature seen from figure 13(d) is that the value of $\epsilon _T|_{\sigma =2\delta }$ for $\gamma =0$, i.e. waves with critical incidence, is the same for surfaces with different roughnesses, even for a smooth surface, which is somewhat unexpected. It would be interesting to explore the relation between the criticality of the slopes of the roughness elements (the definition of the criticality is shown in § 1) and the energy dissipation of waves. However, as already mentioned (see Appendix C for details), this criticality is not a good choice as a control parameter.

Figure 13.

Maximum value in the averaged dissipation profile $\langle \epsilon \rangle _{max}/\epsilon _0$ versus (a) roughness $\lambda$, and (b) off-criticality $\gamma$. Accumulated dissipation rate at $\sigma =2\delta$, $\epsilon _T|_{\sigma =2\delta }/\epsilon _{T0}$, versus (c) roughness $\lambda$, and (d) off-criticality $\gamma$. The symbols represent different off-criticality in (a,c), and different surface roughnesses in (b,d).

It is known that stronger flow strength usually leads to stronger energy dissipation. Thus we also investigate the relationship of wave energy to surface roughness and off-criticality. In figure 14, we plot the profiles of the normalized average kinetic energy $\langle E \rangle /E_0$. Here, the method to obtain the time- and space-averaged kinetic energy $\langle E \rangle$ is the same as that used to obtain $\langle \epsilon \rangle$ in (3.4), and $E_0$ used for the normalization equals $A^{2}w^{2}$. From figures 14(a) and 14(b), we see that the surface roughness has little effect on the strength of the internal wave near the slope except that it shifts the peak position position by a distance roughly equal to the height $h$ of the roughness elements (1 cm). Figures 14(c) and 14(d) show that the wave energy near the slope can be affected by off-criticality. The more quantitative examination is shown in figure 15, where we plot the maximum value in the profile of averaged wave energy $\langle E \rangle _{max}/E_0$ versus surface roughness $\lambda$ in (figure 15a) and vs off-criticality $\lambda$ in (figure 15b). Indeed, the wave energy near the slope has no significant dependence on surface roughness, but is dependent only on the off-criticality. The optimum $\gamma$ equals 0.17 and is the same as that found in figures 13(b) and 13(d).

Figure 14.

Profiles of the normalized averaged kinetic energy $\langle E \rangle /E_0$ at different surface roughnesses $\lambda$, with (a) $\gamma =0.5$, and (b) $\gamma =0.17$; and at different off-criticality $\gamma$, with (c) $\lambda =0$, and (d) $\lambda =0.5$.

Figure 15.

Maximum value in the averaged kinetic energy profile $\langle E \rangle /E_0$ versus (a) roughness, and (b) off-criticality.

In some of the previous studies, critical reflection ($\gamma =0$) has attracted most of the attention as it can result in the strongest wave energy near the slope according to the linear inviscid theory. However, the present study suggests that the strongest wave energy exists at off-criticality 0.17, not 0. In our understanding, the optimum off-criticality should be determined by two effects: the energy concentration effect, which enhances the wave energy near the slope, and the effect induced by the bottom drag, which reduces the wave energy. When the off-criticality decreases to zero, both of these effects are enhanced continuously, and their combination leads to a non-zero optimum off-criticality. In the classical theory, the viscous effect is not incorporated, hence the optimum off-criticality should be zero. The existence of non-zero optimal off-criticality is also found in other studies; e.g. Chalamalla et al. (Reference Chalamalla, Gayen, Scotti and Sarkar2013) reported that the turbulent kinetic energy can be higher for somewhat off-critical reflection compared to exactly critical reflection under certain conditions. A weakly nonlinear model (Kataoka & Akylas Reference Kataoka and Akylas2020) also shows that in some conditions, the viscous effect can be important during the wave reflection from a slope, which leads to the deviation of the optimal off-criticality from zero.

3.4. Mean flow

As mentioned in § 1, the mean flow field induced by the interaction between incident wave and topography can reflect the strength of the nonlinear effect. The previous theoretical and experimental studies have elucidated clearly the two-dimensional mean flow field near slopes with smooth surface (Thorpe & Haines Reference Thorpe and Haines1987; Grisouard Reference Grisouard2010). In this subsection, we focus on the influence of surface roughness on the mean flow field. The vector map of the mean flow velocity $U\boldsymbol {e}_x+W\boldsymbol {e}_z$ for the smooth surface and rough surface cases are shown in figures 16(a) and 16(b), respectively. Here, $U$ and $W$ are long-time-averaged horizontal and vertical velocities, while $\boldsymbol {e}_x$ and $\boldsymbol {e}_z$ are the rightward and upward unit vectors, respectively. A comparison between these two figures clearly shows that the surface roughness changed the structure of the mean flow field: a series of vortices, of size roughly the same as the separation between the tips of roughness elements, are formed near the slope surface. In the case of a smooth surface, the direction of the mean flow is mainly parallel to the slope. However, in the rough surface case, the formation of vortices is accompanied by the enhancement of mean flow in the direction perpendicular to the slope, which is further confirmed in figures 16(c) and 16(d), where we show the contour plots of mean flow velocity in the direction perpendicular to slope surface $V_{\sigma }$. It is found that $V_{\sigma }$ in the rough surface case is much stronger than that in the smooth surface case (notice the difference in the scale bar). For comparison, we also plot the fields of the mean flow velocity in the direction parallel to slope $V_s$ in figures 16(e) and 16( f). Although the roughness elements would induce additional bottom drag force as we mentioned before, it is found that the strength of $V_s$ in the rough surface case is even slightly larger than that in the smooth surface case.

Figure 16.

(a,b) Coarse-grained vector maps of the mean flow velocity $U\boldsymbol {e}_x+V\boldsymbol {e}_z$, with the magnitude $\sqrt {U^{2}+W^{2}}$ coded in both colour and arrow length for the smooth and rough cases, respectively. (c,d) Colour-coded contour plots of mean flow velocity in the direction parallel to slope $V_s$. (e, f) Colour-coded contour plots of the mean flow velocity perpendicular to slope $V_{\sigma }$. In (a,c,e) $\lambda =0$; in (b,d, f) $\lambda =0.5$. Here, $A/D=0.09$, $\gamma =0.50$, and the units are ${\rm cm}\ {\rm s}^{-1}$.

The normalized maximum values of mean flow velocity in the direction normal to slope $(V_{\sigma }/V_0)_{max}$ versus surface roughness $\lambda$ and off-criticality $\gamma$ are plotted in figures 17(a) and 17(b), respectively. Here, $V_0=A\omega$ is used for normalization. It can be seen that $(V_{\sigma }/V_0)_{max}$ first increases and then decreases as $\lambda$ increases continuously. Interestingly, the maximum value of ($V_{\sigma }/V_0)_{max}$, which represents the strongest nonlinear effect, corresponds to the position $\lambda =0.5$, at which the scattering effect becomes strongest, as shown in figure 7, and $\langle \epsilon \rangle _{max}$ reaches a minimum. From figure 17(b), it is found that there also exists an optimal off-criticality ($\gamma =0.17$) for $(V_{\sigma }/V_0)_{max}$. The existence of an optimal off-criticality for mean flow is not difficult to understand, as the wave energy near the slope is largest at that off-criticality, as shown in figure 15(b).

Figure 17.

Normalized maximum value of the mean flow velocity perpendicular to slope, $(V_{\sigma }/V_0)_{max}$, versus (a) roughness $\lambda$, and (b) off-criticality $(\alpha -\beta )/\beta$.