2.1. Background $\beta \text{-}$ effect

When the water in the rotating tank reaches a state of solid-body rotation, the free surface presents a paraboloidal shape due to the centrifugally induced pressure. As shown in figure 2(a), the height of the free surface, $h_s$ , as a function of radius $r$ , is expressed as

(2.1) \begin{equation} h_s(r)=h_0+\frac {\varOmega ^2}{2g}\left(r^2-\frac {R^2}{2}\right), \end{equation}

where $h_0$ is the fluid height at rest, and $ g$ is the gravitational acceleration.

The radial variation of fluid height leads to the so-called $\beta$ -effect, which is in essence a consequence of the local conservation of angular momentum in rotating fluid. The $\beta$ parameter can be written as

(2.2) \begin{equation} \beta _r(r)=-\frac {f_0}{h(r)}\frac {\mathrm{d} h(r)}{\mathrm d r}, \end{equation}

where $f_0=2\varOmega$ is the Coriolis parameter, and $h(r)=h_s(r)-h_b(r)$ , with $h_b(r)$ the bottom height. When the bottom plate is flat (or has a slope), i.e. $h_b=0$ ( $h_b\propto r^1$ ), one can see a strong radial variation in $\beta _r$ , especially at the central region of the tank (see figure 2 b), as in previous studies (Zhang & Afanasyev Reference Zhang and Afanasyev2014; Read et al. Reference Read, Jacoby, Rogberg, Wordsworth, Yamazaki, Miki-Yamazaki, Young, Sommeria, Didelle and Viboud2015; Cabanes et al. Reference Cabanes, Aurnou, Favier and Le Bars2017). In the present work, we focus on examining the impact of small-scale topography on zonal jets. Therefore, it is necessary to eliminate other factors, such as the inhomogeneous $ \beta _r$ , that may affect the morphology and dynamics of zonal jets. Inspired by the experiments of Lemasquerier, Favier & Le Bars (Reference Lemasquerier, Favier and Le Bars2021), we use a similarly designed curved bottom plate $h_b(r)$ such that the fluid height varies exponentially with radius, i.e. $h(r)\propto {\rm e}^{-(\beta _r/2\varOmega )r}$ , which results in a $\beta _r$ independent of $r$ according to (2.2). To reduce experimental complexity and engineering difficulty, here we have only one curved plate (see figures 1(c) and 2(a)), which, at $\varOmega = 5.65$ rad s $^{-1}$ (54 RPM), leads to a constant $ \beta _r$ (without radial variation), i.e. $\beta _r = 36$ m $^{-1}$ s $^{-1}$ , as shown in figure 2(b). With this curved plate for the other rotation rates, i.e. $\varOmega = 6.39, 5.03, 4.08$ rad s $^{-1}$ , $\beta _r$ exhibits only mild variations across the tank when comparing to values with a flat plate (see figure 2 b). The present experiments can thus be considered to reside in a $\beta$ -plane framework. For four tested rotation speeds, the radial mean values of the $\beta _r$ parameter, $\beta =\overline {\beta _r}$ , are $\beta = 45.2$ , 36, 28.4 and 17.2 m $ ^{-1}$ s $ ^{-1}$ , respectively, as listed in table 1.

Table 1.

The parameters of the present experiments in the presence of the small-scale topography with the compact arrangement of roughness ( $\lambda _\eta =l_f$ ). In the column of ’Roughness’, for instance, ’h10d30’ denotes the cones with $h_\eta =10$ mm and $d_\eta =30$ mm, and similarly for the other cones. For the sparse arrangement of roughness ( $\lambda _\eta =2l_f$ ), the values of $\beta _\eta /\beta$ are omitted in this table, which can be easily obtained using (2.4).

2.2. The forcing pattern

There are various ways to force small-scale fluid motions, such as buoyancy-driven convection (Smith et al. Reference Smith, Speer and Griffiths2014; Read et al. Reference Read, Jacoby, Rogberg, Wordsworth, Yamazaki, Miki-Yamazaki, Young, Sommeria, Didelle and Viboud2015; Zhang & Afanasyev Reference Zhang and Afanasyev2016), electromagnetically induced Lorentz forces (Di Nitto et al. Reference Di Nitto, Espa and Cenedese2013; Zhang & Afanasyev Reference Zhang and Afanasyev2014) and the hydraulic sink–source technique (Aubert, Jung & Swinney Reference Aubert, Jung and Swinney2002; Cabanes et al. Reference Cabanes, Aurnou, Favier and Le Bars2017). Here, we utilise the sink–source forcing method since it is more controllable and efficient, and has a well-defined forcing scale, as shown in figure 1(b). The curved bottom plate, made of Bakelite, is supported by five pillars and placed at the base of the tank. On this plate, there are 200 orifices (diameter $d_o=6$ mm) distributed on a square lattice for circulating water. These orifices (100 inlets and 100 outlets) are connected to five submersible pumps (flow rate $Q = 550{-}2100$ l h $ ^{-1}$ ) located beneath the bottom plate via the distributors and a network of silicone tubes. Specifically, each pump controls 40 orifices, consisting of 20 inlets and 20 outlets. Therefore, the spacing of adjacent sinks and sources defines accurately the forcing scale (see figure 1 b) $l_f= 56$ mm ( $2R/l_f=17.5$ ). The output power of the pumps can be finely adjusted by the controller, thus the flow strength changes. Neglecting friction loss in the tubes, the flow rate in a single orifice $Q_o$ ( $=Q/20$ ) is 27.5 ${-}$ 105 l h $ ^{-1}$ . The initial speed at the orifices can be estimated as $u_o=Q_o/A_o$ (where $A_o=\pi d_o^2/4$ is the area of a orifice), and is in the range 0.27 ${-}$ 1.03 m s $^{-1}$ . We can then define the forcing amplitude as the orifice Reynolds number ${\textit{Re}}_o=u_od_o/\nu$ ( $\nu$ is the kinematic viscosity), which ranges from 1500 to 6000, corresponding to the output power of the pumps that we set for different cases.

2.3. Small-scale topography

Considering the geophysically balanced flow over a small-scale topography, the materially conserved PV can be expressed as

(2.3) \begin{equation} q = \frac {\zeta + f_0}{h-h_{\eta }}\approx \frac {\zeta + f_0}{h}(1+h_{\eta }/h), \end{equation}

where $\zeta =\partial u_2/\partial x_1-\partial u_1/\partial x_2={\nabla} ^2\psi$ is the relative vorticity in the horizontal plane, with $\psi$ the eddy streamfunction: $(u_1,u_2)=(-\partial \psi /\partial x_2,\partial \psi /\partial x_1)$ . Here, $h_\eta (x_1,x_2)$ is the small-scale variability in bottom relief and is much smaller than $h$ , which has a spatial distribution corresponding to the roughness elements and the way they are arranged (see figure 1 d).

We attached the small cones on the bottom plate to produce the small-scale topography, as shown in figure 1(d). Four types of cones are used in the experiments, as shown in figure 1(e), and their sizes are $(h_\eta ,d_\eta)= (10, 30), (20, 30), (30, 30), (32, 20)$ mm, respectively, where $h_\eta$ is the height, and $d_\eta$ is the bottom diameter of the cone. It is noted that we refer to the present topography as ’small’ considering that the ratio of cone height to mean fluid height is much smaller than 1.0, i.e. $h_\eta /h_0\lesssim 0.1$ . In this experiment, we have two topography patterns characterised by the spacing of the adjacent cones, i.e. $\lambda _\eta = l_f$ or $\lambda _\eta = 2l_f$ (in the $x_1$ or $x_2$ direction). To be specific, for $\lambda _\eta = l_f$ , the topography pattern is the same as the forcing pattern with a spatial shift, as shown in figure 1(d), whereas for $\lambda _\eta = 2l_f$ , the distribution of the cones is sparser, and the number of the cones is half of that for $\lambda _\eta = l_f$ . In addition, in the central region of the bottom plate ( $r\lt 0.12$ m), we did not arrange the cones considering the large slope there, as shown in figures 1(c,d).

The topographic variation in bottom relief leads to a local disturbance of PV, and the local vorticity has to develop to compensate this variation, as described by (2.3). Therefore, the vorticity variation due to the roughness can be estimated as $\eta =f_0\,h_{\eta }(x_1,x_2)/h(x_1,x_2)$ (Siegelman & Young Reference Siegelman and Young2023; Gulliver & Radko Reference Gulliver and Radko2023) in the limit $h_{\eta }\ll h$ . In the previous numerical simulations (Constantinou & Young Reference Constantinou and Young2017; Siegelman & Young Reference Siegelman and Young2023), the topography was set to present a continuous spectrum in wavenumber space, whereas the present rough topography is discretely distributed. Therefore, a new parameter $\beta _\eta$ with a dimension of PV based on (2.2), called topography-induced PV, is defined to characterise the overall effect of the arranged roughness. By performing a spatial averaging of the roughness effect, the roughness-induced PV is expressed as

(2.4) \begin{equation} \beta _\eta =\frac {f_0}{h_0}\frac {\pi h_\eta d_\eta }{l_f\lambda _\eta }. \end{equation}

One can see that the second fraction on the right-hand side of (2.4) reflects the additional surface solid boundaries (represented nominally by $\pi h_\eta d_\eta$ ) per unit area ( $l_f\lambda _\eta$ ) interacting with the local turbulent flows. In the present experiments, the topography magnitude $\beta _\eta$ ranges from 4.9 m $^{-1}$ s $^{-1}$ to 31.7 m $^{-1}$ s $^{-1}$ . Consequently, the relative PV, defined as $\beta _\eta /\beta$ (Constantinou & Young Reference Constantinou and Young2017), falls in the range $0.11\leqslant \beta _\eta /\beta \leqslant 1.73$ , part of which is listed in table 1. Naturally, for the cases with a smooth bottom plate, i.e. without topography, we have $\beta _\eta =0$ .

2.4. Data acquisition

The 2-D velocity fields are measured by the time-resolved particle image velocimetry (PIV) system that is attached to the rotating frame. The polyamid spheres, of diameter 50 μm, were seeded in water as tracer particles. These tracer particles were illuminated by a horizontal green laser sheet (wavelength 532 nm) of thickness 0.5–1.0 mm, which is located 6 cm above the edge of the bottom plate, as shown in figures 1(a) and 2(a). A CMOS camera (FILR ORX-10G-123S6M) is placed over the co-rotating tank to capture the particle images. The sensor size of the camera is 4096 $\times$ 3000 pixels, and has been cropped into 3200 $\times$ 2650 pixels to fit both the tank borders and the shadow regions (see figures 3 a–f) where measurements are not possible due to the laser refraction by the sides of the cylindrical tank. The optical distortion induced by the paraboloidal fluid surface leads to inhomogeneous image magnification across the measurement plane. Therefore, an extended calibration process was conducted to correct this distortion. Specifically, we use a bi-quadratic shape function to fit a mapping relation with third-order accuracy between the physical coordinates (a snapshot of the coordinate board in water of solid-body rotation) and the corresponding pixel coordinates. For each rotation speed, this calibration process is performed separately under different rotation speeds due to the different mapping relations.

A state-of-the-art cross-correlation algorithm was then used to calculate the velocity fields from the particle images (Zhu et al. Reference Zhu, Xie and Xia2023, Reference Zhu, Xie and Xia2024). The interrogation window of the final pass was 32 $\times$ 32 pixels, with overlap ratio 75 %, which corresponds to approximately 390 $\times$ 320 velocity vectors in the velocity fields. The digital image resolution was approximately 0.35 mm pixel $^{-1}$ , thus the vector spacing in the measured velocity fields was approximately 2.75 mm ( $\sim0.05l_f$ ). The sampling frequency was set to 5–24 Hz according to the flow strength. The velocity components in the measurement plane were denoted as $ u_1(x_1,x_2)$ and $ u_2(x_1,x_2)$ , respectively. Correspondingly, using the polar coordinates $(r, \theta)$ , the radial and azimuthal velocities, i.e. $u_r$ and $u_\theta$ , can be obtained, as shown in figure 1(b).

An example of the experimental runs is as follows, regardless of the presence of the small-scale topography. We gradually increase the rotation rate from rest up to a target one. Then we wait for the fluid to settle in a state of solid-body rotation, which could be considered to be achieved when all the tracer particles in the measurement plane have no relative motions in the rotating frame. After that, we turn on the sink–source forcing, which produces a vortex lattice consisting of cyclones and anticyclones in the initial stage. Subsequently, these vortices break and interact into turbulence, and eventually evolve into zonal flows under the strong background $\beta$ -effect. For all tested cases in the current experiment, at least 2000 rotation periods $ t_R$ ( $=2\pi /\varOmega$ ) are recorded in the steady state of turbulent flows, which corresponds to over 12 000 snapshots and takes approximately 40 min. After the steady state with a sufficient time, we switch off the forcing, and the turbulent flow decays freely. Finally, we decrease the rotation speed to end the experimental run.

2.5. Parameter space

There are three important dimensional variables that we consider in the present experiments, namely, $\beta$ (or $\varOmega$ ), $\beta _\eta$ and $u_{\textit{rms}}$ , which is the total root mean square velocity of turbulence in steady state. As shown below, $u_{\textit{rms}}$ is not only related to the forcing amplitude, but is also affected by the topography magnitude $\beta _\eta$ . Besides the relative PV gradient $\beta _\eta /\beta$ , there are the other three non-dimensional parameters characterising the turbulent flows, i.e. the Ekman number $Ek$ , the forcing-scale-based Reynolds number ${\textit{Re}}_f$ , and the Rossby number $Ro$ , which are defined, respectively, as

(2.5) \begin{equation} Ek=\frac {\nu }{\varOmega h_0^2},\quad {\textit{Re}}_f = \frac {l_fu_{\textit{rms}}}{\nu },\quad Ro=\frac {u_{\textit{rms}}}{\varOmega h_0}. \end{equation}

In the present work, we conducted 91 runs in total, under different rotation speeds, topography parameters and forcing magnitudes. For all cases, the Ekman number $Ek$ is in the range $1.0\times 10^{-6}{-} 4.0\times 10^{-6}$ . The flow strength $u_{\textit{rms}}$ spans the range 3.26–19.9 mm s $ ^{-1}$ , and correspondingly, ${\textit{Re}}_f = 170{-}1100$ and $Ro = 0.003{-}0.02$ . Due to the dominant effect of rotation ( $Ro\ll 1.0$ ), the turbulent flows are quasi-2-D and favour upscale energy transfers via an inverse cascade (Boffetta & Ecke Reference Boffetta and Ecke2012), which results in the formation of large-scale motions. In quasi-2-D turbulence, the upscale energy transfer rate can be estimated as $\epsilon _{\textit{up}} = \alpha u_{\textit{rms}}^2/2$ , with the large-scale damping rate $\alpha = \varOmega \sqrt {Ek}$ , and ranges from $0.35\times 10^{-7}$ to $15.4 \times 10^{-7}$ m $ ^2$ s $ ^{-3}$ in this experiment.

The background $ \beta$ -effect and associated Rossby waves lead to the anisotropisation of the inverse energy transfer towards zonal modes. In $\beta$ -plane turbulence, two characteristic scales are significant, i.e. Rhines wavenumber and transitional wavenumber, which are defined, respectively, as

(2.6) \begin{equation} k_R=\sqrt {\beta /(2u_{\textit{rms}})},\quad k_\beta =C(\beta ^3/\epsilon _{\textit{up}})^{1/5}. \end{equation}

Correspondingly, the length scales are defined as $L_* = 2\pi /k_*$ . The constant coefficient $C$ ( $=(C_Z/C_K)^{3/10}$ ) can be estimated from the zonal and residual energy spectra shown later. In previous studies, $C$ falls into the range 0.29–0.73 (Galperin et al. Reference Galperin, Sukoriansky, Dikovskaya, Read, Yamazaki and Wordsworth2006; Sukoriansky, Dikovskaya & Galperin Reference Sukoriansky, Dikovskaya and Galperin2007; Cabanes et al. Reference Cabanes, Aurnou, Favier and Le Bars2017; Lemasquerier, Favier & Le Bars Reference Lemasquerier, Favier and Le Bars2023). In this experiment, $C$ is approximately 0.35, which is close to the value in the experiments by Lemasquerier et al. (Reference Lemasquerier, Favier and Le Bars2023). The Rhines scale $k_R$ provides a measure of the scale of zonal jets (Rhines Reference Rhines1975; Sukoriansky et al. Reference Sukoriansky, Dikovskaya and Galperin2007). A transitional scale characterises the threshold of turbulence anisotropisation under the $\beta$ -effect. The ratio of these two scales is known as the zonostrophy index (Galperin & Read Reference Galperin and Read2019), i.e.

(2.7) \begin{equation} R_\beta =k_\beta /k_R=\sqrt {2}C\beta ^{1/10}\epsilon _{\textit{up}}^{-1/5}u_{\textit{rms}}^{1/2}, \end{equation}

which is used as an indicator of the strength of the zonation of the turbulent flows.

Figure 3.

Zonal jets without small-scale topography ( $\beta _\eta = 0$ ), with different forcing magnitudes and strengths of background $\beta \text{-}$ effect: (a,g) $\beta =45.2$ m $^{-1}$ s $^{-1}$ , ${\textit{Re}}_f=481$ ; (b) $\beta = 36$ m $^{-1}$ s $^{-1}$ , ${\textit{Re}}_f=458$ ; (c) $\beta =28.4$ m $^{-1}$ s $^{-1}$ , ${\textit{Re}}_f=454$ ; (d,h) $\beta =17.2$ m $^{-1}$ s $^{-1}$ , ${\textit{Re}}_f=515$ ; (e) $\beta = 45.2$ m $^{-1}$ s $^{-1}$ , ${\textit{Re}}_f=1030$ ; (f) $\beta = 36$ m $^{-1}$ s $^{-1}$ , ${\textit{Re}}_f=971$ . Images (a–f) show the instantaneous fields of azimuthal velocity $u_\theta$ overlaid with the down-sampled velocity vectors, and (g,h) show the temporal evolution (Hovmöller diagram) of the azimuthally averaged azimuthal velocity $\langle u_\theta \rangle _\theta$ . The forcing magnitude for (a–d, g–h) is ${\textit{Re}}_o=1940$ , and for (e,f) it is ${\textit{Re}}_o=5900$ . All contours share the same colour bar, and red denotes the prograde jets, whereas blue represents the retrograde ones.

In the absence of the small-scale topography, $R_\beta$ in this experiment lies between 1.24 and 1.53 with prefactor $C\approx 0.35$ , whereas without considering the prefactor for $ k_\beta$ (or $C=1.0$ ), we have $R_\beta = 3.54{-}4.37$ . These values are close to those obtained in the experiments of Cabanes et al. (Reference Cabanes, Aurnou, Favier and Le Bars2017) and Lemasquerier et al. (Reference Lemasquerier, Favier and Le Bars2021, Reference Lemasquerier, Favier and Le Bars2023), where strong zonal jets are observed even in the instantaneous flow fields (see figure 3), and notably larger than values in the other experiments (Di Nitto et al. Reference Di Nitto, Espa and Cenedese2013; Smith et al. Reference Smith, Speer and Griffiths2014; Zhang & Afanasyev Reference Zhang and Afanasyev2014; Read et al. Reference Read, Jacoby, Rogberg, Wordsworth, Yamazaki, Miki-Yamazaki, Young, Sommeria, Didelle and Viboud2015), where the zonal jets are relatively weak and chaotic, and seem to be ’latent’ in the instantaneous fields. In the presence of small-scale topography, the zonostrophy index is found to be almost unchanged. However, the jet strength is observed to be remarkably diminished compared with that in the cases of smooth plates, as is shown later. It implies that the zonostrophy index is not applicable for zonal jets over small-scale topography.

4.1. Jet strength

Figure 7 shows the flow strength represented by the forcing-scale-based Reynolds number ${\textit{Re}}_f$ as a function of the forcing magnitude ${\textit{Re}}_o$ under different topographies and different strengths of background $\beta$ -effect. The flow strength ${\textit{Re}}_f$ against the forcing magnitude ${\textit{Re}}_o$ shows an approximate scaling ${\textit{Re}}_f\propto {\textit{Re}}_o^{0.62\pm 0.07}$ regardless of the presence of topography. In the absence of topography, one can see that for the same forcing magnitude, the flow strengths are different for different $\beta$ . This is reasonable because the flow patterns are notably different, as shown in figure 3. In the presence of topography, the flow strength is remarkably reduced. The reduction of flow strength may be attributed to the additional dissipation by the topography. On the one hand, the small-scale topography interacting with turbulent flows induces more energy transfers at small scales and thus enhances dissipation (Dewar & Hogg Reference Dewar and Hogg2010). On the other hand, the presence of topography introduces an additional form drag (Vanneste Reference Vanneste2000; Marshall et al. Reference Marshall, Ambaum, Maddison, Munday and Novak2017; Radko Reference Radko2023b ). These two effects are further discussed in § 5. Moreover, the reduction of the flow strength ${\textit{Re}}_f$ in the presence of topography is more significant for larger topography magnitudes. For a sparser topography distribution, i.e. $\lambda _\eta =2l_f$ , the reduction is smaller than that for $\lambda _\eta =l_f$ . This indicates that the additional dissipation is associated with the equivalent area of the topography interacting with the turbulent flows, which is related to the cone geometry and the spacing of the cones (or the number of cones), as illustrated in (2.4).

The total kinetic energy (TKE) of the turbulent flows, $E_{uT}=(1/N)\sum _{i=1}^{N}(u_{\theta ,i}^2+u_{r,i}^2)=u_{\textit{rms}}^2$ , can be decomposed as $E_{uT}=E_{uZ}+E_{uR}$ (Huang, Galperin & Sukoriansky Reference Huang, Galperin and Sukoriansky2001; Galperin et al. Reference Galperin, Young, Sukoriansky, Dikovskaya, Read, Lancaster and Armstrong2014b ; Read et al. Reference Read, Jacoby, Rogberg, Wordsworth, Yamazaki, Miki-Yamazaki, Young, Sommeria, Didelle and Viboud2015; Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2021), where $E_{uZ}$ is kinetic energy in the zonal mean flow, i.e. $E_{uZ} = (1/N)\sum _{i=1}^{N}\langle u_\theta \rangle _{\theta ,i}^2$ , and $E_{uR}$ is the residual kinetic energy containing both the radial component and the azimuthal fluctuation part of the zonal component of TKE. The ratio $E_{uZ}/E_{uT}$ represents the fraction of kinetic energy contained in the zonal mean flow, which also reflects the relative jet strength. When $E_{uZ}/E_{uT}$ approaches 1.0, the zonal jets are notably strong, whereas if $E_{uZ}/E_{uT}$ is close to 0, then the turbulent flow becomes isotropic and no zonal flows can be observed. Figure 8(a) shows $E_{uZ}/E_{uT}$ as a function of the dimensionless transitional length scale $L_\beta /l_f$ . The data from the experiments by Lemasquerier et al. (Reference Lemasquerier, Favier and Le Bars2023) without small-scale topography are also shown for comparison. As shown in figure 8, in the absence of small-scale topography, the jet strength $E_{uZ}/E_{uT}$ decreases with the increase of $L_\beta /l_f$ . This trend is consistent with the observation in the instantaneous flow fields shown in figure 3, and is reasonable because $L_\beta /l_f$ essentially reflects the degree of anisotropy of turbulent flow due to the $\beta$ -effect (Read et al. Reference Read, Jacoby, Rogberg, Wordsworth, Yamazaki, Miki-Yamazaki, Young, Sommeria, Didelle and Viboud2015; Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2023).

Figure 8.

The fraction of kinetic energy contained in the zonal flow, $E_{uZ}/E_{uT}$ , as a function of the dimensionless transitional length scale based on (a) $\beta$ and (b) $\beta _{\textit{eff}}$ . The insets show the corresponding results in log-log plots. The magenta asterisks are from the experiments by Lemasquerier et al. (Reference Lemasquerier, Favier and Le Bars2023), and the definitions of the other symbols are the same as in figure 7. The black dash-dotted lines, indicating the scaling $E_{uZ}/E_{uT}=1.52 (L_{\beta _{\textit{eff}}}/l_f)^{-1.95\pm 0.16}$ , are obtained by fitting all the data shown in (b). A reference scaling $E_{uZ}/E_{uT}\sim (L_{\beta _{\textit{eff}}}/l_f)^{-5/3}$ (or $E_{uZ}/E_{uT}\propto \epsilon _{\textit{up}}^{-1/3}\beta _{\textit{eff}}^{1}$ ) is also shown with the grey dotted lines for comparison.

In the presence of small-scale topography, the energy fraction $E_{uZ}/E_{uT}$ is remarkably reduced, and the reduction increases with the increase of topography magnitude $\beta _\eta$ . Namely, the distributed small-scale roughness weakens the strength of the zonal flows. Therefore, considering the impact of small-scale topography, we define an effective (background) $\beta$ -effect, $\beta _{\textit{eff}}$ , which is expressed by

(4.1) \begin{equation} \beta _{\textit{eff}}=\frac {\beta }{1+\beta _\eta /\beta }. \end{equation}

It is noted that the expression form of the effective $\beta$ may be not unique. For example, one may define another form as ${\beta _{\textit{eff2}}} = \beta (1-\beta _\eta /\beta )$ , also showing the weakening background $\beta$ -effect due to topography. However, when $\beta _\eta$ is larger than $\beta$ , i.e. $\beta _\eta /\beta \gt 1$ , which indeed occurs in the present experiment (see table 1) and also real oceanic flows (Constantinou & Young Reference Constantinou and Young2017), $\beta _{\textit{eff2}}$ is negative and thus has no physical meaning.

With the definition of the effective $\beta$ -effect in (4.1), we obtain the correspondingly effective transitional length scale $L_{\beta _{\textit{eff}}}$ by replacing $\beta$ with $\beta _{\textit{eff}}$ in $k_\beta$ (2.6). The jet strength $E_{uZ}/E_{uT}$ as a function of $L_{\beta _{\textit{eff}}}$ is plotted in figure 8(b). It is interesting that the data from all cases, including the experiments of Lemasquerier et al. (Reference Lemasquerier, Favier and Le Bars2023), collapse to a good degree, regardless of the topography magnitude. Furthermore, the data exhibit a unified trend that can be fitted by a scaling $E_{uZ}/E_{uT}=1.52 (L_{\beta _{\textit{eff}}}/l_f)^{-1.95\pm 0.16}$ . This result can be expressed approximately as

(4.2) \begin{equation} E_{uZ}/E_{uT}=1.52(L_{\beta _{\textit{eff}}}/l_f)^{-2}. \end{equation}

Furthermore, using (2.6), the energy fraction in the zonal mean flow described in (4.2) follows the scaling

(4.3) \begin{equation} E_{uZ}/E_{uT}=C_1l_f^2\epsilon _{\textit{up}}^{-2/5}\beta _{\textit{eff}}^{6/5}, \end{equation}

where $C_1=1.52(C/2\pi )^2$ is constant, with $C$ being the same as in (2.6). This indicates that the energy fraction in the zonal mean flow depends on the upscale energy transfer rate and background $\beta$ -effect, which is clearly illustrated in the instantaneous flow fields shown in figures 3, 4 and 5. In addition, the trend described in (4.3) is broadly consistent with previous numerical and theoretical studies (Srinivasan & Young Reference Srinivasan and Young2012; Galperin & Read Reference Galperin and Read2019), although with different scaling behaviours.

4.2. Jet length scale

The jet spacing (or width) $L_{\textit{Jet}}$ is another important characteristic of the zonal jets, which can be estimated from the jet profiles shown in figure 6. We can also calculate the jet scale with the integral method (Zhang & Xie Reference Zhang and Xie2023) as

(4.4) \begin{equation} L_{\textit{Jet}}=\frac {2\pi \int E_Z(k)\,\text{d}k}{\int k E_Z(k)\,\text{d}k}, \end{equation}

where $E_Z(k)$ is the zonal spectrum shown in § 4.3. We have checked that the results obtained from the two methods are consistent. Figure 9 shows the dimensionless jet spacing $L_{\textit{Jet}}/l_f$ as a function of the Rhines scale $L_R$ . In the absence of small-scale topography (see the black symbols in figure 9 a), the jet spacing satisfies a relation $L_{\textit{Jet}}\approx 1.9L_R$ , confirming that the Rhines scale can be used to characterise the length scale of jets. In addition, the length scale of jets reaches a saturated value $L_{\textit{Jet}}/l_f\approx 5.4$ (or $L_{\textit{Jet}}/R = 0.62$ ) due to the confined domain of the experimental tank.

Figure 9.

The length scale of zonal jets as a function of the Rhines scale based on (a) the original background $\beta$ , and (b) the topography-affected $\beta _{\textit{eff}}$ . The definitions of the symbols are the same as in figure 7. The black dash-dotted line shows a relation $L_{\textit{Jet}}=1.9L_{R}$ . The red and magenta dashed lines denote the saturation due to either the system size or the channel effect. The inset in (a) shows the time-averaged azimuthal velocity for $\beta =28$ m $^{-1}$ s $^{-1}$ , ${\textit{Re}}_o=5900$ without topography, indicating the sub-saturated regime.

We replot the jet width as a function of the effective Rhines scale $L_{R_{\textit{eff}}}$ ( $=2\pi /k_R=2\pi \sqrt {2u_{\textit{rms}}/\beta _{\textit{eff}}}$ ) in figure 9(b). It is found that for small topography magnitudes, the jet width is closely related to $L_{R_{\textit{eff}}}$ , following a relation $L_{\textit{Jet}}\approx 1.9L_{R_{\textit{eff}}}$ , similar to that without small-scale topography. However, for large topography magnitudes, especially for cones with $h_\eta =30$ mm and $d_\eta =30$ mm, the jet scale saturates to a limit value that is smaller than the saturated value for a smooth plate ( $L_{\textit{Jet}}/l_f\approx 5.4$ ). The saturation of the jet scale for large topography magnitudes could be attributed to the ‘channel’ effect in which the zonal flows are prone to be separated by the high topographies (see figures 4 and 5).

4.3. Energy spectra and fluxes

We now analyse the impact of small-scale topography on zonal jets from the perspective of energy spectra and energy flux. Because the turbulent flow shows remarkable anisotropy in the azimuthal and meridional directions, the kinetic energy spectra can be decomposed into the zonal-mode ( $E_Z$ , called the zonal spectrum) and residual ( $E_R$ , the residual spectrum) components. In the presence of strong zonal jets (called the zonostrophic regime), the zonal and residual spectra follow different scalings at low wavenumbers, as theoretically analysed in $\beta$ -plane turbulence. The residual spectrum is expected to be associated with the Kolmogorov–Batchelor–Kraichnan theory of 2-D turbulence (Kraichnan Reference Kraichnan1967; Boffetta & Ecke Reference Boffetta and Ecke2012). Specifically, the residual spectrum for $k\lt k_f$ characterises the inverse energy cascade and theoretically follows

(4.5) \begin{equation} E_R^{\textit{theo}}(k)\sim C_K\epsilon _{\textit{up}}^{2/3}k^{-5/3}, \end{equation}

where $C_K\approx 6$ is the universal Kolmogorov–Kraichnan constant, while for $k\gt k_f$ , the residual spectrum follows a scaling $k^{-3}$ , representing the forward enstrophy cascade. In addition, the zonal spectrum, characterising the axisymmetric component of the flow, presents theoretically a steep scaling in the zonostrophic inertial range (Chekhlov et al. Reference Chekhlov, Orszag, Sukoriansky, Galperin and Staroselsky1996; Galperin et al. Reference Galperin, Sukoriansky, Dikovskaya, Read, Yamazaki and Wordsworth2006; Sukoriansky et al. Reference Sukoriansky, Dikovskaya and Galperin2007) as

(4.6) \begin{equation} E_Z^{\textit{theo}}(k)\sim C_Z\beta ^2k^{-5}, \end{equation}

while $C_Z$ is a constant of order unity (Galperin et al. Reference Galperin, Hoemann, Espa and Di Nitto2014a ; Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2023).

Figure 10.

Kinetic energy spectra for zonal jets (a,b) over the smooth bottom plate ( $\beta _\eta =0$ ), and (c,d) over the small-scale topography for different flow strengths. The spectra are separated into (a,c) the residual ( $E_R$ ) and (b,d) the zonal-mode ( $E_Z$ ) contributions. The cases shown here are under the same strength of $\beta$ -effect ( $\beta =36$ m $^{-1}$ s $^{-1}$ ), and the topography magnitude for all cases in (c,d) is $\beta _\eta /\beta =0.60$ .

Considering the rotational symmetry of the velocity fields, we perform a Bessel–Fourier decomposition using the polar coordinates; for details, see the previous studies (Wang, Ronneberger & Burkhardt Reference Wang, Ronneberger and Burkhardt2008; Galperin et al. Reference Galperin, Hoemann, Espa and Di Nitto2014a ; Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2023). Figures 10(a,b) show the residual and zonal energy spectra, respectively, for zonal jets over the smooth bottom plate with different flow strengths. It is noted that the abscissa in figure 10 is the wavenumber scale deduced from the mode numbers (Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2023). We do not use the mode number itself, as in Galperin et al. (Reference Galperin, Hoemann, Espa and Di Nitto2014a , Reference Galperin, Young, Sukoriansky, Dikovskaya, Read, Lancaster and Armstrongb ), because we want to denote specifically the forcing scale $k_f$ in the dimensional form. It is observed that at small scales, most of the energy is concentrated in the residual spectrum, whereas at large scales, energy in the zonal spectrum dominates. Moreover, as shown in figure 10(a), above the forcing wavenumber ( $k\gt k_f$ ), the residual spectra exhibit a scaling close to $k^{-4}$ , which is steeper than the theoretical scaling $k^{-3}$ . The steepening of the spectrum in enstrophy cascade has been commonly observed in previous numerical and experimental studies. The possible causes include viscous dissipation, large-scale friction or the presence of large-scale coherent structures (Boffetta & Ecke Reference Boffetta and Ecke2012; Zhu, Xie & Xia Reference Zhu, Xie and Xia2024).

At large scales ( $k\lt k_f$ ), one sees that for strong flow strengths, the residual energy spectra follow a scaling $k^{-5/3}$ , consistent with the presence of an inverse energy cascade. The scaling range ends at the transitional scale $k_\beta \approx 50$ m $^{-1}$ , indicating that the $\beta$ -effect induces anisotropisation and inhibits the isotropic inverse cascade. With the decrease of flow strength, the scaling range for inverse cascade in the residual spectra becomes narrower, and vanishes for the two lowest flow strengths. As shown in figure 10(b), the zonal spectra for $k\lt k_f$ follow a steep scaling $k^{-5}$ , as predicted in the zonostrophic regime (Cabanes et al. Reference Cabanes, Aurnou, Favier and Le Bars2017; Galperin & Read Reference Galperin and Read2019; Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2023), and this scaling is more prominent for higher upscale energy transfer rates. When the flow strength is relatively low, the zonal spectra present oscillations for $k\lt k_f$ , indicating the presence of alternating jets, as shown in figure 3(a). It is intriguing that the scaling $k^{-5}$ extends below the forcing scale. In the review paper of Galperin, Sukoriansky & Dikovskaya (Reference Galperin, Sukoriansky and Dikovskaya2010), it is demonstrated that the scaling $k^{-5}$ can be extended beyond the zonostrophic inertial range. In addition, the $k^{-5}$ scaling continuing below the forcing scale was also observed in previous experimental studies (Zhang & Afanasyev Reference Zhang and Afanasyev2014; Cabanes et al. Reference Cabanes, Aurnou, Favier and Le Bars2017; Lemasquerier et al. Reference Lemasquerier, Favier and Le Bars2023), for which the underlying theory is unclear.

Figures 10(c,d) show the kinetic energy spectra for zonal jets over small-scale topography under different forcing magnitudes and the same topography magnitude ( $\beta _\eta /\beta = 0.60$ ). Although the shape of both residual and zonal spectra does not change remarkably compared to that for the cases with a smooth bottom plate, the scaling ranges above the forcing scale in both spectra are found to be diminished due to the presence of topography. This observation implies that the presence of small-scale topography hinders the process of inverse energy cascade appreciably.

Figure 11.

Transfer fluxes of (a,c) energy ( $\varPi _E$ ) and (b,d) enstrophy ( $Z_\varOmega$ ) as functions of wavenumber for zonal jets (a,b) over the smooth bottom plate, and (c,d) over the small-scale topography for different flow strengths. The symbols and the corresponding cases are the same as in figure 10. It is noted that the results shown here are based on the divergence-free velocity fields, and the corresponding results based on the uncorrected (divergent) velocity fields are shown in figure 17 for comparison.

The kinetic energy spectra do not provide information on the direction of energy/enstrophy transfers, so we examine the transfer fluxes scale by scale in Fourier space. Using the 2-D PIV velocity field $(u_1,u_2)$ , the energy and enstrophy transfer fluxes as functions of wavenumber scale can be calculated, respectively, by

(4.7) \begin{align}& \varPi _E(k)=-\sum _k^{k_{\textit{max}}}T_k = -\sum _k^{k_{\textit{max}}}\psi ^*(k) \times\hat {J}(\psi , \zeta )_k, \end{align}

(4.8) \begin{align}& Z_{\varOmega }(k)=-\sum _k^{k_{\textit{max}}}S_k =-\sum _k^{k_{\textit{max}}} \zeta ^*(k) \times\hat {J}(\psi , \zeta )_k, \end{align}

where $ J$ is the Jacobian $ J(a,b)=(\partial a/\partial {x_1})(\partial b/\partial {x_2})-(\partial a/\partial {x_2})(\partial b/\partial {x_1})$ , $*$ indicates complex conjugate, and a hat indicates the corresponding Fourier transformed quantities. Here, $T_k$ and $S_k$ are the transfer functions of energy and enstrophy, respectively. As with the previous definition, $\varPi _E(k)\gt 0$ ( $Z_\varOmega (k)\gt 0$ ) indicates energy (enstrophy) flux to smaller length scales at scale $k$ , while $\varPi _E(k)\lt 0$ ( $Z_\varOmega (k)\lt 0$ ) denotes energy (enstrophy) flux towards larger scales.

Figure 11 shows the energy and enstrophy fluxes for turbulent flows with and without the presence of small-scale topography. Due to the possible dual energy cascades in present rotating turbulence (Pouquet et al. Reference Pouquet, Marino, Mininni and Rosenberg2017; Boffetta Reference Boffetta2023), in order to calculate the correct fluxes in 2-D mode of the flow field (Buzzicotti et al. Reference Buzzicotti, Aluie, Biferale and Linkmann2018), we first obtain the horizontal-divergence-free velocity fields using the weighted divergence correction scheme proposed by Wang et al. (Reference Wang, Gao, Wei, Li and Wang2017). For comparison, the corresponding results using the divergent (uncorrected) velocity fields are presented in figure 17 in Appendix A. As shown in figure 11(a), the energy flux $\varPi _E$ below the forcing wavenumber is negative, and its absolute value generally increases with the forcing magnitude, which indicates that the presence of inverse energy transfers facilitates the formation of large-scale coherent structures. It should be mentioned that for $k\lt k_f$ , the scale range with a negative energy flux is relatively wider than the scaling range of inverse cascade in the residual spectrum shown in figure 10. In addition, it is seen that the inverse energy flux calculated from Fourier space is comparable with the estimated upscale transfer rate $\epsilon _{\textit{up}}$ , indicating the self-consistency of our experimental results. Above the forcing wavenumber, the enstrophy flux is positive, as shown in figure 11(b), indicating the existence of forward enstrophy transfers. Moreover, as shown in figures 11(c,d), the presence of small-scale topography suppresses both the inverse energy transfers above the forcing scale and forward enstrophy transfers below the forcing scale.

Figures 12(a,b) show the energy spectra for different topography magnitudes $\beta _\eta$ under the same forcing magnitude and the same degree of background $\beta$ -effect. It is seen that at large scales, i.e. $k\lt k_\beta$ , the energy in the residual spectrum $E_R$ decreases monotonically with increasing $\beta _\eta$ . However, at intermediate and small scales ( $k\gt k_\beta$ ), $E_R$ presents a non-monotonic trend with $\beta _\eta$ . Similar to the energy spectra, the energy fluxes also present a non-monotonic dependence with $\beta _\eta$ , as shown in figures 12(c,e). Specifically, the energy decreases dramatically with $\beta _\eta$ when $\beta _\eta$ is small, although the jet pattern seems to be unchanged, as shown in figures 4(a,b). As mentioned above, this trend may result in additional dissipation and friction drag induced by small-scale topography (Mashayek Reference Mashayek2023; Radko Reference Radko2023b ; Vanneste Reference Vanneste2000). Indeed, the ratio of the time scales of the topographic PV ( $1/\eta \sim 10^0$ s) and the Ekman drag ( $1/\alpha \sim 10^2$ s, see § 2.5) is of the order of $10^{-2}$ , which indicates that the form stress induced by topography is relatively large (Constantinou & Young Reference Constantinou and Young2017). On the other hand, there exist locally induced velocities due to the local variation of PV at the position of small-scale roughness, as described in the theory of PV homogenisation (Rhines & Young Reference Rhines and Young1982; Vallis & Maltrud Reference Vallis and Maltrud1993; Siegelman & Young Reference Siegelman and Young2023), and this process prevents energy from being transferred to large scales.

Figure 12.

The effect of topography magnitude $\beta _\eta$ on energy spectra and scale-to-scale transfer fluxes: (a) residual spectrum; (b) zonal spectrum; (c) energy flux; (d) enstrophy flux. The five cases in (a–d) are under the same forcing magnitude ( ${\textit{Re}}_o=5900$ ) and the same strength of $\beta$ -effect ( $\beta =36$ m $^{-1}$ s $^{-1}$ ). (e,f) The maximum negative and positive values, respectively, of energy (enstrophy) transfer flux $\varPi _{E,max-}$ ( $Z_{\varOmega ,max+}$ ) normalised by $\varPi _{E,max-}(\beta _\eta =0)$ ( $Z_{\varOmega ,max+}(\beta _\eta =0)$ ) as functions of the topography magnitude $\beta _\eta$ . In (e,f), the solid lines represent the forcing magnitude ${\textit{Re}}_o=5900$ , while the dashed lines represent the cases with ${\textit{Re}}_o=1540$ .

When the topography magnitude is moderate, it is interesting that the energy flux presents a mildly increasing trend with $\beta _\eta$ , and this trend is observed for all tested cases. This anomaly may originate from the channel effect of high topography, in which the flows roam in the valleys of the topography (Constantinou & Young Reference Constantinou and Young2017; Siegelman & Young Reference Siegelman and Young2023), as shown in the instantaneous flow fields in figures 4 and 5. As $\beta _\eta$ increases further, the transfer fluxes decrease. It may be a result of the fact that larger topography magnitude induces more form drag and dissipation, which outweighs the enhanced upscale energy fluxes due to the channel effect. In addition, we found that the enstrophy flux for all cases manifests a similar trend with $\beta _\eta$ , as shown in figures 12(d,f).

The enhancement of inverse energy flux due to the presence of small-scale topography has been observed in isotropic quasi-2-D turbulence in the absence of $\beta$ -effect. Zhang & Xie (Reference Zhang and Xie2024) demonstrated that for small topography magnitudes, through the non-local interactions, the topography induced more energy being transferred to larger scales and eventually led to the formation of condensation. They also found that a further increase of topography magnitude led to the suppression of inverse energy transfer. Our experimental results are broadly consistent with the numerical results of Zhang & Xie (Reference Zhang and Xie2024), although in the current study, the turbulent flows are under strong $\beta$ -effect. The above observations indicate that the effect of small-scale topography on the energy transfers in quasi-2-D turbulence, regardless of the presence of $\beta$ -effect, is more complicated than previously thought.

Figure 13.

Time-averaged energy transfer functions $T_k(k_x, k_y)$ for the cases under the same background $\beta$ -effect ( $\beta =45.2$ m $^{-1}$ s $^{-1}$ ) and with different topographies: (a,e) $\beta _\eta /\beta =0$ ; (b,f) $\beta _\eta /\beta =0.22$ ( $\lambda _\eta =l_f$ ); (c,g) $\beta _\eta /\beta =0.67$ ( $\lambda _\eta =l_f$ ); (d,h) $\beta _\eta /\beta =0.47$ ( $\lambda _\eta =2l_f$ ). The forcing magnitude for (a–d) is ${\textit{Re}}_o=1540$ , and for (e–h) it is ${\textit{Re}}_o=5900$ . The black dashed circles denote the forcing wavenumber, while the two outside dashed circles are highlighted considering that there exist the contour ‘spots’ at these two wavenumbers. It is noted that the value ranges of these contour maps are different for the cases shown here to ensure that there is a remarkable contrast in energy transfer at different wavenumbers in each case. Red denotes $T_k\gt 0$ , and blue denotes $T_k\lt 0$ .