2.1.1. Governing equations

A homogeneous fluid layer with constant density $\rho _0$ is considered in a Cartesian plane $(x,y,z)$ . The system rotates with angular velocity $f_0/2$ (with $f_0\gt 0$ the constant Coriolis parameter in the Northern Hemisphere) about the vertical $z$ -axis, which is antiparallel to gravity. The horizontal velocities $u(x,y,t)$ and $v(x,y,t)$ are depth-independent, so the fluid moves in vertical columns that can be stretched or squeezed as they move over the variable bottom topography $h_b(x,y)$ . Under the rigid-lid approximation, the time-independent fluid depth is $h(x,y)=H-h_b(x,y)$ , where $H$ is a reference depth, so gravity waves are filtered out (Grimshaw, Tang & Broutman Reference Grimshaw, Tang and Broutman1994). The continuity equation can be written as

(2.1) \begin{equation} \frac {\partial (hu)}{\partial x} + \frac {\partial (hv)}{\partial y} = 0. \end{equation}

From (2.1), the horizontal velocities are defined using a transport function $\psi (x,y,t)$ such that

(2.2) \begin{align} u \,=\, \frac {1}{h}\frac {\partial \psi }{\partial y}, \hspace {2cm} v \,=\, -\frac {1}{h}\frac {\partial \psi }{\partial x}. \end{align}

The vorticity equation governing the fluid motion is

(2.3) \begin{align} \frac {\partial \omega }{\partial t} + J(q,\psi ) = D + \frac {1}{\rho _0}(\boldsymbol{\nabla }\times \boldsymbol{F})_z, \end{align}

where $\omega (x,y,t)=\partial _xv-\partial _yu$ is the vertical component of the relative vorticity, $J(a,b)=\partial _xa \partial _yb-\partial _ya \partial _xb$ is the Jacobian operator, $D$ is a dissipative term, $\boldsymbol{F}$ is a horizontal force vector per volume unit, ( $\boldsymbol{\nabla }\times \boldsymbol{\cdot })_z$ is the curl vertical component and $q$ is the potential vorticity

(2.4) \begin{equation} q=\frac {f_0+\omega }{h}. \end{equation}

Using (2.2), the vorticity $\omega$ is

(2.5) \begin{equation} \omega \,=\, -\frac {1}{h}{\nabla} ^2\psi + \frac {1}{h^2} \boldsymbol{\nabla } h\boldsymbol{\cdot } \boldsymbol{\nabla }\psi . \end{equation}

2.1.2. Topographic effects

The influence of the bottom topography is better understood in the absence of dissipation and forcing. If the system reaches a quasi-steady state, and assuming $\omega \ll f_0$ , then (2.3) is reduced to

(2.6) \begin{equation} \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }\left ( \frac {f_0}{h}\right ) \approx 0. \end{equation}

This means that the velocity $\boldsymbol{u}$ aligns with the contours of $f_0/h$ or, equivalently, the topographic contours. The fluid moves with shallow water to the right in the Northern Hemisphere, which is the direction of propagation of topographic Rossby waves. These oscillations are equivalent to the westward-travelling planetary Rossby waves caused by the latitudinal variation of the Coriolis parameter or $\beta$ -effect. Thus, the $\beta$ -effect is locally replicated in regions where the bottom topography has a slope $s$ and mean depth $H_m$ , such that the topographic $\beta$ parameter is given by $\beta _t = f_0 s/H_m$ (van Heijst Reference van Heijst1994). In this configuration, the topographic ‘north’ (‘south’) corresponds to shallow (deep) regions, and topographic waves propagate ‘westward’ along the topography. Over abrupt topographies,  $\beta _t$ is at least one order of magnitude larger than the planetary $\beta$ (Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2000).

2.1.3. Dissipative terms

The dissipation term $D$ in (2.3) contains a Laplacian viscous term and bottom friction $D_b$ due to the lower Ekman layer

(2.7) \begin{equation} D= \nu {\nabla} ^2 \omega + D_b, \end{equation}

where $\nu$ is the viscosity coefficient. Usually, Laplacian viscosity in laboratory experiments is much weaker than bottom friction. For numerical simulations using oceanic spatial scales, we will assume that $\nu$ represents a turbulent viscosity coefficient, and bottom friction will not be considered.

We use two forms of the bottom friction $D_b$ . The first one is the classical linear Ekman friction

(2.8) \begin{equation} D_b^{\textit{lin}}= - \frac 12 E^{1/2} f_0 \omega , \end{equation}

where $E$ is the Ekman number $E=2\nu /f_0H^2$ . A second form is the nonlinear Ekman friction model (Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2002)

(2.9) \begin{equation} D_b^{nl} = - \frac {\delta _E}{2h} \left [\omega (\omega +f_0) - \boldsymbol{\nabla }\psi \boldsymbol{\cdot }\boldsymbol{\nabla }q \right ], \end{equation}

where $\delta _E=(2\nu /f_0)^{1/2}$ is the Ekman layer thickness. Note that, for a flat bottom, $\delta _E/H \equiv E^{1/2}$ . In this model, the vorticity (2.5) has to be replaced by

(2.10) \begin{equation} \omega \,=\, -\frac {1}{h}{\nabla} ^2\psi + \frac {1}{h^2} \boldsymbol{\nabla } h\boldsymbol{\cdot } \boldsymbol{\nabla }\psi + \frac {\delta _E}{h^3} J(h,\psi ). \end{equation}

The nonlinear Ekman friction has been used to successfully simulate laboratory experiments in different contexts, e.g. 2-D circular and dipolar vortices (Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2000, Reference Zavala Sansón and van Heijst2002; Tenreiro, Zavala Sansón & van Heijst Reference Tenreiro, Zavala Sansón and van Heijst2006), turbulent flows over abrupt obstacles (Tenreiro et al. Reference Tenreiro, Trieling, Zavala Sansón and van Heijst2013) and rapidly rotating systems (Lemasquerier, Favier & Le Bars Reference Lemasquerier, Favier and Le Bars2023). However, nonlinear Ekman terms have been generally unnoticed, perhaps because they are not straightforward to implement numerically, and their effects become manifest only after a few Ekman periods. Here, we demonstrate that laboratory experiments can be well simulated using either linear or nonlinear friction, and that the results are slightly better with the latter.

2.2.1. Experimental arrangement

To represent an ocean basin in the laboratory, a square acrylic tank with sides of $L=30$ cm was used, mounted at the centre of a rotating table (figure 1). The tank was filled with water to a depth of $H=10$ cm. The bottom of the tank is flat in the central part, with constant slopes adjacent to each wall. The width of the sloping regions is $W=6$ cm and the height at each wall $D_w=6.5$ cm, resulting in a slope of $ s=D_w/W=1.083$ and an average height of approximately $H_m \approx 6.75$ cm. The table rotates counter-clockwise at a constant angular speed of $\varOmega =0.5$ s $^{-1}$ , so the Coriolis parameter is $f_0 = 2\varOmega = 1$ s $^{-1}$ and the rotation period of the table (equivalent to one ‘day’) is $T=2 \pi /\varOmega \approx 12.5$ s. The $\beta$ effect due to the topographic slopes is $\beta _t = s f_0 / H_m \approx 0.16$ (cm s) $^{-1}$ . Table 1 summarises the rotating system parameters in all experiments and the corresponding oceanic values used in numerical simulations (see § 2.3).

Table 1.

Main parameters used in the laboratory experiments. Last column presents oceanic scales used in numerical simulations for an $f$ -plane basin.

Figure 1.

Schematics (side and upper views) of the rotating square tank used in the experiments, where $\varOmega$ is the angular speed, $H$ the maximum water depth and $L$ the side length. The topography is flat in the central region and linear with a slope $s=D_w/W$ next to the walls, where $D_w$ is the height of the topography at the walls and $W$ the slope width.

Due to rotational effects, the fluid moves in vertical columns, making it sufficient to analyse the horizontal motion at a single depth. The experiments were videorecorded at $60$ fps with a camera co-rotating with the tank, capturing the motion of suspended passive particles. As a tracer, polyamide particles with a diameter of $50\unicode{x03BC}$ m, which have nearly neutral buoyancy, were employed. To visualise the particles, the fluid was illuminated 1 cm below the surface with a 532 nm (green light) laser sheet parallel to the surface of the tank. The analysis of the experiments was based on the calculation of instantaneous horizontal velocity fields $(u,v)$ using particle image velocimetry (PIV) with the PIVlab software of MATLAB (Thielicke & Sonntag Reference Thielicke and Sonntag2021). Afterwards, the vertical component of the relative vorticity $\omega$ and other fields of interest (e.g. kinetic energy) are calculated.

2.2.2. Forcing method

The fluid is set in motion by applying a non-intrusive, electromagnetic method based in the Lorentz force, which was introduced in the 1960s (see Bondarenko, Gak & Gak Reference Bondarenko, Gak and Gak2002, and references therein) and used in several works in experiments with shallow fluid layers (Sommeria Reference Sommeria1986; Duran-Matute, Trieling & Van Heijst Reference Duran-Matute, Trieling and Van Heijst2011; González Vera & Zavala Sansón Reference González Vera and Zavala Sansón2015) and rotating systems (Cruz Gómez et al. Reference Cruz Gómez, Zavala Sansón and Pinilla2013). The forcing method uses an electrically conductive fluid to which an electric current and a magnetic field are applied perpendicularly. The resulting Lorentz force per volume unit is

(2.11) \begin{equation} \boldsymbol{F} = \boldsymbol{J} \times \boldsymbol{B}, \end{equation}

where $\boldsymbol{J}$ is the electric current density and $\boldsymbol{B}$ the magnetic field (with magnitudes $J_m$ and $B_m$ , respectively).

In all experiments, saltwater was used as the conducting fluid. The density was 1.03 g cm⁻ $^3$ at an average temperature of 19 $^{\circ }$ C, and the kinematic viscosity was $\nu \approx 0.01$ cm $^2$ s⁻¹. Hereafter, we consider a Cartesian coordinate system with the origin at the centre of the tank. An electric current $I$ is generated in the horizontal $x$ -direction by connecting two graphite electrodes (placed beneath the topographic slopes at $x=\pm 15$ cm) to a voltage source supplying $I=\pm 1.4$ A, the sign depending on the electrodes’ polarity (figure 2). The magnitude of the density current is $J_m=|I|/HL$ . The magnetic field is locally provided by an array of 16 square neodymium magnets (2.54 cm wide) placed under the flat section of the tank. The dominant component of the magnetic field is in the vertical direction, and has an intensity of $B_m \approx \pm 0.3$ T, the sign depending on the magnet orientation. Thus, over each magnet the Lorentz force is directed in the $y$ -direction, $\boldsymbol{F}=\pm J_m B_m \boldsymbol{\hat j}$ . After applying the forcing for some time, the resulting flow acquires a disordered, turbulent state.

To generate different turbulent flows, the magnets were arranged in three different distributions, $A$ , $B$ and $C$ , as shown in figure 2. We obtain six different forcing configurations by applying the electric current with positive or negative polarity. The forcing scale is the separation between magnets (Kelley & Ouellette Reference Kelley and Ouellette2011), which is approximately 5.1 cm. Distribution $A$ consists of $11$ magnets with negative and five with positive polarity, arranged in a random pattern (figure 2 a). This asymmetric array causes the applied forcing to induce a preferential positive or negative circulation in the fluid, depending on the direction of the electric current. The second array $B$ is a different random arrangement of the magnets (figure 2 b). The resulting forcing generates different realisations of the disordered flow compared with previous cases. Finally, configuration $C$ consists of eight magnets with positive polarity and eight with negative polarity arranged alternately, like a chessboard (figure 2 c). This arrangement generates a similar forcing with either sign of the electric current and does not induce a preferential circulation. Additional cases can be defined by varying the magnitude of the electric current or the forcing protocol, as explained in the free-decay (§ 3) and continuously forced (§ 4) experimental sections.

Figure 2.

Distributions of 16 magnets (small squares) placed below the square flat sector of the rotating tank. The sign on each magnet is the polarity of the vertical component of its magnetic field. The two rectangles at $x=\pm L/2$ indicate the electrodes placed below the topography to generate an electric current. (a) Distribution $A$ : random array of 5 positive and 11 negative magnets. (b) Distribution $B$ : different random array of 5 positive and 11 negative magnets. (c) Distribution $C$ : alternate array of 8 positive and 8 negative magnets. The direction of the electric current defines different forcing configurations.

2.2.3. Flow parameters

Typical velocity and length scales in most of experiments were $U\approx 0.5$ cm s⁻¹ and $\ell \approx 5$ cm. The Rossby number based on these scales is $\textit{Ro}=U/f_0\ell \approx 0.1$ , which is small enough to consider the flow’s bidimensionality. The Reynolds number is $\textit{Re}=U\ell /\nu \approx 250$ , so lateral viscous effects are moderate. The Ekman number is $E=2\nu /f_0H^2 =2\times 10^{-4}$ , which gives an Ekman time scale $T_E=2/f_0E^{1/2}=141$ s or approximately $11$ days, which is the decaying time scale associated with bottom friction.

A fundamental factor to mimic oceanic conditions is the topographic $\beta$ -effect due to the slopes (Zavala Sansón & van Heijst Reference Zavala Sansón and van Heijst2000). In non-dimensional form, the $\beta$ -parameter in the laboratory is $\beta '_{\textit{lab}}=\beta _t W/f_0=0.96$ . In a typical mid-latitude ocean basin with Coriolis parameter of $10^{-4}$ s $^{-1}$ , an abrupt bottom slope $10^{-2}$ and mean depth $10^3$ m, the topographic parameter is $\beta _t \approx 10^{-9}$ m $^{-1}$ s $^{-1}$ . Thus, considering a horizontal width of the topography of approximately 100 km, the non-dimensional parameter is $\beta '_{\textit{oce}} \approx 1$ , which is the same order as the experimental value.

The magnitude of the forcing is measured by the Chandrasekhar number defined as $\textit{Ch}=\textit{IBH}/\rho \nu ^2$ , which compares the Lorentz and viscous forces (Duran-Matute et al. Reference Duran-Matute, Trieling and Van Heijst2011). In this definition, it is assumed that the electric current is essentially produced by the potential difference between the electrodes, and not by the much weaker induced currents by Ohm’s law, which are neglected. In most of our experiments, $\textit{Ch} \sim O(10^7)$ . High values of $\textit{Ch}$ indicate the forcing is intense enough to set the fluid in motion against viscous effects.

2.2.4. Type of experiments

Two different type of experiments were performed:

1. (i) Decaying turbulence. Before an experiment, the tank rotates with the forcing turned off (typically, 40 min) until the water reaches solid-body rotation. Then, the electromagnetic force is applied for two to five days, after which the electric current is disconnected. The fluid presents a disordered state, and the experiment starts. The flow evolves while decaying for 30 days, a long enough time scale for rotational effects to become significant but short enough for viscous processes to remain secondary. In most experiments, the fluid surface was exposed to the ambient air, whose effect was negligible because of the strong density difference between water and air, and because the tank rotates at a moderate angular speed. In some cases, a lid was placed over the free surface to ensure the suppression of such effects, but the ultimate result was an increased Ekman friction. The results are presented in § 3.

2. (ii) Randomly forced turbulence. As in the decaying case, the system is led to solid-body rotation before starting. The experiment begins when the electromagnetic forcing is applied to the motionless fluid (in the rotating system). During the first 20–30 days, the energy injection is balanced with dissipative effects, so the system reaches a state of (quasi) equilibrium. From day 30 to day 55, the polarity of the electrodes is reversed every 5 days to maintain the randomness of the forcing and avoid favouring any particular flow pattern. Finally, on day 60, the forcing is turned off, marking the end of the experiment. The results are presented in § 4.

3.1.1. Flow evolution

Figure 4.

Scaled relative vorticity and velocity fields measured in experiment $A^{+}$ from day 0 to day 20.

Figure 4 shows the temporal evolution of the horizontal velocity and scaled relative vorticity fields measured in experiment $A^{+}$ . At time $t=0$ days (immediately after stopping the forcing), the flow appears disorganised, with vortices of various sizes present over both the flat region and the topographic slopes. By $t=3$ days, a counter-clockwise circulation over the slopes starts to form, along with an anticyclone accompanied by a smaller cyclone near the centre of the flat region. At $t=5$ days, the central anticyclone has intensified, and the cyclonic flow over the slopes is well defined, although it exhibits a wavy pattern. By $t=10$ days, the anticyclone has become more circular, occupying most of the flat area, while the cyclonic current continues over the slopes. Finally, at $t=15$ and $t=20$ days, the cyclonic circulation on the slopes and the central anticyclone persist. At these times, small-scale structures develop, which are attributed to unavoidable additional perturbations in the laboratory (e.g. wind drag, oscillations of the rotating table, convective effects). However, these structures are very small and weak, so they do not interfere with the generation of the topographic current and the central anticyclone. Note that the scale of the velocity vectors and the colour bar for vorticity change over time to highlight the flow characteristics clearly, as the system is in continuous decay.

Now consider the kinetic energy $E(t)$ calculated with (3.1). Figure 5(a) shows the $E(t)$ curves measured in experiments $A^{\pm }$ and $A^{\pm }lid$ . For illustrative purposes, the curves start from $t\lt -2$ days (when the energy is almost zero); then, the electromagnetic forcing is switched on at $t=-2$ days, generating the increase in energy until the forcing is stopped at $t=0$ days, when the curves reach a maximum value and define the start of the experiment. For $t\gt 0$ , the energy decays over 25 days. The energy curves in experiments $A^{\pm }$ (red) are very similar, even though the experiments start from different initial conditions. The kinetic energy decays rapidly at first and more gradually at later times. In experiments $A^{\pm }lid$ (blue), the initial energy has a lower value due to the additional friction introduced by the lid. Again, both experiments show the same behaviour during the decay, despite starting from completely different initial conditions.

Figure 5.

(a) Evolution of the kinetic energy per unit mass in experiments $A^{\pm }$ (solid and dashed red lines), and $A^{\pm }lid$ (solid and dashed blue lines). All curves are normalised with the initial energy of experiment $A^{+}$ , $E_0=0.10$ cm $^2$ s⁻². A low-pass filter (frequency cutoff of 12.5 s $^{-1}$ ) was applied to smooth out the noise. The cyan and green solid lines are the theoretical Ekman decay proportional to $\exp (-2t/T_E)$ and $\exp (-4t/T_E)$ , respectively, with $T_E=141$ s. (b) Log–log plots of the curves shown in (a).

The main decay mechanism is bottom friction due to the lower Ekman layer (and the upper layer in the experiments with a lid). This can be verified by comparing the measured energy curves in experiments $A^{\pm }$ with the energy decay due to the standard Ekman friction, $E(t)/E_0=\exp (-2t/T_E)$ , where $T_E$ is the Ekman time scale (see § 2.2.3). For experiments $A^{\pm }lid$ , the upper lid causes the friction to double, so the energy decays as $\exp (-4t/T_E)$ . Both theoretical curves are included in figure 5(a) (cyan and green) and follow closely the energy decay measured in the experiments. This is further verified in the log–log plots in figure 5(b).

3.1.2. Equilibrium state

To examine the flow state towards which the system evolves, figure 6 presents the time-average vorticity fields $\overline {\omega }/f_0$ in experiments $A^{\pm }$ (panels a and b) and $A^{\pm }lid$ (panels c and d). The average fields are calculated from days $5$ to $10$ , which is the period in which the main flow features are already generated. All cases clearly show the formation of an anticyclone in the flat region (clockwise arrows and predominantly negative vorticity), and a cyclonic circulation over the slopes. Other less significant features include the meanders in the current over the topographic slopes and the presence of smaller, weaker eddies in the flat region.

Figure 6.

Time average of the relative vorticity and velocity fields measured in experiments (a) $A^{+}$ , (b) $A^{-}$ , (c) $A^{+}lid$ , (d) $A^{-}lid$ . The average is computed with 63 fields measured between day 5 and 10.

The main average features illustrate the equilibrium state configuration reached by the system regardless of the initial condition (shown in figure 3) and bottom friction magnitude. Indeed, experiments $A^{\pm }$ start with a different vorticity distribution and they end up in a similar state (figures 6 a and 6 b). The same occurs in experiments with an upper lid $A^{\pm }lid$ (figures 6 c and 6 d). The latter equilibrium states are much weaker because of the enhanced friction.

We can also note detailed differences between the average states in figure 6. The velocity fields suggest that the topographic current is more intense in $A^+$ than in $A^-$ , while the central vortex is more intense and bigger in $A^-$ than in $A^+$ . A similar difference is found in experiments $A^{\pm }lid$ , especially regarding the vortex size. Quantitative estimations of the equilibrium features supporting previous assertions are presented in table 3, where columns 2 and 3 show: (i) the maximum velocity $U_{\textit{max}}$ over the slopes to characterise the topographic current, and (ii) the maximum velocity value $V_{\textit{max}}$ in the central anticyclone (columns 4 and 5 contain numerically calculated values to be discussed in § 3.2). These differences are related to the circulation of the initial conditions, as shall be further discussed in numerical simulations (§ 3.2.2).

Table 3.

Characteristic velocity of the topographic current over the slopes ( $U_{\textit{max}}$ ) and the central anticyclone ( $V_{\textit{max}}$ ) measured from the equilibrium states shown in figure 6. Columns 2 and 3: laboratory experiments (approximate error $\pm 20$ %). Columns 4 and 5: numerical simulations using nonlinear Ekman friction (presented in § 3.2.1).

Summarising the laboratory experiments, the decay of an initially disordered flow in the basin rapidly develops an anticlockwise (cyclonic) current over the topographic slopes. Later, an anticyclone forms in the flat central region. This behaviour was repeated in experiments with different initial conditions and varying magnitudes of frictional effects. The characteristics of the anticyclone and the current over the topography may change in other experiments, but their generation and persistence were observed in all the analysed cases.

3.2.1. Simulations of the laboratory experiments

We simulated numerically experiments $A^{\pm }$ and $A^{\pm }lid$ . In these four runs, the flow domain and bottom topography are identical to the experimental square box (figure 1). The simulations replicate the relevant parameters in the laboratory shown in table 1. The initial conditions are the interpolated relative vorticity fields shown in figure 3. To simulate experiments $A^{\pm }$ , we used Ekman friction that is either linear (2.8) or nonlinear (2.9). The experiments with an upper lid, $A^{\pm }lid$ , were simulated by doubling the Ekman friction terms. The evolution time was $30$ days ( $\sim 377$ s).

Figure 7.

Velocity fields (arrows) and relative vorticity (surfaces) calculated at different times in the simulation of experiment $A^+$ using nonlinear Ekman friction (compare with figure 4).

Figure 7 shows the vorticity and velocity fields at different times in the simulation of experiment $A^{+}$ , which can be compared with the panels in figure 4. At time $t=0$ days (panel a), the initial field is virtually identical to the experimental one, as it was specifically constructed that way. The simulated flow closely resembles the stages observed in the experiments: the early generation of a cyclonic current over the slopes and the organisation of the central anticyclone. A difference from the laboratory experiment is that the small structures observed at late times are not generated in the simulation, as expected, since they are due to experimental errors, as discussed for figure 4. In this case, nonlinear Ekman friction was used; the simulation with linear friction (not shown) presented marginal differences.

A quantitative comparison of the energy decay in the laboratory experiments and simulations using linear and nonlinear Ekman friction is presented in figure 8. In all cases, there is a good agreement over at least 10 rotating periods (approximately 120 s), with the comparison slightly better when using nonlinear friction. At later times, the simulated energy continues decreasing, whereas a weak noise remains in the laboratory measurements.

Figure 8.

Evolution of the kinetic energy calculated in laboratory experiments (blue) and numerical simulations using linear (magenta) and nonlinear (red) Ekman friction: (a) $A^{+}$ , (b) $A^{-}$ , (c) $A^{+}lid$ , (d) $A^{-}lid$ . The comparison is slightly better when using nonlinear friction.

Figure 9.

Time average of the relative vorticity and velocity fields measured in the numerical simulations of experiments (a) $A^{+}$ , (b) $A^{-}$ , (c) $A^{+}lid$ , (d) $A^{-}lid$ . The average is computed with 63 fields measured between day 5 and 10 (compare with figure 6). Nonlinear Ekman friction was used.

The simulations also reproduce the average equilibrium state observed in the laboratory, shown in figure 6. To show it, figure 9 presents the average velocity and vorticity fields of the simulated experiments $A^{\pm }$ and $A^{\pm }lid$ using nonlinear friction. The averaged fields denote the anticyclone in the basin centre and the cyclonic slope current. Due to the lower friction in the simulations without the lid (panels a, b), the current and the anticyclone are more intense than in cases with a lid (panels c, d). Also, the stronger decay in the latter inhibits the growth of the anticyclone because the flow organisation is halted. Of course, there are qualitative differences with the laboratory measurements, but the overall equilibrium state is similar. Quantitatively, the experimental and numerical velocity values over the slopes and in the central anticyclone can be compared in table 3. These results show that the numerically simulated fluid tends to the mean state observed in the experiments. Thus, we conclude that the dynamics of the physical model is sufficient to generate the observed phenomena in the laboratory.

3.2.2. Effect of the initial circulation

The numerical simulations allow a more thorough examination of the equilibrium state in ocean-scale cases. In what follows, we perform new runs using a square basin with oceanic dimensions, bottom friction is not considered, and lateral friction is parametrised using a weak turbulent viscosity coefficient. Table 1 includes the rest of the oceanic flow parameters. In these simulations, we extend the analyses to better understand the effects of (i) the initial circulation and (ii) the topographic $\beta$ -effect.

Consider first the effects of the initial circulation. The laboratory experiments $A^{\pm }$ (figure 6 a,b) and the corresponding simulations (figure 9 a,b) showed that the average topographic current is faster and the central anticyclone is weaker in $A^{+}$ than in $A^{-}$ (see estimations in table 3). The only difference in these experiments is the initial flow configuration (figure 3 a,b) and the initial circulation $\varGamma _0$ is more positive in $A^+$ ( $\varGamma ^+_0\gt \varGamma ^-_0$ , see table 2). Therefore, we hypothesise that the more positive the initial circulation, the more intense the topographic current. Equivalently, the more negative the initial circulation, the more intense the central anticyclone. In other words, the strength of the equilibrium state may differ when the initial flow is sufficiently biased to circulate in a pre-established direction (Zavala Sansón Reference Zavala Sansón2025).

Figure 10.

(a) Initial relative vorticity and velocity fields in simulation $P$ ( $\varGamma _0=2.2 \times 10^{5}$ m $^{2}$ s $^{-1}$ ). (b) Average vorticity field in run $P$ calculated with 150 daily fields from day 200 to 350. Panels (c) and (d) show the same fields for simulation $N$ ( $\varGamma _0=-2.2 \times 10^{5}$ m $^{2}$ s $^{-1}$ ). The initial energy is $E_0=0.78 \times 10^{-2}$ m² s⁻² in both runs (or a velocity scale $\sqrt {2E_0}\approx 0.12$ m s⁻¹).

Figure 11.

Five velocity profiles from ensemble $P_i$ (simulations with positive circulation): (a) $u_{\!p}$ -profiles along $y$ , and (b) $v_{\!p}$ -profiles along $x$ . (c–d) Same profiles for ensemble $N_i$ (simulations with negative circulation). All runs start with the same energy $E_0$ and circulation $\varGamma _0$ as in figure 10.

Figure 10 presents two oceanic simulations ( $P$ and $N$ ) with the same circulation but opposite signs. Both runs lasted $365$ days. The initial vorticity fields (panels a and c) consist of a grid of $16 \times 16 = 256$ cyclones and anticyclones with a Gaussian profile and having the same amplitude. The field with positive circulation (run $P$ ) has 160 cyclones and 96 anticyclones arranged randomly (figure 10 a), generating a circulation of $2.2 \times 10^{5}$ m $^{2}$ s $^{-1}$ . The negative configuration (run $N$ ) has the vortices arranged in the same way as $P$ but with opposite signs (figure 10 c); this array generates a circulation of $-2.2 \times 10^{5}$ m $^{2}$ s $^{-1}$ . The two configurations have the same initial energy. Figure 10(b,d) shows the resulting average fields in the two simulations. The equilibrium state is reproduced in both cases: a cyclonic current is established over the topographic slopes, and an anticyclone forms in the central flat region. However, in simulation $P$ (panel b), the central anticyclone is much weaker than in simulation $N$ (panel d); also, the topographic current over the slopes is much more intense in $P$ compared with $N$ . These results confirm that the imposed initial circulation influences the characteristics of the equilibrium state.

Since the initial array in simulations $P$ and $N$ is random, we investigate whether the same equilibrium state is obtained with a different random arrangement having the same energy and circulation. For this purpose, we performed ensembles of simulations with positive and negative circulations, $P_i$ and $N_i$ , with $i=1\ldots 5$ (one of the runs in each group is shown in figure 10). For each member, we calculated the temporal average of the long-term distributions. We found similar equilibrium states as those shown in figure 10(b,d), that is, a strong (weak) topographic current and weak (strong) central anticyclone for ensemble $P_i$ ( $N_i$ ) (not shown).

To show quantitative differences between the equilibrium states in the two ensembles, we calculate the long-term temporal-average velocity $(\overline {u},\overline {v})$ in each simulation and then take the profiles $u_{\!p}(y)$ and $v_{\!p}(x)$ across the domain defined as

(3.3) \begin{align} \text{$u$-profile along $y$}:& \quad u_{\!p}(y) = \overline {u}(x_0,y), \quad \end{align}

(3.4) \begin{align} \text{$v$-profile along $x$}:& \quad v_{\!p}(x) = \overline {v}(x,y_0), \end{align}

where $x_0$ and $y_0$ are the coordinates of the anticyclone centre in each simulation (which is not necessarily the origin). Figure 11 presents the five profiles obtained from each ensemble. The $P_i$ profiles (panels a and b) match between them along the two directions, which indicates that the equilibrium state is practically the same: a strong topographic current and a weak and small anticyclonic structure at the central flat region, as found in figure 10(b). In contrast, the $N_i$ profiles (panels c and d) do not coincide exactly, which means that the equilibrium state might present some variations due to the random initial condition. Also, the $N_i$ -profiles denote a weaker topographic current and a stronger and bigger central anticyclone, as found in figure 10(d).

3.2.3. Simulations with different topographic $\beta$ -effect

To explore the influence of the topographic $\beta$ -effect on the equilibrium state, we conducted simulations with oceanic basins having the exact dimensions as in previous cases ( $L=500$ km and $H=1000$ m) but different slopes. All topographies have a width of $W=100$ km while the maximum height at the walls $D_w$ was 50, 250, 500, 750 and 950 m. Thus, cases range from a gentle slope to one with abrupt topography (small to higher topographic $\beta _t$ ). The simulations are named with the prefix BETA followed by the value of $D_w$ (table 4). The initial vorticity field in all cases consists of an arrangement of 128 cyclones and 128 anticyclones resembling a chessboard, which has zero circulation (figure 12 a). Due to the varying slopes, the volume slightly differs in each basin, so the intensity of the vortices in each simulation was adjusted to maintain the same initial energy.

Table 4.

Topographic parameters in simulations with topographies having the same width $W=100$ km and different height at the wall $D_w$ (second column). In all cases, $f_0=10^{-4}$ s $^{-1}$ .

To illustrate the results, we consider the equilibrium states calculated in simulations BETA50, 500 and 950. Figure 12(b–d) shows the long-term average velocity and vorticity fields denoting the equilibrium states: the cyclonic current over the slopes and the anticyclonic structure on the flat region of the basin. There are some quantitative differences, such as the more intense topographic current in the simulation with higher $\beta _t$ (BETA950, panel d); however, the long-term equilibrium is quite similar regardless of the slope steepness.

Figure 12.

(a) Initial vorticity field in the BETA simulations (table 4). Relative vorticity and velocity fields averaged between days 200 and 350 in simulations: (b) BETA50, (c) BETA500, (d) BETA950. In all experiments, the initial energy is $E_0=1.4 \times 10^{-2}$ m² s⁻² (or a velocity scale $\sqrt {2E_0}\approx 0.17$ m s⁻¹).

The effect of $\beta _t$ is reflected in the phase speed of the topographic waves generated over the slopes and propagating towards the local ‘west’; i.e. with shallow water to the right. Eventually, the waves are organised as the cyclonic topographic current or ‘zonal’ flow (see § 9.1.1 of Vallis Reference Vallis2017). To visualise the waves, we built Hövmoller diagrams of the average cross-slope vorticity field over 300 km along one of the lateral walls. Figure 13(a) presents the Hövmoller diagram for run BETA500. The waves are unclear at the initial stages ( $t\lt 50$ days) because turbulent motions dominate. For $t\gt 50$ days, the waves are clearly identified by the vorticity stripes with negative slope, as shown in the zoom between days 50 and 150 in figure 13(b). In the long term, the waves are damped and become less visible. Using similar diagrams, we also found the signal of topographic waves in simulations BETA50 and BETA250. However, the waves were hardly visible in BETA750 and BETA950, because $\beta _t$ is so high that wavy motions are quickly incorporated into the cyclonic current.

To show that the oscillations shown in figure 13(b) are topographic waves, consider the dispersion relation for a barotropic wave travelling in the local western direction as $\sigma =-\beta _t k/(k^2+R_d^{-2})$ , where $\sigma$ is the frequency, $k$ is the wavenumber along the slope and $R_d=\sqrt {gH_m}/f_0 \approx 852$ km is the deformation radius with $g$ the gravity. Assuming $k\approx 2\pi /W$ , then $k^2\gg R_d^{-2}$ , so the phase speed can be estimated as

(3.5) \begin{equation} c \equiv \frac {\sigma }{k} \approx -\frac {\beta _t}{k^2}. \end{equation}

For simulation BETA500 we find $c\approx -15$ km d⁻¹. The speed of the signals shown in figure 13(b) is given by the slope $\approx -17$ km d⁻¹, which is very close to the calculated value. We verified that the phase speed of the waves detected in runs BETA50 and BETA250 also obey (3.5) (in these cases, $c\approx 5$ km d⁻¹ and 12 km d⁻¹, respectively). The simulations confirm the presence of topographic waves, which were not detected in the experiments, probably because experimental errors mask these oscillations, and damping effects are much more intense.

Figure 13.

(a) Hövmoller diagram for the average cross-slope vorticity along 300 km in simulation BETA500. (b) Zoom between 50 and 100 days. The slope of the vorticity signal is $-17$ km d⁻¹ (the theoretical value is $\approx -15$ km d⁻¹, see text).

4.1.1. Flow evolution

Figure 14.

Scaled relative vorticity (surfaces) and velocity fields (arrows) measured in the continuously forced experiment $C^+_1$ from day 30 to day 55. At these times, the flow has reached a statistically quasi-steady turbulent state.

The fluid starts from rest, and the forcing is applied for 60 days. Then, between days 30 and 60, the system reaches a statistically stationary state where the energy injection is nearly balanced with viscous dissipation. Figure 14 presents six snapshots of the relative vorticity in experiment $C^+_1$ to show the unsteady evolution once the flow has reached a turbulent state. These fields show the unsteadiness and disordered behaviour of the flow within the statistically stationary stage. The flat central region seems to contain larger structures than the periphery. However, closely inspecting the velocity field over the topographic slopes (along the lateral walls) reveals a predominantly cyclonic flow with shallow water to the right. Similar results are obtained in the twin experiment $C^+_2$ (not shown). This behaviour has been observed in previous $f$ and $\beta$ -plane simulations in a similar basin (Zavala Sansón Reference Zavala Sansón2022, Reference Zavala Sansón2025).

Figure 15.

Evolution of the kinetic energy per unit mass in experiments $C^+_1$ and $C^+_2$ (red and green lines, respectively). The curves are normalised with the mean energy between days 30 and 60, $E_m=4.2\times 10^{-6}$ m² s⁻². A low-pass filter (frequency cutoff of 12.5 s $^{-1}$ ) was applied to smooth out the noise. The vertical grey lines indicate the 5-day period at which the Lorentz forcing was reversed to avoid favouring any circulation.

Figure 15 shows the evolution of the total kinetic energy for both realisations, $C^+_1$ and $C^+_2$ . At the beginning of the experiments, the energy continuously increases until reaching its first peak on day 6, after which it decreases, only to rise again and reach a quasi-stable energy value for the rest of the experiments. The vertical grey lines between days 30 and 60 indicate the approximate time the Lorentz forcing was reversed to maintain the randomness and avoid favouring a particular flow pattern. Finally, on day 60, the forcing is turned off, marking the end of the experiment. Both curves present similar oscillations despite the unavoidable subtle differences in the execution of the two realisations. The energy curves demonstrate the reproducibility of the experiments and that the continuously forced system reaches a statistically quasi-stationary turbulent state between days 30 (or even less) and 60.

4.1.2. Equilibrium state

Figure 16.

Time average of the relative vorticity and velocity fields measured in experiments (a) $C^+_1$ , (b) $C^+_2$ . The time average is computed with 376 fields measured between day 30 and 60.

Now we examine the mean state reached by the flow despite the presence of the continuous forcing. Figure 16 shows the averaged velocity and vorticity fields between days $30$ and $60$ in the twin experiments. In both realisations, the cyclonic flow is neatly observed over the slopes, while a robust anticyclone is formed at the flat region. Again, the similarity between the fields demonstrates the reproducibility of the experiments and the consistency of the results. The most remarkable feature is that this equilibrium configuration is similar to the decaying case. Therefore, we conclude that the average formation of a central anticyclone in a basin is a robust phenomenon even in the presence of the forcing.

4.2.1. Equilibrium state in forced simulations

Figure 17.

Time average of the relative vorticity and velocity fields measured in simulations (a) $F_1$ , (b) $F_2$ . The time average is computed with 265 daily fields between days 100 and 365. Both fields have a similar mean energy, $E_m=0.72 \times 10^{-2}$ m $^2$ s⁻², but are generated with a different random forcing.

Figure 18.

Time evolution of (a) the kinetic energy per unit mass (red curve for the total value, purple over the slopes and green over the central flat region), and (b) regional circulations (same colour code) in simulation $F_1$ . All energy curves are normalised with the mean energy $E_m$ calculated between days 100 and 365. Panels (c) and (d) show the same quantities for simulation $F_2$ .

Here, we present the results from two simulations, $F_1$ and $F_2$ , where the fluid is maintained in motion by forcing terms having the same amplitude but different random protocols (defined by different random phases of the forcing harmonic modes), so they develop the same mean energy once they reach statistical equilibrium. The simulated time evolution is similar to that observed in the experiment shown in figure 14: the flow is set in motion from rest, and some time later it reaches a statistically stationary turbulent state (not shown). In what follows, we will focus on the average fields of the turbulent stage.

Figure 17 presents the time-average vorticity and velocity fields in simulations $F_1$ and $F_2$ . Again, despite the continuous action of the random forcing, a preferred state can be observed in all fields. The velocity vectors indicate that, on average, a cyclonic current forms over the slopes near the boundaries, while an anticyclonic flow develops in the basin’s centre. As expected, the averaged vorticity fields present small-scale variability due to the continuous forcing. However, it is still evident that the vorticity is predominantly negative in the centre of the basin with the formation of the cyclonic topographic current with shallow water to its right.

The evolution of the total kinetic energy in simulation $F_1$ , normalised by the mean energy at equilibrium ( $E_m$ ), is presented in figure 18(a) (red curve). The energy measured over the slopes and the flat region is also shown (purple and green curves, respectively). Throughout the simulation, approximately 50 %–60 % of the energy is concentrated over the slopes, while the remaining portion is in the central flat area. We also calculated the circulation in the two regions. In figure 18(b), it can be seen that, during the first days, there is no clear preference to concentrate the circulation in any region. Once the statistically steady state is reached, a clear distinction emerges: circulation is positive over the slopes and negative in the central region. This result is consistent with the averaged fields previously shown. Figure 18(c,d) presents the same curves for simulation $F_2$ . These plots show that a similar evolution of the energy curves and regional circulations occurs when using a different seed for the random forcing.