3. Results

For comparison, we first present flow properties in the steady stage for different damping rates. For all tested cases, no system-sized vortex is observed and, thus, we refer to the present configuration as unbounded turbulence. At the smallest damping rate ( $\gamma =5.2$ ), the coherent vortices coexist with the large-scale streamers produced via inverse cascade (see figure 1 g) and the energy spectrum presents a steepened scaling of $k^{-3}$ in the energy inertial range. At moderate-to-large damping rates, the turbulent vortices have smaller size but more intense vorticity (figure 1 a,d), and the spectral scaling of energy cascade for these cases is $k^{-5/3}$ or flatter than $k^{-5/3}$ (see figure 3). The above-mentioned results are consistent with the extensive study of the energy spectrum and flow pattern in the steady state by Zhu et al. (Reference Zhu, Xie and Xia2024).

As shown in figure 1, after the forcing is switched off, the small-scale vortices disappear quickly. In the (late) decay stage, the flow fields for three damping rates exhibit a very similar pattern that is dominated by several large-scale vortices (the right column). In addition, during the decaying process, the large-scale vortices undergo stretching, resulting in a slender shape (see the green arrows). The stretching behaviour was also observed in the numerical study by Fox & Davidson (Reference Fox and Davidson2010) who suggested that it contributed to the enstrophy fluxes at larger scales. Moreover, it is seen that the length scale of turbulent flow increases during decay. This increase of the length scale in decaying turbulence is not necessarily a sign of an inverse cascade. In 3-D turbulence, the turbulent structures grow during decay due to the preservation of large-scale correlations in the von Karman–Howarth equation (Ishida et al. Reference Ishida, Davidson and Kaneda2006; Esteban et al. Reference Esteban, Shrimpton and Ganapathisubramani2019). This aspect will be further discussed from the perspective of energy fluxes.

Figure 2.

Time evolution of (a) energy and (b) enstrophy–energy ratio for different damping rates. In panel (a), the dotted lines denote correspondingly fitting the results with (3.4) in the initial decay stage for the two cases with $\gamma$ = 13.9 and 5.2, respectively, and the dashed lines are from (3.6). The lower inset in panel (a) plots the deviations defined as $(E_{\textit{fit}}-E_{\textit{exp}})/E_{\textit{exp}}\times 100$ with $(t-t_0)$ for the fit of (3.6) and the upper inset shows $E/E_0$ against dimensionless time, $\alpha (t-t_0)$ . Inset in panel (b) shows the normalised Taylor microscale $\lambda /l_{\!f}$ in semi-log plot.

We consider the incompressible 2-D turbulence described by the Navier–Stokes equation as

(3.1) \begin{equation} \partial _t \boldsymbol{u} + (\boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{\nabla }) \boldsymbol{u} = -\boldsymbol{\nabla }\!p/\rho + \nu{\nabla} ^2 \boldsymbol{u} + \boldsymbol{D} + \boldsymbol{f}, \end{equation}

where $\boldsymbol{f}$ is the external force and $\boldsymbol{D}$ the friction. Multiplying (3.1) by $\boldsymbol{u}$ and integrating over the space, we obtain the time evolution of kinetic energy $E\ (=\langle |\boldsymbol{u}|^2\rangle /2)$ as

(3.2) \begin{equation} {\rm d}E/{\rm d}t = -\nu \langle |\boldsymbol{\nabla }\boldsymbol{u}|^2\rangle + \langle \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{D}\rangle + \langle \boldsymbol{u} \boldsymbol{\cdot }\boldsymbol{f}\rangle. \end{equation}

The frictional term $\boldsymbol{D}$ has different formulations, as proposed in various models. However, it is not straightforward to determine the form of $\boldsymbol{D}$ in forced turbulence, especially in experiments. Many studies of (quasi-)2-D turbulence used a linear damping (Boffetta & Ecke Reference Boffetta and Ecke2012), i.e. $\boldsymbol{D} _0= -\alpha _0 \boldsymbol{u}$ with $\alpha _0$ a constant damping rate. Considering $\boldsymbol{D} _0$ and in the inviscid limit, (3.2) for decaying turbulence ( $\boldsymbol{f=0}$ ) degenerates to

(3.3) \begin{equation} {\rm d}E/{\rm d}t=-2\alpha _0 E, \end{equation}

so the energy should decay exponentially as

(3.4) \begin{equation} E/E_0=e^{-2\alpha _0 (t-t_0)}, \end{equation}

where $E_0=u_{\textit{rms}}^2/2$ is the ensemble-averaged energy in the steady stage.

Figure 2(a) shows the time evolution of normalised energy $E(t)/E_0$ for different damping rates. One can see that energy decays in a more complex manner and the decay rate decreases with time. This phenomenon was noticed by Fang & Ouellette (Reference Fang and Ouellette2017) who identified multiple decay stages. Xia et al. (Reference Xia, Shats and Falkovich2009) considered the exponential decay only in the initial stage. In addition, Grianik et al. (Reference Grianik, Held, Smith and Vallis2004) introduced another formulation called a quadratic drag with an expression of

(3.5) \begin{equation} \boldsymbol{D}_1 = -\beta |\boldsymbol{u}|\boldsymbol{u} = -\alpha _t\boldsymbol{u}, \end{equation}

where $\beta$ has dimensions of inverse length and may be related to the thickness $h$ of the bottom fluid layer in this study. Assuming a linear velocity profile in the fluid layer, the damping rate $\alpha _t=\beta |\boldsymbol{u}|$ is related to the flow strength and can be further expressed as $\alpha _t=C|\boldsymbol{u}|/h$ with $C$ a constant. It is reasonable that $\alpha _t$ is proportional to the velocity gradient $|\boldsymbol{u}|/h$ in the fluid layer. Replacing $\alpha _0$ with $\alpha _t=C|\boldsymbol{u}|/h$ in (3.3), we obtain the energy decaying function for a quadratic damping as

(3.6) \begin{equation} E/E_0=[1+\alpha _{t_0} (t-t_0)]^{-2}, \end{equation}

where $\alpha _{t_0}=Cu_{\textit{rms}}/h$ is the decay rate in the steady stage. In this study, we simply use $\alpha$ to represent $\alpha _{t_0}$ .

We fit the measured energy decay curve with (3.6) and obtain the coefficient $\alpha$ . As shown in figure 2(a) and its lower inset, the experimental data are in reasonably good agreement with the quadratic drag model. Specifically, the fitting errors for the quadratic model are smaller than 10 % across the entire decay stage and for different damping rates. It is noted that the coefficient $\alpha$ for quadratic drag is close to that obtained by fitting the decay curve in the initial decay stage with an exponential form (Zhu et al. Reference Zhu, Xie and Xia2024), because (3.4) and (3.6) are almost equal when $t-t_0$ approaches zero. The upper inset in figure 2(a) plots the energy decay curves against the dimensionless time $\alpha (t-t_0)$ , which demonstrates an excellent collapse across all cases.

Given the presence of viscous dissipation and inherent 3-D motions (discussed later), the level of fitting errors indicates that our experimental data agree reasonably well with the quadratic model. However, the observed deviations of the model from the data suggest that the quadratic model is not fully satisfactory for explaining the energy decay curve. Previous numerical studies (van Kan et al. Reference van Kan, Favier, Julien and Knobloch2022, Reference van Kan, Favier, Julien and Knobloch2024) in 2-D turbulence considered an alternative formulation known as the cubic damping model, i.e. $\boldsymbol{D}_2 = -\beta |\boldsymbol{u}|^2\boldsymbol{u}$ , although its physical meaning remains unclear. This cubic damping has also been used in studies of active turbulence (Wensink et al. Reference Wensink, Dunkel, Heidenreich, Drescher, Goldstein, Löwen and Yeomans2012). In addition, a mixed drag formulation was presented by Marche (Reference Marche2007), which includes the linear and quadratic terms. Moreover, previous studies developed a generalised nonlinear framework (Ray & Fröhlich Reference Ray and Fröhlich2015; Civera et al. Reference Civera, Grivet-Talocia, Surace and Fragonara2021) to model the drag exerted by fluid flow on a moving body. In Appendix A, we discuss the applicability of these different drag models to our experimental data.

Figure 3.

Time evolution of energy spectrum: (a) $\gamma =13.9$ ; (b) $\gamma =8.8$ ; (c) $\gamma =5.2$ . In panels (a–c), the black circles represent energy spectrum in the steady state, and the other five curves denote those in the decay stage. The insets in panels (a–c) show the time evolution of the scaling exponent $\zeta$ obtained by fitting the spectrum in the large-scale (magenta circles) and small-scale (light-blue triangles) ranges, respectively, and the two dashed lines denote $\zeta =-5/3$ and −3, respectively. Panels (d,e, f) show corresponding $E_k(k_i,t)/E_k(k_i,t_0)$ for three typical wavenumbers, representing the large scale, the forcing (intermediate) scale and the small scale, respectively, and the overall decaying curve is duplicated from the inset of figure 2(a) for reference.

Figure 2(b) plots the time evolution of the enstrophy-energy ratio normalised by $\varOmega _0/E_0$ with $\varOmega _0=\omega _{\textit{rms}}^2/2$ the ensemble-averaged value in the steady stage. It is seen that enstrophy decays faster than energy, as is reminiscent of the selective decay mechanism (Mininni et al. Reference Mininni, Montgomery and Pouquet2005). In addition, the enstrophy decay curves do not collapse for different damping rates, indicating the frictional damping has an impact on enstrophy, as further illustrated in figures 3 and 4. The effect of friction on enstrophy cascade has also been reported in previous studies (Bernard Reference Bernard2000; Boffetta et al. Reference Boffetta, Cenedese, Espa and Musacchio2005). The inset of figure 2(b) plots the normalised Taylor microscale $\lambda /l_{\!f}$ ( $\lambda = \sqrt {E/\varOmega }$ ) against decay time. It has been found that the Taylor microscale $\lambda$ increases with decreasing $\gamma$ (Zhu et al. Reference Zhu, Xie and Xia2024). In addition, $\lambda$ becomes larger during decay; namely, the turbulent structures grow, consistent with the observations in figure 1.

Figure 4.

Time evolution of (a–c) spatial-averaged energy and (d–f) enstrophy fluxes as a function of the filtering length scale $r$ for three damping rates: (a,d) $\gamma =13.9$ ; (b,e) $\gamma =8.8$ ; (c, f) $\gamma =5.2$ . The insets in panels (d–f) show the same results as the main figure in a log–log plot and the negative enstrophy fluxes are removed. The black circles represent energy/enstrophy flux in the steady state and the other five curves denote those in the decay stage, the same as in figure 3(a–c).

Figure 3(a–c) shows the time evolution of energy spectra for three different damping rates. In the steady state of quasi-2-D turbulence, the damping rate $\gamma$ plays a significant role in the energy spectra (Zhu et al. Reference Zhu, Xie and Xia2024), as shown by the black circles. In the decaying stage, we obtained the energy spectrum through running average with a time window of $\alpha \Delta t\approx 0.1$ , a treatment similar to that of Mininni & Pouquet (Reference Mininni and Pouquet2013). It may be expected that friction dominates the decaying of large-scale structures, whereas viscosity acts at small scales. However, one can see in figure 3(d–f) that the large-scale motions decay the slowest, while energy at intermediate scales decays the fastest, which can be explained from the energy budget at different scales (see (9) of Boffetta & Ecke (Reference Boffetta and Ecke2012)). In the steady state, the change rate of energy $\partial E/\partial t$ balances the transfer function term $T$ , the forcing term $F$ and two dissipation terms ( $\mathcal{D}_\alpha$ and $\mathcal{D}_\nu$ ) at every scale. When $F$ , limited in a narrow range of scales, as in the present case (see figure 2a in Alexakis & Biferale Reference Alexakis and Biferale2018), is suddenly removed, the three terms $T$ , $\mathcal{D}_\alpha$ and $\mathcal{D}_\nu$ could not respond to this change immediately, and thus the energy at the initial forcing scale decays fastest. As shown later, the energy at this scale is transferred simultaneously to larger and smaller scales.

The different decaying rates at different scales result in changes of the spectral scaling. It is seen that the energy spectrum undergoes two decaying stages. In the initial stage, the shape of spectrum varies with time, whereas in the late stage, it presents a self-preserving decay. Specifically, at large scales, the spectral slope steepens over time when damping is strong (figure 3 a,b), eventually saturating near $k^{-3}$ , while for the weakest damping (figure 3 c), the scaling of $k^{-3}$ persists in the entire decaying stage (see the insets in figure 3 a–c). The steepening of the energy spectrum at large scales during decaying was also reported in previous studies (Tabeling et al. Reference Tabeling, Hansen and Paret1998; Danilov et al. Reference Danilov, Dolzhanskii, Dovzhenko and Krymov2002; Mininni & Pouquet Reference Mininni and Pouquet2013). In contrast, at small scales, the spectrum becomes flatter and saturates to a $k^{-5/3}$ scaling. Mininni & Pouquet (Reference Mininni and Pouquet2013) demonstrated numerically that the spectrum at small scales instead steepened during decaying. We attempt to explain this inconsistency. In decaying 2-D turbulence, the decay rate $\alpha _k$ at a scale $k$ can be expressed as $\alpha + \nu k^2$ and the process of inverse cascade can further increase $\alpha _k$ at small scales since some energy cascades upscale, resulting in a steeper spectrum there during decaying. However, in the presence of forward energy transfers (shown later), energy can be transferred from the intermediate scales to small scales, which decreases the decay rate at small scales and a flatter scaling can be observed.

Recalling that the damping dominates the whole decaying stage, it may thus be expected that the damping rate $\gamma$ also plays a role in the energy spectrum during decay. Surprisingly, on the one hand, the energy spectra in the late stage are independent of $\gamma$ . On the other hand, this spectral behaviour is remarkably similar to that observed in atmospheric flows (Nastrom et al. Reference Nastrom, Gage and Jasperson1984). The interpretation of the Nastrom–Gage spectrum remains an active area of research. Previous studies interpreted the $k^{-3}$ scaling as the enstrophy cascade (Lindborg Reference Lindborg1999; Cho & Lindborg Reference Cho and Lindborg2001) or attributed it to the formation of condensation (Falkovich Reference Falkovich1992; Xia et al. Reference Xia, Punzmann, Falkovich and Shats2008). A consensus on the origin of $k^{-5/3}$ has not been established, and it may be associated with the small-scale convective events (Vallis et al. Reference Vallis, Shutts and Gray1997), the sharp temperature fronts (Tulloch & Smith Reference Tulloch and Smith2006), confinement/rotation-induced inverse energy transfers (Falkovich Reference Falkovich1992; Xia et al. Reference Xia, Shats and Falkovich2009), Rossby number-regulated downscale energy cascade (Vallgren et al. Reference Vallgren, Deusebio and Lindborg2011) or the inertia-gravity waves (Callies et al. Reference Callies, Ferrari and Bühler2014). The present observation combined with the transfer fluxes shown later may provide an alternative explanation for the dynamics of these scalings.

The energy flux $\varPi ^{(r)}$ and enstrophy flux $Z^{(r)}$ are calculated using the filtering space technique based on the filtered velocity/vorticity fields with the filtering length scale $r$ . By definition, $\varPi ^{(r)}\gt 0$ ( $Z^{(r)}\gt 0$ ) indicates that energy (enstrophy) is transferred to smaller length scales, while $\varPi ^{(r)}\lt 0$ ( $Z^{(r)}\lt 0$ ) represents energy (enstrophy) to larger scales. Figure 4 shows the energy and enstrophy fluxes as a function of the filtering scale. Similar to the energy spectrum, the fluxes in the decaying stage are calculated over a finite time interval. In the steady state, energy fluxes are negative at large scales and enstrophy fluxes are positive at small scales (the black circles in figure 4), indicating the presence of inverse energy transfers and forward enstrophy transfers, respectively.

Over the course of decay, the profiles of energy fluxes are shifted to larger length scales. For instance, at a moderate damping rate (see figure 4 b), the peak position of the negative energy flux shifts from 1.5 $l_{\!f}$ to 5 $l_{\!f}$ (although in a somewhat non-monotonic way). This phenomenon was also observed in previous studies (Mininni & Pouquet Reference Mininni and Pouquet2013; Fang & Ouellette Reference Fang and Ouellette2017). It suggests that the increase of the length scale of flow in decaying 2-D turbulence is attributed to the shift of inverse energy fluxes to larger scales. Similarly, the peak position of enstrophy fluxes moves from 0.5 $l_{\!f}$ to larger scales ( $\sim 1.5l_{\!f}$ ).

We now discuss the connection between the scalings of energy spectra and energy/enstrophy cascades. Considering that the $k^{-3}$ scaling (figure 3) and the negative energy fluxes (figure 4 a–c) occur in nearly the same range of scales, we may conclude that the $k^{-3}$ scaling corresponds to the inverse energy transfers. Moreover, the physics underlying the occurrence of the steepened scaling in decaying turbulence may be similar to that for the formation of spectral condensation in forced (unbounded) turbulence. Zhu et al. (Reference Zhu, Xie and Xia2024) demonstrated that in forced 2-D turbulence, the suppression of small-scale vortices by large-scale motions led to the spectral transition to $k^{-3}$ even in unbounded cases. The $k^{-3}$ scaling in decaying 2-D turbulence, especially for strong damping (figure 3 a,b), can be attributed to the disappearance (or rapidly decaying) of the small-scale intense vortices (figure 1).

Figure 5.

(a) Time series of spatial-averaged compressibility ratio $\overline {C}_t$ for different damping rates. (b,c) Instantaneous fields of compressibility ratio $C$ in the steady (cross in panel a) and decaying (circle in panel a) stages for $\gamma = 5.2$ , respectively. The red spots reflect the small-scale 3-D motions. The black bars denote the scale $2l_{\!f}$ . (d,e) Probability density functions of energy flux in different stages for the filtering scale of $r/l_{\!f}=0.4$ and $r/l_{\!f}=2.5$ , respectively.

We observe an intriguing scaling of $k^{-5/3}$ in the late stage of decay for different cases at small scales where energy fluxes are nearly zero, while positive enstrophy fluxes are seen. On the one hand, we would not say that this scaling reflects an enstrophy cascade, which usually corresponds to a spectral scaling not flatter than $k^{-3}$ (Boffetta & Ecke Reference Boffetta and Ecke2012). In addition, the forward enstrophy transfers not only occur at small scales but also are notable at intermediate-to-large scales, as shown in the insets of figure 4(d–f). On the other hand, we cannot rule out the possibility that the $k^{-5/3}$ scaling represents the forward energy transfers considering that the 3-D effect may be weak, but is inherent in thin-layer turbulence (Tithof et al. Reference Tithof, Martell and Kelley2018).

The compressibility ratio defined as $C(x_1,x_2,t)=(\boldsymbol{\nabla }\boldsymbol{\cdot }\boldsymbol{u})^2 /|\boldsymbol{\nabla }\boldsymbol{u}|^2$ can be used to quantify the divergence and thus the two-dimensionality of the 2-D velocity field (Boffetta et al. Reference Boffetta, Davoudi, Eckhardt and Schumacher2004), and $C=0$ means horizontally divergence-free. Figure 5(a) shows the time series of spatial-averaged compressibility ratio $\overline {C}_t$ for different cases. In the steady state ( $t\lt t_0$ ), $\overline {C}_t$ is much smaller than 0.1, indicating that two-dimensionality is well satisfied. After turning off the forcing, we observe a large increase of $\overline {C}_t$ and $\overline {C}_t\approx 0.2$ in the late decaying stage. On the one hand, it indicates that the two-dimensionality of flow is reduced. As shown in figure 5(b,c), the horizontal divergence (an indicator of 3-D motion) in the decaying stage is indeed larger than that in steady state. On the other hand, in free-surface flow above the 3-D homogeneous turbulence, the compressibility is 0.5 (Li et al. Reference Li, Salmon, Hassaini, Chang, Mucignat and Coletti2025). In the present decaying 2-D turbulence, $\overline {C}_t$ is notably smaller than 0.5, suggesting that a certain degree of two-dimensionality is preserved, even in the late decay stage. Moreover, figure 5(d) plots the probability density functions of the energy flux $\varPi ^{(r)}$ in the steady and decaying stages for $r/l_{\!f}=0.4$ . It is seen that the distribution of $\varPi ^{(r)}$ presents a notably negative skewness in the steady stage, while it becomes slightly positive-skewed in the decaying stage. Although this weak positive skewness is not reflected in the profile of energy flux in figure 4, it indicates the enhancement of forward energy transfers. In addition, at large scales ( $r/l_{\!f}=2.5$ ), a negative-skewed distribution of $\varPi ^{(r)}$ remains unchanged in different stages, see figure 5(e) (and figure 4 c). The above-mentioned observations suggest that in the current configuration, the 3-D motion at large scales is suppressed by the effect of spatial confinement (due to large aspect ratio), whereas the small-scale 3-D flow is inhibited by the 2-D Lorentz force in the steady state. Therefore, one can expect that in the decaying stage without the body force, the 3-D flow associated with forward energy transfers prevails at small scales, which may thus cause the $k^{-5/3}$ scaling.