4.1.1. Uncertainty energy

Figure 1 shows the time evolutions of $\langle E_{\varDelta }\rangle$ for each case F1, F2 and F3. The very first thing that happens immediately after the perturbations are seeded is a decrease of $\langle E_{\varDelta }\rangle$ in all three cases. This initial correlating action is caused by the concentration of the initial perturbations at the highest wavenumbers where dissipation is high. The insets of figure 2 show that $\langle \varepsilon _{\varDelta } \rangle$ is orders of magnitude higher than $\langle P_{\varDelta } \rangle$ at the earliest times in all three cases. As time proceeds, the uncertainty's dissipation rate decreases and its production rate increases until production overtakes dissipation (see figure 2) and $\langle E_{\varDelta }\rangle$ begins to grow. This initial growth is shown in the insets of figure 1 and it differs for F1 and F2 on the one hand and F3 on the other. For F1 and F2, $\langle E_{\varDelta }\rangle$ is observed to grow exponentially in the approximate time range $\tau \in [0.2,2.9]$. Previous DNS studies have already observed such exponential growth (Boffetta & Musacchio Reference Boffetta and Musacchio2017; Berera & Ho Reference Berera and Ho2018). For F3, the initial growth is from $\tau \approx 2.5$ to $\tau \approx 12.6$ and is subdivided in two parts. In the time range $\tau \in [2.5,7.5]$, the turbulence and its strain rate are not statistically stationary and the time evolution of $\langle E_{\varDelta }\rangle$ is not exponential. Indeed, the plot of the logarithm of $\langle E_{\varDelta }\rangle$ vs time in the inset of figure 1(c) has a positive curvature in that time range. An exponential growth of $\langle E_{\varDelta }\rangle$ appears to set in at $\tau \approx 7.5$ and lasts until about $\tau \approx 12.6$. It is noteworthy that an exponential growth of uncertainty also exists in F3 and that it starts a little earlier than when stationarity sets in. (The exponential regime's time range is longer for F3 than for F1 and F2 mainly because of F3's lower Reynolds number as argued in Appendix B). The results and analysis in the remainder of this paper confirm these interpretations.

Figure 1.

Time evolutions of average uncertainty energy for different cases: (a) case F1; (b) case F2; (c) case F3; (d) comparison F1–F2. Inset: the initial time evolution of average uncertainty energy in semilogarithmic plot. The uncertainty evolutions of F1 and F2 are presented together in (d). Inset: the uncertainty evolutions during $\tau \in [2.9,6.5]$. The exponential of exponential function fit is indicated by a solid black line.

Figure 2.

Time evolutions of each term in (2.4) for different cases, where $\varepsilon _{{tot}}=\varepsilon ^{(1)}+\varepsilon ^{(2)}$ is the total dissipation of the reference and perturbed fields: (a) case F1; (b) case F2; (c) case F3. Inset: the initial time evolution of the internal production, dissipation and external input/output in semilogarithmic plot.

The growths of $\langle E_{\varDelta }\rangle$ are identical in F1 and F2 (see figure 1d) until the time when $\langle F_{\varDelta }\rangle$ becomes significantly non-zero in F1 (see figure 2a). The regime of exponential growth is followed by what appears to be an exponential of exponential regime from $\tau \approx 2.9$ to $\tau \approx 6.5$. This exponential of exponential growth is the same in F1 and F2 and is highlighted by the fit in the inset of figure 1(d). A similar growth range has been observed in previous DNS that are similar to F1 and go up to even higher Reynolds numbers (Boffetta & Musacchio Reference Boffetta and Musacchio2017; Berera & Ho Reference Berera and Ho2018). This exponential of exponential growth is confirmed by our analysis and further DNS results in § 4.2.

After time $\tau =6.5$, the uncertainty growths for F1 and F2 deviate from each other as shown in figure 1(d) (${|\langle E_{\varDelta }\rangle _{F1}-\langle E_{\varDelta }\rangle _{F2}|}/{\langle E_{\varDelta }\rangle _{F1}}>5\,\%$ and growing as $\tau$ grows above $6.5$) because $\langle F_{\varDelta }\rangle$ starts growing significantly above zero ($\langle F_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle =0.06$ at $\tau =6.5$) in F1 whereas it is identically zero in F2 for all time (see figure 2). The reference and perturbed fields achieve significant decorrelation after the exponential growth of $\langle E_{\varDelta }\rangle$ for both F1 and F2, resulting, in case F1, in non-zero values of $\langle F_{\varDelta }\rangle$ which eventually grow significantly above the reference field's turbulence dissipation rate, but only after $\tau =6.5$. The additional external decorrelating action of the forcing leads to eventually fully decorrelated reference and perturbed fields in case F1 as the ratio $\langle E_{\varDelta }\rangle /\langle E_{{tot}}\rangle$ stops growing and saturates at $0.97\pm 0.07$ after $\tau =10.6$. In case F2 the identical forcing in both fields acts as a perpetual partially correlating (rather than decorrelating) action of the two fields and to a resulting eventual saturation of $\langle E_{\varDelta }\rangle /\langle E_{{tot}}\rangle$ at $0.59\pm 0.05$ for $\tau \ge 8.9$. In Appendix A we provide evidence showing that the early and mid-time evolutions (i.e. the exponential regime and the exponential of exponential regime) of the average uncertainty energy are not very sensitive to the form and amplitude of the initial perturbations.

For case F3, the growth of $\langle E_{\varDelta }\rangle$ following the exponential regime ending at about $\tau \approx 12.6$ can be seen in figure 1(c) and cannot be fitted by the exponential of exponential function detected in cases F1 and F2 nor any clear linear or power-law growth functions. As in F2, the correlating action of the identical external forcing in both the reference and perturbed fields leads to them remaining partially correlated at all times and to an eventual saturation of $\langle E_{\varDelta }\rangle /\langle E_{{tot}}\rangle$ at $0.66\pm 0.01$ for $\tau \ge 25.3$.

We close this subsection by pointing out that the only case of linear growth that we may have detected in our DNS is for F1 in the time range $\tau \in [6.5,10.6]$. A linear growth regime has been predicted by Aurell et al. (Reference Aurell, Boffetta, Crisanti, Paladin and Vulpiani1997), however our simulations suggest that it may depend on the type of forcing. Furthermore, the Reynolds number of our DNS may not be high enough to observe it clearly and the very level of Reynolds number required may itself depend on the external forcing. We examine this issue again in the following subsections. The time ranges of the different uncertainty growth regimes in each case F1, F2 and F3 are summarised in table 2.

Table 2.

Time ranges of different uncertainty growth regimes.

4.1.2. Mechanisms of the uncertainty evolution

The time evolutions of each term in (2.4), including the growth rate $\textrm {d}\langle E_{\varDelta }\rangle /\textrm {d}t$ obtained directly from the DNS, are shown in figure 2. As can be seen in the figure, we started by checking that $\textrm {d}\langle E_{\varDelta }\rangle /\textrm {d}t$ agrees well with its value obtained from (2.4). In all cases F1, F2 and F3, $\langle P_{\varDelta }\rangle > \langle \varepsilon _{\varDelta }\rangle$ when $\textrm {d}\langle E_{\varDelta }\rangle /\textrm {d}t >0$. In cases F2 and F3 where $\langle F_{\varDelta }\rangle = 0$ at all times, the eventual saturation when $\textrm {d}\langle E_{\varDelta }\rangle /\textrm {d}t \approx 0$ is characterised by the balance $\langle P_{\varDelta }\rangle \approx \langle \varepsilon _{\varDelta }\rangle$. This balance reflects the long-time partial correlation between the reference and perturbed fields and the long-time saturation of $\langle E_{\varDelta }\rangle /\langle E_{{tot}}\rangle$ at a value smaller than 1 reported in the previous subsection.

We also observe in figure 2 for all cases F1, F2 and F3 that the long-time saturation is such that $\langle \varepsilon _{\varDelta }\rangle \approx \langle \varepsilon _\textrm {tot}\rangle \equiv \langle \varepsilon ^{(1)}+\varepsilon ^{(2)}\rangle$ which implies $\langle \varepsilon _{{corr}}\rangle \approx 0$. In cases F2 and F3, this means that the long-time saturated non-zero steady state of $\langle P_{\varDelta }\rangle$ is such that $\langle P_{\varDelta }\rangle \approx \langle \varepsilon ^{(1)}+\varepsilon ^{(2)}\rangle \approx \langle F^{(1)}+F^{(2)}\rangle$ (recall $\langle F_{\varDelta }\rangle =0$ and $\langle F_{{corr}} \rangle =0$ in F2 and F3): the correlating action by the identical forcing in both statistically stationary reference and perturbed fields is directly balanced by the decorrelating action of the internal production of uncertainty.

The uncertainty dissipation rate $\langle \varepsilon _{\varDelta }\rangle$ reaches its long-time asymptotic balance with $\langle \varepsilon _{{tot}}\rangle$, i.e. $\langle \varepsilon _{\varDelta }\rangle /\langle \varepsilon _{{tot}}\rangle >0.95$, at about $\tau = 16.1$ for F3 and at about $\tau = 5.6$ for both F1 and F2. This is slightly before but close to the time $\tau = 6.5$ when $\langle F_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle =0.06$ stops being negligible in F1 and the perturbation evolutions start diverging between F1 and F2. The presence of positive $\langle F_{\varDelta }\rangle$ in F1 delays the decay towards $0$ of $\textrm {d}\langle E_{\varDelta }\rangle /\textrm {d}t$ which is reached at about $\tau = 10.6$ for F1 but $\tau = 8.9$ for $F2$. In the case of $F1$ one might even argue that an approximate steady state has resulted for $\textrm {d}\langle E_{\varDelta }\rangle /\textrm {d}t$ between $\tau =6.5$ and $\tau = 10.6$, the time range corresponding to the linear growth regime perhaps observed in figure 1(a) for F1 and also in some previous DNS (Boffetta & Musacchio Reference Boffetta and Musacchio2017; Berera & Ho Reference Berera and Ho2018). After $\tau = 10.6$, $\langle P_{\varDelta }\rangle$ oscillates around zero, corresponding to the saturation of $\langle E_{\varDelta }\rangle /\langle E_{{tot}}\rangle$ at a value $0.97\pm 0.07$ in figure 1(a). This reflects the total decorrelation between the F1 reference and perturbed fields and leads to a long-time saturation balance $\langle F_{\varDelta }\rangle \approx \langle \varepsilon _{\varDelta } \rangle$ in F1 which is to be contrasted with $\langle P_{\varDelta }\rangle \approx \langle \varepsilon _{\varDelta } \rangle$ in F2 and F3. Note that the long-time saturation is such that $\langle F_{{corr}}\rangle \approx 0$ and $\langle F_{\varDelta }\rangle \approx \langle F^{(1)}+F^{(2)}\rangle$ in all cases, including F1. Hence, the long-time saturation balance between $\langle F_{\varDelta }\rangle$ and $\langle \varepsilon _{\varDelta }\rangle$ in case F1 simply reflects $\langle \varepsilon _{{tot}}\rangle \approx \langle F^{(1)}+F^{(2)}\rangle$.

In figure 3 we concentrate on the time evolution of the production–dissipation ratio $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ in all three cases F1, F2 and F3. As highlighted in the insets of this figure's plots, there is, in all three cases, a time range when $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ is about constant, i.e. a time range when the evolutions of $\langle P_{\varDelta }\rangle$ and $\langle \varepsilon _{\varDelta }\rangle$ are similar. In all three cases this time range includes the time range of exponential growth of $\langle E_{\varDelta }\rangle$ identified in the previous subsection; in fact, in case F3 it more or less exactly coincides with it. To be specific, $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle =1.61\pm 0.03$ from $\tau = 0.6$ to $\tau = 2.5$ for F1 and F2, and $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle =1.61\pm 0.08$ from $\tau = 7.2$ to $\tau = 12.8$ for F3. These two values are very close (and the additional case F4 in Appendix B returns a similar value for $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ in F4's similarity regime), indicating that the similarity value of the production–dissipation ratio $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ might be universal and independent of Reynolds number, as the presence of a strange attractor might perhaps imply.

Figure 3.

Time evolutions of ratio $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ for different cases: (a) case F1; (b) case F2; (c) case F3. Inset: the evolution of the ratio in the time range of the exponential growth of the average uncertainty.

4.1.3. Uncertainty spectrum

The uncertainty dissipation rate is the integral over all wavenumbers $k$ of $k^{2} \hat {E}_{\varDelta }(k)$ where $\hat {E}_{\varDelta }(k)$ is the uncertainty spectrum, i.e. the energy spectrum of the velocity difference field. The similarity in the evolutions of uncertainty production and dissipation rates raises the question whether the uncertainty spectrum evolves in some self-similar manner over the time range of that similarity. We answer this question in terms of the integral length scale of the velocity-difference fields considered here which is $L_{\varDelta }=(3{\rm \pi} /4\langle E_{\varDelta }\rangle )\int k^{-1}\hat {E}_{\varDelta }(k)\,\textrm {d}k$ (see Batchelor (Reference Batchelor1953) for an introduction to this length scale for any statistically homogeneous/periodic velocity field). Here $L_{\varDelta }$ is a measure of the length over which the velocity difference field is correlated, i.e. a characteristic length scale of uncertainty containing eddies.

Soon after the initial decay of $\langle E_{\varDelta } \rangle$, the uncertainty spectra collapse with $\langle E_{\varDelta } \rangle (t)$ and $L_{\varDelta } (t)$ at wavenumbers larger than $2/L_{\varDelta }$ as shown in figure 4, i.e. $\hat {E}_{\varDelta }(k,t) = \langle E_{\varDelta }\rangle L_{\varDelta }\, f(kL_{\varDelta })$ for $kL_{\varDelta }\ge 2$, where $f$ is a dimensionless function of dimensionless wavenumber. At wavenumbers $kL_{\varDelta } < 1$ the energy spectra have an approximately power-law dependence on $k$ but do not collapse until soon after the time when the exponential growth of $\langle E_{\varDelta } \rangle (t)$ and the uncertainty's production–dissipation similarity ($\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle \approx 1.6- 1.7$) sets in. Over the time range when $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle \approx 1.6- 1.7$, the uncertainty spectrum is self-similar, i.e. evolves as $\hat {E}_{\varDelta }(k,t) = \langle E_{\varDelta } \rangle L_{\varDelta }\, f(kL_{\varDelta })$ for all wavenumbers (see figure 5). The peak of the spectrum is at $k\approx 2/L_{\varDelta }$ in all three cases F1, F2 and F3. At wavenumbers below $2/L_{\varDelta }$ the uncertainty spectra have an approximately $k^{3.3}$ power law shape, while at wavenumbers above $2/L_{\varDelta }$, they appear to have an exponential shape. Similar uncertainty spectrum shapes have been found in a previous DNS study (Berera & Ho Reference Berera and Ho2018).

Figure 4.

Early time evolution of uncertainty energy spectrum for different cases: (a) case F1; (b) case F2; (c) case F3. The spectra are normalised by $\langle E_{\varDelta }\rangle$ and $L_{\varDelta }$.

Figure 5.

Uncertainty energy spectra for cases (a) F1, (b) F2 and (c) F3 in the similarity regime. The spectra are normalised by $\langle E_{\varDelta }\rangle$ and $L_{\varDelta }$. Inset: semilogarithmic plot of the uncertainty spectra in the wavenumber range higher than $2/L_{\varDelta }$. The collapse of the normalised uncertainty spectra for cases F1, F2 and F3 in the similarity regime is shown in (d).

It is remarkable that the uncertainty spectrum is self-similar in case F3 in the exact same way that it is self-similar in cases F1 and F2 over the time range where $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ is approximately constant. In fact, the self-similar uncertainty spectrum even seems to be the same for F3, F1 and F2 as seen by the collapse in figure 5(d), suggesting a universal shape for the self-similar uncertainty spectrum in HIT. This is remarkable not only because $F3$ has a very different Reynolds number and forcing than F1 and F2, but more importantly because the F3 reference field is not statistically stationary in that time range whereas the F1 and F2 reference fields are. In the F3 case, the uncertainty spectrum reaches its self-similar state at $\tau \approx 5.8$ and the reference field becomes statistically stationary at $\tau =9.4$.

After the time range where $\langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ is approximately constant, the uncertainty spectrum is no longer self-similar (see figure 6). This happens at $\tau =3.5$ for cases F1 and F2 and at $\tau =12.6$ for case F3 when $\hat {E}_{\varDelta }(k_{max})/\hat {E}_{{tot}}(k_{max})>0.95$. These are the times when the reference and perturbed fields decorrelate at the largest resolvable wavenumber (see figure 6). The process of decorrelation between the two fields proceeds by decorrelating them at progressively smaller wavenumbers, causing the uncertainty spectrum to collapse with the reference field's energy spectrum over a progressively wider range of the higher wavenumbers (see figure 6). This progressive decorrelation process from high to small wavenumbers and the uncertainty spectrum's progressive convergence towards the reference field's spectrum prevents the uncertainty spectrum from being self-similar. For F1, the uncertainty spectrum finally collapses with the reference field's energy spectrum at all wavenumbers, indicating that the two fields eventually decorrelate completely at all wavenumbers (see figure 6a). The same happens for F2 and F3 except over the wavenumbers acted by the forcing where a gap always remains between the uncertainty and the reference field spectra, indicating that the two fields retain a degree of correlation at these large scales (see figures 6b and 6c).

Figure 6.

Uncertainty spectra (dashed lines) for different cases after the times when the reference and perturbed fields decorrelate at the largest resolvable wavenumber: (a) case F1; (b) case F2; (c) case F3. The uncertainty spectra are normalised by $\langle E_{{tot}}\rangle$. The dots on the uncertainty spectra represent $k=2/L_{\varDelta }$. The solid line represents the energy spectrum of the reference field when it is statistically steady. The energy spectrum is normalised by $\langle E^{(1)}\rangle$. The uncertainty spectrum shifts gradually along with the arrows representing the direction of time advance.

4.1.4. Characteristic length of uncertainty

The growth of $L_{\varDelta }$ is evident in figure 6. We therefore plot its time evolution in figure 7 and compare it with the integral and Taylor length scales ($L$ and $l_{\lambda }$, respectively) of the reference field for each case F1, F2 and F3. At the very early times when uncertainty dissipation dominates, the velocity-difference field decays and its integral length scale normalised by $L$ is, correspondingly, increasing. In the stationary turbulence F1 and F2 cases, this time regime is followed by the chaotic regime where $\langle E_{\varDelta }\rangle$ grows exponentially and where $L_{\varDelta }/l_{\lambda }^{(1)}$ remains relatively constant at $0.38\pm 0.01$. A constant $L_{\varDelta }/l_{\lambda }^{(1)}$ (though a different constant, $L_{\varDelta }/l_{\lambda }^{(1)} = 0.64\pm 0.03$) is also observed in the non-stationary F3 case during the chaotic regime even though $l_{\lambda }^{(1)}$ grows in time for some of that regime and even though this time regime does not follow immediately after the dissipation-dominated regime. In fact, $L_{\varDelta }$ decreases between the dissipation-dominated and the chaotic regime in the F3 case. It is noteworthy that $L_{\varDelta }$ reaches $l_{\lambda }^{(1)}$ at $\tau =4.9$ for cases F1 and F2 and at $\tau =14.5$ for case F3, a little before the average uncertainty dissipation rate reaches its stationary value in figure 2, i.e. $\tau \approx 5.6$ for F1 and F2 and $\tau \approx 16.1$ for F3. The link between $L_{\varDelta }$ and the Taylor length of the reference field is potentially interesting as the Taylor length is the mean distance between stagnation points in a HIT (Goto & Vassilicos Reference Goto and Vassilicos2009) and therefore tends to represent the average size of turbulent eddies which is highly weighted towards the more numerous smallest ones.

Figure 7.

Time evolution of $L_{\varDelta }/l_{\lambda }^{(1)}$ for different cases: (a) case F1; (b) case F2; (c) case F3. Inset: time evolution of $L_{\varDelta }/L^{(1)}$. The time evolutions of $L_{\varDelta }/l_{\lambda }^{(1)}$ in cases F1 and F2 are plotted together in (d).

Following the exponential growth of $\langle E_{\varDelta }\rangle$, three consecutive time regimes follow for F1 and F2. First, one observes an approximately power-law growth of $L_{\varDelta }$, identical for both F1 and F2 as shown in figure 7(d), more or less coinciding with the exponential of exponential growth of $\langle E_{\varDelta }\rangle$ until $\tau =6.5$. In this time regime, $L_{\varDelta }\sim t^{3/2}$ is a good fit. This fit is reminiscent of the power-law growth of the predictability scale $k_{E}^{-1}\sim t^{3/2}$ obtained in previous numerical simulations (Leith & Kraichnan Reference Leith and Kraichnan1972; Boffetta & Musacchio Reference Boffetta and Musacchio2017) and theoretical arguments (Lorenz Reference Lorenz1969; Frisch Reference Frisch1995; Boffetta & Musacchio Reference Boffetta and Musacchio2017), as a companion conclusion to the linear growth of $\langle E_{\varDelta }\rangle$. However, $L_{\varDelta }$ and $k_{E}^{-1}$ are not equivalent: the predictability scale is defined as the inverse of the minimum wavenumber $k_E$ such that $\widehat {E_{\varDelta }}(k_{E})/\hat {E}_{{tot}}(k_{E})=1$, and $k_{E}^{-1}\sim t^{3/2}$ is obtained on the assumption that the decorrelation process happens in the inertial range. Here $L_{\varDelta }\sim t^{3/2}$ is observed without concurrent linear growth of $\langle E_{\varDelta }\rangle$ but a concurrent exponential of exponential $\langle E_{\varDelta }\rangle$ growth instead.

The second consecutive regime which follows for F1 and F2 is an apparently linear growth of $L_{\varDelta }$ that lasts until the time when $L_{\varDelta }$ saturates to a constant. The third and final regime is this approximately constant $L_{\varDelta }$ regime where $L_{\varDelta }\approx L^{(1)}$ for F1 (see figure 7a) in agreement with the eventual complete decorrelation of the reference and perturbed fields and where $L_{\varDelta }\approx (0.70\pm 0.06)L^{(1)}$ (smaller than $L^{(1)}$) for F2 (see figure 7b) in agreement with the eventual partial correlation between these two fields in this case.

In the F3 case, the chaotic regime where $\langle E_{\varDelta }\rangle$ grows exponentially and $L_{\varDelta }/l_{\lambda }^{(1)} = 0.64\pm 0.03$ is followed by an intermediate regime where $L_{\varDelta }$ grows to eventually reach the final constant regime where $L_{\varDelta } = (0.81\pm 0.03) L^{(1)}$ (smaller than $L^{(1)}$) characterising the final saturation (see figure 7c). As for F2, the fact that $L_{\varDelta }$ is significantly lower than $L$ in the eventual saturation regime reflects the partial long-time correlation imposed by the identical forcing in the reference and perturbed fields.

4.2.1. From similarity to exponential growth

When $\langle F_{\varDelta } \rangle$ is identically zero (as in F2 and F3) or negligibly small compared with $\langle P_{\varDelta }\rangle$ and $\langle \varepsilon _{\varDelta }\rangle$ (as in F1 for $\tau$ smaller than about $6.5$) the evolution equation for $\langle E_{\varDelta } \rangle$ becomes

(4.1)\begin{equation} \frac{{\rm d}}{{\rm d} t}\left\langle E_{\varDelta}\right\rangle=\left\langle P_{\varDelta}\right\rangle-\left\langle \varepsilon_{\varDelta}\right\rangle. \end{equation}

To estimate $\langle P_{\varDelta }\rangle$ in terms of $\langle E_{\varDelta }\rangle$ and obtain an equation of the same form as (1.1), we apply a Reynolds decomposition to (2.12) and write

(4.2)\begin{equation} \left\langle P_{\varDelta}\right\rangle={-}\sum_{i=1}^{3}\left\langle\varLambda^{(1)}_{i}\Delta w_{i}^{2}\right\rangle=\underbrace{-\sum_{i=1}^{3}\left\langle\varLambda^{(1)}_{i}\right\rangle\left\langle\Delta w_{i}^{2}\right\rangle}_{\left\langle P_{\varDelta}\right\rangle_{{Ave}}}\underbrace{-\sum_{i=1}^{3}\left\langle\varLambda_{i}^{(1)\prime}\Delta {w_{i}^{2}}^{\prime}\right\rangle}_{\left\langle P_{\varDelta}\right\rangle_{{Fluc}}}, \end{equation}

where $\varLambda _{i}^{(1)\prime } \equiv \varLambda ^{(1)}_{i}-\langle \varLambda ^{(1)}_{i}\rangle$ and $\Delta {w_{i}^{2}}^{\prime } \equiv \Delta w_{i}^{2}-\langle \Delta w_{i}^{2}\rangle$. In all cases F1, F2 and F3, and at times after the similarity regime, the first term on the right-hand side of (4.2) dominates over the second term and contributes the most to $\langle P_{\varDelta }\rangle$ (see figure 8). During the part of the similarity regime when $\langle E_{\varDelta }\rangle$ grows exponentially, $\beta \equiv \langle P_{\varDelta }\rangle _{{Ave}}/\langle P_{\varDelta }\rangle$ is constant in time and so is $1-\beta =\langle P_{\varDelta }\rangle _{{Fluc}}/\langle P_{\varDelta }\rangle$ (see the insets of figure 8): $\beta$ is a constant equal to $0.53\pm 0.02$ for F1 and F2 and equal to a slightly different value $0.66\pm 0.02$ for F3 where the Taylor length-based Reynolds number is significantly lower than for F1 and F2. One may indeed expect the fluctuation contribution $\langle P_{\varDelta }\rangle _{{Fluc}}$ to increase in magnitude with increasing Reynolds number relative to the mean contribution $\langle P_{\varDelta }\rangle _{{Ave}}$ in (4.2), and $\beta$ to therefore be a decreasing function of Reynolds number.

Figure 8.

Time evolution of $\langle P_{\varDelta }\rangle$ and its decomposition into average and fluctuation parts for different cases: (a) case F1; (b) case F2; (c) case F3. Inset: the early time evolution of the ratio of average term/total production and fluctuation term/total production.

Defining $\gamma ^{(1)}_{i} \equiv \varLambda ^{(1)}_{i}/\sqrt {\langle |S^{(1)}_{ij}|^{2}\rangle }$ (where $|S_{ij}| \equiv \sqrt {S_{ij}S_{ij}}$) and $\theta _{i} \equiv \Delta w_{i}^{2}/2\langle E_{\varDelta }\rangle$, and using $\beta \equiv \langle P_{\varDelta }\rangle _{{Ave}}/\langle P_{\varDelta }\rangle$, we have

(4.3)\begin{equation} \left\langle P_{\varDelta}\right\rangle={-}2\frac{\displaystyle\sum_{i=1}^{3}\left\langle\gamma^{(1)}_{i}\right\rangle\left\langle\theta_{i}\right\rangle}{\beta}\sqrt{\left\langle\left|S^{(1)}_{ij}\right|^{2}\right\rangle}\left\langle E_{\varDelta}\right\rangle. \end{equation}

We now examine the behaviours of $\langle \gamma ^{(1)}_{i}\rangle$ and $\langle \theta _{i}\rangle$.

We start with $\langle \gamma ^{(1)}_{i}\rangle$ which, unlike $\beta$ and $\langle \theta _{i}\rangle$, are properties of the reference field and not of the velocity-difference field: $\langle \gamma ^{(1)}_{i}\rangle$ are the average strain rates along the principal axes of the reference field's strain rate tensor and they are plotted vs time in figure 9. Note the constraints $\sum _{i=1}^{3}\langle \gamma ^{(1)}_{i}\rangle =0$ and $\sum _{i=1}^{3}\langle \gamma _{i}^{(1)2}\rangle =1$. In cases F1 and F2, where the reference flow is statistically stationary, $\langle \gamma ^{(1)}_{i}\rangle$ are constant in time and $\langle \gamma ^{(1)}_{1}\rangle :\langle \gamma ^{(1)}_{2}\rangle :\langle \gamma ^{(1)}_{3}\rangle \approx -0.65:0.12:0.53$ in agreement with Betchov (Reference Betchov1956)'s theoretical demonstration that there must be one principal axis direction which is compressive on average and two which are on average stretching. In case F3, the reference flow is not statistically stationary until about $\tau = 9.4$ but $\langle \gamma ^{(1)}_{i}\rangle$ acquire a stable value before that and are already constant during the similarity period $\tau \approx 5.8$ to $\tau \approx 12.60$ (see figure 9c). In case F3, we observe $\langle \gamma ^{(1)}_{1}\rangle :\langle \gamma ^{(1)}_{2}\rangle :\langle \gamma ^{(1)}_{3}\rangle \approx -0.68:0.13:0.55$, which is very close to F1 and F2, also in agreement with the prediction of Betchov (Reference Betchov1956).

Figure 9.

Time evolution of $\langle \gamma _{i}\rangle$ in the reference flows for different cases: (a) case F1; (b) case F2; (c) case F3.

The average uncertainty energy $\langle E_{\varDelta }\rangle$ consists of three average uncertainty energies $\langle \Delta w_{i}^{2}/2\rangle$ in the principal axes of the reference field's strain rate tensor: $\langle E_{\varDelta }\rangle = \sum _{i=1}^{3} \langle \Delta w_{i}^{2}/2\rangle$. The ratios $\langle \theta _{i}\rangle$ represent the proportion of average uncertainty energy in each principal direction and they of course sum up to 1. Their time evolution is shown in figure 10. Most of the uncertainty energy is concentrated in the compressive direction until $\tau \approx 8- 9$ in cases F1 and F2 and for all time in case F3, in agreement with our observation at the end of § 2.2 that the production of uncertainty occurs by compressive motions. At saturation times there is a tendency for equipartition of average uncertainty energy in the three principal directions, in particular for F1 where the reference and principal fields completely decorrelate in the long term. The tendency remains for F2 and F3 but the average uncertainty energy in the most stretching direction remains significantly below the average uncertainty energy in the other two directions thereby ensuring that $\langle P_{\varDelta }\rangle$ remains positive and the reference and perturbation fields remain partially correlated during eventual saturation.

Figure 10.

Time evolution of $\langle \theta _{i}\rangle$ for different cases: (a) case F1; (b) case F2; (c) case F3. Inset: the time evolution of $\langle \theta _{i}\rangle$ during the similarity regime.

In all three cases F1, F2 and F3, $\langle \theta _{i}\rangle$ are approximately constant during the similarity regime where $\beta$ is also constant in time. During the similarity regime, the $\langle \theta _{i}\rangle$ values are $\langle \theta _{1}\rangle :\langle \theta _{2}\rangle :\langle \theta _{3}\rangle \approx 0.58:0.19:0.23$ for cases F1 and F2 and $\langle \theta _{1}\rangle :\langle \theta _{2}\rangle :\langle \theta _{3}\rangle \approx 0.59:0.18:0.23$ for case F3. The values of $\langle \theta _{i}\rangle$ appear to be universal during the similarity regime whereas $\beta$ seems to be dependent on Reynolds number.

Finally, we discuss the relation between $\langle \varepsilon _{\varDelta }\rangle$ and $\langle P_{\varDelta }\rangle$. The self-similar uncertainty spectrum $\hat {E}_{\varDelta }(k,t) = \langle E_{\varDelta } \rangle L_{\varDelta }\, f(kL_{\varDelta })$ implies that the uncertainty dissipation is

(4.4)\begin{equation} \left\langle\varepsilon_{\varDelta}\right\rangle=2\nu\int k^{2}\hat{E}_{\varDelta}(k)\,\mathrm{d}k=2\nu\left\langle E_{\varDelta} \right\rangle L_{\varDelta}\int k^{2}f(kL_{\varDelta})\,\mathrm{d}k=2\nu\frac{\left\langle E_{\varDelta} \right\rangle}{L_{\varDelta}^{2}}\int x^{2}f(x)\,\mathrm{d}\kern0.7pt x, \end{equation}

where $\int x^{2}f(x)\,{\textrm {d} x}$ is a time constant. Defining $\alpha \equiv \langle \varepsilon _{\varDelta }\rangle /\langle P_{\varDelta }\rangle$, we obtain, from (4.3) and (4.4)

(4.5)\begin{equation} \alpha={-}\left[ \frac{\beta\int_{0}^{\infty}x^{2}f(x)\,\mathrm{d} x}{\displaystyle\sum_{i=1}^{3}\left\langle\gamma^{(1)}_{i}\right\rangle\left\langle\theta_{i}\right\rangle}\right]\frac{\nu}{L_{\varDelta}^{2}\sqrt{\left\langle\left|S^{(1)}_{ij}\right|^{2}\right\rangle}}. \end{equation}

As shown previously in this subsection, the term in square brackets in (4.5) is constant in time. Figure 7(a) suggests that $L_{\varDelta }$ and $l_{\lambda }^{(1)}$ have the same dependence on time but not the same dependence on viscosity. Therefore, the time dependence of $(L_{\varDelta }^{2}\sqrt {\langle |S^{(1)}_{ij}|^{2}\rangle })$ is the same as the time dependence of $((\eta ^{(1)})^{2}(\tau _{\eta }^{(1)})^{-1}{Re}^{(1)}_{\lambda })\sim {Re}^{(1)}_{\lambda }$. For cases F1 and F2, the reference field, and therefore ${Re}^{(1)}_{\lambda }$, are statistically steady and it therefore follows from (4.5) that $\alpha$ is constant in time during the similarity regime. For the same reason, $\alpha$ is constant in time after $\tau =9.4$ in the similarity regime of case F3 because this is when the reference flow reaches the statistically steady state. During $\tau \in [7.5,9.4]$ for F3, $1/{Re}^{(1)}_{\lambda }$ decreases monotonically from $0.0187$ to $0.0156$ as shown in figure 11. This $20\,\%$ decrease is small compared with the variations of $1/{Re}^{(1)}_{\lambda }$ at normalised times $\tau$ smaller than $7.5$ and results in a small decrease of $\alpha$ in the corresponding time period (i.e. a slow increase of $1/\alpha$, as shown in figure 3). Therefore, $\alpha$ can be considered to be approximately constant in the similarity period $\tau \in [7.5,12.5]$ of F3.

Figure 11.

Time evolution of $1/{Re}_{\lambda }$ in case F3.

We have seen at the end of § 4.1.2 and figure 3 that $\alpha = \langle P_{\varDelta }\rangle /\langle \varepsilon _{\varDelta }\rangle$ seems to be independent of viscosity but we also noted two paragraphs above that $\beta$ is not. The dependencies on viscosity of $\beta \nu$ and $(L_{\varDelta }^{2}\sqrt {\langle |S^{(1)}_{ij}|^{2}\rangle })$ in (4.5) must therefore be the same and cancel each other.

Substituting (4.3) and (4.5) into (4.1), we obtain

(4.6)\begin{equation} \frac{\mathrm{d}}{\mathrm{d} t}\left\langle E_{\varDelta}\right\rangle=\varGamma\sqrt{\left\langle\left|S^{(1)}_{ij}\right|^{2}\right\rangle}\left\langle E_{\varDelta}\right\rangle, \end{equation}

where

(4.7)\begin{equation} \varGamma={-}2\frac{1-\alpha}{\beta}\sum_{i=1}^{3}\left\langle\gamma^{(1)}_{i}\right\rangle\left\langle\theta_{i}\right\rangle . \end{equation}

This is a general rewriting of (4.1) with particularly interesting consequences for the similarity regime when $\alpha$, $\beta$, $\langle \gamma ^{(1)}_{i}\rangle$ and $\langle \theta _{i}\rangle$ are constant in time. The dimensionless coefficient $\varGamma$ defined by (4.7) is therefore constant in time during the similarity regime but may depend on Reynolds number (i.e. viscosity) via the dependence of $\beta$ on Reynolds number.

Looking at (4.6), an exponential growth of $\langle E_{\varDelta } \rangle$ with a well-defined Lyapunov exponent $\lambda$ can be derived during the similarity regime because $\varGamma$ is constant in time:

(4.8)\begin{equation} 2\lambda=\varGamma\sqrt{\left\langle\left|S^{(1)}_{ij}\right|^{2}\right\rangle}=\frac{1}{\sqrt{2}}\varGamma\tau_{\eta}^{{-}1}. \end{equation}

The exponential growth of average uncertainty energy is, therefore, a consequence of similarity. How similarity (time-independent $\alpha$, $\beta$, $\langle \theta _{i} \rangle$ and self-similar evolution of the uncertainty spectrum in terms of $\langle E_{\varDelta } \rangle$ and $L_{\varDelta }$) may be a consequence of the presence of a strange attractor is, however, beyond this paper's scope but the question is now posed for future investigations.

The dimensionless coefficient $\varGamma$ obtained from (4.7) and the Lyapunov exponent directly obtained from (1.1) are plotted in figure 12: for all cases F1, F2 and F3, $\varGamma$ is about constant in the time range where exponential growth is present. The actual value of $\varGamma$ in this time range is the same for F1 and F2 but it is different for F3 which has a lower Reynolds number. The scaling $\lambda \tau _{\eta }\sim \varGamma ({Re})$ suggests that the Lyapunov exponent may not scale with the Kolmogorov time $\tau _{\eta }$ (as claimed by Ruelle Reference Ruelle1979) if $\varGamma$ depends on Reynolds number, which it may do on account of a Reynolds number dependence of $\beta$. The coefficient $\varGamma$, as well as the Lyapunov exponent, are also plotted in figure 13 to compare with previous data by Mohan et al. (Reference Mohan, Fitzsimmons and Moser2017). The ratio of $\varGamma$ values in the F1 and F3 cases is $\varGamma _{{F1}}/\varGamma _{{F3}}=1.29$ during the exponential growth time range, whereas $\beta _{{F3}}/\beta _{{F1}}=1.25$ in the same regime. The data of Mohan et al. (Reference Mohan, Fitzsimmons and Moser2017) lead to $\varGamma _{{F1}}/\varGamma _{{F3}} \approx 1.30$ purely on the basis of the Reynolds numbers of F1 and F3 (see figure 13). This confirms the hypothesis that the different values of $\varGamma$ in F1 and F2 on the one hand and F3 on the other are caused by the difference in Reynolds number and nothing else.

Figure 12.

Time evolution of $\varGamma$ and $2\sqrt {2}\lambda \tau _{\eta }$ for different cases: (a) case F1; (b) case F2; (c) case F3. Inset: the time evolution of $\varGamma$ during the similarity regime in semilogarithmic plot. The exponential function fit is indicated by a dash-dotted line for F1 and F2.

Figure 13.

Schematic log–log plot of $\lambda \tau _{\eta }$ with the Taylor length-based Reynolds number ${Re}_{\lambda }$ according to the numerical results (in blue) and models calibrated with Bayesian inference (in red) of Mohan et al. (Reference Mohan, Fitzsimmons and Moser2017). The Lyapunov exponents and the coefficient $\varGamma$ obtained in the present work (in green crosses and red points) are also plotted.

The regime of approximate constancy of $\varGamma$ is followed by a time range $\tau \in [2.9,6.5]$ where $\varGamma$ appears to decay exponentially in the F1 and F2 cases (it is not clear whether such a range does or does not exist in the F3 case), see figure 12. Specifically, the exponential curve fit gives $\varGamma =0.73\exp (-1.26(\tau -2.9))$. Using $\langle E_{{tot}}\rangle$ and $\langle T^{(1)}\rangle _{t}$ to non-dimensionalise equation (4.6), we write

(4.9)\begin{equation} \frac{\mathrm{d} }{\mathrm{d}\tau}\frac{\left\langle E_{\varDelta}\right\rangle}{\left\langle E_{{tot}}\right\rangle}=\varGamma\left\langle T^{(1)}\right\rangle_{t}\sqrt{\left\langle\left|S^{(1)}_{ij}\right|^{2}\right\rangle}\frac{\left\langle E_{\varDelta}\right\rangle}{\left\langle E_{{tot}}\right\rangle} . \end{equation}

For statistically stationary cases F1 and F2 we find $\langle T^{(1)}\rangle _{t}\sqrt {\langle |S^{(1)}_{ij}|^{2}\rangle }=12.77\pm 0.56$ in the time range $\tau \in [2.9,6.5]$. Therefore, (4.9) and our fitting of $\varGamma$ imply $\langle E_{\varDelta }\rangle /\langle E_{{tot}}\rangle \sim \exp (-9.32\exp (-1.26(\tau -2.9)))$, which is approximately consistent with the direct curve fitting in the inset of figure 1(d). Eventually $\varGamma$ tends to $0$ and the average uncertainty energy stops growing in all cases F1, F2 and F3.

4.2.2. Scaling of the Lyapunov exponent during similarity

Our analysis in § 4.2.1 and the data of (Mohan et al. Reference Mohan, Fitzsimmons and Moser2017) presented in figure 13 question the view that $\lambda$ scales with $\tau _{\eta }$ (Ruelle Reference Ruelle1979). If $\lambda$ does not scale with $\tau _{\eta }$ which is the smallest Lagrangian time scale of the turbulence, it may scale with $\tau _{E}=\eta /U$, the shortest Eulerian time scale of the turbulence (Tennekes Reference Tennekes1975), in which case $\lambda \tau _{\eta }\sim \tau _{\eta }/\tau _{E}\sim {Re}_{\lambda }^{1/2}$. The data of Mohan et al. (Reference Mohan, Fitzsimmons and Moser2017) in figure 13 suggests that $\lambda$ grows faster than $\tau _{\eta }^{-1}$ but slower than $\tau _{E}^{-1}$ as Reynolds number increases, perhaps $\lambda \sim \tau _{\eta }^{-(1-c)/2}\tau _{E}^{-(1+c)/2}$, i.e. $\lambda \tau _{\eta }\sim {Re}_{\lambda }^{(1+c)/4}$, where $c\in (-1,1]$. In fact, the results of Mohan et al. (Reference Mohan, Fitzsimmons and Moser2017) suggest that $\lambda \tau _{\eta } \sim {Re}_{\lambda }^{(1+c)/4}$ where the most likely values of $c$ are between $0$ and $1/3$. The large scale random sweeping of the smallest eddies represented in the Eulerian time scale $\tau _E$ appears to influence the growth of uncertainty even though the uncertainty exists only at the smallest scales during the chaotic exponential growth. Interestingly, this large-scale random-sweeping effect is reflected in the decreasing dependence of $\beta$ on Reynolds number (see (4.7) and (4.8)) which implies that $\langle P_{\varDelta }\rangle$ should be increasingly dominated by $\langle P_{\varDelta }\rangle _{{Fluc}}$ rather than $\langle P_{\varDelta }\rangle _{{Ave}}$ in (4.2) as Reynolds number increases. There seems to be a relation between large-scale random sweeping and uncertainty production, and in particular between random sweeping and the way that compression and stretching affect average uncertainty production either through average compression/stretching rates or through the correlations of their fluctuations with uncertainty energy fluctuations in specific stretching/compressive directions. A Lagrangian or some combined Eulerian–Lagrangian description of uncertainty (see, e.g., Boffetta et al. Reference Boffetta, Celani, Crisanti and Vulpiani1997) as advocated by Leith & Kraichnan (Reference Leith and Kraichnan1972) in their introduction might have advantages over the present purely Eulerian approach as it may naturally account for the large-scale sweeping's effect on uncertainty and thereby return a reduced average uncertainty production. The large-scale sweeping's effect on uncertainty might also have some relation with the error in positions of local flow structures that Boffetta et al. (Reference Boffetta, Celani, Crisanti and Vulpiani1997) identified.