2 Instantaneously optimal growth of palinstrophy

Given the dependence of analytic estimate (1.8) on palinstrophy ${\mathcal{P}}$ and Reynolds number $Re={\mathcal{K}}^{1/2}/\unicode[STIX]{x1D708}$ , the sharpness of the estimate is addressed by numerically solving the constrained optimization problem for the objective functional ${\mathcal{R}}$ defined in (1.4):

(2.1) $$\begin{eqnarray}\max _{\boldsymbol{u}\in {\mathcal{S}}_{{\mathcal{K}}_{0},{\mathcal{P}}_{0}}}\,{\mathcal{R}}(\boldsymbol{u}),\end{eqnarray}$$

where

(2.2) $$\begin{eqnarray}{\mathcal{S}}_{{\mathcal{K}}_{0},{\mathcal{P}}_{0}}=\{\boldsymbol{u}\in H^{3}(\unicode[STIX]{x1D6FA})\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,\;{\mathcal{K}}(\boldsymbol{u})={\mathcal{K}}_{0},\;{\mathcal{P}}(\boldsymbol{u})={\mathcal{P}}_{0}\}.\end{eqnarray}$$

In order to investigate the asymptotic behaviour of ${\mathcal{R}}$ as ${\mathcal{P}}_{0}\rightarrow \infty$ , (2.1) is solved numerically for a wide range of values of the energy ${\mathcal{K}}_{0}\in [1,100]$ and some choices of ${\mathcal{P}}_{0}\gg {\mathcal{K}}_{0}/C_{P}^{2}$ , where $C_{P}=1/(2\unicode[STIX]{x03C0})^{2}$ is the Poincaré constant for the unit 2D torus. We compute with the numerical value $\unicode[STIX]{x1D708}=10^{-3}$ for the kinematic viscosity, kept constant in all computations, allowing us to probe the dependence of the optimal rate of growth of palinstrophy for values of the Reynolds number in the range $Re_{0}\in [10^{3},10^{4}]$ . For a given value of $Re_{0}$ , the value of ${\mathcal{P}}_{0}$ defining the constraint manifold ${\mathcal{S}}_{{\mathcal{K}}_{0},{\mathcal{P}}_{0}}$ is chosen so that the optimal vorticity field $\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ of the optimal vector field $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}=\text{argmax}\,{\mathcal{R}}(\boldsymbol{u})$ is sufficiently localized in the computational domain, allowing for the effect of boundaries to be neglected. A family of optimal fields parametrized by their palinstrophy is then constructed by rescaling the optimal field $\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ using the self-similar approach described in appendix B. Details of the numerical methods for the variational problem (2.1) can be found in the work of Ayala & Protas (Reference Ayala and Protas2014a ).

Figures 1(a) and 1(b) show the optimal vorticity fields $\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ corresponding to values of Reynolds number $Re_{0}=10^{3}$ and $Re_{0}=10^{4}$ , respectively, for a fixed value of palinstrophy ${\mathcal{P}}_{0}=10^{8}$ . In each case, the optimal vorticity field consists of a vortex quadrupole of finite area and a localized region of strong vorticity responsible for the extreme growth of palinstrophy. The size of the optimal vortical structure, relative to the size of the domain $\unicode[STIX]{x1D6FA}$ , has a positive correlation with the Reynolds number. A thorough discussion of the properties of the optimal vorticity fields, and their corresponding time evolution under 2D Navier–Stokes dynamics, can be found in Ayala & Protas (Reference Ayala and Protas2014a ,Reference Ayala and Protas b ).

Figure 1. Optimal vorticity field $\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ corresponding to ${\mathcal{P}}_{0}=10^{8}$ and values of Reynolds number (a)  $Re_{0}=10^{3}$ and (b)  $Re_{0}=10^{4}$ . Streamlines corresponding to selected level sets of the streamfunction $\unicode[STIX]{x1D713}=-\unicode[STIX]{x0394}^{-1}\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ are shown for reference.

To verify the power-law nature of the optimal instantaneous rate of production of palinstrophy and assess the sharpness of estimate (1.8) with respect to the predicted behaviour $\text{d}{\mathcal{P}}/\text{d}t\sim {\mathcal{P}}^{3/2}$ , figure 2 shows the dependence on ${\mathcal{P}}_{0}$ of the compensated optimal rate of growth of palinstrophy,

(2.3) $$\begin{eqnarray}\widetilde{{\mathcal{R}}}_{Re_{0},{\mathcal{P}}_{0}}={\mathcal{P}}_{0}^{-3/2}\,{\mathcal{R}}(\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}),\end{eqnarray}$$

for different values of Reynolds number in the interval $Re_{0}\in [10^{3},10^{4}]$ . The data shown here are an extension of the results reported by Ayala & Protas (Reference Ayala and Protas2014a ). To focus our discussion, we only include the portion of the data where a clear power-law behaviour for ${\mathcal{R}}(\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}})$ is observed, corresponding to values of palinstrophy much larger than the Poincaré limit ${\mathcal{P}}_{0}\rightarrow (\unicode[STIX]{x1D708}Re_{0}/C_{P})^{2}$ . As expected from the fact that estimate (1.8) is sharp with respect to the exponent $\unicode[STIX]{x1D6FC}=3/2$ , figure 2 shows that the compensated optimal rate of growth of palinstrophy $\widetilde{{\mathcal{R}}}_{Re_{0},{\mathcal{P}}_{0}}$ , which corresponds to the prefactor $\unicode[STIX]{x1D6FE}(Re)$ , is indeed independent of palinstrophy in the limit ${\mathcal{P}}_{0}\rightarrow \infty$ .

On the other hand, to assess the sharpness of estimate (1.8) with respect to the prefactor

(2.4) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}(Re)=a+b\sqrt{\ln Re+c},\end{eqnarray}$$

figures 3(a) and 3(b) show the dependence of the optimal rate of growth of palinstrophy ${\mathcal{R}}(\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}})$ on $\ln Re_{0}$ and of the compensated optimal rate of growth of palinstrophy $\widetilde{{\mathcal{R}}}_{Re_{0},{\mathcal{P}}_{0}}$ on $\ln Re_{0}$ , respectively, for three different values of palinstrophy, ${\mathcal{P}}_{0}=4.6\times 10^{7}$ , ${\mathcal{P}}_{0}=6.8\times 10^{7}$ and ${\mathcal{P}}_{0}=10^{8}$ . It can be observed in figure 3(b) that all data points collapse onto a single curve of the form

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{Re_{0}}=\tilde{a}+\tilde{b}\sqrt{\ln Re_{0}+\tilde{c}},\end{eqnarray}$$

with fitted parameters

(2.6a-c ) $$\begin{eqnarray}\tilde{a}=-0.093,\quad \tilde{b}=0.128,\quad \tilde{c}=-4.38,\end{eqnarray}$$

shown in the figure as a red dashed curve. The values of $\tilde{a}$ , $\tilde{b}$ and $\tilde{c}$ are obtained by averaging over ${\mathcal{P}}_{0}$ the values of $a_{{\mathcal{P}}_{0}}$ , $b_{{\mathcal{P}}_{0}}$ and $c_{{\mathcal{P}}_{0}}$ corresponding to the parameters that provide the least-squares fit of the data to a model of the same form as estimate (1.8). Although the values of the fitted parameters $\tilde{a}$ , $\tilde{b}$ and $\tilde{c}$ differ from the corresponding values in estimate (1.8), the fundamental dependence of $\unicode[STIX]{x1D6FE}(Re)$ on $Re$ is correctly captured by the behaviour of $\unicode[STIX]{x1D6FE}(Re_{0})=\widetilde{{\mathcal{R}}}_{Re_{0},{\mathcal{P}}_{0}}$ , providing positive evidence for the sharpness of estimate (1.8) with respect to the prefactor $\unicode[STIX]{x1D6FE}(Re)$ . To summarize, the information presented in figures 2 and 3(a,b) indicates that the estimate

(2.7) $$\begin{eqnarray}\frac{\text{d}{\mathcal{P}}}{\text{d}t}\leqslant \left(a+b\sqrt{\ln Re+c}\right){\mathcal{P}}^{3/2}=\unicode[STIX]{x1D6FE}(Re)\,{\mathcal{P}}^{3/2}\end{eqnarray}$$

is indeed sharp with respect to both the exponent $\unicode[STIX]{x1D6FC}=3/2$ and the functional form of the prefactor $\unicode[STIX]{x1D6FE}(Re)=a+b\sqrt{\ln Re+c}$ .

Figure 2. Compensated optimal instantaneous rate of growth of palinstrophy $\widetilde{{\mathcal{R}}}_{Re_{0},{\mathcal{P}}_{0}}={\mathcal{P}}_{0}^{-3/2}{\mathcal{R}}(\widetilde{\boldsymbol{u}}_{{\mathcal{K}}_{0},{\mathcal{P}}_{0}})$ as a function of ${\mathcal{P}}_{0}$ , for values of Reynolds number $Re_{0}\in [10^{3},10^{4}]$ . The arrow indicates the direction of increasing $Re_{0}$ .

Figure 3. (a) Optimal rate of growth of palinstrophy, ${\mathcal{R}}(\widetilde{\boldsymbol{u}}_{{\mathcal{K}}_{0},{\mathcal{P}}_{0}})$ , as a function of $\ln Re_{0}$ , for values of palinstrophy ${\mathcal{P}}_{0}=4.6\times 10^{7}$ (blue stars), ${\mathcal{P}}_{0}=6.8\times 10^{7}$ (red diamonds) and ${\mathcal{P}}_{0}=10^{8}$ (black circles). (b) Compensated optimal rate of growth of palinstrophy, ${\mathcal{P}}_{0}^{-3/2}{\mathcal{R}}(\widetilde{\boldsymbol{u}}_{{\mathcal{K}}_{0},{\mathcal{P}}_{0}})$ , as a function of $\ln Re_{0}$ , for the same values of palinstrophy as in (a). All curves collapse onto a single ‘universal’ curve of the form $\tilde{a}+\tilde{b}\sqrt{\ln Re_{0}+\tilde{c}}$ , with $\tilde{a}=-0.093$ , $\tilde{b}=0.128$ and $\tilde{c}=-4.38$ , shown as a red dashed curve.

Figure 4. Rescaled compensated spectral density $(|\boldsymbol{k}|/\unicode[STIX]{x1D706}_{0})^{2}S(|\boldsymbol{k}|/\unicode[STIX]{x1D706}_{0})$ , for $\unicode[STIX]{x1D706}_{0}$ defined in (2.10), corresponding to $Re=10^{3}$ and palinstrophy values ${\mathcal{P}}_{0}=1.71\times 10^{6}$ , ${\mathcal{P}}_{0}=1.71\times 10^{7}$ and ${\mathcal{P}}_{0}=1.71\times 10^{8}$ . All curves collapse onto a ‘universal’ spectral density. The parameters $P_{1}\approx 0.86$ and $P_{2}\approx 1.88$ defining the Novikov phase diagram are shown as vertical lines.

To complete our analysis of the optimal instantaneous growth of palinstrophy, we now look at the structure of the spectrum of the optimal fields $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}$ . A key step in the derivation of estimate (1.8) (see § A.1) is the choice of cut-off wavenumbers $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ that define the sets $\{|\boldsymbol{k}|\leqslant \unicode[STIX]{x1D6EC}_{1}\}$ , $\{\unicode[STIX]{x1D6EC}_{1}<|\boldsymbol{k}|\leqslant \unicode[STIX]{x1D6EC}_{2}\}$ and $\{|\boldsymbol{k}|>\unicode[STIX]{x1D6EC}_{2}\}$ in wavenumber space, where $\sum |\boldsymbol{k}||\widehat{u}(\boldsymbol{k})|$ displays different behaviour. As discussed in the appendix, $\unicode[STIX]{x1D6EC}_{1}$ and $\unicode[STIX]{x1D6EC}_{2}$ depend on ${\mathcal{K}}$ , ${\mathcal{P}}$ and $\unicode[STIX]{x1D708}$ as

(2.8a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}_{1}^{2}=c_{1}\frac{{\mathcal{P}}^{1/2}}{{\mathcal{K}}^{1/2}}\quad \text{and}\quad \unicode[STIX]{x1D6EC}_{2}^{2}=\frac{1}{c_{2}}\frac{{\mathcal{P}}^{1/2}}{\unicode[STIX]{x1D708}},\end{eqnarray}$$

where $c_{1}$ and $c_{2}$ are dimensionless parameters. Figure 4 shows the rescaled compensated spectral density $(|\boldsymbol{k}|/\unicode[STIX]{x1D706}_{0})^{2}\,S(|\boldsymbol{k}|/\unicode[STIX]{x1D706}_{0})$ corresponding to the optimal fields $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}$ for $Re=10^{3}$ and palinstrophy values ${\mathcal{P}}_{0}=1.71\times 10^{6}$ , ${\mathcal{P}}_{0}=1.71\times 10^{7}$ and ${\mathcal{P}}_{0}=1.71\times 10^{8}$ . The spectral density $S(|\boldsymbol{k}|)$ is computed as

(2.9) $$\begin{eqnarray}S(|\boldsymbol{k}|)=\mathop{\sum }_{2\unicode[STIX]{x03C0}k\leqslant |\boldsymbol{k}|\leqslant 2\unicode[STIX]{x03C0}(k+1)}|\boldsymbol{k}||\widehat{\boldsymbol{u}}(\boldsymbol{k})|^{2},\end{eqnarray}$$

and the scaling factor $\unicode[STIX]{x1D706}_{0}$ is given by

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{0}^{2}=\frac{\displaystyle \int _{0}^{\infty }|\boldsymbol{k}|^{2}S(|\boldsymbol{k}|)\,\text{d}|\boldsymbol{k}|}{\displaystyle \int _{0}^{\infty }S(|\boldsymbol{k}|)\,\text{d}|\boldsymbol{k}|}.\end{eqnarray}$$

The two wavenumbers $\unicode[STIX]{x1D706}_{1}$ and $\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FA}}$ , computed as

(2.11a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D706}_{1}=\frac{\displaystyle \int _{0}^{\infty }|\boldsymbol{k}|S(|\boldsymbol{k}|)\,\text{d}|\boldsymbol{k}|}{\displaystyle \int _{0}^{\infty }S(|\boldsymbol{k}|)\,\text{d}|\boldsymbol{k}|}\quad \text{and}\quad \unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FA}}=\frac{\displaystyle \int _{0}^{\infty }|\boldsymbol{k}|^{3}S(|\boldsymbol{k}|)\,\text{d}|\boldsymbol{k}|}{\displaystyle \int _{0}^{\infty }|\boldsymbol{k}|^{2}S(|\boldsymbol{k}|)\,\text{d}|\boldsymbol{k}|},\end{eqnarray}$$

define, along with the scaling factor $\unicode[STIX]{x1D706}_{0}$ , the Novikov phase diagram of the flow (Santangelo, Benzi & Legras Reference Santangelo, Benzi and Legras1989) in terms of the dimensionless parameters

(2.12a,b ) $$\begin{eqnarray}P_{1}=\frac{\unicode[STIX]{x1D706}_{1}}{\unicode[STIX]{x1D706}_{0}}\quad \text{and}\quad P_{2}=\frac{\unicode[STIX]{x1D706}_{\unicode[STIX]{x1D6FA}}}{\unicode[STIX]{x1D706}_{0}},\end{eqnarray}$$

which satisfy

(2.13a,b ) $$\begin{eqnarray}P_{1}\leqslant 1\quad \text{and}\quad P_{2}\geqslant 1,\end{eqnarray}$$

and are shown in figure 4 as vertical lines located at $P_{1}\approx 0.86$ and $P_{2}\approx 1.88$ . The parameter $P_{1}$ corresponds to the location of the first local maximum of the compensated spectral density $|\boldsymbol{k}|^{2}S(|\boldsymbol{k}|)$ . As expected from the self-similar construction of the optimal fields $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}$ , the rescaled compensated spectral densities collapse onto a single ‘universal’ spectral density, confirming the scale-invariant nature of $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}$ .

3 Finite-time growth of palinstrophy

We now focus our attention on the growth of palinstrophy ${\mathcal{P}}$ over finite time. Time integration of the sharp instantaneous estimate (1.8) and neglect of the time dependence of $Re$ leads to the finite-time estimates

(3.1) $$\begin{eqnarray}{\mathcal{P}}^{1/2}(t)-{\mathcal{P}}^{1/2}(0)\leqslant \left(a+b\sqrt{\ln Re_{0}+c}\right)\left(\frac{{\mathcal{E}}(0)-{\mathcal{E}}(t)}{4\unicode[STIX]{x1D708}}\right)\end{eqnarray}$$

and

(3.2) $$\begin{eqnarray}\max _{t>0}{\mathcal{P}}(t)\leqslant \unicode[STIX]{x1D713}(Re_{0}){\mathcal{P}}(0),\quad \text{with }\unicode[STIX]{x1D713}(Re_{0})=\left(1+\frac{a+b\sqrt{\ln Re_{0}+c}}{4}\,Re_{0}\right)^{2},\end{eqnarray}$$

where the prefactor $\unicode[STIX]{x1D713}(Re_{0})$ depends exclusively on the initial Reynolds number $Re_{0}={\mathcal{K}}^{1/2}(0)/\unicode[STIX]{x1D708}$ , and the values of $a$ , $b$ and $c$ are given in (2.5) and obtained by the fitting procedure described in § 2. Note that although estimate (3.1) is time-dependent, it can still be used to determine to what extent the fields saturating a sharp instantaneous estimate produce time-dependent flows which also saturate a time-dependent estimate. Also note that while (3.2) is an a priori estimate in the form of a power law which has been obtained from a sharp instantaneous estimate, there is no guarantee that it will be sharp with respect to either the exponent or the prefactor in the power law. The derivation of estimates (3.1) and (3.2) can be found in § A.2.

In order to assess the sharpness of the finite-time estimates (3.1) and (3.2), we numerically solve the Cauchy problem (1.1a-c ) using the instantaneously optimal vorticity fields $\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ as initial condition, and carefully monitor the time evolution of different diagnostics, e.g. ${\mathcal{K}}(t)$ , ${\mathcal{E}}(t)$ and ${\mathcal{P}}(t)$ , for sufficiently long times. Time integration is performed using an adaptive Runge–Kutta scheme of order 4, and spatial discretization is performed using a pseudo-spectral Fourier–Galerkin method, with the standard ‘ $2/3$ ’ dealiasing rule. The resolution is increased accordingly, ranging from $512^{2}$ for low- $Re_{0}$ , low- ${\mathcal{P}}_{0}$ simulations, to $4096^{2}$ for high- $Re_{0}$ , high- ${\mathcal{P}}_{0}$ simulations. We refer the reader to the work by Ayala & Protas (Reference Ayala and Protas2014b ) for a thorough discussion of the time evolution of the instantaneously optimal vorticity $\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ .

The sharpness of the pointwise estimate (3.1) can be studied by considering the functions

(3.3a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D711}(t)={\mathcal{P}}^{1/2}(t)-{\mathcal{P}}^{1/2}(0)\quad \text{and}\quad \unicode[STIX]{x1D707}(t)=\left(\frac{a+b\sqrt{\ln Re_{0}+c}}{4\unicode[STIX]{x1D708}}\right)\left({\mathcal{E}}(0)-{\mathcal{E}}(t)\right)\end{eqnarray}$$

and comparing their behaviour as functions of time. For this, consider the characteristic time scale $t_{max}$ and the characteristic palinstrophy scale $\unicode[STIX]{x1D70C}$ defined as

(3.4a,b ) $$\begin{eqnarray}t_{max}=\underset{t>0}{\text{argmax}}~\unicode[STIX]{x1D711}(t)\quad \text{and}\quad \unicode[STIX]{x1D70C}=\max _{t\geqslant 0}\,\unicode[STIX]{x1D711}(t),\end{eqnarray}$$

and the rescaled diagnostics

(3.5) $$\begin{eqnarray}f(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D711}(t_{max}\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D70C}\end{eqnarray}$$

and

(3.6) $$\begin{eqnarray}g(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D707}(t_{max}\unicode[STIX]{x1D70F})/\unicode[STIX]{x1D70C}.\end{eqnarray}$$

With these definitions, the pointwise estimate (3.1) simply reads $f(\unicode[STIX]{x1D70F})\leqslant g(\unicode[STIX]{x1D70F})$ . Figure 5 shows the dependence of $f$ and $g$ on the rescaled time $\unicode[STIX]{x1D70F}=t/t_{max}$ , corresponding to different values of $Re\in [10^{3},10^{4}]$ and ${\mathcal{P}}_{0}\in [1.7\times 10^{6},1.7\times 10^{9}]$ . As expected from the fact that the initial condition is constructed in a self-similar manner, all data collapse onto single ‘universal’ curves for $f$ and $g$ . The data indicate that estimate (3.1) is saturated only over a short interval, as expected from the fact that the fields are optimal only in an instantaneous sense, and the optimal growth of palinstrophy can be sustained only over this short interval.

Figure 5. Rescaled diagnostics $f(\unicode[STIX]{x1D70F})$ (black solid lines) and $g(\unicode[STIX]{x1D70F})$ (blue dashed lines) for different values of ${\mathcal{P}}_{0}$ and $Re_{0}$ . All data collapse onto a ‘universal’ pair of curves.

On the other hand, it follows from estimate (3.1) that

(3.7) $$\begin{eqnarray}Q_{max}:=4\unicode[STIX]{x1D708}\frac{{\mathcal{P}}^{1/2}(t_{max})-{\mathcal{P}}^{1/2}(0)}{{\mathcal{E}}(0)-{\mathcal{E}}(t_{max})}\leqslant a+b\sqrt{\ln Re_{0}+c}.\end{eqnarray}$$

Figure 6(a) shows the dependence of $Q_{max}$ on $Re_{0}$ , for values of palinstrophy in the range ${\mathcal{P}}_{0}\in [10^{6},10^{9}]$ . The curve $\unicode[STIX]{x1D6FE}(Re_{0})=\tilde{a}+\tilde{b}\sqrt{\ln Re_{0}+\tilde{c}}$ , with $\tilde{a}$ , $\tilde{b}$ and $\tilde{c}$ given in (2.5), is included for reference as a red dashed curve. It can be observed from figure 6(a) that, although estimate (3.1) is saturated only over a short time interval, the dependence of the finite-time growth of palinstrophy on $Re_{0}$ , when compensated by the enstrophy dissipation occurring at the maximum palinstrophy time, has a behaviour similar to that predicted by estimate (3.7).

Figure 6(b) shows the dependence on ${\mathcal{P}}(0)$ of the normalized maximum palinstrophy

(3.8) $$\begin{eqnarray}{\mathcal{P}}_{max}={\mathcal{P}}(0)^{-1}\,\max _{t>0}\,{\mathcal{P}}(t)\end{eqnarray}$$

obtained from the time evolution of the optimal vorticity $\widetilde{\unicode[STIX]{x1D714}}_{Re_{0},{\mathcal{P}}_{0}}$ , for fixed $Re_{0}$ . As expected from estimate (3.2), the data indicate that ${\mathcal{P}}_{max}$ is independent of ${\mathcal{P}}(0)$ , with its value depending only on the initial Reynolds number $Re_{0}$ . These results provide compelling evidence supporting the sharpness of the a priori finite-time estimate (3.2) with respect to ${\mathcal{P}}(0)$ .

Figure 6. (a) The quantity $Q_{max}$ , defined in (3.7), as a function of $\ln Re_{0}$ . The prefactor $C_{Re_{0}}=\tilde{a}+\tilde{b}\sqrt{\tilde{c}+\ln Re_{0}}$ , with $\tilde{a}$ , $\tilde{b}$ and $\tilde{c}$ from (2.5), is shown as a red dashed line. (b) Compensated maximum palinstrophy, ${\mathcal{P}}_{max}={\mathcal{P}}_{0}^{-1}\max _{t>0}{\mathcal{P}}(t)$ , as a function of ${\mathcal{P}}_{0}$ , for different values of Reynolds number $Re_{0}\in [10^{3},10^{4}]$ .

4 Discussion and conclusion

We have presented numerical confirmation that the rigorous analytic estimate

(4.1) $$\begin{eqnarray}\frac{\text{d}{\mathcal{P}}}{\text{d}t}\leqslant \left(a+b\sqrt{\ln Re+c}\right){\mathcal{P}}^{3/2}\end{eqnarray}$$

is sharp in its behaviour with respect to both the palinstrophy ${\mathcal{P}}$ and the Reynolds number $Re$ . The power-law dependence on ${\mathcal{P}}$ is predicted by the self-similar analysis from appendix B and confirmed by the data shown in figure 2, where the normalized optimal rate of growth of palinstrophy ${\mathcal{P}}_{0}^{-3/2}{\mathcal{R}}(\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}})$ is plotted against ${\mathcal{P}}_{0}$ . Similarly, the dependence of ${\mathcal{R}}(\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}})$ on $Re_{0}$ is correctly captured by the estimate, although the constants $\tilde{a},~\tilde{b}$ and $\tilde{c}$ that fit the optimal rate of growth in the least-squares sense differ from the analytic values given in § A.1. More careful analysis might give better constants, but it is important to note that the approach used to construct the optimal fields only ensures that solutions to (2.1) are local maximizers: there is no guarantee that they are global maximizers of ${\mathcal{R}}$ . However, the best one can hope for in the search for any other maximizers is to improve the value of the constants $\tilde{a}$ , $\tilde{b}$ and $\tilde{c}$ so that optimal instantaneous production of palinstrophy matches that given by the analytic estimate.

Regarding the finite-time growth of palinstrophy, we have provided evidence supporting the sharpness of the a priori estimate

(4.2) $$\begin{eqnarray}\max _{t>0}\,{\mathcal{P}}(t)\leqslant \unicode[STIX]{x1D713}(Re_{0}){\mathcal{P}}(0)\end{eqnarray}$$

with respect to the initial palinstrophy ${\mathcal{P}}(0)$ . For this, we have used the instantaneously optimal fields $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}$ as initial condition in the 2D incompressible Navier–Stokes equation, and carefully monitored the time evolution of palinstrophy. The sharpness with respect to the prefactor $\unicode[STIX]{x1D713}(Re_{0})$ , on the other hand, is a more subtle issue, and instead we looked at the pointwise estimate

(4.3) $$\begin{eqnarray}{\mathcal{P}}^{1/2}(t)-{\mathcal{P}}^{1/2}(0)\leqslant \frac{\unicode[STIX]{x1D6FE}(Re_{0})}{4\unicode[STIX]{x1D708}}\left({\mathcal{E}}(0)-{\mathcal{E}}(t)\right).\end{eqnarray}$$

This time-dependent estimate was found to be saturated by the instantaneously optimal fields $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}$ only over a short time window, resulting in a sub-optimal dependence of the normalized maximum growth of palinstrophy on $Re_{0}$ . This does not mean, however, that the estimate is not sharp, as the fields $\widetilde{\boldsymbol{u}}_{Re_{0},{\mathcal{P}}_{0}}$ are optimal only at time $t=0$ . A better assessment of the sharpness of the pointwise estimate (3.1) could be performed by solving the finite-time optimization problem:

(4.4) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \max _{\boldsymbol{u}\in {\mathcal{S}}}\,{\mathcal{P}}(\boldsymbol{u}(\cdot ,T)),\quad \text{with}\\ {\mathcal{S}}=\{\boldsymbol{u}\in H^{2}(\unicode[STIX]{x1D6FA})\boldsymbol{ : }\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{u}=0,{\mathcal{K}}(\boldsymbol{u}(\cdot ,0))={\mathcal{K}}_{0},{\mathcal{P}}(\boldsymbol{u}(\cdot ,0))={\mathcal{P}}_{0},\boldsymbol{u}\;\text{solves (1.1)}\},\end{array}\right\}\end{eqnarray}$$

with the constraint manifold ${\mathcal{S}}$ including only the fields that are solutions to the 2D incompressible Navier–Stokes equation defined on the interval $(0,T)$ . This finite-time optimization approach was used by Ayala & Protas (Reference Ayala and Protas2011) to study the maximum finite-time growth of enstrophy of solutions to the one-dimensional Burgers equation, with the results confirmed analytically by Pelinovsky (Reference Pelinovsky2012).

To summarize, the results presented in this paper indicate that it is possible to saturate a priori finite-time estimates using fields that saturate instantaneous estimates. Time-dependent estimates, on the other hand, are saturated only over short time intervals, rendering the finite-time growth of palinstrophy obtained from instantaneously optimal fields suboptimal. This implies that in the context of the 3D Navier–Stokes equation, for which no a priori finite-time estimates for the growth of enstrophy are available and only time-dependent estimates exist, the search for solenoidal fields that are optimal for the growth of enstrophy, the key quantity controlling the regularity of solutions in 3D, must be performed following a finite-time optimization approach where the Navier–Stokes equation is included as part of the constraint, similar to the approach described in (4.4). Although computationally intensive, this task is within reach with current available resources and it is also left as an open question for subsequent work.