2.1 Optimization method

We use a slightly modified version of the nonlinear optimization algorithm of Willis (Reference Willis2012) and Chen et al. (Reference Chen, Herreman and Jackson2015). The objective function $\mathscr{L}$ that will be maximized is

(2.1) $$\begin{eqnarray}\displaystyle \mathscr{L} & = & \displaystyle \ln \langle \boldsymbol{B}_{T}^{2}\rangle -\langle {\it\Pi}_{1}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{U}\rangle -{\it\lambda}_{1}(\langle [\boldsymbol{{\rm\nabla}}\times (\boldsymbol{U}-\overline{\boldsymbol{U}})]^{2}\rangle -s^{2})\nonumber\\ \displaystyle & & \displaystyle -\,\langle {\it\Pi}_{2}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{B}_{0}\rangle -{\it\lambda}_{2}(\langle \boldsymbol{B}_{0}^{2}\rangle -1)\nonumber\\ \displaystyle & & \displaystyle -\,\int _{0}^{T}\langle \boldsymbol{B}^{\dagger }\boldsymbol{\cdot }[\partial _{t}\boldsymbol{B}-\boldsymbol{{\rm\nabla}}\times (\boldsymbol{U}\times \boldsymbol{B})-Rm^{-1}{\rm\Delta}\boldsymbol{B}]\rangle \,\text{d}t.\end{eqnarray}$$

We denote $\langle \ldots \rangle =(1/V)\int _{V}\ldots \text{d}V$ , and the fluid domain is a periodic cube $[0,2{\rm\pi}]^{3}$ , so $V=(2{\rm\pi})^{3}$ . We denote $\boldsymbol{B}_{0}=\boldsymbol{B}(\boldsymbol{x},0)$ , $\boldsymbol{B}_{T}=\boldsymbol{B}(\boldsymbol{x},T)$ , the initial and final magnetic fields. With $\boldsymbol{B}_{T}=\boldsymbol{B}(\boldsymbol{x},T)$ the final time magnetic field, the first term in this functional expresses that we want to maximize the logarithm of magnetic energy at time $T$ . We vary the total flow $\boldsymbol{U}=\overline{\boldsymbol{U}}(\boldsymbol{x})+\boldsymbol{u}(\boldsymbol{x})$ over the space of time-independent solenoidal vector fields (constraint expressed using ${\it\Pi}_{1}$ ) that remain in the vicinity of Kolmogorov flow $\overline{\boldsymbol{U}}=\sin y\,\boldsymbol{e}_{x}$ . We impose this vicinity by constraining the flows such that the shear magnitude of the perturbation flow

(2.2) $$\begin{eqnarray}\displaystyle s=(\langle [\boldsymbol{{\rm\nabla}}\times (\boldsymbol{U}-\overline{\boldsymbol{U}})]^{2}\rangle )^{1/2}=(\langle (\boldsymbol{{\rm\nabla}}\times \boldsymbol{u})^{2}\rangle )^{1/2} & & \displaystyle\end{eqnarray}$$

remains fixed and small, $s\ll 1$ . As in Willis (Reference Willis2012) and Chen et al. (Reference Chen, Herreman and Jackson2015), we fix this perturbation shear magnitude rather than the perturbation root mean square (r.m.s.) speed $u=(\langle \boldsymbol{u}^{2}\rangle )^{1/2}$ to prevent the optimizer from picking up discontinuous flow fields. In the optimization process, we also search for optimal initial magnetic fields $\boldsymbol{B}_{0}=\boldsymbol{B}(\boldsymbol{x},0)$ within the space of solenoidal and normalized vector fields (constraints expressed using multipliers ${\it\lambda}_{1},{\it\Pi}_{2}$ ). At all times $t\in [0,T]$ , the magnetic field $\boldsymbol{B}(\boldsymbol{x},t)$ satisfies the induction equation (constraint expressed using an adjoint magnetic field variable $\boldsymbol{B}^{\dagger }(\boldsymbol{x},t)$ ) that depends on the magnetic Reynolds number, here defined as

(2.3) $$\begin{eqnarray}\displaystyle Rm=\frac{UL}{2{\rm\pi}{\it\eta}}. & & \displaystyle\end{eqnarray}$$

This definition of $Rm$ relates to the dimensional Kolmogorov flow $U\sin (2{\rm\pi}y/L)\,\boldsymbol{e}_{x}$ that fits in an $L$ -periodic box and has amplitude $U$ . As we work inside a $2{\rm\pi}$ -periodic box, $L/2{\rm\pi}$ is the correct length scale; $U$ is the velocity scale and $L/(2{\rm\pi}U)$ the time scale used in (2.1).

At the optimal, we must have ${\it\delta}\mathscr{L}=0$ , for arbitrary variations of ${\it\delta}\boldsymbol{U}$ , ${\it\delta}\boldsymbol{B}$ , ${\it\delta}\boldsymbol{B}_{0}$ , ${\it\delta}\boldsymbol{B}_{T}$ , ${\it\delta}\boldsymbol{B}^{\dagger }$ , ${\it\delta}{\it\bf\Pi}_{1}$ , ${\it\delta}{\it\bf\Pi}_{2}$ , ${\it\delta}{\it\lambda}_{1}$ and ${\it\delta}{\it\lambda}_{2}$ . This generates nine Euler–Lagrange equations. Most of the Euler–Lagrange equations are identical to those of Chen et al. (Reference Chen, Herreman and Jackson2015), only

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}\mathscr{L}}{{\it\delta}{\it\lambda}_{1}}=-\langle [\boldsymbol{{\rm\nabla}}\times (\boldsymbol{U}-\overline{\boldsymbol{U}})]^{2}\rangle +s^{2}=0, & \displaystyle\end{eqnarray}$$

(2.5) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\delta}\mathscr{L}}{{\it\delta}\boldsymbol{U}}=2{\it\lambda}_{1}{\rm\Delta}(\boldsymbol{U}-\overline{\boldsymbol{U}})+\boldsymbol{{\rm\nabla}}{\it\Pi}_{1}+\int _{0}^{T}\boldsymbol{B}\times (\boldsymbol{{\rm\nabla}}\times \boldsymbol{B}^{\dagger })\,\text{d}t=0 & \displaystyle\end{eqnarray}$$

are different. The process of deriving the Euler–Lagrange equations generates boundary terms that here all cancel out due to periodicity. The optimization problem is solved using an iterative method that only differs from the method of Chen et al. (Reference Chen, Herreman and Jackson2015) by the fact that ${\it\lambda}_{1}$ is here fixed by the requirement (2.4). Our code reproduces the results of Willis (Reference Willis2012): putting $\overline{\boldsymbol{U}}=0$ , $s=1$ , $T=8$ we reproduce the threshold $Rm_{c}=2.48$ for the optimized dynamo of Willis (denoted $Rm_{{\it\omega}c}$ in Willis (Reference Willis2012)).

As in Willis (Reference Willis2012) and Chen et al. (Reference Chen, Herreman and Jackson2015), optimal field configurations are degenerate due to symmetries in the objective function. The following eight types of coordinate changes:

(2.6) $$\begin{eqnarray}\displaystyle \boldsymbol{x}\rightarrow \boldsymbol{x}_{0}+\unicode[STIX]{x1D64D}\boldsymbol{x}:\left\{\begin{array}{@{}rcl@{}}(x,y,z)\ & \rightarrow \ & (a,0,c)+(x,y,\pm z),\\ (x,y,z)\ & \rightarrow \ & (a,0,c)+(-x,-y,\pm z),\\ (x,y,z)\ & \rightarrow \ & (a,{\rm\pi},c)+(-x,y,\pm z),\\ (x,y,z)\ & \rightarrow \ & (a,{\rm\pi},c)+(x,-y,\pm z),\end{array}\right. & & \displaystyle\end{eqnarray}$$

with arbitrary $a,c\in [0,2{\rm\pi}]$ , can be denoted in short as $\boldsymbol{x}\rightarrow \boldsymbol{x}_{0}+\unicode[STIX]{x1D64D}\boldsymbol{x}$ , where $\unicode[STIX]{x1D64D}$ is a symmetric orthogonal matrix so that $\unicode[STIX]{x1D64D}^{-1}=\unicode[STIX]{x1D64D}$ . Kolmogorov flow $\overline{\boldsymbol{U}}=\sin y\,\boldsymbol{e}_{x}$ is symmetric with respect to all of these operations, which means that $\overline{\boldsymbol{U}}(\boldsymbol{x})=\unicode[STIX]{x1D64D}\overline{\boldsymbol{U}}(\boldsymbol{x}_{0}+\unicode[STIX]{x1D64D}\boldsymbol{x})$ . By replacing $\overline{\boldsymbol{U}}(\boldsymbol{x})$ with the previous relation in the objective function (2.1) we can, by a change of coordinates, find that $\mathscr{L}$ is invariant: $\mathscr{L}(\boldsymbol{U},\boldsymbol{B},\ldots )=\mathscr{L}(\check{\boldsymbol{U}},\check{\boldsymbol{B}},\ldots )$ , with fields

(2.7a,b ) $$\begin{eqnarray}\displaystyle \check{\boldsymbol{U}}(\boldsymbol{x})=\unicode[STIX]{x1D64D}\boldsymbol{U}(\unicode[STIX]{x1D64D}(\boldsymbol{x}-\boldsymbol{x}_{0})),\quad \check{\boldsymbol{B}}(\boldsymbol{x},t)=\unicode[STIX]{x1D64D}\boldsymbol{B}(\unicode[STIX]{x1D64D}(\boldsymbol{x}-\boldsymbol{x}_{0}),t), & & \displaystyle\end{eqnarray}$$

and similar relations for the Lagrange multipliers $\check{\boldsymbol{B}}^{\dagger },\check{{\it\Pi}}_{1},\check{{\it\Pi}}_{2},\check{{\it\lambda}}_{1},\check{{\it\lambda}}_{2}$ . The same value of $\mathscr{L}$ can therefore be reached with different fields, and, as a consequence, the optimal is degenerate: if $\boldsymbol{U}(\boldsymbol{x})$ is an optimal dynamo that drives a field $\boldsymbol{B}(\boldsymbol{x},t)$ , then $\check{\boldsymbol{U}}(\boldsymbol{x})$ is an optimal dynamo that drives $\check{\boldsymbol{B}}(\boldsymbol{x},t)$ . In the following, we take the liberty of translating, rotating and reflecting optimal field configurations according to (2.6)–(2.7) if this allows a better comparison of independently obtained optima.

3.1 A lower bound in terms of maximal perturbation speed

Proctor (Reference Proctor2004) calculates a lower bound for the magnitude of poloidal perturbation flows that is necessary to trigger dynamo action in dominantly toroidal flows. As suggested by Proctor (Reference Proctor2004), the original method applied to spheres easily extends to Zel’dovich’s planar flow anti-dynamo theorem. We perform this calculation here, for a fluid domain that is a periodic box of size $L$ . We only present essential differences and refer to Proctor (Reference Proctor2004) for all details. The flow and magnetic field are decomposed into toroidal and poloidal parts:

(3.1a,b ) $$\begin{eqnarray}\displaystyle \boldsymbol{U}^{d}=\underbrace{\boldsymbol{{\rm\nabla}}\times {\it\phi}\,\boldsymbol{e}_{z}}_{\boldsymbol{U}_{t}}+\underbrace{\boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times {\it\psi}\,\boldsymbol{e}_{z}}_{\boldsymbol{U}_{p}},\quad \boldsymbol{B}=\underbrace{\boldsymbol{{\rm\nabla}}\times T\,\boldsymbol{e}_{z}}_{\boldsymbol{B}_{t}}+\underbrace{\boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times S\,\boldsymbol{e}_{z}}_{\boldsymbol{B}_{p}}. & & \displaystyle\end{eqnarray}$$

We introduce scalar fields $P,H$ through the relations $P=-{\it\Delta}_{\bot }S$ , $T=-{\it\Delta}_{\bot }H$ , where ${\it\Delta}_{\bot }=\partial _{xx}^{2}+\partial _{yy}^{2}$ . The calculation starts with three lemmas, which become

(3.2a-c ) $$\begin{eqnarray}\displaystyle \hspace{-20.00003pt}\langle \boldsymbol{B}_{t}^{2}\rangle \leqslant \langle (\boldsymbol{{\rm\nabla}}T)^{2}\rangle ,\quad \langle \boldsymbol{B}_{p}^{2}\rangle \leqslant a^{2}\langle (\boldsymbol{{\rm\nabla}}P)^{2}\rangle ,\quad \langle (\boldsymbol{{\rm\nabla}}\times \boldsymbol{{\rm\nabla}}\times H\,\boldsymbol{e}_{z})^{2}\rangle \leqslant a^{2}\langle (\boldsymbol{{\rm\nabla}}T)^{2}\rangle , & & \displaystyle\end{eqnarray}$$

with $a=L/2{\rm\pi}$ in the periodic box of size $L$ . Operating respectively with $P\boldsymbol{e}_{z}$ and $H\boldsymbol{e}_{z}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\times$ on the dimensional induction (1.1), integrating over the volume, using partial integration, Cauchy–Schwartz and Lemmas 1–3 as in Proctor (Reference Proctor2004) leads to two inequalities

(3.3) $$\begin{eqnarray}\displaystyle \hspace{-20.00003pt}\left.\begin{array}{@{}c@{}}\displaystyle \frac{\partial _{t}\left\langle P^{2}\right\rangle }{2}\leqslant \left(aU_{p}-{\it\eta}\right)\left\langle (\boldsymbol{{\rm\nabla}}P)^{2}\right\rangle +U_{p}\left\langle (\boldsymbol{{\rm\nabla}}T)^{2}\right\rangle ^{1/2}\left\langle (\boldsymbol{{\rm\nabla}}P)^{2}\right\rangle ^{1/2},\\ \displaystyle \frac{\partial _{t}\left\langle T^{2}\right\rangle }{2}\leqslant \left(aU_{p}-{\it\eta}\right)\left\langle (\boldsymbol{{\rm\nabla}}T)^{2}\right\rangle +a^{2}\left(U_{t}+U_{p}\right)\left\langle (\boldsymbol{{\rm\nabla}}T)^{2}\right\rangle ^{1/2}\left\langle (\boldsymbol{{\rm\nabla}}P)^{2}\right\rangle ^{1/2}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Here, $U_{p}=\max _{V}\Vert \boldsymbol{U}_{p}\Vert$ and $U_{t}=\max _{V}\Vert \boldsymbol{U}_{t}\Vert$ . Both relations are then combined by writing an equation for $\langle P^{2}\rangle +{\it\mu}a^{-2}\langle T^{2}\rangle$ for some non-dimensional parameter ${\it\mu}>0$ . On the right-hand side, we replace $\langle (\boldsymbol{{\rm\nabla}}T)^{2}\rangle ={\it\tau}^{2}$ and $\langle (\boldsymbol{{\rm\nabla}}P)^{2}\rangle =a^{-2}{\it\Gamma}^{2}{\it\tau}^{2}$ , where ${\it\Gamma}>0$ is non-dimensional and ${\it\tau}>0$ has the dimension of the magnetic field. This yields

(3.4) $$\begin{eqnarray}\displaystyle \hspace{-15.00002pt}\frac{\partial _{t}(\langle P^{2}\rangle +{\it\mu}a^{-2}\langle T^{2}\rangle )}{2}\leqslant \frac{{\it\eta}{\it\tau}^{2}}{a^{2}}\{(Rm_{p}-1)({\it\Gamma}^{2}+{\it\mu})+[Rm_{p}+(Rm_{p}+Rm_{t}){\it\mu}]{\it\Gamma}\} & & \displaystyle\end{eqnarray}$$

in terms of poloidal and toroidal magnetic Reynolds numbers $Rm_{p}=U_{p}a/{\it\eta},Rm_{t}=U_{t}a/{\it\eta}$ . We suppose that $Rm_{p}\leqslant 1$ , in which case the right-hand side has an upper bound. Maximizing with respect to ${\it\Gamma}$ yields this bound:

(3.5) $$\begin{eqnarray}\displaystyle \hspace{-15.00002pt}\frac{\partial _{t}(\langle P^{2}\rangle +{\it\mu}a^{-2}\langle T^{2}\rangle )}{2}\leqslant \frac{{\it\eta}{\it\tau}^{2}}{a^{2}}\left\{\frac{[Rm_{p}+(Rm_{p}+Rm_{t}){\it\mu}]^{2}}{4(1-Rm_{p})}-{\it\mu}(1-Rm_{p})\right\}, & & \displaystyle\end{eqnarray}$$

and it can be tightened by minimizing the right-hand side with respect to ${\it\mu}$ . At the critical value of ${\it\mu}={\it\mu}_{c}$ ,

(3.6) $$\begin{eqnarray}\displaystyle {\it\mu}_{c}=\frac{2\left(1-Rm_{p}\right)^{2}-Rm_{p}(Rm_{p}+Rm_{t})}{(Rm_{t}+Rm_{p})^{2}}, & & \displaystyle\end{eqnarray}$$

we obtain

(3.7) $$\begin{eqnarray}\displaystyle \frac{\partial _{t}(\langle P^{2}\rangle +{\it\mu}_{c}a^{-2}\langle T^{2}\rangle )}{2}\leqslant \frac{{\it\eta}{\it\tau}^{2}}{a^{2}}\frac{(1-Rm_{p})}{(Rm_{p}+Rm_{t})^{2}}[Rm_{p}(Rm_{t}+2)-1]. & & \displaystyle\end{eqnarray}$$

A necessary condition for dynamo action is that the right-hand side is positive definite. This requires $Rm_{p}(Rm_{t}+2)\geqslant 1$ , which for high $Rm_{t}$ simplifies to

(3.8) $$\begin{eqnarray}\displaystyle Rm_{p}\,Rm_{t}\geqslant 1. & & \displaystyle\end{eqnarray}$$

Let us now translate this in terms of the shear norm $s$ of (2.2) and the original magnetic Reynolds number $Rm$ of (2.3). Kolmogorov flow is dominantly toroidal, $\boldsymbol{U}_{t}\approx \boldsymbol{{\rm\nabla}}\times (-UL\cos (2{\rm\pi}y/L)\boldsymbol{e}_{z})$ , so we have $U_{t}/U\approx 1$ . The poloidal flow can only be due to the perturbation flow, which due to (2.2) means that $U_{p}/U\approx s\,v_{max}$ , with $v_{max}$ the maximal speed of the renormalized perturbation flow $\boldsymbol{v}=\boldsymbol{u}/s$ that has $\langle (\boldsymbol{{\rm\nabla}}\times \boldsymbol{v})^{2}\rangle =1$ . We then have $Rm_{t}\approx Rm$ and $Rm_{p}\approx sRm\,v_{max}$ , which leads to the lower bound

(3.9) $$\begin{eqnarray}\displaystyle s\geqslant v_{max}^{-1}\,Rm^{-2}=s_{min,th}^{(1)}. & & \displaystyle\end{eqnarray}$$

The $Rm^{-2}$ power law clearly indicates that this lower bound is well beneath the numerical result $s_{min}\sim Rm^{-1.1}$ in the limit $Rm\rightarrow \infty$ . We measure $v_{max}\in [1.45,2.83]$ for all optimal dynamos. Interpolating more precise values of $v_{max}$ near the threshold $s=s_{min}(Rm)$ , we calculate $s_{min,th}^{(1)}$ which is displayed in table 1. Clearly, $s_{min}\gg s_{min,th}^{(1)}$ for all explored $Rm$ , as it should be, but this also suggests that the theoretical bound is not sharp. Most likely this is the consequence of Lemmas 2 and 3 in (3.2), which seriously overestimate the spatial extent of the magnetic field.

3.2 A lower bound in terms of maximal perturbation shear

Proctor (Reference Proctor2012) calculates an upper bound for kinematic dynamo growth rates at high $Rm$ in shear dominated flows. This analysis can easily be adapted to find a necessary condition for dynamo action. Here, we suppose a flow

(3.10) $$\begin{eqnarray}\displaystyle \boldsymbol{U}^{d}=Sf(y)\boldsymbol{e}_{x}+\boldsymbol{u}^{d}, & & \displaystyle\end{eqnarray}$$

with $f(y)$ arbitrary but $\max |f^{\prime }(y)|=1$ , so that $S$ measures the shear which is supposed to be large. We express the induction equation in Cartesian components, multiply each component with $B_{x},B_{y},B_{z}$ respectively and integrate over the volume. This leads to

(3.11) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\partial _{t}\left\langle B_{x}^{2}\right\rangle \!/2=\langle B_{x}^{2}(\partial _{x}u_{x}^{d})+B_{x}B_{y}(\partial _{y}u_{x}^{d}+Sf^{\prime }(y))+B_{x}B_{z}(\partial _{z}u_{x}^{d})\rangle -{\it\eta}\langle (\boldsymbol{{\rm\nabla}}B_{x})^{2}\rangle ,\\ \partial _{t}\left\langle B_{y}^{2}\right\rangle \!/2=\langle B_{x}B_{y}(\partial _{x}u_{y}^{d})+B_{y}^{2}(\partial _{y}u_{y}^{d})+B_{y}B_{z}(\partial _{z}u_{y}^{d})\rangle -{\it\eta}\langle (\boldsymbol{{\rm\nabla}}B_{y})^{2}\rangle ,\\ \partial _{t}\left\langle B_{z}^{2}\right\rangle \!/2=\langle B_{z}B_{x}(\partial _{x}u_{z}^{d})+B_{z}B_{y}(\partial _{y}u_{z}^{d})+B_{z}^{2}(\partial _{z}u_{z}^{d})\rangle -{\it\eta}\langle (\boldsymbol{{\rm\nabla}}B_{z})^{2}\rangle .\end{array}\right\} & & \displaystyle\end{eqnarray}$$

Proctor (Reference Proctor2012) does not consider the diffusive terms afterwards, but we can keep them in the analysis if we use the inequalities

(3.12) $$\begin{eqnarray}\displaystyle \langle (\boldsymbol{{\rm\nabla}}B_{j})^{2}\rangle \geqslant a^{-2}\langle B_{j}^{2}\rangle & & \displaystyle\end{eqnarray}$$

for $j=x,y,z$ and with $a=L/2{\rm\pi}$ as before in the periodic box of size $L$ . These bounds relate to the fact that the ‘largest scale’ magnetic field has precisely one wavelength in one direction and is homogeneous in the other two directions. Using this and the Cauchy–Schwartz inequality on the flow-related terms we then find that

(3.13) $$\begin{eqnarray}\displaystyle \partial _{t}\left[\begin{array}{@{}c@{}}m_{x}\\ m_{y}\\ m_{z}\end{array}\right]\leqslant \underbrace{\left[\begin{array}{@{}ccc@{}}{\it\gamma}_{xx} & {\it\gamma}_{xy}+S & {\it\gamma}_{xz}\\ {\it\gamma}_{yx} & {\it\gamma}_{yy} & {\it\gamma}_{yz}\\ {\it\gamma}_{zx} & {\it\gamma}_{zy} & {\it\gamma}_{zz}\end{array}\right]\left[\begin{array}{@{}c@{}}m_{x}\\ m_{y}\\ m_{z}\end{array}\right]}_{\boldsymbol{A}}-\frac{{\it\eta}}{a^{2}}\left[\begin{array}{@{}c@{}}m_{x}\\ m_{y}\\ m_{z}\end{array}\right]. & & \displaystyle\end{eqnarray}$$

We denote $m_{j}=\left(\langle B_{j}^{2}\rangle \right)^{1/2}$ , with $j=x,y,z$ and ${\it\gamma}_{ij}=\max _{V}\left(\partial u_{i}^{d}/\partial x_{j}\right)$ the maximal strain rates in the perturbation flow $\boldsymbol{u}^{d}$ . The following steps are identical to the analysis of Proctor (Reference Proctor2012). For high $S$ , it is possible to calculate approximative eigenvalues of the matrix on the right-hand side. Only one of them is positive,

(3.14) $$\begin{eqnarray}\displaystyle {\it\sigma}_{+}\approx (S{\it\gamma}_{yx})^{1/2}, & & \displaystyle\end{eqnarray}$$

and it comes along with a left eigenvector

(3.15) $$\begin{eqnarray}\displaystyle \boldsymbol{a}^{\text{T}}\approx [1,(S/{\it\gamma}_{yx})^{1/2},{\it\gamma}_{yz}/{\it\gamma}_{yx}] & & \displaystyle\end{eqnarray}$$

that is positive definite in all of its components. We multiply (3.13) with $\boldsymbol{a}^{\text{T}}$ and use $\boldsymbol{a}^{\text{T}}\boldsymbol{A}\approx {\it\sigma}_{+}\boldsymbol{a}^{\text{T}}$ to obtain

(3.16) $$\begin{eqnarray}\displaystyle \partial _{t}(\boldsymbol{a}^{\text{T}}\boldsymbol{m})\leqslant \left({\it\sigma}_{+}-{\it\eta}/a^{2}\right)(\boldsymbol{a}^{\text{T}}\boldsymbol{m}). & & \displaystyle\end{eqnarray}$$

A necessary condition for dynamo is that the right-hand side is positive definite. Since the product $\boldsymbol{a}^{\text{T}}\boldsymbol{m}>0$ , this requires that ${\it\sigma}_{+}\geqslant {\it\eta}/a^{2}$ or

(3.17) $$\begin{eqnarray}\displaystyle \frac{SL^{2}}{4{\rm\pi}^{2}{\it\eta}}\,\frac{{\it\gamma}_{yx}L^{2}}{4{\rm\pi}^{2}{\it\eta}}\geqslant 1. & & \displaystyle\end{eqnarray}$$

We express this condition in terms $s$ and $Rm$ of the present study. For the Kolmogorov flow, we have $S=2{\rm\pi}U/L$ . The maximal shear in the perturbation flow is rewritten using ${\it\gamma}_{yx}/S=s{\it\epsilon}_{yx}$ , with ${\it\epsilon}_{yx}=\max _{V}\left(\partial v_{y}/\partial x\right)$ the maximal shear in the renormalized non-dimensional perturbation flow $\boldsymbol{v}=\boldsymbol{u}/s$ . The necessary condition for dynamo action becomes

(3.18) $$\begin{eqnarray}\displaystyle s\geqslant {\it\epsilon}_{yx}^{-1}\,Rm^{-2}=s_{min,th}^{(2)}. & & \displaystyle\end{eqnarray}$$

This second theoretical lower bound also decays as $Rm^{-2}$ with $Rm$ . We measure ${\it\epsilon}_{yx}\in [1.15,2.64]$ for all explored optimal dynamos, and interpolate ${\it\epsilon}_{yx}$ at $s_{min}(Rm)$ to calculate the values of $s_{min,th}^{(2)}$ in table 1. Also here $s_{min}\gg s_{min,th}^{(2)}$ for all explored $Rm$ , as it should be, and also here the theoretical bound is far from sharp in the high- $Rm$ limit. Due to the scaling laws for ${\it\lambda}_{y}^{B},{\it\lambda}_{z}^{B}$ in (2.15), this is certainly due to the inequalities (3.12) which do not take into account the localization of the $y$ and $z$ components of the magnetic field.