2.1. Mean-field setting: $\alpha$-fluctuations

The evolution of the magnetic field in magnetohydrodynamics is described by the induction equation, which, in the non-relativistic limit, can be expressed as

(2.1)\begin{equation} \frac{\partial \boldsymbol{B}}{\partial t} = \boldsymbol{\nabla} \times (\boldsymbol{U} \times \boldsymbol{B}) + \eta \nabla^2 \boldsymbol{B}, \end{equation}

where $\boldsymbol {U}$ is the velocity field, $\boldsymbol {B}$ is the magnetic field and $\eta$ represents the microscopic diffusivity. Dynamo theory postulates that the action of turbulence on the seed magnetic field exponentially amplifies the magnetic field countering the microscopic diffusivity. Furthermore, in a mean-field theory, it is assumed that the typical length scale of large-scale fields is $L$ and that of the small-scale turbulent fields is $\ell _0$, with $L\gg \ell _0$. By averaging over $\ell _0$, and assuming that the Reynolds rules of averaging hold, we can obtain the mean-field induction equation (Steenbeck et al. Reference Steenbeck, Krause and Rädler1966; Moffatt Reference Moffatt1978)

(2.2)\begin{equation} \partial_t\langle{\boldsymbol{B}}\rangle={\boldsymbol\nabla}\times(\langle{\boldsymbol{U}}\rangle\times\langle{\boldsymbol{B}}\rangle+\boldsymbol{\mathcal{E}}) +\eta\nabla^2\langle{\boldsymbol{B}}\rangle, \end{equation}

where $\boldsymbol {\mathcal {E}}=\langle {\boldsymbol {u}\times \boldsymbol {b}}\rangle$ is the turbulent electromotive force (EMF), and $\boldsymbol {u}$ and $\boldsymbol {b}$ are the fluctuating velocity and magnetic fields, respectively.

The calculation of the turbulent EMF involves solving for the fluctuating magnetic field, which in general is a complex task. When $Rm\ll 1$ or when the Strouhal number $St\ll 1$, the quasi-linear approximation (also referred to as the first-order smoothing approximation) is valid, and the fluctuating field is assumed to be generated by the action of the turbulent velocity field on the initial seed mean magnetic field. With the further assumption of homogeneity and isotropy of the turbulence (see, e.g. Brandenburg & Subramanian Reference Brandenburg and Subramanian2005a), an ansatz for the EMF can be obtained as follows:

(2.3)\begin{equation} \mathcal{E}_i=\alpha \langle{B}\rangle_i-\beta \epsilon_{ijk}\partial_j\langle{B}\rangle_k, \end{equation}

where $\alpha =-\tau _u\langle {\boldsymbol {u}\boldsymbol {\cdot }{\boldsymbol \nabla }\times \boldsymbol {u}}\rangle /3$ is related to the mean kinetic helicity $H=\langle {\boldsymbol {u}\boldsymbol {\cdot }{\boldsymbol \nabla }\times \boldsymbol {u}}\rangle$, and $\beta =\tau _u\langle {u^2}\rangle /3$ is the turbulent diffusion in the kinematic regime. Here, $\tau _u$ is the correlation time of the velocity field. At $Rm\gg 1$ as relevant for astrophysical flows, other closure models like the eddy-damped quasi-normal Markovian approximation and the $\tau$-closure give similar results in the kinematic regime (Pouquet, Frisch & Leorat Reference Pouquet, Frisch and Leorat1976; Blackman & Field Reference Blackman and Field2002).

As turbulence is stochastic in nature, the kinetic helicity of the flow also fluctuates in time. To propose a new dynamo mechanism in non-helical turbulence (i.e. when $\alpha$ vanishes on average), Kraichnan (Reference Kraichnan1976) suggested that helicity fluctuations may have a length scale of variation, $\ell _\alpha$, larger than the turbulent scale $\ell _0$, but still smaller than the mean-field scale $L$. The final mean-field equation can be obtained by averaging the mean-field equation (2.2) again over the scale $\ell _\alpha$, i.e. a double-averaging scheme, which yields

(2.4)\begin{equation} \partial_t\langle{\bar{\boldsymbol{B}}}\rangle = (\beta - \alpha^2_{rms}\tau_\alpha)\nabla^2 \langle{\bar{\boldsymbol{B}}}\rangle, \end{equation}

where $\bar {\cdot}$ indicates the second averaging and $\tau _\alpha$ is the correlation time of the $\alpha$ fluctuations (in Kraichnan Reference Kraichnan1976, the terms helicity and $\alpha$ fluctuations are used interchangeably without distinction between them), and $\alpha ^2_{rms}= \overline {\alpha ^2}$ is the average of the square of the $\alpha$-fluctuations. We can define the dynamo number $D_{\!\alpha } = \alpha ^2_{rms}\tau _\alpha /\beta$. When $D_{\!\alpha }>1$, i.e. when there is a negative diffusion in the turbulence, the double-averaged mean magnetic field grows. Thus, $D_{\!\alpha }$ can exceed unity either when the fluctuation strength is stronger or due to the longer correlation time, and hence both these aspects are crucial for determining the dynamo action. However, in the present study, we majorly focus on the duration of the correlation time of $\alpha$-fluctuations as it is essential for the validity of the double-averaged mean-field equation in (2.4). In the shearing-turbulence simulation carried out by Brandenburg et al. (Reference Brandenburg, Rädler, Rheinhardt and Käpylä2008), the total turbulent diffusion coefficient turns out to be positive.

Several studies have investigated the role of zero-mean $\alpha$ fluctuations and shear in explaining the dynamo observed in non-helical shear turbulence, including works by Heinemann et al. (Reference Heinemann, McWilliams and Schekochihin2011), Mitra & Brandenburg (Reference Mitra and Brandenburg2012) and Jingade et al. (Reference Jingade, Singh and Sridhar2018). In all of these models, the success of the approach depended on the existence of the temporal and spatial scale separation between the perceived fluctuations of velocity and $\alpha$, i.e.

(2.5a,b)\begin{equation} \tau_\alpha\gg \tau_u\quad \text{and}\quad \ell_\alpha\gg\ell_0. \end{equation}

In an alternate approach, Jingade & Singh (Reference Jingade and Singh2021) utilized the renovating-flow model (Dittrich et al. Reference Dittrich, Molchanov, Sokolov and Ruzmaikin1984; Sokolov Reference Sokolov1997) to directly solve for (2.1) without explicitly determining the form of the turbulent EMF. The next subsection provides a brief introduction to this approach, and also motivates the need for studying correlation times of individual Fourier modes.

2.2. Renovating-flow model: helicity fluctuations

In the renewing flows, time is divided into intervals $I_n= [n\tau,(n+1)\tau ]$ of length $\tau$ for $n=0,1,2,\ldots\,$. A random flow $\boldsymbol {u}(t,\boldsymbol {x})$ is generated by choosing

(2.6)\begin{equation} \boldsymbol{u}(t,\boldsymbol{x}) = \boldsymbol{u}_\tau(t-n\tau,\boldsymbol{x}), \end{equation}

for each interval $I_n$, where $\boldsymbol {u}_\tau$ is chosen randomly from the ensemble $\varSigma _\tau$ of smooth flows, given by

(2.7)\begin{equation} \varSigma_\tau = \{\boldsymbol{u}_\tau(t,\boldsymbol{x}): 0\leq t< \tau\}. \end{equation}

The velocity fields in each interval are considered to be random and statistically independent realizations of the underlying probability distribution function. The random velocity field of the ensemble $\varSigma _\tau$ is chosen to be a single-scale finite time solution of the Navier–Stokes equation without the Lorentz force (see Jingade & Singh Reference Jingade and Singh2021 for details). The model velocity field $\boldsymbol {u}(t,\boldsymbol {x};S,\boldsymbol {q},h)$ hence depends on a number of parameters: $S$ is a constant shear rate so that the background shear flow is of the form $\boldsymbol {U}^{shear}=Sx\boldsymbol {e}_y$, $\boldsymbol {q} = (q_1,q_2,q_3)$ is the wave vector of $\boldsymbol {u}_\tau$ at initial time and the value of $h$ controls the relative helicity of the flow $\boldsymbol {u}$ which varies in the range $[-1,1]$.

Under the assumption of statistical independence of the velocity field across the intervals, the evolution of the magnetic field equation can be written separately for any given interval $[(n-1)\tau,n\tau ]$ in Fourier space. Specifically, the magnetic field at time $n\tau$ can be written in terms of the initial magnetic field at time $(n-1)\tau$, after averaging over the initial randomness of the magnetic field and the statistical ensemble of the velocity field, as (see Jingade & Singh Reference Jingade and Singh2020 for details)

(2.8)\begin{equation} \boldsymbol{B}(n\tau,\boldsymbol{k}) = \boldsymbol{\mathsf{G}}(\tau,\boldsymbol{k})\boldsymbol{B}_0((n-1)\tau,\boldsymbol{k}), \end{equation}

where $\boldsymbol{\mathsf{G}}$ is a response tensor that describes the average deformation of fluid particle trajectories over the interval $\tau$ due to the velocity field statistics. The response tensor is obtained by neglecting the diffusion term in the induction equation. As a result, the magnetic field at specific realizations exhibits fine structures similar to those at large $Rm$. However, the averaging over the velocity statistics introduces the effective turbulent diffusivity for the growing magnetic field, smoothing the field over the turbulent velocity scale.

In the renovating-flow model presented by Jingade & Singh (Reference Jingade and Singh2021), the idea of helicity fluctuations introduced by Kraichnan (Reference Kraichnan1976) and Sokolov (Reference Sokolov1997) is incorporated. In this model, the helicity of the flow is assumed to take independent and random values in successive time intervals, while the durations of these intervals are fixed. It is also assumed that the renovation time of helicity, $\tau _h$, is an integral multiple of the renovation time of the velocity field, $\tau$, i.e. $\tau _h= m\tau$, where $m \geq 2$ and $\tau$ is the correlation time of the flow. In (2.8), the response tensor is averaged over the statistics of the velocity field, except for the helicity parameter $h$. To incorporate the helicity fluctuation and its time-scale separation in the averaging, the response tensors from adjacent intervals are multiplied together while the helicity is fixed over the $m$ intervals. Later, the product of the response tensor is averaged over the helicity statistics to obtain the double-averaged mean field. Thus, the magnetic field at time $n\tau$ can be expressed in terms of the magnetic field at $(n-m)\tau$ as

(2.9)$$\begin{gather} \boldsymbol{B}(n\tau,\boldsymbol{k}) = {\boldsymbol {\mathcal{G}}}(\tau,\boldsymbol{k})\boldsymbol{B}_0((n-m)\tau,\boldsymbol{k}), \end{gather}$$

(2.10)$$\begin{gather}{\boldsymbol {\mathcal{G}}}(\tau,\boldsymbol{k}) = \langle \boldsymbol{\mathsf{G}}(\tau,\boldsymbol{k}_{n-1})\ldots \boldsymbol{\mathsf{G}}(\tau,\boldsymbol{k}_{n-m}) \rangle_h, \end{gather}$$

where ${\boldsymbol {\mathcal {G}}}(\tau,\boldsymbol {k})$ is the double-averaged response tensor, whose eigenvalues determine the growth rate and cycle period of the dynamo wave (see Jingade & Singh Reference Jingade and Singh2021 for details). It is shown that a non-helical large-scale dynamo can be generated only when the ratio between the time scales is $\geq 3$.

2.3. Sources of helicity fluctuations

In non-helical turbulence, the kinetic helicity fluctuates around zero, and these fluctuations have two primary contributions. The first contribution arises from the non-equivalence of volume averaging and ensemble averaging. In renovating flows, averaging is conducted over the ensemble of the velocity field, strictly adhering to the Reynolds-averaging rule. Conversely, in mean-field theory, averaging is performed over a length scale or a finite volume, leading to an approximate adherence to Reynolds’ rules, as argued by Hoyng (Reference Hoyng1987, Reference Hoyng1988) and Zhou, Blackman & Chamandy (Reference Zhou, Blackman and Chamandy2018). In the work by Hoyng, the resulting error terms manifest as a forcing term in the mean-field equation, giving rise to fluctuations in the transport coefficients, whereas in the work by Zhou et al. (Reference Zhou, Blackman and Chamandy2018), the deviation from Reynolds’ rules introduces additional terms in the mean-field equations, but these appear as spatial gradients of the mean fields. Since volume average is not strictly equivalent to an ensemble average, the volume-averaged mean helicity $H$ fluctuates over time in simulations of non-helical turbulence, although typically with relatively small amplitudes (see figure 1 in the supplementary material available at https://doi.org/10.1017/jfm.2024.1006 for the evolution of the volume-averaged helicity). We denote the correlation time of $H$ as $\tau _H$, which serves as our first analogue to the correlation time of $\alpha$ fluctuations.

Note that the volume-averaged helicity scales as $V^{-1/2}$, where $V$ is the volume of the box, assuming the helicity fluctuations within each eddy remain consistent regardless of the volume. Furthermore, if the helicity fluctuations in each eddy are independent and identically distributed with a uniform correlation time $\tau _0$, then the correlation time of the mean helicity will also be $\tau _0$, independent of the volume. In this study, we examine how the correlation time varies with rotation or shear rate, keeping the volume fixed. The implications of finite volume, such as a scale-dependent dynamo growth rate, could be significant for astrophysical dynamos and will be explored in future work.

The second contribution to helicity fluctuations arises from the intrinsic randomness of turbulence, where the helicity associated with each Fourier mode fluctuates around zero due to the nonlinear interaction of waves. Consequently, our second analogue to the correlation time of helicity fluctuations pertains to individual flow modes rather than volume-averaged quantities. Compared with $\tau _H$, this idea aligns more closely with the renovating-flow model by Jingade & Singh (Reference Jingade and Singh2021), where a single-scale flow is used to investigate the dynamo phenomenon, associating fluctuating helicity at a particular mode $\boldsymbol {q}$ with some correlation time.

However, real turbulence involves multiple scales. The single-scale model of Jingade & Singh (Reference Jingade and Singh2021) can be extended to multi-scale flows by constructing the velocity field from independent non-interacting helical waves, with amplitudes following specific power laws across scales (see Singh & Sridhar Reference Singh and Sridhar2017 for details). In turbulence's steady state, the nonlinear term only cascades energy across scales, leaving the dynamics of individual modes unaffected. This extension justifies our scale-by-scale approach to verifying the dynamo action criteria. Similar methods have been employed by Wilkin, Barenghi & Shukurov (Reference Wilkin, Barenghi and Shukurov2007) and others to study dynamo problems. Consequently, our objective is to determine the correlation times of both velocity and helicity fields across a spectrum of wave vectors $\boldsymbol {k}$ and verify the dynamo criteria scale by scale.

The correlation time of the helicity field is obtained by considering the helicity density in Fourier space, defined as

(2.11)\begin{equation} \tilde g(t,{\boldsymbol{k}})=\tilde{\boldsymbol{u}}^*(t,{\boldsymbol{k}})\boldsymbol{\cdot} {\rm i}{\boldsymbol{k}}\times\tilde{\boldsymbol{u}}(t,{\boldsymbol{k}}), \end{equation}

where $\tilde{\boldsymbol {u}}$ is the Fourier transform of the velocity field. It is important to note that $\tilde g$ is a real quantity and represents the helicity of the velocity field of a particular mode $\boldsymbol {k}$ and at time $t$. We must distinguish $\tilde g$ from the Fourier transform of the helicity density in configuration space, $h=\boldsymbol {u}\boldsymbol {\cdot }{\boldsymbol \nabla }\times \boldsymbol {u}$, which is not used in our analysis.

3.1. Velocity and helicity auto-correlations

In theoretical analysis, ensemble averages are often used to define averages of a quantity in turbulence. However, in simulations, ensembles are often not available, and the turbulent field at different times can be treated as copies of the ensemble, provided the turbulence is stationary. In our simulations, we perform the statistical averaging of the velocity field over time. The equivalence between the ensemble average discussed in the previous section and the temporal average is ensured by the ergodic theorem. The correlation times of velocity and helicity fluctuations are computed through their auto-correlation functions. We define the auto-correlation function of a velocity field $\boldsymbol {u}(t,\boldsymbol {x})$ at the space point location $\boldsymbol {x}$ as

(3.1)\begin{equation} {\mathcal{T}}_{u}(\tau,\boldsymbol{x})\equiv\int_{t_0}^\infty\text{d}t [\boldsymbol{u}(t+\tau,\boldsymbol{x})-\bar{\boldsymbol{u}}(\boldsymbol{x})]\boldsymbol{\cdot} [\boldsymbol{u}(t,\boldsymbol{x})-\bar{\boldsymbol{u}}(\boldsymbol{x})], \end{equation}

where $t_0$ represents the initial time of the simulations when turbulence has become stationary and an overbar denotes a time average. Similarly, the auto-correlation of the Fourier transform of the velocity is defined as

(3.2)\begin{equation} {\mathcal{T}}_{\tilde{u}}(\tau,{\boldsymbol{k}})\equiv\int_{t_0}^\infty\text{d}t [\tilde{\boldsymbol{u}}(t+\tau,{\boldsymbol{k}})-\bar{\tilde{\boldsymbol{u}}}({\boldsymbol{k}})]\boldsymbol{\cdot} [\tilde{\boldsymbol{u}}(t,{\boldsymbol{k}})-\bar{\tilde{\boldsymbol{u}}}({\boldsymbol{k}})]^*, \end{equation}

where the asterisk indicates the complex conjugate.

We also define the auto-correlations of the mean helicity as

(3.3)\begin{equation} {\mathcal{T}}_{H}(\tau) = \int_{t_0}^\infty\text{d}t [H(t+\tau)-\bar{H}]\boldsymbol{\cdot} [H(t)-\bar{H}], \end{equation}

where $H(t) = \int \mathrm {d}^3\kern0.7pt x \boldsymbol {u}(t,\boldsymbol {x})\boldsymbol {\cdot } {\boldsymbol {\omega }}(t,\boldsymbol {x}) = \int \mathrm {d}^3 k \tilde {g}(t,\boldsymbol {k})$, and $\boldsymbol {\omega }={\boldsymbol \nabla }\times \boldsymbol {u}$ is the vorticity field. Here, $H$ has a vanishing ensemble-average mean, but in simulations they fluctuate around zero, as discussed in § 2.3. The auto-correlation of the helicity density $\tilde {g}(\boldsymbol {k}, t)$ in the Fourier space is

(3.4)\begin{equation} {\mathcal{T}}_{\tilde{g}}(\tau,{\boldsymbol{k}})\equiv\int_{t_0}^\infty\text{d}t [\tilde{g}(t+\tau,{\boldsymbol{k}})-\bar{\tilde{g}}({\boldsymbol{k}})]\boldsymbol{\cdot} [\tilde{g}(t,{\boldsymbol{k}})-\bar{\tilde{g}}({\boldsymbol{k}})]^*. \end{equation}

Finally, we define the volume-integrated correlation function as

(3.5)\begin{equation} C_{u}^{}(t)=\int \text{d}^3\kern0.7pt x {\mathcal{T}}_{u}(t,\boldsymbol{x}) = \int \frac{\text{d}^3k}{(2{\rm \pi} )^3} {\mathcal{T}}_{\tilde{u}}(t,\boldsymbol{k}). \end{equation}

The second equality is due to Parseval's theorem.

Figure 1 displays three typical auto-correlation curves of run R3 (further details can be found in § 4). The correlation time of an auto-correlation curve is determined by fitting its initial positive part with an exponentially decaying curve and identifying the time at which its value reaches $1/e$ of its maximum. We denote the correlation time of ${\mathcal {T}}_I$ by $\tau _I$, where $I\in \{u,\tilde {u}, g, \tilde {g}, H\}$.

Figure 1.

Three typical auto-correlation curves for run R3, demonstrating how the correlation times of the velocity and helicity fluctuations vary with the wavenumbers parallel ($k_\parallel$) and perpendicular ($k_\perp$) to the rotation axis.

3.2. Distinguishing different helicity time scales

A volume-integrated correlation function for the helicity modes is defined by integrating the helicity auto-correlation function over the wave vector space, similar to (3.5)

(3.6)\begin{equation} C_{g}^{}(t)=C_{\tilde g}^{}(t)=\int\frac{\text{d}^3k}{(2{\rm \pi} )^3} {\mathcal{T}}_{\tilde{g}}(t,{\boldsymbol{k}}). \end{equation}

It should be noted that, although one might expect the correlation times of $C_{g}^{}$ and $C_H$ to be similar, this is not necessarily the case. To see the reason, we assume that $\tilde g({\boldsymbol {k}})$ has a vanishing mean for the moment and note that

(3.7)$$\begin{gather} C_H=\int\text{d}t'\int\frac{\text{d}^3k_1}{(2{\rm \pi} )^3} \tilde g(t',\boldsymbol{k}_1) \int\frac{\text{d}^3k_2}{(2{\rm \pi} )^3} \tilde g(t'+t,\boldsymbol{k}_2), \end{gather}$$

(3.8)$$\begin{gather}C_{g}^{}=\int\frac{\text{d}^3k}{(2{\rm \pi} )^3}\int \text{d}t' \tilde g(t',\boldsymbol{k})\tilde g(t'+t,\boldsymbol{k}). \end{gather}$$

It is clear from (3.7) and (3.8) that $C_H$ is the auto-correlation of the sum of all modes, whereas $C_{g}^{}$ is proportional to the mean of the auto-correlations over all modes.

Both $\tau _g$ and $\tau _H$ have been utilized in dynamo models, each with its own significance. In the original negative diffusion model proposed by Kraichnan (Reference Kraichnan1976), the time scale for fluctuations of the $\alpha$ coefficient was closely related to $\tau _H$. Similarly, stochastic $\alpha$ dynamo models, such as those by Vishniac & Brandenburg (Reference Vishniac and Brandenburg1997) and Richardson & Proctor (Reference Richardson and Proctor2012), also adopt a time scale associated with $\tau _H$. On the other hand, researchers studying turbulence using random waves, as exemplified by Jingade & Singh (Reference Jingade and Singh2021), focus on the damping time of a single helical wave and therefore utilize $\tau _g$. In our study, we present results for both $\tau _g$ and $\tau _H$, recognizing the importance of considering both perspectives.

3.3. Numerical set-ups and methods

Focusing on the kinematic dynamo regime and neglecting Lorentz forces, we perform compressible isothermal hydrodynamic simulations of isotropically and non-helically forced turbulence using the publicly available Pencil Code (Pencil Code Collaboration et al. 2021). The equations to be solved are

(3.9)$$\begin{gather} \text{D}_t\ln\rho={-}{\boldsymbol\nabla}\boldsymbol{\cdot}\boldsymbol{u}, \end{gather}$$

(3.10)$$\begin{gather}\text{D}_t\boldsymbol{u}=Su_x\boldsymbol{e}_y-c_\text{S}^2{\boldsymbol\nabla}\ln\rho-2\boldsymbol{\varOmega}\times\boldsymbol{u} +\frac{1}{\rho}{\boldsymbol\nabla}\boldsymbol{\cdot}(2\rho\nu\boldsymbol{\mathsf{S}})+\boldsymbol{f}, \end{gather}$$

where $\text {D}_t=\partial _t+\boldsymbol {u}\boldsymbol {\cdot }{\boldsymbol \nabla }+Sx\partial _y$, $S$ is the shear rate of the shear flow $\boldsymbol {U}^{shear}=Sx\boldsymbol {e}_y$, $c_\text {S}$ is a constant sound speed, $\nu$ is the viscosity and ${\mathsf{S}}_{ij}=(\partial _i u_j+\partial _j u_i)/2-\delta _{ij}{\boldsymbol \nabla }\boldsymbol {\cdot }\boldsymbol {u}/3$ is the rate-of-strain tensor. In the burgulence cases, the density equation (3.9) is dropped and the second term on the right-hand side of (3.10) is set identically to zero, turning the Navier–Stokes equations into Burgers’ equation. The rotation rate $\boldsymbol {\varOmega } = \varOmega \boldsymbol {\boldsymbol {e}_{z}}$ is applied along the $z$-direction, and the random force $\boldsymbol {f}$ takes the form of a non-helical plane wave (see Brandenburg Reference Brandenburg2001 for details). The forcing wave vector has a fixed magnitude $k_{f}$, but its phases and directions change at each time step. The simulation box has dimensions $L^3 = (2{\rm \pi} )^3$ with periodic boundary conditions in all the directions or shear-periodic (in the case of shear) boundary conditions in the $x$-direction. In all runs, the forcing wavenumber is $k_{f} = 5$, and the Mach number is approximately $0.1$. A summary of the runs is provided in table 1, which includes the Reynolds number $Re = u_{rms}/(\nu k_{f})$, the Coriolis number $Co = 2\varOmega /(u_{rms} k_{f})$ and the dimensionless shear number $Sh=-S/(u_{rms} k_{f})$. For the rotating turbulence cases R1 to R5, we compute the Coriolis number by excluding the $k< k_{f}$ modes when computing the root-mean-square velocity $u_{rms}$ (see § 4). We denote the Coriolis number computed using this definition of $u_{rms}$ as ${Co}_{f}$. For runs R3 to R5, their Coriolis numbers are sufficiently large to allow for condensation of large-scale vortices, and therefore their Reynolds numbers are apparently higher than those of runs R1 to R2. At even higher values of ${Co}_{f}$, the flow becomes quasi two-dimensional and dynamos will not be supported (Cowling Reference Cowling1933). Therefore, we exclude the exploration of high Coriolis numbers.

Table 1.

A summary of runs, where $N^3$ is the resolution, $Ma=u_{rms}/c_{S}$ is the Mach number, $S$ is the shear rate, $\varOmega$ is the rotation rate, $Re=u_{rms}/\nu k_{f}$ is the Reynolds number, $Co=2\varOmega /u_{rms} k_{f}$ is the Coriolis number, $Sh=-S/u_{rms} k_{f}$ is the dimensionless shear rate and $k_{ze}$ is the Zeman wavenumber. For rotation runs R1 to R5, the root-mean-squared velocity used for computing the Coriolis number excludes large-scale modes with $k< k_{f}$, and therefore their Coriolis numbers will be denoted by ${Co}_{f}$ rather than $Co$. For these runs the Zeman wavenumber $k_{ze}$ is also listed. The runs SB1 to SB4 solve the Burgers’ equation and the Mach number is undefined.

In this study, particular attention is given to ensuring the hydrodynamic stability of the flow in all simulations. The choice of shear rate and rotation rate is made such that the flow remains stable throughout the simulations. This criterion for stability, $-\infty < {-S}/{\varOmega } < 2$, as discussed in Salhi (Reference Salhi2002) and Balbus & Hawley (Reference Balbus and Hawley2006), has been taken into consideration. The presence of instabilities in the flow could lead to vorticity dynamos (Käpylä, Mitra & Brandenburg Reference Käpylä, Mitra and Brandenburg2009), i.e. the generation of additional mean flows. Such mean flows would render the system non-ergodic. To ensure the validity of measurements that require time stationarity such as the temporal average of the correlation functions, it is essential to prevent such instabilities from affecting the measurements.

We use two methods to generate auto-correlation curves of the velocity and helicity fluctuations. In the first method, we output the time series data for $\tilde{\boldsymbol {u}}(t,{\boldsymbol {k}})$ from the simulations and calculate auto-correlation curves during post-processing. The duration of time series for all runs spans at least $200$ times the eddy turnover time $1/(u_{rms} k_{f})$ with a sampling frequency of no less than $10u_{rms} k_{f}$. For runs with the highest shear rates (S4, S5, SB4, K4 and K5), the sampling frequency is increased to a minimum of $20u_{rms} k_{f}$. This approach allows for obtaining auto-correlations for individual Fourier modes. As large-scale flow patterns such as shear or rotation primarily influence the larger scales of the flow, we have limited the output to the range $-2k_{f}\leq k_{x,y,z}\leq 2k_{f}$, which has been confirmed to encompass the energy-dominant modes.

The first method needs high data output frequency and therefore is only applied to a limited range of wave modes. In the second method, correlation functions are computed on the fly and for all the wave modes, although this approach results in a smaller sample size for time averaging. The method is specifically used to obtain shell-averaged correlation functions in the Fourier space. To implement this, we regularly update an time-independent auxiliary field $\tilde {\boldsymbol {v}}({\boldsymbol {k}})$ to match $\tilde{\boldsymbol {u}}(t,{\boldsymbol {k}})$ at specific time intervals (every $\Delta t=50$ in code units, roughly $27$ times the eddy turnover time of the isotropic run). The field $\tilde {\boldsymbol {v}}({\boldsymbol {k}})$ is kept fixed between updates. At a given update time $t_0$, we set $\tilde {\boldsymbol {v}}_0({\boldsymbol {k}})$ to be the value of $\tilde {\boldsymbol {u}}(t_0, {\boldsymbol {k}})$. During the interval $t_0 \leq t < t_0 + \Delta t$, the velocity auto-correlation between $\tilde {\boldsymbol {v}}_0({\boldsymbol {k}})$ and $\tilde {\boldsymbol {u}}(t, {\boldsymbol {k}})$ is computed as

(3.11)\begin{equation} {\mathcal{T}}_{\tilde u}(t,{\boldsymbol{k}})= \tilde{\boldsymbol{v}}^*_0({\boldsymbol{k}})\boldsymbol{\cdot}\tilde{\boldsymbol{u}}(t_0+t,{\boldsymbol{k}})= \tilde{\boldsymbol{u}}^*(t_0,{\boldsymbol{k}})\boldsymbol{\cdot}\tilde{\boldsymbol{u}}(t_0+t,{\boldsymbol{k}}), \quad 0\leq t\leq \Delta t. \end{equation}

Similarly, the helicity correlation ${\mathcal {T}}_{\tilde g}$ is computed. These correlation curves are then averaged at discrete intervals.

While this method allows us to analyse modes at the largest wavenumbers, it limits the number of averaged ensemble members to $T / \Delta t$, where $T$ is the overall time duration of the simulation.

In figure 2, we present correlation times for shell-averaged quantities in isotropic non-helical turbulence. The shell-averaged correlation functions are defined by

(3.12)\begin{equation} C_{\tilde u,\tilde g}(t,k_r)=\frac{1}{4{\rm \pi} }\int \sin\theta\,\text{d}\theta\,\text{d}\phi {\mathcal{T}}_{\tilde u,\tilde g}(t,{\boldsymbol{k}}), \end{equation}

where $(k_r,\theta,\phi )$ are the spherical coordinates in Fourier space. Subsequently, the correlation times for each wavenumber are fitted as $\tau _{\tilde u}^{1D}(k_r)$ and $\tau _{\tilde g}^{1D}(k_r)$ for the velocity and helicity auto-correlations, respectively. The identified $k_r^{-1}$ power law for $\tau _{\tilde u}^{1D}$ in figure 2 suggests that the Eulerian correlation time is primarily influenced by the sweeping effect, where small-scale eddies are transported by larger-scale eddies. Consequently, $\tau _{\tilde u}^{1D}\propto 1/(u_{rms} k_r)$, aligning with findings from prior theoretical and numerical studies (Chen & Kraichnan Reference Chen and Kraichnan1989; Sanada & Shanmugasundaram Reference Sanada and Shanmugasundaram1992; Favier, Godeferd & Cambon Reference Favier, Godeferd and Cambon2010; Clark Di Leoni, Cobelli & Mininni Reference Clark Di Leoni, Cobelli and Mininni2015).

Figure 2.

For run NH, correlation times of shell-averaged auto-correlations for velocity (solid) and helicity (dashed) fields. The vertical line indicates the forcing wavenumber at $k_{f}=5$.

We have also computed the correlation time of the volume-integrated velocity or helicity correlation, along with that of the total helicity using (3.3), (3.5) and (3.6). The obtained values are

(3.13a–c)\begin{equation} \tau_uu_{rms} k_{f}=3.11,\quad \tau_H/\tau_u=2.42,\quad \tau_g /\tau_u=0.61. \end{equation}

These values serve as a set of fiducial references, which we will compare with the results obtained when rotation or shear is introduced. Brandenburg & Subramanian (Reference Brandenburg and Subramanian2005b) previously reported a Strouhal number $St=\tau _uu_{rms} k_{f}$ around unity for the $\boldsymbol {u}\boldsymbol {\cdot }\boldsymbol {\omega }$ correlation, determined by dividing the $\alpha$ coefficient from a test-field method by one third of the mean kinetic helicity. Our value of $3.11$ is not fundamentally at odds with this, mainly due to the different methods and, consequently, the different definitions of $\tau _u$ employed.

In the next two sections, we present the measured correlation times for velocity and helicity fluctuations in rotating and shearing turbulence, respectively. A more detailed discussion of the implications for the shear dynamo problem along with the comparison with previous simulation results is provided in § 6.

5.1. Shearing box and shearing-frame transformation

The shearing box approximation is commonly used in numerical simulations to study a small patch in a differentially rotating flow whose scale is much smaller than the local curvature of the rotating flow (Goldreich & Lynden-Bell Reference Goldreich and Lynden-Bell1965). When considering mean linear shear flows imposed on the turbulent velocity field, the resulting turbulence exhibits inhomogeneity and anisotropy when observed in the laboratory-frame coordinates. In particular, fluid parcels are advected by the background shear flow, making the same-point two-time correlation (3.1) ${\mathcal {T}}_u(\tau,\boldsymbol {X})$ have a shorter correlation time towards the boundary of the box $X=\pm {\rm \pi}$; see the dashed curve in figure 9.

Figure 9.

For run S1, correlation times of $YZ$-averaged velocity auto-correlations in the laboratory frame (dashed) and the shearing frame (solid). Spatial homogeneity is restored in the latter case.

Such decoherence effect can be removed by exploiting a fundamental symmetry of shearing flows. An observer co-moving with the mean flow velocity $SX\boldsymbol {e_Y}$ at location $\boldsymbol {X}$ perceives the same turbulent velocity field as an observer in the laboratory frame at the origin. This principle, known as Galilean invariance, ensures the homogeneity of the velocity field in the shearing coordinates $(x,y,z)$ given by (Sridhar & Subramanian Reference Sridhar and Subramanian2009a,Reference Sridhar and Subramanianb)

(5.1a–c)\begin{equation} x=X; \quad y = Y-St\,X;\quad z = Z. \end{equation}

The general Galilean-invariant time-stationary velocity correlation can be written in the laboratory frame using the symmetry property of shear flow as (see Appendix A)

(5.2)\begin{equation} \overline{\boldsymbol{u}^{lab}(t,\boldsymbol{X})\boldsymbol{\cdot} \boldsymbol{u}^{lab}(t',\boldsymbol{X}')} ={\mathcal{T}}_{u,{lab}}(\boldsymbol{X}-\boldsymbol{X}'-S(t-t')X\boldsymbol{e_Y},t-t'), \end{equation}

where overhead bar denotes the time average in our case. This correlation is time stationary but exhibits spatial inhomogeneity along the $X$-direction. To address this, we can transform the correlator into the shearing frame, where it becomes spatially homogeneous but loses time stationarity. The latter can only be restored in two special slices in the Fourier space (see below for details). The solid curve in figure 9 shows the velocity correlation time of the $yz$-averaged correlators, which restores both spatial homogeneity and time stationarity.

An additional advantage of using the shearing frame is the restoration of normal periodic boundary conditions, allowing for a Fourier transform in the shearwise direction $\hat {\boldsymbol {x}}$. Fourier modes in the shearing frame, denoted as $(k_x,k_y,k_z)$, are equivalent to shearing-wave modes with wave vectors $\boldsymbol {K}=(k_x-St\,k_y,k_y,k_z)$ in the laboratory frame. It is important to note that there would be no difference in the correlation time of the total kinetic helicity, denoted as $\tau _H$, between the two frames. This is due to the fact that $H(t)$, being a volume-averaged quantity, remains constant regardless of the frame of reference.

Without further notice, we shall always refer to shearing-frame coordinates by using $(t,x,y,z)$ in the configuration space and $(k_x,k_y,k_z)$ in the Fourier space. The helicity density at the shearing-wave vector ${\boldsymbol {k}}$ is given by $\tilde g(\boldsymbol {k})=\tilde{\boldsymbol {u}}^*(\boldsymbol {k})\boldsymbol {\cdot }\tilde {\boldsymbol {\omega }}(\boldsymbol {k})$, where the vorticity field $\tilde {\boldsymbol {\omega }} = \boldsymbol {K}(\boldsymbol {k},t)\times \tilde {\boldsymbol {u}}$ is obtained by first taking the curl of $\boldsymbol {u}$ in the laboratory frame, followed by the shearing-frame transformation, and then performing the Fourier transformation.

Sacrificing time stationarity in the Fourier space is unavoidable when reintroducing spatial homogeneity in the shearing frame (see Appendix A). In other words, the unequal-time auto-correlation function $\langle {\tilde {\boldsymbol {u}}(t+t',{\boldsymbol {k}})\boldsymbol {\cdot }\tilde {\boldsymbol {u}}(t',{\boldsymbol {k}})}\rangle$ not only depends on the time difference $t$ but also on $t'$. This outcome is a general consequence of Galilean invariance (Sridhar & Singh Reference Sridhar and Singh2014). In Appendix A, it is demonstrated that the correlation functions ${\mathcal {T}}_{\tilde u,\tilde g}(t,t',\boldsymbol {k})$ can maintain stationarity on two two-dimensional slices in Fourier space, i.e.

(5.3)\begin{equation} C_{\tilde u,\tilde g}^{{I}_x}(t,k_y,k_z)\equiv \int \frac{\text{d}k_x}{2{\rm \pi} } {\mathcal{T}}_{\tilde u,\tilde g}(t,t',\boldsymbol{k}), \end{equation}

and

(5.4)\begin{equation} C_{\tilde u,\tilde g}^{xz}(t,k_x,k_z)\equiv {\mathcal{T}}_{\tilde u,\tilde g}(t,t',k_x,k_y=0,k_z). \end{equation}

Since inverting $x\to -x$ or $y\to -y$ is equivalent to $S\to -S$ for the correlation functions, $C_{\tilde u,\tilde g}^{{I}_x,xz}$ will be invariant under $(k_x,k_y)\to (-k_x,-k_y)$, and so are the corresponding correlation times. We can thus focus ourselves to half of the $(k_x,k_y)$ plane. Furthermore, the quadratic functions $C_{\tilde u,\tilde g}^{{I}_x,xz}$ are also invariant under $k_z\to -k_z$. We therefore only show the results in the $k_{x,y,z}\geq 0$ planes.

In the upcoming subsection, we present $\tau _{\tilde u,\tilde g}^{{I}_x}$ and their corresponding ratios on the ($k_y$, $k_z$) plane. The second quantity (5.4) exclusively involves axisymmetric modes ($k_y=0$) and, as per the anti-dynamo theorem (Cowling Reference Cowling1933; Zel'dovich Reference Zel'dovich1956), cannot solely induce a dynamo. For a comprehensive view, the $k_y=0$ slices (5.4) and the energy densities for both velocity and helicity (corresponding to both quantities (5.3) and (5.4) when $t=0$) are presented in the supplementary material.

5.2. Helicity probability distributions

In figure 10 we show $10$ typical PDF curves of the normalized helicity modes $\tilde g(\boldsymbol {k})/k|\tilde u|^2$ for each of the runs. At large enough shear rate (runs S4 and S5) and low wavenumbers ($k< k_{f}$), the PDFs are typically asymmetric with respect to zero, possibly caused by the shear flow. However, the asymmetry is different to those appear in the low-wavenumber modes in purely rotating cases (cf. figure 7). For the rotating cases, helicity fluctuations appear to be small (i.e. the $\tilde g(\boldsymbol {k})/k|\tilde u|^2$ values are close to zero), with a slight shift of the mean deviating from zero. For the shearing cases, helicity fluctuations with one particular sign are more favoured. In this sense, low-wavenumber modes ($k< k_{f}$) in shearing turbulence would have stronger helicity fluctuations than those in the rotating turbulence. At large wavenumbers ($k>k_{f}$), the PDFs becomes symmetric about zero, and runs with lower shear rates (S1 and S2) exhibit flat PDFs, whereas at higher shear rates (S3 to S5), helicity fluctuations demonstrate less extreme values at $\tilde g/k|\tilde u|^2=\pm 1$, suggesting a decrease in the occurrence of maximally helical fluctuations. This is in contrast to the rotating cases (figure 7) where high-wavenumber modes still exhibit flat PDFs at large rotation rates.

Figure 10.

The PDFs of the normalized helicity $\tilde g/k|\tilde u|^2$ for a few modes for the shearing-turbulence runs.

5.4. Shearing burgulence

Burgers’ equation is often used as a prototype equation to study turbulence, as it simplifies the Navier–Stokes equations by excluding the pressure gradient term. The form of Burgers’ equation that we used to simulate the Burgulence counterpart of the shearing turbulence is given as

(5.5)\begin{equation} \text{D}_t\boldsymbol{u}= Su_x\boldsymbol{e}_y-2\boldsymbol{\varOmega}\times\boldsymbol{u} +\frac{1}{\rho}{\boldsymbol\nabla}\boldsymbol{\cdot}(2\rho\nu\boldsymbol{S})+\boldsymbol{f}, \end{equation}

where $\boldsymbol {u}$ represents the velocity field, $S$ is the shear rate and $\nu$ is the kinematic viscosity. This equation captures essential features of turbulence, such as shock formation and energy dissipation, making it a useful tool for theoretical and numerical studies of turbulent flows.

Theories employing either the second-order correlation approximation or the $\tau$ approximation closures in the shear-current effect argue for the essential role of the pressure gradient term in the Navier–Stokes equation (Squire & Bhattacharjee Reference Squire and Bhattacharjee2015b; Zhou & Blackman Reference Zhou and Blackman2021). Without this term, the required turbulent transport coefficient becomes identically zero. Recent simulations demonstrating shear dynamos in burgulence (Käpylä et al. Reference Käpylä, Rheinhardt and Brandenburg2022) seemingly challenge the shear-current effect as the primary driver in this context. However, this does not automatically confirm the incoherent $\alpha$ or helicity fluctuations as the correct explanation. The theoretical and numerical understanding of the role played by the pressure gradient term in the incoherent $\alpha$ effect is limited, and whether it persists in burgulence remains an open question. In this subsection, we present evidence that the helicity correlation weakens in burgulence, posing a challenge to the efficiency of the incoherent $\alpha$ effect in this scenario.

To avoid simulation crashes in burgulence runs, it was necessary to increase the viscosity (refer to table 1). This adjustment led to slightly lower Reynolds numbers compared with their counterparts in the Navier–Stokes simulations S1 to S5, but still comparable to those in Käpylä et al. (Reference Käpylä, Rheinhardt and Brandenburg2022).

The correlation times of velocity and helicity in the ($k_y$, $k_z$) plane for shear-burgulence simulations are depicted in figure 13. Comparing the first two rows with their hydrodynamics counterparts in figure 11, we observe that both correlation times decrease more strongly with increasing $Sh$ in the case of burgulence, especially in the low-wavenumber ($k< k_{f}$) region. The resulting ratio of the time scales (the third row) shows no substantial scale separation, similar to the shearing hydrodynamic turbulence. The correlation times of volume-averaged correlation functions are presented in figure 14. With increasing $Sh$, $\tau _u$ decreases at a rate similar to the hydrodynamic case, see figure 12. However, both helicity correlation times $\tau _{H,g}$ are suppressed more strongly, resulting in lower magnitudes. This leads to much smaller time scale ratios $\tau _{H,g}/\tau _u$ at $Sh \gtrsim 1$.

Figure 13.

For shearing burgulence, correlation times measured from $k_x$-integrated velocity and helicity correlations in the $(k_y,k_z)$ plane and normalized by the eddy turnover time (a–d and e–h, $\log _{10}$ scale), and their ratios (i–h, linear scale).

Figure 14.

Volume-averaged velocity and helicity correlation times and their ratios for the shearing burgulence cases. The dashed blue and dashed red lines are the fitted power-law relations for $\tau _H/\tau _u$ and $\tau _g/\tau _u$, respectively.

5.5. Keplerian runs

For Keplerian runs, we choose the value $q=S/\varOmega =3/2$ resulting in a Coriolis number of $Co=4Sh/3$. This case is more representative of astrophysical scenarios, such as in galaxies and accretion disks, where $q$ can vary from $1$ to $2$. The methodology applied for these runs remains consistent with the shear runs described in the previous subsection.

In figure 15, Fourier space results reveal a more distinct scale separation compared with both the shearing turbulence (figure 11) and shearing burgulence (figure 13) cases. This enhanced separation is attributed to the additional rotation, known to form condensate through inverse cascade producing large-scale vortices. This time-scale separation persists despite the elongation effect of the vortices in the direction of the shear. The volume-integrated time scales shown in figure 16 also display a larger-scale separation: the magnitudes of both ratios, $\tau _{H,g}/\tau _u$ are similar to the shearing cases (cf. figure 12) at $Sh=0.5$. However, their scalings with $Sh$ become much stronger, with $\tau _H/\tau _u\propto Sh^{0.96}$ and $\tau _g/\tau _u\propto Sh^{1.15}$. Interestingly, these power-law indices are larger than either purely rotating ($\propto {Co}_{f}^{0.85}$ and $\propto {Co}_{f}^{0.09}$, respectively) or purely shearing (with weak counter-rotation, $\propto Sh ^{0.63}$ and $\propto Sh ^{0.38}$, respectively) cases, revealing a constructive collaboration of the rotation and the shear flow.

Figure 15.

For Keplerian turbulence, correlation times measured from the $k_x$-integrated velocity and helicity correlations in the $(k_y,k_z)$ plane and normalized by the eddy turnover time (a–e and f–j, $\log _{10}$ scale), and their ratios (k–o, linear scale).

Figure 16.

For weakly compressible Keplerian turbulence, volume-averaged velocity and helicity correlation times and their ratios. The dashed blue and dashed red lines are the fitted power-law relations for $\tau _H/\tau _u$ and $\tau _g/\tau _u$, respectively.

In these Keplerian runs, we also find that the dynamo number $D_{H,g}$ (see (4.5)) is less than $1.5\times 10^{-3}$, suggesting that the strength of helicity fluctuations is relatively weak. It is important to note that this strength depends on the Mach number of the flows, which in our simulations is around $0.1$. The strength of these helicity fluctuations could increase with higher Mach numbers. Therefore, further exploration of this effectiveness on the dynamo mechanism in different flow regimes, such as transonic and supersonic flows more relevant to astrophysical scenarios, would be of significant interest. However, such an investigation is beyond the scope of the current work.