# Small-scale dynamo with finite correlation times

Yann Carteret<EMAIL_ADDRESS>Laboratoire d’Astrophysique, EPFL,
CH-1290 Sauverny, Switzerland
Dominik Schleicher<EMAIL_ADDRESS>Departamento de Astronomía,
Universidad de Concepción, Casilla 160-C, Concepción, Chile
Jennifer Schober<EMAIL_ADDRESS>Laboratoire d’Astrophysique, EPFL,
CH-1290 Sauverny, Switzerland

###### Abstract

Fluctuation dynamos occur in most turbulent plasmas in astrophysics and are
the prime candidates for amplifying and maintaining cosmic magnetic fields. A
few analytical models exist to describe their behaviour but they are based on
simplifying assumptions. For instance the well-known Kazantsev model assumes
an incompressible flow that is delta-correlated in time. However, these
assumptions can break down in the interstellar medium as it is highly
compressible and the velocity field has a finite correlation time. Using the
renewing flow method developed by Bhat and Subramanian (2014), we aim to
extend Kazantsev’s results to a more general class of turbulent flows. The
cumulative effect of both compressibility and finite correlation time over the
Kazantsev spectrum is studied analytically. We derive an equation for the
longitudinal two-point magnetic correlation function in real space to first
order in the correlation time $\tau$ and for an arbitrary degree of
compressibility (DOC). This generalised Kazantsev equation encapsulates the
original Kazantsev equation. In the limit of small Strouhal numbers
$St\propto\tau$ we use the WKB approximation to derive the growth rate and
scaling of the magnetic power spectrum. We find the result that the Kazantsev
spectrum is preserved, i.e. $M_{k}(k)\sim k^{3/2}$. The growth rate is also
negligibly affected by the finite correlation time; however, it is reduced by
the finite magnetic diffusivity, and the DOC together.

††preprint: APS/123-QED

##  I Introduction

The vast majority of the baryonic matter is in a plasma state, and therefore a
complete description of the Universe needs to include a proper treatment of
the electromagnetic force [1]. From observations it is known that the Universe
is highly magnetised. Indeed, magnetic fields are observed in almost all
astrophysical bodies as for instance in asteroids [2], planets [3], stars [4,
5], galaxies [6, 7] or the intergalactic medium [8, 9, 10]. Due to the broad
range of objects, the typical strength and correlation length of these
magnetic fields are distributed over several orders of magnitude. As an
example in Milky-Way like galaxies, the observed magnetic fields are of a few
tens $\mu$G in strength and correlated on kilo-parsec scales [11].

The most popular mechanism to explain the observed magnetic fields is the
dynamo process which converts the kinetic energy of the flow to magnetic
energy. In the absence of large-scale motions, small-scale or fluctuation
dynamos 111The terms “small-scale dynamo” and “fluctuation dynamo” are used
interchangeably in the literature. amplify the initial magnetic field
exponentially [13]; a process which is most efficient on the smallest scales
of the system. In the kinematic stage of the dynamo, the magnetic field lines
are frozen into the plasma. Due to the turbulent motion of the flow, the
action of the small-scale dynamo is to randomly twist, stretch, and fold these
lines which makes the magnetic field strength grow. However, activating the
dynamo requires an already existing seed field. Although unclear, it is
generally assumed that these seed fields were generated in the early Universe
[14] or through astrophysical processes such as the Biermann battery [15].
Schober  _et al._ [16] also highlighted that the small-scale dynamo can only
amplify the magnetic field for magnetic Reynolds numbers $R_{\mathrm{M}}\sim
UL/\eta$ ($U$ and $L$ are respectively the typical velocity and length scale
of the system) larger than a few hundred. In the non-linear regime after
saturation on the smallest scales, the peak of the magnetic energy shifts from
smaller to larger scales and the magnetic energy increases following a power-
law [17]. The exact behaviour of the dynamo depends on the magnetic Prandtl
number $P_{\mathrm{M}}=\nu/\eta$ and on the type of turbulence [18, 17, 19].

The small-scale dynamo is a key process in astrophysics. Indeed, the strength
of the magnetic fields predicted from the early Universe is not consistent
with the observed typical value of a few $\mu$G in the inter-cluster medium
[20, and references therein] or in high redshift galaxies [21]. Small-scale
dynamos could then provide an explanation for the fast amplification of
magnetic fields in the radiation-dominated phase of the early Universe [22],
in young galaxies [23], and galaxy clusters [13] as they can act on time-
scales much shorter than the age of the system. In the context of supernova-
driven turbulence, it is expected to give rise to the far-infrared-radio
correlation in galaxies [24] and potentially even dwarf galaxies [25]. Small-
scale dynamos might also be involved in the formation of the first stars [26,
27, 28] and black holes [29, 30, 31]; and thus could also affect the epoch of
reionization.

Different approaches have been published along the years in order to model the
complex behaviour of dynamos. For instance Adzhemyan  _et al._ [32] discuss
the turbulent dynamo in the framework of a quantum-field formulation of
stochastic magnetohydrodynamics, where it is described as a mechanism of
spontaneous symmetry breaking. An early theoretical description of the small-
scale dynamo is given by Kazantsev [33]. His equation describes the time
evolution of the two-point magnetic correlation function under the assumption
of a Gaussian incompressible flow that is delta-correlated in time. Its
derivation indicates that the magnetic power spectrum scales as $M_{k}(k)\sim
k^{3/2}$ for $q\ll k\ll k_{\eta}$, where $k_{\eta}$ is the wavenumber above
which diffusion of the magnetic field dominates. Following Kazantsev’s work
many authors have tried to extend this model [see e.g. 34, 35, 36, 37].
Although some astrophysical objects host plasma that is well described by an
incompressible flow (as for instance neutron stars [38]); Kazantsev’s
assumptions strongly simplify the behaviour of most astrophysical bodies.
Indeed the majority of the plasma in the Universe is highly compressible as
indicated by observations of compressive interstellar turbulence [39].
Moreover, in realistic flows the correlation time $\tau$ should be of the
order of the smallest eddy turnover time. Thus the assumptions involved in the
Kazantsev [33] derivation do not allow for an accurate description of all
types of fluctuation dynamos.

In this work we aim to study the small-scale dynamo for the general case of a
flow that is compressible and with finite correlations in time. Zeldovich  _et
al._ [40] pointed out that the so-called renovating flows represent a solvable
analytical model to study the impact of the correlation time on small-scale
dynamos. In this context, Bhat and Subramanian [37] developed a method to
study the dynamo of incompressible flows. They found that the Kazantsev
spectrum was not strongly affected by a finite correlation time, i.e.
$M_{k}(k)\sim k^{3/2}$. However, the growth rate of the dynamo is reduced. On
the other hand, Schekochihin  _et al._ [41] found that a compressible flow
that is delta-correlated in time also preserves the Kazantsev spectrum where
compressibility also reduces the growth rate of the dynamo. As far as we know,
although there are clues that the Kazantsev spectrum should be preserved in
the interstellar medium (compressible and correlated in time flow), there is
no previous theoretical study that demonstrates formally that the combined
actions have no effect on the $M_{k}(k)\sim k^{3/2}$ spectrum. Rogachevskii
and Kleeorin [42] used a path integral method to solve the induction equation
and show that a dynamo can be activated for compressible flows that are
correlated in time. Their results admit solutions consistent with the
Kazantsev spectrum.

The present work assumes a simplified random flow that is compressible and
correlated in time. We present here a generalisation of the previous work by
Bhat and Subramanian [37] by including the effect of compressibility. The
paper is organised as follows: in Sec. II we briefly review the original
Kazantsev theory. In Sec. III we present the renewing flow method used by Bhat
and Subramanian [37]. In Sec. IV we give the derivation of the original
Kazantsev equation (incompressible and delta-correlated in time flow) with the
use of the renovating flow method. In Sec. V we present our generalisation of
the Kazantsev equation for a compressible flow that is correlated in time and
study the WKB solutions in Sec. VI. Finally, we insert our results in the
current context and draw our conclusions in Sec. VII.

##  II Kazantsev theory

Dynamos in the context of an isotropic flow have been hypothesised since the
fifties [see e.g. 43, 44]; however the first one to give a complete
theoretical framework was Kazantsev [33]. In his work an isotropic and
homogeneous flow that is delta-correlated in time was proposed. In this
section we review the basics of the derivation of the Kazantsev equation and
its results, in particular we follow Subramanian [35] for the formalism.

We rewrite the velocity field as

$\bm{u}=\langle\bm{u}\rangle+\delta\bm{u},$ (1)

where $\langle\bm{u}\rangle$ is the mean and $\delta\bm{u}$ the fluctuations.
If we assume the fluctuations to be isotropic, homogeneous, Gaussian random
with zero mean and delta-correlated in time we can set the correlation
function to be

$T_{ij}(r)\delta(t_{1}-t_{2})=\langle\delta u_{i}(\bm{x},t_{1})\delta
u_{j}(\bm{y},t_{2})\rangle,$ (2)

with $r=|\bm{x}-\bm{y}|$. Any two-point correlation function can be expressed
through longitudinal and transverse components [45] as

$T_{ij}(r)=\hat{r}_{ij}T_{\mathrm{L}}(r)+\hat{P}_{ij}T_{\mathrm{N}}(r),$ (3)

with $\hat{r}_{ij}=r_{i}r_{j}/r^{2}$ and
$\hat{P}_{ij}=\delta_{ij}-\hat{r}_{ij}$. For a divergence-free vector field
(in the case of velocity: an incompressible flow $\nabla\cdot\bm{u}=0$) we can
even show that the two components are related by

$T_{\mathrm{N}}=T_{\mathrm{L}}+\frac{r}{2}\frac{\mathrm{d}}{\mathrm{d}r}T_{\mathrm{L}}.$
(4)

A similar decomposition can be performed for the magnetic field. Since
$\bm{B}$ is divergence-free, the magnetic correlation function can be
expressed as

$\displaystyle M_{ij}(r)$ $\displaystyle=$ $\displaystyle\langle\delta
B_{i}(\bm{x})\delta B_{j}(\bm{y})\rangle,$ (5) $\displaystyle=$
$\displaystyle\bigl{(}\hat{r}_{ij}+\hat{P}_{ij}\bigl{)}M_{\mathrm{L}}+\hat{P}_{ij}\frac{r}{2}\frac{\mathrm{d}}{\mathrm{d}r}M_{\mathrm{L}}.$

The time derivative of the two-point magnetic correlation function is thus
given by

$\displaystyle\frac{\partial M_{ij}}{\partial t}$ $\displaystyle=$
$\displaystyle\bigg{\langle}\frac{\partial B_{i}}{\partial
t}B_{j}\bigg{\rangle}+\bigg{\langle}\frac{\partial B_{j}}{\partial
t}B_{i}\bigg{\rangle}-\frac{\partial\langle B_{i}B_{j}\rangle}{\partial t}.$
(6)

Inserting this expression in the induction equation

$\frac{\partial\bm{B}}{\partial
t}=\bm{\nabla}\times(\bm{u}\times\bm{B}-\eta\nabla\times\bm{B}),$ (7)

and using the averaged induction equation

$\frac{\partial\langle\bm{B}\rangle}{\partial
t}=\bm{\nabla}\times(\langle\bm{v}\rangle\times\langle\bm{B}\rangle-[\eta+T_{\mathrm{L}}(0)]\nabla\langle\bm{B}\rangle),$
(8)

Subramanian [35] found an equation for the time evolution of the longitudinal
two-point magnetic correlation function

$\displaystyle\frac{\partial M_{\mathrm{L}}}{\partial t}$ $\displaystyle=$
$\displaystyle
2\kappa_{\mathrm{diff}}M_{\mathrm{L}}^{\prime\prime}+2\bigg{(}\frac{4\kappa_{\mathrm{diff}}}{r}+\kappa_{\mathrm{diff}}^{\prime}\bigg{)}M_{\mathrm{L}}^{\prime}$
(9)
$\displaystyle+\frac{4}{r^{2}}\bigg{(}T_{\mathrm{N}}-T_{\mathrm{L}}-rT_{\mathrm{N}}^{\prime}-rT_{\mathrm{L}}^{\prime}\bigg{)}M_{\mathrm{L}}.$

In this expression
$\kappa_{\mathrm{diff}}\equiv\eta+T_{\mathrm{L}}(0)-T_{\mathrm{L}}(r)$ and a
prime denotes a derivative with respect to $r$. If we further suppose that the
time and spatial dependencies are independent, we can use the ansatz

$M_{\mathrm{L}}(r,t)=\frac{1}{r^{2}\sqrt{\kappa_{\mathrm{diff}}}}\psi(r)e^{2\Gamma
t}.$ (10)

This form is convenient as it highlights a formal similarity to quantum
mechanics. We insert the ansatz into Eq. (9) and find

$-\kappa_{\mathrm{diff}}\frac{\mathrm{d}^{2}\psi}{\mathrm{d}r^{2}}+U(r)\psi=-\Gamma\psi.$
(11)

This equation has the form of a Schrödinger equation and is often referred to
as the Kazantsev equation in the literature, however in this work we will
refer to Eq. (9) as the Kazantsev equation instead. The function $U(r)$ is
equivalent to a potential and is given by

$U(r)\equiv\frac{\kappa_{\mathrm{diff}}^{\prime\prime}}{2}-\frac{(\kappa_{\mathrm{diff}})^{\prime}}{4\kappa_{\mathrm{diff}}}+\frac{2\kappa_{\mathrm{diff}}}{r^{2}}+\frac{2T_{\mathrm{N}}^{\prime}}{r}+\frac{2(T_{\mathrm{L}}-T_{\mathrm{N}})}{r^{2}}.$
(12)

Note that in the derivation of this equation we did not assume at any point
that the flow is incompressible.

Schekochihin  _et al._ [36] studied the Kazantsev equation in Fourier space in
the sub-diffusion limit such that $k_{f}\ll k\ll k_{\eta}$, with $k_{f}$ being
the forcing scale for a single scaled flow and $k_{\eta}$ the Fourier
conjugate of the magnetic diffusion length scale (the scale at which the
magnetic diffusion is important). If incompressibility is assumed, the
Kazantsev equation can be rewritten as [see e.g. 46, 34]

$\frac{\partial M_{k}}{\partial
t}=\frac{\gamma}{5}\bigg{(}k^{2}\frac{\partial^{2}M_{k}}{\partial
k^{2}}-2k\frac{\partial M_{k}}{\partial k}+6M_{k}\bigg{)}-2\eta k^{2}M_{k},$
(13)

where $\gamma$ is a constant that characterises the flow and $M_{k}(k,t)$
represents the magnetic power spectrum. Compared to $M_{\mathrm{L}}(r)$ it
characterises the magnetic correlation function in Fourier space, formally we
have the following relation

$\displaystyle\langle\hat{B}_{i}(\bm{k},t)\hat{B}_{j}^{\ast}(\bm{k}^{\prime},t^{\prime})\rangle=(2\pi)^{3}\hat{M}_{ij}(k,t)\delta^{3}(\bm{k}-\bm{k}^{\prime})\delta(t-t^{\prime})$
(14) $\displaystyle=$ $\displaystyle(2\pi)^{3}\frac{M_{k}(k,t)}{4\pi
k^{2}}\bigg{(}\delta_{ij}-\frac{k_{i}k_{j}}{k^{2}}\bigg{)}\delta^{3}(\bm{k}-\bm{k}^{\prime})\delta(t-t^{\prime}),$

with $\hat{A}^{\ast}$ being the complex conjugate of the Fourier transform
$\hat{A}$. The solution of the Fourier space Kazantsev equation is given by

$\displaystyle M_{k}(k,t)=M_{0}e^{\gamma\lambda
t}k^{3/2}K_{\mathrm{Mc}}(k/k_{0}),$
$\displaystyle\mathrm{Mc}=\sqrt{5(\lambda-\frac{3}{4})},$ (15)

where $K_{\mathrm{Mc}}$ is the Macdonald function, $\lambda$ the normalised
growth rate and $k_{0}=(\gamma/10)^{1/2}$. The magnetic power spectrum thus
scales mostly as $M_{k}(k)\sim k^{3/2}$ in the sub-diffuse limit, which we
refer to as the Kazantsev spectrum. The magnetic spectrum grows exponentially
in time, with a growth rate given by $3\gamma/4$ for an incompressible flow
that is delta-correlated in time.

##  III The renewing flow method

The renewing or renovating flow method was firstly proposed by Steenbeck and
Krause [47]. Zeldovich  _et al._ [40] highlighted that it provides an
alternative to the unphysical assumption of velocities that are delta-
correlated in time but remains analytically solvable. Several authors have
used the method to obtain relevant results with finite correlation times [e.g.
48, 49, 50, 51, 52]. In this work we employ the operator splitting method,
used by Gilbert and Bayly [53] to recover the mean-field dynamo equations.
Following the approach of Bhat and Subramanian [37], in a non-helical flow, we
impose a velocity field of the form

$\bm{u}=\bm{a}\sin{(\bm{q}\cdot\bm{x}+\psi)}.$ (16)

We split the time into intervals of length $\tau$ which is the correlation
time of the flow. In each of these $\tau$-intervals we draw randomly $\bm{a}$,
$\bm{q}$ and $\psi$ such that the flow is overall isotropic, homogeneous, and
with a zero mean222We will discuss the exact way to draw $\bm{a}$, $\bm{q}$
and $\psi$ in future sections (see Sec. IV.1, Sec. V.4.1 and Sec. V.4.2)..
Note that the flow is static only in intervals of the type $[(n-1)\tau,n\tau]$
($n$ being an integer) and renovates for each $\tau$-interval.

In order to apply the operator splitting method we further divide the
$\tau$-intervals into two sub-intervals of duration $\tau/2$. In the first one
we consider that the diffusion of the magnetic field is zero but the velocity
is doubled, in the second one the velocity is now set to zero and diffusion
acts as twice its original value. Using the induction equation (Eq. 7) we need
to solve the following problem

$\displaystyle\begin{aligned} \frac{\partial\bm{B}}{\partial
t}=\bm{\nabla}\times
2\bm{u}\times\bm{B},&&t\in[(n-1)\tau,(n-1)\tau+\tau/2],\end{aligned}$
$\displaystyle\begin{aligned} \frac{\partial\bm{B}}{\partial
t}&=-2\eta\bm{\nabla}\times\bm{\nabla}\times\bm{B},&&t\in[(n-1)\tau+\tau/2,n\tau].\end{aligned}$
(17)

The validity and convergence of the operator splitting method is beyond the
scope of this work, we refer interested readers to Holden  _et al._ [55].

First sub-interval: we consider only the ideal induction equation. In this
case, due to the magnetic flux freezing, the magnetic field is given by the
standard Cauchy solution [see Sec.3.3 of 56]

$B_{i}(\bm{x},t)=\frac{J_{ij}(\bm{x}_{0})}{|J_{ij}|}B_{j}(\bm{x}_{0},t_{0}),$
(18)

where we define $\bm{x}(\bm{x}_{0},t_{0})$ to be the Lagrangian position at a
time $t$ of a fluid element with an initial position $\bm{x}_{0}$ at time
$t_{0}$. The matrix $J_{ij}$ is given by the coordinate transformation, namely

$J_{ij}=\frac{\partial x_{i}}{\partial x_{0,j}},$ (19)

and $|\cdot|$ denotes the determinant of the matrix.

Second sub-interval: we consider only the diffusion of the magnetic field. It
is straightforward to solve the equation of diffusion in Fourier space where
we denote the Fourier transform of $A$ by $\hat{A}$. We find the solution

$\hat{B}_{i}(\bm{k},t)=e^{-\eta\bm{k}^{2}\tau}\hat{B}_{j}(\bm{k},t_{1}),$ (20)

with $t_{1}=t_{0}+\tau/2$.

We express the total magnetic field evolution in Fourier space, from Eq. (18)
and Eq. (20), as

$\hat{B}_{i}(\bm{k},t)=e^{-\eta\bm{k}^{2}\tau}\int
e^{-i\bm{k}\cdot\bm{x}}\frac{J_{ij}(\bm{x}_{0})}{|J_{ij}|}B_{j}(\bm{x}_{0},t_{0})~{}\mathrm{d}^{3}\bm{x},$
(21)

which describes the successive evolution through the two sub-intervals.

We are now ready to give an expression for the two-point correlation function
of the magnetic field in Fourier space

$\displaystyle\left\langle\hat{B}_{i}(\bm{k},t)\hat{B}^{\ast}_{h}(\bm{p},t)\right\rangle=e^{-\eta\tau(\bm{k}^{2}+\bm{p}^{2})}\bigg{\langle}\int\frac{J_{ij}(\bm{x}_{0})}{|J_{ij}|}\frac{J_{hl}(\bm{y}_{0})}{|J_{hl}|}$
$\displaystyle\times
B_{j}(\bm{x}_{0},t_{0})B_{l}(\bm{y}_{0},t_{0})e^{-i(\bm{k}\cdot\bm{x}-\bm{p}\cdot\bm{y})}~{}\mathrm{d}^{3}\bm{x}\mathrm{d}^{3}\bm{y}\bigg{\rangle},$
(22)

where $\langle\cdot\rangle$ denotes an average over the parameter space of the
velocity flow and $A^{\ast}$ is the complex conjugate. We can change the
integration variables
$\\{\bm{x},\bm{y}\\}\rightarrow\\{\bm{x}_{0},\bm{y}_{0}\\}$ such that the
determinants of the two Jacobian matrices cancel. We can also argue that the
initial magnetic field is no longer correlated with the renewing flow in the
next sub-interval, which allows us to split the averages. The final expression
is then given by

$\displaystyle\left\langle\hat{B}_{i}(\bm{k},t)\hat{B}^{\ast}_{h}(\bm{p},t)\right\rangle=e^{-\eta\tau(\bm{k}^{2}+\bm{p}^{2})}\int\left\langle
B_{j}(\bm{x}_{0},t_{0})B_{l}(\bm{y}_{0},t_{0})\right\rangle$
$\displaystyle\times\left\langle
J_{ij}(\bm{x}_{0})J_{hl}(\bm{y}_{0})e^{-i(\bm{k}\cdot\bm{x}-\bm{p}\cdot\bm{y})}\right\rangle~{}\mathrm{d}^{3}\bm{x}_{0}\mathrm{d}^{3}\bm{y}_{0}.$
(23)

Note that in this expression $\bm{x}$ and $\bm{y}$ are functions of the
initial positions.

As the flow is overall isotropic and homogeneous we expect that for an initial
state of the magnetic field, which is also isotropic and homogeneous, these
properties are conserved. Under such assumptions the two-point magnetic
correlation function takes the following form

$\left\langle B_{i}(\bm{x},t)B_{j}(\bm{y},t)\right\rangle=M_{ij}(r,t),$ (24)

where $r=|\bm{x}-\bm{y}|$. We can further introduce a new set of integration
variables
$\\{\bm{x}_{0},\bm{y}_{0}\\}\rightarrow\\{\bm{r_{0}}\equiv\bm{x}_{0}-\bm{y}_{0},\bm{y}_{0}\\}$.
We rewrite the exponential part inside the integral as

$-i\left[\bm{k}\cdot(\bm{x}-\bm{x}_{0})-\bm{p}\cdot(\bm{y}-\bm{y}_{0})+\bm{k}\cdot\bm{r}_{0}+(\bm{k}-\bm{p})\cdot\bm{y}_{0}\right].$
(25)

For now we assume that the evolution tensor, that is given by

$R_{ijhl}\equiv\left\langle
J_{ij}(\bm{x}_{0})J_{hl}(\bm{y}_{0})e^{-i\left[\bm{k}\cdot(\bm{x}-\bm{x}_{0})-\bm{p}\cdot(\bm{y}-\bm{y}_{0})\right]}\right\rangle,$
(26)

is independent of $\bm{y}_{0}$; which is convenient as we can rewrite Eq.
(III) in the following form

$\displaystyle\left\langle\hat{B}_{i}(\bm{k},t)\hat{B}^{\ast}_{h}(\bm{p},t)\right\rangle$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\delta^{3}(\bm{p}-\bm{k})e^{-2\eta\tau\bm{p}^{2}}$
(27) $\displaystyle\times\int
e^{-i\bm{p}\cdot\bm{r}_{0}}R_{ijhl}M_{jl}(r_{0},t_{0})~{}\mathrm{d}^{3}\bm{r}_{0},$

once the integration over $\mathrm{d}^{3}\bm{y}_{0}$ is performed. Note that
the Dirac-Delta function appears from the integration over $\bm{y}_{0}$ since
the exponential is the only dependency on $\bm{y}_{0}$ and can be taken out of
the flow parameters average. We assumed that $R_{ijhl}$ only depends on
$\bm{r}_{0}$ because this form of the equation is more compact; we will show
in further sections (see Sec. IV.3) that this assumption is valid, at least
for the cases we consider.

##  IV Kazantsev equation from the renewing flow method

In his initial work, Kazantsev considered a flow that is delta-correlated in
time and incompressible. This case is the easiest to treat with equations that
are more or less tractable. We use this simplified treatment to present a
detailed calculation in the framework of the renewing flow method. With the
renewing flow method we consider the velocity field to be known; which
constitutes the main difference with previous works on the topic.

###  IV.1 Velocity flow parameters

The first step is to give a suitable parametrisation of $\bm{a}$, $\bm{q}$ and
$\psi$ to ensure the statistical isotropy and homogeneity of the flow. We
further impose an incompressible flow, which translates here to the
requirement that $\bm{a}$ and $\bm{q}$ are orthogonal to each other.

Homogeneity

we draw $\psi$ in each $\tau$-interval from a uniform distribution in the
range $[0,2\pi]$.

Isotropy

we fix the value of $q$ which is the norm of $\bm{q}$. The wavenumber $\bm{q}$
is randomly drawn from a sphere of radius $q$. The velocity orientation
$\bm{a}$ is randomly drawn in the plane perpendicular to $\bm{q}$ such that
$\langle\bm{u}\rangle$=0.

In order to simplify the computations we change the average ensemble. Instead
of averaging over the direction of $\bm{a}$ we prefer to use a new vector
$\bm{A}$ which has a fixed norm and a direction drawn randomly. Then $\bm{A}$
and $\bm{q}$ define a plane where we can project the component of $\bm{A}$
that is orthogonal to $\bm{q}$. This is performed by

$\displaystyle\tilde{P}_{ij}\equiv\delta_{ij}-\hat{q}_{i}\hat{q}_{j},$
$\displaystyle a_{i}=\tilde{P}_{ij}A_{j},$ (28)

where $\hat{q}_{i}\equiv q_{i}/q$ is the normalised component of $\bm{q}$.
Note also that we adopt the Einstein summation rule. Since $\bm{A}$ and
$\bm{q}$ are two independent vectors this parametrisation ensures
$\langle\bm{u}\rangle=0$. We directly see that $a$ is not fixed in this
context, however we can evaluate it from $A$ as

$\displaystyle\left\langle a^{2}\right\rangle$ $\displaystyle=$
$\displaystyle\left\langle
a_{i}a_{i}\right\rangle=\left\langle\tilde{P}_{il}A_{l}\tilde{P}_{ih}A_{h}\right\rangle,$
(29) $\displaystyle\underset{\text{average of }\bm{A}}{=}$
$\displaystyle\frac{A^{2}}{3}\left\langle\tilde{P}_{il}\tilde{P}_{ih}\delta_{lh}\right\rangle\underset{\text{average
of }\bm{q}}{=}\frac{2A^{2}}{3},$

where we used the fact that $\langle A_{i}A_{j}\rangle=\delta_{ij}/3$ for a
random vector.

###  IV.2 Two-point velocity correlation functions

In order to reconstruct the original Kanzatsev equation 9 we only need to
compute the second-order velocity correlator. We use the definition of Bhat
and Subramanian [37]

$T_{ij}=\frac{\tau}{2}\left\langle
u_{i}(\bm{x})u_{j}(\bm{y})\right\rangle\underset{\text{average of
}\psi}{=}\frac{\tau}{4}\left\langle
a_{i}a_{j}\cos{(\bm{q}\cdot\bm{r})}\right\rangle.$ (30)

The factor $\tau/2$ is required here as the flow is correlated in time. It
also ensures that in the limit $\tau\rightarrow 0$ we recover the Kazantsev
equation. The initialisation of $\bm{a}$ and $\bm{q}$ allows us to give an
exact formula for this correlator. Using Eqs. (28) and (29) we average over
the directions of $\bm{A}$ to eliminate it, the remaining average is thus only
over the directions of $\bm{q}$ with

$\displaystyle T_{ij}$ $\displaystyle=$
$\displaystyle\frac{\tau}{4}\left\langle\tilde{P}_{il}A_{l}\tilde{P}_{jh}A_{h}\cos{(\bm{q}\cdot\bm{r})}\right\rangle=\frac{A^{2}\tau}{12}\left\langle\tilde{P}_{ij}\cos{(\bm{q}\cdot\bm{r})}\right\rangle$
(31) $\displaystyle=$
$\displaystyle\frac{a^{2}\tau}{8}\left[\delta_{ij}+\frac{1}{q^{2}}\partial_{i}\partial_{j}\right]\left\langle\cos{(\bm{q}\cdot\bm{r})}\right\rangle,$

where we use the following notation $\partial_{i}\equiv\partial/\partial
r_{i}$. If we recall the proper definition of the average we can write

$\displaystyle\left\langle\cos{(\bm{q}\cdot\bm{r})}\right\rangle$
$\displaystyle\equiv$
$\displaystyle\frac{1}{4\pi}\int^{2\pi}_{0}\int^{\pi}_{0}\sin{(\theta)}\cos{(\bm{q}\cdot\bm{r})}~{}\mathrm{d}\theta\mathrm{d}\phi$
(32) $\displaystyle=$
$\displaystyle\frac{1}{2}\int^{\pi}_{0}\sin{(\theta)}\cos{(qr\cos{(\theta)})}~{}\mathrm{d}\theta$
$\displaystyle=$ $\displaystyle j_{0}(qr),$

where $j_{0}(x)$ is the spherical Bessel function.

###  IV.3 Computation of the evolution tensor

In order to evaluate $R_{ijhl}$ we first need to have an expression for
$J_{ij}$. As we required that $\bm{a}$ and $\bm{q}$ are orthogonal we have

$\frac{\mathrm{d}(\bm{q}\cdot\bm{x}+\psi)}{\mathrm{d}t}\equiv\frac{\mathrm{d}\phi}{\mathrm{d}t}=2\bm{q}\cdot\bm{u}=0,$
(33)

such that $\phi$ is constant along the trajectory of a fluid element. So the
equation $\mathrm{d}\bm{x}/\mathrm{d}t=2\bm{u}$ can be easily integrated333The
factor 2 comes from the fact that in the first sub-interval we have twice the
initial velocity. and gives

$x_{i}-x_{0,i}=a_{i}\tau\sin{(\bm{q}\cdot\bm{x}_{0}+\psi)}$ (34)

for the Lagrangian positions. Using Eq. (19), it is straightforward to
evaluate $J_{ij}$; from the last relation

$J_{ij}(\bm{x}_{0})=\delta_{ij}+\tau
a_{i}q_{j}\cos{(\bm{q}\cdot\bm{x}_{0}+\psi)}.$ (35)

Bhat and Subramanian [37] motivated an expansion of the exponential of the
evolution tensor (Eq. 26) in the limit of small Strouhal numbers $St=qa\tau\ll
1$. In the context of small-scale turbulent dynamos the magnetic spectrum in
the kinematic regime peaks around the resistive scale [58, 59, 60] which can
be evaluated to be $r_{\eta}\sim(l_{0}/R_{\mathrm{M}}^{1/2})$ with $l_{0}$
being the integral scale of the flow444This definition for the resistive scale
is limited to the cases $R_{\mathrm{e}}\simeq 1$. Some authors used a more
general expression $k_{\eta}\propto R_{\mathrm{M}}^{1/2}R_{\mathrm{e}}^{1/4}$
[41, 80, 91]. In the case considered here, the flow has only one typical scale
($1/q$), thus $r_{\eta}\sim 1/(qR_{\mathrm{M}}^{1/2})$. We used
$R_{\mathrm{M}}\sim a/(q\eta)$ for the magnetic Reynolds number which is
usually very high in astrophysical objects [see Tab. 1 of 56] such that
$qr_{\eta}$ is very small and hence
$\sin{(\bm{q}\cdot\bm{r_{\eta}})}\sim\bm{q}\cdot\bm{r_{\eta}}$. The phase of
the exponential in Eq. (26) is then given by $aq\tau p_{\eta}r_{\eta}\sim
qa\tau=St$. Since the terms in the vicinity of the resistive scale will
contribute more to the magnetic spectrum, the expansion of
$\sin{(\bm{q}\cdot\bm{x}_{0}+\psi)}-\sin{(\bm{q}\cdot\bm{y}_{0}+\psi)}=\sin{(\bm{q}\cdot\bm{r}_{0}/2)}\cos{(\bm{q}\cdot(\bm{x}_{0}+\bm{y}_{0})+\psi)}$
is reasonable. In this section, we only keep terms up to second order in
$\tau$ and we will see that it leads to the original Kazantsev equation 9.

The equation 26 for $R_{ijhl}$ can then be rewritten in the form

$\displaystyle
R_{ijhl}=\bigg{\langle}J_{ij}(\bm{x}_{0})J_{hl}(\bm{y}_{0})\big{[}1-i\tau\beta\sigma-\frac{\tau^{2}\beta^{2}\sigma^{2}}{2!}\big{]}\bigg{\rangle},$
(36)

where
$\beta=\sin{(\bm{q}\cdot\bm{x}_{0}+\psi)}-\sin{(\bm{q}\cdot\bm{y}_{0}+\psi)}$
and $\sigma=\bm{a}~{}\cdot~{}\bm{p}$. To continue further we make use of the
average over $\psi$ and we also introduce the notation
$\phi_{x_{0}}=\bm{q}\cdot\bm{x}_{0}+\psi$. In fact if we try to average a
function of the type $\cos{(n\phi_{x_{0}}+m\phi_{y_{0}})}$ or
$\sin{(n\phi_{x_{0}}+m\phi_{y_{0}})}$ with $n$ and $m$ being two integers, we
find that it always goes to zero except when $n=-m$. In particular it
highlights the fact, as we hypothesised in Sec. III, that $R_{ijhl}$ is only
dependent on $\bm{r}_{0}=\bm{x}_{0}-\bm{y}_{0}$.

Term by term evaluation of the average over $\psi$ of Eq. (36), leads to the
following expression

$\displaystyle R_{ijhl}$ $\displaystyle=$
$\displaystyle\bigg{\langle}\delta_{ij}\delta_{hl}+\frac{\tau^{2}a_{i}q_{j}a_{h}q_{l}}{2}\cos{(\bm{q}\cdot\bm{r_{0}})}$
(37) $\displaystyle\hskip
4.26773pt-i\frac{\tau^{2}\sigma}{2}\sin{(\bm{q}\cdot\bm{r_{0}})}(\delta_{hl}a_{i}q_{j}+\delta_{ij}a_{h}q_{l})$
$\displaystyle\hskip
21.33955pt-\frac{\tau^{2}\sigma^{2}}{2}(1-\cos{(\bm{q}\cdot\bm{r_{0}})})\delta_{ij}\delta_{hl}\bigg{\rangle}.$

Each term can then be matched with Eq. (30) to obtain

$\displaystyle R_{ijhl}$ $\displaystyle=$
$\displaystyle\delta_{ij}\delta_{hl}-2\tau\partial_{l}\partial_{j}T_{ih}+2i\tau
p_{m}(\delta_{hl}\partial_{j}T_{im}$ (38)
$\displaystyle+\delta_{ij}\partial_{l}T_{hm})-2\tau
p_{n}p_{m}\delta_{ij}\delta_{hl}(T_{nm}(0)-T_{nm}),$

where we replaced $q_{i}$ by suitable derivatives with respect to the
components of $\bm{r}_{0}$ and $\sigma$ by $a_{m}p_{m}$. This expression for
$R_{ijhl}$ cannot be simplified further and we need to go back to Eq. (27) and
perform the integration.

###  IV.4 Derivation of the Kazantsev equation

The original Kazantsev equation (9 describes the evolution of the two-point
magnetic field correlation function in real space. Instead of evaluating
$\hat{M}_{ih}(\bm{p},t)$ we take its inverse Fourier transform. Formally we
get

$M_{ih}(r,t)=\int
e^{-2\eta\tau\bm{p^{2}}}e^{i\bm{p}(\bm{r}-\bm{r}_{0})}R_{ijhl}M_{jl}(r_{0},t_{0})~{}\frac{\mathrm{d}^{3}\bm{r}_{0}\mathrm{d}^{3}\bm{p}}{(2\pi)^{3}}.$
(39)

In order to further simplify this expression we assume that $\eta$ is small,
such that the exponential can also be expanded giving
$\exp(-2\eta\tau\bm{p}^{2})\sim 1-2\eta\tau\bm{p}^{2}$. This expansion is
justified in the context of negligible $\eta$ or large $R_{\mathrm{M}}$. Terms
like $\eta\tau^{2}$ are also ignored, so the part $-2\eta\tau\bm{p}^{2}$ only
contributes from the $\delta_{ij}\delta_{hl}$ term in the expression of
$R_{ijhl}$ as it is the only term that does not depend on $\tau^{2}$. Once
again we rewrite components of the wavevector (here $\bm{p}$) as derivatives
with respect to the position (here $\bm{r}$) such that
$p_{j}\rightarrow-i\partial_{j}$.

We adopt the notation $[\cdot]_{ij}$ for partial derivatives with respect to
$r_{i}$ and $r_{j}$. In the limit $\tau\rightarrow 0$ we can divide both sides
by $\tau$ and replace $(M_{ih}(r,t)-M_{ih}(r,t_{0}))/\tau\rightarrow\partial
M_{ih}(r,t)/\partial t$ such that from Eq. (39) we arrive at [see 62, for a
detailed calculation]

$\displaystyle\frac{\partial M_{ih}(r,t)}{\partial t}$ $\displaystyle=$
$\displaystyle
2\left[M_{il}T_{jh}\right]_{jl}+2\left[M_{jh}T_{il}\right]_{jl}-2\left[M_{ih}T_{jl}\right]_{jl}$
(40)
$\displaystyle-2\left[M_{jl}T_{ih}\right]_{jl}+2\left[M_{ih}(T_{\mathrm{L}}(0)+\eta)\right]_{jj}.$

Note that $T_{\mathrm{L}}(0)$ appears from
$T_{nm}(0)=\delta_{nm}T_{\mathrm{L}}(0)$. This result is very important for
the formalism as we started from equation 37 that tracks the evolution of an
initial state to the equation 40 for the two-point magnetic field correlation
function that depends on other quantities evaluated at the same space-time
positions.

We can even simplify the computation further by contracting Eq. (40) with
$\hat{r}_{ih}$ on both sides in order to get an equation for
$M_{\mathrm{L}}(\bm{r},t)$. We would like to refer the reader to Tab. B1 for
detailed expressions of different contractions that enter the computation.
Using incompressibility we finally find

$\displaystyle\frac{\partial M_{\mathrm{L}}(r,t)}{\partial t}$
$\displaystyle=$
$\displaystyle\frac{2}{r^{4}}\partial_{r}\left(r^{4}(\eta+T_{\mathrm{L}}(0)-T_{\mathrm{L}})\partial_{r}M_{\mathrm{L}}\right)$
(41)
$\displaystyle-\frac{2}{r}\left(r\partial_{r}^{2}T_{\mathrm{L}}+4\partial_{r}T_{\mathrm{L}}\right)M_{\mathrm{L}},$

which is exactly the incompressible Kazantsev equation 9 in the limit of a
flow that is delta-correlated in time. In comparison to previous works [e.g.
34, 42], the input here is the velocity field that is used to solve directly
the induction equation.

##  V Generalised Kazantsev equation

In this section we will derive the equivalent of the Kazantsev equation in the
context of the renewing flow method. Previous studies have analysed separately
the effects of the finite correlation time [37] and the compressibility [41]
of the flow. By generalised we mean that we relax the incompressibility
assumption used in the previous work of Bhat and Subramanian [37]. Our new
equations then include the contributions from the time correlation of the flow
as well as its degree of compressibility.

###  V.1 Lagrangian positions

In the case of an incompressible flow $\xi\equiv\bm{a}\cdot\bm{q}$ was set to
0 (see Sec. IV). We can introduce a degree of compressibility by relaxing this
condition; allowing $\xi$ to be non-zero with $\xi\in[-aq;aq]$. This allows us
to include the non-trivial contribution from the compressibility of the flow.
We can no longer apply the same reasoning as before (see Sec. IV.3) since this
time we have $\mathrm{d}\phi/\mathrm{d}t=2\xi\sin{(\phi)}$. If we integrate
this expression over the first sub-interval, we find

$|\tan{(\phi/2)}|=e^{\xi\tau}|\tan{(\phi_{0}/2)}|.$ (42)

We defined $\phi=\bm{q}\cdot\bm{x}+\psi$ to be the phase of the velocity field
at the final position (after a time $\tau/2$) and
$\phi_{0}=\bm{q}\cdot\bm{x}_{0}+\psi$ to be the phase of the initial position.

Furthermore by integrating the velocity field we get

$x_{i}-x_{0,i}=\int\frac{\mathrm{d}x_{i}}{\mathrm{d}t}~{}\mathrm{d}t=\frac{a_{i}}{\xi}(\phi-\phi_{0}).$
(43)

However this formula cannot be inverted. The idea is thus to use Eq. (42) to
isolate $\phi$ in order to plug it into Eq. (43) such that we get an
expression of the Lagrangian positions $\bm{x}$ that depends only on the
initial position $\bm{x}_{0}$.

We have imposed a peculiar velocity field that is periodic with respect to the
variable $\phi$ with a period of $2\pi$. It is then expected that the
displacement $\bm{x}-\bm{x}_{0}$ also possesses this periodicity. Furthermore,
the velocity field is static in a $\tau$-interval which means that fluid
elements are permanently pushed in the direction of $\bm{a}$ until they reach
a zero of the velocity field and stop moving. As a result a fluid element with
initial position $\phi_{0}\in[n\pi,(n+1)\pi]$ will have a position after a
time $\tau/2$ such that $\phi\in[n\pi,(n+1)\pi]$ where $n$ is an integer. Eq.
(42) can thus be inverted, leading to

$\phi/2-\pi\lfloor\phi/(2\pi)+1/2\rfloor=\arctan{\left(e^{\xi\tau}\tan{(\phi_{0}/2)}\right)}.$
(44)

Recall that in Sec. IV.3 we motivated an expansion with respect to a small
Strouhal number $St$. We motivate the same idea here as
$|\xi\tau|=|aq\tau\cos{(\gamma)}|<St$ with $\gamma$ being the angle between
$\bm{a}$ and $\bm{q}$. With a similar argument we can show that any new term
depends directly on $St$ raised to some higher powers. In order to include
effects due to finite correlation times we keep terms up to fourth order in
$\tau$. The expansion of the right hand side of Eq. (44) also harbours a floor
function that will cancel the one on the left hand side.

We are now ready to plug the expression of $\phi$ from this expansion into Eq.
(43)

$\displaystyle x_{i}$ $\displaystyle=$ $\displaystyle
x_{0,i}+a_{i}\tau\biggl{(}\sin{(\phi_{0})}+\frac{\xi\tau}{4}\sin{(2\phi_{0})}$
(45)
$\displaystyle+\frac{\xi^{2}\tau^{2}}{12}\bigl{(}\sin{(3\phi_{0})}-\sin{(\phi_{0})}\bigl{)}$
$\displaystyle+\frac{\xi^{3}\tau^{3}}{96}\bigl{(}3\sin{(4\phi_{0})}-4\sin{(2\phi_{0})}\bigl{)}\biggl{)},$

which has the desired limit for $\xi\rightarrow 0$. It is straightforward to
show that the Jacobian is then given by

$\displaystyle J_{ij}$ $\displaystyle=$
$\displaystyle\delta_{ij}+a_{i}q_{j}\tau\biggl{(}\cos{(\phi_{0})}+\frac{\xi\tau}{2}\cos{(2\phi_{0})}$
(46)
$\displaystyle+\frac{\xi^{2}\tau^{2}}{12}\bigl{(}3\cos{(3\phi_{0})}-\cos{(\phi_{0})}\bigl{)}$
$\displaystyle+\frac{\xi^{3}\tau^{3}}{24}\bigl{(}3\cos{(4\phi_{0})}-2\cos{(2\phi_{0})}\bigl{)}\biggl{)}.$

###  V.2 Fourth order velocity two-point function

In order to include finite correlation times we have to consider terms up to
the fourth order in $\tau$. The evolution tensor $R_{ijhl}$ is then given by

$R_{ijhl}=\biggl{\langle}J_{ij}J_{hl}[1-i\tau\beta\sigma-\frac{\tau^{2}\beta^{2}\sigma^{2}}{2!}+i\frac{\tau^{3}\beta^{3}\sigma^{3}}{3!}+\frac{\tau^{4}\beta^{4}\sigma^{4}}{4!}]\biggl{\rangle},$
(47)

where the Jacobian matrices are given in Eq. (46) and

$\displaystyle\beta$ $\displaystyle=$
$\displaystyle\sin{(\phi_{x})}-\sin{(\phi_{y})}+\frac{\tau\xi}{4}(\sin{(2\phi_{x})}-\sin{(2\phi_{y})})$
$\displaystyle+\frac{\tau^{2}\xi^{2}}{12}(\sin{(3\phi_{x})}-\sin{(3\phi_{y})}-\sin{(\phi_{x})}+\sin{(\phi_{y})})$
$\displaystyle+\frac{\tau^{3}\xi^{3}}{96}(3\sin{(4\phi_{x})}-3\sin{(4\phi_{y})}-4\sin{(2\phi_{x})}+4\sin{(2\phi_{y})}).$

The evolution tensor now has dependencies on $a_{i}a_{j}a_{h}a_{l}$ due to the
inclusion of $\tau^{3}$ and $\tau^{4}$ terms. It motivates the introduction of
fourth order two-point correlators that are defined in Bhat and Subramanian
[62] by

$\displaystyle T_{ijhl}^{x^{2}y^{2}}=\tau^{2}\langle
u_{i}(\bm{x})u_{j}(\bm{x})u_{h}(\bm{y})u_{l}(\bm{y})\rangle,$ $\displaystyle
T_{ijhl}^{x^{3}y}=\tau^{2}\langle
u_{i}(\bm{x})u_{j}(\bm{x})u_{h}(\bm{x})u_{l}(\bm{y})\rangle,$ $\displaystyle
T_{ijhl}^{x^{4}}=\tau^{2}\langle
u_{i}(\bm{x})u_{j}(\bm{x})u_{h}(\bm{x})u_{l}(\bm{x})\rangle,$ (49)

where the factor $\tau^{2}$ is included due to the time correlation of the
flow. We can carry out the average over $\psi$ such that the fourth order
correlators are given by

$\displaystyle T_{ijhl}^{x^{2}y^{2}}=\frac{\tau^{2}}{8}\langle
a_{i}a_{j}a_{h}a_{l}(\cos{(2\bm{q}\cdot\bm{r})}+2)\rangle,$ (50)
$\displaystyle T_{ijhl}^{x^{3}y}=\frac{3\tau^{2}}{8}\langle
a_{i}a_{j}a_{h}a_{l}\cos{(\bm{q}\cdot\bm{r})}\rangle,$ $\displaystyle
T_{ijhl}^{x^{4}}=\frac{3\tau^{2}}{8}\langle a_{i}a_{j}a_{h}a_{l}\rangle.$

Note that $\bm{r}$ is still given by $\bm{r}=\bm{x}-\bm{y}$.

Similarly to the second order velocity two-point correlation function we would
like an expression for the fourth order correlators in the case of an
isotropic, homogeneous and non-helical velocity field. Following the ideas of
De Karman and Howarth [45], Batchelor [63], Landau and Lifshitz [64], it can
be shown that 555Note that $\overline{T}_{L/LN/N}$ is only a notation and does
not describe a new average.

$T_{ijhl}(r)=\hat{r}_{ijhl}\overline{T}_{\mathrm{L}}(r)+\hat{P}_{(ij}\hat{P}_{hl)}\overline{T}_{\mathrm{N}}(r)+\hat{P}_{(ij}\hat{r}_{hl)}\overline{T}_{\mathrm{LN}}(r),$
(51)

where $\hat{r}_{ijhl}=r_{i}r_{j}r_{h}r_{l}/r^{4}$ and
$\hat{P}_{ij}=\delta_{ij}-r_{i}r_{j}/r^{2}$. This formula has been derived and
used by Bhat and Subramanian [37]. The bracket $(\cdot)$ operator denotes here
the summation over all the different terms, formally
$\hat{P}_{(ij}\hat{P}_{hl)}=\hat{P}_{ij}\hat{P}_{hl}+\hat{P}_{ih}\hat{P}_{jl}+\hat{P}_{il}\hat{P}_{jh}$
and
$\hat{P}_{(ij}\hat{r}_{hl)}=\hat{P}_{ij}\hat{r}_{hl}+\hat{P}_{ih}\hat{r}_{jl}+\hat{P}_{il}\hat{r}_{jh}+\hat{P}_{hl}\hat{r}_{ij}+\hat{P}_{jl}\hat{r}_{ih}+\hat{P}_{jh}\hat{r}_{il}$.
Knowing that it is straightforward to show that in the case of an
incompressible flow the transverse, longitudinal and mixed terms are related
by

$\displaystyle
6\overline{T}_{\mathrm{LN}}=2\overline{T}_{\mathrm{L}}+r\partial_{r}\overline{T}_{\mathrm{L}},$
$\displaystyle
4\overline{T}_{\mathrm{N}}=4\overline{T}_{\mathrm{LN}}+r\partial_{r}\overline{T}_{\mathrm{LN}}.$
(52)

These two relations will be especially useful to check if our generalised
Kazantsev equation has the right form when assuming incompressibility.

###  V.3 Generalised equation

The compressibility effects are characterised by the introduction of $\xi$ and
$\xi^{2}$ in the evolution tensor and the Jacobian matrices. These factors are
not necessarily fixed between two $\tau$-intervals. To treat them we just need
to recall that $\xi=a_{i}q_{i}$, such that the methodology explained in Secs.
IV.3 & IV.4 can still be applied. Surprisingly we find that the
compressibility only affects the fourth order correlators, and Eq. (40) still
holds for a velocity field that is delta-correlated in time in the
compressible case. The resulting equation is given by

$\displaystyle\frac{\partial M_{ih}}{\partial t}$ $\displaystyle=$
$\displaystyle\left.2[M_{jh}T_{il}]_{jl}-2[M_{ih}T_{jl}]_{jl}+2[M_{il}T_{jh}]_{jl}-2[M_{jl}T_{ih}]_{jl}+2[M_{ih}T_{\mathrm{L}}(0)]_{jj}\right\\}\text{
$\xi^{0}\tau^{2}$ terms}$ (53)
$\displaystyle\left.+2\left[M_{ih}\eta\right]_{jj}\right\\}\text{ term due to
resistive exponential expansion}$
$\displaystyle\left.+\tau\biggl{(}[M_{jl}\widetilde{T}_{ihmn}]_{mnjl}+[M_{ih}(\widetilde{T}_{mnst}+T^{x^{4}}_{mnst}/12)]_{mnst}\right.-[M_{jh}\widetilde{T}_{imns}]_{mnsj}-\left.[M_{il}\widetilde{T}_{hmns}]_{mnsl}\right\\}\text{
$\xi^{0}\tau^{4}$ terms}$ $\displaystyle\hskip
17.07182pt\left.-[M_{jl}\partial_{n}\widetilde{T}_{ihmn}]_{mjl}-[M_{ih}\partial_{t}\widetilde{T}_{mnst}]_{mns}+[M_{jh}\partial_{s}\widetilde{T}_{imns}]_{mnj}+[M_{il}\partial_{s}\widetilde{T}_{hmns}]_{mnl}\right\\}\text{
$\xi^{1}\tau^{4}$ terms}$ $\displaystyle\left.\hskip 14.51074pt\begin{aligned}
&+\frac{2}{3}[M_{jl}\partial_{m}\partial_{n}\widetilde{T}_{ihmn}]_{jl}+\frac{2}{3}[M_{ih}\partial_{s}\partial_{t}\widetilde{T}_{mnst}]_{mn}-\frac{2}{3}[M_{jh}\partial_{n}\partial_{s}\widetilde{T}_{imns}]_{mj}\\\
&-\frac{2}{3}[M_{il}\partial_{n}\partial_{s}\widetilde{T}_{hmns}]_{ml}-\frac{5}{48}[M_{jl}\partial_{m}\partial_{n}T^{x^{2}y^{2}}_{ihmn}]_{jl}-\frac{5}{48}[M_{ih}\partial_{s}\partial_{t}T^{x^{2}y^{2}}_{mnst}]_{mn}\\\
&+\frac{5}{48}[M_{jh}\partial_{n}\partial_{s}T^{x^{2}y^{2}}_{imns}]_{mj}+\frac{5}{48}[M_{il}\partial_{n}\partial_{s}T^{x^{2}y^{2}}_{hmns}]_{ml}+\frac{5}{48}[M_{ih}\partial_{s}\partial_{t}T^{x^{2}y^{2}}_{mnst}|_{0}]_{mn}\biggl{)},\end{aligned}\right\\}\text{
$\xi^{2}\tau^{4}$ terms}$

where we introduced the tensor
$\widetilde{T}_{ijhl}=T^{x^{2}y^{2}}_{ijhl}/4-T^{x^{3}y}_{ijhl}/3$. The
compressibility adds the divergence of the fourth order velocity correlators,
which is evaluated to be zero in the case of an incompressible flow; such that
the incompressible limit ($\xi=0$) gives back Eq. (16) in Bhat and Subramanian
[37].

In order to derive the equation for $M_{\mathrm{L}}$ we contract Eq. (53) with
$\hat{r}_{ih}$. By doing so we need to evaluate every term in the equation
using the velocity correlation functions described in Eq. (3) and Eq. (51).
Even if this computation does not involve very complicated algebra we will not
detail our derivation as it is very extensive. However, we give in Tab. B1 the
main tools and properties to perform the complete calculation. Another very
useful expression to simplify the computation is

$\displaystyle\partial_{i}\partial_{j}\bigl{(}\hat{r}_{ij}f(r)+\hat{P}_{ij}g(r)\bigl{)}$
$\displaystyle=$
$\displaystyle\partial^{2}_{r}f(r)+\frac{2}{r}\partial_{r}(2f(r)-g(r))$ (54)
$\displaystyle\hskip 8.5359pt+\frac{2}{r^{2}}(f(r)-g(r)),$

where $f$ and $g$ are two arbitrary functions of $r$. Using these properties,
simplifications, and denoting $\overline{T}_{L/LN/N}$ the respective
components of $\widetilde{T}_{ijhl}$ we eventually arrive at the generalised
Kazantsev equation

$\displaystyle\frac{\partial M_{\mathrm{L}}}{\partial t}$
$\displaystyle=\partial_{r}^{4}M_{\mathrm{L}}\bigg{(}\tau\bigl{\\{}\overline{T}_{\mathrm{L}}+\frac{T_{\mathrm{L}}^{x^{3}y}(0)}{12}\bigl{\\}}\bigg{)}$
$\displaystyle\hskip
3.41418pt+\partial_{r}^{3}M_{\mathrm{L}}\bigg{(}\tau\bigl{\\{}2\partial_{r}\overline{T}_{\mathrm{L}}+\frac{8}{r}\overline{T}_{\mathrm{L}}+\frac{2T_{\mathrm{L}}^{x^{3}y}(0)}{3r}\bigl{\\}}\bigg{)}$
$\displaystyle\hskip
3.41418pt+\partial_{r}^{2}M_{\mathrm{L}}\bigg{(}-2T_{\mathrm{L}}+2T_{\mathrm{L}}(0)+2\eta+\tau\bigl{\\{}\frac{5}{3}\partial_{r}^{2}\overline{T}_{\mathrm{L}}+\frac{41}{3r}\partial_{r}\overline{T}_{\mathrm{L}}-\frac{8}{3r}\partial_{r}\overline{T}_{\mathrm{LN}}$
$\displaystyle\hskip
46.94687pt+\frac{40}{3r^{2}}\overline{T}_{\mathrm{L}}-\frac{80}{3r^{2}}\overline{T}_{\mathrm{LN}}+\frac{32}{3r^{2}}\overline{T}_{\mathrm{N}}-\frac{5}{48}\partial_{r}^{2}T^{x^{2}y^{2}}_{\mathrm{L}}-\frac{20}{48r}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{L}}$
$\displaystyle\hskip
46.94687pt+\frac{50}{48r}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{LN}}-\frac{10}{48r^{2}}T^{x^{2}y^{2}}_{\mathrm{L}}+\frac{110}{48r^{2}}T^{x^{2}y^{2}}_{\mathrm{LN}}-\frac{80}{48r^{2}}T^{x^{2}y^{2}}_{\mathrm{N}}+\frac{2T_{\mathrm{L}}^{x^{3}y}(0)}{3r^{2}}+K\bigl{\\}}\biggl{)}$
$\displaystyle\hskip
3.41418pt+\partial_{r}M_{\mathrm{L}}\bigg{(}-2\partial_{r}T_{\mathrm{L}}-\frac{8}{r}T_{\mathrm{L}}+\frac{8}{r}T_{\mathrm{L}}(0)+\frac{8\eta}{r}+\tau\bigl{\\{}\frac{2}{3}\partial_{r}^{3}\overline{T}_{\mathrm{L}}+\frac{25}{3r}\partial_{r}^{2}\overline{T}_{\mathrm{L}}-\frac{8}{3r}\partial_{r}^{2}\overline{T}_{\mathrm{LN}}$
$\displaystyle\hskip
46.94687pt+\frac{55}{3r^{2}}\partial_{r}\overline{T}_{\mathrm{L}}-\frac{104}{3r^{2}}\partial_{r}\overline{T}_{\mathrm{LN}}+\frac{32}{3r^{2}}\partial_{r}\overline{T}_{\mathrm{N}}+\frac{8}{3r^{3}}\overline{T}_{\mathrm{L}}-\frac{160}{3r^{3}}\overline{T}_{\mathrm{LN}}+\frac{64}{3r^{3}}\overline{T}_{\mathrm{N}}$
$\displaystyle\hskip
46.94687pt-\frac{5}{48}\partial_{r}^{3}T^{x^{2}y^{2}}_{\mathrm{L}}-\frac{40}{48r}\partial_{r}^{2}T^{x^{2}y^{2}}_{\mathrm{L}}+\frac{50}{48r}\partial_{r}^{2}T^{x^{2}y^{2}}_{\mathrm{LN}}-\frac{70}{48r^{2}}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{L}}+\frac{260}{48r^{2}}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{LN}}$
$\displaystyle\hskip
46.94687pt-\frac{80}{48r^{2}}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{N}}-\frac{20}{48r^{3}}T^{x^{2}y^{2}}_{\mathrm{L}}+\frac{220}{48r^{3}}T^{x^{2}y^{2}}_{\mathrm{LN}}-\frac{160}{48r^{3}}T^{x^{2}y^{2}}_{\mathrm{N}}-\frac{2T_{\mathrm{L}}^{x^{3}y}(0)}{3r^{3}}+\frac{4K}{r}\bigl{\\}}\biggl{)}$
$\displaystyle\hskip
3.41418pt+M_{\mathrm{L}}\bigg{(}-\frac{4}{r}\partial_{r}T_{\mathrm{L}}-\frac{4}{r}\partial_{r}T_{N}-\frac{4}{r^{2}}T_{\mathrm{L}}+\frac{4}{r^{2}}T_{N}+\tau\bigl{\\{}\frac{4}{3r}\partial_{r}^{3}\overline{T}_{\mathrm{L}}+\frac{4}{3r}\partial_{r}^{3}\overline{T}_{\mathrm{LN}}+\frac{20}{3r^{2}}\partial_{r}^{2}\overline{T}_{\mathrm{L}}$
$\displaystyle\hskip
36.98866pt-\frac{12}{3r^{2}}\partial_{r}^{2}\overline{T}_{\mathrm{LN}}-\frac{16}{3r^{2}}\partial_{r}^{2}\overline{T}_{\mathrm{N}}+\frac{8}{3r^{3}}\partial_{r}\overline{T}_{\mathrm{L}}-\frac{104}{3r^{3}}\partial_{r}\overline{T}_{\mathrm{LN}}+\frac{48}{3r^{3}}\partial_{r}\overline{T}_{\mathrm{N}}-\frac{8}{3r^{4}}\overline{T}_{\mathrm{L}}-\frac{56}{3r^{4}}\overline{T}_{\mathrm{LN}}$
$\displaystyle\hskip
36.98866pt+\frac{80}{3r^{4}}\overline{T}_{\mathrm{N}}-\frac{10}{48r}\partial_{r}^{3}T^{x^{2}y^{2}}_{\mathrm{L}}-\frac{10}{48r}\partial_{r}^{3}T^{x^{2}y^{2}}_{\mathrm{LN}}-\frac{50}{48r^{2}}\partial_{r}^{2}T^{x^{2}y^{2}}_{\mathrm{L}}+\frac{30}{48r^{2}}\partial_{r}^{2}T^{x^{2}y^{2}}_{\mathrm{LN}}+\frac{40}{48r^{2}}\partial_{r}^{2}T^{x^{2}y^{2}}_{\mathrm{N}}$
$\displaystyle\hskip
36.98866pt-\frac{20}{48r^{3}}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{L}}+\frac{260}{48r^{3}}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{LN}}-\frac{120}{48r^{3}}\partial_{r}T^{x^{2}y^{2}}_{\mathrm{N}}+\frac{20}{48r^{4}}T^{x^{2}y^{2}}_{\mathrm{L}}+\frac{140}{48r^{4}}T^{x^{2}y^{2}}_{\mathrm{LN}}-\frac{200}{48r^{4}}T^{x^{2}y^{2}}_{\mathrm{N}}\bigl{\\}}\biggl{)},$

where $K=5C(0)/48$ is a constant with
$C(r)=\partial_{r}^{2}(T^{x^{2}y^{2}}_{\mathrm{L}})+4\partial_{r}(T^{x^{2}y^{2}}_{\mathrm{L}})/r-10\partial_{r}(T^{x^{2}y^{2}}_{\mathrm{LN}})/r+2T^{x^{2}y^{2}}_{\mathrm{LN}}/r^{2}-22T^{x^{2}y^{2}}_{\mathrm{L}}/r^{2}+16T^{x^{2}y^{2}}_{\mathrm{N}}/r^{2}$.
This equation has the most generic form if we assume only isotropy,
homogeneity and non-helicity of the velocity flow in the vicinity of small
$St$. In order to solve this equation we should define the boundary
conditions. The magnetic field correlation function should go to zero for
infinitely large space scales. Also we would require $M_{\mathrm{L}}$ to be
finite in $r=0$ such that the auto-correlation of the magnetic field is a
local maxima. These two conditions can be summarised by

$\displaystyle\lim_{r\rightarrow 0}\partial_{r}M_{\mathrm{L}}(r,t)=0,$
$\displaystyle\lim_{r\rightarrow\infty}M_{\mathrm{L}}(r,t)=0.$ (56)

Note that if we assume incompressibility in Eq. (V.3) we retrieve Eq. (17) in
Bhat and Subramanian [37]. Except for its length, the general aspect of the
equation is unchanged for an arbitrary degree of compressibility (DOC). The
most interesting difference arises in terms that depend on $M_{\mathrm{L}}$.
In the incompressible case, these terms cancel perfectly but not when the DOC
is non-zero. We can already get the intuition that these terms will control
the time growth rate of the magnetic correlation function.

###  V.4 Small-scale limit

In this section we discuss the limit of length scales much smaller than the
turbulent forcing scale (i.e. $z~{}\equiv~{}qr~{}\ll~{}1$) of Eq. (V.3). The
Kazantsev spectrum $M_{k}(k)\sim k^{3/2}$ is predicted in the range $q\ll k\ll
k_{\eta}$. Since we consider large $R_{\mathrm{M}}$, it is sufficient to
expand our generalised equation in the limit of small $z$. We introduce two
different cases, which correspond to two initialisation for $\bm{a}$ and
$\bm{q}$. The first case is used to give a detailed derivation. However, the
second case is more general and gives rise to a lengthy calculation so we will
only present the results.

####  V.4.1 Two independent vectors

First consider the case where $\bm{a}$ and $\bm{q}$ are perfectly independent.
It is straight forward to evaluate Eq. (30) and Eq. (50) knowing that $\langle
a_{i}a_{j}a_{h}a_{l}\rangle=\delta_{(ij}\delta_{hl)}/15$ for a random vector
and we can directly plug the expansion of the correlators’ components in Eq.
(V.3). These considerations simplify strongly our generalised Kazantsev
equation such that it reduces to the following expression in the limit $z\ll
1$

$\displaystyle\frac{\partial M_{\mathrm{L}}}{\partial t}$ $\displaystyle=$
$\displaystyle
q^{2}T_{\mathrm{L}}(0)\bigg{[}\bigl{(}\frac{2\eta}{T_{\mathrm{L}}(0)}+\frac{z^{2}}{3}\bigl{)}\partial_{z}^{2}M_{\mathrm{L}}+\bigl{(}\frac{8\eta}{zT_{\mathrm{L}}(0)}+2z\bigl{)}\partial_{z}M_{\mathrm{L}}+\frac{8}{3}M_{\mathrm{L}}\bigg{]}$
(57)
$\displaystyle+\frac{a^{4}q^{4}\tau^{3}}{160}\bigg{[}\frac{z^{4}}{10}\partial_{z}^{4}M_{\mathrm{L}}+\frac{8z^{3}}{5}\partial_{z}^{3}M_{\mathrm{L}}+\frac{958z^{2}}{135}\partial_{z}^{2}M_{\mathrm{L}}+\frac{404z}{45}\partial_{z}M_{\mathrm{L}}+\frac{32}{27}M_{\mathrm{L}}\bigg{]}.$

We can further assume that $\tilde{M}_{\mathrm{L}}$ is independent of space
such that we use the ansatz
$M_{\mathrm{L}}(r,t)=\tilde{M}_{\mathrm{L}}(z)e^{\gamma\tilde{t}}$ where
$\tilde{t}=tT_{\mathrm{L}}(0)q^{2}$ and $\gamma$ is a normalised growth rate.
We also set $\bar{\tau}=\tau T_{\mathrm{L}}(0)q^{2}$ and rename
$T_{\mathrm{L}}(0)=\eta_{t}$ to stick to the conventions used in Bhat and
Subramanian [37]. After some algebra we end up with

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\bigl{(}\frac{2\eta}{\eta_{t}}+\frac{z^{2}}{3}\bigl{)}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}+\bigl{(}\frac{8\eta}{z\eta_{t}}+2z\bigl{)}\partial_{z}\tilde{M}_{\mathrm{L}}+\bigl{(}\frac{8}{3}-\gamma\bigl{)}\tilde{M}_{\mathrm{L}}$
(58)
$\displaystyle+\frac{9\bar{\tau}}{10}\bigg{[}\frac{z^{4}}{10}\partial_{z}^{4}\tilde{M}_{\mathrm{L}}+\frac{8z^{3}}{5}\partial_{z}^{3}\tilde{M}_{\mathrm{L}}+\frac{958z^{2}}{135}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}+\frac{404z}{45}\partial_{z}\tilde{M}_{\mathrm{L}}+\frac{32}{27}\tilde{M}_{\mathrm{L}}\bigg{]}.$

We now focus on the range $z_{\eta}=qr_{\eta}~{}\ll~{}z~{}\ll~{}1$, where
$\bar{\tau}$ terms cannot be neglected. Here, we use a Landau-Lifshitz
approximation [66] and consider $\bar{\tau}$ to be a small parameter. In order
to derive approximated expressions for the high order derivatives of
$\tilde{M}_{\mathrm{L}}$ as a function of the first and the second order
derivatives, we neglect $\bar{\tau}$ and $\sqrt{\eta/\eta_{t}}$ compared to
$z$

$\displaystyle
z^{3}\partial_{z}^{3}\tilde{M}_{\mathrm{L}}=-8z^{2}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}+(3\gamma_{0}-8)\partial_{z}\tilde{M}_{\mathrm{L}},$
$\displaystyle
z^{4}\partial_{z}^{4}\tilde{M}_{\mathrm{L}}=(3\gamma_{0}+58)z^{2}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}-10(3\gamma_{0}-14)\partial_{z}\tilde{M}_{\mathrm{L}},$
(59)

where $\gamma_{0}$ is the growth rate for a delta-correlated in time flow. As
a first approach (we will give in Sec. VI a more rigorous treatment) we
neglect $\sqrt{\eta/\eta_{t}}$. The two expressions for high order derivatives
can be plugged into Eq. (58) to obtain

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\frac{9\bar{\tau}}{10}\bigg{[}z^{2}\bigl{(}\frac{3\gamma_{0}}{10}+\frac{13}{135}\bigl{)}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}+z\bigl{(}\frac{9\gamma_{0}}{5}+\frac{78}{135}\bigl{)}\partial_{z}\tilde{M}_{\mathrm{L}}$
(60)
$\displaystyle+\frac{32}{27}\tilde{M}_{\mathrm{L}}\bigg{]}+\frac{z^{2}}{3}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}+2z\partial_{z}\tilde{M}_{\mathrm{L}}+\bigl{(}\frac{8}{3}-\gamma\bigl{)}\tilde{M}_{\mathrm{L}}.$

It is pretty obvious that this equation admits a power law solution
$\tilde{M}_{\mathrm{L}}\sim z^{-\lambda}$. Solving for $\lambda$ we find

$\lambda=\frac{5}{2}\pm\frac{i}{2}\bigg{[}4\frac{8+16\bar{\tau}/5-3\gamma}{1+81\gamma_{0}\bar{\tau}/100+13\bar{\tau}/50}-25\bigg{]}^{1/2}.$
(61)

We find that the real part of $\lambda$ is $5/2$, which is exactly the same as
Bhat and Subramanian [37] and is expected for a Kazantsev spectrum. Gruzinov
_et al._ [67] have argued that the growth rate, in the limit of
$R_{\mathrm{M}}\rightarrow\infty$ is given by finding a value of $\lambda$
such that $\mathrm{d}\gamma/\mathrm{d}\lambda=0$. We can then plug that value
into Eq. (61), $\gamma$ is thus given by

$\gamma=\frac{7}{12}-\frac{147}{320}\bar{\tau},$ (62)

where we used the self-consistent value for $\gamma_{0}=7/12$. Note that the
complete expression for the growth rate of the dynamo is then
$\gamma_{\mathrm{tot}}=\gamma T_{\mathrm{L}}(0)q^{2}$.

####  V.4.2 Arbitrary degree of compressibility

The main problem with the initialisation that we just presented is that it
does not include any parameter to control the degree of compressibility (DOC).
We define the DOC by

$\sigma_{c}\equiv\frac{\langle(\nabla\cdot\bm{u})^{2}\rangle}{\langle(\nabla\times\bm{u})^{2}\rangle}=\frac{\langle
a_{i}a_{j}q_{i}q_{j}\rangle}{\langle\epsilon_{ijk}\epsilon_{ihl}a_{k}a_{l}q_{j}q_{h}\rangle},$
(63)

where $\epsilon_{ijk}$ is the Levi-Civita symbol; the third expression is
obtained after averaging over $\psi$. The DOC is then zero for an
incompressible flow and goes to infinity for a fully irrotational flow. In
order to derive an equation for an arbitrary DOC we set $\bm{q}$ to be a
random vector with norm $q$ and $\bm{a}$ defined by

$a_{i}=b(\tilde{P}_{ij}\hat{A}_{j}\sin{(\theta)}+\hat{q}_{j}\hat{A}_{j}\hat{q}_{i}\cos{(\theta)}),$
(64)

where as before $\bm{A}$ is a random vector of norm $A$ and
$\hat{A}_{j}=A_{j}/A$. The two parameters $b$ and $\theta$ (that are constant)
allow to control, respectively, the norm of $\langle\bm{a}^{2}\rangle$ and the
DOC. In such a parametrisation the component of $\bm{A}$ along $\bm{q}$ is
always rescaled by $\cos{(\theta)}$ whereas the component of $\bm{A}$
orthogonal to $\bm{q}$ in the plane described by $\bm{A}-\bm{q}$ is always
recaled by $\sin{(\theta)}$. This parametrisation is taken for convenience,
and $\theta$ can be interpreted as the mean absolute angle between $\bm{a}$
and $\bm{q}$. Although this parametrisation might seem arbitrary, we can show
that the results we derive here are independent on the exact evaluation of
$a_{i}$ as long as $\sigma_{c}$ is uniquely defined (see Appendix A). Under
such considerations

$\displaystyle\sigma_{c}=\frac{1}{2\tan{(\theta)}^{2}},$
$\displaystyle\langle\bm{a}^{2}\rangle=b^{2}\bigg{(}\frac{2}{3}\sin{(\theta)}^{2}+\frac{1}{3}\cos{(\theta)}^{2}\bigg{)}.$
(65)

We directly see that the value $\theta=\pi/2$ represents the incompressible
case and $\theta=0$ the fully irrotational one. Note that due to the random
behavior of $\bm{A}$ we have $\langle\xi\rangle=0$.

We apply the exact same methodology as for the first initialisation, which
means we expand the two-point correlators, use the ansatz, and re-express the
high order derivatives with the first and the second order ones. The
expressions for the velocity correlators with this parametrisation can be
found in the Appendix B (Eq. B). Furthermore, we define the two functions

$\displaystyle\epsilon(\theta)=\frac{216}{5(\Omega+3)^{2}}\biggl{[}\frac{\Omega+3}{5-\Omega}\bigg{(}\frac{1}{24}\Omega_{1}-\frac{3}{28}\Omega_{2}+\frac{3}{40}\Omega_{3}\bigg{)}$
$\displaystyle\hskip
14.22636pt\times\bigg{(}5\gamma_{0}-\frac{40}{\Omega+3}\bigg{)}+\frac{157}{420}\Omega_{1}-\frac{599}{630}\Omega_{2}+\frac{121}{180}\Omega_{3}\biggl{]},$
$\displaystyle\zeta(\theta)=\frac{216}{5(\Omega+3)^{2}}\biggl{[}\frac{2}{3}\Omega_{1}-\frac{14}{9}\Omega_{2}+\frac{8}{9}\Omega_{3}\biggl{]}.$
(66)

The set of five parameters $\Omega_{\mathrm{i}}$ appears very naturally in the
derivation of Eq. (B) and depends only on $\theta$; exact expressions are
given in Eq. (B). From the velocity correlators we have $\eta_{t}=\tau
b^{2}(\Omega+3)/72$. However the parameter $b$ is still free and we can set it
to $b^{2}=6a^{2}/(\Omega+3)$ such that $\langle\bm{a}^{2}\rangle=a^{2}$. As a
result $\eta_{t}=\tau a^{2}/12$, and the normalised correlation time can be
evaluated to $\bar{\tau}=St^{2}/12$. The normalised correlation time is then
fully controlled by $St$; independently on the choice of DOC.

The resulting equation is

$\displaystyle\left(\frac{2\eta}{\eta_{t}}+z^{2}\frac{5-\Omega}{5(\Omega+3)}\right)\partial_{z}^{2}\tilde{M}_{\mathrm{L}}$
$\displaystyle+\left(\frac{8\eta}{z\eta_{t}}+6z\frac{5-\Omega}{5(\Omega+3)}\right)\partial_{z}\tilde{M}_{\mathrm{L}}+\left(\frac{8}{\Omega+3}-\gamma\right)\tilde{M}_{\mathrm{L}}$
$\displaystyle+\bar{\tau}\bigg{[}\epsilon(\theta)z^{2}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}+6\epsilon(\theta)z\partial_{z}\tilde{M}_{\mathrm{L}}+\zeta(\theta)\tilde{M}_{\mathrm{L}}\bigg{]}$
$\displaystyle=$ $\displaystyle 0,$

Similarly to the first case we compute the growth rate and scale factor of the
power law solution in the limit of large $R_{\mathrm{M}}$, we find that the
real part of $\lambda$ is still $5/2$. The growth rate is given this time by

$\displaystyle\gamma_{0}=\frac{7+5\Omega}{4(\Omega+3)},$
$\displaystyle\gamma=\gamma_{0}+\bar{\tau}\bigl{[}\zeta(\theta)-\frac{25}{4}\epsilon(\theta)\bigl{]}.$
(68)

We already see from this quick evaluation that the Kazantsev spectrum seems to
be preserved even with an arbitrary DOC and correlation time. However to
confirm this first approach and include the effects of finite magnetic
Reynolds numbers, we need to study more carefully the solutions to Eq.
(V.4.2).

##  VI Finite magnetic resistivity solutions

The scaling solution that has been derived in the previous section only works
if the term $\sqrt{\eta/\eta_{t}}$ is neglected. However to include effects
due to a finite magnetic resistivity we should not systematically neglect it.
A WKB approximation can be used to evaluate the solution of Eq. (V.4.2)
including the finite resistivity. An explicit derivation of the WKB solutions
can be found in Appendix C. We only review the main results obtained for the
magnetic power spectrum and the growth rate of the dynamo including a finite
magnetic Reynolds number.

Parameters | Incompressible | Intermediate | Irrotational
---|---|---|---
$\theta$ | $\frac{\pi}{2}$ | $\frac{\pi}{4}$ | 0
$\sigma_{c}$ | $0$ | $\frac{1}{2}$ | $\infty$
$\lambda_{k}$ | $\frac{3}{2}$ | $\frac{3}{2}$ | $\frac{3}{2}$
$\gamma_{0}$ | $\frac{3}{4}$ | $\frac{7}{12}$ | $\frac{1}{4}$
$\gamma_{1}$ | $-\frac{135}{224}$ | $-\frac{147}{320}$ | $-\frac{1017}{2240}$
$\gamma_{R_{\mathrm{M}}}$ | $\frac{1}{5}+\bar{\tau}\frac{27}{280}$ | $\frac{1}{3}+\bar{\tau}\frac{293}{1200}$ | $\frac{3}{5}+\bar{\tau}\frac{3429}{2800}$
$R_{\mathrm{M},\mathrm{thresh}}$ | $\sim 3\cdot 10^{5}$ | $\sim 1.5\cdot 10^{5}$ | $\sim 7\cdot 10^{4}$
Table 1: Presentation of the velocity field and magnetic spectrum parameters
for three types of flow: incompressible, irrotational, and intermediate (see
Sec. V.4).

###  VI.1 Growth rate

The normalised growth rate of the dynamo that includes contributions from the
magnetic resistivity (through $R_{\mathrm{M}}$), compressibility (through
$\Omega$ and $\theta$) and finite correlation time (through $\bar{\tau}$) is
found to be

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{\pi}{\ln{(R_{\mathrm{M}})}}\bigg{)}^{2}\bigg{[}\frac{5-\Omega}{5(\Omega+3)}+\bar{\tau}\epsilon(\theta)\bigg{]}+\frac{7+5\Omega}{4(\Omega+3)}$
(69)
$\displaystyle+\bar{\tau}\bigg{[}\zeta(\theta)-\frac{25}{4}\epsilon(\theta)\bigg{]},$
$\displaystyle\equiv-\bigg{(}\frac{\pi}{\ln{(R_{\mathrm{M}})}}\bigg{)}^{2}\gamma_{R_{\mathrm{M}}}+\gamma_{0}+\bar{\tau}\gamma_{1},$

where the functions $\epsilon(\theta)$ and $\zeta(\theta)$ are given in Eq.
(V.4.2) and we have introduced the different components of the growth rate
$\gamma_{R_{\mathrm{M}}}$, $\gamma_{0}$, and $\gamma_{1}$. In Tab. 1 we list
the different parameters of the flow and the magnetic spectrum for three
regimes, namely incompressible ($\nabla\cdot\bm{u}=0$), irrotational
($\nabla\times\bm{u}=0$), and the intermediate case treated in Sec. V.4.1. To
get a better intuition on the results presented here, we display in Fig. 1 the
DOC dependency of the two main contributions to the growth rate. From the
evolution of $\gamma_{0}$ it is very clear that the compressibility tends to
decrease the growth rate of the magnetic energy spectrum of the dynamo.
Moreover, $\gamma_{0}$ is comprised between 0.75 and 0.25 which indicates that
the dynamo action always exists. It is interesting to note that $\gamma_{0}$
is a monotonously decreasing function of the DOC, whereas $\gamma_{1}$ has a
maximum around $\sigma_{c}\sim 2$.

Figure 1: Evolution of the two main contributions to the magnetic spectrum
growth rate with respect to the DOC for large magnetic Reynolds number. The
left axis represents the main contribution $\gamma_{0}$ and the right axis the
contribution to the growth rate related to the correlation time $\gamma_{1}$
of Eq. (69). Dashed lines correspond to the value for a fully irrotational
flow ($\sigma_{c}\rightarrow\infty$).

In Fig. 2 we study more precisely the dependence of the growth rate on
$R_{\mathrm{M}}$. As expected from Eq. (69), $\gamma$ increases as
$R_{\mathrm{M}}$ increases. In practice, due to the WKB approximation, there
is a limiting value on the magnetic Reynolds number
($R_{\mathrm{M},\mathrm{thresh}}$) for which Eq. (69) is valid such that we
need to keep $R_{\mathrm{M}}>R_{\mathrm{M},\mathrm{thresh}}$. The impact on
the total growth rate of the DOC is stronger when $R_{\mathrm{M}}$ is small.
In this work, we only considered first order corrections; and discrepancies
can already represent $\sim~{}85\%$ for the lowest values of $R_{\mathrm{M}}$
presented. However, the correlation time has a negligible impact on the total
growth rate in the limit $St\ll 1$. Indeed, the correlation time enters the
computation through $St$ which itself contributes through $\bar{\tau}\propto
St^{2}\ll 1$.

Figure 2: Ratio of the growth rate and its main contribution for a few values
of $R_{\mathrm{M}}$ and $St=10^{-2}$. Dashed lines correspond to the value for
a fully irrotational flow ($\sigma_{c}\rightarrow\infty$).

###  VI.2 Magnetic power spectrum

In the range of interest, $z_{\eta}~{}\ll~{}z~{}\ll~{}1$, we find the solution
for the longitudinal two-point magnetic correlation function

$M_{\mathrm{L}}(z,t)=e^{\gamma\tilde{t}}z^{-5/2}M_{0}\cos{\bigg{[}\frac{\pi}{\ln{(R_{\mathrm{M}})}}\ln{\bigg{(}\frac{z}{z_{0}}\bigg{)}}\bigg{]}},$
(70)

where $\gamma$ is given by Eq. (69). This result is already very interesting
as we can identify a power law $z^{-5/2}$ independent of the correlation time
that dominates the spectrum compared to the slowly varying $\cos{[\ln{(z)}]}$
function. However the Kazantsev spectrum we are interested in predicts that
the magnetic power spectrum scales as $M_{k}(k)\sim k^{-3/2}$ in the range
$q\ll k\ll k_{\eta}$. We can show (see Appendix D) that the magnetic power
spectrum and the longitudinal two-point correlation function are related by

$M_{k}(k,t)=\frac{1}{\pi}\int(kr)^{3}M_{\mathrm{L}}(r,t)j_{1}(kr)~{}\mathrm{d}r.$
(71)

The Bessel function $j_{1}(x)$ is very peaked around $x\sim 2$, so the
dominant part of the integral is around $k\sim 1/r$. Thus for
$M_{\mathrm{L}}(r)\sim r^{\lambda}$ we have $M_{k}(k)\sim k^{\lambda_{k}}$
with $\lambda_{k}=-(1+\lambda)$. Plugging $\lambda=-5/2$ gives the well-known
Kazantsev spectrum $M_{k}(k)\sim k^{3/2}$ even for the compressible and time-
correlated flow considered here. Note that the main contribution to the power
spectrum derived does not depend on the any of the parameters of the flow. The
last row of Tab. 1 corresponds to the minimal value of $R_{\mathrm{M}}$ for
which the WKB approximation holds (see Appendix C.3). We see that for most
astrophysical application of the small-scale dynamo our derived results remain
valid.

##  VII Discussion and conclusions

Several authors have previously modeled the kinematic phase of the small-scale
dynamo with the Kazantsev theory [see e.g. 34, 41, 36, 68]. They found that
the Kazantsev spectrum is preserved, even for a compressible flow. However,
they often assumed Gaussian statistics of the velocity field, such that the
flow is delta-correlated in time. There are several examples of analytic
treatments that include a finite correlation time: Kolekar  _et al._ [51] and
Lamburt and Sokoloff [69] used a similar approach to our work but in the
context of the mean-field dynamos, Schekochihin and Kulsrud [70] and Kleeorin
_et al._ [71] considered a general case of fluctuation dynamo, and Bhat and
Subramanian [62] solved the incompressible case. Most of the theoretical
studies, if not all, have found that the Kazantsev spectrum is preserved even
if compressibility, finite correlation time, or finite resistivity are
considered. Our work shows that the combined effect of the three on the
Kazantsev spectrum is negligible, as Eq. 70 scales mostly with the power-law
$z^{-5/2}$. However, our results are only derived for first order corrections
from the correlation time; as higher order corrections are usually hard to
treat.

Besides the shape of the magnetic energy spectrum, the dynamo growth rate
$\gamma$ is of particular interest. Our results for $\gamma$ are similar to
the ones obtained by Kulsrud and Anderson [34] and Schekochihin  _et al._ [36]
for the limit of incompressibility. Schekochihin  _et al._ [41] also derived a
formula for the growth rate for an arbitrary DOC for a delta-correlated in
time flow. They found that $\gamma$ ranges between $3/4$ for an incompressible
flow and $1/8$ for a fully irrotational flow. It does not match our results by
a factor of two in the limit of the fully irrotational flow. Similarly, the
growth rate related to the finite resistivity $\gamma_{R_{\mathrm{M}}}$
matches their result in the incompressible case but is overestimated by a
factor of 2 in the fully compressible limit. The discrepancy can be solved
when we consider instead of $\gamma$ alone the complete growth rate, namely
$\gamma\eta_{t}q^{2}$. In their paper Schekochihin  _et al._ [41] defined the
initial growth rate from the velocity correlators in Fourier space while we
define it from real space. If we transfer a factor $(\Omega+3)/4$ from
$\eta_{t}$ to $\gamma$, our growth rate matches theirs in both limits. It is
worth mentioning that Illarionov and Sokoloff [72] derived a growth rate that
depends on the Strouhal number but does not match ours exactly. However, they
also found that $\gamma_{\mathrm{tot}}\propto aqSt$ and that $\bar{\tau}$
reduces the growth rate. Although, we derived a expression for the growth rate
that includes the DOC, a finite resistivity and time correlations our
treatment of turbulence is very simple due to the imposed velocity field. A
more rigorous treatment [16] highlights that the growth rate might also be a
power law of the Reynolds number. Rogachevskii and Kleeorin [42] used a
different approach to the problem as they impose directly a velocity
correlation function instead of the velocity field itself. Their results match
ours for the magnetic power spectrum as long as $T_{\mathrm{L}}(r)\sim r^{2}$;
but the growth rate differs as their velocity spectrum is different from ours.
This highlights that the Kazantsev spectrum should be preserved for a large
class of flows. Using a similar approach, some authors [73, 74, 75] studied
more deeply the influence of the Prandtl number on the growth rate or magnetic
spectrum while we fix the Reynolds number in our work. These papers suggest
that the growth rate we derived should also depend on the Prandtl number if an
even more general case is considered; this relation is beyond the scope of
this paper as we want to focus on the effect of compressibility.

From a numerical point of view it seems indeed that a slope close to $3/2$ in
the magnetic energy spectrum can be observed in both incompressible and
compressible MHD simulations at large length scales [see e.g. 76, 77, 78, 79,
80]. Although the slope measured in simulations is often close to the
theoretical prediction, small discrepancies can still arise. One of the
possible explanations can directly appear from the size of the simulation box.
Indeed $k$ has to be small but simulations are limited in resolution. This
often can lead to an insufficient separation of spatial scales. A related
problem is the assumption of very large hydrodynamic and/or magnetic Reynolds
numbers in theoretical models. The required large values of these two numbers
make a comparison between numerical simulations and theory hard. Kopyev  _et
al._ [81] also found that time irreversible flows can generate a nontrivial
deviation to the Kazantsev spectrum. Regarding the growth rate of the dynamo,
its reduction by the correlation time has also been observed in numerical
studies [82]. Further discussions of the current state of dynamo numerical
simulations can be found in Brandenburg  _et al._ [83].

In conclusion, we have given an example of an analytical treatment for the
fluctuation dynamo in the most generic case of a compressible flow with a
finite correlation time. To this end, we proposed a framework to study the
cumulative effects of a finite correlation time and an arbitrary degree of
compressibility by generalising the former work of Bhat and Subramanian [37].
We used the renovating flow method which assumes a very crude flow that does
not allow for a very complex modelling of turbulence but keep the analytical
treatment tractable. We derived a generalisation to the Kazantsev equation in
real space (Eq. V.3) that is valid at any scale. We note however that if we
assume an incompressible flow that is delta-correlated in time at this point
we retrieve the original Kazantsev equation. This equation describes the time
evolution of the two-point magnetic correlation function $M_{\mathrm{L}}$ from
the velocity correlators and the spatial derivatives of $M_{\mathrm{L}}$ up to
the fourth order. We then studied solutions for length scales much smaller
than turbulent forcing scale (i.e. $qr\ll 1$).

By the use of the WKB approximation, we derived formulas for the growth rate
and slope of the magnetic power spectrum $M_{k}(k)$ for large magnetic
Reynolds number $R_{\mathrm{M}}\gg 1$ and small Strouhal number $St\ll 1$. In
particular, it allowed to capture the effect of finite magnetic diffusivity.
Furthermore we could define a lower bound on $R_{\mathrm{M}}$ for which our
results should hold, $R_{\mathrm{M},\mathrm{thresh}}\sim 10^{5}$, which is
smaller than most of the typical values in astrophysical objects. Although the
growth rate showed dependencies on both the degree of compressibility and the
correlation time, the Kazantsev spectrum seemed to be preserved, i.e.
$M_{k}(k)\sim k^{3/2}$, independently of $\tau$ or $\sigma_{c}$. Our results
are derived in a very special context, namely for a renovating flow. But our
predictions regarding the magnetic field spectrum seem robust in the sense
that both numerical and theoretical studies agree with the conservation of the
Kazantsev spectrum for compressible and time-correlated flows.

###### Acknowledgements.

DRGS gratefully acknowledges support by the ANID BASAL projects ACE210002 and
FB210003, as well as via the Millenium Nucleus NCN19-058 (TITANs). YC and DRGS
thank for funding via Fondecyt Regular (project code 1201280). JS acknowledges
the support by the Swiss National Science Foundation under Grant No. 185863.

##  Appendix A General initialisation for delta-correlated in time flow

We would like to review an even more general initialisation than the one
presented in Sec. V.4.2. We only consider a delta-correlated in time flow but
this discussion could in principle be generalised to a finite correlation
time. A very general expression for $\bm{a}$ that preserves isotropy is

$a_{i}=b(\tilde{P}_{ij}\hat{A}_{j}f_{1}+\hat{q}_{j}\hat{A}_{j}\hat{q}_{i}f_{2}),$
(72)

where $f_{1}$ and $f_{2}$ are two constants. In order to control the norm $a$
we should impose that $f_{1/2}$ are between minus one and one. It is then
straightforward to show that $\sigma_{c}=f_{2}^{2}/(2f_{1}^{2})$. Once again
we compute the velocity correlators and plug in Eq. (V.3). To simplify this
derivation we also neglect the resistivity $\eta$. We find the equation

$\displaystyle\frac{z^{2}}{5}\frac{2f_{1}^{2}+3f_{2}^{2}}{2f_{1}^{2}+f_{2}^{2}}\partial_{z}^{2}\tilde{M}_{\mathrm{L}}+\frac{6z}{5}\frac{2f_{1}^{2}+3f_{2}^{2}}{2f_{1}^{2}+f_{2}^{2}}\partial_{z}\tilde{M}_{\mathrm{L}}$
$\displaystyle\hskip
42.67912pt+\big{(}4\frac{f_{1}^{2}+f_{2}^{2}}{2f_{1}^{2}+f_{2}^{2}}-\gamma\big{)}\tilde{M}_{\mathrm{L}}=0,$
(73)

that again allows some power law solution. If we follow the same approach than
in Sec. V.4.2, we find

$\displaystyle\lambda=\frac{5}{2}\pm ig(f_{1},f_{2},\gamma),$
$\displaystyle\gamma=\frac{6f_{1}^{2}+f_{2}^{2}}{4(2f_{1}^{2}+f_{2}^{2})},$
(74)

with $g(f_{1},f_{2},\gamma)$ a function that characterises the growth rate.
Once again the power spectrum slope is constant and $\gamma=3/4$ for an
incompressible flow, $\gamma=1/4$ for a fully irrotational one and
$\gamma=7/12$ if $f_{1}=f_{2}$. If we define

$\displaystyle f_{1}=\sin{\theta},$ $\displaystyle f_{2}=\cos{\theta},$ (75)

we retrieve the initialisation presented. This is convenient as we reduced the
number of parameters to only $\theta$ to completely and uniquely define
$\sigma_{c}$. Also it presents the option to work with another more natural
parameter as in this case $f_{1}^{2}$ and $f_{2}^{2}$ are related by
$f_{1}^{2}+f_{2}^{2}=1$. In fact, we just showed that the exact initialisation
does not matter as long as $\sigma_{c}$ is uniquely defined and that we can
choose the most convenient one. Note also that $f_{2}=\sqrt{2}\cos{(\theta)}$
is also an option that keeps the norm of $\bm{a}$ independent of the DOC.

##  Appendix B Complementary expressions

We display in Tab. B1 the main tools used to contract Eq. (53) with
$\hat{r}_{ij}$.

Expression | Reduced form
---|---
$\hat{r}_{ii}$ | $1$
$\hat{P}_{ii}$ | $2$
$\hat{P}_{ij}\hat{r}_{il}$ | $0$
$\hat{P}_{ij}\hat{P}_{il}$ | $\hat{P}_{jl}$
$\partial_{i}$ | $\hat{r}_{i}\partial_{r}$
$\partial_{i}\partial_{j}$ | $\hat{r}_{ij}\partial_{r}^{2}+\hat{P}_{ij}\frac{\partial_{r}}{r}$
$\partial_{i}\hat{r}_{j}$ | $\frac{1}{r}\hat{P}_{ij}$
$\partial_{i}\hat{r}_{jl}$ | $\frac{1}{r}(\hat{r}_{l}\hat{P}_{ij}+\hat{r}_{j}\hat{P}_{il})$
$\partial_{i}\hat{P}_{jl}$ | $-\frac{1}{r}(\hat{r}_{l}\hat{P}_{ij}+\hat{r}_{j}\hat{P}_{il})$
$T_{ii}$ | $2T_{\mathrm{N}}+T_{\mathrm{L}}$
$\hat{r}_{i}T_{ij}$ | $\hat{r_{j}}T_{\mathrm{L}}$
$\hat{r}_{ij}T_{ij}$ | $T_{\mathrm{L}}$
$\hat{P}_{ij}T_{ij}$ | $2T_{\mathrm{N}}$
$M_{ij}T_{il}$ | $\hat{P}_{jl}M_{\mathrm{N}}T_{\mathrm{N}}+\hat{r}_{jl}M_{\mathrm{L}}T_{\mathrm{L}}$
$T_{iijj}$ | $\overline{T}_{\mathrm{L}}+8\overline{T}_{\mathrm{N}}+4\overline{T}_{\mathrm{LN}}$
$T_{iijl}$ | $\hat{r}_{jl}\overline{T}_{\mathrm{L}}+4\hat{P}_{jl}\overline{T}_{\mathrm{N}}+(\hat{P}_{jl}+2\hat{r}_{jl})\overline{T}_{\mathrm{LN}}$
$\hat{r}_{i}T_{ijhl}$ | $\hat{r}_{jhl}\overline{T}_{\mathrm{L}}+(\hat{P}_{jl}\hat{r}_{h}+\hat{P}_{jh}\hat{r}_{l}+\hat{P}_{hl}\hat{r}_{j})\overline{T}_{\mathrm{LN}}$
$\hat{r}_{ij}T_{ijhl}$ | $\hat{r}_{hl}\overline{T}_{\mathrm{L}}+\hat{P}_{hl}\overline{T}_{\mathrm{LN}}$
$\hat{P}_{ij}T_{ijhl}$ | $4\hat{P}_{hl}\overline{T}_{\mathrm{N}}+2\hat{r}_{hl}\overline{T}_{\mathrm{LN}}$
$\partial_{i}T_{ijhl}$ | $\hat{r}_{jhl}(\partial_{r}\overline{T}_{\mathrm{L}}+\frac{2}{r}\overline{T}_{\mathrm{L}}-\frac{6}{r}\overline{T}_{\mathrm{LN}})+(\hat{r}_{j}\hat{P}_{hl}+\hat{r}_{h}\hat{P}_{jl}+\hat{r}_{l}\hat{P}_{jh})(\partial_{r}\overline{T}_{\mathrm{LN}}+\frac{4}{r}\overline{T}_{\mathrm{LN}}-\frac{4}{r}\overline{T}_{\mathrm{N}})$
$\partial_{i}\partial_{j}T_{ijhl}$ | $\hat{r}_{hl}(\partial_{r}^{2}\overline{T}_{\mathrm{L}}+\frac{4}{r}\partial_{r}\overline{T}_{\mathrm{L}}-\frac{10}{r}\partial_{r}\overline{T}_{\mathrm{LN}}+\frac{2}{r^{2}}\overline{T}_{\mathrm{L}}-\frac{22}{r^{2}}\overline{T}_{\mathrm{LN}}+\frac{16}{r^{2}}\overline{T}_{\mathrm{N}})$
$+\hat{P}_{hl}(\partial_{r}^{2}\overline{T}_{\mathrm{LN}}+\frac{8}{r}\partial_{r}\overline{T}_{\mathrm{LN}}-\frac{4}{r}\partial_{r}\overline{T}_{\mathrm{N}}+\frac{12}{r^{2}}\overline{T}_{\mathrm{LN}}-\frac{12}{r^{2}}\overline{T}_{\mathrm{N}})$
Table B1: Summary of the most basic tools to contract Eq. (53) with
$\hat{r}_{ih}$.

If we carry out all the algebra of Sec. V.4.2 we get the following expressions
for the two-point velocity correlators

$\displaystyle T_{ij}=\frac{\tau
b^{2}}{12}\bigg{\\{}\hat{r}_{ij}(\frac{\Omega+1}{2}+\Omega\partial_{z}^{2})+\hat{P}_{ij}(\frac{\Omega+1}{2}+\Omega\frac{\partial_{z}}{z})\bigg{\\}}j_{0}(z),$
$\displaystyle
T^{x^{2}y^{2}}_{ijhl}=\frac{\tau^{2}b^{4}}{120}\bigg{\\{}\hat{r}_{ijhl}\bigl{[}\frac{3}{16}\Omega_{1}\partial_{z}^{4}+\frac{3}{2}\Omega_{2}\partial_{z}^{2}+3\Omega_{3}+6\frac{\Omega_{\mathrm{tot}}}{j_{0}(2z)}\bigl{]}$
$\displaystyle\hskip
2.84544pt+\hat{r}_{(ij}\hat{r}_{hl)}\bigl{[}\frac{3}{16}\Omega_{1}(\frac{\partial_{z}^{2}}{z^{2}}-\frac{\partial_{z}}{z^{3}})+\frac{1}{2}\Omega_{2}\frac{\partial_{z}}{z}+\Omega_{3}+2\frac{\Omega_{\mathrm{tot}}}{j_{0}(2z)}\bigl{]}$
$\displaystyle\hskip
2.84544pt+\hat{P}_{(ij}\hat{r}_{hl)}\bigl{[}\frac{3}{16}\Omega_{1}(\frac{\partial_{z}^{3}}{z}-2\frac{\partial_{z}^{2}}{z^{2}}+2\frac{\partial_{z}}{z^{3}})+\frac{1}{4}\Omega_{2}(\partial_{z}^{2}+\frac{\partial_{z}}{z})$
$\displaystyle\hskip
85.35826pt+\Omega_{3}+2\frac{\Omega_{\mathrm{tot}}}{j_{0}(2z)}\bigl{]}\bigg{\\}}j_{0}(2z),$
$\displaystyle
T^{x^{3}y}_{ijhl}=\frac{\tau^{2}b^{4}}{40}\bigg{\\{}\hat{r}_{ijhl}\bigl{[}3\Omega_{1}\partial_{z}^{4}+6\Omega_{2}\partial_{z}^{2}+3\Omega_{3}\bigl{]}$
$\displaystyle\hskip
2.84544pt+\hat{r}_{(ij}\hat{r}_{hl)}\bigl{[}3\Omega_{1}(\frac{\partial_{z}^{2}}{z^{2}}-\frac{\partial_{z}}{z^{3}})+2\Omega_{2}\frac{\partial_{z}}{z}+\Omega_{3}\bigl{]}$
$\displaystyle\hskip
2.84544pt+\hat{P}_{(ij}\hat{r}_{hl)}\bigl{[}3\Omega_{1}(\frac{\partial_{z}^{3}}{z}-2\frac{\partial_{z}^{2}}{z^{2}}+2\frac{\partial_{z}}{z^{3}})+\Omega_{2}(\partial_{z}^{2}+\frac{\partial_{z}}{z})$
$\displaystyle\hskip 113.81102pt+\Omega_{3}\bigl{]}\bigg{\\}}j_{0}(z),$ (76)

Where the set of five parameters is defined as follow

$\displaystyle\begin{aligned}
\Omega=2\sin{(\theta)}^{2}-1,&&\Omega_{1}=\Omega^{2},&&\Omega_{2}=\Omega\frac{\Omega+1}{2},\end{aligned}$
$\displaystyle\begin{aligned}
\Omega_{3}=\bigl{(}\frac{\Omega+1}{2}\bigl{)}^{2},&&\Omega_{\mathrm{tot}}=\frac{\Omega_{1}}{5}-\frac{2\Omega_{2}}{3}+\Omega_{3}.\end{aligned}$
(77)

These five parameters appear very naturally in the derivation of the velocity
correlators that is why we decided not to reduce the expressions to a single
dependency on $\Omega$. Note that $\Omega=1$ for an incompressible flow and
$\Omega=-1$ for a fully irrotational one.

##  Appendix C WKB solutions derivation

The scaling solution that has been derived in the previous section only works
if the term $\sqrt{\eta/\eta_{t}}$ is neglected. However to include effects
due to a finite magnetic resistivity we should not systematically neglect it.
A WKB approximation can be used to evaluate the solution of Eq. (V.4.2)
including the finite resistivity.

###  C.1 The WKB approximation

The WKB (Wentzel–Kramers–Brillouin) approximation is first introduced in 1926
[84, 85]. In particular this approximation method has been extensively used in
quantum mechanics to solve the Schrödinger equation [86, 87]. Formally the
method can be used to solve equations of the type

$\frac{\mathrm{d}^{2}\Theta}{\mathrm{d}x^{2}}+p(x)\Theta=0,$ (78)

where the WKB solutions to this equation are linear combinations of

$\Theta=\frac{1}{p^{1/4}}\exp\\!\bigg{(}\pm i\int
p^{1/2}~{}\mathrm{d}x\bigg{)}.$ (79)

We call turning points the value of $x$ where $p(x)$ is zero. In a given
interval if $p(x)<0$ the solution is in the form of growing and decaying
exponential however if $p(x)>0$ we have an oscillatory regime. Moreover the
solutions need to satisfy boundary conditions, especially it is common to
impose $\Theta(x)\rightarrow~{}0$ for $x\rightarrow\pm\infty$.

###  C.2 Magnetic spectrum and growth rate at finite magnetic Reynolds number

In the context of dynamos the WKB approximation is commonly used to derive the
growth rate of the two-point correlation function of magnetic fluctuations.
Reconsider Eq. (V.4.2), which is valid in the limit $z\ll 1$. To apply the WKB
approximation we define a new coordinate, which is more convenient to use
[35], as $e^{x}=\bar{z}\equiv\sqrt{\eta_{t}/\eta}z$. With this new coordinate
Eq. (V.4.2) becomes

$\displaystyle\bigg{(}\frac{\mathrm{d}^{2}\tilde{M}_{\mathrm{L}}}{\mathrm{d}x^{2}}-\frac{\mathrm{d}\tilde{M}_{\mathrm{L}}}{\mathrm{d}x}\bigg{)}\bigg{(}\bar{\tau}\epsilon(\theta)+\frac{5-\Omega}{5(\Omega+3)}+\frac{2}{\bar{z}^{2}}\bigg{)}$
$\displaystyle\hskip
14.22636pt+\frac{\mathrm{d}\tilde{M}_{\mathrm{L}}}{\mathrm{d}x}\bigg{(}6\bar{\tau}\epsilon(\theta)+6\frac{5-\Omega}{5(\Omega+3)}+\frac{8}{\bar{z}^{2}}\bigg{)}$
$\displaystyle\hskip
36.98866pt+\tilde{M}_{\mathrm{L}}\bigg{(}\bar{\tau}\zeta(\theta)+\frac{8}{\Omega+3}-\gamma\bigg{)}=0.$
(80)

To simplify notations we rewrite

$\bigg{(}\frac{\mathrm{d}^{2}\tilde{M}_{\mathrm{L}}}{\mathrm{d}x^{2}}-\frac{\mathrm{d}\tilde{M}_{\mathrm{L}}}{\mathrm{d}x}\bigg{)}A(x,\theta)+\frac{\mathrm{d}\tilde{M}_{\mathrm{L}}}{\mathrm{d}x}B(x,\theta)+\tilde{M}_{\mathrm{L}}C(x,\theta)=0,$
(81)

where the three functions are simply

$\displaystyle A(x,\theta)$
$\displaystyle=\bar{\tau}\epsilon(\theta)+\frac{5-\Omega}{5(\Omega+3)}+\frac{2}{\bar{z}^{2}},$
$\displaystyle B(x,\theta)$
$\displaystyle=6\bar{\tau}\epsilon(\theta)+6\frac{5-\Omega}{5(\Omega+3)}+\frac{8}{\bar{z}^{2}},$
$\displaystyle C(x,\theta)$
$\displaystyle=\bar{\tau}\zeta(\theta)+\frac{8}{\Omega+3}-\gamma.$ (82)

We further assume that $\tilde{M}_{\mathrm{L}}$ can be expressed as a product
of two functions $\tilde{M}_{\mathrm{L}}=g(x)W(x)$. The idea is to impose
certain relations on $g(x)$ such that all first order derivatives of $W(x)$
are cancelled, leading us to an equation that has the desired form. If we take

$\frac{\mathrm{d}g}{\mathrm{d}x}=g\frac{A(x,\theta)-B(x,\theta)}{2A(x,\theta)},$
(83)

we find the desired equation Eq. (78) for $W(x)$ with

$\displaystyle
p(x)=\frac{1}{A^{2}}\bigg{[}AC-\frac{1}{2}(B^{\prime}A-A^{\prime}B)-\frac{1}{4}(A-B)^{2}\bigg{]},$
(84)

where primes denote derivative with respect to $x$ and the three functions are
given by Eq. (C.2). After some computation we can even show that

$p(x)=\frac{A_{0}\bar{z}^{4}-B_{0}\bar{z}^{2}-9}{(2+F\bar{z}^{2})^{2}},$ (85)

where, for convenience, we set the following three functions of the DOC

$\displaystyle A_{0}$ $\displaystyle=$
$\displaystyle\bigg{(}\bar{\tau}\epsilon(\theta)+\frac{5-\Omega}{5(\Omega+3)}\bigg{)}$
$\displaystyle\hskip
14.22636pt\times\biggl{\\{}\frac{7+5\Omega}{4(\Omega+3)}+\bar{\tau}\bigl{[}\zeta(\theta)-\frac{25}{4}\epsilon(\theta)\bigl{]}-\gamma\biggl{\\}},$
$\displaystyle B_{0}$ $\displaystyle=$ $\displaystyle
2\gamma+19\bar{\tau}\epsilon(\theta)-2\bar{\tau}\zeta(\theta)+\frac{15-19\Omega}{5(\Omega+3)},$
$\displaystyle F$ $\displaystyle=$
$\displaystyle\bar{\tau}\epsilon(\theta)+\frac{5-\Omega}{5(\Omega+3)}.$ (86)

Recall that we are interested in the solution for the range
$z_{\eta}~{}\ll~{}z~{}\ll~{}1$ which implies roughly that
$1~{}\ll~{}\bar{z}~{}\ll~{}R_{\mathrm{M}}^{1/2}$. If we take the limit of very
small $\bar{z}$, $x\rightarrow-\infty$, we see that $p\rightarrow-9/4$. As
$\bar{z}$ increases $p(x)$ increases too, let’s call the first turning point
$\bar{z}_{0}$. We can guess from the evaluation of $\gamma$ in Sec. V.4.2 that
$A_{0}$ is very small compared to $B_{0}$. Indeed when plugging in the value
for $\gamma$ we found previously, we obtain that $A_{0}$ goes to zero while
$B_{0}$ has a part independent on $\bar{\tau}$. In particular it implies that
$\bar{z}_{0}$ is large enough to neglect the constant terms in the equation of
$p(x)$ (i.e. $\bar{z}_{0}\gg 1$). The opposite limit of very large $\bar{z}$,
$x\rightarrow\infty$, is not described by Eq. (C.2) as it is valid only in the
small $z$ limit. We need to go back to Eq. (V.3) and use that in the limit of
very large $z$ the velocity correlators and their derivatives should go to
zero. After some computation we obtain for the highest contribution

$p(x)\sim-2e^{2x}\frac{(1+\eta_{t}/\eta)\gamma_{0}}{V(\theta,\eta_{t},\eta,\bar{\tau})^{2}},$
(87)

such that $p(x)<0$ in this limit. Note that we do not need to specify the
exact form of $V(\theta,\eta_{t},\eta,\bar{\tau})$ as the denominator is
always positive. In this formula we also neglected terms that depend on
$\bar{\tau}$ in the numerator as they should always be smaller than
$\eta_{t}/\eta$ or $\gamma_{0}$ which are both positive. Such a form means
that $p(x)$ must have gone through another zero at some point that we call
$\bar{z}_{1}$. To simplify the treatment we will say that Eq. (85) is valid
for $z<1$ and Eq. (87) is valid for $z>1$. The boundary between the two can be
taken to be $z_{1}$ such that $\bar{z}_{1}\sim R_{\mathrm{M}}^{1/2}$. In fact
we will find that the final results have a small dependence on the exact value
of $z_{1}$ such that we can approximate it without changing the conclusions
[41, 56, 62]. To summarise we consider that we have damped solutions for
$\bar{z}\ll\bar{z}_{0}$ and $\bar{z}_{1}\ll\bar{z}$ and an oscillatory one for
$\bar{z}_{0}\ll\bar{z}\ll\bar{z}_{1}$. The exponentially growing solutions are
discarded as $M_{\mathrm{L}}(z)$ must remain finite at both $z=0$ and
$z=\infty$.

In order for the oscillatory solution to match the two damped regime we have
to require [88, 89]

$\int_{x_{0}}^{x_{1}}p(x)^{1/2}~{}\mathrm{d}x=\frac{(2n+1)\pi}{2},$ (88)

where $n$ is an integer. This condition is key to determine the growth rate
$\gamma$ of the two-point correlation of the magnetic field. In the context of
this work we only consider the fastest eigen-mode given by $n=0$. As we
already mentioned the constant terms in Eq. (85) can be neglected which makes
the solution to Eq. (88) exact. Evaluating the integral gives

$\displaystyle\int_{x_{0}}^{x_{1}}p(x)^{1/2}~{}\mathrm{d}x$ $\displaystyle=$
$\displaystyle\int_{\bar{z}_{0}}^{\bar{z}_{1}}\frac{p(z)^{1/2}}{z}~{}\mathrm{d}z,$
(89)
$\displaystyle\simeq\int_{\bar{z}_{0}}^{\bar{z}_{1}}\frac{\sqrt{A_{0}z^{2}-B_{0}}}{Fz^{2}}~{}\mathrm{d}z,$
$\displaystyle=\frac{\sqrt{A_{0}}}{F}\biggl{\\{}\ln{\bigg{(}\frac{z_{1}}{z_{0}}+\sqrt{\frac{z_{1}^{2}}{z_{0}^{2}}-1}\bigg{)}}-\sqrt{1-\frac{z_{0}^{2}}{z_{1}^{2}}}\biggl{\\}},$

where to go from first to second line we used that
$\bar{z}_{0}\sim\sqrt{B_{0}/A_{0}}>0$. We can thus use the condition of Eq.
(88), square both sides, and isolate the growth rate. The growth rate is
finally given by

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle-\bigg{(}\frac{\pi}{\ln{(R_{\mathrm{M}})}}\bigg{)}^{2}\bigg{[}\frac{5-\Omega}{5(\Omega+3)}+\bar{\tau}\epsilon(\theta)\bigg{]}$
(90) $\displaystyle\hskip
17.07182pt+\frac{7+5\Omega}{4(\Omega+3)}+\bar{\tau}\bigg{[}\zeta(\theta)-\frac{25}{4}\epsilon(\theta)\bigg{]},$

where again we plugged the self-consistent value for $\gamma_{0}$. Note that
in this equation we also used the self-consistent evaluations
$\bar{z}_{0}\sim\ln{(R_{\mathrm{M}})}$ and $\bar{z}_{1}\sim
R_{\mathrm{M}}^{1/2}$, such that we neglected $\bar{z}_{0}$ compared to
$\bar{z}_{1}$.

In the oscillatory range
$1~{}\ll~{}\bar{z}_{0}~{}\ll~{}\bar{z}~{}\ll~{}\bar{z}_{1}$ the WKB solution
is thus given by

$W(x)\sim\bigg{(}\frac{\ln{(R_{\mathrm{M}})}}{\pi}\bigg{)}^{1/2}\cos{\bigg{[}\frac{\pi}{\ln{(R_{\mathrm{M}})}}\ln{\bigg{(}\frac{z}{z_{0}}\bigg{)}}\bigg{]}}.$
(91)

In this limit we see that Eq. (83) can be simplified such that
$g^{\prime}(x)\rightarrow-5g(x)/2$ which gives $g(x)\sim e^{-5x/2}$. The two-
point magnetic correlation function is then also scaling as $z^{-5/2}$. So
finally we find the equation for the longitudinal two-point magnetic
correlation function in the region $z_{\eta}~{}\ll~{}z~{}\ll~{}1$

$M_{\mathrm{L}}(z,t)=e^{\gamma\tilde{t}}z^{-5/2}M_{0}\cos{\bigg{[}\frac{\pi}{\ln{(R_{\mathrm{M}})}}\ln{\bigg{(}\frac{z}{z_{0}}\bigg{)}}\bigg{]}},$
(92)

where $\gamma$ is given by Eq. (90).

###  C.3 Validity of the WKB approximation

Figure 3: $p_{\mathrm{lim}}$ as a function of the magnetic Reynolds number
$R_{\mathrm{M}}$ for $St=10^{-2}$ and a couple of DOC $\sigma_{c}$. The dashed
line corresponds to the threshold over $p_{\mathrm{lim}}$ such that the
intersections with the curves give the threshold
$R_{\mathrm{M},\mathrm{thresh}}$ for which the WKB approximation holds.

It can be showed that if we plug the solutions of the WKB approximation into
Eq. (78) we arrive at the following equation

$\frac{\mathrm{d}^{2}\Theta}{\mathrm{d}x^{2}}+\bigg{(}1+\frac{1}{4p(x)^{2}}\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}-\frac{3}{16p(x)^{3}}\bigl{(}\frac{\mathrm{d}p}{\mathrm{d}x}\bigl{)}^{2}\bigg{)}p(x)\Theta=0,$
(93)

such that we retrieve the initial problem to solve only if

$\displaystyle p_{\mathrm{lim}}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{4p(x)^{2}}\frac{\mathrm{d}^{2}p}{\mathrm{d}x^{2}}-\frac{3}{16p(x)^{3}}\bigl{(}\frac{\mathrm{d}p}{\mathrm{d}x}\bigl{)}^{2}$
(94)
$\displaystyle=\frac{z^{2}p^{\prime\prime}(z)+zp^{\prime}(z)}{4p(z)^{2}}-\frac{3z^{2}p^{\prime}(z)^{2}}{16p(z)^{3}}$

is very small compared to 1. Here primes denote derivatives with respect to
the $z$ variable. Furthermore, in a similar way to Schober  _et al._ [16], we
consider that the criterion of validity for our WKB approximation is
$|p_{\mathrm{lim}}|<0.1$. We find that $p_{\mathrm{lim}}$ depends not only on
the magnetic Reynolds number but also on $St$, $\sigma_{c}$ and $z_{c}$. We
define here $z_{c}$ to be the scale at which we evaluate $p(z)$ and it
derivatives. As the WKB approximation is valid between the two zeros of $p(z)$
we must impose $z_{0}\ll z_{c}\ll z_{1}$. Until now, we only ask
$R_{\mathrm{M}}$ to be very large, but the latter criterion gives us a way to
quantify it. In particular, we use the expressions derived earlier for $p(z)$
and $\bar{\tau}$ to define a threshold on $R_{\mathrm{M}}$ for which we
consider that the derived results are valid666Note that this threshold on
$R_{\mathrm{M}}$ is completely unrelated from the critical value of magnetic
Reynolds number for which the dynamo can exist.. In order to respect the
conditions imposed on $z_{c}$, we take $z_{c}=(z_{0}+z_{1})/2$. Although the
scale can seem arbitrary, we find only a slight dependency on it as long as
$z_{c}$ is not too close to $z_{0}$ or $z_{1}$. In Fig. 3 we present
$p_{\mathrm{lim}}$ for a fixed $St=10^{-2}$ and a few DOC. Again, $St$ being
tiny its exact value does not highly impact $R_{\mathrm{M},\mathrm{thresh}}$.
It appears that the threshold of this work, regarding $R_{\mathrm{M}}$ is
around $5\cdot 10^{5}$. More precisely, the $R_{\mathrm{M},\mathrm{thresh}}$
threshold decreases until it reaches $\sim 7\cdot 10^{4}$ when the DOC goes to
infinity. The results derived in this work concerning the magnetic field are
thus valid for most astrophysical objects where the fluctuation dynamo plays a
major role. Note that from Fig. 3 we also have a valid WKB approximation for
very small $R_{\mathrm{M}}$. We can exclude this range of validity as we
derived our generalised Kanzantsev equation (i.e. the expansion with respect
to $St$) with the condition that $R_{\mathrm{M}}$ was a large number.

##  Appendix D Proof of Eq. (71)

Let’s start by expressing the magnetic power spectrum as the Fourier transform
of the magnetic two-point correlation and take the Fourier transform of this
expression

$M_{k}(k)=2\pi k^{2}\hat{M}_{ii}(k)=\frac{k^{2}}{(2\pi)^{2}}\int
M_{ii}(r)e^{i\bm{k}\cdot\bm{r}}~{}\mathrm{d}^{3}\bm{r}.$ (95)

Now use the properties of $M(r)$ to derive the following

$\displaystyle M_{k}(k)$ $\displaystyle=$ $\displaystyle\frac{k^{2}}{2\pi}\int
r^{2}\sin{(\theta)}M_{ii}(r)e^{ikr\cos{(\theta)}}~{}\mathrm{d}r\mathrm{d}\theta,$
(96) $\displaystyle=\frac{ik}{2\pi}\int
rM_{ii}(r)\bigg{(}e^{-ikr}-e^{ikr}\bigg{)}~{}\mathrm{d}r,$
$\displaystyle=\frac{1}{\pi}\int
kr\bigg{(}3M_{\mathrm{L}}(r)+r\partial_{r}M_{\mathrm{L}}(r)\bigg{)}\sin{(kr)}~{}\mathrm{d}r,$
$\displaystyle=\frac{1}{\pi}\int
krM_{\mathrm{L}}(r)\bigg{(}\sin{(kr)}-\cos{(kr)}\biggl{)}~{}\mathrm{d}r,$
$\displaystyle=\frac{1}{\pi}\int(kr)^{3}M_{\mathrm{L}}(r)j_{1}(kr)~{}\mathrm{d}r,$

where to go from the third to fourth line we integrated by parts. It is pretty
obvious from the definition of $rM_{\mathrm{L}}(r)$ that the boundary terms
just go to zero.

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