3.1. Hot, highly ionised discs

The evolution of the surface density of an accretion disc is described by a diffusion equation, with the diffusion parameter proportional to the disc kinematic viscosity, $\nu$, that is, proportional to the Shakura–Sunyaev parameter $\alpha$ (see, for example, Lynden-Bell & Pringle Reference Lynden-Bell and Pringle1974; Pringle Reference Pringle1981). During the decay from a dwarf nova outburst, the mass drains from the disc onto the central white dwarf. The time scale for this decay depends directly on the value of $\alpha$. The decay time scale of the outburst allows for a measurement of $\alpha$ in the hot state from modelling the outburst light curve. The disc size is known from the properties of the system and the disc temperature is obtained from the spectra. A simple calculation by Bath & Pringle (Reference Bath and Pringle1981) suggested that $\alpha \approx 1$. Since then, there has been extensive and more detailed modelling of dwarf nova outburst behaviour, and all models point to relatively large values of $\alpha$. The models imply that $\alpha \approx 0.2\unicode{x2013}0.3$ (e.g. Pringle, Verbunt & Wade Reference Pringle, Verbunt and Wade1986; Smak Reference Smak1998, Reference Smak1999; Buat-Ménard, Hameury & Lasota Reference Buat-Ménard, Hameury and Lasota2001; Cannizzo Reference Cannizzo2001a,Reference Cannizzob; Schreiber, Hameury & Lasota Reference Schreiber, Hameury and Lasota2003, Reference Schreiber, Hameury and Lasota2004; Balman & Revnivtsev Reference Balman and Revnivtsev2012; Kotko & Lasota Reference Kotko and Lasota2012; Coleman et al. Reference Coleman, Kotko, Blaes, Lasota and Hirose2016).

It should be noted that these measurements are in line with the values of $\alpha$ deduced from time-dependent disc behaviour in other systems with highly ionised accretion discs, for example, the soft X-ray transients and the Be stars (see the discussion by Martin et al. Reference Martin, Nixon, Pringle and Livio2019).

If, as we assume here, the dominant stresses that give rise to $\alpha$ are magnetic, this implies (see (2.5)) that the magnetic pressure in the disc is comparable to the gas pressure. Indeed, formally, since for the MHD dynamo driven by ($R \phi$) shear we expect that $\langle B_\phi ^2 \rangle \gg \langle B_R^2 \rangle$, it is evident that we require $\langle B_\phi ^2 \rangle / \rho c_s^2 \gtrsim 1$. In other words, the observations seem to imply that the $\beta$ parameter of the disc plasma is such that $\beta \lesssim 1$.

From the observed disc properties, we may estimate the physical value of the magnetic diffusivity, $\eta$, and hence the magnetic Reynolds number defined as

(3.1)\begin{equation} {\mathcal{R}}_{m} = c_s H /\eta, \end{equation}

at a typical point in the plane of a dwarf nova accretion disc in outburst.

We take the central star to have a mass $M = 1 M_\odot = 2 \times 10^{33}$ g, and consider a typical radius in the disc, $R = 10^{10}$ cm. For a highly ionised disc, in the bright state of a dwarf nova, we take a typical accretion rate of ${\dot M} = 10^{18}$ g s$^{-1}$ and we take the dimensionless viscosity parameter to be $\alpha = 0.3$ in line with observations.

We evaluate $\varOmega _0$ as

(3.2)\begin{equation} \varOmega_0 = \left(\frac{GM}{R^3} \right)^{1/2} = 1.2 \times 10^{{-}2}\,{\rm radian}\,{\rm s}^{{-}1}. \end{equation}

For the magnetic diffusivity, we assume the usual Spitzer value of $\eta = 3.5 \times 10^{12}\,T^{-3/2}$ cm$^2$ s$^{-1}$ (Spitzer Reference Spitzer1962; Potter & Balbus Reference Potter and Balbus2014). From Frank et al. (Reference Frank, King and Raine2002), we find that the temperature in the plane of the disc is $T_{c} = 7.1 \times 10^4$ K. Thus, we have that $\eta = 1.9 \times 10^5$ cm$^2$ s$^{-1}$. Similarly, we find that the disc scale height is given by $H = 3.8 \times 10^8$ cm, so that the disc opening angle is $H/R = 0.038$.

From these, we are able to estimate a typical magnetic Reynolds number as

(3.3)\begin{equation} {\mathcal{R}}_{m} = 9.1 \times 10^{9}. \end{equation}

In summary, we note that, in these discs, whatever the origin of the turbulent behaviour within the disc that gives rise to the observed effective viscosity, whether it is purely hydrodynamic or (as is generally believed) magnetohydrodynamic, the mechanism that produces it is able to drive the fluid motions only up to, or close to, the sound speed. The fact that $\alpha$ is always found to be close to this limit (for these discs) implies that whatever instability might give rise to the driving mechanism for the turbulence, the turbulent velocities grow until they become trans-sonic.

Thus, in agreement with the original conjecture of Shakura & Sunyaev (Reference Shakura and Sunyaev1973), saturation of the turbulence is achieved once the motions become trans-sonic. This must be the result of the fact that once the motions approach the sound speed, the nature of the turbulence changes in a fundamental fashion. In line with the ideas of Shakura & Sunyaev (Reference Shakura and Sunyaev1973, Reference Shakura and Sunyaev1976), it is evident that the change in the nature of the turbulence might occur for one, or both, of two physical reasons. First, in the case of hydrodynamic turbulence, as the turbulence becomes trans-sonic, shocks begin to dominate the dissipative process. Second, in the case of MHD turbulence, once the Alfvén speed approaches the sound speed (i.e. once $\beta \approx 1$), not only do the turbulent velocities become trans-sonic, but, in addition, the time scale for the Parker instability (leading to loss of magnetic flux from the disc) becomes comparable with the shearing time scale (growth time scale for magnetic flux) $\approx 1/\varOmega$ (cf. Tout & Pringle Reference Tout and Pringle1992).

The corollary of this basic finding is that numerical simulations of disc turbulence (for highly ionised discs) which do not find that the strength of the turbulence grows until limited by the sound speed (and which therefore do not find the large values of $\alpha$ implied by the observational data) cannot provide an adequate description of observed accretion disc dynamos. It seems likely that such models must be missing some fundamental physics (King et al. Reference King, Pringle and Livio2007).

3.2. Cool discs

The value of $\alpha$ in the low state dwarf nova accretion disc is difficult to determine, as the disc in that state shows little in the way of time evolution, and what time evolution there is appears to be dominated by the continued addition of mass to the disc from the companion star. In addition, estimates of $\alpha$ in the quiescent disc require modelling of the complete outburst cycle. However, all models of dwarf nova outburst cycles seem to require that in the quiescent disc, the value of $\alpha$ is less than that found in the outburst disc by at least a factor of ${\approx }10$ (Meyer & Meyer-Hofmeister Reference Meyer and Meyer-Hofmeister1983; Pringle et al. Reference Pringle, Verbunt and Wade1986; Cannizzo Reference Cannizzo1993, Reference Cannizzo2001b; Coleman et al. Reference Coleman, Kotko, Blaes, Lasota and Hirose2016).

These findings are in line with the idea (suggested for the dwarf novae quiescent discs by Gammie & Menou Reference Gammie and Menou1998) that the driving force for MHD disc turbulence, namely the MRI, is suppressed once the magnetic diffusivity becomes too large. A number of (local, shearing box) simulations suggest that the MRI becomes inoperative once ${\mathcal {R}}_{m} \lesssim 10^3\unicode{x2013}10^4$ (Hawley, Gammie & Balbus Reference Hawley, Gammie and Balbus1996; Stone et al. Reference Stone, Hawley, Gammie and Balbus1996; Fleming, Stone & Hawley Reference Fleming, Stone and Hawley2000; Davis, Stone & Pessah Reference Davis, Stone and Pessah2010). These results strengthen the argument that the anomalous viscosity (at least in dwarf novae) is an MHD effect (Martin et al. Reference Martin, Nixon, Pringle and Livio2019).

4.1. Local simulations – the shearing box

Most simulations of disc dynamos make use of the shearing box approximation (Hawley, Gammie & Balbus Reference Hawley, Gammie and Balbus1995). Here the computational domain is a Cartesian box of size ${\sim }H \ll R$ co-moving with the fluid at some fixed radius, $R_0$, where the angular velocity is $\varOmega _0$. Thus, $(R, \phi, z) \rightarrow (x = R - R_0, y = \phi - \varOmega _0 t, z)$. The base flow in the box is a linear shear $u_y = U^\prime x$, where in a Keplerian disc, $U^\prime = \frac {3}{2} \varOmega _0$ and the box rotates with angular velocity $\varOmega _0$. In the simulations, the disc may, or may not, be stratified in the $z-$direction.

Typically, the simulations start with a small initial field. For example, Brandenburg et al. (Reference Brandenburg, Nordlund, Stein and Torkelsson1995) and Hawley et al. (Reference Hawley, Gammie and Balbus1996) take an initial poloidal field, with zero net flux, of the form $B_z \propto \sin k x$. The early results are summarised in a review by Balbus & Hawley (Reference Balbus and Hawley1998). The most important finding was that with small initial seed fields, a steady, turbulent MHD disc dynamo could occur. However, for seed fields which had no externally imposed net field (i.e. for those simulations relevant to astrophysical discs), the magnitude of the $R \phi$ magnetic stress was around $\alpha \sim 0.01$, approximately an order of magnitude smaller than that required by observations. Balbus & Hawley (Reference Balbus and Hawley1998) conceded that while the dynamo saturation with $\alpha \approx 1$ postulated by Shakura & Sunyaev (Reference Shakura and Sunyaev1973) was an attractive physical possibility, this was not what was found in the simulations. They concluded, ‘It appears likely, therefore, that there is a dynamo regime that is characterised by unstable growth continuously balancing dissipation scale losses. This leads to subthermal magnetic fields and a dimensionless stress tensor of order $\alpha \approx 0.01$. Whether there are other modes of dynamo operation that arise naturally in accretion disks – at different magnetic Prandtl numbers, for example – is a fascinating and completely openquestion.’

In the 25 years or so since then, there have been many shearing box simulations of the accretion disc dynamo, using a variety of codes both grid-based and spectral, and a variety of physical parameters, with steadily increasing sophistication and resolution. The net result has been a much deeper understanding of the inner workings of the shearing box dynamo process, but the conclusions are unchanged. The simulations that assume zero externally imposed net magnetic flux typically find values of $\alpha$ at most around $\alpha \approx 0.01$ and often less. For example, Fromang et al. (Reference Fromang, Papaloizou, Lesur and Heinemann2007) using both grid-based and spectral codes obtained $\alpha \approx 0.001$ in their best resolved simulations, Heinemann & Papaloizou (Reference Heinemann and Papaloizou2009) use a spectral code and find $\alpha \approx 0.005$, Davis et al. (Reference Davis, Stone and Pessah2010) using a grid-based code find $\alpha \approx 0.01$, Salvesen et al. (Reference Salvesen, Simon, Armitage and Begelman2016) using a grid-based code find $\alpha \approx 0.01$, Walker, Lesur & Boldyrev (Reference Walker, Lesur and Boldyrev2016) using a spectral code find that with no net imposed flux dynamo, the activity decays, so that $\alpha = 0$, and Mamatsashvili et al. (Reference Mamatsashvili, Chagelishvili, Pessah, Stefani and Bodo2020) using a spectral code find $\alpha \approx 0.006\unicode{x2013}0.01$. It is clear that such low values are not in agreement with observations. However, there is a discussion of this tension by Shi, Stone & Huang (Reference Shi, Stone and Huang2016) who present grid-based shearing box simulations. There they find that higher values ($\alpha \approx 0.1$) can be obtained in an unstratified shearing box with an unusually large height-to-width ratio. Whether these larger values of alpha in tall, unstratified boxes carries over to the more realistic stratified case has been questioned (e.g. Ryan et al. Reference Ryan, Gammie, Fromang and Kestener2017).

In summary, while such numerical simulations have successfully demonstrated the existence of a self-sustaining disc dynamo, they have not been able to produce magnetic fields of sufficient magnitude to agree with the observational data. Typically, they produce values of $\alpha \approx 0.01$ or less, much smaller than the values of $\alpha \approx 0.2\unicode{x2013}0.3$ required to account for the observational data. So far, there is no explanation of why the value of $\alpha \approx 0.01$ emerges from the majority of the shearing box dynamo simulations. A physical understanding of what gives rise to the saturation of the dynamo process in numerical simulations would be of great value in understanding this difference between simulated and observed accretion discs.

4.2. Global simulations

King et al. (Reference King, Pringle and Livio2007) drew attention to the discrepancy between the magnitudes of the disc magnetic fields that are required by observations and those that can be achieved in numerical, shearing box simulations. They discussed various reasons as to why the numerical simulations at that time might be inadequate. They noted that the shearing box approximation limits azimuthal structures to azimuthal wavenumbers $m=0$ or $m \gtrsim R/H \gg 1$, and suggested that this might play a role in suppressing large-scale azimuthal fields. For example, Tout & Pringle (Reference Tout and Pringle1992) note that in their semi-analytic dynamo model, which produces fields with $\beta \sim 1$, magnetic buoyancy plays an important role in the conversion of toroidal field to poloidal and acts at toroidal wavelengths $\lambda _\phi \gg H$. Thus, it may be that it is the localisation of dynamo activity in the shearing box approximation that is responsible for the small values of $\alpha$ that are obtained in such simulations. We therefore need to consider global simulations.

We consider two recent global, grid-based, numerical simulations presented by Pjanka & Stone (Reference Pjanka and Stone2020) and Oyang, Jiang & Blaes (Reference Oyang, Jiang and Blaes2021), which are specifically designed to simulate the accretion discs in cataclysmic variables. Our main interest here is the numerical diffusivity present in such simulations. In both of these simulations, material is introduced at the outer edge of the disc (to model the stream of material from the low mass star) and is taken to contain small loops (size ${\sim }H$) of magnetic flux, of magnitude $\beta \gg 1$ which can then initialise dynamo action. This seems reasonable, since the low mass star is sun-like and so presumably has surface magnetic fields (Livio & Pringle Reference Livio and Pringle1994). Oyang et al. (Reference Oyang, Jiang and Blaes2021) also experiment with initially introducing small loops of flux at all radii in the disc.

The disc thicknesses in dwarf novae are typically such that $H/R \approx 0.02\unicode{x2013}0.05$ (see, for example, Frank et al. Reference Frank, King and Raine2002). The thinner disc in the simulations of Pjanka & Stone (Reference Pjanka and Stone2020) has $H/R = 0.1$ and for that disc, they find in the body of the disc that $\alpha \approx 0.003$. The disc of Oyang et al. (Reference Oyang, Jiang and Blaes2021) has $H/R = 0.044$ and they find values of $\alpha \approx 0.01$. Thus, neither of these global disc dynamo simulations is capable of accounting for the observed values of $\alpha$.

6.1. Straight (radial) field lines

We take an initial magnetic field, at time $t=0$, of the form ${\boldsymbol B} = B_0 (0, \cos (kx))$. This can be thought of as an approximation to a magnetic loop of size $l = {\rm \pi}/k$ in a shearing (but not rotating) box of size ${\sim }l$. We assume the fluid to be incompressible and to obey the standard MHD equations with a magnetic diffusivity $\eta$.

In this case, we can define the magnetic flux function $A(x,y, t)$ such that (Davidson Reference Davidson2001; Yeates & Hornig Reference Yeates and Hornig2011)

(6.1)\begin{equation} {\boldsymbol B} = \left( \frac{\partial A}{\partial y}, - \frac{\partial A}{\partial x} \right), \end{equation}

which, in this case, obeys the equation

(6.2)\begin{equation} \frac{\partial A}{\partial t} + U^\prime y \frac{\partial A}{\partial x} = \eta \nabla^2 A. \end{equation}

The flux function is such that the magnetic field lines are given by $A =$ const.

At time $t=0$, we have that

(6.3)\begin{equation} A(x,y,t=0) ={-} k^{{-}1} B_0 \sin (kx). \end{equation}

If we have ideal MHD, so that $\eta = 0$, we expect the field lines to be simply advected by the shear flow, so that

(6.4)\begin{equation} A(x,y,t) ={-} k^{{-}1} B_0 \sin k(x - U^\prime t y). \end{equation}

In this case, the magnetic field grows indefinitely in a linear fashion, viz.

(6.5)\begin{equation} {\boldsymbol B} = B_0 (U^\prime t, 1) \cos k (x - U^\prime ty). \end{equation}

For non-ideal MHD, with magnetic diffusivity $\eta > 0$, we can see that the solution is separable, and of the form

(6.6)\begin{equation} A = f(t) \sin k(x - U^\prime ty). \end{equation}

Substituting this into (6.2), we find that

(6.7)\begin{equation} \frac{{\rm d} f}{{\rm d} t} ={-} \eta f k^2 [1 + (U^\prime)^2 t^2]. \end{equation}

Thus, we have that

(6.8)\begin{equation} A(x,y,t) ={-} k^{{-}1} B_0 \sin [k(x - U^\prime ty)] \exp \left\{- \eta k^2 \left[t + \tfrac{1}{3} (U^\prime)^2 t^3\right] \right\}, \end{equation}

and, therefore, that

(6.9)\begin{equation} {\boldsymbol B} = B_0 (U^\prime t, 1) \cos [k(x-U^\prime ty)] \exp \left\{- {\mathcal{R}}_{m}^{{-}1} \left[U^\prime t + \tfrac{1}{3} (U^\prime t)^3\right] \right\}, \end{equation}

where ${\mathcal {R}}_{m} = U^\prime /(k^2 \eta )$ is the relevant magnetic Reynolds number.

Thus, the spatially averaged (e.g. over a box of size $-l \le x,y \le l$) value of the magnetic energy (${\mathcal {E}}_B$) is of the form (cf. Pringle, McMillan & Teaca Reference Pringle, McMillan and Teaca2017)

(6.10)\begin{equation} {\mathcal{E}}_B(t) / {\mathcal{E}}_B(0) = [1 + (U^\prime t)^2] \exp \left\{- 2{\mathcal{R}}_{m}^{{-}1} \left[U^\prime t + \tfrac{1}{3} (U^\prime t)^3\right]\right\}. \end{equation}

We plot this in figure 1 for various values of ${\mathcal {R}}_{m}$. From this, we can see that the evolution is as follows.

1. (i) Initially, the magnetic field energy starts to decay at a rate of $2U^\prime /{\mathcal {R}}_{m}$ due to magnetic diffusivity. This initial decay phase is brief.

2. (ii) Provided that ${\mathcal {R}}_{m}$ is large enough, the shear is strong enough to provide an enhancement of the magnetic field strength. For large ${\mathcal {R}}_{m}$, field growth occurs after a time $U^\prime t \approx {\mathcal {R}}_{m}^{-1}$. In this phase, the magnetic energy grows linearly and this is a direct consequence of the shear.

3. (iii) Eventually, the shear decreases the spatial length scale of the magnetic field sufficiently that diffusivity wins, and, as is well known, the field ultimately decays (cf. Weiss Reference Weiss1966; Moffatt Reference Moffatt1978). The exponential decay is quicker than the simple $\exp (- \eta k^2 t)$, which is what happens if there is no shear, because the combination of shear and diffusivity hastens the process.

Figure 1.

Evolution of the magnetic energy, scaled to the initial value, with time for several values of the magnetic Reynolds number, ${\mathcal {R}}_{m}$, for the case of initial radial field lines (6.10).(a) Values of $0.1 \le {\mathcal {R}}_{m} \le 10$ with the magnetic energy on a linear scale. (b) Values of ${\mathcal {R}}_{m}$ up to $10^7$ with the magnetic energy on a log scale. For small ${\mathcal {R}}_{m}$, the energy decays rapidly, while for large ${\mathcal {R}}_{m}$, the field initially decays before exhibiting growth and final decay.

The maximum possible enhancement of the magnetic field energy, ${\mathcal {E}}_B(t_{\rm max})/{\mathcal {E}}_B(0)$, depends on ${\mathcal {R}}_{m}$. We plot this in figure 2. For ${\mathcal {R}}_{m} \gg 1$, the maximum enhancement occurs when $U^\prime t \approx {\mathcal {R}}_{m}^{1/3}$, and is equal to ${\mathcal {E}}_B(t_{\rm max})/{\mathcal {E}}_B(0) \approx ({\mathcal {R}}_{m}/e)^{2/3}$.

Figure 2.

The maximum growth factor of the magnetic field energy plotted as a function of the magnetic Reynolds number, ${\mathcal {R}}_{m}$, for the case of initial radial field lines (i.e. the maximum value attained from (6.10)). The red-dashed line indicates the prediction appropriate to the limit of large ${\mathcal {R}}_{m}$, which is accurate for ${\mathcal {R}}_{m} \gtrsim 10$. For ${\mathcal {R}}_{m} \lesssim 3.8$, the magnetic energy never grows back above the original value.

In figure 3, we plot the geometric evolution of the magnetic field lines at different times as an illustration; note that the value of ${\mathcal {R}}_{m}$ does not affect the geometry and only affects the field strength in this case. Panels (a)–(d) correspond to times of $U^\prime t = 0,0.1,1,10$, respectively.

Figure 3.

Evolution of the magnetic field lines from an initially linear radial ($y$-direction) configuration. Time increases as $U^\prime t$ is: (a) 0; (b) 0.1; (c) 1 and (d) 10. The colour bar denotes the value of $A$, with lines of constant $A$ delineating the field lines. For clarity, we also plot five field lines, which at $U^\prime t = 0$ correspond to $A = -k^{-1}B_0 \sin (k\gamma )$ with $k={\rm \pi}$, $B_0=1$, and $\gamma = -2/3, -1/3, 0, 1/3, 2/3$ (the solid black lines represent those field lines with $A \ge 0$ and the dashed lines represent those with $A < 0$). In subsequent panels, only these field lines are plotted, with the apparent number increased due to stretching of the field lines in the $x$-direction. This figure illustrates the conversion of radial field ($y$-direction) into azimuthal field ($x$-direction). The strength of the field varies in time and the evolution of the strength depends on ${\mathcal {R}}_{m}$ (see figure 1). Here we have plotted the case where $\eta \rightarrow 0$ (corresponding to ${\mathcal {R}}_{m} \rightarrow \infty$) and, as such, the range of values for $A$ remains fixed (cf. (6.4)).

6.2. Loops of magnetic field

We now explore the behaviour of loops of magnetic flux. To stay with one Fourier mode in each direction, we take

(6.11)\begin{equation} A(x,y,t=0) = \sin k_x x \sin k_y y. \end{equation}

In the region $0 \le k_x x,\ k_y y \le {\rm \pi}$, the magnetic field takes the form of loops around the point $({\rm \pi} /2, {\rm \pi}/2)$ with the field pointing in the anti-clockwise direction. The rest of space is populated with similar rectangular tiles containing loops of magnetic field of alternating chirality.

We note that we may rewrite the flux function as

(6.12)\begin{equation} A(x,y,t=0) = \tfrac{1}{2} [ \cos (k_x x- k_y y) - \cos (k_x x+k_y y) ]. \end{equation}

Thus, with zero diffusivity, we have simple advection of the field by the shear so that

(6.13)\begin{equation} A(x,y,t) = \tfrac{1}{2} [ \cos (k_x x - U^\prime t k_x y - k_y y) - \cos (k_x x - U^\prime t k_x y + k_y y)]. \end{equation}

Because we have have a combination of two Fourier modes, for $\eta > 0$, we now look for solutions of the form

(6.14)\begin{equation} A(x,y,t) = \tfrac{1}{2} [ f_1(t) \cos (k_x x - U^\prime t k_x y - k_y y) - f_2(t) \cos (k_x x - U^\prime t k_x y + k_y y)], \end{equation}

which, as before, can be substituted into the evolution equation (6.2) and solved for $f_1(t)$ and $f_2(t)$. Using the initial conditions that at $t=0$, $f_1(0) = f_2(0) = 1$, we have

(6.15)\begin{equation} f_1(t) = \exp \left\{ - \eta \left[ (k_x^2 + k_y^2)t + k_x k_y U't^2 + \tfrac{1}{3}k_x^2 (U')^2 t^3 \right] \right\}, \end{equation}

and

(6.16)\begin{equation} f_2(t) = \exp \left\{ - \eta \left[ (k_x^2 + k_y^2)t - k_x k_y U't^2 + \tfrac{1}{3}k_x^2 (U')^2 t^3\right] \right\}. \end{equation}

Putting this all together, we have

(6.17)\begin{align} A(x,y,t) & = [ \sin k_x(x - U'ty) \sin k_yy \cosh ( \eta k_x k_y U't^2) \nonumber\\ & \quad -\cos k_x (x - U'ty) \cos k_y y \sinh ( \eta k_x k_y U't^2) ] \nonumber\\ & \quad \times \exp \left\{ - \eta \left[(k_x^2 + k_y^2)t + \tfrac{1}{3} k_x^2 (U')^2 t^3\right] \right \}. \end{align}

In this case, the evolution of the spatially averaged magnetic field energy (e.g. over a box of size $-{\rm \pi} \le k_x x, k_y y \le {\rm \pi}$) is

(6.18)\begin{equation} \frac{{\mathcal{E}}_B(t)}{{\mathcal{E}}_B(0)} = \frac{[k_x^2 + (k_y + k_x U't)^2] f^2_1(t) + [ k_x^2 + (k_y - k_x U't)^2] f^2_2(t)}{2 [k_x^2 + k_y^2]}. \end{equation}

As before, we define the magnetic Reynolds number as ${\mathcal {R}}_{m} = U^\prime /(k_x^2\eta )$, but here, a secondary number ($k_y/k_x$) is required to define the flow. For illustration, we make the following figures with $k_y/k_x = 1$. First we plot the evolution of the magnetic energy (6.18) in figure 4. The result is similar to that for an initial straight field. For small ${\mathcal {R}}_{m}$, the field energy decays rapidly. For large ${\mathcal {R}}_{m}$, there is a period of significant growth of the field energy before the field later decays. For loops, the critical ${\mathcal {R}}_{m}$ at which the field exhibits any growth is significantly larger than for the straight field lines, requiring ${\mathcal {R}}_{m} \gtrsim 14.7$ for loops rather than just ${\mathcal {R}}_{m} \gtrsim 3.8$ for lines.

Figure 4.

Evolution of the magnetic energy, scaled to the initial value, with time for several values of the magnetic Reynolds number, ${\mathcal {R}}_{m}$, for the case of initial loops of magnetic field (6.18). (a) Values of $0.1 \le {\mathcal {R}}_{m} \le 32$ with the magnetic energy on a linear scale. (b) Values of ${\mathcal {R}}_{m}$ up to $10^7$ with the magnetic energy on a log scale. For small ${\mathcal {R}}_{m}$, the energy decays rapidly, while for large ${\mathcal {R}}_{m}$, the field initially decays before exhibiting growth and final decay. The behaviour is similar, particularly at large ${\mathcal {R}}_{m} \gg 1$, with larger ${\mathcal {R}}_{m} \gtrsim 14.7$ required to exhibit field energy growth.

In figure 5, we plot the maximum growth of the magnetic energy against ${\mathcal {R}}_{m}$. In this figure, the black solid line corresponds to maximum growth from initial field loops, while the red-dashed lines are for the case of field lines plotted in figure 2. Here we can see that the maximum growth is weaker for loops of field compared to field lines for the same value of ${\mathcal {R}}_{m}$. However, we can also see that both cases exhibit the same functional dependence of the maximum growth rate on ${\mathcal {R}}_{m}$; for field loops with $k_x = k_y = 1$, the maximum growth in magnetic energy is ${\mathcal {E}}_B(t_{\rm max})/{\mathcal {E}}_B(0) \approx (1/2)({\mathcal {R}}_{m}/e)^{2/3}$. This is to be expected as at late times, where the maximum is reached for large ${\mathcal {R}}_{m}$, the initial field loops have already been sheared out into lines of field. To illustrate this, we plot the magnetic field lines resulting from the initial loops of field in figure 6. This figure again shows the conversion of radial to azimuthal field, and that for $U^\prime t \gg 1$, the field structure is similar to the case with initial lines of field as the loops become strongly stretched in the $x$-direction.

Figure 5.

The maximum growth factor of the magnetic field energy plotted as a function of the magnetic Reynolds number, ${\mathcal {R}}_{m}$, for the case of initial loops of magnetic field (i.e. the maximum value attained from (6.18)). The red-dashed line shows the solution presented in figure 2 for the case of initial radial field lines. Here, for ${\mathcal {R}}_{m} \lesssim 14.7$, the magnetic energy never grows back above the original value, which contrasts with the value of ${\mathcal {R}}_{m} \lesssim 3.8$ in the initial radial field line case.

Figure 6.

Evolution of the magnetic field lines from initial loops of magnetic field with $k_x = k_y = {\rm \pi}$. Time increases as $U^\prime t$ is: (a) 0; (b) 0.1; (c) 1 and (d) 10. The colour bar denotes the value of $A$, with lines of constant $A$ delineating the field lines. The regions with $A > 0$ (yellow) correspond to regions where the magnetic field lines are oriented in the counterclockwise direction, while for regions with $A < 0$ (blue), the magnetic field lines are oriented with the opposite chirality. For clarity, we also plot field lines corresponding to $A = -3/4, -1/2, -1/4, 1/4, 1/2, 3/4$ (the solid black lines represent those field lines with $A \ge 0$ and the dashed lines represent those with $A < 0$) with the exception of panel (d) where only field lines with $A=-1/2$ and $1/2$ are plotted for clarity. The strength of the field varies in time and the evolution of the strength depends on ${\mathcal {R}}_{m}$ (see figure 4). Here we have plotted the case where $\eta \rightarrow 0$ (corresponding to ${\mathcal {R}}_{m} \rightarrow \infty$) and, as such, the range of values for $A$ remains fixed (cf. (6.4)). As with the initial linear field case (see figure 3), the field lines are stretched due to the shear. As time proceeds, the solution for initial loops of field takes on a similar geometry to the case with initial lines of field for $U^\prime t \gg 1$.