Turbulent Magnetic Dynamos with Halo Lags, Winds, and Jets

It helps when interpreting the magnetic field topology that we find to recall that scale-invariant fields must appear self-similar at all resolutions and so they normally only close at infinity. This is true even though the divergence of the field is everywhere zero, so that a field line is never broken. Exceptional cases do arise wherein the field lines close through a singularity, but we do not study these here.

In this paper we study the structure of scale-invariant (or self-similar) magnetic dynamo fields, which contain, for the first time, both a rotating halo that lags behind the disk rotation and an accelerating outflow/inflow. Observational examples of halos that are best fit with accelerating outflows are NGC 3556 (Miskolczi et al. 2019) NGC 891 (Schmidt et al. 2019), and NGC 5775 (G. Heald et al. 2021, in preparation). We also allow the (scaled) subscale (also called "fast" elsewhere) turbulent vorticity (sometimes described as hydrodynamic helicity) to increase with latitude in an attempt to imitate an AGN. The usual scale-invariant variation of this quantity is proportional to cylindrical radius and hence it decreases near the axis.

The "small-"scale (i.e., sub-resolution-scale) turbulence must have net hydrodynamic vorticity for the α effect to be present, so we often write subscale vorticity as synonymous with the α effect. Because of a notational clash with our temporal scale factor α, we refer to this effect as the α_h effect.

The inclusion of a rotating galactic halo that lags the disk is an important observed feature that we include in the dynamo action. Rotational velocity in the galactic halo gas significantly declines with height above the disk of spiral galaxies (e.g., Rand 2000; Heald et al. 2007). This "halo lag" inspired a model (Henriksen & Irwin 2016, = H–I) wherein the halo magnetic field is generated through coupling to the distant halo and/or the intergalactic medium. This model accounts for both the lagging halo rotation and a global magnetic field. The model is dynamical in the sense that pressure and gravity also contribute to the steady state, but it did not include the subscale turbulent dynamo.

The present formalism follows that in Henriksen et al. (2018a) so the formalism is only briefly introduced in the next section. The scaled lag and wind/accretion are allowed to be variable with the latitude in the halo. However, we do not solve for the dynamics of the halo gas. Rather, according to our basic assumption, the halo lag, outflow/inflow, and subscale helicity are determined in functional form by the scaling symmetry. The time dependence is determined by the global invariants under the scaling symmetry, much as the scale-invariant dissipation rate determines the Kolmogorov scaling law in hydrodynamic turbulence.

The H–I model led to a rather coherent field structure. The celebrated "X-type" projected magnetic field (Beck 2015; Krause 2015) was present, as were also spiraling field lines (axially symmetric) rising and expanding into the galactic halo. In this paper we wish to determine whether there are characteristic observable galactic magnetic topologies that may distinguish between a dominant α_h effect or a dominant macroscopic dynamo. We compare some of our results to the recent catalog of magnetic fields in the CHANG-ES survey given in Krause et al. (2020). All of these images are freely available for download in FITS format at queensu.ca/changes.

In this section, we use the logarithmic time, T, as defined in Equation (7) ⁵ . When we recall that $R={{re}}^{-\delta T}$, we see from Equations (10), (13), and (7) that

$$\begin{eqnarray}&&{\boldsymbol{b}}\propto {e}^{(2-a)\delta T}\propto {\left(1+\alpha t\right)}^{(2/a-1)}.\end{eqnarray} \tag{ 16 }$$

It is now clear that 1/α (or $1/{\tilde{\alpha }}_{h}\alpha$ when ${\tilde{\alpha }}_{h}$ multiplies α in the definition of T in Equation (18)) may be identified with a characteristic time (growth or decay) for the self-similar dynamo, but that a determines the form of the growth function.

This is where global conserved quantities enter the scale-invariant picture. If for example a = 1, corresponding to a constant global velocity or initial constant "seed" magnetic field, the field increases linearly in time at a rate given by α or ${\tilde{\alpha }}_{h}\alpha$. If we suppose instead that the usual magnetic helicity integrated over a volume V is conserved, that is

$$\begin{eqnarray}&&\int \,A\cdot b\,{dV}=\mathrm{constant},\end{eqnarray} \tag{ 17 }$$

then the global constant has Dimensions L⁶/T². This requires 6δ − 2α = 0 for invariance in time, and hence a = α/δ = 3. We conclude from Equation (16) that the magnetic field must decline as (1 + α t)^−1/3. Such a decline agrees with previous suggestions (e.g., Shukurov et al. 2006; Blackman 2015).

Conserving the local magnetic helicity density (Moffat 1978) itself suggests L³/T² as the physical dimensions of a conserved quantity everywhere (gauge-invariant over a closed volume however small), hence a = 3/2 (a Keplerian value) from 3δ − 2α = 0 and so the field grows slowly as (1 + α t)^1/3.

If the macroscopic "current" helicity b · ∇ × b is conserved locally, then the local constant has Dimensions L/T² so that a = 1/2, which gives rapid growth ∝ (1 + α t)³. The volume-integrated version of this helicity implies L⁴/T² whence a = 2, which implies a steady state. In our numerical studies, we normally use a = 1, which corresponds to either an invariant global velocity or a constant initial "seed" magnetic field. Such a velocity may be set by a flat rotation curve, or possibly by a typical constant outflow velocity. An initial seed field must be determined by cosmology and a theory of galaxy formation (e.g., Wang & Abel 2009; Beck et al. 2012; Pakmor et al. 2014).

A limiting exponential growth in the dynamo field is given by supposing that there is a global constant with a fixed dimension of reciprocal time. This might be a fixed angular velocity Ω or possibly a constant rotational lag rate, but in any case the assumption requires α → 0 so that the relevant quantity is conserved in time. In that case the limit of Equation (16) as α → 0 gives exponential growth as b ∝ e^{2δ t} . Here, δ is a reciprocal time characteristic of the magnetized galaxy. If this is r/v_ϕ , then the e-folding time is essentially a rotation period, as might be expected.

The auxiliary function ${\tilde{\alpha }}_{h}$ is first taken constant and in this section, w = 0. Because of the radial dependence, the turbulent dynamo vanishes at small radius. This explicitly avoids any effect of an AGN.

It is convenient to write the logarithmic time T according to

$$\begin{eqnarray}&&{e}^{\alpha T}\equiv (1+{\tilde{\alpha }}_{h}\alpha t),\end{eqnarray} \tag{ 18 }$$

rather than the usual choice taken in Equation (7).

This choice of logarithmic time removes ${\tilde{\alpha }}_{h}$ from the equation to be solved for the azimuthal vector potential, which becomes simply (the prime indicates differentiation with respect to ζ )

$$\begin{eqnarray}&&(1+{\zeta }^{2}){\tilde{A}}_{{\rm{\Phi }}}^{\prime\prime} +({K}^{2}-v^{\prime} ){\tilde{A}}_{{\rm{\Phi }}}=0.\end{eqnarray} \tag{ 19 }$$

The constant ${\tilde{\alpha }}_{h}$ is constrained not to be strictly zero with this temporal scaling. Its value has an effect only on the time dependence of the field, once the reciprocal scale α (a = 1 ≥ α = δ) has been chosen for macroscopic reasons. We treat the pure flux conservation (sometimes referred to as "flux freezing") dynamo with ${\tilde{\alpha }}_{h}=0$ separately below.

Subsequently the other components of the vector potential follow as (the middle expression restates Equation (19) when combined with the first and third)

$$\begin{eqnarray}\begin{array}{rcl}K{\tilde{A}}_{R} & = & v{\tilde{A}}_{{\rm{\Phi }}}-(1+v\zeta ){\bar{A}}_{{\rm{\Phi }}}^{{\prime} }\\ K{\tilde{A}}_{{\rm{\Phi }}} & = & {\tilde{A}}_{R}^{{\prime} }+\zeta {\tilde{A}}_{Z}^{{\prime} },\\ K{\tilde{A}}_{Z} & = & {\tilde{A}}_{{\rm{\Phi }}}+(v-\zeta ){\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ 20 }$$

We have set

$$\begin{eqnarray}&&K\equiv 2-a.\end{eqnarray} \tag{ 21 }$$

The scaled magnetic field can be shown to have the simple form

$$\begin{eqnarray}&&{\tilde{b}}_{R}=-{\tilde{A}}_{{\rm{\Phi }}}^{{\prime} },\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{\tilde{b}}_{{\rm{\Phi }}}={\tilde{A}}_{R}^{{\prime} }+\zeta {\tilde{A}}_{z}^{{\prime} },\end{eqnarray} \tag{ 23 }$$

$$\begin{eqnarray}&&{\tilde{b}}_{Z}={\tilde{A}}_{{\rm{\Phi }}}-\zeta {\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }.\end{eqnarray} \tag{ 24 }$$

These equations agree with those in Henriksen et al. (2018a; the combined Equations (27) and (28) of that paper) when Δ = 1, u = w = 0, and all diffusion terms are removed.

If we set v = 0 in the equations of Section 3.1, we obtain the pure α_h dynamo with a radially increasing subscale turbulent vorticity. Equation (19) becomes quite simple and is manifestly invariant under a change of sign of ζ, although the derivative will change sign on crossing zero if ${\bar{A}}_{\phi }$ does not change sign. This occurs in what we call the dipolar solution below. The vector potential ${\bar{A}}_{\phi }$ can also change sign, which implies, by Equation (24), that ${\tilde{b}}_{Z}$ passes through zero at the disk. This occurs for the quadrupole solution. The equations for the magnetic field (Equations (22)–(24)) show that consequently only the radial field component changes sign on crossing the equator for the dipole solution, but that both b_Z and b_Φ change sign for the quadrupole case.

The solution of Equation (19) is given in terms of hypergeometric functions (as described below for v ≠ 0 in Section 3.3), although at small ζ there is a simple oscillatory limit. In terms of the independent solution amplitudes {C₁, C₂}, the solution {1, 0} is the "dipolar solution" (b_z continuous across the disk) and {0, 1} is the "quadrupole solution" (b_z zero at the disk). In the quadrupole case, ${\bar{A}}_{\phi }$ passes through zero and changes sign at the disk. Hence, b_R does not change sign on crossing the disk but b_Z and b_ϕ do.

Conical symmetry is a characteristic of our version of self-similarity/scale invariance within a given galaxy. ⁶ Hence, we should not be surprised to find evidence of this symmetry in each base solution of Equation (3). Nevertheless, the behavior can be unexpectedly complex. We show some of the possibilities in Figures 1 and 2, together with the projections in Figure 3.

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As a general comment on our figures, when field lines are shown they are accurate in 3D given the point at which we start them. In 3D vector plots, each vector shows the local direction of the magnetic field where the local point is either at the tail or the head of the vector. When looking at 2D planar projections, the vectors are the component of the local magnetic field in that plane. Components perpendicular to that plane do not appear. Because the plotting routine plots the relative strength of vectors, these projections can give a false impression of the strength of the projected field. This must be taken into account when one tries to reconcile the planar field with a 3D representation of the same field. The problem is not present when considering different views of the same 3D field lines.

Figure 1 demonstrates that, unless one is very near the galactic axis, the spirally rising field lines are very open. The dipole solution shows some interesting structure with height because near the axis, the vertical and azimuthal field components go through zero and reverse direction on the same side of the disk, normally referred to as a "parity change." The field line at the lower left is shown descending from a large height, but this is not necessary. The dipole field line ensemble in Figure 2 begins at z = 1.8. The quadrupole solution at the upper right of Figure 1 is a slowly rising spiral. The single field line is assembled in a cluster spaced around a circle in Figure 2, the different lines corresponding to different colors. Only close to the axis does the quadrupole show the "X-field" behavior in the {r, z} plane. However, the dipole solution shows a strong "X field" throughout each quadrant. These poloidal projections are shown in Figure 3 below.

Figure 3 shows the {r, z} cuts through the upper half of the upper row images of Figure 1. These figures correspond to the images in Figure 2. Only the dipole shows a convincing high-latitude "X field" in projection. The axial behavior of the magnetic fields shows the field coming down at z ≳ 0.3, and then rising at larger radii. At the same radius below z ∼ 0.3, the field is rising as part of a field that descended at smaller radii.

The quadrupole image on the right shows a magnetic field parallel to the disk plus an axial rising (inside r ∼ 0.15) magnetic field. We should recall Equation (9) at this point, which indicates that the true subscale turbulent vorticity goes as ${\tilde{\alpha }}_{h}\delta {{re}}^{-\alpha T}$. When K = a = 1 and α = δ, and the quantity δ r is generally set so as to give a macroscopic velocity in order of magnitude. That leaves ${\tilde{\alpha }}_{h}$ to determine the magnitude of the alpha effect.

One can only expect the pure α_h dynamo in galaxies for which there is little evidence of macroscopic jets or winds (unless the flow is parallel to the magnetic field). The macroscopic (α/Ω) dynamo in our model occurs either due to the lagging rotation in the halo or because of a gradient in the outflow/inflow velocities. There are essentially no parameters in the magnetic field of this section beyond the choice of the self-similar class, a, which we have set equal to 1, reflecting a constant global velocity. This encourages us to search for this coherent magnetic field behavior in the observations. One might expect this behavior to be correlated with vigorous star formation.

Referring to the observed fields displayed in the CHANG-ES catalog of Krause et al. (2020), NGC 3044 may be a candidate for a pure star formation (i.e., subscale turbulence resulting in α_h ) dynamo (see the top of Figure 4 and the figures of this section). The base function would be predominantly dipole. The characteristic magnetic behavior is observed to be predominantly high-latitude "X field" with no parallel equatorial component. Moreover, the field falls off rapidly radially but extends far into the halo. The very strong axial field is a formal feature of Equation (3) when α_h → 0 as r → 0, because the spatial gradient times α_h is required to match the time derivative. There is also the 1/r field dependence that reflects the self-similar global symmetry. Figures 1–3 all illustrate these properties for the α_h dipole. The rotation measure (RM) given in Krause et al. (2020) and displayed in rudimentary form by contours in Figure 4 varies as might be expected from rising, spiraling, field lines. In between the double lines in the galaxy figures (solid and dashed curves close to each other) is where the RM sign changes. On the solid curve side, the RM is positive (field lines pointing toward us) and on the dashed curve side, the RM is negative (field lines pointing away from us). The RM also shows a sign change across the disk in some regions which can correspond to a change in sign of radial field. Figures 9 and 10 below show that the α_h dynamo with wind or even pure flux freezing with wind may also fit the data. NGC 3448 (not shown) may be a more depolarized second example.

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NGC 3556 is another extreme where a coherent galactic field is scarcely established. It is likely that it is dominated by a series of local dynamos driven by local star formation, so we are actually seeing the turbulent nature of subscale vorticity. That is, there are multiple α_h dynamos acting. The data are shown for comparison in the bottom image of Figure 4.

The general solution to Equation (19) with $v^{\prime} =-{v}_{1}$ for ${\tilde{A}}_{{\rm{\Phi }}}(\zeta )$ becomes

$$\begin{eqnarray}\begin{array}{rcl}{\tilde{A}}_{{\rm{\Phi }}}(\zeta ) & = & C1\,(1+{\zeta }^{2})F({\alpha }_{1},{\beta }_{1},{\gamma }_{1},-{\zeta }^{2})\\ & & +C2\,\zeta (1+{\zeta }^{2})F({\alpha }_{2},{\beta }_{2},{\gamma }_{2},-{\zeta }^{2}),\end{array}\end{eqnarray} \tag{ 25 }$$

where F(α, β, γ, z) is the hypergeometric function. The arguments are (with n = 1, 2)

$$\begin{eqnarray}\begin{array}{rcl}{\alpha }_{n} & = & (\displaystyle \frac{3}{4}+\displaystyle \frac{n-1}{2})+\displaystyle \frac{\sqrt{1-4({K}^{2}+{v}_{1})}}{4},\\ {\beta }_{n} & = & (\displaystyle \frac{3}{4}+\displaystyle \frac{n-1}{2})-\displaystyle \frac{\sqrt{1-4({K}^{2}+{v}_{1})}}{4},\\ {\gamma }_{n} & = & \displaystyle \frac{1}{2}+(n-1).\end{array}\end{eqnarray} \tag{ 26 }$$

In our examples, the solution {C1, C2} = {1, 0} is "dipolar" in form with b_Z nonzero at the disk, while the solution {C1, C2} = {0, 1} is quadrupolar and has zero b_Z at the disk.

From Equation (9), we note that when ${\tilde{\alpha }}_{h}$ is constant we may use its value to define the subscale turbulent velocity < 1 and so let the velocity scale δ r be macroscopic. That is, δ r reflects mainly lagging rotational velocity v or wind velocity w (although we do not use a "wind" velocity in this section). Consequently, Equation (8) shows that the magnitude of v and w should be chosen $O(1/{\tilde{\alpha }}_{h})$ so as to reflect the larger macroscopic rotation and wind magnitudes. It is likely that ${\tilde{\alpha }}_{h}=O(0.1)$ (e.g., Tamburro et al. 2009) so that v₁ = O(1) (Heald et al. 2007) ⁷ and w_o and w₁ may be O(10) in extreme cases (e.g., Heesen et al. 2018). Extensive supernova activity may lead to larger subscale velocities locally.

In Figure 5, we show a dipole solution and a quadrupole solution in 3D on the top row and a cluster of corresponding field lines for each case in the bottom row. We show only the upper half-plane for clarity, but b_Φ and b_Z change sign on crossing the disk in the quadrupole solution, while only b_R does so in the dipole solution.

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We see that the quadrupole solution produces a projected "X-type" geometry (recall the axial symmetry) at a small polar angle that includes the axis, while the dipole solution produces the X geometry at much larger polar angles. The dipole field lines appear to be straight in this representation because the poloidal field is much larger than the toroidal field. However, they do turn in azimuth as is shown in Figure 6.

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The dipole has an interesting structure at a large height where the field lines become tightly wound near the axis. The winding sense is opposite to the sense of the exterior winding, which implies that the azimuthal field passes through zero at some height and radius. This leads to a sign change in the RM with latitude that can be seen in an RM plot (Henriksen et al. 2018a), but we do not show this here for brevity.

The addition of rotational halo lag has not greatly changed the topology from that of the pure α_h dynamo as seen in Figures 2 and 3. The dipole solution changes the rotation sense at height near the axis but remains similar at a large radius, showing a strong X field. The quadrupole field is less tightly wound in the presence of halo lag and shows an "X field" only in a very limited range of latitudes. At this point, X-field magnetic topology favors very much a dipole symmetry across the galactic equator.

Figure 6 shows the poloidal and toroidal cuts of the same fields shown in Figure 5. The quadrupole solution shows very similar behavior to the pure alpha solution although it is less tightly wound. The dipole solution has very tight winding near the galactic axis as does the pure turbulent field and has a very strong X-field behavior generally. The sign of the pitch angle of the dipole magnetic field changes sign with height and radius. Such behavior is more in accordance with the average CHANG-ES behavior, as shown in Krause et al. (2020).

In terms of the observations reported in Krause et al. (2020), Figures 6 and 3 show that it would be difficult to detect the effect of moderate halo rotational lag on the α_h dynamo for the dipole, except near the axis. However, near the axis, tight winding can lead to beam depolarization as for the dipole in Figure 5. This often seems to be the case observationally. The quadrupole does not deliver a convincing X field.

We turn in the next section to a similar study of the magnetic effects of a combined accelerating wind and halo rotational lag.

Given the (subscale) turbulent dynamo with constant ${\tilde{\alpha }}_{h}$ plus either outflow or accretion and rotational halo lag, Equations (3) plus scale invariance lead to the equations for the scaled vector potential as

$$\begin{eqnarray}\begin{array}{rcl}K{\tilde{A}}_{R} & = & -{\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }(1+\zeta v(\zeta ))+v(\zeta ){\tilde{A}}_{{\rm{\Phi }}}-w(\zeta )({\tilde{A}}_{R}^{{\prime} }+\zeta {\tilde{A}}_{Z}^{{\prime} }),\\ K{\tilde{A}}_{{\rm{\Phi }}} & = & {\tilde{A}}_{R}^{{\prime} }+\zeta {\tilde{A}}_{Z}^{{\prime} }-w(\zeta ){\tilde{A}}_{{\rm{\Phi }}}^{{\prime} },\\ K{\tilde{A}}_{Z} & = & {\tilde{A}}_{{\rm{\Phi }}}+(v(\zeta )-\zeta ){\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ 27 }$$

These may be combined to give the equation for ${\tilde{A}}_{{\rm{\Phi }}}$ as

$$\begin{eqnarray}&&\begin{array}{l}(1+w{\left(\zeta \right)}^{2}+{\zeta }^{2}){\tilde{A}}_{{\rm{\Phi }}}^{\prime\prime} +2w(\zeta )(K+w(\zeta )^{\prime} ){\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }\\ \quad +(K(K+w(\zeta )^{\prime} )-v(\zeta )^{\prime} ){\tilde{A}}_{{\rm{\Phi }}}=0.\end{array}\end{eqnarray} \tag{ 28 }$$

The scaled magnetic field components are given as in Section 3.1. These can be expressed solely in terms of the azimuthal vector potential and its derivative plus the auxiliary velocities, by using Equations (27).

The forms for the auxiliary scaled vertical velocity and the scaled rotational lag (the physical velocity is given in Equation (8)) are taken as

$$\begin{eqnarray}&&w(\zeta )={w}_{o}+{w}_{1}\zeta ,\end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray}&&v(\zeta )=-{v}_{1}\zeta ,\end{eqnarray} \tag{ 30 }$$

where w_o gives the scaled vertical velocity at the disk, w₁ is the scaled rate of acceleration with ζ and v₁ gives the scaled halo lag. If w_o and w₁ are negative, then we have a decelerating flow onto the disk. If w₁ > 0 with negative w_o then we may have an accelerating inflow. The rotational lag rate v₁ is generally positive. One should note that near the axis of the galaxy ζ → ∞ . Either scaled velocity is then singular above the nucleus, but the physical velocity (8) simply increases with height.

Equation (28) may now be solved in terms of hypergeometric functions with complicated arguments, according to MAPLE 2019. We proceed to express a few results for "typical" parameters in Figures 7 and 8.

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In Figure 7, we show sample "dipole" and "quadrupole" magnetic fields with and without a strong wind acceleration. The cuts are in the {r, z} plane and because the field is axially symmetric, the 3D topology is easily inferred. The macroscopic motions act on the subscale magnetic field that is present due to turbulence (α_h dynamo). The constant auxiliary function ${\tilde{\alpha }}_{h}$ does not appear in the equations except in the temporal evolutionary Equation (18). The magnetic topology is very sensitive to an accelerating wind velocity. Such a gradient generates an α_h /Ω (macroscopic) dynamo just as a halo rotational lag or a gradient of angular velocity in the disk.

It is not shown here but we find that a strong disk wind (w_o ≤ ∼ 10) without strong acceleration only convects the boundary conditions at the disk to large heights, with little change in the topology. For this reason, we have focused on the acceleration in the examples.

What this figure emphasizes overall is the strong dependence on the wind gradient of the magnetic topology in the halo with a moderate initial disk wind. The "quadrupole" solution shows an "X-field" structure only with the wind acceleration. The dipole field gives a strong "X field," particularly without wind acceleration.

By varying the initial disk wind w_o and the acceleration rate w₁, intermediate cases can be found. The field is also combed out by accelerating accretion because there is no barrier at the disk in the model. Decelerating accretion can be arranged with both w_o and w₁ negative.

The quadrupole column in Figure 8 compares a moderate accelerating wind with no halo rotational lag (upper panel) to one with a halo rotational lag, comparable to the accelerating wind velocity at the disk. The flatter spiral generation with the presence of a halo lag is quite marked because the lines in the lower panel rise only to a height of 0.7 rather than ∼1. We recall that the unit of length is the radius of the galactic disk.

A similar flattening effect is observed with the same change in rotational halo lag for the dipole, as shown in the left-hand column panels. Because it appears (Figure 7) that the dipole is the principal source of strong X fields, we can recognize the effects of halo lag in the increasing polar angle (decreasing halo latitude) of the magnetic polarization with increasing lag rate.

Adding a halo rotational lag adds to the complexity of the magnetic topology. In particular, it can lead most noticeably to a reversal of the magnetic helicity with height, which leads to the observed reversal in "parity" in the RM. This is similar to the behavior found in Henriksen & Irwin (2016). The lag also increases the extent of the projected "X field" near the disk.

Figure 9 shows the CHANG-ES polarization (with the Faraday depth sign change) data for the galaxy NGC 4013. The X magnetic field is wound very tightly and is really only visible very close to the disk. This might suggest, according to Figures 7 and 8, that the coherent magnetic field in NGC 4013 is a quadrupole with considerable halo rotational lag. However, close inspection of Figure 9 reveals that the field is more like an inverted V than an X. That is, the field converges on the axis above the disk. There is a weak indication of this for the quadrupole in Figure 6.

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We conclude from this section that a vigorous constant wind can carry equatorial boundary conditions to large heights. However, a gradient in the wind velocity is more important for the magnetic complexity such as strong deviations from a pure α_h dynamo. The main effect of a strong wind is to transport the magnetic field to greater heights and to remove the "X field" in favor of a poloidal field perpendicular to the galactic disk. The behavior is particularly pronounced near the axis. The rotational halo lag, along with the wind acceleration, acts as an α/Ω dynamo given the α_h subscale turbulent average dynamo. The rotational lag gives a field strongly wound near the disk while the wind acceleration stretches the field vertically.

Reviewing the observations in the CHANG-ES catalog, we note that NGC 3044 (top of Figure 4) may be compatible with a moderate disk wind and acceleration as seen for the dipole in Figure 7 at the upper left. This is more consistent with the extended halo in NGC 3044 than is the pure α_h dynamo of Section 3.2.

Quadrupole behavior seems rarer but may be illustrated in NGC 4013 (Figure 9). We refer to both the CHANG-ES catalog (Krause et al. 2020) and Figure 12 in Stein et al. (2019). At the upper right in Figure 7, we see a quadrupole with a field largely parallel to the disk (i.e., a very low-latitude X field) that falls off rapidly into the halo. The model therefore predicts a large-angle but weak X field, which corresponds to Figure 12 in Stein et al. (2019). The data from Krause et al. (2020) is shown in Figure 9. The sign change in the RM across the disk in NGC 4013 can also be due to a quadrupole.

We consider the special case when there is no active α_h effect combined with macroscopic motion, as there has been in all previous cases. It should be described nevertheless as an α/Ω dynamo because there must be an initial field for the velocity to act on. With a = 1 and K = 1, flux density is conserved (equivalent Dimensionally to a magnetic field or velocity constant) and a local field grows linearly according to Section 2.1.

The existence of a global magnetic flux constant requires a = 3 and hence K = −1. Global flux conservation can only produce topological enhancements in the magnetic field which will decline slowly in time. We recall the discussion of time dependence in Section 2.1, which shows the magnetic field to decrease in time as (1 + α t)^−1/3 with a = 3.

We set ${\tilde{\alpha }}_{h}=0$ in this section and so Equation (7) defines the logarithmic time. Because we also suppress the diffusion in this paper, we are left with the pure flux freezing during the macroscopic motion as described by the second term in Equation (3). We continue with our scale-invariant formulation, which prescribes the velocity field according to Equation (8) and the magnetic field according to Equations (10) and (13). We proceed with only rotational and outflow/inflow velocities so that the radial velocity u = 0. This limit is related to the study performed by Henriksen & Irwin (2016) using dynamical equipartition and a Mestel disk, but here the assumption of scale invariance dictates the velocity field, rather than a dynamical model.

The reason for studying this limiting case is due to the predictive nature of the analytic magnetic fields with regard to scale heights of the magnetic field. This is related to the recent results for CHANG-ES galaxies reported in Krause et al. (2018).

The surviving term in Equation (3) requires solving a simple set of equations for the scale-invariant vector potential namely

$$\begin{eqnarray}\begin{array}{rcl}K{\tilde{A}}_{R} & = & v({\tilde{A}}_{{\rm{\Phi }}}-\zeta {\tilde{A}}_{{\rm{\Phi }}}^{{\prime} })-w({\tilde{A}}_{R}^{{\prime} }+\zeta {\tilde{A}}_{z}^{{\prime} }),\\ K{\tilde{A}}_{{\rm{\Phi }}} & = & -w{\tilde{A}}_{{\rm{\Phi }}}^{{\prime} },\\ K{\tilde{A}}_{Z} & = & v{\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }.\end{array}\end{eqnarray} \tag{ 31 }$$

Note that it is the outflow/accretion that drives the whole system. Setting w = 0 renders the system trivial.

Equations (31) are easily solved analytically when we take the halo rotational lag in the pattern frame and the outflow velocity in the familiar forms

$$\begin{eqnarray}\begin{array}{rcl}v & = & -{v}_{1}\zeta ,\\ w & = & {w}_{o}+{w}_{1}\zeta .\end{array}\end{eqnarray} \tag{ 32 }$$

However, even the case where v and w are both constant (so that physical velocities are proportional to r from Equation (8)) is of interest as it is both analytic and simple. The field components follow ultimately from Equations (22) to (24). These imply with v and w constant and Equations (31) that

$$\begin{eqnarray}\begin{array}{rcl}{b}_{R} & = & -\displaystyle \frac{{e}^{(2\delta -\alpha )T}}{{rw}}{{KC}}_{1}{e}^{-\tfrac{K}{w}\tfrac{z}{r}},\\ {b}_{{\rm{\Phi }}} & = & \displaystyle \frac{{e}^{(2\delta -\alpha )T}}{{rw}}{e}^{-\tfrac{K}{w}\tfrac{z}{r}}({C}_{1}v-{C}_{2}K),\\ {b}_{Z} & = & \displaystyle \frac{{e}^{(2\delta -\alpha )T}}{r}{C}_{1}{e}^{-\tfrac{K}{w}\tfrac{z}{r}}(1+\displaystyle \frac{K}{w}\zeta ).\end{array}\end{eqnarray} \tag{ 33 }$$

From these relations we observe that the exponential decline of the magnetic field in z at fixed r is slower with larger wind velocity, so that the scale height increases. The exponential decline is also slower at larger r for a constant outflow, which gives the magnetic (or radio) halo a "butterfly" shape. The azimuthal field increases with increasing v and decreases with increasing w at the disk. The radial field also decreases with increasing w at the disk. Boundary conditions at the disk are "lifted" into the halo. This behavior has also been found in the previous section.

In general, an individual field line always extends in the same sense in radius, azimuth, and height so that it appears as a rising, expanding helix, much as shown in Henriksen & Irwin (2016). It differs from the pure α_h dynamo of Section 3.2 in that the spirals are detectable at much larger radii. This topology already gives an X-type field near the disk when projected into the {r, z} plane.

We note from the solution that b_R changes sign with w on crossing the disk. However, there is no halo rotational lag in z with v constant, but there is in radius by Equation (8). Therefore, there is no need for v to change sign on crossing the disk. Moreover, b_Φ also changes sign on crossing the disk. The component b_Z does not change sign because ζ also changes sign. Thus the field obeys a dipole symmetry, but with a change in sign of both tangential field components.

As a result of these symmetries, there is a change in the magnetic helicity crossing the disk. The term in C₂ comes from the homogeneous part of the equation so we normally set it equal to zero. It would be necessary in order to confront a specified boundary condition. The only quadrupole-type solution is with C₁ = 0, which gives only a purely azimuthal field, possibly discontinuous at the disk

When the outflow and lag functions have the forms of Equations (32), rather than being constant, the solution for the field becomes

$$\begin{eqnarray}\begin{array}{rcl}{b}_{R} & = & \displaystyle \frac{{e}^{(2\delta -\alpha )T}}{{rw}(\zeta )}K{\tilde{A}}_{{\rm{\Phi }}}(\zeta ),\\ {b}_{{\rm{\Phi }}} & = & -\displaystyle \frac{{e}^{(2\delta -\alpha )T}}{{rw}(\zeta )}\left(\zeta {v}_{1}(1+\displaystyle \frac{K\zeta }{w(\zeta )})\right.\\ & & \left.-\displaystyle \frac{{{Kv}}_{1}}{{w}_{1}}(\zeta +\displaystyle \frac{{w}_{o}}{{w}_{1}}\mathrm{ln}w(\zeta ))+{{KC}}_{2}/{C}_{1}\right){\tilde{A}}_{{\rm{\Phi }}}(\zeta ),\\ {b}_{Z} & = & \displaystyle \frac{{e}^{(2\delta -\alpha )T}}{r}\left(1+\displaystyle \frac{K}{w}\zeta \right){\tilde{A}}_{{\rm{\Phi }}}(\zeta ).\end{array}\end{eqnarray} \tag{ 34 }$$

The azimuthal vector potential has the power-law form

$$\begin{eqnarray}&&{\tilde{A}}_{{\rm{\Phi }}}={C}_{1}{e}^{-\tfrac{K}{{w}_{1}}\mathrm{ln}w(\zeta )}.\end{eqnarray} \tag{ 35 }$$

In these formulae w₁ ≠ 0 and, if w_o = 0, there is a logarithmic singularity at ζ = 0. Once again the general behavior is that of a rising, widening spiral as was first seen in Henriksen & Irwin (2016). On crossing the equator, we might expect w_o and v₁ to change sign, but not w₁ so that w changes sign. Hence, b_R and b_ϕ change sign but b_z does not, implying dipole behavior as in the constant-velocity case. We intend the real part of the logarithm in these expressions.

We append a relatively generic example in Figure 10 in order to fix ideas. Depending mainly on the relative size of the halo lag v₁ relative to the wind acceleration, w₁, the spirals are flatter when the ratio is large and more extended when the ratio is small. Recall that the units in the figures are the disk radius. The constant wind velocity merely convects boundary conditions as we have seen before. This is because it does not contribute to a differential twisting of the field.

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The "X-type" field behavior is found in projection as a characteristic of rising, spiraling fields. It will be at an angle close to the disk when the halo lag rate is dominant and closer to the galactic axis when the wind acceleration is stronger. A strong wind acceleration case (not shown) has almost straight field lines in the halo rather than flatly spiraling field lines. This is very similar to the quadrupole α_h behavior.

We summarize this section by comparing it to the pure α_h dynamo of Section 3.2. We have found a widening, spiraling halo structure in both limits as elaborated below.

The pure α_h subscale turbulent average dynamo produces tightly wound spirals near the axis of the galaxy, where they are unlikely to be observed. The large radius spirals are very slow and the projections into the {r, z} plane show an "X-type" topology only near the disk for the quadrupole solution. The dipole α_h solution shows a rapid decline in field strength with height in the azimuthal and vertical field components when the lag and wind are weak.

The macroscopic flux-freezing scale-invariant dynamo of the present section produces rising spiral magnetic fields with a large polar angle when the halo rotational lag velocity dominates the wind velocity. The X-type topology is then projected into the {r, z} plane in a well-defined low-latitude sector. The wind-dominated case shown at the upper left gives a more typical X-field topology. There is also scope for varying the sign of the Faraday depth above the disk when the spirals are rapidly turning inside the galactic halo. This is especially likely when the halo rotational lag dominates.

Referring once more to Krause et al. (2020), the flux-freezing "dynamo" alone cannot give both a high-latitude X field and a parallel field in the disk. However, it shows clearly the association between the X-field topology and rising magnetic spirals. The wind-dominated field is a natural explanation for the observed magnetic topology in NGC 3044 (see Figure 4).

The C-band image of NGC 4631 indicates similar poloidal behavior, but the multiple sign changes in RM in this galaxy suggest the presence of non-axially symmetric spiral arms (see e.g., Woodfinden et al. 2019) in the halo.

We turn to consider a subscale turbulent dynamo varying with latitude and acted on by a rotationally lagging halo. That is, we allow the auxiliary turbulent quantity ${\tilde{\alpha }}_{h}$ to be a function of ζ ≡ z/r. By concentrating the scaled subscale turbulent helicity on the galactic axis in this way, we imitate the effect that an AGN may have on the global galactic magnetic field. The effects of an axial wind must be left to another work.

The suggestion of an axially symmetric "AGN dynamo" assumes that the AGN axis is approximately aligned with the axis of the galaxy. In galaxies where this is not the case, we can expect more asymmetric galactic magnetic fields, at least near the nucleus. It is not necessary for a galaxy to have an axial concentration of magnetic fields, such as may be produced by an AGN, in order to display a large-scale coherent magnetic field. We have seen this in previous sections, especially with the quadrupole mode where the field may be tightly wound about the axis.

If we choose ${\tilde{\alpha }}_{h}$ to be linear in the tangent of the latitude, ζ, according to

$$\begin{eqnarray}&&{\tilde{\alpha }}_{h}={\alpha }_{h1}\zeta ,\end{eqnarray} \tag{ 36 }$$

then we must use T as defined in Equation (7). Consequently, the constant ${\tilde{\alpha }}_{h1}$ no longer vanishes from the equations for the vector potential.

We have, with Equation (36), assumed (recalling Equation (9)) that the physical subscale turbulent helicity increases linearly with height in the halo. However, the scale-invariant solution may be expected to react to the latitude dependence in ${\tilde{\alpha }}_{h}$. The linear form satisfies the antisymmetry across the disk (e.g., Klein & Fletcher 2015).

This form for ${\tilde{\alpha }}_{h}$ parallels the assumption regarding the form of the halo rotational lag (see Equation (14)) relative to a pattern frame. In fact, if we assume that our calculation is done in a uniformly rotating (Ω) pattern frame, we can associate 2Ωr with the base hydrodynamic vorticity to which we add the value (Equation (36)).

Equations (3) for the components of the vector potential become, when ${\tilde{\alpha }}_{h}$ varies generally with ζ,

$$\begin{eqnarray}\begin{array}{rcl}K{\tilde{A}}_{R} & = & -{\tilde{\alpha }}_{h}(\zeta ){\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }+v{\tilde{A}}_{{\rm{\Phi }}}-\zeta v{\tilde{A}}_{{\rm{\Phi }}}^{{\prime} },\\ K{\tilde{A}}_{{\rm{\Phi }}} & = & {\tilde{\alpha }}_{h}(\zeta )({\tilde{A}}_{R}^{{\prime} }+\zeta {\tilde{A}}_{Z}^{{\prime} }),\\ K{\tilde{A}}_{Z} & = & {\tilde{\alpha }}_{h}(\zeta )({\tilde{A}}_{{\rm{\Phi }}}-\zeta {\tilde{A}}_{{\rm{\Phi }}}^{{\prime} })+v{\tilde{A}}_{{\rm{\Phi }}}^{{\prime} },\end{array}\end{eqnarray} \tag{ 37 }$$

and these combine to give the equation for the azimuthal vector component. Once this equation is solved, the other components and the magnetic field components follow. The equation to be solved takes the general form

$$\begin{eqnarray}&&\begin{array}{l}(1+{\zeta }^{2}){\tilde{A}}_{{\rm{\Phi }}}^{\prime\prime} +\displaystyle \frac{d\mathrm{ln}{\tilde{\alpha }}_{h}(\zeta )}{d\zeta }(1+{\zeta }^{2}){\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }\\ \quad +(\displaystyle \frac{{K}^{2}}{{\tilde{\alpha }}_{h}{\left(\zeta \right)}^{2}}-\zeta \displaystyle \frac{d\mathrm{ln}{\tilde{\alpha }}_{h}(\zeta )}{d\zeta }-v^{\prime} ){\tilde{A}}_{{\rm{\Phi }}}=0.\end{array}\end{eqnarray} \tag{ 38 }$$

Using Equations (36) and (14), this last equation simplifies to

$$\begin{eqnarray}&&(1+{\zeta }^{2}){\tilde{A}}_{{\rm{\Phi }}}^{\prime\prime} +\left(\zeta +\displaystyle \frac{1}{\zeta }\right){\tilde{A}}_{{\rm{\Phi }}}^{{\prime} }+\left(\displaystyle \frac{{K}^{2}}{{\alpha }_{h1}^{2}{\zeta }^{2}}-1+{v}_{1}\right){\tilde{A}}_{{\rm{\Phi }}}=0.\end{eqnarray} \tag{ 39 }$$

This equation has a logarithmic singular point at ζ = 0. However, in the limit ζ → 0, the solution is proportional to ζ^p , where ${p}^{2}=-{K}^{2}/{\alpha }_{h1}^{2}$. Consequently, the singularity is an oscillation in $\mathrm{ln}\zeta$ and is easily recognized as α_h1 → 0. We observe in our examples below that this oscillation is associated with the subscale turbulence becoming dominant, as it extends to smaller scales.

The magnetic field is no longer invariant under a change in sign of ζ according to Equation (39) because of the term in v₁, which changes sign on crossing the disk. The sign of α_h1 is irrelevant as one might expect because only the magnitude of the vorticity should produce the alpha dynamo.

A solution of Equation (39) at positive ζ can be pasted into the negative region (keeping the sign of ${\tilde{A}}_{{\rm{\Phi }}}$ unchanged, but remembering to change the sign of v₁) to establish symmetry. This yields, by Equation (37) and the equations for b, a change in sign of b_R and b_Φ but not in b_Z . This would construct local dipole boundary conditions, but a quadrupole boundary follows by changing the sign of ${\tilde{A}}_{{\rm{\Phi }}}$. However, the oscillations in sign near the disk due to small-scale turbulence confuse the issue. These fluctuations become smoothed to a coherent quadrupole for the {0, 1} solution and to a coherent dipole for the {1, 0} solution when v₁ = α_h1 = O(1).

We observe, from Equations (8) and (9), that if indeed ${\tilde{\alpha }}_{h}={\alpha }_{h1}\zeta$, and v = −v₁ ζ, then

$$\begin{eqnarray}&&{v}_{s}=-\delta {{rv}}_{1}\zeta {e}^{(-\alpha )T}\,\,\,\,\,\,{\alpha }_{h}={\alpha }_{h1}\zeta {e}^{(-\alpha T)}.\end{eqnarray} \tag{ 40 }$$

It is convenient to choose δ r to be the actual lag rate so that v₁ = O(1). This means that the value of α_h1 is likely to be also O(1).

The magnetic field components take the same form as in Section 3.1 except for ${\bar{b}}_{{\rm{\Phi }}}$. Equation (23) becomes, using the expression for ${\bar{A}}_{{\rm{\Phi }}}$ in Equation (37) divided by ${\tilde{\alpha }}_{h}(\zeta )$,

$$\begin{eqnarray}&&{\tilde{b}}_{{\rm{\Phi }}}=\displaystyle \frac{K{\tilde{A}}_{{\rm{\Phi }}}}{{\tilde{\alpha }}_{h}(\zeta )}.\end{eqnarray} \tag{ 41 }$$

Recalling the linear form ${\tilde{\alpha }}_{h}$ (Equation (36)), we see that the sign of α_h1 affects the sign of ${\tilde{b}}_{{\rm{\Phi }}}$ as one expects from helical (vortical) turbulence, but it does not affect the magnetic topology.

3.6.1. Examples

The two independent solutions of Equation (39) ({C₁, C₂} = {0, 1} or {1, 0}) are complex conjugates. We choose the real part of the {0, 1} solution and the imaginary part of the {1, 0} solution. This is because these choices best fit the expected quadrupole and dipole boundary conditions.

The oscillatory logarithmic singularity at small ζ is quite obvious in the solutions for small α_h1 near the disk for small values of α_h1. This is formally because whenever ${K}^{2}/{\alpha }_{h1}^{2}$ dominates the factor of ${\tilde{A}}_{\phi }$ in Equation (39), the oscillatory approximation extends to larger ζ. However, ζ must still be less than 1 (polar angle greater than 45°), so that the oscillations occur near the galactic disk. Physically, this is of interest because the α_h term in Equation (3) is due to subscale turbulence. As this term becomes dominant at smaller scales (i.e., smaller z at fixed r), we can expect turbulence to manifest itself in the magnetic field.

The {r, z} plane for the {0, 1} (quadrupole) solution is shown at the upper right in Figure 11 and some corresponding field lines are shown at the lower right. The field lines are constrained to the same scales on each axis. The images in the left column of the figure are for the {1, 0} (dipole) solution. Strong field lines descend rapidly near the axis and then rise. The "X-type" projected field lines are confined to a range of latitudes near 45° latitude.

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One should remember that in all images of the {r, z} plane, the diagonal, apparently rectilinear magnetic vectors, are actually turning in the {r, ϕ} plane. Thus, they represent rising or falling spirals as confirmed in Figure 12. The twist rate differs depending on the parameters and frequently varies with height above the disk and radius in the disk.

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The quadrupole behavior is strongest parallel to the axis and also parallel to the equator. The transition in field direction occurs near 45° and removes any X-type topology in favor of a nearly vertical field at large latitude. The corresponding field lines in the lower-right panel illustrate that the field near the axis is nearly vertical with a small toroidal field. The upper-right panel shows how field lines starting at higher latitudes fill the space of the lower panel.

The dipole behavior in the left column of Figure 11 has a descending strong vertical field near the axis that rises almost rectilinearly, as seen in the field lines in the lower panel. These produce the X-field topology near 45°. The toroidal plot in Figure 12 indicates only a very small azimuthal field compared to the quadrupole. These figures are for nearly equal lag velocity and subscale turbulent velocity.

Figure 12 shows, at the upper right, the 3D version of the {r, z} cut at the upper right in Figure 11. The toroidal cut of the same field is shown at the lower right. The field lines descend from the halo at a large radius and then rise again at latitudes greater than π/4. The very slight toroidal field near the axis compared to the vertical field is shown in the lower-right panel.

At the upper left in Figure 12, we have the 3D image of the dipole with the same parameters. The field lines descend while turning slowly and subsequently rise rectilinearly at larger radius. This develops in a self-similar fashion at all heights. The transition region contains a projected "X field" present mainly near 45°. The azimuthal components of the vectors change sign with radius.

In Figure 13, we show the effects of reducing the scale and strength of the subscale turbulence. The halo rotational lag has been kept constant but ${\tilde{\alpha }}_{h1}=0.25$ in both the quadrupole and dipole panels. Both modes now show X-field topology in two narrow but separated sectors. The quadrupole actually shows three X-field sectors, but one is very close to the equator of the galaxy. The more interesting development is the increasing turbulence in the disk of the galaxy for both modes. This tendency increases in latitude extent as ${\tilde{\alpha }}_{h1}$ decreases further.

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This small-scale turbulence is to be expected when the alpha dynamo dominates. Its appearance actually represents a confirmation of the form for this subscale dynamo, as used in the classical Equation (3).

An interesting result of this section is the appearance in the dipole of a strong axial magnetic field, together with the X-field topology that is intermediate between the axis and the equator. Although we have not included a "jet" or a nuclear "wind" directly, the dipole produces an axial field that we know from previous sections would not be changed by a constant vertical outflow velocity required in a jet or wind. This is achieved by assuming a strong source of scaled subscale vorticity on the galactic axis. This can be produced by an AGN, that is, by a supermassive black hole or a burst of star formation. Such nuclear flows may therefore be an important contributor to the global galactic magnetic field.

If jets or nuclear starbursts do contribute to the coherent galactic magnetic field, we can expect a correlation between the presence of a galactic AGN and the strength of the magnetic field. In fact, the disk field is also strong and turbulent (see Figure 13). Both the correlation and the turbulence would indicate an important subscale dynamo. Similar global magnetic topology can be produced without an AGN but the strong axial field will be absent. This prediction may be readily tested for galaxies where the AGN axis and galactic axis coincide.

Perhaps the best candidates from CHANG-ES results are NGC 4388 and NGC 3079. Recent images of these two galaxies are shown in Figure 14. There is indeed a correlation between the strength of the coherent galactic polarization and the presence of AGN activity as a nuclear wind. The RM sign changes in NGC 4388 tend to indicate spiraling fields above the disk. This is less clear in NGC 3079 but also present. NGC 4388 shows rapid changes in field direction in the disk region, which may be turbulent.

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In Section 3.2, we studied the α_h dynamo alone and displayed the results for the quadrupole and dipole in Figure 1. There are very few parameters in this part because we chose ${\tilde{\alpha }}_{h}$ to be constant with latitude. This allowed us to remove it from the equations by placing it in the time dependence and as a factor in the velocity.

In Figure 1, one sees in the dipole a very slow spiral twist except near the axis, as is shown in the lower row by an axial field line. There is also a characteristic oscillation in the direction of the azimuthal field with height in the dipole. The quadrupole on the right has slowly spiraling field lines as indicated in the lower row of the figure. Clusters of these field lines would project only weakly as an "X-type" topology above the plane as indicated in Figures 2 and 4. Notice that no strong vertical field is produced near the axis of the galaxy and the turbulent dynamo produces only a weak, nearly vertical, X field.

It should be remarked that our model generally has no absolute measure of the strength of the magnetic field unless we have a measure of such a field at a given time. We can say that we expect ${\tilde{\alpha }}_{h}\delta r\leqslant \approx 0.1$ if δ r is comparable to the rotation speed of the disk. The growth timescale is then $\approx 1/(\delta {\tilde{\alpha }}_{h1})$, that is, perhaps equal to 10 rotation periods. The period would be measured at the start of the flat rotation curve of the galaxy.

In Section 3.3, we added a halo lag to the α_h dynamo. The lag is a form of the macroscopic α/Ω dynamo. The rotational lag is linear with height, and we take the lag rate in our examples to be typically dv_ϕ /dz = −α_d δ ≈ 10–20 km s⁻¹ kpc⁻¹. In Figure 5, we show clusters of field lines and 3D vector plots for the quadrupole and dipole. Figure 6 shows the ({r, z} and {r, ϕ}) cuts. The rotational lag increases the twist of the quadrupole relative to the zero-lag case, but the poloidal field is not greatly changed. The dipole has been more strongly affected. It shows a stronger X-field topology and continues to show what would be a sign change in the RM with height.

In Section 3.4, we have added an outflow/inflow to the turbulent dynamo with rotational lag. Figure 7 shows, for both the quadrupole and the dipole, the strong and weak wind, zero rotational halo lag, {r, z} vectors. It is clear that the strong wind gradient "straightens" (more vertical) the spirals in both cases and migrates the field to greater heights. The dipole field lines are rendered almost vertical. Moderate wind disk velocity and gradient enhance the X-field topology.

Figure 8 shows sample field lines for the quadrupole on the left and the dipole on the right. The upper row has no rotational halo lag but moderate wind parameters corresponding to the upper row in Figure 7. The lower row adds a typical rotational halo lag, and we see the field lines spiraling at lower heights.

In Figure 9, we have shown the peculiar polarization of the galaxy NGC 4013 as a possible example of a quadrupole turbulent field distorted by a lagging halo.

In Section 3.5, we have focused on the effect of pure flux-freezing scale invariance. There is no current turbulent dynamo so that at some earlier time, there must be a seed field. The field will eventually decay according to the discussion in Section 2.1, depending on global constants. The reason for dwelling on this rather restricted "dynamo" is that (a) it is wholly analytic, and (b) it creates at least a temporary magnetic field similar to that of a real dynamo. An analytic solution also exists when the magnetic field is not axially symmetric but rather forms magnetic spiral arms. We do not pursue this in this paper.

The results of this section are summarized in Equations (33) and (34). However, Figure 10 shows a cluster of field lines for the dipole with moderate acceleration and lag compared to the outflow velocity. This shows beautifully the rising spiral topology of the magnetic field. The {r, z} cuts in the upper row are for the same parameters on the left. The X field is strongly present. When there is strong rotational lag and weak wind, the low-latitude X field at the upper right is found.

The forms of the explicit equations for the magnetic field are noteworthy for their description of the {r, z} dependence. In the constant scaled velocity (there is a linear radial dependence for the physical velocity) case we find an exponential decline as a function of −z/(rw). The scale height is therefore wr and increases both with w and r. This produces a higher magnetic halo at a larger radius and with a stronger wind. This can be compared to the CHANG-ES paper conclusions in this regard (Krause et al. 2018).

The behavior found here affords an explanation of the correlation between scale height and diameter found in Krause et al. (2018), provided that galaxies scale similarly as well as self-similarly. A superposition of the coherent magnetic fields in CHANG-ES galaxies (Krause et al. 2020) encourages this view.

The solution with variable wind and halo rotational lag is more complicated but varies in the halo essentially with the power ${({w}_{o}+{w}_{1}z/r)}^{-K/{w}_{1}}$. For accelerating outflow, when w_o and w₁ are positive, the magnetic halo has a similar structure to that given by constant velocity although it is not exponential. The halo structure in z evidently can be more subtle with w₁ < 0.

Section 3.6 illustrates how the subscale turbulent dynamo might appear when the turbulence is strong near the galactic axis. Figure 11 shows an {r, z} cut at the upper right for the quadrupole and corresponding field lines at the lower right. The concentration of field to the axis is evident, but there is no X field. The left column shows the dipole for the same parameters. The field is again very strong at height and there is an X field at 45°.

Almost unique to this case is the dominant axial poloidal field. Something similar is found with constant ${\tilde{\alpha }}_{h}$ but it is not as pronounced. The "X-field" region is extensive in the dipole. This morphology of the X field and "jet" axial field is quite distinct from the other examples in this paper.

The upper 3D and {r, z} images in Figure 12 have the same parameters as the upper panels in Figure 11. These confirm the structure found in the {r, z} plane. The lower images show the field lines for the same parameters. The quadrupole shows the extreme axial field.

Figure 13 shows an unexpected result of reducing the amplitude and vertical scale of the subscale turbulence while it becomes dominant. The turbulence becomes visible in the disk of the galaxy.

This section led us to suggest that the AGN (jet or nuclear wind) may have a part to play in the large-scale coherent galactic magnetic field. We would predict an isolated X field, a strong parallel equatorial field, possibly turbulent, and a strong field parallel to the axis as indicative of an "AGN dynamo" with halo rotational lag.

The only element missing in this study of the "parts" that make up an axially symmetric scale-invariant galactic dynamo is diffusion. The removal of the increasing magnetic flux is accomplished here by either outflow or implicitly by a globally conserved quantity. This was discussed in Section 2.1. We believe that this catalog can help in identifying some of the processes at work in galactic magnetism. Magnetic arms have been discussed elsewhere and lead to quite recognizable RM structures in the halo.

The recent numerical study, Butsky et al. (2017), is remarkable for its detailed simulation of what we refer to as the subscale turbulent dynamo. A widely distributed, naturally occurring, set of supernovae are used to pump energy and magnetic field at stellar scale into the interstellar medium. A numerically established, unmagnetized, spiral galaxy is the initial condition. In their Figure 2, the evolved magnetic field is shown as a spiral in the disk but with no convincing "X field" in the halo. The field does not generally extend into the halo as far as indicated by observations. We conclude that additional lag or wind effects are required as we have indicated in this paper. There is a remarkable convergence in the conclusion stemming from two quite different methods.

In the paper Pakmor et al. (2020), the authors examine a suite of cosmological simulations to determine the outflow of magnetic flux from a disk galaxy into its halo and more distant surroundings. Their conclusion is similar to our own, in that coherent fields require coherent gas flows. However, they also require subscale turbulence to amplify an initial seed field, to the point where coherent flow can operate. This turbulence operates even in the circumgalactic gas, just as we assume in Equation (36).

Their Figure 8 shows the turbulence actually present in the gas and magnetic field. In our approach, this turbulence must produce an electric field parallel to the magnetic field for amplification to occur Steenbeck et al. (1966). We have found such turbulence to appear at small ζ, but with the turbulent velocity of Equation (36), the energy spectrum is ∝ ζ³, that is, k⁻³ with k ∝ ζ⁻¹. A Kolmogorov spectrum requires α_h ∝ ζ^1/3m which changes Equation (38) slightly. This does not change the qualitative amplification ability of the turbulence.

More recent papers, already cited in the introduction, study the driving of the galactic wind and the evolution of the magnetic field starting from early magneto genesis. The important conclusion from our perspective is the successful amplification of plausible primordial fields to those galactic fields observed today. This seems to require the action of a distributed turbulent (α/Ω) dynamo during galaxy formation. Wind-driving has been shown to be associated with vigorous star formation and the resulting α_h turbulence provided that the supernovae are inserted discretely. None of these results conflict fundamentally with the results of this and our earlier papers that take the scale-invariant asymptotic view, i.e., the limiting consequence of all the physical interactions in the numerical developments is the scale-invariant solution.

There is a kind of consensus that the "X-field" topology is due to the action of galactic outflow on a preexisting field of stellar scale origin. It is easy to show that an extended wind in spherical geometry (e.g., Chevalier 1985), centered on the nucleus of a galaxy and acting on a preexisting radial field component, will produce an "X field" globally given a realistic rotation law. This is most readily shown using a Lagrangian approach to the field evolution, but we must leave the demonstration to other work.