Galactic Magnetic Fields. I. Theoretical Model and Scaling Relations

Magnetic fields in galaxies are inferred from observations to usually be dominated by a random small-scale component (Beck et al. 2019). Such random fields can be produced by fluctuation (small-scale) dynamo action, as well as the turbulent tangling of the mean magnetic field. It is also possible that part of the random component detected in observations is actually unresolved mean field. Below we neglect the influence of the mean field on the random field for simplicity, leaving such effects for future study.

Random galactic fields are inferred from observations to be anisotropic (e.g., Fletcher et al. 2011). Below, we assume that this anisotropy is solely produced by the large-scale galactic shear arising from differential rotation. Other sources of anisotropy, such as shear and compression in spiral density waves, could also be important. The galactic differential rotation stretches the radial component of the random field, leading to a linear increase with time of the azimuthal component. This can lead to a cumulative effect for a duration of about the correlation time of turbulence τ. Let the standard deviation of any component of the random field be given by ${\sigma }_{i}=\sqrt{\overline{{b}_{i}^{2}}}$, with i = (r, ϕ, z). For a random magnetic field in a statistically steady (saturated) state, we can estimate (Shukurov & Subramanian 2021, Section 4.4.1)

$$\begin{eqnarray}&&{\sigma }_{\phi }={\sigma }_{r}(1+q{\rm{\Omega }}\tau ).\end{eqnarray} \tag{ 4 }$$

Galactic outflows (winds or fountain flow) may also contain large-scale shear, which would enhance σ_z relative to σ_r . We neglect galactic outflows for simplicity, but general expressions that include the mean vertical outflow speed U₀ are given in Appendix A.1. With U₀ = 0, we obtain

$$\begin{eqnarray}&&{\sigma }_{z}={\sigma }_{r}.\end{eqnarray} \tag{ 5 }$$

The strength of the random field is thus given by

$$\begin{eqnarray}&&b\equiv \sqrt{\overline{{{\boldsymbol{b}}}^{2}}}=\sqrt{{\sigma }_{r}^{2}+{\sigma }_{\phi }^{2}+{\sigma }_{z}^{2}}={\sigma }_{r}\sqrt{2+{\left(1+q{\rm{\Omega }}\tau \right)}^{2}}\end{eqnarray} \tag{ 6 }$$

and that of the isotropic background by

$$\begin{eqnarray}&&{b}_{\mathrm{iso}}=\sqrt{3}{\sigma }_{r}.\end{eqnarray} \tag{ 7 }$$

The degree of anisotropy can be expressed as

$$\begin{eqnarray}&&\varepsilon =\displaystyle \frac{{b}_{\mathrm{ani}}}{b},\end{eqnarray} \tag{ 8 }$$

where

$$\begin{eqnarray}&&{b}_{\mathrm{ani}}\equiv \sqrt{{b}^{2}-{b}_{\mathrm{iso}}^{2}}=\displaystyle \frac{{b}_{\mathrm{iso}}}{\sqrt{3}}{\left[2q{\rm{\Omega }}\tau \left(1+\displaystyle \frac{q{\rm{\Omega }}\tau }{2}\right)\right]}^{1/2}.\end{eqnarray} \tag{ 9 }$$

Simulations of fluctuation dynamos in periodic boxes with isotropic forcing can be used to estimate the strength of the random field in the saturated state. Since large-scale shear is typically not included in such simulations, they are best used to estimate b_iso, rather than b. In particular, they can help to constrain the ratio ξ of energy density of the saturated magnetic field ${b}_{\mathrm{iso}}^{2}/8\pi$ to that of the turbulent motions $\tfrac{1}{2}\rho {u}^{2}$, for a range of magnetic Reynolds and magnetic Prandtl numbers; in galaxies, one expects Rm ≫ 1 and Pm ≫ 1. These simulations suggest that $\xi ={b}_{\mathrm{iso}}^{2}/4\pi \rho {u}^{2}$ ranges from ≈0.4 for solenoidally forced subsonic turbulence, to values that are considerably smaller for compressively forced or supersonic turbulence. Based loosely on results from Federrath et al. (2011) (see also Seta & Federrath 2021) we choose

$$\begin{eqnarray}&&{b}_{\mathrm{iso}}=\displaystyle \frac{{\xi }_{0}^{1/2}{B}_{\mathrm{eq}}}{\max (1,{ \mathcal M }/{A}_{1})},\end{eqnarray} \tag{ 10 }$$

where A₁ is constant of order unity,

$$\begin{eqnarray}&&{ \mathcal M }\equiv \displaystyle \frac{u}{{c}_{{\rm{s}}}}\end{eqnarray} \tag{ 11 }$$

is the turbulent sonic Mach number, c_s is the speed of sound, and ξ₀ = 0.4. Here,

$$\begin{eqnarray}&&{B}_{\mathrm{eq}}=\beta {\left(4\pi \rho \right)}^{1/2}u\end{eqnarray} \tag{ 12 }$$

is the field strength corresponding to energy equipartition with turbulence. A parameter, β, has been inserted to account for imprecision in both theory and observational inference.

Spiral galaxies also contain magnetic field components that are coherent on scales up to the system size. This large-scale component (sometimes called the mean or regular field) can generally be explained by appealing to mean-field dynamo action (Beck et al. 2019). Below we apply α-Ω dynamo theory including a nonlinear back reaction that quenches the dynamo and leads to saturation (Pouquet et al. 1976; Krause & Rädler 1980; Kleeorin & Ruzmaikin 1982; Ruzmaikin et al. 1988; Blackman & Field 2002; Rädler et al. 2003; Brandenburg & Subramanian 2005; Shukurov et al. 2006; Subramanian & Brandenburg 2006; Sur et al. 2007; Chamandy et al. 2014; Gopalakrishnan & Subramanian 2023).

We make use of an analytic steady-state solution of the mean-field dynamo equations that assumes a thin disk and employs the no-z approximation to replace z-derivatives by divisions by h(r), with coefficients chosen to produce good agreement with solutions that retain the z dependence (Subramanian & Mestel 1993; Moss 1995; Phillips 2001; Chamandy et al. 2014). General expressions that allow for a finite mean vertical velocity are derived in Chamandy et al. (2014) and can be found in Section A.1. In the absence of mean vertical outflow or inflow, the strength of the mean magnetic field in the saturated state is given by

$$\begin{eqnarray}&&\overline{B}\equiv | \bar{{\boldsymbol{B}}}| =K\displaystyle \frac{\pi l}{h}{\left[\left(\displaystyle \frac{D}{{D}_{{\rm{c}}}}-1\right){R}_{\kappa }\right]}^{1/2}{B}_{\mathrm{eq}},\end{eqnarray} \tag{ 13 }$$

where K is a factor of order 0.1–1 that accounts for theoretical uncertainty (Chamandy & Singh 2018), D is the dynamo number, subscripts "k" and "c" refer to the kinematic ($\overline{B}\ll {B}_{\mathrm{eq}}$) and critical (no growth or decay) values, and D > D_c (supercritical dynamo); if D ≤ D_c, we instead adopt $\overline{B}=0$. The critical dynamo number is given by

$$\begin{eqnarray}&&{D}_{{\rm{c}}}=-{\left(\displaystyle \frac{\pi }{2}\right)}^{5},\end{eqnarray} \tag{ 14 }$$

and the kinematic dynamo number by

$$\begin{eqnarray}&&D={R}_{\alpha }{R}_{{\rm{\Omega }}},\end{eqnarray} \tag{ 15 }$$

with Reynolds-type dimensionless numbers

$$\begin{eqnarray}&&{R}_{\alpha }\equiv \displaystyle \frac{{\alpha }_{{\rm{k}}}h}{\eta },\qquad {R}_{{\rm{\Omega }}}\equiv -\displaystyle \frac{q{\rm{\Omega }}{h}^{2}}{\eta },\qquad {R}_{\kappa }\equiv \displaystyle \frac{\kappa }{\eta }.\end{eqnarray} \tag{ 16 }$$

where

$$\begin{eqnarray}&&{\alpha }_{{\rm{k}}}=-\displaystyle \frac{1}{3}\tau \,\overline{{\boldsymbol{u}}\cdot {\boldsymbol{\nabla }}\times {\boldsymbol{u}}},\end{eqnarray} \tag{ 17 }$$

the turbulent diffusivity of $\overline{{\boldsymbol{B}}}$ is given by

$$\begin{eqnarray}&&\eta =\displaystyle \frac{1}{3}\tau {u}^{2},\end{eqnarray} \tag{ 18 }$$

and κ is the turbulent diffusivity of the quantity

$$\begin{eqnarray}&&{\alpha }_{{\rm{m}}}=\displaystyle \frac{1}{3}\tau \displaystyle \frac{\overline{{\boldsymbol{b}}\cdot {\boldsymbol{\nabla }}\times {\boldsymbol{b}}}}{4\pi \rho },\end{eqnarray} \tag{ 19 }$$

which becomes important in the nonlinear regime owing to the back reaction of the field onto the flow. The total contribution to the α effect is obtained by summing α_k and α_m. We approximate α_k as (Ruzmaikin et al. 1988; Chamandy et al. 2016)

$$\begin{eqnarray}{\alpha }_{{\rm{k}}}=\left\{\begin{array}{cc}\displaystyle \frac{{C}_{\alpha }{\tau }^{2}{u}^{2}{\rm{\Omega }}}{h},\quad & \mathrm{if}\min \left(1,\displaystyle \frac{{C}_{\alpha }^{{\prime} }h}{{C}_{\alpha }\tau u}\right)\geqslant {\rm{\Omega }}\tau ;\\ \displaystyle \frac{{C}_{\alpha }\tau {u}^{2}}{h}, & \mathrm{if}\min \left({\rm{\Omega }}\tau ,\displaystyle \frac{{C}_{\alpha }^{{\prime} }h}{{C}_{\alpha }\tau u}\right)\geqslant 1;\\ {C}_{\alpha }^{{\prime} }u, & \mathrm{if}\min \left({\rm{\Omega }}\tau ,1\right)\geqslant \displaystyle \frac{{C}_{\alpha }^{{\prime} }h}{{C}_{\alpha }\tau u},\end{array}\right.\end{eqnarray} \tag{ 20 }$$

where C_α and ${C}_{\alpha }^{{\prime} }$ are constants of order unity that account for a lack of precision in the modeling; note that ${C}_{\alpha }^{{\prime} }\,u$ acts as a ceiling on α_k. The pitch angle of $\bar{{\boldsymbol{B}}}$ is given by

$$\begin{eqnarray}&&\tan {p}_{B}=\displaystyle \frac{{\overline{B}}_{r}}{{\overline{B}}_{\phi }}=\displaystyle \frac{{\pi }^{2}}{4{R}_{{\rm{\Omega }}}}=-\displaystyle \frac{{\pi }^{2}\,\tau \,{u}^{2}}{12\,q\,{\rm{\Omega }}\,{h}^{2}},\end{eqnarray} \tag{ 21 }$$

defined such that −π/2 < p_B ≤ π/2 with p_B < 0 for trailing spirals (with respect to the galactic rotation).

For f_SB = 0, Equation (A10) for the turbulent correlation length becomes

$$\begin{eqnarray}&&l=\left(\displaystyle \frac{{\rm{\Gamma }}-1}{{\rm{\Gamma }}}\right){C}_{l}{l}_{\mathrm{SN}}=\displaystyle \frac{3}{10}{l}_{\mathrm{SN}},\end{eqnarray} \tag{ 22 }$$

where we have assumed Γ = 5/3 and C_l = 3/4 (see Section A.2.1 and Table 1), and where ${l}_{\mathrm{SN}}$ is the driving scale of turbulence driven by isolated SNe. The quantity ${l}_{\mathrm{SN}}$ is equal to the radius of the SN remnant (SNR) when its age is ${t}_{{\rm{s}}}^{\mathrm{SN}}$, defined as the time at which the SNR expansion speed becomes equal to the ambient sound speed, ${\dot{R}}_{\mathrm{SN}}={c}_{{\rm{s}}}$. At this time, the SNR is assumed to fragment and merge with the interstellar medium (ISM), transferring its energy to the latter. This gives

$$\begin{eqnarray}&&{l}_{\mathrm{SN}}={R}_{\mathrm{SN}}({t}_{{\rm{s}}}^{\mathrm{SN}})=0.14\,\mathrm{kpc}\ \psi {E}_{51}^{16/51}{n}_{0.1}^{-19/51}{c}_{10}^{-1/3},\end{eqnarray} \tag{ 23 }$$

where ${E}_{51}={E}_{\mathrm{SN}}/{10}^{51}\,\mathrm{erg}$ is the SN energy (excluding that in neutrinos), n_0.1 = n/0.1 cm⁻³ is the gas number density, and c₁₀ = c_s/10 km s⁻¹. A dimensionless parameter of order unity, ψ, is introduced in the present work to account for the uncertainty in the model. To convert from mass density to number density, we have used the expression

$$\begin{eqnarray}&&n=\displaystyle \frac{\rho }{\mu {m}_{{\rm{H}}}},\end{eqnarray} \tag{ 24 }$$

with mean molecular mass μ = 14/11.

The rms turbulent velocity, u, is estimated by equating the energy injection rate per unit volume from SNe with the dissipation rate per unit volume ∼ρ u³/2l owing to the turbulent energy cascade. With f_SB = 0, Equation (A16) becomes

$$\begin{eqnarray}&&u={\left(\displaystyle \frac{4\pi }{3}{l}_{\mathrm{SN}}^{3}{{lc}}_{{\rm{s}}}^{2}\nu \right)}^{1/3},\end{eqnarray} \tag{ 25 }$$

with l given by Equation (22) and ${l}_{\mathrm{SN}}$ by Equation (23).

We estimate the correlation time τ as the minimum of the turnover time τ_e of energy-carrying eddies, and the time τ_r for the flow to renovate due to the passage of an SN blast wave,

$$\begin{eqnarray}&&\tau =\min ({\tau }_{{\rm{r}}},{\tau }_{{\rm{e}}}).\end{eqnarray} \tag{ 26 }$$

The eddy turnover time (comparable to the lifetime of the largest eddies) is estimated as

$$\begin{eqnarray}&&{\tau }_{{\rm{e}}}=\displaystyle \frac{l}{u},\end{eqnarray} \tag{ 27 }$$

where l is given by Equation (22) and u is given by Equation (25) for the case f_SB = 0. For this case, the renovation time is given by

$$\begin{eqnarray}&&{\tau }_{{\rm{r}}}={\tau }_{\mathrm{SN}}^{{\rm{r}}}=\displaystyle \frac{3}{4\pi {l}_{\mathrm{SN}}^{3}\nu }=6.8\,\mathrm{Myr}\ \displaystyle \frac{1}{4}{\nu }_{50}^{-1}{E}_{51}^{-16/17}{n}_{0.1}^{19/17}{c}_{10}.\end{eqnarray} \tag{ 28 }$$

Equations (22) (or A10), (25) (or A16), and (26), can be used wherever the quantities l, u, and τ appear in the equations of Section 3.

Turbulence driving in galaxies is still not very well understood and there is still a lack of consensus, e.g., about the driving scale(s). Thus, we choose to explore the implications of making different assumptions about turbulence driving, as summarized in Table 2. Model S is our fiducial model and uses the framework summarized above for turbulence driven by expanding isolated SNRs. We include two simpler models for comparison. Model Alt1 is our minimalistic model, which assumes l ∝ h, u ∝ c_s, and τ ∝ l/u (in deriving the scaling relation exponents, equalities can be replaced with proportionalities). Model Alt2 is our hybrid model and assumes ${l}_{\mathrm{SN}}\propto h$ and τ ∝ l/u, but retains Equation (25) for u, which permits two asymptotic regimes (${ \mathcal M }\ll A$ and ${ \mathcal M }\gg A$), as in Model S. The python tool mentioned in Section 2 can be used to explore a wider set of combinations of assumptions about turbulence.

Scaling relations for Models S, Alt1, and Alt2 are given in Tables 3 and 4. Where reduced fractional forms of exponents are cumbersome to write, we have rounded to decimal values. We choose to write the scaling relations in terms of Σ_tot rather than Σ_⋆, but we refer to them somewhat interchangeably below. Note that the dependency on ζ is the same as the dependency on 1/Σ_tot.

Table 2. Versions of the Model for Which Scaling Relations are Derived

Model	Driving scale	Regime	Correl. time	Mach no.
S	$\propto {R}_{\mathrm{SN}}({t}_{{\rm{s}}})$	a	τe < τr ⇒ τ = τe	${ \mathcal M }\ll A$
		b	τe < τr ⇒ τ = τe	${ \mathcal M }\gg A$
		c	τr < τe ⇒ τ = τr	${ \mathcal M }\ll A$
		d	τr < τe ⇒ τ = τr	${ \mathcal M }\gg A$
Alt1	∝h		τ = τe	${ \mathcal M }={\rm{const}}$
Alt2	∝h	a	τ = τe	${ \mathcal M }\ll A$
		b	τ = τe	${ \mathcal M }\gg A$

Note. See Section 6.1 for details.

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Table 3. Predicted Scaling Relations for Parameters h, l, u, and τ for our Fiducial SN-driven Turbulence Model (Model S) as well as Two Simpler Alternative Models, Alt1 and Alt2

Model	h ∝	l ∝	u ∝	τ ∝
Sa	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}^{-0.37}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.37}{T}^{0.21}$	${{\rm{\Sigma }}}^{-0.50}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.16}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{0.27}$	${{\rm{\Sigma }}}^{0.12}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.21}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-0.07}$
Sb	${{\rm{\Sigma }}}^{-1.48}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1.49}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.99}{T}^{0.33}$	${{\rm{\Sigma }}}^{-0.92}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.55}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.37}{T}^{-0.04}$	${{\rm{\Sigma }}}^{-0.74}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.24}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.50}{T}^{0.17}$	${{\rm{\Sigma }}}^{-0.18}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.31}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.13}{T}^{-0.21}$
Sc	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}^{-0.37}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.37}{T}^{0.21}$	${{\rm{\Sigma }}}^{-0.50}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.16}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{0.27}$	${{\rm{\Sigma }}}^{19/17}{{\rm{\Sigma }}}_{\mathrm{tot}}^{2/17}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1}{T}^{13/34}$
Sd	${{\rm{\Sigma }}}^{-1.48}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1.49}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.99}{T}^{0.33}$	${{\rm{\Sigma }}}^{-0.92}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.55}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.37}{T}^{-0.04}$	${{\rm{\Sigma }}}^{-0.74}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.24}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.50}{T}^{0.17}$	${{\rm{\Sigma }}}^{1.29}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.17}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1.12}{T}^{0.46}$
Alt1	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	T1/2	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}{T}^{1/2}$
Alt2a	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{4/3}$	${{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1/3}$
Alt2b	${{\rm{\Sigma }}}_{\mathrm{tot}}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-2/3}{T}^{-2/3}$	${{\rm{\Sigma }}}_{\mathrm{tot}}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-2/3}{T}^{-2/3}$	${{\rm{\Sigma }}}_{\mathrm{tot}}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1/3}$	${{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1/3}$

Note. To derive scaling relations, the models are subdivided into the asymptotic regimes ${ \mathcal M }\ll A$ and ${ \mathcal M }\gg A$ as well as τ_e/τ_r < 1 and τ_e/τ_r > 1. See Section 6 for details.

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Table 4. Predicted Scaling Relations for the Strengths of the Isotropic (b_iso) and Anisotropic (b_ani) Components of the Random Magnetic Field, and the Strength ($\overline{B}$) and Pitch Angle (p_B ) of the Mean Magnetic Field

Model	biso ∝	bani ∝	$\overline{B}\propto$	$\tan {p}_{B}\propto$
Sa	${{\rm{\Sigma }}}^{0.003}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.34}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{-0.23}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}^{0.07}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.23}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/6}{T}^{-0.26}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}^{0.13}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.13}{T}^{-0.29}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}^{-0.87}{{\rm{\Sigma }}}_{\mathrm{tot}}^{1.46}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{-1.52}$
Sb	${{\rm{\Sigma }}}^{1.24}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.74}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.50}{T}^{0.33}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}^{1.15}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.59}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.56}{T}^{0.23}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}^{0.32}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.19}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.13}{T}^{-0.21}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}^{1.29}{{\rm{\Sigma }}}_{\mathrm{tot}}^{2.17}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1.12}{T}^{-0.54}$
Sc	${{\rm{\Sigma }}}^{0.003}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.34}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{-0.23}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}^{0.56}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.40}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/6}{T}^{-0.03}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}^{0.13}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.13}{T}^{-0.29}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}^{0.12}{{\rm{\Sigma }}}_{\mathrm{tot}}^{1.79}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1.07}$
Sd	${{\rm{\Sigma }}}^{1.24}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.74}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.50}{T}^{0.33}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}^{1.88}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.83}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1.05}{T}^{0.57}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}^{0.32}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.19}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.13}{T}^{-0.21}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}^{2.77}{{\rm{\Sigma }}}_{\mathrm{tot}}^{2.66}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-2.11}{T}^{0.13}$
Alt1	${{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{1/2}$	q1/2Ω1/2Σ1/2 T1/4	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{T}^{1/2}$	q−1Ω−1Σtot T−1/2
Alt2a	${{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{5/6}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/6}{T}^{2/3}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{T}^{1/2}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{1/3}$
Alt2b	${{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{5/6}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/6}{T}^{2/3}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1/3}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{1/3}$

Note. Expressions for b_ani assume qΩτ/2 ≪ 1 and those for $\overline{B}$ assume $\min (1,C{{\prime} }_{\alpha }h/{C}_{\alpha }\tau u)\geqslant {\rm{\Omega }}\tau$ and D ≫ D_c.

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An important caveat is that the observables q(r), Ω(r), Σ(r), Σ_SFR(r), Σ_⋆(r), and T(r) are not mutually statistically independent. For example, the SFR and gas surface densities are typically related by the Kennicutt–Schmidt (KS) law,

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{\mathrm{SFR}}\propto {{\rm{\Sigma }}}^{N},\end{eqnarray} \tag{ 38 }$$

with N ≈ 1.4 (Kennicutt & Evans 2012). As this relation is rather tight and universal, we have used the KS law to replace Σ by ${{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/N}$ in Tables 5 and 6, eliminating the explicit dependence on Σ. We focus on Tables 5 and 6 in the discussion below.

Table 5. As Table 3, but with the Additional Assumption That Σ_SFR ∝ Σ^1.4

Model	h ∝	l ∝	u ∝	τ ∝
Sa	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.37}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.27}{T}^{0.21}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.16}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.021}{T}^{0.27}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.21}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.24}{T}^{-0.07}$
Sb	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1.49}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.064}{T}^{0.33}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.55}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.29}{T}^{-0.04}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.24}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.032}{T}^{0.17}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.31}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.26}{T}^{-0.21}$
Sc	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.37}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.27}{T}^{0.21}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.16}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.021}{T}^{0.27}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{2/17}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.20}{T}^{13/34}$
Sd	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1.49}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.064}{T}^{0.33}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.55}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.29}{T}^{-0.04}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.24}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.032}{T}^{0.17}$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{0.17}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.19}{T}^{0.46}$
Alt1	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	T1/2	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}{T}^{1/2}$
Alt2a	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}T$	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{4/3}$	${{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1/3}$
Alt2b	${{\rm{\Sigma }}}_{\mathrm{tot}}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-2/3}{T}^{-2/3}$	${{\rm{\Sigma }}}_{\mathrm{tot}}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-2/3}{T}^{-2/3}$	${{\rm{\Sigma }}}_{\mathrm{tot}}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1/3}$	${{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{T}^{-1/3}$

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Table 6. As Table 4, but with the Additional Assumption That Σ_SFR ∝ Σ^1.4

Model	biso ∝	bani ∝	$\overline{B}\propto$	$\tan {p}_{B}\propto$
Sa	${{\rm{\Sigma }}}_{\mathrm{tot}}^{0.34}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.34}{T}^{-0.23}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.23}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.21}{T}^{-0.26}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.13}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.091}{T}^{-0.29}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{tot}}^{1.46}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.29}{T}^{-1.52}$
Sb	${{\rm{\Sigma }}}_{\mathrm{tot}}^{0.74}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.39}{T}^{0.33}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.59}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.26}{T}^{0.23}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.19}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.10}{T}^{-0.21}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{tot}}^{2.17}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.19}{T}^{-0.54}$
Sc	${{\rm{\Sigma }}}_{\mathrm{tot}}^{0.34}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.34}{T}^{-0.23}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.40}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.23}{T}^{-0.03}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.13}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.091}{T}^{-0.29}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{tot}}^{1.79}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.24}{T}^{-1.07}$
Sd	${{\rm{\Sigma }}}_{\mathrm{tot}}^{0.74}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.39}{T}^{0.33}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.83}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.29}{T}^{0.57}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.19}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.10}{T}^{-0.21}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{tot}}^{2.66}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.13}{T}^{0.13}$
Alt1	${{\rm{\Sigma }}}_{\mathrm{tot}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.36}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.36}{T}^{1/4}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.36}{T}^{1/2}$	q−1Ω−1Σtot T−1/2
Alt2a	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.69}{T}^{5/6}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.52}{T}^{2/3}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.36}{T}^{1/2}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{1/3}$
Alt2b	${{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.69}{T}^{5/6}$	${q}^{1/2}{{\rm{\Omega }}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.52}{T}^{2/3}$	${q}^{1/2}{\rm{\Omega }}{{\rm{\Sigma }}}_{\mathrm{tot}}^{1/2}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.024}{T}^{-1/3}$	${q}^{-1}{{\rm{\Omega }}}^{-1}{{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{T}^{1/3}$

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Other correlations between our observables have also been measured, like the so-called spatially resolved star-forming main sequence relating Σ_SFR and Σ_⋆ (e.g., Cano-Díaz et al. 2016; Baker et al. 2022). However, this relation shows quite a lot of scatter. Moreover, the power-law exponent, $\partial \mathrm{ln}{{\rm{\Sigma }}}_{\star }/\partial \mathrm{ln}{{\rm{\Sigma }}}_{\mathrm{SFR}}$, is not well constrained, and may be sensitive to resolution (Hani et al. 2020). For this reason, we treat Σ_tot ≈ Σ_⋆ and Σ_SFR as mutually independent quantities.

In Model S (SN-driven model), the exponents in Tables 5 and 6 are remarkably consistent when transitioning between the four different physical regimes. Thus, the results for Model S are fairly insensitive to variations in the Mach number and the ratio of the eddy turnover and renovation times. By contrast, in Model Alt2 (hybrid model), the rms turbulent velocity u has very different power-law exponents for the two limiting cases (${ \mathcal M }\ll A$ and ${ \mathcal M }\gg A$). This implies a sharp transition in the transonic regime, which seems unlikely from a physical standpoint, lending support to the claim that Model S is the most realistic among the models.

We now compare our results with previously published observations, continuing to focus on Tables 5 and 6, which take into account the KS law, i.e., the empirical relation between Σ and Σ_SFR (Equation (38)).

6.5.1. Turbulent Velocity

6.5.2. Relationship between Field Strength and Σ_SFR

It has been observed in many galaxies that the total magnetic field strength correlates with Σ_SFR, and many authors have obtained values for j in the relation B∝Σ_SFR ^j. Niklas & Beck (1997) find j = 0.34 ± 0.08, assuming energy equipartition between magnetic field and cosmic rays. Chyży (2008) finds a tight correlation looking at the spatial variation of B and Σ_SFR in the Virgo Cluster spiral galaxy NGC 4254, and obtains j = 0.18 ± 0.01. They also find a tight correlation between the random component (calculated from the unpolarized emission) and Σ_SFR, with a power-law exponent of 0.26 ± 0.01. Chyży et al. (2011) find j = 0.30 ± 0.04 for dwarf irregular galaxies. Heesen et al. (2014) infer j = 0.30 ± 0.02 for 17 THINGS galaxies. Van Eck et al. (2015) find j = 0.19 ± 0.03, using results that assume energy equipartition. Wang & Lilly (2022) interpret Heesen et al. (2014) and other observations and assume energy equipartition to obtain j = 0.15 ± 0.06. They show how an exponential profile for Σ_SFR(r)—seen in many disk galaxies—can be explained by their model as long as 0.1 ≲ j ≲ 0.2.

For the galaxy NGC 6946, Tabatabaei et al. (2013) obtain j = 0.14 ± 0.01 and, if B is replaced by the turbulent component (calculated from the unpolarized emission), they obtain an equally strong correlation with a slightly different exponent of 0.16 ± 0.01. Vollmer et al. (2020) obtain j values between 0.03 and 0.92, with a lot of scatter and a best-fit value of either 0.13 ± 0.09 or 0.44, depending on the statistical technique adopted. There has also been a flurry of very recent work. Heesen et al. (2023) find j = 0.182 ± 0.004, but with considerable scatter, or j = 0.22 when they use the mean values of each of the 12 galaxies considered. However, when they account for cosmic-ray transport, they infer a value of j = 0.284 ± 0.006 instead of 0.182 ± 0.004. These authors also report that B is found to be more tightly correlated with the gas surface density ${\rm{\Sigma }}={{\rm{\Sigma }}}_{\mathrm{HI}}+{{\rm{\Sigma }}}_{{{\rm{H}}}_{2}}$ than with Σ_SFR. Tabatabaei et al. (2022) study the galaxy M33 and find bimodal behavior. For the high-SFR regime they obtain j = 0.25–0.26 and, for the low-SFR regime, they obtain j = 0.10–0.12. Basu et al. (2017) obtains j = 0.35 ± 0.03 for the dwarf irregular galaxy IC 10. Finally, Manna & Roy (2023) obtain j = 0.31 ± 0.06 for a sample of seven galaxies, using the equipartition assumption.

In the present work we have derived expressions for b_iso, b_ani and $\overline{B}$, and scaling relations are presented in Table 6. Observers generally infer that the isotropic random component of the magnetic field dominates in most galaxies. Moreover, when a mean component is modeled from the observations, this component is usually determined to be quite weak, suggesting that most of the polarized emission is not due to the mean field, but to an anisotropic small-scale component (Beck et al. 2019). At face value, this would imply that, typically, ${b}_{\mathrm{iso}}\gt {b}_{\mathrm{ani}}\gt \overline{B}$. However, the mean field may often be underestimated if, as seems likely, it contains significant power at scales that cannot be resolved.

Examining the scaling relations for b_iso, b_ani, and $\overline{B}$ in Table 6 for Model S, we see that they are related to Σ_SFR by power laws with exponents in the ranges 0.34–0.39, 0.21–0.29, and 0.09–0.10, respectively. Thus, if b _iso is the dominant field component, a suitable weighted average might give j ∼ 0.30, whereas if $\overline{{\boldsymbol{B}}}$ is the dominant component, we might expect j ∼ 0.15. While a detailed comparison between theory and observation is clearly not possible here, the theoretical and empirical estimates for j agree rather well. Furthermore, the prediction of our model that j is higher for the isotropic random component than for the mean component aligns with the findings of Chyży (2008) and Tabatabaei et al. (2013) (though the latter work finds only a marginal difference), as well as with those of Tabatabaei et al. (2022) if the mean field is relatively stronger in the low-SFR regions.

On the other hand, Model Alt1 predicts j = 0.36, which is reasonably consistent with observations, but Model Alt2 predicts an exponent 0.69 for b_iso, 0.52 for b_ani, and a range 0.02–0.36 for $\overline{B}$, which are too high to explain observations.

These results are summarized in the left panel of Figure 2, where it can be seen that the range predicted by Model S corresponds reasonably well to the range spanned by the data. Keep in mind that in our model j would be a weighted average of the j values for b_iso, b_ani, and $\overline{B}$, so the observed values of j would be expected to lie somewhere between the highest and lowest predictions. Taking our Model S at face value, the upper cluster of points between 0.25 and 0.35 is just what might be expected if b_iso dominates but b_ani and/or $\overline{B}$ are significant, whereas the lower cluster of points between about 0.1 and 0.2 could arise if the mean magnetic field is particularly strong.

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A theoretical model by Schleicher & Beck (2013) (see also Schleicher & Beck 2016) predicts $B\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}$, whereas the model by Schober et al. (2023) (see also Schober et al. 2016) predicts $B\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{{\rm{\Sigma }}}^{1/6}$, which gives $B\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.45}$ if Σ_SFR ∝ Σ^1.4 is assumed. Those models have similarities with one another. They are also somewhat similar to Model Alt2a of the present work, but their prescriptions for the gas scale height are different from ours.

6.5.3. Relationship between Field Strength and Density

6.5.4. Relationship between Field Alignment and Σ_SFR

6.5.5. Other Dependencies of Magnetic Field Properties

6.5.6. Relation between Mean and Total Field Strengths

6.5.7. Relation between Mean-field Strength and Pitch Angle

6.5.8. Cases Where No Empirical Scaling Relation Was Found

6.5.9. Arm-interarm Contrast of Random and Mean Fields

Elstner & Gressel (2012) is the only work of which we are aware that obtains such scaling relations from direct numerical simulations (Gressel et al. 2008a, 2011). These authors employ simulations of SN-driven turbulence to study dynamo action in a small section of a galaxy. They provide scaling relations for a few quantities, based on nine simulations. In seven of their runs, the final magnetic field energy is of the order of the turbulent energy, but in the other two the simulation ends when B ≪ B_eq, so those runs are less relevant. They also find that both vertical outflows and diamagnetic pumping are important, but these effects have been neglected in deriving the scaling relations found in the present work.

Given the differences between their model and ours, it is perhaps most useful to compare results for the turbulent diffusivity η because this quantity is not expected to be sensitive to details such as the dynamo saturation mechanism (Brandenburg & Subramanian 2005). Elstner & Gressel (2012) find η ∝ σ^0.4 n^0.4Ω^−0.55, where σ is the SFR density in dimensions of [T]⁻¹[L]⁻², and, as above, n and Ω are the number density of gas and the rotation angular speed, respectively. The value of σ is determined from their input SN rate assuming a constant initial mass function, so we can replace σ with Σ_SFR, resulting in $\eta \propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.4}{n}^{0.4}{{\rm{\Omega }}}^{-0.55}$.

To compare this result with our models, we derive scaling relations using Equation (18), the scaling relations of Table 3, and Equation (30), and assume $\rho /n={\rm{const}}$. We use Table 3 rather than Table 5 because the overall SN rate density is varied as a parameter in the Elstner & Gressel (2012) simulations. ¹¹ Neglecting the temperature dependence, Models Sa-d give $\eta \propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{n}^{-0.9}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.4}$, $\eta \propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.2}{n}^{-0.7}{{\rm{\Sigma }}}_{\mathrm{tot}}^{0.2}$, $\eta \propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}{n}^{0.1}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.3}$, and $\eta \propto {n}^{-0.1}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-0.4}$, respectively (where, in the last relation, we have neglected the very weak dependence on Σ_SFR). Models Alt1, Alt2a, and Alt2b give $\eta \propto {{\rm{\Sigma }}}_{\mathrm{tot}}^{-1}$, $\eta \propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{1/3}{{\rm{\Sigma }}}_{\mathrm{tot}}^{-2}$, and $\eta \propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1}{{\rm{\Sigma }}}_{\mathrm{tot}}^{2}$, respectively. Given that Ω depends on the gravitational potential, Ω and Σ_tot are not completely independent from one another. Thus, the absence of Ω in our scaling relations for η does not necessarily imply disagreement with Elstner & Gressel (2012). For this reason, let us focus on Σ_SFR and n. In Models Sa, Sb, and Alt2a, we find Σ_SFR exponents of 1/3, 0.2, and 1/3, respectively, which are close to the value of 0.4 obtained by Elstner & Gressel (2012). In the other models, the agreement is poorer, with exponents −1/3, ≈0, 0, and −1, for Models Sc, Sd, Alt1, and Alt2b, respectively. Now turning to the dependence on n, we obtain a positive exponent only for Model Sc (0.1), whereas other models give exponents ranging from −0.9 to 0, which is in disagreement with the value of 0.4 obtained by Elstner & Gressel (2012). We can conclude that the level of agreement between our predicted scaling relation for η and that found by Elstner & Gressel (2012) is rather poor. However, the dependence on Σ_SFR agrees quite well in Model S whenever τ_e < τ_r and in Model Alt2a, where ${ \mathcal M }\ll A$ (although Gressel et al. 2008b report mildly supersonic turbulence for the simulations of that paper, so Alt2b is probably more relevant than Alt2a, but there the model predicts a dependence on Σ_SFR different from that seen in the simulations).

In addition, Gressel et al. (2008a) report that the ratio of the strength of the mean field to that of the random field is related to the SN rate (SFR density) as $\overline{B}/b\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.38\pm 0.01}$, while for similar simulations by Bendre et al. (2015), the authors report $\overline{B}/b\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{-0.30\pm 0.07}$. Generally, we expect b_iso > b_ani (Beck et al. 2019), so we focus on the ratio $\overline{B}/{b}_{\mathrm{iso}}$, using Table 4. For Model Alt1, we find no dependence of $\overline{B}/{b}_{\mathrm{iso}}$ on Σ_SFR. For Models Alt2a and Alt2b, we find $\overline{B}/{b}_{\mathrm{iso}}\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}$ and $\overline{B}/{b}_{\mathrm{iso}}\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{-2/3}$, respectively. For Models Sa and Sc, we find $\overline{B}/{b}_{\mathrm{iso}}\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{-1/3}$, and for Models Sb and Sd, we find $\overline{B}/{b}_{\mathrm{iso}}\propto {{\rm{\Sigma }}}_{\mathrm{SFR}}^{0.37}$. Thus, for Models Alt2a, Sa, and Sc (all of which have ${ \mathcal M }\ll A$) the agreement is good, for Model Alt2b the agreement is fair, and for Models Sb and Sd the agreement is poor. Differences may be partly attributable to the different physical assumptions between our models and those of Elstner & Gressel (2012). However, there is another important caveat, namely that the separation of the magnetic field into random and mean components may be sensitive to the method of averaging. The mean-field dynamo theory on which our models are founded assumes ensemble averaging, while Gressel et al. (2008a) use horizontal averaging.

Thus far we have focused on the scaling relation exponents. We now make an order-of-magnitude estimate of the proportionality coefficients in the magnetic field strength versus Σ_SFR relations. This is important for checking that the magnitude of the magnetic field strengths predicted by the model roughly agree with observational and theoretical expectations. We must first fix the values of the other observables; this is done by choosing a typical value based on data for the galaxies M31, M33, M51, and NGC 6946, which are the galaxies studied in Paper II (in preparation). We choose Ω = 20 km s⁻¹ kpc⁻¹, Σ = 5 M_⊙ pc⁻², and Σ_tot = 175 M_⊙ pc⁻², respectively. In addition we choose T = 10⁴ K and q = 1. Further, the adjustable parameters of the model are chosen to be values that generally produce good agreement with the velocity dispersion and magnetic field data in the preliminary analysis of Paper II: ζ = 10, β = 8, C_α = 4, and K = 0.3.

The scaling law can be written as

$$\begin{eqnarray}&&B=\widetilde{B}{\left(\displaystyle \frac{{{\rm{\Sigma }}}_{\mathrm{SFR}}}{\widetilde{{\rm{\Sigma }}}}\right)}^{j},\end{eqnarray} \tag{ 39 }$$

where $\widetilde{{\rm{\Sigma }}}={10}^{-3}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$ is a typical value of the SFR surface density and $\widetilde{B}$ is the coefficient we want to estimate. To calculate $\widetilde{B}$, we also need a proportionality coefficient for the KS relation Σ_SFR = CΣ^N . Using the data for Σ_SFR and Σ used in Paper II, and assuming N = 1.4, we obtain C≈10⁻¹⁵ in cgs units. As an alternative, we estimated the coefficient from Leroy et al. (2008), who found that the star formation efficiency is, on average, ∼3 × 10⁻¹⁰ yr⁻¹ for Σ ≈ 5–10 M_⊙ pc⁻². For N = 1.4, this leads to C≈10⁻¹⁶ in cgs units. With these values, we obtain the values in Table 7. These rough estimates are consistent, at the order-of-magnitude level, with those of Heesen et al. (2023), who estimate a range 4.7–7.4 μG for $\widetilde{B}$. Moreover, we generally find ${b}_{\mathrm{iso}}\gt {b}_{\mathrm{ani}}\gt \overline{B}$, which is consistent with the usual ordering inferred by Beck et al. (2019). We note, however, that this ordering can be sensitive to the parameters of our model.

Table 7. Typical Numerical Coefficients ${\widetilde{B}}_{i}$ for Scaling Relations of the Form ${B}_{i}={\widetilde{B}}_{i}{({{\rm{\Sigma }}}_{\mathrm{SFR}}/\widetilde{{\rm{\Sigma }}})}^{j}$, Where $\widetilde{{\rm{\Sigma }}}={10}^{-3}\,{M}_{\odot }\,{\mathrm{yr}}^{-1}\,{\mathrm{kpc}}^{-2}$, for Two Different Estimates of the Numerical Coefficient in the Kennicutt–Schmidt Law Relating Σ_SFR and Σ (see Section 6.7 for details)

	Model S			Model Alt1			Model Alt2
KS coef.	biso	bani	$\overline{B}$	biso	bani	$\overline{B}$	biso	bani	$\overline{B}$
10−15	10	2	1	8	3	3	12	4	3
10−16	44	16	2	20	7	6	28	10	6

Notes. Fitting data for the total magnetic field strength in multiple galaxies (estimated assuming energy equipartition with cosmic rays), Heesen et al. (2023) obtain $\widetilde{B}$ in the range 4.7–7.4 μG. Our model coefficients are generally of the same order of magnitude as theirs. All field strengths are in μG and the KS relation coefficient is in cgs units.

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We used the model to derive scaling relations for the gas scale height, key turbulence parameters, and magnetic field properties, in terms of the observables. These are relations of the form X ∝ x^a y^b z^c ⋯ , where X is a quantity of interest, x, y, and z are observable quantities, and a, b, and c are constants. Scaling relations are useful for placing priors on missing information in data sets and for providing physical insight, and are ubiquitous in astrophysics. The scaling relations can be derived analytically, but we provide a link to a general numerical tool in Section 2. To reduce the solutions to scaling relations, we focused on certain plausible asymptotic regimes. Most importantly, we assumed that turbulence is driven purely by isolated SNe and that mean (large-scale) vertical and radial gas motions can be neglected. In deriving scaling relations for the mean magnetic field strength, we assumed that the Coriolis number is small and that the dynamo is strong (Section 6), but these assumptions were not necessary for deriving scaling relations for the pitch angle of the mean field.

These scaling relations can be found in Tables 3 and 4. Our fiducial model considers turbulence to be driven by isolated SNRs, as they decelerate to the sound speed and merge with the ISM. Given the lack of consensus about turbulence driving in the ISM, we also considered two simpler prescriptions for calculating the turbulence parameters, as summarized in Table 2.

The predictions in Tables 3 and 4 do not consider possible correlations between the observed quantities on the right side of the scaling relations. Given the well-known empirical relation between Σ_SFR and Σ (the Kennicutt–Schmidt law), we used Σ_SFR ∝ Σ^N with N = 1.4 to eliminate the dependence on Σ, and the results are presented in Tables 5 and 6.

The theoretical scaling relations in Tables 5 and 6 were then compared with empirical scaling relations in the literature (Section 6.5). The level of agreement between the model predictions and observations is remarkably good for our fiducial model, Model S, given the various theoretical and observational uncertainties. The level of agreement is generally poorer for the alternative turbulence prescriptions we tried (Models Alt1 and Alt2). This can be taken as further evidence that turbulence driving in nearby galaxies is dominated by SN feedback. It also suggests that assuming that turbulence is driven at the disk scale height can lead to incorrect results, and points to a need for modeling turbulence driving in more detail, taking into account the dynamics of SNRs.

Our model should be thought of as an adaptable tool for understanding galactic magnetic fields rather than a fixed set of formulae. This tool could be extended by including additional physical effects. Outflows and SBs (which tend to cause outflows) are already included in the most general version of our model, summarized in the Appendix, but including these effects introduces extra parameters that are challenging to constrain. SBs may be unlikely to dominate turbulence driving because they tend to blow out of the disk, which limits the amount of energy they transfer to the ISM (Boomsma et al. 2008; Chamandy & Shukurov 2020). Likewise, rough estimates suggest that mean outflow speeds may often be too small to strongly affect mean-field dynamo action (Chamandy et al. 2016). However, such findings are preliminary and more work is needed on the roles of SBs and outflows in turbulence driving and magnetic field evolution. Furthermore, our model does not include radial inflow (e.g., Schmidt et al. 2016; Trapp et al. 2022), which may affect the mean-field dynamo (Moss et al. 2000) and play a role in driving interstellar turbulence, though primarily at high redshift (Krumholz et al. 2018).

On the MHD side, we have not included the turbulent tangling of the mean field to produce random field, nor additional effects—still not very well understood—involving the influence of the random field on the mean-field dynamo (e.g., Chamandy & Singh 2018; Gopalakrishnan & Subramanian 2023). Furthermore, the equations for the turbulence parameters, gas scale height, etc., do not take into account the magnetic field. This is hard to avoid given the lack of current knowledge about such feedback effects, and attempting to include them would make the model more complicated without providing much benefit, given the various uncertainties. Nevertheless, such effects may be important (e.g., Evirgen et al. 2019). In some cases, they can be roughly included in our existing model by choices of parameter values; for example, the effect of the magnetic pressure on the gas scale height can be included heuristically by increasing the value of ζ in Equation (32).

Our model does not consider cosmological evolution, but there is scope for extending the model to include high redshift galaxies (Schleicher & Beck 2013; Krumholz et al. 2018; Schober et al. 2023).

There is scope for combining our model with galaxy models that rely on certain magnetic field parameters as input in order to calculate those parameters self-consistently. For example, as mentioned in the introduction, the galaxy model of Wang & Lilly (2022) depends critically on the exponent in the scaling relation between total magnetic field strength B and SFR surface density Σ_SFR, as do theoretical models of the infrared-radio correlation (e.g., Schleicher & Beck 2013). In our model, there is—in the strict sense, at least—no scaling relation between total field strength B and Σ_SFR because $B={({b}_{\mathrm{iso}}^{2}+{b}_{\mathrm{ani}}^{2}+{\overline{B}}^{2})}^{1/2}$, with each component having its own separate scaling relation (for b_ani and $\overline{B}$, that too only in certain plausible asymptotic limits). However, the exponents in each of these relations lie in the range 0.1–0.4 for our fiducial model, Model S (Table 6), and this range is broadly consistent with the values inferred from observations (Section 6.5.2 and the left panel of Figure 2). Thus, our model can be seen as a possible refinement to models that assume that B simply scales with B_eq (defined in Equation (12)) without other scalings. Even if $B/{B}_{\mathrm{eq}}={\rm{const}}$ is assumed, one still needs a relation between the rms turbulent speed u and Σ_SFR. Our fiducial model, Model S, predicts a very weak dependence of u on Σ_SFR, with power-law exponent in the range −0.03 to −0.02 (Table 5). Here again, our model, which includes a detailed model of turbulence driving by SNe, can be seen as a possible refinement to other approaches.

Modeling scaling relations can be complicated by correlations between the observables. While we used the KS relation to remove Σ as an independent variable, we made no attempt to incorporate the resolved star-forming main sequence relation between Σ_⋆ and Σ_SFR, for instance. If this could be used to eliminate Σ_⋆, it would affect the Σ_SFR dependence of the scaling relations derived. Nor did we attempt to relate the angular rotation speed Ω with the disk surface density Σ_tot, for example, though they are not completely independent. Another caveat is that parameters like ζ may vary somewhat between galaxies, which would introduce scatter.

Observationally derived scaling relations for magnetic fields are now plentiful and will improve as new instruments like the Square Kilometre Array become available. This increases the urgency of studying such scaling relations theoretically and, in our view, various complementary approaches can be utilized. For instance, one could make use of a population synthesis model that solves for the magnetic fields of galaxies using the output of a semi-analytic model of galaxy formation as input (Rodrigues et al. 2015, 2019). Second, one could run several local ISM simulations to explore the parameter space, building on the work begun by Elstner & Gressel (2012). Third, one could explore scaling relations using cosmological zoom MHD simulations (e.g., Pakmor et al. 2024). Given the heterogeneous nature of galaxies, such a multipronged statistical approach may be very useful for learning about galactic magnetic fields and the various processes that shape them.

This section generalizes the magnetic field model of Section 3 to include a galactic outflow, with outflow speed U₀.

Using an expression from Hollins et al. (2017), we can write

$$\begin{eqnarray}&&{\sigma }_{z}\simeq {\sigma }_{r}{\left[1+\displaystyle \frac{{K}_{U}{U}_{0}\tau }{l}\left(1+\displaystyle \frac{1}{{\left(1+q{\rm{\Omega }}\tau \right)}^{2}}\right)\right]}^{1/2},\end{eqnarray} \tag{ A1 }$$

where U₀ is defined to be equal to the mean vertical outflow speed at the disk surface z = ± h, with h the scale height of the gaseous disk, and K_U a constant of order unity. As it represents an average for the entire disk, the quantity U₀ should be taken as the mass-weighted, area-averaged outflow speed, and has been estimated to be in the range 0.2–2 km s⁻¹ for spiral galaxies (Shukurov et al. 2006).

Substituting Equations (4) and (A1) into the expressions for b and b_iso, and setting K_U = 1, we obtain

$$\begin{eqnarray}\begin{array}{rcl}{b}_{\mathrm{ani}} & \equiv & \sqrt{{b}^{2}-{b}_{\mathrm{iso}}^{2}}=\displaystyle \frac{1}{\sqrt{3}}\left[2q{\rm{\Omega }}\tau \left(1+\displaystyle \frac{q{\rm{\Omega }}\tau }{2}\right)\right.\\ & & {\left.+\,\displaystyle \frac{{U}_{0}\tau }{l}\left(1+\displaystyle \frac{1}{{\left(1+q{\rm{\Omega }}\tau \right)}^{2}}\right)\right]}^{1/2}.\end{array}\end{eqnarray} \tag{ A2 }$$

The mean magnetic field strength in the saturated state is now given by

$$\begin{eqnarray}&&\overline{B}\equiv | \bar{{\boldsymbol{B}}}| ={{KB}}_{\mathrm{eq}}\,\chi (p){\left(\displaystyle \frac{D}{{D}_{{\rm{c}}}}-1\right)}^{1/2}\displaystyle \frac{l}{h}{\left({R}_{U}+{\pi }^{2}{R}_{\kappa }\right)}^{1/2}.\end{eqnarray} \tag{ A3 }$$

where we make use of the Reynolds-type dimensionless number

$$\begin{eqnarray}&&{R}_{U}\equiv \displaystyle \frac{{U}_{0}h}{\eta }.\end{eqnarray} \tag{ A4 }$$

The critical dynamo number is now given by

$$\begin{eqnarray}&&{D}_{{\rm{c}}}=-{\left(\displaystyle \frac{\pi }{2}\right)}^{5}{\left(1+\displaystyle \frac{1}{{\pi }^{2}}{R}_{U}\right)}^{2}.\end{eqnarray} \tag{ A5 }$$

Additionally, we have

$$\begin{eqnarray}&&\chi (p)={\left(2-\displaystyle \frac{3{\cos }^{2}p}{2\sqrt{2}}\right)}^{-1/2},\end{eqnarray} \tag{ A6 }$$

where

$$\begin{eqnarray}&&\tan {p}_{B}=\displaystyle \frac{{\overline{B}}_{r}}{{\overline{B}}_{\phi }}=\displaystyle \frac{{\pi }^{2}+{R}_{U}}{4{R}_{{\rm{\Omega }}}}=-\displaystyle \frac{{\pi }^{2}\,\tau \,{u}^{2}+6\,h\,{U}_{0}}{12\,q\,{\rm{\Omega }}\,{h}^{2}},\end{eqnarray} \tag{ A7 }$$

and p_B is the pitch angle of $\bar{{\boldsymbol{B}}}$, defined such that −π/2 < p_B ≤ π/2 with p_B < 0 for trailing spirals. Note that χ ≈ 1, depending only weakly on p_B . Given the uncertainties of the model, this dependence is inconsequential, and thus we set χ = 1. The quantity K in Equation (A3) is an adjustable parameter of the model.

This section summarizes the SN-driven turbulence model of Chamandy & Shukurov (2020). For an overall SN rate per unit volume ν, the rate per unit volume of isolated SNe is given by

$$\begin{eqnarray}&&{\nu }_{\mathrm{SN}}=(1-{f}_{\mathrm{SB}})\nu ,\end{eqnarray} \tag{ A8 }$$

where f_SB is the fraction of SNe in SBs, and that of SBs is given by

$$\begin{eqnarray}&&{\nu }_{\mathrm{SB}}=\displaystyle \frac{{f}_{\mathrm{SB}}\nu }{{N}_{\mathrm{SB}}},\end{eqnarray} \tag{ A9 }$$

where N_SB is the mean number of SNe in a given SB. Higdon & Lingenfelter (2005) estimate that the fraction of SNe occurring in OB associations is ∼3/4 for the Milky Way, which suggests f_SB ∼ 3/4 for our Galaxy.

A.2.1. Turbulent Correlation Length l

The turbulent correlation length is estimated as

$$\begin{eqnarray}&&l=\left(\displaystyle \frac{{\rm{\Gamma }}-1}{{\rm{\Gamma }}}\right){C}_{l}{l}_{\mathrm{SB}}\left(\displaystyle \frac{1+({l}_{\mathrm{SN}}/{l}_{\mathrm{SB}}){\dot{E}}_{\mathrm{SN}}^{{\rm{i}}}/{\dot{E}}_{\mathrm{SB}}^{{\rm{i}}}}{1+{\dot{E}}_{\mathrm{SN}}^{{\rm{i}}}/{\dot{E}}_{\mathrm{SB}}^{{\rm{i}}}}\right),\end{eqnarray} \tag{ A10 }$$

where ${l}_{\mathrm{SN}}$ and l_SB are the driving scales for isolated SNe and SBs, respectively, and ${\dot{E}}_{\mathrm{SN}}^{{\rm{i}}}$ and ${\dot{E}}_{\mathrm{SB}}^{{\rm{i}}}$ are their respective energy injection rates. For Kolmogorov turbulence, Γ = 5/3, and C_l is a constant of order unity that comes from turbulence theory: below, we adopt C_l = 3/4 (Monin & Yaglom 2007).

For SBs, we identify the driving scale of turbulence with the final radius reached by an SB in the midplane of the galaxy,

$$\begin{eqnarray}&&{l}_{\mathrm{SB}}\approx \min \left[{R}_{\mathrm{SB}}({t}_{{\rm{s}}}^{\mathrm{SB}}),\lambda h\right],\end{eqnarray} \tag{ A11 }$$

where the first case corresponds to deceleration to c_s and the second case to blowout from the disk. Here ${t}_{{\rm{s}}}^{\mathrm{SB}}$ is the SB age for which the SB expansion has slowed to the ambient sound speed (if it has not blown out), λ is a parameter of order unity, and

$$\begin{eqnarray}&&{R}_{\mathrm{SB}}({t}_{{\rm{s}}}^{\mathrm{SB}})=0.53\,\mathrm{kpc}{\epsilon }_{0.1}^{1/3}{N}_{100}^{1/3}{E}_{51}^{1/3}{n}_{0.1}^{-1/3}{c}_{10}^{-2/3},\end{eqnarray} \tag{ A12 }$$

where 0.1_0.1 is the fraction of the SB energy that is mechanical and N₁₀₀ = N_SB/100.

The ratio of the rates of energy per unit volume injected by isolated SNe and SBs is ${l}_{\mathrm{SN}}^{3}{\nu }_{\mathrm{SN}}/{l}_{\mathrm{SB}}^{3}{\nu }_{\mathrm{SB}}$, which gives

$$\begin{eqnarray}&&\displaystyle \frac{{\dot{E}}_{\mathrm{SN}}^{{\rm{i}}}}{{\dot{E}}_{\mathrm{SB}}^{{\rm{i}}}}=\left\{\begin{array}{l}0.63\left(\displaystyle \frac{3(1-{f}_{\mathrm{SB}})}{{f}_{\mathrm{SB}}}\right){\epsilon }_{0.1}^{-1}{E}_{51}^{-1/17}{n}_{0.1}^{-2/17}{c}_{10},\\ \qquad \qquad \mathrm{if}\,{t}_{{\rm{s}}}^{\mathrm{SB}}\leqslant {t}_{{\rm{b}}}^{\mathrm{SB}};\\ 1.47\left(\displaystyle \frac{3(1-{f}_{\mathrm{SB}})}{{f}_{\mathrm{SB}}}\right){N}_{100}{E}_{51}^{16/17}{n}_{0.1}^{-19/17}{c}_{10}^{-1}{\lambda }^{-3}{h}_{0.4}^{-3},\\ \qquad \qquad \mathrm{if}\,{t}_{{\rm{b}}}^{\mathrm{SB}}\lt {t}_{{\rm{s}}}^{\mathrm{SB}}.\end{array}\right.\end{eqnarray} \tag{ A13 }$$

Here ${t}_{{\rm{b}}}^{\mathrm{SB}}$ is the age of the SB when it blows out of the disk (if it in fact blows out). The similarity solution for SBs yields, from ${\dot{R}}_{\mathrm{SB}}({t}_{{\rm{s}}}^{\mathrm{SB}})={c}_{{\rm{s}}}$,

$$\begin{eqnarray}&&{t}_{{\rm{s}}}^{\mathrm{SB}}=31\,\mathrm{Myr}\ {\epsilon }_{0.1}^{1/3}{N}_{100}^{1/3}{E}_{51}^{1/3}{n}_{0.1}^{-1/3}{c}_{10}^{-5/3}.\end{eqnarray} \tag{ A14 }$$

From ${R}_{\mathrm{SB}}({t}_{{\rm{b}}}^{\mathrm{SB}})=\lambda h$, we find

$$\begin{eqnarray}&&{t}_{{\rm{b}}}^{\mathrm{SB}}=15\,\mathrm{Myr}\ {\epsilon }_{0.1}^{-1/2}{N}_{100}^{-1/2}{E}_{51}^{-1/2}{n}_{0.1}^{1/2}{\lambda }^{5/2}{h}_{0.4}^{5/2},\end{eqnarray} \tag{ A15 }$$

where h_0.4 = h/(0.4 kpc).

A.2.2. Root-mean-square Turbulent Velocity u

Applying the method briefly outlined in Section 4.2 to the case f_SB ≠ 0, we obtain the general expression

$$\begin{eqnarray}&&u={\left[\displaystyle \frac{4\pi }{3}{{lc}}_{{\rm{s}}}^{2}\nu \left((1-{f}_{\mathrm{SB}}){l}_{\mathrm{SN}}^{3}+\displaystyle \frac{{f}_{\mathrm{SB}}}{{N}_{\mathrm{SB}}}{l}_{\mathrm{SB}}^{3}\right)\right]}^{1/3}.\end{eqnarray} \tag{ A16 }$$

A.2.3. Turbulent Correlation Time τ

The quantity τ_r is the time for the flow to renovate due to the passage of an SN or SB blast wave. The renovation rate is equal to the sum of the rates from isolated SNe and SBs, so the renovation time is given by

$$\begin{eqnarray}&&{\tau }_{{\rm{r}}}={\left(\displaystyle \frac{1}{{\tau }_{\mathrm{SN}}^{{\rm{r}}}}+\displaystyle \frac{1}{{\tau }_{\mathrm{SB}}^{{\rm{r}}}}\right)}^{-1}.\end{eqnarray} \tag{ A17 }$$

A generalization of Equation (28) to f_SB ≠ 0 gives

$$\begin{eqnarray}&&{\tau }_{\mathrm{SN}}^{{\rm{r}}}=6.8\,\mathrm{Myr}\ \left(\displaystyle \frac{1}{4(1-{f}_{\mathrm{SB}})}\right){\nu }_{50}^{-1}{E}_{51}^{-16/17}{n}_{0.1}^{19/17}{c}_{10},\end{eqnarray} \tag{ A18 }$$

where ν₅₀ = ν/(50 kpc⁻³ Myr⁻¹). The renovation time for SBs is equal to $3/(4\pi {l}_{\mathrm{SB}}^{3}{\nu }_{\mathrm{SB}})$, which gives

$$\begin{eqnarray}&&{\tau }_{\mathrm{SB}}^{{\rm{r}}}=\left\{\begin{array}{l}4.3\,\mathrm{Myr}\ {\left(\displaystyle \frac{{f}_{\mathrm{SB}}}{3/4}\right)}^{-1}{\nu }_{50}^{-1}{\epsilon }_{0.1}^{-1}{E}_{51}^{-1}{n}_{0.1}{c}_{10}^{2},\\ \qquad \qquad \mathrm{if}\,{t}_{{\rm{s}}}^{\mathrm{SB}}\leqslant {t}_{{\rm{b}}}^{\mathrm{SB}};\\ 9.9\,\mathrm{Myr}\ {\left(\displaystyle \frac{{f}_{\mathrm{SB}}}{3/4}\right)}^{-1}{\nu }_{50}^{-1}{N}_{100}{\lambda }^{-3}{h}_{0.4}^{-3},\\ \qquad \qquad \mathrm{if}\,{t}_{{\rm{s}}}^{\mathrm{SB}}\gt {t}_{{\rm{b}}}^{\mathrm{SB}}.\end{array}\right.\end{eqnarray} \tag{ A19 }$$