Energy equipartition is a powerful theoretical tool for understanding astrophysical plasmas. It is invoked, for example, to measure magnetic fields in the interstellar medium (ISM), as evidence for small-scale turbulent dynamo action, and, in general, to estimate the energy budget of star-forming molecular clouds. In this study we motivate and explore the role of the volume-averaged root-mean-squared (rms) magnetic coupling term between the turbulent, δB and largescale, B 0 fields, (δBV . By considering the second moments of the energy balance equations we show that the rms coupling term is in energy equipartition with the volume-averaged turbulent kinetic energy for turbulence with a sub-Alfvénic large-scale field. Under the assumption of exact energy equipartition between these terms, we derive relations for the magnetic and coupling term fluctuations, which provide excellent, parameter-free agreement with time-averaged data from 280 numerical simulations of compressible MHD turbulence. Furthermore, we explore the relation between the turbulent, mean-field and total Alfvén Mach numbers, and demonstrate that sub-Alfvénic turbulence can only be developed through a strong, large-scale magnetic field, which supports an extremely super-Alfvénic turbulent magnetic field. This means that the magnetic field fluctuations are significantly subdominant to the velocity fluctuations in the sub-Alfvénic large-scale field regime. Throughout our study, we broadly discuss the implications for observations of magnetic fields and understanding the dynamics in the magnetised ISM.

The turbulent dynamo is a powerful mechanism that converts turbulent kinetic energy to magnetic energy. A key question regarding the magnetic field amplification by turbulence, is, on what scale, kp, do magnetic fields become most concentrated? There has been some disagreement about whether kp is controlled by the viscous scale, kν (where turbulent kinetic energy dissipates), or the resistive scale, kη (where magnetic fields dissipate). Here we use direct numerical simulations of magnetohydrodynamic turbulence to measure characteristic scales in the kinematic phase of the turbulent dynamo. We run 104-simulations with hydrodynamic Reynolds numbers of 10 ≤ Re ≤ 3600, and magnetic Reynolds numbers of 270 ≤ Rm ≤ 4000, to explore the dependence of kp on kν and kη. Using physically motivated models for the kinetic and magnetic energy spectra, we measure kν, kη and kp, making sure that the obtained scales are numerically converged. We determine the overall dissipation scale relations $k_\nu = (0.025^{+0.005}_{-0.006})\, k_\text{turb}\, \mbox{Re}^{3/4}$ and $k_\eta = (0.88^{+0.21}_{-0.23})\, k_\nu \, \mbox{Pm}^{1/2}$, where kturb is the turbulence driving wavenumber and Pm = Rm/Re is the magnetic Prandtl number. We demonstrate that the principle dependence of kp is on kη. For plasmas where Re ≳ 100, we find that $k_p= (1.2_{-0.2}^{+0.2})\, k_\eta$, with the proportionality constant related to the power-law ‘Kazantsev’ exponent of the magnetic power spectrum. Throughout this study, we find a dichotomy in the fundamental properties of the dynamo where Re > 100, compared to Re < 100. We report a minimum critical hydrodynamic Reynolds number, Recrit = 100 for bonafide turbulent dynamo action.

Energy equipartition is a powerful theoretical tool for understanding astrophysical plasmas. It is invoked, for example, to measure magnetic fields in the interstellar medium (ISM), as evidence for small-scale turbulent dynamo action, and, in general, to estimate the energy budget of star-forming molecular clouds. In this study we motivate and explore the role of the volume-averaged root-mean-squared (rms) magnetic coupling term between the turbulent, $\delta \mathrm{{\boldsymbol {\mathit {B}}}}$ and large-scale, $\operatorname{\mathrm{{\boldsymbol {\mathit {B}}}}_0}$ fields, $\operatorname{\left\langle (\delta \mathrm{{\boldsymbol {\mathit {B}}}}\cdot \operatorname{\mathrm{{\boldsymbol {\mathit {B}}}}_0})^{2} \right\rangle ^{1/2}_{\mathcal {V}}}$. By considering the second moments of the energy balance equations we show that the rms coupling term is in energy equipartition with the volume-averaged turbulent kinetic energy for turbulence with a sub-Alfvénic large-scale field. Under the assumption of exact energy equipartition between these terms, we derive relations for the magnetic and coupling term fluctuations, which provide excellent, parameter-free agreement with time-averaged data from 280 numerical simulations of compressible MHD turbulence. Furthermore, we explore the relation between the turbulent, mean-field and total Alfvén Mach numbers, and demonstrate that sub-Alfvénic turbulence can only be developed through a strong, large-scale magnetic field, which supports an extremely super-Alfvénic turbulent magnetic field. This means that the magnetic field fluctuations are significantly subdominant to the velocity fluctuations in the sub-Alfvénic large-scale field regime. Throughout our study, we broadly discuss the implications for observations of magnetic fields and understanding the dynamics in the magnetised ISM.

The turbulent small-scale dynamo (SSD) is likely to be responsible for the magnetisation of the interstellar medium (ISM) that we observe in the Universe today. The SSD efficiently converts kinetic energy Ekin into magnetic energy Emag, and is often used to explain how an initially weak magnetic field with Emag ≪ Ekin is amplified, and then maintained at a level Emag ≲ Ekin. Usually, this process is studied by initialising a weak seed magnetic field and letting the turbulence grow it to saturation. However, in this Part I of the Growth or Decay series, using three-dimensional, visco-resistive magnetohydrodynamical turbulence simulations up to magnetic Reynolds numbers of 2000, we show that the same final state in the integral quantities, energy spectra, and characteristic scales of the magnetic field can also be achieved if initially Emag ∼ Ekin or even if initially Emag ≫ Ekin. This suggests that the final saturated state of the turbulent dynamo is set by the turbulence and the material properties of the plasma, independent of the initial structure or amplitude of the magnetic field. We discuss the implications this has for the maintenance of magnetic fields in turbulent plasmas and future studies exploring the dynamo saturation.