Abstract:The Saffman helicity invariant of Hosking & Schekochihin (Phys. Rev. X, vol. 11, issue 4, 2021, 041005), which we here call the Hosking integral, has emerged as an important quantity that may govern the decay properties of magnetically dominated non-helical turbulence. Using a range of different computational methods, we confirm that this quantity is indeed gauge invariant and nearly perfectly conserved in the limit of large Lundquist numbers. For direct numerical simulations with ordinary viscosity and ma… Show more
“…As already noted by Zhou et al. (2022), there is an initial increase in . This is explained by the fact that the magnetic field obeys Gaussian statistics initially, but not during the later evolution.…”
Section: Comparison With Simulations For Different Diffusivitiessupporting
confidence: 52%
“…Such large values can currently only be obtained with magnetic hyper-diffusivity (Hosking & Schekochihin 2021; Zhou et al. 2022), but this has not been attempted in the present work. Figure 4.Evolution of , showing first a slight increase and then a decline proportional to for Run B1 and proportional to for Run B2.…”
Section: Comparison With Simulations For Different Diffusivitiesmentioning
confidence: 96%
“…Earlier work by Zhou et al. (2022) showed that decreases as the Lundquist number increases, and is, in MHD, approximately 0.2 for , and decreases to for . Such large values can currently only be obtained with magnetic hyper-diffusivity (Hosking & Schekochihin 2021; Zhou et al.…”
Section: Comparison With Simulations For Different Diffusivitiesmentioning
confidence: 96%
“…In practice, it is chosen empirically and must still be small compared with the size of the domain; see Hosking & Schekochihin (2021) for various examples and Zhou et al. (2022) for a comparison of different computational techniques for obtaining .…”
Section: Hosking Integral and Scaling For The Hall Cascadementioning
confidence: 99%
“…The decay properties of the non-helical Hall cascade were not yet theoretically understood at the time. In the last one to two years, however, significant progress has been made in describing the decay of magnetically dominated turbulence, where a new conserved quantity has been identified, which is now called the Hosking integral (see Zhou, Sharma & Brandenburg (2022), for details regarding the naming). The purpose of the present paper is to propose the scaling of the Hall cascade under the assumption that it is governed by the constancy of the Hosking integral, which now has different dimensions than in MHD.…”
The Hosking integral, which characterizes magnetic helicity fluctuations in subvolumes, is known to govern the decay of magnetically dominated turbulence. Here, we show that, when the evolution of the magnetic field is controlled by the motion of electrons only, as in neutron star crusts, the decay of the magnetic field is still controlled by the Hosking integral, but now it has effectively different dimensions than in ordinary magnetohydrodynamic (MHD) turbulence. This causes the correlation length to increase with time
$t$
like
$t^{4/13}$
instead of
$t^{4/9}$
in MHD. The magnetic energy density decreases like
$t^{-10/13}$
, which is slower than in MHD, where it decays like
$t^{-10/9}$
. These new analytic results agree with earlier numerical simulations for the non-helical Hall cascade.
“…As already noted by Zhou et al. (2022), there is an initial increase in . This is explained by the fact that the magnetic field obeys Gaussian statistics initially, but not during the later evolution.…”
Section: Comparison With Simulations For Different Diffusivitiessupporting
confidence: 52%
“…Such large values can currently only be obtained with magnetic hyper-diffusivity (Hosking & Schekochihin 2021; Zhou et al. 2022), but this has not been attempted in the present work. Figure 4.Evolution of , showing first a slight increase and then a decline proportional to for Run B1 and proportional to for Run B2.…”
Section: Comparison With Simulations For Different Diffusivitiesmentioning
confidence: 96%
“…Earlier work by Zhou et al. (2022) showed that decreases as the Lundquist number increases, and is, in MHD, approximately 0.2 for , and decreases to for . Such large values can currently only be obtained with magnetic hyper-diffusivity (Hosking & Schekochihin 2021; Zhou et al.…”
Section: Comparison With Simulations For Different Diffusivitiesmentioning
confidence: 96%
“…In practice, it is chosen empirically and must still be small compared with the size of the domain; see Hosking & Schekochihin (2021) for various examples and Zhou et al. (2022) for a comparison of different computational techniques for obtaining .…”
Section: Hosking Integral and Scaling For The Hall Cascadementioning
confidence: 99%
“…The decay properties of the non-helical Hall cascade were not yet theoretically understood at the time. In the last one to two years, however, significant progress has been made in describing the decay of magnetically dominated turbulence, where a new conserved quantity has been identified, which is now called the Hosking integral (see Zhou, Sharma & Brandenburg (2022), for details regarding the naming). The purpose of the present paper is to propose the scaling of the Hall cascade under the assumption that it is governed by the constancy of the Hosking integral, which now has different dimensions than in MHD.…”
The Hosking integral, which characterizes magnetic helicity fluctuations in subvolumes, is known to govern the decay of magnetically dominated turbulence. Here, we show that, when the evolution of the magnetic field is controlled by the motion of electrons only, as in neutron star crusts, the decay of the magnetic field is still controlled by the Hosking integral, but now it has effectively different dimensions than in ordinary magnetohydrodynamic (MHD) turbulence. This causes the correlation length to increase with time
$t$
like
$t^{4/13}$
instead of
$t^{4/9}$
in MHD. The magnetic energy density decreases like
$t^{-10/13}$
, which is slower than in MHD, where it decays like
$t^{-10/9}$
. These new analytic results agree with earlier numerical simulations for the non-helical Hall cascade.
The Saffman helicity invariant of Hosking & Schekochihin (Phys. Rev. X, vol. 11, issue 4, 2021, 041005), which we here call the Hosking integral, has emerged as an important quantity that may govern the decay properties of magnetically dominated non-helical turbulence. Using a range of different computational methods, we confirm that this quantity is indeed gauge invariant and nearly perfectly conserved in the limit of large Lundquist numbers. For direct numerical simulations with ordinary viscosity and magnetic diffusivity operators, we find that the solution develops in a nearly self-similar fashion. In a diagram quantifying the instantaneous decay coefficients of magnetic energy and integral scale, we find that the solution evolves along a line that is indeed suggestive of the governing role of the Hosking integral. The solution settles near a line in this diagram that is expected for a self-similar evolution of the magnetic energy spectrum. The solution will settle in a slightly different position when the magnetic diffusivity decreases with time, which would be compatible with the decay being governed by the reconnection time scale rather than the Alfvén time.
In decaying magnetohydrodynamic (MHD) turbulence with a strong magnetic field, the spectral magnetic energy density is known to increase with time at small wavenumbers
$k$
, provided the spectrum at low
$k$
is sufficiently steep. This process is called inverse cascading and occurs for an initial Batchelor spectrum, where the magnetic energy per linear wavenumber interval increases like
$k^4$
. For an initial Saffman spectrum that is proportional to
$k^2$
, however, inverse cascading has not been found in the past. We study here the case of an intermediate
$k^3$
spectrum, which may be relevant for magnetogenesis in the early Universe during the electroweak epoch. This case is not well understood in view of the standard Taylor expansion of the magnetic energy spectrum for small
$k$
. Using high resolution MHD simulations, we show that, also in this case, there is inverse cascading with a strength just as expected from the conservation of the Hosking integral, which governs the decay of an initial Batchelor spectrum. Even for shallower
$k^\alpha$
spectra with spectral index
$\alpha >3/2$
, our simulations suggest a spectral increase at small
$k$
with time
$t$
proportional to
$t^{4\alpha /9-2/3}$
. The critical spectral index of
$\alpha =3/2$
is related to the slope of the spectral envelope in the Hosking phenomenology. Our simulations with
$2048^3$
mesh points now suggest inverse cascading even for an initial Saffman spectrum.
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