# Improved upper bound on the Frank number of $3$-edge-connected graphs

János Barát
Alfréd Rényi Institute of Mathematics
University of Pannonia, Department of Mathematics
8200 Veszprém, Egyetem utca 10., Hungary
<EMAIL_ADDRESS>
and
Zoltán L. Blázsik
Alfréd Rényi Institute of Mathematics
MTA–ELTE Geometric and Algebraic Combinatorics Research Group
SZTE Bolyai Institute
<EMAIL_ADDRESS>
Research supported by ERC Advanced Grant ”GeoScape” and the National Research,
Development and Innovation Office, grant K-131529. The author was supported by
the ÚNKP-22-4-SZTE-480 New National Excellence Program of the Ministry for
Culture and Innovation from the source of the National Research, Development
and Innovation Fund. The research was supported by the Hungarian National
Research, Development and Innovation Office, OTKA grant no. SNN 132625.

###### Abstract

In an orientation $O$ of the graph $G$, an arc $e$ is deletable if and only if
$O-e$ is strongly connected. For a $3$-edge-connected graph $G$, the Frank
number is the minimum $k$ for which $G$ admits $k$ strongly connected
orientations such that for every edge $e$ of $G$ the corresponding arc is
deletable in at least one of the $k$ orientations. Hörsch and Szigeti
conjectured the Frank number is at most $3$ for every $3$-edge-connected graph
$G$. We prove an upper bound of $5$, which improves the previous bound of $7$.

##  1 Introduction

The graphs in this paper are finite and without loops or multiple edges. We
recommend the excellent book by Bondy and Murty [2] for the concepts and
notations used here.

A graph $G$ is defined by its vertex set $V$ and edge set $E$. An orientation
of $G$ is a directed graph $D=(V,A)$ such that each edge $uv\in E$ is replaced
by exactly one of the arcs $(u,v)$ or $(v,u)$. A circuit is a directed cycle.
A graph is cubic if every vertex has degree $3$. A graph is $k$-edge-connected
if and only if the removal of any $k-1$ edges leaves a connected graph. A
$2$-edge-connected graph is often called bridgeless.

A directed graph is strongly connected if and only if selecting two arbitrary
vertices $x$ and $y$, there is a directed $(x,y)$-path. An orientation of $G$
is k-arc-connected if and only if the removal of any $k-1$ arcs leaves a
strongly connected directed graph.

######  Theorem 1.1 (Robbins).

A graph has a strongly connected orientation if and only if it is $2$-edge-
connected.

The following theorem is a fundamental result in the theory of directed graphs
[10].

######  Theorem 1.2 (Nash-Williams).

A graph has a $k$-arc-connected orientation if and only if it is $2k$-edge-
connected.

These results served as a motivation for Hörsch and Szigeti [7] to investigate
the following concepts. In an orientation $O$ of $G$, the arc $e$ is deletable
if and only if $O{-}e$ is strongly connected. For a $3$-edge-connected graph
$G$, the Frank number, denoted by $F(G)$, is the minimum $k$ for which $G$
admits $k$ strongly connected orientations such that for every edge $e$ of $G$
the corresponding arc is deletable in at least one of the $k$ orientations.
Why $3$-edge-connected graphs? Suppose $G$ has a cut of size at most $2$. In
any orientation of $G$, the removal of any of these edges results in either a
directed cut or a directed graph that is not even connected. Hence no set of
orientations can satisfy the conditions. On the other hand, if $G$ is
$4$-edge-connected, then $G$ admits a $2$-arc-connected orientation by Theorem
1.2. This orientation yields that $F(G)=1$ by definition. Consequently,
$F(G)=1$ if and only if $G$ is $4$-edge-connected. Thus the problem is
interesting only if the edge-connectivity of $G$ is 3, i.e.
$\kappa^{\prime}(G)=3$. In the sequel, we consider only graphs with edge-
connectivity $3$.

Hörsch and Szigeti [7] showed that any $3$-edge-connected graph $G$ satisfies
$F(G)\leq 7$. Prior to that, DeVos et al. proved a more general result with a
weaker bound [4]: For every 3-edge-connected graph $G$, there exists a
partition of $E(G)$ into at most nine sets $\\{X_{1},X_{2},\dots,X_{m}\\}$ so
that $G\setminus X_{i}$ is 2-edge-connected for every $1\leq i\leq m$. Our
main result improves the best known upper bound on the Frank number.

######  Theorem 1.3.

For every $3$-edge-connected graph the Frank number is at most $5$.

The paper is organised as follows. In the second section, we introduce the
main tools and results, which we use in our proof. The third section is
dedicated to the proof of our main result. We conclude by discussing the
limits of our proof technique.

##  2 Preliminaries

For some integer $k$, a $k$-flow $(o,v)$ on a graph $G$ consists of an
orientation $o$ of the edges of $G$ and a valuation $v:E(G)\mapsto\\{0,\pm
1,\pm 2,\dots,\pm(k-1)\\}$ such that at every vertex the sum of the values on
incoming edges equals the sum on the outgoing edges. A $k$-flow $(o,v)$ is
nowhere-zero if the value of $v$ is not $0$ for any edge of $G$. A nowhere-
zero $k$-flow on $G$ is all-positive if the value $v(e)$ is positive for every
edge $e$ of $G$. Every nowhere-zero $k$-flow can be transformed to an all-
positive nowhere-zero $k$-flow by changing the orientation of the edges with
negative $v(e)$ and changing negative values of $v(e)$ to $-v(e)$. Inspired by
their ideas and approach, we use the following result by Goedgebeur et al.
[6].

######  Lemma 2.1.

Let $G$ be a $3$-edge-connected graph, and let $(o,v)$ be an all-positive
nowhere-zero $k$-flow on $G$. Any edge of $G$, which receive value $1$ in
$(o,v)$ is deletable in $o$.

Indeed, an orientation arising from a flow is always strongly connected, and
the removal of any arc of value 1 cannot create a directed cut since the flow
is nowhere-zero. Using a slightly stronger Lemma, Goedgebeur et al. [6] proved
the following result, which we state without proof.

######  Theorem 2.2.

If a graph $G$ admits a nowhere-zero $4$-flow, then $F(G)=2$.

It is well-known that every $3$-edge-colorable cubic graph admits a nowhere-
zero $4$-flow. Tutte posed the following stronger claim, which inspired a vast
amount of research.

######  Conjecture 2.3 (Tutte’s $4$-flow conjecture).

Every bridgeless graph without a Petersen-minor has a nowhere-zero $4$-flow.

Everyone believes the validity of this conjecture. This partly explains why
the only known examples of graphs with Frank number 3 are created from the
Petersen graph using certain operations [1]. Each of the constructed graphs
contains the Petersen graph as a minor.

Since we consider 3-edge-connected graphs only, we can use the following
result by Jaeger [8], which was later improved by Seymour [11].

######  Theorem 2.4 (Jaeger).

Every bridgeless graph has a nowhere-zero $8$-flow.

######  Theorem 2.5 (Seymour).

Every bridgeless graph has a nowhere-zero $6$-flow.

Let $H$ denote an Abelian group. An $H$-flow on an oriented graph $D$ is an
assignment of values of $H$ to the arcs of $D$ such that for each vertex $v$,
the sum of the values on the incoming arcs is the same as the sum of the
values on the outgoing arcs. For a graph $G$, an $H$-flow is defined
accordingly. A nowhere-zero $H$-flow on $G$ is an $H$-flow, where $0\in H$ is
not assigned to any edge. The following is a useful corollary of a theorem by
Tutte:

######  Fact 2.6.

If $H$ and $H^{\prime}$ are two finite Abelian groups of the same order, then
the graph $G$ has an $H$-flow if and only if $G$ has an $H^{\prime}$-flow.

In what follows, we particularly use a
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-flow on a $3$-edge-
connected graph $G$. We use $0/1$ vectors with three coordinates to denote
elements of $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. The flow
condition at a vertex $v$ implies the following property: in each coordinate,
the sum of values on edges incident to $v$ is $0$. Hence there are an even
number of $1$’s in each coordinate, on the edges incident to a fixed vertex.

##  3 Improvement of the upper bound

Surprisingly, Theorem 2.4, the weaker flow result of Jaeger is the useful for
our purposes. The main idea of the proof is the following. We fix a nowhere-
zero 8-flow of a $3$-edge-connected graph $G$ existing by Theorem 2.4. We
create 5 other nowhere-zero $k$-flows of $G$ in such a way that we control the
set of edges with value 1, and we apply Lemma 2.1. Since every edge of $G$
receives value 1 in at least one of the flows, we are done. Let us recall our
main theorem.

Theorem 1.3 For every $3$-edge-connected graph the Frank number is at most
$5$.

###### Proof.

Combining Theorem 2.4 and Fact 2.6, we consider a nowhere-zero
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-flow on $G$. For
$i\in\\{1,2,3\\}$, let $G_{i}$ denote the subgraph of $G$ induced by those
edges of $E$, which have value 1 at the $i$th coordinate. Note that these
subgraphs might have some edges in common, more precisely the number of
nonzero coordinates of each edge is the same as the number of subgraphs to
which it belongs. By the nowhere-zero property, every edge is contained in at
least one of these subgraphs.

By the flow condition, the subgraphs $G_{1},G_{2},G_{3}$ are Eulerian. We can
think of an Eulerian trail as a directed graph. Thus we can partition each
edge set $E(G_{i})$ into edge-disjoint circuits. We may use some vertices in
different Eulerian trails. For $i=1,2,3$, let us fix an orientation $o_{i}$ of
$E(G_{i})$. At this point, it might occur that an edge of $G$ has different
orientations in different subgraphs $G_{i}$. The next aim is to select, for
each $i\in\\{1,2,3\\}$, an appropriate positive value $v_{i}$ to send along
$o_{i}$.

Let us emphasize that after fixing the pair $(o_{i},v_{i})$ for each $G_{i}$,
we define another flow $(O_{1},f_{1})$ in the next step. The orientation and
value of an arc in $(O_{1},f_{1})$ is determined by the superposition of the
chosen $(o_{i},v_{i})$ pairs or triples. We always assign a positive value. If
the orientations go opposite, then we let the largest value determine the
direction and subtract the values going in opposite direction.

Let us construct an all-positive nowhere-zero 8-flow $(O_{1},f_{1})$ by
sending values $v_{1}=1$, $v_{2}=2$ and $v_{3}=4$ along the fixed orientations
$o_{i}$ of $G_{i}$, respectively. Indeed, this is a flow, and there can be no
arcs of value 0. The maximum value of an arc is 7, if this edge was of type
$(1,1,1)$ in the original
$\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-flow, and the edge
received the same orientation in all three orientations $o_{i}$. Since in
every edge-cut, the sum of the flow values in the two directions is the same,
there are no directed cuts if we take the superposition of the flows
$o_{1},o_{2},o_{3}$. Thus $O_{1}$ is strongly connected.

In order to prove $F(G)\leq 5$, we create a set of strongly-connected
orientations $\mathcal{O}$ of $G$ such that $|\mathcal{O}|=5$, and any edge of
$G$ is deletable in at least one of the orientations of $\mathcal{O}$. Let
$O_{1}$ be the first member of $\mathcal{O}$. By Lemma 2.1, the arcs of value
1 in $(O_{1},f_{1})$ are deletable with respect to $O_{1}$. What are these
arcs? We summarize the possible final orientations and values of the arcs in
Table 1.

Value of $e$ in the original $\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$-flow | Final orientation and value of $e$ in $(O_{1},f_{1})$
---|---
$(1,0,0)$ | same as in $o_{1}$, 1
$(0,1,0)$ | same as in $o_{2}$, 2
$(0,0,1)$ | same as in $o_{3}$, 4
$(1,1,0)$ | $\left\\{\begin{tabular}[]{cc}same as in $o_{2}$, 3&\qquad if $e$ has the same orientation in $o_{1}$ and $o_{2}$,\\\ same as in $o_{2}$, 1&\qquad if $e$ has different orientations in $o_{1}$ and $o_{2}$.\end{tabular}\right.$
$(1,0,1)$ | $\left\\{\begin{tabular}[]{cc}same as in $o_{3}$, 5&\qquad if $e$ has the same orientation in $o_{1}$ and $o_{3}$,\\\ same as in $o_{3}$, 3&\qquad if $e$ has different orientations in $o_{1}$ and $o_{3}$.\end{tabular}\right.$
$(0,1,1)$ | $\left\\{\begin{tabular}[]{cc}same as in $o_{3}$, 6&\qquad if $e$ has the same orientation in $o_{2}$ and $o_{3}$,\\\ same as in $o_{3}$, 2&\qquad if $e$ has different orientations in $o_{2}$ and $o_{3}$.\end{tabular}\right.$
$(1,1,1)$ | $\left\\{\begin{tabular}[]{cc}same as in $o_{3}$, 7&\qquad\noindent\hbox{}\hfill{{\hbox{\begin{tabular}[c]{@{}c@{}}if $e$ has the same orientation\\\ in all three of $o_{1},o_{2},o_{3}$,\end{tabular}}\hbox{}\hfill}}\\\ same as in $o_{3}$, 5&\qquad\noindent\hbox{}\hfill{{\hbox{\begin{tabular}[c]{@{}c@{}}if the orientation of $e$ in $o_{1}$ is different\\\ from that in $o_{2}$ and $o_{3}$,\end{tabular}}\hbox{}\hfill}}\\\ same as in $o_{3}$, 3&\qquad\noindent\hbox{}\hfill{{\hbox{\begin{tabular}[c]{@{}c@{}}if the orientation of $e$ in $o_{2}$ is different\\\ from that in $o_{1}$ and $o_{3}$,\end{tabular}}\hbox{}\hfill}}\\\ same as in $o_{3}$, 1&\qquad\noindent\hbox{}\hfill{{\hbox{\begin{tabular}[c]{@{}c@{}}if the orientation of $e$ in $o_{3}$ is different\\\ from that in $o_{1}$ and $o_{2}$.\end{tabular}}\hbox{}\hfill}}\end{tabular}\right.$
Table 1: The possible orientations and values of $e$ in $(O_{1},f_{1})$.

There are three types of arcs of value 1 in $(O_{1},f_{1})$. We create 4 other
flows such that the larger valued arcs of $O_{1}$ receive value 1 in at least
one orientation. We refer to any edge of $G$ as one which belongs to the
corresponding row of Table 1. The next goal is to create 4 other strongly
connected orientations such that every type of edge defined by Table 1 gets
flow value 1 in at least one of those orientations, and hence is deletable in
those orientations by Lemma 2.1. We achieve this by combining the following
two things: we reverse the orientations $o_{i}$ in $G_{i}$ for each
$i\in\\{1,2,3\\}$ if necessary, and we change the values $v_{i}$
appropriately. By reversing the orientations, we can switch the role of a
fixed edge with respect to its role described in Table 1, and by changing the
values we can change the final values of the edges.

We define the following two $10$-flows ($O_{2}$,$O_{3}$) and two $8$-flows
($O_{4}$,$O_{5}$) based on the orientations $o_{1},o_{2},o_{3}$. Again, if an
edge is present in several circuits, we superimpose the values as before111the
larger value deciding the direction except if 2+3 goes in one direction and 4
in the opposite to make the flow $(O_{i},f_{i})$ all-positive for all
$i\in\\{1,2,3,4,5\\}$. We define $O_{2}$ by keeping the same orientations as
in $O_{1}$, but sending value 4 along $o_{1}$, value 2 along $o_{2}$, and
value 3 along $o_{3}$. The only difference between $O_{3}$ and $O_{2}$ is that
the orientations of the circuits in $o_{3}$ are reversed. We get $O_{4}$ from
$O_{1}$ by reversing the orientation of the circuits in $o_{2}$ and sending
value 2 along $o_{1}$, value 1 along $o_{2}$ and value 4 along $o_{3}$. The
only difference between $O_{5}$ and $O_{4}$ is that the values along $o_{2}$
and $o_{3}$ are swapped. We indicate in Table 2 which edges receive value 1 in
the corresponding orientations of those edges, which did not receive value 1
with respect to $O_{1}$.

original value | $(O_{2},f_{2})$ | $(O_{3},f_{3})$ | $(O_{4},f_{4})$ | $(O_{5},f_{5})$
---|---|---|---|---
$(0,1,0)$ |  |  | value 1 |
$(0,0,1)$ |  |  |  | value 1
$(1,1,0)$ |  |  | value 1 |
$(1,0,1)$ row 6 |  | value 1 |  |
$(1,0,1)$ row 7 | value 1 |  |  |
$(0,1,1)$ row 8 |  | value 1 |  |
$(0,1,1)$ row 9 | value 1 |  |  |
$(1,1,1)$ row 10 |  |  |  | value 1
$(1,1,1)$ row 11 | value 1 |  |  |
$(1,1,1)$ row 12 |  | value 1 |  |
Table 2: Cross-table for clearing the remaining cases referring to Table 1

By Lemma 2.1, it is clear that
$\mathcal{O}=\\{O_{1},O_{2},O_{3},O_{4},O_{5}\\}$ yields $F(G)\leq 5$. ∎

Let us continue with some comments and observations on the proof.

######  Remark 3.1.

It does not really matter what kind of flow we use to construct an orientation
of $\mathcal{O}$. However, there are two key observations: since the
orientation arises from flows, it cannot have a directed cut. Therefore, it is
a strongly connected orientation, and by the nowhere-zero property and the
$3$-edge-connectivity of $G$ an arc of value 1 cannot be the only arc going in
the opposite direction in any cut.

######  Remark 3.2.

Note that we do not claim that these are the only deletable edges of
$O_{i}\in\mathcal{O}$. It may happen that some arcs with higher values are
also deletable. Hence our result is slightly stronger than just proving
$F(G)\leq 5$ since we only relied on the arcs of value 1.

## Discussion

We do not see how to use directly Theorem 2.5 to improve the Frank number of
$3$-edge-connected graphs. A natural question arises, can we use a similar
proof technique to achieve an even better upper bound? Goedgebeur et al. [6]
proved that if $G$ admits a nowhere-zero all-positive 4-flow, then $F(G)\leq
2$. On the other hand, if we start with an arbitrary all-positive $k$-flow for
some $k\geq 5$, then we either need to have some control over the arcs with
value 1 or need to know some underlying structure, which we can exploit. In
this proof technique, it seems vital to have those Eulerian subgraphs. To find
them, we had to use an equivalent $H$-flow for an appropriate group $H$.
However, the even degrees within those subgraphs was guaranteed by the fact
that the corresponding coordinate was 0/1 valued. Hence it is unclear to us,
what kind of $H$ can be used here other than $\mathbb{Z}^{m}_{2}$.

Suppose that $H=\mathbb{Z}^{m}_{2}$ for some $m\geq 2$. Let us introduce the
notation $A_{k}$ for those arcs of the original $H$-flow, which have exactly
$k$ non-zero coordinates ($k\in\\{1,2,\dots,m\\}$). Similarly to our proof, an
all-positive $H$-flow gives rise to $m$ Eulerian subgraphs $G_{i}$ (for
$i\in\\{1,2,\dots,m\\}$) and we can fix an orientation $o_{i}$ on each of
them. How many different types of arcs can get value 1? We carefully choose
values $v_{i}$ and consider orientations $(O_{j},f_{j})$ as the superposition,
while no arc gets value 0 in the process.

The existence of the arcs of $A_{2}$ ensures that the values $v_{i}$ must be
pairwise different, otherwise there is a possibility that an arc gets value 0
in the superposition. Thus at most one of the families in $A_{1}$ can get
value 1 (the ones within $G_{i}$ if and only if $v_{i}=1$) in each
orientation. Hence with this technique, we definitely need at least $m$
orientations, consequently $m\leq F(G)$.

There are $2\binom{m}{2}$ possible types of arcs corresponding to the arcs of
$A_{2}$. At most two of them can have value 1 in a fixed orientation $(O,f)$.
Indeed, if there were at least three types of such arcs, then we would choose
two such that the first one is non-zero at coordinates $i_{1},i_{2}$, the
second one at coordinates $j_{1},j_{2}$ such that
$v_{i_{1}}-v_{i_{2}}=1=v_{j_{1}}-v_{j_{2}}$, and $i_{1},i_{2},j_{1},j_{2}$ are
pairwise different. This could possibly lead to some arcs with final value 0,
whenever an arc $xy$ of $A_{4}$ has $1$’s at coordinates
$i_{1},i_{2},j_{1},j_{2}$. In addition, the corresponding four orientations
are chosen in such a way that $xy$ is oriented in the same direction in
$G_{i_{1}}$ and $G_{j_{2}}$, but in the opposite direction in the other two
Eulerian subgraphs. That yields $\binom{m}{2}\leq F(G)$. Thus if one would
like to improve our result, then $m$ should be at most 3. Since $m=2$
corresponds to the $4$-flows, that circles back to Conjecture 2.3, the only
possibility is $m=3$.

From the previous observations, one can deduce that we need at least three
orientations to deal with the arcs of $A_{1}$. In each of these orientations,
there is an Eulerian subgraph $G_{i}$ for $i\in\\{1,2,3\\}$ such that
$v_{i}=1$. These orientations cannot have two types of arcs with value 1 from
$A_{2}$, since that would be possible only if the three values were $1$, $2$
and $3$. Again this contradicts the non-zero property of our flow ($1+2-3=0$).
Consequently, at least three types of arcs are missing after these three
orientations from the arcs of $A_{2}$. Hence at least two other orientations
are needed. Therefore our upper bound of 5 is best if
$H=\mathbb{Z}_{2}\times\mathbb{Z}_{2}\times\mathbb{Z}_{2}$. We would like to
conclude by posing a natural problem.

######  Question 3.3.

Is it possible to improve the upper bound on $F(G)$ using a similar flow
technique but starting with a different group $H$?

## References

  * [1] J. Barát, Z.L. Blázsik Quest for graphs of Frank number $3$. submitted
https://arxiv.org/pdf/2209.08804

  * [2] J.A. Bondy, U.S.R. Murty Graph Theory. Springer-Verlag, London, XII+663 pages, (2008).
  * [3] G. Brinkmann, K. Coolsaet, J. Goedgebeur, H. Mélot House of Graphs: a database of interesting graphs. Discrete Applied Mathematics, 161(1-2):311–314, (2013). Available at http://hog.grinvin.org
  * [4] M. DeVos, T. Johnson, P.D. Seymour Cut Coloring and Circuit Covering. manuscript https://web.math.princeton.edu/~pds/papers/cutcolouring/paper.pdf
  * [5] R. Diestel Graph Theory. Springer GTM 173, 5th edition (2016), XVIII+447 pages.
  * [6] J. Goedgebeur, E. Mácajová, J. Renders On the Frank number and nowhere-zero flows on graphs. manuscript https://arxiv.org/pdf/2305.02133.pdf
  * [7] F. Hörsch, Z. Szigeti Connectivity of orientations of $3$-edge-connected graphs. European J. Combin. 94 (2021)
  * [8] F. Jaeger Flows and generalized coloring theorems in graphs. J. Combinatorial Theory Ser. B 26 (1979), 205–216.
  * [9] L.M. Lovász Tutte’s flow conjectures. (2012) 24 pages. https://tlovering.files.wordpress.com/2012/06/laszloessay.pdf
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# Turbulent magnetic decay controlled by two conserved quantities

Axel Brandenburg1,2,3,4<EMAIL_ADDRESS>Aikya Banerjee5  1Nordita, KTH
Royal Institute of Technology and Stockholm University, Hannes Alfvéns väg 12,
SE-10691 Stockholm, Sweden 2The Oskar Klein Centre, Department of Astronomy,
Stockholm University, AlbaNova, SE-10691 Stockholm, Sweden 3McWilliams Center
for Cosmology & Department of Physics, Carnegie Mellon University, Pittsburgh,
PA 15213, USA 4School of Natural Sciences and Medicine, Ilia State University,
3-5 Cholokashvili Avenue, 0194 Tbilisi, Georgia 5Department of Physical
Sciences, Indian Institute of Science Education and Research Kolkata, Mohanpur
741246, West Bengal, India

###### Abstract

The decay of a turbulent magnetic field is slower with helicity than without.
Furthermore, the magnetic correlation length grows faster for a helical than a
nonhelical field. Both helical and nonhelical decay laws involve conserved
quantities: the mean magnetic helicity density and the Hosking integral. Using
direct numerical simulations, we show quantitatively that in the fractionally
helical case the magnetic energy and correlation length are approximately
given by the maximum of the values for the purely helical and purely
nonhelical cases. The time of switch-over from one to the other decay law can
be obtained on dimensional grounds and is approximately given by
$I_{\mathrm{H}}^{1/2}I_{\mathrm{M}}^{-3/2}$, where $I_{\mathrm{H}}$ is the
Hosking integral and $I_{\mathrm{M}}$ is the mean magnetic helicity density.
An earlier approach based on the decay time is found to agree with our new
result and suggests that the Hosking integral exceeds naive estimates by the
square of the same resistivity-dependent factor by which the turbulent decay
time exceeds the Alfvén time.

###### keywords:

astrophysical plasmas, plasma simulation, plasma nonlinear phenomena

##  1 Introduction

In recent years, there has been significant interest in the study of decaying
turbulent magnetic fields. One of the main applications has been to the
understanding of the magnetic field evolution during the radiation-dominated
era of early universe (Brandenburg et al., 1996; Christensson et al., 2001;
Banerjee & Jedamzik, 2004). The special case with finite magnetic helicity has
been studied and understood for a long time (Hatori, 1984; Biskamp & Müller,
1999). It is the prime example of large-scale magnetic field growth due to an
inverse cascade. The possibility of such an inverse cascade is explained by
the conservation of magnetic helicity (Frisch et al., 1975). However, even in
the absence of magnetic helicity, an inverse cascade can develop (Kahniashvili
et al., 2013; Zrake, 2014; Brandenburg et al., 2015), and it is well explained
by the conservation of what is now called the Hosking integral (Hosking &
Schekochihin, 2021, 2023a; Zhou et al., 2022; Brandenburg et al., 2023b),
which is the correlation integral of the magnetic helicity density.

In all the cases mentioned above, the spectral magnetic helicity was either
zero or it had the same sign at all wavenumbers. A special situation was
studied in the work of Brandenburg et al. (2023a), where the magnetic helicity
was finite, but it was balanced by fermion chirality of the opposite sign, so
that the net chirality was vanishing. For such a system, the decay was again
successfully explained by the conservation of the Hosking integral, which was
adapted to include the chirality from the fermions.

We have seen that the Hosking integral can be applied to broad ranges of
systems where magnetic helicity is still important locally, but globally, the
net magnetic helicity vanishes. However, there is an important class of
astrophysically relevant systems, where the magnetic field is not generated by
magnetogenesis, as in the early universe, but by dynamo action. This means
that some of the kinetic energy of turbulent motions is converted into
magnetic energy. It is important to stress that, even if the velocity field
were to be helical, i.e., if there is kinetic helicity in the system, as is
generally the case when there is rotation and stratification of density and/or
velocity, magnetic helicity conservation still precludes the generation of
magnetic helicity, at least on dynamical timescales (Ji, 1999).

In the aforementioned helically driven large-scale dynamos, magnetic helicity
can be generated at small scales, but it is then balanced by magnetic helicity
at large scales so as to conserve magnetic helicity. Alternatively, we can
also say that magnetic helicity is produced at large scales, for example by
the tilting of buoyantly rising magnetic flux tubes in cyclonic convective
events, as envisaged by Parker (1955). Magnetic helicity conservation then
implies magnetic twist of opposite sign at smaller scales. In practice,
because there is always finite magnetic diffusivity, which acts especially on
small scales, the magnetic helicity from large scales will, after some time,
dominate the total magnetic helicity owing to the loss at small scales where
it has the opposite sign. Therefore, there is always a small imbalance between
the contributions from small and large length scales. It is therefore a
situation that is only partially suited to the phenomenology involving the
conservation of the Hosking integral.

If we now were to turn off the driving, the turbulence would gradually decay.
This decay should then be governed by the conservation of both the Hosking
integral and the mean magnetic helicity density. This idea is purely
theoretical and has never been tested under conditions that we expect to be
relevant to astrophysical settings. Such conditions could apply to the decay
of a magnetic field produced in a proto-neutron star. There, we expect a
turbulent dynamo to occur that is driven by convection (Thompson & Duncan,
1993). This would happen when the neutrino opacity is large enough to prevent
neutrinos from escaping freely (Epstein, 1979; Burrows & Lattimer, 1986).

Another source of turbulence in proto-neutron stars could be the
magnetorotational instability that results from the radially outward
decreasing angular velocity gradient associated with collapsed material having
an approximately constant angular momentum density (Guilet et al., 2022). In
both cases, the turbulence itself has kinetic helicity of opposite signs in
the northern and southern hemispheres (negative in the north and positive in
the south). In each hemisphere, this leads to dynamo action of the type
described above, but the magnetic helicities have opposite signs not only in
the two hemispheres, but also on small and large length scales. One would thus
focus only on one hemisphere and ignore the interaction between north and
south. The magnetic helicities from small and large length scales would then
nearly cancel. Such fields have been called “bihelical” and their decay
properties were first studies by Yousef & Brandenburg (2003). They found that
the positive and negative contributions rapidly mix and annihilate and that
the ratio of the magnetic helicity spectrum to the magnetic energy spectrum
has local extrema both at small and large scales, although the latter is
dominant in an absolute sense.

Since the net magnetic helicity of a bihelical magnetic field does not vanish
exactly, and since the mean magnetic helicity itself is an important conserved
quantity, we are confronted with a situation where the magnetic decay is
governed by two conserved quantities. Investigating this aspect in a more
controlled fashion is the main purpose of this paper.

In an earlier paper, Tevzadze et al. (2012) did already study a case with
fractional helicity. They found that the correlation length developed a
steeper growth (indicative of magnetic helicity domination) at a specific
moment that depends on the value of the magnetic helicity as well as on the
initial values of the magnetic energy and the magnetic correlation length.

Meanwhile, we know about the Hosking integral $I_{\mathrm{H}}$ and can ask
whether the time of switch-over from a decay controlled by $I_{\mathrm{H}}$ to
one controlled by the mean magnetic helicity density $I_{\mathrm{M}}$ can be
computed based on dimensional arguments. Indeed, given that the quantity
$I_{\mathrm{H}}$ has dimensions $\,{\rm cm}^{9}\,{\rm s}^{-4}$ and
$I_{\mathrm{M}}$ has dimensions $\,{\rm cm}^{3}\,{\rm s}^{-2}$ (see
Brandenburg, 2023, and note that the magnetic field is here understood to be
in Alfvén units with dimensions $\,{\rm cm}\,{\rm s}^{-1}$), a combination of
$I_{\mathrm{H}}$ and $I_{\mathrm{M}}$ that yields a time would be
$I_{\mathrm{H}}^{1/2}I_{\mathrm{M}}^{-3/2}$. It will turn out that this is
indeed the time of switch-over between the two regimes.

##  2 Our model

###  2.1 Basic equations

We simulate the compressible magnetohydrodynamic (MHD) equations with an
isothermal equation of state with constant sound speed $c_{\rm s}$, so the
pressure $p$ and the density $\rho$ are related by $p=\rho c_{\rm s}^{2}$. The
equations for the magnetic vector potential, $\bm{A}$, the velocity $\bm{U}$,
and the logarithmic density $\ln\rho$, are

$\frac{\partial\bm{A}}{\partial t}=\bm{U}\times\bm{B}+\eta\nabla^{2}\bm{A},$
(1) $\displaystyle\frac{{\rm D}{}\bm{U}}{{\rm
D}{}t}=\mbox{\boldmath$f$}{}-c_{\rm
s}^{2}{\bm{\nabla}}\ln\rho+\frac{1}{\rho}\left[\bm{J}\times\bm{B}+{\bm{\nabla}}\cdot(2\rho\nu\mbox{\boldmath${\sf
S}$}{})\right],$ (2) $\frac{{\rm D}{}\ln\rho}{{\rm
D}{}t}=-{\bm{\nabla}}\cdot\bm{U},$ (3)

where ${\rm D}{}/{\rm D}{}t=\partial/\partial t+\bm{U}\cdot{\bm{\nabla}}$ is
the advective derivative, $\bm{B}={\bm{\nabla}}\times\bm{A}$ is the magnetic
field, $\bm{J}={\bm{\nabla}}\times\bm{B}/\mu_{0}$ is the current density,
$\mu_{0}$ is the vacuum permeability, $\nu$ is the viscosity, ${\sf
S}_{ij}=(\partial_{i}U_{j}+\partial_{j}U_{i})/2-\delta_{ij}{\bm{\nabla}}\cdot\bm{U}/3$
are the components of the traceless rate-of-strain tensor ${\sf S}$, and $f$
is a monochromatic, spatially periodic forcing function with wavenumber $k$,
where $k=|\bm{k}|$ is the length of the wavevector that is taken randomly at
each time step from a shell of thickness $\delta k$ such that $k_{\rm
f}-\delta k/2\leq k<k_{\rm f}+\delta k/2$ with $k_{\rm f}$ being the mean
forcing wavenumber. Our computational domain is a periodic cube of size
$L^{3}$, and $k_{1}=2\pi/L$ is the smallest wave number.

###  2.2 Parameters of the model

The amplitude $f_{0}$ of the forcing function is chosen such that the rms
velocity $u_{\rm rms}$ is about 10% of the sound speed. The system is governed
by the rms Mach number as well as the kinetic and magnetic Reynolds numbers,

$\mbox{\rm Ma}=u_{\rm rms}/c_{\rm s},\quad\mbox{\rm Re}=u_{\rm rms}/\nu k_{\rm
f},\quad\mbox{\rm Re}_{\rm M}=u_{\rm rms}/\eta k_{\rm f}.$ (4)

When the value of $\mbox{\rm Re}_{\rm M}$ is sufficiently large, there is
dynamo action, i.e., the rms magnetic field strength, $B_{\rm rms}$, increases
exponentially. The initial magnetic field is chosen to be a random, Gaussian-
distributed field of sufficiently low amplitude so that $B_{\rm rms}$ grows
initially exponentially before it saturates at a value that is comparable to
the equipartition field strength $B_{\rm eq}=\sqrt{\rho_{0}\mu_{0}}\,u_{\rm
rms}$.

To recover information about the turbulent decay that is independent of the
size and shape of the computational domain, we must choose the value of
$k_{\rm f}/k_{1}$ to be sufficiently large. The sensitivity of this on the
results has been studied on various occasions (e.g., Zhou et al., 2022). A
reasonable compromise that still allows for sufficiently large Reynolds
numbers seems to be $k_{\rm f}/k_{1}=60$. This is the value that will be used
for the main run in the present paper, but we also present some results with
$k_{\rm f}/k_{1}=30$.

###  2.3 Initial conditions

So far, the model is equivalent to that of Brandenburg (2001), who studied the
large-scale dynamo action associated with helical turbulence. The new aspect
is that we now also simulate the case where the turbulent driving is turned
off, i.e., $f_{0}=0$. In those cases, we use as initial condition a snapshot
from the driven simulation at a time when the magnetic field has still not yet
reached saturation, i.e., the magnetic field is still in the kinematic regime.
In the more idealized studies where we focus on the decay govern by two
conserved quantities (Hosking integral and mean magnetic helicity density), we
construct an initial magnetic vector potential in Fourier space as
$\tilde{\mathbf{A}}(k)=\mbox{\boldmath${\sf
R}$}{}(k;\varsigma)\tilde{\mathbf{A}}^{\mathrm{nhel}}$, where

$\mathsf{R}_{ij}(\bm{k};\varsigma)=\delta_{ij}-\hat{k}_{i}\hat{k}_{j}+{\rm
i}\hat{k}_{\ell}\varsigma\epsilon_{ij\ell}$ (5)

is a matrix with $\hat{k}_{i}$ being the components of the unit vector
$\hat{\bm{k}}=\bm{k}/k$, $|\varsigma|\leq 1$ is a nondimensional parameter
that quantifies the fractional helicity, and
$\tilde{\mathbf{A}}^{\mathrm{nhel}}$ is a nonhelical field with random phases
and possesses the desired spectrum for the magnetic field $\mbox{\rm
Sp}(\bm{B})=k^{2}\mbox{\rm Sp}(\bm{A})$, i.e.,

$\mbox{\rm
Sp}(\bm{B})=\frac{A_{0}k^{\alpha}}{1+(k/k_{\mathrm{p}})^{5/3+\alpha}},$ (6)

where $A_{0}$ is an amplitude, $k_{\mathrm{p}}$ denotes the initial position
of the spectral peak, $\alpha$ is the subinertial range slope (usually
$\alpha=4$), and the inertial range has a $k^{-5/3}$ spectrum. For
$\varsigma\neq 0$, we have a finite magnetic helicity and expect them the
decay to be governed by both the Hosking integral and the mean magnetic
helicity density.

In Eq. (6), $\mbox{\rm Sp}(\cdot)$ denotes a shell integrated spectrum. This
operation will also be applied to the local, gauge-dependent magnetic helicity
density $h=\bm{A}\cdot\bm{B}$, so that $\mbox{\rm
Sp}(h)=(k^{2}/8\pi^{3}L^{3})\oint_{4\pi}|\tilde{h}|^{2}\,{\rm d}{}\Omega_{k}$.
The tilde marks a quantity in Fourier space, and $\Omega_{k}$ is the solid
angle in Fourier space, so that $\int\mbox{\rm Sp}(h)\,{\rm d}{}k=\langle
h^{2}\rangle$, and likewise for $\int\mbox{\rm Sp}(\bm{B})\,{\rm
d}{}k=\langle\bm{B}^{2}\rangle$. Owing to the integration over shells in
three-dimensional wavenumber space, the spectrum of a spatially random
($\delta$ correlated) field is proportional to $k^{2}$. This is indeed the
case for a globally non-helical field, where $\langle h\rangle=0$.

###  2.4 Diagnostic parameters

The relevant information that quantifies the Hosking integral is the first
nonvanishing coefficient in the Taylor expansion,

$\left.\mbox{\rm Sp}(h)\right|_{k\to
0}=\frac{I_{\mathrm{H}}}{2\pi^{2}}k^{2}+...\;;$ (7)

see Hosking & Schekochihin (2021), Schekochihin (2022), and Zhou et al. (2022)
for details. This is also the method used here to determine the value of
$I_{\mathrm{H}}$. We confirm that $2\pi^{2}\mbox{\rm Sp}(h)/k^{2}$ has a flat
part for small values of $k$ and use its value at $k=k_{1}$ to measure
$I_{\mathrm{H}}$. Below, we also confirm that $I_{\mathrm{H}}$ is nearly
independent of time; Zhou et al. (2022) for quantitative assessment of its in
variance in the ideal limit. Note also that, since $\int\mbox{\rm Sp}(h)\,{\rm
d}{}k=\langle h^{2}\rangle$, which has dimensions $(\,{\rm cm}^{3}\,{\rm
s}^{-2})^{2}$, $\mbox{\rm Sp}(h)$ has dimensions, $\,{\rm cm}^{7}\,{\rm
s}^{-4}$ and therefore $I_{\mathrm{H}}$ has dimensions $\,{\rm cm}^{9}\,{\rm
s}^{-4}$, as expected.

Our model is spatially homogeneous and we consider in all cases periodic
boundary conditions. The model can then be characterized by the magnetic
energy and helicity spectra, $E_{\rm M}(k,t)$ and $H_{\rm M}(k,t)$,
respectively. They are normalized such that their integrals give the mean
magnetic energy and helicity densities, ${\cal E}_{\rm M}\equiv\int E_{\rm
M}(k,t)\,{\rm d}{}k=\langle\bm{B}^{2}\rangle/2\mu_{0}$ and
$I_{\mathrm{M}}\equiv\int H_{\rm M}(k,t)\,{\rm
d}{}k=\langle\bm{A}\cdot\bm{B}\rangle$, respectively.111The name
$I_{\mathrm{M}}$ has been chosen here to mark its important role as an ideal
invariant and to highlight its usage analogously with that of the Hosking
integral $I_{\mathrm{H}}$. Here and below, angle brackets denote volume
averaging.

The position of the peak of the spectrum is characterized by the inverse
magnetic integral scale, $k_{\mathrm{peak}}=\xi_{\rm M}^{-1}$, where $\xi_{\rm
M}$ is here defined as

$\xi_{\rm M}=\left.\int k^{-1}\,E_{\rm M}(k,t)\,{\rm d}{}k\,\right/\int E_{\rm
M}(k,t)\,{\rm d}{}k.$ (8)

Of particular importance are the time dependencies $\xi_{\rm M}(t)$ and ${\cal
E}_{\rm M}(t)$, which, in turn, are characterized by the instantaneous scaling
exponents $q(t)={\rm d}{}\ln\xi_{\rm M}/{\rm d}{}\ln t$ and $p(t)=-{\rm
d}{}\ln{\cal E}_{\rm M}/{\rm d}{}\ln t$.

To facilitate comparison with other work, it is useful to present our results
in nondimensional form. The time used in the numerical simulations is made
nondimensional by plotting the evolution versus $c_{\rm s}k_{1}t$, which is
convenient for numerical reasons, because $c_{\rm s}$ and $k_{1}$ are constant
in time. However, physically more meaningful would be a nondimensionalization
by using the Alfvén speed and the inverse correlation length. Both are time
dependent, but the values $v_{\mathrm{Ae}}$ and $k_{\mathrm{e}}$ at the end of
the simulations seem to be most meaningful.

##  3 Results

In the present context, we have to deal with two conserved quantities, namely
the Hosking integral $I_{\mathrm{H}}$ and the mean magnetic helicity density
$I_{\mathrm{M}}=\langle\bm{A}\cdot\bm{B}\rangle$. The former case has been
studied extensively in recent years. Specifically, Brandenburg & Larsson
(2023) and Brandenburg et al. (2023b) found

$\xi_{\rm M}(t)\approx 0.12\,I_{\rm H}^{1/9}\,t^{4/9},\quad{\cal E}_{\rm
M}(t)\approx 3.7\,I_{\rm H}^{2/9}\,t^{-10/9},\quad E_{\rm
M}(k,t)\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}0.025\,I_{\rm
H}^{1/2}\,(k/k_{0})^{3/2}.$ (9)

The hope is that the coefficients in these expressions are universal, but it
should be noted that they have not yet been verified in other contexts.

###  3.1 Decay controlled by magnetic helicity

In the helical case with $I_{\mathrm{M}}\neq 0$, we have $\xi_{\rm M}\propto
t^{2/3}$ and ${\cal E}_{\rm M}\propto t^{-2/3}$ (Hatori, 1984; Biskamp &
Müller, 1999; Brandenburg & Kahniashvili, 2017). In the present context, the
pre-factors are important. Using the data from Fig. 2(c) of Brandenburg &
Kahniashvili (2017), we find

$\xi_{\rm M}(t)\approx 0.12\,I_{\rm M}^{1/3}\,t^{2/3},\quad{\cal E}_{\rm
M}(t)\approx 4.3\,I_{\rm M}^{2/3}\,t^{-2/3},\quad E_{\rm
M}(k,t)\mathrel{\mathchoice{\vbox{\offinterlineskip\halign{\hfil$\displaystyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\textstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptstyle#$\hfil\cr<\cr\sim\cr}}}{\vbox{\offinterlineskip\halign{\hfil$\scriptscriptstyle#$\hfil\cr<\cr\sim\cr}}}}0.7\,I_{\rm
M}.$ (10)

Generally, we can write

$\xi_{\rm M}(t)=C_{i}^{(\xi)}\,I_{i}^{\sigma}t^{q},\quad{\cal E}_{\rm
M}(t)=C_{i}^{({\cal E})}\,I_{i}^{2\sigma}t^{-p},\quad E_{\rm
M}(k)=C_{i}^{(E)}\,I_{i}^{(3+\beta)\sigma}(k/k_{0})^{\beta},$ (11)

where the index $i$ in the integrals $I_{i}$ and the coefficients
$C_{i}^{(\xi)}$, $C_{i}^{({\cal E})}$, and $C_{i}^{({E})}$ stand for M or H
for magnetic helicity and Hosking scalings, respectively, and $\sigma$ is the
exponent with which length enters in $I_{i}$: $\sigma=1/3$ for the magnetic
helicity density ($i={\rm M}$) and $\sigma=1/9$ for the Hosking integral
($i={\rm H}$); see Table LABEL:TExpSummary for a summary of the
coefficients.222In Eq. (3.3c), we have here corrected a typo in Eq. (12c) of
Brandenburg & Larsson (2023) and Eq. (3.1c) of Brandenburg et al. (2023b),
where the exponent on $I_{i}$ was incorrectly stated as $(3+\beta)/\sigma$
instead of $(3+\beta)\sigma$, but the calculations were done correctly.

Table 1:  Summary of the coefficients characterizing the decays governed by
the conservation of magnetic helicity ($i=\mathrm{M}$) and the Hosking
integral ($i=\mathrm{H}$).

$i$ $\beta$ $q$ $p$ $\sigma$ $C_{i}^{(\xi)}$ $C_{i}^{({\cal E})}$
$C_{i}^{(E)}$ M 0 2/3 2/3 1/3 0.13 4.1 0.7 H 3/2 4/9 10/9 1/9 0.12 3.7 0.025

Figure 1:  (a) Magnetic energy spectra, as well as compensated evolutions of
(b) $\xi_{\rm M}(t)$ and (c) ${\cal E}_{\rm M}(t)$ for the maximally helical
run of Fig. 2(c) of Brandenburg & Kahniashvili (2017). In (a), the red symbols
denote the spectral peaks.

In Figure 1, we show the magnetic energy spectra, as well as compensated
evolutions of $\xi_{\rm M}(t)$ and ${\cal E}_{\rm M}(t)$ for the maximally
helical run of Fig. 2(c) of Brandenburg & Kahniashvili (2017). We see that the
peak of ${\cal E}_{\rm M}(t)$ remains underneath a nearly flat envelope (its
slope is $\beta=0$), as is expected for a fully helical turbulent decay.
Although the compensated evolutions of $\xi_{\rm M}(t)$ and ${\cal E}_{\rm
M}(t)$ are not yet fully converged toward the end of that run (the lines are
now yet flat), we can already read off the approximate values
$C_{\mathrm{M}}^{(\xi)}\approx 0.12$ and $C_{\mathrm{M}}^{({\cal E})}\approx
4.3$ toward the end of the run. The values of these coefficients will be
revisited later in this paper.

Figure 2:  Evolution of $\xi_{\rm M}(t)$ (upper row) and ${\cal E}_{\rm M}(t)$
(lower row) for the run with $k_{\rm f}/k_{1}=30$ and $\varsigma=0.003$,
compensated by the expected evolution if the decay is controlled either by
$I_{\mathrm{H}}$ (left column) or by $I_{\mathrm{M}}$ (right column). The
dashed line denotes the use of $I_{\mathrm{M}}$ at the end of the run, while
for the solid line, the time-dependent value was taken.  Figure 3:  Evolutions
of $(2\pi^{2}/k^{2})\,\mbox{\rm Sp}(h)$, normalized by
$v_{\mathrm{Ae}}^{4}/k_{\mathrm{e}}^{5}$, for $k/k_{1}=1$ (solid line), 2
(dashed-dotted line), and 3 (dashed line), for the nearly nonhelical run with
$\varsigma=0.003$ and $k_{\rm f}/k_{1}=30$.

In Figure 2, we again show the compensated evolutions of $\xi_{\rm M}(t)$ and
${\cal E}_{\rm M}(t)$, but now for a nearly perfectly nonhelical new run
($\varsigma=0.003$) with $k_{\rm f}/k_{1}=30$. The resulting values
coefficients are close to those estimated previously, namely
$C_{\mathrm{H}}^{(\xi)}\approx 0.12$ and $C_{\mathrm{H}}^{({\cal E})}\approx
4.7$. This supports the previous hypotheses of Brandenburg & Larsson (2023)
and Brandenburg et al. (2023b) that these coefficients may indeed be
universal.

To determine the value of $I_{\mathrm{H}}(t)$, we plot in Figure 3 the
evolutions of $(2\pi^{2}/k^{2})\,\mbox{\rm Sp}(h)$ (normalized by
$v_{\mathrm{Ae}}^{4}/k_{\mathrm{e}}^{5}$) for $k/k_{1}=1$, 2, and 3 for the
run with $k_{\rm f}/k_{1}=30$ and $\varsigma=0.003$. We see that for
$k/k_{1}=1$, the result shows a nearly negligible decline proportional to
$t^{0.07}$. Note that, in units of $v_{\mathrm{Ae}}^{4}/k_{\mathrm{e}}^{5}$,
the value of $I_{\mathrm{H}}$ is about 500.

###  3.2 Decay controlled by $I_{\mathrm{M}}$ and $I_{\mathrm{H}}$

If both $I_{\mathrm{M}}$ and $I_{\mathrm{H}}$ control the decay, we have a
combination of the two decay laws such that the late times are always
controlled by the more strongly conserve quantity, i.e., by $I_{\mathrm{M}}$,
so

$\xi_{\rm M}\approx
0.12\,I_{\mathrm{M}}^{1/3}\,t^{2/3}+0.12\,I_{\mathrm{H}}^{1/9}\,t^{4/9},$ (12)
${\cal E}_{\rm M}\approx
4.3\,I_{\mathrm{M}}^{2/3}\,t^{-2/3}+3.7\,I_{\mathrm{H}}^{2/9}\,t^{-10/9}.$
(13)

Since the second term involving $I_{\mathrm{H}}$ is initially larger, one
expects their contributions to be dominant after some time. Thus, the magnetic
helicity will always survive and be the dominant contributed to explaining the
decay.

Figure 4:  Magnetic energy spectra for $k_{\rm f}/k_{1}=60$ and
$\varsigma=0.01$ at times $v_{\mathrm{Ae}}k_{\mathrm{e}}\,t=0.07$, 0.18, 0.40,
0.82, 1.65, 3.3, 6.1, 11.1, and 20.7.

To examine now a run where the decay is controlled both by $I_{\mathrm{M}}$
and $I_{\mathrm{H}}$, we now increase the initial fractional helicity slightly
from 0.003 to 0.01; see Figure 4 for magnetic energy spectra at different
times for a run with $k_{\rm f}/k_{1}=60$ and $\varsigma=0.01$. Note that the
peaks of the spectra evolve at first underneath an envelope with the slope
$\beta=3/2$, as expected for a decay controlled by $I_{\mathrm{H}}$. At later
times, however, the envelope becomes flat (slope $\beta=0$), as expected for a
decay controlled by $I_{\mathrm{M}}$. $I_{\mathrm{H}}$.

Figure 5:  Similar to Figure 2, but for $k_{\rm f}/k_{1}=60$ and
$\varsigma=0.01$.

In Figure 5, we show the evolutions of $\xi_{\rm M}(t)$ and ${\cal E}_{\rm
M}(t)$ for $k_{\rm f}/k_{1}=60$ and $\varsigma=0.01$ compensated by $t^{-2/3}$
and $t^{2/3}$, respectively, as well as $t^{-4/9}$ and $t^{10/9}$,
respectively. We see that now the curves compensated by $t^{-2/3}$ and
$t^{2/3}$, respectively, become nearly constant, as expected for a decay that
is governed by magnetic helicity conservation. Specifically, we find $\xi_{\rm
M}/(I_{\rm M}^{1/3}\,t^{2/3})\approx 0.14$ and ${\cal E}_{\rm M}/(I_{\rm
M}^{2/3}\,t^{-2/3})\approx 4.0$. During a short intermediate interval,
however, we see that the curves compensated by $t^{-4/9}$ and $t^{10/9}$,
respectively, show brief plateaus around $v_{\mathrm{Ae}}k_{\mathrm{e}}t=1$
with $\xi_{\rm M}/(I_{\rm H}^{1/9}\,t^{4/9})\approx 0.12$ and ${\cal E}_{\rm
M}/(I_{\rm H}^{2/9}\,t^{-10/9})\approx 4.0$.

Figure 6:  Decay of magnetic energy (black line) and the fit given by Eq. (13)
(dotted blue line) as well as Eq. (15) with $q=2$ (dashed orange line) and
$q=10$ (red). The red dotted line corresponds to the limit $q\to\infty$, as
realized by Eqs. (16) and (17).

###  3.3 Improved fits with $I_{\mathrm{M}}$ and $I_{\mathrm{H}}$

We have seen that the limiting cases where the decay is controlled either by
$I_{\mathrm{M}}$ or by $I_{\mathrm{H}}$ are well reproduced by Eq. (11). It
turns out, however, that the combined fits given by Eqs. (12) and (13) are not
very accurate. Improved fits can be obtained by using large weighting
exponents for both contributions, i.e.,

$\xi_{\rm
M}\approx\left[\left(0.12\,I_{\mathrm{M}}^{1/3}\,t^{2/3}\right)^{q}+\left(0.14\,I_{\mathrm{H}}^{1/9}\,t^{4/9}\right)^{q}\right]^{1/q},$
(14) ${\cal E}_{\rm
M}\approx\left[\left(4.0\,I_{\mathrm{M}}^{2/3}\,t^{-2/3}\right)^{q}+\left(4.0\,I_{\mathrm{H}}^{2/9}\,t^{-10/9}\right)^{q}\right]^{1/q}.$
(15)

The result is shown in Figure 6, where we show that $q=10$ yields satisfactory
fits, while $q=2$ and $q=1$ (our original hypothesis) are poor. The fact that
the coefficients for both parts are different from those of the individual
fits and that they happen to be $4.0$ in Eq. (15), but different from each
other in Eq. (14) is probably just by chance and reflect that degree of
uncertainty of these values.

It is important to emphasize that the limit $q\to\infty$ corresponds to

$\xi_{\rm
M}\approx\max\left(0.12\,I_{\mathrm{M}}^{1/3}\,t^{2/3},\;0.14\,I_{\mathrm{H}}^{1/9}\,t^{4/9}\right),$
(16) ${\cal E}_{\rm
M}\approx\max\left(4.0\,I_{\mathrm{M}}^{2/3}\,t^{-2/3},\;4.0\,I_{\mathrm{H}}^{2/9}\,t^{-10/9}\right).$
(17)

These expressions yield discontinuities in the derivative. An advantage of
such expressions is that one can clearly see the regimes of validity of both
expressions. The critical times when characterizing the cross-over from
Hosking scaling to magnetic helicity scaling are given by

$t_{\xi}\approx(0.12/0.14)^{9/2}\left(I_{\mathrm{H}}/I_{\mathrm{M}}^{3}\right)^{1/2}\approx
0.50\,I_{\mathrm{H}}^{1/2}I_{\mathrm{M}}^{-3/2},$ (18) $t_{\cal E}\approx
I_{\mathrm{H}}^{1/2}I_{\mathrm{M}}^{-3/2}.$ (19)

It would be plausible to assume that both times should equal each other. The
fact that they are not equal to each other might hint, again, at the
possibility that the precise values of these coefficients are still uncertain.
On the other hand, looking at Figure 5, it is actually true that $\xi_{\rm M}$
approaches the $I_{\mathrm{M}}$-dominated scaling by a factor two earlier than
${\cal E}_{\rm M}$.

###  3.4 Comparison with earlier work

As mentioned in the introduction, the switch-over time from nonhelically to
helically dominated decay has been studied by Tevzadze et al. (2012) under the
assumption that $p=1$ and $q=1/2$ (Christensson et al., 2001) instead of
$p=10/9$ and $q=4/9$, as now motivated by the conservation of the Hosking
integral. The basic idea is to assume that at the time of switch-over,
$t_{\ast}$, the real-space realizability condition (Biskamp, 2003;
Kahniashvili et al., 2010) is saturated, i.e., $2\xi_{\rm M}(t_{\ast}){\cal
E}_{\rm M}(t_{\ast})=I_{\mathrm{M}}$. Next, inserting $\xi_{\rm
M}(t_{\ast})=\xi_{\rm M}(t_{0})\,(t_{\ast}/t_{0})^{q}$ and ${\cal E}_{\rm
M}(t_{\ast})={\cal E}_{\rm M}(t_{0})\,(t_{\ast}/t_{0})^{-p}$, we find
$2\xi_{\rm M}(t_{0}){\cal E}_{\rm
M}(t_{0})\,(t_{\ast}/t_{0})^{-(p-q)}=I_{\mathrm{M}}$, and therefore

$t_{\ast}=t_{0}\,\left[2\xi_{\rm M}(t_{0}){\cal E}_{\rm
M}(t_{0})/I_{\mathrm{M}}\right]^{1/(p-q)}.$ (20)

For $p=1$ and $q=1/2$, we have $1/(p-q)=2$ and recover the result of Tevzadze
et al. (2012), while for $p=10/9$ and $q=4/9$, we have $1/(p-q)=3/2$.
Comparing with Eq. (19), we see that $I_{\mathrm{M}}$ enters with the same
exponent $3/2$, and that the remainder can be identified with

$I_{\mathrm{H}}=\left[2\xi_{\rm M}(t_{0}){\cal E}_{\rm
M}(t_{0})\right]^{3}t_{0}^{2}.$ (21)

This suggests that $I_{\mathrm{H}}$ is related to $\xi_{\rm M}$ and ${\cal
E}_{\rm M}$, but the problem is that $t_{0}$ is not straightforwardly related
to the Alfvén time $\xi_{\rm M}/v_{\rm A}$, where $v_{\rm A}^{2}=2{\cal
E}_{\rm M}$. Indeed, Brandenburg et al. (2024) found that there is a pre-
factor $C_{\mathrm{M}}$ that increases with increasing magnetic Reynolds
number. Thus, inserting $t=C_{\mathrm{M}}\xi_{\rm M}/v_{\rm A}$, we find

$I_{\mathrm{H}}=C_{\mathrm{M}}^{2}\xi_{\rm M}^{5}v_{\rm A}^{4}.$ (22)

The facts that $C_{\mathrm{M}}$ enters quadratically and approaches values in
the range 20–50 for large magnetic Reynolds numbers explains why
$I_{\mathrm{H}}$ strongly exceeds the naive estimate $\xi_{\rm M}^{5}v_{\rm
A}^{4}$. Interestingly, Zhou et al. (2022) found that part of the large excess
over the naive estimate is related to non-Gaussianity. Another smaller part
has to do with the spectral shape. Linking the value of $C_{\mathrm{M}}$ to
non-Gaussianity provides a new clue to the question of why there is a
resistivity-dependent relation between decay and Alfvén times in hydromagnetic
turbulence.

##  4 Conclusions

The present work has shown that the decay laws for the combined case of two
conserved quantities is best represented not simply by the sum of the
individual laws, but that a good description of the numerical results is
obtained by taking the maximum between the two individual decay laws. The
switch-over from one to the other decay law occurs earlier for $\xi_{\rm
M}(t)$ than for ${\cal E}_{\rm M}(t)$. This behavior is surprising, but also
confirmed by direct inspection of the two time traces in Figure 5(b) and (d).

Comparing with earlier work on the switch-over from one to the other regime
suggests that the decay time enters in such a relation. This is remarkable,
because in hydromagnetic turbulence the decay time is known to be longer than
the Alfvén time by a resistivity-dependent factor of up to 50. This property
might also explain why the value of the Hosking integral is always found to
strongly exceeds the naive estimate $\xi_{\rm M}^{5}v_{\rm A}^{4}$. However,
is shown in Zhou et al. (2022), also other factors enter that involve the
spectral shape. It would therefore be interesting to revisit this question.

As alluded to in the introduction, an obvious astrophysical application of our
work is the decay of an initially bihelical magnetic field. Such situations
are important in proto-neutron stars after the neutrino-driven convection
ceases. Although this was actually our initial motivation, we have not
analyzed this case any further, because the most important aspect turned out
to be the fact that the magnetic field has always fractional helicity in such
cases, which we have now addressed.

A more general question is that of a decay governed by two decay laws and
whether there are other useful examples where the physics discussed in the
present paper can be studied. As far as turbulence is concerned, one might
think of the Loitsyansky and Saffman integrals, which represent the
coefficients of the $k^{4}$ and $k^{2}$ terms in the Taylor expansion of the
kinetic energy spectrum (see, e.g., Davidson, 2000). Here, the Saffman
integral might become more important at later times when long-range
interactions have occurred (Hosking & Schekochihin, 2023b). This could also be
applied to the magnetic case.

## Funding

This work was supported in part by the Swedish Research Council
(Vetenskapsrådet, 2019-04234), the National Science Foundation under Grant No.
NSF PHY-2307698, and a NASA ATP Award 80NSSC22K0825. We acknowledge the
inspiring atmosphere during the program on the “Generation, evolution, and
observations of cosmological magnetic fields” at the Bernoulli Center in
Lausanne. We acknowledge the allocation of computing resources provided by the
Swedish National Allocations Committee at the Center for Parallel Computers at
the Royal Institute of Technology in Stockholm and Linköping.

## Declaration of Interests

The authors report no conflict of interest.

## Data availability statement

The data that support the findings of this study are openly available on
http://norlx65.nordita.org/~brandenb/projects/Two-Conserved/. All calculations
have been performed with the Pencil Code (Pencil Code Collaboration et al.,
2021); DOI:10.5281/zenodo.3961647.

## Authors’ ORCIDs

A. Brandenburg, https://orcid.org/0000-0002-7304-021X

A. Banerjee, https://orcid.org/0009-0004-6288-0362

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