#  Local Thermodynamics and Entropy for
Relativistic Hydrostatic Equilibrium

Shuichi Yokoyama,
∗ Department of Physical Sciences, College of Science and Engineering,
Ritsumeikan University, Shiga 525-8577, Japan syr18046[at]fc.ritsumei.ac.jp

By the method proposed in arXiv:2010.07660, the entropy current and entropy
density for a relativistic hydrostatic equilibrium system with spherical
symmetry are constructed in the Einstein gravity with cosmological constant.
It is shown that the constructed entropy density satisfies both the local
Euler relation and the first law of thermodynamics non-perturbatively with
respect to the Newton constant. Finally the established relativistic
thermodynamics is applied to one of the systems with uniform energy density as
a crude model of a degenerate star and its local thermodynamic observables are
determined analytically.

##  1 Introduction

Celestial bodies have attracted people from ancient time by not only their
aesthetic appearances but also their physical properties. The fact that
astronomical bodies are visible implies that they exist stably enough to be
regarded as an equilibrium state, which is locally achieved by the balance
between the attractive force of gravity and the repulsive force of matter
inside. This suggests that astronomical bodies can be investigated as local
thermodynamical objects described by local thermodynamical variables such as
pressure and temperature whose dynamics obeys the laws of local
thermodynamics.

The most innovative concept in thermodynamics is entropy [1]. The laws of
thermodynamics can be described most concisely by employing entropy. Then it
is fundamental to ask whether there exists entropy for stable astronomical
bodies, and, if it exists, what its precise form is. Although more than a
century has passed since the notion of entropy was created by Clausius [1] and
astronomical bodies were studied as thermodynamic objects by Helmholtz and
Kelvin [2, 3], a definite answer for this question seems to have remained
unknown. The main reason for this will be that local thermodynamics has not
been established on curved spacetime. However, this question is very basic and
must be answered because the existence of entropy provides the ground to apply
basic results derived by assuming statistical physics to local thermodynamic
variables as seen in classic arguments for the understanding of physics of
astronomical bodies.

Recently the author and collaborators have proposed a method to construct
entropy current and entropy density as a conserved current and a conserved
charge density, respectively, for general field theory defined on general
curved spacetime with covariantly conserved energy momentum tensor even
without any global symmetry [4]. (See also [5].) In this paper, the author
aims at answering the above question adopting the proposed method by
simplifying a situation in such a way to approximate an astronomical body as a
relativistic hydrostatic equilibrium system with spherical symmetry. Then the
question is positively answered and the expression of the entropy density
endowed with laws of local thermodynamics is explicitly determined for an
arbitrary value of the Newton constant as shown below.

##  2 Relativistic hydrostatic equilibrium system

Consider a $d$-dimensional system of fluid of spherically symmetric
configuration in hydrostatic equilibrium [6, 7]. Such a fluid may be described
by a so-called perfect fluid, whose energy stress tensor is given by

$T_{\mu\nu}=(\rho+p)u_{\mu}u_{\nu}+pg_{\mu\nu},$ (2.1)

where $\rho$ is the total energy density, $p$ is the pressure, $u^{\mu}$ is
the fluid velocity normalized by $g_{\mu\nu}u^{\mu}u^{\nu}=-1$, $g_{\mu\nu}$
is a metric tensor in a general rotationally symmetric form

$\displaystyle g_{\mu\nu}\mathrm{d}x^{\mu}\mathrm{d}x^{\nu}=$
$\displaystyle-f\mathrm{d}t^{2}+h\mathrm{d}r^{2}+r^{2}\tilde{g}_{ij}\mathrm{d}x^{i}\mathrm{d}x^{j}$
(2.2)

where $\tilde{g}_{ij}$ is the metric for the $d-2$ dimensional internal space
whose the Ricci curvature tensor is $\tilde{\rm
Ric}_{ij}=(d-3)\tilde{g}_{ij}$, such as the unit sphere. The rotational
symmetry constrains all the functions $u^{\mu},\rho,p,f,h$ to depend only on
the radial coordinate $r$.

For the purpose to understanding sufficiently low energy physics of the above
hydrostatic equilibrium system, it is sufficient to consider gravitational
interaction described by the Einstein-Hilbert action. Then the hydrostatic
equilibrium is described by the Einstein equation. In order for fluid to
couple to the Einstein gravity consistently, its energy stress tensor is
required to satisfy the covariantly conservation equation in the gravitational
background, $\nabla_{\mu}T^{\mu}\\!_{\nu}=0$. For the perfect fluid (2.1), the
covariant conservation equation leads to the following relativistic fluid
equations

$\displaystyle\check{\nabla}\rho+(\rho+p)\vartheta=$ $\displaystyle 0,$ (2.3)
$\displaystyle(\rho+p)\check{\nabla}u_{\mu}+\bar{\nabla}_{\mu}p=$
$\displaystyle 0,$ (2.4)

where
$\check{\nabla}:=u^{\mu}\nabla_{\mu},\bar{\nabla}_{\mu}:=\nabla_{\mu}+u_{\mu}\check{\nabla}$
and $\vartheta=\bar{\nabla}_{\mu}u^{\mu}$ is the so-called expansion of the
fluid. Although the system is finally analyzed in the comoving frame of the
fluid velocity $(u^{\mu})=(1/\sqrt{f},0,\vec{0})$, the fluid equations here
are rewritten in a slightly more general frame with the fluid radially moving,
$(u^{\mu})=(u^{t},u^{r},\vec{0})$, for later use. Then the normalization
condition is

$-f(u^{t})^{2}+h(u^{r})^{2}=-1.$ (2.5)

In this radially moving frame, the relativistic fluid equations (2.3), (2.4)
reduce to

$\displaystyle\rho^{\prime}+(\rho+p)(\log(\sqrt{|g|}u^{r}))^{\prime}=0,$ (2.6)
$\displaystyle p^{\prime}+(\rho+p)(\log(fu^{t}))^{\prime}=0,$ (2.7)

where the prime means the differentiation with respect to the radial
coordinate. Note that (2.7) is obtained from the time component of (2.4),
while the radial component of (2.4) reduces to (2.7) by using (2.5).

In the comoving frame of no energy motion, $u^{r}\to 0$, the condition of the
covariant conservation (2.7) reduces to

$p^{\prime}=-\frac{(p+\rho)}{2}(\log f)^{\prime},$ (2.8)

and the Einstein equation

$\displaystyle(\log h)^{\prime}=\frac{2rh}{d-2}(8\pi
G_{N}\rho+\Lambda)-\frac{(d-3)\left(h-1\right)}{r},$ (2.9) $\displaystyle(\log
f)^{\prime}=\frac{2rh}{d-2}(8\pi
G_{N}p-\Lambda)+\frac{(d-3)\left(h-1\right)}{r},$ (2.10)

where $G_{N}$ is the Newton constant and $\Lambda$ is the cosmological
constant. Introducing a new variable $M_{r}$ defined by

$M_{r}:={r^{d-3}\over 2G_{N}}\left(1-{1\over h}-{2\Lambda
r^{2}\over(d-2)(d-1)}\right),$ (2.11)

one can rewrite (2.9) as

$\displaystyle\rho=$ $\displaystyle\frac{(d-2)M_{r}^{\prime}}{8\pi r^{d-2}},$
(2.12)

while (2.10) as

$\displaystyle(\log f)^{\prime}=$
$\displaystyle\frac{r^{d-1}\frac{1}{(d-2)}(8\pi
G_{N}p-\Lambda)+(d-3)G_{N}M_{r}}{r^{d-2}-2rG_{N}M_{r}}.$ (2.13)

Plugging this into (2.8) with $d=4$ and $\Lambda=0$, one obtains the so-called
TOV equation given in [6].

##  3 Local thermodynamic relations and entropy

As mentioned in introduction, any equilibrium system will be expected to be
described by a set of thermodynamic variables obeying thermodynamic relations.
Then how does one find the entropy for a hydrostatic equilibrium system in
curved spacetime as described above? The author answers this question adopting
the method proposed in [4], which provides a general prescription to construct
a conserved quantity for a closed system with its energy momentum tensor
covariantly conserved and even without any global symmetry. The prescription
is to find a vector field to satisfy what is called a conservation equation
$T^{\mu}\\!_{\nu}\nabla_{\mu}\xi^{\nu}=0$. Once such a vector field
$\xi^{\nu}$ is found, an associated conserved current is constructed as
$J^{\mu}[\xi]=\sqrt{|g|}T^{\mu}\\!_{\nu}\xi^{\nu}$ and a conserved charge
$Q[\xi]=\int d^{d-1}xJ^{t}[\xi]$. In order to construct entropy current
associated with fluid, an associated vector field is assumed to be
proportional to its fluid velocity, $\xi^{\nu}=-\zeta u^{\nu},$ where $\zeta$
is a scalar function, so that the ansatz respects general covariance.
Substituting this ansatz into the conservation equation leads to
$-\rho\check{\partial}\zeta+\zeta p\vartheta=0.$ In a radially moving frame,
this becomes $u^{r}\zeta^{\prime}={p\vartheta\over\rho}\zeta$, where $\zeta$
is assumed to depend only on $r$ for a spherically symmetric equilibrium
system. In this frame, the expansion is computed as
$\vartheta=u^{r}\frac{-\rho^{\prime}}{\rho+p}$, where (2.6) was used.
Substituting this back, one obtains

$\zeta^{\prime}=-{p\over\rho}\frac{\rho^{\prime}}{\rho+p}\zeta.$ (3.1)

This determines $\zeta$ in terms of macroscopic observables. Then the
conserved current associated with the vector field is computed as

$\displaystyle s^{\mu}=\sqrt{|g|}\rho\zeta u^{\mu}.$ (3.2)

This is constructed to satisfy the continuity equation
$\partial_{\mu}s^{\mu}=0$. This can be confirmed as follows. In the
spherically symmetric equilibrium system,
$\partial_{\mu}s^{\mu}=\partial_{r}s^{r}$. On the other hand,

$\displaystyle\partial_{r}s^{r}=$
$\displaystyle\sqrt{|g|}u^{r}((\log(\sqrt{|g|}u^{r}))^{\prime}\rho\zeta+\rho^{\prime}\zeta+\rho\zeta^{\prime})$
$\displaystyle=$
$\displaystyle\sqrt{|g|}u^{r}(\frac{-\rho^{\prime}}{\rho+p}\rho\zeta+\rho^{\prime}\zeta-{p}\frac{\rho^{\prime}}{\rho+p}\zeta)=0,$
(3.3)

where (2.6) and (3.1) were used. This completes the confirmation.

As a result, $s^{\mu}$ is a conserved current and its time component
$s^{t}=:s$ is a conserved charge density. These conserved quantities are
claimed to be the entropy current and the entropy density of the system,
respectively [4]. In order to provide strong evidence, let me show below that
this identification leads to the local Euler’s relation and the first law of
thermodynamics concurrently in the comoving frame, which are respectively
described by

$\displaystyle Ts=$ $\displaystyle u+pv,$ (3.4) $\displaystyle Tds=$
$\displaystyle du+pdv,$ (3.5)

where $v$ is the measure of the volume element at a constant time slice,
$v=r^{d-2}\sqrt{h\tilde{g}}$, and $u$ is the internal energy density, $u=\rho
v$, and $T$ is a scalar corresponding to the temperature defined to satisfy
both (3.4) and (3.5). To show this, first solve (3.1) as

$\zeta=\beta_{0}u^{t}f(1+\frac{p}{\rho}),$ (3.6)

where $\beta_{0}$ is an integration constant. Indeed, assuming (3.6) one can
calculate as
$\zeta^{\prime}=\zeta\partial_{r}\log\left((u^{t}f)(1+\frac{p}{\rho})\right)=\zeta(-\frac{p\rho^{\prime}}{\rho(\rho+p)}),$
where (2.7) was used. This satisfies (3.1). Plugging (3.6) into (3.2),
$s^{\mu}$ can be written as
$s^{\mu}=\beta_{0}u^{t}f^{\frac{3}{2}}(u+pv)u^{\mu}$. Therefore the Euler
relation (3.4) holds if and only if the scalar $T$ or its inverse $\beta:=1/T$
is given by $\beta=\beta_{0}(u^{t})^{2}f^{\frac{3}{2}}.$ In the comoving
frame, $u^{t}\to 1/\sqrt{f}$, this becomes

$\beta=\beta_{0}\sqrt{f}.$ (3.7)

In what follows, the frame is fixed by the comoving one. Then the derivative
of $\beta$ with respect to $r$ is computed as

$\beta^{\prime}=-\frac{p^{\prime}}{p+\rho}\beta,$ (3.8)

where (2.8) was used, and thus that of $s=s^{t}$ is
$s^{\prime}=\beta^{\prime}(u+vp)+\beta(u+vp)^{\prime}=\beta(u^{\prime}+pv^{\prime}).$
This is the first law of thermodynamics (3.5). This completes the proof. As a
result, it concludes that $s^{\mu},s,\beta$ are interpreted as the entropy
current, the entropy density, and the inverse temperature, respectively.

Comments are in order. The (inverse) temperature introduced to satisfy the
local Euler relation and the first law of thermodynamics in (3.7) exactly
matches the one derived by Tolman as the ’proper temperature’ for a local
observer [8]. This temperature was also derived by different ways in standard
textbooks [9, 10]. On the other hand, the ’entropy vector’ and ’proper entropy
density’ were introduced to discuss hydrostatic equilibrium in [8] without
explicit expression, while the entropy density and the entropy current are
explicitly determined in this paper. From the entropy density, one can compute
the local entropy inside the sphere of radius $\bar{r}$ denoted by
$\mathbf{S}_{\bar{r}}$ as
$S_{\bar{r}}=\int_{\mathbf{S}_{\bar{r}}}\mathrm{d}^{d-1}xs=\int_{0}^{\bar{r}}dr(\rho+p)V_{r}^{\prime}/T$,
where $V_{\bar{r}}=\int_{\mathbf{S}_{\bar{r}}}\mathrm{d}^{d-1}xv$ is the
volume of the spherical region within the radius $r$. These results were
obtained without restricting the value of the Newton constant. Therefore the
thermodynamic quantities and thermodynamic relations hold non-pertubatively
with respect to the Newton constant.

##  4 Application to a spherical star with uniform density

In order to see how the relativistic thermodynamics work, it is applied to a
perfect fluid with spherical configuration of radius $R$ and uniform energy
density in the dark energy specified by the cosmological constant $\Lambda$.
This could be a crude model of a degenerate core deep inside of a massive star
and a compact star of degenerate matter. The analysis is performed with the
general dimension $d\geq 3$ and assumes that the cosmological constant is
sufficiently small.

Inside the star or its core, $r\leq R$, the energy density is constant,
$\rho=\rho_{0}$, so that $M_{r}$ can be determined from (2.12) as

$\displaystyle M_{r}={8\pi\rho_{0}r^{d-1}\over(d-2)(d-1)}=:M(r)$ (4.1)

where the integration constant was determined for $G_{N}M_{r}$ to vanishes at
the center. Then from (2.11), $h=\frac{r_{0}^{2}}{r_{0}^{2}-r^{2}}$, where
$r_{0}=\sqrt{\frac{(d-2)(d-1)}{2(8\pi G_{N}\rho_{0}+\Lambda)}}$, and from
(2.13)

$\displaystyle(\log f)^{\prime}=\frac{(d-3)8\pi G_{N}\rho_{0}r+(d-1)r(8\pi
G_{N}p-\Lambda)}{(d-2)(d-1)-16\pi G_{N}\rho_{0}r^{2}}.$ (4.2)

Plugging this into (2.8), one can solve the TOV equation as

$p=\frac{-C\sqrt{r_{0}^{2}-r^{2}}+d-3-2\widetilde{\Lambda}}{C\sqrt{r_{0}^{2}-r^{2}}-(d-1)}\rho_{0},$
(4.3)

where $C$ is an integration constant and $\widetilde{\Lambda}:=\Lambda/(8\pi
G_{N}\rho_{0})$. Fixing the integration constant so as for the pressure to
vanish at the surface of the star, one can determine the radial dependence of
pressure as

$\displaystyle p=$
$\displaystyle\frac{\left(\sqrt{r_{0}^{2}-r^{2}}-\sqrt{r_{0}^{2}-R^{2}}\right)\left(d-3-2\widetilde{\Lambda}\right)}{-\left(d-3-2\widetilde{\Lambda}\right)\sqrt{r_{0}^{2}-r^{2}}+(d-1)\sqrt{r_{0}^{2}-R^{2}}}\rho_{0},$
(4.4)

Note that in order for the configuration to be stable, the pressure has to be
positive and finite, which is satisfied if and only if
$d-3-2\widetilde{\Lambda}>0,~{}-\left(d-3-2\widetilde{\Lambda}\right)r_{0}+(d-1)\sqrt{r_{0}^{2}-R^{2}}>0$.
This condition is rewritten as follows.

$R<\sqrt{\frac{(d-2-\widetilde{\Lambda})(d-2)}{(d-1)4\pi
G_{N}\rho_{0}}}=:R_{cr},~{}~{}~{}{\rm max}\\{\frac{\Lambda}{(d-3)4\pi
G_{N}},\frac{-\Lambda}{8\pi G_{N}}\\}<\rho_{0}.$ (4.5)

The temperature is also determined analytically by solving (3.8) as

$T=T(R){p+\rho_{0}\over\rho_{0}}=T(R)\frac{2\sqrt{r_{0}^{2}-R^{2}}(1+\widetilde{\Lambda})}{-(d-3-2\widetilde{\Lambda})\sqrt{r_{0}^{2}-r^{2}}+(d-1)\sqrt{r_{0}^{2}-R^{2}}},$
(4.6)

where an integration constant appears as the surface temperature of the star
$T(R)$. This is fixed by input of observational data. In particular, the
temperature at the center is given by the one at the surface as

$T(0)=\frac{2(1+\widetilde{\Lambda})\sqrt{1-(R/r_{0})^{2}}}{-(d-3-2\widetilde{\Lambda})+(d-1)\sqrt{1-(R/r_{0})^{2}}}T(R).$
(4.7)

Note that the core maximum temperature becomes higher and higher as the radius
approaches the critical value $R_{cr}$. Substituting (4.4) and (4.6) into
(3.4) one obtains the entropy density as $s=\frac{u}{T(R)}$, where
$u=\frac{\rho_{0}r_{0}}{\sqrt{r_{0}^{2}-r^{2}}}\sqrt{\tilde{g}}$.
Interestingly, this shows that while the temperature monotonically decreases
as it approaches the surface away from the core, the internal density and thus
the entropy one monotonically increase for a uniform stellar object in
hydrostatic equilibrium.

Outside the star, $r\geq R$, there is no fluid. Thus the density and pressure
of fluid vanish, so is the entropy density from (3.4). Therefore, from (2.12),
$M_{r}$ has to be a constant, which is determined as $M_{r}=M(R)$ from the
continuity condition at the surface of the star. On the other hand, from (2.9)
and (2.10), $(\log f)^{\prime}=-(\log h)^{\prime}$, so that $f$ is
proportional to $1/h$. Setting the boundary condition at infinity for the
metric to be asymptotic to the standard (anti-) de Sitter space metric, one
can fix the proportional coefficient as $1$, so that $f,h$ are determined as
$f=1/h={-2\Lambda r^{2}\over(d-2)(d-1)}+1-{2G_{N}M(R)\over r^{d-3}}=:f(r)$.
Then the metric (2.2) in the matter-empty region is given by the Kottler
metric with the so-called gravitational mass $M(R)$. In the matter-empty
region the thermodynamic relations (3.4) and (3.5) are trivial, though the
temperature is non-trivially determined from (3.7) as
$T=T(R)\sqrt{f(R)/f(r)}$. It can be seen that the local temperature is
formally divergent at the location of the horizon of the black hole whose mass
is identical to the stellar gravitational mass. Note that in order for $f(r)$
to be always positive, the radius of the star is required to be greater than
that of any event horizon of the Schwarzschild black hole with the same mass
$M(R)$. For $\Lambda=0,d=4$ this condition is in fact satisfied by the above
constraint (4.5), which reduces to $M(R)<4R/(9G_{N})$.

It is not difficult to compute local additive quantities inside the sphere of
radius $r$ with $r\leq R$. The local internal energy $U_{r}$ is computed as
$U_{r}=\rho_{0}V_{r}$, where
$V_{r}=\tilde{V}\int_{0}^{r}\mathrm{d}r^{\prime}r^{\prime
d-2}{r_{0}\over\sqrt{r_{0}^{2}-r^{\prime 2}}}$. Particularly if one chooses
$d=4$ and the internal manifold is the unit sphere, then $V_{r}=2\pi
r_{0}\left(r_{0}^{2}\arctan\frac{r}{\sqrt{r_{0}^{2}-r^{2}}}-r\sqrt{r_{0}^{2}-r^{2}}\right)$.
The local entropy is
$S_{r}=\int_{\mathbf{S}_{r}}\mathrm{d}^{d-1}xs=\frac{U_{r}}{T(R)}$.

##  5 Discussion

Entropy current and entropy density for a relativistically hydrostatic
equilibrium system with spherical symmetry have been constructed by the method
in [4]. It has been shown that the entropy density concurrently satisfies the
local Euler relation and the first law of thermodynamics with the local
temperature identical to the one determined by Tolman [8] non-perturbatively
with the Newton gravitational constant. After the foundation of relativistic
thermodynamics has been established, it has been applied to a spherically
symmetric hydrostatic system of a perfect fluid with uniform energy density as
a crude model of a degenerate core deep inside a massive star and a compact
star. It has been shown that local thermodynamic observables are precisely
determined even deep inside the core in this model.

An important virtue of the presented method is that this is applicable not
only to an equilibrium system but also to a non-equilibrium one. In
particular, the constructed entropy density in this paper is ensured to be the
one left after an evolutionary dynamical system reaches the equilibrium with
spherical symmetry. It would be very interesting to apply the presented method
to a non-equilibrium system.

Further applications of the established relativistic thermodynamics to a more
realistic situation and to the construction of a model of celestial bodies
will be reported in the near future.

## Acknowledgement

This work is supported in part by the Grant-in-Aid of the Japanese Ministry of
Education, Sciences and Technology, Sports and Culture (MEXT) for Scientific
Research (No. JP22K03596).

## References

  * [1] R. Clausius, T. Hirst, and J. Tyndall, The Mechanical Theory of Heat: With Its Applications to the Steam-engine and to the Physical Properties of Bodies.  J. Van Voorst, 1867.
  * [2] H. L. F. von Helmholtz, “Lxiv. on the interaction of natural forces,” Philosophical Magazine Series 1 11 489–518.
  * [3] W. Kelvin, Popular Lectures and Addresses.  Nature series. Macmillan and Company, 1889.
  * [4] S. Aoki, T. Onogi, and S. Yokoyama, “Charge conservation, entropy current and gravitation,” Int. J. Mod. Phys. A 36 (2021), no. 29, 2150201, 2010.07660.
  * [5] S. Aoki, T. Onogi, and S. Yokoyama, “Conserved charges in general relativity,” Int. J. Mod. Phys. A 36 (2021), no. 10, 2150098, 2005.13233.
  * [6] J. R. Oppenheimer and G. M. Volkoff, “On massive neutron cores,” Phys. Rev. 55 (Feb, 1939) 374–381.
  * [7] O. Heckmann, “REVIEW: Relativity, Thermodynamics and Cosmology, by R. C. Tolman,” apj 82 (Dec., 1935) 435.
  * [8] R. C. Tolman, “On the weight of heat and thermal equilibrium in general relativity,” Phys. Rev. 35 (Apr, 1930) 904–924.
  * [9] I. B. I. B. Zel’dovich, Relativistic astrophysics.  University of Chicago Press, Chicago, 1971 - 1983.
  * [10] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation.  W. H. Freeman, San Francisco, 1973.


compare due to differences in their definitions used by different research
groups). It has been recently found in studies of forced statistically
stationary turbulence (Sahoo & Biferale, 2018; Alexakis, 2017; Alexakis &
Biferale, 2018) that the PPP helical triads contribute with an inverse-cascade
flux at wavenumbers in the inertial range and with a direct-cascade flux in
the dissipative range. Our results, being based on freely decaying turbulence
from extreme or generic initial conditions, confirm this, even quantitatively,
if one assumes (see, for example, Frisch (1995)) that the state of the maximum
dissipation in a transient evolution is a proxy for a statistically stationary
turbulence in a forced evolution, at least considering energy spectra and
energy fluxes. Regarding the flux towards the region with $k>2$, which sits in
the inertial range, we find that the PPP contribution is negative and of the
order of 10% of the total flux in absolute terms near the time of the maximum
dissipation, which is in quantitative agreement with the works cited above.
Regarding the flux towards the region with $k>10$, which sits in the
dissipative range, we find that the PPP contribution is positive and of the
order of 30% of the total flux in absolute terms near the time of the maximum
dissipation, again in quantitative agreement with the works cited above. It is
worth mentioning that the case of the flow with the Taylor-Green initial
condition leads to qualitatively different results in the inertial-range
fluxes (namely, the fluxes towards the region with $k>2$): the PPP
contribution to the inverse cascade is there but is small and the main
contribution comes from the PMM helical triad type (Class II in Waleffe’s
notation), cf. figures 22(b)(a,c,e).

It is well known that appearance of small scales and singularity formation in
solutions of time-dependent PDE problems can be conveniently characterized in
terms of the evolution of singularities in the complex extensions of these
fields (Weideman, 2003; Matsumoto et al., 2008; Siegel & Caflisch, 2009). In
particular, blow-up of the solution is signaled by the collapse of some of
these complex-plane singularities onto the real axis (i.e., when the width of
the analyticity strip shrinks to zero). As regards the diagnostics employed in
the present study, it is an interesting open question how properties of triad
phases in a flow can be deduced from the evolution of complex-plane
singularities characterizing these solutions, and vice versa.

## Acknowledgments

DK, BP and MDB acknowledge financial support from University College Dublin
via Seed Funding project SF1568, and the hospitality of this institution where
a part of this research was carried out. DK and BP were also partially funded
through an NSERC (Canada) Discovery Grant. Computational resources were
provided by Compute Canada under its Resource Allocation Competition.

Declaration of Interests. The authors report no conflict of interest.

##  Appendix A Minimizing the Blow-up Time in the 1D Inviscid Burgers Equation

In this appendix we demonstrate that the problem of finding an initial
condition $u_{0}\in H^{1}(\Omega)$ for the inviscid Burgers system (4) with a
fixed enstrophy ${\mathcal{E}}_{0}>0$ such that the corresponding solution
blows up in the shortest time $t^{*}(u_{0})$ does not in fact admit a
solution. In other words, it is possible to find initial data $u_{0}$ with
${\mathcal{E}}(u_{0})={\mathcal{E}}_{0}$ such that the corresponding solution
of (4) blows up arbitrarily quickly. We thus have

###### Theorem 1

The minimization problem

$\min_{u_{0}\in H^{1}(\Omega)}t^{*}(u_{0})\quad\text{subject
to}\quad{\mathcal{E}}(u_{0})={\mathcal{E}}_{0}$ (37)

has no solution.

###### Proof 1

Since for the inviscid Burgers equation (4) we have an explicit expression for
the blow-up time $t^{*}=-1/\left[\inf_{x\in\Omega}{\partial u_{0}\over\partial
x}(x)\right]$, problem (37) is equivalent to

$\max_{g\in L^{2}(\Omega)}\|g\|_{L^{\infty}}\quad\text{subject
to}\quad\frac{1}{2}\|g\|_{L^{2}}^{2}={\mathcal{E}}_{0},$ (38)

where $g:={\partial u_{0}\over\partial x}$. It is easy to construct a sequence
of functions $g_{n}$ with a fixed $L^{2}$ norm and vanishing support
$\lim_{n\rightarrow\infty}|\operatorname{supp}_{x\in\Omega}g_{n}(x)|=0$ such
that their $L^{\infty}$ norms increase without bound,
$\lim_{n\rightarrow\infty}\|g_{n}\|_{L^{\infty}}=\infty$, thus demonstrating
that a maximum is not attained in problem (38).

##  Appendix B Proof of the Symmetry of Conjugate Triads under Zero Helicity
Assumption

In this appendix we state and prove the following theorem implying the
symmetry of conjugate triads.

###### Theorem 2

If the incompressible velocity field ${\mathbf{u}}$ satisfies the condition of
oddity under the parity transformation, namely,

$\mathbf{u}(-\mathbf{x},t)=-\mathbf{u}(\mathbf{x},t)\qquad\text{for
all}\qquad\mathbf{x}\in\mathbb{R}^{3},\quad t\in\mathbb{R},$ (39)

then:

  1. 1.

the helical modes $u_{\mathbf{k}}^{+},u_{\mathbf{k}}^{-}$ satisfy the identity

$u_{-\mathbf{k}}^{s}=-u_{\mathbf{k}}^{-s},\qquad\text{for
all}\qquad\mathbf{k}\in\mathbb{Z}^{3}\setminus\\{\mathbf{0}\\},\quad s=\pm,$

  2. 2.

the helicity spectrum is zero: ${\mathcal{H}}_{{\mathbf{k}}}=0,\quad\text{for
all}\quad{\mathbf{k}}\in\mathbb{Z}^{3}\setminus\\{\mathbf{0}\\}$, cf. (22b),
and

  3. 3.

the generalized helical triad phases (27) exhibit the following invariance
with respect to the transformations $s_{j}\rightarrow-s_{j}$, $j=1,2,3$:

$\Phi_{{\mathbf{k}_{1}}{\mathbf{k}_{2}}{\mathbf{k}_{3}}}^{-s_{1}\,-s_{2}\,-s_{3}}=-\Phi_{{\mathbf{k}_{1}}{\mathbf{k}_{2}}{\mathbf{k}_{3}}}^{s_{1}\,\,s_{2}\,\,s_{3}},\qquad\text{whenever}\qquad{\mathbf{k}}_{1}+{\mathbf{k}}_{2}+{\mathbf{k}}_{3}={\mathbf{0}}.$
(40)

###### Proof 2

Assuming condition (39) is satisfied, let us first establish the corresponding
relation for the Fourier components of the velocity field. From equation (18)
we have

$\mathbf{u}(\mathbf{x},t)=\sum_{\mathbf{k}\in\mathbb{Z}^{3}\setminus\mathbf{0}}\mathbf{\widehat{u}}_{\mathbf{k}}(t)\exp(2\pi
i\mathbf{k}\cdot\mathbf{x})\,,$

so

$\mathbf{u}(-\mathbf{x},t)=\sum_{\mathbf{k}\in\mathbb{Z}^{3}\setminus\mathbf{0}}\mathbf{\widehat{u}}_{\mathbf{k}}(t)\exp(-2\pi
i\mathbf{k}\cdot\mathbf{x})=\sum_{\mathbf{k}\in\mathbb{Z}^{3}\setminus\mathbf{0}}\mathbf{\widehat{u}}_{-\mathbf{k}}(t)\exp(2\pi
i\mathbf{k}\cdot\mathbf{x})\,,$

where in the last equality we relabelled the wavevectors from $\mathbf{k}$ to
$-\mathbf{k}$. Equating this expression with the Fourier expansion of
$-\mathbf{u}(\mathbf{x},t)$ we obtain, from uniqueness of the expansion,

$\mathbf{\widehat{u}}_{-\mathbf{k}}(t)=-\mathbf{\widehat{u}}_{\mathbf{k}}(t),\qquad\text{for
\,\, all}\qquad\mathbf{k}\in\mathbb{Z}^{3}\setminus\\{\mathbf{0}\\}\,.$ (41)

Now we prove each of the claims in the theorem:

  1. 1.

The helical mode decomposition (19) gives

$\mathbf{\widehat{u}}_{\mathbf{k}}(t)=\mathbf{h}_{\mathbf{k}}^{+}u_{\mathbf{k}}^{+}(t)+\mathbf{h}_{\mathbf{k}}^{-}u_{\mathbf{k}}^{-}(t)\,$

and noting that $\mathbf{h}_{-\mathbf{k}}^{s}=\mathbf{h}_{\mathbf{k}}^{-s}$ we
obtain

$\mathbf{\widehat{u}}_{-\mathbf{k}}(t)=\mathbf{h}_{-\mathbf{k}}^{+}u_{-\mathbf{k}}^{+}(t)+\mathbf{h}_{-\mathbf{k}}^{-}u_{-\mathbf{k}}^{-}(t)=\mathbf{h}_{\mathbf{k}}^{-}u_{-\mathbf{k}}^{+}(t)+\mathbf{h}_{\mathbf{k}}^{+}u_{-\mathbf{k}}^{-}(t).$

Invoking uniqueness of the helical decomposition, relation (41) is equivalent
to

$u_{-\mathbf{k}}^{-}(t)=-u_{\mathbf{k}}^{+}(t)\,,\qquad\mathbf{k}\in\mathbb{Z}^{3}\setminus\\{\mathbf{0}\\}\,.$

This of course also implies that
$u_{-\mathbf{k}}^{+}(t)=-u_{\mathbf{k}}^{-}(t)$.

  2. 2.

The helicity spectrum is defined as

$\mathcal{H}_{\mathbf{k}}(t)=|u_{\mathbf{k}}^{+}(t)|^{2}-|u_{\mathbf{k}}^{-}(t)|^{2}\,.$

Since the original velocity field is real-valued, we obtain
$u_{-\mathbf{k}}^{s}(t)=[u_{\mathbf{k}}^{s}(t)]^{*}$, such that

$|u_{\mathbf{k}}^{-}(t)|=|u_{\mathbf{-k}}^{-}(t)|=|{-u_{\mathbf{k}}^{+}(t)}|=|u_{\mathbf{k}}^{+}(t)|,$

implies
$\mathcal{H}_{\mathbf{k}}(t)=0\,,\quad\mathbf{k}\in\mathbb{Z}^{3}\setminus\\{\mathbf{0}\\}$.

  3. 3.

The generalized helical triad phases are obtained, mod $2\pi$, from combining
equations (27) and (28):

$\Phi_{{\mathbf{k}_{1}}{\mathbf{k}_{2}}{\mathbf{k}_{3}}}^{s_{1}\,\,s_{2}\,\,s_{3}}=\phi_{\mathbf{k}_{1}}^{s_{1}}+\phi_{\mathbf{k}_{2}}^{s_{2}}+\phi_{\mathbf{k}_{3}}^{s_{3}}+\arg[\mathbf{h}_{\mathbf{k}_{1}}^{s_{1}}\times\mathbf{h}_{\mathbf{k}_{2}}^{s_{2}}\cdot\mathbf{h}_{\mathbf{k}_{3}}^{s_{3}}]+\begin{cases}\arg(s_{2}k_{2}-s_{1}k_{1}),&\text{if}\quad\mathbf{k}_{3}\in\mathcal{C};\mathbf{k}_{1},\mathbf{k}_{2}\in\mathcal{U}\setminus\mathcal{C}\\\
\arg(s_{2}k_{2}-s_{3}k_{3}),&\text{if}\quad\mathbf{k}_{3},\mathbf{k}_{2}\in\mathcal{C};\mathbf{k}_{1}\in\mathcal{U}\setminus\mathcal{C}\end{cases}\,,$

where $\phi_{\mathbf{k}}^{s}=\arg u_{\mathbf{k}}^{s}(t)$. We now analyze the
behaviour of this expression under the transformation $s_{j}\to-
s_{j},\,\,j=1,2,3$. First, using part 1 of the theorem, we get:

$\phi_{\mathbf{k}}^{-s}=\arg
u_{\mathbf{k}}^{-s}(t)=\arg(-u_{-\mathbf{k}}^{s}(t))=\pi+\phi_{-\mathbf{k}}^{s}=\pi-\phi_{\mathbf{k}}^{s}\,,$

where the last equality follows from the reality condition. Second, from the
definition of the helical basis vectors we have

$\mathbf{h}_{\mathbf{k}}^{-s}=[\mathbf{h}_{\mathbf{k}}^{s}]^{*}\,.$

Third, we have, mod $2\pi$,
$\arg(-s_{2}k_{2}-(-s_{1})k_{1})=\pi-\arg(s_{2}k_{2}-s_{1}k_{1})$, and so on.
Therefore, we obtain

$\Phi_{{\mathbf{k}_{1}}{\mathbf{k}_{2}}{\mathbf{k}_{3}}}^{-s_{1}\,\,-s_{2}\,\,-s_{3}}=\phi_{\mathbf{k}_{1}}^{-s_{1}}+\phi_{\mathbf{k}_{2}}^{-s_{2}}+\phi_{\mathbf{k}_{3}}^{-s_{3}}+\arg[\mathbf{h}_{\mathbf{k}_{1}}^{-s_{1}}\times\mathbf{h}_{\mathbf{k}_{2}}^{-s_{2}}\cdot\mathbf{h}_{\mathbf{k}_{3}}^{-s_{3}}]+\begin{cases}\arg(-s_{2}k_{2}+s_{1}k_{1}),&\text{if}\quad\mathbf{k}_{3}\in\mathcal{C};\mathbf{k}_{1},\mathbf{k}_{2}\in\mathcal{U}\setminus\mathcal{C}\,,\\\
\arg(-s_{2}k_{2}+s_{3}k_{3}),&\text{if}\quad\mathbf{k}_{3},\mathbf{k}_{2}\in\mathcal{C};\mathbf{k}_{1}\in\mathcal{U}\setminus\mathcal{C}\,,\end{cases}$
$=3\pi-\left(\phi_{\mathbf{k}_{1}}^{s_{1}}+\phi_{\mathbf{k}_{2}}^{s_{2}}+\phi_{\mathbf{k}_{3}}^{s_{3}}\right)+\arg[\mathbf{h}_{\mathbf{k}_{1}}^{s_{1}}\times\mathbf{h}_{\mathbf{k}_{2}}^{s_{2}}\cdot\mathbf{h}_{\mathbf{k}_{3}}^{s_{3}}]^{*}+\pi-\begin{cases}\arg(s_{2}k_{2}-s_{1}k_{1}),&\text{if}\quad\mathbf{k}_{3}\in\mathcal{C};\mathbf{k}_{1},\mathbf{k}_{2}\in\mathcal{U}\setminus\mathcal{C}\,,\\\
\arg(s_{2}k_{2}-s_{3}k_{3}),&\text{if}\quad\mathbf{k}_{3},\mathbf{k}_{2}\in\mathcal{C};\mathbf{k}_{1}\in\mathcal{U}\setminus\mathcal{C}\,,\end{cases}$
$=4\pi-\left(\phi_{\mathbf{k}_{1}}^{s_{1}}+\phi_{\mathbf{k}_{2}}^{s_{2}}+\phi_{\mathbf{k}_{3}}^{s_{3}}\right)-\arg[\mathbf{h}_{\mathbf{k}_{1}}^{s_{1}}\times\mathbf{h}_{\mathbf{k}_{2}}^{s_{2}}\cdot\mathbf{h}_{\mathbf{k}_{3}}^{s_{3}}]-\begin{cases}\arg(s_{2}k_{2}-s_{1}k_{1}),&\text{if}\quad\mathbf{k}_{3}\in\mathcal{C};\mathbf{k}_{1},\mathbf{k}_{2}\in\mathcal{U}\setminus\mathcal{C}\,,\\\
\arg(s_{2}k_{2}-s_{3}k_{3}),&\text{if}\quad\mathbf{k}_{3},\mathbf{k}_{2}\in\mathcal{C};\mathbf{k}_{1}\in\mathcal{U}\setminus\mathcal{C}\,,\end{cases}$

so that, mod $2\pi$, we finally arrive at

$\Phi_{{\mathbf{k}_{1}}{\mathbf{k}_{2}}{\mathbf{k}_{3}}}^{-s_{1}\,\,-s_{2}\,\,-s_{3}}=-\Phi_{{\mathbf{k}_{1}}{\mathbf{k}_{2}}{\mathbf{k}_{3}}}^{s_{1}\,\,s_{2}\,\,s_{3}}\,.$

##  Appendix C Fourier-Lagrange Formula for the Inviscid Burgers Equation

In order to determine the solutions of the inviscid Burgers system (4) with
initial conditions (9) and (11), we will use an adaptation of the Fourier-
Lagrange formula originally proposed by Fournier & Frisch (1983) which
provides an exact expression for the Fourier coefficients $\widehat{u}_{k}(t)$
of the solution for $0\leq t<t^{*}$. One difference with respect to the
approach in Fournier & Frisch (1983) is that here we work on a periodic,
rather than unbounded, spatial domain. As a starting point, we write the
solution of system (4) in terms of the flowmap $\xi=\xi(a,t)$, where
$a\in[0,1]$ is the Lagrangian coordinate, as Kreiss & Lorenz (2004)

$u(\xi(a,t),t)=u_{0}(a),\quad\xi(a,t)=a+tu_{0}(a),\quad a\in[0,1],\quad 0\leq
t<t^{*}.$ (42)

Then, the Fourier coefficients of the solution can be expressed as, cf. (12),

$\displaystyle\widehat{u}_{k}(t)$ $\displaystyle=\int_{0}^{1}e^{-2\pi
ikx}u(x,t)\,dx=\int_{0}^{1}e^{-2\pi ik\xi(a,t)}u(\xi(a,t),t)\,d\xi$
$\displaystyle=\int_{0}^{1}\frac{d}{d\xi}\left(\frac{-1}{2\pi ik}e^{-2\pi
ik\xi(a,t)}\right)u(\xi(a,t),t)\,d\xi=\frac{1}{2\pi ik}\int_{0}^{1}e^{-2\pi
ik\xi(a,t)}\frac{du}{d\xi}(\xi(a,t),t)\,d\xi$ $\displaystyle=\frac{1}{2\pi
ik}\int_{0}^{1}e^{-2\pi ik\left[a+tu_{0}(a)\right]}\frac{du_{0}}{da}(a)\,da,$

where we used (42), performed integration by parts with respect to $\xi$ and
employed the identity $\frac{du}{da}(\xi(a,t),t)=\frac{du_{0}}{da}(a)$ which
follows from (42) together with the chain rule. Next, defining
$\eta(a,t):=\xi(a,t)-a=tu_{0}(a)$, we have

$\displaystyle\widehat{u}_{k}(t)$ $\displaystyle=\frac{1}{2\pi
ikt}\int_{0}^{1}{\partial\eta\over\partial a}(a,t)e^{-2\pi
ik\left[a+\eta(a,t)\right]}\,da$ $\displaystyle=\frac{1}{2\pi
ikt}\int_{0}^{1}\left(\frac{-1}{2\pi ik}\right){\partial\over\partial
a}\left[e^{-2\pi ik\eta(a,t)}\right]e^{-2\pi ika}\,da$
$\displaystyle=\frac{-1}{2\pi ikt}\int_{0}^{1}e^{-2\pi
ik\left[a+tu_{0}(a)\right]}\,da,\quad k\neq 0,\quad 0<t<t^{*},$ (43)

where we again performed integration by parts, this time with respect to $a$.

For a given initial condition $u_{0}$, relation (43) allows for a
straightforward evaluation of the Fourier coefficients of the solution of (4)
by performing an integral. For example, using the unimodal initial condition
(9), we obtain

$\displaystyle\widehat{u}_{k}(t)$ $\displaystyle=\frac{-1}{2\pi
ikt}\int_{0}^{1}e^{-2\pi ik\left[a+tA\sin(2\pi a)\right]}\,da,$
$\displaystyle=\frac{-1}{2\pi ikt}J_{k}(-kAt),\quad k\neq 0,\quad 0<t<t^{*},$
(44)

where we have used the integral representation of the Bessel functions of the
first kind
$J_{k}(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}e^{i[x\sin(\tau)-n\tau]}\,d\tau$
(DLMF, 2020). An expression for the Fourier coefficients of the solution
corresponding to the extreme initial condition (11) can be obtained in a
similar manner.

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