# Rough neutron fields and nuclear reactor noise

Eric Dumonteil  Institut de Recherche sur les Lois Fondamentales de l’Univers
CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France

###### Abstract

Nuclear reactor cores achieve sustained fission chain reactions through the
so-called ’critical state’ -a subtle equilibrium between their material
properties and their geometries. Observed at macroscopic scales during
operations, the resulting stationary neutron field is tainted by a noise term,
that hinders various fluctuations occurring at smaller scales. These
fluctuations are either of a stochastic nature (whenever the core is operated
at low power) or related to various perturbations and vibrations within the
core, even operated in its power regime. For reasons that are only partially
understood using linear noise theory, incidental events have been reported,
characterized by an increase of the power noise. Such events of power noise
growth, sometimes up to seemingly unbounded levels, have already led in the
past to volontary scramming of reactors.

In this paper, we will extend the findings of [1] (where a statistical field
theory of critical processes was employed to model stochastic neutron noise)
by incorporating the effects of power noise. We will show that the evolution
of the neutron field in a reactor is intimately connected to the dynamic of
surface growths given by the Kardar-Parisi-Zhang equation. Recent numerical
results emerging from renormalization group approaches will be used to
calculate a threshold in the amplitude of the reactor noise above which the
core could enter a new criticality state, and to estimate the critical
exponents characterizing this phase transition to rough neutron fields. The
phenomenology of this roughening transition will be correlated and compared to
data of misunderstood reactor noise levels and reactor instabilities, and will
be shown to provide both qualitative and quantitative insights into this long-
standing issue of reactor physics.

††preprint: APS/123-QED

##  I  Introduction

Neutron transport theory aims at studying the behavior of neutron gaz evolving
in multiplicative media. It is key to reactor physics, as it constitutes the
foundation upon which simulation codes are developed and used to support all
industrial activities revolving around reactors, such as radiation shielding
and criticality safety studies, fuel cycle management, and reactor operations,
just to name a few. It supposes that the average behavior of the dilute
neutron gas is faithfully described by a linear Boltzmann equation, clearing
the path to the development and use of various approximation schemes. The most
widespread of these schemes is the so-called diffusion theory: neutrons are
considered at thermal equilibrium so that the spatiotemporal characteristics
of their population -and the associated core power distribution- are properly
reproduced by a simple heat equation 111sometimes the two-groups diffusion
theory is also used, that couples two diffusion equations in two energy
domains as to grasp spectral phenomena. With proper adjustments of the
coefficients of diffusion equations, the numerical schemes are so accurate
that they are implemented in the automatic protection systems of many
commercial reactors for core operations and scramming procedures.

However, power fluctuations can affect the overall statistics of neutrons
within the reactor. At start-up, for example, the so-called zero-power state
of the core can be subjected to stochastic fluctuations, arising from the
random path of neutron flights and from the stochastic nature of the fission
process, producing variable number of neutrons each time heavy nuclei are
fissioned. The fission process in fact equips the random walk with a branching
structure [3, 4, 5]. Building on a number of theoretical studies [6, 7, 8],
recent experimental results [9] have provided a concrete example of such low-
power phenomena developing in the core: the emergence of spatio-temporal
correlations, dubbed ’neutron clustering’, offer a concrete example of an in
situ mechanism ultimately leading to deviations from the expected neutron
distribution.

At high reactor power, the need to cool the core imposes high coolant flow
rates, ultimately leading to various vibrations of its components such as the
internals, the fuel rods, or the assemblies. The water serving as a neutron
moderator, these vibrations therefore translate in a random noise on local
reactivities (i.e. related to the net production of neutrons) because induced
fissions are extremely sensitive to the incoming neutron energies. The
observable consequence is that the neutron flux, measured by in-core or ex-
core detectors, also develops fluctuations around its mean value. Being used
as a diagnostic tools to characterize and eventually locate reactor core
perturbations [10], neutron noise is also perceived as a safety concern.
Indeed, on the one hand, reactor automatic protection systems calculate in
real-time a derivative of the flux to appreciate its variations. This enables
detection, for instance, of accidental power transients: blinding such a
protection system may jeopardize the safety strategy. On the other hand, more
worryingly, high reactor noise levels and instabilities sometimes appear:
power fluctuations and oscillations progressively or abruptly might increase
to require, ultimately, reactor scramming procedures [11]. Such events have
attracted a great deal of attention in the last 50 years to the present day
[12] and have triggered theoretical [13] and numerical [14] investigations.
Although known to be associated with fluctuations in coolant temperature and
vibrations of reactor components, these events and the parameters that
influence them are still only partially understood [15].

Although formal approaches to describing fluctuations are generally provided
in the context of linear noise theory, recent developments in the statistical
field theory of reaction-diffusion processes [16] paved the way to describing
the stochastic fluctuations of the branching random walk using critical
phenomena: in particular, in the presence of stabilizing feedbacks (such as
the Doppler effect 222when the temperature in the fuel increases, the Doppler
broadening of neutron cross-sections tends to increase capture resonances and
hence constitutes one of the main feedback mechanism), these fluctuations were
proved to be related to a time-directed percolation process and to shift the
effective random walk of neutrons from a diffusive to a super-diffusive
behavior [1]. While such counter-reactions do not occur at zero power, this
modeling described, however, very well the behavior of simulated population of
neutrons: implementing population control mechanisms [18, 8], Monte Carlo
simulations of reactors (even when deployed on peta-scale architectures) imply
neutron statistics only met during the first stages of reactor startup.

This paper now addresses the modeling of real neutron noise occurring in the
power regime of reactors, using a statistical field theory of critical
processes. In Section II we will describe the statistics of neutrons in
fissile media using a stochastic heat equation with multiplicative noise.
Relating this equation to the celebrated Kardar-Parisi-Zhang (KPZ) equation,
we will show that the reactor might be subjected to a roughening phase
transition, where the neutron field becomes rough and is described by the
exponents of the KPZ universality class, even when reactor feedbacks are
considered. Using results from both perturbative and non-perturbative
numerical approaches to estimate both the phase transition threshold and the
fixed point value, Section III will first briefly present the phenomenology of
reactor instabilities and then select two reactor instabilities events so as
to correlate the roughening transition model with the reactor experimental
data.

##  II  Nuclear reactor model and the KPZ equation

At start-up, the power in nuclear reactors can be extremely small -around 1
kW-, and the associated neutron statistics is therefore close to $~{}10^{6}$
cm-3, lending itself to stochastic fluctuations. The power regime is instead
associated to high neutron densities: indeed, when the reactor reaches its
nominal power, the neutron densities are $~{}10^{13}$ cm-3 and stochastic
fluctuations have been totally washed out. At this point, fission reactions
release gigawatts of thermal power : such a heat source requires to be
transferred from the primary to the secondary loop, through increased flow
rates of the coolant primary pumps. The subsequent temperature fluctuations
and induced vibrations are therefore intrinsically associated to high neutron
densities and high fuel temperatures, and feedback effects might need to be
taken into account for a proper description of the neutron gas. In this
section, we will first build a simple reactor model in the intermediate
temperature regime -where thermal feedback mechanisms can safely be
neglected-, and then in the high temperature regime, adding a quadratic field
to model the Doppler temperature effect.

###  II.1 The nuclear reactor model

Industrial schemes used to operate light-water reactors often rely on the two-
group diffusion approximation: thermal neutrons induce fissions and outgoing
neutrons belong to a fast energy group prior to their slowing down to the
thermal group. Both groups are simple diffusion equations with proper
coefficients. Even a one-group diffusion equation allows to grasp the main
spatial features of the neutron field. The temporal behavior is also
adequately described by a double time scale: most neutrons produced during
fissions are called prompt neutrons and are emitted on a short ($10^{-14}$ s)
time scale, while a small fraction is produced following electroweak
interactions within fission fragments on rather long time scales (from 1 s to
1 min). However, the study of stationary spatial distribution can safely be
based on only one effective time scale, denoted $\Lambda$. Finally, as is
usually done, the resulting one-group diffusion equation is supplemented by a
stochastic term that models the random temporal and spatial fluctuations
within the core. The resulting equatio can be written as

$\frac{\partial}{\partial t}n(\vec{x},t)=D\Delta
n(\vec{x},t)+\frac{\rho}{\Lambda}n(\vec{x},t)+\sigma\xi(\vec{x},t)n(\vec{x},t),$
(1)

where $n(\vec{x},t)$ denotes the neutron density field, $D$ is the diffusion
coefficient, $\rho$ is the dimensionless reactivity (a net balance between
neutron productions by fissions and neutron disappearance by captures), and
where $\sigma\xi(\vec{x},t)$ is an uncorrelated white noise of strength
$\sigma^{2}$ given by

$\langle\xi(\vec{x},t)\xi(\vec{x}^{\prime},t^{\prime})\rangle=\delta^{3}(\vec{x}-\vec{x}^{\prime})\delta(t-t^{\prime}).$
(2)

The multiplicative white noise term reflects the various periodic fluctuations
that occur in the power regime (see, for instance, Ref. [19] among many
others), occurring on different spatial scales: for instance, in pressurized
water reactors (PWR) the smallest perturbation size $a$ is such that $a\sim 1$
cm, and is associated to temperature fluctuations in the coolant flow [20],
while mechanical vibrations of the core components occur at intermediate
scales and can range up to the whole reactor typical size $L$ -between two and
three orders of magnitude above. In fact, both scales introduce natural
regularization of the white noise: $a$ in the UV sector and $L$ in the IR
sector. Noticeably, stochastic fluctuations occurring on microscopic scales
cannot be adequately grasped by such a multiplicative noise model but can
instead be described by additive noise [1].

###  II.2 The roughening transition model

Reactor power noise and dynamic of surface growths Known in statistical
physics literature as the Stochastic Heat Equation (SHE) with multiplicative
noise, Eq. 1 can be reformulated using the following change of variables

$\displaystyle\begin{split}&\nu=D,\\\ &\eta=\frac{2\nu}{\lambda}\sigma,\\\
&c=2D\frac{\rho/\Lambda}{\lambda},\\\ \end{split}$ (3)

where a free positive parameter $\lambda$ was introduced. Then, associating
the neutron density field $n(\vec{x},t)$ to a height field $h(\vec{x},t)$
through a Cole-Hopf transformation
$n(\vec{x},t)=\exp[\frac{\lambda}{2\nu}h(\vec{x},t)]$ allows to map our
stochastic heat equation to a $\lambda$-parameterized family of Kardar-Parisi-
Zhang (KPZ) equations supplemented by an additive term $c$

$\frac{\partial}{\partial t}h(\vec{x},t)=\nu\Delta
h+\frac{\lambda}{2}\left(\vec{\nabla}h\right)^{2}+\eta\,\xi(\vec{x},t)+c.$ (4)

Whenever the reactor is stationary, $\rho$ is close to 0 and the constant term
$c$ also becomes small. Conversely, the KPZ equation is mapped onto the SHE
through

$\displaystyle\begin{split}&\rho/\Lambda=\frac{\lambda c}{2\nu},\\\ &D=\nu,\\\
&\sigma=\frac{\lambda}{2\nu}\eta.\\\ \end{split}$ (5)

This connexion between the power noise equation of reactor physics and the KPZ
equation, which relates the power fluctuations of a reactor to the dynamics of
surface growths, went largely unnoticed in the reactor physics community. To
better understand this connection, and since the choice of a particular Cole-
Hopf transform fixes the relative amount of stochasticity and of nonlinearity
in the KPZ equation, it is possible to choose the natural gauge
$\lambda/\nu=2$ which leads to the equation

$\frac{\partial}{\partial t}h(\vec{x},t)=D\Delta
h+D\left(\vec{\nabla}h\right)^{2}+\sigma\,\xi(\vec{x},t)+\rho/\Lambda.$ (6)

This particular form underlines a simple alphabet connecting the neutron
density field to surface growth. In fact, the speed of surface growth is given
by the reactivity state of the reactor: when the reactor is supercritical with
constant positive reactivity $\rho$, the surface grows at a speed
$\rho/\Lambda$. Also, while the neutron diffusion coefficient $D$ drives both
the smoothering by evaporation and the lateral growth of the surface in the
same proportions (in this particular gauge), the multiplicative noise of the
reactor in the power regime becomes an additive noise in the KPZ equation. In
a spirit of generality, in the following we will not fix any particular gauge
and use Eq. 3 and Eq. 5 to connect the KPZ equation and the SHE equation.

KPZ renormalization in dimension $d=3$ Surprisingly, when looking at
characterizing the KPZ universality class, it is customary to use the inverse
Cole-Hopf transformation so as to get ride of $\lambda$ nonlinearity in the
KPZ equation, so that the multiplicative noise equations describing reactor
fluctuations are used, in a mirror approach, as a mean to solve surface growth
equations. Merely looking for briefly summarizing a large literature on the
subject, the study of fluctuations in statistical field theory can be
decomposed in three steps: the white noise is first regularized introducing a
momentum scale $\kappa$, then a field theory à la Janssen-De Dominicis-Martin-
Siggia-Rose formalism (JDMSR) [21, 22, 23, 24] is employed to relate the
calculation of observables to an effective $\kappa$-regularized action
$\mathcal{S}_{\kappa}$, and finally a renormalization group (RG) analysis is
led to define hypothetical fixed points in the RG flow procedure and to
calculate the critical exponents of these fixed points. In our case, the white
noise regularization can be done following Ref. [25], by taking the UV limit
$\kappa\to 0$ of a spatially colored noise

$\langle\xi(\vec{x},t)\xi(\vec{x}^{\prime},t^{\prime})\rangle_{\kappa}=\delta(t-t^{\prime})V_{\kappa}(\vec{x}-\vec{x}^{\prime}),$
(7)

with
$V_{\kappa}(\vec{r}=\vec{x}-\vec{x}^{\prime})=(\kappa/\sqrt{\pi})^{3}e^{-(\kappa\vec{r})^{2}}$,
such that $V_{\kappa}(\vec{r})\xrightarrow[\kappa\to\infty]{}\delta(\vec{r})$.
Interestingly enough, this noise regularization is extremely close to physical
noise model build to reproduce flow temperature fluctuations in reactor
physics [20]. Observables $\langle\mathcal{O}[n]\rangle$ can then be expressed
as functional integral over noise configurations through

$\langle\mathcal{O}(n)\rangle_{\kappa}=\int\mathcal{D}\xi(t)\mathcal{O}[n]P[\xi],$
(8)

where the normalization constant has been intentionally omitted, and which,
still following the JDMSR approach, is shown to be equal to

$\langle\mathcal{O}[n]\rangle_{\kappa}=\int\mathcal{D}n\mathcal{D}\bar{n}\,\mathcal{O}[n]\exp^{-\mathcal{S}_{\kappa}[n,\bar{n}]}.$
(9)

In this expression, the auxiliary field $\bar{n}(\vec{x},t)$ has been
introduced and the effective action $\mathcal{S}_{\kappa}$ is given by

$\mathcal{S}_{\kappa}[n,\bar{n}]=\int d\tau
d\vec{x}\,\bar{n}(\vec{x},\tau)\left[\partial_{\tau}-\Delta\right]n(\vec{x},\tau)-\frac{g}{4}\int
d\tau
d\vec{x}d\vec{x}^{\prime}\bar{n}(\vec{x},\tau)n(\vec{x},\tau)V_{\kappa}(\vec{x}-\vec{x}^{\prime})\bar{n}(\vec{x}^{\prime},\tau)n(\vec{x}^{\prime},\tau)$
(10)

with a time re-scaling $\tau=2Dt$, and using the notation
$g=\frac{\sigma^{2}}{D}$ for the bare coupling constant. Taking the UV limit
and neglecting the power of the fields higher than quadratic (considered to be
irrelevant under the renormalization group [25]), finally leads to

$\mathcal{S}_{\kappa}[n,\bar{n}]\xrightarrow[\kappa\to\infty]{}\int d\tau
d\vec{x}\,\Bigg{[}\bar{n}(\vec{x},\tau)\left[\partial_{\tau}-\Delta\right]n(\vec{x},\tau)-\frac{g}{4}(n\bar{n})^{2}\Bigg{]}.$
(11)

Roughening phase transition Introducing the dimensionless running coupling
$\hat{g}=\frac{\sigma^{2}}{D}\kappa$, the renormalization of $\hat{g}$ is
achieved by requiring this coupling to be $\kappa$ (i.e., cutoff) independent,
which takes the form of fixed points in the dimension $d=3$ Calan-Symanzik
equation

$\frac{\partial\hat{g}}{\partial\kappa}=\hat{g}-\frac{\hat{g}^{2}}{(4\pi)^{3/2}\Gamma(3/2)}.$
(12)

This equation possesses two fixed points. The fixed IR point $\hat{g}=0$
corresponds to $\kappa\to 0$, fixed in practice by the IR cutoff
$\kappa^{-1}=L$. The fixed UV point $\hat{g}_{c}=(4\pi)^{3/2}\Gamma(3/2)$
corresponds to $\kappa\to\infty$, obtained in practice for $\kappa^{-1}=a$. In
coherence with References [26, 27, 28], when considering numerical
applications, we will set the minimum spatial scale $a$ close to unity [28],
since $a=\zeta/\pi$, where $\zeta$ is the cross-over length [28] below which
no fluctuations are included [29]. As discussed in Subsection II.1, in the
power regime, the minimal spatial scale of fluctuations is considered to
correspond to temperature perturbations with $\zeta\simeq 1$ cm [20]. Using
these notation, the bare noise coupling $\sigma_{rt}$ corresponding to the
roughening transition threshold can be expressed in terms of the bare
diffusion coefficient of Eq. 1 following

$\sigma_{rt}^{2}=4\sqrt{\pi}D\,\zeta.$ (13)

Renormalized parameters and critical exponents Below this neutron noise
threshold, the behavior of the reactor should be given trustably by the mean
field. Above this threshold, the RG flow pushes the coupling $\hat{g}$ and the
coefficients of Eq. 1 toward renormalized values associated with the so-called
strong noise regime. Therefore, the critical state within the reactor shifts
to the KPZ universality class. The renormalized value $\hat{g}^{*}$ of the UV
fixed point in dimension $d=3$ is shown to be close to $\hat{g}^{*}\simeq
5/2\,\hat{g}_{c}$ [29], which leads to a (spatial) scale dependent expression
of the renormalized parameters $\sigma_{\kappa}$ and $D_{\kappa}$ given by

$\sigma_{\kappa}^{2}=10\pi^{3/2}\frac{D_{\kappa}}{\kappa}.$ (14)

When $\kappa^{-1}$ increases from the microscopic scale $a$ to the macroscopic
scale $L$, the parameters acquire effective meaning: $\sigma_{\kappa}$ and
$D_{\kappa}$ are, respectively, the observed/effective noise and diffusion
coefficient at the spatial scale $\kappa^{-1}$. $\sigma_{\kappa}$ and
$D_{\kappa}$ can be expressed in the form of power laws characterized by
critical exponents

$\displaystyle D_{\kappa}\sim\kappa^{-(2-z)},$ (15)
$\displaystyle\sigma_{\kappa}^{2}\sim\kappa^{-(3\chi+1)},$ (16)

with $z+\chi=2$. Numerically, different estimates of critical exponents are
found in the literature, such as $\chi=0.33$ [30], $\chi=0.3$ [31] or
$\chi=0.17$ [29] and in the following we will retain $\chi\simeq 0.3$ thus
fixing $z\simeq 1.7$. From a reactor physics point of view, a scaling
following $(\Delta x)^{z}\propto\Delta t$ with $z$ sensitively less than 2
tends to shift the random walk of neutrons from a diffusive behavior to a
semi-ballistic one and might drastically change the spatial distribution of
neutrons within the core, thus unambiguously signing its strong noise
criticality regime.

Rough neutron fields Also, the exponent $\chi$ describes the roughness $W_{h}$
of the interface in the KPZ equation, which scales as $W_{h}\sim L^{\chi}$,
for sufficiently large $L$ and for large times. Although no equivalent
exponent emerges from the study of the SHE equation with multiplicative noise,
it is possible to estimate the ’roughness’ of the neutron field under mild
assumptions. Indeed, most nuclear reactors are considered decoupled in the
sense that their typical size $L$ (few meters) is very large compared to the
neutron’s typical traveled path length before capture which is a few
centimeters 333The neutron’s path length in a diffusive medium with diffusion
with a (macroscopic) absorption cross section $\Sigma_{a}$ is given by
$6\frac{D}{\Sigma_{a}}$ , ensuring the validity of the $L^{\chi}$ scaling of
the roughness of the KPZ interface. In the parametric sector where
$\lambda<<\nu/h$ (which can be set by tuning $\lambda$ since the SHE is
connected to a $\lambda$-parameterized family of KPZ equations, and assuming
that h is bounded), the relation between the neutron field $n(\vec{x},t)$ and
$h(\vec{x},t)$ becomes linear, thus ensuring a linearity (at first order)
between the roughness of the neutron field $W_{n}$ and the roughness of the
KPZ interface through
$W_{n}=(\frac{\lambda}{2\nu})^{2}W_{h}=(\frac{\lambda}{2\nu})^{2}L^{\chi}$.
The noise term $\eta$ is, however, enhanced by this re-parameterization (we
have chosen a small non-linearity / high noise gauge), which acts as a
prefactor but leaves the scaling law valid: at first order, the roughness of
the neutron field should also scale as $L^{\chi}$. In the general case,
independently of any particular approximation, and without calculating a
precise scaling law, the neutron density field is by definition an exponential
function of a potentially rough height field. This indicates that decoupled
nuclear reactors develop in the power regime the equivalent of neutron
clustering [9] in the stochastic regime, and that the neutron field of the SHE
also present a roughness if the roughening transition of the KPZ equation is
attained.

###  II.3 SHE equation with feedbacks

Multiplicative noise with a wall When the reactor power is increased,
stabilizing feedbacks kick into action. The most prominent one is the Doppler
broadening of the cross-sections: in thermal reactor, where neutrons are
slowed down by elastic collisions in the moderator (water plus additional
nuclei such as boron), the broadening of these cross-sections under
temperature increase enlarges the heavy nuclei resonances and constitutes a
trap for neutrons that are captured by the nuclei. Unable to participate in
the chain reaction, the reactivity $\rho$ is sensitively diminished. Most
models capture this effect using a term proportional to $-n^{2}$ [33, 34, 35].
In [1], a stochastic noise model was built taking feedbacks into account.
Paradoxically, this zero-power noise model of nuclear reactors adequately
describes the maximal power attained by Monte Carlo simulations which, as of
today, can simulate -in terms of neutrons- only a few Watts on modern parallel
computing architectures [9]. This model was therefore convenient for
understanding simulation artifacts. Now, the same approach can be applied to
the power noise model of a real reactor, given by Eq. 1 and supplemented by
the $-\alpha n^{2}$ term. This model -of a stochastic heat equation with
multiplicative noise plus powers of the field- has already been extensively
studied in the literature, and, in the following, the main results
demonstrated in Ref. [30, 36] are recalled. Still through the Cole-Hopf
transform, the stochastic equation with multiplicative noise

$\frac{\partial}{\partial t}n(\vec{x},t)=D\Delta
n(\vec{x},t)+\sigma\xi(\vec{x},t)N(\vec{x},t)+\frac{\rho}{\Lambda}n(\vec{x},t)-\alpha
n(\vec{x},t)^{2}$ (17)

becomes the KPZ equation with a wall

$\frac{\partial}{\partial t}h(\vec{x},t)=D\Delta
h+D\left(\vec{\nabla}h\right)^{2}+\sigma\,\xi(\vec{x},t)+\frac{\rho}{\Lambda}-\alpha
e^{h}.$ (18)

This wall acts as a pinning structure constraining the evolution of the
neutron density field, and the criticality of the reactor would correspond to
a pinning/depinning phase transition.

Common features with the KPZ equation A renormalization group analysis of the
multiplicative noise equation (with a wall) and the simple KPZ equation (i.e.
without a wall) however indicates that they share several common points [37]:
both have two fixed points separating a low noise Gaussian phase and a strong
noise phase. Also, the transition point to the strong noise regime is still
given by the same value of the dimensionless running coupling constant
$\hat{g}_{c}$ calculated in Eq 13 and, in the case of the strong noise phase,
the critical exponent $z$ that governs the spatio-temporal scaling is also
unchanged w.r.t. the KPZ equation, indicating an effective superdiffusive
behavior of the neutron random walk. Consequently, $\chi$ should also be
unaffected, as well as the $\kappa$-scaling of the effective parameters
$\sigma_{\kappa}$ and $D_{\kappa}$. But supplementary critical exponents are
associated with the pinning/depinning phase transition (via the presence of
the wall), and the reader is refered to [30, 36] for more information.

##  III  Reactor noise and instabilities

We have seen that decoupled reactors might reach a new criticality regime
dictated by the universality classes of the KPZ equation and the
multiplicative noise SHE equation, for high noises given by equation Eq. 13.
In this section, we will give more quantitative insight on the previous
findings and we will correlate them with two particular events that exhibited
strong noise patterns and power instabilities.

###  III.1 Phenomenology

More than 430 nuclear reactors are currently in operation worldwide, the vast
majority of them being thermal water-moderated reactors: neutrons are slowed
down through successive scattering on protons and induce fissions on heavy
nuclei once thermalized. Among these reactors, the two main technologies are
pressurized water reactors (PWR) and boiling water reactors (BWR). While both
technologies benefit from a long and extensive experience feedback -dating
back to the 1970s- neutron flux fluctuations measured by in-core and ex-core
instrumentation have attracted much attention for two opposite reasons. On the
one hand, neutron noise is a powerful diagnostic tool for probing a reactor
during operation and even to characterize and sometimes locate perturbations
in the core. On the other hand, large noise baselines in PWR reactors and
instability events with sometimes sudden increases in noise in BWRs have
themselves been a recurrent cause for concern. Originally termed reactor
instabilities, the phenomenology of these largely unexplained sudden neutron
noise patterns has soon caught the attention of reactor physicists (see, for
instance, the OECD international benchmark on the Swedish BWR Ringhals 1 core
[38]). Since then, several such events have occurred in BWRs, noticeably in
Sweden and Mexico [39]. The attention devoted to neutron noise has recently
been renewed, taking the form of international collaborations [12], as recent
measurements in various PWR reactors have identified drifts and contained
increases of their ’neutron noise’ during operations. Furthermore, many of
these reactors exhibit a high baseline of noise during normal operation, such
as in pre-Konvoi and Konvoi PWRs in Germany [15], Switzerland [40] or Spain
[41, 20], and even more recently in the first operating EPRs [42]. Although
normal noise levels depend on reactors (they are close to $0.2\,\%$ 444in
terms of standard deviation of the core power for Westingouse PWRs, $2\,\%$
for Konvoi and $5\,\%$ for preKonvoi [44]), both progressive increase
exceeding $10\,\%$ [45] and sudden unstable events developing on $\sim 1$
minute time scale have been recorded, particularly on BWRs [39, 46]. In these
last cases, fluctuations would swiftly exceed these values, leading to
immediate reactor scramming [11]. Depending on local regulations and
industrial procedures, some reactors now have a reduced nominal power to avoid
the development of such instabilities, with a threshold usually set between 8
and 10$\%$ [44]. In addition, as said in the previous section, many reactor
automatic protection systems are based on the calculation of the time
derivative of the core power to eventually prevent unwanted power excursion.
This strategy can be affected by high neutron noise, thus translating a
commercial matter into a nuclear safety concern.

The state of the art in the understanding of the neutron noise relies on the
analysis of the characteristic frequencies at which fluctuations occur. The
typical frequency spectrum is different on BWRs and PWRs: while peaked at
frequencies below 1 Hz (often close to 1/2 Hz), the spectrum is rather flat
for BWRs, which develop stronger and more frequent noise events-, and the
white noise hypothesis is hence more realistic. In contrast, the spectrum of
PWRs is closer to a decaying exponential (still in the frequency domain) [47].
For both reactors, fluctuations can be either out-of-phase (or regional)
whenever they have opposite sign in different regions of the core (e.g.
upper/lower part of the core) or in-phase (or global) when they are
synchronized throughout the core. This militates in favor of a C- or S-shaped
vibration of the assemblies [48], connecting the neutron noise to mechanical
vibrations of the core components between 1 Hz and 50 Hz (of fuel rods and
assemblies and thermal shield below 25 Hz, and of the reactor vessel and
internals below 50 Hz). Below 1Hz, they are believed to be connected to
temperature fluctuations in the primary loop. The two mechanisms have
different control pathways on the neutron population: in the case of coolant
temperature fluctuations, both density and Doppler effect affect the
probability to slow down neutrons and hence their capacity to reproduce
through fissions, while in the case of mechanical vibrations, it is the inter-
assembly coolant thickness that is modified.

In the next two subsections, a neutron noise model relying on the strong noise
hypothesis will be built, and correlated on two sets of experimental data -one
corresponding to a PWR and the other to a BWR-, so as to grasp at a minimal
cost the wide variety of neutron noise events.

###  III.2 Almaraz Trillo PWR noise data

Being related to commercial issues, data released from power plants in general
and concerning neutron noise in particular are relatively scarce. Among
published data available, experimental measurements associated with increased
noise levels from the 1000 MW Almaraz Trillo PWR reported in 2011 [49, 45]
present an interesting scheme, as can be observed in Figure 1, giving the time
evolution of the root mean square of the neutron noise measured by ex-core
detectors (left plot), as proposed by these references, and a projection of
the noise amplitudes in a histogram aiming at presenting both the average
noise and its typical deviations (right plot). In fact, throughout the cycle,
this noise is progressively amplified by a factor of 2 between cycles 8 and
25, with intracycle variations ranging at their extremum from 4 to 14$\%$. In
this section, this data sample will be used for comparison with the roughening
phase transition model derived above, looking first at the overall amplitude
of the noise and then at the variations of the noise.

Figure 1: Almaraz-Trillo PWR data from [49, 45]. Left plot: RMS of the neutron
noise amplitude (in $\%$) as a function of time (in arbitrary units). Local
minima correspond to BOC measurements and local maxima to EOC measurements.
Right plot: noise amplitude histogram of all measurements. The estimated
average noise is $8\%$.

At cycle 20, no more general trend seems to affect cycle-to-cycle
fluctuations, and the noise response of the reactor seems stationary. A
histogram that reports the noise amplitude from that cycle is shown in Figure
1. Despite large fluctuations in the noise itself, which can be attributed to
intracycle effects 555So as to compensate the reactivity loss due to fuel
burnup effects, boron is used at beginning of cycle in an amount adjusted in
such a way that the reactivity through the cycle stays constant. However, the
average absorption cross sections of the moderator decrease linearly with time
during the cycle, in the first order, leading to the observed linear increase
in noise induced by the cross sections, the average noise value extracted from
this graph is estimated to be close to 10$\%$.

Noise amplitude from the roughening transition Making the assumption that
reactor instabilities are related to the roughening transition, the noise
threshold at which the transition occurs can also be numerically evaluated, in
a context where spatial patterns of noise, while presenting involved
structures, as analyzed, for example, in the linear noise theory [48]\- are
neglected. The noise given by Eq. 13 is associated with Eq. 1 which, once
solved, gives access to the neutron density field $n(\vec{x},t)$. Closely
following the approach detailed in [13], and since we are only interested in
temporal fluctuation of a spatially averaged noise, Eq. 1 can be rewritten
factorizing the purely time-dependent amplitude $P(t)$ out of a remaining
spatial part of the neutron density that is considered constant $n(x,t)\approx
n_{0}$. Eq. 1 hence takes the simplified form

$\frac{\partial}{\partial
t}P(t)=\frac{\rho}{\Lambda}P(t)+\sigma\xi(\vec{x},t)P(t).$ (19)

In addition, the spatiotemporal noise term
$\eta(\vec{x},t)=\sigma\xi(\vec{x},t)$ and its intensity $\sigma$ were
implicitly related through the definition

$\langle\eta(\vec{x},t)\eta(\vec{x}^{\prime},t^{\prime})\rangle=\sigma^{2}\delta^{3}(\vec{x}-\vec{x}^{\prime})\delta(t-t^{\prime}),$
(20)

which spatial dependence can readily be integrated over the whole reactor of
volume $V=\int dx^{3}$ still assuming that we are only interested in the
temporal component of the noise. Therefore, the resulting temporal noise takes
the form

$\langle\eta(t)\eta(t^{\prime})\rangle\int
dx^{3}=\sigma^{2}\delta(t-t^{\prime})\int dx^{3}\delta^{3}(\vec{x}),$ (21)
$\langle\eta(t)\eta(t^{\prime})\rangle=\frac{\sigma^{2}}{V}\delta(t-t^{\prime}),$
(22)

hence recasting equation Eq. 19 in a classical white noise equation

$\frac{\partial}{\partial
t}P(t)=\frac{\rho}{\Lambda}P(t)+\frac{\delta\rho}{\Lambda}P(t)$ (23)

(where we have used the notation
$\delta\rho/\Lambda=\frac{\sigma}{\sqrt{V}}\xi(t)$ for the volume-rescaled
noise coupling). To derive an equation for the noise from this equation as
suggested by Ref. [13], $P(t)$ is also split in a (time) fluctuating component
a constant term $P_{0}$, second order terms are neglected, and the system is
considered to be close to criticality so that one obtains the equation for the
fluctuations

$\frac{\partial}{\partial t}\delta
P(t)=\frac{\delta\rho}{\Lambda}P_{0}=\frac{\sigma}{\sqrt{V}}\xi(t)P_{0},$ (24)

or equivalently in the frequency domain, after a Fourier transform

$\Big{|}\frac{\delta P(w)}{P_{0}}\Big{|}=\frac{\sigma^{2}}{Vw}.$ (25)

This equation connects the fluctuations in the amplitude factor of the neutron
density, i.e., the neutron noise $\delta P/P_{0}$, to the reactivity noise
$\delta\rho$, which was formulated as a function of the noise coupling term
$\sigma$. Following Eq. 13, the neutron noise at which the roughening
transition occurs, noted $\Big{|}\delta P/P_{0}\Big{|}_{rt}$, is given by

$\Big{|}\frac{\delta
P}{P_{0}}\Big{|}_{rt}=\frac{\sigma_{rt}^{2}}{Vw}=4\sqrt{\pi}\frac{D\zeta}{Vw}.$
(26)

A numerical estimation of this term can be worked out by setting the
observation frequency $w$ to $1$Hz - close to its maximum- and using a
cylindrical reactor volume of typical radius $r=150~{}cm$ and $H=400~{}cm$ to
calculate $V$. The bare diffusion coefficient $D$ is set to
$10^{5}~{}cm^{2}.s^{-1}$ as suggested by the thermal energy PWR data of [20].
Also in accordance with this reference, and as stated in the previous section,
$\zeta\simeq 1$ cm is the minimal spatial scale below which no power
fluctuations occur and can be associated with a temperature-induced change of
the fuel absorption cross section. For these parameters, the RMS of the noise
amplitude is $\sqrt{\Big{|}\delta P/P_{0}\Big{|}_{rt}}\simeq 15\%$. As can be
observed on the right plot of Figure 1, the experimental data present a
bimodal distribution, the noise amplitude progressively drifting within the
cycle from a local minimum value to a local maximum value. The average value
(over all available cycles) is, as discussed in the previous paragraph, close
to $10\%$. Given the level of detail of the roughening transition model, its
value can be considered to be very close to these experimental data. Indeed,
any increase/decrease in the minimal correlation length $\zeta$ linearly
affects the noise threshold estimation. But this length differs sensitively
from few mm to few cm depending on scarce experimental measurements and widely
varying theoretical modeling [13, 20]. Beyond this agreement, it should be
noted that Eq. 26 sets the dependence of the noise at which the transition
occurs on other physical parameters: the bigger the reactor volume $V$ -or the
smaller the diffusion coefficient $D$-, and the smaller the noise (i.e. the
vibration level) at which the reactor meets the rough regime. This prediction
could also lend itself to experimental confirmation.

In this subsection, we considered a -close to- stationary reactor with a long-
term high noise level and showed that its noise amplitude was compatible with
the noise associated with the roughening transition. To better assess a
potential switch from a Gaussian behavior to a rough behavior in a nuclear
reactor, the ideal situation would be to consider a hypothetical event where
the reactor switches, on small time scales, from standard operating conditions
(assumed to belong to the Gaussian fixed point) to a degraded state of the
reactor with a high neutron level (that could be correlated to the strong
noise regime) and to try to associate the shift in the critical parameters to
the universal exponents of the phase transition. Such situations of abrupt
neutron noise increase are more frequently met with BWRs and therefore, in the
next subsection, the Laguna Verde BWR incident is discussed.

###  III.3 The Laguna Verde emergency incident

BWRs Noisy baselines and instability events occur more frequently in BWRs than
in PWRs and also seem to develop on smaller time scales or equivalently with
heavier tails in the frequency spectra. By comparison to PWR, BWR have a
slightly harder neutron spectrum: a proper modeling of their noise would
fairly require the so-called two-group approximation, where two coupled noisy
diffusion equations should be used for each energy group. But their
particularity is believed to be associated with the two-phase coolant flow:
liquid water is injected at the bottom of the reactor and subject to a liquid-
vapor phase transition during its travel through the core. The resulting
scale-free structure of bubble sizes during the boiling crisis has recently
been experimentally put into evidence and shown to belong to the percolation
universality class [51]. The minimal size of the bubbles is considered to be
$\approx 0.01$ cm [52, 53] and defines the minimal correlation length $\zeta$,
which, recalling Eq. 26, lowers the threshold in the noise amplitude
associated with the roughening transition from an order of magnitude compared
to PWR, thus eventually occurring at $\sqrt{\Big{|}\delta
P/P_{0}\Big{|}_{rt}}\simeq 1.5\%$. This is relatively consistent with
empirical observations that relate more frequent and abrupt noisy events in
BWR than in PWR.

The event To provide more quantitative insight into such phenomena and to
illustrate such an event occurring on short time scales, the 1995 Laguna Verde
emergency incident is now considered [54, 52, 55, 39]. It happened in January
1995, during the start-up of the reactor, and lasted 723 s during which power
oscillations grew apparently unbounded, even though the power was slightly
decreased in an attempt to control their amplitude (see bottom plot of Figure
2, where an extraction of visible data points from the previous references has
been performed). As the maximum peak-to-peak value of the oscillations reached
$10\%$, a scramming procedure was triggered. The top plot of Figure 2 presents
the growth of the noise during the event, using an interpolation of the raw
data after subtracting the average decrease trend in power. As can be
observed, the initial noise amplitude was below $1\%$, indicating a stable
behavior of the reactor, but initiated an apparent exponential growth with a
period close to 100 s.

Figure 2:  Bottom plot: experimental data of the Laguna Verde BWR 1995
instability event extracted from [54, 52, 39], presenting the average power
level versus time, as recorded by APRM detectors. Top plot: linear
interpolation of the neutron noise amplitude during the event.

Shift in power spectra due to roughening transition The roughening transition
to a strong noise regime could once again provide information on this
particular phenomenology. Indeed, the scale-free behavior of the boiling [51]
discussed above mirrors a scale-free correlation length $\kappa^{-1}$ and
could enforce the scale invariance hypothetized by the renormalization
procedure. As discussed before, the roughening transition threshold is lower
due to the minimal size of the bubbles, explaining that BWRs might be proner
to enter a rough noise regime. In the case of Laguna Verde, the tripping of
the reactor has occurred during a change of equilibrium where the power was
rather constant on average (it was initially lowered during the first stage of
the incident), but where the noise amplitude was promptly increasing. Unlike
the PWR case in the previous subsection, where a strong noise equilibrium was
hypothetized, the emergency procedure most probably prevented such a regime
(if any) in Laguna Verde. Since noise amplitudes characteristic from this
regime cannot consequently be estimated, interpreting this change of the
behavior of the reactor requires to use other tools, such as a frequency
spectrum analysis. Eq. 25 can be rewritten

$\Big{|}\frac{\delta P(w)}{P_{0}}\Big{|}=\delta\rho(w)G_{0}(w),$ (27)

where the reactivity perturbation $\delta\rho$ is defined as before, and where
the $G_{0}(w)=\frac{1}{w\Lambda}$ is the so-called zero-power reactor transfer
function 666which is modified as
$G_{0}(w)=\frac{1}{w*(1+\frac{\beta/\Lambda}{iw+\lambda})}$ when delayed
neutron are taken into account, with $\beta$ the delayed neutron fraction and
$\lambda$ the precursor decay rate. For more realistic transfer functions that
take into account complex space-dependent thermal-hydraulic and neutronic
phenomena, the differential equation Eq. 23 is sensitively modified, and the
effective transfer function $G(w)$ could present substantial deviations from
$G_{0}$. In any case, auto-power spectrum density (APSD) signals are often
characterized by power laws for the $w$ spectrum, which allow one to
understand rapid kinetic changes in the reactor. However, trying to appreciate
the hypothetical change in APSD power laws associated to the roughening
transition can be based on the fact that the change from normal behavior to a
strong noise regime is only dictated by the shift in the tails of
$\delta\rho(w)$. For a white noise signal, $\delta\rho$ is constant. This
corresponds to the temporal white noise $\eta(\vec{x},t)$ associated with the
bare noise coupling $\sigma$. In the strong noise regime, $\sigma_{\kappa}$
acquires an effective meaning and depends on the observation scale
$\kappa^{-1}$: the strength of the effective noise coupling is crucially
dependent on this scale. The scale-dependent change in the effective noise
amplitude is given as a power law associated to a critical exponent and, using
Eq. 16, takes the form

$\sigma^{2}(\kappa)\propto\kappa^{-\eta_{D}^{*}},$ (28)

with $\eta_{D}^{*}=3\chi+1$ and $\chi\simeq 0.3$ (see Subsection II.2). Now,
since perturbations in the case of BWR are believed to be mainly correlated
with void effects arising from the two-phase flow, a simplified model can be
postulated that inversely relates the frequency scale $w$ at which the reactor
is observed to the spatial scale $\kappa^{-1}$ at which it is observed. In
fact, the coolant transports perturbations at a constant velocity $v\approx
100$ cm.s-1. At a given position, the duration associated with a perturbation
is hence given by $w^{-1}=2\pi\kappa^{-1}/v$, from which the proportionality
between $w$ and $\kappa$ is hypothesized. In this model, a small perturbation
of $\kappa^{-1}\simeq 1$ cm corresponds to 10 Hz, whereas a quarter-core
perturbation of $\kappa^{-1}\simeq 100$ cm corresponds to 1 Hz: these
perturbations will define the dynamic frequency range for the observations.
Finally, it is possible to appreciate the change in the spectrum of the
reactor transfert function $G(w)$ when the core shifts from normal operating
conditions to a strong noise regime, using

$\Big{|}\frac{\delta P(w)}{P_{0}}\Big{|}\propto\frac{G(w)}{w^{3\chi+1}}.$ (29)

Therefore, the RMS of the noise amplitude should approximately scale as
$w^{-1/2}$ under normal conditions (with $G(w)=G_{0}(w)\propto 1/w)$), while
the scaling associated with a strong noise regime should be given by
$w^{-(1/2+(3\chi+1)/2)}\approx w^{-3/2}$. To appreciate whether such a change
had occurred during the Laguna Verde incident, data were reanalyzed according
to the methodology of Reference [52]. This paper compares a standard sequence
of recorded data of Laguna Verde at nominal power (NP), to the data recorded
during the incident. Since the reactor was operated at $30\%$ NP at the
outbreak of the incident and, furthermore, during a power decrease, Alvarez-
Ramirez et al. proposed a detrended noise analysis of the sequence. The
results of the analysis, consistent with Reference [52], appear in Figure 3,
where the fluctuation values of the experimental data are extracted from
Reference [52] but presented as a function of the frequency $w$ and adjusted
only by focusing on large values $w$ (over a 700 s signal). The blue points
represent the fluctuation values associated with the Laguna Verde reactor
operating under normal conditions (the $G(w)$ function), while the orange data
points are experimental data recorded during the incident. The noise amplitude
estimated by the detrended fluctuation function clearly exhibits different $w$
scalings. The results of the adjustment using power laws of both signals are
indicated in the plot. The normal sequence, labeled ”stationary operating
conditions”, indeed exhibits a frequency power law of 0.4, close to the
$1/\sqrt{(}w)$ associated with a white noise signal. This anticorrelation most
probably signs the presence of feedbacks that keep the signal stationary [52].
Concerning the incidental signal, labeled ”emergency incident”, the fit of the
slope on a log-log scale is also coherent with [52], and equal to $1.89$ with
a $2\%$ confidence interval, when observed on the dynamic range of $[1-10]$
Hz. This value is sensitively higher than the value calculated for the strong
noise regime but still coherent given the level of simplifications of the
model. It should also be noted that $1/w^{2}$ noise spectra were also
characterized in this range for the APSD of high-noise events observed in PWRs
[15], which could indicate that the insights provided by the strong noise-
induced shift in the frequency power law calculated for BWRs could also
contribute to understanding open issues of PWR noise measurements.

Figure 3: Experimental data of the Laguna Verde BWR 1995 instability event
[54, 52, 39]. Fluctuation function as proposed by Reference [52] presented
versus frequency $w$, during a stationary operation of the Laguna Verde
reactor (data: blue solid line, fit: blue dashed line) and during the
emergency incident (data: orange solid line, fit: orange dashed line).

##  IV Conclusion

Small and limited power fluctuations are key to the safety of nuclear
reactors. On the one hand, automatic protection systems rely on them to assess
the reactor power time derivative and take appropriate actions. On the other
hand, neutron noise measurements are sometimes used to assess the condition of
operating reactor cores. However, for reasons that are only partially
understood using linear noise theory (on which both theoretical and numerical
modeling of neutron power noise are based), the fluctuations themselves can
sometimes increase to unusual levels, as in some PWRs, or grow unboundedly, as
has been observed in some BWRs, and hence pose direct safety concerns.

In this paper, exploiting the formal proximity between such reactor
fluctuations models and the stochastic heat equation, a field-theoretic
approach to the temporal and spatial behavior of reactors has been proposed.
Through a Cole-Hopf transformation, the one-group diffusion equation with
stochastic multiplicative noise is indeed mapped onto the KPZ equation, plus a
wall if feedback mechanisms are taken into account. This observation paved the
way for the study of neutron power noise using renormalization techniques,
which predicted that, under certain conditions, the reactor could switch to a
new criticality state, profoundly different from standard criticality, and
that the neutron fields could become rough, in direct analogy to the dynamics
of surface growth, even when feedback mechanisms are taken into account.

The phenomenology of such rough neutron fields was correlated with two
specific events. First, the noise amplitude measured during a noise increase
of the Almaraz-Trillo PWR was compared to the noise threshold of the
roughening transition. The proximity of both values thus confirmed the
relevance of interprations of PWR high noise sequences with the help of the
strong-noise universality class. Second, in BWR, the small minimal correlation
length related to the two-phase flow was shown to lower this threshold in the
noise amplitude down to values currently met during operations, hence
eventually explaining why these reactors are proner to develop high noise
events. Data from the Laguna Verde BWR accident in 1995 were used to
characterize the spectral frequency shift that separates the standard
operation of the reactor from its incidental functioning. This shift was also
evaluated from the roughening transition model, showing a good agreement with
the experimental value.

From the perspective of reactor physics, this theory, if further supported by
data, could help characterize and understand instabilities, especially those
occurring in BWRs, and hopefully design diagnostic tools to prevent their
appearance. For example, the use of largest Lyapunov exponents has been
proposed [46] in the context of real-time analysis of BWR plant data, which
typically implies fractional exponents similar to those derived using the
roughening transition model. This seems to be all the more feasible as the
Lyapunov exponents of the SHE with multiplicative noise have recently been
derived, albeit in lower dimensions [57]. For PWRs, long-term trends (over
many cycles) in noise amplitude have been measured in various reactors. Such
phenomena may also benefit from interpretation in light of the roughening
phase transition. Indeed, if, as has been calculated, temperature fluctuations
and core component vibrations put the reactor close to the phase transition
limit, any small-scale persistent perturbation would explain such long-lasting
trends.

Finally, on a different note, from the perspective of statistical physics, BWR
noise events could represent a first measurement of critical exponents of the
KPZ universality class in dimension $d=3$, thus confirming the adage that
every cloud has a silver lining.

## Acknowledgments

The author is grateful to Benjamin Dechenaux and Léonie Canet for very helpful
comments and illuminating discussions.

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