# Assessing non-Oberbeck-Boussinesq effects of convection in cryogenic helium

Michal Macek1 Georgy Zinchenko2 Věra Musilová1 Pavel Urban1 Jörg Schumacher2,3
1The Czech Academy of Sciences, Institute of Scientific Instruments,
Královopolská 147, CZ-61264 Brno, Czech Republic
2Institut für Thermo- und Fluiddynamik, Technische Universität Ilmenau,
Postfach 100565, D-98684 Ilmenau, Germany
3Tandon School of Engineering, New York University, New York City, NY 11201,
USA

(August 27, 2024)

###### Abstract

The present study investigates the non-Oberbeck-Boussinesq (NOB) effects which
arise due to the temperature dependence of material properties in cryogenic
helium experiments of turbulent Rayleigh-Bénard convection. They are manifest
as a difference of the measured mean temperature at the center of the closed
cell, $T_{c}$, from the arithmetic mean temperature obtained from the
prescribed fixed and uniform temperatures at the top and bottom copper plates
of the apparatus, $T_{m}=(T_{\rm bot}+T_{\rm top})/2$. Therefore, the material
properties such as specific heat at constant pressure, dynamic viscosity,
thermal conductivity, the isobaric expansivity, and the mass density are
expanded into power series with respect to temperature up to the quadratic
order with coefficients obtained from the software package HEPAK. A subsequent
nonlinear regression that uses deep convolutional networks delivers a
dependence of the strength of non-Oberbeck-Boussinesq effects in the
pressure–temperature parameter plane. Strength of the NOB effects is evaluated
via the deviation of the mean temperature profile $\xi_{\mathrm{NOB}}\equiv
T_{m}-T_{c}$ from the top/bottom-symmetric Oberbeck-Boussinesq case
$\xi_{\mathrm{NOB}}=0$. Training data for the regression task are obtained
from 236 individual long-term laboratory measurements at different Rayleigh
numbers which span 8 orders of magnitude.

##  I Introduction

Controlled laboratory experiments of turbulent Rayleigh-Bénard convection
(RBC) are one pillar of turbulence research to obtain a deeper understanding
of the physical transfer processes and their coupling to statistical
properties and structures, both in the bulk and the boundary layers of
buoyancy-driven flows Kadanoff2001 ; Ahlers2009 ; Chilla2012 ; Verma2018 . The
highest Rayleigh numbers $Ra$ for fluid flows at Prandtl numbers $Pr\simeq 1$
are obtained in two gases, either compressed sulphur hexafluoride, SF6, He2012
; He2012a  or cryogenic helium, 4He, the latter of which is cooled down to a
few Kelvin Chavanne1997 ; Niemela2000 ; Niemela2003 ; Roche2010 ; Urban2011 ;
Urban2012 ; Urban2019 ; Roche2020 . While the Rayleigh number $Ra$ quantifies
the thermal driving of convective turbulence, the Prandtl number $Pr$ is the
ratio of molecular momentum to temperature diffusion. Together with a third
parameter, the aspect ratio of the exclusively used cylindrical closed vessels
$\Gamma=D/H$ with cell diameter $D$ and cell height $H$, these three
dimensionless numbers determine the control parameters of the experiments and
are subsequently used to quantify the response of the apparatus in the form of
power laws of the turbulent momentum and heat transfer. The latter are
quantified by the dimensionless Reynolds and Nusselt numbers, $Re$ and $Nu$
Ahlers2009 ; Chilla2012 . The present study is focused to the experiments in
cryogenic helium 4He.

The Rayleigh-Bénard convection model incorporates the Oberbeck-Boussinesq (OB)
approximation Chilla2012 ; Verma2018  which considers the working fluid as
incompressible. In addition, the mass density field of the fluid,
$\rho({\bm{x}},t)$ is taken as a linear function of the temperature field
$T({\bm{x}},t)$ and given by

$\rho({\bm{x}},t)=\rho_{\rm ref}[1-\alpha(T({\bm{x}},t)-T_{\rm
ref})]\quad\mbox{with}\quad\alpha=-\frac{1}{\rho}\frac{\partial\rho}{\partial
T}\Big{|}_{p}\,,$ (1)

with the isobaric thermal expansion coefficient or expansivity $\alpha$. This
dependence is incorporated in the volume forces and thus couples the
temperature field to the momentum balance. Quantities $\rho_{\rm ref}$ and
$T_{\rm ref}$ are reference magnitudes of density and temperature,
respectively. One important consequence of the OB approximation is that
statistical properties, such as mean profiles, in the lower and upper halves
of the convection cell including the corresponding viscous and thermal
boundary layers, are symmetric with respect to the midplane at $z=H/2$.
Consequently,

$T_{c}:=\Big{\langle}T\left(z=\frac{H}{2}\right)\Big{\rangle}=\frac{T_{\rm
bot}+T_{\rm top}}{2}=:T_{m}\,,$ (2)

in laboratory experiments with the prescribed fixed and uniform temperatures
at the top and bottom, $T_{\rm top}$ and $T_{\rm bot}$.

Cryogenic RBC experiments at the highest Rayleigh numbers have to be operated
close to the critical point (CP) of He Niemela2000 ; Urban2011 ; Urban2012 ;
Urban2019 . At this point, where the saturated vapor curve (SVC) representing
the phase boundary between the gas and liquid state ends, the material
properties of the working fluid such as specific heat at constant pressure,
$C_{p}$, dynamic viscosity $\mu$, or thermal conductivity $\lambda$ fluctuate
strongly. This is considered as one possible source of the deviations from the
Boussinesq limit, which are experimentally probed by a violation of (2). In
other words, non-Boussinesq (NOB) effects are detected as $T_{c}\neq T_{m}$.
It is exactly this deviation which we want to explore in detail in the present
work for cryogenic 4He. Therefore, we define the non-Oberbeck-Boussinesq
parameter

$\xi_{\rm NOB}(p_{m},T_{m},\chi_{k}):=T_{m}-T_{c}\,,$ (3)

where $\chi_{k}$ are for now a short-hand notation for material properties
which will be specified further below. Figure 1 (panel a) summarizes the
operating points within the $p$–$T$ diagram for cryogenic experiments
conducted in the apparatus of the group in Brno (Czech Republic) Urban2010 .
We indicate the mean temperature as well as the range of the applied outer
temperature difference $\Delta T=T_{\rm bot}-T_{\rm top}>0$ at the mean
pressure $p=p_{m}$. It is seen that a number of measurements are close to the
phase boundary (solid black curve) and a few even in the vicinity of the
critical point (magenta star). In panel (b), the measured values of $\xi_{\rm
NOB}$ are plotted in the phase diagram of the two control parameters, $Ra$ and
$Pr$. It is important to note that $Ra$ and $Pr$ are control parameters
characterizing OB convection, thus ideally $\xi_{\rm NOB}=0$ independent of
$Ra$ and $Pr$. The experimentally observed values $\xi_{\rm NOB}\neq 0$
unambiguously indicate presence of NOB effects and must be captured
introducing additional control parameters. We define and discuss a suitable
set below.

In this work, we will systematically investigate the non-Boussinesq effects in
cryogenic helium experiments at high Rayleigh numbers spanning a range of
$10^{7}\lesssim Ra\lesssim 10^{15}$ performed in Brno. We quantify the
deviation of the center temperature $T_{c}$ from $T_{m}$ in the $p$–$T$
diagram by means of nonlinear regression applying deep neural networks, i.e.,
determine $\xi_{\rm NOB}(p_{m},T_{m})$. This regression proceeds in three
different levels of refinement, partly based on a perturbative expansion of
the temperature dependence of essential material parameters and state
variables $p$, $T$ as described in the next section. Furthermore, we aim to
identify which of the variations of the material parameters are the most
important ones for the magnitude of the NOB parameter. The complex material
dependencies including discontinuities at the phase boundary and the
singularities at the critical point are tabulated in the software package
HEPAK written by V. Arp et al. Arp1998  and retailed by Horizon Technologies
Inc Horizon . They allow us to quantify the prefactors of the polynomial
expansions of the material parameters and state variables at different orders.
Starting point is the set of fully compressible equations of motion which was
outlined by Gray and Giorgini Gray1976 . Material parameter dependencies have
been also systematically discussed for high-Rayleigh-number experiments in
compressed SF6 in refs. Shishkina2016 ; Weiss2018 ; Yik2021 .

Figure 1: Summary of the operating points $(p_{m},T_{m})$ of the cryogenic
Rayleigh-Bénard experiments. (a) The mean temperatures $T_{m}$ and the
temperature range $\Delta T=T_{\rm bot}-T_{\rm top}$ at a mean pressure
$p_{m}$ are provided in the $p$–$T$ parameter plane. The ’errorbars’ stemming
from the $(p_{m}$,$T_{m}$) points denote the ranges of $\Delta T$ between the
cold (blue) and hot (red) plate temperatures $T_{t}$, $T_{b}$, respectively.
The solid line marks the saturated vapor curve (SVC) and the star symbol in
the center of the figure indicates the critical point (CP) with $T_{\rm
cri}=5.195$ K, $p_{\rm cri}=0.228$ MPa. The different colors of experimental
$(p_{m}$,$T_{m}$) points correspond to new data (green), and data published in
Refs. Urban2014  (black) and Urban2021  (brown). (b) Color-coded non-
Boussinesq parameter $\xi_{\rm NOB}$ at the $Ra,Pr$ control parameters for the
experiments shown in panel (a), given in units of Kelvin.

Our paper is organized as follows. Section II introduces the fully
compressible equations of motion and discusses the resulting polynomial
expansions for the material properties. In Sec. III, we explore the basic
state and transport properties of 4He as a function of pressure $p$ and
temperature $T$ in connection with the ($p_{m},T_{m}$) operating points of all
realized RBC experiments within the respective $\Delta T$ region. Here, we
also discuss the second mechanism to NOB convection, the compressibility
effects which are shown to be negligible in the present setup. In Sec. IV, we
briefly overview the essential features of the experimental set-up for
cryogenic RBC. Section V discusses the nonlinear regression results. In the
last section, we give the conclusions and outlook. Technical details of the
deep neural networks and an error analysis of the machine learning procedures
are discussed in the appendices A and B.

##  II Perturbative expansion of the equations of motion

Convective motions in a fluid layer are described by the set of three balance
equations involving the continuity equation for the mass balance, the Navier-
Stokes equations for the momentum balance, and the energy balance equation.
Following here the textbook by Batchelor Batchelor1960  and the seminal work
by Gray and Giorgini Gray1976 , they are given by

$\displaystyle\frac{D\rho}{Dt}$ $\displaystyle=-\rho\frac{\partial
u_{j}}{\partial x_{j}},$ (4) $\displaystyle\rho\frac{Du_{i}}{Dt}$
$\displaystyle=-\frac{\partial p}{\partial x_{i}}-\rho g_{i}\alpha
T+\frac{\partial}{\partial x_{j}}(\mu\Gamma_{ij}),$ (5) $\displaystyle\rho
C_{p}\frac{DT}{Dt}-\alpha T\frac{Dp}{Dt}$
$\displaystyle=\frac{\partial}{\partial x_{j}}\left(\lambda\frac{\partial
T}{\partial x_{j}}\right)+\mu\Phi,$ (6)

where $D\bullet/Dt=\partial\bullet/\partial t+{\bm{u}}\cdot\nabla\bullet$ is
the material derivative and

$\Gamma_{ij}=\frac{\partial u_{i}}{\partial x_{j}}+\frac{\partial
u_{j}}{\partial x_{i}}-\frac{2}{3}\frac{\partial u_{k}}{\partial
x_{k}}\delta_{ij}$ (7)

is the rate of strain tensor in the compressible case. Furthermore,

$\Phi=\frac{1}{2}{\Gamma_{ij}}\left(\frac{\partial u_{i}}{\partial
x_{j}}+\frac{\partial u_{j}}{\partial x_{i}}\right),$ (8)

is the dissipation function and $g_{i}=\left(0,0,-g\right)$ the vector that
contains the acceleration due to gravity $g$ in vertical direction. The bulk
viscosity is set to zero.

A simplified form of the convection equations (4)–(6) is obtained in the form
of the OB approximation, once the dynamical viscosity $\mu$ , the thermal
conductivity $\lambda$, the isobaric thermal expansivity $\alpha$, and the
specific heat $C_{p}$ at constant pressure are taken as constants.
Furthermore, the mass density is taken to be constant $\rho=\rho_{0}$, such
that the flow is basically incompressible, and heating due to pressure
variations remains subdominant, which results to

$\displaystyle\frac{\partial u_{j}}{\partial x_{j}}$ $\displaystyle=0,$ (9)
$\displaystyle\rho_{0}\frac{Du_{i}}{Dt}$ $\displaystyle=-\frac{\partial
p}{\partial x_{i}}-\rho_{0}g_{i}\alpha T+\mu\frac{\partial^{2}u_{i}}{\partial
x_{j}^{2}},$ (10) $\displaystyle\rho_{0}C_{p}\frac{DT}{Dt}$
$\displaystyle=\lambda\frac{\partial^{2}T}{\partial x_{j}^{2}}.$ (11)

This implies that the speed of sound is much larger than the typical
convection velocity, the free-fall velocity $U_{\rm f}=\sqrt{g\alpha\Delta
TH}$ for small $\alpha\Delta T$. In all expressions above we have used the
Einstein summation convention.

Our objective is to evaluate the importance of individual terms of the
compressible equations (4)–(6) beyond the OB limit systematically and analyze
their effects on the mean temperature profiles in the thermal convection
experiments in cryogenic helium. We will therefore assume that all material
parameters are functions of the temperature only and that their pressure
dependence is much less significant for the present experimental conditions.
They can then be approximated by Taylor expansions with respect to $T$, which
we will follow up to the quadratic expansion term. This results to the
following expressions

$\displaystyle\rho$ $\displaystyle={{\rho}_{0}}\left(1-{{\alpha}_{0}}\delta
T+{{\beta}_{0}}\delta{{T}^{2}}\right),$ (12) $\displaystyle\mu$
$\displaystyle={{\mu}_{0}}\left(1+{{m}_{10}}\delta
T+{{m}_{20}}\delta{{T}^{2}}\right),$ (13) $\displaystyle{{C}_{p}}$
$\displaystyle={{C}_{p0}}\left(1+{{c}_{10}}\delta
T+{{c}_{20}}\delta{{T}^{2}}\right),$ (14) $\displaystyle\lambda$
$\displaystyle={{\lambda}_{0}}\left(1+{{l}_{10}}\delta
T+{{l}_{20}}\delta{{T}^{2}}\right),$ (15) $\displaystyle\alpha$
$\displaystyle={{\alpha}_{0}}\left(1+{{a}_{10}}\delta
T+{{a}_{20}}\delta{{T}^{2}}\right).$ (16)

where $\delta T=T-T_{m}$. The first index in coefficients defines the term
number in Taylor series decomposition, index $0$ refers to the mean
temperature $T_{m}$ and the minus in (12) follows a convention usual in fluid
dynamics. All constants in the expansions (12)–(16) are determined from
quadratic interpolations of the respective material properties at three
temperature values $T_{\rm top}$, $T_{m}$, and $T_{\rm bot}$ at a given
pressure value $p$ using HEPAK Arp1998 .

The substitution of Eqs. (12)–(16) into the compressible fluid equations
(4)–(6) and the subsequent performance of a number of transformations leads to
a set of a dimensionless convection equations which include the NOB effects up
to second order with respect to temperature. They are given by

$\displaystyle-\varepsilon_{1}\frac{D\theta}{Dt}+2\varepsilon_{2}\theta\frac{D\theta}{Dt}$
$\displaystyle=-\left(1-\varepsilon_{1}\theta+\varepsilon_{2}\theta^{2}\right)\frac{\partial
u_{j}}{\partial x_{j}},$ (17)
$\displaystyle\left(1-\varepsilon_{1}\theta+\varepsilon_{2}\theta^{2}\right)\frac{Du_{i}}{Dt}$
$\displaystyle=-\frac{\partial\left(p-p_{s}\right)}{\partial
x_{i}}+\left[\left(\theta-\theta_{s}\right)-\frac{\varepsilon_{2}}{\varepsilon_{1}}\left(\theta^{2}\\!-\\!\theta_{s}^{2}\right)\right]k_{i}+$
$\displaystyle\frac{1}{Re_{\rm
f}}\left(1+\varepsilon_{3}\theta+\varepsilon_{4}\theta^{2}\right)\frac{\partial\Gamma_{ij}}{\partial
x_{j}}+\frac{1}{Re_{\rm
f}}\left(\varepsilon_{3}+2\varepsilon_{4}\theta\right)\Gamma_{ij}\frac{\partial\theta}{\partial
x_{j}},$ (18)
$\displaystyle\left(1-\varepsilon_{1}\theta+\varepsilon_{2}\theta^{2}\right)\left(1+\varepsilon_{5}\theta+\varepsilon_{6}\theta^{2}\right)\frac{D\theta}{Dt}$
$\displaystyle=\frac{\tilde{D}}{Re_{\rm
f}}\left(1+\varepsilon_{3}\theta+\varepsilon_{4}\theta^{2}\right)\Phi+$
$\displaystyle\frac{1}{Re_{\rm
f}Pr}\left(1+\varepsilon_{7}\theta+\varepsilon_{8}\theta^{2}\right)\frac{\partial^{2}\theta}{\partial
x_{j}^{2}}+\frac{1}{Re_{\rm
f}Pr}\left(\varepsilon_{7}+2\varepsilon_{8}\theta\right)\left(\frac{\partial\theta}{\partial
x_{j}}\right)^{2}+$
$\displaystyle\tilde{D}\left(1+\varepsilon_{9}\theta+\varepsilon_{10}\theta^{2}\right)\left[\varepsilon_{1}\frac{D\left(p-p_{s}\right)}{Dt}-\left(1-\varepsilon_{1}\theta_{s}+\varepsilon_{2}\theta_{s}^{2}\right)u_{3}\right](\theta+\tilde{T}_{m})\,.$
(19)

Here,

$\theta=\frac{\delta T}{\Delta T}=\frac{T-T_{m}}{T_{\rm bot}-T_{\rm
top}}\quad\mbox{and}\quad\tilde{T}_{m}=\frac{T_{m}}{\Delta
T}=\frac{T_{m}}{T_{\rm bot}-T_{\rm top}}\,,$ (20)

and $\theta_{s}$ and $p_{s}$ are temperature and pressure equilibrium (static
heat conduction) profiles, respectively. Furthermore, we define in the energy
balance a dimensionless parameter

$\tilde{D}=\frac{g\alpha_{0}H}{C_{p0}}\,,$ (21)

which is denoted as the dissipation number Verhoeven2015 . This number relates
the dry adiabatic lapse rate $g/C_{p0}$ to the characteristic temperature drop
$\alpha_{0}H$. The velocity was made non-dimensional by $U_{\rm f}$. The
Prandtl number reads $Pr=\nu_{0}/\kappa_{0}$ with the temperature diffusivity
$\kappa_{0}=\lambda_{0}/(\rho_{0}C_{p0})$, the free-fall Reynolds number
$Re_{\rm f}=U_{\rm f}H/\nu_{0}$, and the unit vector in the momentum balance
points into the positive $z$–direction, $k_{i}=(0,0,1)$. The free-fall
Reynolds number follows to $Re_{\rm f}=\sqrt{Ra/Pr}$ where $Ra$ is the
Rayleigh number

$Ra=\frac{\alpha_{0}}{\nu_{0}\kappa_{0}}g\Delta TH^{3}\,.$ (22)

Quantity | First order expansion | Second order expansion
---|---|---
Mass density $\rho$ | ${{\varepsilon}_{1}}={{\alpha}_{0}}\Delta T$ | ${{\varepsilon}_{2}}={{\beta}_{0}}{{\Delta T}^{2}}$
Dynamic viscosity $\mu$ | ${{\varepsilon}_{3}}={{m}_{10}}\Delta T$ | ${{\varepsilon}_{4}}={{m}_{20}}{{\Delta T}^{2}}$
Specific heat $C_{p}$ | ${{\varepsilon}_{5}}={{c}_{10}}\Delta T$ | ${{\varepsilon}_{6}}={{c}_{20}}{{\Delta T}^{2}}$
Thermal conductivity $\lambda$ | ${{\varepsilon}_{7}}={{l}_{10}}\Delta T$ | ${{\varepsilon}_{8}}={{l}_{20}}{{\Delta T}^{2}}$
Isobaric expansion coefficient $\alpha\quad$ | ${{\varepsilon}_{9}}={{a}_{10}}\Delta T$ | ${{\varepsilon}_{10}}={{a}_{20}}{{\Delta T}^{2}}$
Table 1: List of all Taylor expansion parameters $\varepsilon_{i}$ for
$i=1,...,10$.

Finally, in table 1, we list all parameters $\varepsilon_{i}$ which were used
in Eqs. (17)–(II) for first and second order expansions. Setting all these
expansion parameters $\varepsilon_{i}$ and $\tilde{D}$ to zero recovers the OB
equations (9)–(11) Ahlers2009 ; Chilla2012 . Expansion parameters with an odd-
number index are for the linear expansion while those with an even index are
for the quadratic order. Notice that $\tilde{D}$, similar to the OB control
parameters $Ra$ and $Pr$ depends only on local values of the material
properties at the reference temperature $T_{m}$, and—like the Prandtl number
$Pr$—is independent of $\Delta T$. In this sense it differs from
$\varepsilon_{i}$, which are “non-local” and depend on the temperature
derivatives at $T_{m}$ as well as $\Delta T$. The regression algorithms in
Sec. V will proceed in incremental steps, i.e., consider linear expansions
only at first and incorporate second order subsequently.

##  III State and transport properties of cryogenic 4He

Figure 2 shows the accurate values of the mass density $\rho$, the specific
heat at constant pressure $C_{p}$, the dynamic viscosity $\mu$, and the
thermal conductivity $\lambda$ with respect to the $p$–$T$ plane in a region
of gaseous helium phase including the regions near to the vapor liquid
saturation curve and the critical point. These surfaces are obtained from the
HEPAK code Arp1998 , which is based on high-order interpolation of all
available and reliable measurements. The plots give us a first guidance in the
selection of the appropriate the functional form of the state and transport
properties in the equations beyond the OB approximation. All displayed
quantities, $\rho$, $C_{p}$, $\lambda$, and $\mu$ develop a discontinuity at
the phase boundary in the pressure-temperature plane which corresponds to a
first-order gas–liquid phase transition. Furthermore, $C_{p}$ and $\lambda$
develop a divergence at the critical point (CP), which is given by $p_{\rm
CP}=227$ kPa and $T_{\rm CP}=5.2$ K. The precursors of this divergence are
visible here. Panel (a) displays in fact the equation of state $\rho(T,p)$. In
the vicinity of this point, most of the RBC experiments in cryogenic helium
have been performed as indicated by the black dots in all panels of the
figure. They indicate mean pressure and temperature (and a view from the top
would reproduce the points of Fig. 1). Notice that the density $\rho$ and the
specific heat $C_{p}$ vary by several orders of magnitude over the domain
displayed, while the dynamic viscosity $\mu$ and the thermal conductivity
$\lambda$ vary by a factor of smaller than 4 only.

Finally, in order to assess possible contribution of compressibility to the
NOB effects, we evaluate the Mach number $M=U_{\rm{f}}/c=\sqrt{g\alpha\Delta
TH}/c$, with isobaric thermal expansivity $\alpha$ and the speed of sound $c$
obtained from HEPAK (not shown). The results show that for all experiments
considered, the Mach numbers are in the range $M\lesssim 10^{-2}$ and
guarantee that possible breaking of OB conditions due to compressibility can
be neglected. Thus the NOB effects in RBC experiments with cryogenic helium
stem solely from temperature dependencies of the fluid properties.

Figure 2: Basic state quantities (a,b) and molecular transport properties
(c,d) which characterize the complex material properties of cryogenic helium
4He near its critical point (CP) of $p_{\rm CP}=227$ kPa and $T_{\rm CP}=5.2$
K as a function of pressure $p$ and temperature $T$. Panel (a) shows the
density $\rho$ which thus corresponds with the equation of state $\rho(T,p)$.
Panel (b) plots the specific heat $C_{p}$, panel (c) the dynamical viscosity
$\mu$, and panel (d) the thermal conductivity $\lambda$. The black bullets
indicate the mean values $p_{m}$ and $T_{m}$ for all experiments shown in Fig
1. The data are obtained from the HEPAK code Arp1998 .

##  IV Experiments in cryogenic 4He

Cryogenic helium 4He has been used to reach extreme turbulence intensity in
“tabletop” RBC experiments on one hand thanks to the peculiar material
properties near the CP allow to reach high Rayleigh numbers $Ra$, on the other
hand, due to technical advantages, as heat leaks and many other parasitic
effects are naturally highly suppressed in cryogenic conditions. The first
advantage goes side-by-side with caveats of inevitably varying the Prandtl
number $Pr=\nu/\kappa$ Niemela2000 ; Urban2011  as well as NOB effects
stemming from (21) and the dependencies (12)–(16). Here we re-analyze RBC data
obtained at the Brno cryogenic turbulence facility Urban2010 .

The Brno experiment comprises a cryostat with a helium cryogenic experimental
cell with the height $H=0.3$ m and diameter $d=0.3$ m (aspect ratio
$\Gamma=d/H=1$) with particular effort to minimize the influence of the cell
structure and materials on the observed convection. The cell has been designed
to withstand pressures of $3.5$ bars to cover a range of Rayleigh numbers
$10^{7}\leq Ra\leq 10^{15}$. Here, we list the main features of the
experimental cell only, including all crucial recent upgrades. In an ideal RBC
experiment, the top and bottom plates should maintain non-fluctuating constant
temperatures $T_{\rm top}$ and $T_{\rm bot}$ and the cell sidewalls should be
adiabatic. The top and bottom plates of our cell are made of 28 mm thick
annealed OFHC copper of very high thermal conductivity of at least
$\lambda_{\rm Cu}=2$ kW m-1 K-1 at 5 K. Parasitic heat fluxes from the
sidewalls into the working fluid are minimized by using very thin stainless
steel sidewalls with thickness $\delta=0.5$ mm and a thermal conductivity
$\lambda_{w}$. A special design of the cell corners is used, see Fig. 4 of
Urban2010 . One way to estimate the influence of the sidewall on the heat
transport is via the wall parameter $W=4\delta\lambda_{w}/(\lambda_{\rm
He}D)$; for our cell $0.22>W>0.15$ depending on actual value of the thermal
conductivity for each data point. By correction, we mean a subtraction of the
heat conducted by sidewalls from the heat that passes through the working
fluid by convection Roche2001 . We also paid attention to employ good thermal
shielding and minimize other external parasitic heat flows into the cell which
could substantially influence the convection dynamics.

A temperature correction which needs to be addressed in cryogenic experiments
is that due to adiabatic gradient, $g/C_{p}$. It is given by $\Delta
T_{\rm{ad}}=g\alpha HT_{m}/C_{p}=\tilde{D}T_{m}$, see also Eq. (21), which has
to be subtracted from the measured temperature difference $\Delta T$ before
evaluating $Ra$ and comparing to the results of DNS based on Eqs. (4)–(6),
otherwise the experimental data points would be very much off the expected
$Nu(Ra)$ dependencies. In typical large RBC cryogenic helium experiments,
including the experiments in Brno, $\Delta T_{\rm{ad}}$ is of order 1 mK. In
the largest RBC cell at Oregon, it was about 3 mK Niemela2000 . In contrast,
in turbulent RBC in air at room temperature the adiabatic gradient correction
is not important, as $g/C_{p}=0.01$ K/m; thus the scale height across which
$T$ drops by an order of magnitude is 1 km.

Cryogenic helium does not absorb thermal radiation, which leaves the radiation
corrections to the Nusselt number negligibly small. We point to a
straightforward evaluation in ref. Urban2020rad  and to the discussion of an
extreme case in Lepot18 .

The top and bottom plates of the cell are equipped with four germanium (Ge)
thermometers, calibrated at Physikalisch-Technische Bundesanstalt Berlin
(Germany) up to the best currently available precision of $\pm 2$ mK absolute
accuracy over the entire temperature range of interest. These Ge thermometers
are embedded in the middle and on the sides of both plates. We see no
horizontal temperature gradients within Ge sensors accuracy of 2 mK in both
copper plates. Following the last upgrade, both copper plates contain a pair
of fast response Lakeshore DT-670 silicon (Si) diode thermometers, allowing to
resolve the plate temperature fluctuations, which on one hand enable much
better control of the temperature boundary conditions Urban2021  compared to
previous experiments via Proportional-Integrative-Derivative feedback loops
and, on the other hand, can be used to correlate dynamics of the turbulent
large-scale circulation in the bulk with temperature fluctuations affecting
the boundary layers. Additional sensors are placed inside the cell, calibrated
in situ by us against the primary four Ge thermometers, and can be adjusted to
measure directly the absolute turbulent core temperature $T_{c}$ or the
fluctuations $\delta T_{c}$.

##  V Nonlinear regression of non-Boussinesq parameters

###  V.1 Experimental data base and regression procedure

We are interested in the dependence of the NOB parameter on state variables
and material properties, i.e., most generally this results to a function
$\xi(p_{m},T_{m},\Delta
T,\alpha(T,p),\lambda(T,p),\mu(T,p),C_{p}(T,p),\rho(T,p))$. The perturbative
expansion of Sec. II has reduced this high-dimensional function to one of 13
input parameters. These are the mean pressure $p_{m}$, the mean temperature
$T_{m}$, the temperature difference between bottom and top boundaries $\Delta
T$, and the expansion parameters $\varepsilon_{i}$ for $i=1,...,10$. The
latter ones determine the first and second temperature derivatives of the
material properties and state variables as summarized in Table 1. The
nonlinear regression proceeds in three levels of increasing complexity. We aim
at reconstructing the following functions,

$\displaystyle\xi_{1}$ $\displaystyle:=\xi_{\rm NOB}(p_{m},T_{m},\Delta T)\,,$
(23) $\displaystyle\xi_{2}$ $\displaystyle:=\xi_{\rm NOB}(p_{m},T_{m},\Delta
T,\varepsilon_{2k+1})\quad\mbox{for}\quad k=0,...,4\,,$ (24)
$\displaystyle\xi_{3}$ $\displaystyle:=\xi_{\rm NOB}(p_{m},T_{m},\Delta
T,\varepsilon_{k})\quad\mbox{for}\quad k=1,...,10\,.$ (25)

In the first approach $\xi_{1}$, no expansion parameter $\varepsilon_{i}$ is
used for the training of the neural network. It is explored how the NOB
parameter depends on mean pressure and temperature as well as imposed
temperature difference only. The result can serve as a baseline to forecast
NOB effects on the temperature profile asymmetry $\xi_{\rm
NOB}(p_{m},T_{m},\Delta T)$ in future experiments. In the more detailed
successive steps, we are interested in a finer resolution of the effects of
individual fluid properties on the temperature asymmetry. In the second
(third) approach $\xi_{2}$ ($\xi_{3}$), sets of all linear-order (linear and
quadratic-order) expansion coefficients $\varepsilon_{2k+1}$
($\varepsilon_{2k}$ ) are taken together as input data for a slightly deeper
neural network architecture since the feature extraction proceeds in a higher-
dimensional feature space. In this case, before calculating the non-Boussinesq
parameter, it was necessary to perform preliminary calculations to determine
$\varepsilon_{k}$ at each point of the $p$–$T$-space within the given $\Delta
T$ by means of HEPAK. Values of the NOB parameters (23)-(25) are given in
units of K throughout the paper.

Figure 3: Contour plots of the reconstruction of the non-Boussinesq parameter
$\xi_{1}(p,T)$ for different $\Delta T$ is shown in panels (a), (b), (d), and
(e). Panels (c) and (f) display the corresponding root mean square error
(RMSE) for two of the four cases. The RMSE is given by Eq. (26). Unit for the
color code is Kelvin.

###  V.2 First reconstruction method without expansion parameters

In the first reconstruction method, there are three input parameters, the mean
pressure $p_{m}$ and the mean temperature $T_{m}$ which define the operating
point of a particular laboratory measurements. The third input parameter is
the applied temperature difference between the bottom and top plates. The only
output parameter is the difference between mean temperature $T_{m}$ and
temperature $T_{c}$ in the center of the layer, $\xi_{1}$, as already defined
in the last subsection, see Eq. (23). For details on the network architecture,
see table 2 in Appendix A.

Figure 3, shows the reconstruction of $\xi_{1}(T,P)$ for 4 different outer
temperature differences $\Delta T$ which are indicated in the title of panels
(a), (b), (d), and (e). In panels (c) and (f), we display in addition the root
mean square error (RMSE) contour plots for two cases which arises when taking
the $i=1,...,N_{\rm rec}=100$ individual reconstructions. It is defined as

${\rm RMSE}(p,T)=\sqrt{\frac{1}{N_{\rm rec}}\sum_{i=1}^{N_{\rm
rec}}|\xi_{k}(i,T,p)-\overline{\xi}_{k}(T,p)|^{2}}\quad\mbox{for}\quad
k=1,2,3\,,$ (26)

where $\overline{\xi}_{k}(T,p)$ the mean reconstructed surface. In panels (a),
(b), (d), and (e), it can be seen that the maximum value of the NOB parameter
is always reached in the vicinity of the phase boundary and at the critical
point. As the temperature difference $\Delta T$ increases, the NOB parameter
also increases. In panels (c) and (f), it is seen that the RMSE also increases
for bigger $\Delta T$. One reason is that there are less corresponding
experimental data. The largest errors occur near the critical point at the end
of the phase boundary.

The non-Oberbeck-Boussinesq parameter $\xi_{1}$ is also shown in $Ra$–$Pr$
parameter space in Fig. 4 for three outer temperature differences $\Delta T$,
cf. Fig. 1(a) for an analogous display of experimental values. All branches
for each $\Delta T$ start at $Pr=0.7$ for the smallest Rayleigh numbers. When
the Rayleigh number increases by 8 orders of magnitude, the Prandtl number
increases monotonically up to $Pr\simeq 15$. The data points are color-coded
by $\xi_{1}$. The NOB parameter also grows along the curves up to its highest
values at $Pr\simeq 2$. With a further increase of the Prandtl number, the NOB
parameter $\xi_{1}$ remains however nearly unchanged.

Figure 4: Plot of $\xi_{1}$ in the $Ra$–$Pr$ parameter space for three
different outer temperature differences $\Delta T$. The data points are color-
coded by $\xi_{1}$ as given by the legend to the right. Unit for the color
code is Kelvin.

###  V.3 Second reconstruction method including linear-order
$\varepsilon$-parameters

For the next level, we reconstruct the parameter field $\xi_{2}$ with the
linear order expansion parameters $\varepsilon_{i}$. Input parameters for the
deep neural network are now mean pressure $p_{m}$, mean temperature $T_{m}$,
outer temperature difference $\Delta T$ (as in the last case) together with
all $\varepsilon_{2k+1}$ for $k=0,...,4$. The output is now the NOB parameter
$\xi_{2}$ of Eq. (23). The neural network is detailed in table 3 of Appendix
A.

The results are shown in Fig. 5, in analogy with Fig. 3, for four different
temperature differences. Also, we add again two root mean square error plots
in panels (c, f) for $\Delta T=0.1$ and $0.4$, respectively. It can be seen
that the qualitative behavior is similar to the results of the first
reconstruction method. The difference between the reconstructions $\xi_{1}$
and $\xi_{2}$ is highlighted in Fig. 6, showing the deviation
$|\xi_{1}-\xi_{2}|$. Maximum deviations are up to $\xi\simeq 0.005$ K,
observed for $\Delta T=0.2$ K near the critical point and for $\Delta T=0.4$ K
in the lower-$T$ part near the saturation curve (maximum relative value of
$|\xi_{1}-\xi_{2}|/\xi_{1}\sim 40\%$). The graphs show that the influence of
linear coefficients near the critical point grows as the temperature
difference increases.

The $Ra$–$Pr$ plots related to the $|\xi_{1}-\xi_{2}|$ differences are shown
in Fig. 7 and display the largest deviation near the critical point and at the
saturation line when $Pr\gtrsim 2$ and $Ra$ varying from $10^{12}$ to
$10^{14}$.

Figure 5: Contour plots of the reconstruction of the non-Boussinesq parameter
$\xi_{2}(p,T)$ for different $\Delta T$ is shown in panels (a), (b), (d), and
(e). Panels (c) and (f) display the corresponding root mean square error
(RMSE) for two of the four cases. The RMSE is given by Eq. (26). The unit for
the color code is Kelvin. Figure 6: Contour plots of the absolute difference
$|\xi_{1}-\xi_{2}|$ between the 1st and 2nd reconstruction method in the
$p$–$T$ parameter plane for four different $\Delta T$ values which correspond
to those in panels (a), (b), (d), and (e) of Figs. 3 and 5. The unit for the
color code is Kelvin. Figure 7: The absolute difference $|\xi_{1}-\xi_{2}|$
between the first and the second reconstruction method in the $Ra$–$Pr$
parameter space for three different outer temperature differences $\Delta T$
(cf. Fig. 6). The unit for the color code is Kelvin.

###  V.4 Third reconstruction method including quadratic-order
$\varepsilon$-parameters

The last reconstruction method includes 13 input parameters to obtain
$\xi_{3}$. This comprises linear and quadratic expansions with respect to the
temperature encoded by $\varepsilon_{i}$ for $i=1,\dots,10$. Our analysis
shows that the absolute difference $|\xi_{2}-\xi_{3}|$ remains very small for
all the $p_{m}$, $T_{m}$, $\Delta T$ values considered. This can be seen in
Fig. 8, which corresponds to four ’sections’ through the $p-T$-plane at
$T_{m}=4.6,4.91,5.2$ and $5.5$ K showing the $p$-dependencies of $\xi_{1}$,
$\xi_{2}$ and $\xi_{3}$. Finally, we refer to Appendix B, where we have
summarized further error analysis for all three reconstruction methods. This
analysis illustrates how strongly the 100 individual reconstructions of
$\xi_{k}$ vary when selecting different subsets as training and test data, see
Fig. 10.

Figure 8: Comparison of individual reconstructions of the NOB parameter
$\xi_{1}$ (red), $\xi_{2}$ (blue) and $\xi_{3}$ (green) plotted as functions
of pressure $p$ at four values of $T_{m}=4.6$, 4.91, 5.2, and 5.5 K (see
insets in panels) and four values of $\Delta T=0.01$, 0.1, 0.2, and 0.4 K (cf.
Figs. 3 and 5). The vertical red dashed lines in panels with $T_{m}=4.6$ K and
$4.91$ K denote the $p$ values at the SVC.

Figure 9: Correlation of the reconstructed values of the NOB parameters
$\xi_{1}$ (a), $\xi_{2}$ (b) and $\xi_{3}$ (c) of Eqs. (23)-(25) with the
experimental values of $\xi_{\rm NOB}$ of Eq. (3). The gray line is the ideal
fit for visual guidance. The red and yellow curves at the bottom of each panel
represent the standard deviation $s$ and the sum of residuals $|r|$ for the
ensembles of $N_{\rm rec}=100$ neural network runs to obtain each point,
respectively; see Appendix B for details.

###  V.5 Discussion of the reconstruction methods

The results in Secs. V.2–V.4 display a general trend for the magnitude of NOB
effects in RBC in cryogenic helium gas quantified by the response parameters
$\xi_{i}$, $i=1,2,3$. All of them grow significantly as we increase pressure
$p_{m}$ and decrease temperature $T_{m}$ towards the phase boundary at SVC and
towards the CP. This is seen throughout Figs. 3 and 5 for different $\Delta T$
values, chosen here to cover the ranges of $\Delta T$ taken in high-$Ra$
turbulent RBC experiments. Figure 8 shows in more detail the differences
between individual reconstructions obtained by different neural network
architectures. We can observe that while the differences between $\xi_{2}$ and
$\xi_{3}$ are practically negligible, the reconstruction calculating
$\xi_{1}$, involving only the basic experimental parameters, namely
$p_{m}$,$T_{m}$ and $\Delta T$, significantly differ from $\xi_{2}$ and
$\xi_{3}$, which in addition take into account the linear and quadratic terms
in the expansion of the material properties of helium as a function of
temperature, expressed by the dimensionless NOB control parameters
$\varepsilon_{i}$, $i=1,...,10$, given in Tab. 1. In particular, $\xi_{2}$ and
$\xi_{3}$ show much more pronounced growth at the phase boundary near the SVC
and at the CP as a function of pressure than $\xi_{1}$. Further, the $\xi_{2}$
and $\xi_{3}$ curves are on average markedly more convex, while $\xi_{1}$
often grows in a more concave fashion towards the SVC and/or CP, see e.g. the
most pronounced case for $\Delta T=0.2$ K and $T_{m}=5.2$ K (very close to the
critical temperature $T_{\rm cri}=5.195$ K). Note that the occurrence of
crossings of the concave regions of $\xi_{1}$ with convex parts of $\xi_{2}$
apparent in Fig. 8 explains appearance of tongue-shaped features (contours of
$|\xi_{1}-\xi_{2}$ =0) seen in Fig. 6.

In addition to the three reconstructions (23)-(25) detailed in Secs. V.2–V.4,
we performed also several four-parameter reconstructions taking individual
$\varepsilon_{i}$, $i=1,...,10$ one-by-one in addition to $p_{m}$, $T_{m}$ and
$\Delta T$ within the appropriate neural network. Each of the results differed
from the $\xi_{1}$ surfaces shown in Fig. 3 by less then the experimental
accuracy of $\leavevmode\nobreak\ 3$mK. Thus only the combination of the
linear expansion parameters, resulting in $\xi_{2}$ of Eq. (24), can be
considered significant.

In Fig. 9, we finally plot the $\xi_{\mathrm{NOB}}$ results obtained by the
machine learning ($y$-axis) in comparison to the experimental data ($x$-axis)
for $\xi_{1}$ (a), $\xi_{2}$ (b), and $\xi_{3}$ (c). Each point thus
corresponds to an experimental value and a value obtained from the neural
network. In addition, a line $y=x$ for an ideal fit to the data is shown. The
presented spread of points shows a fairly good reconstruction of the non-
Boussinesq parameters by the ML algorithm.

##  VI Conclusion and outlook

In this work, we investigated the non-Oberbeck-Boussinesq behavior in high-
Rayleigh-number laboratory experiments of turbulent Rayleigh-Bénard convection
in cryogenic helium 4He. The NOB effects in this experimental setup are shown
to be caused by the temperature dependence of the material properties at the
molecular level, while the compressibility effects can be neglected with a
Mach number of $M\lesssim 10^{-2}$. The temperature $T_{c}$ measured at the
center of the RBC cell is found to deviate from the arithmetic mean of the
temperatures of the copper plates at the top and bottom, $T_{m}$. This is an
indicator of an asymmetry of the statistical properties between the top and
bottom in the cell, which unambiguously signifies breaking of the OB
approximation. The deviations have been determined in a series of experiments
which provide a sparse data set to reconstruct the function $\xi_{\rm
NOB}(p,T)=T_{m}-T_{c}$ by a nonlinear regression. The experiments are
characterized by the operating point in the pressure–temperature plane,
$(p_{m},T_{m})$, and the outer temperature difference, $\Delta T$. We thus
provide a smooth approximation (at different levels of accuracy) for the
strength of the NOB effects with respect to two state variables.

In detail, we performed reconstructions of the NOB parameter by deep neural
networks in three different ways. The first approach is based on the operating
point $p_{m},T_{m}$ and $\Delta T$. The second and third approaches
incorporate the expansion coefficients up to the linear and quadratic orders
of the Taylor expansion with respect to $\delta T=T-T_{m}$ of the material
properties and mass density, respectively. The comparison of the different
methods can be summarized as follows: (i) The inclusion of the linear-order
temperature expansion alters the reconstruction results by up to 40%. (ii) The
inclusion of the second expansion order does not alter the magnitude of the
NOB effects significantly. Our study provides a first systematic
reconstruction of the NOB effects in experiments with cryogenic helium, and
renders a set of maps for expected NOB effects in a wide area of the
pressure–temperature plane for different values of $\Delta T$ in future
experiments. A systematic analysis of the impact of the expansions on the heat
and momentum transfer could be a next step which would require numerical
investigations of the OB configuration for these parameters.

###### Acknowledgements.

The joint project is supported by grant no. 21-06012J of the Czech Science
Foundation (GAČR) for M.M. and by grant no. SCHU 1410/31-1 of the Deutsche
Forschungsgemeinschaft (DFG) for G.Z. The work of G.Z. is co-funded by the
European Union (ERC, MesoComp, 101052786). Views and opinions expressed are
however those of the authors only and do not necessarily reflect those of the
European Union or the European Research Council. Neither the European Union
nor the granting authority can be held responsible for them. Finally, we thank
Ladislav Skrbek, Olga Shishkina, and Valentina Valori for discussions.

##  Appendix A Details of the deep neural networks for nonlinear regression

In the following, we will provide some technical details of the machine
learning methods that were used for the nonlinear regression analysis, see
e.g. Goodfellow2016 . In order to calculate all results, we had to use network
models with different input and output parameters. They are provided in the
following three tables together with some information. In the 1st model we
trained a neural network without any information about the perturbative
expansion as explained in the main text. The model consists of a total of 6
layers. These are (1) linear layers which apply a linear transformation of the
incoming data followed by (2) a rectified linear unit (ReLU) function which
performs an element-wise nonlinear activation. ReLU and the parametric ReLU
(PReLU) are given by

${\rm ReLU}(x)=\max_{x}(0,x)\quad\mbox{and}\quad{\rm
PReLU}(x)=\max(0,x)+a\min(0,x)\,,$ (27)

for input data $x$ with parameter $a>0$. Furthermore, (3) the BatchNorm1d
function of the PyTorch library is applied as a batch normalization to fix
expectation value E and variance Var of the input in order to accelerate the
training. This function is given by

${\rm BatchNorm1d}(x)=\frac{x-{\rm E}[x]}{\sqrt{{\rm Var}[x]+\epsilon}}\,,$
(28)

for input data $x$ in the form of a mini-batch. Finally, (4) the sigmoid
applies another element-wise nonlinear activation which is given by

$\sigma(x)=\frac{1}{1+\exp(-x)}\,.$ (29)

This first neural network in table 2 obtains $\xi_{1}$ as the output resulting
from three quantities at input, see Eq. (23).

Layer | Output shape |  Number of parameters
---|---|---
Linear | [16, 400] | 1600
ReLU | [16, 400] | 1
BatchNorm-1d | [16, 400] | 800
Linear | [16, 1] | 401
BatchNorm-1d | [16, 1] | 2
Sigmoid | [16, 1] | 0
Total |  | 2804
Table 2: Details of the deep neural network which is used for the first
approach. The output shape consists of the batch size and the number of
weights.

For our nonlinear regression task, there are however only 236 data points
available to train the neural network. This is a small number for a full
training of a deep neural network in the considered pressure-temperature
interval. It can lead to the fact that in the randomly chosen input data for
the training-testing procedure, no data point near the critical line/point is
chosen at all. This in turn can result in an incorrect approximation of the
non-Boussinesq parameter $\xi_{\rm NOB}$. In order to remove this effect of an
insufficient amount of experimental data, training and testing were performed
in a cycle of 100 runs at each of the three levels. Each run takes a randomly
chosen subsample of the RBC data and uses the rest for testing. The deviation
of each calculation from the mean is shown in appendix B. In this way, we
obtained more robust regression results for a lacking number of RBC
experiments.

The second reconstruction method includes the parameters $\varepsilon$ of the
linear order expansion. These parameters, however, cannot be obtained close to
the phase boundary from HEPAK and were thus reconstructed first by a neural
network. The corresponding architecture is specified in the right fraction of
table 3. Input is again $p_{m}$, $T_{m}$, and $\Delta T$. With these input
parameters, we obtain 5 outputs, namely $\varepsilon_{1}$, $\varepsilon_{3}$,
$\varepsilon_{5}$, $\varepsilon_{7}$, and $\varepsilon_{9}$. Subsequently, we
add them as further input parameters, i.e., we have a total of 8 inputs to
calculate the contour plot of $\xi_{2}$ in the $p$–$T$ plane. The recursion is
now more complicated in comparison to the first approach because all property
gradients in the experimental data fluctuate strongly, particularly close to
the phase boundary. Also, the model is more complex since it includes more
parameters. The architecture of the neural network is basically the same for
both substeps.

| Reconstruction of $\xi_{2}$ |  | Evaluation of $\varepsilon_{2i+1}$
---|---|---|---
Layer | Output shape |  Number of parameters |  | Output shape |  Number of parameters
Linear | [16, 400] | 3600 |  | [16, 500] | 2000
PReLU | [16, 400] | 1 |  | [16, 500] | 1
BatchNorm-1d | [16, 400] | 800 |  | [16, 500] | 1000
Linear | [16, 1] | 401 |  | [16, 5] | 2505
BatchNorm-1d | [16, 1] | 2 |  | [16, 5] | 10
Sigmoid | [16, 1] | 0 |  | [16, 5] | 0
Total |  | 4804 |  |  | 5516
Table 3: Details of the deep neural network which is used for the second
approach. The output shape consists of the batch size and the number of
weights for each network layer.

The third reconstruction method is similar to the second. Now, we use linear
and quadratic orders, reconstruct the 10 expansion coefficients first, and
obtain $\xi_{3}$ from a network with 13 inputs. The architecture is specified
in table 4. It is basically similar to the second reconstruction approach.

| Reconstruction of $\xi_{3}$ |  | Evaluation of $\varepsilon_{i}$
---|---|---|---
Layer | Output shape |  Number of parameters |  | Output shape |  Number of parameters
Linear | [16, 400] | 5600 |  | [16, 500] | 2000
PReLU | [16, 400] | 1 |  | [16, 500] | 1
BatchNorm-1d | [16, 400] | 800 |  | [16, 500] | 1000
Linear | [16, 1] | 401 |  | [16, 10] | 5010
BatchNorm-1d | [16, 1] | 2 |  | [16, 10] | 20
Sigmoid | [16, 1] | 0 |  | [16, 10] | 0
Total |  | 6804 |  |  | 8031
Table 4: Details of the deep neural network which is used for the third
approach. The output shape consists of the batch size and the number of
weights.

##  Appendix B Error analysis of the nonlinear regression

Fig. 10 shows the deviations of the $N_{\rm rec}=100$ individual
reconstructions, which are indexed with $i$, of the non-Boussinesq parameter
$\langle\xi_{k}(i)\rangle_{p,T}$ from the mean of all calculations for $k=1,2$
and 3. We therefore first average the reconstructed field $\xi_{k}(T,P)$ over
the $p$–$T$ plane which is indicated by $\langle\cdot\rangle_{p,T}$. The mean
non-Boussinesq parameter is eventually obtained by

$\overline{\langle\xi_{k}\rangle}_{p,T}=\frac{1}{N_{\rm
rec}}\sum_{i=1}^{N_{\rm
rec}}\langle\xi_{k}(i)\rangle_{p,T}\quad\mbox{for}\quad k=1,2,3\,.$ (30)

Figure 10: Reconstructions of $\xi_{i}(T,p)$. Deviation from the mean of the
100 individual reconstructions. Points are $\langle\xi_{i}(i)\rangle_{p,T}$,
the solid line is $\overline{\langle\xi_{i}\rangle}_{p,T}$.

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