# Reconstructing Rayleigh-Bénard flows out of temperature-only measurements
using nudging

Lokahith Agasthya<EMAIL_ADDRESS>Department of Physics, University of
Rome “Tor Vergata” and INFN, Via della Ricerca Scientifica 1, 00133, Rome RM,
Italy  Angewandte Mathematik und Numerische Analysis, Bergische Universität
Wuppertal, Gaußstrasse 20 D-42119 Wuppertal, Germany  Computation-based
Science and Technology Research Center, The Cyprus Institute, 20 Kavafi Str.,
Nicosia 2121, Cyprus  Patricio Clark Di Leoni  Department of Mechanical
Engineering, Johns Hopkins University, Baltimore, MD 21218, USA  Departmento
de Física, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos
Aires, Buenos Aires 1428, Argentina  Luca Biferale
https://people.roma2.infn.it/~biferale/ Department of Physics, University of
Rome “Tor Vergata” and INFN, Via della Ricerca Scientifica 1, 00133, Rome RM,
Italy

(August 29, 2024)

###### Abstract

Nudging is a data assimilation technique that has proved to be capable of
reconstructing several highly turbulent flows from a set of partial
spatiotemporal measurements. In this study we apply the nudging protocol on
the temperature field in a Rayleigh-Bénard Convection system at varying levels
of turbulence. We assess the global, as well as scale by scale, success in
reconstructing the flow and the transition to full synchronization while
varying both the quantity and quality of the information provided by the
sparse measurements either on the Eulerian or Lagrangian domain. We asses the
statistical reproduction of the dynamic behaviour of the system by studying
the spectra of the nudged fields as well as the correct prediction of the heat
transfer properties as measured by the Nusselt number. Further, we analyze the
results in terms of the complexity of the solutions at various Rayleigh
numbers and discuss the more general problem of predicting all state variables
of a system given partial or full measurements of only one subset of the
fields, in particular temperature. This study sheds new light on the
correlation between velocity and temperature in thermally driven flows and on
the possibility to control them by acting on the temperature only.

††preprint: AIP/123-QED

##  I Introduction

Thermally driven flows are of fundamental importance in several geophysical as
well as industrial processes, including but not limited to atmospheric
convectionHartmann, Moy, and Fu (2001); Suselj, Kurowski, and Teixeira (2019),
convection in the ocean Våge  _et al._ (2018), mantle convection Kronbichler,
Heister, and Bangerth (2012) and melting of pure metals Brent, Voller, and
Reid (1988). The intensity of the thermal instabilities in these flows is
characterized by the Rayleigh number, which is the ratio between the buoyant
and viscous forces scaled by the ratio of the momentum and thermal
diffusivity. At increasing Rayleigh number, natural convection displays a rich
set of different dynamical behaviour, including transition to chaos, formation
of patterns and finally fully developed turbulence Ahlers, Grossmann, and
Lohse (2009). Despite being omnipresent in nature, several facts about
convective flows remain unsettled. For example, in Rayleigh-Bénard convection,
it is conjectured that there exists a cross-over length $L_{B}$, the so-called
Bolgiano lengthscale, such that for $L\gg L_{b}$ the Bolgiano-Obukhov (BO59)
Bolgiano Jr (1959); Obukhov (1959) scaling is expected while for $L\ll L_{b}$
, the scaling is thought to comply with Kolmogorov (K41) prediction Kolmogorov
(1941); Lohse and Xia (2010). The two regimes entail very different velocity-
temperature correlations: in the BO59 case temperature is strongly active at
all scales and kinetic energy cascades to the small scales via interactions
with the thermal component, while in K41, kinetic energy is mainly injected by
buoyancy in the bulk and transferred to high wavenumbers via the nonlinear
Navier-Stokes terms. Similarly, the interplay between large-scale thermal
plumes and small-scale and strongly intermittent fluctuations is also not
entirely understood Frisch (1995). Thus, the exact nature of the expected
energy spectrum and the particular scaling regimes is still a subject of
active research.

Related to the question of the nature of the velocity-temperature
correlations, lies an old conjecture stated by Charney Charney, Halem, and
Jastrow (1969), which has strong implications to both our fundamental
understanding of convective flows and to their applications. The conjecture
states that temperature measurements alone are sufficient to predict all the
state variables of the atmosphere. Early studies Ghil, Shkoller, and Yangarber
(1977); Ghil, Halem, and Atlas (1979) showed that the conjectures is true in
simple systems, but also presented numerical evidence to the contrary. In the
past few years the conjecture has been studied in different models, such as in
a 3D Planetary Geostrophic model Farhat, Lunasin, and Titi (2016), a Rayleigh-
Bénard flow in the infinite Prandtl number limit Farhat  _et al._ (2020), and
a Rayleigh-Bénard flow in the non-turbulent regime Altaf  _et al._ (2017). The
first two studies were successful in reconstructing the whole state out of
thermal measurements, but the third study struggled to do so when the prior on
the velocity field was not set correctly. So while the conjecture seems to
hold in simple systems, it is not clear what can happen in highly-turbulent
scenarios with large scale separation. In these cases the correlations between
temperature and velocity at the smaller scales may not be strong enough for
the temperature to fully enslave the evolution of the velocity field. The
conjecture has started to gain even more importance in past years, as weather
forecast centers are starting to be able to resolve scales where three-
dimensional turbulent convection becomes relevant Leutwyler  _et al._ (2016);
Yano  _et al._ (2018). Outside the realm of convective flows, the question of
what fields or components carry enough information to allow for full
reconstruction of an entire system is an open-ended problem that sits at the
heart of all disciplines related to Data Assimilation (DA) Kalnay (2003);
Bauer, Thorpe, and Brunet (2015).

\begin{overpic}[width=433.62pt]{Fig1.pdf} \end{overpic} Figure 1: Snapshots of
the temperature fields for the reconstruction experiments for flow with
$\rm{Ra}=7.2\times 10^{7}$. Panel (a) shows the ground-truth while (b), (c),
and (d) show the constructed nudging field $T_{n}(\boldsymbol{r},t)$ with
$k_{l}=1/14$, $k_{l}=1/31$ and $1/97$, respectively. The temperatures are
normalised to range between $-1$ (blue) and $+1$ (red) by dividing by $T_{d}$.
The individual squares here correspond to the nudging squares $S_{i}$ with
side length $\chi=6$. The typical length $l$ between the probes is shown in
panel (d).

The main aim of this paper is to address the above issues in turbulent flows
head on. We focus on Rayleigh-Bénard flows in 2D and utilize a Data
Assimilation technique called Nudging Hoke and Anthes (1976); Lakshmivarahan
and Lewis (2013) as a means to reconstruct the whole state of the system given
only information on temperature. Nudging consists of adding a relaxation
feedback term to the system that penalizes the flow when it strays away from a
reference set of data. When the nudged flow synchronizes to the reference
flow, missing data from either missing variables or missing scales can be
recovered. References Farhat, Lunasin, and Titi (2016); Farhat  _et al._
(2020); Altaf  _et al._ (2017) listed above, used nudging to reconstruct the
velocity fields using only thermal data, as per the Charney conjecture. In a
similar vein, Farhat, Jolly, and Titi (2015); Farhat, Lunasin, and Titi (2017)
studied the problem of nudging a Rayleigh-Bénard flow using only velocity
information from a purely theoretical approach and Farhat  _et al._ (2018) did
it using only vorticity information. In non-thermal flows, Lalescu, Meneveau,
and Eyink (2013); Clark Di Leoni, Mazzino, and Biferale (2020a) demonstrated
that nudging can accurately reconstruct missing scales in highly turbulent
flows, and Clark Di Leoni, Mazzino, and Biferale (2018); Buzzicotti and Clark
Di Leoni (2020) showed that nudging is sensitive to discrepancies in
parameters between the nudged and reference model. Nudging has also been used
in finite dimensional dynamical systems and weather models Hoke and Anthes
(1976); Auroux and Blum (2008); Du and Smith (2013); Pazó, Carrassi, and López
(2016), and for boundary condition matching von Storch, Langenberg, and Feser
(2000); Waldron, Paegle, and Horel (1996); Miguez-Macho, Stenchikov, and
Robock (2004). Our focus in this paper is on turbulent 2D Rayleigh-Bénard
flows, where we performed numerical experiments to investigate the effects of
using different amounts of information (i.e., distance between measuring
probes), different types of information (whether the data is taken from moving
Lagrangian probes or fixed Eulerian ones) and the dependence of the results on
the Rayleigh number (i.e., the intensity of the turbulence). The study is a
step towards understanding the feasibility of data assimilation techniques on
real-world observational data when information on one or more state variables
are missing. Our main result concerns the limit up to which the velocity field
can be inferred from the temperature field. In particular, we show that there
exist a transition  for some characteristic Rayleigh number, $\rm{Ra}\sim
10^{7}$ where the capability to construct the whole velocity configuration
deteriorates suddenly.

This article is organised in the following way. In Sec. II we give an overview
of the nudging technique and the specific protocol employed in this study.
Sec. III details the numerical experiments and the parameter space explored.
Sec. IV describes the key results of the study. Sec. V presents our
conclusions and future research directions.

##  II Nudging the Rayleigh-Bénard flow

\begin{overpic}[width=433.62pt]{Fig2.pdf} \end{overpic} Figure 2: (a)
$E_{\mathbf{u}}$ and (b) $E_{T}$ for the two reference flows. Panel (c) shows
$v(x,z_{0},t)$ at various times during the run for the lower $\rm{Ra}$
reference flow with $\rm{Ra}=7.2\times 10^{7}$ where each curve represents a
particular instant of time. Panel (d) shows the same for the higher $\rm{Ra}$
reference flow with $\rm{Ra}=36.3\times 10^{7}$.  Figure 3: (a) Evolution of
the kinetic energy $K(t)(\times 10^{5})$ in simulation units for the reference
flow with $\rm{Ra}=7.2\times 10^{7}$ and the corresponding reconstructed flows
for various $k_{l}$. (b) Evolution of the kinetic energy $K(t)(\times 10^{5})$
in simulation units for the Reference flow with $\rm{Ra}=36.3\times 10^{7}$
and the corresponding reconstructed flows for various $k_{l}$. In both panels,
the vertical red line shows the time when the reconstructed flows are
considered to have attained the statistically stationary regime in which the
various measurements and analysis are performed.

A Rayleigh-Bénard flow can be described as a compressible fluid bounded at the
top and bottom by two plates, each at a specific and constant temperature.
Under the Boussinesq approximation, density fluctuations are assumed to be
small, such that the flow can be solved in the incompressible regime and
density enters the equations only through the buoyancy term. The full 2D
equations for the velocity $\mathbf{u}=(u,v)$, temperature $T$, and pressure
$p$ take the form

$\displaystyle\partial_{t}\mathbf{u}+(\mathbf{u}\cdot\mathbf{\nabla})\mathbf{u}=-\mathbf{\nabla}p+\nu\nabla^{2}\mathbf{u}-\beta
T\bf g,$ (1) $\displaystyle\frac{\partial T}{\partial
t}+\mathbf{u}\cdot\mathbf{\nabla}T=\kappa\nabla^{2}T,$ (2)
$\displaystyle\nabla\cdot\mathbf{u}=0,$ (3)

where the average temperature of the fluid is set to zero and the density to
unity. Here, $\beta$ is the thermal expansion coefficient of the fluid at
temperature $T_{0}=0$, $\nu$ is the kinematic viscosity of the fluid, $\kappa$
is the thermal conductivity of the fluid and ${\bf g}$ is the acceleration due
to gravity. The domain is a rectangle with horizontal and vertical length
$L_{x}$ and $L_{z}$, respectively, and is periodic in the horizontal $x$
direction. The vertical temperature and velocity boundary conditions are
given, respectively, by

$\begin{gathered}T(z=0)=T_{d},\qquad T(z=L_{z})=-T_{d},\\\
\mathbf{u}(z=0)=\mathbf{u}(z=L_{z})=0.\\\ \end{gathered}$ (4)

where $T_{d}$ is positive. We also define a velocity scale $u_{0}=\sqrt{|{\bf
g}|L_{z}\beta\Delta T}$, where $\Delta T=2T_{d}$ is the temperature difference
between the top and bottom walls, and a turnover time scale
$\tau_{0}=2L_{z}/u_{0}$. The two dimensionless numbers that characterize the
flow are the Rayleigh number $\rm{Ra}$ and the Prandtl number $\rm Pr$
defined, respectively, as

$\begin{gathered}{\rm{Ra}}=|{\bf g}|\beta\frac{\Delta
TL_{z}^{3}}{\nu\kappa},\qquad{\rm Pr}=\frac{\nu}{\kappa}.\end{gathered}$ (5)

An important response parameter measured in the Rayleigh-Bénard system is the
dimensionless Nusselt number, given by

${\rm{Nu}}(z)=\frac{\langle
vT-\kappa\partial_{z}T\rangle_{x,t}}{\frac{\kappa\Delta T}{L_{z}}},$ (6)

where $\langle\cdot\rangle_{x,t}$ indicates the time and spatial averages at a
given height $z$. The Nusselt number measures the ratio of vertical heat
transfer due to convection to the vertical heat transfer due to conduction. In
the Rayleigh-Bénard convection, the Nusselt number is constant and has the
same value at all heights so,

$\langle{\rm{Nu}}\rangle={\rm{Nu}}(z),\qquad\text{for }0\leq z\leq L_{z},$ (7)

where $\langle\cdot\rangle$ indicates the average over the entire domain and
time.

We also define the Kolmogorov length scale as
$\eta_{\kappa}=\left({\nu^{3}}/{\epsilon}\right)^{\frac{1}{4}}$, where
$\epsilon=({\nu\kappa^{2}}/{L_{z}^{4}})\rm{Ra}(\langle\rm{Nu}\rangle-1)$ is
the the average rate of energy dissipation Siggia (1994).

In this study, we assume to have a set of passive probes at positions
$\boldsymbol{X}^{i}(t)$, with $i=1,\dots,N_{p}$, suspended in a reference
Rayleigh-Bénard flow. The probes make periodic measurements of the fluid
temperature $T^{i}(t)=T(\boldsymbol{X}^{i}(t),t)$ with constant sampling
frequency $f=1/\tau$ where $\tau$ is the time difference between two
successive temperature measurements by the probes. We set $f$ high enough so
that a simple linear interpolation in time results in an error smaller than
$1$ part in $10^{5}$. This is done by considering the evolution of the
temperature at individual grid points and performing a linear interpolation in
time with varying sampling frequency. $f$ was chosen as the smallest value so
the sampling was smooth and faster than the fastest time scale of the system.
This way additional complexities of errors arising from temporal interpolation
are avoided and we can assume to know $T^{i}(t)$ at every time-step. In the
main section of the paper, the probes are considered to be Eulerian, i.e.
their locations $\boldsymbol{X}^{i}(t)$ are kept fixed in the laboratory
reference frame. In Appendix B, we discuss and detail the case of Lagrangian
probes which do not remain fixed in time but follow trajectories of tracers
particles in the flow instead.

The idea behind nudging is to run a new system where we steer the evolution of
the flow onto the path set by the data. To do this, we construct a nudging
field $T_{n}(\boldsymbol{r},t)$ using the measurements $T^{i}(t)$ and run
another Rayleigh-Bénard system, denoted by velocity
$\bm{\mathcal{U}}=(\mathcal{U},\mathcal{V})$ and temperature $\mathcal{T}$,
where an extra heat source term $-\alpha(\mathcal{T}-T_{n})$ is added to the
the evolution equation for $\mathcal{T}$. This term penalizes the nudged
temperature when it deviates from the reference values - if the temperature is
higher (lower) than the nudging field, you locally absorb (release) heat. For
each probe, we define a nudging square $S_{i}$ with centre at
$\boldsymbol{X}^{i}$ and fixed side length $\chi$ typically chosen of the same
order as $\eta_{\kappa}$, (see also Table 2) and we nudge only in regions
belonging to the sub-domain $\mathcal{S}\coloneqq\cup_{i=1}^{N_{p}}S_{i}$. We
do this by defining $\alpha$ as

$\alpha(\boldsymbol{r},t)=\begin{cases}\alpha_{0},&\text{for
}\boldsymbol{r}\in\mathcal{S},\\\ 0,&\text{otherwise,}\end{cases}$ (8)

and the nudging field is defined as

$T_{n}(\boldsymbol{r},t)=T^{i}(t),\qquad\text{for }\boldsymbol{r}\in S_{i},$
(9)

that is, the measured temperature $T^{i}(t)=T(\boldsymbol{X}^{i}(t),t)$ is set
as the nudging temperature uniformly in a square of length $\chi$ centred at
the probe-location. Thus, the full equations for the nudged fields read

$\displaystyle\partial_{t}\bm{\mathcal{U}}+(\bm{\mathcal{U}}\cdot\nabla)\bm{\mathcal{U}}=-\mathbf{\nabla}\mathcal{P}+\nu\nabla^{2}\bm{\mathcal{U}}-\beta\mathcal{T}\bf{g},$
(10) $\displaystyle\frac{\partial\mathcal{T}}{\partial
t}+\bm{\mathcal{U}}\cdot\mathbf{\nabla}\mathcal{T}=\kappa\nabla^{2}\mathcal{T}-\alpha(\mathcal{T}-T_{n}),$
(11) $\displaystyle\mathbf{\nabla}\cdot\bm{\mathcal{U}}=0,$ (12)

where $\alpha(\boldsymbol{r},t)$ sets the strength of the coupling between the
reconstructed flow and the nudging field and has dimension $1/t$. It is very
important to stress that the only energy input into the nudged system is via
the nudging field, i.e. we impose adiabatic boundary conditions at the top and
bottom walls.

$\partial_{z}\mathcal{T}|_{z=0}=\partial_{z}\mathcal{T}|_{z=L_{z}}=0.$ (13)

In other words, the basic line for the reconstruction has no prior - if
$T_{n}=0$, the flow is zero everywhere. Since the Rayleigh number of the
Rayleigh-Bénard convection is set by the temperature boundaries, it is likely
that sampling preferentially from near the walls near the thermal boundary
would result in the most accurate reconstruction. However, in the interest of
a fair assessment of the nudging protocol, we consider only temperature probes
on a uniform array without supposing we know anything about the nature of the
thermal boundary. As a result the only information about the $\rm{Ra}$
available is that which is encoded in the probe measurements.

Here and hereafter we characterize the spatial density of the probes by a
characteristic wavenumber $k_{l}$ given by

$k_{l}=\frac{\Delta r}{l},$ (14)

where $\Delta r=1$ is the grid-spacing in the numerical algorithm to evolve
the flow and $l$ is the typical distance between the nearest probes. As the
number of probes increases, $l$ decreases and $k_{l}$ increases as
$\sqrt{N_{p}}$. The case of $k_{l}=1$ corresponds to the situation where we
have complete information, since there are probes located at every grid-point.
In Fig. 1 we show the temperature snapshot of a temperature field as well as
the nudging fields $T_{n}(\boldsymbol{r},t)$ constructed from this field for
three different values of $k_{l}$. From the figures, it is clear that for
$k_{l}\gtrsim 0.1$ the density of point measurements should be enough to be
able to accurately interpolate the temperature field while for $k_{l}\lesssim
0.03$ or smaller, the input information is rather limited. The problem then is
two-fold — to understand how much information about the temperature field is
needed to reconstruct it to a given degree of accuracy and to investigate the
extent up to which the velocity field can be reconstructed from temperature
data.

##  III Numerical Experiments

\begin{overpic}[width=433.62pt,keepaspectratio]{Fig4.pdf} \par\end{overpic}
Figure 4: Snapshots of the (a) normalised ground truth field $T/T_{d}$ for
$\rm{Ra}=7.2\times 10^{7}$, (b) the normalised reconstructed temperature
field, $\mathcal{T}/T_{d}$ for $k_{l}=1/14$ corresponding to a distance
between probes $l\sim 7\eta_{\kappa}$ and (c) the error,
$\delta_{T}(\boldsymbol{r},t)$ at a given instant of time. The yellow curve in
panel (c) shows the time-averaged vertical profile of $\delta_{T}$ with the
scale shown on the top right. The lower panels show the same quantities for
the higher $\rm{Ra}$ flow with $\rm{Ra}=36.3\times 10^{7}$.
\begin{overpic}[width=433.62pt,keepaspectratio]{Fig5.pdf} \par\end{overpic}
Figure 5: Snapshots of the (a) normalised ground truth field $v/u_{0}$ for
$\rm{Ra}=7.2\times 10^{7}$, (b) the normalised reconstructed field,
$\mathcal{V}/u_{0}$ for $k_{l}=1/14$ and (c) the error,
$\delta_{v}(\boldsymbol{r},t)$ at a given instant of time. The yellow curve in
panel (c) shows the time-averaged vertical profile of $\delta_{v}$ with scale
shown on the top right. The lower panels show the same quantities for the
higher $\rm{Ra}$ flow with $\rm{Ra}=36.3\times 10^{7}$.

###  III.1 Reference flows

Equations (1) - (3) are evolved using the Lattice-Boltzmann method (details in
Appendix A) until the average kinetic energy of the flow becomes statistically
stationary. Once a stationary state is reached, the $N_{p}$ passive probes are
initialised in the domain on a uniform and equally spaced grid and the data
from the probes, that is the fluid temperatures $T^{i}(t)$ at
$\boldsymbol{X}^{i}(t)$, is obtained for $\sim 250$ turn-over times
$\tau_{0}$. This measured data is used to construct the nudging field $T_{n}$
as detailed in section II. The flows studied have $\rm Pr=1$ and $\rm{Ra}$
between $10^{7}$ \- $10^{9}$, which is the well-known transition to turbulence
regime in Rayleigh-Bénard convection. Two flows in particular, one with a
moderate value of $\rm{Ra}=7.2\times 10^{7}$ and a second one with a higher
value of $\rm{Ra}=36.3\times 10^{7}$ (see Table. 1) are studied and presented
in more detail.

Reference Flows
---
$\boldsymbol{\rm{Ra}}$ | $\boldsymbol{7.206\times 10^{7}}$ | $\boldsymbol{36.27\times 10^{7}}$
$\rm Pr$ | 1.0 | 1.0
Grid | $864\times 432$ | $1200\times 600$
$\nu$ | $6.67\times 10^{-4}$ | $6.67\times 10^{-4}$
$\rm{Re}$ | 2091 | 6092
$\langle\rm{Nu}\rangle$ | 24.835 | 38.802
$\eta_{\kappa}$ | 2.12 | 1.75
$T_{d}$ | 0.025 | 0.015
$f$ | 164 | 1130
Table 1: The parameters used for the reference flows. Frequency of sampling
$f$ is in units of probes measurements per turnover time $\tau_{0}$ while all
other parameters are in simulation units. $\rm{Re}$ is defined as
${\rm{Re}}=u_{\rm{rms}}L_{z}/\nu$ where $u_{\rm{rms}}$ is the root mean square
velocity of the flow.

We define two important spectral quantities to understand the small-scale
structure of the flow better. First, the time-averaged spectrum (all spectra
in this study are measured only in the horizontal direction in a narrow band
about the line $z=z_{0}=L_{z}/2$) of the velocity field $\mathbf{u}$, or the
kinetic energy spectrum, given by

$E_{\mathbf{u}}(k_{x})=\frac{1}{2}\left\langle\left|\hat{\mathbf{u}}(k_{x},z_{0},t)\right|^{2}\right\rangle_{t},$
(15)

where $\hat{\mathbf{u}}(k_{x},z_{0},t)$ are the Fourier coefficients of the
field $\mathbf{u}$ and $\langle.\rangle_{t}$ denotes the time averaging.
Similarly, the spectrum of the temperature field is defined as

$E_{T}(k_{x})=\left\langle|\hat{T}(k_{x},z_{0},t)|^{2}\right\rangle_{t},$ (16)

where $\hat{T}(k_{x},z_{0},t)$ are the Fourier coefficients of the field $T$.

The magnitude of the problem can be appreciated by looking at the reference
flow characteristics, as shown in Fig. 2 where we present $E_{\mathbf{u}}$,
$E_{T}$ and instantaneous horizontal cuts of the vertical velocity at
$z=z_{0}$, that is, $v(x,z_{0},t)$ for both reference flows. We see in panel
(b) that the spectrum of the temperature field becomes steeper when increasing
Rayleigh number, indicating the presence of less well defined temperature
plumes when turbulence increases. The slope of the temperature spectrum for
the moderate $\rm{Ra}$ case shows that the temperature field is better
correlated over a broader range of scales compared to the higher $\rm{Ra}$
flow. The energy spectra for both flows are close to a power law as shown in
panel (a). On closer inspection, one can see that the spectrum for the lower
$\rm{Ra}$ flow is indeed flatter much like the temperature spectrum. A large
contrast can also be seen in the energy contained in the first few Fourier
modes, where for the lower $\rm{Ra}$ flow, the first mode contains far greater
energy compared to the successive modes. In the higher $\rm{Ra}$ flow the
first mode is still the most energetic, but the successive modes still contain
a significant amount of energy. This contrast is borne out more clearly in the
horizontal cuts of the vertical velocity for both the flow. In the lower
$\rm{Ra}$ flow (panel (c)), even the instantaneous velocity field is smooth
and highly structured with a regular, large-scale pattern with only small
fluctuations whereas for the higher $\rm{Ra}$ flow (panel (d)), the velocity
field in the bulk is far more chaotic and rugged, with a large-scale flow
pattern not immediately discernible.

###  III.2 Nudging experiments

Once $T_{n}$ is obtained, another flow is initialised with velocity
$\bm{\mathcal{U}}(\boldsymbol{r},t)=0$ and temperature
$\mathcal{T}(\boldsymbol{r},t)=0$ everywhere and evolved according to
equations (10) – (12). As already described, the thermal boundary conditions
at the top and bottom walls are set to a no-thermal flux boundary condition so
that the only energy inputs into the nudged system arise from the nudging
term. The case of $k_{l}=1$ is a special case scenario where it is assumed
that the temperature data is available on every point at every time-step. In
this case we use the fixed temperature boundary conditions of the Rayleigh-
Bénard equations and we set $T_{n}(\boldsymbol{r},t)=T(\boldsymbol{r},t)$
everywhere.

The nudged simulations are evolved until they attain a statistically
stationary kinetic energy. All measurements and further analysis are made in
this stationary state. Corresponding to each reference flow, several nudging
experiments are performed by varying $k_{l}$, $\alpha_{0}$ and $\chi$ (see
Table I and Appendix C). Identically to the spectra already defined for the
reference flows (eqns. (15) - (16)), we define the kinetic energy spectra
$E_{\bm{\mathcal{U}}}$ and the thermal energy spectra $E_{\mathcal{T}}$ for
the nudged simulations as well.

The results presented in the rest of the study are for parameters listed in
Table. 2. Varying these parameters does not change the qualitative results
greatly and an overview of the behaviour of the system on changing
$\alpha_{0}$ and $\chi$ can be found in Appendix C. In the main text we focus
on the effects of varying $k_{l}$ and how these are affected by varying the
Rayleigh number of the reference flow. The domain-averaged kinetic energy
$K(t)$ of the flow is given by

$K(t)=\frac{1}{2}\langle|\mathbf{u}(\boldsymbol{r},t)|^{2}\rangle_{V},$ (17)

where $\langle\cdot\rangle_{V}$ denotes the average over the entire domain at
a given instant of time. The evolution of the kinetic energy for the lower
$\rm{Ra}$ reference flow and the higher $\rm{Ra}$ reference flows along with
those of the reconstructions for different $k_{l}$ with the parameters from
Table. 2 is shown in Figure 3. It is clear that the kinetic energy of the
reference flows (black curves) has attained a stationary kinetic energy. The
vertical red lines correspond to the time when the reconstructed flows attain
a statistically stationary kinetic energy.

Reconstructions
---
$\boldsymbol{\rm{Ra}=7.206\times 10^{7}}$ | $\boldsymbol{\rm{Ra}=36.27\times 10^{7}}$
$\boldsymbol{N_{p}}$ |  $\boldsymbol{k_{l}}$ |  $\boldsymbol{\chi}$ |  $\boldsymbol{\alpha_{0}}$ |  $\boldsymbol{N_{p}}$ |  $\boldsymbol{k_{l}}$ |  $\boldsymbol{\chi}$ |  $\boldsymbol{\alpha_{0}}$
$3.7\times 10^{5}$ |  1 |  - |  0.01 |  $7.2\times 10^{5}$ |  1 |  - |  0.01
7688 |  1/7 |  6 |  0.01 |  14792 |  1/7 |  6 |  0.08
3872 |  1/10 |  6 |  0.01 |  7442 |  1/10 |  6 |  0.08
1922 |  1/14 |  6 |  0.01 |  3698 |  1/14 |  6 |  0.08
800 |  1/22 |  6 |  0.01 |  1485 |  1/22 |  6 |  0.08
392 |  1/31 |  6 |  0.01 |  735 |  1/31 |  6 |  0.08
162 |  1/48 |  6 |  0.01 |  300 |  1/49 |  6 |  0.08
81 |  1/68 |  6 |  0.01 |  150 |  1/69 |  6 |  0.08
40 |  1/97 |  6 |  0.01 |  72 |  1/100 |  6 |  0.08
- |  - |  - |  - |  36 |  1/141 |  6 |  0.08
Table 2: The parameters used for the reconstructed (nudged) flows in
simulation units.

###  III.3 Error quantification

\begin{overpic}[width=433.62pt]{Fig6.pdf} \end{overpic} Figure 6: The global
reconstruction errors (a) $\Delta_{T}$, and (b) $\Delta_{v}$ for
reconstructions of the two reference flows are varying $k_{l}$. Also shown is
the baseline reconstruction error (black) for the higher $\rm{Ra}$ flow. The
arrows indicate the value of $k_{l}$ presented in detail in this study and
whose corresponding nudging fields $T_{n}$ for the moderate $\rm{Ra}$ case are
shown in Figure 1.

To compare the reconstructed configurations from the nudging experiments with
the ground-truth, we use various measures to quantify the efficacy of the
nudging technique applied here. First, we define a point-wise error for
temperature and vertical velocity given by

$T_{\Delta}(\boldsymbol{r},t)=\mathcal{T}(\boldsymbol{r},t)-T(\boldsymbol{r},t),$
(18)

and

$v_{\Delta}(\boldsymbol{r},t)=\mathcal{V}(\boldsymbol{r},t)-v(\boldsymbol{r},t).$
(19)

Next, we define the global L2-error on the temperature and vertical velocity
respectively given by

$\Delta_{T}=\frac{\langle T_{\Delta}^{2}(\boldsymbol{r},t)\rangle}{\langle
T^{2}(\boldsymbol{r},t)\rangle},$ (20)

and

$\Delta_{v}=\frac{\langle v_{\Delta}^{2}(\boldsymbol{r},t)\rangle}{\langle
v^{2}(\boldsymbol{r},t)\rangle},$ (21)

where $\langle\cdot\rangle$ indicates the average over the entire domain and
the entire (stationary) run-time. We note that perfect synchronisation would
lead to a value of $\Delta\sim 0$ while if the reconstructed flow is
statistically correct but uncorrelated with the ground truth, we would have
$\Delta\sim 2$. $\Delta_{T}$ for the reconstructed flows is compared with a
“Baseline Reconstruction” where we calculate $\Delta_{T}$ by assuming
$\mathcal{T}(\boldsymbol{r},t)=T_{n}(\boldsymbol{r},t)$ for $\boldsymbol{r}\in
S$, and setting uniformly $\Delta_{T}=1$ otherwise - that is setting the
temperature uniformly identical to the nudged field where the flow is nudged
and equal to the mean temperature, $\mathcal{T}(\boldsymbol{r},t)=0$, every
where else.

\begin{overpic}[width=433.62pt,keepaspectratio]{Fig7.pdf} \end{overpic} Figure
7: Energy and temperature spectra for the higher $\rm{Ra}$ reference flow with
$\rm{Ra}=36.3\times 10^{7}$ and the reconstructions at various $k_{l}$ \- (a)
$E_{\mathbf{u}}(k_{x})$ (black) and $E_{\bm{\mathcal{U}}}(k_{x})$. (b)
$E_{T}(k_{x})$ (black) and $E_{\mathcal{T}}(k_{x})$. The dotted vertical line
shows the wavenumber corresponding to the length $\chi$ of the nudging
squares.

To visualise the local errors at each instant, we also define an instantaneous
L2-error on the temperature and vertical velocity given by

$\delta_{T}(\boldsymbol{r},t)=\frac{T^{2}_{\Delta}(\boldsymbol{r},t)}{\langle
T^{2}(\boldsymbol{r},t)\rangle_{x,t}},$ (22)

$\delta_{v}(\boldsymbol{r},t)=\frac{v^{2}_{\Delta}(\boldsymbol{r},t)}{\langle
v^{2}(\boldsymbol{r},t)\rangle_{x,t}}$ (23)

respectively where $\langle\cdot\rangle_{x,t}$ indicates the time and spatial
averages at a given height $z$.

For a comparison of the scale-by-scale reconstruction, we also define the
spectrum of the errors of temperature and vertical velocity respectively given
by

$E^{\Delta}_{T}(k_{x})=\left\langle|\hat{T}_{\Delta}(k_{x},z_{0},t)|^{2}\right\rangle_{t},$
(24)

and

$E^{\Delta}_{v}(k_{x})=\left\langle|\hat{v}_{\Delta}(k_{x},z_{0},t)|^{2}\right\rangle_{t},$
(25)

where $\hat{T}_{\Delta}(k,t)$ and $\hat{v}_{\Delta}(k,t)$ are respectively the
fourier coefficients of the fields $T_{\Delta}$ and $v_{\Delta}$.

##  IV Results

###  IV.1 Instantaneous and Global L2-Errors

We start by showing visualizations of the reference temperature field $T$, the
nudged temperature field $\mathcal{T}$ and the temperature error field
$\delta_{T}$ at one given instant for both the flows in Fig. 4 and likewise
for $v$, $\mathcal{V}$ and $\delta_{v}$ in Fig. 5. Further, the vertical time-
averaged profile of the errors, that is $\langle\delta_{T}\rangle_{x,t}$ and
$\langle\delta_{v}\rangle_{x,t}$ are also overlaid on the snapshots of the
instantaneous errors (right-most panels, solid yellow line). The nudged fields
correspond to a case with $k_{l}=1/14$, i.e., a high density of probes as seen
in Fig. 1(b). Both reference flows are characterised by a rising hot plume and
a falling cold plume and two large-scale counter-rotating vortices. The
visualisations show clearly that the higher $\rm{Ra}$ flow (bottom row) shows
an abundance of fine-scale structure and further, the plumes are poorly-
defined, in contrast to the moderate $\rm{Ra}$ flow, where while the flow is
not completely laminar, the plumes are clearly demarcated. The nudging
protocol is able to accurately reconstruct the temperature field, even at
higher Rayleigh number. In this case the maximum error is of the order of
$10\%$ and is concentrated on the plume and vortex structures while the
thermal boundary layers show larger errors as clear from the average profile.
This profile for the moderate $\rm{Ra}$ case is smooth and the value of the
error doesn’t change very sharply depending on whether the particular region
is nudged or not (that is, whether $\boldsymbol{r}\in S$ or not). In contrast,
in the reconstruction of the higher $\rm{Ra}$ flow, the profile of the error
shows highly oscillatory behaviour, indicating that while errors are lower in
the nudged regions, they are relatively higher even in adjoining regions which
are not nudged. This is a manifestation of the ruggedness of the temperature
field when the flow is more turbulent. This is an indication that the low
magnitude of error is solely due to the high density of probes in the system
and not due to a complete synchronization of the reconstruction with the
reference flows. Closer inspection of this averaged profile (yellow curves)
near the top and bottom walls for both the flows shows that the error reaches
the highest value close to the thermal boundaries. The large error near the
thermal boundary is due to the relatively small thickness of the thermal
boundary layer and the steep temperature gradient away from the wall. It is
clear that to capture the precise behaviour of the temperature near the
boundary walls needs a higher density of probes immersed near the walls.

\begin{overpic}[width=433.62pt]{Fig8.pdf} \end{overpic} Figure 8: The spectra
of the errors (a) $E_{v}^{\Delta}$ and (b) $E_{T}^{\Delta}$ for various
$k_{l}$ for reconstructions of the higher $\rm{Ra}$ flow with
$\rm{Ra}=36.3\times 10^{7}$. For comparison, the reference spectra (a) $E_{v}$
and (b) $E_{T}$ are also shown (black squares). The inset in panel (a) shows
$E_{v}$ for the moderate $\rm{Ra}$ reference flow with $\rm{Ra}=7.2\times
10^{7}$ and $E^{\Delta}_{v}$ for the corresponding nudging experiments. The
dotted vertical line shows the wavenumber corresponding to the length $\chi$
of the nudging squares.

The situation is different for the velocity field, which is reconstructed
indirectly through the thermal forcing and depends on the response of the
velocity field to the temperature field. We see from the reference velocity
fields (Panels (a) and (d) in Fig. 5), the regions with rising fluid (red) and
falling fluid (blue) are more clearly delineated in the lower $\rm{Ra}$ flow
unlike in the higher $\rm{Ra}$ flow. The nudged field is in good agreement
with the truth for the lower Rayleigh number case (top row) while large
systematic errors on the whole volume develop for the higher $\rm{Ra}$ flow
(bottom row) where the reconstructed velocity fails to capture the precise
instantaneous shape of the reference while still broadly capturing the regions
of rising and falling fluid accurately. The average profile of the errors for
both cases are smooth. The invisibility of the nudging squares to the velocity
field is a clear indication that the velocity field is not set locally by the
local temperature but rather by the large-scale interplay between the
temperature and velocity field. The reconstruction of the vertical velocity is
most accurate in the bulk. Notice that due to the no-slip boundary condition
($\mathbf{u}=0$) imposed at the top and bottom walls for the reference as well
as reconstructed flows, the particular form of L2-error chosen here ceases to
be well-defined at the vertical boundaries.

In Fig. 6 we show the main quantitative summary of our study, where we plot
$\Delta_{T}$ and $\Delta_{v}$ as a function of $k_{l}$ for the reconstructions
of both flows. As seen in panel (a), the reconstruction error for temperature
decreases when increasing the number of probes. A transition to
synchronization-to-data can be seen at $k_{l}\sim 1/14$, corresponding to a
typical wavenumber $k_{x}\sim 0.036$ or $1/28$ in Fig. 2. Equivalently, this
corresponds to providing information at a separation of $\sim 7\eta_{\kappa}$
for the moderate $\rm{Ra}$ flow and $\sim 8\eta_{\kappa}$ for the higher
$\rm{Ra}$ flow. The transition to perfect synchronization of the temperature
fields occurs at similar scales in the two flows studied, and is similar to
that observed in homogeneous and isotropic flows Lalescu, Meneveau, and Eyink
(2013); Clark Di Leoni, Mazzino, and Biferale (2020a). The baseline
reconstruction error approaches $0$ for $k_{l}=1/7$, the scale at which for
the chosen $\chi$, the nudging squares cover the entire domain and thus, the
baseline error includes only the small-scale temperature errors introduced by
the application of a constant nudging temperature within each nudging square.
Setting $\chi$ to be larger would cause these small-scale errors to be far
larger while setting $\chi$ too small would lead to a reconstructed flow with
significantly smaller kinetic energy and increase the effective distance
between the probes, thus leading to larger global errors. The behaviour for
changing $\chi$ for a given $k_{l}$ is discussed in Appendix C.

As for the reconstruction of the velocity field shown in panel (b), while the
velocity field can be accurately reconstructed in the lower Rayleigh number
case, the higher Rayleigh number case does not synchronize even for full
information on the temperature ($k_{l}=1$), showing a plateau at close to
$40\%$ for the average minimum error committed, the plateau being reached
already at $k_{l}=1/48$. The result is not completely surprising, indicating
that for high Rayleigh number the slaving of the velocity field is less and
less effective, and the imposition of more and more information on the forcing
field (here the temperature) is not sufficient to recover full synchronization
when turbulence is intense enough.

\begin{overpic}[width=433.62pt]{Fig9.pdf} \end{overpic} Figure 9: Profile of
the Nusselt number for the ground truth and the reconstructions of the
reference flow for (a) moderate $\rm{Ra}$ flow with $\rm{Ra}=7.2\times 10^{7}$
and (b) higher $\rm{Ra}$ flow with $\rm{Ra}=36.3\times 10^{7}$. The reference
Nusselt number $\rm{Nu}(z)$ and the reconstructed Nusselt number
$\textit{Nu}(z)$ are divided by the corresponding reference Nusselt number
$\langle\rm{Nu}\rangle$ with magnitude (a) $\langle\rm{Nu}\rangle=24.835$ and
(b) $\langle\rm{Nu}\rangle=38.802$ respectively.

###  IV.2 Scale-by-scale spectral properties and spectral errors

In Figure 7 we show the statistical reconstruction of the flow properties by
plotting the spectra for the reconstructed higher $\rm{Ra}$ flows -
$E_{\bm{\mathcal{U}}}$ and $E_{\mathcal{T}}$ \- the same quantities as shown
in the first two panels of Fig. (2) - for various $k_{l}$ along with the
ground truth in black. Panel (a) shows that the nudging experiments retrieve a
flow with a velocity field with similar dynamic properties as the reference at
all scales. In panel (b) showing the spectrum of the thermal energy, as
expected the case with $k_{l}=1$ follows the ground truth exactly while the
case with $k_{l}=1/14$ captures the correct large-scale dynamics and at the
same time introduces spurious correlations at smaller scales. This is due to
the finite sizes of the nudging squares within which the nudging temperature
is imposed uniformly, leading to systematic errors of small magnitudes for
points close to $\boldsymbol{X}^{i}$.

Further insight into the scale-wise behaviour is provided in Fig. 8, which
shows the error spectra $E_{v}^{\Delta}$ and $E_{T}^{\Delta}$ and the
corresponding reference spectra $E_{T}$ and $E_{v}$. The scale-wise relative
error can be gauged by the vertical displacement between the reference
spectrum (black) and the reconstructed spectra. In panel (b) the error
spectrum is nearly flat at large-scales with a value strongly dependent on
$k_{l}$ while $E_{T}$ falls off gradually, indicating an increasing relative
error for increasing $k_{x}$ with the lowest relative errors at the largest
scales. For larger $k_{x}$ (smaller length-scales) we see a clear
manifestation of the errors introduced by the finite nudging squares for
$k_{l}=1/14$ and $k_{l}=1/31$. The error-spectrum for $k_{l}=1/100$ is
identical to the temperature spectrum of the ground truth, a result of the
fact that the probes are extremely sparse and hence the reconstructed
temperature field is nearly $0$ everywhere. The black curve in panel (a) is
the spectrum of the vertical velocity $E_{v}(k_{x})$. The relative error is
smallest at the largest scale and at smaller scales (larger $k_{x}$), we see
that the error spectra and reference spectrum decay almost identically and
hence shows a nearly constant relative error. It is in this aspect that we see
the greatest contrast between the two $\rm{Ra}$ flows. The inset shows that
reconstructions of the moderate $\rm{Ra}$ flow has far smaller relative error
at the largest scales and this even persists for larger values of $k_{x}$. The
small-scale behaviour is similar to what is observed for the higher $\rm{Ra}$
case in the main panel.

Another important point to note is that the value of the error spectrum
$E_{v}^{\Delta}$ shows little dependence on $k_{l}$ and that the only scales
of the velocity field that truly synchronize are the largest ones. The largest
scale structure present are the hot and cold thermal plumes, which are
captured correctly by the reconstruction. When the degree of turbulence is
lower, there is a relative absence of fine-scale structures in the flow and
most of the energy is contained in the large-scale alone - this large-scale
synchronisation persists at some mid-range and smaller scales as well. For
more turbulent flows where the plumes are less well-defined and the velocity
field is more rugged, the synchronisation at the largest scales exists but to
a smaller degree and it does not persist at smaller scales. This leads
eventually to a much larger global error. These observations are explored in
further detail below.

###  IV.3 Inferring Nusselt Number

The Nusselt number defined in equation (6) measures the heat transfer due to
convection relative to that due to conduction in the Rayleigh-Bénard system.
Similarly, for the reconstructed flows we define the Nusselt number as

$\textit{Nu}(z)=\frac{\mathcal{V}\mathcal{T}-\kappa\partial_{z}\mathcal{T}}{\frac{\kappa\Delta
T}{L_{z}}},$ (26)

where $\Delta T$ is the temperature difference between the horizontal walls
for the corresponding reference flow. As already noticed, given only a set of
temperature measurements it would be impossible to infer neither the Rayleigh
number nor the Nusselt number. In Fig. 9 we show the average profile of the
Nusselt number as a function of the distance from the wall for the reference
as well as the reconstructed flows for the moderate $\rm{Ra}$ flow and the
higher $\rm{Ra}$ flow in panels (a) and (b) respectively for different
$k_{l}$. As expected, for the moderate Rayleigh number flow, the Nusselt
number is perfectly reconstructed as soon as we are close to the transition to
perfect synchronization, except at the walls where we force $\rm{Nu}\to 0$ by
setting an adiabatic boundary condition. In the $k_{l}=1$ case where the walls
have fixed temperature identical to the reference, the Nusselt profile is
reconstructed perfectly. In the higher $\rm{Ra}$ case on the other hand, even
when the temperature field is perfectly synchronised, there is a large
discrepancy between the reconstructed and the reference values. It is
interesting to notice that the main source of errors comes from the region
just after the thermal boundary layer, where the correlation between plumes
and vertical velocity drafts are particularly important. In the bulk the
reconstruction is accurate when the probes reach a high enough density
($k_{l}\geq 1/14$).

###  IV.4 Varying Rayleigh Number

\begin{overpic}[width=433.62pt]{Fig10.pdf} \end{overpic} Figure 10: (a)
$\Delta_{v}$ for the reconstructed flows with $k_{l}=1$ as a function of
$\rm{Ra}$ of the reference flows (cyan, filled triangles) along with
$1-E_{1}/E_{\rm{tot}}$ and $1-C_{s.t}$for the reference flows. Inset shows the
same plot without the log-scale on the y-axis (b) Relative scale-wise error
$E_{v}^{\Delta}/E_{v}$ for the reconstructed flows.

To further corroborate our findings reported thus far regarding the large-
scale synchronization of the reconstructed flows with the reference flows, we
conduct another series of nudging experiments for varying values of $\rm{Ra}$
with $k_{l}=1$, that is, complete information on temperature. For all the
flows, we have $\Delta_{T}\lesssim 10^{-3}$. We show the corresponding values
of $\Delta_{v}$ as a function $\rm{Ra}$ in Fig. 10(a). The inset of the Figure
shows the same but in the semilog scale. In Fig. 10(b) we show the spectra of
the errors $E^{\Delta}_{v}$ compensated by the spectra of the reference
$E_{v}$ for a selection of cases. At $\rm{Ra}\sim 10^{7}$ the velocity field
is reconstructed correctly by the nudging protocol, but at around $\rm{Ra}\sim
10^{8}$ a transition occurs, and the nudged flow is not able to synchronize to
the reference flow anymore.

In order to get a better understanding of this phenomenon, we calculate the
strength of the Large Scale Circulation (LSC) by looking at the ratio of the
energy in the lowest wavenumber (first Fourier mode) $E_{1}$ and the energy
$E_{\rm{LS}}$ contained in the first four Fourier modes Xi  _et al._ (2016)
given by

$E_{\rm{LS}}=E_{1}+E_{2}+E_{3}+E_{4},$ (27)

where $E_{i}$ is the average energy contained in the $i$-th Fourier mode. In
Fig. 10(a), we plot (blue triangles) the deviations from the LSC strength for
the various reference flows, i.e. $1-E_{1}/E_{\rm{LS}}$. At low $\rm{Ra}$ most
of the energy is contained in the largest mode, something that is suggested by
Fig. 2(c), while when $\rm{Ra}$ is increased, the flow becomes more turbulent
and disordered, as seen in Figs. 2(a) and (d) and the energy contained in the
smaller scales is more and more significant. We see that the deviations from
the LSC scale similarly as $\Delta_{v}$. The results are qualitatively the
same even if we consider $E_{\rm{LS}}$ as the sum of all Fourier modes of the
energy instead of just the first four, with the mean of $1-E_{1}/E_{\rm{LS}}$
merely shifted upward.

Further, we consider a continuous saw-tooth function $\lambda$ with period
$L_{x}$ and translated by a distance $a$ given by

$\lambda(x,a)=\begin{cases}\frac{4}{L_{x}}(x-a),&0\leq
x\leq\frac{L_{x}}{4}+a\\\
-\frac{4}{L_{x}}(x-a)+2,&\frac{L_{x}}{4}+a<x\leq\frac{3L_{x}}{4}-a\\\
\frac{4}{L_{x}}(x-a)-4,&\frac{3L_{x}}{4}-a<x\leq L_{x}\end{cases}$ (28)

with $a$ chosen at each instant such that

$\int_{x=0}^{x=L_{x}}v(x,L_{z}/2,t)\cdot\lambda(x,a)dx$

is maximum. The similarity between the saw-tooth function and the vertical
velocity signal is then calculated as the time-averaged Pearson Correlation
Coefficient, denoted $C_{s.t}$. In Fig. 10(a) we also show the value
$1-C_{s.t}$ which quantifies the deviations from the saw-tooth mode for the
vertical velocity. We see again that $\Delta_{v}$ scales similar to the
deviations from the saw-tooth function.

This picture indicates that at low $\rm{Ra}$ supplying the location of the hot
and cold plumes is enough to reconstruct the flow accurately, as the plumes
set the structure of the largest scales with a single dominant mode that can
be approximated as a cosine or a saw-tooth. When the flow becomes more
turbulent the number of structures that can exist in the flow increases and,
thus, information on the plumes positions alone cannot uniquely determine the
proper solution. This, further coupled with the fact that there exist several
features of the flow at smaller scales leads to a poor reconstruction of the
Rayleigh-Bénard Convection at larger Rayleigh numbers.

##  V Conclusion

We have presented here a study of nudging applied to a thermally driven fluid
in the presence of a bulk external forcing (gravity), where the only control
variable is the temperature. We have varied the quantity of information as
well as the quality of information to be used. We have shown that given a
greater quantity of information on the temperature, the nudging method used
here yields a superior reconstruction of the temperature field, with a
transition to full-synchronisation around $k_{l}\sim 0.07$ corresponding to a
distance between probes of around $7\eta_{\kappa}$ for both flows. The
reconstruction of the velocity field on the other hand saturates from a
relatively lower $k_{l}$ and supplying more information, even the full
temperature field with near-perfect reconstruction of temperature fails to
further improve the velocity reconstruction. The degree of synchronisation
between the reconstructed and the reference velocity fields depend on the
Rayleigh number and the degree of turbulence in the reference flow, with the
largest scales of the flow synchronising the most effectively while at smaller
scales, the velocity fields remain largely asynchronous.

The quality of information was varied by following a Lagrangian approach as
well as an Eulerian approach and in the construction of the nudging field
$T_{n}$ given the information on the temperature. While the different
approaches yielded slightly different reconstructions of the temperature
field, the accuracy of the reconstruction of the velocity field remains nearly
the same.

Rayleigh Bénard convection is driven by the response of the vertical velocity
to the local temperature. The correlation between these two fields however
manifests first at the largest scales in the velocity and local temperature
measurements alone fails to predict the local velocity. Providing sufficient
information on the temperature provides information of these largest scale
features - the plumes - and hence the large-scale flow. Our study thus helps
to understand the fundamental role played by the temperature field alone in
the Rayleigh-Bénard convection and answers the question – how much does the
velocity field depend on the quantity and quality of information available on
the temperature field? The accurate reconstruction of the velocity field at
lower Rayleigh numbers is a result of the relative absence of smaller and fine
scale structure in such flows. At higher $\rm{Ra}$, when the thermal flow is
more turbulent, richer in small-scale structures, with thermal plumes less
well-defined and rapidly fluctuating velocity field, the indirect
reconstruction of the velocity field reproduces the correct flow dynamics and
synchronises relatively poorly even at the largest scale, though the relative
error on the first Fourier mode is still of the order $\sim 10^{-1}$ for the
flow with ${\rm{Ra}}\sim 10^{9}$ and highly turbulent. To accurately
reconstruct the smaller and smaller scales would require additional inputs
into the system - it is conceivable, for example, that even a small amount of
data on the velocity field could drastically improve the accuracy of the
reconstructed velocity field and that there exists an optimum way to supply a
combination of temperature and velocity data. In this study we focus
exclusively on the reconstructions using the temperature field alone and leave
the possibility of using a combination of temperature and velocity fields for
future work.

## Acknowledgements

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 765048. This work was supported by the European Research Council
(ERC) under the European Union’s Horizon 2020 research and innovation
programme (Grant Agreement No. 882340).

## Data Availability Statement

Data available on request from the authors - The data that support the
findings of this study are available from the corresponding author upon
reasonable request.

##  Appendix A Lattice Boltzmann Method

All fields are obtained by solving the Rayleigh-Bénard and the corresponding
nudging equations using the Lattice-Boltzmann method in 2D with a standard
D2Q9 grid. Two sets of populations $f$ and $g$ which represent the fluid phase
and the thermal phase respectively are introduced with their time evolution
governed by the BGK collision operator. He, Chen, and Doolen (1998) The
populations solve the Lattice Boltzmann equations

$\begin{gathered}f_{i}(\boldsymbol{r}+\mathbf{c}_{i},t+1)=f_{i}(\boldsymbol{r},t)+\frac{f_{i}-f^{\rm
eq}}{\tau_{f}},\\\
h_{i}(\boldsymbol{r}+\mathbf{c}_{i},t+1)=h_{i}(\boldsymbol{r},t)+\frac{h_{i}-h^{\rm
eq}}{\tau_{h}}\end{gathered}$ (29)

where the vectors $\mathbf{c}_{i}$ for $i=1,\dots,9$ are the discrete particle
velocities, $\Delta t$ is the lattice time-step, so that $\mathbf{c}_{i}\Delta
t$ go from each lattice point to the 8 nearest neighbouring lattice points in
the uniform 2D grid and $\mathbf{c}_{0}=0$. We have the time-step and the grid
spacing respectively $\Delta t=\Delta r=1$, as is the standard practice.
$f^{\rm eq}$ and $h^{\rm eq}$ are the equilibrium population distributions as
defined in He, Chen, and Doolen (1998) given by

$\begin{gathered}f^{\rm
eq}=w_{i}\rho\left(1+\frac{\mathbf{u}\cdot\mathbf{c}_{i}}{c^{2}_{s}}+\frac{(\mathbf{u}\cdot\mathbf{c}_{i})^{2}}{2c^{4}_{s}}-\frac{\mathbf{u}\cdot\mathbf{u}}{2c^{2}_{s}}\right)\\\
h^{\rm
eq}=w_{i}T\left(1+\frac{\mathbf{u}\cdot\mathbf{c}_{i}}{c^{2}_{s}}+\frac{(\mathbf{u}\cdot\mathbf{c}_{i})^{2}}{2c^{4}_{s}}-\frac{\mathbf{u}\cdot\mathbf{u}}{2c^{2}_{s}}\right)\end{gathered}$
(30)

where $w_{i}$ are the weights for each population set by the grid used, D2Q9
in this study. $c_{s}=1/\sqrt{3}$ is the lattice speed of sound set by the
choice of $\mathbf{c}_{i}$. $\tau_{f}$ and $\tau_{h}$ are respectively the
fluid and the thermal relaxation times which set the values for kinematic
viscosity $\nu$ and thermal conductivity $\kappa$ as

$\begin{gathered}\nu=c_{s}^{2}(\tau_{f}-0.5);\\\
\kappa=c_{s}^{2}(\tau_{h}-0.5)\end{gathered}$ (31)

To account for the buoyancy force term, the Guo-forcing scheme Guo, Zheng, and
Shi (2002) is employed and the equation for $f_{i}$ is modified as

$f_{i}(\boldsymbol{r}+\mathbf{c}_{i},t+1)=f_{i}(\boldsymbol{r},t)+\frac{f_{i}-f^{\rm
eq}}{\tau_{f}}+M_{i}$ (32)

with

$M_{i}=\left(1-\frac{1}{2\tau_{f}}\right)w_{i}\left(\frac{\mathbf{c_{i}}-\mathbf{u}}{c^{2}_{s}}+\frac{(\mathbf{c_{i}}\cdot\mathbf{u})\mathbf{c_{i}}}{c^{4}_{s}}\right)\cdot\mathbf{F}$
(33)

where $\mathbf{F}$ is the force vector. The fluid hydrodynamic quantities at
each point in space and time are obtained from the various moments of the
populations as

$\displaystyle\rho$ $\displaystyle=\sum_{i}f_{i};$ (34)
$\displaystyle\mathbf{u}$
$\displaystyle=\frac{1}{\rho}\sum_{i}f_{i}\mathbf{c}_{i}+\frac{\mathbf{F}}{2\rho};$
(35)

The ease of implementation of the Guo-forcing scheme is from the fact that the
velocity $\mathbf{u}$ that enters the expression for $f^{\rm eq}$ in equation
(30) is the same as the hydrodynamic velocity obtained in equation (35). This
isn’t the case for other forcing schemes.

The nudging experiments require the addition of the nudging term (see equation
(11)) to the temperature evolution equation. The two-population method is
ideal for this study since it allows us to set the Prandtl number by varying
$\tau_{f}$ and $\tau_{h}$ independently and makes the addition of a heat
source term (nudging term) in equation 10 simpler. The equation for $h_{i}$ is
modified as Seta (2013)

$\begin{split}h_{i}(\boldsymbol{r}+\mathbf{c}_{i},t+1)&=h_{i}(\mathbf{r},t)+\frac{h_{i}-h^{\rm
eq}}{\tau_{h}}\\\ &+\left(1-\frac{1}{2\tau_{h}}\right)w_{i}Q\end{split}$ (36)

where $Q=-\alpha(\mathcal{T}-T_{n})$ is the required source term. For the
ground truth, $Q=0$. The temperature is then obtained at each lattice grid
point from the thermal populations $h_{i}$ as

$\mathcal{T}=\sum_{i}h_{i}+\left(1-\frac{1}{2\tau_{h}}\right)Q$ (37)

Figure 11: The global reconstruction error $\Delta_{T}$ as a function of
$k_{l}$ with Eulerian and Lagrangian nudging for reconstructions of the lower
$\rm{Ra}$ flow with $\rm{Ra}=7.2\times 10^{7}$ Figure 12: (a) $\Delta_{T}$ for
reconstructions of the lower $\rm{Ra}$ flow with $\rm{Ra}=7.2\times 10^{7}$
for $k_{l}=1/14$ as a function of $\alpha_{0}\tau$, where only $\alpha_{0}$ is
varied. (b) $\Delta_{T}$ for reconstructions of the lower $\rm{Ra}$ flow for
different configurations of the nudging squares for $k_{l}=1/14$ . The blue
arrows in both panels indicate the value chosen for more detailed analysis for
this study.

##  Appendix B Lagrangian Nudging

Apart from nudging using fixed temperature probes on a uniform grid as shown
in figure 1, another novel approach to nudging is to nudge along the
trajectories of tracer particles as in Clark Di Leoni, Mazzino, and Biferale
(2020b), which in our case is the same as having probes as passive tracer
particles. Thus, the probe locations $\boldsymbol{X}^{i}(t)$ for
$i=1,2,\dots,N_{p}$ satisfy

$\frac{d\boldsymbol{X}^{i}(t)}{dt}=\mathbf{u}(\boldsymbol{X}^{i}(t),t)$ (38)

Consecutive measurements of temperature and position are made with the same
frequency $f$ as the Eulerian case and both quantities are again interpolated
linearly to obtain a reading at each time-step. Off-grid temperatures are
obtained using a bi-linear interpolation. The nudging-squares for each probe
is constructed with the nearest grid point to the interpolated probe location
as the centre. In the regions where the nudging squares of multiple probes
intersect, $T_{n}(\boldsymbol{r},t)$ is set as the mean of the $T^{i}(t)$ of
the intersecting probes while $\alpha(\boldsymbol{r},t)$ is set to be
$n\alpha_{0}$, where $n$ is the number of probes whose nudging squares
intersect at $\boldsymbol{r}$.

As shown in Figure. 11, the Lagrangian approach to nudging shows only a
marginal improvement in reconstruction of the temperature field for smaller
$k_{l}$ when compared to the Eulerian approach while the reconstruction of the
velocity field remains unchanged and identical to the Eulerian case for the
corresponding $k_{l}$.

##  Appendix C Selection of the Parameters

The results presented in the main text have fixed $\chi=6$ and
$\alpha_{0}\tau=4$. These results are representative of results for a larger
family of parameters. The main quantitative result of the study - once
$\Delta_{T}$ reaches a low enough value for a given flow, $\Delta_{v}$ reaches
a saturation value - still holds true. Here we report the effect of changing
the two parameters, namely the value of $\alpha$ and the size of the nudging
squares $\chi$. The results are summarized in Figure 12. In panel (a) we see
that $\Delta_{T}$ is far larger for smaller values of $\alpha_{0}$ with a
transition to a saturation value of $\Delta_{T}\sim 0.1$ at
$\alpha_{0}\tau\sim 0.4$. In panel (b) we see the variation of $\Delta_{T}$
for changing size of the nudging squares $\chi$. Additionally, the “Gaussian”
case refers a nudging experiment where we set

$\alpha(\boldsymbol{r},t)=\alpha_{0}\exp{\left(-\frac{|\boldsymbol{r}-\boldsymbol{X}^{i}(t)|^{2}}{2}\right)},\qquad\text{for
$\boldsymbol{r}\in S_{i}$}$ (39)

to examine whether the discontinuity of the forcing term has a significant
impact on the reconstruction of the temperature field. There exists an optimum
value of $\chi$ for the given flow, but the change in $\Delta_{T}$ is not very
significant. On one hand, increasing the size of nudging squares introduces
more temperature errors at the smallest scale while on the other hand, very
small nudging squares lead to large regions between probes which aren’t nudged
and hence decorrelate from the known temperature of the probe location. The
reconstruction error in velocity for changing $\alpha_{0}$ as well as changing
$\chi$ behaves as anticipated already in figure 6 \- $\Delta_{v}$ decreases
with decrease in $\Delta_{T}$ upto a certain point and then saturates. This
behaviour is qualitatively similar for the higher $\rm{Ra}$ flow as well.

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