text:
\section{Introduction}

Turbulent flows of a compressible viscous fluid are mostly described by the Navier-Stokes (NS) equations in the conservation form
\begin{equation}
\frac{\partial\mathbf{U}}{\partial t}+\nabla\cdot\mathbf{F}_a=\nabla\cdot\mathbf{F}_d,
\label{NS-eq}
\end{equation}
subject to a given proper boundary condition and the initial condition
\begin{equation}
\mathbf{U}|_{t=0}=\mathbf{U}_0(\mathbf{r}), \hspace{1.0cm} \mathbf{r}\in\Omega\hspace{0.1mm},
\label{NS-ini}
\end{equation}
where $\mathbf{U}(\mathbf{r},t)=(\rho,\,\rho\hspace{0.3mm}\mathbf{u}\hspace{0.1mm},\,E)^\mathbf{T}$ is a vector of conserved variables, $\nabla$ is the Hamiltonian/gradient operator,
\begin{equation}
\mathbf{F}_a=
\begin{pmatrix}
    \rho\hspace{0.3mm}\mathbf{u} \\
    \rho\hspace{0.3mm}\mathbf{u}\otimes\mathbf{u}+P\hspace{0.3mm}\mathbf{I} \\
    (E+P)\hspace{0.3mm}\mathbf{u}
\end{pmatrix},
\hspace{1.0cm}
\mathbf{F}_b=
\begin{pmatrix}
    0 \\
    \bm{\tau} \\
    \mathbf{u}\cdot\bm{\tau}+\mathbf{q}
\end{pmatrix}
\label{NS-ad}
\end{equation}
are the advection/hyperbolic and diffusion fluxes, $\mathbf{r}\in\Omega$ is the vector of spatial position, $t$ denotes the time, $\mathbf{U}_0(\mathbf{r})$ is a given vector, $\rho$ is the density of mass, $\mathbf{u}$ is the velocity vector, $\mathbf{u}\otimes\mathbf{u}$ denotes the tensor product of $\mathbf{u}$ and $\mathbf{u}$, $P$ is the pressure, $\mathbf{I}$ denotes the unit tensor, $E$ is the total energy per unit mass, $\bm{\tau}$ denotes the tensor of viscous stress and $\mathbf{q}$ is the vector of heat diffusion flux, respectively.

Note that tiny stochastic disturbances in fluid flows resulting from either small-scale thermal fluctuations or environmental noises are unavoidable in practice and the triggered deviations might exponentially increase \citep{leith1972predictability, boffetta2001predictability, boffetta2017chaos}. However, these tiny stochastic disturbances are {\em not} considered in the above-mentioned NS equations (\ref{NS-eq})-(\ref{NS-ad}) that are deterministic. \citet{LLNS1959} proposed a stochastic form of the NS equations that models the effect of thermal fluctuations via an additional stochastic stress tensor \citep{graham1974hydrodynamic, swift1977hydrodynamic, bell2007numerical, donev2010accuracy, donev2014low}. However, the direct numerical simulation (DNS) of this stochastic model is rather difficult due to the extremely fine resolution that is required to accurately measure the velocities at dissipation-range length scales. By contrast, molecular dynamics (MD) provides molecular-level simulation techniques \citep{bird1998molecular, donev2011enhancement, smith2015molecular, mcmullen2022navier} for directly investigating the role of thermal fluctuations in turbulent flows.

Currently, \citet{mcmullen2022navier} investigated a decaying turbulent flow and found that, due to thermal fluctuations, the molecular-gas-dynamics spectra grow quadratically with wave number in the dissipation range, while the Navier-Stokes spectra decay exponentially.
Furthermore, the transition to quadratic growth occurs at a length scale much larger than the gas molecular mean free path, namely in a regime that the NS equations are widely believed to describe. Thus, they provided the first direct evidence
that ``the Navier-Stokes equations do not describe turbulent gas flows in the dissipation range because they neglect thermal fluctuations'' \citep{mcmullen2022navier}, which is in agreement with the results given by \citet{bandak2022dissipation}, \citet{bell2022thermal}, \citet{Eyink2022} and so on.
\citet{Gallis2021-PRF} separately used the direct simulation Monte Carlo (DSMC) method (molecular gas dynamics) and direct numerical simulation (DNS) of the NS equations to simulate a freely decaying turbulent flow, i.e. the compressible Taylor-Green vortex flow, and found that both methods produce basically the same energy decay for the Mach and Reynolds numbers they examined, but the molecular fluctuations in DSMC (and in experiments) can break symmetries, which in turn can cause the flows different from but basically similar to those given by the DNS.
Note that the above-mentioned investigations mainly focus on the influence of tiny stochastic disturbances resulting from thermal fluctuations on {\em small-scale} properties of freely {\em decaying} turbulence. Might the micro-level stochastic disturbances have huge influences on {\em large-scale} properties of some {\em sustained} turbulent flows? This is the primary motivation for our investigation of this paper.

Some turbulent flows might have multiple states according to \citet{frisch1986fully} and his well-known monograph on turbulence \citep{frisch1995turbulence} that ``it is typical for dissipative dynamical systems to have more than one attractor'' and ``each attractor has an associated basin'', and thus ``the statistical properties of the solution will then depend on to which basin the initial condition belongs''. The first evidence of this phenomenon in a turbulent flow is given by \citet{huisman2014multiple} for their study on the highly turbulent Taylor-Couette flow. Note that the above-mentioned work mainly focuses on the influence of initial data on the flow state of turbulence as well as its statistical properties. Considering that there are some previous investigations \citep{knobloch1989effect, kraut1999preference, masoller2002noise, de2007noise} in the effects of noises (in general whose levels are much larger than the stochastic disturbances considered later in this paper) on the dynamical systems with multiple attractors, in this paper we aim to demonstrate that not only the initial conditions but also the weak, small-scale stochastic noises/disturbances can determine in which basin the solution of a sustained turbulent flow will reside for a long time.

In theory, the influence of {\em tiny} stochastic disturbances on turbulent flows is fundamental, since this kind of stochastic disturbances is often in a micro-level of magnitude and is {\em unavoidable} in practice.
There are two kinds of stochastic disturbances: natural and artificial. The thermal fluctuation belongs to the former, whereas the latter can be caused by many sources. Note that the background numerical noises, i.e. truncation errors and round-off errors, always exist for all numerical algorithms. Besides, it is widely believed that turbulence has a close relationship to chaos and thus the background numerical noises should increase exponentially (and quickly) until to the same level of ``true'' physical solution. Therefore, a computer-generated simulation of the NS equations is a kind of mixture of the ``true'' physical solution and ``false'' numerical noises. Note that the background numerical noise is tiny and random, dependent on different numerical algorithms and data accuracy, which itself is a kind of artificial stochastic disturbance. Therefore, in a natural way, the background numerical noises can be regarded as the sum of all kinds of tiny artificial stochastic disturbances.
In this paper, we investigate the {\em large-scale} influences of numerical noises as tiny artificial stochastic disturbances on a {\em sustained} turbulence by means of numerically solving the above-mentioned NS equations via a traditional numerical algorithm (Runge-Kutta's method with double precision, marked by RKwD) and the so-called ``Clean Numerical Simulation'' (CNS) \citep{Liao2009, Liao2013A, Liao2013B, hu2020risks, qin2020influence, xu2021accurate, Li2021SCPMA, Liao2022NA, AAMM-14-799}, separately, with the same initial/boundary conditions using the same values of physical parameters. The result given by the traditional numerical algorithm RKwD is a mixture of the ``true'' physical solution and ``false'' numerical noises, which are mostly at the same order of magnitude, where the background numerical noise is regarded as the sum of all tiny artificial stochastic disturbances. By contrast, the CNS can greatly reduce the background numerical noise to any a required level so that the numerical noises are {\em negligible} compared with the ``true'' physical solution, and thus the corresponding numerical result is convergent (reproducible) in an interval of time long enough for statistics, as described below. In other words, results given by the CNS can be regarded as a ``clean'' benchmark solution. Thus, we can investigate the influences of tiny artificial stochastic disturbances on turbulent flows by means of comparing the RKwD simulations with the CNS benchmark solution.

Then, let us briefly talk about the motivation and basic ideas of the CNS. Strictly speaking, all numerical simulations are not ``clean'', since background numerical noises (i.e. truncation errors and round-off errors) always exist there. Indeed, for non-chaotic systems, the magnitude of numerical noises can be controlled on a tiny level much smaller than ``true'' physical solution so that the influences of the numerical noises can be neglected.  However, for chaotic dynamical systems \citep{Li1975Period, Parker1989Practical, Lorenz1993The, PeterSmith1998Explaining, sprott2010}, numerical noises increase exponentially due to the ``sensitivity dependence on initial condition'' (SDIC), which was first discovered by \citet{poincare1890probleme} and later rediscovered by \citet{lorenz1963deterministic} with a more famous name ``butterfly-effect'': a hurricane happening in North America might be created by a flapping of the wings of a distant butterfly in South America several weeks earlier. More importantly, it was further found by \citet{lorenz1989computational, lorenz2006computational} that a chaotic dynamical system has the sensitivity dependence {\em not only} on initial condition (SDIC) {\em but also} on numerical algorithms (SDNA) in single/double precision. Such kind of uncertainty certainly raises serious doubt about the reliability of numerical simulations of chaotic systems. For example, \citet{Teixeira2007Time} carefully investigated the time-step sensitivity of three nonlinear atmospheric models by means of some traditional numerical algorithms (in single/double precision), and made a rather pessimistic conclusion that ``for chaotic systems, numerical convergence {\em cannot} be guaranteed {\em forever}''.

To overcome the above-mentioned limitations/restrictions of traditional numerical algorithms (in single/double precision) for chaotic dynamical systems, \citet{Liao2009} suggested a numerical strategy, namely the ``Clean Numerical Simulation'' (CNS). The basic idea of the CNS \citep{Liao2013A, Liao2013B, hu2020risks, qin2020influence, Li2021SCPMA, Liao2022NA, AAMM-14-799} is to greatly decrease the background numerical noises, i.e. truncation errors and round-off errors, to such a tiny level that the influence of numerical noises can be neglected in an interval of time $0\leq t \leq T_{c}$ that is long enough for statistics, where $T_{c}$ is the so-called ``critical predictable time''.
The CNS is based on such a well-known phenomenon: for a numerical/computer-generated simulation of chaotic dynamical system, the level of simulation-deviation (in an average meaning) from its (``true'')  physical  solution increases {\em exponentially} to a macroscopic one (at $t = T_{c}$), i.e.
\begin{equation}
{\cal E}(t) =  {\cal E}_0 \exp(K\hspace{0.3mm}t), \hspace{1.0cm}  t\in[0, T_{c}],  \label{def:delta-t}
\end{equation}
where $K > 0$ is the so-called ``noise-growing exponent'', ${\cal E}_0$ denotes the level of background numerical noise, ${\cal E}(t)$ is the level of simulation-deviation (in an average meaning) from the physical solution, respectively.  In theory, the critical predictable time $T_c$ is determined by a critical level ${\cal E}_c$ of simulation-deviation from its physical solution, say,
\begin{equation}
T_{c} = \frac{1}{K} \ln \left( \frac{{\cal E}_c}{ {\cal E}_0 }\right).    \label{def:delta_c}
\end{equation}
Obviously, for a given critical level ${\cal E}_c$, the smaller the level of the background numerical noise ${\cal E}_0$, the larger the critical predictable time $T_c$. This is the reason why in the frame of the CNS we have to greatly decrease the background numerical noises, i.e. truncation errors and round-off errors, to a tiny enough level.
So, different from the Taylor series method,  the key point of the CNS is the so-called  ``critical predictable time''    $T_c$  that determines a temporal interval  $[0,T_c]$ in which the numerical simulations are ``reliable''  and  ``clean'',  since their ``false'' numerical noises are much smaller than the ``true'' physical solution and thus are negligible.
For more details about the CNS, please refer to \citet{Liao2009, Liao2013A, Liao2013B} and his coauthors \citep{hu2020risks, qin2020influence, xu2021accurate, Li2021SCPMA, Liao2022NA}.

The CNS has been successfully applied to many chaotic dynamical systems. For example, by means of traditional numerical algorithms (in double precision), one can get convergent (i.e. reproducible) numerical simulations of the famous Lorenz equations just in a rather short interval of time, say, approximately $t\in [0,32]$. However, using the CNS, a convergent numerical simulation of the Lorenz equations was obtained first by \citet{Liao2009} in $t\in [0,1000]$ and then by \cite{LIAO2014On} in a much longer interval of time, i.e. $t\in[0, 10000]$. Besides, since the background numerical noises of the CNS can be much smaller even than the micro-level physical uncertainty, \cite{lin2017origin} successfully applied the CNS to provide a direct rigorous evidence that the micro-level  thermal fluctuation is the origin of macroscopic randomness of the turbulent Rayleigh-B{\'e}nard convection. Especially, it is worth noting that the CNS has been successfully applied to find more than $2000$ new families of periodic orbits of three-body system \citep{Li2017More, li2018over, li2019collisionless, Li2021SCPMA, Liao2022NA}. The discovery of these new periodic orbits was reported twice in the famous popular magazine {\em New Scientist} \citep{NewScientist2017, NewScientist2018}, because only three families of periodic orbits of the three-body problem had been reported in three hundred years after Newton mentioned this problem in 1687. All of these illustrate the validity, novelty, and great potential of the CNS.

Recently, an efficient CNS algorithm has been proposed to solve the spatio-temporal chaotic systems, i.e. the complex Ginzburg-Landau equation \citep{hu2020risks} and the damped driven sine-Gordon equation \citep{qin2020influence}, respectively. Using the CNS result as a benchmark solution, one can investigate the influence of numerical noises on the computer-generated simulation of a spatio-temporal chaotic system. It was found \citep{hu2020risks, qin2020influence} that numerical noises might lead to huge deviations of computer-generated simulations of some spatio-temporal chaotic systems not only in trajectories but also even in {\em statistics}.

In this paper, we apply the CNS and a traditional algorithm (based on the 4th-order Runge-Kutta's method with double precision, marked by RKwD), {\em separately}, to solve a sustained turbulence, i.e. the two-dimensional (2D) turbulent Rayleigh-B{\'e}nard convection (RBC). Note that, using the CNS, the background numerical noises can be decreased to such a tiny level that the numerical noises are negligible in a long enough interval of time, so that the CNS result can be regarded as a benchmark solution of the ``true'' physical result to investigate the influence of tiny artificial stochastic disturbances by comparing the RKwD simulations with the CNS benchmark solution.
In this way, we provide a rigorous evidence that tiny artificial stochastic disturbances have huge influences on large-scale properties of the turbulent Rayleigh-B{\'e}nard convection not only in statistics but also even in flow types: the CNS benchmark solution always keeps the vortical/roll-like turbulent convection, however, for the RKwD simulations, the shearing convection occurs and its corresponding flow field turns to zonal flow thereafter. This phenomenon is reasonable if the boundaries of different attractor basins (mentioned above) in this multistable system are intricately interwoven as has been observed in other cases \citep{shrimali2008nature}.

\section{Mathematical model for 2D turbulent RBC}    \label{sec2}

The buoyancy-driven convection in a fluid layer between two horizontal parallel plates heated from below and cooled from above, known as the Rayleigh-B{\'e}nard convection (RBC) for compressible viscous fluids, is one of the most fundamental and classic paradigms of nonlinear dynamics in fluid mechanics. It was first investigated by \cite{Rayleigh1916On} and the continuous efforts devoted to the study of this problem have greatly enriched our understanding \citep{Chandrasekhar1961Hydrodynamic, Schluter1965On, Heslot1987Transitions, Kadanoff2001Turbulent, Niemela2006Turbulent, Ahlers2009Heat, lohse2010small, Zhou2013Thermal, Goluskin2014Convectively, wang2020From}.

As illustrated in Figure~\ref{schematic_drawings}, a thin layer of fluid is confined between two horizontal plates that are separated by a distance $H$, where $T_{0}$ and $T_{0}+\Delta T$ denote the temperatures of the top and bottom boundary surfaces, $L$ is the horizontal length of computational domain and $g$ is the gravity acceleration, respectively.
The typical vortical/roll-like motions of two-dimensional (2D) RBC, as illustrated in Figure~\ref{schematic_drawings}(a), and their corresponding turbulent states, have been studied extensively by means of the direct numerical simulation (DNS) \citep{saltzman1962finite, Fromm1965Numerical, Veronis1968Large, moore1973two, curry1984order, zienicke1998bifurcations, johnston2009comparison, Huang2013Counter, Zhang2017Statistics, Zhu2018Transition}. However, there is another type of flow in the 2D turbulent RBC in the case of the free-slip boundary conditions imposed on two horizontal parallel plates and the periodic boundary conditions on the left and right sides, namely zonal flow \citep{Goluskin2014Convectively, van2014Effect, Hardenberg2015Generation, wang2020From}, as shown in Figure~\ref{schematic_drawings}(b). It is worth noting that such kind of turbulent zonal flow has been widely found in nature and laboratory such as in the atmosphere of Jupiter \citep{Heimpel2005Simulation, Kaspi2018Jupiter} and some Jovian planets \citep{Sun1993Banded, Cho1996The, Yano2003Outer}, in the oceans \citep{Maximenko2005Observational, Richards2006Zonal}, in the Earth's outer core \citep{Miyagoshi2010Zonal}, in toroidal tokamak devices \citep{Diamond2005Zonal} and so on. Thus, here we choose the 2D turbulent RBC with the free-slip boundary conditions at the upper and lower plates and the periodic boundary conditions in the horizontal direction as our mathematical model for the sustained turbulence, governed by the NS equations.

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
             \subfigure[]{\includegraphics[width=2.5in]{Fig1a.eps}}
             \subfigure[]{\includegraphics[width=2.5in]{Fig1b.eps}}
        \end{tabular}
 \end{center}
\caption{Schematic drawings of two-dimensional turbulent Rayleigh-B{\'e}nard convections in two totally different flow types: (a) typical vortical/roll-like flow and (b) zonal flow. The fluid layer between two parallel plates which are separated by a height $H$ obtains heat from the bottom boundary surface because of the constant temperature difference $\Delta T>0$, where $L$ is the horizontal length of computational domain and the downward direction of gravity acceleration $g$ is indicated.}
\label{schematic_drawings}
\end{figure}

Using the length scale $H$, velocity scale $\sqrt{g\alpha H\Delta T}$ and temperature scale $\Delta T$ as the characteristic scales, the corresponding dimensionless NS equations, combined with the Boussinesq approximation \citep{saltzman1962finite},  read
\begin{equation}
\frac{\partial}{\partial t}\nabla^{2}\psi+\frac{\partial(\psi,\nabla^{2}\psi)}{\partial(x,z)}-\frac{\partial\theta}{\partial x}-\sqrt{\frac{Pr}{Ra}}\nabla^{4}\psi=0,       \label{RB_psi}
\end{equation}
\begin{equation}
\frac{\partial\theta}{\partial t}+\frac{\partial(\psi,\theta)}{\partial(x,z)}-\frac{\partial\psi}{\partial x}-\frac{1}{\sqrt{PrRa}}\nabla^{2}\theta=0,       \label{RB_theta}
\end{equation}
where $\psi$ is a stream-function with the definition
\begin{equation}
u=-\frac{\partial\psi}{\partial z}, \hspace{1.0cm} w=\frac{\partial\psi}{\partial x},       \label{psi}
\end{equation}
in which $u$ and $w$ are the horizontal and vertical velocities, $\theta$ is the temperature departure from a linear variation background, i.e. the temperature is expressed as $T=\theta-z+1$ in the case of $T_0=0$, $t$ denotes the time,  $x\in[0,\Gamma]$ and $z\in[0,1]$ are the horizontal and vertical position coordinates, $\Gamma=L/H$ denotes the aspect ratio, $\nabla^{2}$ is the Laplace operator and thus $\nabla^{4}=\nabla^{2}\nabla^{2}$,
\begin{equation}
\frac{\partial(a,b)}{\partial(x,z)}=\frac{\partial a}{\partial x}\frac{\partial b}{\partial z}-\frac{\partial b}{\partial x}\frac{\partial a}{\partial z}       \label{Jacobi}
\end{equation}
is the Jacobi operator, the Rayleigh number $Ra$ and Prandtl number $Pr$ are defined by
\begin{equation}
Ra=\frac{g\alpha H^{3}\Delta T}{\nu\kappa}, \hspace{1.0cm} Pr=\frac{\nu}{\kappa},       \label{Ra_Pr}
\end{equation}
respectively, in which $\alpha$ is the thermal expansion coefficient and $\nu=\mu/\rho$ is the kinematic viscosity.

Note that the free-slip boundary conditions are adopted at the upper and lower plates where temperatures are assumed to be constant. Hence, the NS equations (\ref{RB_psi})-(\ref{RB_theta}) have the following boundary conditions:
\begin{equation}
\psi=\frac{\partial^{2}\psi}{\partial z^{2}}=\theta=0       \label{free-slip}
\end{equation}
at $z=0$ and $z=1$. On the other hand, since the fluid layer can extend to infinity in the horizontal direction, we adopt the periodic boundary conditions for $\psi$ and $\theta$ at the lateral boundaries in the horizontal direction, i.e. at $x=0$ and $x = \Gamma$.

Without loss of generality, in this paper let us consider the case with the aspect ratio $\Gamma=L/H=2\sqrt{2}$, which is large enough for the approximation of heat flux at an infinite aspect ratio \citep{saltzman1962finite,curry1984order,lin2017origin}, the Prandtl number $Pr=6.8$ (corresponding to the water at room temperature 20$^{\circ}$C), and the Rayleigh number $Ra=6.8\times10^{8}$ corresponding to a turbulent state, respectively. In addition, the initial temperature and velocity fields are randomly generated by the thermal fluctuations in Gaussian white noises \citep{lin2017origin}, with the temperature standard deviation $\sigma_T=10^{-10}$ and the velocity standard deviation $\sigma_u=10^{-9}$, respectively.

\section{Influence of numerical noises as artificial stochastic disturbances}    \label{ssec}

The deterministic NS equations (\ref{RB_psi})-(\ref{RB_theta}) are numerically solved here, separately, by means of a traditional algorithm (Runge-Kutta's method with double precision, marked by RKwD) whose numerical noises are mostly in the same order of magnitude as the ``true'' physical solution, and the CNS whose numerical noises are much less than the physical solution and thus negligible. By means of comparing the RKwD simulations with the CNS benchmark solution, it is found that the numerical noises indeed might lead to huge large-scale differences even in statistics and flow types of the 2D turbulent RBC, as described below.

First, we apply the clean numerical simulation (CNS) to greatly decrease the background numerical noises, i.e. the truncation errors and round-off errors, to such a tiny level that the numerical noises are much smaller than and thus negligible compared with the ``true'' physical solution of the 2D turbulent RBC in an interval of time that is long enough for statistics. In this way, a convergent (reproducible) solution of the 2D turbulent RBC can be obtained, which is used here as the ``clean''  benchmark solution. On the other hand, with the {\em same} initial/boundary conditions and the {\em same} physical parameters as described in the previous section, the NS equations (\ref{RB_psi})-(\ref{RB_theta}) are also numerically solved by a traditional algorithm, i.e. the fourth-order Runge-Kutta's method with double precision (marked by RKwD) using the time-step $\Delta t = 1\times10^{-4}$, whose numerical noises increase exponentially until to the same level of the ``true'' physical solution and thus are {\em not} negligible. By comparing these RKwD simulations with the CNS benchmark solution, we can investigate the influence of the numerical noises as tiny artificial stochastic disturbances on the 2D turbulent RBC in detail. In this section, we only briefly show some results of comparison. For details about the CNS algorithms, please refer to Appendix A.

Briefly speaking, to decrease the spatial truncation-error to a small enough level, we discretize the spatial domain of flow field by a uniform mesh $N_x\times N_z =1024\times1024$, and besides apply the Fourier spectral method with the $3/2$ rule for dealiasing \citep{pope2001turbulent}.
The corresponding spatial resolution is high enough for the considered turbulent RBC: the horizontal (maximum) grid spacing $\Delta_x=L/N_x=0.00276$ is less than the minimum Kolmogorov scale \citep{pope2001turbulent}, which will be shown later in details.
Besides, to decrease the temporal truncation-error to a small enough level for the CNS, we use the 45th-order (i.e. $M=45$) Taylor expansion with a time step $\Delta t=10^{-3}$. In addition, to decrease the round-off error to a small enough level, we use 70 significant digits (i.e. $N_{s}=70$) in multiple precision for all physical/numerical variables and parameters.
Similarly, we get another CNS result using the Fourier spectral method on the same uniform mesh $N_x\times N_z =1024\times1024$ with the even smaller background numerical noises by means of a higher order (i.e. $M=47$) Taylor expansion with the same time step ($\Delta t=10^{-3}$) and the higher multiple precision with more significant digits (i.e. $N_{s}=72$). Comparing these two CNS results, it is found that they have no distinct differences in an interval of time $0 \leq t\leq 500$, which is long enough for statistics. This verifies the convergence and reliability of our CNS result in $t\in[0,500]$ given by means of $M=45$, $\Delta t=10^{-3}$, and $N_{s}=70$, which is therefore used in the following parts as the ``clean'' benchmark solution.

As shown in Figures~\ref{Contour} and \ref{Contour-2}, the numerical simulation given by the RKwD is compared with the ``clean'' benchmark solution given by the CNS, and the corresponding movie is available at \href{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4}{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4}. Note that these two simulations have exactly the same initial conditions caused by the micro-level thermal fluctuations. For both of the CNS and RKwD simulations, the tiny initial disturbances of velocity and temperature evolve progressively from micro-level to macro-level until $t\approx 25$ when the transition from laminar flow to turbulence occurs, then the strong mixing occurs in $t\in[25,36]$ and the typical vortical/roll-like convection appears at $t\approx 50$, as shown in Figure~\ref{Contour}.
Thereafter, as the time increases, the RKwD simulation deviates from the CNS benchmark solution larger and larger, so that a distinct difference in large-scale between them can be observed, indicating that the numerical noises (as artificial stochastic disturbances) could indeed lead to some {\em large-scale} differences of flow fields of velocity and temperature in a macroscopic level, for example, as shown in Figure~\ref{Contour}  at $t=100$ and $t=185$. However, even so, the flow type of these two simulations keeps the same (i.e. vortical/roll-like turbulent convection) until $t\approx 188$ when the shearing convection occurs for the RKwD simulation and its corresponding flow field turns to a kind of zonal flow thereafter, as shown in Figures~\ref{Contour} and \ref{Contour-2}. On the contrary, the CNS benchmark solution thereafter always sustains the non-shearing vortical/roll-like convection during the whole process of simulation. Therefore, the RKwD simulation and the CNS benchmark solution have different {\em types} of turbulent convections after $t>188$. It should be emphasized that such kind of {\em qualitative} difference in large-scale is triggered only by the numerical noises (as artificial stochastic disturbances). All of these highly suggest that the numerical noises (as artificial stochastic disturbances) have quantitatively and qualitatively {\em large-scale} influences on the {\em sustained} turbulence, i.e. the two-dimensional turbulent RB convection considered in this paper.

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
            \includegraphics[width=2.5in]{Fig2L1.eps}
            \includegraphics[width=2.5in]{Fig2R1.eps} \\
            \includegraphics[width=2.5in]{Fig2L2.eps}
            \includegraphics[width=2.5in]{Fig2R2.eps} \\
            \includegraphics[width=2.5in]{Fig2L3.eps}
            \includegraphics[width=2.5in]{Fig2R3.eps} \\
            \includegraphics[width=2.5in]{Fig2L4.eps}
            \includegraphics[width=2.5in]{Fig2R4.eps} \\
            \includegraphics[width=2.5in]{Fig2L5.eps}
            \includegraphics[width=2.5in]{Fig2R5.eps} \\
            \includegraphics[width=2.5in]{Fig2L6.eps}
            \includegraphics[width=2.5in]{Fig2R6.eps} \\
            \includegraphics[width=2.5in]{Fig2L7.eps}
            \includegraphics[width=2.5in]{Fig2R7.eps} \\
            \includegraphics[width=2.5in]{Fig2L8.eps}
            \includegraphics[width=2.5in]{Fig2R8.eps}
        \end{tabular}
    \caption{Evolutions of the $\theta$ (dimensionless temperature departure from a linear variation background) field in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$. Left: the CNS benchmark solution; Right: the RKwD simulation given by the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$. The corresponding movie is available at \href{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4}{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4}.}    \label{Contour}
    \end{center}
\end{figure}

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
            \includegraphics[width=2.5in]{Fig3L1.eps}
            \includegraphics[width=2.5in]{Fig3R1.eps} \\
            \includegraphics[width=2.5in]{Fig3L2.eps}
            \includegraphics[width=2.5in]{Fig3R2.eps} \\
            \includegraphics[width=2.5in]{Fig3L3.eps}
            \includegraphics[width=2.5in]{Fig3R3.eps} \\
            \includegraphics[width=2.5in]{Fig3L4.eps}
            \includegraphics[width=2.5in]{Fig3R4.eps} \\
            \includegraphics[width=2.5in]{Fig3L5.eps}
            \includegraphics[width=2.5in]{Fig3R5.eps} \\
            \includegraphics[width=2.5in]{Fig3L6.eps}
            \includegraphics[width=2.5in]{Fig3R6.eps} \\
            \includegraphics[width=2.5in]{Fig3L7.eps}
            \includegraphics[width=2.5in]{Fig3R7.eps} \\
            \includegraphics[width=2.5in]{Fig3L8.eps}
            \includegraphics[width=2.5in]{Fig3R8.eps}
        \end{tabular}
    \caption{Evolutions of the $\theta$ (dimensionless temperature departure from a linear variation background) field in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$. Left: the CNS benchmark solution; Right: the RKwD simulation given by the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$. The corresponding movie is available at \href{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4}{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4}.}    \label{Contour-2}
    \end{center}
\end{figure}

How about the influence of numerical noises (as artificial stochastic disturbances) on statistical results? The heat transport of 2D turbulent RBC can be typically quantified by the Nusselt number defined by
\begin{equation}
N\hspace{-0.3mm}u(t)=1-\left. \frac{\partial\langle\theta(x,z,t)\rangle_{x}}{\partial z}\right|_{z=1},    \label{Nu}
\end{equation}
where $\langle a \rangle_{x}=\int_{0}^{\Gamma}a dx/\Gamma$ denotes the spatial average in horizontal direction. As shown in Figure~\ref{Nu_t}(a), the distinct deviation between the two time histories of $N\hspace{-0.3mm}u(t)$ given respectively by the CNS benchmark solution and the RKwD simulation happens at $t\approx80$
when the numerical noises of the RKwD simulation have been enlarged to a macroscopic level because of the butterfly-effect of chaos.
Note that the $N\hspace{-0.3mm}u(t)$ given by the RKwD simulation drops down greatly at $t\approx188$ until $N\hspace{-0.3mm}u(t) \approx$ 30 after $t>300$, one order of magnitude less than that of the CNS benchmark solution. Such a huge difference is not only quantitative but also qualitative, which is definitely due to the appearance of the zonal flow at $t \approx 188$ that is triggered by the numerical noises of the RKwD simulation (as artificial stochastic disturbances).
On the other hand, the Reynolds number is also calculated to measure the global convection strength, which is obtained via the root-mean-square (rms) velocity $U_{rms}$ \citep{sugiyama2009flow,Zhang2017Statistics}, i.e.
\begin{equation}
Re(t)=\sqrt{\frac{Ra}{Pr}}\hspace{0.03cm}U_{rms}    \label{Nu}
\end{equation}
with $U_{rms}=\sqrt{\langle u^2+w^2\rangle_{A}}$, where $\langle a\rangle_{A}=\int_{0}^{\Gamma}\int_{0}^{1}adxdz/\Gamma$ denotes the spatial average. As shown in Figure~\ref{Nu_t}(b), the CNS benchmark solution and RKwD simulation give almost the same values of $Re(t)$ when t < 188. However, the departure begins  at $t \approx 188$ when the shearing convection occurs, and thereafter the deviation between the RKwD simulation and the CNS benchmark solution becomes more and more obvious. Here, it should be emphasized that the Reynolds number $Re$ of the CNS benchmark solution at $t=500$ is about 3500 larger than that of the RKwD simulation! This is indeed a huge difference. Besides, the similar phenomena are observed in the comparisons of the spatially averaged heat flux $\langle wT\rangle_{A}$ and kinetic energy $\langle E_V\rangle_{A}$, given respectively by the CNS benchmark solution and RKwD simulation, as shown in Figure~\ref{wT_t}(a) and (b), where the maximum ratio of kinetic energy reaches about 2.5. Indeed, the numerical noises (as artificial stochastic disturbances) might lead to qualitatively huge deviations of the 2D turbulent RBC.

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
            \subfigure[]{\includegraphics[width=2.55in]{Fig4a.eps}}
            \subfigure[]{\includegraphics[width=2.55in]{Fig4b.eps}}
        \end{tabular}
    \caption{Comparisons of the instantaneous Nusselt number $N\hspace{-0.3mm}u$ and Reynolds number $Re$ in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$. (a) The Nusselt number $N\hspace{-0.3mm}u$; (b) The Reynolds number $Re$. Solid line in red: the CNS benchmark solution; Dashed line in black: the RKwD simulation given by the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$.}    \label{Nu_t}
    \end{center}
\end{figure}

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
            \subfigure[]{\includegraphics[width=2.55in]{Fig5a.eps}}
            \subfigure[]{\includegraphics[width=2.55in]{Fig5b.eps}}
        \end{tabular}
    \caption{ Comparisons of the spatially averaged heat flux $\langle wT\rangle_{A}$ and kinetic energy $\langle E_V\rangle_{A}$ in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$, where $\langle a\rangle_{A}=\int_{0}^{\Gamma}\int_{0}^{1}adxdz/\Gamma$ denotes the spatial average. (a) The heat flux $\langle wT\rangle_{A}$; (b) The kinetic energy $\langle E_V\rangle_{A}$. Solid line in red: the CNS benchmark solution; Dashed line in black: the RKwD simulation given by the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$.}    \label{wT_t}
    \end{center}
\end{figure}

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
             \subfigure[]{\includegraphics[width=2.55in]{Fig6a.eps}}
             \subfigure[]{\includegraphics[width=2.55in]{Fig6b.eps}}
        \end{tabular}
    \caption{Comparisons of the spatially averaged kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{A}$ and thermal dissipation rate $\langle\varepsilon_{T}\rangle_{A}$ in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$, where $\langle a\rangle_{A}=\int_{0}^{\Gamma}\int_{0}^{1}adxdz/\Gamma$ denotes the spatial average. (a) The kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{A}$; (b) The thermal dissipation rate $\langle\varepsilon_{T}\rangle_{A}$. Solid line in red: the CNS benchmark solution; Dashed line in black: the RKwD simulation given by the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$.}    \label{e_eT_t}
    \end{center}
\end{figure}

Besides the above-mentioned large-scale quantities, the comparisons of some small-scale properties of fluid flows such as the kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{A}$ and the thermal dissipation rate $\langle\varepsilon_{T}\rangle_{A}$ of this 2D turbulent RBC given by the CNS solution and the RKwD simulation are as shown in Figure~\ref{e_eT_t}, where
\begin{equation}
\varepsilon_{V}(x,z,t)=\frac{1}{2}\sqrt{\frac{Pr}{Ra}}\sum_{ij}\left[ \partial_iu_j(x,z,t)+\partial_ju_i(x,z,t) \right]^2    \label{e}
\end{equation}
and
\begin{equation}
\varepsilon_{T}(x,z,t)=\frac{1}{\sqrt{Pr\,Ra}} \hspace{0.03cm} \Big| \nabla \left[ \theta(x,z,t)-z \right] \Big|^2    \label{eT}
\end{equation}
with $i,\,j=1,\,2$, $u_1(x,z,t)=u(x,z,t)$, $u_2(x,z,t)=w(x,z,t)$, $\partial_1=\partial /\partial x$, $\partial_2=\partial /\partial z$, and that $\nabla$ is the Hamilton operator and $\langle \;\; \rangle_{A}$ denotes the spatial average, respectively.  Note that, for the RKwD simulation,  both of the kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{A}$ and thermal dissipation rate $\langle\varepsilon_{T}\rangle_{A}$  greatly drop down at $t\approx 188$ when the shearing convection (i.e. the zonal flow) occurs, which is triggered only by the numerical noises (as artificial stochastic disturbances).

By the way, as shown in Figure~\ref{e_eT_t}(a), we have the maximum kinetic energy dissipation rate $(\langle\varepsilon_{V}\rangle_{A})_{max}=0.00646$ at $t=26.9$ when the transition from the laminar flow to turbulence occurs, corresponding to the minimum Kolmogorov scale
\[\left(\langle\eta\rangle_{A}\right)_{min}\approx(Pr/Ra)^{3/8} \big[ (\langle\varepsilon_{V}\rangle_{A})_{max} \big] ^{-1/4}=0.00353.\]
Thus, the criteria condition that the maximum horizontal grid spacing $\Delta_x=\Gamma/N_x=0.00276 < 0.8(\langle\eta\rangle_{A})_{min}=0.00282$ is indeed satisfied, say, the spatial resolution used in this paper is fine enough for the 2D turbulent RBC under consideration.

Figure~\ref{e_wT_z} shows the comparisons of the kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{x,t}(z)$ and the heat flux $\langle wT\rangle_{x,t}(z)$, where $\langle a\rangle_{x,t}=\int_{0}^{\Gamma}\int_{0}^{500}a \; dxdt/\Gamma/500$ denotes the horizontally spatial and temporal average. Obviously, both of the kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{x,t}$ and the heat flux $\langle wT\rangle_{x,t}$ of the CNS benchmark solution are significantly larger than those given by the RKwD simulation. Especially, near the lower and upper plates, the $\langle\varepsilon_{V}\rangle_{x,t}$ given by the CNS benchmark solution has a much sharper peak than that given by the RKwD simulation, as shown in Figure~\ref{e_wT_z}(a). These indicate that the numerical noises (as artificial stochastic disturbances) can lead to large-scale deviations in statistics of the 2D turbulent RBC under consideration.

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
            \subfigure[]{\includegraphics[width=2.55in]{Fig7a.eps}}
            \subfigure[]{\includegraphics[width=2.55in]{Fig7b.eps}}
        \end{tabular}
    \end{center}
\caption{Comparisons of the horizontally and temporally averaged kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{x,t}(z)$ and the heat flux $\langle wT\rangle_{x,t}(z)$ in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$, where $\langle a\rangle_{x,t}=\int_{0}^{\Gamma}\int_{0}^{500}adxdt/\Gamma/500$ denotes the horizontal and temporal average. (a) The kinetic energy dissipation rate $\langle\varepsilon_{V}\rangle_{x,t}(z)$; (b) The heat flux $\langle wT\rangle_{x,t}(z)$. Solid line in red: the CNS benchmark solution; Dashed line in black: the RKwD simulation given by the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$.}    \label{e_wT_z}
\end{figure}

The comparison of the probability density functions (PDFs) of the stream function $\psi(x,z,t)$ in $0 \leq x < \Gamma$, $0 \leq z \leq 1$ and $0 \leq t \leq 500$ given by the CNS benchmark solution and the RKwD simulation is as shown in Figure~\ref{PDF_psi}. Unlike the PDF of the CNS benchmark solution that has a kind of asymmetry about $\psi = 0$, the PDF of the RKwD simulation has no such kind of asymmetry but two peaks at $\psi\approx0$ and $\psi \approx 0.25$. Furthermore, the comparison of the PDFs of $\theta(x,z,t)$ given by the CNS benchmark solution and RKwD simulation is as shown in Figure~\ref{PDF_theta}. As shown in Figure~\ref{PDF_theta}(a), except at $\theta\approx0$, the PDF of $\theta(x,z,t)$ given by the CNS benchmark solution remains almost the same value. In contrast, the PDF of $\theta$ given by the RKwD simulation is relatively more typical, as shown in Figure~\ref{PDF_theta}(b). Thus, the numerical noises (as artificial stochastic disturbances) indeed lead to large-scale deviations even in the PDFs of the 2D turbulent RBC under consideration.

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
             \subfigure[]{\includegraphics[width=2.55in]{Fig8a.eps}}
             \subfigure[]{\includegraphics[width=2.55in]{Fig8b.eps}}
        \end{tabular}
    \caption{Probability density functions (PDFs) of $\psi(x,z,t)$ in $0 \leq x < \Gamma$, $0 \leq z \leq 1$ and $0 \leq t \leq 500$ in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$. (a) The PDF given by the CNS benchmark solution; (b) The PDF given by the RKwD simulation obtained via the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$.}     \label{PDF_psi}
    \end{center}

    \begin{center}
        \begin{tabular}{cc}
             \subfigure[]{\includegraphics[width=2.55in]{Fig9a.eps}}
             \subfigure[]{\includegraphics[width=2.55in]{Fig9b.eps}}
        \end{tabular}
    \caption{Probability density functions (PDFs) of $\theta(x,z,t)$ in $0 \leq x < \Gamma$, $0 \leq z \leq 1$ and $0 \leq t \leq 500$ in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$. (a) The PDF given by the CNS benchmark solution; (b) The PDF given by the RKwD simulation obtained via the Runge-Kutta's method with double precision using $\Delta t = 10^{-4}$.}     \label{PDF_theta}
    \end{center}
\end{figure}

All of the above-mentioned comparisons indicate that the micro-level background numerical noises as a kind of artificial stochastic disturbances can lead to large-scale differences not only in spatio-temporal trajectories but also even in {\em flow types} of the 2D turbulent RBC, which further affects the statistics of the Nusselt number, the Reynolds number, the kinetic energy, the kinetic energy dissipation rate, the thermal dissipation rate and so on.
Note that it is currently reported by \cite{mcmullen2022navier} that ``the Navier-Stokes equations do not describe turbulent gas flows in the dissipation range because they neglect thermal fluctuations'', say, tiny stochastic disturbances resulting from thermal fluctuations might influence the {\em small-scale} properties of the freely {\em decaying} turbulent flows under their consideration. In this paper, the detailed comparisons between the CNS benchmark solution and the RKwD simulation provide us the  rigorous  evidence that numerical noises as a kind of small-scale artificial stochastic disturbances might influence the {\em large-scale} properties of a {\em sustained} turbulence, i.e. the 2D turbulent RB convection considered in this paper.

\section{Concluding remarks and discussions}

All numerical algorithms have the background numerical noises, i.e. the truncation errors and round-off errors, which are tiny and random. It was reported that, for a chaotic dynamic system, the random numerical noises increase exponentially due to the butterfly-effect of chaos, until up to the {\em same} order of magnitude as its ``true'' physical solution \citep{hu2020risks, qin2020influence}. Therefore, numerical simulations of a {\em deterministic} chaotic system are mostly a mixture of the ``true'' physical solution, which is {\em deterministic} in physics, and the  ``false'' numerical noises, which is however {\em stochastic}. This is the reason why numerical simulations of a {\em deterministic} chaotic system given by traditional algorithms in single/double precision often look like {\em stochastic}, and why it has been wrongly believed that a {\em deterministic} chaotic system can lead to {\em randomness}. This is also the reason why \citet{Teixeira2007Time} made a rather pessimistic conclusion that ``for chaotic systems, numerical convergence {\em cannot} be guaranteed {\em forever}''. Obviously, even given an accurate initial condition, this kind of randomness of chaotic system comes from the randomness of artificial background numerical noises. Thus, the background numerical noises can be naturally regarded as a kind of {\em artificial stochastic disturbances} to a chaotic system when it is solved numerically.

In this paper, the so-called clean numerical simulation (CNS) is adopted to {\em accurately} investigate the influence of numerical noises as a kind of tiny artificial stochastic disturbances on the two-dimensional turbulent RB convection under consideration. This is mainly because the CNS can reduce the background numerical noises (i.e. round-off errors and truncation errors) to any a required level, which can be so small that the ``false'' numerical noises are negligible compared with its ``true'' physical solution and thus the numerical simulations of turbulence are convergent/reproducible in an interval of time long enough for statistics, as illustrated in this paper. Strictly speaking, the CNS solution is also a mixture of the ``true'' physical solution and the ``false'' numerical noises. However, unlike the simulation given by a traditional algorithm in single/double precision, the ``false'' numerical noises of a CNS solution are often several orders of magnitude smaller than its ``true'' physical solution in a region $t\in[0,T_{c}]$ long enough for statistics, so that its numerical noises are negligible and the CNS result is ``convergent'' and ``reproducible''. Such kind of CNS solution in  $t\in[0,T_{c}]$ can be used as a ``clean'' benchmark solution for comparison with those given by traditional algorithms in single/double precision so as to investigate the influence of numerical noises as tiny artificial stochastic disturbances on the two-dimensional turbulent RB convection under consideration.  So, unlike the Taylor series method,  the key point of the CNS is the so-called  ``critical predictable time''    $T_c$  that determines a temporal interval  $[0,T_c]$ in which the numerical simulations are ``reliable''  and  ``clean'',  since their ``false'' numerical noises are much smaller than the ``true'' physical solution and thus are negligible.

It was currently reported by \citet{mcmullen2022navier} that tiny stochastic disturbances resulting from thermal fluctuations might influence the {\em small-scale} properties of the freely {\em decaying} turbulence under their consideration, which is in agreement with the conclusions given by \citet{Gallis2021-PRF}, \citet{bandak2022dissipation}, \citet{bell2022thermal}, \citet{Eyink2022} and so on.
In this paper, we investigate the large-scale influence of numerical noises as a kind of tiny artificial stochastic disturbances on a sustained turbulence. Using the two-dimensional (2D) turbulent Rayleigh-B{\'e}nard convection (RBC) as an example, we illustrate that the numerical noises as a kind of micro-level artificial stochastic disturbances could indeed lead to {\em large-scale} deviations not only in spatio-temporal trajectories but also even in statistics of the {\em sustained} turbulence considered in this paper. Especially, such kind of tiny artificial stochastic disturbances even leads to different types of flows: the shearing convection occurs for the RKwD simulations, and its corresponding flow field turns to a kind of zonal flow thereafter, however the CNS benchmark solution always sustains the non-shearing vortical/roll-like convection during the whole process of simulation, as shown in Figures~\ref{Contour} and \ref{Contour-2}. Thus, we provide a {\em rigorous}  evidence that numerical noises as a kind of tiny artificial stochastic disturbances have not only quantitatively but also qualitatively {\em large-scale} influences on a {\em sustained} turbulence, i.e. the 2D turbulent RB convection considered in this paper.  Of course, for various types of turbulent flows governed by the NS equations, more investigations are needed in the future.

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
            \subfigure[]{\includegraphics[width=2.55in]{Fig10a.eps}}
            \subfigure[]{\includegraphics[width=2.55in]{Fig10b.eps}}
        \end{tabular}
    \caption{Comparison between the CNS benchmark solution $\theta_{CNS}$ and the numerical noises $\varepsilon_{n} = \theta_{CNS} - \theta_{RKwD}$ at the point $x=1$ and $y=1/2$, where $\theta_{RKwD}$ is the RKwD simulation.  Solid line: the CNS benchmark solution; Dashed line: the  numerical noises.  (a)  $1\leq t\leq 500$ ; (b) $75\leq t\leq 100$.}    \label{comparison-error}
    \end{center}
\end{figure}

\begin{figure}
    \begin{center}
        \begin{tabular}{cc}
            \subfigure[]{\includegraphics[width=2.55in]{Fig11a.eps}}
            \subfigure[]{\includegraphics[width=2.55in]{Fig11b.eps}}
        \end{tabular}
    \caption{Temporal evolution of the Nusselt number $N\hspace{-0.3mm}u$ and the Reynolds number $Re$ in the case of $Pr = 6.8$, $Ra = 6.8 \times 10^{8}$, and $L/H = 2\sqrt{2}$. (a) Nusselt number $N\hspace{-0.3mm}u$; (b) Reynolds number $Re$; Solid line in red: the CNS benchmark solution; Dashed line in black: the RKwD simulation (with $\Delta t =  10^{-4}$); Dashed line in blue:  the RKwD$^{\prime}$ simulation (with $\Delta t = 2 \times 10^{-4}$).}    \label{Nu_t-2}
    \end{center}
\end{figure}

Why? We try to give an explanation. This 2D turbulent RBC system might have two possible final states: the vortical/roll-like flow and the zonal flow, which can be seen as two minima of a double-well potential. Once the CNS benchmark solution falls in one of these two minima (e.g. the roll-like flow), it remains there forever, since the corresponding ``false'' numerical noises are much less than the ``true'' physical solution and thus can not trigger off the transition to another minimum. On the contrary, the RKwD simulation can perform the transition from one state to the other, because its ``false'' numerical noises as artificial stochastic disturbances might be at the same order of magnitude as the ``true'' physical solution, as shown in Figure~\ref{comparison-error}, so that the RKwD simulation might depart very far away from its ``true'' physical solution and thus fall in another minimum.
If so, the transition of the RKwD simulation from the vortical/roll-like flow to the zonal flow of the 2D turbulent RBC should occur at random times for different numerical noises as a kind of artificial stochastic disturbances. In order to check this hypothesis, we perform a new RKwD simulation of the 2D turbulent RBC with the identical initial/boundary conditions and physical parameters of the previous one, but using a different time-step $\Delta t = 2\times10^{-4}$, marked by RKwD$^{\prime}$. The corresponding numerical noises as a different kind of artificial stochastic disturbances are verified to be at the same tiny level as the previous ones. As shown in Figure~\ref{Nu_t-2} (that describes the temporal evolutions of the Nusselt number $N{u}$ and the Reynolds number $Re$), the transition from the vortical/roll-like flow to the zonal flow of the RKwD$^{\prime}$ simulation with $\Delta t = 2 \times 10^{-4}$ occurs indeed at a different time $t\approx 230$, compared to $t\approx 188$ of the previous RKwD simulation with $\Delta t = 10^{-4}$. Thus, our above-mentioned explanation in terms of a stochastic dynamical system in the double-well potential should be reasonable. This test provides useful information to better understand the origin of the phenomenon reported in this paper.

Thus, generally speaking, if a chaotic system has $N$-well potential, where $N\geq 2$, its numerical simulations given by traditional algorithms in single/double precision should fall {\em randomly} in one of the $N$ minimums, and besides the transition between different minimums should occur frequently and randomly. Similarly, if a turbulent flow has multiple states in a large scale, small disturbances might lead to the transition between different states. Such kind of small disturbances can be either natural (such as the environmental perturbations) or artificial (such as the background numerical noises). In this case, it is {\em impossible} to make a correct prediction of flow states by means of traditional algorithms in single/double precision. By contrast, if a turbulent flow has an {\em unique} state, the background numerical noises should not have a large-scale influence on the flow. This should be a piece of good news for researchers in the field of CFD.

In theory, even if a turbulent flow has multiple states in large-scale, we can still give a {\em deterministic} prediction of its large-scale states by means of the CNS, since the {\em artificial} background numerical noises are negligible in a long enough interval of time for CNS solution. However, a {\em natural} disturbance, say, thermal fluctuation, always exists in fluid flows, which is random, small-scale, but {\em unavoidable} in practice. It is an open question whether or not, like tiny artificial numerical noises, the micro-scale thermal fluctuation might have a large-scale influence on some turbulent flows with multiple states. Note that the influence of thermal fluctuation is {\em not} considered in the NS equations. So, it is highly suggested to investigate the large-scale influence of thermal fluctuation on turbulent flows with multiple states by means of the Landau-Lifshitz-Navier-Stokes (LLNS) equations \citep{LLNS1959}. It is also worthwhile seriously discussing which one is better for turbulent flows (especially those with large-scale multiple states), either the {\em deterministic} NS equations or the {\em stochastic} LLNS equations. Obviously, more investigations are necessary in the future.

It should be emphasized that the external noises considered in most of related articles are mostly about ten orders of magnitude larger than the numerical noises considered in this paper, which might greatly change the characteristics of the chaotic/turbulent systems under their consideration. Besides, even when their external noises are zero, their numerical simulations are in fact a mixture of the ``true'' physical solution and the ``false'' numerical noises: both of them might be at the same level, as shown in this paper. Thus, strictly speaking, the conclusions based on this kind of ``mixtures'' should be {\em not} rigorous in theory. Thus, the CNS provides us, for the first time, a {\em rigorous} way to investigate the influence of external disturbances and very tiny artificial numerical noises on chaotic dynamical systems and turbulent flows.

In summary, numerical noises as weak, small-scale stochastic perturbations increase exponentially to a macro level of numerical simulations, and besides might have a large influence on the macroscopic statistics of turbulent flows. Therefore, we should pay more attentions to the influences of small-scale stochastic perturbations on turbulence. Finally, this work also illustrates the validity, novelty, and great potential of the CNS as a reliable and accurate tool in theoretical studies of turbulence. Hopefully, the CNS might provide a brand-new, extremely accurate numerical tool to study turbulent flows.

\backsection[Acknowledgements]{Thanks to the anonymous reviewers for their valuable suggestions and constructive comments.}

\backsection[Funding]{This work is partly supported by the National Natural Science Foundation of China (No. 91752104) and Shanghai Pilot Program for Basic Research - Shanghai Jiao Tong University (No. 21TQ1400202).}

\backsection[Declaration of Interests]{The authors report no conflict of interest.}

\backsection[Data availability statement]{The data that support the findings of this study are available on request from the corresponding author.   The movie related to Figures~\ref{Contour} and \ref{Contour-2} is available at \href{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4}{https://github.com/sjtu-liao/RBC/blob/main/RBC-mv.mp4} }

\backsection[Author ORCID]{Shijie Qin, https://orcid.org/0000-0002-0809-1766; Shijun Liao, https://orcid.org/0000-0002-2372-9502}

\backsection[Author contributions]{Liao conceived and designed the analysis. Qin performed the analysis. Both wrote the manuscript.}

redpajama_set_name: RedPajamaArXiv


At Stanford, doctors, chefs (and doctor-chefs!) came together to create a course for medical students to learn to prepare delicious food that supports and promotes health. The goal is for students to live healthier themselves and to use food as medicine.


text:
Appeal launched to find missing 29-year-old woman
Jodi Creevey who is known to frequent Southport often wears a number of gold rings and costume jewellery, including a ring reading 'mum' on her right hand
Andrew BrownEditor
Missing from home: Jodi Creevey
Merseyside Police is becoming increasingly concerned about the whereabouts of 29-year-old Jodi Creevy who has been reported missing from her home in Huyton.
Jodi, who is known to frequent Southport, was reported missing on Friday, August 28 and enquiries have been carried out since to try to locate her.
She is the second female police are asking for help in tracing, after detectives issued a missing persons appeal for 16-year-old Samantha Josephine Nutt yesterday .
Jodi is described as being white, of fair complexion, around 5ft 8in tall, and of slim build with long, straight dark brown hair and blue eyes.
She is known to wear a number of gold rings and costume jewellery, including a ring reading 'mum' on her right hand. Jodi sometimes wears hair extensions and has her hair in a pony tail.
She was Last seen on 8 or 15 August in Huyton wearing navy blue Ugg boots and a knee length beige/brown coat.
Jodi is known to frequent the Huyton, Kirkby, Southport, Larkhill, Bootle, Formby, Old Swan, St Helens and Ellesmere Port areas.
Officers are urging Jodi, anyone who knows where she is or anyone who has seen a person matching the description or photograph to call Merseyside Police on 101.
NewsSefton and West Lancs snow photos

redpajama_set_name: RedPajamaCommonCrawl


text:
Asia market round-up: Goldman Sachs - Check out but don't leave
By Chris Leahy
Critics of Goldman Sachs enjoy accusing the firm of being a hedge fund. But such accusations look wide of the mark given that Goldman's Asia's equity guru, Hyder Ahmad, has just quit to start his own fund.
As head of risk arbitrage and co-head of Asia securities, Ahmad built Goldman Sachs's equity operation in Asia, described by one insider as a "very well-oiled money-making machine". Moreover, Ahmad achieved this while remaining one of the most urbane and likeable of bankers.
Keen to return to active trading, Ahmad is believed to have raised more than $1 billion already for an Asia-focused, multi-strategy fund.
Goldman might have lost some star bankers to hedge fund start-ups lately but the firm can take solace that many never leave the fold completely. Ahmad is no exception: once his fund is closed he will be returning to Goldman Sachs, this time as a client of the firm's prime brokerage division.
Banking BankingGoldman SachsAugust 2006
Chris Leahy


text:
Home Australia News
Model Grace Athanatos pleads guilty for internally supplying cocaine
A 24-year-old model has thanked her legal team after she avoided a criminal conviction for attempting to carry a condom filled with 5.6g of cocaine into a Sydney concert.
Student and Only Fans creator Grace Athanatos was caught by drug-detection dogs at a DJ MaRLo concert at Sydney Olympic Park on May 14.
Police found a knotted condom inside the Melbourne woman's vagina via a private room strip search. She was then charged with supplying drugs (less than a commercial quantity).
Want a streaming service dedicated to news? Flash lets you stream 25+ news channels in 1 place. New to Flash? Try 1 month free. Offer ends 31 October, 2022 >
While her Instagram is normally filled with glamourous outfit shots, and photos of her dining out at high end restaurants, Ms Athanatos had a slight change of tone this week, when she thanked her legal team for her result in court.
"HUGE shout out to my saviour @ahmeddob and his team at @dibassociateslawyers for helping me today," she shared to her 9000-plus followers.
"Honestly such an amazing result and I wouldn't have been able to do it without them. Highly recommend seeing them if you're ever naughty or find yourself in trouble like me."
Appearing at Burwood Local Court on Monday, Ms Athanatos avoided a formal conviction and was given a 12 month conditional behaviour bond. She was represented by criminal lawyer Ahmed Dib from the Kogarah-based Dib & Associates Lawyers.
Speaking to Daily Mail, the burgeoning influencer insisted she was a "nice girl" and "never intended to supply" the drug. She said she had planned to share her stash among her girlfriends and had only taken a single 'bump' of cocaine prior to the bust.
"It was really scary!" she told the publication.
"I was so shaken up and I just wanted to do the right thing and be nice.
"Obviously I wanted to have a little fun, I had a small amount and yes I tried to hide it – like everyone else – but I was caught."
Under NSW law, the maximum penalty for offence two years imprisonment and/ or an $11,000 fine if dealt in at a Local Court. If the offence proceeds to the district court, it carries a maximum sentence of 15 years and/ or a $220,000 fine.
Speaking about her brush with the law, Ms Athanatos said the case had been "blown up" but the judge was on her side.
"Their case was weak and the judge was so, so understanding and literally saw right through it," she said.
In 2018, Sydney woman Grace Poo was caught smuggling 1600 MDMA pills internally and in a "huge bra" into the Midnight Mafia music, which also took place at Sydney Olympic Park.
Ms Poo was found with several condoms full of the drugs and jailed for two years, with a minimum sentence of a year. Court proceedings revealed Ms Poo was given $200, a ticket into the festival and 40 free pills for supplying the drugs.
On July 20, 2022, NSW Police uncovered 125kg of cocaine which had been hidden in the base of an industrial generator. Valued at $50 million, a 29-year-old man was arrested and charged with importing a commercial quantity of border-controlled drug, supplying a prohibited drug (more than large commercial quantity) and participating in a criminal group and contributing to criminal activity.
A 32-year-old man was also arrested and charged with the above offences, as well as the use of an offensive weapon to prevent lawful detention.
Assisted by UK and Australian crime authorities, a Branch Commander for the UK National Crime Agency, Lydia Bloomfield said the investigation "targeted a network we suspect was providing a rare and highly sophisticated service to other organised crime groups," she said.
"A shipment of this quantity of cocaine from the UK to Australia is unusual but demonstrates the global scale of the class A drugs trade – which causes violence and intimidation throughout its supply chain."
Read related topics:Sydney
The numbers show the U.S. economy is at least teetering on a recession
Willoughby Homes building licence suspended, hasn't paid staff superannuation
Chatswood, Sydney: Group attempt to break in with machete
Flight attendant, boyfriend not happy with girlfriend's in-flight meal
Jane Hume tells Insiders opposition has no policies


Recently traded in. F150 3.5L V6 Ecoboost. It's equipped with, power locks, power front seats, navigation, air, 2 keyless remotes, towing package, SYNC, 18 wheels, running boards, accident free, one owner and much more. Thank you for considering George Stockfish Ford.


text:
Promote the need for sincere information for a democracy based on trust
DESCARTES FOUNDATION
CIRCLE OF experts
EXPERTS' BLOG
Thematic overviews
Syntheses of research papers
About usTeamExecutive boardScientific CommitteeCircle of expertsJoin the FoundationContact the FoundationMedia Relations
Our reportsExpert's blogThematic OverviewsSyntheses of research papersNewletters
BibliographyPodcasts and videosEventsActorsInitiatives
Production vs. Consumption of information on Twitter
Synthesis produced by the Fondation Descartes of the following research paper:
Mukerjee, S., Jaidka, K., & Lelkes, Y. (2020). The Ideological Landscape of Twitter: Comparing the Production versus Consumption of Information on the Platform.
This article reconsiders two highly debated issues concerning the social network Twitter: the formation of echo chambers 1, on the one hand, and the ideological fragmentation of this social network, on the other. The authors of this article argue that these two harmful phenomena are not necessarily linked to the use of Twitter. Indeed, they consider that previous studies that have demonstrated the presence of echo chambers and strong ideological fragmentation on Twitter have largely focused their analysis on small groups of users, often interested in politics — groups that do not reflect the true diversity of Twitter users. Moreover, according to the authors, most existing studies have focused on the content of political tweets, without paying sufficient attention to the actual importance of political content on Twitter as a whole.
Based on these observations, the authors establish the following two main objectives:
To gain a better understanding of the nature of the content that primarily circulates on Twitter in the United States. To do this, the authors analyzed the content of tweets published by opinion leaders within a randomly selected group of users (the term "opinion leaders" refers here to accounts that have significantly more followers than the average user).
To estimate the degree of "ideological diversity" to which each account in the randomly selected group of users is exposed.
The authors observe that only 10% of the opinion leaders in their sample group are politicians, whereas 44% of them are from the world of entertainment. Moreover, most tweets are published by non-partisan accounts. It is therefore difficult to talk about the polarization of Twitter as a whole, since only a small proportion of users have an ideological stance. Lastly, the majority of users do not consume content that is ideologically homogenous. On the other hand, there exists a group of users and opinion leaders who are highly interconnected and closed off from the outside world. This group, which constitutes an echo chamber, is that of American conservatives.
The authors conclude that analyzing the ways in which ordinary Americans follow opinion leaders on Twitter does not support the widespread notion that users of this social network systematically lock themselves into echo chambers, nor that Twitter contributes to ideological fragmentation and the polarization of opinions.
An echo chamber refers to a virtual space in which information, ideas, or beliefs are amplified or reinforced through communication and repetition. Echo chambers are mainly used to designate spaces on the Internet (forums, threads, social networks, Facebook pages, etc.) where users exchange profusely on the same subject. For more information, we recommend this article composed by the Fondation Descartes devoted to echo chambers.[⇡]
Topic : Social Media
Format : Syntheses of research papers
Edition : OSF Preprints
Country : United-States
Keywords : Echo chamber Ideology Polarization Social media
Consult the resource
Endowment fund for the creation of the Fondation Descartes
A DONATION


text:
Q: I'm having trouble with this sql query html/php Hello I am new to web design so please excuse me if this looks like crap!
Ok what I am trying to do here is set up comments for each post.I figured a good way to do that would be to run a loop and assign $save_id[] to store the post id and a assign
$r <input type='text' name = *'".$r."'* style = 'width:478px; margin: 0 auto; margin-top:-9px;'id='butt_box' placeholder='Respond?' title = 'Leave a response.'  /> which seems to work fine and returns the post id. When I submit the comment $_POST['$s'] its not registering and the database is not updateing so I guess my question is can anyone see what I am doing wrong or specificly why  if(isset($_POST['$s'])) fails?
enter code here $result =  mysql_query("SELECT * FROM user_posts WHERE user ='$name'  ORDER BY time_date DESC LIMIT 100");
        $save_id =array();
        while($row = mysql_fetch_assoc($result))
        {
            $save_id[] = $row["id"];
            $r = $row["id"];
            echo '
';
            echo '
';echo $name.'

';echo $row["time_date"].'

';
            echo $row["post"]. '
';
            echo "<form method='post' action=''>
            <input type='text' name = '".$r."' style = 'width:478px; margin: 0 auto; margin-top:-9px;'id='butt_box' placeholder='Respond?' title = 'Leave a response.'  />

            <input type='submit' value='Post' class='tfbutton' style =' float:right;'>
            </form>";
        }

        if(!$result)
        {
            echo mysql_error();
        }
        foreach($save_id as $s)
        {

        if(isset($_POST['$s']))
        {
           echo $_POST['$s'];
            $mypost = isset($_POST['$s']) ?
            $_POST['$s'] : '';

            if(isset($mypost) && strlen($mypost) > 0)
            {
                $text = addLinks($mypost);
                $text = CheckEmotes($text);

                $p = $text;
                $emote = "none";
                $dt = date('Y-m-d H:i:s');

                $sql = "INSERT INTO  `user_post_comments`(poster,post_id,responder,response,post_date)
                VALUES('$name','$s','$name',$text',$dt')";
                $rg = mysql_query($sql);

                if(!$rg)
                {
                    echo mysql_error();
                }
            }
            //header("Refresh:5; url=refresh.php");
        }
    }

A: $_POST['$s'] should be $_POST[$s] ($s without the quotes)

A:
$_POST['$s'] should be $_POST[$s] ($s without the quotes)

This is true. The reason behind this is because your $_POST is seeing your variable ($s) as text. Because of the quotes it thinks it is not a variable. Removing this should fix the problem.
Update, This line looks a bit odd:
 mysql_query("SELECT * FROM user_posts WHERE user ='$name'  ORDER BY time_date DESC LIMIT 100");

I hope your issue is fixed when the quotes are removed but if that is not the case id suggest changing your query like this:
$selectAllFromUser = "SELECT * FROM user_posts WHERE user ='".$name."' ORDER BY time_date DESC LIMIT 100";
mysql_query($selectAllFromUser);

It might be possible that your query sees your $name variable as text as well. (Although i do not know the specifics of your version etc.)
(edit; it has been a long time but i think this should get you going).
enter code here
$selectAllFromUser= "SELECT * FROM user_posts WHERE user ='".$name."'  ORDER BY time_date DESC LIMIT 100";
$result =  mysql_query($selectAllFromUser);
        $save_id =array();
        while($row = mysql_fetch_assoc($result))
        {
//not working with arrays often, but shouldnt it be required to specify a // position?
            $save_id[] = $row["id"];
            $r = $row["id"];
            echo '
';
            echo '
{$row["id"]}

{ $row["time_date"]}

{$row["post"]}
';
            echo "<form method='post' action=''>
            <input type='text' name = '".$r."' style = 'width:478px; margin: 0 auto; margin-top:-9px;'id='butt_box' placeholder='Respond?' title = 'Leave a response.'  />

            <input type='submit' value='Post' class='tfbutton' style =' float:right;'>
            </form>";
        }

        if(!$result)
        {
            echo mysql_error();
        }
        foreach($save_id as $s)
        {

        if(isset($_POST[$s]))
        {
           echo $_POST[$s];
            $mypost = isset($_POST[$s]) ?
            $_POST[$s] : '';

            if(isset($mypost) && strlen($mypost) > 0)
            {
                $text = addLinks($mypost);
                $text = CheckEmotes($text);

                $p = $text;
                $emote = "none";
                $dt = date('Y-m-d H:i:s');
//again using the $s without setting a position, is this valid?
                $sql = "INSERT INTO  `user_post_comments`(poster,post_id,responder,response,post_date)
                VALUES('{$name}','{$s}','{$name}','{$text}','{$dt}')";
                $rg = mysql_query($sql);

                if(!$rg)
                {
                    echo mysql_error();
                }
            }
            //header("Refresh:5; url=refresh.php");
        }
    }

redpajama_set_name: RedPajamaStackExchange


LizPR: Are you out of your mind? How does this kind of schedule happen?
LizPR: So a lot of the decision-making comes from the orchestra side?
JKP: Certainly. They are the ones who need to sell tickets. The communication between conductor, orchestra, manager and artist can feel like a delicate dance. Sometimes everything is decided in a day. On other occasions I've had months of back and forth negotiations, and once or twice, after offering 8 different concerti and having none of them fit, we've decided to postpone until the following season.
LizPR: But I've heard stories of some artists insisting on playing the same concerto for a whole season. Wouldn't that make your life easier?
JKP: I once played Beethoven 3rd Concerto dozens of times in one season: with Christopher Hogwood and the Saint Paul Chamber Orchestra on a tour of Europe, and with Gunther Herbig and the Toronto Symphony on an Asian tour. Every concert was a unique experience, and it wasn't just because of playing a new piano every night, or being in a different city. I was inspired by my colleagues, and by a musical masterpiece that simply changes every time I play it.
But having said that, I've never voluntarily insisted on playing the same concerto for a whole season. It would limit my chances to explore the vast repertoire of piano concerti, and I think it would limit my opportunities with orchestras, who have specific repertoire needs from week to week.
Of course, all of this decision-making should be invisible to the concert-going public. Their first contact with repertoire choice is when they see next season's brochure, and everything (usually) is perfectly in place. But somewhere in the background, an orchestra's Artistic Administrator has been pulling his or her hair out…!
LizPR: How are you going to pace yourself through this grueling season of concerti?
The killer month for me will be February 2011. In a three-week period, I'm performing Rachmaninoff 3rd with Andrew Litton and the Minnesota Orchestra, Tchaikovsky 1st with Daniel Hege and the Wichita Symphony, and then Brahms 2nd with Larry Rachleff and the Rhode Island Philharmonic. That's going to be an interesting month!
LizPR: How will you manage your practicing? Clearly you can't wait for one work to be 'performed' before practicing the next.
JKP: True. In fact, on some concert nights I have been known to return to the concert hall at midnight, set myself up at the upright piano in my dressing room, and practice for next week's concerto. I usually have to do it that way: earlier in the day I feel a bit guilty practicing Brahms if I'm performing Tchaikovsky that evening.
LizPR: Your season encompasses everything from Mozart to Rachmaninoff and Gershwin. Have you ever confined your repertoire to a certain period or style?
JKP: One of my friends, the great hornist William VerMeulen, when asked what kind of music he likes, replies "Good Music." I've always believed in that philosophy and consequently have played and listened to music ranging from jazz to rock, as well as almost every corner of the classical repertoire.
There are times when I feel more comfortable with the elegance of Mozart, and other times when I need to let loose with a Brahmsian flurry of octaves. Pianists are blessed with a depth and richness of repertoire not equaled for any other solo instrument; it would be a shame to specialize too much and miss out!
LizPR: What kind of skills do you take to a concerto performance that you wouldn't need in a recital setting?
JKP: That's a great question…the first prerequisite of concerto playing is to be a great listener, and that often comes with chamber music experience. You have to be able to react to minute changes that are constantly happening around you.
It certainly helps to have a 'big' sound, especially if you're playing almost any concerto by a Russian composer. Tchaikovsky, Rachmaninoff, Prokofieff and others wrote symphonic orchestra parts that can be sonically overwhelming and occasionally you really do have to play "vs." the orchestra. On the other hand, if you're playing a Mozart Concerto, you might want your sound to blend with the orchestra to create a more harmonious whole.
LizPR: Thank you, and good luck on managing this remarkable load of works next season!


text:
Q: Pass selected value in dropdownlist from view to controller when form post I have these models and a view model. When form posting to create the entries, the selected value for WardID and  DoctorID becomes zero in the controllers POST method.
View code
@model NewClient.HospitalMgt.ViewModel.PatientAdmissionViewModel
....
@using (Html.BeginForm()) {
    @Html.AntiForgeryToken()
    @Html.ValidationSummary(true)

    <fieldset>

            @Html.LabelFor(model => model.Admission.WardID, "Ward")

            @Html.DropDownList("WardID", String.Empty)
            @Html.ValidationMessageFor(model => model.Admission.WardID)

            @Html.LabelFor(model => model.Admission.DoctorID, "Doctor")

            @Html.DropDownList("DoctorID", String.Empty)
            @Html.ValidationMessageFor(model => model.Admission.DoctorID)

            @Html.LabelFor(model => model.Patient.PatientName)

            @Html.EditorFor(model => model.Patient.PatientName)
            @Html.ValidationMessageFor(model => model.Patient.PatientName)

        .... //more controls for properties of Patient

            @Html.LabelFor(model => model.Admission.AdmissionNumber)

            @Html.EditorFor(model => model.Admission.AdmissionNumber)
            @Html.ValidationMessageFor(model => model.Admission.AdmissionNumber)

        .... //more controls for properties of Admission

            <input type="submit" value="Create" />

    </fieldset>
}

Controller code
public ActionResult Create()
{
    ViewBag.WardID = new SelectList(db.Wards, "WardID", "WardNumber");
    ViewBag.DoctorID = new SelectList(db.Doctors, "DoctorID", "DoctorName");
    return View();
}

[HttpPost]
[ValidateAntiForgeryToken]
public ActionResult Create(PatientAdmissionViewModel Id)
{
    if (ModelState.IsValid)
    {
        var c = new Patient()
        {
            Address = Id.Patient.Address,
            ContactNumber = Id.Patient.ContactNumber,
            EmergencyContact = Id.Patient.EmergencyContact,
            PatientName = Id.Patient.PatientName,
        };
        var a = new Admission()
        {
            AdmissionNumber = Id.Admission.AdmissionNumber,
            AdmissionDate = Id.Admission.AdmissionDate,
            **WardID = Id.Admission.WardID, //becomes zero
            DoctorID = Id.Admission.DoctorID //becomes zero**
        };
        db.Patients.Add(c);
        db.Admissions.Add(a);
        db.SaveChanges();
        return RedirectToAction("Index");
    }

    ViewBag.WardID = new SelectList(db.Wards, "WardID", "WardNumber", Id.Admission.WardID);
    ViewBag.DoctorID = new SelectList(db.Doctors, "DoctorID", "DoctorName", Id.Admission.DoctorID);
    return View(Id);
}

Models and View models
//[NotMapped]
public class PatientAdmissionViewModel
{
    public Patient Patient { get; set; }
    public Admission Admission { get; set; }
}

public class Patient
{
    public int PatientID { get; set; }
    [Required(ErrorMessage = "A Patient Name is required")]
    public string PatientName { get; set; }
    public string Address { get; set; }
    [Required(ErrorMessage = "A Contact Number is required")]
    public string ContactNumber { get; set; }
    public string EmergencyContact { get; set; }
    public virtual Ward Ward { get; set; }
}

public class Admission
{
    public int AdmissionID { get; set; }
    public string AdmissionNumber { get; set; }
    public Nullable<int> PatientID { get; set; }
    public int WardID { get; set; }
    public int DoctorID { get; set; }
    public System.DateTime AdmissionDate { get; set; }
    public virtual Ward Ward { get; set; }
    public virtual ICollection<PatientPayment> PatientPayments { get; set; }
}

A: You use of @Html.DropDownList("WardID", String.Empty) and @Html.DropDownList("DoctorID", String.Empty) are generating inputs with name="WardID" and name="DoctorID" respectively, but you model does not contain properties with those names (it does however contain a property named Admission which contains those properties).
You 'view model' is not really a view model (a view model should not contain properties which are data models when editing). Your view model should also contain properties for the SelectList's so that you can strongly bind to your model. Your view model needs to be
public class PatientAdmissionViewModel
{
    [Display(Name = "Ward"]
    [Required(ErrorMessage = "Please select a ward")]
    public int? SelectedWard { get; set; }
    [Display(Name = "Doctor"]
    [Required(ErrorMessage = "Please select a doctor")]
    public IEnumerable<SelectListItem> WardList { get; set; }
    public int? SelectedDoctor { get; set; }
    public IEnumerable<SelectListItem> DoctorList { get; set; }

    public string PatientName { get; set; }
    ... // other properties of Patient that you need in the view

    public string AdmissionNumber{ get; set; }
    ... // other properties of Admission that you need in the view
}

and in the GET method, initialize an instance of the view model, set its properties and return it to the view
public ActionResult Create()
{
    var model = new PatientAdmissionViewModel()
    {
        WardList = new SelectList(db.Wards, "WardID", "WardNumber"),
        DoctorList = new SelectList(db.Doctors, "DoctorID")
    };
    return View(model);
}

and in the view
@Html.DropDownListFor(m => m.SelectedWard, Model.WardList, "-Please select-")
....
@Html.DropDownListFor(m => m.SelectedDoctor, Model.DoctorList, "-Please select-")

and finally, in the POST method, you initialize new instances of your data models, set their properties based on the values in the view models and save the data models.
I also recommend you read the answers to What is ViewModel in MVC?.

A: instead of this
@Html.DropDownList("WardID", String.Empty)
@Html.DropDownList("DoctorID", String.Empty)

use this on view
@Html.DropDownListFor(model => model.Admission.WardID, (SelectList)ViewBag.WardID, "--Select One--")
@Html.DropDownListFor(model => model.Admission.DoctorID, (SelectList)ViewBag.DoctorID, "--Select One--")

use this on controller
 WardID = Id.Admission.WardID,
 DoctorID = Id.Admission.DoctorID


A Queen Anne period house with stylish modern interiors in a very quiet London location overlooking Ham Common. The house is close to Richmond Park, Ham House and River Thames. Free on road parking for multiple vehicles. The original Georgian features, such as sash windows and wooden floors, meet a creative and unusual interior. Art and colourful objects adorn the rooms, several rooms offer modern builds inside them that mean some areas are set over two levels. The rooms are large, including a substantial dining area. Production catering facilities are available at the location.


text:
Cameron Ely, Son Of 'Tarzan' Actor Ron Ely And Harvard Quarterback, Murders Mom Valerie Ely — What Happened?
Entertainment And News
By Amy Lamare — Written on Oct 22, 2019
Tragedy hit the home and family of Tarzan actor Ron Ely, 81, last week. His wife, Valerie Ely, 62, was fatally stabbed to death in their Santa Barbara home on October 15. A 911 call was made by the couple's 30-year-old son Cameron during which he said his father did it. In fact, that's how the Los Angeles news outlets originally reported the crime. Police responded to the scene and found that that was not at all the case. In fact, it was Cameron who murdered his mother.
The incident happened in the family's home in Hope Ranch in Santa Barabara County. When the police arrived at the Ely home, they found Ron frail and unable to speak clearly because of a disability. Ron told the officers that Cameron had driven off after the murder. He was found 90 minutes later on the family's property. Police found him to be a threat when they arrived on the scene and shot and killed him. Now Ron Ely is left mourning his wife and son and wondering how it all went so wrong.
Which begs the question: Who is Cameron Ely and what caused this horrible tragedy to happen?
1. His education
Cameron Ely attended San Marcos High School in California as well as the prestigious Phillips Exeter Academy in New Hampshire. Phillips Exeter is regularly on the list of the best boarding schools in America. Tuition costs $49,000 a year. Cameron was a highly-recruited football player out of high school. 24/7 Sports profiled him in 2005 saying, "The sky is the limit for San Marcos high school QB Cameron Ely." He was recruited by Harvard University where he played Quarterback during the 2007 season. During that year, he had a GPA of about 4.0. He's not listed on the football roster for Harvard for any other year. He graduated in 2012 with a BA in Psychology.
RELATED: Wife Of 'Tarzan' Star Ron Ely Stabbed To Death By Son
2. Close-knit family
Ron Ely married Valerie Lundeen in 1984. They have two daughters, Kirsten and Kaitland, both 32 as well as Cameron, who was 30 at the time of his death. They were a close-knit family. In fact, Ron Ely stepped away from his acting career to focus on his family. In 2013, Ron Ely told the Daily Express: "Late in life I had a young family. I decided to stop acting and work at home, as an author, that way I could be with the kids all through school and be able to attend their sports games and things."
I pray for the Ely family. Cameron and I played on the same travel ball team as kids and competed against eachother in HS. I don't know details other than what's online but mental illness needs more attention. It's affecting every one of us personally and/or someone we know. https://t.co/EreCX7YFJS
— Andrew [Andy] Gil (@CoachAGil) October 16, 2019
3. Cameron's job
Cameron went to some of the best schools in the country in Phillips Exeter and Harvard. Presumably, he could have done nearly anything he wanted. According to The Los Angeles Times, for the past two years, Cameron Ely worked as a licensed security guard.
RELATED: Man Shows 'No Mercy,' Guns Down 5 Of His Chicago Neighbors For No Apparent Reason
4. Only a pretty girl could draw him away
In a 2012 interview with Fanboy Planet while at ComicCon, Ron Ely said in reference to Cameron wandering away from him during the interview: "Must have been [drawn away by] a pretty girl, because that's the only thing that could draw him away." During the same interview, he elaborated on stepping away from his acting career to focus on his family. "I am fully committed to my wife and my children. That's why I stepped out of the business because I wanted to spend more time to raise my kids," Ely said.
The deputies identified the couple's son, 30-year-old Cameron Ely, as the suspect, and found him outside the home after a search. Four of the deputies opened fire and killed him. https://t.co/jnbkuC8dnE
— WTVG 13abc (@13abc) October 17, 2019
5. Ron transported to the hospital
When police arrived at the scene of Valerie Ely's murder, they found the former Tarzan star very frail and having difficulty speaking. According to multiple reports, he had a medical condition or disability. Even without that, the shock of what happened in his own home may have contributed to his state. He was transported to a local hospital for evaluation.
RELATED: Glamour Model Esmeralda Gonzalez Found Encased In Concrete In Las Vegas Desert — What We Know
6. His sisters
In the aftermath of Cameron's crime, his sisters Kirsten and Kaitland will have to deal with the destruction he wrought and the loss he brought upon their family. Kirsten's Facebook page clearly shows how close the family was and what an integral role their mom played in their lives, even as adults. Valerie Ely is survived by her husband Ron, her daughters Kirsten and Kaitland and her grandson, Jaxon.
7. The motive
At this time, there is no known motive. Some Twitter users have speculated that Ron's failing health and mounting bills led to Cameron's parents putting the family home on the market. Others have speculated Cameron may have had an opoid issue. We may never know why he snapped and fatally stabbed his mom while trying to blame it on his father. We'll update this story if and when more information comes out.
Ron Ely's house was for sale, Ron's health failing needing more care, bills stacking up, were things going to change for Cameron more than he could cope with... https://t.co/g1qtoseKir?
— bil willard (@WillardDunn) October 16, 2019
A Mom Is Told She Must Cut Her 3 Daughters' 'Knee & Waist Length' Hair Or They Can't Be In Sister-In-Law's Wedding
Woman Sees Male Colleague's 'Nasty' Messages About Her In Group Chat After He Shares His Screen During A Virtual Meeting
Woman Asks If It's 'Normal' To Call Her Husband 'Daddy' Even When Their Child Isn't Around 'Out Of Habit'
How long before the opioid evidence is released? We all can see where this is going! #opioidcrisis #opioid #CameronEly https://t.co/8jnamFkb56 via @nypost
— Tom Leykis (@tomleykis) October 19, 2019
Did OJ Simpson's Son Jason Kill Nicole Brown & Ron Goldman?
11 Strict Rules That Elon Musk Makes His Girlfriends & Wives Follow
Justin Bieber Suspected As The Father Of Kourtney Kardashian's Son Reign (Instead Of Scott Disick)
Why Kate Moss Was 'Scared' & 'Couldn't Get Out Of Bed For Weeks' After Working With Mark Wahlberg
Amy Lamare is a Los Angeles based writer and editor covering entertainment, pop culture, beauty, fashion, fitness, technology, and the intersection of technology, business, and philanthropy. You can find her on Instagram and Facebook.


text:
Burma's Stubborn State
How to Curtail the Military Junta
By Michael Green
MICHAEL GREEN is Associate Professor of International Relations at the Edmund A. Walsh School of Foreign Service at Georgetown University and a Senior Adviser and Japan Chair at the Center for Strategic and International Studies. He served on the staff of the National Security Council from 2001–5.
MORE BY Michael Green
Courtesy Reuters
In November 2007, Derek Mitchell and I published an essay in Foreign Affairs ("Asia's Forgotten Crisis," November/December 2007) arguing that U.S. policy toward Burma (renamed Myanmar by the country's military junta) needed to move beyond the debate over whether to place sanctions on the country's repressive military junta or engage it. We also asserted that Washington must form a comprehensive strategy that leverages regional relationships and uses a mix of incentives to nudge the isolated regime toward democracy.
Three years later, the junta has released the Nobel Peace Prize laureate Aung San Suu Kyi, the longtime inspirational leader of Burma's democratic opposition who had been under some form of house arrest or detention for the last 14 years. Although the release of Suu Kyi has to be welcomed by her supporters worldwide, it falls far short of objectives set for Burma policy by the Obama administration and others in the international community. They sought not only Suu Kyi's unconditional and permanent release but also the inclusion of the democratic opposition and ethnic minorities in Burma's political process, as well as the establishment of international standards and monitoring for the country's elections on November 7.
None of this happened. Meanwhile, the regime has opened a dangerous new path by partnering with North Korea to explore nuclear capabilities, a potentially explosive issue given recent revelations that Pyongyang has successfully moved forward with its own program for highly enriched uranium. Although freeing Suu Kyi may allow Burma's leaders to escape scrutiny for now, their budding nuclear ambitions could rejuvenate international interest in placing pressure on their regime.
Suu Kyi was released in the wake of Burma's early November elections, which solidified the rule of the regime's Union Solidarity and Development Party. The international community rightly dismissed the elections as fraudulent. The election stuffed the legislature with former military men who had swapped their uniforms for suits. Burma's constitution -- passed overwhelmingly in a referendum staged
This article is a part of our premium archives.
To continue reading and get full access to our entire archive, please subscribe.
Southeast Asia Burma & Myanmar Security WMD & Proliferation GW Bush Administration Obama Administration


text:
Clips Playlist Rock Inglês
Two Door Cinema Club "Something Good Can Work"
Two Door Cinema Club are an indie rock band from Northern Ireland, formed in 2007, the band is composed of band members Sam Halliday, Alex Trimble and Kevind Baird. They released their debut album, Tourist History on 1 March 2010. It won the Choice Music Prize for Irish Album of the Year 2010 the following year.
One my favorite tunes over here. I'm completly addict to that song!!
Café Tacvba [O2 Shepherd's Bush Empire/JUL 18] / Playlist
Franz Ferdinand e George Erza no O2 Academy
Los Fabulosos Cadillacs em Londres [O2 Shepherd's Bush Empire/JUL 17] / Playlist
The 80's e the Goldbergs (playlist 04/2017)
Café Tacvba/México (playlist 12/2016)


Prepare yourself to enter the digital realm! Do you like cartoons? Do you like sequential art? Do you like video-games? Do you want to learn how to draw on a computer? Then this class is for YOU!
Students will explore many aspects of New Media in this year long course. Together, we will learn elements of Sequential Art, Character Design, Animation, Web Design, and the magic behind many different computer programs. If you like zines & comics, concept art, animation, game design, or are even interested in learning how to take your art to the "digital realm," this is the place for you! Don't have a lot of experience creating art on computers? No problem, we will cover the basics of every program we use before we dive in. By the end of the year all students will have a diverse digital portfolio on a personal website designed by themselves!
Our year is essentially divided into two halves, first semester we concentrate on learning the basics of Photoshop, Drawing Tablet Aptitude, Character Design, World Building, and Sequential Art. Our second semester consists entirely of moving image based projects; animation and basic game design.
Students who take our course for a second year, equipped with knowledge in many basic programs, are able to spread their wings and concentrate on more independent, self directed work within a concentration of their choosing.
Emily studied film at Columbia College and Art Education at the University of Illinois at Chicago. At UIC, Emily received numerous honors, including the Chancellor's Student Service award of 2007 for her volunteer work with Chicago orphans and help running the student gallery. Emily dabbled in a variety of jobs; as a showroom and studio assistant at Mars Gallery, as a Production Assistant for Warner Bros. based film productions, as a build shop worker for Redmoon Theater, managing large scale paint teams, and designing designer suit catalogs. She has also taught in a variety of settings; teaching fine arts at the Meskill Senior Center, Ceramics at the D.P.D, New Media at Deerpath Middle School West, sculpture out of her studio in Lacuna Lofts, and now at Gallery 37. She currently works in freelance web development, social media management and education for creatives, and as an organizer for Chicago Zinefest. Emily is also an active artist. Emily's work spans numerous mediums; to the point that there are few areas she has yet to delve into. However she enjoys multi-medium drawing and sculpture the most. Her subjects are diverse and can range from whimsical twists on reality to full on fantasy or science fiction. Her work has been shown throughout Chicago and suburban galleries, San Francisco, Michigan, at the University of Chicago Children's Hospital Wing, and the Chicago Botanic Gardens.
Caitlin Stich is a multidisciplinary artist currently teaching Digital Media within the Career & Technical Education department for Chicago Public Schools. After learning photography composition and development skills in her high school darkroom, she began working for a small photography business as an event photographer and videographer and has never stopped documenting the human experience. Along with a love for photo/video, Caitlin continues to develop large scale community murals, works as a freelance graphic designer, and is excited to join the Gallery 37 team assisting the 3D Modeling and Videography AAP courses.


text:
Galathea rostrata is een tienpotigensoort uit de familie van de Galatheidae. De wetenschappelijke naam van de soort is voor het eerst geldig gepubliceerd in 1880 door A. Milne Edwards.

Galatheidae

redpajama_set_name: RedPajamaWikipedia