% $Id: notes.tex,v 1.22 2019/04/15 10:29:35 brandenb Exp $
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\title{The spectral nature of solar phototurbulence}
\author{}
\date{\today,~ $ $Revision: 1.22 $ $}
\begin{document}
\maketitle

\section{Preliminaries}

In the Sun, the radiative diffusivity is at least four orders of magnitude
larger than the viscosity.
This means that the Prandtl number, i.e., the ratio of kinematic viscosity
to thermal or radiative diffusivity is always below $10^{-4}$.
One would therefore expect the temperature field to be always much
smoother than the velocity field.
However, on small length scales---comparable to the viscous diffusive
cutoff or Kolmogorov scale---the radiative diffusion approximation
breaks down.
The gas experiences Newtonian cooling rather than photon diffusion.
Such cooling is independent of the scale of the temperature structures
and is thus no longer increasing toward smaller scales, as in Fickian
diffusion.
This means that our intuition about a smooth temperature field owing to
a low Prandtl number may need to be reconsidered.
Here we explore some of the consequences of this idea using simple
numerical experiments.

In thinking about a minimalistic numerical experiment, it is useful
to remember that turbulence is constantly dissipating energy through
viscosity---even if the viscosity is extremely small.
This source of heating is always present and can be
significant in high Reynolds number turbulence.
This heating may already be sufficient for a simple numerical experiment.
Thus, we consider turbulence, here sustained by volume forcing in
a small domain within the Sun.
The effects of gravity and surface cooling, which can give rise
convection, will be omitted, so we restrict ourselves to isotropic
turbulence using triply-periodic boundary conditions.

In the Sun, the photon mean-free path is given by
$\ell_\gamma=(\kappa\rho)^{-1}$, where $\kappa$ is the specific opacity
and $\rho$ is the density.
We find it convenient to work here instead with the wavenumber or inverse
length scale and therefore refer to the critical photon wavenumber
$k_\gamma=\kappa\rho$, above which radiation works predominantly through
Newtonian heat exchange.
The Kolmogorov wavenumber depends on the viscosity $\nu$ and
the mean rate of energy dissipation $\epsilon$ and is given by
$k_\nu=(\epsilon/\nu^3)^{1/4}$.
This is the largest wavenumber of the inertial range, beyond which the
spectral kinetic energy begins to fall off exponentially.
Analogously, we can define a radiative diffusion cutoff wavenumber,
$k_\chi=(\epsilon/\chi^3)^{1/4}$, but it ignores that aforementioned
complication that on small length scales, radiation works through
Newtonian heating or cooling.
Finally, we have the wavenumber of the energy-carrying eddies $\kf$,
which, in the Sun, is usually being associated with the inverse
pressure scale height.
In our numerical experiments, on the other hand, we assume
volumetric random stirring with a wavenumber comparable to the
lowest wavenumber in the computational domain.

We also consider a series of experiments where large-scale heating is
accomplished by imposing a uniform temperature gradient, enabling
temperature fluctutions to be generated by tangling.

\subsection{Governing equations}

We solve the hydrodynamics equations for logarithmic density $\ln\rho$,
velocity $\uu$, and specific entropy $s$, in the form
\EQA
{\DD \ln \rho \over \DD t}&=&-\nab\cdot\uu, \\
 \rho{\DD \uu\over \DD t}&=&-\nab p + \rho\ff + \nab\cdot(2\rho\nu\SSSS), \\
 \rho T {\DD s \over \DD
  t}&=&-\nab\cdot\FF_{\rm rad}-\beta u_z+2\iota\rho\nu \SSSS^2,
\label{sRT}
\ENA
where $p$ is the gas pressure, $\nu$ is the viscosity,
$\SSSS=\half[\nab\uu+(\nab\uu)^T]-\onethird\IIII\nab\cdot\uu$
is the traceless rate-of-strain tensor, $\IIII$ is the unit tensor,
$T$ is the temperature, 
$\iota$ and $\beta$ are optional parameters ($\beta=0$ and $\iota=1$
is the standard case without large-scale heating and with viscous
heating included), $\FF_{\rm rad}$ is the radiative flux and
energy supply is provided by the forcing function $\ff = \ff(\xx,t)$,
which is random in time and defined as
\EQ
\ff(\xx,t)={\rm Re}\{N\ff_{\kk(t)}\exp[\ii\kk(t)\cdot\xx+\ii\phi(t)]\},
\label{ForcingFunction}
\EN
where $\xx$ is the position vector.
The wavevector $\kk(t)$ and the random phase
$-\pi<\phi(t)\le\pi$ change at every time step, so $\ff(\xx,t)$ is
$\delta$-correlated in time.
Therefore, the normalization factor $N$ has to be proportional to $\delta t^{-1/2}$,
where $\delta t$ is the length of the time step.
On dimensional grounds it is chosen to be
$N=f_0 c_{\rm s}(|\kk|c_{\rm s}/\delta t)^{1/2}$, where $f_0$ is a
nondimensional forcing amplitude.
At each timestep we select randomly one of many possible wavevectors
in a certain range around a given forcing wavenumber with average value
$k_{\rm f}$.
In the following, we choose $k_{\rm f}/k_1=1.5$.
The parameter $\iota$ in \Eq{sRT} has been introduced to allow us turning
off viscous heating in some of our numerical experiments.

As equation of state, we adopt a perfect gas with
$p=({\cal R}/\mu)T\rho$, where ${\cal R}$ is the universal gas constant
and $\mu$ is the mean molecular weight.
The pressure is related to $s$ via $p=\rho^\gamma\exp(s/\cv)$, where
the adiabatic index
$\gamma=\cp/\cv$ is the ratio of specific heats at constant pressure
and constant volume, respectively, and $\cp-\cv={\cal R}/\mu$.
To obtain the radiative flux, we adopt the gray approximation,
ignore scattering, and assume that the source function $S$ (not
to be confused with the rate-of-strain tensor $\SSSS$) is given
by the frequency-integrated Planck function, so $S=(\sigmaSB/\pi)T^4$,
where $\sigmaSB$ is the Stefan--Boltzmann constant.
The negative divergence of the radiative flux is then given by
\EQ
-\nab\cdot\FF_{\rm rad}=\kappa\rho \oint_{4\pi}(I-S)\,\dd\Omega,
\label{fff}
\EN
where $\kappa$ is the opacity per unit mass
(assumed independent of frequency) and $I(\xx,t,\nnn)$
is the frequency-integrated specific intensity corresponding to
the energy that is carried by radiation per unit area, per unit time,
in the direction $\nnn$, through a solid angle $\dd\Omega$.
We obtain $I(\xx,t,\nnn)$ by solving the radiative transfer equation,
\EQ
\nnn\cdot\nab I=-\kappa\rho\, (I-S),
\label{RT-eq}
\EN
along a set of rays in different directions $\nnn$ using the
method of long characteristics.
We adopt a Kramers opacity given by
\EQ
\kappa=\kappa_0\rho^a T^b,
\label{kappa}
\EN
where $a=1$ and $b=-7/2$.
The radiative conductivity $K(\rho,T)$ is given by
\EQ
K(\rho,T)={16\sigmaSB T^3\over 3\kappa\rho}
={16\sigmaSB T^{3-b}\over 3\kappa_0\rho^{a+1}}.
\label{K-model}
\EN

We are particularly interested in spectra of specific entropy,
$E_s(k)$, which are normalized such that
\EQ
\int_0^\infty E_s(k)\,\dd k=\half\bra{s^2}.
\EN

\section{Choice of parameters}

We consider a cubic domain of size $L^3$, where $L=1\Mm$.
With our choice of $k_{\rm f}/k_1=1.5$, this implies
$k_{\rm f}\approx10\Mm^{-1}$.
We use $576^3$ meshpoints.
For the viscosity we choose $\nu=10^{-3}\Mm\km\s^{-1}$.
We choose $f_0=0.1$, which results in an rms velocity of about
$4\km\s^{-1}$.
The Reynolds number is then
\EQ
\Rey=\urms/\nu\kf=420.
\EN
The initial density is $\rho=4\times10^{-4}\g\cm^{-3}$,
$T=36,000\K$ (initially $39,000\K$, but can cool down to $34,000\K$),
$\cs=30\km\s^{-1}$, $p=0.2\g\cm^{-3}\km^2\s^{-2}$.
For the radiation transport, we use 6 rays.
This turns out to be sufficient, because a comparison with
14 and 22 rays did not result in noticeable differences,

\section{Cooling times}

To provide background for the interpretation of our results, we discuss
first the dependence of the radiative cooling rate $\lambda$ on the
mean-free path and the wavenumber $k$ of the temperatures structures.
It is given by
\cite[see Appendix~A of][ for details]{BB14}
\begin{equation}
\lambda={c_\gamma\ell k^2/3\over1+\ell^2 k^2/3}=\left\{
\begin{array}{ll}
c_\gamma\ell k^2/3\cr
c_\gamma/\ell
\end{array}
\right.
\end{equation}
Optically thick if $\ell^2 k^2/3\ll1$.
\begin{equation}
\lambda=(c_\gamma/\ell)\,\min(1,\ell^2 k^2/3).
\end{equation}
See \Tab{Tcool}.

\begin{table*}[b!]\caption{Cooling times based on
$c_\gamma=3\km\s^{-1}$, which is relevant for
$\rho=4\times10^{-4}\g\cm^{-3}$ and $T=36,000\K$.
Here, $\Pe=\urms/\chi\kf$.
}\vspace{12pt}\centerline{\begin{tabular}{lrrllr}
      & $\ell\qquad$ & $\lambda\qquad$ & $\quad\tau$ & $\quad\chi$ \\
$\Pr$ & $k_\gamma=\kappa\rho\;$ & $c_\gamma k_\gamma\quad$ &
$(c_\gamma k_\gamma)^{-1}$ & $c_\gamma/3k_\gamma$ & $\Pe$ \\
\hline
0.004 & $  4\Mm^{-1}$ & $   12\ks^{-1}$ & $0.1\ks$    & 0.25    &$1.6\!\!\!\!\!$ \\
0.04  & $ 40\Mm^{-1}$ & $  120\ks^{-1}$ & $0.01\ks$   & 0.025   &   16 \\
0.4   & $400\Mm^{-1}$ & $ 1200\ks^{-1}$ & $0.001\ks$  & 0.0025  &  160 \\
4     &$4000\Mm^{-1}$ & $12000\ks^{-1}$ & $0.0001\ks$ & 0.00025 & 1600 \\
\label{Tcool}\end{tabular}}\end{table*}

\section{Advection tests}

Advection tends to sharpen the temperature jump, while radiation tends
to smoothen it.
Thus, we expect that with decreasing for opacity and thus increasing
mean-free path, the thermal diffusivity, given by $\chi=c_\gamma\ell/3$,
the temperature profile should become smoother.
However, when the opacity decreases further, the mean-free path begins to
exceed the viscous length scale, $\sim\nu/U_0$, where $U_0$ is the magnitude
of the advection velocity.
When that is the case, the medium becomes essentially optically thin
on those small length scales, and radiation thus loses its diffusing
properties.
We thus expect ...

\section{Results}

\FFig{pplam} shows the results of a numerical experiment using
a sinusoidal initial condition for the specific entropy.

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pvisc}
\end{center}\caption[]{
Critical wavenumbers for the solar convection zone
model of \cite{Spr74}.
}\label{pvisc}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pplam}
\end{center}\caption[]{
Numerically determined decay rates for $k_\ell=40$, $400$, and
$1000\Mm^{-1}$ using 288 meshpoints in one dimension.
Note that  cooling becomes inefficient at wavenumbers above $\kappa\rho$.
}\label{pplam}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pspec_comp576}
\end{center}\caption[]{
Series~A.
Spectra for $k_\gamma=4$ (black), $40$ (red), $400$ (blue), and $4000$ (green).
}\label{pspec_comp}\end{figure}

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pspec_comp}
%\includegraphics[width=.49\columnwidth]{pspec_comp_time}
%\includegraphics[width=.49\columnwidth]{pspec_comp_time1}
%\end{center}\caption[]{
%OLD: Series~A.
%Spectra for $k_\gamma=8$ (black), 40 (red), $400$ (blue),
%and $400$ optically thick (orange).
%The lower panel shows time series of $E_s(k_\ast,t)$ at $k_\ast=100\Mm^{-1}$.
%}\label{pspec_compOLD}\end{figure}

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pspec_comp_grad}
%\includegraphics[width=.49\columnwidth]{pspec_comp_grad_time}
%\includegraphics[width=.49\columnwidth]{pspec_comp_grad_time1}
%\end{center}\caption[]{
%Series~B.
%Spectra for $k_\gamma=40$ (red), $400$ (blue), and $400$ optically thick (orange).
%$\beta=5\times10^{-3}$ code units.
%The lower panel shows time series of $E_s(k_\ast,t)$ at $k_\ast=100\Mm^{-1}$.
%}\label{pspec_comp_grad}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pspec_comp_grad_novheat576}
\end{center}\caption[]{
Series~C.
Spectra for $k_\gamma=4$ (black), $40$ (red), $400$ (blue), and $4000$ (orange).
$\beta=5\times10^{-3}$ code units, but viscous heating turned off.
%The lower panel shows time series of $E_s(k_\ast,t)$ at $k_\ast=100\Mm^{-1}$.
%For both $k_\gamma=40$ (red) and $400$ (blue), there seems to be
%persistant heating even after $3\ks$, although the cooling times
%are only $0.01\ks$ and $0.001\ks$; see \Tab{Tcool}.
}\label{pspec_comp_grad_novheat}\end{figure}

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pspec_comp_grad_novheat}
%\includegraphics[width=.49\columnwidth]{pspec_comp_grad_novheat_time}
%\includegraphics[width=.49\columnwidth]{pspec_comp_grad_novheat_time1}
%\end{center}\caption[]{
%OLD: Series~C.
%Spectra for $k_\gamma=40$ (red), $400$ (blue), and $400$ optically thick (orange).
%$\beta=5\times10^{-3}$ code units, but viscous heating turned off.
%The lower panel shows time series of $E_s(k_\ast,t)$ at $k_\ast=100\Mm^{-1}$.
%For both $k_\gamma=40$ (red) and $400$ (blue), there seems to be
%persistant heating even after $3\ks$, although the cooling times
%are only $0.01\ks$ and $0.001\ks$; see \Tab{Tcool}.
%}\label{pspec_comp_grad_novheatOLD}\end{figure}

%%288c400_thickt_grad6_novheat
%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{pspec_comp_grad_chit}
%\end{center}\caption[]{
%Comparison of spectra between the optically thick approximation (black)
%and just specific entropy diffusion (red).
%}\label{pspec_comp_grad_chit}\end{figure}

%\begin{figure}[t!]\begin{center}
%\includegraphics[width=\columnwidth]{grad40_img_0015}
%\includegraphics[width=\columnwidth]{grad400_img_0015}
%\end{center}\caption[]{
%Images for 40 (top) and 400 (bottom).
%}\label{pspec_comp_grad_chit}\end{figure}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pslices}
\end{center}\caption[]{
Images for 4, 40, 400, and 4000.
\red{add x [Mm]}
}\label{pslices}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pcomp_TT}
\end{center}\caption[]{
Temperature evolution for 4, 40, 400, and 4000.
}\label{pcomp_TT}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pcomp_pdf}
\end{center}\caption[]{
PDFs for 4, 40, 400, and 4000.
}\label{pcomp_pdf}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pspec_comp576}
\includegraphics[width=\textwidth]{pspec_comp576b}
\end{center}\caption[]{
Spectra \url{pspec_comp576} and \url{pspec_comp576b}.
}\label{pspec_comp_time576_100}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pspec_comp_time576_6}
\includegraphics[width=\textwidth]{pspec_comp_time576_100}
\end{center}\caption[]{
Temporal evolution at $k=6$ (top) and 100 (bottom)
}\label{pspec_comp_time576_100}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pspec_comp576a_grad_novheat}
\end{center}\caption[]{
Spectra \url{pspec_comp576a_grad_novheat}
}\label{pspec_comp576a_grad_novheat_time576_100}\end{figure*}

\begin{figure*}[t!]\begin{center}
\includegraphics[width=\textwidth]{pspec_comp_thickthin576}
\end{center}\caption[]{
Spectra in the optically thin cases with $k_\gamma=40$ (black) and 400 (red)
and the optically thick cases with $k_\gamma=40$ (blue) and 400 (green).
}\label{pspec_comp_time576_100}\end{figure*}

%plot,ts.urms/1e-3/1.5/(2*!pi)
\begin{table}[htb]\caption{Parameters.
}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
Series & A & B & C \\
\hline
visc heat & yes & yes & no \\
$\beta$ & $0$ & $5\times10^{-3}$ \\
$\urms$  & 3.5 & 3.8 \\
$\Rey$  & 1800 & 400 \\
$\epsK$ & .... & ....\\
\label{Ttimescale}\end{tabular}}\end{table}

We expect {\em insulation} and inefficient cooling at wavenumbers
above $\kappa\rho$.
This is not obvious from the numerical results which show that the
spectral specific entropy is actually independent of the opacity.
Furthermore, the spectral specific entropy increases with wavenumber,
which is opposite to the behavior spectral kinetic energy.
This is clearly a consequence of the somewhat unusual source of
heating which is viscous heating and therefore naturally a small-scale
phenomenon.

\begin{verbatim}
\end{verbatim}

Discuss instability of descending blobs.
Visualize them.
Do decay simulation.

%r e f
\begin{thebibliography}{}

\bibitem[Barekat \& Brandenburg(2014)]{BB14}
Barekat, A., \& Brandenburg, A.\yana{2014}{571}{A68}
%{Near-polytropic stellar simulations with a radiative surface}

\bibitem[Spruit(1974)]{Spr74}
Spruit, H. C.\ysph{1974}{34}{277}
%{290}{A model of the solar convection zone}

\end{thebibliography}

\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/tex/notes/photo/notes.tex,v 1.22 2019/04/15 10:29:35 brandenb Exp $
\end{verbatim}

\end{document}



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