% $Id: notes.tex,v 1.7 2020/03/18 18:06:02 brandenb Exp $
%\documentclass{article}
\documentclass[twocolumn]{article}
\setlength{\textwidth}{160mm}
\setlength{\oddsidemargin}{-0mm}
\setlength{\textheight}{220mm}
\setlength{\topmargin}{-8mm}
%\usepackage{graphicx,natbib,bm,url,color,slashed,amsmath,mathtools}
\usepackage{graphicx,natbib,bm,url,color,comment}
\newcommand{\fsl}[1]{\ensuremath{\mathrlap{\!\not{\phantom{#1}}}#1}}
\graphicspath{{./fig/}{./png/}}
\thispagestyle{empty}
\input macros
\def\red{\textcolor{red}}
\def\blue{\textcolor{blue}}
\title{Stellar atmospheric photo-overshoot}
\author{}
\date{\today,~ $ $Revision: 1.7 $ $}
\begin{document}
\maketitle

\section{Introduction}

Convection is a standard form of energy transport in regions of a star
where the specific entropy decreases with height, i.e., the stratification
is superadiabatic.
Overshoot tends to occur on the boundaries between convective and
radiative zones, and thus cannot occur in on its own.
Overshoot is characterized by an oppositely oriented transport of
energy, because overshooting plumes are heavier and cooler than their
surroundings, so the product of velocity and temperature fluctuations
is negative.

We report here the results of a numerical study of a stably stratified
layer of gas whose opacity increases with decreasing pressure and
therefore with height.
Adiabatically upward moving blobs of fluid therefore become more opaque
and can be elevated by radiation pressure.
The elevated fluid becomes even more opaque and will be elevated even further.
Likewise, downward moving fluid will become more transparent and will
sink further, and can fill the place left by the upward moving fluid.
This can lead to instability and the development of overturning motions,
or even continuous mass loss.
Similar effects also occur when the temperature dependence of the opacity
has a local maximum, which can lead to turbulence in those layers where
the temperature dependence of the opacity has a local maximum.

Our goal is to study basic properties of such flows in an idealized system.
We perform a sequence of numerical experiments using a simple power law
prescription for the opacity of the form
\EQ
\kappa=\kappa_0\,(\rho/\rho_0)^a\,(T/T_0)^b,
\label{Kramers}.
\EN
where the exponents $a$ and $b$ and the pre-factor $\kappa_0$ will be
varied, and the coefficients $\rho_0$ and $T_0$ will be kept fixed.

The numerical treatment of such a system poses numerical difficulties
owing to the requirement of very short time steps.
This is because the radiative cooling time becomes short at high
temperatures, which is when the radiation pressure tends to play an
important role.
This was recently investigated in detail in a separate paper \citep{BD19}.
To cope with this difficulty, we simply reduce the speed of light.
This increases the effect of radiation pressure, but has no direct
effect on the rest of the equations.
However, since $c$ is related to the Stefan-Boltzmann constant
through $\sigmaSB=a_{\rm rad}c/4$, there are two possibilities.
Either we keep $a_{\rm rad}$ fixed and vary $\sigmaSB$,
or we vary $a_{\rm rad}$ and keep $\sigmaSB$ fixed.
In that case, however, it would therefore gives the system more
radiation energy than it actually has.

\section{The model}

\subsection{Governing equations}

We solve the basic equations for the logarithmic density $\ln\rho$,
the velocity $\uu$, and the specific entropy $s$, in the form
\EQ
{\DD\ln\rho\over\DD t}=-\nab\cdot\uu,
\label{dlnrho}
\EN
\EQ
\rho{\DD\uu\over\DD t}=-\nab p + \rho\grav + {\rho\kappa\over c}\FF_{\rm rad}
+\nab\cdot(2\rho\nu\SSSS),
\EN
\EQ
\rho T{\DD s\over\DD t}=-\nab\cdot\FF_{\rm rad}
+2\rho\nu\SSSS^2,
\EN
\EQ
\nnn\cdot\nab I=-\kappa\rho\,(I-S),
\EN
\EQ
\FF_{\rm rad}=\int_{4\pi}\nnn I\,\dd\Omega,
\EN
\EQ
\nab\cdot\FF_{\rm rad}=\int_{4\pi}(I-S)\,\dd\Omega,
\label{ngradI}
\EN
where $\grav=(0,0,-g)$ is the gravitational acceleration in
Cartesian coordinates $(x,y,z)$,
$c$ is the speed of light, $\FF_{\rm rad}$ is the radiative flux,
${\sf S}_{ij}=\half(\partial_i u_j+\partial_j u_i)-\onethird
\delta_{ij}\nab\cdot\uu$ are the components of the traceless
rate-of-strain tensor,
and $\nu$ is the kinematic viscoity.

\subsection{Nondimensional form of the equations}

It is useful to write the equations in non-dimensional form
by normalizing temperature and density by some representative
values that characterize the surface.
In a non-convecting gray atmosphere with constant exponents
$a$ and $b$, the temperature falls of linearly with height
until it reaches a constant value, $T_0$.
In the deeper, optically thick part, the density falls off
like a polytrope, so $\rho\propto T^n$ where $n=(3-b)/(1+a)$.
In the upper optically thin part, which is isothermal,
the density falls off exponentially with the pressure scale height
$\Hpz=\cp T\nabad/g$, which is also equal to the density scale height
in this isothermal part.
It is then convenient to measure lengths in units of $\Hpz$, i.e.,
\EQ
[x]=\Hpz.
\EN
Since $g=\const$, it is convenient to measure time in units of
$[t]=\sqrt{\Hpz/g}$, and velocity in units of
\EQ
u_0\equiv\sqrt{\Hpz g}.
\EN
Finally, we measure density and pressure in units of the values
$\rho_0$ and $p_0$ at the crossover between polytropic and
isothermal stratifications.
They are related to each other through an equation of state
\EQ
p_0/\rho_0=\Rgas T_0/\mu_0,
\EN
where $\Rgas$ is the universal gas constant and $\mu_0$ is
the mean molecular weight.

To determine $\rho_0$, it is useful to recall the analytic
solution for a gray atmosphere with constant $a$ and $b$
in the form \citep{Bra16}
\EQ
T/T_0=\left[1+(n+1)\nabla_{\rm rad}^{(0)}(p/p_0)^{1+a}\right]^{1/(4+a-b)}
\EN
with $\nabla_{\rm rad}^{(0)}=\cp\Frad\nabad/(gK_0)$ being the usual
radiative double-logarithmic temperature gradient and
$K_0=16\sigma T_0^3/(3\kappa_0\rho_0)$ the radiative conductivity
evaluated for our representative values $T_0$ and $\rho_0$.
Since $\nabla_{\rm rad}^{(0)}$ itself depends on $\rho_0$, we can
determine $\rho_0$ such that $(n+1)\nabla_{\rm rad}^{(0)}=1$.
Using $\Frad=\sigmaSB\Teff^4=2\sigmaSB T_0^4$, where $\Teff$ is the
effective temperature, this yields
\EQ
\rho_0=8/(3\kappa_0\Hpz).
\EN

\subsection{Stratification without radiation pressure}

In thermodynamic equilibrium, the radiative flux must be constant, i.e.,
\EQ
F_{\rm rad}=-K\,\dd T/\dd z=\const,
\label{FradEqn}
\EN
where $K=16\sigmaSB T^3/(3\kappa\rho)$ is the radiative conductivity
with $\sigmaSB$ being the Stefan--Boltzmann constant, and $z$ is the
vertical coordinate in a Cartesian coordinate system.
We have then a polytropic stratification with $\rho\propto T^n$, where
\EQ
n=(3-b)/(1+a)
\label{n_Def}
\EN
is the polytropic index.

The double-logarithmic temperature gradient is obtained by dividing
the two equations through each other, i.e.,
\EQ
\nabla={\dd\ln \meanT\over\dd\ln \meanP}
={F\meanP\over K\meanT\,\meanrho g}={F\cP\nabad \over Kg},
\label{nabla}
\EN
where we have used the perfect gas equation of state in the form
$\meanP/\meanT\,\meanrho=\cP-\cV=\cP\,(1-1/\gamma)=\cP\nabad$.
We can also define a {\em hypothetical} radiative temperature gradient
$\nabrad$ that would result if all the energy were carried by radiation,
so we can write
\EQ
F_{\rm tot}={Kg\over \cP\nabad}\,\nabrad,
\label{nabrad_Def}
\EN
which follows from \Eq{nabla}.

Dividing \Eq{FradEqn} by the equation for hydrostatic equilibrium,
$\dd P/\dd z=-\rho g$, we have
\EQ
{\dd T\over\dd P}
={F_{\rm rad}\over K_0\rho_0 g} {(P/P_0)^a\over (T/T_0)^{3+a-b}},
\label{dTdP}
\EN
where $K_0=16\sigmaSB T_0^3/(3\kappa_0\rho_0)$ is a constant
and $P/P_0=(\rho/\rho_0)(T/T_0)$ is the ideal gas equation with a
suitably defined constant $P_0=(\cp-\cv)\rho_0 T_0$.
Here, $\rho_0$ and $T_0$ are reference values that were
defined in \Eq{Kramers}.
\EEq{dTdP} can be integrated to give
\EQ
(T/T_0)^{4+a-b}=(n+1)\nabrad^{(0)}(P/P_0)^{1+a}+(T_{\rm top}/T_0)^{4+a-b},
\label{ToverT0}
\EN
where $\nabrad^{(0)}=F_{\rm rad}P_0/(K_0T_0\rho_0 g)$, which is
defined analogously to the $\nabrad$ without superscript $(0)$ in
\Eqs{nabla}{nabrad_Def}, and $T_{\rm top}$ is an integration constant
that is specified such that $T\to T_{\rm top}$ as $P\to0$.
Note also that $4+a-b=(n+1)(1+a)$, where $n$ was defined in \Eq{n_Def}
as the polytropic index, so the ratio of $4+a-b$ to $1+a$ is just $n+1$,
which enters in front of the $\nabrad^{(0)}$ term in \Eq{ToverT0}.
Since $K\propto T^{3-b}/\rho^{1+a}\propto T^{4+a-b}/P^{1+a}$,
we have $K\to\const=K_0$ for $T\gg T_{\rm top}$.

\begin{equation}
\end{equation}

%\begin{comment}
\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{pvar_s144x72a}
\end{center}\caption[]{
\url{pvar_s144x72a}
The yellow horizontal line around $z\approx8\Mm$ marks the loation
of the $\tau=1$ surface.
}\label{pvar_s144x72a}\end{figure}

\begin{figure}[t!]\begin{center}
\includegraphics[width=\columnwidth]{ptt_s144x72a}
\end{center}\caption[]{
\url{ptt_s144x72a}
}\label{ptt_s144x72a}\end{figure}
%\end{comment}

%\begin{table}[htb]\caption{
%}\vspace{12pt}\centerline{\begin{tabular}{lccccccc}
%\hline
%\label{Ttimescale}\end{tabular}}\end{table}

\begin{verbatim}
\end{verbatim}

Hot stars are observed to exhibit macroturbulence.
Those stars have either no or only very shallow subsurface convection
zones that are associated with local maxima in the temperature dependence
of the opacity.
Photoconvection is a form of convection that is driven by super-Eddington
luminosities where the fluid remains suspended.

%r e f
\begin{thebibliography}{}

\bibitem[Brandenburg(2016)]{Bra16}
Brandenburg A.\yapj{2016}{832}{6}
%{Stellar mixing length theory with entropy rain}

\bibitem[Brandenburg \& Das(2019)]{BD19}
Brandenburg, A. \& Das, U.\sgafd{2019}
%{The time step constraint in radiation hydrodynamics}
{1901.06385}

\bibitem[Arons(1992)]{Aro92}
Arons, J.\yapj{1992}{388}{561}
%{578}{Photon bubbles - Overstability in a magnetized atmosphere}

\end{thebibliography}

\vfill\bigskip\noindent\tiny\begin{verbatim}
$Header: /var/cvs/brandenb/tex/radiation/radpressure/notes.tex,v 1.7 2020/03/18 18:06:02 brandenb Exp $
\end{verbatim}

\end{document}

Begelman 2001
Magnetic field important: constrains motions to be 1-D.
Flux ~ 1/rho.

Aron (1992)

Gammie (1998). Vertical energy transport will be enhanced.


Photon bubble instability discovered by Gammie (1998).

