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<!DOCTYPE html>
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<head>
<!-- 2025-02-20 Thu 09:47 -->
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<title>400A - Convection & MLT</title>
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<p>
<b>Materials</b>: Kippenhahn's book Ch. 6 and 7, Cox & Giuli vol. 1 Ch. 14,
Schwarzschild Ch. 7, <a href="https://ui.adsabs.harvard.edu/abs/2022ApJS..262...19J/abstract">Jermyn et al. 2022</a>, <a href="https://ui.adsabs.harvard.edu/abs/2022ApJ...926..169A/abstract">Anders et al. 2022</a>, <a href="http://online.kitp.ucsb.edu/online/stars17/cantiello2">Matteo
Cantiello's talk at KITP in 2017</a>, review by <a href="https://ui.adsabs.harvard.edu/abs/2023Galax..11...75J/abstract">Joyce & Tayar 2023</a>.
</p>
<div id="outline-container-orgf14d0c7" class="outline-2">
<h2 id="orgf14d0c7"><a href="#orgf14d0c7">Energy transport in stars 2/2</a></h2>
<div class="outline-text-2" id="text-orgf14d0c7">
<p>
In the <a href="./notes-lecture-ETransport.html">previous lecture on energy transport</a> we have dealt with the
situation where energy diffusion (carried by photons, i.e., radiative
energy transport, or carried by electrons, i.e., conductive energy
transport) is sufficient to sustain the star and carry all the flux to
maintain <i>local</i> energy conservation. However, this is <i>not</i> always the
case! For example, we know this is not the case in the outer layers of
the Sun, we can <a href="https://apod.nasa.gov/apod/ap200203.html">directly see that</a>:
</p>
<iframe width="600" height="400" src="https://www.youtube.com/embed/CCzl0quTDHw?si=1h3tkmpwsi4w9uRz" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
<p>
We can also see this is not the case in the envelopes of red
supergiants, as for example Betelgeuse, VY Canis Majoris, etc. Less
directly, we can also infer this to happen in the cores of massive
stars: <i>some stellar layers are not stably stratified and energy
transport is not only diffusive</i>.
</p>
</div>
</div>
<div id="outline-container-orgc5b332f" class="outline-2">
<h2 id="orgc5b332f"><a href="#orgc5b332f">Convection</a></h2>
<div class="outline-text-2" id="text-orgc5b332f">
<p>
In this lecture we will deal with convection, which allows for <i>energy
transport through macroscopic motion of matter</i> resulting in a
<i>non-zero energy flux with a zero matter flux</i>.
</p>
<p>
Convection involves turbulence (the "last" open problem of classical
physics), and is because of this inherently multi-dimensional. Because
of this, convection is usually one of the most approximate ingredients
in stellar calculations, and often the root of many open problems.
</p>
<p>
Convection is a <i>thermal</i> instability (as we will see it kicks in if the
temperature gradient is steeper than some threshold condition),
although it results in <i>local</i> motion of gas/plasma: as we will see the
velocities are very small compared to thermal velocities. Moreover, it
is <i>not</i> unique to stars: the monsoon clouds in Tucson's summer are also
manifestation of convection in the Earth atmosphere!
</p>
<p>
Before discussing the details of the physics of convection by using
oversimplified pictures dating back to <a href="https://en.wikipedia.org/wiki/Ludwig_Prandtl">Prandtl</a>, have a look at the
following animations of multi-dimensional simulations of convection,
these are probably/hopefully closer to reality than many of the
oversimplifications we will use later in this lecture.
</p>
<p>
<b>N.B.:</b> It is important to keep in mind that we adopt modeling
simplifications to make the simulation of stars tractable, but these
introduce systematic uncertainties which are active topic of research.
Approximations are present also in the multidimensional simulations
shown below, and therefore they should <i>not</i> be taken as the truth!
However, these multi-dimensional simulations do not need to make the
same assumptions we will discuss in this lecture, which makes them
informative on how rough these approximations are.
</p>
<p>
What all these simulations of convection in different settings show is
that the morphology of the convective flow is <i>more complicated</i> than
what we will assume. You can imagine "thermal flux tubes" carried by
the gas that transport energy in all these situations, but the way we
describe them in spherically symmetric models of stars <i>cannot</i> (and
does not attempt to) capture all the details we can see in the Sun and
or a few nearby red supergiants, and we can simulate in restricted
time and spatial domains.
</p>
</div>
<div id="outline-container-org823ea96" class="outline-3">
<h3 id="org823ea96"><a href="#org823ea96">Core convection in a massive star (<a href="https://ui.adsabs.harvard.edu/abs/2022ApJ...926..169A/abstract">Anders et al. 2022b</a>)</a></h3>
<div class="outline-text-3" id="text-org823ea96">
<p>
This is a simulation of the temperature fluctuations (right) and
vertical velocities (left) in "code units" using the code <a href="https://github.com/DedalusProject/dedalus">Dedalus</a>.
</p>
<iframe width="600" height="400" src="https://player.vimeo.com/video/684826914" frameborder="0" allow="autoplay; encrypted-media" allowfullscreen=""></iframe>
</div>
</div>
<div id="outline-container-orgc5ba8d4" class="outline-3">
<h3 id="orgc5ba8d4"><a href="#orgc5ba8d4">Envelope convection in a red super-giant (<a href="https://ui.adsabs.harvard.edu/abs/2022ApJ...929..156G/abstract">Goldberg et al. 2022</a>)</a></h3>
<div class="outline-text-3" id="text-orgc5ba8d4">
<p>
This is a 3D radiation-hydrodynamics simulation of large portions of
the envelope of a red supergiant (not that different from
Betelgeuse), computed with the <a href="https://www.athena-astro.app/">ATHENA++</a> code.
</p>
<iframe width="600" height="400" src="https://www.youtube.com/embed/Cq5EqDkXFhQ?si=m4hGwqZy_YPeOcxC" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe>
</div>
</div>
<div id="outline-container-org686e685" class="outline-3">
<h3 id="org686e685"><a href="#org686e685">Thermonuclear runaway during a Nova explosion (<a href="https://ui.adsabs.harvard.edu/abs/2018A%26A...619A.121C/abstract">Casanova et al. 2018</a>)</a></h3>
<div class="outline-text-3" id="text-org686e685">
<p>
This is a 2D simulation of convection developing during a nova
explosion using the <a href="https://flash.rochester.edu/site/">FLASH</a> code. The movie shows the (log<sub>10</sub>) mass
fraction of \(^{20}\mathrm{Ne}\), and is taken from <a href="http://www.fen.upc.edu/users/jjose/">J. Jordi's personal webpage</a>
(one of the co-authors).
</p>
<video controls width="600" height="400" src="./images/125M-ONe-3.mp4" </video>
</div>
</div>
</div>
<div id="outline-container-orgd60934c" class="outline-2">
<h2 id="orgd60934c"><a href="#orgd60934c">The timescale for convection</a></h2>
<div class="outline-text-2" id="text-orgd60934c">
<p>
We can define a timescale associated to the macroscopic motion of
matter with net zero mass flux (but non-zero energy flux) with
\(\tau_\mathrm{conv} = \Delta r/v_\mathrm{conv}\). This is the "convective
turnover timescale", where \(\Delta r\) is the spatial extent of a convective
region, and \(v_\mathrm{conv}\) is the velocity of the blobs of matter
moving around.
</p>
<p>
In the following we will see how to find \(\Delta r\) and get an estimate of
\(v_\mathrm{conv}\).
</p>
<ul class="org-ul">
<li><b>Q</b>: We have already encountered two <i>global</i> timescales relevant to
stars, the free fall timescale τ<sub>ff</sub> and the Kelvin-Helmholtz
timescale \(\tau_{KH}\). \(\tau_\mathrm{conv}\) is instead a <i>local</i> timescale, relevant only
to the convective layers. How do you think it compares to the two
global timescales we have defined?</li>
</ul>
<p>
Since convection is an <i>instability</i> we expect that it kicks in when the
stability of the matter stratification and transport of energy by the
other mechanisms we have already discussed breaks down: <i>convection
occurs if energy diffusion is insufficient to carry the flux</i>.
</p>
<p>
How convection exactly turns on/off and especially how to treat this
in stellar evolution models is still a debated issue - but we can
construct a <a href="#org0247a33">stability criterion</a>: when this is violated, convection
develops. <a href="#orga677a3d">Later on</a> we will develop a theoretical model for the <i>time-
and spatially- averaged steady state</i> which we expect convection to
reach when the instability saturates, glossing over the growth phase
of the instability. Hopefully, stellar <i>evolution</i> timescales are long
enough that describing the averaged steady state resulting from
convection is sufficient for our purposes (but there are exceptions,
for example during explosions of when looking at phenomena on
timescales shorter than \(\tau_\mathrm{conv}\)).
</p>
</div>
</div>
<div id="outline-container-org0247a33" class="outline-2">
<h2 id="org0247a33"><a href="#org0247a33">Convective stability criterion: Schwarzschild & Ledoux criterion</a></h2>
<div class="outline-text-2" id="text-org0247a33">
<figure id="org4a7c55c">
<img src="./images/conv_blobs.png" alt="conv_blobs.png" width="100%">
<figcaption><span class="figure-number">Figure 1: </span>Skematic picture of the convective stability based on Prandtl oversimplified "bubble picture". The right panel shows schematically the \(\rho(P)\) track, in the left panel the bubble is moved upwards (i.e. towards lower pressure), so we expect adiabatic expansion of the bubble to maintain pressure equilibrium with the environment. The This is Fig. 5.3 in Onno Pols' lecture notes.</figcaption>
</figure>
<p>
To derive a stability criterion, let's assume to start from a stable
situation, where the temperature gradient is determined by the
(radiative) diffusion of energy: \(dT/dr \propto \kappa L/(4\pi r^{2})\).
</p>
<p>
Let's consider a parcel of gas initially in equilibrium with its
surroundings at pressure \(P_{1}\) and density \(\rho_{1}\). To determine a
stability criterion, let's perturb such parcel by displacing it by a
certain (small) amount \(\Delta\) and discuss how the parcel reacts: if
things act to move the parcel of gas back towards its original
position we have a stable situation, if instead a seed initial
displacement \(\Delta\) result in more displacement, we have an unstable
situation.
</p>
<p>
As the gas parcel moves, we can assume it maintains hydrostatic
equilibrium with the surrounding: we are looking for a thermal
instability that acts on a longer timescale than dynamical timescale.
Moreover, if this were not the case, any pressure imbalance would be
equalized through sound-waves. Therefore, throughout the path \(\Delta\) and
at the final position we have \(P_\mathrm{bubble} = P_\mathrm{environment}\).
</p>
<p>
Let's calculate the density. Since we assume \(\Delta\) to be "small" (w.r.t.
the relevant spatial scales in the star), a first order approximation
is sufficient: \(\rho_\mathrm{bubble} = \rho_{1} + (d\rho/dr)_\mathrm{ad} \Delta\). The
relevant way to calculate \(d\rho/dr\) here is to assume that the gas parcel
moves <i>adiabatically</i>: there is no time for heat exchange, \(dq=0\), and the
specific entropy of the bubble remains constant \(ds = 0\) (recall
thermodynamics!).
</p>
<p>
<b>N.B.:</b> We can expect that convection occurs when it is energetically
convenient for a parcel of gas to keep its internal energy (\(dq=0\))
and move elsewhere to deposit that internal energy. We will revisit at
the end of this lecture this assumption.
</p>
<p>
We want to compare \(\rho_\mathrm{bubble}\) after the displacement by \(\Delta\)
to the environment density. Once again we can use a first order
approximation, but for the environment we have assumed an initially
stable stratification, meaning \(d\rho/dr\) is not adiabatic, but the
radiative gradient. Therefore \(\rho_{2} = \rho_{1} + (d\rho/dr)_\mathrm{env}\Delta \equiv \rho_{1} +
(d\rho/dr)_\mathrm{rad }\Delta\), because we are assuming the surrounding
environment to be in radiative equilibrium (and assessing the
stability of that equilibrium).
</p>
<p>
<b>N.B.:</b> we have derived an equation for \(dT/dr\) in radiative equilibrium
(i.e., when the energy is transported by the diffusion of photons),
which combined with the EOS can be turned into \((d\rho/dr)_\mathrm{rad}\).
</p>
<p>
<b>N.B.:</b> In reality, the "bubble" picture is a gross oversimplification.
In a convective layer what really moves around are "flux tubes" of
thermal energy carried by gas, but there is not a true "bubbling". A
common misconception is that water boiling is exhibiting convection:
that is not exactly correct. Water boiling is <i>by definition</i> a phase
transition from liquid to gas, however, shortly before the phase
transition occurs, it is true that conduction in the water is
typically insufficient to carry the energy release at the bottom, and
convective motion can be spotted as a small simmering of the water
breaking down into small waves at the surface.
</p>
</div>
<div id="outline-container-org59f8599" class="outline-3">
<h3 id="org59f8599"><a href="#org59f8599">Instability condition</a></h3>
<div class="outline-text-3" id="text-org59f8599">
<p>
Comparing the density of the displaced bubble \(\rho_\mathrm{bubble}\) to
the density of the environment \(\rho_{2}\) we can determine a condition for
instability. If \(\rho_\mathrm{Bubble} \leq \rho_{2}\) then there will be a buoyant
force acting to displace it further up (Archimedes force, the bubble
displaces a volume of fluid heavier than its own weight!):
</p>
<div class="latex" id="orgf1ad53b">
\begin{equation}\label{eq:instability_crit}
\mathrm{Instability\ if:} \ \rho_\mathrm{bubble} \leq \rho_{2} \Rightarrow
\left(\frac{d\rho}{dr}\right)_\mathrm{ad} \leq
\left(\frac{d\rho}{dr}\right)_\mathrm{env} \equiv \left(\frac{d\rho}{dr}\right)_\mathrm{rad} \ \ .
\end{equation}
</div>
<p>
Therefore, the density stratification for radiative energy transport
is <i>unstable w.r.t. buoyancy forces</i> <b>if</b> the density gradient
\((d\rho/dr)_\mathrm{rad}\) is larger than the adiabatic gradient
\((d\rho/dr)_\mathrm{ad}\): if the gradient is <i>too steep</i> then radiative
diffusion is not sufficient to carry the energy flux and convection
kicks in, and the threshold defining what is <i>too steep</i> is the
adiabatic gradient. This is the criterion derived by <a href="https://en.wikipedia.org/wiki/Martin_Schwarzschild">Martin
Schwarzschild</a>, however, in stellar physics textbooks it is often
re-written in a slightly different form.
</p>
</div>
<div id="outline-container-org44bcd61" class="outline-4">
<h4 id="org44bcd61"><a href="#org44bcd61">Schwarzschild criterion</a></h4>
<div class="outline-text-4" id="text-org44bcd61">
<p>
For the same reason why we tend to use the Lagrangian enclosed mass as
independent coordinate in stellar calculation, it is impractical to
use the derivatives w.r.t. radius when trying to determine whether the
stratification of gas is stable or not. For example, a red supergiant
envelope (which is convective!) has a radial extent of ∼ few
100s-1000s of \(R_{☉}\). Moreover, since we are only dealing with
properties of the stellar gas here, it is more practical to use as
independent coordinate something else that is more directly related to
the gas itself. The common choice is to use the pressure \(P\) itself.
</p>
<p>
Thus, rewriting \(d\rho/dr = (d\rho/dP)/dP/dr\), using the ideal gas EOS, and
defining the pressure scale height as the e-folding length of the
pressure:
</p>
<div class="latex" id="org9e3c76e">
\begin{equation}
H_{p} = - \frac{dr}{d \log(P)} \Leftrightarrow P(r) \simeq P_{0} e^{-r/H_{p}} \ \ ,
\end{equation}
</div>
<p>
we can rewrite:
</p>
<div class="latex" id="orge760caf">
\begin{equation}
\frac{d\rho}{dr}= -\frac{P}{H_{p}} \frac{d\rho}{dP} \ \ .
\end{equation}
</div>
<p>
<b>N.B.:</b> because of the minus sign in the definition of pressure scale
height (which is there to make \(H_{p}\) a positive quantity), the signs
change when going from \((d\rho/dr)_{i}\) to \(\nabla_{i}\).
</p>
<p>
Furthermore, it is helpful to rewrite this in terms of temperature
gradients instead of density gradients. These changes of variables,
assuming an ideal gas EOS with constant mean molecular weight \(\mu\) allow
to re-write the instability condition \ref{eq:instability_crit} in the
form most commonly called Schwarzschild criterion:
</p>
<div class="latex" id="orgcef348a">
\begin{equation}\label{eq:schwarzschild_crit}
\mathrm{Instability\ if:} \ \frac{\partial \log(T)}{\partial \log(P)}_\mathrm{rad} = \nabla_\mathrm{rad} > \frac{\partial \log(T)}{\partial \log(P)}_\mathrm{ad} = \nabla_\mathrm{ad} \ \ ,
\end{equation}
</div>
<p>
with \(\mu\) = constant. Note that from the radiative transport equation we
can directly obtain \(\nabla_\mathrm{rad}\).
</p>
<div class="latex" id="org989e57c">
\begin{equation}\label{eq:nabla_rad}
\nabla_\mathrm{rad} = \frac{3\kappa L P}{64\pi G m \sigma T^{4}} \propto \kappa L\ \ ,
\end{equation}
</div>
<p>
So we see immediately that this is going to be large and prone to
convective instability in regions where there is a large opacity \(\kappa \equiv
\kappa(m)\) and/or regions with a large luminosity \(L \equiv L(m)\).
</p>
</div>
</div>
<div id="outline-container-org2314657" class="outline-4">
<h4 id="org2314657"><a href="#org2314657">Ledoux criterion</a></h4>
<div class="outline-text-4" id="text-org2314657">
<p>
In a star, \(\mu\) is <i>not</i> always constant: as we have already seen there
can be regions of <i>partial ionization</i> where \(\mu\) changes as we move
through them, and we already know that in the fully ionized inner
regions of the star we have \(\mu \simeq 1/(2X+3Y/4+Z/2)\) so as we convert
hydrogen into helium in the core (and later on helium into metals), we
also expect \(\mu\) to change. We can thus rewrite the instability condition
\ref{eq:instability_crit} without assuming \(\mu\) = constant.
</p>
<p>
To do this, it may be helpful to write the EOS functional dependence
in a very general form \(P\equiv P(\rho, T, {X_{i}}) \equiv P(\rho, T, \mu)\). By
differentiating this we obtain
</p>
<div class="latex" id="orgc5210f7">
\begin{equation}
\frac{d\rho}{\rho} = \alpha \frac{dP}{P} -\delta \frac{dT}{T} + \varphi \frac{d\mu}{\mu} \ \ ,
\end{equation}
</div>
<p>
where \(\alpha\), \(\delta\), and \(\varphi\) are coefficients that depend on the details of
the EOS, but known as long as the EOS is known.
</p>
<p>
We can then rewrite \ref{eq:instability_crit} as
</p>
<div class="latex" id="org3d7c01d">
\begin{equation}\label{eq:ledoux_crit}
\mathrm{Instability\ if:} \nabla_\mathrm{rad} \geq \nabla_\mathrm{ad} + \frac{\varphi}{\delta} \nabla_{\mu} \ \ ,
\end{equation}
</div>
<p>
with \(\nabla_{\mu} = \partial log(\mu)/\partial\log(P)\). Eq.
\ref{eq:ledoux_crit} is usually referred to as the Ledoux criterion
(<a href="https://ui.adsabs.harvard.edu/abs/1947ApJ...105..305L/abstract">Ledoux 1947</a>).
</p>
</div>
</div>
<div id="outline-container-org747da7f" class="outline-4">
<h4 id="org747da7f"><a href="#org747da7f">Secular mixing processes: semiconvection and thermohaline mixing</a></h4>
<div class="outline-text-4" id="text-org747da7f">
<p>
For stellar layers that are stable according to the Ledoux criterion
but unstable according to the Schwarzschild criterion as we have
defined them above, that is
</p>
<div class="latex" id="orgb87f453">
\begin{equation}\label{eq:semiconv_crit}
\nabla_\mathrm{ad} \le \nabla_\mathrm{rad} \leq \nabla_\mathrm{ad} + \frac{\varphi}{\delta} \nabla_{\mu} \ \ ,
\end{equation}
</div>
<p>
the chemical potential gradient acts as a stabilizing force. There
will <i>not</i> be a full blown instability, but rather, in the approximate
toy model we have used to derive the instability criterion, the gas
parcels will experience small oscillations where the μ gradient acts
as a damping force. This is the so called <b>semiconvection</b>.
If during the oscillations gas with different \(\mu\) mix, this will
decrease \(\nabla_{\mu}\) which is the ingredient damping the oscillation:
semiconvection oscillation will slowly grow in amplitude.
</p>
<p>
Viceversa, if a layer is Schwarzschild stable but Ledoux unstable
(this can occur depending on δ and φ, that is depending on the EOS and
the chemical composition):
</p>
<div class="latex" id="orgca89e3f">
\begin{equation}\label{eq:thermohaline_crit}
\nabla_\mathrm{ad} + \frac{\varphi}{\delta} \nabla_{\mu} \le \nabla_\mathrm{rad} \leq \nabla_\mathrm{ad} \ \ ,
\end{equation}
</div>
<p>
then the mean molecular weight gradient acts to <i>destabilize</i> the layer.
In this case, in our simplistic picture, a blob of gas will slowly
start moving because of \(\nabla_{\mu}\) but there will be no restoring forces,
and we obtain the so called <b>thermohaline mixing</b> or <b>double diffusion
instability</b>. The name double diffusion comes from the fact that for
the gas parcel to move it has to diffuse thermal energy to its
environment (which otherwise would stabilize it), as its different
chemical composition also diffuses. This leads to the formation of
long "fingers", as you can <a href="https://www.stellarphysics.org/thermohaline-mixing">prove in a kitchen experiment</a>:
</p>
<figure id="orgc3b0bad">
<img src="./images/thermohaline.jpg" alt="thermohaline.jpg" width="40%">
<figcaption><span class="figure-number">Figure 2: </span>Double-diffusive fingers in hot salty water on top of cold fresh water. Credits: <a href="https://www.stellarphysics.org/">M. Cantiello</a>.</figcaption>
</figure>
<p>
The thermohaline mixing is obviously not only a stellar phenomenon: it
can occur for example in the sea close to the equator, where the
surface is heated by the Sun and evaporates faster, leading to a layer
with hotter and saltier water (higher \(\mu\)) on top of colder and less
salty water below.
</p>
<p>
An example where it occurs in stars are accretors in binaries which
may receive helium enriched material from the outer layers of the core
of the donor star, putting helium rich higher μ gas on top of the lower μ
envelope.
</p>
</div>
</div>
<div id="outline-container-org090df05" class="outline-4">
<h4 id="org090df05"><a href="#org090df05">Which instability criterion should one use?</a></h4>
<div class="outline-text-4" id="text-org090df05">
<p>
Naively, one may think that the Ledoux convection is more physically
accurate, since it requires one less hypothesis (which we know to not
always be correct inside a star). However, when calculating stellar
models, what we are interested in is the <i>long-term</i> evolution of the
star: as you can see from the thermohaline mixing figure above, this
is not a 1D process (the "fingers" end with "mushrooms"), and we
typically care about timescales very long compared to the timescales
for these processes. These are in fact <i>thermal</i> processes and their
timescales are proportional to the <i>local</i> thermal timescale, which as
we have already seen is generally short compared to the evolutionary
timescale.
</p>
<p>
Especially for convection in the stellar cores (where we will see
energy is generated, therefore L can be very large and drive
convection, especially in massive stars), convection may flatten the μ
gradient on a timescale short compared to the main sequence lifetime,
therefore erasing the ingredient that differentiates the two (see for
instance <a href="https://ui.adsabs.harvard.edu/abs/2022ApJ...928L..10A/abstract">Anders et al. 2022a</a>). This is an active topic of debate in
the recent literature!
</p>
<p>
Nevertheless, once either the Schwarzschild or Ledoux criterion is
assumed, given the \(T(m) \equiv T(m(P)) = T(P)\) profile of a star one can determine the
radial extent \(\Delta r\) of the unstable region.
</p>
</div>
</div>
</div>
</div>
<div id="outline-container-orga677a3d" class="outline-2">
<h2 id="orga677a3d"><a href="#orga677a3d">Mixing length theory</a></h2>
<div class="outline-text-2" id="text-orga677a3d">
<p>
Let's now consider what happens in an unstable layer: we need to model
how the energy is transported in these layers, were radiative
diffusion is insufficient and the gas will start moving. An ideal
solution to this problem would follow the dynamics of buoyant parcels
of gas over the (long) thermal timescale, which is in general <i>not</i>
possible: hydrodynamic simulations can only compute the much shorter
<i>dynamical</i> timescales!
</p>
<p>
Physically, in the unstable situation we have described above, we
should expect macroscopic motion of gas (the "bubbles") to start, and
these "bubbles" would move upward adiabatically, maintaining
hydrostatic balance with the surroundings, until they release their
excess heat, cooling down and finally falling back. This obviously is
<i>not</i> a one dimensional problem, since we have some bubbles moving
upwards and some moving downwards simultaneously so that the net mass
flux is zero, but the net energy flux is non-zero. Moreover this
typically leads to turbulence in the flow which is an inherently
multi-dimensional problem.
</p>
<p>
<a href="https://en.wikipedia.org/wiki/Erika_B%C3%B6hm-Vitense">Erika Bohm-Vitense</a> developed in the late 1950s an effective mean-field
theory to describe the space- and time-averaged steady state at which
convective energy transport would saturate. This is the so-called
<i>mixing length theory</i> (MLT) that is widely applied in stellar evolution
still today, and is based on the simplified "bubble picture" from
Prandtl we already used to derive a stability criterion.
</p>
<p>
Before deriving the energy flux in a convective region according to
MLT, let's try to get an intuition for why this is very successful,
despite being a very rough approximation of what it tries to describe.
We can consider a more familiar example of convection for that, such
as a self-sustaining flame:
</p>
<figure id="org041e9f9">
<img src="./images/fire.jpg" alt="fire.jpg" width="50%">
<figcaption><span class="figure-number">Figure 3: </span>A flame sustains itself by driving convection that brings in more oxygen to allow combustion to happen.</figcaption>
</figure>
<p>
MLT is meant to describe the spatial and temporal average of the gas
flow in the convective region driven by the (chemical) energy release
from the flame. Intuitively, it's like taking a picture (assuming any
snapshot in time is statistically equivalent to any other), and then
averaging across the horizontal cross section of this flame to obtain
an approximation to the time- and space- averaged vertical flow of
energy and temperature. With all the limitations that this entails,
MLT is a very successful theory that is sufficient for <i>most</i> stellar
evolution applications since those typically are concerned with
timescales that are very long w.r.t. the convective turnover timescale
(i.e., in the fire analogy, very long compared to the "flickering" of
the flames).
</p>
<ul class="org-ul">
<li><b>Q</b>: based on this, can you guess where/when MLT will be an
insufficient approximation?</li>
</ul>
</div>
<div id="outline-container-org1a8f10f" class="outline-3">
<h3 id="org1a8f10f"><a href="#org1a8f10f">Convective energy flux</a></h3>
<div class="outline-text-3" id="text-org1a8f10f">
<p>
To calculate the energy flux carried by convection within the
framework of MLT, let's consider the difference in temperature between
a bubble that is displaced upwards by an amount ℓ in an unstable layer
w.r.t. the surrounding environment:
</p>
<div class="latex" id="org0280b21">
\begin{equation}
\Delta T = T_\mathrm{bubble} - T_\mathrm{env} \simeq \left(T_{1} + \frac{dT}{dr}\rvert_\mathrm{bubble}\ell \right) - \left(T_{1} + \frac{dT}{dr}\rvert_\mathrm{env}\ell \right) = \left(\frac{dT}{dr}\rvert_\mathrm{bubble} - \frac{dT}{dr}\rvert_\mathrm{env} \right)\ell \ \ .
\end{equation}
</div>
<p>
<b>N.B.</b>: Earlier we have considered an infinitesimal radial displacement
\(\Delta\) to perform a linear stability analysis and obtain a stability
criterion. Now \(\Delta\rightarrow\ell\) does not need to be small a priori.
</p>
<p>
To put this in the form of the gradients that we have defined above
for the stability, we notice that \(dT/dr = T \times d \log(T)/ d \log(P) \times
d \log(P)/dr\), and assume that \(T_\mathrm{bubble} \simeq T_{env} \equiv T\),
that is effectively consider only the zeroth order of the Taylor
series in temperature, and rewrite for the temperature difference:
</p>
<div class="latex" id="org5505631">
\begin{equation}
\Delta T = \frac{\ell}{H_{p}} T \left(\nabla_\mathrm{rad} - \nabla_\mathrm{ad}\right) \ \ ,
\end{equation}
</div>
<p>
where we use the assumption that the environment is characterized by a
radiative gradient and the bubble by an adiabatic gradient.
</p>
<p>
<b>N.B.:</b> The sign has changed becaue of the minus in the definition of
the pressure scale height \(H_{p}\). The stability criterion obtained above
says that the unstable situation is when \(\nabla_\mathrm{rad}\) is steeper
than \(\nabla_\mathrm{ad}\), so the form above also guarantees that \(\Delta T\) is
actually positive, as we expect for a bubble raising and carrying an
excess thermal energy compared to the background.
</p>
<p>
The excess energy per unit volume carried by the raising bubble is
then \(c_{p}\rho\Delta T\), where \(c_{p}\) is the specific heat at constant
pressure. Multiplying by the velocity of the bubble we get the
<b>convective flux</b> (as you can verify by dimensional analysis!):
</p>
<div class="latex" id="orgdbbde97">
\begin{equation}
F_\mathrm{conv} = c_{p} \rho \frac{\ell}{H_{p}} T \left(\nabla_\mathrm{rad} - \nabla_\mathrm{ad}\right) v_\mathrm{conv} \ \ .
\end{equation}
</div>
<ul class="org-ul">
<li><b>Q</b>: why do we use the constant pressure \(c_{p}\) here? <b>Hint</b>: think of the
assumptions we have discussed above.</li>
</ul>
<p>
Here there are two things we don't know yet: how far the bubble goes \(\ell\)
and the convective velocity.
</p>
<p>
<b>N.B.:</b> In general, to maintain the net-zero mass flux, for each bubble
of mass \(\delta m\) raising there is one of the same mass sinking. The raising
one carries excess thermal energy w.r.t. the radiative background, and
the sinking one carries a deficiency in thermal energy w.r.t. the
background so the total convective flux should be twice what we have
derived. On the other hand, by taking the difference in the gradients
across the entire (as of yet unknown) travel path \(\ell\), we are
overestimating the gradient difference, and on average it should be
roughly half of that, canceling out the mistake we make by neglecting
the sinking bubbles.
</p>
</div>
</div>
<div id="outline-container-org477d7c7" class="outline-3">
<h3 id="org477d7c7"><a href="#org477d7c7">Convective velocity</a></h3>
<div class="outline-text-3" id="text-org477d7c7">
<p>
To estimate the convective velocity \(v_\mathrm{conv}\) we can consider
the work done by the buoyancy forces (per unit volume) on the bubble.
</p>
<ul class="org-ul">
<li><b>Q</b>: before we even do this calculation, can you imagine an
upper-limit for v<sub>\mathrm</sub>{conv} in the approximated picture we are developing?
(<b>Hint</b>: we have assumed that any pressure imbalance between the
bubble and the environment would be quickly be washed out)</li>
</ul>
<p>
The buoyancy force per unit volume acts in the direction opposite of
gravity and has amplitude equal to weight of the displaced fluid minus
the weight of the bubble itself, \(B = - \Delta\rho \times |g|\), where \(\Delta\rho=
\rho_\mathrm{bubble} - \rho_\mathrm{env}\) is the difference in density
between the rising fluid element and the environment. Doing a Taylor
expansion and keeping only the first order in ℓ we have:
</p>
<div class="latex" id="org9723e23">
\begin{equation}
\Delta\rho \simeq \left(\rho_{1} +\frac{d\rho}{dr}\rvert_\mathrm{ad} \ell\right) - \left(\rho_{1} +\frac{d\rho}{dr}\rvert_\mathrm{rad} \ell\right) \ \ ,
\end{equation}
</div>
<p>
This is the difference in density between the bubble and the
environment at the end of the whole (yet unknown) travel path \(\ell\). On
average throughout the path, since the difference was \(\Delta\rho = 0\) at the
beginning, we only have half of that, so let's just consider 1/2 of
this to estimate the work done by buoyancy. We can further express the
density gradients as a function of \(\nabla_{i} = \partial log(T)/\partial log(P)\rvert_{i}\).
</p>
<p>
By energy conservation, the work done by buoyancy on the bubble is
equal to the kinetic energy (per unit volume) acquired by the bubble,
which is what we will use to make \(v_\mathrm{conv}\) appear:
</p>
<div class="latex" id="orgd4e562e">
\begin{equation}\label{eq:v_conv_MLT}
E_\mathrm{kin, Bubble} = B\cdot\ell \Rightarrow \frac{1}{2}\rho v_\mathrm{conv}^{2} = \frac{\rho}{4H_{p}}(\nabla_\mathrm{rad}-\nabla_\mathrm{ad})\ell^{2}g \ \ .
\end{equation}
</div>
<p>
<b>N.B.:</b> the buoyancy force and the displacement vector are antiparallel,
which, together with the definition of \(H_{p}\), adjusts the minus signs.
</p>
<p>
Conveniently in Eq. \ref{eq:v_conv_MLT} both \(v_\mathrm{conv}\) and \(\ell\)
are squared: the same exact reasoning applies to the bubbles sinking
and those rising! Eq. \ref{eq:v_conv_MLT} is a relation between the
two unknowns we have in the convective energy flux, \(v_\mathrm{conv}\)
and \(\ell\), which allows us to eliminate one for the other:
</p>
<div class="latex" id="org7a3ea81">
\begin{equation}\label{eq:v_conv_MLT_estimate}
v_\mathrm{conv} = \sqrt{\frac{\ell^{2}g}{2H_{P}} (\nabla_\mathrm{rad} - \nabla_\mathrm{ad})}\ \ ,
\end{equation}
</div>
</div>
</div>
<div id="outline-container-orgbd20962" class="outline-3">
<h3 id="orgbd20962"><a href="#orgbd20962">The mixing length and α<sub>MLT</sub></a></h3>
<div class="outline-text-3" id="text-orgbd20962">
<p>
At this point enters the heuristic hypothesis proposed by <a href="https://ui.adsabs.harvard.edu/abs/1958ZA.....46..108B/abstract">Bohm-Vitense
1958</a> (<b>N.B.:</b> the original paper is in German): let's assume that the
length scale \(\ell\) traveled <i>on average</i> by a convectively moving bubble
before losing its identity by releasing its excess heat to the
surroundings (or absorbing the amount of heat it was lacking, in the
case of a sinking bubble), is proportional to the local pressure scale
height. This heuristic hypothesis is sensible, since the pressure
scale height tells us something about the thermal stratification of
the gas, and we are discussing an instability that needs to transport
energy when diffusion is insufficient, and it is still very widely
used today. It gives us the central hypothesis of MLT:
</p>
<div class="latex" id="orgcb8d249">
\begin{equation}\label{eq:alpha_MLT}
\ell = \alpha_{MLT} H_{p}
\end{equation}
</div>
<p>
The average length traveled by a bubble \(\ell\) is the so-called mixing
length that gives the name to this "theory", and the proportionality
constant \(\alpha_\mathrm{MLT}\) is one of the most infamous free parameters
in stellar evolution that is calibrated on stellar observations. If
the heuristic hypothesis underpinning this approach holds, it should
be a quantity of order 1.
</p>
<p>
Putting all things together, we can now express the convective energy
flux as a function of known quantities and this free parameter
\(\alpha_\mathrm{MLT}\):
</p>
<div class="latex" id="orgc47be40">
\begin{equation}\label{eq:conv_flux_MLT}
F_\mathrm{conv} = \rho c_{P} T \alpha_\mathrm{MLT}^{2} \sqrt{\frac{1}{2} g H_{P}} (\nabla_\mathrm{rad}-\nabla_\mathrm{ad})^{3/2} \ \ .
\end{equation}
</div>
<p>
<b>N.B.:</b> The convective flux predicted by MLT in Eq.
\ref{eq:conv_flux_MLT} is proportional to a power of the
<i>superadiabaticity</i> \((\nabla_\mathrm{env}-\nabla_\mathrm{ad}) \equiv (\nabla_\mathrm{rad} -
\nabla_\mathrm{ad})\), because of the assumption of an initially radiative background
environment.
</p>
</div>
</div>
<div id="outline-container-org8c7c3d3" class="outline-3">
<h3 id="org8c7c3d3"><a href="#org8c7c3d3">Efficiency of convection</a></h3>
<div class="outline-text-3" id="text-org8c7c3d3">
<p>
Convection is an <i>instability,</i> meaning once it kicks in, it grows
exponentially fast towards a saturated state. We have neglected the
growth phase (see also <a href="#org0afb088">below</a>), and found an approximate description
for the steady state depending on a free parameter \(\alpha_\mathrm{MLT}\). We can now
ask, in such steady state, how big is the superadiabaticity needed
such that the convective flux carries <i>all</i> the energy? We can estimate
this equating the convective flux \(F_\mathrm{conv}\) to the entire flux that needs
to be carried throughout a layer at radius \(r\):
</p>
<div class="latex" id="orgad5f852">
\begin{equation}
F_\mathrm{conv} \equiv \frac{L(r)}{4\pi r^{2}} \ \ .
\end{equation}
</div>
<p>
To obtain an order of magnitude estimate, we can substitute in
\(F_\mathrm{conv}\) the average density of the star, \(T\) from the virial
theorem estimate, assume a monoatomic gas for \(c_{P}\), and using \(L(r)/4\pi
r^{2} \sim L/R^{2}\) we obtain:
</p>
<div class="latex" id="org20560c4">
\begin{equation}
\nabla_\mathrm{rad} - \nabla_\mathrm{ad} \simeq \left(\frac{LR}{M}\right)^{2/3}\frac{R}{GM} \simeq 10^{-8} \ \ ,
\end{equation}
</div>
<p>
where in the last one we use the numerical values for the Sun. This of
course is an estimate valid in the interior of the Sun (because we
have used implicitly assumptions of LTE, and used averaged values).
</p>
<p>
The fact that the superadiabaticity is so small implies that <i>when
convection is efficient</i>, <i>the temperature gradient in the star can be
approximated with adiabatic</i>. This comes from a relatively rough
estimate, and validates a posteriori many of the questionable
assumptions we have made in this lecture: since in the end the
gradient is very nearly adiabatic when convection is efficient, the
details do not matter that much and our assumption that the gradient
of \(T\) or \(\rho\) in the "bubble" are <i>adiabatic</i> (\(dq=0\)) is justifiable a
posteriori.
</p>
<p>
<b>N.B.:</b> In the outer layers of the star, where \(\rho \ll \langle \rho \rangle\) and \(T\ll \langle T
\rangle\), this estimate breaks down, convection is not necessarily efficient
and the gradient is not necessarily adiabatic. This is important for
many stellar applications, for example eruptive mass loss of massive
stars, and dynamical stability of mass transfer in binaries.
</p>
</div>
</div>
<div id="outline-container-org02992ee" class="outline-3">
<h3 id="org02992ee"><a href="#org02992ee">On the convective velocity and chemical mixing</a></h3>
<div class="outline-text-3" id="text-org02992ee">
<p>
Eq. \ref{eq:v_conv_MLT_estimate} derived above tells us that