Convection II

Ch. 5.5 of Pols here & HKT, Ch. 5; Ch. 5.3.3: LeBlanc 2011, Kippenhahn Ch. 6.5.

Day 9 - September, 25, 2025

Agenda:

• Updates/Reminders - Presentation Tips - Schedule on D2L (2m)
• Lecture (25m)
• HW1 Review - Corrections Due: EoD, Sept. 30, 2025 (20m)
• Report out on HW1 using white boards (20m)

Recap - Criteria for stability against convection

For convection to occur, you typically will need one of the following:

• a large , convection occurs in opaque regions of a star.
• region with large energy flux, stars with values near the center are expected to have convective cores - massive stars.
• a small value of such as at partial ionization zones at relatively low temperatures. stars of all masses have shallow surface convection zones at temperatures where hydrogen and helium are partially ionized.

Recap - Schwarzschild and Ledoux Criterion

Because normally increases inward, , the result is that gradients can lead to a stabilizing effect.

For an ideal gas, with all energy transported by radiation we have () and :

For , this reduces to the Schwarzschild Criterion.

Secular mixing processes

• We have asked about dynamical stability: what happens when hydrostatic equilibrium is perturbed?
• We can also ask about thermal (secular) stability: what happens when the thermal equilibrium situation is perturbed?

Convective Energy Transport

• We still have not addressed how much energy can by transported by convection?
• Or, a way to determine other properties of the convection.

Secular mixing processes

Secular mixing processes are long-term and typically non-periodic.

Semi-Convection: For stellar layers that are stable according to the Ledoux criterion but unstable according to the Schwarzschild criterion as we have

and the region is said to be semi-convective.

Secular mixing processes

Semi-Convection:

• The gas parcels will experience small oscillations where the gradient acts as a damping force.

• If during the oscillations, gas with different mix, will descrease which is damping the oscillation: semiconvection oscillation will slowly grow in amplitude. Example Heger et al 2000

Secular mixing processes

Viceversa, if a layer is Schwarzschild stable but Ledoux unstable (this can depend on the EOS and the chemical composition - ):

then the mean molecular weight gradient acts to destabilize the layer.

• The blob of gas will slowly start moving because of but there will be no restoring forces, and we obtain the so called thermohaline mixing or double diffusion instability.

Secular mixing processes

Thermohaline Mixing/Double Diffusion Instability:

• 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 prove in a kitchen experiment:

Double-diffusive fingers in hot salty water on top of cold fresh water. Credits: M. Cantiello.

Secular mixing processes

Thermohaline Mixing/Double Diffusion Instability:

• 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 ) on top of colder and less salty water below.

• 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 gas on top of the lower envelope.

A Model for Convection?

Simulating numerically requires:

• solving the equations of hydrodynamics in three dimensions
• over a huge range of length scales and time scales,
• and of pressures, densities and temperatures.

Streamlines of the velocity field of a star from Fields 2022.

Mixing Length Theory (MLT)

We have to resort to a simple one-dimensional theory that is based on rough estimates, and is known as the mixing length theory (MLT).

• Erika Bohm-Vitense 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 mixing length theory (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.

Mixing Length Theory (MLT)

• 1. one approximates the complex convective motions by blobs of gas that travel up or down over a radial distance (the mixing length), after which they dissolve in their surroundings and lose their identity
• 2. As the blob dissolves it releases its excess heat to its surroundings (or absorbs for sinking blobs)
• 3. The mixing length is a free parameter, of the order of the local pressure scale height: in HSE.

The convective energy flux

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 in an unstable layer w.r.t. the surrounding environment:

• Earlier we have considered an infinitesimal radial displacement to perform a linear stability analysis and obtain a stability criterion. Now does not need to be small a priori.

The convective energy flux

• we first notice that
• we also assume that , that is effectively consider only the zeroth order of the Taylor series in temperature, and rewrite for the temperature difference:

where we use the assumption that the environment is characterized by a radiative gradient and the bubble by an adiabatic gradient.

The convective energy flux

The excess internal energy per unit volume carried by the raising bubble is then , where is the specific heat at constant
pressure.

• If the blob moves with convective velocity , we can compute the convective flux:

Convective Velocity According to MLT

To estimate the convective velocity we can consider the work done by the buoyancy forces (per unit volume) on the bubble.

If the difference in density between a gas element and its environment is , then the buoyancy force will give an acceleration:

• The blob is acceralated over a distance for time given by

Convective Velocity According to MLT

We find that , we can now substitute in our equation for above to find

and for the convective flux:

The mixing length and

The central hypothesis of MLT gives the mixing length as

• 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.

The mixing length and

Putting all things together, we can now express the convective energy
flux as a function of known quantities and this free parameter :

• The convective flux predicted by MLT is proportional to a power of the superadiabaticity , because of the assumption of an initially radiative background environment.

Efficiency of convection

Convection is an instability, meaning once it kicks in, it grows exponentially fast towards a saturated state. We have neglected the growth phase, and found an approximate description for the steady state depending on a free parameter .

• Q: We can now ask, in such steady state, how big is the superadiabaticity needed such that the convective flux carries all the energy?

Efficiency of convection

We can estimate this equating the convective flux to the entire flux that needs to be carried throughout a layer at radius :

We wish to find an order of magnitude estimate:

• substitute in the average density of the star, from the virial theorem estimate, assume a monoatomic gas for , and using .

Efficiency of convection

Making these assumptions and plugging in typical solar values we find

This is an estimate valid in the interior of the Sun (because we have used implicitly assumptions of LTE, and used averaged values).

• In this estimate, convection is so efficient at transporting energy that only a tiny superadiabaticity is required.

Efficiency of convection

We can conclude that in the deep stellar interior the actual temperature stratification is nearly adiabatic, and independent of the details of the theory.

• In the outer layers of the star, where and , this estimate breaks down, convection is not necessarily efficient and the gradient is not necessarily adiabatic.

As the surface is approached, convection becomes very inefficient at transporting energy. Then so that radiation effectively transports all the energy, and despite convection occuring.

Convective mixing

Besides being an efficient means of transporting energy, convection is also a very efficient mixing mechanism.

• We found where the second term is .

• We have just seen that for efficient convection, the superadiabaticity is small, implying that the convective velocities are much smaller than the local sound speed

• Nevertheless, even a velocity of is sufficient to mix material very efficiently over the evolutionary timescales

Convective mixing

We can compute the mixing timescale as

which is about for solar values.

For the Sun, the convective mixing timescale is of the order of weeks to months:

over a thermal timescale, and certainly over a nuclear timescale, a convective region inside a star will be mixed homogeneously.

Convective mixing

• Similar argument apply to semiconvection and thermohaline mixing (and even the kitchen experiment can clearly show that thermohaline mixing can result in mixing of the composition).

• This is not always important, as we will see: in the Sun's envelope for example, convection mixes homogeneous material. However, in the core of a massive star, it mixes the material in the burning region (where hydrogen is turned into helium) into material that is non-burning and thus initially more hydrogen rich.

Convective mixing

One can derive from MLT a diffusion coefficient for the mixing of chemicals by convection

(and similarly for thermohaline and semiconvective mixing), allowing for a diffusive approximation of convective mixing.

• In reality convective mixing is an advective process: the macroscopic motion of the fluid carries around chemicals, and then they diffuse from the "bubble" into the environment after having being advected.

Consequences of Convective Efficiency

• A star in which nuclear burning occurs in a convective core will homogenize the region inside the core by transporting burning ashes (e.g. helium) outwards and fuel (e.g. hydrogen) inwards.

• A star with a deep convective envelope, such that it extends into regions where nuclear burning has taken place, will mix the burning products outwards towards the surface. This Dredge-Up can modify the surface composition, and provide a window into nuclear processes that have taken place deep inside the star.

Summary of the key assumptions

• The gradient of the background environment is radiative (meaning energy is transported by radiation diffusion)
• We model the thermal flux tubes carrying energy up and down with no mass flux as bubbles moving adiabatically. These are an idealization of realistic convective flows which are turbulent and thus space-and time-dependent on very short timescales.
• The "bubbles" maintain hydrostatic equilibrium w.r.t. the environment at any point in their travel

Summary of the key assumptions

• We assume that the average distance travelled by a bubble is proportional to the local pressure scale height .
• is the free parameter of this approach that effectively exists because convection is inherently a 3D phenomenon that we are trying to approximate in spherical symmetry.

HW1 Review

In class: 5 Groups, Discuss solutions, Converge. Choose scribe to draw up the solution legibly and explain to the class.

After Class: End of day, September, 30, 2025

• (Optional) Submit as a PDF, corrected solutions for incorrect HW questions for up to half credit on original assigment on D2L on original HW1.