Notes on Mixing Length Theory


These notes are mostly inspired from reading Cox & Giuli (“Principles of stellar structure”); some insights are from A.Maeder (“Physics, formation and evolution of rotating stars”) and Kippenhahn & Weigert (“Stellar Structure and Evolution”)

Pressure Scale Height

In hydrostatic equilibrium we can define the total pressure scale height, \({H_{\mathrm{P}}}\), as \[-\frac{{{\mathrm d}}\ln {\mathrm{P}}}{{{\mathrm d}}r} \equiv \frac{1}{{H_{\mathrm{P}}}} = \frac{\rho g}{{\mathrm{P}}}\] where \({\mathrm{P}}\) is the total pressure (\({\mathrm{P}}_{gas} + {\mathrm{P}}_{rad}\)). The pressure scale height is a measure of the distance over which the pressure changes by an appreciable fraction of itself.


We can define the gradients \(\nabla \equiv \frac{{{\mathrm d}}\ln {T}}{{{\mathrm d}}\ln {\mathrm{P}}}\) and \(\nabla' \equiv \frac{{{\mathrm d}}\ln {T}'}{{{\mathrm d}}\ln {\mathrm{P}}}\). Here \(\nabla\) represents the average temperature gradient with respect to pressure of all matter at a given level, while \(\nabla'\) is the temperature gradient with respect to pressure of a rising/falling fluid element. At first order the temperature excess of such fluid elements can therefore be written as \[\Delta {T}(\Delta r) = {T}'(r + \Delta r) - {T}(r + \delta r) \simeq \Delta r \bigg[ \frac{{{\mathrm d}}{T}'}{{{\mathrm d}}r}-\frac{{{\mathrm d}}{T}}{{{\mathrm d}}r}\bigg].\] Assuming \({T}' \simeq {T}\), i.e. the temperature is not changing drastically within the distance \(\Delta r\), one can write \[\Delta {T}(\Delta r) = \Delta r \, {T}\bigg[ -\frac{{{\mathrm d}}\ln {T}}{{{\mathrm d}}r}- \bigg(-\frac{{{\mathrm d}}\ln {T}'}{{{\mathrm d}}r}\bigg)\bigg],\] and using the assumption of pressure equilibrium, the definitions of pressure scale height \({H_{\mathrm{P}}}\) and the gradients \(\nabla\) and \(\nabla'\) we obtain \[\label{eq:deltat} \Delta {T}(\Delta r) = - \frac{{{\mathrm d}}\ln {\mathrm{P}}}{{{\mathrm d}}r} \Delta r \, {T}\bigg[ \frac{{{\mathrm d}}\ln {T}}{{{\mathrm d}}\ln {\mathrm{P}}}- \bigg(\frac{{{\mathrm d}}\ln {T}'}{{{\mathrm d}}\ln {\mathrm{P}}}\bigg)\bigg] = \Delta r \frac{{T}}{{H_{\mathrm{P}}}}\, (\nabla - \nabla').\] Note that