Linear Advection

In astrophysics, many systems (stars, the ISM, gas in galaxies) is modeled as a fluid, following the equations of hydrodynamics:

\[\begin{align*} \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho {\bf U}) &= 0 \\ \frac{\partial (\rho {\bf U})}{\partial t} + \nabla \cdot (\rho {\bf U}{\bf U}) + \nabla p &= 0 \\ \frac{\partial (\rho E)}{\partial t} + \nabla \cdot (\rho {\bf U} E + {\bf U}p) &= 0 \end{align*}\]

These express conservation of mass, momentum, and energy. If we expand out the diveragence in the first equation, we get:

\[\frac{\partial \rho}{\partial t} + {\bf U} \cdot \nabla \rho + \rho \nabla \cdot {\bf U} = 0 \]

This has the form of an advection equation.

Note

Our model equation for this, in one-dimension will be the linear advection equation:

\[\frac{\partial a}{\partial t} + u \frac{\partial a}{\partial x} = 0\]

Looking at the second equation (momentum), and expanding out and simplifying, we have:

\[\frac{\partial {\bf U}}{\partial t} + {\bf U} \cdot \nabla {\bf U} + \frac{1}{\rho} \nabla p = 0\]

This is a nonlinear equation (because of the \({\bf U}\cdot \nabla {\bf U}\) term). This nonlinearity admits a rich diversity of solutions: rarefactions, shocks, and turbulence.

Note

Our model equation for this will be the inviscid Burgers’ equation:

\[\frac{\partial u}{\partial t} + u \frac{\partial u}{\partial x} = 0\]