ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence

Project description November 14, 2016, Axel Brandenburg

Project on stratified shear flows

Background

The stabilizing buoyancy force due to shear is an important aspect to shear-flow instabilities and is particularly relevant in meteorology. Taylor, G. I. (1931) and Goldstein (1931) published the first work on this problem in the same issue of Proc. Roy. Soc.

In Handout 9 on stratified shear flows we discussed the Goldstein-Taylor equation as a straightforward generalization of Rayleigh’s instability equation. We arrived at the equation

c_{i} ∫ \frac{N^{2} − \frac{1}{4}(U′)^{2}}{|U − c|^{2}} |ψ|^{2} = −c_{i} ∫ (|ψ′| + k^{2}|ψ|^{2}) dx  (1)

with c = c_{r} + ic_{i}. This shows that the flow cannot be unstable if N^{2} − \frac{1}{4}(U′)^{2} > 0 everywhere in the domain. In terms of the gradient Richardson number Ri(z) = N^{2}/(U′)^{2}, this means Ri > 1/4 for stability.

Project details

The idea of the project is to solve this equation numerically. To solve the fully nonlinear problem, we can use a hydrodynamics code such as Athena^{1}, Dedalus^{2}, or the Pencil Code^{3}. The Rayleigh instability problem is a limiting case (ν → 0) of the Kelvin-Helmholtz instability, which is used as a common test problem for numerical codes. It has been used last Summer during the Bootcamp for

Computational Fluid Dynamicshttp://www.nordita.org/~brandenb/teach/PencilCode/LCDworkshop2016/

at the Laboratory for Computational Dynamics (LCD), which is upstairs in the third floor. See ? for details, but in the absence of stratification.

1. Review the literature on numerical studies of the stratified shear-flow problem.

2. Start by solving the usual Kelvin-Helmholtz instability and check the validity of the inflectionpoint theorem and the more stringent Fjørtoft (1950) theorem.

3. Add constant gravity to the problem. Verify that the resulting stratification obeys the hydrostatic solution. You may consider either the fully compressible case (e.g., with an isothermal equation of state) or the incompressible Boussinesq problem.

4. Check numerically the validity of the Richardson criterion, which says that the flow cannot be unstable if N^{2} − \frac{1}{4}(U′)^{2} > 0 everywhere in the domain. Here, N is the buoyancy frequency. In terms of the gradient Richardson number Ri(z) = N^{2}/(U′)^{2}, this means Ri > 1/4 for stability.

5. Consider the same problem in the presence of turbulence. Here, the turbulence can be driven by an external monochromatic body force (forcing wavenumber k_{f}). However, you may also consider the problem of turbulence as a result of the shear-flow instability itself.

1http://www.astro.princeton.edu/~jstone/athena.html

2http://dedalus-project.org/

3http://pencil-code.nordita.org/

6. Review your results in view of recent discussions of the validity of Richardson criterion in the literature; see Zilitinkevich et al. (2007) who claim that “The proposed model permits the existence of turbulence at any gradient Richardson number”.

References

Fjørtoft, R., “Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex,” Geofys. Publ. Oslo 17, 1-52 (1950).

Goldstein, S., “On the stability of superposed streams of fluids of different densities,” Proc. Roy. Soc. Lond. 132, 524-548 (1931).

Taylor, G. I., “Effect of variation in density on the stability of superposed streams of fluid,” Proc. Roy. Soc. Lond. 132, 499-523 (1931).

Zilitinkevich, S. S., Elperin, T., Kleeorin, N., & Rogachevskii, I., “Energy- and flux-budget (EFB) turbulence closure model for stably stratified flows. Part I: steady-state, homogeneous regimes,” Boundary-Layer Meteorology 125, 167-191 (2007).