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

Project description November 11, 2016, Axel Brandenburg

Project on solitons

Background

The f-mode is just like deep water waves. In that case it is well known that the presence of surface tension leads to additional modes known as capillary waves (Dias & Kharif, 1999). This leads to characteristic alterations of the f-mode, that tell us something about surface tension. This might also be astrophysically interesting, because magnetic fields leads to an extra tension-like force that could mimic the effects of surface tension.

Project details

1. Derive the linenar dispersion relation

ω^{2} = (g|k| + T |k|^{3}) tanh |k|h.  (1)

2. Following the review by Dias & Kharif (1999), explain the possibility of instabilities through resonance wave interactions in the form

nk = k_{1} + k_{2},  (2)
nω(k) = ω(k_{1}) + ω(k_{2}),  (3)

where n = 2, 3, ...

3. Verify the effective wave equation given by Dias & Kharif (1999) in the form

2i\frac{∂A}{∂t} + p\frac{∂^{2}A}{∂x^{2}} + q \frac{∂^{2}A}{∂y^{2}} + γA|A|^{2} = −iϵ (+s\frac{∂^{3}A}{∂x∂y^{2}} + r \frac{∂^{3}A}{∂x^{3}} + uA^{2} \frac{∂A^{∗}}{∂x} − vA^{2} \frac{∂A}{∂x}) (4)

4. Solve this equation numerically under simplifying assumptions.

5. Explain the concept of the bifurcation of waterwaves when phase and group velocities are nearly equal and derive

2i\frac{∂A}{∂t} + p\frac{∂^{2}A}{∂x^{2}} + q \frac{∂^{2}A}{∂y^{2}} + γA|A|^{2} = 0  (5)

for this case.

References

Dias, F., & Kharif, C., “Nonlinear gravity and capillary-gravity waves,” Ann. Rev. Fluid Dyn. 31, 301-346 (1999).