The projects are open-ended questions that you need to research, and then present to the rest of the class (+ other attendees) during a 20-minute talk. If you have other ideas for projects that you would like to investigate, do not hesitate to propose them to the instructor.

Projects can be done and presented anytime during the quarter, ideally before Final's week. Good presentation times are GAFD seminars.

Projects:

Projects related to waves.

• Resonant wave interactions. See Section 51 of the Drazin and Reid book on hydrodynamic stability.
• The Korteveg-de-Vries equation. This equation describes the propagation of nonlinear surface waves (for instance). The project involves deriving the equation, and discussing the soliton solutions. See the GFD notes for instance
• A generalized model for waves in stratified fluids with a free surface. This combines the study of internal waves and of surface waves. See me for the model setup.
• Waves propagating along a fluid interface in a 2-layer fluid. See me for the model setup.

Projects related to instabilities

• Rayleigh-Benard convection : comparison of numerical simulations with theory: This project requires you to have or find an online code, and compare the predictions from linear theory and weakly nonlinear theory to the outcomes of the code. For the code, a good start could be Dedalus for instance.
• The effect of shear on Rayleigh Benard convection (linear theory): This project requires you to add shear and re-do the linear stability analysis for RBC. There are several possibilities, including doing this, including forcing the shear through the boundaries (which combines plane Couette flow with RBC) or forcing the shear through a body-force. Please see me to discuss the model setup and how to proceed (you will likely have to solve a two-point BVP to do the problem).
• Benard-Marangoni convection (i.e. convection driven by surface tension) (linear theory): This project requires you to study the effects of surface tension on driving fluid motion (the Marangoni effect). See Nield 1964 for a starting reference.
• Weakly nonlinear theory for thermocompositional convection
• The Howard and Krishnamurti reduced model for convection and shear (see Krishnamurti and Howard 1981 for the experiment, and Howard and Krishnamurti 1986 for the theory.
• Energy stability for double-diffusive convection. See me for the model setup.
• The Taylor instability (centrifugal instability), linear theory. See, e.g. Section 3 of the Drazin and Reid "Hydrodynamic Stability" book.
• Energy stability analysis for diffusive shear instabilities. See Garaud et al. 2015
• MHD instabilities (e.g. MRI, magnetic buoyancy instabilities, instabillities of toroidal fields), linear theory. See Chandrasekhar, "Hydrodynamic and hydromagnetic stability".