PhD thesis proposal: Anomalous-scattering induced instabilities in an open channel flow (Poitiers, France)

Scientific context

Anomalous scattering, a process where incident waves are scattered and amplified due to the emission of waves with negative energy [1], leads to interesting and novel predictions. Certain such processes are analogous to gravitational ones, especially the analogue Hawking effect [1-3], the cornerstone of the Analogue Gravity program that seeks to mimic wave propagation in curved spacetime (such as a black hole where the flow passes from sub- to super-critical [4]) with waves in moving media. Of equal importance, however, are processes that are very different from those of standard gravity. In particular, short-wave dispersion, which is ubiquitous in condensed matter systems but absent in the Lorentz invariant description of spacetime, opens up a rich landscape of new behaviours. This includes the related phenomena of white-hole (or white-fountain, the time-reverse of a black hole) Hawking radiation, undulations dressing white-fountain horizons, and the black-hole laser (BHL) effect.

The BHL effect occurs in a flow with two horizons, one a black-hole and the other a white-fountain horizon [5]. This can lead to the presence of trapped modes living in the “cavity” between the two horizons. Through the same anomalous scattering processes responsible for the Hawking effect, these trapped modes are continuously amplified, leading to exponential growth and thus to a dynamical instability. Eventually, this growth will saturate, being regulated by nonlinearities. The exponential growth involved means that the BHL effect is expected to yield a strong signal, making it a potential observational precursor to the Hawking effect. But it is not necessarily clear what signal to search for, as the signal evolves in time and the end state is not a priori known, depending on the nonlinearities of the system.

Similarly, downstream from a white-fountain horizon (where the flow passes from super- to sub-critical), the flow is unstable against the appearance of an undulation [6]. It has been shown that the initial growth of this undulation can be related to the Hawking effect [7], which in this scenario yields a macroscopic occupation of short waves. However, its final form is also governed by nonlinearities. It is unclear to what extent the mere occurrence of this undulation can be considered as a manifestation of the Hawking effect. It is also unclear whether the growth of the undulation is experimentally observable.

The open channel flows realised at Institut Pprime routinely contain both black-hole and white-fountain horizons, the latter with their associated undulations. Fluctuations associated with the undulations have been observed [6], but remain poorly understood. As yet, a clear black-hole laser effect has not been observed. However, given the exponential growth rate and the unknown end state, we can expect the signal to be sensitive to the experimental parameters. There is thus much scope for theoretical guidance and optimization of such experiments.


Project

In the context of the CNRS Chair in Physical Hydrodynamics, we would like to recruit a PhD student to work with us at Institut Pprime. The primary goal of the thesis will be to provide theoretical input into the expected behaviour of instabilities induced by anomalous scattering (namely, undulations and the BHL effect) in an open channel flow. It will involve analytical treatments and numerical simulations, as well as data analysis. Ultimately, it is hoped that the understanding gained will be used in the guidance of future experiments, to optimize the visibility of growth and/or saturation effects.

Analysis of instabilities falls into two well-defined parts. Firstly, there is the linear wave analysis, which should provide an exponential growth rate associated with a particular background flow. Various complicating factors can be included, such as friction- and viscosity-induced dissipation, which will tend to reduce the growth rate. The second part involves a determination of the end state of the instability, governed by the growing importance of nonlinearities in the wave dynamics. This determines an overall timescale – the time to reach saturation – which will be critical in the design of experiments.

The primary application of the analysis will be to the case of one-dimensional open channel flows over one or two obstacles. However, other applications are possible. In the two-dimensional circular jump experiment, a flow is obtained which may well have two horizons in the radial direction, with a cavity in between. This is a very different regime of very shallow water with a different dispersion relation, but a similar BHL instability may well be at play.  


Candidate profile

• Comfortable with both analytical and numerical approaches to theoretical problems
• Some familiarity with Matlab and/or Mathematica is desirable
• Good level in English


References

[1] G. Rousseaux et al., Observation of negative-frequency waves in a water tank: a classical analogue to the Hawking effect?, New J. Phys. 10, 053015 (2008) 
[2] S. Weinfurtner et al., Measurement of stimulated Hawking emission in an analogue system, Phys. Rev. Lett. 106, 021302 (2011)
[3] L.-P. Euvé et al., Observation of noise correlated by the Hawking effect in a water tank, Phys. Rev. Lett. 117, 121301 (2016)
[4] L.-P. Euvé et al., Scattering of co-current surface waves on an analogue black hole, Phys. Rev. Lett. 124, 141101 (2020)
[5] S. Corley and T. Jacobson, Black hole lasers, Phys. Rev. D 59, 124011 (1999)
[6] J. Fourdrinoy et al., Correlation on weakly time-dependent transcritical white-hole flows, Phys. Rev. D 105, 085022 (2022)
[7] A. Coutant and R. Parentani, Undulations from amplified low frequency surface waves, Phys. Fluids 26, 044106 (2014)