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Stable doubly diffusive convectons

Dr Cedric Beaume (SoM), Prof Alastair Rucklidge (SoM)

Contact email: c.m.l.beaume@leeds.ac.uk

Doubly diffusive convection is frequently encountered in natural sciences. For example, solar radiations heat the oceans making their surface warmer. In addition, due to evaporation, the density of salt in the oceans (salinity) increases towards the surface. This doubly diffusive configuration where salinity and temperature diffuse in the ocean is called thermohaline convection and gives rise to interesting phenomena. Indeed, the oceans are structured into thermohaline staircases in which the salinity remains mostly constant but jumps at specific depth levels. Thermohaline convection in these staircases is responsible for an instability called salt finger instability whereby the interface between two layers of different salinities becomes unstable and produces vertically elongated structures (fingers) of salty fluid sinking within the purer layer. This instability has been widely studied and was found to play a major role in the mixing of the oceans at low latitude and to strongly interact with large scale oceanic currents.

In this project, we will investigate the properties of spatially localised doubly diffusive convection states named convectons. These states consist in one or few convection rolls surrounded by motionless fluid and persist in spite of the fact that the fluid is homogeneously forced. An example is shown in figure 1. Convectons have recently been studied and revealed interesting properties among which: (i) a large number of such states co-exists in the same physical conditions and (ii) they are found below the instability threshold.

Figure 1: Convecton represented by two opposite contours of the rightward velocity. Gravity is to the left.

We will study the role of such states on the global dynamics of the system. To that aim, we will consider different configurations to find stable convectons. These will constitute the first ever computation of stable spatially localised fluid flows in three dimensions. The characterisation of these convectons will provide invaluable information on their role in the typically chaotic dynamics observed in nature and advance the theory of pattern formation.

Objectives

The student will study spatially localised states of doubly diffusive convection, namely convectons, in three dimensions. Important tasks will be undertaken to characterise their stability:

  • Stable convectons: the student will compute stable convectons in an elongated closed container similar to that of figure 1. These solutions have always been found unstable to a tilting instability. Computing stable convectons will be achieved by varying the size of the cross section of the closed container. Two-dimensional computations revealed stable convectons and the absence of twist instability which hints at the success of this method.
  • Convecton cartology: once stable convectons are found, the student will work on a map showing where these convectons are stable as a function of the various parameters of the system. This will provide a better understanding on pattern formation in high dimensions and guide the design of future experiments.
  • Onset of chaos: the mechanisms leading to chaos in doubly diffusive convection are not under­stood yet but stem from the emergence of convectons and of the twist instability. By choosing parameter values close to the twist instability threshold, the student will unravel the mechanisms behind the emergence of chaos in doubly diffusive convection.

Potential for high impact outcome

This project will provide a detailed stability analysis of doubly diffusive convectons. In particular, it will shed light on the conditions required for the twist instability, an instability that has been found to destabilise convectons. Finding the region in parameter space where this instability is absent is crucial for three reasons: (i) computing a stable convecton would represent the first ever computation of a stable three-dimensional localised flow state, (ii) the parameter region in which this solution is found will guide experiments to obtain the first ever experimental realisation of a steady spatially localised convection state and (iii) studying the region where the twist instability emerges will provide key understanding on the chaotic dynamics typically observed in doubly diffusive convection.

Training

The student will work under the supervision of Dr C´edric Beaume and Prof Alastair Rucklidge within the Department of Applied Mathematics. This project provides a high level of specialist scientific training in: (i) fluid dynamics, in particular in coupled convection problems; (ii) dynamical systems and more specifically in spatially localised pattern formation and (iii) high-performance computing with the handling of large scale simulations and data related to fluid flows. Co-supervision will involve weekly formal meetings with Dr Beaume and Prof Rucklidge as well as more frequent meetings with Dr Beaume and/or Prof Rucklidge when necessary. An extended visit to the University of Toulouse (France) in Prof Alain Bergeon’s group will also be planned during the end of the second year. Prof Alain Bergeon is a collaborator of Dr Beaume and leading author on convectons. The PhD student will have access to a broad spectrum of training workshops put on by the Organisational Development and Professional Learning team at the University of Leeds and will also be provided with dedicated training on fluid dynamics and numerical methods by Dr Beaume and pattern formation by Prof Rucklidge.

Student profile

The student should have an interest for fluid dynamics and dynamical systems, a taste for mathematics and numerics and be curious and motivated.

References

A. S. Alnahdi, J. Niesen & A. Rucklidge, SIAM J. Appl. Dyn. Sys. 13, 1311–1327 (2014)

C. Beaume, A. Bergeon & E. Knobloch, Phys. Fluids 23, 094102 (2011)

C. Beaume, A. Bergeon, H.-C. Kao & E. Knobloch, J. Fluid Mech. 717, 417–448 (2013)

C. Beaume, A. Bergeon & E. Knobloch, Phys. Fluids 25, 024105 (2013)

C. Beaume, A. Bergeon & E. Knobloch, preprint, (2017)

C. Beaume, E. Knobloch & A. Bergeon, Phys. Fluids 25, 114102 (2013)

A. Bergeon, J. Burke, E. Knobloch & I. Mercader, Phys. Rev. E 78, 046201 (2008)

J. Burke & E. Knobloch, Phys. Rev. E 73, 056211 (2006)

E. Knobloch, Annu. Rev. Condens. Matter Phys. 6, 325–359 (2016)

D. Lo Jacono, A. Bergeon & E. Knobloch, J. Fluid Mech. 730, R2 (2013)

I. Mercader, O. Batiste, A. Alonso & E. Knobloch, Phys. Rev. E 80, 025201(R) (2009) T. Watanabe, M. Iima & Y. Nishiura, SIAM J. Appl. Dyn. Sys. 15, 789–806 (2016)

Related undergraduate subjects:

  • Applied mathematics
  • Computer science
  • Computing
  • Mathematics
  • Mechanical engineering
  • Physics