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Numerical and experimental modelling of strongly rotating convection and phase changes, with applications to Numerical Weather Prediction

Prof Onno Bokhove (SoM), Pro Steve Tobias (SoM), Prof Michael Fairweather (SCE)

Project partner(s): Met Office potential

Contact email:

Accurate prediction of (rotating) convection is important in Numerical Weather Prediction (NWP). Convection is also coupled to the phase changes of water in the atmosphere and, hence, the notoriously difficult prediction of cloud dynamics and precipitation (e.g., Romps, 2015). However, rather than modelling the dynamics of air in the atmosphere with atmospheric water in all its different phase changes and forms (water vapour, rain, ice, hail and several types of snow crystals), we propose in this project to only consider an idealized second phase (e.g., water) in solid or gaseous form. This is akin to the fast auto-conversion and evaporation/sublimation models by Hernandes-Duenas et al. (2015) in which water only appear as water vapour or liquid water. A material with phase changes limited predominantly to the solid and gaseous phase is iodine. In addition, we aim to focus and therefore simplify the modelling of this complex dynamics by focusing on large-scale atmospheric flows with phase changes, i.e. in the limit where the effects of the Earth’s rotation are dominant (Julien et al. 2006, 2012).

The first aim of the project is therefore to consider the theoretical formulation of convection with phase changes between solid and gaseous phases in this limit of strong rotation, in combination with a linear and nonlinear stability analysis. The second aim is to numerically model atmospheric convection under strong rotation and phase changes. Both spectral and finite element type numerical discretisations will be extended/investigated, for models with and without phase changes, in idealised domains. The third, tentative aim of the project is to compare our modelling outcomes with findings in a preliminary laboratory Rayleigh-Benard experiment, using a rotating platform in a fume cupboard with a tank filled with an inert gas (dry air) as well as iodine crystals and iodine vapor, with heating from below and cooling from the top (Fig. 1). Alternatively, an interesting alley of further research concerns data assimilation for our strongly rotating flows with phase changes, which is also of direct interest to weather-prediction centres around the world. Which of the latter options is chosen depends on your interest.

Finally, we aim to collaborate on this project with existing weather institutes and consulting firms in the UK and USA, by extending our current contacts with the UK Met Office as well as other partners. A research visit to Prof Keith Julien, Department of Mathematics, University of Colorado, Colorado, USA is anticipated. The work proposed is inspired by and related to convective (cloud) modelling in NWP. In essence the experimental set-up and corresponding mathematical modelling concerns a small-scale and idealized weather experiment.

Figure 1: Left: Sketch of the experimental set-up for rotating Rayleigh-Benard convection with iodine phase changes with heating from below and cooling from the top. The iodine is a simplification of the more complex phase changes of water incorporated in NWP. Right: sample of the clearly visible iodine vapor.


The desired outcomes of the project are consequently as follows:

  1. Extend, study and analyse the rotating convection model of Julien et al. (2006, 2012), Sprague et al. (2006) to include phase changes (see, e.g., also Bokhove, Tobias, Kent, 2015; Zerroukat et al. 2015).
  2. Adapt and verify the numerical models of this new model, using and adapting (existing) spectral and/or finite element models.
  3. Set up preliminary experiments of rotating convection with phase changes on our rotating platform in a fume cupboard or explore data assimilation methods in our newly developed models.
  4. Compare the numerical and analytical modelling against the preliminary experimental results.

Potential for high impact outcome

NWP including the dynamics of phase changes is an active area of research and breakthroughs will have high impact in NWP. We are in a unique position at Leeds to answer important unresolved questions about how phase changes can be modelled accurately with atmospheric convection. The research topic has immediate relevance to NWP as well as data assimilation techniques within NWP, and we therefore anticipate the project to generate several papers with at least one for submission to a very high impact journal.


The student will work under the direct supervision of Profs Onno Bokhove and Steve Tobias within the Astrophysical and Geophysical Fluid Dynamics group of the School of Mathematics, and Prof Michael Fairweather in the School of Chemical and Process Engineering.  This project provides a high level of specialist scientific training in: (i) state-of-the-science application and analysis of atmospheric models; (ii) high-level numerical techniques for atmospheric convection and phase changes; (iii) the use of cutting-edge supercomputers and (iv) experimental or data-assimilation methods. Co-supervision will involve regular meetings between all partners. The successful PhD student will have access to a broad spectrum of training workshops put on by the University of Leeds that include an extensive range of training workshops in numerical modelling, through to managing your degree, to preparing for your viva (

Student profile

The student should have a strong interest in applied mathematics, dynamic meteorology, meso-scale environmental problems, a strong background in a quantitative science (math, physics, engineering, environmental sciences) and a flair for, and good familiarity with, programming and scientific computing.


The project includes a strong collaboration with Prof Keith Julien in the USA and further collaboration with the Met Office will evolve during the project, also in conjunction with a current NERC DTP Case project of the supervisors with the Met Office.


  • Bokhove, O., Tobias, S. & Kent, T. 2015 On a 3D Model with Anisotropic, Rotating Convection and Phase Changes for DA. Poster:
  • Hernandes-Duenas, L.M. Smith and S.N. Stechmann 2015: Stability and instability criteria for idealized precipitating hydrodynamics. J. Atmos. Sci. 72, 2379–2393.
  • Julien, K. Knobloch, E., Milliff, R., Werne, J. 2006: Generalized quasi-geostrophy for anisotropic rotationally constrained flows. J. Fluid Mech. 555.
  • Julien, K., Knobloch, E., Rubio, A. M. & Vasil, G. M. 2012 Heat transport in Low-Rossby-number Rayleigh–Bénard Convection. Phys. Rev. Lett. 109, 254503.
  • D.M. Romps 2015: MSE minus CAPE is the true conserved variable for an adiabatically lifted parcel. J. Atmos. Sci. 72, 3639–3645.
  • Sprague, M., Julien, K., Werne, J. 2006: Numerical simulation of an asymptotically reduced system for rotationally constrained convection. J. Fluid Mech 551.
  • Zerroukat, M. Allen, T. 2015: A moist Boussinesq shallow water equations set for testing atmospheric models. J. Comp. Physics 290.

Related undergraduate subjects:

  • Computer science
  • Engineering
  • Environmental science
  • Mathematics
  • Meteorology
  • Physical geography