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# PhD Position at Northumbria University, UK

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Rotating Keplerian flows in helical magnetic fields: Global stability analysis

(RDF17/MPEE/KIRILLOV)

https://www.findaphd.com/search/ProjectDetails.aspx?PJID=81660

Project DescriptionDifferentially rotating flows, i.e. the flows in which the fluid parcels move with a velocity that depends on the distance from the axis of rotation, widely occur both in natural and industrial environments. Gas flow orbiting a gravitating centre, like a star, tends to acquire a form of a flat disk that rotates differentially according to the laws of Kepler. The Keplerian rotating flow should be laminar according to the centrifugal Rayleigh criterion because the angular momentum of the Keplerian flow decreases with the increase in the distance from the star. Nevertheless, observations show that the matter in the accretion disks is in the turbulent state. The apparent paradox can be resolved if one takes into account that the matter of the disk is usually ionized and thus electrically conducting and that the stars usually have magnetic fields. Even weak magnetic fields turn out to be capable to destabilize a Keplerian disk if the fluid is a good electrical conductor. When the conductance is poor, the destabilizing properties depend on the spatial distribution of the magnetic field. Recently, it has been shown theoretically in the short-wavelength WKB approximation that helical and azimuthal magnetic fields are capable to induce the magnetorotational instability even in poorly electrically conducting Keplerian flows. The aim of the PhD project is to prove this property by solving a boundary value problem with a variety of boundary conditions. The non-self-adjoint boundary value problem will require a perturbative, asymptotic and numerical treatment and will involve such concepts as negative energy waves, dissipation-induced instabilities, and Hamiltonian mechanics. As a first step, a magnetized Couette-Taylor flow with the quasi-Keplerian shear profile is planned to consider with the subsequent transition to more realistic models of accretion disks. The theory is mathematically challenging but highly demanded by both astrophysical and magnetohydrodynamics communities.

Eligibility and How to Apply

Please note eligibility requirement:

• Academic excellence of the proposed student i.e. 2:1 (or equivalent GPA from non-UK universities [preference for 1st class honours]); or a Masters (preference for Merit or above); or APEL evidence of substantial practitioner achievement.

• Appropriate IELTS score, if required (evidence required by 1 August 2017).

For further details of how to apply, entry requirements and the application form, see

https://www.northumbria.ac.uk/research/postgraduate-research-degrees/how-to-apply/

Please ensure you quote the advert reference above on your application form.

Deadline for applications: 20 January 2017

Start Date: 2 October 2017

Northumbria University is an equal opportunities provider and in welcoming applications for studentships from all sectors of the community we strongly encourage applications from women and under-represented groups.Funding Notes

This project is being considered for funding in competition with other projects, through one of two types of funding packages available:

• Fully funded studentships include a full stipend, paid for three years at RCUK rates for 2017/18 (this is yet to be set, in 2016/17 this is £14,296 pa) and fees (Home/EU £4,350 / International £13,000 / International Lab-based £16,000), and are available to applicants worldwide.

• As Northumbria celebrates its 25th anniversary as a University and in line with our international outlook, some projects may also be offered to students from outside of the EU supported by a half-fee reduction.

References

O.N. Kirillov (2016) Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics. Proc. R. Soc. London A, subm., arXiv:1610.06970

F. Stefani, O.N. Kirillov (2015) Destabilization of rotating flows with positive shear by azimuthal magnetic fields. Phys. Rev. E, 92: 051001(R)

O.N. Kirillov, F. Stefani, Y. Fukumoto (2014) Local instabilities in magnetized rotational flows: A short-wavelength approach. Journal of Fluid Mechanics, 760: 591- 633

O.N. Kirillov, F. Stefani (2013) Extending the range of the inductionless magnetorotational instability. Physical Review Letters, 111(6): 061103

O.N. Kirillov (2013) Stabilizing and destabilizing perturbations of PT -symmetric indefinitely damped systems. Philosophical Transactions of the Royal Society A, 371: 20120051

Kirillov ON. 2013. Nonconservative stability problems of modern physics, De Gruyter Studies in

Mathematical Physics 14. Berlin, Boston: De Gruyter.

O.N. Kirillov, F. Stefani, Y. Fukumoto (2012) A unifying picture of helical and azimuthal MRI, and the universal significance of the Liu limit. The Astrophysical Journal, 756(83): 6pp

O.N. Kirillov, D.E. Pelinovsky, G. Schneider (2011) Paradoxical transitions to instabilities in hydromagnetic Couette-Taylor flows. Physical Review E, 84(6): 065301(R)

O.N. Kirillov, F. Stefani (2010) On the relation of standard and helical magnetorotational instability. The Astrophysical Journal, 712: 52-68

O.N. Kirillov (2010) Eigenvalue bifurcation in multiparameter families of non-self-adjoint operator matrices. Zeitschrift fur angewandte Mathemtik und Physik-ZAMP, 61(2): 221-234

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