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Prof Steven M Tobias

Professor of Applied Mathematics
Head of Applied Maths

Contact details

Room: 8.22
Tel: +44 (0)113 3435172
Email: S.M.Tobias @ leeds.ac.uk

Keywords

Dynamo Theory
The solar cycle and solar dynamo
Turbulence
Convection
Direct Statistical Simulation

Research interests

I am primarily interested in the dynamics of astrophysical and geophysical fluids and the generation and behaviour of magnetic fields.

My research includes investigations into the following topics:

Solar Dynamo Theory:

The magnetic field of the sun is believed to be generated by a hydromagnetic dynamo operating at the base of the solar convection zone in a thin region of shear called the tachocline. This magnetic field leads to the formation of the well-known 11-year solar activity cycle. The strong magnetic field has profound terrestrial implications, leading as it does to solar flares, coronal mass ejections and changes in the total solar irradiance. I conduct research into how this systematic magnetic field is generated deep within the sun and its dynamics in reaching the solar surface.

Dynamics of the Solar Tachocline:

As noted above, the solar tachocline is a region of strong shear at the base of the solar convection zone. The fundamental question concerning the tachocline is "Why is it still there?". The answer to this problem lies in an understanding of stably stratified magnetohydrodynamic (MHD) turbulence. Current research includes quantifying the role of magnetic field in modifying transport in the tachocline and analysing the consequences for angular momentum transport in the Sun.

Statistical Approaches to Fluid Mechanics and MHD:

An exciting new approach to fluid mechanics comes from solving equations for the statistics of the fluid rather than the fluid equations themselves. This approach seems to work well when strong shear flows or large-scale magnetic fields are present. I am currently interested in researching how these methods perform in problems such as

(i) The generation of astrophysical and geophysical jets, such as the jet-stream or the jets found on Jupiter and Saturn.

(ii) The interaction of turbulence with magnetic fields in stably stratified layers.

(iii) The instability of magnetic fields in the tachocline, accretion discs, extra-solar planets and experiments

(iv) The driving of mean flows in Tokamaks.

Stably Stratified Turbulence

I am currently researching the dynamics of stably stratified turbulence, to examine whether "layering" ever occurs in a realistic setting. This turbulence is important for transport in the oceans and atmosphere and also in many situations of astrophysical interest.

Useful links

More detailed homepage
Publications
I am an editor of Proceedings of the Royal Society A

Current postgraduate students

Christopher Mccabe (2014)
Anton Pershin (2017)

Publications

Márquez-Artavia X, Jones CA, Tobias SM Rotating magnetic shallow water waves and instabilities in a sphere Geophysical and Astrophysical Fluid Dynamics, 111, 282-322, 2017
DOI:10.1080/03091929.2017.1301937
View abstract

Calkins MA, Julien K, Tobias SM Inertia-less convectively-driven dynamo models in the limit of low Rossby number and large Prandtl number Physics of the Earth and Planetary Interiors, 266, 54-59, 2017
DOI:10.1016/j.pepi.2017.03.003
View abstract

Tobias SM, Marston JB Three-dimensional rotating Couette flow via the generalised quasilinear approximation Journal of Fluid Mechanics, 810, 412-428, 2017
DOI:10.1017/jfm.2016.727
View abstract

Nigro G, Pongkitiwanichakul P, Cattaneo F, Tobias SM What is a large-scale dynamo? MONTHLY NOTICES OF THE ROYAL ASTRONOMICAL SOCIETY, 464, L119-L123, 2017
DOI:10.1093/mnrasl/slw190

Calkins MA, Long L, Nieves D, Julien K, Tobias SM Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers PHYSICAL REVIEW FLUIDS, 1, 2016
DOI:10.1103/PhysRevFluids.1.083701

Raynaud R, Tobias SM Convective dynamo action in a spherical shell: Symmetries and modulation Journal of Fluid Mechanics, 799, R6, 2016
DOI:10.1017/jfm.2016.407
View abstract

Pongkitiwanichakul P, Nigro G, Cattaneo F, Tobias SM SHEAR-DRIVEN DYNAMO WAVES in the FULLY NONLINEAR REGIME Astrophysical Journal, 825, 2016
DOI:10.3847/0004-637X/825/1/23
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Dallas V, Tobias SM Forcing-dependent dynamics and emergence of helicity in rotating turbulence Journal of Fluid Mechanics, 798, 682-695, 2016
DOI:10.1017/jfm.2016.341
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Kondić T, Hughes DW, Tobias SM THE DECAY of A WEAK LARGE-SCALE MAGNETIC FIELD in TWO-DIMENSIONAL TURBULENCE Astrophysical Journal, 823, 2016
DOI:10.3847/0004-637X/823/2/111
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Marston JB, Chini GP, Tobias SM Generalized Quasilinear Approximation: Application to Zonal Jets Physical Review Letters, 116, 2016
DOI:10.1103/PhysRevLett.116.214501
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Child A, Hollerbach R, Marston B, Tobias S Generalised quasilinear approximation of the helical magnetorotational instability Journal of Plasma Physics, 82, 2016
DOI:10.1017/S0022377816000556
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Weiss NO, Tobias SM Supermodulation of the Sun's magnetic activity: The effects of symmetry changes Monthly Notices of the Royal Astronomical Society, 456, 2654-2661, 2016
DOI:10.1093/mnras/stv2769
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Gilbert AD, Mason J, Tobias SM Flux expulsion with dynamics Journal of Fluid Mechanics, 791, 568-588, 2016
DOI:10.1017/jfm.2016.60
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Calkins MA, Julien K, Tobias SM, Aurnou JM, Marti P Convection-driven kinematic dynamos at low Rossby and magnetic Prandtl numbers: Single mode solutions Physical Review E, 93, 2016
DOI:10.1103/PhysRevE.93.023115
View abstract

Currie LK, Tobias SM Mean flow generation in rotating anelastic two-dimensional convection Physics of Fluids, 28, 2016
DOI:10.1063/1.4939300
View abstract

Miesch MS, Matthaeus WH, Brandenburg A, Petrosyan A, Pouquet A, Cambon C, Jenko F, Uzdensky D, Stone J, Tobias SM, Toomre J, Velli M Large-Eddy Simulations of Magnetohydrodynamic Turbulence in Heliophysics and Astrophysics Space Science Reviews, 194, 97-137, 2015
DOI:10.1007/s11214-015-0190-7
View abstract

Calkins MA, Julien K, Tobias SM, Aurnou JM A multiscale dynamo model driven by quasi-geostrophic convection Journal of Fluid Mechanics, 780, 143-166, 2015
DOI:10.1017/jfm.2015.464
View abstract

Teed RJ, Jones CA, Tobias SM The transition to Earth-like torsional oscillations in magnetoconvection simulations Earth and Planetary Science Letters, 419, 22-31, 2015
DOI:10.1016/j.epsl.2015.02.045
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Hsu PC, Diamond PH, Tobias SM Nonperturbative mean-field theory for minimum enstrophy relaxation Physical Review E - Statistical, Nonlinear, and Soft Matter Physics, 91, 2015
DOI:10.1103/PhysRevE.91.053024
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Tobias SM, Cattaneo F The electromotive force in multi-scale flows at high magnetic Reynolds number Journal of Plasma Physics, 81, 2015
DOI:10.1017/S0022377815001063
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Zhdankin V, Boldyrev S, Perez JC, Tobias SM Energy dissipation in magnetohydrodynamic turbulence: Coherent structures or "nanoflares"? Astrophysical Journal, 795, 2014
DOI:10.1088/0004-637X/795/2/127
View abstract

Cattaneo F, Tobias SM On large-scale dynamo action at high magnetic reynolds number Astrophysical Journal, 789, 2014
DOI:10.1088/0004-637X/789/1/70
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Cattaneo F, Tobias SM On the measurement of turbulent magnetic diffusivities: the three-dimensional case Journal of Fluid Mechanics, 735, 457-472, 2013
DOI:10.1017/jfm.2013.506
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Tobias SM, Cattaneo F Shear-driven dynamo waves at high magnetic Reynolds number Nature, 497, 463-465, 2013
DOI:10.1038/nature12177
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Bartello P, Tobias SM Sensitivity of stratified turbulence to the buoyancy Reynolds number Journal of Fluid Mechanics, 725, 1-22, 2013
DOI:10.1017/jfm.2013.170
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Tobias SM, Marston JB Direct statistical simulation of out-of-equilibrium jets Physical Review Letters, 110, 2013
DOI:10.1103/PhysRevLett.110.104502
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Mizerski KA, Tobias SM Large-scale convective dynamos in a stratified rotating plane layer Geophysical and Astrophysical Fluid Dynamics, 107, 218-243, 2013
DOI:10.1080/03091929.2012.672567
View abstract

Tobias SM, Cattaneo F On the measurement of the turbulent diffusivity of a large-scale magnetic field Journal of Fluid Mechanics, 717, 347-360, 2013
DOI:10.1017/jfm.2012.575
View abstract

Teed RJ, Jones CA, Tobias SM The dynamics and excitation of torsional waves in geodynamo simulations Geophysical Journal International, 196, 724-735, 2013
DOI:10.1093/gji/ggt432
View abstract

Dritschel DG, Tobias SM Two-dimensional magnetohydrodynamic turbulence in the small magnetic Prandtl number limit JOURNAL OF FLUID MECHANICS, 703, 85-98, 2012
DOI:10.1017/jfm.2012.195

Weliwita JA, Rucklidge AM, Tobias SM Skew-varicose instability in two-dimensional generalized Swift-Hohenberg equations PHYS REV E, 84, 2011
DOI:10.1103/PhysRevE.84.036201

Tobias SM, Cattaneo F, Brummell NH ON THE GENERATION OF ORGANIZED MAGNETIC FIELDS ASTROPHYS J, 728, 2011
DOI:10.1088/0004-637X/728/2/153

Tobias SM, Cattaneo F, Brummell NH On the generation of organized magnetic fields Astrophysical Journal, 728, 2011
DOI:10.1088/0004-637X/728/2/153
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Tobias SM, Dagon K, Marston JB Astrophysical fluid dynamics via direct statistical simulation Astrophysical Journal, 727, 2011
DOI:10.1088/0004-637X/727/2/127
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Tobias SM, Dagon K, Marston JB ASTROPHYSICAL FLUID DYNAMICS VIA DIRECT STATISTICAL SIMULATION ASTROPHYS J, 727, 2011
DOI:10.1088/0004-637X/727/2/127

Mizerski KA, Tobias SM The effect of stratification and compressibility on anelastic convection in a rotating plane layer GEOPHYSICAL AND ASTROPHYSICAL FLUID DYNAMICS, 105, 566-585, 2011
DOI:10.1080/03091929.2010.521748

Byington BM, Brummell NH, Tobias SM The effect of small scale motion on an essentially-nonlinear dynamo Proceedings of the International Astronomical Union, 6, 367-368, 2010
DOI:10.1017/S1743921311017820
View abstract

Jones CA, Thompson MJ, Tobias SM The Solar Dynamo SPACE SCI REV, 152, 591-616, 2010
DOI:10.1007/s11214-009-9579-5

Houghton SM, Knobloch E, Tobias SM, Proctor MRE Transient spatio-temporal chaos in the complex Ginzburg-Landau equation on long domains PHYS LETT A, 374, 2030-2034, 2010
DOI:10.1016/j.physleta.2010.02.078

Livermore PW, Hughes DW, Tobias SM Nonlinear generation of large-scale magnetic fields in forced spherical shell dynamos PHYS FLUIDS, 22, 2010
DOI:10.1063/1.3313930

Berkoff NA, Kersalé E, Tobias SM Comparison of the anelastic approximation with fully compressible equations for linear magnetoconvection and magnetic buoyancy Geophysical and Astrophysical Fluid Dynamics, 104, 545-563, 2010
DOI:10.1080/03091929.2010.521747

Berkoff NA, Kersale E, Tobias SM Comparison of the anelastic approximation with fully compressible equations for linear magnetoconvection and magnetic buoyancy GEOPHYS ASTRO FLUID, 104, 545-563, 2010
DOI:10.1080/03091929.2010.521747

Brummell NH, Tobias SM, Cattaneo F Dynamo efficiency in compressible convective dynamos with and without penetration GEOPHYS ASTRO FLUID, 104, 565-576, 2010
DOI:10.1080/03091929.2010.495067

Tobias SM, Cattaneo F, Boldyrev S MHD dynamos and turbulence In Ten Chapters in Turbulence, 351-404, 2010
DOI:10.1017/CBO9781139032810.010
View abstract

Courvoisier A, Hughes DW, Tobias SM Mean induction and diffusion: the influence of spatial coherence J FLUID MECH, 627, 403-421, 2009
DOI:10.1017/S0022112009005783

Tobias SM The Solar Dynamo: The Role of Penetration, Rotation and Shear on Convective Dynamos SPACE SCI REV, 144, 77-86, 2009
DOI:10.1007/s11214-008-9442-0

Cattaneo F, Tobias SM Dynamo properties of the turbulent velocity field of a saturated dynamo J FLUID MECH, 621, 205-214, 2009
DOI:10.1017/S0022112008004990

Houghton SM, Tobias SM, Knobloch E, Proctor MRE Bistability in the complex Ginzburg-Landau equation with drift PHYSICA D, 238, 184-196, 2009
DOI:10.1016/j.physd.2008.09.011

Tobias SM The Solar Dynamo: The Role of Penetration, Rotation and Shear on Convective Dynamos. 32, 77-86 2009
DOI:10.1007/978-1-4419-0239-9_6

Mason J, Hughes DW, Tobias SM The effects of flux transport on interface dynamos MON NOT R ASTRON SOC, 391, 467-480, 2008
DOI:10.1111/j.1365-2966.2008.13918.x

Abreu JA, Beer J, Steinhilber F, Tobias SM, Weiss NO For how long will the current grand maximum of solar activity persist? GEOPHYS RES LETT, 35, 2008
DOI:10.1029/2008GL035442

Brummell NH, Tobias SM, Thomas JH, Weiss NO FLUX PUMPING AND MAGNETIC FIELDS IN THE OUTER PENUMBRA OF A SUNSPOT ASTROPHYS J, 686, 1454-1465, 2008

Tobias SM, Cattaneo F, Brummell NH Convective dynamos with penetration, rotation, and shear ASTROPHYS J, 685, 596-605, 2008

Tobias SM, Cattaneo F Limited role of spectra in dynamo theory: Coherent versus random dynamos PHYS REV LETT, 101, 2008
DOI:10.1103/PhysRevLett.101.125003

Rashid FQ, Jones CA, Tobias SM Hydrodynamic instabilities in the solar tachocline ASTRON ASTROPHYS, 488, 819-827, 2008
DOI:10.1051/0004-6361:200810039

Tobias SM, Cattaneo F Dynamo action in complex flows: the quick and the fast J FLUID MECH, 601, 101-122, 2008
DOI:10.1017/S002211200800044X

Tobias SM, Diamond PH, Hughes DW beta-plane magnetohydrodynamic turbulence in the solar tachocline ASTROPHYS J, 667, L113-L116, 2007

Tobias SM, Diamond PH, Hughes DW β-plane magnetohydrodynamic turbulence in the solar tachocline Astrophysical Journal, 667, 2007
View abstract

Kersalé E, Hughes DW, Tobias SM The Nonlinear Evolution of Instabilities Driven by Magnetic Buoyancy: A New Mechanism for the Formation of Coherent Magnetic Structures The Astrophysical Journal, 663, L113-L116, 2007
View abstract

Bushby PJ, Tobias SM On predicting the solar cycle using mean-field models ASTROPHYS J, 661, 1289-1296, 2007

Livermore PW, Hughes DW, Tobias SM The role of helicity and stretching in forced kinematic dynamos in a spherical shell PHYS FLUIDS, 19, 2007
DOI:10.1063/1.2717947

Tobias SM, Weiss N The solar dynamo and the tachocline In The Solar Tachocline, 319-350, 2007
DOI:10.1017/CBO9780511536243.014
View abstract

Livermore PW, Hughes DW, Tobias SM The role of kinetic helicity in spherical shell dynamos European Space Agency, (Special Publication) ESA SP, 2006
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Thomas JH, Weiss NO, Tobias SM, Brummell NH On the fine structure of magnetic fields in sunspot penumbrae ASTRON ASTROPHYS, 452, 1089-1090, 2006
DOI:10.1051/0004-6361:20064967

Kersalé E, Hughes DW, Ogilvie GI, Tobias SM Global Magnetorotational Instability with Inflow. II. The Nonlinear Development of Axisymmetric Wall Modes The Astrophysical Journal, 638, 382-390, 2006
DOI:10.1086/498704
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Courvoisier A, Hughes DW, Tobias SM α effect in a family of chaotic flows Physical Review Letters, 96, 2006
DOI:10.1103/PhysRevLett.96.034503
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Cattaneo F, Tobias SM Interaction between dynamos at different scales Physics of Fluids, 17, pp.127105, 2005
DOI:10.1063/1.2145755
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Tobias SM The solar tachocline: Formation, stability and its role in the solar dynamoFLUID DYNAMICS AND DYNAMOS IN ASTROPHYSICS AND GEOPHYSICS, 193-+ 2005

Tobias SM, Hughes DW The Influence of Velocity Shear on Magnetic Buoyancy Instability in the Solar Tachocline The Astrophysical Journal, 603, 785-802, 2004
DOI:10.1086/381492

Kersale E, Hughes DW, Ogilvie GI, Tobias SM, Weiss NO Global magnetorotational instability with inflow. I. Linear theory and the role of boundary conditions ASTROPHYS J, 602, 892-903, 2004

Weiss NO, Thomas JH, Brummell NH, Tobias SM The origin of penumbral structure in sunspots: Downward pumping of magnetic flux ASTROPHYS J, 600, 1073-1090, 2004

Llewellyn-Smith SG, Tobias SM Vortex dynamos Journal of Fluid Mechanics, 498, 1-21, 2004
DOI:10.1017/S0022112003007006

Julien K, Knobloch E, Tobias SM Highly supercritical convection in strong magnetic fields In Advances in nonlinear dynamos, 192-223, 2003

Julien K, Knobloch E, Tobias SM Nonlinear magnetoconvection in the presence of a strong oblique field In Stellar Astrophysical Fluid Dynamics, 345-356, 2003

Tobias SM The solar dynamo. Philos Trans A Math Phys Eng Sci, 360, 2741-2756, 2002
DOI:10.1098/rsta.2002.1090
View abstract

Thomas JH, Weiss NO, Tobias SM, Brummell NH Downward pumping of magnetic flux as the cause of filamentary structures in sunspot penumbrae Nature, 420, 390-393, 2002
DOI:10.1038/nature01174

Mason J, Hughes DW, Tobias SM The competition in the solar dynamo between surface and deep-seated alpha-effects The Astrophysical Journal, 580, L89-L92, 2002
DOI:10.1086/345419

Thomas JH, Weiss NO, Tobias SM, Brummell NH Magnetic flux pumping and the structure of a sunspot penumbra ASTRON NACHR, 323, 383-386, 2002

Tobias SM Modulation of solar and stellar dynamos Astronomische Nachrichten, 2002

Hughes DW, Tobias SM On the instability of magnetohydrodynamic shear flows P ROY SOC LOND A MAT, 457, 1365-1384, 2001

Brummell NH, Cattaneo F, Tobias SM Linear and nonlinear dynamo properties of time-dependent ABC flows Fluid Dynamics Research, 28, 237-265, 2001
DOI:10.1016/S0169-5983(00)00027-7
View abstract

Tobias SM, Brummell NH, Clune TL, Toomre J Transport and storage of magnetic field by overshooting turbulent compressible convection ASTROPHYS J, 549, 1183-1203, 2001

Tobias SM, Brummell NH, Clune TL, Toomre J Transport and Storage of Magnetic Field by Overshooting Turbulent Compressible Convection The Astrophysical Journal, 549, pp.1183–1203, 2001
DOI:10.1086/319448

Brummell NH, Cattaneo F, Tobias SM Small- and large-scale dynamo action in simple forced flows None, 145-152, 2001

Weiss NO, Knobloch E, Tobias SM Modulation and symmetry-breaking in low-order models of the solar dynamo. In Dynamo and Dynamics , a mathematical challenge, 26, 381-390, 2001

Tobias SM; Brummell NH; Toomre J Magnetic pumping at the base of the solar convection zoneIAU SYMP, 156-158 2001
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Tobias SM, Weiss NO Resonant Interactions Between Solar Activity and Climate Journal of Climate, 13, 3745-3759, 2000
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Julien K, Knobloch E, Tobias SM Strongly Nonlinear Magnetoconvection in the Presence of Oblique Fields Journal of Fluid Mechanics, 410, 285-322, 2000
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Proctor MRE, Tobias SM, Knobloch E Noise sustained structures due to convective instability in finite domains Physica D - Nonlinear Phenomena, 145, 191-206, 2000

Proctor MRE, Tobias SM, Knobloch E Noise sustained structures due to convective instability in finite domains Physica D - Nonlinear Phenomena, 145, 191-206, 2000

Julien K, Knobloch E, Tobias SM Nonlinear magnetoconvection in the presence of strong oblique fields Journal of Fluid Mechanics, 410, 285-322, 2000

Tobias SM, Proctor MRE, Knobloch E Convective and Absolute Instabilities of Fluid Flows in Finite Geometry Physica D - Nonlinear Phenomena, 113, 43-72, 1998
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Tobias SM The Solar Cycle: Parity Interactions and Amplitude Modulation Astronomy and Astrophysics, 322, 1007-1017, 1997
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Kent T, Bokhove O, Tobias S Dynamics of an Idealized Fluid Model for Investigating Convective-scale Data Assimilation Tellus A: Dynamic Meteorology and Oceanography