# School of Mathematics

## 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

**Kent T, Bokhove O, Tobias S** Dynamics of an idealized fluid model for investigating convective-scale data assimilation *TELLUS SERIES A-DYNAMIC METEOROLOGY AND OCEANOGRAPHY*, **69**, 2017

DOI:10.1080/16000870.2017.1369332

**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

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**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

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**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

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**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

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**Currie LK, Tobias SM** Mean flow generation in rotating anelastic two-dimensional convection *Physics of Fluids*, **28**, 2016

DOI:10.1063/1.4939300

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**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

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**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

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**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

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**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

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**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 dynamo*FLUID 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

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**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 zone*IAU 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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**Beer J, Tobias SM, Weiss NO** On long-term modulation of the Sun’s magnetic cycle *Monthly Notices of the Royal Astronomical Society*

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**Tobias SM, Marston JB** Direct Statistical Simulation of Jets and Vortices in 2D Flows *Physics of Fluids*

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