A list of the presentations I have given can be found here. Below is a my publication list.

## Publications

For each paper, click on Abstract to show/hide the abstract.

**Referred publications**

- S. M. Houghton & E. Knobloch
*Swift-Hohenberg equation with broken cubic-quintic nonlinearity*Phys. Rev. E, Volume 84, Issue 1 (2011), pg. 016204 (10 pages). Abstract.

The cubic-quintic Swift-Hohenberg equation (SH35) provides a convenient order parameter description of several convective systems with reflection symmetry in the layer midplane, including binary fluid convection. We use SH35 with an additional quadratic term to determine the qualitative effects of breaking the midplane reflection symmetry on the properties of spatially localized structures in these systems. Our results describe how the snakes-and-ladders organization of localized structures in SH35 deforms with increasing symmetry breaking, and show that the deformation ultimately generates the snakes-and-ladders structure familiar from the quadratic-cubic Swift-Hohenberg equation (SH23). Moreover in nonvariational systems like convection odd-parity convectons necessarily drift when the reflection symmetry is broken, permitting collisions among moving localized structures. Collisions between both identical and nonidentical traveling states are described.

- S. M. Houghton & P. J. Bushby
*Localised plumes in three-dimensional compressible magnetoconvection*MNRAS, Volume 412 (2011), pg 555-560. Abstract.

Within the umbrae of sunspots, convection is generally inhibited by the presence of strong vertical magnetic fields. However, convection is not completely suppressed in these regions: bright features, known as umbral dots, are probably associated with weak, isolated convective plumes. Motivated by observations of umbral dots, we carry out numerical simulations of three-dimensional, compressible magnetoconvection. By following solution branches into the subcritical parameter regime (a region of parameter space in which the static solution is linearly stable to convective perturbations), we find that it is possible to generate a solution which is characterised by a single, isolated convective plume. This solution is analogous to the magnetohydrodynamic convectons that have previously been found in two-dimensional calculations. These results can be related, in a qualitative sense, to observations of umbral dots.

- S. M. Houghton, E. Knobloch, S. M. Tobias & M. R. E. Proctor
*Transient spatiotemporal chaos in the complex Ginzburg-Landau equation on long domains*Phys. Lett. A, Volume 374 (2010), pg 2030-2034. Abstract.

Numerical simulations of the complex Ginzburg-Landau equation in one spatial dimension on periodic domains with sufficiently large spatial period reveal persistent chaotic dynamics in large parts of parameter space that extend into the Benjamin-Feir stable regime. This situation changes when nonperiodic boundary conditions are imposed, and in the Benjamin-Feir stable regime chaos takes the form of a long-lived transient decaying to a spatially uniform oscillatory state. The length of the transient has Poisson statistics and no domain length is found sufficient for persistent chaos.

- J. Burke, S. M. Houghton & E. Knobloch
*Swift-Hohenberg equation with broken reflection symmetry*Phys. Rev. E, Volume 80 (2009), Number 3, 036202. Abstract.

The bistable Swift-Hohenberg equation possesses a variety of time-independent spatially localised solutions organised in the so-called snakes-and-ladders structure. This structure is a consequence of a phenomenon known as homoclinic snaking, and is in turn a consequence of spatial reversibility on the snakes-and-ladders structure. We examine here the consequences of breaking spatial reversibility on the snakes-and-ladders structure. We find that generically the localized states both drift and dilate, and show that the snakes-and-ladders structure breaks up into a stack of isolas. We explore the evolution of this new structure with increasing reversibility breaking and study the dynamics of the system outside of the snaking region using a combination of numerical and analytical techniques.

- S. M. Houghton & E. Knobloch
*Homoclinic snaking in bounded domains*Phys. Rev. E, Volume 80 (2009), Number 2, 026210. Abstract.

Homoclinic snaking is a term used to describe the back and forth oscillation of a branch of time-independent spatially localized states in a bistable, spatially reversible system as the localized structure grows in length by repeatedly adding rolls on either side. On the real line this process continues forever. In finite domains snaking terminates once the domain is filled but the details of how this occurs depend critically on the choice of boundary conditions. With periodic boundary conditions the snaking branches terminate on a branch of spatially periodic s$However, with non-Neumann boundary conditions they turn continuously into a large amplitude filling state that replaces the periodic state. This behavior, shown here in detail for the Swift-Hohenberg equation, explains the phenomenon of "snaking without bistability", recently observed in simulations of binary fluid convection by Mercader, Batiste, Alonso and Knobloch (preprint).

- S. M. Houghton, S. M. Tobias, E. Knobloch & M. R. E. Proctor
*Bistability in the Complex Ginzburg-Landau Equation with Drift*Physica D, Volume 238 (2009), pg 184-196. Abstract.

Properties of the complex Ginzburg-Landau equation with drift are studied focusing on the Benjamin-Feir stable regime. On a finite interval with Neumann boundary conditions the equation exhibits bistability between a spatially uniform time-periodic state and a variety of nonuniform states with complex time dependence. The origin of this behavior is identified and contrasted with the bistable behavior present with periodic boundary conditions and no drift.

- P. J. Bushby, S. M. Houghton, M. R. E. Proctor & N. O. Weiss
*Convective intensification of magnetic fields in the quiet Sun*Monthly Notices of the Royal Astronomical Society, Volume 387 (2008), Issue 2, pg 698-706. Abstract.

Kilogauss-strength magnetic fields are often observed in intergranular lanes at the photosphere in the quiet Sun. Such fields are stronger than the equipartition field

*Be*, corresponding to a magnetic energy density that matches the kinetic energy density of photospheric convection, and comparable with the field*Bp*that exerts a magnetic pressure equal to the ambient gas pressure. We present an idealised numerical model of three-dimensional compressible magnetoconvection at the photosphere, for a range of values of the magnetic Reynolds number. In the absence of a magnetic field, the convection is highly supercritical and is characterised by a pattern of vigorous, time- dependent, granular, motions. When a weak magnetic field is imposed upon the convection, magnetic flux is swept into the convective down flows where it forms localised concentrations. Unless this process is significantly inhibited by magnetic diffusion, the resulting fields are often much greater than*Be*, and the high magnetic pressure in these flux elements leads to their being partially evacuated. Some of these flux elements contain ultra-intense magnetic fields that are significantly greater than*Bp*. Such fields are contained by a combination of the thermal pressure of the gas and the dynamic pressure of the convective motion, and they are constantly evolving. These ultra-intense fields develop owing to nonlinear interactions between magnetic fields and convection; they cannot be explained in terms of convective collapse within a thin flux tube that remains in overall pressure equilibrium with its surroundings. - P. J. Bushby & S. M. Houghton
*Spatially intermittent fields in photospheric magnetoconvection*Monthly Notices of the Royal Astronomical Society, Volume 362 (2005), Issue 1, pg 313-320. Abstract.

Motivated by recent high-resolution observations of the solar surface, we investigate the problem of non-linear magnetoconvection in a three-dimensional compressible layer. We present results from a set of numerical simulations which model the situation in which there is a weak imposed magnetic field. This weak-field regime is characterized by vigorous granular convection and spatially intermittent magnetic field structures. When the imposed field is very weak, magnetic flux tends to accumulate at the edges of the convective cells, where it forms compact, almost `point-like' structures which are reminiscent of those observed in the quiet Sun. If the imposed field is slightly stronger, there is a tendency for magnetic flux to become concentrated into `ribbon-like' structures which are comparable to those observed in solar plages. The dependence of these simulations upon the strength of the imposed magnetic field is analysed in detail, and the concept of the fractal dimension is used to make a further, more quantitative comparison between these simulations and photospheric observations.

- S. M. Houghton
*Localised solutions in Magnetoconvection*October 2005. PhD dissertation, Summary. If you would like further details, or a copy of the dissertation then please contact me.

Within the dark central umbra of sunspots we can observe localised bright convective features, called umbral dots. It is believed that sunspots, and therefore umbral dots, are formed by the interactions between convection and strong magnetic fields. Numerically localised convective features have been observed in two-dimensional, and highly truncated three-dimensional, simulations of convection in the presence of an imposed magnetic field.

The focus of this dissertation is to consider convection in the presence of a strong magnetic field, as present in the central umbra of a sunspot, to gain an understanding of the formation and structure of localised convective patterns.

We begin with a brief survey of solar observational results and previous results from mathematical modelling of the system. We then reconsider the two-dimensional problem through the formation of discrete models that allow very large systems to be considered. Results from the discrete models agree well with results from the full system. In Chapter 3, the numerical methods used to model magnetoconvection are described and full details of the mathematical problem are given. A study of compressible convection in the presence of a strong imposed magnetic field, in three dimensions is then completed and the results presented. To allow large domains to be investigated a Swift-Hohenberg type model for Boussinesq magnetoconvection is developed in Chapter 5. The thermal conductivity is allowed to vary with depth. This vertical variation breaks the up-down symmetry of Boussinesq convection and allows hexagonal patterns to form.

It is found that while localised convective regions can be found in a reasonably generic way for the two-dimensional problem, they typically appear only as transients in the three- dimensional problem. Instead we find relatively weak convection patterns whose amplitude varies on a long length-scale.

**In preparation/submitted**

These are papers I am working on, or that have been submitted to a journal for publication.

- S. M. Houghton, S. M. Tobias, E. Knobloch & M. R. E. Proctor
*Boundary effects and pattern interactions for the Kuramoto-Sivashinsky equation with drift in finite domains*. - S. M. Houghton & M. R. E. Proctor
*Discrete models for localised states in magnetoconvection*.

**Other papers**

These are the papers where I am not on the author list, but am thanked for my contributions.

- J. Knobloch & T. Wagenknecht
*Snaking of multiple homoclinic orbits in reversible systems*Accepted for publication in SIADS, August 2008. - L. J. Silvers & M. R. E. Proctor
*The Interaction Of Multiple Convection Zones In A-type Stars*Monthly Notices of the Royal Astronomical Society, Volume 380 (2007) Issue 1, pg 44-50. - Paul J. Bushby
*Super-equipartition fields in simulations of photospheric magnetoconvection*Proceedings of the International Astronomical Union (2006), Volume 2 pg 514-516. - M. R. E. Proctor
*An extension of the Toroidal Theorem*Geophysical & Astrophysical Fluid Dynamics, Volume 98 (2004), Issue 3, pg 235-240.