| Summary: | Most dynamo models agree that large-scale, weak poloidal magnetic field is wound up into strong toroidal field through differential rotation. Therefore, the strong radial gradients of angular velocity present in the solar tachocline are extremely likely to play a crucial role in the solar dynamo. Motivated by this, we focus on the instability mechanism thought to be responsible for the break up of a horizontal field in a stably stratified layer, namely magnetic buoyancy instability, and the effect a velocity shear has upon said instability. To study this interaction, we derive a new set of equations incorporating velocity shear and magnetic buoyancy into the Boussinesq approximation. These equations not only provide us with the ability to study the effects of a velocity shear on magnetic buoyancy instability, but also allow us to study magnetic buoyancy instability in the presence of a magnetic field varying on a short O(d) length scale, compared to the equations of Spiegel & Weiss (1982), which are restricted to field variations on a longer O(Hp) scale. Stability criteria for this new system is obtained through a linear analysis on the ideal (diffusionless) system. Motivated by the work of Mizerski et al. (2013) we use the newly derived equations to study the short-wavelength linear magnetic buoyancy instability. We first study this problem in the absence of a velocity shear and dissipation, deriving asymptotic results analogous to Mizerski et al. The governing set of equations are then solved numerically to verify the asymptotic results. A velocity shear is then added into the analysis; we derive new asymptotic results and use them to comment on the influence the velocity shear has on the instability. Upholding the short-wavelength limit, we individually introduce each diffusive parameter into the analysis, observing the role each has on the instability. Finally, we solve the newly derived system of linearised equations numerically, whilst including all diffusive parameters. We comment on the role of the diffusive parameters and investigate how they influence the instability.
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