## 1. Introduction

[2] Ice streams are a conspicuous patterning in ice-sheets, both in the surface topography and the surface velocity [*Bamber et al.*, 2001]. Their significance is due to their discharging a very substantial proportion of the accumulation of an ice-sheet, and exhibiting flow variability that possibly extends in scale up to the very large surges that played major roles in operation of the climate system (e.g., Heinrich Events [*MacAyeal*, 1993]). Consequently, the successful computation of ice-stream location and flow is one of the key goals of theoretical glaciology. However, even computing ice-stream location has been problematical up to now, with different models under the same forcing predicting different stream geometries [*Payne et al.*, 2000]. We consider the use of a more complex mechanical model, and demonstrate that this step allows consistent prediction of ice stream location.

[3] While many ice-streams are clearly associated with topographic lows in the bedrock (e.g., Pine Island Glacier [*Vaughan et al.*, 2006]), evidence from former ice-sheets, in particular the existence of cross-cutting lineations [*Clark*, 1993] shows that basal topography is not the only control on ice-stream location. The existence of thermo-viscous instabilities [e.g., *Payne*, 1995] in plane flow and related “hydraulic runaway” mechanisms [e.g., *Sayag and Tziperman*, 2008] has led to the suggestion and numerical demonstration that ice-streams might be generated by a map-plane fingering instability caused by coupling of the ice flow and the thermal field [*Fowler and Johnson*, 1996; *Payne and Dongelmans*, 1997]. Using the shallow-ice approximation (SIA), the latter authors presented calculations generated by a time-dependent numerical model that solved the flow and heat equations with a temperature-dependent ice viscosity, showing that certain parameter combinations (principally surface temperature and accumulation rate) generated ice-stream-like features. They argued that a fingering mechanism operated such that if a warmer area propagated upstream locally, it would draw flow in which generated localized heating, leading to less viscous ice, increased speeds, drawdown and further channelling of ice - an ice-flux capture mechanism. However, an intercomparison experiment between models from ten different groups [*Payne et al.*, 2000] although exhibiting instabilities, failed to generate comparable patterning, with considerable variation in detail shown in particular at short wavelengths. This raised the issue of whether the computed stream generation was a numerical artefact.

[4] In an effort to answer this question, *Hindmarsh* [2004] examined the linearized instability of the shallow ice approximation (SIA) in thermo-viscous calculations, and showed that its use was ill-posed at short wavelengths. *Hindmarsh* [2006a] showed that using mechanical models which incorporated horizontal stress gradients damped instabilities at short wavelengths, removing the ill-posing. He also showed that use of the membrane stress approximation (MSA) (i.e., the three-dimensional version of the longitudinal stress approximation, which includes horizontal stress gradients) [*Blatter*, 1995; *MacAyeal*, 1989; *Hindmarsh*, 2006b] was as accurate as use of the full Stokes equations for this application. *Sayag and Tziperman* [2008] found similar results using a related but different mechanism for generating instabilities. Previous workers [e.g., *Hulbe and MacAyeal*, 1999; *Marshall and Clarke*, 1997] have used membrane-stress approximations in thermo-mechanically coupled calculations, but not to address the issue of spontaneous ice-stream genesis.

[5] Using the SIA, *Hindmarsh* [2006a] presented some finite-amplitude (i.e., fully non-linear) calculations which showed that the patterning of basal temperature depended strongly on the discretization method used, arguing that this was a symptom of the ill-posedness of the SIA. *Bueler et al.* [2007] have argued against this, suggesting that the problem lay in flaws in the construction of the numerical schemes. Some support for this idea comes from the work of *Saito et al.* [2006], who found that the use of higher-order stress models did not improve calculations of streams.

[6] Solving a well-posed system is essential to the success of numerical methods, so if we are to accept the reasoning of *Hindmarsh* [2006a], the implication is that incorporating horizontal stress gradients in thermo-mechanically coupled computations of ice-sheet evolution should produce results which, in the broad patterning at least, are independent of the grid-size and discretization method used. This paper examines this issue, adopting the simplest possible MSA due to *MacAyeal* [1989]. A set of parameters which generate steady streams using the SIA are used, and results are shown to depend on the grid-size used under conditions of grid refinement. The same set of parameters are then used to generate instabilities with the MSA, and here the results do not depend substantially on the grid used; moreover, the streams are realistically sized.