# Higher Order Physics

## Pattyn-Bocek Diagnostic Model

We have integrated into CISM an improved version of Frank Pattyn's incomplete 2nd order model (Pattyn 2003). This model modifies the full Stokes equations by neglecting vertical resistive stress (Pattyn). This assumption that the pressure at any point in the ice is due only to the weight of the ice above it and not due to resistance to motion is called the hydrostatic approximation (cite?). The model also assumes that the vertical gradients in the horizontal velocity field are much greater in magnitude than the horizontal gradients in the vertical velocity field (Pattyn). These assumptions allow the model to be simplified by solving for only two rather than three components of the velocity vector field.

### Stress Field

Main Article: Stress Field Equations for Pattyn 2003 Model

### Boundary Conditions

Main Article: Boundary Conditions for Pattyn 2003 Model

### Software Integration

Main Article: Integration of Pattyn 2003 Model

## Payne-Price Diagnostic Model

### Stress Field

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### Boundary Conditions

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## Prognostic Model

An experimental prognostic scheme using the existing thickness evolution model from Glimmer has been developed. The scheme, similar to the one that Pattyn describes, simply converts the higher-order velocities to vector diffusivities and passes them to the thickness evolution routine. However, it has not been extensively tested on prognostic experiments, and is likely to lack stability.

Alternative prognostic schemes have been suggested, such as the use of higher-order velocities as a sliding law for shallow ice (Bueler) or an incremental remapping scheme (Lipscomb) that has been developed in a stand-alone model and is awaiting integration with CISM.

## References

• Hooke, R, Principles of Glacier Mechanics. Upper Saddle River, NJ: Prentice Hall, 1998.
• MacAyeal, et. al., "An ice-shelf model test based on the Ross Ice Shelf, Antarctica", Annals of Glaciology, vol. 23, 1996.
• Pattyn, F, "A new three-dimensional higher-order thermomechanical ice sheet model: Basic sensitivity, ice stream development, and ice flow across subglacial lakes", Journal of Geophysical Research, Vol. 108, No. B8, 2003.
• Versteed, H and Malalasekera, W, An introduction to Computational Fluid Dynamics: The Finite Volume Method, Edinburgh Gate, Essex, England: Pearson Education, 1995.