# Presentation: 092459

## When iceberg calving matters:  A cautionary tale

**J. N. Bassis**
**Institute of Geophysics and Planetary Physics, Scripps Institution of Oceanography**
**Department of the Geosciences, University of Chicago**

**Project supported by NSF, Thanks to Julie Palais**

## Overview

- What we want:  Formulate a calving law that can be implemented into large scale numerical models
- Method of approach:  Try to implement a calving “law” into an ice sheet/shelf model and see what happens
- Key lesson:  Most calving laws result in unstable ice shelves
-  Original intent was to focus on what happens to grounded ice
-  But very concerned that results easily contaminated by “bad” calving law
- Working hypothesis:  Ice shelves are unstable once they become highly fractured: generalize calving law hypothesis proposed by Benn et al., [2006]
- Present preliminary results:  Implement calving law into numerical model to study the behavior

## Calving Laws

- Traditional approach:
- Correlate everything in sight with calving rate:

- Calving Rate = Terminus Velocity – Rate of Terminus Advance

## Calving Laws

- Traditional approach
- Correlate everything in sight with calving rate:

- Calving Rate = Terminus Velocity – Rate of Terminus Advance

- Taylor Expand

## Calving Laws

- Traditional approach
- Correlate everything in sight with calving rate:

- Calving Rate = Terminus Velocity – Rate of Terminus Advance

- Zeroth order calving law:
*V**c* ≈ *V**term*

- Taylor Expand

## Calving Laws

- Traditional approach
- Correlate everything in sight with calving rate:

- Calving Rate = Terminus Velocity – Rate of Terminus Advance

- Taylor Expand

- Zeroth order calving law:
*V**c* ≈ *V**term*

- Write in terms of rate of terminus advance

## Calving Laws

- Take home points:
-   The rate of terminus advance contains more physics than the calving rate (by itself)
-   Need to look at calving glaciers/ice shelves that _aren’t_ in steady-state
-   Calving law may be sensitive to many parameters

## Calving Laws

- Take home points:
-   The rate of terminus advance contains more physics than the calving rate (by itself)
-   Need to look at calving glaciers/ice shelves that _aren’t_ in steady-state
-   Calving law may be sensitive to many parameters

- Need to look for scaling laws – i.e. formulate in terms of non-dimensional criterion

## A simple calving rule

- Assume ice shelf becomes unstable when ratio of crevasse penetration depth to ice thickness exceeds a critical ratio, *r**c*

## A simple calving rule

- Assume ice shelf becomes unstable when ratio of crevasse penetration depth to ice thickness exceeds a critical ratio, *r**c*

- Calculate crevasse depth assuming viscous flow

## A simple calving rule

- Assume ice shelf becomes unstable when ratio of crevasse penetration depth to ice thickness exceeds a critical ratio, *r**c*

- Generalize slightly:  Iceberg of size L detaches when integral of rc from ice front inwards normalized by L exceeds a critical value
- Future work:  Assume Poisson process with probability determined by the integral

- Calculate crevasse depth assuming viscous flow

- L

## A simple calving rule

- Assume ice shelf becomes unstable when ratio of crevasse penetration depth to ice thickness exceeds a critical ratio, *r**c*

- Generalize slightly:  Iceberg of size L detaches when integral of rc from ice front inwards normalized by L exceeds a critical value
- Future work:  Assume Poisson process with probability determined by the integral

- Calculate crevasse depth assuming viscous flow

- L

- Equivalent to applying a critical strain rate that varies between ice shelves

## A simple calving rule

- Assume ice shelf becomes unstable when ratio of crevasse penetration depth to ice thickness exceeds a critical ratio, *r**c*

- Calculate crevasse depth assuming viscous flow

- L

- Equivalent to applying a critical strain rate that varies between ice shelves

- Model depends on three parameters:
- 	(i) rcrit,
- 	(ii) meltwater,
- 	(iii) time between calving events

## A simple test case

- Ice shelf spreading in one direction with no lateral shear:

- Two possibilities:
- Grow without limit
- No stable ice shelf possible

## A simple test case

- Ice shelf spreading in one direction with no lateral shear:

- Two possibilities:
- Grow without limit
- No stable ice shelf possible

- Not very interesting!

## A more complicated test case

- Assume ice shelf is in channel with friction along the walls

## A more complicated test case

- Assume ice shelf is in channel with friction along the walls

**Growth phase with no iceberg calving**

## A more complicated test case

- Assume ice shelf is in channel with friction along the walls

**Growth phase with no iceberg calving**

**Steady calving front with constant calving rate**

## A more complicated test case

- Assume ice shelf is in channel with friction along the walls

**Growth phase with no iceberg calving**

**Steady calving front with constant calving rate**

- Position of calving front determined by critical ratio dc and input flux

## A simulated embayment

- Combine previous two experiments:  assume friction between 0 and 50 km and zero friction beyond 50 km

## A simulated embayment

- Combine previous two experiments:  assume friction between 0 and 50 km and zero friction beyond 50 km

- Embayment

## A simulated embayment

- Embayment

**Growth phase with no iceberg calving**

- Combine previous two experiments:  assume friction between 0 and 50 km and zero friction beyond 50 km

## A simulated embayment

- Embayment

**Growth phase with no iceberg calving**

**“****Sawtooth” oscillations in calving front**

- Combine previous two experiments:  assume friction between 0 and 50 km and zero friction beyond 50 km

## A simulated embayment

## A simulated finite embayment

- Amplitude and period of oscillations depend on:
- 	1) Critical ratio, rc
- 	2) Time between calving events (i.e. depends on model 		time step)
- 	3) Flux into the ice shelf

- Combine previous two experiments:  assume friction between 0 and 50 km and zero friction beyond 50 km

## A simulated finite embayment

- Amplitude and period of oscillations depend on:
- 	1) Critical ratio, rc
- 	2) Time between calving events (i.e. model time step)
- 	3) Flux into the ice shelf

- We are implicitly assuming calving events are time-predictable

- Combine previous two experiments:  assume friction between 0 and 50 km and zero friction beyond 50 km

## A simulated finite embayment

- Amplitude and period of oscillations depend on:
- 	1) Critical ratio, rc
- 	2) Time between calving events (i.e. model time step)
- 	3) Flux into the ice shelf

- We are implicitly assuming calving events are time-predictable

- More trouble: If we truncated the ice shelf at grid points, we would generate irregular oscillations

- Combine previous two experiments:  assume friction between 0 and 50 km and zero friction beyond 50 km

## Influence of inflow velocity

**channel**

- Long channel:  Ice front position increases with inflow velocity

## Influence of inflow velocity

**channel **

**embayment**

- Long channel:  Ice front position increases with inflow velocity

- With embayment: Terminus position remains constant with increasing inflow velocity

**Length of Embayment**

## Influence of inflow velocity

**channel**

**embayment**

- Long channel:  Ice front position increases with inflow velocity

- With embayment: Terminus position remains constant with increasing inflow velocity

**Length of Embayment**

- Suggests:  Embayment geometry has strong control on calving and position of calving front

## Effect of meltwater on stability

- Assume crevasses are partially filled with fresh water

## Effect of meltwater on stability

- Assume crevasses are partially filled with fresh water

- Increasing meltwater results in retreat of ice shelves

## Effect of meltwater on stability

**No stable ice shelf possible**

- Assume crevasses are partially filled with fresh water

- Increasing meltwater results in retreat of ice shelves
- Beyond a critical amount of meltwater no stable ice shelf configurations

## Effect of meltwater on stability

**Ice shelf collapse**

**Add basal melting of 0-3 m/year**

- Assume crevasses are partially filled with fresh water

- Increasing meltwater results in retreat of ice shelves
- Beyond a critical amount of meltwater no stable ice shelf configurations
- Region of instability grows with basal/surface melting

## Conclusions

- The good:  Implementation of a calving law qualitatively predicts stable ice fronts, cyclic calving and ice shelf disintegration
- The bad:  Law depends on combination of numerical parameters and variables that are not measurable in the lab
- The ugly: Fun and games with models isn’t good enough – need to compare with real data
- The hopeful: Can be easily generalized to two dimensions (integrate over area) and to include random component
- The slightly unrealistic but still possible:  Seismicity may provide estimate of crevasse density/critical penetration depth

## Why is iceberg calving a hard problem?

**Grain size/Laboratory sample**
**<10 cm**

**Micro-scale**

**Meso-scale**

**Meso-fracture and rift scale**
**100 – 1000 m**

**Macro-scale**

**Ice shelf/Glacier scale**
**> 10, 000 m**

**Think:  Individual fault strands**

**Think:  Large scale tectonics**

## Why is iceberg calving a hard problem?

**Grain size/Laboratory sample**
**<10 cm**

**Micro-scale**

**Meso-scale**

**Meso-fracture and rift scale**
**100 – 1000 m**

**Macro-scale**

**Ice shelf/Glacier scale**
**> 10, 000 m**

**Think:  Individual fault strands**

**Think:  Large scale tectonics**

## Why is iceberg calving a hard problem?

**Grain size/Laboratory sample**
**<10 cm**

**Micro-scale**

**Meso-scale**

**Meso-fracture and rift scale**
**100 – 1000 m**

**Macro-scale**

**Ice shelf/Glacier scale**
**> 10, 000 m**

- Physics of fracture is complicated and poorly understood
- Numerically very expensive to compute

## Why is iceberg calving a hard problem?

**Grain size/Laboratory sample**
**<10 cm**

**Micro-scale**

**Meso-scale**

**Meso-fracture and rift scale**
**100 – 1000 m**

**Macro-scale**

- Large-scale stability of calving glaciers

**Ice shelf/Glacier scale**
**> 10, 000 m**

- Physics of fracture is complicated and poorly understood
- Numerically very expensive to compute

## The model:  a cheap trick

## The model:  a cheap trick

- “kernel” or “shape” function

## The model:  a cheap trick

- “kernel” or “shape” function

- Depth integrate

## The model:  a cheap trick

- “kernel” or “shape” function

- Depth integrate

## The model:  a cheap trick

- “kernel” or “shape” function

- Depth integrate

- Reproduces asymptotic approximations for ice sheets (SIA), ice streams and ice shelves
- Ice sheet, ice stream, ice shelf flow regimes triggered by basal shear stress through *ε**(x) *term

## A brief word about the model

- The model is more complicated than it needs to be to study ice shelves
- Flowline model
- Includes longitudinal stresses and vertical shear (when basal friction is present)
- Use trick to get vertical shear
- Write velocity in terms of shape function

- Choose shape function so that it reduces to ice sheet and ice shelf cases under appropriate basal stress conditions

## A brief word about the model

- For this study:

- Solve pseudo-spectrally (i.e. solution is sum of Chebyshev polynomials)
- Use shock following coordinates near the calving front
- Allow calving from anywhere in the interior of model (not just grid points)