# Ice Dynamics Part 1: Idealized Experiments

# Ice Dynamics, Part 1: idealized experiments

In this notebook we are going to explore the dynamics of a blob of ice under various forcing conditions and how to use the basic functionalities of OGGM flowline model(s). For this purpose we are going to use simple and “idealized” glaciers models. We will consider the slope of the glacier’s bed, the temperature of the ice, and its basal friction.

This notebook has been adapted from https://edu.oggm.org/en/latest/notebooks_ice_flow_parameters.html and https://edu.oggm.org/en/latest/notebooks_flowline_intro.html

You can find and run the notebook from https://www.github.com/skachuck/clasp474_w2021.

```
# import oggm-edu helper package
import oggm_edu as edu
```

```
# import modules, constants and set plotting defaults
import matplotlib.pyplot as plt
plt.rcParams['figure.figsize'] = (9, 6) # Default plot size
# Scientific packages
import numpy as np
# Constants
from oggm import cfg
cfg.initialize_minimal()
# OGGM models
from oggm.core.massbalance import LinearMassBalance
from oggm.core.flowline import FluxBasedModel, RectangularBedFlowline, TrapezoidalBedFlowline, ParabolicBedFlowline
# There are several numerical implementations in OGGM core. We use the "FluxBasedModel"
FlowlineModel = FluxBasedModel
```

## Basics

First we set-up a simple run with a constant flat bedrock:

### Glacier bed

```
# define horizontal resolution of the model:
# nx: number of grid points
# map_dx: grid point spacing in meters
nx = 400
map_dx = 100
```

```
# define glacier top and bottom altitudes in meters
top = 3400
bottom = 3400
```

```
# create a linear bedrock profile from top to bottom
bed_h, surface_h = edu.define_linear_bed(top, bottom, nx)
```

```
# calculate the distance from the top to the bottom of the glacier in km
distance_along_glacier = edu.distance_along_glacier(nx, map_dx)
```

```
bed_h = bed_h.copy()
```

We’ll start with a roundish blob of ice.

```
surface_h[:100] = bed_h[:100] + 50*np.sqrt(100 - distance_along_glacier[:100]**2)
```

```
# plot the glacier bedrock profile and the initial glacier surface
plt.plot(distance_along_glacier, surface_h, label='Initial glacier')
edu.plot_xz_bed(x=distance_along_glacier, bed=bed_h);
```

Now we have to decide how wide our glacier is, and what the *shape* of its bed is. For a start, we will use a rectangular “u-shaped” bed (see the documentation), with a constant width of 300m:

```
initial_width = 300 #width in meters
# Now describe the widths in "grid points" for the model, based on grid point spacing map_dx
widths = np.zeros(nx) + initial_width/map_dx
# Define our bed
init_flowline = RectangularBedFlowline(surface_h=surface_h, bed_h=bed_h, widths=widths, map_dx=map_dx)
```

The init_flowline variable now contains all geometrical information needed by the model. It can give access to some attributes, which are quite useless for a non-existing glacier:

```
print('Glacier length:', init_flowline.length_m)
print('Glacier area:', init_flowline.area_km2)
print('Glacier volume:', init_flowline.volume_km3)
```

### Mass balance

Then we will need a mass balance model. In our case this will be zero, because we’re looking at how a lump of ice deforms under it’s own weight. We should, we hope, see that the volume of ice doesn’t change over time:

```
# ELA at 3000m a.s.l., gradient 4 mm m-1
ELA = 3400 #equilibrium line altitude in meters above sea level
altgrad = 0 #altitude gradient in mm/m
mb_model = LinearMassBalance(ELA, grad=altgrad)
```

### Model run

Now that we have all the ingredients to run the model, we just have to initialize it:

```
# The model requires the initial glacier bed, a mass balance model, and an initial time (the year y0)
model = FlowlineModel(init_flowline, mb_model=mb_model, y0=0.)
```

Let’s first run the model for one year:

```
runtime = 1
model.run_until(runtime)
edu.glacier_plot(x=distance_along_glacier, bed=bed_h, model=model, mb_model=mb_model, init_flowline=init_flowline)
```

What can you see? Not much? Let’s print some things out.

```
print('Year:', model.yr)
print('Glacier length (m):', model.length_m)
print('Glacier area (km2):', model.area_km2)
print('Glacier volume (km3):', model.volume_km3)
```

**Q1** How has the glacier changed since the initial state? Has the glacier’s volume actually been conserved? Explain why / why not.

We can now run the model for 150 years and see how the output looks like:

```
runtime = 150
model.run_until(runtime)
edu.glacier_plot(x=distance_along_glacier, bed=bed_h, model=model, mb_model=mb_model, init_flowline=init_flowline)
```

Let’s print out a few infos about our glacier:

```
print('Year:', model.yr)
print('Glacier length (m):', model.length_m)
print('Glacier area (km2):', model.area_km2)
print('Glacier volume (km3):', model.volume_km3)
```

Note that the model time is now 150 years. Running the model with the same input again, calls the already calculated results but does not execute the method `model.run_until`

another time, which safes computational time.

```
model.run_until(150)
print('Year:', model.yr)
print('Glacier length (m):', model.length_m)
```

If we want to compute longer, we have to set the desired date. Hereby, the model computes only the additional missing years.

```
runtime = 500
model.run_until(runtime)
edu.glacier_plot(x=distance_along_glacier, bed=bed_h, model=model, mb_model=mb_model, init_flowline=init_flowline)
```

```
print('Year:', model.yr)
print('Glacier length (m):', model.length_m)
print('Glacier area (km2):', model.area_km2)
print('Glacier volume (km3):', model.volume_km3)
```

However, it is important to note, that the model will not calculate back in time. Once calculated for 500 years, the model will not run again for 450 years and remains at 500 years. Try running the cell below. Does the output match what you expected?

```
model.run_until(450)
print('Year:', model.yr)
print('Glacier length (m):', model.length_m)
```

It might be useful to store some intermediate steps of the evolution of the glacier. We make a loop so that the model reports to us several times.

```
# Reinitialize the model
model = FlowlineModel(init_flowline, mb_model=mb_model, y0=0.)
# Year 0 to 10000 in 10 years step
yrs = np.arange(0, 10001, 10)
# Array to fill with data
nsteps = len(yrs)
length = np.zeros(nsteps)
vol = np.zeros(nsteps)
# Loop
for i, yr in enumerate(yrs):
model.run_until(yr)
length[i] = model.length_m
vol[i] = model.volume_km3
# I store the final results for later use
flat_glacier_h = model.fls[-1].surface_h
```

We can now plot the evolution of the glacier length and volume with time:

```
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
ax1.plot(yrs, length);
ax1.set_xlabel('Years')
ax1.set_ylabel('Length (m)');
ax2.plot(yrs, vol);
ax2.set_xlabel('Years')
ax2.set_ylabel('Volume (km3)');
```

**Q2** How does the glacier’s length change over time? Is it constant? linear? accelerating? decelerating? Can you explain the pattern of the glacier’s extension by referring to the rheology of ice?

**Q3** Having seen now 10,000 years of the model’s volume, how did the code do in mass conservtion?

## Adding a slope

We can now add a slope to the glacier to see how flowing downhill changes things. We’ll make the bottom 1 km lower than the top, but keep the same initial thickness profile.

```
# define glacier top and bottom altitudes in meters
top = 3400
bottom = 2400
# create a linear bedrock profile from top to bottom
bed_h_slope, surface_h_slope = edu.define_linear_bed(top, bottom, nx)
bed_h_slope = bed_h_slope.copy()
surface_h_slope[:100] = bed_h_slope[:100] + 50*np.sqrt(100 - distance_along_glacier[:100]**2)
```

```
# plot the glacier bedrock profile and the initial glacier surface
plt.plot(distance_along_glacier, surface_h_slope, label='Initial glacier')
edu.plot_xz_bed(x=distance_along_glacier, bed=bed_h_slope);
```

We’ll perform the same time loop as we did for the flat glacier, making sure to save things to nbew variables so that we can compare them. But first:

**Q4** What do you expect will happen? Will the glacier flowing downhill extend faster or slower? Why?

```
# Reinitialize the model
slope_flowline = RectangularBedFlowline(surface_h=surface_h_slope, bed_h=bed_h_slope, widths=widths, map_dx=map_dx)
model = FlowlineModel(slope_flowline, mb_model=mb_model, y0=0.)
# Year 0 to 10000 in 10 years step
yrs = np.arange(0, 10001, 10)
# Array to fill with data
nsteps = len(yrs)
length_slope = np.zeros(nsteps)
vol_slope = np.zeros(nsteps)
# Loop
for i, yr in enumerate(yrs):
model.run_until(yr)
length_slope[i] = model.length_m
vol_slope[i] = model.volume_km3
# I store the final results for later use
slope_glacier_h = model.fls[-1].surface_h
```

```
edu.glacier_plot(x=distance_along_glacier, bed=bed_h_slope, model=model, mb_model=mb_model, init_flowline=slope_flowline)
```

```
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
ax1.plot(yrs, length, label='Flat bed');
ax1.plot(yrs, length_slope, label='Sloped bed');
ax1.set_xlabel('Years')
ax1.set_ylabel('Length (m)');
ax1.legend()
ax2.plot(yrs, vol);
ax2.plot(yrs, vol_slope);
ax2.set_xlabel('Years')
ax2.set_ylabel('Volume (km3)');
```

**Q5** Revaluate your expectations. Was your hypothesis borne out by the model?

## Temperature

We start with the internal deformation which results in creeping of a glacier. To describe it we use Glens’s creep parameter. The default in OGGM is to set Glen’s creep parameter `A`

to the “standard value” defined by Cuffey and Paterson¹.

The parameter relates shear stress to the rate of deformation and is assumed to be constant. It depends on crystal size, fabric, concentration and type of impurities, as well as on ice temperature² (you can find a more detailed description of it here). In the following we will change it and see what happens.

Therefore we have to define the bedrock and a grid. (We do similiar steps like in flowline_model to generate a set-up for our experiment.)

```
# Glenn flow parameter
A_big = cfg.PARAMS['glen_a'] * 5
# Reinitialize the model
warm_flowline = RectangularBedFlowline(surface_h=surface_h_slope, bed_h=bed_h_slope, widths=widths, map_dx=map_dx)
model = FlowlineModel(warm_flowline, mb_model=mb_model, y0=0., glen_a=A_big)
# Year 0 to 10000 in 10 years step
yrs = np.arange(0, 10001, 10)
# Array to fill with data
nsteps = len(yrs)
length_warm = np.zeros(nsteps)
vol_warm = np.zeros(nsteps)
# Loop
for i, yr in enumerate(yrs):
model.run_until(yr)
length_warm[i] = model.length_m
vol_warm[i] = model.volume_km3
# I store the final results for later use
warm_glacier_h = model.fls[-1].surface_h
```

**Q6** Before plotting the result, try to hypothesize what the result will be in terms of length and thickness.

```
edu.glacier_plot(x=distance_along_glacier, bed=bed_h_slope, model=model, mb_model=mb_model, init_flowline=warm_flowline)
```

```
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
ax1.plot(yrs, length, label='Flat bed');
ax1.plot(yrs, length_slope, label='Sloped bed');
ax1.plot(yrs, length_warm, label='Warm ice');
ax1.set_xlabel('Years')
ax1.set_ylabel('Length (m)');
ax1.legend()
ax2.plot(yrs, vol);
ax2.plot(yrs, vol_slope);
ax2.plot(yrs, vol_warm);
ax2.set_xlabel('Years')
ax2.set_ylabel('Volume (km3)');
```

## Sliding

Basal sliding occurs when there is a water film between the ice and the ground. In his seminal paper, Hans Oerlemans uses a so-called “sliding parameter” (here: fs), representing basal sliding. In OGGM this parameter is set to 0 per default, but it can be modified:

```
# Sliding parameter
fs = 5.7e-20
# Reinitialize the model
slide_flowline = RectangularBedFlowline(surface_h=surface_h_slope, bed_h=bed_h_slope, widths=widths, map_dx=map_dx)
model = FlowlineModel(slide_flowline, mb_model=mb_model, y0=0., fs=fs)
# Year 0 to 10000 in 10 years step
yrs = np.arange(0, 10001, 10)
# Array to fill with data
nsteps = len(yrs)
length_slide = np.zeros(nsteps)
vol_slide = np.zeros(nsteps)
# Loop
for i, yr in enumerate(yrs):
model.run_until(yr)
length_slide[i] = model.length_m
vol_slide[i] = model.volume_km3
# I store the final results for later use
slide_glacier_h = model.fls[-1].surface_h
```

**Q7** Before plotting the result, try to hypothesize what the result will be in terms of length and thickness.

```
edu.glacier_plot(x=distance_along_glacier, bed=bed_h_slope, model=model, mb_model=mb_model, init_flowline=slope_flowline)
```

```
f, (ax1, ax2) = plt.subplots(1, 2, figsize=(14, 5))
ax1.plot(yrs, length, label='Flat bed');
ax1.plot(yrs, length_slope, label='Sloped bed');
ax1.plot(yrs, length_warm, label='Warm ice');
ax1.plot(yrs, length_slide, label='Sliding bed');
ax1.set_xlabel('Years')
ax1.set_ylabel('Length (m)');
ax1.legend()
ax2.plot(yrs, vol);
ax2.plot(yrs, vol_slope);
ax2.plot(yrs, vol_warm);
ax2.plot(yrs, vol_slide);
ax2.set_xlabel('Years')
ax2.set_ylabel('Volume (km3)');
```

We can, finally plot all these glacier thickness together to see how the thickness profiles depend on things like slope of bedrock, temperature of ice, and sliding at the bed.

```
# Plot the final result:
plt.plot(distance_along_glacier, flat_glacier_h-bed_h, label='Flat glacier')
plt.plot(distance_along_glacier, slope_glacier_h-bed_h_slope, label='Sloped glacier')
plt.plot(distance_along_glacier, warm_glacier_h-bed_h_slope, label='Sloped+warm glacier')
plt.plot(distance_along_glacier, slide_glacier_h-bed_h_slope, label='Sloped+sliding glacier')
plt.legend()
plt.title('Glacier Thicknesses after 10,000 years');
```

**Q8** Describe how the thicknesses with different temperatures and basal friction differ from each other visually.

## Play and Explore

Now play around with things and make some predictions. Definitely copy and paste code, but make sure to save variables under different names so you can plot them together.

Go back into the previous loops and save the maximum ice thickness (which is accessible via `model.thick_stag[0].max()`

) making lists by initializing them before the loop with `maxthk_case = np.zeros(nsteps)`

(where you replace `case`

with things like `flat`

, `slope`

, etc) and then filling them each timestep with `maxthk_case[i] = model.thick_stag[0].max()`

, just like with the length and the volume. Plot maximum thickness against glacier length. What do you expect the relationship will be? How do you expect this relationship is affected by slope, temperature, and sliding?

**Q9** Record your expectations and findings for at least one of these experiments.

```
```

```
```

```
```

```
```

```
```

```
```

```
```

```
```

```
```

```
```

```
```

```
```

```
```

## Unfinished Module: Stress / Strain

OGGM computes the velocities, which can be accessed in `model.u_stag[0]`

(`[0]`

because the velocity is a vector with 2 components in this 2D model,

```
plt.plot(distance_along_glacier, model.u_stag[0][1:])
```

```
plt.plot(distance_along_glacier, np.gradient(model.u_stag[0], map_dx)[1:]);
plt.ylim([0,1e-12]);
```

```
plt.plot(distance_along_glacier, np.gradient(slope_glacier_h, map_dx));
```

```
tana = np.gradient(slope_glacier_h, map_dx)
tau = cfg.PARAMS['ice_density']*(slope_glacier_h-bed_h_slope)*tana*9.8
plt.plot(distance_along_glacier, tau, label='Sloped glacier')
```

```
plt.plot(distance_along_glacier, cfg.PARAMS['glen_a']*tau**3, label='Sloped glacier')
```

```
cfg.PARAMS
```

```
(top - bottom)/(nx*map_dx)
```

## References

¹ Cuffey, K., and W. S. B. Paterson (2010). The Physics of Glaciers, Butterworth‐Heinemann, Oxford, U.K.

² Oerlemans, J. (2001). *Glaciers and climate change.* CRC Press. (p. 59)