# Introduction to Ice Sheet Modeling

## Contents |

## Introduction

Ice sheets are key components of Earth's climate system. They contain nearly all of the planet's fresh water, changes in their volume have an immediate effect on sea level, changes in their area and surface characteristics affect global albedo, and they play a role in the circulation of both the atmosphere and the ocean. Through the latter half of the Quaternary, ice sheets have modulated the planetary response to orbitally-driven insolation cycles. Looking forward, the Greenland and Antarctic ice sheets have the potential to play important roles in climate change.

A numerical model is a discrete approximation of a continuous process. The approximation is discrete due to the finite nature of a computer's precision. The underlying process is continuous because it is commonly formulated in terms of partial or ordinary differential equations. Numerical models can not be solved until the boundary conditions and initial conditions are specified. Such models may be very simple such as a Wikipedia:Harmonic oscillator, or very complex, as in a Wikipedia:Global climate model.

The model we will use is called CISM. It has its basis in a set of programs and climate "drivers" built over many years by a research group at the University of Bristol, led by Tony Payne. Historically, the effort was called Glimmer. More recently, a collaborative project has extended the effort in many novel and exciting ways. This is called the Community Ice Sheet Model, or CISM.

## Conservation equations

The mathematical descriptions of physical systems we encounter in Earth science problems begin from statements requiring the conservation of energy, momentum, or mass.

### Integral form

The mathematical formulation of conservation can be arrived at by considering the change in a quantity that is known in a **control** volume . The Wikipedia:control volume is enclosed by a surface , with outward positive unit vector normal to .

The value of within may change over time if

- there is a flux through . The flux is partitioned into two parts, one due to Wikipedia:diffusion and another due to Wikipedia:advection.
- Creation or destruction of within .

Formally, time rate of change in within is written:

where represents the flux due to diffusion (), represents velocity field *advecting* , represents a source (or sink) of . Vector quantities are represented in **boldface**. The negative signs in front of the first two terms on the right-hand side indicate that an outward flux results in decrease of in the volume enclosed by .

This statement of conservation of in the unit volume is always true, independent of the size of and even if the fields enclosed by are not continuous. This is the case because we integrate over . It is important to note that that information on spatial scales smaller than is lost in the integration.

### Derivative form

Numerical models are often easier to formulate from the derivative form of the conservation equation. This requires the derivatives of to exist within . This requirement allows the integral form of the conservation equation to be written as partial differential equations which are upheld with in the control volume.

Begin with the terms describing diffusive and advective fluxes into or out of the control volume. The Wikipedia:divergence theorem states that

Or, adapting the notation that the subscript indicates a single component of a vector, and that repeated subscript indices in a single term are to be summed,

Using the divergence theorem, the surface integrals over fluxes may be replaced,

Assuming that the coordinate system is stationary with respect to the velocity field (Eularian reference frame), it is possible to write

The end result is that the integral form of the conservation equation can now be written

Because is an arbitrary volume, this equation can only be true if the term in brackets is zero for the volume. Hence, for any volume having continuously differentiable ,

This is the general form for all conservation laws in continuum mechanics.

### Applications of the conservation equation

Having established a generalized conservation law, it is now applied to the three quantities which are conserved in an ice sheet model; mass, energy, and momentum.

#### Conservation of mass

In this case is the mass , or more conveniently , the integral of the density over the volume. Assuming that there are no sources or sinks of mass in the volume ( = 0), the conservation equation is written

Ice is incompressibile, meaning that the density does not change in time, and the equation for local mass continuity is

Applying the operator in a cartesian coordinates produces

To make use of this statement, we need to integrate

from the base to the upper surface of the ice mass. The integral of is simply the difference between the vertical component of the velocity at the upper and lower surfaces, so

Changing the order of integration using Leibnitz rule

The vertical velocity at the upper surface is the result of motion down the surface slope, the rate of new accumulation and any time-change in surface height

recognizing that a negative accumulation rate indicates ablation. Similarly, the vertical velocity at the lower surface is

in which represents the basal accumulation rate.

Substituting equations we find that many terms cancel

Finally, making the simplification we have

The vertically-integrated form

in which represents the vertically averaged velocity, i.e. . This equation is prognostic. We use the current geometry of the ice to compute a future time-change in that geometry.

#### Conservation of energy

The first law of thermodynamics is used to make a basic statement of conservation of energy in a volume of ice V within a surface S is

in which represents the energy of the volume, is a flux due to diffusion, and represents any sources or sinks of energy within the volume. The term is a flux through due to advection. Following the steps laid out earlier, we use the divergence theorem and the assumptions of continuous fields and incompressibility, such that

Our goal is to use the first law of thermodynamics in order to compute the temperature of the ice and any change it may undergo over time.

The energy is the product of density and the specific internal energy of the ice , which is itself the product of the specific heat capacity and temperature because there is no transfer between internal energy and pressure for an incompressible fluid. Thus,

The flux due to diffusion follows Fourier's "law" for heat conduction so

in which represents the thermal diffusivity of ice and we assume gradients in its magnitude to be negligible.

Using progress made above and assuming that is small with respect to other terms, we can write the advection term

Two quantities must be considered as energy sources, the work done on the system by internal deformation and the latent heat associated with phase changes. The former is the product of strain rate and the deviatoric stress . The latter is the product of the latent heat of fusion and the amount of material subject to melting (freezing) per unit volume per unit time, .

At last, we are able to write equation \ref{equation:enbal2} in terms of temperature

It is often the case that horizontal terms and are small enough to be ignored.

#### Conservation of momentum

Starting from Newton's second law of motion, conservation of momentum is

where represents time, represents density, represents velocity, represents the stress tensor, represents the acceleration due to gravity, represents the volume of an arbitrary fluid element, and in a cartesian coordinate system. Equation \ref{equation:mobal1} tells us that a fluid element of arbitrary size experiences a "body force" due to gravity and a force due to the surrounding fluid.

Making the assumptions that we have continuous fields and that ice is incompressible (that is, its density does not change under conditions of interest to us), we can write

in which is a material derivative. Due to the fact that Froude number Wikipedia:Froude number for ice flow is extremely small, the acceleration term (the first term on the left hand side) could be neglected and we arrive to a steady-state form.

We are left with the very simple statement that the gravitational driving force is balanced by forces resulting from the stresses .

The stress tensor has nine components in our three dimensional cartesian coordinate system

The components along the diagonal are called normal stresses and the off-diagonal components are called shear stresses. Deformation results not from the full stress but from the deviatoric stress

in which is the Kroneker delta.

### Constitutive relationship

Strain rates are related to the stress tensor by the generalized Glen flow law

in which is the absolute temperature corrected for the pressure dependence of the melt temperature, is the second invariant of the stress tensor and the exponent is 3. The rate factor follows the Arrhenius relationship

in which is a constant, represents the activation energy for crystal creep, is the gas constant, and is a tuning parameter used to account for the effects of impurities and anisotropic ice fabrics. The homologous temperature is

in which is 9.8 10 K Pa, about 8.7 10 K m. The pressure-dependent melt temperature is simply the triple point temperature less the product .

## Numerical solutions of field equations

3D, thermo-mechanically coupled ice sheet models. 3D refers to the explicit vertical layering of the model for computing temperature. Thermo-mechanical means that the ice viscosity is sensitive to temperature, and an iterative procedure must be used to find the flow rates from temperature. Both models exploit the often used shallow ice approximation \citep{hutter83}, which accounts for the membrane like nature of ice sheets by reducing the stress tensor to only leading order terms resulting from simple shear. The shallow ice approximation, when combined with the non-linear constituative relation given by Glen's flow law \citep{paterson94}, leads to the following expression for horizontal velocities

All parameters and symbols used in this paper appear in table \ref{symbols}. It is worth noting that all quantities used to find the horizontal velocities are computed locally. The shallow ice approximation eliminates terms resulting from transverse or lateral stresses. The temperature sensitivity of ice flow is given by an Arrhenius relation,

Which, in typical ice temperature ranges (-50 -- 0 C) varies over 3 orders of magnitude. However because most shear occurs at the base, and the base is warmed by dissipation and geothermal heat flow, a more appropriate range is -20 -- 0 C. This gives a range of over about a factor of 30.

Assuming that horizontal diffusion is negligible, again due to the membrane nature or very small aspect ratio (ratio of vertical to lateral extent) of ice sheets, the ice temperature field is found from the conservation of energy

The terms on the right hand side from left to right are vertical diffusion, horizontal advection, vertical advection, and dissipation (using only the shallow ice stress tensor). Equation \ref{temp} is subject to the boundary conditions

The upper surface is set to a mean annual temperature (), and the lower surface is accounting for the heat sources from both geothermal heat () flux, and frictional heat generated when ice slides over the bed. The temperatures are constrained by the melting point corrected for pressure via the Clausius--Clapeyron gradient

The vertical advection term in equation \ref{temp} requires vertical velocities. They are found from incompressibility,

by integrating with respect to , giving

Were the complete accounting must include the melt rate and bed topography. Basal melt rates computed from the jump boundary condition at the bed,

Having solved for the temperature dependent velocity fields, changes in the ice sheet's geometry are computed from the continuity equation

Both models use the finite difference methods for descritization of the partial differential equations. GLIMMER differs from PISM in that it offers a choice of implicit schemes for solving equation \ref{continuity}, whereas PISM uses an explicit scheme \citep{press92}. Further, GLIMMER utilizes a rescaled, or vertical coordinate \citep{lliboutry87} and PISM does not. PISM uses the PETSc \citep{petsc-user-ref} library to achieve parallelism and has the capacity to include a more complete stress formulation, but that capacity was not used here. Additional discussion of the field equations and numerical methods used in GLIMMER can be found in \citet{payne97} and \citet{payne99}.

The ice sheet model CISM uses a finite difference method to solve the governing thermodynamic equations for ice using the {\it shallow ice approximation}. This is the approach generally adopted for modeling large ice masses. The assumption is made that slopes at the upper and lower surfaces are sufficiently small that normal stress components can be neglected. This leads to a {\it local} balance between the gravitational driving stress and the basal shear stress and expressions for the shear stresses

Evolution of the ice thickness uses equation \ref{equation:mabalfinc} and the temperature solver uses a version of equation \ref{equation:enbalfin} simplified to neglect horizontal diffusion (a typical simplification).

The model equations are solved on a regular grid using the Glen flow law (equation \ref{equation:Glen}) and appropriate boundary conditions for the upper and lower surfaces. These include the surface ice accumulation rate and temperature and a geothermal gradient (applied at the base of a bedrock layer with specified thermal properties). Basal traction may also be specified, in the situation where ice is at the melt temperature at the base. Isostatic adjustment of the land surface beneath the ice sheet, not discussed here, is also included.

### numerical scheme

The continuous functions represented by the model governing equations cannot be solved exactly. Instead, they are discretized so that finite approximations of their solutions may be made. There are a variety of numerical techniques available for this purpose, GLIMMER makes use of a finite difference method.

In brief, the model domain (a region of Earth's surface, for example, Greenland) is subdivided into a regularly-spaced horizontal grid and derivatives are approximated along the grid directions. The grid is fixed in space over the course of the model run. Model variables such as ice thickness are updated at each time step according to the numerical approximations of the governing equations. The vertical dimension is treated using a non-dimensional "stretch" coordinate so that an evolving ice thickness may be accommodated. The scaling is:

so that at the surface and at the base. The governing equations must be re-written in the new, coordinate system.

If you would like to read more about the inner workings of GLIMMER, its documentation is available at the class website. This is not necessary for the present lab exercise.

### ISIS

The Interactive System for Ice Sheet Modeling (ISIS) is a user interface to GLIMMER. It is still in development, as part of an {\it International Polar Year} collaboration among groups at the University of Montana, Portland State, UC Santa Cruz, Auburn, and the University of Texas at El Paso. We will use ISIS as a means to set up and run experiments involving the Greenland Ice Sheet. Most of our analysis of model output will be done using {\scshape Matlab} scripts.

ISIS installs with a number of pre-defined model domains and experiments. We will use a few of the Greenland setups. The Greenland initialization, used to generate a "modern" steady state, from which experiments may be started, requires about two hours of run time. To save time, the results of that simulation, stored in netCDF-format files titled \lstinline{gland-ClimateEvo.2ka.nc}, \lstinline{gland-ClimateEvo.hot2.nc}, and \lstinline{gland-ClimateEvo.1ka.nc}, have been prepared for you and will be distributed in class. You should store them in the ISIS UserOutput folder.

The basic procedure for initiating a pre-defined model run are outlined here. You are encouraged to dig more deeply into the ISIS interface.

- start ISIS: you will see a File pull-down menu, a Help pull-down menu and four tabbed pages. The tabbed pages are Configuration, Excecution, Visualization, and Analysis. If you click on the Configuration tab you will see a nested list of model parameters that may be set by the user.

- Choose {\bf Select Scenario} in the {\bf File} pull-down menu. This opens a new interface window with a list of pre-defined scenarios. Open the Greenland scenario list. The first option, Greenland Climate Evolution, has already been run for you. Choose the third setup in the list, {\bf Greenland 500 year climate warming.}

- Choose {\bf save as} in the {\bf File} pull-down menu and save the configuration file in the {\bf UserConfig} directory. {\it This step is important. If you don't do this, you will overwrite the ISIS configuration file for this simulation when you run the model.}

- Click the {\bf Configuration} tab again. Open {\bf CF output} from the list of options and save each of the three output files for this model run to the {\bf UserOutput} directory. You will need to do this each time you set up a new model experiment.

- Choose the {\bf Excecution} tab and push the run button. You will see text from the runtime log file appear in the window at the right.

- {\bf Visualization} provides some simple tools for inspecting the model output.

==={\scshape Matlab=== scripts for data visualization: steps for you to follow}

A group of {\scshape Matlab} scripts are available at the class website. Using these tools, with a few simple modifications, will allow you to answer all of the questions associated with this lab. You are welcome to modify and expand the scripts.

ISIS stores data at selected time steps as the model runs. With the default settings, ISIS produces output files with two temporal resolutions, 20 years and 100 years. The output file names indicate the temporal resolution, so that \lstinline{gland-500-1.100.nc} is a 500 year long run with saved data at every 100th year.

#### reading the output data

ISIS generates netCDF output data files. NetCDF (Network Common Data Form) is a machine-independent, self-describing, binary data format that is a widely-used standard for exchanging scientific data. You will read these files into the {\scshape Matlab} workspace using two scripts, \lstinline {netcdf.m} and \lstinline{prepare_output.m}, provided at the class website. \lstinline{prepare_output.m} uses the first to read a netCDF file and pull out variables to be used in the later analysis. The values are stored in a data structure, a compact but somewhat obscure framework. You can read more about data structures in the {\scshape Matlab} help. You will need to make sure that two data file names are correct in \lstinline{prepare_output.m}: the model output file, with a \lstinline{.nc} extension, and a {\scshape Matlab} native format file, with a \lstinline{.mat} extension. The latter file will contain the set of variables to be used by the next script. The lines you will need to modify are

\begin{lstlisting}[basicstyle=\ttfamily,commentstyle={},stringstyle={}] test=netcdf(`gland-500-1.100.nc'); outname=`gland-500-1-100.mat'; \end{lstlisting}

Save and run the script. This is the list of variables stored in the {\scshape Matlab} file generated for \lstinline{gland-500-1.100.nc}: \vspace{0.5 cm}

\begin{center}

\begin{tabular}{llcl} name & dimension & description & units\\ \hline

N & 1x1 & saved time steps\\ T & 6x11x141x83 & ice temperature & C \\ b & 6x141x83 & bed elevation & m\\ bdot & 6x141x83 & basal melt rate & m a\\ h & 6x141x83 & ice thickness & m\\ ux & 6x11x141x83 & velocity, x-component & m a \\ uy & 6x11x141x83 & velocity, y-component & m a \\ uz & 6x11x141x83 & velocity, z-component & m a \\ x1 & 83x1 & x grid node locations & m \\ y1 & 141x1 & y grid node locations & m \\ zeta & 11x1 & vertical coordinate & \\

\hline \end{tabular} \end{center}

\vspace{0.5cm}

As you can see, the output variables are multi-dimensional, including both space and time. Some variables, such as the thickness \lstinline{h} have three dimensions: time and two horizontal dimensions while others, such as the temperature \lstinline{T} have four: time, vertical coordinate, and two horizontal coordinates. This 500 year run with data saved every 100 years yields 6 saved times, the initial time and 5 steps during the model run. The horizontal coordinates correspond to the horizontal grid that defines the model domain. The vertical coordinate is in the system, which itself is described in the variable \lstinline{zeta}.

#### a simple map

Next, make a simple color map of ice thickness at the start of the model run by typing the following commands at the prompt in the {\scshape Matlab} command window. You may notice that the array \lstinline{h} requires some special handling before we can render the thickness field. The {\scshape Matlab} function \lstinline{squeeze} is used to convert the an array with three dimensions, one a sigleton, into an array with only two dimensions that can be handled by the \lstinline{pcolor} routine. The {\scshape Matlab} function \lstinline{double} is used to convert the data type from single to double, also a requirement for rendering the color map. The data type single requires less storage space than does the type double. Here are the lines you need to type:

\begin{lstlisting}[basicstyle=\ttfamily,commentstyle={},stringstyle={}] >> clear >> load gland-500-1-100.mat >> figure(1) >> pcolor(double(squeeze(h(1,:,:)))), axis equal >> xlabel(`column number') >> ylabel(`row number') >> colorbar \end{lstlisting}

The black lines in the figure you just created show you the resolution of the numerical model. To eliminate them, type: \begin{lstlisting}[basicstyle=\ttfamily,commentstyle={},stringstyle={}] >> shading flat \end{lstlisting}

Here's another set of commands to plot the surface temperature at the start of a model run:

\begin{lstlisting}[basicstyle=\ttfamily,commentstyle={},stringstyle={}] >> clear >> load gland-500-1-100.mat >> figure(2) >> pcolor(double(squeeze(T(1,1,:,:)))), axis equal >> xlabel(`column number') >> ylabel(`row number') >> colorbar \end{lstlisting}

The plotting scripts provided for you at the class website select a particular latitude along which to plot several sections through the ice sheet. You may wish to investigate other sections. If so, the map in the figure you just created can be used to find new row and column numbers for a different section.

## Questions

Select a Greenland future warming scenario in ISIS and run the ice sheet model. When the model run is complete, extract output data from the resulting netCDF file using \lstinline{prepare_output.m}, as described in section \ref{section:Mlab}.

- Run the scripts \lstinline{plot_slices.m} and \lstinline{plot_vertprof.m} with the output from the model initialization. To do this you will need to set that data file \lstinline{gland-ClimateEvo-2ka.mat} as the file to load in line 6 of the script. Change the plotting time step number \lstinline{n} to 126 at line 26 of \lstinline{plot_slices.m}.

##Describe and explain the differences in the horizontal speed profiles at sites near the center of the ice sheet and near the ice sheet margin. ##Describe and explain the differences in the temperature profiles at sites near the center of the ice sheet and near the ice sheet margin.

- What's up with the temperature of the ice near the base of the ice sheet ice at the margin site?

- Run the complete suite of plotting routines with the output from your climate warming model. You should run the script \lstinline{plot_slices.m} first, followed by the scripts \lstinline{plot_vertprof.m} and \lstinline{plot_change.m}. Be sure to set the file name for the file you wish to load and the time step number to plot in the script \lstinline{plot_slices.m}. Please indicate at the start of your answers which ISIS scenario you chose.

##How does ice speed change over the course of the simulation along the two sections across the ice sheet? What changes in the ice are responsible for the change you observe? ##What physical processes are responsible for the thick core of cold ice in the interior of the ice sheet at the end of the 500 year warming run? ##Suppose you continued running forward in time from here with no further change in the surface temperature. Describe what would happen to the temperature profiles at the center and margin sites.

- Why is the ice divide not in the center of the ice sheet?

- {\bf graduate students} Keeping the warming scenario the same, how do the temperature and velocity fields change when the basal traction is reduced (in essence, the basal traction parameter in the model scales the effectiveness of basal water at lubricating the bed and facilitating fast flow)? Explain the physical processes responsible for the changes you document.