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<!DOCTYPE html>
<html>
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0, minimum-scale=1.0, maximum-scale=1.0">
<meta name="author" content="Simulating the Melting of Ice Caps">
<meta property="og:url" content="http://imaginary.github.io/melting-ice-caps/">
<meta property="og:type" content="website">
<meta property="og:title" content="Simulating the Melting of Ice Caps">
<link rel="icon" type="image/ico" href="assets/img/favicon.png">
<link rel="stylesheet" href="vendor/bootstrap/css/bootstrap.min.css" media="all" type="text/css">
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<title>Simulating the Melting of Ice Caps
</title>
</head>
<body>
<div id="main">
<div class="header">
<div class="title">Simulating the Melting of Ice Caps
</div>
</div>
<div class="slideshow-wrapper">
<div class="slideshow" data-slideshow-id="main">
<div class="slide" data-slide-id="1">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<div class="subtitle">Authors: Maëlle Nodet and Jocelyne Erhel
</div>
</div>
</div>
<div class="row">
<div class="col-md-12 col-no-padding">
<div class="embed-responsive embed-responsive-half16by9 front-video">
<video class="embed-responsive-item" loop>
<source data-path="tl-hill.mp4" type="video/mp4">
</video>
</div>
</div>
</div>
<div class="row">
<div class="col-md-8 col-md-push-2">
<div class="large text-center">
<p>
Sea levels are rising for various reasons related to global warming. The glaciers of Antarctica and Greenland,
known as ice caps or ice sheets, play a major role in changes in sea level. Is it possible to predict future
changes in these ice caps, and particularly the calving of icebergs into the ocean?
</p>
</div>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="2">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>Contribution to sea level rise
</h1>
</div>
</div>
<div class="row">
<div class="col-md-6 col-md-push-3">
<p>
The contribution of ice caps to sea level rise is essentially a combination of three phenomena: snowfalls and
melting of ice on the surface, due to climate change and variations in glacier altitude; thinning of floating ice
shelves, which are melting from below because the ocean is warming up; and the discharge of ice into the sea when
ice flowing towards the sea breaks off, calving icebergs.
</p><img class="img-responsive" src="assets/img/calving.jpg">
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="3">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>Modelling and simulation
</h1>
</div>
</div>
<div class="row">
<div class="col-md-3 col-md-push-1"><img class="img-responsive" src="assets/img/subglacial-river.jpg">
Subglacial river frozen in winter<br>
Photo : Yves Chaux
</div>
<div class="col-md-3 col-md-push-1"><img class="img-responsive" src="assets/img/snow-covered-surface.jpg">
Snow-covered surface of a glacier<br>
Photo : Yves Chaux
</div>
<div class="col-md-4 col-md-push-1">
<p>
The physical processes in an ice cap, both inside and on the surface, are being studied by glaciologists, in
collaboration with climatologists, oceanologists, mathematicians and computer scientists. A numerical model based
on mathematical equations can be used to simulate the ice cap dynamics, particularly the calving of icebergs.
Algorithms solve the equations and are used in the computer codes.
</p>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="4">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>Physical processes at work in an ice sheet
</h1>
</div>
</div>
<div class="row">
<div class="col-md-6">
<div class="switch-display" data-option-show="default"><img class="img-responsive" src="assets/img/physical-processes-inside.jpg"></div>
<div class="switch-display" data-option-show="2"><img class="img-responsive" src="assets/img/physical-processes-base.jpg"></div>
<div class="switch-display" data-option-show="3"><img class="img-responsive" src="assets/img/en/physical-processes-surface.jpg"></div>
<div class="switch-display" data-option-show="4"><img class="img-responsive" src="assets/img/physical-processes-sea.jpg"></div>
</div>
<div class="col-md-4">
<div class="row">
<div class="col-md-12">
<div class="switch-display" data-option-show="default">
<div class="panel">
<h2>Inside</h2>
<p>
Ice is a fluid that flows under the influence of gravity. It flows much more slowly than liquid water because it
has different characteristics. In particular, ice is a plastic fluid, i.e. it changes shape when it flows. The
physical and mechanical properties of ice change with pressure and temperature. Pressure increases with depth
because of the weight of the ice, and temperature, which is very low on the surface, also increases with depth.
Consequently, the velocity of the glacier varies in intensity and direction inside the ice cap.
</p>
</div>
</div>
<div class="switch-display" data-option-show="2">
<div class="panel">
<h2>At the base</h2>
<p>
When the ice is cold, at the base it sticks to the bedrock and the friction forces reduce its velocity. The
temperature at the base, as inside the glacier, increases with the geothermal flux, which is the heat given off by
the earth’s crust. When the ice is at its melting point (0°C), a film of meltwater enables it to slide and
increases its velocity.
</p>
<p>
In Antarctica and Greenland, the areas where sliding is greatest are on the coast, in valleys resembling canyons
or fjords. These rivers of ice, known as outlet glaciers, can have spectacular velocities, above 10 km per year.
</p>
</div>
</div>
<div class="switch-display" data-option-show="3">
<div class="panel">
<h2>On the surface</h2>
<p>
On the surface, ice is added to the existing glacier by precipitations and by the refreezing of water: this is
accumulation. However, ice also disappears because it melts or because of the wind: this is ablation. The weather
conditions and temperature at the surface of the ice cap have an effect on the amount of precipitation as well as
on melting and wind transportation, and therefore on accumulation and ablation and consequently on the glacier’s
altitude. When the surface temperature increases because of global warming, the ice melts and the altitude
decreases through ablation. But the temperature increases when the altitude decreases, so the glacier continues to
melt and thin. This phenomenon can therefore escalate and the ice cap can quickly disappear.
</p>
<p>
This instability of small ice caps is an example of feedback loop, where a cause produces an effect, which in turn
has a reinforcing impact on that cause.
</p>
</div>
</div>
<div class="switch-display" data-option-show="4">
<div class="panel">
<h2>At sea level</h2>
<p>
Fast outlet glaciers lose large amounts of ice in the form of icebergs and thus contribute to sea level rise.
Floating ice shelves are an extension of ice caps over the ocean. When seawater temperature rises, they melt from
the base, raising the sea level. This sea level rise in turn affects the velocity of the glacier: feedback can be
observed between the different physical processes. The sea level rise changes the grounding line, which is the line
where the ice cap starts to float. This modifies the geometry of the ice shelf, and consequently the flow of ice.
</p>
</div>
</div>
</div>
</div>
</div>
<div class="col-md-2">
<div class="switch-group">
<h2 class="title">Select an area
</h2>
<ul class="options">
<li>
<button data-option="default">Inside
</button>
</li>
<li>
<button data-option="2">At the base
</button>
</li>
<li>
<button data-option="3">On the surface
</button>
</li>
<li>
<button data-option="4">At sea level
</button>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="5">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>The mathematical model
</h1>
</div>
</div>
<div class="row">
<div class="col-md-5 vpull-over-title">
<div class="switch-display" data-option-show="default"><img class="img-responsive" src="assets/img/en/model-inside.png"></div>
<div class="switch-display" data-option-show="2"><img class="img-responsive" src="assets/img/en/model-base.png"></div>
<div class="switch-display" data-option-show="3"><img class="img-responsive" src="assets/img/en/model-surface.png"></div>
<div class="switch-display" data-option-show="4"><img class="img-responsive" src="assets/img/en/model-sea.png"></div>
</div>
<div class="col-md-5">
<div class="switch-display" data-option-show="default">
<div class="panel">
<h2>Inside</h2>
<p>
The ice flows because it is a fluid. The model expresses two fundamental laws of physics as equations. On the
one hand, the ice is an incompressible fluid, so when a cube of ice changes shape, its volume and mass remain the
same: this is the law of conservation of mass. On the other hand, the ice is at equilibrium, so the sum of the
forces acting on a cube of ice is zero: this is the law of conservation of momentum, or Newton’s second law, when
acceleration is zero.
</p>
<p>
The forces acting on a cube of ice are due to gravity, pressure and viscosity. To compute the force due to
viscosity, a relationship must be defined between the velocity of the fluid and the mechanical stresses due to the
plasticity of the fluid. Glaciologists use Glen’s law, which brings temperature into the equation and models the
characteristics of the ice crystals.
</p>
<p>
An equation is also needed to define the temperature inside the ice: this is the law of conservation of energy,
which accounts for geothermal flux and surface temperature. In practice, the temperature is often computed
separately with a simplified model. Thus a set of equations is obtained in which the unknowns are the velocity of
the fluid (intensity and direction) and the pressure, throughout the volume of the glacier. This system of
equations is known as the Stokes Equations. They are a simplified version of the notorious Navier-Stokes Equations,
for which a 1 million dollar prize has been offered to anyone who can solve them. The Stokes Equations describe the
relationships between the variations in velocity and pressure in each direction: they are partial differential
equations.
</p>
</div>
</div>
<div class="switch-display" data-option-show="2">
<div class="panel">
<h2>At the base</h2>
<p>
The friction and sliding conditions are complex and are often modelled using a law of friction (another partial
differential equation) which involves a friction coefficient specific to the glacier being studied. This
coefficient varies in space and is very difficult to measure.
</p>
<p>
In addition, the altitude of the bedrock can vary. The bedrock slowly sinks under the weight of the ice and slowly
rises again when the ice melts. There is feedback between the surface mass balance, the glacier flow, and the
altitude of the bedrock. This rebound, known as isostasy, happens after around 10,000 years and can therefore be
ignored when changes in the ice sheet over a century are being studied.
</p>
</div>
</div>
<div class="switch-display" data-option-show="3">
<div class="panel">
<h2>On the surface</h2>
<p>
The surface altitude can increase through accumulation or decrease through ablation. It can also vary because of
the glacier’s movement. The equation expressing the change over time in the surface altitude is the surface mass
balance. Accumulation and ablation are computed using a climate model, while the change due to movement is
calculated with the glacier’s surface velocity.
</p>
<p>
The surface mass balance is also a partial differential equation, with not only spatial variations in the altitude
but also variations over time.
</p>
</div>
</div>
<div class="switch-display" data-option-show="4">
<div class="panel">
<h2>At sea level</h2>
<p>
The grounding line is the place where the ice cap starts to float on the ocean. The floating ice shelf is also
subject to the buoyancy force exerted by the water. There is an equation expressing the balance of forces for the
boundary between the ice cap and the ocean, accounting for the difference in density between ice and water.
</p>
<p>
There is also an equation for the ice cap’s other boundaries, to complete the system.
</p>
</div>
</div>
</div>
<div class="col-md-2">
<div class="switch-group">
<h2 class="title">Select an area
</h2>
<ul class="options">
<li>
<button data-option="default">Inside
</button>
</li>
<li>
<button data-option="2">At the base
</button>
</li>
<li>
<button data-option="3">On the surface
</button>
</li>
<li>
<button data-option="4">At sea level
</button>
</li>
</ul>
</div>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="6">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>Computing the ice sheet dynamics
</h1>
</div>
</div>
<div class="row">
<div class="col-md-2 col-md-push-3"><img class="img-responsive" src="assets/img/en/computing-dynamics.png"></div>
<div class="col-md-3 col-md-push-4">
<p>
To predict the ice sheet behaviour, it is necessary to compute the surface altitude at regular intervals, e.g.
every year. To do this, it is necessary to update the Surface Mass Balance (SMB), which depends on velocity and
weather conditions. It is also necessary to adjust the glacier’s velocity, which depends on the altitude.
</p>
<p>
Since the surface altitude for year 0 is known from measurements, the velocity for year 0 can be computed using the
Stokes Equations. The Surface Mass Balance can then be computed for year 0 and the surface altitude for year 1 is
worked out from this.
</p>
<p>
The velocity for year 1 can then be computed, followed by the Surface Mass Balance, and then the surface altitude
for year 2, etc.
</p>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="7">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>Computing the velocity of a glacier
</h1>
</div>
</div>
<div class="row">
<div class="col-md-4 col-md-push-2"><img class="img-responsive" src="assets/img/greenland-mesh.jpg"><small>
<p>
Greenland mesh and simulated velocity. There are more triangles in the outlet glacier areas so that rapid
variations in velocity can be computed precisely.
</p></small></div>
<div class="col-md-4 col-md-push-2">
<div class="row">
<div class="col-md-12">
<p>
If the surface altitude for year n is known, it is possible to compute the velocity for year n by solving the
Stokes partial differential equations.
</p>
<p>
The first step is to go from unknowns that are everywhere defined to unknowns defined by a finite number of values.
This step is known as spatial discretisation. The surface of the glacier is cut up into small triangles, and the
thickness of the glacier into slices, which results in the division of the glacier’s volume into prisms (kinds of
ice cubes), creating what is known as a mesh.
</p>
</div>
</div>
<div class="row">
<div class="col-md-8">
<p> </p><small>
<p>
Image by Gillet-Chaulet, F., Gagliardini, O., Seddik, H., Nodet, M., Durand, G., Ritz, C., Zwinger, T., Greve, R.,
and Vaughan, D. G.: Greenland ice sheet contribution to sea-level rise from a new-generation ice-sheet model, The
Cryosphere, 6, 1561-1576, 2012. Licence CC-BY 3.0.
</p></small>
</div>
</div>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="8">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>Solving the stokes equations
</h1>
</div>
</div>
<div class="row">
<div class="col-md-5 col-md-push-1"><img class="img-responsive" src="assets/img/vitesse-simule.jpg">
<p>
Simulated velocity in 3D of the Greenland ice cap. Outlet glaciers are much faster (in red, 12 km per year) than
other areas (in blue, 1 meter per year).
</p><small>
<p>
Image by Fabien Gillet-Chaulet, CNRS & LGGE, Grenoble.
</p></small>
</div>
<div class="col-md-4 col-md-push-1">
<p>
The Stokes equations are then converted to a system of complicated algebraic equations (without any derivatives),
with as many equations as unknowns. It is not possible to solve exactly such a system; therefore the solution is
approximated in a sequence of steps. The iterative algorithm is stopped when a quantity related to the error is
small enough.
</p>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="9">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>The future for Greenland
</h1>
</div>
</div>
<div class="row">
<div class="col-md-6 col-md-push-3">
<p>
A computer code can simulate changes to the ice shelf by applying the weather conditions at the surface, supplied
for example by the IPCC’s climate models. These weather data depend on the socioeconomic scenario chosen.
</p>
<div class="embed-responsive embed-responsive-16by9">
<video class="embed-responsive-item" controls controlsList="nodownload nofullscreen noremoteplayback">
<source data-path="elmerice_simulation.mp4" type="video/mp4">
</video>
</div>
</div>
</div>
</div>
</div>
<div class="slide" data-slide-id="10">
<div class="container-fluid">
<div class="row">
<div class="col-md-12">
<h1>Quiz
</h1>
</div>
</div>
<div class="row">
<div class="col-md-2 col-md-push-2"><img class="img-responsive" src="assets/img/quiz-l.png"></div>
<div class="col-md-4 col-md-push-2">
<p>
To simulate the ice sheet dynamics, a mesh is defined by cutting the volume into small ice cubes. What happens
when the number of ice cubes is reduced?
</p>
<div class="switch-group switch-group-quiz text-center">
<h2 class="title">
</h2>
<ul class="options" data-option-once>
<li>
<button class="switch-group-quiz-wrong" data-option="1">1. The computation is quicker and more accurate
</button>
</li>
<li>
<button class="switch-group-quiz-right" data-option="2">2. The computation is quicker and less accurate
</button>
</li>
<li>
<button class="switch-group-quiz-wrong" data-option="3">3. The computation is slower and more accurate
</button>
</li>
<li>
<button class="switch-group-quiz-wrong" data-option="4">4. The computation is slower and less accurate
</button>
</li>
</ul>
</div>
<div class="switch-display switch-group-quiz-wrong" data-option-show="1">
<h2>No, the correct answer is the 2</h2>
<p>
To compute the velocity and altitude with a good level of accuracy, a high resolution mesh is required, especially
in areas of high velocity. But the number of equations increases as the mesh resolution increases and the
computation time increases with the number of equations. As with most numerical simulations, a tradeoff has to be
found between the accuracy of the result and the computation time.
</p>
</div>
<div class="switch-display switch-group-quiz-right" data-option-show="2">
<h2>Bravo!</h2>
<p>
To compute the velocity and altitude with a good level of accuracy, a high resolution mesh is required, especially
in areas of high velocity. But the number of equations increases as the mesh resolution increases and the
computation time increases with the number of equations. As with most numerical simulations, a tradeoff has to be
found between the accuracy of the result and the computation time.
</p>
</div>
<div class="switch-display switch-group-quiz-wrong" data-option-show="3">
<h2>No, the correct answer is the 2</h2>
<p>
To compute the velocity and altitude with a good level of accuracy, a high resolution mesh is required, especially
in areas of high velocity. But the number of equations increases as the mesh resolution increases and the
computation time increases with the number of equations. As with most numerical simulations, a tradeoff has to be
found between the accuracy of the result and the computation time.
</p>
</div>
<div class="switch-display switch-group-quiz-wrong" data-option-show="4">
<h2>No, the correct answer is the 2</h2>
<p>
To compute the velocity and altitude with a good level of accuracy, a high resolution mesh is required, especially
in areas of high velocity. But the number of equations increases as the mesh resolution increases and the
computation time increases with the number of equations. As with most numerical simulations, a tradeoff has to be
found between the accuracy of the result and the computation time.
</p>
</div>
</div>
<div class="col-md-2 col-md-push-2"><img class="img-responsive" src="assets/img/quiz-r.png"></div>
</div>
</div>
</div>
<div class="slide" data-slide-id="credits">
<div class="container-fluid">
<div class="row">
<div class="col-md-4 col-md-push-1">
<h1>Credits
</h1><small>
<h3>Authors:</h3>
<p>
Maëlle Nodet (Université Grenoble Alpes)<br>
Jocelyne Erhel (Inria)
</p>
<h3>Design / Production:</h3>
<p>
Interstices (interstices.info)<br>
IMAGINARY (www.imaginary.org)
</p>
<h3>Graphic Design:</h3>
<p>Victoria Denys</p>
<h3>Images:</h3>
<p>
Calving Ice Berg in Illulissat Icefjord (Greenland), Göran Ingman (CC-BY). (p.2)<br>
Yves Chaux (p. 3);<br>
d'après Gillet-Chaulet, F., Gagliardini, O., Seddik, H., Nodet, M., Durand, G., Ritz, C., Zwinger, T., Greve, R.,
and Vaughan, D. G. (p. 7) ;<br>
Fabien Gillet-Chaulet, CNRS & LGGE, Grenoble (p. 8)
</p>
<h3>Videos:</h3>
<p>
Martin Funk, VAW / ETHZ (p. 1) ;<br>
eSTICC Elmer ICE, Simulation Fabien Gillet-Chaulet LGGE, Grenoble, Visualisation Jyrki Hokkanen CST-IT Center For
Science (p. 9)
</p>
<p>This presentation is distributed under a Creative Commons BY-NC-SA license.</p></small>
</div>
<div class="col-md-4 col-md-push-3">
<h1>Bibliography
</h1><small>
<p>
Maëlle Nodet, Jocelyne Erhel. Modéliser et simuler la fonte des calottes polaires, Interstices, 2015.
</p>
<p>
Maëlle Nodet, Jocelyne Erhel. Des outils mathématiques pour prévoir la fonte des calottes polaires, Interstices,
2015.
</p>
<p>
Maëlle Nodet. De la glace à la mer. Matapli, SMAI, 2013.
</p>
<p>
Guillaume Jouvet. L'évolution des glaciers, modélisation et prédiction. Accromath, Vol 8.2, 2013.
</p>
<p>
Guillaume Jouvet. The future of glaciers, Imaginary, 2013
</p>
<p>
Fabien Gillet-Chaulet. Elmer/Ice, un modèle de calotte polaire de nouvelle génération. Journée Mésochallenge
Equip@meso, 2013.
</p>
<p>
Catherine Ritz, Tamsin L. Edwards, Gaël Durand, Antony J. Payne, Vincent Peyaud & Richard C. A. Hindmarsh:
Potential sea-level rise from Antarctic ice-sheet instability constrained by observations, Nature, 528, 115-118,
2015.
</p>
<p>
Gillet-Chaulet, F., Gagliardini, O., Seddik, H., Nodet, M., Durand, G., Ritz, C., Zwinger, T., Greve, R., and
Vaughan, D. G.: Greenland ice sheet contribution to sea-level rise from a new-generation ice-sheet model, The
Cryosphere, 6, 1561-1576, 2012.
</p></small>
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