Glaciers and Ice sheets
A.C. Fowler
Mat.hemat.ical Inst.itut.e
Oxford University
24-29 St Giles'
Oxford OX1 3LB
ENGLAND
1 Dynamic phenomena
Glaciers are huge and slow moving rivers of ice which exist. in various parts of the world:
Alaska, the Rockies, t.he Alps, Spit.sbergen, China, for example. They drain areas in which
snow accumulates, much as rivers drain catchment areas where rain falls. Glaciers also
flow in the same basic way t.hat rivers do. Although glacier ice is solid, it can deform by
the slow creep of dislocations wit.hin the lattice of ice crystals which form the fabric of
the ice. Thus, glacier ice effectively behaves like a viscous material, with. however, a very
large viscosity: a typical value of ice viscosity is 1 bar year (in t.he metre-bar-year system
of units!). Since 1 bar = 10 5 Pa, 1 year ~ 3 x 10 7 s, this is a viscosity of some 10 12 Pa s,
about. 10 15 times that of wat.er. As a consequence of t.heir enormous viscosit.y, glaciers move
slowly - a typical velocity would be in the range 10-100 m y-1 (met.res pel year), cert.ainly
measurable but hardly dramatic. More awesome are the dimensions of glaciers. Dept.hs of
hundreds of metres are typical, widt.hs of kilomet.res, lengths of t.ens of kilometres. Thus
glaciers can have an important effect. on t he human environment in their vicinity. They
are also indirect monitors of climate; for example, many lit.hographs of Swiss glaciers show
that. they have been receding since the nineteent.h century. a phenomena thought to be
due t.o the terminat.ion of the 'lit.tle ice age' in the middle ages.
Where glaciers are the glacial equivalent. of rivers, i.e. channeled flow, -ice sheets are
the equivalent of droplets, but alt.ogether on a grander scale. When an entire cont.inent,
or at least a substantial portion thereof, has a polar climat.e, t.hen snow accumulat.es on
the uplands, is compressed int.o ice, and flows out t.o cover the cont.inent., much as a drop
of fluid on a t.able will spread under the act.ion of gravit.y. However, whereas droplets can
reach a st.eady state through t.he contractile effect of surface tension, this is not. relevant
t.o large ice sheets. In them, equilibrium can be maint.ained through a balance between
accumulation in the cent.re and ablaf'ion at. the margins. This can occur eit.her t.hrough
melting of the ice in the warmer climate at. the ice margin, or t.hrough calving of icebergs.
(Indeed, the same balance of accumulation at. higher elevat.ions with ablation at. lower
elevat.ions is responsible for the normal quasi-st.eady profile of valley glaciers.)
NATO ASI Series, Vol. J 48
The Mathematic" of Model..:: for Climatology
and Envi ronment
Edited by Jcslis IIdcfonso Diaz
© Springer- Verlag: Berlin Heidelberg 1997
A.C. Fowler
Mat.hemat.ical Inst.itut.e
Oxford University
24-29 St Giles'
Oxford OX1 3LB
ENGLAND
1 Dynamic phenomena
Glaciers are huge and slow moving rivers of ice which exist. in various parts of the world:
Alaska, the Rockies, t.he Alps, Spit.sbergen, China, for example. They drain areas in which
snow accumulates, much as rivers drain catchment areas where rain falls. Glaciers also
flow in the same basic way t.hat rivers do. Although glacier ice is solid, it can deform by
the slow creep of dislocations wit.hin the lattice of ice crystals which form the fabric of
the ice. Thus, glacier ice effectively behaves like a viscous material, with. however, a very
large viscosity: a typical value of ice viscosity is 1 bar year (in t.he metre-bar-year system
of units!). Since 1 bar = 10 5 Pa, 1 year ~ 3 x 10 7 s, this is a viscosity of some 10 12 Pa s,
about. 10 15 times that of wat.er. As a consequence of t.heir enormous viscosit.y, glaciers move
slowly - a typical velocity would be in the range 10-100 m y-1 (met.res pel year), cert.ainly
measurable but hardly dramatic. More awesome are the dimensions of glaciers. Dept.hs of
hundreds of metres are typical, widt.hs of kilomet.res, lengths of t.ens of kilometres. Thus
glaciers can have an important effect. on t he human environment in their vicinity. They
are also indirect monitors of climate; for example, many lit.hographs of Swiss glaciers show
that. they have been receding since the nineteent.h century. a phenomena thought to be
due t.o the terminat.ion of the 'lit.tle ice age' in the middle ages.
Where glaciers are the glacial equivalent. of rivers, i.e. channeled flow, -ice sheets are
the equivalent of droplets, but alt.ogether on a grander scale. When an entire cont.inent,
or at least a substantial portion thereof, has a polar climat.e, t.hen snow accumulat.es on
the uplands, is compressed int.o ice, and flows out t.o cover the cont.inent., much as a drop
of fluid on a t.able will spread under the act.ion of gravit.y. However, whereas droplets can
reach a st.eady state through t.he contractile effect of surface tension, this is not. relevant
t.o large ice sheets. In them, equilibrium can be maint.ained through a balance between
accumulation in the cent.re and ablaf'ion at. the margins. This can occur eit.her t.hrough
melting of the ice in the warmer climate at. the ice margin, or t.hrough calving of icebergs.
(Indeed, the same balance of accumulation at. higher elevat.ions with ablation at. lower
elevat.ions is responsible for the normal quasi-st.eady profile of valley glaciers.)
NATO ASI Series, Vol. J 48
The Mathematic" of Model..:: for Climatology
and Envi ronment
Edited by Jcslis IIdcfonso Diaz
© Springer- Verlag: Berlin Heidelberg 1997