Fowler A. Mathematical Geoscience

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770 11 Jökulhlaups

and Υ and N satisfy the boundary layer equations



|Υ

(n−1)/n



=−N,

=Υ,

(11.134)

with boundary conditions

N =N

,Υ=

,Υ

=0onX =0,

Υ →0asX →∞.

(11.135)

Suppose now that

(

t −t)

,t<

t. (11.136)

We seek a similarity solution in the form

N =N

ψ(η), Υ =

φ(η), η =

(

t −t)

; (11.137)

substituting these forms into the boundary layer equations, we ﬁnd that we require

b =

(n −1)α −1

2(n +1)

, (11.138)

and then φ and ψ satisfy the boundary value problem





|φ



(n−1)/n





=−Γψ,

εηψ



+ψ =φ,

(11.139)

where

ε =

(n −1)α −1

2(n +1)α

,Γ=



n−1

αm

2(n+1)



1/n

; (11.140)

Γ can be chosen arbitrarily through the choice of m. To be precise, we choose

Γ =1, so that

2(n+1)

n−1

; (11.141)

it follows from this that

η =





2(n+1)

X. (11.142)

The boundary conditions for φ and ψ are

φ =1,ψ=1,φ



=0onη =0,

φ →0asη →∞.

(11.143)

To solve these equations, it is convenient to deﬁne the subsidiary function χ via

χ =



|φ



(n−1)/n

, (11.144)

11.6 Cauldrons and Calderas 771

Fig. 11.10 Solution of

(11.146)and(11.147)forφ,

with n =1(asin(11.148))

and n =3

so that



=χ |χ|

n−1

,χ



=−φ. (11.145)

We note that if n =3 and α =3, the parameter ε ≈0.2 and is small. This suggests a

perturbation solution for the equations. Although apparently a singular perturbation,

we see from the boundary conditions that the perturbation is actually regular; the

neglect of the term in ε suppresses a singular solution ψ ∝η

−1/ε

which is precluded

by the boundary condition for ψ. At leading order ψ = φ, and the problem thus

reduces to



=χ |χ|

n−1



=−φ,

(11.146)

with

φ =1,χ=0onη =0,

φ →0,χ→0asη →∞.

(11.147)

When n =1, the problem can be solved analytically, and we obtain

φ =e

−η/

√

cos



η/

√



,χ=e

−η/

√

sin



η/

√



(11.148)

(see also Question 11.10). For n>1, a numerical solution is necessary. Figures

11.10 and 11.11 show the solution of (11.146) with (11.147)forn =1, as well as a

numerical solution for n =3.

To ﬁnd the subsidence proﬁle, we solve (11.98), using the deﬁnition of w in

(11.132) and (11.137), thus

∂s

∂t

=−μ



1 −φ(η)



. (11.149)

Figure 11.12 shows a simulation of this model.

772 11 Jökulhlaups

Fig. 11.11 Solution of

(11.146)and(11.147)forχ ,

with n =1(asin(11.148))

and n =3. When n =3, the

maximum value of χ = 0.615

at η =1.6

It is of interest to calculate the maximum stress and bending moment in the beam.

From (11.85) and (11.125), the maximum stress is at the surface and base of the ice,

and is given by

|τ

max



1/n

. (11.150)

Fig. 11.12 Solution of (11.149) (starting at t = 0 days) at times (a) t = 350 days, (b) t = 371

days, (c) t =382 days, (d) t = 393 days using (11.136)forN

,with

t = 1, c = 2.5 ×10

−3

, α = 3,

γβ/ν

=15.8. The dimensional blow-up time is 416 days, and the units of vertical subsidence are

metres, corresponding to a choice of μd =N

/ρ

g = 3 m, and thus N

=0.27 bars. The results are

plotted as if for a lake of width 20 km, in which case if d =4km,thenν =0.2 and thus γβ =0.63,

corresponding to a value of τ

=0.43 bars

11.7 Floods from Ice Sheets 773

After some algebra, we ﬁnd that the maximum stress is

|τ

max





1/n



γβ



1/(n+1)

|χ|, (11.151)

and the bending stress M in (11.133) is given by

M =



γβ



1/(n+1)

χ. (11.152)

When n = 1, the maximum stress and bending stress are at η = 1.11, where χ =

0.32. For n =3, the maximum dimensionless bending stress is χ =0.62 at η =1.6.

Ring Fractures

Since (via (11.66), (11.76) and (11.77)) the longitudinal stress is scaled with τ

,we

see from (11.151) that when n = 3, the maximum stress in the overlake ice has a

dimensional value of

max

≈0.76



γβ



1/4

, (11.153)

where we take H =A = 1 and use the deﬁnition of D

in (11.127) and the maxi-

mum of χ ≈0.62; the maximum occurs at a distance from the margin of

x =1.6





1/8

1/2

(γβ)

1/4

, (11.154)

where l

is the lake width. If τ

max

reaches the yield stress τ

of ice, then the ice will

fracture, forming a crevasse, and the overlake ice will reset itself to the application

of effective boundary conditions at the position of this ring fracture. Thereafter,

continued rise of effective pressure will allow a new ring fracture to occur in-lake

of the old fracture, and in this way a sequence of such fractures may form, with a

spacing indicated by (11.154), as seen in Fig. 11.9. If we equate τ

max

=τ

, and use

(11.153) to eliminate

, we ﬁnd that the fracture spacing should be

x ≈1.84





1/2



γβ



3/8

1/2

. (11.155)

Estimates of this are typically less than the lake width, but do not appear con-

sistent with the small scale (tens of metres) cracks visible in Fig. 11.9,forwhich

some further development of the theory would appear to be necessary, for example

associated with the ﬁnite strain of the surface ice.

11.7 Floods from Ice Sheets

We have mostly focussed our attention on subglacial ﬂoods from beneath glaciers or

small ice caps. There is no intrinsic reason why ﬂoods should not occur from lakes

774 11 Jökulhlaups

beneath ice sheets. It is now known that there are many lakes beneath the Antarctic

ice sheet, for example, and it seems reasonable to suppose that these might also

drain semi-periodically. There is in fact now a good deal of evidence for this, and

there is plenty of evidence of ﬂoods from ice sheets during the last ice ages. We now

gather together some of this story.

Badlands According to Webster’s dictionary, badlands are regions marked by

intricate erosional sculpturing, scanty vegetation, and fantastically formed hills. In

eastern Washington State in the U.S.A., the Channelled Scablands are an example

of such a landscape. They were formed as a result of massive ﬂoods from glacial

Lake Missoula, a pro-glacial lake which formed south of the Cordilleran Ice Sheet

(the western part of the Laurentide Ice Sheet). It is thought that the drainage of

Missoula was blocked by a lobe of the ice sheet, and that the resulting massive build-

up of the lake to a volume in excess of 2,000 cubic kilometres, led to a sequence

of ﬂoods (at least 40, probably more, at intervals of years to decades) of enormous

magnitude (estimates for peak discharge are in the region of 10

−1

); these

ﬂoods caused the massive erosion which formed the scablands. Flow speeds of the

order of 25 m s

−1

and ﬂow depths of up to 300 m caused the erosion of channels

into the solid basaltic rock, and the formation of gigantic forms of lateral bars and

ripples.

The 8,200 Year B.P. Cooling Event We have already discussed the sudden cli-

matic cooling event at 8,200 years B.P. (before present, the ‘present’ being taken

as 1950). It is thought that this is due to the catastrophic ﬂooding of glacial Lake

Agassiz, which formed south of the dwindling Laurentide ice sheet, in the vicinity

of Hudson Bay, as the ice melted. The meltwater builds up behind the ice, but is

blocked from escaping to the south. Eventually, a ﬂood is initiated, probably un-

der the ice sheet, causing massive inﬂux of fresh water to the Hudson Strait, and

thence to the North Atlantic, where, as discussed in Chap. 2, it can temporarily shut

down the oceanic thermohaline circulation, and thus cause a sudden cooling in the

northern hemisphere.

Floods from the Antarctic Similar scabland landscapes have been reported on

high terrain in Antarctica, which indicate that massive ﬂoods occurred there also. It

is thought that these ﬂoods are subglacial (because they are at such high elevation),

and this suggests that massive sub-Antarctic jökulhlaups have occurred in the past.

The likely candidate for the source of such a massive ﬂood is Lake Vostok, con-

taining some 5,400 cubic kilometres of water, and situated under the central part of

East Antarctica. There is in fact little to distinguish sub-ice sheet ﬂoods mathemat-

ically from sub-ice cap ﬂoods, beyond the different scales. If there is net drainage

towards the lake, and this is not taken up by basal freeze-on to the base of the ice,

then inevitably, it seems, a ﬂood will occur. Modelling of the ﬁlling of Lake Vostok,

for example, suggests that ﬂoods with peak discharges of the order of 10

−1

lasting for a year can occur, with a period of the order of 40,000 years.

Much smaller ﬂoods are known to occur (from other lakes) at present, and ap-

pear to constitute the natural way in which drainage takes place beneath the ice

11.7 Floods from Ice Sheets 775

sheet. Satellite imagery has revealed relatively rapid ice surface collapses (of order

of metres in a year), which are presumably due to one subglacial lake discharging

into another. We can imagine drainage under the ice sheet as consisting of a pseudo-

porous ﬂow effected by short term drainage events between the lakes, which act as

the pores of the medium.

A question which arises is whether the Nye–Röthlisberger theory can predict

these small amplitude ﬂuctuations. Returning to the theory, we see that (11.27)im-

plies that, since the total discharge ∼Q

is of the order of A

, where h

is the

drawdown depth, we have

∼

. (11.156)

This suggests that if the drawdown is, as frequently observed, of the order of a metre,

then N

∼ρ

∼0.1 m. A drawdown of 4 metres is consistent with N

∼0.4m,

similar to values inferred on the Whillans ice stream B.

In turn, our discussion of drainage mechanics in Chap. 10 suggests that such low

effective pressures are consistent with a canal-type drainage over sediments. Two

questions now arise: ﬁrst, can our previous theory reproduce such differently scaled

ﬂoods, and second, is such a theory consistent with canal-style drainage?

To answer the ﬁrst, we consider the dependence of the effective pressure scale

on the volume ﬂux scale Q

.From(11.16), we can deduce



11/8

KL(fρ

3/8



1/n

1/4n

(11.157)

(cf. (10.226)). If a 20 km by 20 km lake deﬂates by 4 m in a year, we can estimate

the volume ﬂux as 50 m

−1

. Even if this is two orders of magnitude less than the

ﬂoods found in Fig. 11.6, it changes N

by less than a factor of two. Although there

are many parameters in (11.157), few of them are adjustable. The simplest way to

reduce N

in (11.157) from 32 bars (see (11.20)) to < 1 bar is to increase the closure

coefﬁcient K by (say) four orders of magnitude. In terms of viscosity, this means

replacing an ice viscosity of 2 ×10

Pa s by one of 2 ×10

Pa s, coincidentally

similar to early estimates of till ‘viscosity’. Recalling that Q

is calculated inde-

pendently via (11.28), we see that increasing K by 10

does indeed reduce Q

, and thus reduces N

by 10

3/2

≈ 31.6toavalue≈ 1 bar. The time scale also

becomes a little longer, although not apparently as much as one would like. Never-

theless, if one simply takes the Grímsvötn model and increases K by 10

, together

with a suitable reﬁlling rate to give a period of ten years, we obtain the ﬁt shown in

Fig. 11.13 to data measured on the Lambert Glacier in Antarctica, where there is an

apparent ﬂood every ten years or so.

Thus it seems we can replicate these short term, small amplitude ﬂoods; but there

are issues in the data in Fig. 11.13 which give cause for concern. The recovery phase

shows that surface elevation accelerates as it rises, completely at odds with the Nye

theory. And, while the abrupt phase of the ﬂood is well represented by the model,

the slow down of the subsidence towards the end is also problematic.

In fact, one should have misgivings about applying the Nye theory as it stands.

Apparently, (11.157) indicates the usual Röthlisberger balance which presumably

776 11 Jökulhlaups

Fig. 11.13 The Nye theory for surface elevation (in metres) (computed from the effective pressure

variation), assuming geometric parameters appropriate to the Lambert Glacier. The solid line in-

dicates surface elevation, from ERS satellite altimeter data provided by Andy Shepherd, while the

dashed line comes from the solution of the Nye model. The principal changes in the parameters

are that we take K =0.5 ×10

−20

−3

−1

, which enables the relatively small amplitude change,

and the lake reﬁlling rate is taken to be 7 m

−1

, in order to obtain a period of ten years

is inappropriate for drainage through sediments, where the equilibrium of (11.17)

ought to correspond to N being a decreasing function of Q. What is needed is a

theory of ﬂoods through sediment-ﬂoored canals, but that has yet to be constructed.

Dansgaard–Oeschger Events Dansgaard–Oeschger events were described in

Sect. 2.5.5. They are rapid changes of northern hemisphere climate (by ﬁve to

ten degrees Celsius) which occurred on a time scale of decades during the last ice

age, and which occur semi-periodically, with a rough period of some 1,500 years.

In Sect. 2.5.5, we raised the idea that sub-Laurentide jökulhlaups might provide a

mechanism for the freshwater release, but postponed a detailed discussion. We pro-

vide some further discussion now.

The main problem with sub-Laurentide lake ﬂoods is the problem of where

do you store the water. Since we know that there were massive pro-glacial lakes

(Glacial Lake Agassiz, Glacial Lake Missoula, for example), we can certainly con-

template the existence of massive sub-ice sheet lakes. But why would they form?

Lake Vostok in Antarctica lies in a deep tectonic basin, but such tectonic features

are largely absent under the Laurentide Ice Sheet.

A clue may lie in the contemplation of Fig. 11.14, which shows the association

of eskers with the area of exposed bedrock (shield), suggesting that drainage beyond

this region might have taken the form of a distributed, canal-like system at a lower

effective pressure. If that were the case, then the subglacial streams making their

way from the central ice domes would encounter a transition at the edge of the

bedrock where they would face a virtual escarpment. The hydraulic head (cf. (11.8))

of the subglacial water is

φ =−N +ρ

gs +ρ

gb, (11.158)

11.7 Floods from Ice Sheets 777

Fig. 11.14 The Laurentide Ice Sheet during the last glaciation (heavy line). The inner solid curve

marks the boundary of the mostly exposed crystalline bedrock. Note the gap to the upper right

of the Great Lakes, where the St. Lawrence ice stream ﬂowed into the Gulf of St. Lawrence. The

short ﬁne lines, mostly within the bedrock region, represent eskers. The black regions represent

major lakes, ranging westwards from the Great Lakes through Lake Winnipeg, Great Slave Lake,

and Great Bear Lake. Figure copyright Geological Society of America, reproduced from Clark and

Walder (1994), Fig. 5, and kindly provided by Peter Clark

so, presuming the ice surface s is continuous, a jump down in effective pressure of

N corresponds to a jump up in bed elevation b of

b =

N

ρ

. (11.159)

If the effective pressure were 10 bars on the bedrock side and 1 bar on the

sediment side, then in effect the streams would encounter a barrier of elevation

≈1100 m. This is a fairly signiﬁcant barrier, requiring an ice surface drop of 100 m

to overcome it. If the transition takes place over 100 km, then the ice surface slope

can probably compensate, but if it occurs more rapidly, then water may accumulate

at the base of the virtual scarp, causing a subglacial lake to form. This is a runaway

process, because the formation of the lake drops the basal shear stress to zero, thus

tending to ﬂatten the ice surface and removing its slope.

This raises the possibility that the present lakes surrounding the Canadian Shield

are the remnant of former subglacial lakes. The lakes might have been strung to-

gether like a necklace, allowing for circumferential drainage between them, ending

with discharge down the St. Lawrence spillway.

778 11 Jökulhlaups

There are a number of effects of such putative lakes on the ice dynamics. The

ice above a lake becomes an ice shelf, controlled by longitudinal stresses. There is

a grounding line at each margin, and the upstream grounding line, at least, has an

enhancement of shear stresses (see (10.133), (10.147) and (10.148)) by a factor of

(

)

2/(n+2)

≈ 6.3. This suggests that the ice will dig itself a hole upstream of the

lake, providing for the continued existence of the lake after the ice sheet has gone. It

seems possible that the Great Lakes, indeed all of these shield-marginal lakes, could

have been constructed in this way.

The dynamics at the downstream grounding line are less clear, partly because the

effective pressure is low. Indeed, the transition to low N heralds the condition for

ice stream dynamics, and it may thus be no coincidence that several subglacial lakes

in Antarctica have been found at the heads of ice streams. The presence of a lake

promotes faster ﬂow, and this may well provide the necessary seeding for ice stream

formation downstream.

Floods on Mars As we discussed in Chap. 10, there is ice on Mars, both in the

polar ice caps, and in the soil. Many geomorphological features show evidence of

the past presence of liquid water or ice on the surface, and in particular, there are

many indications of massive outﬂow channels. These features have led people to

suppose that at one time water was plentiful on the surface, perhaps forming an

ocean in the northern lowlands,

and that there may have been large ice caps, from

beneath which powerful jökulhlaups emerged to carve the Martian surface features.

The question arises, how could this come about? One attractive idea views the

hydrological cycle on Mars in the following way. Just as on Earth, planetary out-

gassing from volcanoes produces water vapour and carbon dioxide. And just as on

Earth, volcanoes erupt periodically. The larger the volcano, the larger the eruption

and the longer the period between eruptions. The largest eruptions on Earth (the

basaltic ﬂood eruptions, described in Sect. 9.1) are thought to be associated with

the arrival at the crust of giant thermal plumes in the Earth’s mantle (see Chap. 8),

associated with time-periodicity of mantle convection. On Mars, which hosts the

largest volcano in the Solar System, Olympus Mons, it is reasonable to suppose that

extremely massive eruptions may occur at intervals of perhaps tens of millions of

years. The resulting greenhouse-induced heating of permafrost causes massive out-

ﬂows and the collapse of the source terrain: essentially a hydraulic volcano. These

ﬂoods ﬂow into the northern lowlands, forming an ocean, whose evaporation and

subsequent precipitation forms large ice sheets in the southern highlands, whence

form the glacially sculpted features which are observed. The outgassing produces

enough greenhouse gas to warm the atmosphere (allowing water to form), and also

increases atmospheric pressure above the triple point pressure so that water is sta-

ble. As the volcanic activity subsides, the atmosphere cools, and the climate reverts

to its present cold, dry interglacial type. As with much else concerning Mars and

The hypsometry of Mars is odd: the southern hemisphere is elevated, and the northern hemisphere

is relatively ﬂat, and much lower.

The study of Martian landforms is called areomorphology.

11.8 Notes and References 779

the other planets, data are scarce, hypotheses are cheap, and speculation is rife. But

then, it is the imaginative aspect of science that makes it so much fun.

11.8 Notes and References

Much of the early literature on Icelandic jökulhlaups is in Icelandic (!), for example

in the journal Jökull; a more accessible introduction to the subject are the papers

by Björnsson (1974, 1992), and in particular the same author’s book (Björnsson

1988) gives a thorough review, and is an essential classic. The theoretical literature

is somewhat sparse, but shows signs of maturation with the increasing interest in

palaeo-ﬂoods on Earth and Mars. The seminal paper is by Nye (1976), who pre-

sented the basic theory. Variants on the model are discussed by Spring and Hutter

(1981) and Clarke (1982); in particular, they emphasise the importance of the tem-

perature equation. Clarke (2003) critically reviews ﬂood modelling, and in particu-

lar discusses and recalibrates the ﬂow resistance of the channel. There is also a brief

discussion in the book by Paterson (1994).

The recent 1996 eruption and its aftermath was described at the time on a number

of web pages; for example see http://www.spri.cam.ac.uk/jok/jok.htm. A descrip-

tion is given in the paper by Gudmundsson et al. (1997). A recent review which

gives a very complete historical account of the subject, and which also emphasises

the differences between the Nye-type melt-opening jökulhlaup and the Gjálp-type

hydrofracturing jökulhlaup, is that by Roberts (2005).

Nye’s Model Nye’s (1976) model is a fairly astonishing tour de force. In partic-

ular, his derivation of the energy equation (11.4) is succinct and to the point. It is,

however, a physicist’s derivation (there is work done here and here, energy gained

there, etc.), and it is not something that in the nature of things satisﬁes a mathe-

matician, who wants to see a formal derivation. Spring and Hutter (1982) provided

such a derivation, and this exercise is repeated in Appendix E, which attempts to

draw a middle line between Nye’s direct approach, and the very abstract approach

of Spring and Hutter. Although Eq. (11.4) appears ‘obvious’, this appendix suggests

that it is far from being so.

Wide Channels and Other Problems Despite its quite astonishing success, the

Nye model (and our solution of it) has a number of difﬁculties associated with it. The

ﬁrst of these is the assumption of a semi-circular channel. With an assumption of

easy lateral slip of ice at the bed, the resulting closure can be calculated from that for

a circular channel. But why should the channel be semi-circular? This assumption

seems completely arbitrary, yet the resulting theory appears to work exceptionally

well.

Particularly when one considers the peak size of S (≈600 m

for Fig. 11.6,

At least during ﬂoods. In normal circumstances, wide channels may be preferred (Hooke et al.

1990). See also the discussion on hybrid channels in the notes for Chap. 10.