Fowler A. Mathematical Geoscience

Подождите немного. Документ загружается.

780 11 Jökulhlaups

and ≈2,900 m

for Fig. 11.8), which imply maximum channel radii in the region

of 20 m and 43 m, respectively, it is pertinent to wonder whether the assumption of

semi-circular cross section is likely to be correct.

We can give some plausibility arguments to suggest why a circular shape is

adopted. For ‘weak’ ﬂoods in which the channel effective pressure is always pos-

itive, channel closure is that of a more viscous ‘ﬂuid’ pushing back a less viscous

one. For this situation, viscous ﬁngering does not occur, and a circularly deforming

boundary is stable. Further arguments in this direction are that, since the ice pres-

sure is larger at the glacier bed, the contracting effective stress is larger there: this

mitigates against wider channels, while taller channels are not favoured because of

the consequent increased lateral melting.

None of these arguments work for ‘strong’ (ﬂotation driven) ﬂoods, which as we

have discussed, propagate as viscous fractures at the tip, and in which the channels

must therefore be wide (Jóhannesson 2002a, 2002b).

Temperature The assumption of equilibrium temperature (and thus the removal

of θ from the Nye model) seems rather unfortunate, and it would seem that at least it

would affect the ﬂood dynamics in a quantitative manner. Let us therefore reconsider

the energy equation. Neglecting ε in (11.17), we have

∂θ

∂x

Φ −m, (11.160)

where we write the hydraulic gradient as

Φ =Φ +

∂N

∂X

(11.161)

(here x =δX). Since

Q|Q|=S

8/3

Φ, m ≈



|Q|

1/2



0.8

, (11.162)

the temperature equation simpliﬁes to

∂θ

∂x

Φ −

, (11.163)

where

λ =

γ |Q|

0.5

sgnQ

Φ|

0.15

. (11.164)

Our simpliﬁcation of the Nye model to the form (11.42) was based on the neglect

of the advective term in (11.163). As we stated, between ﬂoods this seems entirely

reasonable, since in equilibrium θ ≈ λ

Φ  1. During ﬂoods, if we take

Φ to be

constant, we see that the advective term θ

= exp(−x/λ), and in fact this is small

away from the lake, since for weak ﬂoods, we have computed the peak Q ∼10

−2

thus λ ∼ 0.1, and θ

∼ 10

−4

. However, the neglect of the advective term is inac-

curate near the lake where x =δX, and in this region the equilibrium assumption

leads to the approximation for the advective term

≈

√



0.85





, (11.165)

11.8 Notes and References 781

where the prime denotes differentiation of

0.85

with respect to X: γ and δ are de-

ﬁned in (11.18), with typical values given in (11.21). This term is of order γ

√

Q/δ,

and becomes signiﬁcant at peak discharge. Its effect on the melt rate m which ap-

pears in the channel closure equation (11.17)

is to replace m =

ΦQ by

m ≈Q[

Φ −θ

] (11.166)

(see (11.160)). Inspection of our numerical results shows that S

< 0atthelake

at peak discharge, and thus (since

Φ = Q

8/3



> 0, and thus θ

> 0. The

effect of including temperature adjustment should therefore be to reduce the peak

discharge.

This assumes the lake temperature is essentially equal to the freezing tempera-

ture, i.e., θ = 0atx = 0. If the lake is superheated, as might be the case following

an eruption, then we expect θ

< 0, and the peak discharge will be enhanced. As we

have seen, the ﬂoods may be rather different in this case. In particular, we associate

warm water temperature with rapid lake ﬁlling, and a likely strong ﬂood propagating

down glacier as a wide fracture channel. Now consideration of the empirical heat

transfer relation (11.5) shows that the term raised to the power 0.8 is essentially

the Reynolds number for a semi-circular channel. For a wide channel, the relevant

length scale is the channel depth, and this implies that to use (11.5) for a wide chan-

nel, we should multiply the constant a

by (h/w)

0.4

, where h is channel depth and

w is channel width. For a wide channel, this cause an increase in γ , which enhances

the importance of the thermal advective term.

However, our problem is really the opposite of this. According to (11.163) and

(11.164), the exit temperature of the water ought to be θ ∼ γ

√

Q. For a peak

discharge of Q = 0.033 (corresponding to 6,000 m

−1

) and with γ ∼ 2.5asin

(11.21), we have θ ∼ 0.45 corresponding to an exit temperature of 1.6°, using the

scale for θ

in (11.20). This is about thirty times higher than is observed (Clarke

2003), suggesting that heat transfer should be much more efﬁcient than that given

by the Dittus–Boelter relation assumed by Nye (1976). Two possible reasons are

that the measurements used in establishing the Dittus–Boelter relation were done at

Reynolds numbers between 10

and 10

, which is two or three orders of magnitude

less than our situation. It may simply be that the heat transfer parameterisation is

not very accurate. Possibly more likely is that other physical processes contribute to

a larger effective heat transfer; for example, mechanical erosion of ice by the turbu-

lent, sediment-laden ﬂow. In either event, it seems there are good reasons to suppose

that a practical value of γ may be much smaller than given in (11.21).

It has to be said in any case that inclusion of the temperature equation in the form

(11.163) is not a straightforward addition to the numerical problem of solving the

Nye equations. The reason for this is that when X

∗

> 0, the θ equation (11.163)

must be solved in a direction away from the seal: that is, we prescribe (or in fact

require) θ =0atX =X

∗

, and must solve for θ by stepping backwards into X<X

∗

and forwards into X>X

∗

. This is not an insurmountable problem, but it is at least

awkward, because of the singularity at X = X

∗

. A reasonable alternative would

be to ignore advection if X

∗

< 0, and only solve (11.163)ifX

∗

> 0. If the lake

temperature is positive, this gives a discontinuity in θ at ﬂood initiation, and this

would cause further numerical awkwardness.

782 11 Jökulhlaups

Snout Closure and Open Channel Flow If we reconsider the channel closure

equation

∂S

∂t

|Q|

8/3

−S|N|

n−1

N, (11.167)

it is clear that the boundary condition N =0 at the snout x =1 is problematic, since

it predicts indeﬁnite opening of the channel. The problem is that the closure rate

term is based on Nye’s (1953) calculation of closure of a cylindrical borehole in an

inﬁnite medium, which becomes increasingly irrelevant at the snout. It is simple to

modify Nye’s analysis to consider the effect of a stress-free outer boundary at radius

, and the effect of this is to modify the closure term in (11.167) so that the closure

equation becomes

∂S

∂t

|Q|

8/3

−

S|N|

n−1



1 −





1/n



, (11.168)

where S

=πR

. This limits the growth of S because S ≤S

everywhere; (11.168)

applies if S<S

, and is replaced by the condition S =S

if S ≥S

. In practice we

might take S

proportional to ice depth (perhaps to some power).

A further complication is that in practice the channel may reach atmospheric

pressure at some distance up stream from the snout. In this case, the location of the

position where N =0 is a free boundary, with the extra condition being that the ﬂux

becomes equal to the open channel value.

Inertia Terms One of the subtleties that Spring and Hutter introduced was the

inertial acceleration terms in the water momentum equation. Later, Clarke (2003)

found a way to solve this formulation of the problem numerically. The inertial terms,

when scaled, are multiplied by a form of Froude number squared, but one computed

with the ice depth, not the channel depth (see Question 11.5). In general, this term

is small and can be safely neglected. Inclusion of inertia terms simply introduces

complication without any advantage.

Channel Roughness In his seminal paper, Nye (1976) used a Manning roughness

of n



= 0.12 m

−1/3

s in order to ﬁt the rising limb of the 1972 Grímsvötn hydro-

graph. This represents a very rough channel, and Clarke (2003) thought that such a

value was too high. Nye only used this to ﬁt the rising limb, and did not otherwise

solve his model. In fact, it is perfectly possible to ﬁt the whole 1972 hydrograph

using lower choices of roughness; for example, Fowler (2009) ﬁtted the 1972 peak

discharge and duration using values A

= 10 km

(cf. Björnsson 1992, Fig. 2)

and n



= 0.04 m

−1/3

s: see also Question 11.7. On the other hand, we might well

suppose that a sediment-laden torrent at Reynolds number 10

might well be very

rough.

Our previous choice of A

= 30 km

(after (11.30)) corresponded to the maximum area before

1940, since when the lake area has declined; 10 km

is approximately the minimum lake area in

1972. A

is an approximately linearly increasing function of lake level, and hence a decreasing

function (see (11.25)) of N

11.8 Notes and References 783

The Viscous Beam Thin ﬁlms abound in applied mathematics, having applica-

tions in, for example, glass blowing, foam drainage, and coating ﬂows, and in the

ﬁrst two of these, where there is no shear stress applied at the upper and lower sur-

faces, the problem is essentially the same as that describing ice shelves, or where

there is a load, ice cauldrons. An entry into this literature is through the papers by

Howell (1996) and Teichman and Mahadevan (2003), for example.

Floods from Ice Sheets The description of the Channelled Scablands of Eastern

Washington State and their origin by massive ﬂoodwaters is due to Bretz (1923,

1969). Initially the ﬂoods were associated with glacial meltwater, and later with the

ice-dammed glacial Lake Missoula. The sequential nature of the Missoula ﬂoods is

described by Waitt (1984), for example.

The 8,200 year cold event is discussed by Alley et al. (1997), and a theoretical

discussion is given by Clarke et al. (2004). Commentary on this paper by Sharpe

(2005) and the authors’ reply (Clarke et al. 2005) focusses on the related work by

John Shaw and his co-workers (e.g., Shaw 1983; Shaw et al. 1989). Shaw’s central

thesis is that drumlins (see Chap. 10) were formed subglacially by ﬂowing meltwa-

ter. The analogy with ﬂuvial erosional forms then dictates high Reynolds number

ﬂows, and consequently that such erosion could only have occurred in huge ﬂoods

(so the argument goes). Shaw and his followers have been somewhat messianic in

their pursuit of this thesis, but it has to be said that many of the central pillars of the

argument are coming to be accepted: the existence of subglacial lakes, the existence

of pro- and subglacial ﬂoods. Like a Shakespearean tragedy, the single ﬂaw in the

argument stems from the unnecessary assumption that because drumlins look ‘like’

ﬂuvial erosion forms, they must be such forms. Most scientists are put off the Shaw

theory because of the apparently unrealistic constraints which the theory seeks to

impose: vast lakes, enormous ﬂoods, beyond the ability of a reputable theory to

explain as being physically possible.

Sub-Antarctic lakes are described by Siegert et al. (1996), for example, who give

an inventory of such lakes. A more recent review is by Siegert (2005). The largest

and best known is Lake Vostok (Siegert et al. 2001), which has an approximate vol-

ume of 5,000 km

. Lake Vostok may ‘drain’ by basal freeze on, which currently

appears to remove the lake water. For the sorts of subglacial ﬂoods described by

Denton and Sugden (2005) (see also Sugden and Denton 2004

), some such huge

body of water must drain to the coast, although whether this is possible for Vostok

is not known. Goodwin (1988) describes a jökulhlaup of six months duration ob-

served near the coast in East Antarctica. More recently, Wingham et al. (2006)have

observed small scale ﬂoods of a subglacial lake in the Adventure Trough of East

Antarctica. Erlingsson (2006) raises the possibility of a Vostok jökulhlaup.

Dansgaard–Oeschger events were already discussed in Sect. 2.5.5. Ganopolski

and Rahmstorf (2001) provide the model result that freshwater ﬂux oscillations of

magnitude 0.1 Sv (10

−1

) can cause switching of the circulatory state. Similar

Note: in Fig. 1 of this paper, the two marks of 130° E should both read 163° E (David Sugden,

private communication).

784 11 Jökulhlaups

results are given by Stocker and Wright (1991). Alley et al. (2001) and Ganopolski

and Rahmstorf (2002) suggest that a very weak periodicity may be ampliﬁed to

produce the 1,500 year cycle by means of stochastic resonance.

Mars An elegant, if slightly manic, discussion of recent theories about Mars is

the semi-popular book by Kargel (2004). The list of contents is off-putting, but the

material is stimulating, and there are many excellent photographs from the vari-

ous Mars orbiting spacecraft. The literature on Mars is mostly based around these

images and their interpretation. An early paper on massive ﬂoods is by Baker and

Milton (1974), and a more recent review is that by Baker (2001). Two contrasting

views (CO

or water?) are described by Hoffman (2000) and Coleman (2003), and

the possibility of jökulhalups on Mars, possibly associated with sub-ice volcanoes,

is considered by Chapman et al. (2003).

11.9 Exercises

11.1 In the absence of lake reﬁlling at the inlet, a suitable boundary condition for

(11.17)ats =0isQ =0. Assuming Q

is chosen so that Ω =1 (why?), ﬁnd

typical values of the dimensionless parameters, and hence derive an approx-

imate ordinary differential equation for N describing steady state drainage.

Show further that, if δ is small, a further simpliﬁcation is possible, and in this

case derive an approximate drainage law in the form N =cQ

, and give ex-

plicit expressions and typical values for c and ν. The approximation δ →0is

a singular perturbation; where does if fail, why, and what is the resolution?

11.2 Suppose the hydraulic gradient near X =0 is given by

Φ =1 −ae

−bX

and N and Q are determined on 0 <X<∞ by

Φ +

∂N

∂X

=N sgnQ,

Q =ω(X −X

∗

with

N →1asX →∞,

N =N

on X =0,

=−(ν +ωX

∗

where ω and ν are small. Show that

(0) =2e

∗

−



1 +

b −1



−1

−(b−1)X

∗

and deduce that the seal is weak (seal breaking occurs when the lake level is

below ﬂotation: N

(0)>0) if a<b+1.

11.9 Exercises 785

11.3 Use the relations



fρ



3/8

=(Kt

)

−1/n

to ﬁnd explicit relations for the scales S

, m

, etc. in terms of Q

, and hence

show that the parameters

ε =

,δ=

are given by

ε =

,δ=



1/4

11/8

KL(fρ

3/8



1/n

11.4 Use the deﬁnitions



fρ



3/8

=(Kt

)

−1/n

to estimate values for the peak discharge Q

max

∼Q

and the total discharge

max

∼ Q

(assuming A

is constant). By varying in turn A

, K, f and

, show that

max

∝V

max

and compare the values of b with the Clague–Mathews result b =

, or that

of Björnsson, b =1.84.

[The Clague and Mathews (1973) peak discharge/volume relationship is

a venerable curiosity of subglacial hydrology, and appears to have no simple

explanation. Other studies include those by Björnsson (1992), Walder and

COsta (1996), and Ng and Björnsson (2003), the last of which provides a

detailed analytic investigation.]

11.5 The momentum equation in the model for subglacial drainage is modiﬁed to

include inertial acceleration terms, thus

+uu

) =Φ +

∂N

∂x

−

fρ

gQ|Q|

8/3

where Φ =

is the basic hydraulic gradient, d

is ice depth, l is glacier

length, N is the effective pressure, Q is volume ﬂux, S is cross-sectional

area, and u =

is the mean velocity. By scaling the variables as

Φ =Φ

∗

,N∼N

,x∼l, Q ∼Q

S ∼



fρ



3/8

,t∼

786 11 Jökulhlaups

show that the dimensionless form of this equation can be written as

(εu

+uu

) =Φ

∗

+δ

∂N

∂x

−

Q|Q|

8/3

where

δ =

,ε=

and u

. Show that, if Q

= 10

−1

, S

= 10

, g = 10 m s

−2

and d

=500 m, then F

∼0.02, and explain why more generally the accel-

eration terms can be neglected. What complications do they cause if they are

included?

11.6 In an approximate model for the channel effective pressure between ﬂoods,

N satisﬁes the Riccati equation

Φ(X) +

∂N

∂X

sgn(X −X

∗

with boundary conditions

N →1asX →∞,

N =N

at X =0,

where we assume Φ(∞) =1.

Assume that

Φ =1 −ae

−bX

and use appropriate substitutions in X ≶ X

∗

of the form N = w



/cw to

reduce the model to a pair of linear equations for w. Hence show that for

X>X

∗

N =−



,v=J



λe

−X/ν



where

λ =

√

,ν=

and for X<X

∗

N =



,w= J

iν



iλe

−X/ν



+αY

iν



iλe

−X/ν



where α must be chosen so that N is continuous at X

∗

Hence show that

(0) =

λJ



(λ)

νJ

(λ)

and deduce that the seal is weak if



(λ)

> 0, and thus if

a<a



(2/b),1



where j



ν,1

is the ﬁrst zero of J



(z).

11.9 Exercises 787

Use tables to evaluate j



ν,1

for ν =

, 1, 2, 3, 4, 5, and compare the conse-

quent values of a

for b<4 with those obtained from the asymptotic result



ν,1

≈ ν + 0.81ν

1/3

when ν  1, and deduce that the small b approxima-

tion is accurate in this range. By plotting the results numerically, show that a

useful approximation to a

is then

≈1.2 +b,

for b<4.

An even better approximate model is to take

Φ +

∂N

∂X

sgn(X −X

∗

where σ = ω

1/11

≈ 0.44 (since Q ∼ ω)forω = 1.2 × 10

−4

. In this case,

modify the analysis above to show that a seal is weak if

a<a



σbj



(2/σ b),1



and deduce that



≈1.2 +0.44b

for b<9.

11.7 The dimensionless, reduced Nye model for jökulhlaups depends on dimen-

sionless parameters ω =δΩ and ν deﬁned by

ω =



11/8

1/4

KL(fρ

3/8



1/n

,ν=

and the dimensional results depend on the volume ﬂux scale Q

and time

scale t

deﬁned by

(fρ

3/8

1/4







fρ



3/8



n+1



4/(3n−1)

The periodic solutions therefore have dimensional period

P =

∗

and the ﬂoods have peak discharge

max

∗

and duration

∗

788 11 Jökulhlaups

In general, the starred quantities are functions of ω and ν, but let us assume

they depend only on the quantity

α =

4n/(4n−1)





4n/(4n−1)



11/8

KL(fρ

3/8



4/(4n−1)

Suppose that a particular numerical solution, with particular choices for

the scales and parameters (and in particular f =f

, A

and m

produces ﬂoods of (dimensional) duration t

, peak discharge Q

and with

period P

, whereas the actual observed ﬂoods have corresponding quantities

, Q

and P

. Show how, by choosing new values f = f

, A

= A

and

, the numerical solution can be made to ﬁt the data, and give explicit

expressions for the ratios

and

11.8 The Nye model for jökulhlaups can be reduced to the system

∂S

∂t

|Q|

8/3

−S|N|

n−1

Q =ω(X −X

∗

Φ +

∂N

∂X

Q|Q|

8/3

with the boundary conditions that

∂N

∂t

(0,t) = Q(0,t)−ν,

∂N

∂X

→0asX →∞,

assuming that the effect of thermal advection is ignored.

Suppose that ω ∼ν  1. Show that between ﬂoods, the variables can be

rescaled (explaining how) so that the equations take the form

∂s

∂T

|q|

8/3

−|Π |

n−1

Π,

q =ξ −ξ

∗

Φ +Π

|q|q

8/3

subject to

=−1 +αq on ξ =0,

→0asξ →∞,

where

α =

4n/(4n−1)

,β=

(n+1)/(4n−1)

and show that for values of ω, ν ∼10

−3

, β 1butα =O(1) (e.g., if n =3).

11.9 Exercises 789

Show that if β is put to zero, then the Nye model between ﬂoods (when

∗

> 0 and Π>0) can be solved by computing the solution to the system

dξ



|ξ −ξ

∗

1/4n



8n/11

sgn(ξ −ξ

∗

) −Φ(ξ),

subject to

R →Φ

11/8n

∞

as ξ →∞

and



∗(1/4n)



ξ=0



=−1 −αξ

∗

Consider and explain the difﬁculties in solving this problem numerically.

Explain why this approximation can break down if ν ω.

11.9 The vertically integrated dimensionless viscous ice beam equations are given

∂T

∂x

+ν

∂U

∂x

=γβNh

+β(s −h)s

∂M

∂x

−ν

∂L

∂x

+S =−γβNhh

−



−h



∂S

∂x

=−

γβN

where M, S, T , U and L are deﬁned by

M =−



2zτ

dz, S =



dz, T =



dz,

U =



dz, L =



zσ

dz,

s(x,t) is the ice surface elevation, h(x, t) is the lake roof, and p + τ

−ν

Show that if

−p +τ

=β(s −z), τ

on x =x

, which denote the subglacial lake margins, then

2T +ν

U =

β(s −h)

−M +ν

L =

β(s −h)

(s +2h),

S =0,

at x =x

. Show that



−

Ndξ=0,