Fowler A. Mathematical Geoscience

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250 4 River Flow

Fig. 4.16 Asketchmapof

the river Severn

Fig. 4.17 Proﬁle of the Severn during passage of a bore. Note that high water occurs someway

below the bore (the tide continues to come in after the passage of the bore), but that the tide near

Sharpness already starts to ebb before the bore reaches Maisemore

wave speed,

a smooth wave cannot occur, and a shock must form, as indicated in

Fig. 4.18.

We want to study this phenomenon in the context of the St. Venant equations

(4.49), where for a wide channel, we choose the hydraulic radius and cross sectional

area to be

R =h, A =wh, (4.130)

where w is the width, and is taken to be a prescribed function of s. The phenomenon

of concern occurs over the length of the river, so that long wave theory is appropri-

ate. From Fig. 4.17, a suitable length scale is of the order of 45 km, where the

We assume the Froude number F is less than one at low stage, which is the realistic condition; in

that case, one wave travels upstream. If F>1, a standing wave would form at the boundary.

4.5 Nonlinear Waves 251

Fig. 4.18 Idealised (and

highly exaggerated) river

basin geometry

length scale used in writing (4.49)isd/sinα, and is 2 km if we take d =2 m and

S = sin α = 10

−3

. If we take a typical velocity upstream as 2 m s

−1

, then the cor-

responding time scale is 10

s, or 15 minutes, and the Froude number is about 0.3.

The scale up in distance is thus of order 22, while that in time to the half-period of

tidal oscillations is similar. This suggests that we rescale both time and space as

t ∼

,s∼

, (4.131)

where a plausible value of ε may be of order 0.05. In this case (4.49) can be writ-

ten in the form (where now, because u will be negative during inﬂow, we take the

friction term in the corrected form ∝|u|u)

+(wuh)

=0,

εF

+uu

) =1 −

|u|u

−εh

(4.132)

or equivalently in the form



±F

∂

∂t



√

h ±Fu



∂

∂s





√

h ±Fu



=1 −

|u|u

∓

εFw



√

, (4.133)

which shows explicitly that the characteristic wave speeds are

√

+u, (4.134)

as we found before. Finally, we wish to study the situation shown in Fig. 4.18, where

the tidal range is signiﬁcantly larger than the river depth. The simplest choice is to

suppose the tidal amplitude is also O(1/ε), so that appropriate boundary conditions

for (4.132)are

wuh =1ats =0,

h =

(t)

at s =1,

(4.135)

representing a constant upstream volume ﬂux, and a prescribed tidal range.

252 4 River Flow

The assumption that ε 1 allows us to solve (4.132) asymptotically. The solu-

tion has two parts, river and estuary, joined at a front which we denote by s =s

Upstream, for s<s

, the ﬂow is quasi-stationary, and we have, to leading order,

wuh ≈1, 1 −

u|u|

≈1, (4.136)

whence

u ≈w

−1/3

,h=w

−2/3

. (4.137)

The steady solution of (4.132) is appropriate, because the sub-characteristic wave

propagates downstream, and after any initial transient, the upstream boundary con-

dition leads to a steady ﬂow.

Downstream, for s>s

, we write

h =

, (4.138)

so that

+(wuH )

≈0,

1 −H

≈0

(4.139)

(the surface is ﬂat); from this we have

H ≈s −1 +H

(t), (4.140)

and from this there follows

u ≈

−



wds

, (4.141)

where we choose the integration constant for matching purposes at s

.Alsotomatch

the solution to that in s<s

, we need to take

=1 −H

. (4.142)

Transition Region

At the front, we deﬁne

s =s

+εX, ˙s

=c, w



(t)



; (4.143)

then to leading order we have

−cw

+(w

hu)

=0,

(u −c)u

=1 −

u|u|

−h

(4.144)

with boundary conditions

h →h

−

−2/3

,u→u

−

−1/3

as X →−∞,

h ∼X, u ∼c as X →∞,

(4.145)

4.5 Nonlinear Waves 253

in order to match to the upstream and downstream solutions. Note that this transition

region, like that for the monoclinal ﬂood wave, is mediated by the full St. Venant

equations, but without a diffusive term. Only the conditions on h in (4.145) are nec-

essary, those on u following automatically. A ﬁrst integral of the mass conservation

equation (4.144)

gives

(u −c)h =K =



−1/3

−c



−2/3

, (4.146)

and from this we ﬁnd

−|K +ch|(K +ch)

−K

. (4.147)

This can be compared with (4.100). The difference in the present case is that c and

K in (4.143) and (4.146) are given, and the question is only whether a solution exists

joining h = h

−

−2/3

upstream to the downstream solution h ∼X. Note that as

X →−∞, K +ch →w

−1

, so that h →w

−2/3

can consistently be satisﬁed.

Let us suppose that the tide is coming in, thus c<0. We suspect that a smooth

solution in the transition region may not be possible if −c is greater than the up-

stream wave speed. Using (4.134) and (4.145), this condition can be written in the

form

−c>w

−1/3



−1



(4.148)

(assuming F<1). If we suppose that the opposite inequality holds, i.e., −c<

−1/3

(

−1), then a little algebra shows that this is precisely the criterion that

−

−2/3

>(KF)

2/3

, (4.149)

i.e., the denominator of (4.147) is positive. To see that there is a solution of this

problem in this case, we need to show that the numerator of the right hand side

(4.147) is also positive, for then h will increase indeﬁnitely as required.

The numerator, N , is given by

N =



−w

−2



−





−1



h −w

−2/3







−1



h −w

−2/3



−w

−2



(4.150)

Both expressions in curly brackets are zero when h =h

−

at X =−∞;forh slightly

greater than h

−

, the left curly bracketed expression is positive, while the right curly

bracketed expression decreases, since c<0. The numerator is thus positive for

h −h

−

small and positive, and remains so. From this it follows that a solution of

the transition problem exists if −c<w

−1/3

(

−1), and thus a bore will not form.

It remains to be shown that no solution exists if the opposite inequality, (4.148),

holds. In this case the denominator of the right hand side of (4.147) is initially

negative. As before, the numerator is positive if h>h

−

, and equivalently negative

if h<h

−

, thus implying h

< 0ifh>h

−

, and h

> 0ifh<h

−

. This means

solutions of (4.147) can only approach h

−

as X →∞, and no transition solution

254 4 River Flow

Fig. 4.19 Bore formation occurs for large tides and rapidly widening rivers with reasonably sized

Froude numbers. If the tide oscillates sinusoidally and the river slope is constant, then the front

position s

will trace an ellipse as shown in the (s

, ˙s

) plane. For a funnel-shaped river, the width

w decreases as s

decreases, so that (

−1)w

−1/3

is a decreasing function of s

,asshown.Bore

formation therefore occurs according to (4.148) for the solid tidal curve, but not for the smaller

amplitude dotted one

exists. This suggests another form of solution, one in which a discontinuity forms at

the critical condition

−˙s

=w(s

)

−1/3



−1



, (4.151)

and thereafter propagates upstream as a shock front. This is the bore. Figure 4.19

shows a schematic illustration of the criterion (4.151) for bore formation.

Propagation of the Bore

The outer river and estuary solutions (4.137), (4.140) and (4.141) remain valid after

the formation of a shock, but the transition region is replaced by a shock at s

, where

the values of h

−

and u

−

(given by (4.137) with w = w

) jump (up) to values h

and u

, which have to be determined along with s

. Initially h

and u

are O(1),

and we anticipate that this remains true; in this case s

is still given by

=1 −H

+O(ε); (4.152)

the location of the bore is essentially determined by the tidal range. Jump conditions

of mass and momentum across the developing bore then imply that the bore speed

˙s

=c satisﬁes

c =

[hu]

−

[h]

−

]

−

[hu

]

−

[hu]

−

, (4.153)

and these two relations serve to determine h

and u

, since c =−

4.5 Nonlinear Waves 255

Shock Structure

We can use the transition equations (4.144), modiﬁed by the addition of the diffusive

term in (4.127), to study the shock structure of the bore. The equations then take the

form

−cw

+(w

hu)

=0,

(u −c)u

=1 −

u|u|

−h

κF

∂

∂X



∂u

∂X



(4.154)

and the boundary conditions are still (4.145). The difference with the preceding

analysis is that when a bore forms, we expect the diffusive term to act as a singular

perturbation which allows the matching of two distinct outer solutions through an

interior shock (the bore). Writing

u =c +

, (4.155)

we ﬁnd that h satisﬁes

∂h

∂X

−|K +ch|(K +ch)

−K

−

κF

−K

∂

∂X



∂h

∂X



. (4.156)

As discussed before (4.151), the only way h can approach h

−

as X →−∞in

bore-forming conditions is if the outer solution (where κ =0) in X<0is

h ≡h

−

,X<0. (4.157)

We suppose that h jumps through the shock to a value h

−

. According to

the argument following (4.150), the numerator of (4.147) for the outer solution in

X>0 is then positive, and so, providing h

, the outer solution for h will

increase monotonely from h

, and h ∼X as X →∞. It only remains to show that

a shock structure exists connecting h

−

to h

>(KF)

2/3

Supposing without loss of generality the shock to be at X =0, we deﬁne

X =κKF

ξ (4.158)

(noting that K>0), so that to leading order (4.156) becomes

∂h

∂ξ

=−

−K

∂

∂ξ



∂h

∂ξ



. (4.159)

Integrating this, we ﬁnd

∂h

∂ξ

=−h





−h

−





−



. (4.160)

Consideration of the right hand side of this equation shows that if h

−

then −



is zero at h =h

−

, negative for h>h

−

until it becomes positive for large

h. Thus there is one further zero of h



at h

−

, and h



> 0 between these two

values, always assuming that h

−

, which is guaranteed by (4.149). Thus the

shock layer structure takes h monotonically from h

−

to h

, given by



−h

−





−



, (4.161)

256 4 River Flow

and it only remains to check that h

>(KF)

2/3

, so that the outer solution to (4.147)

in X>0 does indeed increase as X →∞. This is clear from the deﬁnition of −



given by (4.160), which shows that −



is a convex upwards function G(h), and in

particular shows that G



)>0. Since from (4.159),



(h) =

−K

, (4.162)

we can deduce that indeed h

>(KF)

2/3

This analysis shows that in bore-forming conditions, the diffusive term in (4.154)

does indeed allow a shock structure to exist, and this describes what is known as a

turbulent bore, appropriate at reasonably large Froude numbers. The Severn bore

showninFig.4.15 is an example of an undular bore, appropriate at lower Froude

numbers, and consisting of an oscillatory wave train. The St. Venant equations do

not appear to be able to describe this kind of bore, where the oscillations have a

wavelength comparable to the depth, and the vertical velocity structure may need to

be considered in attempting to model it. This is discussed further below.

4.6 Notes and References

A preliminary version of the material in this chapter is in my own book on modelling

(Fowler 1997), although with much less detail than presented here. The general

subject of river ﬂow is treated in its contextual, geographical aspect by books on

hydrology, such as those of Chorley (1969) or Ward and Robinson (2000). Ward and

Robinson’s book, for example, deals with precipitation, evaporation, groundwater

and other topics as well as the dynamics of drainage basins, but is less concerned

with detailed ﬂow processes in rivers. For these, we turn to books on hydraulics,

such as those by French (1994)orChow(1959). A nice book, which bridges the

gap, and also includes a discussion of sediment transport and channel morphology

and pattern, is that by Richards (1982).

Roll Waves Flood waves and roll waves have been discussed from the present

perspective by Whitham (1974). The linear instability at Froude number greater than

two was analysed by Jeffreys (1925), and the ﬁnite amplitude form of roll waves

was described by Dressler (1949), whose presentation we follow here. The book by

Stoker (1957) gives a nice discussion, as well as a useful photograph of roll waves

on a spillway in Switzerland. The eddy viscous diffusive term in (4.127) was added

by Needham and Merkin (1984). Balmforth and Mandre (2004) provide a thorough

review, and also provide a discussion of the mechanics of wavelength selection.

They also, following Yu and Kevorkian (1992), provide a weakly nonlinear model

for roll wave evolution when F −2  1; a strongly nonlinear model would be more

relevant at higher F . Their experiments are consistent with the idea that the form of

the inlet condition is instrumental in determining the roll wavelength.

4.6 Notes and References 257

Tidal Bores The effect of tidal variations on river ﬂow is discussed by Pugh

(1987); in particular, he describes the phenomenon of the river bore. Another use-

ful little book is that by Tricker (1965). The literature on bores seems to be rather

sparse, although the phenomenon itself has been well known for a (very) long time.

Chanson (2005) refers to the fact that the mascaret of the Seine river in France was

documented in the ninth century. Lord Rayleigh, while president of the Royal So-

ciety, wrote down the jump conditions for the bore velocity over a hundred years

ago (Rayleigh 1908). There is a very informative article by Lynch (1982), prior to

which the principal analysis is that of Abbott and Lighthill (1956), who analyse the

St. Venant equations, and apply their results to the Severn bore. The presentation is

extremely opaque, however. The little book by Rowbotham (1970) is a gem, and has

many other striking photographs besides that shown in Fig. 4.14.

More recently, there has been an upsurge of interest in modelling bores. Su et al.

(2001) construct a numerical model of the turbulent bore of the Hangzhou Gulf and

Qiantangjiang river in China using the St. Venant equations. In a number of papers,

Chanson and co-workers have studied the dynamics of undular bores (Wolanski et

al. 2004; Chanson 2005), both observationally and experimentally. Chanson (2009)

reviews the observational and experimental literature, with numerous illustrations.

In order to obtain an oscillatory wave train (such as one also ﬁnds in capillary

waves), it seems that a higher derivative term in (4.160) might be necessary, either

as h

ξξξ

or from a term u

XXX

in (4.154). Such terms are commonly found in higher-

order approximations to water wave equations, as for example in the Korteweg–de

Vries equation. To get a ﬂavour of such an analysis, we consult the derivation of

the Korteweg–de Vries equation by Ockendon and Ockendon (2004, pp. 106 ff.).

Reverting to dimensional coordinates, their derivation of the Korteweg–de Vries

equation takes the form, assuming a backwards travelling wave,

+···=

√

gd d

sss

. (4.163)

If we simply suppose that such a term can be added to the St. Venant equation, then,

using the scales in (4.48), the St. Venant equations (4.50)or(4.127) become

+(wuh)

=0,

+uu

) +h

=1 −

|u|u

κF

∂

∂s



∂u

∂s



sss

(4.164)

Repeating the shock structure analysis, (4.156) is replaced by

−

FKS





XXX

+κF





+P(h)h

−N(h)=0, (4.165)

where

P(h)=h

−K

,N(h)=h

−|K +ch|(K +ch) (4.166)

(N(h) is the numerator in (4.147) discussed following (4.150)).

We write

h =h

−

φ, N =h

−

n(φ), P =h

−

p(φ), c =−



−

V, (4.167)

258 4 River Flow

Fig. 4.20 Model of a

turbulent bore. Solution of

(4.165) in the form (4.171),

using values F =1.5, V =0,

β =0.1, δ =0.01. The time

step used is 10

−5

,andthe

plot takes h

−

=1 in its scales

for X and h

whence (4.145) and (4.146)imply

K =h

3/2

−

(1 +V), (4.168)

and hence

n(φ) =φ

−|1 +V −Vφ|(1 +V −Vφ), p(φ)=φ

−(1 +V)

. (4.169)

Lastly we put

X =h

−

Z. (4.170)

Then (4.165) becomes

−δ





ZZZ

+βφ





+p(φ)φ

−n(φ) =0, (4.171)

where

δ =

F(1 +V)S

11/2

−

,β=

κF

(1 +V)

3/2

−

, (4.172)

and both are small.

The boundary conditions for φ are that

φ →1asZ →−∞,φ∼ Z as Z →∞. (4.173)

Figures 4.20 and 4.21 show numerical solutions of the transition equation (4.171)

for two different values of β. The ﬁrst corresponds to a relatively high value of β,

when δ is sufﬁciently small to be ignored, and the preceding shock structure analysis

(following (4.154)) is valid. Formally this requires δ β

At lower values of β, however, it is inadmissible to neglect the third derivative

term. To analyse what happens in this case, write

Z =

√

δζ, (4.174)

and deﬁne

μ =

√

. (4.175)

4.6 Notes and References 259

Fig. 4.21 Model of an

undular bore. Solution as for

Fig. 4.20, except that

β =0.001

Assuming δ  1, we can neglect the term in n within the transition zone, so that

−





ζζζ

+μφ





+p(φ)φ

≈0. (4.176)

The turbulent bore is regained if μ 1. For the case μ  1, deﬁne

ψ =1 −

, (4.177)

whence



(1 −ψ)





(1 −ψ)







1 −ψ





(1 −ψ)

=0. (4.178)

Suppose ﬁrst that μ is small; then a ﬁrst integral of (4.178) with μ =0is





(ψ) =0, (4.179)

where



(ψ) =



1−ψ

p(φ) dφ, W(0) =0. (4.180)

Integrating and changing the order of integration, we can write

W(ψ)=



1−ψ



ψ −



1 −



p(φ) dφ. (4.181)

As a function of ψ , W(0) = W



(0) = 0, and (since p(1)<0, equivalent to the

bore-forming condition (4.148)) W



(0)<0; thus W is negative for small ψ>0.

Since W



(ψ) =

p(φ)

(1−ψ)

, and p is an increasing function of φ, we see that W reaches

a negative minimum, and thereafter increases, tending towards ∞ as ψ → 1 and

φ →∞.

(4.179) is the equation of a nonlinear oscillator, and shows that φ increases from

zero at Z =−∞, and then oscillates about the minimum of W . In fact with μ =0,

there would be precisely one oscillation, with φ returning to zero at Z =+∞.This

does not happen for two reasons. The term in μ is a damping term (this is clear

in (4.176) if the coefﬁcient φ

is ignored; alternatively one can view (4.176)asa