Fowler A. Mathematical Geoscience

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190 3 Oceans and Atmospheres

with boundary conditions

ψ =0atX =0,ψ→

as X →∞; (3.230)

the solution is

ψ =

1 −e

βX

, (3.231)

and represents a northwards current of magnitude v ∼

at the western boundary.

Thus the circulation is highly skewed.

There are also boundary layers adjoining the upper and lower boundaries, and

these are similar to each other. For example, near the lower boundary, we put

y =ε

1/2

Y, (3.232)

so that

−βψ

≈1 +ψ

, (3.233)

with boundary conditions

ψ =0atY =0,ψ→

1 −x

as X →∞; (3.234)

the appropriate ‘initial’ condition for the parabolic equation (3.233) is that

ψ =0atx =1. (3.235)

(3.233) has a similarity solution, given by

ψ =

1 −x



1 −f(η)



,η=



1 −x



1/2

, (3.236)

where f satisﬁes the differential equation



+2ηf



−4f =0, (3.237)

with boundary conditions

f(0) =1,f(∞) =0. (3.238)

The solution is the error function integral

f(η)=i

erfc η. (3.239)

See Abramowitz and Stegun (1964). The error function integrals are deﬁned iteratively by

erfcη = i

n−1

erfcη,i

erfcη = erfcη, and satisfy the equations f



+ 2ηf



− 2nf

= 0, where

(η) =i

erfcη; this is easily shown inductively by differentiating the equation, which shows that



n−1

3.9 Western Boundary Currents: The Gulf Stream 191

The assumption of a square box is irrelevant to the method of solution. An ar-

bitrary domain will have a solution structure of the same form, with an attached

western boundary layer of thickness O(ε), in which ψ

∼ 1/ε. We can now as-

sess the neglect of the lateral drag term in E

. The size of this term in the western

boundary layer is O(E

/ε

), and therefore the boundary layer structure above is

valid, providing E

ε

, i.e.,





2sinλ



3/2

. (3.240)

If we use our estimates, then we have E

∼ 0.4 × 10

−7

, (

2sinλ

)

3/2

∼

0.8×10

−8

(2sinλ)

3/2

and (3.240) is barely feasible. This suggests that it may be more realistic to suppose

that the lateral drag term in E

controls the western boundary layer structure, and

we now consider its effect. In any case, the basal drag term can only allow the no-

ﬂow-through condition, and the lateral term is necessary to bring the velocity to

zero.

3.9.2 Effects of Lateral Drag

For simplicity, we neglect the basal drag term, so that the model for the stream

function is

βψ

=−1 +E

∇

ψ, (3.241)

together with the conditions

ψ =

∂ψ

∂n

=0on∂B. (3.242)

The outer solution ψ ∼

1−x

is as before, and the appropriate rescaling in the

western boundary layer is

x =

1/3

, (3.243)

and then the boundary layer equation is

βψ

=ψ

XXXX

, (3.244)

together with the boundary conditions

ψ =ψ

=0atX =0,ψ→

as X →∞. (3.245)

The solution of this is

ψ =



1 −exp



−

1/3



cos

1/3

√

sin

1/3

√

3 X



; (3.246)

192 3 Oceans and Atmospheres

note the oscillatory decay away from the boundary layer. We leave the solution in

the horizontal boundary layers as an exercise (see Question 3.13).

3.10 Global Thermohaline Circulation

While the surface winds drive an oceanic circulation which is conﬁned to the rel-

atively near surface, there is a deeper circulation which is driven ultimately by the

same source as that which drives the weather systems, that is to say, the radiatively

induced poleward temperature gradient. While the atmospheric circulation can be

viewed as a form of thermal convection mediated by the effects of a strong rotation,

the deep oceanic circulation can be viewed as a form of thermal convection medi-

ated by the strong effects of salinity. As such, this large scale convection is called the

global thermohaline circulation, and it is often, slightly misleadingly, described as a

conveyor belt, with descending water in the North Atlantic travelling southwards as

North Atlantic Deep Water (NADW) to the Antarctic, where the conveyor sends it

to the Indian and Paciﬁc Oceans. There it rises, and eventually returns to the North

Atlantic as surface water.

The poleward convection in the oceans is not affected by rotation in the same way

as it is in the atmosphere, because of the presence of continents. In particular, con-

vection in the Atlantic is channelled by the conﬁning continents of the Americas to

the west, and Europe and Africa to the east, and so it runs north to south. However,

the oceans are saline, and this has a signiﬁcant effect on the convection, because of

the large contribution of salt to the density. While there is no source or sink of salt,

salinity gradients are generated either by (stabilising) freshwater inputs via conti-

nental river outﬂow, or by (destabilising) evaporation, which provides a freshwater

vapour ﬂux to the atmosphere and a consequent saliniﬁcation of the ocean surface.

If we remove the wind-driven circulation from the picture entirely, we think of

competing forms of thermal and saline convection, for example in the North At-

lantic. A purely thermal convection is produced by the equator to pole temperature

gradient, and will cause a convective circulation in the form of a large scale roll. The

Rayleigh number is so enormous that the steady roll may be unstable, with intermit-

tent plumes developing out of the surface boundary layer, but one would expect the

convective style to be essentially circulatory.

If, on the other hand, one removes the thermal buoyancy entirely, then the evap-

oration of the surface waters near the equator will lead to a destabilising surface

salinity, but the consequent convection will be more ﬁnger-like, and localised, since

there is no large scale imposed salinity gradient.

Superimposing these two notions, we might suppose a circulatory thermal con-

vection, with the unstable saline surface boundary layer providing a series of lo-

calised downwelling plumes. In practice, such deep water formation regions do in-

deed exist, but there are not many of them. The two principal ones are in the North

Atlantic, which forms the North Atlantic Deep Water, and in the Weddell Sea in

the Antarctic, which forms the Antarctic Bottom Water (ABW). Enormous mix-

ing takes place at the interface between these two water masses, and the Antarctic

3.11 Tides and Tsunamis 193

circumpolar current, which rotates west to east round Antarctica, acts as a kind of

mixer, spraying out the NADW into the Paciﬁc and Indian oceans, where it even-

tually wells up and returns to the North Atlantic surface water by various routes:

through the Drake Passage between South America and Antarctica, from the Arctic

via the Bering Strait, through Indonesia and round South Africa.

Although the origin of the thermohaline circulation may reside in the poleward

thermal gradient, its nature may be largely salinity driven. The Atlantic surface wa-

ters are more saline than those of the Paciﬁc, there being a net freshwater vapour

ﬂux from the Atlantic basin towards the Paciﬁc. As was discussed in Sect. 2.5.7,it

is thought that the rapid climate changes indicated by Dansgaard–Oeschger events

may be associated with switches in the strength of the North Atlantic circulation—

the so-called North Atlantic salt oscillator. The idea of this is that when the circula-

tion is strong, it is warmer in the north, so that ice sheet melting is increased. The

increased freshwater ﬂux to the North Atlantic reduces the salinity of the surface

ocean, thus reducing the air temperature, until eventually the circulation may even

switch off. As the air temperature is reduced, however, melting on the ice sheets

decreases and may cease entirely, allowing the ice sheets to regrow. The consequent

decreased freshwater ﬂux can then allow the oceanic circulation to restart.

3.11 Tides and Tsunamis

We go to the beach, and if we are paying attention, we notice that the tide comes

in twice a day. Most of us know that tides are due to the gravitational attraction

of the Sun and the Moon, and this seems to make sense. The Moon (which has the

dominant effect) exerts an attraction on the water envelope of the oceans, pulling the

water towards the Moon. Since the Earth rotates once a day, the high water remains

stationary with respect to the Moon, and so we get the diurnal tide, apparently. But

why are there then two tides a day? Worse, why is there only one tide a day in

some places, and worst of all why is there sometimes almost no tide at all in certain

locations, for example in the Mediterranean?

The answer to the most obvious of these problems, that of the semi-diurnal tide,

is indicated in Fig. 3.8. Intuitively, we think that the pull of the Moon will cause

a bulge in the oceans only on the side nearest to the Moon. This is because we are

thinking at laboratory scale, and are forgetting the variation of gravity with distance.

The Moon pulls the centre of the Earth with a certain force. On side N of the Earth

in Fig. 3.8, this force is greater, because N is nearer to the Moon; consequently the

ocean surface is pulled towards the Moon. So also is the Earth’s surface, but: the

Fig. 3.8 The attractive effect

of the Moon on the Earth’s

oceans

194 3 Oceans and Atmospheres

Fig. 3.9 Tide-generating

force diagram

Earth is essentially rigid, and this deformation is inconsequential. On the far side of

the Earth, the force of attraction is correspondingly weaker, and relative to the force

on the Earth, the oceans experience a repulsion. Hence the bulge is as shown, and

thus as the Earth rotates, there are two tides a day.

3.11.1 The Tidal Equations

Suppose at a point P on the Earth, the centre of the Moon is at distance r,asshown

in Fig. 3.9. The distance of the centre of the Earth from the centre of the Moon

is denoted d

, and the radius of the Earth is r

. The ﬂuid envelope of the Earth

experiences the gravitational force due to the Earth, but in addition there is a force

towards the Moon. However, to compute the tide-generating force, we must subtract

from this the attractive force of the Moon on the Earth. Thus the tide-generating

force per unit mass at P is

=∇





−





i, (3.247)

where i is the unit vector from the centre of the Earth to the centre of the Moon, G

is the gravitational constant, and M is the mass of the Moon. We can equivalently

write this force as the gradient of a potential,

=GM∇



−

cosξ



. (3.248)

We can simplify this by using the expansion



1 −

cosξ



−1/2

∞



n=0





(cosξ), (3.249)

where P

is the nth Legendre polynomial. Now r

d

; substituting (3.249)into

(3.248) and retaining the ﬁrst signiﬁcant term, we obtain

≈

GMr

∇



(cosξ)



. (3.250)

3.11 Tides and Tsunamis 195

Fig. 3.10 Spherical

trigonometry relating the

angle ξ to the hour angle H ,

the declination δ,andthe

latitude λ. M indicates the

position of the

tide-generating body (e.g., the

Moon), and P is the local

position on the Earth

The second Legendre polynomial is deﬁned by

(cosξ)=



3cos

ξ −1



. (3.251)

Next we need to identify the angle ξ in terms of the normal angles of spherical

polar coordinates. To do this we need a little spherical trigonometry. The geometry

of the situation is indicated in Fig. 3.10, where we take the sphere radius to be one,

without loss of generality. We want to relate the angle ξ to the declination of the

Moon δ, the latitude λ, and the so-called hour angle H . This is simply longitude,

except that the rotation of the Earth causes it to increase with time, speciﬁcally

H =ωt +φ, (3.252)

where ω is the angular speed of rotation of the Earth. The bare bones of Fig. 3.10

are shown in Fig. 3.11. To relate ξ to the other variables, we consider triangles on

the unit sphere, such as that shown in Fig. 3.12. If the lengths of the sides are a, b,

c, and the corresponding opposite angles are α, β and γ , then we have the following

formulae, which are, respectively, the ﬁrst cosine rule and the sine rule:

cosa =cos b cos c +sin b sin c cos α,

sin α

sin a

sinβ

sinb

sin γ

sin c

(3.253)

Applying these formulae to the two triangles in Fig. 3.11 which constitute the

quadrilateral, and bearing in mind that the two basal angles are right angles, we

196 3 Oceans and Atmospheres

Fig. 3.11 The spherical

quadrilateral

derive the formulae

sin ζ =

sin δ

cosX

cosξ = cosζ cos λ +sin δ sin λ, (3.254)

cosζ =cosδ cos H,

and from these we ﬁnd

cosξ =sin λ sin δ +cos λ cos δ cos H. (3.255)

Finally, the tide-generating force can be written as

=D∇χ, (3.256)

where

D =

3GMr

(3.257)

Fig. 3.12 Sides and angles of

a spherical triangle

3.11 Tides and Tsunamis 197

is known as the Doodson number (although it has dimensions), and χ = 2cos

ξ ,

whence

χ =cos

λ cos

δ cos 2H +sin 2λ sin 2δ cos H +



cos

λ cos

δ +2sin

λ sin



(3.258)

The time dependence of the forcing is expressed in the hour angle H, and we see that

the three components represent, respectively, a semi-diurnal forcing (∝ cos 2H ),

a diurnal forcing (∝ cos H ), and a ‘long period’ forcing, independent of Earth’s

rotation, but dependent on longer term orbital variations. Evidently the comparable

but smaller effect of the Sun can be considered in the same way, and will add further

ingredients to the tide-generating force.

Our model for tides is based on the Eqs. (3.1), except applied to an incompress-

ible ocean. We use a depth integrated shallow water theory with a free upper bound-

ary, but it is convenient to write the shallow water equations in vector form, delaying

the intricacies of spherical polar coordinates until later. We deﬁne a vertical coordi-

nate

z =r −r

, (3.259)

and we denote the ocean surface as z =η, and the ocean ﬂoor as z =b. The ocean

depth is thus h =η −b, and we suppose that the depth-averaged horizontal velocity

ﬁeld is u. From ﬁrst principles, mass conservation yields the equation

∂h

∂t

+∇.(hu) =0, (3.260)

where ∇ denotes the horizontal gradient vector. The (horizontal) momentum equa-

tion is obtained from (3.1), and is



+2

×u



=−∇p +ρD∇χ. (3.261)

In deriving this (note that u and ∇ are horizontal) we have integrated over the depth

and then used the mass conservation equation. The term 2

×u is the horizontal

component of the Coriolis force, which is obtained by deﬁning 

to be the vertical

(in the z direction) component of the Earth’s angular velocity. In addition, shallow

water theory implies that

p ≈ρg(η −z), (3.262)

and thus

∇p ≈ρg∇η. (3.263)

Next we scale the equations. We denote the horizontal distance vector on the

sphere as x, and we deﬁne a dimensionless parameter ε as

ε =

, (3.264)

198 3 Oceans and Atmospheres

where d is mean ocean depth. Values of D/g are 0.27 m for the Moon, and 0.12 m

for the Sun, while d ≈3,800 m, so the parameter ε is very small, having a typical

value of order 10

−4

. We scale the variables as follows:

x ∼r

,η∼εd, b, h ∼d, u ∼ε



gd, t ∼

√

, (3.265)

and this yields the non-dimensional system

+∇.(hu) = 0,

h =−b +εη, (3.266)

+ε(u.∇)u +2S sinλ k ×u =−∇η +∇χ,

where k is the unit vector in the vertical,

S =

ωr

√

(3.267)

is a Strouhal number, and χ is given by (3.258), with now

H =φ +St. (3.268)

With values ω = 7.27 ×10

−5

−1

, r

= 6.37 × 10

m, g = 9.8ms

−2

, d = 3.8 ×

m, we ﬁnd S ≈2.4. Neglecting terms of O(ε) in (2.19), we have the tidal model

+∇.(hu) =0,

+2S sin λ k ×u =−∇η +∇χ,

(3.269)

in which we can take h(x) independent of time.

3.11.2 Ocean Tides

We begin by taking uniform depth h = 1 and ignoring the Coriolis force, thus we

put S =0in(3.269)

(but not in the deﬁnition of χ). From this there follows

+∇. u =0,

=−∇η +∇χ,

(3.270)

whence

=∇

η −∇

χ. (3.271)

In the spherical polar coordinates φ and λ,

∇ =



cosλ

∂

∂φ

∂

∂λ



, cos

λ ∇

∂

∂φ

∂

∂ν

, (3.272)

3.11 Tides and Tsunamis 199

(cf. (3.28)), where we deﬁne

ν =ln



1 +tan(λ/2)

1 −tan(λ/2)



∂

∂ν

=cos λ

∂

∂λ

. (3.273)

The tide-generating potential given by (3.258) contains separate components due to

semi-diurnal, diurnal, and long period variations. The combined effect of these (and

of the tidal effects of the Sun) can be obtained by linear superposition. For simplicity

we will consider only the semi-diurnal lunar tide, denoted M

, and suppose that the

tide-generating potential is just

χ =cos

δ cos

λ cos 2(φ +St). (3.274)

Newton’s equilibrium theory (illustrated in Fig. 3.8) assumes that η = χ,butev-

idently this could only be approximately valid for slowly varying χ, i.e., S  1.

This is not the case on the Earth, and consequently the times of high tides lag the

times of maximum attractive force.

The simplest case to consider is that of a narrow canal at a ﬁxed latitude, which

circumtraverses the globe. To obtain a solution in this case, we write (3.271)inthe

form (using (3.268) and (3.272))

cos

λη



∂

∂H

∂

∂ν



(η −χ), (3.275)

where we assume that η depends only on the combination H =φ +St. Supposing

the variation of ν is small, we write η =η

(0)

+η

(1)

+···,etc.,

and then we have

to leading order

(0)

−χ

(0)

≈f(H), (3.276)

and at the next order

∂

(η

−χ

(1)

)

∂ν



cos

λ −1





cos

λχ

. (3.277)

The boundary conditions of no ﬂow through the side walls require η

−χ

=0, and

therefore integration of (3.277) between the walls gives an integrability condition

for its solution; this determines f and thus η (omitting the superscript zero), and the

result is

η =

1 −S

cos

. (3.278)

This represents a westward travelling wave of speed −S (since χ ∝φ +St), whose

amplitude is modulated by latitude.

At the equator, λ =0 and (since S>1) the canal tides are out of phase with the

tide-generating potential (the lag time is one quarter of a lunar day, slightly over

To be more formal, we would write ν =ν

+ε ˜ν,takeε (here denoting the dimensionless canal

width) to be small, expand as η =η

(0)

+ε

(1)

+···, and so on, but the end point is the same.