Fowler A. Mathematical Geoscience

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180 3 Oceans and Atmospheres

Fig. 3.6 Contours of

temperature (dashed lines)

and potential temperature

(solid lines) in a forming front

As the front develops, the baroclinic instability also distorts the ﬂow in a wave-

like pattern. The effect of this is to bend the front round, as illustrated in Fig. 3.7,

forming a series of vortex-like rings. In the atmosphere, these are the cyclonic dis-

turbances which form the mid-latitude low pressure storm systems, with typical

dimensions of 2000 km. They also occur in the ocean, forming coherent rings of

some 50 km diameter.

The description above is a little idealistic. On the Earth, fronts are an intrinsic

consequence of the difference in properties between different air masses. The mid-

latitude cells, for example, are bounded north and south by fronts across which the

wind direction and the temperature changes. The warm mid-latitude westerlies are

bounded polewards by the cold polar easterlies. The situation is complicated by

continents and oceans. Continental air is dry, whereas oceanic air is moist. As a

consequence of these geographic variations there are a number of different types of

air masses, and the boundaries between these provide the seeds for frontal develop-

ment. The fronts move and distort as shown in Fig. 3.7, but it is more sensible to

think of the roll-up of a planar front and the formation of storm systems as a result of

(Kelvin–Helmholtz like) instability of a linear vortex sheet, rather than as a conse-

quence of shock formation in the nonlinear wave evolution of the quasi-geostrophic

potential vorticity (QGPV) equation. In fact, the QGPV equation does not do a very

good job of numerical weather front prediction.

3.7.1 Depressions and Hurricanes

The storm systems which develop as shown in Fig. 3.7 are called cyclones. They

are like vortices which rotate anti-clockwise, and are associated with low pressure

at their centres (thus they are also called depressions). Conversely, a high pressure

vortex rotating clockwise is called an anti-cyclone. A severe storm with central pres-

sure of 960 millibars represents a dimensionless amplitude of 0.04 ∼ε

, and is thus

within the remit of the quasi-geostrophic scaling.

This comment is due to Peter Lynch.

3.7 Frontogenesis 181

Fig. 3.7 Two views of the formation of cyclonic depressions from a baroclinically unstable front.

The illustration resembles the Kármán vortex street which forms at moderate Reynolds number in

the ﬂow past a cylinder. The upper diagram shows isobars, the front, and cloud cover (stippled);

the lower diagram shows isotherms, and ﬂow of cold air (solid arrows)andwarmair(dashed

arrows). From Barry and Chorley (1998), page 162; their image is derived from a ﬁgure in Shapiro

and Keyser (1990), and is reproduced with permission of the American Meteorological Society

In the tropics, tropical cyclones occur, and the most severe of these is the hur-

ricane, or typhoon. In essence, the hurricane is very similar to the mid-latitude de-

pression, consisting of an anti-clockwise rotating vortex, with wind convergence at

the surface, and divergence at the tropopause. It is, however, fuelled by convection,

and can be thought of as the result of a strong convective plume interacting with the

Coriolis force, which causes the rotation, and in fact organises it into a spiral wave

structure, as can be seen in satellite images by the spiral cloud formations.

The hurricane is distinguished by its high winds, high rainfall and relatively small

size (hundreds rather than the thousands of kilometres of a mid-latitude cyclone).

The strongest hurricane on record was hurricane Gilbert in 1988, where the central

pressure fell to 888 mbar, and maximum windspeeds were in excess of 55 m s

−1

(200 km hr

−1

). The strong convection is a consequence of evaporation from a warm

ocean, and it is generally thought that hurricane formation requires a sea surface

temperature above 27° centigrade, or 300 K. Relative to a mean surface temperature

of 288 K, this is an amplitude of 12 K, and dimensionlessly 12/288 ≈ 0.04, of

O(ε

). In the tropics, the Rossby number ε is higher, and near the equator the quasi-

geostrophic approximation breaks down, but hurricanes do not form in a band near

the equator.

182 3 Oceans and Atmospheres

Hurricanes typically move westwards in the prevailing tropospheric winds, and

dissipate as they move over land, where the fuelling warm oceanic water is not

present, and surface friction is greater. They develop a central eye, which is rela-

tively calm and cloud free, and in which air ﬂow is downwards. In hurricanes, this

eye is warm.

3.8 The Mixed Layer and the Wind-Driven Oceanic Circulation

Much of what we have said concerning the dynamics of the atmosphere applies to

the world’s oceans. The oceans form a thin layer of mean depth (slightly less than)

four kilometres, spread over the globe. The dynamics of the oceans are thus those of

a shallow layer of ﬂuid on a sphere, just as for the atmosphere. There are, however,

some differences. Water is essentially incompressible, though in fact the density

dependence on temperature and salinity causes the oceans to be stably stratiﬁed,

just as the atmosphere is. The Brunt–Väisälä frequency is about ten times smaller in

the ocean than in the atmosphere.

More importantly, the ocean is blocked by continents. The atmosphere is blocked

by mountains, but can ﬂow over them; the oceans have to ﬂow round continents.

This causes boundary layer effects in the oceanic circulation which are distinctive.

The other major difference between oceans and atmosphere is in the driving

mechanism for the ﬂow. Differential heating between equator and pole drives the

atmospheric ﬂow, and this also drives the global thermohaline circulation (see

Sect. 3.10) of the ocean, but the atmospheric circulation itself drives a circulation

by means of wind stress at the surface. The global convective circulation due to dif-

ferential heating and the wind-driven circulation interfere with each other, and it is

not even clear which, if either, is dominant in determining the ﬂow. In this sense,

oceanic ﬂow is much less well understood than atmospheric ﬂow.

The vertical structure of the oceans is as follows. Near the surface there is a mixed

layer, of typical thickness of the order of 50–100 metres, in which the density is

uniform. This layer exists by virtue of the atmospheric wind stress, which mixes

the surface waters. Below the mixed layer, the density begins to increase, and there

is a thermocline over which the temperature changes from its warm surface value

to the cooler deep ocean value. The thermocline has a thickness of the order of a

kilometre, and the temperature contrast (warm at the surface, cool at depth) exists

throughout temperate latitudes. It does not exist at the poles, but here the thermal

structure is determined by the presence of sea ice. At the poles, salinity is of greater

importance in determining the density proﬁle. The thermal structure of the oceans is

consistent with the concept of a thermally driven convective ﬂow, which we describe

later. First, we describe the wind-driven circulation.

The principal feature of the near-surface circulation of the oceans is the pres-

ence of circulatory ﬂows with strong western boundary currents. In the North At-

lantic, there is a clockwise circulation, with a strong current running up the Eastern

seaboard of the United States. This Gulf Stream separates and ﬂows across towards

Europe, and is instrumental in providing Northern Europe with its anomalously

3.8 The Mixed Layer and the Wind-Driven Oceanic Circulation 183

warm climate. A similar current (the Kuroshio) occurs in the Western Paciﬁc. These

strong currents are due to the effects of the wind-driven circulation. In order to un-

derstand them, we need to formulate a model for ocean circulation in response to

surface forcing.

Our starting point is with the dimensionless shallow water equations of (3.43)

and (3.44). While the dimensionless variables are deﬁned the same way as for the

atmosphere, the scales are somewhat different. Typical velocities in the ocean are of

order U ∼0.1ms

−1

, a typical ocean horizontal length scale is l ∼ 3,000 km, while

the ocean depth is taken as h ∼ 4 km. Assuming these values, we ﬁnd δ ∼ 10

−3

Σ ∼0.5, Ro ∼0.25×10

−3

and F ∼0.5×10

−3

. We immediately assume that δ and

Ro are negligible. We also assume that sea water is (approximately) incompressible,

so that ∇. u =0, and we suppose that the ocean depth is uniform, of dimensionless

depth one. Our coordinate system assumes that z = 0 deﬁnes mean sea level. From

(3.43), we thus have

cosλ

∂u

∂x

cosλ

∂(vcos λ)

∂y

∂w

∂z

=0. (3.180)

The third component of the momentum equation implies hydrostatic equilibrium,

and in view of the smallness of F

/Ro, we write the solution as

p =−z +

η(x,y). (3.181)

η represents sea surface elevation, and for F

/Ro ∼10

−3

, values of η ∼O(1) cor-

respond to elevations of order four metres. The two horizontal components of the

momentum equations thus take the form

−v sin λ =−

cosλ

∗

u sin λ =−η

∗

(3.182)

and the friction terms are prescribed as in (3.46), (3.36), (3.37) and (3.38), which

yields

∗

∇

u +E

∂

∂z

, (3.183)

where ∇

denotes the horizontal Laplacian, and

2Ωl

2Ωh

(3.184)

deﬁne horizontal and vertical Ekman numbers, respectively. If we take ε

∼

−2

−1

, ε

∼10

−1

, then we ﬁnd E

∼0.4 ×10

−5

, E

∼0.4 ×10

−7

(3.180), (3.182) and (3.183) constitute the system we want to solve, subject to

the conditions of no ﬂow through the surface or base,

w =0atz =0 and z =−1, (3.185)

184 3 Oceans and Atmospheres

and subject to an applied (dimensional) surface wind stress τ

, which implies

∂u

∂z

hτ

ρε

at z =0. (3.186)

In addition we apply a no-slip condition at the base, thus

u =0 at z =−1. (3.187)

The velocity scale U must be chosen by a suitable balance of the driving bound-

ary condition (3.186). It is in fact not quite obvious how to do this. To do so, we

need to anticipate the nature of the solution.

The Ekman numbers E

and E

are very small. Therefore they can be neglected

except in boundary layers, and the ﬂow is approximately geostrophic. If we integrate

the mass conservation equation upwards from the bottom, we ﬁnd that as z →0−,

the vertical velocity w will be non-zero, and of O(1), if we suppose (by choice of

U ) that u, v ∼ O(1) in the bulk ﬂow. Therefore in order that w decrease to zero,

we require a boundary (Ekman) layer near the surface, where the vertical Ekman

viscous term becomes important. Evidently, this layer is of thickness O(E

1/2

), and

in order for w to decrease by O(1) in the boundary layer, we need u ∼ v ∼ E

−1/2

in the Ekman layer. Hence we must have

∂u

∂z

∼

in the surface Ekman layer, and

this allows us to deﬁne the velocity scale. In view of (3.184), this suggests that if τ

is a scale for the wind stress, and we deﬁne

=τ

τ, (3.188)

then we should choose

=2ρUΩh. (3.189)

The boundary condition (3.186) then becomes

∂u

∂z

=τ at z =0. (3.190)

We may now proceed to a solution. Away from all boundaries, we have the outer

geostrophic solution u =u

(x, y), v =v

(x, y), where

−v

sin λ =−

cosλ

sin λ =−η

(3.191)

By eliminating η and using the fact that

∂

∂y

=Σ

∂

∂λ

, we can show that the mass

conservation equation takes the form

∂w

∂z

=Σv

cotλ, (3.192)

3.8 The Mixed Layer and the Wind-Driven Oceanic Circulation 185

and thus

outer

→w

+Σv

cotλ as z →0−, (3.193)

where w

is an apparent surface vertical velocity due to Ekman pumping, which we

now calculate.

In the surface Ekman layer, we put

z =−E

1/2

ζ, u =E

−1/2

U, (3.194)

so that, if we deﬁne the complex velocity and stress

S =U +iV, τ =τ

+iτ

, (3.195)

where τ =(τ

,τ

) and U =(U, V ), then approximately

ζζ

=iS sin λ, (3.196)

together with

−

∂S

∂ζ

=τ on ζ =0,S→0asζ →∞. (3.197)

The solution is

S =

−Bζ

, (3.198)

where

B =

(1 ±i)

√

(±sin λ)

1/2

, (3.199)

where we select the upper or lower sign depending on whether λ>0orλ<0,

respectively.

The continuity equation in the Ekman layer is

∂w

∂ζ

cosλ

∂U

∂x

cosλ

∂(V cos λ)

∂y

; (3.200)

integrating from ζ = 0toζ =∞and matching to the outer solution (3.193) then

requires

+Σv

cotλ =

cosλ

∂

∂x



∞

Udζ+

cosλ

∂

∂y



cosλ



∞

Vdζ



. (3.201)

Simple calculation gives



∞

Sdζ =−

iτ

sin λ

, (3.202)

186 3 Oceans and Atmospheres

and therefore (3.201) becomes

+Σv

cotλ =k.∇ ×



sinλ



, (3.203)

where the vertical (k) component of ∇ ×F in the present pseudo-spherical coordi-

nates is deﬁned as

k.∇×F =

cosλ

∂F

∂x

−

cosλ

∂

∂y

cosλ). (3.204)

If we ignore the small Ekman pumping velocity w

, then

Σv

cotλ =k.∇ ×



sin λ



(3.205)

describes the so-called Sverdrup ﬂow of the oceans due to the applied wind stress.

Being purely algebraic, it pays no attention to continents. Therefore, the no-slip

condition that we would like to apply at a continental margin cannot be applied;

to do this we need to bring back the horizontal friction terms involving the hori-

zontal Ekman number E

. In fact, the basal Ekman pumping term also allows a

regularisation, and we consider its form ﬁrst.

Near the base, we write

z =−1 +E

1/2

ζ, (3.206)

and with s =u +iv, s satisﬁes

ζζ

=i(s −s

) sin λ, (3.207)

where s

= u

+ iv

. The solution satisfying s = 0atζ = 0 and with s → s

ζ →∞is

s =s



1 −e

−Bζ



, (3.208)

where B is given by (3.199). Mass conservation then implies

∂w

∂ζ

=−E

1/2



cosλ

∂u

∂x

cosλ

∂(vcos λ)

∂y



, (3.209)

and in turn this implies that

w ∼w

+ΣE

1/2

ζ cotλ as ζ →∞, (3.210)

where

=−E

1/2



cosλ

∂

∂x



∞

(u −u

)dζ +

cosλ

∂

∂y



cosλ



∞

(v −v

)dζ



(3.211)

3.8 The Mixed Layer and the Wind-Driven Oceanic Circulation 187

From (3.208),



∞

(s −s

)dζ =−

, (3.212)

and therefore we ﬁnd





cosλ

∂

∂x



√

sin λ

)



cosλ

∂

∂y



cosλ

√

sinλ

−u

)



(3.213)

This expression is written for λ>0 in the northern hemisphere. We give the corre-

sponding recipe for the southern hemisphere below.

Next, we consider how to include the horizontal friction terms. To see how to do

this, we reconsider (3.182), which we write, using (3.183), in the form (away from

the vertical Ekman layers)

−v

sinλ =−

cosλ

∇

sinλ =−η

∇

(3.214)

The solution of (3.200), integrated through the surface boundary layer, is still

ζ =∞

=k.∇ ×



sinλ



, (3.215)

and the solution of (3.180) still implies that this must be equal to

z=0−

−



cosλ

∂u

∂x

cosλ

∂(v

cosλ)

∂y



. (3.216)

Equating these two results, and eliminating η in (3.214), we ﬁnd after some algebra

that (3.203) generalises to

+Σv

cotλ =k.∇ ×



sin λ



cosλ

cosλ sin λ



∂

∂x

∇

−

∂

∂y

∇



. (3.217)

In most derivations of this model, the ﬁxation with latitude λ has long since dis-

appeared, and when we look at the form of the Ekman terms in E

and E

,itis

easy to see why. There are two things that help us. Both are based on the fact that

the Ekman term will be completely negligible, except in boundary layers. Therefore

the geostrophic approximation (3.191) is appropriate outside boundary layers. How-

ever, if we only require the velocity ﬁeld to satisfy the no-ﬂow-through condition at

continents (and not the no-slip condition), then only the gradient of the velocity ﬁeld

will change in the continental margin boundary layer, and to leading order (3.191)

will still apply. This enables us to use the geostrophic approximation in the friction

term. We cannot use this argument if we wish to apply the no-slip boundary condi-

tions, but we will ignore this subtlety here, and suppose that in both friction terms,

and v

are given by (3.191).

188 3 Oceans and Atmospheres

The other assistance comes from the fact that because the viscous term is only

relevant in thin boundary layers, then since λ will be approximately constant in

such boundary layers, it is valid to ignore the derivatives of λ which arise in the

Laplacian. Speciﬁcally, the deﬁnition (ignoring terms of O(δ) and the like) of the

Laplacian is

∇

∂

∂x

cosλ

∂

∂y



cosλ

∂

∂y



, (3.218)

and can in boundary layers be taken to be ∇

∂

∂x

∂

∂y

. Adopting the geostrophic

approximation (3.191), we then ﬁnally obtain an equation for a ‘stream function’ ψ,

which we deﬁne by

ψ =

sin λ

, (3.219)

which is

βψ

= sin λ k.∇ ×



sin λ



−



2sinλ

cosλ



cosλ

+ψ



cosλ



cosλ

∇

+∇



. (3.220)

In the southern hemisphere, the corresponding equation can be shown to have the

same form, providing we write

√

sin λ as

√

|sin λ| and take y as pointing polewards

(though evidently this is redundant in (3.220) since only second derivatives in y

appear). The parameter β is deﬁned here

β =Σ cosλ

. (3.221)

We will study the boundary layer structure of this equation, and the formation of the

western boundary currents, in the following section.

3.9 Western Boundary Currents: The Gulf Stream

Now at last we will ignore the largely irrelevant latitude terms in (3.220), and we

will consider the case of an ocean in a box B:0<x<1, 0 <y<1, and we will

require no ﬂow through each side of the box; we may also require no slip if we

consider the sides as representing continents. We have in mind a representation of

the North Atlantic, with x =0 representing the North American coastline, and x =1

representing Africa and Europe.

We assume that the wind stress is purely meridional, but varying linearly with

latitude, thus τ = (

+ y,0), and we inconsequentially ignore the trigonometric

The parameter β appears as the same coefﬁcient in other derivations, but is usually dimensional.

3.9 Western Boundary Currents: The Gulf Stream 189

terms in the deﬁnition of ∇ × τ . This wind ﬁeld provides a representation of pre-

vailing westerlies in mid-latitudes, and the easterly trade winds near the equator.

The version of Eq. (3.220) we aim to solve is thus

βψ

=−1 −ε∇

ψ +E

∇

ψ, (3.222)

where we deﬁne

ε =



2sinλ

, (3.223)

and the boundary conditions of no ﬂow through imply that the stream function is

constant, i.e.,

ψ =0on∂B. (3.224)

If, in addition, we prescribe no slip at the boundary, then also

∂ψ

∂n

=0on∂B. (3.225)

We are interested in the boundary layer structure of the solution for small E

and

small ε.

3.9.1 Effects of Basal Drag

Both of the small terms, in ε and E

, represent singular perturbations to the basic

Sverdrup ﬂow, and we will consider their regularising effects separately. First, we

suppose ε 1 and neglect E

. The equation to be solved is thus

βψ

=−1 −ε∇

ψ. (3.226)

We will be able with this model to satisfy only the no-ﬂow-through condition ψ =0

on ∂B, since (3.226) is second order and elliptic.

The sub-characteristics go to the left, and therefore any boundary layer will exist

at the left of the domain; this is the western boundary current, and the cause of the

Gulf Stream. The outer solution is the Sverdrup ﬂow, and is given by

ψ =

1 −x

, (3.227)

which represents a southerly ﬂow v =−1/β (since u ≈−ψ

, v ≈ ψ

). There is a

boundary layer of thickness O(ε) adjoining the western boundary x = 0, in which

we put

x =εX, (3.228)

so that

βψ

≈−ψ

, (3.229)