Fowler A. Mathematical Geoscience

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140 3 Oceans and Atmospheres

Atmospheric winds (and thus weather) are driven by heating from the Sun. The

Sun heats the Earth non-uniformly, because of the curvature of the Earth’s surface,

but the outgoing long wave radiation is much more uniform. Consequently, there is

an energy imbalance between the equator and the poles. The equator is differentially

heated, and the poles are differentially cooled. It is important to realise that the

primary climatic energy balance (which, as we saw in Chap. 2, determines the mean

temperature of the Earth) is between net incoming short wave radiation and outgoing

long wave radiation; the Earth’s weather systems and general circulation arise as a

consequence of spatial variation in this balance, and as such are a perturbation to

the basic energy balance. Weather is a detail.

The oceans are similar. The ﬂuid is water and not air, but the oceans also lie in a

thin layer on the Earth. For various reasons, their motion is more complicated and

less well understood. For a start, their motion is baulked by continents. The great

oceans lie in basins, and their global circulation is dictated to some extent by the

topography of these basins. The atmosphere may have to ﬂow over mountains, but

it can do so: oceans have to ﬂow round continents.

In addition, the oceans are driven not only by the same differential heating which

drives the atmosphere, but also by the atmospheric winds themselves; this is the

wind-driven circulation. It is not even clear whether this is the primary driving force.

A ﬁnal complication is that the density of ocean water depends on salinity as well

as temperature, so that oceanic convection is double-diffusive in nature. (One might

say in compensation that cloud formation in the atmosphere means that atmospheric

convection is multi-phase convection, but this is not conceived of as being funda-

mental to the nature of atmospheric motion.)

The basic nature of the atmospheric general circulation is thus that it is a con-

vecting ﬂuid. Hot air rises, and so the equatorial air will rise at the expense of the

cold polar air. In the simplest situation, the Earth’s differential heating would drive

a convection cell with warm air rising in the tropics and sinking at the poles; this

circulation is called the Hadley circulation.

In reality, the hemispheric circulation consists of three cells rather than one.

The tropical cell (terminating at about 30° latitude) is still called the Hadley cell,

then there is a mid-latitude cell and a polar cell. This basic circulation is strongly

distorted by the rotation of the Earth, which as we shall see is rapid, so that the

north/south Hadley type circulations are ﬂung to the east (at mid-latitudes): hence

the prevailing westerly winds of common European experience.

This eastwards wind is called the zonal wind. And it is unstable: a phenomenon

called baroclinic instability causes the uniform zonal wind to form north to south

waves, and these meandering waves form the weather systems which can be seen

on television weather forecast charts. At a smaller scale, such instabilities lead to

weather fronts, essentially like shocks, and in the tropics these lead to cyclones and

hurricanes. In order to begin to understand how this all works, we need a mathemat-

ical model, and this is essentially a model of shallow water theory (or shallow air

theory) on a rapidly rotating sphere.

A westerly wind is one coming from the west. It will be less confusing to call such a wind

eastwards, and vice versa for easterlies, i.e., westwards winds.

3.2 The Geostrophic Circulation 141

3.2 The Geostrophic Circulation

The basic equations describing atmospheric (or indeed, oceanic) motion are those

of mass, momentum and energy in a rotating frame, and can be written in the form

dρ

+ρ∇. u = 0,



+2 ×u



=−∇p −ρ∇Φ +F, (3.1)

ρc

−βT

= ∇. q +Q.

In these equations, ρ is the density, u is the velocity, p is the pressure, T is the

temperature. d/dt is the material derivative following a ﬂuid element, i.e., d/dt =

∂/∂t + u.∇.  is the angular velocity of the Earth, and the equations have been

written with respect to a set of coordinates ﬁxed in the (rotating) Earth.

Φ is called the geopotential; it is the gravitational potential corrected for the

effect of centrifugal force, and is deﬁned by

Φ =Φ

−

| ×r|

, (3.2)

where Φ

is the gravitational potential. The surface Φ = 0 is called sea level; the

surface of the oceans would be this geopotential surface in the absence of motion.

We take z to be the coordinate normal to Φ = 0; essentially it is in the radial di-

rection, and to a good approximation we can take Φ = gz, where g is called the

gravitational acceleration (although in fact it includes a small component due to

centrifugal force).

(3.1) must be supplemented by an equation of state. In the atmosphere, we take

the perfect gas law

p =

ρRT

(3.3)

(cf. (2.35)), where R is the gas constant and M

is the molecular weight of dry

air.

3.2.1 Eddy Viscosity

The force F represents the effects of friction. Molecular viscosity is insigniﬁcant in

the atmosphere and oceans, but the ﬂows are turbulent, and the result of this is that

The effect of the rotating coordinate system is that time derivatives of vectors a are transformed

ﬁx

rot

+ ×a, because in differentiating a =a

, both the components a

and the unit

vectors e

change with time, and

= ×e

142 3 Oceans and Atmospheres

momentum transport by small scale eddying motion is often modelled by a diffusive

frictional term of the form ρε

∇

u, where ε

is an ‘eddy’ (kinematic) viscosity.

More generally, ε

varies with distance from rough boundaries.

Some discussion

of eddy viscosity is given in Appendix B. A complication in the atmosphere is that

the vertical motion is much smaller than the horizontal, and this leads to the idea

that different eddy viscosities are appropriate for horizontal and vertical momentum

transport. We denote these coefﬁcients as ε

and ε

, and take them as constants.

To be precise, we then represent the frictional terms in the form

F =ρε

∇

u +ρε

∂

∂z

, (3.4)

where ∇

∂

∂x

∂

∂y

, and x, y are ‘horizontal’ coordinates, z is the vertical co-

ordinate. Later we discuss a more precise deﬁnition of the relation of these local

Cartesian coordinates to the appropriate spherical coordinates of the system. Since

the friction terms will only be important in boundary layers where the sphericity is

unimportant, we need not concern ourselves with such niceties in deﬁning F.

Estimates of the eddy coefﬁcients are given later. Frictional effects are generally

relatively small. In the atmosphere, they are conﬁned to a ‘boundary layer’ adjoining

the surface, having a typical depth of 1000 metres, and bulk motion above this layer

is effectively inviscid.

3.2.2 Energy Transport

The catch-all term Q in the energy equation represents internal heating due both

to absorption of short wave radiation and to latent heat release by condensation,

while q includes both sensible and radiative heat ﬂux. As for viscosity, molecular

thermal conductivity is negligible, but there is an eddy diffusive transport which is

larger. In addition, there is a radiative transport term, which was deﬁned in (2.27).

For an opaque medium, this also takes the form of a diffusive ﬂux, as given by

(2.30). The principal internal heating terms are due to absorption of solar short wave

radiation, and to condensation and cloud formation. In a saturated atmosphere, this

was represented in Chap. 2 by the term −ρ

in (2.57). As we shall see, more

care is required if we wish to provide boundary conditions for the temperature. The

term Q is generally small but not insigniﬁcant in the troposphere; it is dominant in

the stratosphere.

Because our main concern is with the dynamics generated by the momentum

equation, it is tempting simply to prescribe Q (for example, from measured atmo-

spheric temperature proﬁles). We wish to avoid doing this, because it is precisely

the local imbalance in incoming and outgoing radiative transport which drives the

And then, we write F = ∇. τ

, τ

(∇u +∇u

3.2 The Geostrophic Circulation 143

atmospheric motions we are describing. In particular, we need to understand how

these imbalances are manifested in the boundary conditions for (3.1)

To be speciﬁc, we will write the energy equation for a saturated, grey, opaque

atmosphere. Certainly the greyness and opacity are inaccurate, and the atmosphere

is by no means always saturated. Nevertheless, our discussion serves an important

pedagogic function. The energy equation is similar to (2.57) (including a source

term due to absorption of short wave radiation, see Fig. 2.3), but we deal more

speciﬁcally with the source term associated with water vapour. We write the energy

equation as

ρc

−

=∇.(k

∇T)−∇.q

+ρLC, (3.5)

where Q

is the short wave absorption term, and C is the condensation rate of water

vapour, with the units being measured as (minus) the rate of change of mixing ratio

(ρ

/ρ) per unit time. We have used the fact that, for the perfect gas law, the thermal

expansion coefﬁcient β =1/T . The long wave radiative ﬂux is given, from (2.27)

and (2.28), by





I(r, s) s dω(s), (3.6)

and (using (2.10))

I =

σT

−

ρκ

s.∇I. (3.7)

I satisﬁes the boundary condition at the top of the atmosphere (taken to be z =∞)

that

I =0asz →∞, s.k < 0, (3.8)

i.e., there is no incoming long wave radiation (k is a unit vector pointing upwards).

At the ground surface, we assume that temperature is continuous, so that

I =B =

σT

on z =0, s.k > 0. (3.9)

The effective thermal conductivity k

represents the eddy conductivity associated

with turbulent ﬂow; generally, the eddy thermal diffusivity will be similar or equal

to the eddy kinematic viscosity (and thus also will have different values in the hori-

zontal and the vertical).

The opaque limit treats the derivative in (3.7) as a small perturbation, and a reg-

ular expansion then leads to (2.29), i.e., I =σT

/π +···, from which we derive

(2.30):

≈−k

∇T, (3.10)

144 3 Oceans and Atmospheres

where the effective radiative conductivity (see (2.51)) is

16σT

3κρ

. (3.11)

Because this is a singular perturbation, the approximate solution does not necessar-

ily satisfy the boundary conditions (3.8) and (3.9) (though in fact it does approx-

imately satisfy the latter), and thus the effective ﬂux given by (3.10) may not be

accurate near the top or bottom of the atmosphere. This should not seriously affect

its use in the energy equation (3.5).

In fact, the opaque limit can only apply, if at all, in the troposphere, where the

density is reasonably high. So the issue of its use in the stratosphere, and the asso-

ciated boundary condition (3.8), is somewhat irrelevant. What in fact we will wish

to do is to apply an effective thermal boundary condition for (3.5) at the tropopause.

We return to the consideration of this below.

The term Q

represents the volumetric absorption within the troposphere of the

short wave radiation which is received at the top of the atmosphere. Denoting this

latter term (a ﬂux) as q

, we see from Fig. 2.4 that, over the whole atmosphere,

the absorbed short wave radiation is some 0.2q

. Only a small amount of this is

absorbed in the stratosphere, so that the quantity absorbed within the troposphere,

∼ 0.2q

. The reﬂected short wave radiation (via the planetary albedo) q

≈

0.3q

, and the amount absorbed at the ground q

≈0.5q

Evaporation provides another important energy ﬂux at the surface. If we denote

the rate of evaporation (as a velocity, metres of water per second) as E, then the

latent heat ﬂux due to evaporation is ρ

LE, where ρ

is the density of water, and

this is comparable to the short wave absorption, ≈0.2q

, as shown in Fig. 2.3. Iden-

tifying evaporative mass transport as a latent heat ﬂux is confusing even if conve-

nient, and we now discuss its interpretation in detail. To do this, we have to write a

moisture mass balance equation, and associated mass and energy balance boundary

conditions.

Let q

−

denote the heat ﬂux delivered to the ocean/atmosphere surface from the

ocean. (A similar discussion can be made for the continents, but evaporation is then

negligible.) q

−

consists of both a radiative part and a convective part, which can

be prescribed following a prescription of energy transfer in the ocean. Let q

de-

note the radiative and convective heat ﬂux at the surface to the atmosphere. We can

write

=−

∂T

∂z

, (3.12)

where

k =k

, (3.13)

(and there would be a similar expression for q

−

, in terms of the oceanic temperature

gradient). We can then write a boundary condition of Stefan type for (3.5)atthe

surface as

LE =q

−q

−

; (3.14)

3.2 The Geostrophic Circulation 145

this represents the net evaporation E, measured as a velocity, at the ocean surface

due to the net energy received at the surface. (On continents, E = 0 and this condi-

tion determines q

The evaporation term is a surface source term for atmospheric moisture content

(measured as mixing ratio m) which thus satisﬁes a conservation equation of the

form

=∇.(ε

∇m) −C, (3.15)

where ε

is the eddy diffusivity (again, anisotropic as discussed previously), and

C is the same condensation rate which appears in the energy equation (3.5). The

interpretation of this equation in relation to the discussion in Chap. 2, where the

assumption of saturation leads, via (2.55), (2.56) and (2.58), to an expression for m

as a function of T and p, m =m

(T , p), is that when (3.15) applies for unsaturated

air, i.e., while m<m

(T , p), then C =0. For saturated air, we have m =m

(T , p),

and (3.15) determines the condensation rate C.

The boundary condition for (3.15)

at the surface is

−ρε

∂m

∂z

=ρ

E (3.16)

(and that at the tropopause can be taken to be that the evaporative ﬂux is zero).

Lastly, we attempt to write a boundary condition for (3.5) at the tropopause,

z = h. We denote the upwards radiative and convective heat ﬂux there as (q

−

∇T).k = q

. At the tropopause, if we suppose the radiant heat ﬂux is given by

Stefan’s law, the radiative boundary condition would be

=−

∂T

∂z

=σT

, (3.17)

presuming a vacuum beyond.

This condition would be suitable if the atmosphere really was all contained in

the troposphere, and then the temperature at the tropopause would be the effec-

tive long wave emission temperature of the Earth, 255 K. However, as discussed in

Chap. 2 (see Sects. 2.2.7 and 2.2.8) some ten per cent of the atmosphere lies above

the troposphere, most of it in the stratosphere. Ozone and other gases absorb solar

radiation, particularly ultra-violet radiation, in the stratosphere, and the absorption

of this radiation in the stratosphere raises its temperature. In fact, the temperature at

the tropopause dips to about 220 K, and then rises in the stratosphere to about 270 K

at a height of about 50 km (the stratopause). According to Stefan’s fourth power law,

if 70% of incoming short wave radiation is emitted as long wave radiation at 255 K,

The discussion could be elaborated to include a conservation equation for cloud density, i.e., for

water density conservation, including a source term due to condensation and a sink term due to

precipitation. It is not necessary, at least as regards energy conservation, since there is little energy

transport associated with precipitation.

146 3 Oceans and Atmospheres

then 114% is emitted at 288 K (consistent with Fig. 2.3), and only 39% at 220 K.

A realistic tropopause thermal boundary condition requires some description of the

stratosphere.

The condition we need to apply is that the radiative and convective heat ﬂux is

continuous at the tropopause. If the opaque approximation (3.10) could be applied

throughout the atmosphere, then this would simply imply continuity of temperature

gradient. The temperature structure shown in Fig. 2.5 would then imply accumu-

lation of heat at the tropopause, which makes no sense. In fact the approximation

(3.10) can only apply where the atmosphere is optically deep, and if at all, this is in

the troposphere. More generally, the radiative part of the ﬂux is given by an integral

of the emission density B. For example, in a one-dimensional, grey atmosphere, the

radiative ﬂux is given by (see Question 2.7)

=2π



(τ ) +





(τ





|τ



−τ |



dτ



, (3.18)

where the exponential integrals are given by

(z) =



∞

−zt

, (3.19)

and B

= B(0). From this we can derive the opaque approximation (see Ques-

tion 2.7), and if we suppose vacuum beyond the stratopause, so that q

= πB

at τ =0, then (since E

(0) =





(τ



(τ



)dτ



=0, (3.20)

which shows that the temperature structure must have a structure like that in Fig. 2.5.

Despite this, the radiative heat ﬂux at the tropopause is upwards. We apply the

boundary condition

−

∂T

∂z

at z =h, (3.21)

and will suppose that the radiative ﬂux above the tropopause is known, although in

fact we should have to solve an integral equation (such as (3.18), if q

is known in

the stratosphere) in order to ﬁnd it.

The discussion above appears to be consistent with the concept of the troposphere

as the dense, well-mixed ﬂuid layer in which radiative and sensible heat transport

terms are small, and thus the temperature is adiabatic, as described in Chap. 2; above

this lies the much less dense stratosphere in which radiative transport dominates, as a

consequence of which ∂q

/∂z ≈Q

, and the net radiative ﬂux rises from its value at

the tropopause. Effectively, the tropopause is like an interface between well-mixed,

vigorously convecting ﬂuid, and quiescent, stably stratiﬁed ﬂuid.

3.2 The Geostrophic Circulation 147

3.2.3 Global Energy Balance

Integration of (3.5) over a vertical tropospheric column 0 <z<hyields (approxi-

mately) the energy balance equation

(I +P)=q

−q



dz +



ρLC dz, (3.22)

where q

and q

are the combined radiative and sensible heat ﬂuxes upwards at sea

level and the tropopause, respectively, I =



ρc

Tdzis the internal enthalpy and

P =



ρΦ dz is the potential energy. From (3.15) and (3.16), we have



ρLC dz =ρ

LE −



ρLmdz. (3.23)

Deﬁning the latent heat of moisture as

M =



ρLmdz, (3.24)

we thus have

(I +P +M) =q

−q



dz +ρ

LE. (3.25)

The four terms on the right are delineated in Fig. 2.3; q

is the combined radia-

tive and sensible surface heat ﬂux, q

is the long wave radiative heat lost from

the atmosphere,



dz is the internal heating due to short wave absorption, and

LE is the latent heat ﬂux. Although latent heat appears as a source term in the

global equation, it occurs in the boundary conditions of the point forms of the equa-

tions. Approximate (globally averaged) values of these quantities (see Fig. 2.3)are

≈0.3q

, q

≈0.7q



dz ≈0.2q

, ρ

LE ≈0.2q

, where q

is the received

short wave radiation, having a globally averaged value of 342 W m

−2

In summarising this discussion, we see that (3.5) is a convective diffusion equa-

tion for the temperature, with terms on the right hand side representing both sources

and transport. The temperature ﬁeld is driven by heating from below (q

in (3.12))

and cooling from above (q

in (3.17)). Estimated values of these are q

≈ 0.7q

≈0.3q

, while the internal condensation term is ρLCh ≈ 0.2q

, and the absorbed

radiation term is Q

h ≈ 0.2q

. Because the troposphere is being heated both from

below and within, it is convectively unstable. Because the heating is differential

and thus q

decreases from equator to poles), there is a secondary circulation

(Hadley, mid-latitude and polar cells) directed polewards; and because of the rapid

rotation of the Earth, this secondary circulation is diverted zonally, along lines of

latitude. To see all this, we need to non-dimensionalise the equations.

The approximation assumes an approximate hydrostatic balance, and an approximately one-

dimensional atmosphere.

148 3 Oceans and Atmospheres

3.2.4 Choosing Coordinates

We now write the equations in terms of spherical polar coordinates. We take r to be

the radius measured from the Earth’s centre, λ to be the angle of latitude, and φ to

be the angle of longitude. In terms of the more usual deﬁnition of spherical polar

coordinates (r,θ,φ), r and φ are the same, and λ =

− θ. We denote velocity

components in φ,λ,r directions as u, v, w (because we are setting up φ,λ,r, i.e.,

east, north, upwards, as future x,y,z Cartesian variables), and we denote the vector

velocity u =(u,v,w).

Then the material derivative takes the form

∂

∂t

+u .∇, (3.26)

and conservation of mass can be written in the form

dρ

+ρ∇. u =0, (3.27)

where the deﬁnitions of the vector derivatives are

∇. u =

r cosλ

∂u

∂φ

r cosλ

∂(vcos λ)

∂λ

∂

∂r





∇ =



r cosλ

∂

∂φ

∂

∂λ

∂

∂r



(3.28)

The momentum equations have the form

−

tanλ −2Ωv sin λ +2Ωwcos λ =−

ρr cosλ

∂p

∂φ

tanλ +2Ωusinλ =−

ρr

∂p

∂λ

, (3.29)

−

)

−2Ωucos λ =−

∂p

∂r

−g +

The energy equation in (3.1)is

ρc

−

=∇.(

k∇T)+Q

+ρLC, (3.30)

It should be pointed out that the Earth deviates noticeably from being a sphere; it is more nearly

an oblate spheroid, whose radius varies by some 20 km between pole and equator. This is of some

conceptual importance, since gravity is the most important force, and the use of a purely spherical

coordinate system would yield large ‘horizontal’ forces in the momentum equations. The correct

procedure is to deﬁne the level ‘horizontal’ surfaces to be geopotential surfaces, so that there are

no horizontal gravitational forces. But the geometric deviation from sphericity is so small that in

effect we regain the form of the equations in spherical polars, as presented here.

3.2 The Geostrophic Circulation 149

where we assume that

k represents a combined effective radiative and sensible ther-

mal conductivity.

These are awkward equations, but they can be simpliﬁed by scaling and approx-

imation. One of the features of the Earth’s weather systems is that they have a hor-

izontal length scale which, though large, is not global in extent. The description of

such systems is facilitated by using a local, near Cartesian coordinate system.

However, there is a difﬁculty in doing this. It is necessary to choose a particular

latitude on which to put the Cartesian origin, and this then limits the applicability of

the resulting approximate model to phenomena appropriate to this latitude. Luckily,

as we have seen, there is a natural division of the global circulation into three bands

(one in each hemisphere): tropical, mid-latitude and polar. We associate these three

latitudes with values of λ near zero, of O(1), and near ±

. Particularly, the polar

régime is an awkward one, because of the degeneracy of the equations near λ =

We will concentrate our discussion on mid-latitude phenomena, and take λ =λ

deﬁne the x–z plane.

Speciﬁcally, we deﬁne east, north and vertical coordinates x, y and z by the

relations

x =φr cos λ

,y=(λ −λ

)r, z =r −r

, (3.31)

where r

is the radius at sea level. We then have

r cosλ

∂

∂φ

=μ

∂

∂x

∂

∂λ

∂

∂y

∂

∂r

∂

∂z



∂

∂x

∂

∂y



, (3.32)

where

μ =

cosλ

, (3.33)

so that

∇ =



∂

∂x

∂

∂y

∂

∂z





∂

∂x

∂

∂y



∇. u =μ

∂u

∂x

+μ

∂(v/μ)

∂y

∂w

∂z



∂w

∂x

∂w

∂y

+2w



(3.34)

The mass and energy equations are still (3.27) and (3.30), and the momentum

equations are then

−2Ωvsin λ +2Ωwcos λ +



uw −uv tan λ



=−

∂p

∂x

+2Ωusin λ +



vw +u

tanλ



=−

∂p

∂y

, (3.35)

−2Ωucos λ −

)

=−



∂p

∂z



∂p

∂x

∂p

∂y



−g +f