James I.N. Introduction to Circulating Atmospheres

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4.5 A Lagrangian view of the meridional circulation

109

(a)

(b)

Fig. 4.16. (a) Schematic view of the passage of a fluid parcel through a sinuous jet

in the upper troposphere, (b) Projection of the same fluid trajectory on to the y-p

plane.

activity. Hence, it will rise slightly more quickly in the northern part of its

orbit than it sinks in the southern part. The result is shown schematically

in Fig. 4.17; each successive orbit of the parcel will be slightly higher than

its predecessor. To the north of the maximum, the opposite effect will occur

and the orbits will descend. This slow drift of the fluid parcel over many

periods is called 'Stokes drift' and it is well known in the case of surface

waves in shallow water. The total mass transport is the sum of the Stokes

drift and the indirect Ferrel circulation induced by the eddy temperature

and momentum fluxes, and, clearly, there will be a degree of cancellation

between the two components.

It is not easy to give simple dynamical arguments concerning which effect

will dominate. We will not present a detailed quantitative argument in this

book; the interested reader is referred to texts such as that by Andrews

et. al. (1987) for a more advanced account of these matters. However, we

note that the discussion of Chapter 3 led us to conclude on thermodynamic

grounds that a net thermally direct circulation is required to maintain the

kinetic energy associated with the general circulation against the dissipative

effects of friction. So we anticipate that the Stokes drift will prove stronger

than the induced indirect circulation. In fact, careful analysis shows that in

'nonacceleration conditions', the Stokes drift balances the Ferrel circulation

exactly, so there is no net mass circulation. Nonacceleration conditions

require the waves to be frictionless, adiabatic and steady. In the midlatitude

troposphere, none of these conditions holds, and it can be deduced that a

substantial direct circulation of mass must exist. It is this circulation which

helps to account for the spring maximum of ozone in the lower stratosphere.

Recognizing this thermally direct mass flux is important in discussing the

110

The zonal mean meridional circulation

Fig. 4.17. Schematic illustration of the tropospheric distribution of eddy kinetic

energy, and the Stokes drift of air parcels to north and south of the storm track

latitudes.

transport of any conserved tracer (which may include potential vorticity),

and is needed to complete a thermodynamically consistent account of the

global circulation.

4.6 Problems

4.1 Using the data in Fig. 4.1, estimate typical values of

[v]

in the Hadley

cell. Hence estimate the time taken for a parcel of air to circulate in the

Hadley cell. Compare this time to the seasonal timescale.

4.2 Repeat this calculation for the Ferrel circulation.

4.3 Compare 6 at

100

kPa and 30kPa at selected latitudes. Estimate the

Brunt-Vaisala frequency N at these latitudes, where

Discuss which processes might determine N.

4.4 By comparing the wind speed at the tropopause with the poleward

4.6 Problems 111

temperature gradient at selected latitudes, verify that the [u] and [9] fields

are in thermal wind balance.

4.5 Using the data in the following table, calculate the radiative equi-

librium temperatures at the pole, equator and for the globe. Use the results

to confirm the estimate of Y which was derived following Eq. (4.12).

Globe

Pole

Equator

Albedo

0.29

0.72

0.27

Insolation W m~

344

182

436

4.6 The annual mean precipitation over the Pacific is about 3000 mm

per year, concentrated in a strip 7° of latitude wide. Estimate the latent

heat release implied and compare this with the heating rate at the equator

according to the Held-Hou model. Use this estimate to evaluate w. Hence,

deduce the typical meridional wind speed in a Hadley cell which includes

condensation.

4.7 From Fig. 4.7(a), showing the variation of

<j>

</>

, etc., plot a graph

showing the strength of the subtropical jet as a function of

yo-

4.8 Estimate typical values of the static stability parameter s

in the tro-

posphere. Derive a relationship between s

and the Brunt-Vaisala frequency

N. State the units of s

4.9 Using the cross sections of [v'T'] and [wV] from Figs. 5.5 and 5.7,

estimate the magnitude of terms typical of the forcing of the mean merid-

ional circulation by midlatitude heat and momentum fluxes in the winter

midlatitudes. Which term dominates? Estimate a typical poleward velocity

[v] induced by midlatitude eddies, and compare this with the meridional

velocity associated with the Ferrel cell, as shown in Fig. 4.1.

4.10 It has been suggested that breaking, vertically propagating gravity

waves have an effect equivalent to a drag on the zonal flow, with a maximum

value just above the midlatitude tropopause. Using the meridional circulation

equation, Eq. (4.31), sketch the nature of the circulation you expect to be

induced by breaking gravity waves. What deceleration due to breaking

gravity waves would be required to induce a meridional wind [v] of

m s"

at the breaking level at 50°N?

Transient disturbances in the midlatitudes

5.1 Timescales of atmospheric motion

The circulation of the atmosphere is intrinsically unsteady; fluctuations on

all timescales are observed. In the last chapter, it was shown that the

fluxes of temperature, momentum, and so on, carried by such transients

play an important part in determining the time mean circulation of the

atmosphere. Our task in this chapter will be to describe the transients

on various timescales, and to discuss the mechanisms which can give rise

to transient behaviour. Just as the atmosphere contains a wide range

of spatial scales, from the molecular to the global, so the atmospheric

circulation exhibits a wide range of timescales, ranging from timescales of

just a few seconds associated with the overturning of small turbulent eddies,

to geological timescales for major climate changes.

Some of the frequencies observed are directly related to the frequencies

of periodic forcings. For example, diurnal and semi-diurnal variations of

temperature and wind are associated with the diurnal variation of solar heat

input. These 'thermal tides' are important at high levels in the atmosphere

and can be detected in the lower atmosphere. More importantly for our

purposes, the annual cycle of radiative forcing has a profound effect on

large scale atmospheric circulation. This seasonal cycle of meteorological

quantities affects nearly all parts of the globe.

But in addition to these externally imposed periods, the atmospheric flow

itself generates all kinds of timescales internally. Various wave motions have

characteristic frequencies, while irregular, aperiodic fluctuations arise from

the chaotic, quasi-turbulent character of much atmospheric flow. Why such

irregular fluctuations arise in the first place is an intriguing question. The

superficial answer is that the time mean flow is unstable to small amplitude

112

5.1 Timescales of atmospheric motion 113

disturbances. But how is the mean flow maintained in an unstable state?

Why do the developing instabilities not wipe out the shears and temperat-

ure gradients which make instability possible, leaving the atmosphere in a

quiescent state of near neutral stability? These are questions to which we

will return in Section 7.4.

In this chapter, we will examine these 'transient' features of

the

atmospheric

flow. We will discuss how they are generated and how they contribute to the

transport of heat and other quantities across the globe. We will attempt to

illustrate how individual weather systems contribute to the global circulation

of the atmosphere. In the last chapter, we noted that the Hadley circulation

reduced the temperature gradients in the tropics, but that it actually increased

the temperature gradi ent and horizontal wind shears in the midlatitudes. The

transients of the midlatitudes are responsible for reducing these temperature

gradients and transporting heat and momentum out of the subtropical

latitudes and into the higher latitudes.

Before turning to a more detailed discussion of the dominant scales and

shapes of atmospheric transients, consider the frequency and wavenumber

characteristics of the observed transients. We recall the notation introduced

in Chapter 2, in which the time mean of any quantity Q is Q, while the

deviation from the time mean is denoted Q

. The kinetic energy associated

with the transient eddies is:

K = i (V

+ i/

) . (5.1)

The pressure-latitude section in Fig. 5.1 shows the zonal mean distribution of

eddy kinetic energy on all timescales. The transients are small in the tropics

but become much more important in the midlatitudes, with maximum values

near the tropopause and somewhat poleward of the subtropical jet core. The

transients become weaker towards the pole. The winter season has stronger

transients than the summer; this seasonal cycle is a good deal more marked

in the northern hemisphere. Larger values are also seen in the stratosphere,

close to the equator and at high latitudes in the winter hemisphere.

The transient eddy kinetic energy varies not only with height and latit-

ude,

but also with longitude. In the northern hemisphere winter, maxima

of the eddy kinetic energy are located over the western side of the ocean

basins, with minima over North America and Asia. The pattern is shown in

Fig. 5.2a. In the southern hemisphere winter, a single maximum is observed

over the south Atlantic and Indian Oceans, with lower values over the Pacific.

These maxima are coincident with the regions where developing and mature

midlatitude depressions occur most frequently. On the eastern side of the

114

Transient disturbances in the midlatitudes

. • i . i • • , i • . , • , , •

LATITUDE

Fig. 5.1. Latitude-pressure sections of transient eddy kinetic energy for (a)

December-February and (b) June-August. Contour interval 25m

; shading

indicates values in excess of

300m

s~~

Based on six years of ECMWF data.

ocean basins, systems tend to be occluded and decaying, and so are much

less vigorous over the continents.

This pattern is accentuated considerably if the time series of each velocity

component is filtered so as to remove the lower frequencies before construct-

ing the variances. In general, numerical filtering of a time series involves

5.1

Timescales

atmospheric

motion 115

replacing the nth member of the time series with a suitably weighted average

of the neighbouring members:

Qn = Z

™iQn+i-

(5.2)

The properties of the filter are defined by the choice of the filter weights w

For example, a crude low pass filter, which removes the higher frequencies

but leaves the lower frequency variance to survive intact, results if the

weights are constant and simply equal to l/(2j +1). In this case, the mean is

a simple running mean of the time series. Conversely, by defining

simply

to be the deviation from some such running mean, a crude high pass filter is

established. More sophisticated filters are designed to produce an extremely

sharp response, so that only a precisely defined range of frequencies is

passed. But since the time spectrum of atmospheric motions is smooth and

continuous with no obvious spectral gaps, there is little physical reason to use

a particularly elaborate filter for our purposes. The high pass filtered eddy

kinetic energy (shown in Fig. 5.2b) was constructed by the use of a running

mean filter such as that suggested above. It had the effect of removing much

of the variance associated with transients with periods longer than about

six days. The filtered transient eddy kinetic energy is a good deal less than

the unfiltered equivalent, but the maxima are very clearly defined. They

form zonally elongated areas running across the Atlantic from the North

American coast, and across the Pacific from the Asian coast. Similar maxima

are found if other measures of transient eddy activity, such as geopotential

height variance or low level temperature flux, are plotted. The tracks of

developing depression centres tend to run along the long axes of these areas,

for which reason they are commonly called 'storm tracks'. These storm track

regions can be seen in the heating fields in Fig. 3.8, especially in the Northern

Hemisphere winter. Chapter 7 contains a fuller discussion of the midlatitude

storm tracks.

The higher frequency transients of the midlatitudes, by which we mean

transients with periods between one and ten days or so, owe their existence

to a hydrodynamic instability of the zonal flow called 'baroclinic instability'.

In the first part of this chapter, we will discuss the mean size and structure

of the transient eddies. Then we will discuss the energy associated with the

transients, and how it can be generated. The theory of baroclinic instability

will be introduced in Section 5.4, while the extensions and limitations of this

theory will be described in Section 5.5.

The lower frequency transients have a more complex origin, which is

116

Transient disturbances in the midlatitudes

(a)

Fig. 5.2. Transient eddy kinetic energy at 25 kPa for the period December-February,

northern hemisphere, (a) Total transient eddy kinetic energy, contour interval

50 m

s~~

with values in excess of 300 m

shaded.

still imperfectly understood. Low frequency behaviour results in part from

the internal dynamics of the atmospheric flow, particularly the nonlinear

interactions between different scales of motion which lead to a cascade

of energy towards lower frequencies. But low frequencies are also forced

externally, by interactions between the atmospheric circulation and more

slowly varying systems, such as the ocean. As the frequency becomes lower,

the internal dynamics is generally supposed to become less important, while

the role of external forcing becomes dominant. Describing low frequency

variability is difficult because the data records are not generally long enough

to produce statistically reliable patterns of variances and covariances for the

lower frequencies. Gaining greater insight into the underlying mechanisms

5.2 The structure of transient eddies

111

Fig. 5.2 (cont). (b) High frequency transient eddy kinetic energy, contour interval

25 m

s~~

with values in excess of 150 m

shaded. The eddies contributing to

(b) have periods less than about 6 days. Based on six years of ECMWF data.

is a goal of much current research. We will look in more detail at low

frequency variability in Chapter 8. In the remainder of this chapter, we will

focus on the higher frequency transients.

5.2 The structure of transient eddies

Scale analysis shows that large scale motions in the Earth's midlatitudes

are nearly nondivergent; the geostrophically balanced rotational part of the

wind dominates over the small, mainly divergent ageostrophic wind. The

ratio of the geostrophic to ageostrophic wind speeds is of order Rossby

number. So it is useful to model the horizontal structure of transient eddies

by a geostrophic streamfunction which is supposed to vary sinusoidally in

118

Transient disturbances

in the

midlatitudes

the x- and ^-directions:

l{kx+ly

(5.3)

More properly, Eq. (5.3) should be regarded as one component of a Fourier

summation over all permitted k and /. In this chapter, we will interpret the

equation as describing the typical transient eddy; the values of k and / are

regarded as some kind of average for all the observed transients. We may

regard the amplitude f as a sinusoidal function of time; in that case, the

average over a complete period of the wave will be the same as the average

over a complete wavelength. The dimension of the individual vortices is half

a wavelength in each direction, that is, n/k in the x-direction and n/l in the

y-direction.

The northward and eastward components of the wind are then related to

the streamfunction by:

= iWe (5 4a)

(5.4b)

The appearance of i in these expressions indicates that the velocity wave

and the streamfunction wave are

TT/2

out of phase. The variance of u and

v indicate the typical magnitudes of u! and i/. The streamfunction is not

measured directly, but it is closely related to the geopotential height Z, which

can be observed and analysed. The geostrophic streamfunction is related to

the geopotential height by:

= Zz. (5.5)

Hence, the velocity variances and the geopotential height variance are related

by:

assuming that / is constant over the width n/l of the eddy. Thus, a

comparison of velocity variance and geopotential height variance will lead

to an estimate of the typical wavenumber of the transients, that is, to their

spatial scale. A comparison of the u- and i;-variance leads to a measure of

the typical shape of the eddies, since:

v* k