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L ND=f

will have QGPV dynamics uncoupled from the surface dynamics, and

cannot be expected to cohere and grow. Thus, short-wave instabilities must also be

shallow, and deep modes ﬁlling out a density scale height must have horizontal

scales comparable to the radius of deformation L

¼NH=f

, if not longer. These

arguments are quite general and carry over to strongly nonlinear motions.

The simplest possible model that satisﬁes the necessary conditions for instability

in a continuous atmosphere is the Eady model. The Eady model has r ¼constant,

b ¼0, @

¼L ¼constant, and rigid lids at z ¼0 and H

. While this scenario

provides only a crude model of the atmosphere, it allows for insights into the

dependence of a growing disturbance on horizontal scale and stability. The primary

simpliﬁcation is the vanishing potential vorticity gradient in the interior of the ﬂuid.

Since the QGPV is zero initially in the interior, it will remain so for all time.

Therefore, all the dynamics involve the time-dependent boundar y condition (11)

applied at each boundary. Despite the vanishing of the QGPV gradient, the Eady

model satisﬁes the necessary conditions for instability because vertical shear of the

basic state ﬂow at the upper boundary provides an additional term in (24) that is

equal and opposite to the lower boundary integral.

For this situation, (11) and the condition that q

vanish in the interior lead to wave

solutions on each boundary (considered in isolation), whose effective restoring force

is the meridional temperature g radient. Such waves are called Rossby edge waves,

where edge waves are boundary-tr apped disturbances. The wave at the lower boun-

dary propagates eastward (relative to the surface ﬂow), and the wave at the lid

propagates westward. Localized warm surface air is associated with low pressure

and cyclonic ﬂow, and is effectively a positive QGPV perturbation. Cold air at the

surface features high pressure and anticyclonic ﬂow, analogous to a negative QGPV

perturbation .

Baroclinic instability occurs in this model when the waves at each boundary

amplify the temperature at the opposite boundary. For this to happen, the ‘‘deforma-

tion depth’’ deﬁned above must be at least comparable to the distance between the

boundaries. If the disturbance horizontal length scale is too short, or if the bound-

aries are too far apart, unstable modes do not exist.

Figure 1 shows the disturbance structure for a wave with equal zonal and meri-

dional wavenumber (k ¼l). For this situation, the wavelength of the perturbation that

grows fastest is

ﬃﬃﬃ

ðH

 4000 km ð26Þ

similar to the length scale of observed synoptic-scale transients. The growth rate of

the instability is approximately given by

46 EXTRATROPICAL ATMOSPHERIC CIRCULATIONS

ImðoÞ

0:25f

ð27Þ

This leads to an e-folding time of 2 or 3 days for typical extratropical values of the

shear and N, again consistent with observed synoptic eddy growth rates. Figure 1b

shows that for this particular wavelength, as for all unstable baroclinic waves, the

trough and ridge axes slope westward with height. This phase displacement allows

Figure 1 Properties of the most unstable Eady wave. (a and d ) Geopotential height and

temperature patterns on the upper and lower boundaries, respectively. (b) The stream function

for the ageostrophic ﬂow, showing the overturning in the x-z plane. Ascent is just to the east of

the surface trough and descent is just to the west of the surface ridge. (c) Contours of

meridional geostrophic wind v

(solid) and isentropes (dashed) in the x-z plane. H and L

indicate the ridge and trough axes; W and C denote the phase of the warmest and coldest air.

Circles with enclosed crosses indicate the axis of maximum poleward ﬂow, and circles with

enclosed dots indicate the axis of maximum equatorward ﬂow (from Gill, 1982).

3 BAROCLINIC INSTABILITY AND FRONTOGENESIS 47

the velocity and temperature per turbations to be partly in phase at each boundary,

thus leading to a poleward heat ﬂux. The axes of the warmest and coldest air,

however, tilt eastward with height. East of the trough, where the perturbation meri-

dional velocity is poleward, Figure 1c shows that the ageostrophic vertical motion is

upward. Thu s, parcel motion is poleward and upward in the region where tempera-

ture perturba tions are positive, and vice versa in regions where the temperature

perturbation s are negative. Through condensation in ascending, adiabatically cooling

trajectories, this vertical velocity pattern provides a link to the hydrological cycle in

the extratropical troposphere and, in particular, provides the ‘‘comma cloud’’ signa-

ture of surface troughs.

The scaling of the growth rate with N in (27) suggests that baroclinic instability

would produce a statically stable extratropical atmosphere on Earth even in the

absence of moisture. Loc al radiative convective equilibrium would drive N to zero

but would have a strong equator–pole temperature gradient. Such a ﬂow would be

violently unstable to baroclinic instability. It is presumed that baroclinic instability

increases N so as to adjust toward a more stable state. Eddies could do this by taking

cold air manufactured in polar regions and sliding it under warmer air, and vice versa

for warm air produced near the tropics. Either by vertical heat ﬂuxes that increase N

or by horizontal heat ﬂuxes that reduce the equator–pole temperature gradient (and

hence the vertical shear) , baroclinic instability acts to adjust the extratropical ﬂow

toward a state that is neutrally stable to baroclinic instabil ity.

In addition to their role in the tropospheric general circulation, synoptic-scale

eddies also drive weather-related events at smaller scales. It has long been under-

stood that extratropical cyclones have cold frontal regions in which the temperature

typically changes rapidly, often accompanied by shifts in the wind and by precipita-

tion. In general, these fronts are considered to form in a growing nonlinear baroclinic

wave, though this secondary role in no way diminishes their importance in practical

meteorology. Rather, the theoretical emphasis shifts toward the mechanism for

generating fronts, namely frontogenesis.

The basis of frontogenesis lies in the advection of surface temperature (7). The

kinematics of frontogenesis can be understood by considering a certain class of

horizontal velocity ﬁelds called deformation ﬁelds. Such deformation ﬁelds, which

are local features of large-scale horizontal wave motions, contain conﬂuent regions

that tend to concentrate a large-scale preexisting temperature gradient, squeezing

isotherms together. A prototypical deformation ﬁeld is given by u ¼gx, v ¼gy,

where g is a constant with dimension (time)

1

. Suppose the potential temperature

ﬁeld is initially oriented so the isotherms are parallel to the y axis. Then at the

ground, where w vanishes, the time evolution of the potential temperature must

satisfy

y ¼u@

y ¼ gx@

y ð28Þ

since y is independent of y. The solution to (28) is easily veriﬁed to be

yðtÞ¼y

ðxe

Þð29Þ

48 EXTRATROPICAL ATMOSPHERIC CIRCULATIONS

where y

(x) is the surface distribution of potential temperature at the initial instant.

The temperature gradient at the surface is

y ¼ e

yyðxe

Þð30Þ

where the dot indicates derivative with respect to the argument. Therefore, the

surface temperature gradient increases exponentially with time in regions of conﬂu-

ence in the deformation ﬁeld.

Naturally, as time goes by and the temperature ﬁeld changes as a resul t of this

conﬂuence, the changing ther mal wind in the y direction will produce, by Coriolis

accelerations, ﬂow in the x direction that will alter the initial deformation velocity

ﬁeld. Thus, the above solution will be valid only initially and even then describes the

structure of the temperature gradient at the ground. However, this solution does show

that large-scale ﬂow features can generat e strong density gradients.

A more complete theoretical explanation of frontogenesis lies outside of the class

of motions described by quasigeostrophic theor y. This is because fronts are inher-

ently anisotropic, with a very narrow frontal zone compared to the spatial extent

along the front. A different scaling of the primitive equations, namely the semi-

geostrophic theory originally developed by Hoskins and Bretherton, explicitly

recognizes this anisotropy and provides a more complete dynamical theory for the

formation of fronts [see Hoskins (1982) for a review]. While such a theor y lies

beyond the scope of our treatment, it does presen t a number of features absent in

the deformation model of frontogenesis examined above. Foremost among these

features is that the frontogenetic mechanism is so powerful in semigeostrophic

theory that inﬁnite temperature gradients can be generated in a ﬁnite amount of

time. Of course, in reality instabilities can be expected to occur in regions of such

gradients, and accompanying mixing of surface temperature will tend to limit the

sharpness of the frontal zone. However, the prediction of a discontinuity indicates

the vigor of the frontogenetic process.

4 STATIONARY PLANETARY WAVES

Another striking component of the observed extratropical atmospheric circulation

are the large-scale stationary planetary (Rossby) waves that dominate the zonally

asymmetric ﬂow during the Northern Hemisphere winter. Flow over larger-scale

surface features (e.g., the Tibetan plateau), along with longitudinally varying

diabatic heating force these waves, which subsequently propagate zonally and meri-

dionally throughout the troposphere, and vertically into the upper atmosphere. In the

troposphere, these waves inﬂuence the regional climates of Earth, as they affect the

position and strength of the midlatitude storm tracks. In the stratosphere, breaking

vertically propagating Rossby waves instigate strong changes in the stratospheric

ﬂow, called sudden warmings (Andrews et al., 1987, Section 6). More generally, the

breaking of these vertically propagating waves plays an important role in the overall

4 STATIONARY PLANETARY WAVES 49

exchange of air between the stratosphere and the troposphere, as this breaking forces

planetary-scale vertical ascent and descent of air (Holton et al., 1995).

First, let us consider the zonal and meridional propagation of stationary planetary

waves. This problem is most easily treated using a barotropic model on the sphere,

linearized about a zonally symmetric but latitudinally varying mean ﬂow. In this

model, stationary waves are solutions of the equation

g 0

ða cos fÞ

1

þ v

ða

1

þ bÞ¼u

f ðh

a cos fÞ

1

 rz

ð31Þ

where z

a

1

þða cos fÞ

1

is the relative perturbation vorticity, primes

indicate deviation from zonal mean background states, and h

is the resting depth of

the barotropic ﬂuid. The damping coefﬁcient r is taken to be independent of latitude.

The response of this system with h

equal to the Nor thern Hemisphere topogra-

phy is shown in Figure 2 , where u

and u

equal the zonally averaged Northern

Hemisphere winter time 300 mb and surface winds, respectively. The response ﬁeld

appears to be comprised of two wavetrains propagating equatorwards, one produced

by the Rockies and the other by the Tibetan plateau.

These structures can be explained by Rossby wave ray-tracing theory. If we

transform the linearized, unforced inviscid vorticity equation into Mercator coordi-

nates x ¼ al; a

1

dy ¼ðcos fÞ

1

df, then (30) takes the form

uu@

ð@

þ @

Þc

¼

bb@

ð32Þ

where

uu  u

g 0

ðcos fÞ

1

and

bb  cos fðb þ a

1

Þ. For a wave of the form

¼ Re

cc expðikxÞ, one has

cc ¼ðk



bb=

uuÞ

cc ð33Þ

Given a source localized in latitude, the structure of (32) requires that zonal wave-

numbers k > k

ð

bb=

uuÞ

1=2

remain meridionally trapped in the vicinity of the source,

while wavenumbers k < k

are free to propagate away from the source.

A source localized in longitude as well as latitude can be regarded as producing

two rays for each n < n

, where n ¼ak is the zonal wavenumber on the sphere, and

¼ak

. These rays correspond to the two possible meridional wavenumbers,

al ¼ðn

 n

1=2

. Since the meridional group velocity of nondivergent Rossby

waves c

¼ @o=@l ¼ 2

bbklðk

þ l

2

has the same sign as l, the positive (negative)

sign corresponds to poleward (equatorward) propagation. The zonal group velocity

¼ @o=@k ¼ u

bbðk

 l

Þðk

þ l

2

is eastward for stationary planetary

waves with k l. Therefore, the rays can be traced downstream from their source,

noting that the zonal wavenumber remains unchange d b ecause the mean ﬂow is

independent of x, whereas the meridional wavenumber adjusts to satisfy the local

dispersion relation. The ray path is tangent to the vector group velocity (c

, c

) for a

particular k and l, and is always refracted toward the larger ‘‘refraction index’’

7 n

)

1=2

, i.e., rays turn toward larger n

in accordance with Snell’s law for

50 EXTRATROPICAL ATMOSPHERIC CIRCULATIONS

Figure 2 (a) Topography of the Northern Hemisphere. (b) The stream function response of

the Northern Hemisphere of a barotropic model, forced using the zonal mean winds and

Northern Hemisphere topography. Solid contours are positive or zero; dashed contours are

negative (from Held, 1982).

4 STATIONARY PLANETARY WAVES 51

optics. The atmosphere tends on average to exhibit low values of the index of

refraction toward the polar regions, with high values toward the subtropical regions.

Therefore, a wave excited somewhere in extratropics will refract away from the polar

latitudes and propagate in the direction of the tropics.

When the zonal mean ﬂow varies with latitude, two important complications

occur. The ﬁrst is the existence of a turning latitude where n ¼n

. At this latitude,

both the meridional wavenumber and meridional group velocity go to zero; thus, for

a wave propagating poleward and eastward out of the tropics, it would continue to be

refracted until at some point its troughs and ridges are aligned entirely north=south.

After this point, the wave continues to turn and begins to propagate back toward the

tropics.

The second complication is posed by the existence of a critical latitude, where

¼0. Critical latitudes are generally marked by complicated dynamics because the

implicit linear assumption that particle position deviations from rest are small does

not hold. For the case of a Rossby wave train, as it approaches its critical latitude,

linear theory predicts that the meridional wavenumber l !1, while the meridional

group velocity vanishes. The development of small scales while propagation slows

leads linear theory to predict the absorption of the Rossby wave train at critical

latitudes. However, the inclusion of nonlinearities opens a whole set of possibilities,

including the reﬂection of the equatorward propagating wave train back into the

extratropics. More generally, down gradient ﬂuxes of potential vorticity associated

with stationary wave breaking (or for that matter, with synoptic transient wave

breaking) at critical latitudes located on the ﬂanks of the upper tropospheric or

stratospheric jets plays an important role in the momentum budget of the extra-

tropical atmospheric general circulation. Speciﬁcally, this wave breaking results in

signiﬁcant wave-induced forcing of the zonal basic state ﬂow.

Stationary Rossby waves not only propagate zonally and meridionally but also

vertically. To understand this vertical propagation, let us return to the continuously

stratiﬁed QG equations of Se ction 2. We are free to seek stationary solutions to Eqs.

(14) and (15) with c ¼0. For the case where u

and N are constant, but including

topography h

¼ exp(ily)exp(ikx), the dispersion relation (17) leads to a condition on

the vertica l wavenumber m. Provided that 0 < u

< b (k

þ l

þ e(2H)

2

)

1

, m will

be real and given by

m ¼Nf

1

ðbu

1

ðk

þ l

Þeð2HÞ

2

1=2

ð34Þ

Examination of the vertical group velocity (@o=@m) reveals that it has the same sign

as m, so choosing the positive branch of (34) corresponds to a wave propagating

upward from a source at the ground and is an acceptable solution. For winds outside

this range, m will be imaginary, and the waves will be evanescent rather than

propagate with height. The most important fact highlighted by this simple system

is that only sufﬁciently weak westerlies permit vertical stationary wave propagation.

For parameters relevant to Earth’s atmosphere, stationary wavenumbers three and

greater will not readily propagate into the stratosphere, thus accounting for the

52 EXTRATROPICAL ATMOSPHERIC CIRCULATIONS

predominance of wavenumbers one and two in the winter stratosphere. Additionally,

the summer stra tospheric easterlies effectively block the propagation of all stationary

waves, accounting for the observed zonal character of the summer stratospheric

circulation.

Of particular importance is the external Rossby wave, deﬁned as the evanescent

mode with the gravest vertical structure, which tends to dominate the response

downstream of a localized forcing. For realistic ﬂows such as the one shown in

Figure 3a, the external mode is equivalent barotropic, with geopotential amplifying

with height in the troposphere and decaying with height in the stratosphere. The

stationary Gre en’s function response to a ‘‘spike’’ imposed topographic forcing

¼sin(ly)d(x), where d(x) is the Dirac delta function, is shown in Figure 3c. The

emergence of the external mode in the far-ﬁeld is apparent, due to the rapid vertical

propagation of longer wavelength stationary Rossby waves and destructive interfer-

ence among shorter wavelength stationary Rossby waves as they propagate upward

and downward, conﬁned by their midtropospher ic turning levels.

5 ISENTROPIC POT ENTIAL VORTICITY

In any ﬂuid dynamical problem, the existence of conserved quantities provides a

foundation on which to build ﬁrm theoretical descriptions of dynamical phenomena.

The usefulness of having such a conserved quantity has been apparent throughout

the above discussion, as, for example, the development of the necessary conditions

for stability in Section 3 explicitly used the conservation of QGPV to develop the

global pseudomomentum conservation relation (24). In the atmospher e, the Rossby–

Ertel isentropic potential vorticity (IPV) goes beyond QGPV in that it is conserved

by the full dynamics of the atmosphere in the absence of forcing and dissipation,

rather than by just the simpliﬁed QG subset of those dynamics.

The conservation of IPV in the full dynamical equations of motion can be shown

as follows. In a frictionless, adiabatic atmosphere a close material contour on an

isentropic surface remains on that surface and its absolute circulation

u  dl ¼

z  n dS ð35Þ

is conser ved. For an elemental cylinder of cross-sectional area S between the isen-

tropic surfaces y and y þ dy; C

¼ z  nS, and the mass m ¼rS dh and dy are also

conserved. Here n is the unit normal to the isentropic surface and h is the distance in

this direction. Therefore

dy ¼ r

1

z  n

ð36Þ

is also conserved. In the inﬁnitesimal limit this gives material conservation of the IPV

P ¼ r

1

z  Hy ð37Þ

5 ISENTROPIC POTENTIAL VORTICITY 53

At the level of the hydrostatic approximation, z

 n ¼ z

, a quantity resembling the

vertical component of absolute vorticity but with horizontal derivatives evaluated on

isentropic surfaces. Then (37) is similarily approximated by

P ¼ s

1

ð38Þ

Figure 3 (a) Idealized zonal wind and static stability proﬁle. (b) Vertical structure of the

external Rossby wave for this ﬂow proﬁle. (c) Stationary response to ‘‘spike’’ topography for

this ﬂow proﬁle, where the ar row marks the location of the source. Solid lines are positive

contours and dashed lines are negative (from Held, 1982).

EXTRATROPICAL ATMOSPHERIC CIRCULATIONS

where s ¼g

1

@p=@y plays the role of density in isentropic coordinates.

In a frictionless, adiaba tic atmosphere IPV is conserved by the two-dimensional

(2D) mot ion on an isentropic surface; and similarly y is conserved by the 2D motion

on an iso-IPV surface. Most important ly, if the interior IPV and boundary y distribu-

tions are known, then balanced motions may be determined by inversion of an

elliptic operator. The simplest balance assumption is geostrophy; under this assump-

tion, given the IPV=surface y distributions one could deduce the structure of the

waves and instabilities that comprise the large-scale extratropical atmospheric circu-

lation. However, much more general balance assumpti ons can be made; for example,

the assumption of some form of semigeostrophic balance would allow the study of

IPV-dynamical interactions in the vicinity of fronts.

Provided the elliptic operator used for the IPV inversion is linear, a straightfor-

ward examination of the relative roles of various IPV disturbances in a given dyna-

mical situation is possible. For example, the use of a QG inversion operator to study

observed cyclogenesis leads to a similar dynamical picture to that outlined for the

Eady model above, namely the intensiﬁcation of the surface temperature anomalies

by winds that stem from the upper tropospheric IPV anomalies, and vice versa.

However, IPV allows more general investigations within this framework than does

QGPV; for example, the importance of diabatic production of IPV by latent heat

release and surface friction during cyclogenesis can be using such a decomposition.

Hoskins et al. (1985) review a number of applications of IPV inversion, along with

an in-depth review of the associated concepts underlying IPV conservation and

atmospheric dynamics.

A cross section of the time-averaged IPV (Fig. 4) shows that the strongest gradi-

ents of IPV along isentropic surfaces occur at the tropopause. The other region of

Figure 4 Climatology of isentropic potential vorticity (IPV) and potential temperature y for

the Northern Hemisphere winter. Isentropes are dashed and drawn in intervals of 30 K; IPV

has units of 10

6

K (kg s)

1

¼1 PV unit. Contours of IPV are 0, 0.5, 1, 2, 5, and 10 PV

units, with the contour at 2 PV units stippled to indicate the approximate position of the

dynamical tropopause.

5 ISENTROPIC POTENTIAL VORTICITY 55