Salby M.L. Fundamentals of Atmospheric Physics

Подождите немного. Документ загружается.

380

12 Large-Scale

Motion

geostrophic equilibrium, couples the circulation to the stratification and makes

vertical shear a measure of the departure from barotropic stratification.

From (12.11), an equatorward temperature gradient is accompanied by

positive shear of the zonal flow. Motion then becomes increasingly westerly

with elevation. Examples are found in the troposphere and winter stratosphere

(Fig. 1.7). Radiative heating at low latitude and cooling at middle and high

latitudes establish an equatorward temperature gradient in the troposphere in

both hemispheres. Thermal wind balance then implies westerlies that intensify

upward to form the subtropical jets (Fig. 1.8). In the winter stratosphere,

ozone heating at low latitude and longwave (LW) cooling to space at high

latitude (e.g., in polar night) establish a temperature gradient of the same

sense over a deep layer. Westerlies then intensify upward in the polar-night jet

to a maximum near the stratopause. Opposite behavior occurs in the summer

stratosphere, where solar insolation maximizes at high latitudes (Fig. 1.28)

to produce a poleward temperature gradient. The zonal flow then becomes

easterly above the tropopause and intensifies upward to a maximum near the

stratopause.

These features can also be inferred directly from geostrophic equilibrium

and the hypsometric relationship, which is a statement of hydrostatic balance.

In the presence of a horizontal temperature gradient, vertical spacing of iso-

baric surfaces is compressed in cold air and expanded in warm air (Fig. 6.2).

The meridional gradient of isobaric height in the troposphere then steepens

with elevation, which, through geostrophic equilibrium, produces zonal mo-

tion that intensifies upward. Strong zonal motion favored by the distribution

of radiative heating (Fig. 1.29) and geostrophic equilibrium transfers neither

heat nor chemical constituents meridionaUy. Consequently, the earth's rota-

tion tends to stratify properties meridionally, just as gravity tends to stratify

them vertically.

Thermal wind balance also applies to zonally asymmetric motion. The po-

lar front, which separates warm tropical air from cold polar air (Fig. 6.3),

coincides with strong westerly shear and the instantaneous position of the jet

stream (Fig. 1.9a). Inside a cold-core low, as typifies extratropical cyclones

(Problem 6.7), cyclonic shear reinforces circumferential motion, which then

intensifies upward to a maximum near the tropopause. The reverse occurs in-

side a warm-core low, as typifies hurricanes (Problem 6.9). Anticyclonic shear

then opposes circumferential motion to produce maximum winds near the

surface (Problem 12.34).

Geostrophic equilibrium governs motion that is sufficiently steady, slow, and

weakly curved for material acceleration to be negligible relative to the Coriolis

and pressure gradient forces (i.e., for

< < 1). It also requires frictional drag

to be negligible. These conditions are well satisfied by large-scale extratropical

motion away from the surface, which implies that

v h

is nearly nondivergent.

However, the conditions for geostrophic equilibrium break down as the equa-

tor is approached or as the horizontal scale of advection becomes small, which

12.3 Frictional Geostrophic Motion 381

render terms O(Ro) nonnegligible. By introducing an ageostrophic component

to vh, these effects introduce divergence and hence vertical motion (12.9).

12.3 Frictional Geostrophic Motion

Inside the planetary boundary layer, frictional drag is large enough to inval-

idate geostrophic equilibrium. Turbulent eddy motions within a kilometer of

the surface, which are produced by strong vertical shear (see Fig. 13.3), mix

momentum between bodies of air. The accompanying momentum flux exerts

shear stresses on individual air parcels that render D of the same order as the

Coriolis and pressure gradient forces (Sec. 10.6).

If drag is represented as Rayleigh friction with linear drag coefficient K,

= KVh,

(12.12)

then mechanical equilibrium is expressed by

x 13 h -- -Vpf~ - KVh,

(12.13)

which can be solved for the components of horizontal motion (Problem 12.13).

Friction modifies both the magnitude and the direction of vh. The ensuing

mechanical equilibrium can be motivated by gradually introducing drag into

the geostrophic balance in Fig. 12.1. Introducing D reduces the velocity of the

air parcel, which reduces the Coriolis force -fk x vh acting on it, which, in

turn, allows the pressure gradient force to drive the parcel's motion across

isobars toward low pressure (Fig. 12.6). Eventually, the motion achieves an

angle 6 from isobars such that the resultant of the three forces acting on the

parcel vanishes. The air parcel is then in frictional geostrophic equilibrium.

Unlike simple geostrophic motion, frictional geostrophic motion is not

solenoidal. The divergent component of Vh then introduces vertical motion

via continuity (12.9). Frictional convergence into a center of low surface pres-

sure (Fig. 12.7a) is compensated by rising motion overhead. By organizing

moisture near the surface and reinforcing upward displacements, such motion

favors cloud formation in a cyclone. It also reduces vertical stability (Sec. 7.4.4),

which likewise provides conditions favorable to convection. Conversely, fric-

tional divergence out of a center of high surface pressure (Fig. 12.7b) is com-

pensated by subsidence. By opposing upward displacements, sinking motion

inhibits cloud formation and favors the formation of a subsidence inversion

that traps pollutants near the surface.

The foregoing behavior is a consequence of vertical motion, which does

not develop under simple geostrophic equilibrium. Divergence accompanying

frictional geostrophic motion is inversely proportional to f (Problem 12.14).

Therefore, vertical motion is favored in the tropics, where small f allows large

departures from geostrophic equilibrium. The latter support thermally direct

circulations in which air ascends over low pressure and descends over high

pressure (Fig. 1.30).

382

Large-Scale Motion

9

-Vp~ ~'fk

x v h Vh

-D = -KVh

Figure 12.6 Frictional geostrophic balance in a flow characterized by nearly straight height

contours and containing Rayleigh friction.

tional Convergence

tionol Divergence

Figure

12.7 Vertical motion introduced by (a) frictional convergence into surface low pressure

and (b) frictional divergence out of surface high pressure.

12.4 Curvilinear Motion

The other factor driving the circulation out of geostrophic equilibrium is ma-

terial acceleration. If its trajectory is sufficiently curved, an air parcel will

experience a centripetal acceleration (which is embodied in the advective con-

tribution

Oh" VpVh)

that renders the Rossby number nonnegligible. Unsteadi-

12.4

Curvilinear Motion

383

ness (which can introduce the same effect through the local time derivative

3Vh/3t )

is typically less important.

Consider steady motion in which parcels move along curved trajectories

characterized by the local radius of curvature R. Parcel trajectories then coin-

cide with streamlines. It is convenient to introduce the

trajectory coordinates s

and n, which increase along and orthogonal to the path of an individual parcel

(Fig. 12.8). Then the coordinate vectors s and n are tangential and orthogo-

nal to the local velocity

Vh, with n pointing to the left of s. R is defined to be

positive if the center of curvature lies in the positive n direction. Curvature is

then cyclonic if R is of the same sign as

f (fR > 0) and anticyclonic if it is of

opposite sign

(fR < 0).

In this coordinate system, the horizontal velocity is described by

where the speed is simply

Vh = VS, (12.14.1)

v= dt" (12.14.2)

9

V 2

-fk x v

V h

Figure

12.8 Trajectory coordinates s and n, which increase along and orthogonal to the path

of an individual parcel. The coordinate vectors s and n are then tangential and orthogonal to the

local velocity

Vh, with n pointing to the left of s to form a right-handed coordinate system: s xn = k.

The trajectory's curvature is represented in the radius of curvature R, which is defined to be pos-

itive (negative) if the center of curvature lies in the positive (negative) n direction. The local ve-

locity is then described by

Vh = VS, with speed v = ds/dt. Centripetal acceleration (v2/R)n follows

from an imbalance between the pressure gradient force

-3r and the Coriolis force -fk • Vh.

384

12 Large-Scale Motion

Then the material acceleration of an individual parcel follows as

dVh dv ds

dt = --dt s + v dt"

(12.15)

The first term on the right-hand side represents longitudinal acceleration along

the parcel's trajectory. The second represents centripetal acceleration, which

acts transverse to the parcel's motion and follows from curvature of its trajec-

tory.

By the chain rule,

ds ds

ds dt

---'O~

ds"

(12.16)

Analysis similar to that in Sec. 10.6.2 shows that the unit vector s changes at

the rate

d$ v

= --n (12.17)

dt R

(Problem 12.21). Substituting (12.16) then results in

dv h dv v 2

d t = -[t7 s + -~ n

(12.18)

for the material acceleration in terms of components longitudinal and trans-

verse to the motion. As R --+ oo, the centripetal acceleration approaches zero

and the parcel's motion becomes rectilinear.

Incorporating (12.18) transforms the horizontal momentum equation into

1) 2 OrI ) 0~

d-~S + ---Rn - -fvn - ~Sos - On n,

(12.19)

which implies the component equations in the s and n directions:

dv Orb

dt = 0s' (12.20.1)

v 2 o~

-- + fv

= -~. (12.20.2)

R On

Consider now motion that is tangential to contours of height, which then

serve as streamlines. Then

O~/Os

vanishes and (12.20.1) implies that v =

const along an individual streamline. Accordingly, we focus on the momentum

budget in the normal direction. The centripetal acceleration

v2/R

reflects the

material acceleration in (11.37) and is

O(Ro),

where the Rossby number

Ro-

12.4 Curvilinear Motion

385

is based on the length scale R. According to (12.20.2), centripetal acceleration

follows as an imbalance between the Coriolis and pressure gradient forces.

12.4.1 Inertial Motion

Consider a balance between the centripetal and Coriolis accelerations alone

l) 2

-~ + fv - O, (12.21.1)

which results in the absence of a pressure gradient. The radius of curvature

follows as

R- f. (12.21.2)

On an f plane, R = const, so streamlines form circular orbits that are anticy-

clonic. The period of revolution is given by

77"

r - 12 sin q~' (12.22)

which is half a

sidereal day:

the time for a Foucalt pendulum to sweep through

180 ~ of azimuth. The motion (12.21) describes an

inertial oscillation,

in which

parcels revolve opposite to the vertical component of planetary vorticity. In-

ertial oscillations have been observed in the oceans, but they do not play a

major role in the atmospheric circulation.

12.4.2 Cyclostrophic Motion

Consider motion in the limit of either R or f approaching zero. Then

Ro~ c~,

so the material acceleration dominates the Coriolis acceleration. Under these

conditions, (12.20.2) reduces to the balance

v 2 d~

--R = On'

(12.23.1)

which defines

cyclostrophic equilibrium.

Motion with the

cyclostrophic wind

speed

v c -- ~-R

OdP

(12.23.2)

can be either cyclonic or anticyclonic, but always about low pressure (e.g.,

O~/On

of opposite sign to R).

Cyclostrophic equilibrium applies to (1) strong flow curvature, wherein the

advective acceleration is large, or (2) slow planetary rotation, wherein the

Coriolis force is weak. The former is typical of small-scale vortices such as

tornadoes, whereas the latter is important in tropical cyclones. Slow rotation is

386

12 Large-Scale Motion

also a feature of the Venusian atmosphere, which is governed by cyclostrophic

equilibrium instead of geostrophic equilibrium.

12.4.3 Gradient Motion

To first order in

Ro,

all three terms in (12.20.2) are retained, which defines

gradient equilibrium.

Gradient balance describes horizontal motion when no

two of the terms dominate. This often applies to tropical motions, for which

is small.

Equation (12.20.2) is quadratic in v and can be solved for the

gradient wind

speed

1 __

fR zlz/f2_R 2 ddp

2 W 4 R--. (12.24)

1) gr =

Not all roots of (12.24) are physically meaningful, real wind speed requiring

{< f-~ (R>O)

9 (12.25)

3n >

__f4 R

( R < O)

We restrict subsequent analysis to regular cyclonic or anticyclonic motion in

the Northern Hemisphere, for which

3~/3n

< 0 (Fig. 12.8). The second in-

equality in (12.25) limits the pressure gradient that can be sustained by an-

ticyclonic motion. Because

d~/dn

< 0 and R < 0,

~/dn

is sandwiched

between 0 and

f2R/4.

Near the center of an anticyclone, where R ~ 0, the

height distribution must therefore become fiat and the accompanying motion

weak. No corresponding limitation applies to a cyclone because, with R > 0,

~d~/dn

< 0 automatically satisfies the first inequality in (12.25). Hence, cy-

clones are free to intensify, but amplification of anticyclones is limited. This

distinction, which explains why winds are comparatively light inside an anticy-

clone, follows from the fact that the pressure gradient force acts in the same

direction as the centripetal acceleration in a cyclone but opposite to it in an

anticyclone (Problem 12.32).

Letting

Ro ~ 0

recovers geostrophic equilibrium from (12.20.2), whereas

Ro --+ c~

recovers cyclostrophic equilibrium. Intermediate to these extremes,

the geostrophic wind speed only approximates the gradient wind. Substituting

the geostrophic wind speed

Vg --

1 o~

f c~n

for the pressure gradient transforms (12.20.2) into

Vg _ __

= 1 -t Vgr.

(12.26)

Vg r

12.5 Weakly

Divergent Motion

387

The geostrophic wind speed overestimates the gradient wind speed in cyclonic

motion

(fR

> 0) and underestimates it in anticyclonic motion

(fR

< 0), the

discrepancy being proportional to

Ro.

For large-scale extratropical motions,

the discrepancy is of order 10 to 20%. However, discrepancies as large as 50 to

100% can occur inside intense cyclones and in tropical systems characterized

by small

fR.

12.5 Weakly Divergent Motion

Geostrophic balance and the idealized curvilinear balance considered above

are useful for describing the structure of large-scale motion, but they provide

no information on how the circulation evolves (Problem 12.22). Such equations

are termed

diagnostic.

Unsteadiness and divergence, although comparatively

small, allow the circulation to change from one state to another. Equations

describing such changes are termed

prognostic. A

simple framework for de-

scribing how the circulation evolves can be developed in the absence of vertical

motion.

12.5.1 Barotropic Nondivergent Motion

Consider motion that is horizontally nondivergent and under barotropic strat-

ification. Continuity (12.9) then implies that vertical motion vanishes. By ther-

mal wind balance (12.11),

is independent of height, so the motion is also

two-dimensional. The governing equations in physical coordinates then re-

duce to

dVh

d---t- + fk x v h = -aV p, (12.27.1)

V. Vh -- O,

(12.27.2)

where friction is ignored and V is understood to refer to the horizontal gradi-

ent. With vector identity (D.14) from Appendix D, (12.27.1) can be written

3v h

at + ~V(Vh " Vh) + (~ + f)k • Vh -- -aVp,

(12.28.1)

where

C-(v

•

vh).k

Ov Ou

ax ay

(12.28.2)

is the vertical component of

relative vorticity.

Consistent with prior convention,

relative vorticity is cyclonic if it has the same sign as the planetary vorticity

388 12 Large-Scale

Motion

f and anticyclonic otherwise. Applying the curl to (12.28.1) and making use

of other vector identities in Appendix D yields the vorticity budget for an air

parcel

o~t

-- + v h 9 V~ + v h 9

Vf = (Vp x Va).k

Pv3f =0 (12.29)

because a = a(p) under barotropic stratification. Since

af af

"0-- =

3y dt'

the vorticity budget can be expressed

d(~ + f)

= O. (12.30)

Equation (12.30) asserts that the

absolute vorticity (~ + f)

(e.g., that appar-

ent to an observer in an inertial reference frame) is conserved for an individual

air parcel. Thus, under barotropic nondivergent conditions, absolute vorticity

behaves as a tracer of horizontal air motion and is rearranged by the circula-

tion. Air moving southward must compensate for decreasing f by spinning up

cyclonically (see the outbreak of cold polar air in the cyclone west of Africa

in Figs. 1.23 and 1.24). Air moving northward must evolve in the opposite

manner. On large scales, the circulation tends to remain close to barotropic

stratification, which makes absolute vorticity approximately conserved.

Because it is nondivergent, this motion can be represented in terms of a

streamfunction (12.6), which automatically satisfies the continuity equation

(12.27.2). Substituting the vorticity (1217.3) then transforms (12.30) into the

barotropic nondivergent vorticity equation:

~q, ~ ~q, ~ )

Ot Oy Ox + -~X-X ~yy {Vzq' + f} - O, (12.31)

where qJ is defined by (12.8). Equation (12.31) provides a closed prognostic

system for the single unknown qJ that describes how the motion field evolves.

Advection terms in the material derivative make (12.31) quadratically nonlin-

ear in the dependent variable q~. With suitable initial conditions and boundary

conditions, the barotropic nondivergent vorticity equation can be integrated

for the motion at some later time.

12.5 Weakly Divergent Motion 389

12.5.2 Vorticity Budget under Baroclinic Stratification

Under more general circumstances, absolute vorticity is not conserved. How-

ever, the governing equations still possess a conservation principle, albeit more

complex. Baroclinic stratification renders the motion fully 3-dimensional. Even

though vorticity is then a 3-dimensional vector quantity, the quasi-horizontal

nature of large-scale motion makes its vertical component dominant (Prob-

lem 12.41).

Exclusive of friction, the horizontal momentum equations in physical coor-

dinates are

ou @

0--7 + v. Vu - fv - -a Ox '

(12.32.1)

ov @

-- + v. Vv + fu -- -a~, (12.32.2)

Ot Oy

where v and V denote the full 3-dimensional velocity and gradient. Cross

differentiating (12.32) with respect to x and y and subtracting results in

( )( )

O 3v 3u Ov. Vv- ~ 9 Vu

~+v.V ~x

Oy + 0-7 Oy

( Ou O_~y ) Of Op Oa Op Oa

+f ~xx + +v--=

Oy Ox Oy Oy Ox

d(~ + f)

Or 3v

)

+fVz'Vh+

~x'VV----'Vu =(VzpxVza ).k.

The third term in (12.33) can be written

3v .

Vv 9 Vu- +

Ox Oy 3x 3x Oy -~y Ox

3w 3v Ow Ou

= ffV~

.v h -~

Ox 3z 03, Oz

Similarly, the last term in (12.34) can be expressed

OwOv OwOu= ( Ow 0~)

O x 3 z o y O z - ~ --O-xx + rl ,

3w 3v

Oy 3z

(12.33)

where

3u'~ 3w 3v 3w 3u

Oy -t Ox Oz Oy Oz

)

(12.34)

(12.35.1)

(12.35.2)

Ou ON

3z 3x

(12.35.3)