Salby M.L. Fundamentals of Atmospheric Physics
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400
12 Large-Scale
Motion
12.20.
12.21.
12.22.
12.23.
12.24.
12.25.
12.26.
12.27.
12.28.
12.29.
Show that, under barotropic stratification,
Vp(I) is
invariant with eleva-
tion.
Show that the Lagrangian derivative of the coordinate vector s tan-
gential to a parcel's trajectory is given by (12.17).
(Hint:
Express the
vectorial change of s during an interval
dt
in terms of its angular ve-
locity to, which, in turn, can be expressed in terms of v and R; see
Fig. 10.7.)
Determine the circumstances under which the special class of curvilin-
ear motion considered in Sec. 12.4 is solenoidal.
Show that the relative error introduced by treating steady motion
geostrophically is O(R o).
A midlatitude cyclone centered at 45~ produces a radial height gra-
dient at 500 mb of 0.16 m km -1 at a characteristic radial distance of
10 ~ If the motion is sufficiently steady, calculate the corresponding
(a) geostrophic wind and (b) gradient wind.
As in Problem 12.24, but for a tropical cyclone centered at 10~ which
produces a radial height gradient at 850 mb of 1.88 m km -1 at a char-
acteristic radial distance of 1 ~
Estimate the relative error introduced by treating motion geostrophi-
cally for (a) an
omega block,
in which the jet stream is deflected pole-
ward at 40 ~ where it has a velocity of 30 m s -1 and assumes a radius
of curvature of 2000 km, (b) a tropical depression at 10 ~ characterized
by a radius of 200 km and a velocity of 30 m s -1, and (c) the undis-
turbed polar-night vortex at 60 ~ , wherein the motion is zonal and has
a velocity of 80 m s -1.
The
Kelvin wave
plays an important role in tropical circulations, having
the form of a vertical overturning in the equatorial plane. Horizontal
motion in the Kelvin wave is zonal and described by
i(y) 2]
Vh(X,
y) = exp --
exp[i(kx - trt)]i,
where y is measured on an equatorial/3 plane (~b 0 = 0) and Y corre-
sponds to 10 ~ of latitude. (a) Sketch the velocity field over one wave-
length in the x direction. (b) Calculate the vorticity and divergence of
the wave, sketching each on the equatorial/3 plane. (c) Derive a system
of equations that determines the streamfunction and velocity potential
of the wave. Outline the solution of this system.
Meteorologists refer to the vector change of horizontal wind across a
layer as the
thermal wind.
Show that the thermal wind is proportional
to the horizontal gradient of layer-mean temperature.
Derive an expression for thermal wind balance in height coordinates.
Problems
401
12.30. A vertical sounding at 45~ reveals isothermal conditions with T = 0~
below 700 mb, but vertical shear characterized by the following wind
profile:
10m s-1 S 950mb
v= 14.14m s -1 SW 850mb,
10 m s -1 S 750 mb
where the wind blows from the direction indicated. (a) Calculate the
mean horizontal temperature gradient in the layers between 950 and
850 mb and between 850 and 750 mb. (b) Estimate the rate at which
the preceding layers would warm/cool locally through temperature ad-
vection (e.g., were temperature approximately conserved). (c) Use the
result of part (b) to estimate the lapse rate created between those layers
after 1 day, noting the physical implications.
12.31. In terms of horizontal forces, discuss why cyclostrophic motion can be
either cyclonic or anticyclonic, but always about low pressure.
12.32. Use the horizontal force balance to explain why, physically, the strength
of curvilinear motion is limited for anticyclonic curvature but not for
cyclonic curvature.
12.33. Demonstrate that
I < vg
cyclonic motion
l) gr
I
> vg
anticyclonic motion.
12.34. A circular hurricane has azimuthal velocity
v(r, p) - vo(p) r > ro,
where r is measured from the center of the system, v0(1000 mb) = 50
m s -1, and the hurricane's characteristic dimension, r 0 = 100 km, is
small enough for it to be treated on an f plane with f0 = 5.0 x 10 -5 s -1.
(a) As a function of p, determine the range of r where the motion can
be treated as geostrophic, providing justification for this approximation.
(b) In the foregoing range of r, determine the geopotential field O(r, p),
which approaches an undisturbed value ~ as r --+ o0. (c) Hurricanes
are warm-core depressions (see Problem 6.9), so the radial temperature
gradient is negative. It follows that the azimuthal velocity decreases with
height. If the system has an average horizontal temperature gradient of
1 K km -1, in the foregoing range of r, estimate the level
p(r)
where v
becomes negligible.
12.35. Consider the core of a tornado that is isothermal and rotates with
uniform angular velocity f~. Derive an expression for the radial profile
of pressure inside the tornado's core in terms of the pressure P0 at its
center.
402 12 Large-Scale Motion
12.36. Isobaric height in the Venusian atmosphere varies locally as
9 (x,y)--~ 1 +e L--------g--- ,
where the length scale L is of arbitrary dimension. (a) If the motion is
steady and frictionless, provide an expression for the horizontal velocity.
(b) Can a similar result be obtained if the flow is transient? Why?
12.37. Satellite measurements reveal the following thermal structure"
( x2 +,2)
T(x, y, p) - T- T - AT(p) exp
- L2
where y is measured from 45 ~ a is the radius of the earth, the length
scale L is of order 1000 km or longer, T and T are constants that refer
to the undisturbed zonally symmetric state, and
AT(p)-A+Bln(~-~)
describes a cold-core cyclonic disturbance with A and B constants and
P00 - 1000 mb. (a) Determine the height of the 500-mb surface as a
function of position. Sketch the distribution of zs00(x, y), presuming AT
is not large enough to reverse the equatorward temperature gradient.
(b) Determine the horizontal motion if meridional excursions are small
enough for it to be treated on an f plane. Justify the validity of this
S -1
expression based on a characteristic velocity of 10 m . Sketch the
motion. (c) For undisturbed conditions (AT -- 0), describe the vertical
profile of horizontal wind. If motion vanishes at the surface, determine
the pressure at which the horizontal wind speed reaches 20 m s -1.
12.38. A stratospheric sudden warming is accompanied by a reversal of the
westerly circumpolar flow corresponding to the polar-night vortex
(Fig. 1.10). Developing within a couple of days, this dramatic change
in motion is attended by sharply warmer temperatures over the polar
cap that follow from a rearrangement of air by planetary waves (see
Fig. 14.26). If the initial motion at middle and high latitudes is ap-
proximated by rigid body rotation with angular velocity 0.251~, if the
reversed circumpolar flow at high latitude is also characterized by rigid
body rotation, but with zonal easterlies of 20 m s -1 at 75~ and if the
motion leading to this state is approximately nondivergent, estimate
where air prevailing over the polar cap must have originated from.
12.39. Consider steady, barotropic, nondivergent motion on a midlatitude fl
plane. The motion is zonal and uniform with v - u-/ upstream of
an obstacle at x - 0, which deflects streamlines meridionally from
their undisturbed latitudes. If the disturbed motion downstream devi-
ates from the upstream motion by a small meridional perturbation v',
Problems
403
(a) construct a differential equation governing the trajectory
y(x)
of an
air parcel initially at y = 0, (b) obtain the general solution downstream
of the obstacle, (c) determine the trajectory if the parcel is deflected at
(0, 0) with a slope of tan a, and (d) interpret the motion downstream
of the obstacle in terms of a restoring force.
12.40. Derive the vertical vorticity budget for barotropic nondivergent motion:
+ Vh" V~ + v h 9
Vf = (Vp x Va)-k.
dt
12.41.
12.42.
12.43.
12.44.
12.45.
12.46.
12.47.
12.48.
12.49.
Perform a scale analysis to show that the vertical component of vortic-
ity dominates horizontal components for large-scale, three-dimensional
motion under generally baroclinic conditions.
Derive the vorticity budget (12.33).
Show that a state of static equilibrium (v = 0) can persist only under
barotropic stratification.
Perform a scale analysis to show that vertical stretching dominates other
mechanisms forcing absolute vorticity in (12.36).
Rising motion is an essential feature of organized convection in the
tropics, where f is small enough to permit comparatively large hori-
zontal divergence yet still dominates relative vorticity. In terms of vor-
ticity forcing, explain why motion in convective centers is anticyclonic
near the tropopause (see Fig. 15.8).
Explain why cyclogenesis would be favored in a cold continental air
mass that has moved over a warmer maritime region, such as occurs in
the north Pacific and Atlantic storm tracks (refer to Fig. 9.38).
The zonal circulation is disrupted by a planetary wave that advects
air northward from 10~ to 50~ If winds at 10~ are easterly with
speed of 5 m s -1 and relative vorticity of 1.2 • 10 -5 s -1, estimate the
column-averaged vorticity at 50~ if (a) air motion is barotropic non-
divergent and (b) air motion is confined between an isentropic surface
near 1000 mb and another near the tropopause, which slopes downward
towards the pole from 100 mb in the tropics to 300 mb at 50~
(a) Explain why solenoidal production of vorticity vanishes in isentropic
coordinates (12.38).
(Hint:
How many thermodynamic degrees of free-
dom are present in general? Along an isentropic surface?) (b) Apply
this argument to the budget of vorticity in isobaric coordinates.
Use (11.27) to derive the equivalent expression for potential vorticity
(V
X
l~i). %70
Q=
p
which is likewise conserved under inviscid adiabatic conditions.
404 12 Large-Scale
Motion
12.50.
12.51.
12.52.
12.53.
12.54.
12.55.
Consider a material element which is bounded above and below by
(nearly horizontal) isentropic surfaces that are separated incrementally
by
dO.
The element has cross-sectional area
dA
on those surfaces.
(a) Express the element's incremental mass in terms of
~p/~O.
(b) Use
potential vorticity together with Stokes' theorem to derive an expression
governing the circulation
F = f Vh" ds
about the element.
Consider an idealized model of the jet stream and its interaction with
a transverse mountain range like the Rockies: A two-dimensional ridge
oriented north-south on a midlatitude/3 plane presents a topographic
barrier to an upstream zonal flow. The motion is inviscid and adiabatic.
An isentropic surface coincides with the topography (which displaces it
to smaller pressure) and another remains level near the tropopause. If
the upstream flow is uniform, use conservation of potential vorticity to
(a) sketch a vertical section of the flow and topography, noting the sign
of relative vorticity for an air column as it is advected past different
stations, and (b) determine the horizontal trajectory of an air column
over and downstream of the barrier.
Derive the conservation principle (12.48) for quasi-geostrophic motion.
Vertical stretching, which forces changes in absolute vorticity, appears
in the numerator of
Qg,
rather than the position it assumes in the
isentropic potential vorticity Q. Argue this form from the shallowness of
vertical displacements, which makes relative changes of an air parcel's
depth small. See Andrews
et al.
(1987) for a formal comparison of
Qg
and Q.
Satellite observations provide thermal structure from the surface up-
ward and the mixing ratio of a long-lived chemical species above the
tropopause--both at successive times. (a) Construct an algorithm to
deduce the three-dimensional trajectories of individual air parcels on
the timescale of a week, noting the level of approximation and sources
of error. (b) Where does this approximation break down? How could
this limitation be averted with additional chemical tracers and what
property would then be required of those tracers? (c) How would the
application of this algorithm be complicated in the troposphere?
(Hint:
See Fig. 13.2.)
In practice, satellite observations are available
asynoptically,
that is,
different positions are observed at different instants. (a) How might
this feature of tracer observations be accommodated to implement the
algorithm in Problem 12.54? (b) Discuss the practical limitations of
inferring global air motion in light of the timescale for advection versus
that for the earth to be sampled completely.
Chapter 13
The Planetary Boundary Layer
Previous development treats motion as a smoothly varying field property.
However, atmospheric motion is always partially turbulent. Laboratory ex-
periments provide a criterion for the onset of turbulence in terms of the
dimensionless
Reynolds number
LU
Re
= ~, (13.1)
v
where L and U are scales characterizing a flow and v =
tz/p
is the kinematic
viscosity (Sec. 10.6). For
Re
greater than a critical value of about 5000, smooth
laminar motion undergoes a transition to turbulent motion, which is inherently
unsteady, three-dimensional, and involves a spectrum of space and time scales.
Even for a length scale as short as 1 m, the critical Reynolds number with
v = 10 -5 m 2 s -1 is exceeded for velocities of U > 0.05 m s -1. Consequently,
atmospheric motion inevitably contains some turbulence. ~
Until now, turbulent transfers have been presumed small enough to treat
an individual air parcel as a closed system. This approximation is valid away
from the surface and outside convection because the timescale for turbu-
lent exchange in the
free atmosphere
is much longer than the 1-day timescale
characterizing advective changes of an air parcel. Budgets of momentum, in-
ternal energy, and constituents for an individual parcel can then be treated
exclusive of turbulent exchanges with the parcel's environment~at least over
timescales comparable to advection. This feature of the circulation is reflected
in the spectrum of kinetic energy (Fig. 13.1). In the free atmosphere, kinetic
energy is concentrated at periods longer than a day, where it is associated with
large-scale disturbances and seasonal variations. However, in a neighborhood
of the surface, as much as half of the kinetic energy lies at periods of order
minutes. Inside the
planetary boundary layer,
turbulent eddies are generated
mechanically from strong shear as the flow adjusts sharply to satisfy the no-slip
condition at the ground (see Fig. 13.3). Turbulence is also generated thermally
through buoyancy when the stratification is destabilized.
~Stratification of mass leads to v increasing exponentially with height. Therefore, a level is even-
tually reached where v is sufficiently large to drive Re below the critical value and suppress
turbulence (Problem 13.3). This transition occurs at the turbopause, as pictured in Fig. 1.4.
405
406
13
The Planetary Boundary Layer
700
%
600
500
400
300
~r 200
(~ 100
Ii
0
Figure 13.1
I I I I I I I /
Free Atmosphere
(20- 30 km)
Ground
1 yr 1 mo 1 day 1 hr 1 min 1 sec
I I I I I I I I I
10 -3 10 -2 10" 1 100 101 102 103 104 105
FREQUENCY (cpd)
Power spectrum of atmospheric kinetic energy. Adapted from Vinnichenko
(1970). Copyright (1970) Munksgaard International Publishers Ltd., Copenhagen, Denmark.
Mixing between air parcels cannot be ignored inside the boundary layer
because the timescale for turbulent exchange there is comparable to that for
advection by the large-scale flow. Turbulent exchanges of mass, heat, and
momentum then make an air parcel an open system and render its behavior
diabatic. Turbulent mixing transfers heat and moisture between the surface and
the atmosphere. Because it destroys gradients inherent to large-scale motion,
turbulent mixing also acts to dissipate the circulation.
13.1 Description of Turbulence
Turbulent fluctuations need not be hydrostatic, so the equations of motion
are expressed most conveniently in physical coordinates. The limited vertical
scale of turbulent eddies makes compressibility nonessential. However, vari-
ations of density associated with temperature fluctuations play a key role in
stratified turbulence because, through buoyancy, they couple the motion to
gravity. These features are embodied in the Boussinesq approximation (Sec.
12.5). Equivalent to incompressibility, the Boussinesq approximation allows
13.1
Description of Turbulence
407
density to be treated as constant, except where accompanied by gravity in the
buoyancy force.
Consider the motion-related stratification (Sec. 11.4), which is regarded as
a small departure from static conditions. In terms of the static density po(Z),
the specific buoyancy force experienced by an air parcel (11.35) is
lop p
fb = g- (13.2)
Po 3z Po
The gas law and Poisson's relation for potential temperature, along with the
neglect of compressibility, then imply
p T p
~--
[
P0 To P0
0
~-
. (13.3)
00
Hence, the vertical momentum budget becomes
dw 10p 0
dt Po Oz t- g-oo Dz" (13.4)
Similarly, the thermodynamic equation becomes
dO dOo Oo
-}- W-- -- --qnet,
(13.5)
dt dz cpT o
where Oo(Z) describes the basic stratification and 0 the small departure from
it. Under the Boussinesq approximation, the continuity equation reduces to a
statement of three-dimensional nondivergence. Then the equations governing
turbulent motion become
du 1 Op
dt fv - Po Ox D x, (13.6.1)
dv 1 31)
d----t -~- fu - Oy,
(13.6.2)
Po ay
dw 1 31) 0
dt = Po Oz + g-Oo Dz' (13.6.3)
3u 3v Ow
-- - -- =0, (13.6.4)
3x +--~y q Oz
dO
dt
dOo Oo
D
dz cpT o
-- qnet,
(13.6.5)
408 13 The
Planetary Boundary Layer
where
d
= --+v.V
dt 3t
~ o~ o~
= ~t + u--- + v--- + w-- (13.6.6)
dx
ay
oz
includes three-dimensional advection. The frictional drag D- -(1/p)V.r,
which follows from the deformation tensor e (10.23), reduces under the Boussi-
nesq approximation to
D
= -l,,vzv.
(13.6.7)
Compared to the timescale of turbulent eddies, radiative heating in
0net
is
slow enough to be ignored. However, thermal diffusion (e.g., conduction) op-
erates efficiently on small scales that are created by turbulent deformations
(refer to Fig. 12.4). Diffusion of momentum and heat are both involved in
dissipation of turbulence, which takes place on the smallest scales of motion.
Sharp gradients produced when anomalies are strained down to small dimen-
sions are acted on efficiently by molecular diffusion.: When those gradients
are destroyed, so too is the energy transferred to small scales. The foregoing
process reflects a cascade of energy from large scales of organized motion to
small scales, where it is dissipated by molecular diffusion; see Tennekes and
Lumley (1972) for a formal development of turbulent energetics.
13.1.1 Reynolds Decomposition
Of relevance to the large-scale circulation is the time-averaged motion and
how it is influenced by turbulent fluctuations. To describe such interactions,
each field variable is separated into a slowly varying mean component, which
is denoted by an overbar, and a fluctuating component, which is denoted by a
prime, for example,
v = fi + v', (13.7.1)
with
v --7 - 0. (13.7.2)
For this decomposition to be meaningful, timescales of the two components
must be widely separated. Only then can averaging be performed over an inter-
val long enough for a stable mean to be recovered from fluctuating properties
yet short enough for mean properties to be regarded as steady.
2The
scale dependence of diffusion can be inferred from the viscous drag (13.6.7), which
increases quadratically with the inverse scale of motion anomalies.
13.1 Description of Turbulence
409
The continuity equation (13.6.4) allows the material acceleration to be ex-
pressed in terms of the divergence of a momentum flux
du Ou
= -- + v. (vu),
(13.8.1)
dt dt
dv dv
= + v. (vv),
(13.8.2)
dt 3t
dw dw
= ~ + V. (vw),
(13.8.3)
dt at
where terms on the far right describe the three-dimensional momentum car-
ried by the velocity v. Likewise, the material rate of change of temperature in
(13.6.7) can be expressed in terms of the divergence of a heat flux
dO 30
= + V. (vO).
(13.9)
dt dt
Expanding as in (13.5) and averaging then obtains the equations governing
the time-mean motion
( )
d -~ 1 ?-~ 3u'u' ?u'v' 3u'w'
dt f-6- + +
Po Ox 3x 3y 3z
(13.10.1)
( )
d ~ 1 3-ff du'v' Ov'v' Ov'w'
d--t + f-~ -- + +
Po ?Y Ox Oy Oz
(13.10.2)
( )
d -~ 1 ?-p 0 Ou'w' 3v'w' Ow'w'
dt = PodZ t-g-~o- dx + ?y + 3z
(13.10.3)
OK O~ 3N
.... ~ ~ 0
Ox t- dy 4 3z
(13.10.4)
( -)
d 0 _dOo _ 3u'O' 3v'O' Ow'O'
dt. t-w dz 3x + Oy + Oz
(13.10.5)
where
m
d 0
= -- + ~. V (13.10.6)
dt 3t
is the time rate of change following the mean motion, and diffusive transfers
of momentum and heat are slow enough, on scales of the mean flow, to be
ignored.
Terms in (13.10.1) through (13.10.3) involving time-averaged products of
fluctuating velocities represent the convergence of an eddy momentum flux
(e.g., of eddy momentum carried by the eddy velocity). Mean x momentum