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Chapter 6 Hydrostatic equilibrium
Chapter 6
Hydrostatic Equilibrium
Changes of thermodynamic state accompanying vertical motion follow from the distribution of atmospheric mass, which is determined ultimately by gravity. In the absence of motion, Newton's second law applied to the vertical reduces to a statement of hydrostatic equilibrium (1.16), wherein gravity is balanced by the vertical pressure gradient. This simple form of mechanical equilibrium is accurate even in the presence of motion because the acceleration of gravity is much greater than vertical accelerations of individual air parcels for nearly all motions of dimensions greater than a few tens of kilometers. Only inside deep convective towers and other small-scale phenomena are vertical accelerations large enough to invalidate hydrostatic equilibrium. Because it is such a strong body force, gravity must be treated with some care. Complications arise from the fact that the gravitational acceleration experienced by an air parcel does not act purely in the vertical and varies with location. According to the discussion above, gravity is large enough to overwhelm other contributions in the balance of vertical forces. The same is true for the horizontal force balance. Horizontal components of gravity introduced by the earth's rotation and additional sources must be balanced by other horizontal forces, which unnecessarily complicate the description of atmospheric motion and can overshadow those forces actually controlling the motion of an air parcel.
6.1
Effective Gravity
The acceleration of gravity appearing in (1.16) is not constant. Nor does it act purely in the vertical. In the reference frame of the earth, the gravitational acceleration experienced by an air parcel follows from three basic contributions: 1. Radial gravitation by the planet's mass 2. Centrifugal acceleration due to rotation of the reference frame 3. Anisotropic contributions, as illustrated in Fig. 6.1. Radial gravitation by the planet's mass is the dominant contribution. Under idealized circumstances, that component of g acts 143
144
6 Hydrostatic Equilibrium
---'---
z = con st. = const.
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f
Figure 6.1 Effective gravity g illustrated in relation to its contributions from radial gravitation and centrifugal acceleration. Also shown are surfaces of constant geometric altitude z and corresponding surfaces of constant geopotential ~, to which g is orthogonal. perpendicular to surfaces of constant geometric altitude z, which are concentric spheres. The rotation of the earth introduces another contribution. Because it is noninertial, the reference frame of the earth includes a centrifugal acceleration that acts perpendicular to and away from the axis of rotation. Last, departures of the planet from sphericity and homogeneity (e.g., through variations of surface topography) introduce anisotropic contributions to gravity that must be determined empirically. Collectively, these contributions determine the effective gravity, which can be expressed as follows: a2 g(A, q~, z) = (a + z) 2gOk + ~2(a + z)cos dper + ~(A, q~, z), (6.1) where a is the mean radius of the earth, go is the radial gravitation at mean sea level, k is the local upward unit normal, l~ is the planet's angular velocity, A and ~b are longitude and latitude, respectively, e r is a unit vector directed outward from the axis of rotation, and ~(A, ~b, z) represents all anisotropic contributions. Even if the planet were perfectly spherical, effective gravity would not act uniformly across the surface of the earth, nor entirely in the vertical. According to (6.1), the centrifugal acceleration deflects g into the
6.2
Geopotential Coordinates
145
horizontal, which introduces a component of gravity along surfaces of constant z. The centrifugal acceleration also causes the vertical component of g to vary from a minimum of 9.78 m s -2 at the equator to a maximum of 9.83 m s -2 at the poles. Although more complicated, anisotropic contributions have a similar effect. These contributions introduce horizontal components of gravity that must be balanced by other forces acting along surfaces of constant geometric altitude. Thus, (1.16) represents but one of several component equations that are required to describe hydrostatic balance. Horizontal forces introduced by gravity unnecessarily complicate the description of atmospheric behavior because they can overshadow the forces actually controlling the motion of an air parcel. For instance, rotation introduces a horizontal component that must be balanced by a variation of pressure along surfaces of constant geometric altitude. That horizontal pressure gradient exists even under static conditions, for which an observer in the reference frame of the earth would anticipate Newton's second law to reduce to a simple hydrostatic balance in the vertical. To avoid such complications, it is convenient to introduce a coordinate system that consolidates horizontal components of gravity into the vertical coordinate.
6.2
Geopotential Coordinates
Because gravity is conservative, the specific work performed during a cyclic displacement of mass through the earth's gravitational field vanishes: Awg = - f g(A, 4', z ) - d ~ - 0,
(6.2)
where ds denotes the incremental displacement of an air parcel. According to Sec. 2.1.4, (6.2) is a necessary and sufficient condition for the existence of an exact differential dap = 6wg - - g . dg,
(6.3)
which defines the gravitational potential or geopotential ~. In (6.3), d ~ equals the specific work performed against gravity to complete the displacement d~. Then, per the aforementioned discussion, gravity is an irrotational vector field that can be expressed in terms of its potential function g = -V~.
(6.4)
Surfaces of constant geopotential are not spherical like surfaces of constant geometric altitude. Centrifugal acceleration in the rotating reference flame of the earth distorts geopotential surfaces into oblate spheroids (Fig. 6.1). The component of XT~ along surfaces of constant geometric altitude introduces a
146
6
Hydrostatic Equilibrium
horizontal component of gravity that must be balanced by a horizontal pressure gradient force. Representing gravity is simplified by transforming from pure spherical coordinates, in which elevation is fixed along surfaces of constant geometric altitude, to geopotential coordinates, in which elevation is fixed along surfaces of constant geopotential. Geopotential surfaces can be used for coordinate surfaces because 9 increases monotonically with altitude (6.3), which ensures a one-to-one relationship between those variables. Introducing the aforementioned transformation is tantamount to measuring elevation z along the line of effective gravity, which will be termed height to distinguish it from geometric altitude. Defining z in this manner agrees with the usual notion of local vertical being plumb with the line of gravity and will be adopted as the convention hereafter. In geopotential coordinates, gravity has no horizontal component. Consequently, in terms of height, (6.4) reduces to simply (6.5)
ddO - gdz,
where g denotes the magnitude of g, which is understood to act in the direction of decreasing heightmnot geometric altitude. Using mean sea level as a reference elevation I yields an expression for the absolute geopotential r
=
f0 Zgdz.
~
(6.6)
Having dimensions of specific energy, the geopotential represents the work that must be performed against gravity to raise a unit mass from mean sea level to a height z. Evaluating 9 from (6.6) requires measured values of effective gravity throughout. However, since it is a point function, 9 is independent of path by (6.2) and the exact differential theorem (2.7). Therefore, 9 depends only on height z and surfaces of constant geopotential coincide with surfaces of constant height. Using surfaces of constant geopotential for coordinate surfaces consolidates gravity into the vertical coordinate, height. However, g still varies with z through (6.5). To account for that variation, it is convenient to introduce yet another vertical coordinate in which the dependence on height is absorbed. Geopotential height is defined as Z = ml f0 ~'g d z go 1
=
(6.7)
go 1Mean sea level may be used to define a reference value of 9 because it nearly coincides with a surface of constant geopotential.
6.3 Hydrostatic Balance
147
where go - 9.8 m s - 2 is a reference value reflecting the average over the surface of the earth. Then d Z - ml d ~ go
(6.8)
= g dz.
go Since it accounts for the variation of gravity, geopotential height simplifies the expression for hydrostatic balance. In terms of Z, (1.16) reduces to dp - - p g d z
(6.9)
= -pgodZ.
When elevation is represented in terms of geopotential height, gravity can be described by the constant value go, which simplifies the description of atmospheric motion. Although it has these formal advantages, geopotential height is nearly identical to height because g ~ go throughout the homosphere. In fact, the two measures of elevation differ by less that 1% below 60 km and only 3% below 100 km. Therefore, Z can be used interchangeably with z, which will be our convention hereafter. Then g is understood to refer to the constant reference value go.
6.3
Hydrostatic Balance
In terms of geopotential height, hydrostatic balance can be expressed (6.10)
gdz = -vdp.
The ideal gas law transforms this into RT
dz = - ~ d l n = - H d l n p,
p (6.11)
where the scale height H is defined in (1.17). According to (6.11), the change of height is proportional to the local temperature and to the change o f - In p. Since changes of temperature and thus H are small compared to changes of pressure, the quantity - I n p can be regarded as a dimensionless measure of height. Consider a layer bounded by two isobaric surfaces p = p l ( x , y, z) and p = p2(x, y, z), as illustrated in Fig. 6.2. By integrating (6.11) between those
148
6 Hydrostatic Equilibrium
-.
Zl: z(x,y, Pl)
I
,,
fx
Figure 6.2 Isobaric surfaces (dark shading) in the presence of a horizontal temperature gradient. Vertical spacing of isobaric surfaces is compressed in cold air and expanded in warm air, which makes the height of an individual isobaric surface low in the former and high in the latter. Because pressure decreases upward monotonically, a surface of constant height (light shading) then has low pressure where isobaric height is low and high pressure where isobaric height is high. Therefore, contours of height for an individual isobaric surface can be interpreted similarly to isobars on a constant height surface. Also indicated is the geostrophic velocity vg (Sec. 12.1), which is directed parallel to contours of isobaric height and has magnitude proportional to its horizontal gradient.
surfaces, we obtain A z - - z 2 -- z I = - ( n ) I n
~1
'
(6.12.1)
where (H) = R,_ ,IT~ g
(6.12.2)
6.3
Hydrostatic Balance
149
and
(T) - fpp2 T d l n p fp7 d l n p _- fpPl~ Tdln p ln(~)
(6.12.3) '
define the layer-mean scale height and temperature, respectively, and Zl = z(x, y, Pl) and z2 = z(x, y, P2) denote the heights of the two isobaric surfaces. Known as the hypsometric equation, (6.12) asserts that the thickness of a layer bounded by two isobaric surfaces is proportional to the mean temperature of that layer and the pressure difference across it. In regions of cold air, the e-folding scale H is small, so pressure in (1.17) decreases with height sharply and the vertical spacing of isobaric surfaces is compressed. By comparison, H is large in regions of warm air, so pressure decreases with height more slowly, and the spacing of isobaric surfaces is expanded. If zl =0, the hypsometric equation provides the height of the upper surface z(x, y, P2). For example, the height of the 500-mb surface in Fig. 1.9 can then be expressed as a vertical integral of temperature. By (6.12), the height of an isobaric surface can be determined from measurements of temperature between that surface and the ground or, alternatively, between that surface and a reference isobaric surface, the height of which is known. Such temperature measurements are made routinely by ground-launched rawinsondes, which also measure air motion and humidity and ascend via balloon to as high as 10 mb. Launched twice a day, rawinsondes provide dense coverage over populated landmasses, but only sparse coverage over oceans and in the tropics. Temperature measurements are also made remotely by satellite. Satellite measurements of temperature provide continuous global coverage of the atmosphere, albeit asynoptically. ~ Satellite retrievals also have coarser vertical resolution (~ 4 to 8 km) than that provided by rawinsondes (~1 km). Both are assimilated operationally in synoptic analyses of the instantaneous circulation, like those shown in Figs. 1.9 and 1.10. According to Fig. 1.9, the height of the 500-mb surface slopes downward toward the pole at midlatitudes, where zs00 decreases from 5900 to 5000 m. That decrease of 500-mb height mirrors the poleward decrease of layer-mean temperature between the surface and 500 mb, (T)500, which is shown in Fig. 6.3 for the same time as that in Fig. 1.9a. The wavy region where (T)500 and zs00 change abruptly delineates the polar front, which separates cold polar air from warmer air at lower latitudes. As is apparent from Fig. 1.9a, that region 2The term synoptic refers to the simultaneous distribution of some property, for example, a snapshot over the earth of the instantaneous height and motion on the 500-mb surface (Fig. 1.9a). However, a polar-orbiting satellite necessarily observes different regions at different instants. Termed asynoptic, such measurements sample the entire globe only after 12 to 24 hr.
150
6
Hydrostatic Equilibrium ~ooOO.oOOO~ ......... .o,.,oOr
~. ..., 9...
....~.
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...'"
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281
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:..." :... ."
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Figure 6.3 Mean temperature (Kelvin) of the layer between 1000 mb and 500 mb for March 4, 1984. The bold solid line marks a meridional cross section displayed in Fig. 6.4.
of sharp temperature change coincides with the strong circumpolar flow of the jet stream, which is tangential to contours of isobaric height. Meridional gradients of temperature and height are particularly steep east of Asia and North America, where they mark the North Pacific and North Atlantic storm tracks that are manifested in the time-mean pattern (Fig. 1.9b) as locally intensified jets. Anomalies of cold air that punctuate the polar front in Fig. 6.3 introduce synoptic-scale depressions of the 500-mb surface in Fig. 1.9a. By deflecting contours of isobaric height, those features disrupt the motion of the jet stream, which, in their absence, is nearly circumpolar. The hypsometric equation provides a theoretical basis for using pressure to describe elevation. Because H is positive, (6.12) implies a one-to-one relationship between height and pressure: a given height corresponds to a single
6.4 Stratification
151
pressure. Therefore, pressure can be used as an alternative to height for the vertical coordinate. Isobaric coordinates, which are developed in Chapter 11, use surfaces of constant pressure for coordinate surfaces. Although they evolve with the circulation, isobaric coordinates afford several advantages. In isobaric coordinates, pressure becomes the independent variable and the height of an isobaric surface becomes the dependent variable, for example, z2 = z(x, y, P2). Because pressure decreases monotonically with height, low height of an isobaric surface corresponds to low pressure on a surface of constant height and just the reverse for high values (Fig. 6.2). For the same reason, contours of isobaric height resemble isobars on a surface of constant height. Consequently, the horizontal distribution of isobaric height may be interpreted analogously to the distribution of pressure on a surface of constant height. From hydrostatic equilibrium, the latter represents the weight of the atmospheric column above a given height. Therefore, the distribution of isobaric height reflects the horizontal distribution of atmospheric mass above the mean elevation of that isobaric surface. Contours of 500-mb height that delineate the polar front in Fig. 1.9a correspond to isobars on the constant height surface: z ,~ 5.5 km, with pressure decreasing toward the pole. Thus, less atmospheric column lies above that elevation poleward of the front than equatorward of the front. Figure 6.4 shows a meridional cross section of the thermal structure and motion in Figs. 6.3 and 1.9a. The horizontal distribution of mass just noted results from compression of isobaric surfaces poleward of the front, which is centered near 40 N, and expansion equatorward of the front. This introduces a tilt to isobaric surfaces at midlatitudes, one that steepens with height. The jet stream is found in the same region and likewise intensifies vertically, to a maximum near the tropopause. Similar behavior at high latitudes marks the base of the polar-night jet. 6.4
Stratification
Expressions (2.33) and (5.32) for the dry adiabatic and saturated adiabatic lapse rates, which describe the evolution of a displaced air parcel under unsaturated and saturated conditions, respectively, hold formally with z representing geopotential height and g constant. Neither Fa nor Fs has a direct relationship to the temperature of the surroundings because a displaced parcel is thermally isolated under adiabatic conditions. Thermal properties of the environment are dictated by the history of air residing at a given location, for example, by where that air has been and what thermodynamic influences have acted on it. The environmental lapse rate is defined as F=
dT dz'
(6.13)
where T refers to the ambient temperature. Like parcel lapse rates, F > 0 corresponds to temperature decreasing with height. Conversely, F < 0 cor-
152
6 Hydrostatic
Equilibrium
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10
20
30
40
50
60
70
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90
LATITUDE
Figure 6.4 Meridional cross section of thermal structure and motion in Figs. 1.9a and 6.3 at the position indicated in the latter. Vertical spacing of isobaric surfaces (bold solid curves) is expanded at low latitude and compressed at high latitude, introducing a slope at midlatitudes that steepens with height. Zonal wind speed (contoured in m s -1) delineates the subtropical jet, which coincides with the steep slope of isobaric surfaces, both intensifying upward to a maximum at the tropopause. Contour increment: 5 m s -1.
responds to temperature increasing with height, in which case the profile of environmental temperature is said to be inverted. The compressibility of air leads to atmospheric mass being stratified, as is reflected in the vertical distributions of density and pressure. Since the environment is in hydrostatic equilibrium, the distribution of pressure may be related to the thermal structure through the hypsometric relation. Incorporating (6.11) transforms (6.13) into 1
dT
H dlnp
=F
or
din T dln p
=~r g F Fd
(6.14)
6.4
6.4.1
Stratification
153
Idealized Stratification
For certain classes of thermal structure, (6.14) can be integrated analytically to obtain the distributions of pressure and other thermal properties inside an atmospheric layer. LAYER OF CONSTANT LAPSE RATE
If
F
-
const ~- 0, (6.14) yields TTs = ( - ~ ~ ) ~ K ,
(6.15)
where the subscript s refers to the base of the layer. Because (6.16)
T(z) = Ts - rz,
(6.15) can be used to obtain the vertical distribution of pressure: PPs - [ 1 -
K-~F (ZHss)] (~r) .
(6.17)
Then (2.31) implies the vertical profile of potential temperature: Fd
O(z, - ( T s - F z ) [ l - K F---aF( ~ s s ) ]
r ,
(618).
where p~ - P0 = 1000 mb has been presumed for the base of the layer. As shown in Fig. 6.5, the compressibility of air makes pressure decrease with height for all F. The same is true of density. However, potential temperature decreases with height only for F > F a, whereas it increases with height for F < Fa. In both cases, 0 varies sharply with height because the pressure term dominates over the temperature term in determining the height dependence in (6.18). As a result, environmental potential temperature can change by several hundred degrees in just a few scale heights, as is typical in the stratosphere (refer to Fig. 17.19). By contrast, the potential temperature of a displaced air parcel is independent of its height under adiabatic conditions. The thermal structure described by (6.16) through (6.18) can apply locally (e.g., inside a certain layer), even if temperature does not vary linearly throughout the atmosphere. Should it apply for the entire range of height, (6.16) has one further implication. For F > 0, positive temperature requires the atmosphere to have a finite upper bound
L
Ztop -- -~-,
where p vanishes. For F < 0, no finite upper bound exists.
(6.19)
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6.5 Lagrangian Interpretation of Stratification
155
ISOTHERMAL LAYER
For the special case F - 0, (6.17) is indeterminate. Reverting to the hydrostatic relation (6.11) yields z=-Hln
(-~~)
or
P -- e ~, Ps z
(6.20)
which is identical to (1.17) under these circumstances. Then the vertical profile of potential temperature is simply O(z) = Tse"~
: 0se~.
(6.21)
Under isothermal conditions, environmental potential temperature increases by a factor of e every K-a --- 3 scale heights. ADIABATIC LAYER
For the special case F = Fa, (6.15) reduces to T K -T-~- ( - ~ s )
(6.22,
and the distribution of pressure (6.17) becomes t
P
1-
.
(6.23)
Equation (6.22) is the same relationship between temperature and pressure implied by Poisson's equation (2.30.2), which defines potential temperature (2.31). Accordingly, the vertical profile of potential temperature (6.18) reduces to O(z) -
O~ -
const
(6.24)
inside this layer.
6.5
Lagrangian Interpretation of Stratification
As noted earlier, hydrostatic equilibrium applies in the presence of motion as well as under static conditions. Therefore, each of the stratifications in
156
6 Hydrostatic Equilibrium
Sec. 6.4 is valid even if a circulation is present, as is invariably the case. Under those circumstances, vertical profiles of temperature, pressure, and potential temperature [(6.16) through (6.18)] correspond to the horizontal-mean thermal structure and hence to averages over many ascending and descending air parcels. Interpreting thermal structure in terms of the behavior of individual air parcels provides some insight into the mechanisms controlling mean stratification. For a layer of constant lapse rate, the relationship between temperature and pressure (6.15) resembles one implied by Poisson's equation (2.30.2), but for a polytropic process (Sec. 2.5.1) with ( F / F d ) K in place of K. Consequently, we can associate the thermal structure in (6.16) through (6.18) with a vertical rearrangement of air in which individual parcels evolve diabatically according to a polytropic process. In that description, air parcels moving vertically exchange heat with their surroundings in such proportion for their temperatures to vary linearly with height. For an individual air parcel, the foregoing process has a polytropic specific heat that satisfies R
F
(cp -c)
rd
or
Cp
C-
1 - --F--- .
(6.25)
The corresponding heat transfer for the parcel is then given by (2.37) and its change of potential temperature follows from (2.40).
6.5.1
Adiabatic Stratification
Consider a layer characterized by V = Va,
(6.26.1)
the stratification of which is termed adiabatic. Then (6.25) implies C=0,
(6.26.2) 6q - dO - O,
so individual air parcels evolve adiabatically. Under these circumstances, air parcels move vertically without interacting thermally with their surroundings. Such behavior can be regarded as the limiting situation when vertical motion occurs on a timescale that is short compared to diabatic effects, as is typical of cumulus and sloping convection. Although their vertical motion is adiabatic, individual parcels may still exchange heat at the boundaries of the layer. In
Lagrangian Interpretation of Stratification
6.5
157
fact, such heat transfer is necessary to maintain vertical motion in the absence of mechanical forcing. The evolution of an individual parcel can be envisaged in terms of a circuit that it follows between the upper and lower boundaries of the layer, where heat transfer is concentrated. Such a circuit is depicted in Fig. 6.6 for a layer representative of the troposphere. At the lower boundary, an individual parcel moves horizontally long enough for heat transfer to occur. Isobaric heat absorption [e.g., through absorption of longwave (LW) radiation and transfer of sensible heat from the surface] increases the parcel's potential temperature (Fig. 6.7a). The accompanying increase of temperature (Fig. 6.7b) causes the parcel to become positively buoyant and rise. If the timescale of vertical motion is short compared to that of heat transfer, the parcel ascends along an adiabat in state space until it reaches the upper boundary, where it again moves horizontally. Isobaric heat rejection (e.g., through LW emission to space) then decreases the potential temperature and temperature of the parcel, which therefore becomes negatively buoyant. The parcel then sinks along a different adiabat until it has returned to the surface and completed a thermodynamic cycle. If it remains unsaturated [e.g., below its lifting condensation level (LCL)], each air parcel comprising the layer preserves its potential temperature away from the boundaries. Consequently, each parcel traces out a uniform profile of 0 as it traverses the layer. The horizontal-mean potential temperature O(z) is equivalent to an average over all such parcels at a given elevation, so it is likewise independent of height, that is, n
O(z) - const.
~
(6.27.1)
q
--~m,-~--
~
~
~
-
- 1 ~
~
~
~
--
- - q ~
~
- - ~ m
~
I I
I I
I I
I I
.~._
m
__..~.._
__
~.~[...
m
m-4b--
m
~ . . ~ -
~
~
-
~
-
- - - . ~ - - ~
q>O
Figure 6.6 Idealized circuit followed by an air parcel during which it absorbs heat at the base of a layer and rejects heat at its top, with adiabatic vertical motion between.
158
6
Hydrostatic
Equilibrium
#q
--
-- . ~ - t ~ . ~ -
I
I
I I i I I I I
0
! I
(a)
~ q>O #q
\. . . .
_-~-~. \ '.....
\ IL rd~ \
\
',, = --T
\\
",.',,
\
\.
",,
\ Tv
\
"...
\
"--,, '....
\
(b)
\~
--
~ - 1 ~
\ \
:
-1~
.
.
.
.
~ q>O
Figure 6.7 Thermodynamic cycle followed by the air parcel in Fig. 6.6 in terms of (a) potential temperature and (b) temperature. Horizontally averaged behavior for a layer composed of many such parcels is also indicated by the dotted lines.
As is true for an individual parcel (Sec. 2.4.2), a uniform distribution of potential temperature corresponds to a constant environmental lapse rate F - F a.
(6.27.2)
Consequently, the thermal structure associated with (6.24) can be interpreted as the horizontal mean for a layer in which air is being actively and adiabatically rearranged in the vertical. This interpretation can be extended to saturated conditions (e.g., inside cloud). An air parcel's evolution is then pseudo-adiabatic, so its equivalent potential temperature is conserved. Active vertical rearrangement_of air will then make the horizontal-mean equivalent potential temperature Oe(z) independent of height B
E m
Oe(Z ) = const.
(6.28.1)
6.5
Lagrangian Interpretation of Stratification
From Sec. 5.4.3, this uniform distribution of environmental lapse rate V = F,,
159
Oe(Z) corresponds to a mean
(6.28.2)
the corresponding stratification being termed saturated adiabatic. Even though all parcels inside the layer evolve in like fashion, local behavior will differ from the horizontal mean because conserved properties may still vary from one parcel to another (e.g., due to different histories experienced by those parcels at the layer's boundaries). An ascending parcel will have values of 0 and 0e, which are greater than the corresponding horizontalmean values because that parcel will have recently absorbed heat at the lower boundary. Conversely, a descending parcel will have values of 0 and 0e, which are less than the corresponding horizontal-mean values because that parcel will have recently rejected heat at the upper boundary. Therefore, the actual stratification of the layer will vary with horizontal position. If heat transfer at the boundaries were to be eliminated, convectively driven motions would then "spin down" through turbulent and molecular diffusion toward a limiting state of no motion. Since diffusion destroys gradients between individual parcels, this limiting state is characterized by homogeneous distributions of 0 and 0e and thus, everywhere, by the environmental lapse rate F -- F d or F s. Hence, this limiting homogeneous state has stratification identical to the horizontal-mean stratification of the layer being convectively overturned, which may therefore be regarded as "statistically well mixed." The stratification (6.28) is characteristic of the troposphere. Close to the saturated adiabatic lapse rate (Fig. 1.2), the mean thermal structure in the troposphere follows from efficient vertical exchange of moist air inside cumulus and sloping convection. Air rearranged by convection is continually replenished with moisture through contact with warm ocean surfaces. Because convective motions operate on timescales of a day or shorter, they make the troposphere statistically well mixed and, in the mean, described approximately by (6.28). In traversing the circuit, the parcel in Fig. 6.7 absorbs heat at high temperature and rejects heat at low temperature. By the second law (Sec. 3.2), net heat is absorbed over a cycle. Then the first law (2.13) implies that the parcel performs net work during its traversal of the circuit. It follows that an individual parcel in the above circulation behaves as a heat engine. In fact, the thermodynamic cycle in Fig. 6.7 is analogous to the Carnot cycle pictured in Fig. 3.2, except that heat transfer occurs (approximately) isobarically instead of isothermally. More expansion work is performed by the parcel during ascent than is performed on the parcel during descent. The area circumscribed by the parcel's evolution in Fig. 6.7b reflects the net work it performs during the
160
6
Hydrostatic Equilibrium
cycle and is proportional to the change of the parcel's potential temperature at the upper and lower boundaries (i.e., to the heat transfer there). Net heat absorption and work performed by individual air parcels make the general circulation of the troposphere behave as a heat engine, one driven thermally by heat transfer at its lower and upper boundaries. Work performed by individual parcels is associated with a redistribution of mass when air that is effectively warmer and lighter (namely, when compressibility is taken into account) at the lower boundary is exchanged with air that is effectively cooler and heavier air at the upper boundary. The latter represents a conversion of potential energy into kinetic energy, which maintains the general circulation against frictional dissipation.
6.5.2
Diabatic Stratification
The idealized behavior just described relies on heat transfer being confined to the lower and upper boundaries of the layer, where an air parcel resides long enough for diabatic effects to become important. On longer timescales, as are typical of vertical motions in the stratosphere, the evolution of an individual air parcel is not adiabatic. Radiative transfer, which is the primary diabatic influence outside the boundary layer and clouds, is characterized by cooling rates of order 1 K day -1 in the troposphere (see Fig. 8.24). Cooling rates as large as 10 K day -1 occur in the stratosphere and near cloud (Fig. 9.33). By comparison, the cooling rate associated with adiabatic expansion follows from (2.33) as
= -F'w,
(6.29.1)
where T and z refer to an individual parcel, F' denotes its lapse rate, and w=
dz dt
(6.29.2)
is the parcel's vertical velocity. A vertical velocity of 0.1 m s -1, which is representative of cumulus and sloping convection, gives (dT/dt)a d = 84 K day -1 under unsaturated conditions and of order 50 K day -1 under saturated conditions. Both are much larger than radiative cooling rates, so heat transfer in those motions can be ignored to a good approximation. On the other hand, a velocity of 1 mm s -~, which is representative of vertical motion in the stratosphere, is associated with an adiabatic cooling rate of only (dT/dt)a d = 0.84 K day -1. Heat transfer in such motions cannot be ignored and may even dominate temperature changes associated with expansion and compression of individual air parcels.
6.5
Lagrangian Interpretation of Stratification
161
Under those circumstances, an air parcel moving vertically interacts thermally with its environment. Because 0 is no longer conserved, diabatic motion implies a different thermal structure. Consider a layer characterized by Fd > F = const,
(6.30.1)
the stratification of which is termed subadiabatic: Temperature decreases with height slower than dry adiabatic. By (6.18), 0 increases with height (compare Fig. 6.5). Then (6.25) together with (2.37) gives c>O 6q > 0
dT > 0
6q < 0
dT < 0
(6.30.2)
for F < 0 (temperature increasing with height) and c and 6q with reversed inequalities for F > 0. Both imply that an air parcel absorbs heat during ascent and rejects heat during descent, so its potential temperature increases with height. As illustrated in Fig. 6.8, heat is rejected at high temperature in upper levels and absorbed at low temperature in lower levels. By the second law, net heat is rejected over a cycle. Then the first law implies that net work must be performed on the parcel for it to traverse the circuit. More work is performed on the parcel during descent than is performed by the parcel during ascent. 3 Opposite to the troposphere, this behavior is characteristic of the stratosphere. Net heat rejection and work performed on individual air parcels make the general circulation of the stratosphere behave as a refrigerator, one driven mechanically by waves that propagate upward from the troposphere. By rearranging air, planetary waves exert an influence on the stratosphere analogous to paddle work. When dissipated, they cause air at middle and high latitudes to cool radiatively, so individual parcels sink across isentropic surfaces to lower 0 (see Figs. 8.27 and 17.11). Conversely, air at low latitudes warms radiatively, so parcels then rise to higher 0. For both, vertical motion is slow enough to make diabatic effects important and to allow the temperature of individual parcels to increase with height. This feature of the stratospheric circulation follows from buoyancy. Contrary to the troposphere, buoyancy in the stratosphere strongly restrains vertical motion because, on average, warmer (high 0) air overlies cooler (low 0) air. Work performed on the stratosphere against the opposition of buoyancy leads to a redistribution of mass, when heavier air at lower levels is exchanged with lighter air at upper levels. The latter represents a conversion of kinetic energy 3In practice, this paradigm can be pushed only so far because stratospheric air must eventually enter the troposphere to complete the thermodynamic cycle (refer to Fig. 17.9), which makes the stratosphere an open system.
162
6
Hydrostatic
Equilibrium
(a) /
I
,~'
I
....."/ ...:
/
!
...........'"
#
q>0
I
I
...I
q <0/'~
I
../"
/
#
-~;,
.... ......i~
/ /
t q
.....
OIm'
//
/ / /~.~. _ ~/_ .~_ / t q>O
t q
(b) I
§
/=
/
.'
j
1
/
IF<0
~q>O
..:
'
I I I
/
"~,.~ q
/ / ...
I
§
/
:
/
"
/ ~
-t~ "--~ -~ t
I !
T
....
q>O
/ J
T
it,
I I
F i g u r e 6.8 As in Fig. 6.7, b u t for an air parcel w h o s e vertical m o t i o n is d i a b a t i c a n d whose temperature increases with height.
into potential energy, which is dissipated thermally through LW emission to space. Stratification with r > r~,
(6.31)
which is termed superadiabatic, can also be interpreted in terms of diabatic motion of individual air parcels. However, such stratification will be seen to be mechanically unstable. Buoyancy then reinforces vertical motion, so it occurs on timescales too short for diabatic effects to be important. Through convective overturning, a superadiabatic layer quickly adjusts to adiabatic stratification, for reasons developed in Chapter 7.
Problems
163
Suggested Reading
Atmospheric Thermodynamics (1981) by Iribarne and Godson contains a thorough discussion of atmospheric statics. Problems 6.1. Calculate the discrepancy between geometric and geopotential height, exclusive of anisotropic contributions to gravity, up to 100 km. 6.2. Show that the polytropic potential temperature in Problem 2.9 is conserved under the diabatic conditions described in Sec. 6.5.2. 6.3. Use (2.27) and replace z by - I n p to relate the area circumscribed in Fig. 6.7 to the work performed by the air parcel during one thermodynamic cycle. 6.4. Approximate the thermodynamic cycle in Fig. 6.7 by one comprised of two isobaric legs and two adiabatic legs to show that (a) net work is performed during a complete cycle and (b) the work performed is proportional to the change of temperature associated with isobaric heat transfer. 6.5. Show that an atmosphere of uniform negative lapse rate need not have an upper bound. 6.6. For reasons developed in Chapter 12, large-scale horizontal motion is nearly tangential to contours of geopotential height and given approximately by the geostrophic velocity
1( o ) 8~
,
where v = (u, v) denotes the horizontal component of motion in the eastward (x) and northward (y) directions and 9 is the geopotential along an isobaric surface. Suppose that satellite measurements provide the three-dimensional distributions of temperature and of the mixing ratio of a long-lived chemical species. From those measurements, construct an algorithm to determine 3-dimensional air motion on large scales at elevations great enough to ignore variations in surface pressure. 6.7. An upper level depression, typical of extratropical cyclones during early stages of development, affects isobaric surfaces aloft but leaves the surface pressure distribution undisturbed. Consider the depression of isobaric height associated with the temperature distribution
=
--
[ ~2 -'~ (~ -- ~0)2]
T(A, 4~, ~) -- T + T cos(4~) + T' exp -
L2
9f(~),
where T, T, and T' reflect global-mean, zonal-mean, and disturbance contributions to temperature, respectively, ~ - - l n ( p / p o ) , with P0 =
164
6 Hydrostatic Equilibrium
{-cos
1000 mb, is a measure of elevation,
0 ~>~T, with ~r = --In(100/1000) reflecting the tropopause elevation. The surface pressure Ps = Po remains constant, ~b0 = 7r/4, L = 0.25 rads, and T = 250 K, T = 30 K, and T' -- 5 K. (a) Derive an expression for the 3-dimensional distribution of geopotential height. (b) Plot the vertical profiles of disturbance temperature and height in the center of the anomaly. (c) Plot the horizontal distribution of geopotential height for the midtropospheric level ~ = ~r/2, as a function of A and q~ > 0. (d) Plot a vertical section at latitude ~b0, showing the elevations ~: of isobaric surfaces as functions of A.
6.8. Under the conditions of Problem 6.7, how large must the anomalous temperature T' be for the disturbance to form a cut-off low at the midtropospheric level ~: = ~r/2? 6.9. Unlike midlatitude depressions, hurricanes are warm-core cyclones that are invariably accompanied by a trough in surface pressure. Consider a hurricane associated with the temperature distribution T(A, ~b, ~:) = T + T'(A, q~)e-'~, where T and T' reflect zonal-mean and disturbance contributions to temperature, ~: = - l n ( p / p o ) , with P0 = 1000 mb, is a measure of elevation, m
T'(A, ~ b ) - Tar
~b)exp[asrs(A, ~b)],
in which ~:,(A, 4~) = 0 . 1 exp
-
L2
,
with a -1 = ~:r/2, and ~:r = In(100/1000) reflects the surface and tropopause pressures, respectively, and L = 0.1 rads. (a) Derive an expression for the 3-dimensional distribution of geopotential height. (b) Plot the vertical profiles of disturbance temperature and height in the center of the anomaly. (c) Plot the horizontal distribution of geopotential height for the midtropospheric level ~: = ~:r/2. (d) Plot a vertical section at the equator, showing the elevations ~: of isobaric surfaces as functions of A. 6.10. Show that subadiabatic stratification requires an individual parcel moving vertically to, in general, absorb (reject) heat as it ascends (descends). 6.11. Surface analyses produced by the weather service plot pressure adjusted to the common reference level, mean sea level (MSL), so that different locations can be compared meaningfully. (a) Derive an approximate
Problems
165
expression for the pressure at MSL in terms of the local surface height Zs, surface pressure Ps, and the global-mean surface temperature T 288 K. (b) A surface analysis shows a ridge with a maximum of 1040 mb over Denver, which lies at an elevation of 5300 ft above MSL. What surface pressure was actually recorded there?
6.12. Deep convection inside a region of the tropics releases latent heat to achieve a column heating rate of 250 W m -2. (a) If, uncompensated, that heating would produce a temperature perturbation that is invariant with height up to the tropopause, calculate the implied change of thickness for the 1000- to 100-mb layer after 1 day (compare Fig. 1.30). (b) What process, in reality, compensates for the tendency of tropospheric thickness implied in part (a)?
Hydrostatic Equilibrium
Changes of thermodynamic state accompanying vertical motion follow from the distribution of atmospheric mass, which is determined ultimately by gravity. In the absence of motion, Newton's second law applied to the vertical reduces to a statement of hydrostatic equilibrium (1.16), wherein gravity is balanced by the vertical pressure gradient. This simple form of mechanical equilibrium is accurate even in the presence of motion because the acceleration of gravity is much greater than vertical accelerations of individual air parcels for nearly all motions of dimensions greater than a few tens of kilometers. Only inside deep convective towers and other small-scale phenomena are vertical accelerations large enough to invalidate hydrostatic equilibrium. Because it is such a strong body force, gravity must be treated with some care. Complications arise from the fact that the gravitational acceleration experienced by an air parcel does not act purely in the vertical and varies with location. According to the discussion above, gravity is large enough to overwhelm other contributions in the balance of vertical forces. The same is true for the horizontal force balance. Horizontal components of gravity introduced by the earth's rotation and additional sources must be balanced by other horizontal forces, which unnecessarily complicate the description of atmospheric motion and can overshadow those forces actually controlling the motion of an air parcel.
6.1
Effective Gravity
The acceleration of gravity appearing in (1.16) is not constant. Nor does it act purely in the vertical. In the reference frame of the earth, the gravitational acceleration experienced by an air parcel follows from three basic contributions: 1. Radial gravitation by the planet's mass 2. Centrifugal acceleration due to rotation of the reference frame 3. Anisotropic contributions, as illustrated in Fig. 6.1. Radial gravitation by the planet's mass is the dominant contribution. Under idealized circumstances, that component of g acts 143
144
6 Hydrostatic Equilibrium
---'---
z = con st. = const.
(o cos ,/,) !i!i!i!i!i!i!!!i!i!i!i!i!i!i!i!i iiiiiiiiiiil....................
..............
. . . . . . . . . . . . . . . .
...................•.....•.p.......................................................................................................................•.•
i:i:i:i:i:i:i:i:i:i:~:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:i:!:i:!:i:!:!:i:!:!:: !ii!i!iiii~i~i!i ~ iiii!iii!i!i!iii i!!ii! i!!iii!ii!iii!iii!ii!iiiiii!!ii!iiii!iiiiiiiiiiiiiiiiiiiiiii!iiiii!iiii!i!ii!!!iii!i!i!ii!iiiiiii!iiiiii!!iii!iiiii!!iii!ii!!i !i i!iii! iiii!i!i!!ii!i!ii!ii
f
Figure 6.1 Effective gravity g illustrated in relation to its contributions from radial gravitation and centrifugal acceleration. Also shown are surfaces of constant geometric altitude z and corresponding surfaces of constant geopotential ~, to which g is orthogonal. perpendicular to surfaces of constant geometric altitude z, which are concentric spheres. The rotation of the earth introduces another contribution. Because it is noninertial, the reference frame of the earth includes a centrifugal acceleration that acts perpendicular to and away from the axis of rotation. Last, departures of the planet from sphericity and homogeneity (e.g., through variations of surface topography) introduce anisotropic contributions to gravity that must be determined empirically. Collectively, these contributions determine the effective gravity, which can be expressed as follows: a2 g(A, q~, z) = (a + z) 2gOk + ~2(a + z)cos dper + ~(A, q~, z), (6.1) where a is the mean radius of the earth, go is the radial gravitation at mean sea level, k is the local upward unit normal, l~ is the planet's angular velocity, A and ~b are longitude and latitude, respectively, e r is a unit vector directed outward from the axis of rotation, and ~(A, ~b, z) represents all anisotropic contributions. Even if the planet were perfectly spherical, effective gravity would not act uniformly across the surface of the earth, nor entirely in the vertical. According to (6.1), the centrifugal acceleration deflects g into the
6.2
Geopotential Coordinates
145
horizontal, which introduces a component of gravity along surfaces of constant z. The centrifugal acceleration also causes the vertical component of g to vary from a minimum of 9.78 m s -2 at the equator to a maximum of 9.83 m s -2 at the poles. Although more complicated, anisotropic contributions have a similar effect. These contributions introduce horizontal components of gravity that must be balanced by other forces acting along surfaces of constant geometric altitude. Thus, (1.16) represents but one of several component equations that are required to describe hydrostatic balance. Horizontal forces introduced by gravity unnecessarily complicate the description of atmospheric behavior because they can overshadow the forces actually controlling the motion of an air parcel. For instance, rotation introduces a horizontal component that must be balanced by a variation of pressure along surfaces of constant geometric altitude. That horizontal pressure gradient exists even under static conditions, for which an observer in the reference frame of the earth would anticipate Newton's second law to reduce to a simple hydrostatic balance in the vertical. To avoid such complications, it is convenient to introduce a coordinate system that consolidates horizontal components of gravity into the vertical coordinate.
6.2
Geopotential Coordinates
Because gravity is conservative, the specific work performed during a cyclic displacement of mass through the earth's gravitational field vanishes: Awg = - f g(A, 4', z ) - d ~ - 0,
(6.2)
where ds denotes the incremental displacement of an air parcel. According to Sec. 2.1.4, (6.2) is a necessary and sufficient condition for the existence of an exact differential dap = 6wg - - g . dg,
(6.3)
which defines the gravitational potential or geopotential ~. In (6.3), d ~ equals the specific work performed against gravity to complete the displacement d~. Then, per the aforementioned discussion, gravity is an irrotational vector field that can be expressed in terms of its potential function g = -V~.
(6.4)
Surfaces of constant geopotential are not spherical like surfaces of constant geometric altitude. Centrifugal acceleration in the rotating reference flame of the earth distorts geopotential surfaces into oblate spheroids (Fig. 6.1). The component of XT~ along surfaces of constant geometric altitude introduces a
146
6
Hydrostatic Equilibrium
horizontal component of gravity that must be balanced by a horizontal pressure gradient force. Representing gravity is simplified by transforming from pure spherical coordinates, in which elevation is fixed along surfaces of constant geometric altitude, to geopotential coordinates, in which elevation is fixed along surfaces of constant geopotential. Geopotential surfaces can be used for coordinate surfaces because 9 increases monotonically with altitude (6.3), which ensures a one-to-one relationship between those variables. Introducing the aforementioned transformation is tantamount to measuring elevation z along the line of effective gravity, which will be termed height to distinguish it from geometric altitude. Defining z in this manner agrees with the usual notion of local vertical being plumb with the line of gravity and will be adopted as the convention hereafter. In geopotential coordinates, gravity has no horizontal component. Consequently, in terms of height, (6.4) reduces to simply (6.5)
ddO - gdz,
where g denotes the magnitude of g, which is understood to act in the direction of decreasing heightmnot geometric altitude. Using mean sea level as a reference elevation I yields an expression for the absolute geopotential r
=
f0 Zgdz.
~
(6.6)
Having dimensions of specific energy, the geopotential represents the work that must be performed against gravity to raise a unit mass from mean sea level to a height z. Evaluating 9 from (6.6) requires measured values of effective gravity throughout. However, since it is a point function, 9 is independent of path by (6.2) and the exact differential theorem (2.7). Therefore, 9 depends only on height z and surfaces of constant geopotential coincide with surfaces of constant height. Using surfaces of constant geopotential for coordinate surfaces consolidates gravity into the vertical coordinate, height. However, g still varies with z through (6.5). To account for that variation, it is convenient to introduce yet another vertical coordinate in which the dependence on height is absorbed. Geopotential height is defined as Z = ml f0 ~'g d z go 1
=
(6.7)
go 1Mean sea level may be used to define a reference value of 9 because it nearly coincides with a surface of constant geopotential.
6.3 Hydrostatic Balance
147
where go - 9.8 m s - 2 is a reference value reflecting the average over the surface of the earth. Then d Z - ml d ~ go
(6.8)
= g dz.
go Since it accounts for the variation of gravity, geopotential height simplifies the expression for hydrostatic balance. In terms of Z, (1.16) reduces to dp - - p g d z
(6.9)
= -pgodZ.
When elevation is represented in terms of geopotential height, gravity can be described by the constant value go, which simplifies the description of atmospheric motion. Although it has these formal advantages, geopotential height is nearly identical to height because g ~ go throughout the homosphere. In fact, the two measures of elevation differ by less that 1% below 60 km and only 3% below 100 km. Therefore, Z can be used interchangeably with z, which will be our convention hereafter. Then g is understood to refer to the constant reference value go.
6.3
Hydrostatic Balance
In terms of geopotential height, hydrostatic balance can be expressed (6.10)
gdz = -vdp.
The ideal gas law transforms this into RT
dz = - ~ d l n = - H d l n p,
p (6.11)
where the scale height H is defined in (1.17). According to (6.11), the change of height is proportional to the local temperature and to the change o f - In p. Since changes of temperature and thus H are small compared to changes of pressure, the quantity - I n p can be regarded as a dimensionless measure of height. Consider a layer bounded by two isobaric surfaces p = p l ( x , y, z) and p = p2(x, y, z), as illustrated in Fig. 6.2. By integrating (6.11) between those
148
6 Hydrostatic Equilibrium
-.
Zl: z(x,y, Pl)
I
,,
fx
Figure 6.2 Isobaric surfaces (dark shading) in the presence of a horizontal temperature gradient. Vertical spacing of isobaric surfaces is compressed in cold air and expanded in warm air, which makes the height of an individual isobaric surface low in the former and high in the latter. Because pressure decreases upward monotonically, a surface of constant height (light shading) then has low pressure where isobaric height is low and high pressure where isobaric height is high. Therefore, contours of height for an individual isobaric surface can be interpreted similarly to isobars on a constant height surface. Also indicated is the geostrophic velocity vg (Sec. 12.1), which is directed parallel to contours of isobaric height and has magnitude proportional to its horizontal gradient.
surfaces, we obtain A z - - z 2 -- z I = - ( n ) I n
~1
'
(6.12.1)
where (H) = R,_ ,IT~ g
(6.12.2)
6.3
Hydrostatic Balance
149
and
(T) - fpp2 T d l n p fp7 d l n p _- fpPl~ Tdln p ln(~)
(6.12.3) '
define the layer-mean scale height and temperature, respectively, and Zl = z(x, y, Pl) and z2 = z(x, y, P2) denote the heights of the two isobaric surfaces. Known as the hypsometric equation, (6.12) asserts that the thickness of a layer bounded by two isobaric surfaces is proportional to the mean temperature of that layer and the pressure difference across it. In regions of cold air, the e-folding scale H is small, so pressure in (1.17) decreases with height sharply and the vertical spacing of isobaric surfaces is compressed. By comparison, H is large in regions of warm air, so pressure decreases with height more slowly, and the spacing of isobaric surfaces is expanded. If zl =0, the hypsometric equation provides the height of the upper surface z(x, y, P2). For example, the height of the 500-mb surface in Fig. 1.9 can then be expressed as a vertical integral of temperature. By (6.12), the height of an isobaric surface can be determined from measurements of temperature between that surface and the ground or, alternatively, between that surface and a reference isobaric surface, the height of which is known. Such temperature measurements are made routinely by ground-launched rawinsondes, which also measure air motion and humidity and ascend via balloon to as high as 10 mb. Launched twice a day, rawinsondes provide dense coverage over populated landmasses, but only sparse coverage over oceans and in the tropics. Temperature measurements are also made remotely by satellite. Satellite measurements of temperature provide continuous global coverage of the atmosphere, albeit asynoptically. ~ Satellite retrievals also have coarser vertical resolution (~ 4 to 8 km) than that provided by rawinsondes (~1 km). Both are assimilated operationally in synoptic analyses of the instantaneous circulation, like those shown in Figs. 1.9 and 1.10. According to Fig. 1.9, the height of the 500-mb surface slopes downward toward the pole at midlatitudes, where zs00 decreases from 5900 to 5000 m. That decrease of 500-mb height mirrors the poleward decrease of layer-mean temperature between the surface and 500 mb, (T)500, which is shown in Fig. 6.3 for the same time as that in Fig. 1.9a. The wavy region where (T)500 and zs00 change abruptly delineates the polar front, which separates cold polar air from warmer air at lower latitudes. As is apparent from Fig. 1.9a, that region 2The term synoptic refers to the simultaneous distribution of some property, for example, a snapshot over the earth of the instantaneous height and motion on the 500-mb surface (Fig. 1.9a). However, a polar-orbiting satellite necessarily observes different regions at different instants. Termed asynoptic, such measurements sample the entire globe only after 12 to 24 hr.
150
6
Hydrostatic Equilibrium ~ooOO.oOOO~ ......... .o,.,oOr
~. ..., 9...
....~.
::-......v:i:;,~:.,.,.:~ "~'
/
'
"~',,
...'"
J
Z
/
J
........ "
:::
-:;> \ ./ \
I !
'
....
O,.o ..":......
281
"'" i /
9.....
~"~i
...;?"-/ /
:..." :... ."
\
\
P
I
!~. ...ii... 9"............-""
I #
/ e
/ /
,/ s~ ........... ".........
\
.... "'~',
.... ...... 9 j ..9
"'"-... H .........
"', ...... 9......... ""~...
~ .
2{34. """"--.
. . . . . . . . . . . . .
-
.::"
".....
.... ,.~:;.....
....
""
~~176176
Figure 6.3 Mean temperature (Kelvin) of the layer between 1000 mb and 500 mb for March 4, 1984. The bold solid line marks a meridional cross section displayed in Fig. 6.4.
of sharp temperature change coincides with the strong circumpolar flow of the jet stream, which is tangential to contours of isobaric height. Meridional gradients of temperature and height are particularly steep east of Asia and North America, where they mark the North Pacific and North Atlantic storm tracks that are manifested in the time-mean pattern (Fig. 1.9b) as locally intensified jets. Anomalies of cold air that punctuate the polar front in Fig. 6.3 introduce synoptic-scale depressions of the 500-mb surface in Fig. 1.9a. By deflecting contours of isobaric height, those features disrupt the motion of the jet stream, which, in their absence, is nearly circumpolar. The hypsometric equation provides a theoretical basis for using pressure to describe elevation. Because H is positive, (6.12) implies a one-to-one relationship between height and pressure: a given height corresponds to a single
6.4 Stratification
151
pressure. Therefore, pressure can be used as an alternative to height for the vertical coordinate. Isobaric coordinates, which are developed in Chapter 11, use surfaces of constant pressure for coordinate surfaces. Although they evolve with the circulation, isobaric coordinates afford several advantages. In isobaric coordinates, pressure becomes the independent variable and the height of an isobaric surface becomes the dependent variable, for example, z2 = z(x, y, P2). Because pressure decreases monotonically with height, low height of an isobaric surface corresponds to low pressure on a surface of constant height and just the reverse for high values (Fig. 6.2). For the same reason, contours of isobaric height resemble isobars on a surface of constant height. Consequently, the horizontal distribution of isobaric height may be interpreted analogously to the distribution of pressure on a surface of constant height. From hydrostatic equilibrium, the latter represents the weight of the atmospheric column above a given height. Therefore, the distribution of isobaric height reflects the horizontal distribution of atmospheric mass above the mean elevation of that isobaric surface. Contours of 500-mb height that delineate the polar front in Fig. 1.9a correspond to isobars on the constant height surface: z ,~ 5.5 km, with pressure decreasing toward the pole. Thus, less atmospheric column lies above that elevation poleward of the front than equatorward of the front. Figure 6.4 shows a meridional cross section of the thermal structure and motion in Figs. 6.3 and 1.9a. The horizontal distribution of mass just noted results from compression of isobaric surfaces poleward of the front, which is centered near 40 N, and expansion equatorward of the front. This introduces a tilt to isobaric surfaces at midlatitudes, one that steepens with height. The jet stream is found in the same region and likewise intensifies vertically, to a maximum near the tropopause. Similar behavior at high latitudes marks the base of the polar-night jet. 6.4
Stratification
Expressions (2.33) and (5.32) for the dry adiabatic and saturated adiabatic lapse rates, which describe the evolution of a displaced air parcel under unsaturated and saturated conditions, respectively, hold formally with z representing geopotential height and g constant. Neither Fa nor Fs has a direct relationship to the temperature of the surroundings because a displaced parcel is thermally isolated under adiabatic conditions. Thermal properties of the environment are dictated by the history of air residing at a given location, for example, by where that air has been and what thermodynamic influences have acted on it. The environmental lapse rate is defined as F=
dT dz'
(6.13)
where T refers to the ambient temperature. Like parcel lapse rates, F > 0 corresponds to temperature decreasing with height. Conversely, F < 0 cor-
152
6 Hydrostatic
Equilibrium
. 's ~ii 10 -
\
,
m
,
/
l~ I
-
l
"
~, llI\\\\\\\",,,",V-JlIIIIlflTT-TT"
x I1111\-
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-,,
, I
/
L,oo_, ,'/!1 k
~g ,, ' , / / t \ ' , , \ ~ / / /
",,
) {,"lll '-/'jj/- / 8$Omb~.
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: ._ ,,
7"
r
t
/ /
I
I
I
I
I
I
I
10
20
30
40
50
60
70
/i ~ ' " 80
90
LATITUDE
Figure 6.4 Meridional cross section of thermal structure and motion in Figs. 1.9a and 6.3 at the position indicated in the latter. Vertical spacing of isobaric surfaces (bold solid curves) is expanded at low latitude and compressed at high latitude, introducing a slope at midlatitudes that steepens with height. Zonal wind speed (contoured in m s -1) delineates the subtropical jet, which coincides with the steep slope of isobaric surfaces, both intensifying upward to a maximum at the tropopause. Contour increment: 5 m s -1.
responds to temperature increasing with height, in which case the profile of environmental temperature is said to be inverted. The compressibility of air leads to atmospheric mass being stratified, as is reflected in the vertical distributions of density and pressure. Since the environment is in hydrostatic equilibrium, the distribution of pressure may be related to the thermal structure through the hypsometric relation. Incorporating (6.11) transforms (6.13) into 1
dT
H dlnp
=F
or
din T dln p
=~r g F Fd
(6.14)
6.4
6.4.1
Stratification
153
Idealized Stratification
For certain classes of thermal structure, (6.14) can be integrated analytically to obtain the distributions of pressure and other thermal properties inside an atmospheric layer. LAYER OF CONSTANT LAPSE RATE
If
F
-
const ~- 0, (6.14) yields TTs = ( - ~ ~ ) ~ K ,
(6.15)
where the subscript s refers to the base of the layer. Because (6.16)
T(z) = Ts - rz,
(6.15) can be used to obtain the vertical distribution of pressure: PPs - [ 1 -
K-~F (ZHss)] (~r) .
(6.17)
Then (2.31) implies the vertical profile of potential temperature: Fd
O(z, - ( T s - F z ) [ l - K F---aF( ~ s s ) ]
r ,
(618).
where p~ - P0 = 1000 mb has been presumed for the base of the layer. As shown in Fig. 6.5, the compressibility of air makes pressure decrease with height for all F. The same is true of density. However, potential temperature decreases with height only for F > F a, whereas it increases with height for F < Fa. In both cases, 0 varies sharply with height because the pressure term dominates over the temperature term in determining the height dependence in (6.18). As a result, environmental potential temperature can change by several hundred degrees in just a few scale heights, as is typical in the stratosphere (refer to Fig. 17.19). By contrast, the potential temperature of a displaced air parcel is independent of its height under adiabatic conditions. The thermal structure described by (6.16) through (6.18) can apply locally (e.g., inside a certain layer), even if temperature does not vary linearly throughout the atmosphere. Should it apply for the entire range of height, (6.16) has one further implication. For F > 0, positive temperature requires the atmosphere to have a finite upper bound
L
Ztop -- -~-,
where p vanishes. For F < 0, no finite upper bound exists.
(6.19)
0
tn .
--
o
.
.
~
_
!
LO-
~: ~
.
L
i
~4 '
o
i
~ '
~
~ ~
I
co '-
o
'
oi '
~
sH Z
..--..._
II
q7
I
I"
o~ '
o
~'-
.I
,~ '
~
,
~ '
o
I
.,
i
cl '
Ln
0
--
--
-
-
0
6
0 0
CO
0
0
0 h0
o 0 0 0 0 cO
0
0 0 @d
0
9
9
0 -,-9
0 0 IX.
0 0
0
oo o o
o
(2::)
~-
I~
~
0
+--,
0
0 ~J
~J 0 ." ,...~
o l~ =
~.
(lJ
,==.,
"0
"~
~J
~J
0~
0
0
~
~=
6.5 Lagrangian Interpretation of Stratification
155
ISOTHERMAL LAYER
For the special case F - 0, (6.17) is indeterminate. Reverting to the hydrostatic relation (6.11) yields z=-Hln
(-~~)
or
P -- e ~, Ps z
(6.20)
which is identical to (1.17) under these circumstances. Then the vertical profile of potential temperature is simply O(z) = Tse"~
: 0se~.
(6.21)
Under isothermal conditions, environmental potential temperature increases by a factor of e every K-a --- 3 scale heights. ADIABATIC LAYER
For the special case F = Fa, (6.15) reduces to T K -T-~- ( - ~ s )
(6.22,
and the distribution of pressure (6.17) becomes t
P
1-
.
(6.23)
Equation (6.22) is the same relationship between temperature and pressure implied by Poisson's equation (2.30.2), which defines potential temperature (2.31). Accordingly, the vertical profile of potential temperature (6.18) reduces to O(z) -
O~ -
const
(6.24)
inside this layer.
6.5
Lagrangian Interpretation of Stratification
As noted earlier, hydrostatic equilibrium applies in the presence of motion as well as under static conditions. Therefore, each of the stratifications in
156
6 Hydrostatic Equilibrium
Sec. 6.4 is valid even if a circulation is present, as is invariably the case. Under those circumstances, vertical profiles of temperature, pressure, and potential temperature [(6.16) through (6.18)] correspond to the horizontal-mean thermal structure and hence to averages over many ascending and descending air parcels. Interpreting thermal structure in terms of the behavior of individual air parcels provides some insight into the mechanisms controlling mean stratification. For a layer of constant lapse rate, the relationship between temperature and pressure (6.15) resembles one implied by Poisson's equation (2.30.2), but for a polytropic process (Sec. 2.5.1) with ( F / F d ) K in place of K. Consequently, we can associate the thermal structure in (6.16) through (6.18) with a vertical rearrangement of air in which individual parcels evolve diabatically according to a polytropic process. In that description, air parcels moving vertically exchange heat with their surroundings in such proportion for their temperatures to vary linearly with height. For an individual air parcel, the foregoing process has a polytropic specific heat that satisfies R
F
(cp -c)
rd
or
Cp
C-
1 - --F--- .
(6.25)
The corresponding heat transfer for the parcel is then given by (2.37) and its change of potential temperature follows from (2.40).
6.5.1
Adiabatic Stratification
Consider a layer characterized by V = Va,
(6.26.1)
the stratification of which is termed adiabatic. Then (6.25) implies C=0,
(6.26.2) 6q - dO - O,
so individual air parcels evolve adiabatically. Under these circumstances, air parcels move vertically without interacting thermally with their surroundings. Such behavior can be regarded as the limiting situation when vertical motion occurs on a timescale that is short compared to diabatic effects, as is typical of cumulus and sloping convection. Although their vertical motion is adiabatic, individual parcels may still exchange heat at the boundaries of the layer. In
Lagrangian Interpretation of Stratification
6.5
157
fact, such heat transfer is necessary to maintain vertical motion in the absence of mechanical forcing. The evolution of an individual parcel can be envisaged in terms of a circuit that it follows between the upper and lower boundaries of the layer, where heat transfer is concentrated. Such a circuit is depicted in Fig. 6.6 for a layer representative of the troposphere. At the lower boundary, an individual parcel moves horizontally long enough for heat transfer to occur. Isobaric heat absorption [e.g., through absorption of longwave (LW) radiation and transfer of sensible heat from the surface] increases the parcel's potential temperature (Fig. 6.7a). The accompanying increase of temperature (Fig. 6.7b) causes the parcel to become positively buoyant and rise. If the timescale of vertical motion is short compared to that of heat transfer, the parcel ascends along an adiabat in state space until it reaches the upper boundary, where it again moves horizontally. Isobaric heat rejection (e.g., through LW emission to space) then decreases the potential temperature and temperature of the parcel, which therefore becomes negatively buoyant. The parcel then sinks along a different adiabat until it has returned to the surface and completed a thermodynamic cycle. If it remains unsaturated [e.g., below its lifting condensation level (LCL)], each air parcel comprising the layer preserves its potential temperature away from the boundaries. Consequently, each parcel traces out a uniform profile of 0 as it traverses the layer. The horizontal-mean potential temperature O(z) is equivalent to an average over all such parcels at a given elevation, so it is likewise independent of height, that is, n
O(z) - const.
~
(6.27.1)
q
--~m,-~--
~
~
~
-
- 1 ~
~
~
~
--
- - q ~
~
- - ~ m
~
I I
I I
I I
I I
.~._
m
__..~.._
__
~.~[...
m
m-4b--
m
~ . . ~ -
~
~
-
~
-
- - - . ~ - - ~
q>O
Figure 6.6 Idealized circuit followed by an air parcel during which it absorbs heat at the base of a layer and rejects heat at its top, with adiabatic vertical motion between.
158
6
Hydrostatic
Equilibrium
#q
--
-- . ~ - t ~ . ~ -
I
I
I I i I I I I
0
! I
(a)
~ q>O #q
\. . . .
_-~-~. \ '.....
\ IL rd~ \
\
',, = --T
\\
",.',,
\
\.
",,
\ Tv
\
"...
\
"--,, '....
\
(b)
\~
--
~ - 1 ~
\ \
:
-1~
.
.
.
.
~ q>O
Figure 6.7 Thermodynamic cycle followed by the air parcel in Fig. 6.6 in terms of (a) potential temperature and (b) temperature. Horizontally averaged behavior for a layer composed of many such parcels is also indicated by the dotted lines.
As is true for an individual parcel (Sec. 2.4.2), a uniform distribution of potential temperature corresponds to a constant environmental lapse rate F - F a.
(6.27.2)
Consequently, the thermal structure associated with (6.24) can be interpreted as the horizontal mean for a layer in which air is being actively and adiabatically rearranged in the vertical. This interpretation can be extended to saturated conditions (e.g., inside cloud). An air parcel's evolution is then pseudo-adiabatic, so its equivalent potential temperature is conserved. Active vertical rearrangement_of air will then make the horizontal-mean equivalent potential temperature Oe(z) independent of height B
E m
Oe(Z ) = const.
(6.28.1)
6.5
Lagrangian Interpretation of Stratification
From Sec. 5.4.3, this uniform distribution of environmental lapse rate V = F,,
159
Oe(Z) corresponds to a mean
(6.28.2)
the corresponding stratification being termed saturated adiabatic. Even though all parcels inside the layer evolve in like fashion, local behavior will differ from the horizontal mean because conserved properties may still vary from one parcel to another (e.g., due to different histories experienced by those parcels at the layer's boundaries). An ascending parcel will have values of 0 and 0e, which are greater than the corresponding horizontalmean values because that parcel will have recently absorbed heat at the lower boundary. Conversely, a descending parcel will have values of 0 and 0e, which are less than the corresponding horizontal-mean values because that parcel will have recently rejected heat at the upper boundary. Therefore, the actual stratification of the layer will vary with horizontal position. If heat transfer at the boundaries were to be eliminated, convectively driven motions would then "spin down" through turbulent and molecular diffusion toward a limiting state of no motion. Since diffusion destroys gradients between individual parcels, this limiting state is characterized by homogeneous distributions of 0 and 0e and thus, everywhere, by the environmental lapse rate F -- F d or F s. Hence, this limiting homogeneous state has stratification identical to the horizontal-mean stratification of the layer being convectively overturned, which may therefore be regarded as "statistically well mixed." The stratification (6.28) is characteristic of the troposphere. Close to the saturated adiabatic lapse rate (Fig. 1.2), the mean thermal structure in the troposphere follows from efficient vertical exchange of moist air inside cumulus and sloping convection. Air rearranged by convection is continually replenished with moisture through contact with warm ocean surfaces. Because convective motions operate on timescales of a day or shorter, they make the troposphere statistically well mixed and, in the mean, described approximately by (6.28). In traversing the circuit, the parcel in Fig. 6.7 absorbs heat at high temperature and rejects heat at low temperature. By the second law (Sec. 3.2), net heat is absorbed over a cycle. Then the first law (2.13) implies that the parcel performs net work during its traversal of the circuit. It follows that an individual parcel in the above circulation behaves as a heat engine. In fact, the thermodynamic cycle in Fig. 6.7 is analogous to the Carnot cycle pictured in Fig. 3.2, except that heat transfer occurs (approximately) isobarically instead of isothermally. More expansion work is performed by the parcel during ascent than is performed on the parcel during descent. The area circumscribed by the parcel's evolution in Fig. 6.7b reflects the net work it performs during the
160
6
Hydrostatic Equilibrium
cycle and is proportional to the change of the parcel's potential temperature at the upper and lower boundaries (i.e., to the heat transfer there). Net heat absorption and work performed by individual air parcels make the general circulation of the troposphere behave as a heat engine, one driven thermally by heat transfer at its lower and upper boundaries. Work performed by individual parcels is associated with a redistribution of mass when air that is effectively warmer and lighter (namely, when compressibility is taken into account) at the lower boundary is exchanged with air that is effectively cooler and heavier air at the upper boundary. The latter represents a conversion of potential energy into kinetic energy, which maintains the general circulation against frictional dissipation.
6.5.2
Diabatic Stratification
The idealized behavior just described relies on heat transfer being confined to the lower and upper boundaries of the layer, where an air parcel resides long enough for diabatic effects to become important. On longer timescales, as are typical of vertical motions in the stratosphere, the evolution of an individual air parcel is not adiabatic. Radiative transfer, which is the primary diabatic influence outside the boundary layer and clouds, is characterized by cooling rates of order 1 K day -1 in the troposphere (see Fig. 8.24). Cooling rates as large as 10 K day -1 occur in the stratosphere and near cloud (Fig. 9.33). By comparison, the cooling rate associated with adiabatic expansion follows from (2.33) as
= -F'w,
(6.29.1)
where T and z refer to an individual parcel, F' denotes its lapse rate, and w=
dz dt
(6.29.2)
is the parcel's vertical velocity. A vertical velocity of 0.1 m s -1, which is representative of cumulus and sloping convection, gives (dT/dt)a d = 84 K day -1 under unsaturated conditions and of order 50 K day -1 under saturated conditions. Both are much larger than radiative cooling rates, so heat transfer in those motions can be ignored to a good approximation. On the other hand, a velocity of 1 mm s -~, which is representative of vertical motion in the stratosphere, is associated with an adiabatic cooling rate of only (dT/dt)a d = 0.84 K day -1. Heat transfer in such motions cannot be ignored and may even dominate temperature changes associated with expansion and compression of individual air parcels.
6.5
Lagrangian Interpretation of Stratification
161
Under those circumstances, an air parcel moving vertically interacts thermally with its environment. Because 0 is no longer conserved, diabatic motion implies a different thermal structure. Consider a layer characterized by Fd > F = const,
(6.30.1)
the stratification of which is termed subadiabatic: Temperature decreases with height slower than dry adiabatic. By (6.18), 0 increases with height (compare Fig. 6.5). Then (6.25) together with (2.37) gives c>O 6q > 0
dT > 0
6q < 0
dT < 0
(6.30.2)
for F < 0 (temperature increasing with height) and c and 6q with reversed inequalities for F > 0. Both imply that an air parcel absorbs heat during ascent and rejects heat during descent, so its potential temperature increases with height. As illustrated in Fig. 6.8, heat is rejected at high temperature in upper levels and absorbed at low temperature in lower levels. By the second law, net heat is rejected over a cycle. Then the first law implies that net work must be performed on the parcel for it to traverse the circuit. More work is performed on the parcel during descent than is performed by the parcel during ascent. 3 Opposite to the troposphere, this behavior is characteristic of the stratosphere. Net heat rejection and work performed on individual air parcels make the general circulation of the stratosphere behave as a refrigerator, one driven mechanically by waves that propagate upward from the troposphere. By rearranging air, planetary waves exert an influence on the stratosphere analogous to paddle work. When dissipated, they cause air at middle and high latitudes to cool radiatively, so individual parcels sink across isentropic surfaces to lower 0 (see Figs. 8.27 and 17.11). Conversely, air at low latitudes warms radiatively, so parcels then rise to higher 0. For both, vertical motion is slow enough to make diabatic effects important and to allow the temperature of individual parcels to increase with height. This feature of the stratospheric circulation follows from buoyancy. Contrary to the troposphere, buoyancy in the stratosphere strongly restrains vertical motion because, on average, warmer (high 0) air overlies cooler (low 0) air. Work performed on the stratosphere against the opposition of buoyancy leads to a redistribution of mass, when heavier air at lower levels is exchanged with lighter air at upper levels. The latter represents a conversion of kinetic energy 3In practice, this paradigm can be pushed only so far because stratospheric air must eventually enter the troposphere to complete the thermodynamic cycle (refer to Fig. 17.9), which makes the stratosphere an open system.
162
6
Hydrostatic
Equilibrium
(a) /
I
,~'
I
....."/ ...:
/
!
...........'"
#
q>0
I
I
...I
q <0/'~
I
../"
/
#
-~;,
.... ......i~
/ /
t q
.....
OIm'
//
/ / /~.~. _ ~/_ .~_ / t q>O
t q
(b) I
§
/=
/
.'
j
1
/
IF<0
~q>O
..:
'
I I I
/
"~,.~ q
/ / ...
I
§
/
:
/
"
/ ~
-t~ "--~ -~ t
I !
T
....
q>O
/ J
T
it,
I I
F i g u r e 6.8 As in Fig. 6.7, b u t for an air parcel w h o s e vertical m o t i o n is d i a b a t i c a n d whose temperature increases with height.
into potential energy, which is dissipated thermally through LW emission to space. Stratification with r > r~,
(6.31)
which is termed superadiabatic, can also be interpreted in terms of diabatic motion of individual air parcels. However, such stratification will be seen to be mechanically unstable. Buoyancy then reinforces vertical motion, so it occurs on timescales too short for diabatic effects to be important. Through convective overturning, a superadiabatic layer quickly adjusts to adiabatic stratification, for reasons developed in Chapter 7.
Problems
163
Suggested Reading
Atmospheric Thermodynamics (1981) by Iribarne and Godson contains a thorough discussion of atmospheric statics. Problems 6.1. Calculate the discrepancy between geometric and geopotential height, exclusive of anisotropic contributions to gravity, up to 100 km. 6.2. Show that the polytropic potential temperature in Problem 2.9 is conserved under the diabatic conditions described in Sec. 6.5.2. 6.3. Use (2.27) and replace z by - I n p to relate the area circumscribed in Fig. 6.7 to the work performed by the air parcel during one thermodynamic cycle. 6.4. Approximate the thermodynamic cycle in Fig. 6.7 by one comprised of two isobaric legs and two adiabatic legs to show that (a) net work is performed during a complete cycle and (b) the work performed is proportional to the change of temperature associated with isobaric heat transfer. 6.5. Show that an atmosphere of uniform negative lapse rate need not have an upper bound. 6.6. For reasons developed in Chapter 12, large-scale horizontal motion is nearly tangential to contours of geopotential height and given approximately by the geostrophic velocity
1( o ) 8~
,
where v = (u, v) denotes the horizontal component of motion in the eastward (x) and northward (y) directions and 9 is the geopotential along an isobaric surface. Suppose that satellite measurements provide the three-dimensional distributions of temperature and of the mixing ratio of a long-lived chemical species. From those measurements, construct an algorithm to determine 3-dimensional air motion on large scales at elevations great enough to ignore variations in surface pressure. 6.7. An upper level depression, typical of extratropical cyclones during early stages of development, affects isobaric surfaces aloft but leaves the surface pressure distribution undisturbed. Consider the depression of isobaric height associated with the temperature distribution
=
--
[ ~2 -'~ (~ -- ~0)2]
T(A, 4~, ~) -- T + T cos(4~) + T' exp -
L2
9f(~),
where T, T, and T' reflect global-mean, zonal-mean, and disturbance contributions to temperature, respectively, ~ - - l n ( p / p o ) , with P0 =
164
6 Hydrostatic Equilibrium
{-cos
1000 mb, is a measure of elevation,
0 ~>~T, with ~r = --In(100/1000) reflecting the tropopause elevation. The surface pressure Ps = Po remains constant, ~b0 = 7r/4, L = 0.25 rads, and T = 250 K, T = 30 K, and T' -- 5 K. (a) Derive an expression for the 3-dimensional distribution of geopotential height. (b) Plot the vertical profiles of disturbance temperature and height in the center of the anomaly. (c) Plot the horizontal distribution of geopotential height for the midtropospheric level ~ = ~r/2, as a function of A and q~ > 0. (d) Plot a vertical section at latitude ~b0, showing the elevations ~: of isobaric surfaces as functions of A.
6.8. Under the conditions of Problem 6.7, how large must the anomalous temperature T' be for the disturbance to form a cut-off low at the midtropospheric level ~: = ~r/2? 6.9. Unlike midlatitude depressions, hurricanes are warm-core cyclones that are invariably accompanied by a trough in surface pressure. Consider a hurricane associated with the temperature distribution T(A, ~b, ~:) = T + T'(A, q~)e-'~, where T and T' reflect zonal-mean and disturbance contributions to temperature, ~: = - l n ( p / p o ) , with P0 = 1000 mb, is a measure of elevation, m
T'(A, ~ b ) - Tar
~b)exp[asrs(A, ~b)],
in which ~:,(A, 4~) = 0 . 1 exp
-
L2
,
with a -1 = ~:r/2, and ~:r = In(100/1000) reflects the surface and tropopause pressures, respectively, and L = 0.1 rads. (a) Derive an expression for the 3-dimensional distribution of geopotential height. (b) Plot the vertical profiles of disturbance temperature and height in the center of the anomaly. (c) Plot the horizontal distribution of geopotential height for the midtropospheric level ~: = ~:r/2. (d) Plot a vertical section at the equator, showing the elevations ~: of isobaric surfaces as functions of A. 6.10. Show that subadiabatic stratification requires an individual parcel moving vertically to, in general, absorb (reject) heat as it ascends (descends). 6.11. Surface analyses produced by the weather service plot pressure adjusted to the common reference level, mean sea level (MSL), so that different locations can be compared meaningfully. (a) Derive an approximate
Problems
165
expression for the pressure at MSL in terms of the local surface height Zs, surface pressure Ps, and the global-mean surface temperature T 288 K. (b) A surface analysis shows a ridge with a maximum of 1040 mb over Denver, which lies at an elevation of 5300 ft above MSL. What surface pressure was actually recorded there?
6.12. Deep convection inside a region of the tropics releases latent heat to achieve a column heating rate of 250 W m -2. (a) If, uncompensated, that heating would produce a temperature perturbation that is invariant with height up to the tropopause, calculate the implied change of thickness for the 1000- to 100-mb layer after 1 day (compare Fig. 1.30). (b) What process, in reality, compensates for the tendency of tropospheric thickness implied in part (a)?