2.6.2 Adiabatic Lapse Rate and Stability
Meteorologists use the concept of air parcel as a body of air sufficiently small to be defined by a single state (p, T), yet sufficiently large to preserve its identity during transport over some distance of interest. Applying the laws of thermodynamics to such an idealized air parcel gives valuable insight into atmospheric motions. The temperature of an air parcel changes as its pressure changes. A less variable measure of the heat content of an air parcel is the potential temperature θ [K]:
(2.26)
where p0 = 1000 hPa is the reference pressure, κ = R/cp = 0.286, and cp = 1005 J K–1 kg–1 is the specific heat of dry air at constant pressure. The potential temperature is the temperature that an air parcel (p, T) would reach if it were brought adiabatically (without external input or loss of energy) to the reference pressure p0. Under adiabatic conditions we have dθ/dz = 0. One can show from a simple thermodynamic cycle analysis that under these adiabatic conditions the temperature must decrease linearly with altitude
(2.27)
where Γd is called the dry adiabatic lapse rate. “Dry” refers to an air parcel sub-saturated with respect to water vapor; in the case of a saturated air parcel, latent heat release/loss during cloud condensation/evaporation complicates the analysis. The case of a saturated air parcel is discussed next.
An atmosphere left to evolve without exchanging energy with its surroundings will eventually achieve an adiabatic lapse rate due to the motion of air parcels up and down. Input or output of energy will force the actual lapse rate Γ = –dT/dz to differ from the adiabatic lapse rate. In the stratosphere, for example, absorption of solar radiation by ozone causes the temperature to increase with altitude, a situation called a temperature inversion. The value of Γ relative to Γd diagnoses the stability of an air parcel relative to vertical motions (Figure 2.6).
Figure 2.6 Schematic representation of the vertical temperature profile corresponding to a dry adiabatic lapse rate (9.8 K km–1 in black) and two hypothetical actual lapse rates (stable conditions in blue and unstable conditions in red). If an air parcel moves upwards under adiabatic conditions (green arrow), it will experience a buoyancy force that opposes the displacement if the temperature of the parcel T′ is lower than the temperature T of the local environment (T′ < T). In this case, the buoyancy force will restore the air parcel to its original position (stable conditions). If T′ > T, the buoyancy force reinforces the displacement and drives the parcel further away from its original position (unstable conditions).
Consider an air parcel located initially at an altitude z in an atmosphere with temperature T(z) and lapse rate Γ. Let us apply an elemental push upward to this air parcel so that its altitude increases by δz. This motion takes place adiabatically so that the new temperature of the air parcel is T′(z + δz) = T(z) – Γd δz. The temperature of the surrounding air at that altitude is T(z + δz) = T(z) – Γδz. If Γ > Γd, then the air parcel at z + δz is warmer and hence lighter than the surrounding air; it is therefore accelerated upward by buoyancy, amplifying the initial upward displacement. A similar reasoning can be made if the air parcel is initially pushed downward; at altitude z – δz it will be colder than its surroundings and buoyancy will accelerate the downward motion. Thus an atmosphere with Γ > Γd is said to be unstable with respect to vertical motions. Rapid convective vertical mixing takes place in such an atmosphere. If on the contrary the lapse rate is smaller than the adiabatic lapse rate such that Γ < Γd, an air parcel displaced adiabatically toward higher altitude will become colder and denser than the surrounding air. As a result, it will return to its original level; the atmosphere is said to be stable. If Γ = Γd, the air parcel will continue its upward or downward motion with no acceleration, and the atmosphere is said to be neutral.
If air is saturated with water vapor, the stability conditions are modified: Ascending motion results in water condensation, which releases heat within the air parcel even under the adiabatic assumption. Similarly, in such an atmosphere, downward motion results in water evaporation and hence internal cooling. There results a decrease in stability. The lapse rate for saturated air parcels moving adiabatically up or down, called the wet (or moist or saturated) adiabatic lapse rate Γw, can be derived from the energy balance equation and the Clausius–Clapeyron equation
(2.28)
Γw is smaller than the dry adiabatic lapse rate Γd. Its value depends on the water vapor condensation rate, which is determined by the water vapor mass mixing ratio rw under the saturated conditions of the air parcel. Since rw under these conditions is a strong function of temperature (Clausius–Clapeyron equation), it follows that Γw depends strongly on temperature. It typically ranges from 2 to 8 K km–1. Under saturated conditions, stability requires that Γ < Γw. Buoyant motions in clouds occur when Γ > Γw and are referred to as wet convection. The atmosphere is said to be conditionally unstable if Γw < Γ < Γd. Such an atmosphere is stable unless sufficient water vapor is supplied to it to make it saturated, in which case it becomes unstable (Figure 2.7).
Figure 2.7 Effect of atmospheric humidity on the static stability of the atmosphere. The solid lines represent the dry (Γd) and wet (Γw) adiabatic lapse rates. As the actual lapse rate increases, the atmosphere evolves from a state of absolute stability to a state of conditional stability (shaded region for which saturated parcels are unstable but unsaturated parcels are not) and to a state of absolute instability.
Unstable conditions in the atmosphere can be triggered by solar heating of the ground. The heat is communicated by conduction from the ground to the overlying atmosphere, leading to Γ > Γd. Under such conditions, rapid vertical motions maintain an effective adiabatic lapse rate for the atmosphere, so that Γ > Γd is not practically observed. In fact, observation of Γ = Γd is generally a reliable diagnostic of an unstable atmosphere; the unstable lapse rate continually re-adjusts to Γd through the motion of air parcels up and down. Conversely, cooling of the ground at night produces a stable atmosphere (Γ < Γd). Particularly stable conditions are encountered when the temperature increases with height and produces a temperature inversion. In the troposphere, such an inversion often occurs during compressional heating associated with large-scale descent of air (a process called subsidence) or when the ground is particularly cold. In the stratosphere, the temperature increases with height due to the absorption of solar UV radiation by ozone. Thus the whole stratosphere is characterized by an inversion.
Buoyant convection as described above is the principal driver for vertical transport of trace constituents in the troposphere. Because it is driven by local temperature gradients, it occurs at scales too small to be resolved by regional or global atmospheric models. It therefore needs to be parameterized by using the model-scale information on temperature gradients and water vapor to estimate the resulting model-scale vertical motions. Such convective parameterizations rely on approximation of the actual physics and often include empirical or adjustable coefficients to better reproduce observations. They are crucial to the representation of vertical motions and cloud formation in atmospheric models. See Chapter 8 for further discussion.
2.7 Geostrophic Balance
We now turn to the forces driving horizontal motions in the atmosphere. Horizontal pressure gradients resulting from differential heating produce motions directed from high- to low-pressure areas. A complication is that the Earth is a rotating sphere, where different points have different translational velocities in a fixed frame of reference. The useful frame of reference for us is one that rotates with the sphere, since we measure all air motions with respect to this frame of reference. From the perspective of this rotating frame of reference, any motion taking place in the fixed frame of reference (such as driven by a pressure-gradient force) will be deflected due to the rotation. The deflection accelerates the air parcel away from its original direction and thus behaves as a fictitious force, called the Coriolis force. The Coriolis force operates in three dimensions but is negligibl
e in the vertical relative to the acceleration of gravity. It is of critical importance for large-scale motions in the horizontal direction.
To understand the Coriolis effect, consider that the translational Earth’s rotation velocity VT (directed eastward)
VT = a Ω cos φ
(2.29)
decreases with increasing latitude φ. With an Earth’s radius a = 6378 km and an angular rotation velocity Ω = 7.292 × 10–5 rad s–1 (or 2 π rad d–1), the velocity VT is 1672 km h–1 at the Equator and 836 km h–1 at 60° latitude. If we consider an air parcel that is displaced poleward in the northern hemisphere, starting from latitude φ1, the conservation of angular momentum in the absence of external forces requires that the product ρa V(φ) a cos φ remain constant at its initial value ρa VT(φ1) a cos φ1 during the displacement of the parcel. Here, V is the absolute eastward velocity of the air parcel in the fixed frame of reference, and ρa is the air density. Since VT(φ) decreases with latitude φ, this condition can only be fulfilled if, for an observer located at the Earth’s surface, the air parcel acquires a gradually increasing eastward velocity. For the same reason, an air parcel moving toward the Equator in the northern hemisphere will be displaced westward (see Figure 2.8).
Figure 2.8 Trajectory of an object (such as an air parcel) directed from the north pole toward the Equator at 90° W. (a) Case of a non-rotating planet. (b) Deflection of the trajectory toward the right due to the rotation of the Earth. The arrival point at the Equator is displaced to the west of the original target point.
Reproduced with permission from Lutgens et al. (2013). Copyright © Pearson Education, Inc.
The same Coriolis effect also applies to motions in the longitudinal direction. In that case it can be understood in terms of the centrifugal force exerted on air parcels in the rotating frame of reference of the Earth. An air parcel at rest at a given latitude is subject to a centrifugal acceleration that would make it drift toward the Equator were it not for the oblate geometry of the Earth (Figure 2.9). The resultant force of gravity (oriented toward the center of the Earth) and reaction (oriented normal to the surface) exactly cancels the centrifugal force, as shown in Figure 2.9. This should not be surprising considering that the oblate geometry is actually a consequence of the centrifugal force applied to the solid Earth. Consider now an eastward motion applied to the air parcel so that V > VT. This motion increases the centrifugal force and deflects the air parcel equatorward. Conversely, a westward motion with V < VT weakens the centrifugal force and deflects the air parcel poleward. In both cases the deflection is to the right in the northern hemisphere and to the left in the southern hemisphere.
Figure 2.9 Equilibrium of forces for an air parcel at rest in the frame of reference of the rotating Earth. The centrifugal force is directed away from the axis of rotation, gravity is directed toward the center of the Earth, and the reaction force is perpendicular to the surface. The centrifugal force would cause the air parcel to drift toward the Equator if the Earth were a perfect sphere. The oblate geometry of the Earth (greatly exaggerated for the purpose of this figure) results in equilibrium in the triangle of forces.
Reproduced from Jacob (1999).
In summary, for an observer on the rotating Earth, air parcels moving horizontally are subject to a Coriolis force that is perpendicular to the direction of motion and proportional to the parcel’s velocity; this force deflects air parcels to the right in the northern hemisphere and to the left in the southern hemisphere. It can be shown that the corresponding Coriolis acceleration is
(2.30)
where v represents the velocity vector in the rotating frame of reference and Ω is the Earth angular velocity vector directed from the south to the north pole. When expressed in Cartesian coordinates and considering the zonal and meridional wind components (u, v), the Coriolis acceleration becomes
(2.31)
(2.32)
where f = 2 Ω sin φ is the Coriolis parameter. It is positive in the northern hemisphere and negative in the southern hemisphere. Its amplitude increases with latitude. Thus, the Coriolis acceleration, which is zero at the Equator, increases with latitude and with the velocity of the flow. Its effect is substantial for large-scale motions (~1000 km, the synoptic scale).
For an observer attached to the rotating Earth, the large-scale motions in the extratropical atmosphere can be represented by a balance between the Coriolis and the pressure-gradient forces, called the geostrophic approximation:
(2.33)
or in a Cartesian projection (x and y being the geometric distances in the zonal and meridional directions, respectively)
(2.34)
(2.35)
From (2.33) we see that the geostrophic motions on a horizontal surface are parallel to the isobars (lines of constant pressure). In the northern hemisphere (f > 0), air parcels rotate clockwise around high pressure (anti-cyclonic) cells, and counter-clockwise around low pressure (cyclonic) cells (see Figure 2.10). The situation is reversed in the southern hemisphere (f < 0).
Figure 2.10 Flow of air in the northern hemisphere between anti-cyclonic (high) and cyclonic (low) regions. The motion originally directed from the high- to low-pressure cells is deflected to the right by the Coriolis force.
When formulated using pressure rather than geometric altitude as the vertical coordinate, the geostrophic balance takes the form:
(2.36)
(2.37)
where Φ is the geopotential. Thus, on isobaric surfaces, the geostrophic motions follow the contours of the geopotential fields. Replacing yields the thermal wind equations
(2.38)
(2.39)
These show that the vertical shear in the horizontal (constant pressure level) wind field is proportional to the horizontal temperature gradient. In both hemispheres, the zonal wind component u increases with height when temperature decreases with latitude and decreases with height when temperature increases with latitude (Figure 2.11). The strong decrease of temperature with latitude in the troposphere produces intense subtropical jet streams, seen in Figure 2.11 as westerly wind maxima centered at about 40° latitude and 10 km altitude.
Figure 2.11 Zonal mean temperature (a) and zonal wind velocity (b) as a function of latitude and altitude for January, from the COSPAR International Reference Atmosphere (CIRA).
Reprinted with permission from Shepherd (2003), Copyright © American Chemical Society.
Near the surface, the geostrophic flow is modified by friction resulting from the loss of momentum as the flow encounters obstacles (vegetation, ocean waves, buildings, etc.). The friction force is directed in the direction opposite to the flow (slowdown of the wind), effectively weakening the Coriolis force. This deflects the flow toward areas of low pressure (or low geopotential areas on isobaric surfaces), as shown in Figure 2.12.
Figure 2.12 Winds around low- and high-pressure cells in the northern hemisphere. Geostrophic balance dominates aloft and the flow is directed along isobaric lines. Near the surface, friction deflects the flow toward low pressure.
Reproduced from Ahrens (2000), Copyright © Cengage Learning EMEA.
2.8 Barotropic and Baroclinic Atmospheres
A barotropic atmosphere is one in which changes in air density are driven solely by changes in pressure. It is a good approximation in the tropics, where horizontal temperature gradients are small. In a barotropic atmosphere, isobaric (uniform pressure) surfaces coincide with isopycnic (uniform air density) surfaces. From the ideal gas law, they must also coincide with isothermal (uniform temperature) and isentropic (uniform potential temperature) surfaces. Since there is no temperature gradient on isobaric surfaces, the geostrophic wind is independent of height (see (2.38) and (2.39)). Under adiabatic conditions (dθ/dt = 0), air parcels remain on isentropic surfaces, and since no pressure gradient exists along these surfaces to drive atmospheric motions, no potential energy is available for conversion into kinetic energy.
Outside the tropics, where meridional temperature gradients are large (Fi
gure 2.11), the temperature varies along the isobars, and the atmosphere is said to be baroclinic. Isobars and isentropes do not coincide. In this case, pressure gradients can drive adiabatic displacement along isentropic surfaces. Conversion of potential energy into kinetic energy becomes possible. Temperature gradients along isobars cause vertical shear in the geostrophic wind (see thermal wind equation), leading to a strong jet stream in the upper troposphere as discussed in Section 2.7. The axis of the jet stream is located in the 30°–60° latitudinal band characterized by a pronounced meridional temperature gradient separating cold and dense air of polar origin from warmer, less dense tropical air. In the presence of strong velocity shears, the jet stream may be unstable with respect to small perturbations, and disturbances may amplify, producing the so-called baroclinic instability.
Figure 2.13 illustrates baroclinic instability. The meridional gradient in temperature causes the isentropic surfaces (isentropes) to slope upward with increasing latitude. A poleward motion at constant altitude or with an upward slope shallower than the isentropes produces an unstable atmosphere even though the isentropes imply a vertically stable atmosphere (∂θ/∂z >
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