Download Changes in Thickness and Vertical Motions

GENERAL CIRCULATION OF THE ATMOSPHERE
Now that you have become acquainted with radiation and heat, let us examine
how these processes act to produce the general motions of the atmosphere.
You must surely be familiar with the general geometry and motions of the earth
and sun. Despite some notable historical attempts by organized religion to suppress
the truth, the relative motions of the earth and sun are now well known. They lead to the
observed spatial and temporal distributions of radiation balance at the planet surface.
The result is a distinct decrease in received solar radiation with increasing
latitude. Recall that net radiation is energy received minus energy emitted. Figure 10
illustrates the approximate distribution of received solar and net emitted infrared versus
latitude. This assumes a spatial average over an entire year for each latitude band.
Average Radiation Absorbed and Emitted vs Latitude
Absorbed
Solar
Radiation
Surplus
Emitted
Infrared
Radiation
Energy
Deficit
Deficit
90
40
0
40
90
Latitude
Figure 10. The absorbed and emitted radiation vs. latitude
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Notice that:
1.
In the tropics, solar gains are greater than infrared losses. There exists a
surplus of energy or positive net radiation.
2.
The middle and high latitudes lose more infrared than they receive from
solar. This deficit results in a net loss of radiation.
The cross over points lie at about 38 N and 38 S. So a majority of the Earth’s
surface suffers a net loss of radiation, while the tropics are the continued beneficiary of
a surplus. If no other processes were available to mediate this situation, the tropics
would grow incredibly hot while the rest of the planet would experience extreme cold.
Of course, this does not occur. Such a gradient in temperature cannot exist, as heat will
flow in response. There must exist mechanisms to transport heat poleward from tropical
regions. In fact, such transport is conducted through both atmospheric and oceanic
processes. Let's consider the atmosphere first.
Idealized Atmospheric Motions For a Non-rotating Earth:
It is best to start with a simple situation, even if it isn't realistic. So assume for a
moment that the Earth did not rotate. Of course the unequal distribution of radiation is
still present.
Energy Balance and Transport in the Tropics
Consider the tropics where a large surplus of radiation is available. What
happens to this free energy? Earlier it was noted that most net radiation was
consumed to evaporate water over the globe. So most available energy in the tropics is
used to evaporate water from the land and oceans (mainly oceans). Some of the
energy is also absorbed by the surface waters, and used to heat them. The energy
used in evaporation is not lost, but where does it go? It is stored and resides in a
potential state in the water vapor itself. Since it is effectively hidden in such a state, it is
called latent energy or latent heat. When water vapor condenses back into liquid, the
same value of energy is released as heat.
Since such a large value of energy is required to evaporate water, the process of
converting radiation to latent heat is very important to climate. The main processes
involved in the energy balance of the surface in the tropical oceans are depicted below.
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Tropics
Large
Radiation
Values
Rnet
Evaporate Water
Heat Surface
Moist and
Unstable Air
Evaporation
Heat
The warm and moist air near the surface is very unstable, leading to rising
motion. The low density of the lower atmosphere in these regions is not only related to
the warm temperatures, but also to the high water vapor contents. Water vapor is less
dense than dry air. So adding water vapor to air reduces the density, and promotes
vertical motion.
We can now look at the mechanisms that actually transfer energy upward and
poleward in the tropics. Figure 11 is an illustration of the main transport processes in
the tropics. The warm, evaporating ocean surface results in a vertical movement of very
moist air. As this air rises, it expands (due to decreasing pressure) and cools. As the
air cools, the maximum amount of water vapor that could be present in the air rapidly
reduces. Recall that the maximum or saturation vapor pressure of air is a function only
of temperature.
Eventually the rising air cools to a value where condensation begins, forming
clouds. As condensation occurs, the heat energy absorbed during evaporation is now
released, warming the air and causing further vertical motion. The radiation energy that
was initially diverted into latent heat and resided in the water vapor is now released as
heat. Eventually the air dries out, and diverges aloft towards the poles.
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Clouds
Release
of Latent
Heat
Warm, Moist
Air Rises
Intertropical Convergence Zone -- ITCZ
Figure 11. Transport of water vapor and energy from tropical waters
It is critical to note that heat has been transported vertically by two mechanisms:


sensible heat flow from the warm water to the air
vertical transport of `latent' heat of evaporation, later released in the clouds
The latter process is by far the most important. The amount of radiant energy at the
surface is greatest in the tropics, and most of it is diverted into latent heat. Conversion
of radiant energy into evaporation of water and subsequent transport, is the most
important energy conduit that nature employs to drive the climate system. Latent
heat and its transport is the most critical process that defines the climate of the planet.
Since the atmosphere is a continuous fluid, rising motions at the surface require
a transport of air from the surroundings to replace the vertically moving air. This is
required to conserve mass. The only way that this can happen is to have a flow into the
region from the sides. As a result, a zone of converging air must exist in this region of
rising air. This is called the Intertropical Convergence Zone (ITCZ). It is characterized
by copious amounts of cloudiness and rainfall.
As the rising air reaches the top of the troposphere it diverges horizontally, and
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moves poleward. It gradually cools by radiation loss, and subsequently sinks back to
the surface at latitudes of about 30N and 30S. This defines a set of Hadley Cells. Of
course the latitudinal position of these cells varies with time of year. The ITCZ will
generally follow the location of maximum radiation. Figure 12 illustrates the structure of
the Hadley Cells.
Radiational
Cooling
Hadley Cells
Radiational
Cooling
Cold and Dry Air
Latent Heat
Released
Clouds
Warm and
Dry Air
Warm and
Dry Air
Moist
Air
~ 30 N
ITCZ
~ 30 S
Figure 12. The circulation and processes defining the Hadley Cells
Now let's consider the poles, which suffer large radiation losses. The resulting
cold, dense air slides equatorward. This flow must meet the poleward surface flow from
the descending limbs of the Hadley Cells. As a result, two other cells are defined in
each hemisphere, the Polar and Ferrel Cells. Figure 13 depicts this idealized
circulation for a non-rotating earth. Although this simple view helps illustrate some of
the important forces driving atmospheric motion, in fact this is not what is observed!
Things are complicated by the rotation of the planet.
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Idealized Circulation for Non-rotating Earth
Polar Cell
Ferrel Cell
Hadley Cell
ICE
ITCZ
Figure 13. Simplified circulation if Earth did not rotate
SPECIAL TOPIC III.
Temperature Change of Rising Air
As air rises in the atmosphere, the pressure decreases. This reduction in
pressure results in expansion of the air parcel. The kinetic energy of the air molecules is
being spread into a larger volume. Hence, it makes sense that the temperature of this
expanding air will decrease. For the moment let us consider unsaturated air. Using
basic thermodynamic properties of air, we can express the rate of change of
temperature in terms of the fundamental variables.
The 1st Law of Thermodynamics can be expressed for the atmosphere as:
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dq = c p dT - dP
where q is heat flow into or out of the parcel, T is temperature, P is pressure,  is
specific volume, which is 1/density or 1/, and cp is specific heat capacity of air.
Generally a rising parcel of air can be assumed to have essentially no heat exchange
with the surroundings. This is called an adiabatic process. In this case dq = 0. We can
also substitute for  or 1/ using the Gas Law defined in Special Topic I. This leads to:
0 = c p dT - RT
dP
P
dividing both sides of the above expression by T yields:
cp
dT
dP
=R
T
P
Recall from Special Topic I. that the atmosphere is hydrostatic, meaning that pressure
is defined by the weight of the air column above any point. This was expressed as:
dp = -  g dz
This can be substituted for dP in the earlier expression, and the Gas Law used to
substitute for  as follows:
cp
dT
R (  gdz)
=
T
P
P
=
RT
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c p dT = - gdz
dT
g
=dz
cp
So we now have a relationship for changes of temperature with height for a rising parcel
of air that has not reached saturation. This turns out to be about -10 C / km, and is
called the dry adiabatic lapse rate.
As the air rises, the amount of water that can exist in state of water vapor will
also decrease. This is because the amount of water vapor present under saturated
conditions rises rapidly with temperature. This value is usually expressed as the
saturation vapor pressure. The reduction in temperature for rising air means a
simultaneous reduction in the saturation vapor pressure. Of course, the actual water
vapor content or vapor pressure of the air is constant during this process. At some
height the saturation vapor pressure, which is reducing due to the decrease in
temperature, will reach the actual vapor pressure. The air is now saturated, and
condensation of water vapor into liquid water will begin.
During condensation, latent heat energy is now released. This is the same
energy that was stored in the water vapor during evaporation. As a result, rising air that
is saturated, cools more slowly than rising air that is unsaturated. Since the amount of
condensation will slowly diminish with further rising motion, the actual rate of cooling will
vary with height. This is termed the moist adiabatic lapse rate. This rate is not
constant with height, since the rate of condensation changes with height. Eventually,
when all the water vapor has condensed into liquid, the rate of cooling returns to the dry
adiabatic lapse rate.
Effects of Earth Rotation:
Reality or Hallucination? A Question of Reference Frames
In order to interpret positions and motions in space, we always employ some
frame of reference, whether we realize it or not. There are two main classes of
reference frames. An inertial frame is absolute and might be considered fixed and
unchanging. The background of deep space might be considered as an example. A
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non-inertial frame is relative and not fixed. It may be moving and changing constantly.
For obvious reasons the reference frame of choice for most macroscopic
processes is the surface of our rotating planet. This is of course a moving reference
frame, which presents difficulties in interpreting motions within such a system. Because
of this curved and moving reference frame, motions in the atmosphere will appear to
experience accelerations and changes in direction. Motions that actually occur in a
straight line in absolute space, will appear to be deflected in our relative space.
The above phenomenon is called the Coriolis Effect. The name honors Gustav
Gaspard de Coriolis, who first laid out a mathematical solution for moving reference
frames in the early 19th century.
The Coriolis Effect is not a real physical force. It is an apparent acceleration or
deflection due to our insistence upon using a moving frame of reference. The most
important properties of this effect are:
1.
It is 3-dimensional in nature. However, the horizontal components are
much larger than the vertical ones.
2
The apparent acceleration is small in actual value. Hence, the effect is
only important for motions that occur over long time periods. In other
words, it is critical for large-scale motions.
3.
The magnitude of the apparent acceleration varies with latitude. It is zero
at the equator and increases with the sin of the latitude. Hence, it has a
maximum value at the poles.
4.
The apparent deflection is always to the right in the Northern Hemisphere,
and to the left in the Southern Hemisphere.
So motions in tropical regions will experience small deflections by the Coriolis
Effect, while those of the middle and high latitudes will appear to be deflected to a
much greater extent. The above rules allow one to qualitatively understand the effects
of Coriolis on motions over the planet. A more quantitative understanding will require
some attention to the mathematical expressions. These are discussed briefly in the
following section.
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SPECIAL TOPIC IV.
The Coriolis Effect
The mathematical solution to the moving reference frame problem of concern
here turns out to be fairly simple. The apparent acceleration vector due to the rotating
Earth is given by:
2 x V
where  is the Earth rotation vector, and V is the velocity vector of the wind. This
describes the 3-dimensional acceleration applied by the Coriolis effect. It is always
operating at right angles to the direction of motion.
When the above cross product is carried out and the magnitude of each of the
terms is examined, it turns out that the vertical component as well as terms involving
the vertical wind are small. Hence, only the horizontal components are of interest here.
The horizontal component is given as:
2 sin 
where  is the rotation rate of the Earth, or 7.27 x 10 -5 s-1, and  is latitude. The
acceleration at right angles to the motion is calculated by simply multiplying the above
term by the appropriate horizontal velocity component.
2 sin  . u
2 sin  . v
Where u is the x or east-west component of velocity, and v is the y or north-south
component of velocity. The top expression represents the acceleration in the y direction
imposed on air moving in the x direction. The lower expression represents acceleration
in the x direction imposed upon air moving in the y direction. In the N. Hemisphere, the
acceleration will always act to the right, while in the S. Hemisphere the acceleration
always acts to the left of the direction of motion. Please note that the above
expressions are accelerations.
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As an example, consider air moving northward at 10 m s-1, at 45N. The above
expression would result in a calculated acceleration of about 10 -3 m s-2 acting to the
right or eastward. Hence, over time the moving air will appear to be deflected eastward
from the original path.
In order for the trajectory of the air to be deflected any substantial amount, a
significant amount of time will be required. This is because the actual value of the
acceleration is so small, that it must operate for a long enough time period to
appreciably alter the direction of the wind.
Clearly the acceleration term is zero at the equator and maximum at the pole. In
the middle and upper latitudes the size of the term is around 10 -4 s-1. This is a very
small value of acceleration, which means that it will be important only for large-scale
motions that persist for long periods of time.
Resulting Average Surface Circulations:
By applying the properties of the Coriolis Effect to the motions depicted in our
simplistic circulation model of Figure 13, the mean surface circulation for a rotating
planet can be depicted. The results are illustrated in Figure 14.
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Polar Easterlies
Westerlies
Subtropical High
Subtropical High
Trade Winds
ITCZ
Subtropical High
Subtropical High
Figure 14. Resulting mean flows for the rotating Earth
We can examine the results for the tropics and middle and high latitudes
separately.
Tropics
Near the equator the Coriolis effect is small, and gently deflects the flows
converging on the ITCZ. They become the NE and SE Trade Winds. Note that there is
still convergence of flow at the ITCZ.
The zones of subsiding warm and dry air at about 30 N and 30 S induce an arid
region of high pressure. Because of thermal contrasts between continents and oceans,
the subsiding air does not define a broad region of high pressure. Instead fairly distinct
cells of high pressure are observed. These are called subtropical highs. Their locations
are fairly predictable, and they move with the seasons. The dry and subsiding air
associated with these subtropical highs, produces very warm and arid climates. Indeed,
most of the major deserts on the Earth are located in these regions.
Middle and High Latitudes
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When one moves to the middle and high latitudes, the Coriolis effect increases
significantly. When we apply this to the idealized surface winds, the deflections produce
zones of middle latitude westerlies and polar easterlies, as shown in Figure 14. There
are two points to be made here:
1.
These are mean or temporal average circulations. The actual winds at any
moment in time could differ from these.
2.
The mean flows are parallel to one another. Hence, the cold air in polar regions
flows parallel to the warm air from the subtropical regions.
The 2nd observation is very significant. It indicates that there is no direct
transport of heat between the subtropical and polar regions by the mean flow. Yet the
observed temperatures on the planet indicate that a substantial poleward heat transport
must be occurring in the middle and high latitudes. We will discuss oceans later, but
they cannot account for this much transport. So there must be a different mechanism
for meridional heat exchange in these regions.
Horizontal Temperature Gradients and Jetstreams:
Because of the east-west nature of the flows in the middle and polar latitudes the
cold and warm air are not mixed by the mean flow. Hence, there must exist a zone of
very large horizontal temperature gradient in each hemisphere. In order to understand
how these temperature gradients affect the winds, we must first look at the forces that
govern wind.
The unequal heating of the Earth surface produces horizontal variations in
pressure. The resulting horizontal pressure gradient imposes acceleration, causing the
wind to blow initially towards the low pressure. Of course, the Coriolis effect will induce
a deflection on the moving air. Figure 15 illustrates the balance of forces on the air. As
the air moves towards lower pressure it increases in velocity in response to the
acceleration from the pressure gradient. Recall that the Coriolis effect is proportional to
the velocity. Hence, the magnitude of the Coriolis effect increases, causing a further
increase in the deflection, until it eventually increases to a value equal to the
acceleration caused by the pressure gradient. There is a then balance between the two.
The resulting wind blows parallel to lines of constant pressure. This is called the
geostrophic wind. This wind is often observed in the middle and upper troposphere,
when the flow is not greatly curved.
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Geostrophic Wind
Balance Between Pressure Gradient and Coriolis Acceleration
V
P
L
P
V
C
P
P
V
V
C
C
H
C
Figure 15. The evolution towards the balance of forces for the Geostrophic Wind
Since the pressure field at the surface often has centers of high and low
pressure, then the wind will often follow a curved or even circular path. Thus, we must
consider another factor that is present when motions are along curved paths. Any
circular motion, requires that a centripetal acceleration act inward to maintain the
circular path. Hence, in addition to pressure gradient and Coriolis effect, we must
include centripetal acceleration for flow around centers of high and low pressure.
The resulting wind is called the gradient wind. The balance of forces in this
case is shown in Figure 16 and 17. Note that centripetal acceleration is always directed
inward by definition. However, pressure gradient is directed outward near a high
pressure and inward near a low pressure. Coriolis of course is always to the right of the
direction of motion. The centripetal acceleration can be viewed as the imbalance
between the pressure gradient and Coriolis terms.
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Gradient Wind Around Low Pressure
Coriolis Acceleration does not balance Pressure
Gradient and Centripetal Acceleration
V
Ce
L
P
C
Figure 16. Gradient wind around low pressure
Gradient Wind Around High Pressure
Pressure Gradient does not balance Centripetal
Acceleration and Coriolis Acceleration
Ce
H
P
C
V
Figure 17. Gradient wind around high pressure
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______________________________
SPECIAL TOPIC V.
Geostrophic and Gradient Winds:
The geostrophic wind consists of a balance between the Coriolis effect and the
pressure gradient, which can be expressed as:
1 P
 x
1 P
f ug =  y
f vg =
where f is the Coriolis acceleration or 2 sin ,  is density, P is pressure, and ug and
vg are the x and y components of the geostrophic wind.
Note that the wind in the x direction is defined by the pressure gradient in the y
direction. This is because the Coriolis effect acts perpendicular to the wind.
If we consider the flow around pressure centers, the centripetal acceleration
must be considered. This term acts inward to hold the motion in a circle. The
appropriate expression for the balance of forces are now given as:
2
V h 1 P
f Vh+ =
r
 n
40
where Vh is the horizontal wind vector, r is the radius of curvature of the circle, and n is
the direction perpendicular to the flow.
______________________________
Horizontal Temperature Gradient and Thermal Wind:
We need to consider the relationship between horizontal pressure gradients with
height under the presence of horizontal temperature gradients. First let us look at a
simple case where there are no horizontal variations in temperature. This is illustrated
in Figure 18. Note that the pressure surfaces are parallel to each other as well as the
surface.
Change of Pressure With Height When
Horizontal Temperature is Constant
P4 = 200 mb
P3 = 400 mb
Height
P2 = 600 mb
P1 = 800 mb
T = constant
P0 = 1000 mb
Horizontal Distance
Figure 18. Pressure surfaces vs. height for uniform surface temperature
When there is a gradient of temperature in the horizontal, the horizontal gradient
of pressure and its variation with height are altered. An example is provided in Figure
19. Notice that the pressure lines become slanted with height, indicating a horizontal
gradient of pressure. Also note the increase in the slope of the lines with height. The
magnitude of the horizontal pressure gradient is now increasing with height.
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P4
P3
Height
P2
P1
Cold
Warm
Horizontal Distance
Figure 19. Pressure surfaces vs. height under horizontal temperature gradient
Recall that warmer air is less dense than cold air. Hence, the vertical thickness
of atmosphere between any two pressure levels will be greater for a warmer air column.
It requires a greater depth of low density air to provide the necessary mass between
two pressure levels. The figure illustrates that the height of a given pressure level
increases above warmer air, and decreases above the more dense colder air. So cold
air is associated with a thinner atmosphere, while warm air is associated with a
thicker atmosphere.
So an increase in horizontal pressure gradient with elevation results from the
horizontal temperature gradient. The slope of the lines of constant pressure increases
with height. This indicates that the horizontal pressure gradient is increasing with
height above a temperature gradient. To understand this you must recall that the
pressure decreases exponentially with height. As a result, the density of air also
reduces rapidly with height. So as one goes higher in the atmosphere, the change in
height for a given drop in pressure increases rapidly. As a result, the difference in
thickness between two pressure levels for air columns of different density increases
with height.
If pressure gradient increases with height, then so must the wind velocities. So
the net result of a horizontal temperature gradient is to induce an increase in the
geostrophic wind with height. This phenomenon is called the thermal wind. Clearly the
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greater the horizontal temperature gradient, the greater the increase of winds with
height. Such an atmosphere, characterized by horizontal gradients in temperature and
density, is termed baroclinic.
The horizontal temperature gradient is greatest in the middle latitudes, where it is
cold air from the poles moves adjacent to the warm air from the subtropics. This
gradient in temperature produces a belt of westerlies in the upper troposphere. At the
actual interface between warm and cold air masses, the gradient of temperature is
especially large, as shown in Figure 20.
P4
P3
Height
P2
P1
Cold
Warm
Horizontal Distance
Figure 20. Pressure surfaces over narrow horizontal temperature gradient
Over this sharp zone of temperature contrast the thermal wind process
produces narrow zones of very high velocity winds within the westerlies in the upper
troposphere. These are called jetstreams, and they exist in each hemisphere.
Jetstreams and sharp horizontal temperature gradients in the middle latitudes are
mutually arising features. That is to say, the existence of one defines the existence of
the other.
The upper air westerlies and embedded jetstreams move over each hemisphere
in wave-like patterns. When the flow is mainly oriented east-west it is called zonal flow.
In this case there is little wave structure to the flow. On the other hand, when there are
large waves present, much of the flow is oriented north-south or parallel to the meridian
lines. Hence, this is referred to as meridional flow. These waveforms are denoted by
the terms troughs and ridges. Figure 21 shows examples of a trough and a ridge.
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Upper Air Flow
Ridge
Trough
Figure 21. Illustration of trough and ridge in upper air flow
When the waves shown in Figure 21 are rather large in size, they are referred to
as Rossby Waves. There may be a variable number of these waves across a
hemisphere. The air flows in the direction indicated in these upper air winds. But the
waves also move from west to east. So the troughs and ridges will move eastward at a
phase velocity which is different from the velocity of air moving along the wave. The
speed is variable, but in general the larger the wavelength the smaller rate of
movement. Of course, the size and structure of the waves often changes as they move.
It is a very dynamic system.
Another way to view the flow of the upper air is to consider the spatial variation of
height for a given pressure level. In other words at what height in the atmosphere a
given pressure is reached. When this value is plotted for each location, the resulting
map is that of a height field. The higher values in the field represent regions of the
atmosphere where a large height value or thickness is required to reach a certain
pressure level. It turns out that the flow of air is parallel to the lines of constant height,
or contours. In fact, the velocity of the air will be proportional to the strength of the
gradient of the height field. It is completely analogous to the relationship between
horizontal pressure gradient and winds.
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Linkage of Jetstreams to Surface Pressure Pattern
The patterns of the upper air westerlies such as illustrated in Figure 21 have a
connection to the pressure fields and winds of the surface. How can wave-like patterns
in the upper air flow connect with surface pressure systems? There are several
processes that explain the connections. We can treat one of the processes in a
somewhat descriptive fashion, while the other process directly applies mathematics and
physics.
Changes in Thickness and Vertical Motions
First consider the eastward movement of a trough in the upper air flow, and a
location upwind of this trough. As the trough approaches the height of any pressure
surface is decreasing. Thus, the depth of atmosphere is lowering with time at this
location. The atmosphere is getting thinner. If this is true, then the atmosphere must
also be getting more dense and colder.
If the atmosphere ahead of the advancing trough is getting colder, there must be
a process or mechanism at work to reduce the temperature. Given that we have already
shown that the atmosphere obeys the hydrostatic and adiabatic approximations, there
is only one possible process that could result in cooling the atmosphere in this case.
This process is rising motion. So the lowering heights of pressure surfaces with time are
associated with upward vertical motions of the air.
Vorticity and Vertical Motions
Another approach to examine the connections between upper air waves and
vertical motions involves looking at changes in the curvature of the wind. The rotation of
a fluid is expressed by a mathematical property called vorticity. A rotation in a
counterclockwise direction is defined as positive. We call this flow cyclonic in the
Northern Hemisphere, since it is the same direction as flow around a low pressure.
Conversely, a rotation in a clockwise direction is defined as negative. This flow is called
anticyclonic.
Since the Earth spins, it imparts a natural rotation to air. This means the Coriolis
Effect will result in the presence of vorticity. The amount will depend upon the sin of the
latitude. The direction of rotation caused by Earth rotation will be counterclockwise in
the Northern Hemisphere, and it will induce a positive vorticity. So moving air always
has a background value of positive vorticity at all locations in the Northern Hemisphere
save the equator. Any rotations that exist due to the curvature of the flow, will either add
or subtract from this background value.
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SPECIAL TOPIC V.
The mathematical expression for rotation of a fluid is given by:
   V
where V is the velocity vector. The usual notation for the x, y, and z components of wind
are u, v, and w. So the above can be expanded as:
iˆ
ˆj
kˆ

x
u

y
v

z
w
v 
 u
w 
 
 w
 v
u 
  
 iˆ   
 ˆj    kˆ

y

z

z

x
  x y 

 
As we will be using vorticity to explain why air horizontally diverges and converges, the
term of most interest is the horizontal rotation around a vertical axis. This is the last
term in the above expression. So we are left with;

v u

x y
This contribution to vorticity by the gradients in wind is called relative vorticity.
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Vorticity
Measure of the spin or rotation of a fluid
We are interested in the rotation around vertical axis
Anticyclonic or
Negative Vorticity
Cyclonic or Positive
Vorticity
The diagram below shows how horizontal changes of u and v can indeed induce a
rotation. The change in v with x is positive, while u is decreasing with y, and is negative.
Since the second term is subtracted from the first, they add to produce positive relative
vorticity.
Y
Induces Positive Vorticity
u decreasing with y
v increasing with x
X
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The vorticity can be separated into the effects of the Earth’s rotation and the
relative vorticity due to the gradients in the wind. The sum of these is called the
absolute vorticity. Recall that the vorticity induced by the Coriolis in the Northern
Hemisphere is always positive, and increases with the latitude. The relative vorticity can
be either zero, positive, or negative, depending on the degree of curvature of the flow.
Now let us return to an upper air flow with a trough and a ridge. As the air flows
over the top of a ridge the relative vorticity reaches a maximum negative value. So the
absolute vorticity is at the lowest value at that point. As the air moves towards the
trough, the relative vorticity is increasing from negative to positive. This means the
absolute vorticity is increasing.
As the air flows around the bottom of the trough, the relative vorticity reaches the
maximum positive value, and absolute vorticity reaches its largest value. As the air
moves ahead of the trough, the relative vorticity is decreasing as is the value of
absolute vorticity. So ahead of a ridge the rotation is increasing. However, ahead of a
trough the rotation is decreasing. This is illustrated in Figure 22.
Negative
Vorticity
Absolute
Vorticity
Decreasing
Divergence
Convergence
Absolute
Vorticity
Increasing
Positive
Vorticity
Figure 22.
Changes in vorticity and associated divergence and convergence
Notice that there is a zone of convergence ahead of the ridge, and divergence
ahead of the trough. How are these connected to vorticity changes? This can be
discussed in a descriptive way by recalling the principle of conservation of angular
momentum. This states that the product of the angular velocity and the radius squared
is conserved during rotation.
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Ahead of the ridge the vorticity or rotation rate is increasing, which implies the
radius of curvature of the flow must decrease. In other words, the flow must be
converging in order to spin faster. So convergence is observed ahead of the ridge. The
opposite effect is observed ahead of the trough. Here the vorticity is decreasing, and
the rate of rotation is slowing. The radius must be growing larger, which means the air
is spreading out or diverging. So divergence is observed in front of the trough.
In the Southern Hemisphere the same physical processes apply, but the signs
change. The Coriolis parameter is negative, so the vorticity values produced by the
Earth’s rotation are negative. For example, the structure we define as a ridge, when
present in the upper air flow in the Southern Hemisphere is physically the same as a
trough in the Northern Hemisphere. Since cold air now comes from the south, the
center of a ridge is actually an indication of cold air and low heights. At the ridge center
the background vorticity and relative vorticity are both negative. Ahead of the ridge to
the east, the relative vorticity gradually increases to positive values at the trough. So the
absolute vorticity value is changing from very negative to less negative. This means that
the total rotation is growing smaller, and divergence should be observed in the upper air
flow ahead of the ridge. So the physics are the same in the Southern Hemisphere as
they must be. But the change in sign of the Coriolis and opposite meaning of critical
flow patterns, require careful attention.
Increasing Vorticity Values -- Relate to Convergence
Decreasing Vorticity Values -- Relate to Divergence
As area or radius gets smaller due to
convergence -- rotation rate increases
Convergence
So how do the divergence and convergence in the upper air lead to vertical
motion? If air is converging aloft the collision will force air outwards. There must be a
flow of air directed downwards. Conversely, if the air is diverging aloft, then it must be
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replaced by air rising from below. So we now have vertical motions resulting from the
changes in rotation in the upper air. Note that it is changes in vorticity with distance that
cause divergence and convergence aloft, which induces vertical motions. Another way
to express this is that the advection or transport of vorticity towards a region,
causes vertical motions.
So waves in the upper air induce vertical motions by two processes:


Changes in thickness with time
Transport or advection of vorticity
These are the processes by which waves in the upper-air flow cause vertical motions.
Figure 23 shows the relationship between the resulting vertical motions and surface
pressure.
Convergence
Upper Air
Divergence
High Pressure
Divergence
Convergence
Low Pressure
Surface
Figure 23. Connections between vertical motions and surface pressures
The rising and sinking air will induce low and high pressures at the surface. The
resulting pressure gradients will often cause the formation of a center of surface high
pressure in front of a ridge, and center of surface high pressure in front of a trough.
Let us look at the situation in advance of an upper air trough. The resulting
gradient winds near the surface will blow counterclockwise around the low pressure
system. Considering for a moment the N. Hemisphere, this means that warm air will be
moved northward, while cold air will move southward. A cold front is likely to form at the
southern boundary of the cold air mass, and a warm front at the northern boundary of
the warm air mass. This is depicted in Figure 24.
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Cold
Initial
Formation
of Low
Low
Upper Air Flow
Developed
Low
Pressure
System
Figure 24.
L
Surface
Warm
Overunning
Clouds-Precip
Latent Heat
Relationship between trough and developing low pressure system
It should be clear that the wave-like behavior of the upper air in the vicinity of the
jetstream, is crucial to vertical motions surface pressure systems. The key issues that
we have discussed in the last few sections can be summarized as follows:
Divergence ahead of upper air trough induces surface
convergence and low pressure -- rising motion
Divergence aloft related to curvature of flow or vorticity
Rising motion also related to atmosphere becoming
thinner and colder
Upper air ridge causes convergence, which leads to
surface divergence and high pressure -- sinking motion
Related to curvature or vorticity changes in upper air
flow
Sinking motion also related to atmosphere becoming
thicker and warmer
The surface weather near the low will be unsettled, with precipitation and strong
winds common due to the rising motion associated with low pressure and the collisions
of cold and warm air. The condensation of water vapor into liquid water occurring during
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the forced rise of air masses, releases additional latent heat. This latent heat release is
very substantial, and provides further energy for the storms. Hence, latent heat plays a
significant role in energy transport, even in the middle latitudes.
These are baroclinic storms, which are the necessary result of strong horizontal
temperature gradients. As the trough in the jetstream moves eastward, so does the
surface low and the associated flow pattern and weather. Of course, the lifetime of the
upper air feature is limited to a matter of days, since the upper air wave pattern is
always changing. Hence, these systems result in a transient movement of heat
towards the pole.
If we consider the situation upstream of an advancing ridge in the jetstream, the
opposite effects will be observed. Now heights will be rising, implying that the
atmosphere must be growing thicker and less dense. Such warming of the atmosphere
is associated with subsiding air, leading to higher surface pressure. The surface winds
will then flow clockwise around the high pressure. This circulation also moves cold air
south, and warm air north.
Importance to Global Heat Transport
So, low pressure disturbances or baroclinic storms, and high pressure centers,
both caused by the waves in the upper air flow and jetstream induce transient
meridional flows of heat. These flows are very transient, but they are large! In addition,
a substantial part of the heat transport is in the form of latent heat released during the
storms. This is the indirect mechanism that transfers heat poleward in the middle
latitudes.
Note that this has been a very simplified and qualitative approach to these
processes. We have only considered horizontal temperature gradients, and not
discussed the angular momentum imparted to the atmosphere by the rotating planet.
This momentum is transported poleward, and plays a role in formation of jetstreams.
But that topic is beyond our scope in this class.
We have considered what is normally called the polar front or polar jetstream.
This is associated with the largest horizontal temperature gradient that exists along the
boundary between the cold polar air and warmer air from the subtropics. In reality, there
is sometimes a second jetstream. The subtropical jetstream exists just poleward of
the Hadley cells. It is much weaker than the polar jet, but still induces baroclinic
disturbances and transports heat meridionally.
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