Download Chapter 12: Atmospheric Structure - Atmospheric Sciences

Weather Observation and Analysis
John Nielsen-Gammon
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Chapter 12. ATMOSPHERIC STRUCTURE
12.1 Structure of the Troposphere and Stratosphere
The very names “troposphere” and “stratosphere” refer to the
contrast between turbulence and vertical mixing in the troposphere and the
stable, layered stratosphere. This chapter examines the basic structure of
the troposphere and stratosphere, the two lowest layers of the atmosphere,
and considers the importance of the interface between them, the
tropopause, in the structure and behavior of weather systems.
Without the Sun, the Earth would be a cold, boring ball. Sunlight
drives the temperatures and winds of the atmosphere. How much of the
typical state of the atmosphere is due directly to the effect of Sun, and how
much is due to adjustments and reactions taking place in the atmosphere?
In the stratosphere, the basic picture given by the direct effects of
the sun is rather close to reality. Quite simply, the top of the stratosphere
is warm because a relatively large amount of solar radiation gets absorbed
there.
One might think that if solar radiation is absorbed at the top of the
stratosphere, there ought to be a whole lot more solar radiation absorbed at
the bottom of the stratosphere or in the troposphere, simply because the air
is a lot denser there. Well, sure, if the same amount of radiation that
reaches the top of the stratosphere made it down into the troposphere,
there’d be a lot of absorption down low. But only the most energetic
wavelengths of radiation are subject to absorption by the gases in the
atmosphere, and once you get past the top of the stratosphere, most of
those wavelengths have already been absorbed. Because of this
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absorption, the amount of energy still to be absorbed farther down in the
stratosphere or troposphere gets smaller and smaller as the radiation
penetrates lower and lower into the atmosphere.
If absorption of solar radiation by the atmosphere were the only
important process heating the air, the temperature would be quite cold at
ground level, because there would be essentially no energy left that the air
could absorb. As it is, the warmest part of the stratosphere is at its top,
and the temperature decreases downward toward the troposphere,
becoming quite cold as low as 15 km above ground level. But lower
down, there’s something else that absorbs solar radiation: the ground.
Remember that the air at low levels is heated by coming in contact
with the ground. This heating is responsible for the relatively warm
temperatures we observe at ground level. Temperatures at the ground are
typically 50 K or more warmer than temperatures at the tropopause. So
there are two main places where heat is added to the atmosphere: at the top
of the stratosphere and at the bottom of the troposphere.
The differences in where heat is added are responsible for the
fundamental differences between the troposphere and stratosphere. In the
troposphere, the heating makes the atmosphere less stable. As you warm
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up a low layer, the rate at which the temperature decreases with height
becomes larger and larger. If left unchecked, eventually the lapse rate
would exceed the dry adiabatic lapse rate, and unstable overturning of the
affected air would result. In contrast, adding heat to the top of the
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stratosphere helps to generate and perpetuate a deep temperature
inversion, with warmer temperatures aloft, making the stratosphere very
stable.
On a sounding, the tropopause is usually detectable as a sudden
transition from temperature rapidly cooling with height to temperature
staying constant or warming with height. In order to cover all possible
circumstances, the World Meteorological Organization has arbitrarily
defined the tropopause as the base of the first deep layer above 500 mb
where the temperature cools at less than 2C/km.
12.2 The Real Tropospheric Lapse Rate
The simple description in the previous section has one critical
inconsistency. If the vertical temperature profile in the troposphere is
determined by air becoming unstable and mixing vertically, the lapse rate
in the troposphere should be dry adiabatic. Instead, we observe it to be
much stabler than dry adiabatic. There must be some additional process,
heretofore not discussed, that causes the middle and upper troposphere to
be warmer than we would otherwise expect.
One such additional process is called latent heating. If air
containing water vapor rises past its lifting condensation level, the water
vapor will condense and the air will not cool as rapidly anymore as it
ascends. While dry vertical instability would drive the atmosphere toward
the dry adiabatic lapse rate, moist convection drives the atmosphere
toward the moist adiabatic lapse rate. Indeed, in the tropics the observed
lapse rate is very close to the moist adiabatic lapse rate throughout the
troposphere. Convection, in the tropics, is the primary driving factor
controlling the overall vertical temperature structure of the troposphere.
In mid-latitudes, particularly during wintertime, we don’t observe
moist convection very often. Yet the observed lapse rate is still fairly
close to the moist adiabatic lapse rate. This turns out to be a consequence
of winter storms. Such storms tend to be fiercer if the stratification in the
troposphere is weak. The net effect of winter storms is to carry warm air
northward and upward, and cold air downward and southward.
Preferentially causing cold air to sink and warm air to rise, quite apart
from any upright instability, has the net effect of decreasing the lapse rate
and increasing the stability. So while moist convection rules in the
tropics, midlatitude cyclones (low pressure systems, not tornadoes) rule
closer to the pole.
Both effects give us a lapse rate in the lower troposphere that is
close to 5 K/km rather than 10 K/km. So while this is stable for upward
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and downward motion, it’s not very stable. Furthermore, vertical motions
are fundamental to the processes controlling the lapse rate, so such vertical
motions must be common features of the wind field in the troposphere.
All this makes the troposphere quite unlike the stratosphere.
12.3 Response to Vertical Motion
We’ve already noted that vigorous vertical motion must be more
common in the troposphere than the stratosphere. Now let’s utilize the
concepts of the previous chapter to see what vertical motion does to the
troposphere and stratosphere.
First, a review. The change in temperature at a particular level due
to upward or downward motion is a combination of two effects: vertical
temperature advection and adiabatic expansion or compression. The
quantity known as potential temperature nicely wraps both of these effects
into a single package. With potential temperature, the change in
temperature at a given level due to vertical motion is directly proportional
to the vertical derivative of potential temperature and the amount of
vertical displacement: − w(T θ ) ∂θ ∂z . If potential temperature is
uniform with height, upward and downward motion don’t change the
temperature pattern at all. If potential temperature increases with height,
downward motion will raise the temperature at a particular level and
upward motion will lower the temperature.
Graphically, it’s easy to imagine potential temperature. Think
about contours of potential temperature in a vertical section. The contours
predominantly run horizontally. The closer together the contours, the
stabler the atmosphere and the bigger the effect of vertical motion. Since
potential temperature is conserved, it is perfectly okay to imagine that the
potential temperature lines (or “isentropes”) are actually being carried
along by the air as it moves up or down. The opposite situation would be
air that is weakly stable or neutral. In that situation the isentropes would
be far apart and there could be little change of potential temperature (or
temperature, for that matter) caused by any vertical motion.
So, the only other thing you need to know is: because the
stratosphere is stable, its isentropes are tightly packed in the vertical. In
the troposphere, the isentropes are far apart and the air is much less stable.
Now imagine that there is some ascent. In the troposphere, that
ascent will cause some cooling. The pressure at the base of the ascent
would rise a bit, gradually cutting off the horizontal convergence that must
be present at low levels beneath the ascent. In the stratosphere, the ascent
would cause a much greater and more rapid cooling. The pressure would
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rise quickly, and the convergence (and associated ascent) would be easily
stifled. One could say that the stratosphere is much more resistant to
upward motion.
You’d have the same problem with descent. In the stratosphere,
the downward motion would cause rapid warming, lowering the pressure
at the base of the descending air and cutting off the horizontal divergence
that would otherwise be present.
Individual air parcels, too, suffer the same fate. Imagine an
unstable air parcel ascending through the upper troposphere. At altitudes
corresponding to the upper troposphere, there’s not much water vapor left
in any air parcel, so the lapse rate would have to pretty close to dry
adiabatic for that air parcel to still be unstable and continue rising. But
once it hits the stratosphere, by definition the surrounding air is stable, and
a deep temperature inversion is probably present. Any ascending air
parcel would quickly become much cooler than the surrounding
stratospheric air and would sink back down.
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These processes have an obvious effect during moist convection.
A classic deep thunderstorm has a broad anvil aloft and an overshooting
top. The anvil is typically near the tropopause level, because that’s where
the air ceases to be unstable and reaches its level of neutral buoyancy.
Above the anvil, over the updraft core, a few knobs of higher cloud can
often be seen. This is the overshooting top, and it is caused by air hitting
the stratosphere while ascending fairly rapidly. As the air rises farther, it
becomes much cooler than its surroundings, and proceeds to sink back
down.
The presence of an overshooting top is an excellent marker for
vigorous convection. The lower stratosphere tends to be pretty close to
isothermal no matter where you look, so the size of an overshooting top
will almost entirely depend on the strength of the updraft beneath it and
the upward momentum of the air as it reaches the stratosphere.
12.4 Convergence at the Tropopause
Not only is the tropopause the interface between the troposphere
and stratosphere, it’s also a place where the tropospheric jet stream tends
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to be strongest. In earlier chapters we noted that variations in the height
pattern along the jet stream are associated with accelerations of the jet
stream winds, and that those accelerations in turn imply ageostrophic
winds and certain patterns of convergence and divergence. We will now
consider the vertical motion response to jet-level convergence and
divergence.
For brevity we will only consider divergence. If you reverse all
the verbs and adjectives, the same arguments apply to convergence.
If we think about divergence at the tropopause as being associated
with jet accelerations, there are certain basic areas that are associated with
divergence, as discussed in Chapter 11. One is the region between a
trough and a downstream ridge. Another is to the right of the entrance to a
jet streak or to the left of the exit of a jet streak.
If there’s horizontal divergence at the tropopause, there must be
some combination of downward motion in the lower stratosphere and
upward motion in the upper troposphere. Since the stratosphere is
resistant to vertical motion, in reality most of the vertical motion must take
place in the troposphere. So divergence in the upper troposphere is almost
always associated with upward motion beneath it.
Again, because of the vertical structure of the atmosphere,
convergence at near the tropopause is mostly associated with downward
motion in the troposphere.
An even greater generalization is possible. Because the wind
patterns present at jet stream level tend to extend down at least to the
middle troposphere, we might expect that the vertical motion associated
with convergence and divergence diagnosed at the tropopause might
actually extend through most of the troposphere. And indeed it does!
Most large-scale vertical motion begins close to the ground and ends close
to the tropopause.
Across the spectrum of vertical motion, from an ascending air
parcel to large-scale ascent, the tropopause acts like a squishy lid to
vertical motion. To a first approximation, the tropopause can be
considered to be a rigid lid, a solid barrier to upward and downward
motion. All the weather action takes place in the troposphere.
12.5 Vertical Wind Shear
Is it windy in the basket of a hot air balloon?
The answer is no. A balloon has so little mass relative to its size
that it drifts along with the wind. Someone riding with the balloon feels
almost no wind whatsoever. This is true even if the wind is blowing 60
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miles per hour. The landing might be a tad rough, but while in the air the
balloon is racing along with the wind, and if you were riding with the
balloon and closed your eyes, you wouldn’t even know a breeze was
blowing.
Now consider a slightly different situation. Suppose you tie a
small helium balloon to a string and attach the string to your car’s antenna,
then take it out on the highway at 60 mph. The string of the balloon will
be nearly horizontal and the balloon will be dangling behind the antenna.
If there’s a mouse hanging on to the balloon, that mouse is feeling a strong
wind. Of course, it’s not “wind” that the mouse is feeling; the actual wind
might be calm. Instead, it’s the air rushing past the car. But the balloon
doesn’t know the difference. It’s behaving just as if the car were parked
but pointing into a 60 mph wind.
Ready for the third situation? Imagine that the small helium
balloon is now tied to the top of the large hot air balloon. Will the balloon
string be pointed straight up, or will the small balloon be trailing behind
the big balloon, just as it trailed behind the car’s antenna?
The answer is that, more than likely, the small balloon will be
straight above the big balloon. Even though the big balloon is moving at
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60 mph, so’s the air around the big balloon. If the small balloon feels that
same 60 mph wind, it will drift along at the same speed as the big balloon
and there will be nothing to keep it from pointing straight up.
The key to all these examples is the relative motion of the air. In
the case of the car, the balloon is tied to something that’s moving at a
speed 60 mph different from the air around it, whether we consider the
moving car example or the parked car example. In the case of the hot air
balloon, the helium balloon is tied to something that’s moving at the same
speed as the air around it, so there’s no extra wind pushing on any side of
the balloon to keep it from pointing straight up.
Or is it? Actually, we have only assumed that the wind at the level
of the small balloon is identical to the wind at the level of the big balloon.
In the real atmosphere, there could be a difference. The wind at one
height is not necessarily going to be equal to the wind at another height.
When there’s a difference, when the wind is blowing at a different speed,
or a different direction, or both, at one level compared to another level,
that phenomenon is called vertical wind shear, or wind shear for short.
Suppose the wind at small balloon level is weaker than the wind at
big balloon level. That situation is sort of like the car example: the small
balloon is being pulled faster than the air in which it is embedded, so it
will lag behind the big balloon. Conversely, if the wind at small balloon
level is stronger, the small balloon will be blown out in front of the big
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balloon. If the speed stays the same but the wind changes direction with
height, the small balloon will be blown to one side or the other. Anytime
the small balloon would not be pointing straight up, we have wind shear.
12.6 Computing Wind Shear from Wind Vectors
Wind shear can be thought about in two equally valid but different
and therefore contradictory ways. The first way is probably the simplest:
wind shear is just the difference between the winds at two different levels.
So if the big balloon was at a level where the wind was 60 kt, and the
small balloon above was where the wind was only 50 kt, the magnitude of
the wind shear between those two levels is 10 kt. We’ll call this quantity
the difference shear, just to give it a name.
Mathematically, if we consider some low level a and some higher
level b, the wind shear is the vector wind difference: the wind vector at
level b minus the wind vector at level a. Shear, being a vector, has both
magnitude and direction. So, for example, if the wind at big balloon
height (level a) is 60 kt from the west and at small balloon height it is 70
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kt from the west, the wind shear would be a vector of 10 kt pointing from
the west. Conversely, if the wind at small balloon height is 50 kt from the
west, the wind shear would be a vector of 10 kt pointing toward the west
rather than from it.
More complex examples might require doing the subtraction
graphically or with a calculator. Sometimes it’s easiest with wind
components. Suppose, for example, there’s mainly a change in direction.
At big balloon level, say the wind is from the west at 65 mph. That means
u = 65 mph and v = 0. Meanwhile, at small balloon level, the wind is from
the west-southwest at 65 mph, with u = 60 mph and v = 25 mph. To
compute the wind shear, subtract the components. Doing so, we get u = -5
mph and v = 25 mph, making a vector that is oriented toward the northnorthwest. Using the same terminology as for wind itself, we would say
that the vertical shear is south-southeasterly. So while big balloon is
drifting toward the east, small balloon is leaning off toward the northnorthwest.
The other, and more fundamental, meaning of vertical wind shear
is as the vertical derivative of the horizontal wind. Rather than the amount
the wind changes between two particular altitudes, it is the rate at which
the wind changes with height. We’ll call this quantity the derivative
shear.
If you’re still with me, you may be wondering, how do you take
the derivative of a vector? The answer is, you do it one component at a
time. The x component of the vertical shear is the vertical derivative of
the u component of the wind, and the y component of the vertical shear is
the vertical derivative of the y component of the wind. So
r
∂vh ∂u
∂v
=
i+
j
∂z
∂z
∂z
The subscript h signifies that only the horizontal wind is included
in wind shear. (A little bit strange-sounding, I know: the vertical shear
only involves the horizontal wind.)
In pressure coordinates, there’s a similar definition of the wind
shear; we’ll add a minus sign so that the shear vector corresponds to the
change in wind in the upward direction, whatever the coordinate system:
r
∂vh ∂u
∂v
=
i+
j
∂p ∂p
∂p
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To evaluate the derivative form of vertical shear with real data,
whether in height coordinates or pressure coordinates, you’d have to
determine the values of u and v at two different levels, subtract the higher
from the lower, and then divide by the vertical distance between the two
levels to get the x and y components of the shear. But notice that our
example above (with the 65 mph wind) does all that except divide by the
vertical distance. This, then, is the difference between the first definition
of vertical shear and the second definition: whether or not you divide by
the vertical interval.
This brings up a related point. When expressing the shear as the
difference in wind between two levels, the answer is relatively
meaningless unless everyone knows which two levels are being used in
the computation. On the other hand, the derivative version of shear is
valid at a single level, just like a horizontal derivative is valid at a single
point on a graph. So it makes sense to talk about the 1000 to 500 mb
vertical shear on one hand, and to talk about the vertical shear at 500 mb
on the other. The former would be the difference shear, and the latter
would be the derivative shear.
12.7 The Mathematics of Thermal Wind
The concept of “thermal wind” is perhaps the most alien of
concepts considered in this course. It requires sophisticated spatial
reasoning, advanced mathematical reasoning, or both. Some are better at
spatial reasoning than others, but the necessary spatial powers can be
acquired through repeated critical examination of weather maps.
Similarly, extensive instruction in calculus supplies the relevant
mathematical background, but most will be able to follow the logical
reasoning immediately.
Either way, thermal wind is the key to understanding the
atmosphere as a complete unit rather than a set of discrete, seemingly
independent levels. It is key to looking at one map and instantly knowing
what must be happening at other levels. It is key to looking at a sounding
from one location and knowing what will happen there in the next 24
hours. Finally, it is a reality check: if how you imagine the atmosphere to
work is inconsistent with thermal wind, your imagination is wrong.
Despite its name, thermal wind, while a vector, is not a true wind.
That’s why I’ve put it within quotation marks occasionally. Instead, it is a
vertical wind shear, representing the change of horizontal wind with
height. More specifically, it is the geostrophic vertical wind shear, the rate
of change of the imaginary geostrophic wind with height. As we’ve seen,
on normal weather-map scales, the wind is close to geostrophic balance,
so the true vertical wind shear will be close to the geostrophic vertical
wind shear too. So the thermal wind is not a wind that blows. Instead, it
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is a wind shear, the sort of thing that makes a small helium balloon tilt
away from vertical when suspended from a large hot-air balloon.
The thermal wind, being just the geostrophic vertical wind shear,
can be written in pressure coordinates as
−
r
∂v g
∂p
=−
∂u g
∂p
i−
∂v g
∂p
j
The thing about the thermal wind is not what it is, but what it
relates to. Geostrophic wind, of course, is directly related to the horizontal
pressure gradient by definition as
r
g
vg = 0 k × ∇h Z
f
To get the vertical shear of the geostrophic wind, we’ll
differentiate both sides with respect to pressure and throw in our minus
sign:
−
r
∂v g
∂p
=−
g0 ∂
(k × ∇ h Z ) = g 0 k × ∇ h ⎛⎜⎜ − ∂Z ⎞⎟⎟
f ∂p
f
⎝ ∂p ⎠
Okay, so the geostrophic wind shear is proportional to the vertical
variation of the horizontal pressure gradient force, or its equivalent in
height coordinates. So?
It turns out (and this is key) that the right hand side of that equation
can be simplified into something much more meaningful. I have taken the
liberty of rearranging the order of differentiation on the right hand side of
the previous equation to relate it directly to the hypsometric equation,
which relates thickness to temperature:
−
∂Z Rd T
=
pg
∂p
Now follow this. If the vertical variation of height with respect to
pressure is proportional to temperature, then the vertical variation of the
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horizontal height gradient is proportional to the horizontal temperature
gradient. Let’s say that again:
If
Vertical variation of height
is proportional to
temperature
Then
Vertical variation of height gradient
is proportional to
temperature gradient.
Taking the horizontal gradient of both sides of the previous
equation,
⎛ ∂Z ⎞ Rd
∇ h ⎜⎜ −
∇ hT
⎟⎟ =
⎝ ∂p ⎠ pg
and k-crossing that equation gives us
⎛ ∂Z ⎞ Rd
k × ∇ h ⎜⎜ −
k × ∇ hT
⎟⎟ =
⎝ ∂p ⎠ pg
And since (according to what was said a few paragraphs ago) the
vertical variation of the horizontal pressure gradient is proportional to the
geostrophic vertical wind shear, that must mean that the horizontal
temperature gradient must also be proportional to the geostrophic wind
shear. One more time: the horizontal temperature gradient is proportional
to the geostrophic vertical wind shear. Combining our altered
hypsometric equation with the equation for geostrophic wind shear gives:
−
r
∂v g
∂p
=
Rd
k × ∇ hT
pf
So, for one more time: the geostrophic vertical wind shear is
proportional to the horizontal temperature gradient.
Now that you’ve made it this far, go back and read this section
again. Slowly. And make sure that you understand each step.
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12.8 Interpreting Thermal Wind
One nice thing about the thermal wind equation is it is similar to
the geostrophic wind equation, so the same techniques for estimating
geostrophic wind from height contours apply to estimating geostrophic
wind shear (thermal wind) from temperature contours (isotherms).
Compare the equation for thermal wind with the equation for geostrophic
wind:
−
r
∂v g
∂ ln p
=
Rd
k × ∇ hT
pf
r
g
vg = 0 k × ∇h Z
f
Aside from a few constants, they are identical: geostrophic wind
shear is to horizontal temperature gradients as the geostrophic wind itself
is to horizontal height gradients. Consider the example below.
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This figure is a horizontal map, so north is to the top of the page
and east is to the right. If the contours are height contours, the vectors
would represent the geostrophic wind. If the contours are temperature
contours, the vectors would represent the thermal wind, that is, the
geostrophic vertical wind shear. Either way, the relationship between the
contours and the direction and magnitude of the vectors is the same.
Where the contours are close together, the gradient is strong, so the wind
(if they’re height contours) or wind shear (if they’re temperature contures)
must be strong too.
The implications of the concept of thermal wind are substantial.
Because pressure is strongly tied to large-scale winds through geostrophic
balance, and pressure is also strongly tied to temperature variations
through hydrostatic balance, the winds and temperature must also be
strongly tied. Specifically, they’re connected through the thermal wind.
Wherever there’s a strong temperature gradient, the geostrophic wind
vector must be changing rapidly with height.
Take the typical midlatitude wintertime situation of weak winds
near the ground and strong westerlies at jet stream level. The vertical
wind shear is the strong westerly vector minus the weak vector, so the
wind shear is also westerly and strong. Just as westerly geostrophic winds
imply lower heights to the north, westerly geostrophic vertical wind shear
implies lower temperatures to the north. Indeed, if you have weak winds
near the ground and strong winds at jet stream, the strong winds aloft will
always be nearly parallel to the mid-tropospheric isotherms, with colder
temperatures to the left if you are facing downwind.
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Note that this technique for inferring the temperature distribution
only works so easily if the winds aloft are really strong or the winds near
the ground are really weak. Otherwise, you’d have to be careful and
subtract the two vectors to determine the magnitude and direction of the
vertical wind shear through the troposphere, and therefore the magnitude
and direction of the mean temperature gradient through the troposphere.
12.9 Warm and Cold Cores
Another useful application of thermal wind is in understanding the
vertical structure of high and low pressure systems. For simplicity, let’s
consider only situations in which the wind direction is constant with
height, so there is no directional shear, only speed shear. In other words,
the shear vector is parallel to the wind vector at both levels.
Suppose you have a low. According to thermal wind, the colder
temperatures should be to the left of the wind shear. If the wind speed
increases with height, the wind shear is in the same direction as the wind
itself, so the colder temperatures must be to the left of the wind, that is,
near the center of the low. Conversely, if the wind speed decreases with
height, the wind shear is pointed in the direction opposite the wind itself,
so colder temperatures would be to the right of the wind and warmer
temperatures would be to the left near the center of the low.
The first of these is called a ‘cold-core low’, and the second is a
‘warm-core low’. A typical upper-level cutoff low pressure system is a
cold-core low, while a hurricane is a warm-core low. We see from this
that the fact that hurricanes are strongest near the ground is directly related
to the fact that hurricanes are warmest in the eye. Conversely, cold-core
lows are strongest aloft, since (as we noted) wind speeds are stronger aloft.
The same principles apply to highs. An anticyclonic circulation
that increases in strength upward must have cold temperatures to the left
of the wind, outside the anticyclone, and warm temperatures to the right of
the wind, inside the anticyclone. Conversely, an anticyclone that weakens
with height has cold temperatures in its core.
Warm and cold highs and lows are best visualized in vertical
sections. The next diagram shows the height of various pressure surfaces
in a slice through a warm-core low. Since this is a vertical section, the
strength of the wind is (emphatically) not related to the spacing of the
pressure surface. Instead, the slope of the pressure surfaces is proportional
to the strength of the geostrophic wind. The spacing between the pressure
surfaces is the thickness, and a larger spacing corresponds to warmer
temperatures. So in this figure, we have warmer temperatures near the
center, and therefore the slope of the pressure surfaces (and the strength of
the geostrophic wind) decreases upward.
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The 250 mb map would have a weak low, with heights barely
lower inside the low than outside. The 500 mb map would have a
somewhat stronger low, and would also feature warm temperature within
the low. The 850 mb map would have the strongest low, and strongest
winds, of all.
By now you’re probably so indoctrinated to geostrophic wind that
you’re looking at the diagram above and envisioning geostrophic winds
going from west to east. If so, you’re wrong. Geostrophic winds are only
parallel to height contours on a constant pressure map, and this is a
vertical section. Instead, in a vertical section, the spacing of pressure
surfaces is directly related to the temperature distribution.
Now you can infer geostrophic winds from this vertical section,
but perhaps not in the way you might expect. Remember that the
geostrophic wind is proportional to the height gradient, or the slope of
constant pressure surfaces. The vertical section above shows a part of that
slope: the component of the height gradient in the direction of x.
According to the geostrophic wind relations, the stronger slope
corresponds to stronger geostrophic wind; one of the stronger winds in the
preceding figure is directly above the label x.
What about direction? Since the geostrophic wind (in the northern
hemisphere) blows with higher heights to the right, we can look at the
diagram and see that to the right of the low, the geostrophic wind must be
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blowing into the page, and to the left of the low, the geostrophic wind
must be blowing out of the page.
You can use diagrams such as this to determine the temperature
pattern associated with any arbitrary height pattern. Start with a sketch of
a couple of constant pressure surfaces. Slope them strongly where you
want strong winds, slope them gradually where you want weak winds.
Now look at how far apart your two lines are. Where they are close
together, the thickness (and therefore the temperature) is low; where they
are far apart the thickness (and therefore the temperature) is high.
12.10 Temperature Advection from Sounding and Profiler
Data
We’ve already looked at soundings and wind profiler output as
providing essential information about the vertical structure of the
atmosphere. There’s also some horizontal structure information in there
too: if the wind is close to geostrophic balance, one can look at the winds
and determine which way the pressure gradient or height gradient points.
Now, with thermal wind, we can look at the vertical variation of
wind in a sounding or profiler plot and determine the wind shear and
therefore the horizontal temperature gradient. If westerly winds increase
with height, for example, there must be colder temperatures to the north
and warmer temperatures to the south.
Suppose we take the case of directional shear. Suppose at one
level, say 850 mb, the wind is from the northwest at 30 kt, and at a higher
level, say 700 mb, the wind is from the southwest at 30 kt. Whip out a
pencil and paper to verify that the 850 to 700 mb wind shear is a vector
from the south at 42 kt. If the wind is in approximate geostrophic balance,
the colder temperatures must be to the west and the warmer temperature
must be to the east.
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We also must have cold advection. Note that at both 850 mb and
700 mb, the wind has a component from the west, from colder air toward
warmer air. Temperature advection is proportional to the temperature
gradient multiplied by the component of wind parallel to that temperature
gradient. So the stronger the westerly component of wind and the stronger
the vertical wind shear, the stronger the temperature advection.
In this particular example, the wind “backed” with height. That is,
with increasing altitude, the wind vector rotated counterclockwise. (Note
that we’re not saying anything about the horizontal wind pattern, just the
vertical variation of the wind vector itself.) It turns out that whenever the
wind backs with height, to the extent that geostrophic balance holds, there
must be cold advection, and whenever the wind veers with height (rotates
clockwise), there must be warm advection. You can verify this for
yourself by sketching a few examples.
This was a prime forecasting tool in the days before the
widespread real-time exchange of weather observations. The most up-todate information a forecaster has is the observation that forecaster makes.
If the forecaster checks the clouds and notices a significant difference
between the wind direction at ground level and the wind direction aloft,
beyond what friction alone would produce, he or she can predict with
confidence an impending change in temperature.
And although it’s a bit off-topic, I must interject a few words about
severe weather. Meteorologists know that supercells and tornadoes
require low-level wind shear. On TV, the conditions favoring such
supercells are often referred to as “a clash of hot and cold air masses”, or
some such verbiage. Well, it’s true that you need warm and cold air
masses next to each other to get a strong temperature gradient and thus
strong wind shear. But the two air masses don’t need to be battling each
other; the mere fact of them sitting next to each other is good enough.
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12.11 Recap
The vertical wind shear is the vertical variation of the horizontal
wind. It can be expressed in units of wind itself, by subtracting the wind
vector at a higher level from the wind vector at a lower level, or in units of
wind speed per unit height (or unit pressure), obtained by computing the
derivatives of each of the wind components.
The thermal wind, despite its name, is not properly a wind. Rather,
it is defined as the geostrophic vertical wind shear. The “thermal” part of
the name comes in because the geostrophic vertical wind shear magnitude
is proportional to the horizontal temperature gradient, and the shear vector
is oriented parallel to the isotherms with colder temperatures 90 degrees to
the left of the direction toward which the shear vector points. Wherever
the wind is nearly in geostrophic balance, the thermal wind will nearly
correspond to the actual vertical wind shear.
From these basic facts one can conclude that beneath the jet stream
should tend to be a strong temperature gradient, since the vertical wind
shear tends to be strongest there. The colder temperatures will be below
and to the left of the jet stream, while the warmer temperatures will be
below and to the right of the jet stream. At the jet stream level itself,
temperature gradients will be weak because the wind is neither increasing
with height (that happens below jet stream level) nor decreasing with
height (that happens above jet stream level).
Closed circulations whose intensity varies with height imply
temperature variations as well, since the intensity changes mean there
must be vertical wind shear. Specifically, a cold-core circulation will be
stronger aloft if it’s a low and be weaker aloft if it’s a high, and a warmcore circulation will be stronger aloft if it’s a high and be weaker aloft if
it’s a low. A very cold core might have a high at the ground and a low
aloft.
The vertical wind shear can also be used to diagnose temperature
advection. Cold advection implies winds backing with height, while warm
advection implies winds veering with height.
Questions
1. On several sounding diagrams, identify the location of the
tropopause.
2. Suppose the lowest 3 km of the atmosphere has convergence at
the rate of 3x10-5 s-1. Compute the rate of change of temperature at
heights of 1 km, 2 km, and 3 km due to the resulting vertical motion if the
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atmosphere is (a) weakly stratified ( ∂T ∂z = -8 C/km) and (b) strongly
stratified ( ∂T ∂z = 0 C/km).
3. Given two weather maps, one with winds on the 500 mb surface
and one with winds on the 850 mb surface, sketch a map with vectors
depicting the difference shear between the two levels. The difference
shear in this case is defined as the 500 mb wind minus the 850 mb wind.
4. Given a map of temperatures at 700 mb, sketch the vector field
that represents the geostrophic vertical wind shear at that level.
5. If your answers to (3) and (4) are for the same time, compare
your two vector fields? How well do the patterns of direction and relative
magnitudes agree? What reasons are there for them to not match exactly?
6. Using a map of jet stream winds at 250 mb and the knowledge
that winds at the ground are likely to be much weaker, determine the areas
where the troposphere is relatively warm and where it is relatively cool.
Look at some maps of temperature within the troposphere and evaluate
how well the 250 mb wind field allowed you to estimate the overall
temperature pattern.
7. Sketch imaginary horizontal maps of 850 mb height and 500 mb
height that would be associated with (a) a cold core high and (b) a cold
core low.
8. A sounding shows winds from the east in the lower troposphere
and winds from the west in the upper troposphere. What is the
temperature advection? Explain.
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