Conservation of Angular Momentum,
Vorticity, and Divergence: An Overview of Topics and
Teaching Methods.
Meteorology 280 Project
Jeff Gawrych
May 2004
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I Introduction
Several concepts in meteorology are so important, that they often become
the focus of study in both synoptic and dynamic meteorology. Among these crucial
topics are the conservation of angular momentum, vorticity, and divergence. These
topics underlie the basic movements and actions of the atmosphere, and as a result
become weather forecasting tools. That is, meteorologists quantify the atmosphere, form
analytic expressions, and solve for such variables as wind, temperature, moisture, and
pressure. It seems obvious that these would be the topics covered in all meteorology
courses, but that is not the case.
When teaching meteorology, one must first examine whom one is speaking to,
and adjust one’s instruction style accordingly. Quantitative analysis is the standard
teaching method for major students, who possess the science background to
mathematically solve the equations that describe the atmospheric motions. In this
method, the math comes first, so that when later doing qualitative analysis, the proof is in
the equations. Take a highly simplified case as an example, if x+2=0, the algebraic
solution is –2. Qualitatively, this can be explained by “if I have two apples, but then two
are taken away, how many are left?” General education (GE) or other non-major
students often do not have the math skills to take a quantitative approach, so the “normal”
teaching method is often ineffective. For the sake of argument, all non-meteorology
students will be referred to as GE students. In most cases, some fundamental aspects
explained by mathematical expressions are omitted altogether. The result is that GE
students taking a meteorology survey course miss out on the fundamentals of atmospheric
circulation.
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II Conservation of Angular Momentum
The Conservation of Angular Momentum (COAM) explains many basic principles
of rotating objects. Since we live on a rotating planet, COAM’s effects are critical in
meteorology. Angular momentum is constantly being transported to and from the
atmosphere, and its effects are noticed in many weather phenomena such as jet streams,
extra-tropical cyclones, hurricanes and tornadoes. Mid-latitude westerlies and tropical
easterlies seen in the general circulation also owe their existence to the COAM. These
topics will be addressed, but first it is essential to review some basic topics.
Physics gives us several fundamental conservation laws. 1. Conservation of
Energy: states energy cannot be created nor destroyed. 2. Conservation of Mass: states
mass cannot be created nor destroyed. 3. Conservation of Linear Momentum: states
linear momentum cannot be created nor destroyed.
The fourth fundamental pertaining
to fluid dynamics is the Conservation of Angular Momentum (COAM). This
conservation law is not as well known, although we experience it every day. One of the
reasons angular momentum is important is because we live on a rotating planet. The
equation: L = mvr, where L is the angular momentum, m is the mass of the small object,
v is the magnitude of its velocity, and r is the separation between the objects. This
formula explains that for a fixed angular inertia, the radius r and the angular velocity are
inversely proportional. Assume a certain radial distribution of angular velocity as an
initial condition. Any change in this distribution must obey COAM. By altering the
initial conditions, the radius and velocity change accordingly. For a constant mass, if the
velocity increases, the radius must decrease, and vice-versa.
Another way to view COAM is to picture the earth from space. One can see it
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rotating about its axis at some angular velocity. Because the earth has mass and it rotates,
it therefore contains some amount of angular momentum. The entire earth system
(includes atmosphere and solid earth) must conserve angular momentum (L) unless an
additional outside torque is applied to the system. Again, L = mvr = constant.
In meteorology, angular momentum is broken down into components, most
importantly atmospheric, oceanic, and solid body. These three components must balance
each other out so that the total system (earth) conserves angular momentum.
Consider the simple situation where the total system AM has only two components:
earth and atmosphere.
L system = L atm + L earth, = constant.
This means that L earth is inversely proportional to L atm. Any changes in mass, velocity,
or radial distance will cause changes somewhere else, to satisfy COAM.
The winds in the tropics are easterly, so the atmosphere gains AM (L earth < L atm).
Conversely, the atmosphere gives up AM in the mid-latitudes where surface winds are
westerly (flow in the opposite direction of the rotating planet). So, the atmosphere
conserves AM by balancing westerly and easterly winds. If they were not in balance,
Learth would have to adjust, leading to a longer or shorter length of day.
COAM in the general circulation
The general circulation depends heavily on COAM, since angular momentum
depends on the distance form this axis of rotation and the rotational velocity. At the
equator, the distance, r from the axis of rotation is a maximum, whereas r is at a
minimum value at the poles. COAM explains why a single Hadley cell circulation is not
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physical.
In a single-cell model, the rising/falling branches of the Hadley cell are at the
equator/poles, due to more/less solar insolation. Upper level winds move from the
equator to both poles, but to conserve AM, as r decreases, v must increase. This would
create extremely large westerly winds that are dynamically impossible. To conserve
angular momentum, a 3-cell approach is now used consisting of a Hadley cell, a Ferrel
cell, and a Polar cell.
COAM in hurricanes
COAM is seen in hurricanes, where the strongest winds are noticed just outside the
eye. As an air parcel spirals inward towards the center of a hurricane, the radius of
curvature decreases, so therefore the velocity increases, and higher wind speeds are
noticed. Angular momentum tells us that the maximum wind speeds should be noticed
where the radius is the smallest. Conversely, weaker winds are found further away from
the center of the hurricane where the radius is much larger. For rotating bodies, if their
radii decrease they must spin faster in order to conserve angular momentum. The same
theory is applicable to tornadoes as well.
The Coriolis effect
The Coriolis effect is a consequence of the principal of conservation of angular
momentum. The Coriolis force is an apparent force that only acts when air is moving. For
example, take a particle of air at 30° S is rotating west to east with the earth’s surface at a
tangential velocity of about 1450 km/hour. If that particle of air starts to move towards
the equator the conservation principle requires that the particle continue to rotate
eastward at 1450 km/hour even though the rotational speed of the earth’s surface below it
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is accelerating as the particle closes with the equator, which is rotating at 1670 km/hour.
Thus a fluid moving towards the equator is deflected westward relative to the earth’s
surface, but not deflected relative to space. Conversely, air moving from low latitudes,
with high rotational speed and momentum, is deflected eastward, i.e. as a westerly wind,
when moving to higher latitudes with lower rotational speeds.
Table 1: Tangential eastward velocity at the earth’s surface
Equator
1670 km/hour
464 meters/sec
15° North
1613 km/hour
448 meters/sec
30° North
1446 km/hour
402 meters/sec
45° North
1181 km/hour
328 meters/sec
60° North
835 km/hour
232 meters/sec
75° North
432 km/hour
120 meters/sec
90° North
0 km/hour
0 meters/sec
Addition of Torque:
Mountain torques arise from differences in pressure and friction and their
interaction with variations in topography. Generally, AM oscillations occur often where
mid-latitude jet streams interact with topography (e.g. polar jet over the Rocky
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mountains). Lott et al (year?) found that mountain-induced torques can anticipate the
beginnings of weather regimes, as well as the breakup of large-scale systems.
During an El Nino year, the southern oscillation kicks in and the tropical
easterlies become more westerly. This oscillation essentially speeds up the atmosphere,
so to conserve AM; the earth adjusts by slowing down (rotating slower), so the length–ofday increases. Tsing-Chang et al (year?) found that AM anomalies are greatest during
warm ENSO years (winds are westerly), and weakest during cold ENSO years.
Schmiit-Hubsch et al (year) found that length of day variations caused by
variations in atmospheric and oceanic AM. Seasonal AM variations (100-500 days) are
due to global winds and pressure fields, and sub-seasonal variations (5-150 days) are
mainly due to ocean tides.
Seasonal variation of AM that involves the distribution of winds and pressure
fields has also been measured. Studies show that there is a measurable difference in
length of day from season to season. Most notably, the length of day increases in the
Northern Hemisphere winter where 1) stronger westerlies blow due to factors such as a
stronger north-south temperature gradient (dT/dy), 2) more topography creates more
mountain torque 3) an increases of mass of the atmosphere at lower latitudes.
COAM also shows up in several other areas. The Chandler wobble is the change
in the spin of the earth on its axis. Seitz et al (year?) found that even slight atmospheric
or oceanic forcing can excite and intensify a Chandler Wobble. The Madden-Julian
oscillation, the oscillation of surface and upper-level winds in the tropics. Large-scale
eddies forced by tropical convection were found to play a dominant role in forcing the
AAM response during northern winter (November-March). (Winkler, 2001) The
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movement of ocean currents also plays a pivotal role in COAM. Ocean currents are to
the earth as winds are to the atmosphere, and therefore transport angular momentum
around the globe.
Overall, the addition of torques cause angular momentum to increase in some
areas and decrease in other areas. A balance of angular momentum must always exist
between the atmosphere, oceans, and the solid earth, and any change to one component
triggers a change in the other.
III Teaching about the Conservation of Angular Momentum
To GE students, explain qualitatively first. Lectures should begin with explaining
how the planet rotates at some angular velocity  around its axis. The most common
example illustrating this law is ice skaters and angular momentum. This concept is
familiar intuitively to the ice skater who spins faster when arms are drawn in, and slower
when arms are extended; although most ice skaters don't think about it explicitly, this
method of spin control is nothing but an invocation of the law of angular momentum
conservation. Another example is watching water go down the sink. The water starts
slowly but speeds up, as it gets closer to the drain. Also, bring it a yo-yo to illustrate that
as one shortens the string, the yo-yo’s velocity increases. From these examples, it is
obvious that the principle of angular momentum relates to all fields, ranging from the
science to the arts.
Laboratory experiments can involve looking at pressure maps. Have them
diagnose wind speeds in a certain location given a mass to work with. They should find
that exciting weather phenomena such as tornadoes and hurricanes must conserve angular
momentum. Also, students can learn about COAM from the general circulation. Easterly
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winds dominate in the tropics, where the atmosphere gains angular momentum from the
earth (i.e. where angular momentum of atmosphere < angular momentum of earth)
The atmosphere gives up angular momentum in the mid-latitudes where surface winds
are westerly. Students can point this out and thus prove that there is a net poleward
transport of angular momentum w/in atmosphere.. Students should also notice how
easterly and westerly winds tend to balance each other out to conserve angular
momentum
IV Vorticity
Mathematically, vorticity is the curl of the velocity vector; qualitatively it is a
measure of spin. Either way, this is a difficult topic to conceptualize, because of its
intangible quality. The equation in Cartesian coordinates is ∂v/∂x - ∂u/∂y, where u is the
east-west component of the wind and v is the north south component. So, it is the N-S
variation in the wind in the E-W direction minus the E-W variation in the wind in the N-S
direction. Another way is to describe vorticity is natural coordinates: ∂V/∂n + V/Rs,
where there is a shear and curvature term. This coordinate system allows a more tangible
approach to studying vorticity. Shear vorticity is caused by wind shear of straight line
flow. The movement of air in a curved flow causes changing in wind direction or
magnitude with height, and curvature vorticity. The combination of wind shear and
curvature determine the vorticity.
Stokes theorem is another visual way to study vorticity. In this case, Vorticity =
Circulation /Area, and shows that the circulation around a contour that contains a group
of vortices is just equal to the sum of the enclosed vortex strengths. In other words,
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summing up numerous vorticities will eventually form a circulation that is far easier to
visualize and understand.
Figure 1: Stokes theorem:
Circulation is the sum of
many vorticities around an
area A
V Teaching about Vorticity
Lecture should involve wind shear and general westerly flow in the northern
hemisphere. Once this is established, show how curvature and wind shear combine to
make vorticity. This is normally what is done in an undergraduate meteorology course,
but can be applied to GE students. Lab setting should be analyzing maps to illustrate
shear and changes in curvature. Prerequisite instruction should involve explaining wind
shear and illustrating curved flow, by looking at height contours. The “big picture” is
that large-scale disturbances are better understood because of this quantity.
Explaining vorticity is best if the quantity can be visualized. First, look at model
output and describe that vorticity does a good job at telling you where mid-latitude
weather may occur. For a GE class, I would use the natural coordinate system to explain
how vorticity is due to wind shearing and curved flow. Also, one can explain how
vorticity is simply the mean circulation of air divided by an area. For example, illustrate
how a typical extra-tropical cyclone has a counter-clockwise circulation. This circulation
is noticed because of the aggregation of sums of vorticity.
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Figure 2: Shear and Curvature Vorticity
Shear Vorticity
Curvature Vorticity
VI Convergence and Divergence
Convergence and divergence are much easier to explain to a GE, and are just as
important in meteorology. Convergence is simply the accumulation of air over a specific
area. Convergence leads to increased pressure at the location of the convergence. For
example, during rush hour, many automobiles use the onramp to merge, or converge onto
the freeway. The result is that traffic congestion (pressure) builds up. Divergence occurs
when the congestion is letting up and the cars are able to speed up. As they speed up, the
freeway becomes less congested. In the atmosphere, if air starts building up at a specific
elevation, the atmosphere responds by either forcing rising or sinking motion.
Figure 3 demonstrates how divergence and convergence cause rising or sinking motion,
which in turn can lead to cloud formation and precipitation. Notice that if divergence
begins at upper levels, that area is left with a relatively lower pressure. The system reacts
by replenishing the lost air with air from below, or rising air. This rising air will cool
adiabatically, condense and form clouds, and possibly precipitation. . The response in
this case is air to converge at the surface. On the other hand, upper-level convergence
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leads to sinking motion, which leads to surface high-pressure, which leads to divergence
at the surface.
Mathematically, convergence and divergence are explained by the continuity
equation/conservation of mass. These laws basically explain, that what goes in, must
come out, which demonstrates the same idea illustrated in figure 2.
Mass convergence
will occur if more mass is coming into a certain area than is exiting. Mass divergence
occurs if more mass exits a region than enters.
Figure 3: Convergence and divergence and their relationship to pressure.
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VII Teaching about Convergence and Divergence
Lecture material should maybe start with conservation of mass, and then how
temperate fluctuation leads to pressure deviations. Show a non-science example: What
goes in must come out. Take a bottle of marbles with a hole on top and one on the side
bottom. Show how stuffing marbles into the bottom forces the column to rise
synonymous with rising air. Letting marbles out the bottom forces the marbles on top to
fill their place sinking motion (subsidence). A helpful tool is to explain air inside a
parcel, and how it must stay in balance.
If mass is forced in or out from side or bottom,
the system must react to stay in balance.
Illustrating a simple sea-breeze circulation is helpful when teaching about
convergence and divergence, while also illustrating how temperature differences affect
pressure fields. Figure 3 shows how non-uniform heating of the surface can lead to sea
breeze via convergence and divergence.
Figure 3: Simple Sea-breeze circulation
a) Initial conditions: No
wind, equal pressure in both
columns, and at both levels
(1000 mb and 950 mb).
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b) Sea-breeze development:
Daytime heating in inland
area leads to rising air and an
increase in the thickness of
column. Pressure surface
begins to slope. Area near
ocean does not heat up as
rapidly as inland area.
c) Sea-breeze circulation: At
surface, low-pressure forms at
SAC. PGF is from high to low,
so sea breeze forms and wind
travels from high to low pressure.
At 950 mb, rising air in SAC
causes elevated high-pressure, so
return flow develops aloft.
VIII Conclusion
Difficulty emerges when teaching qualitative for several reasons. Most likely,
this is the opposite way that a meteorologist learned the material, and now he/she is
trying to reverse it. Also, basic math still has to be used to explain certain problems.
Also, meteorology is built on a foundation of many different concepts, which are all
connected and interrelate. With that said, certain changes have to be made. Concepts in
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a survey course must take on a new focus. Appendix I is a course outline for a 15-week
semester lecture.
Meteorology, like chemistry, physics, and biology should be accompanied by a
lab course, where the topics are reinforced. This depends on growth in interest in the
earth sciences as well as creation of new labs that are relevant to daily life. Most people
understand the importance of weather and life, but need to be shown this explicitly.
Laboratory classes should use weather maps as guides, but should also use basic
newspaper, or mainstream media weather maps to better reinforce the “real world”
significance of these concepts.
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References:
Bluestein, H.B., 1992: Synoptic-Dynamic Meteorology in Midlatitudes . Vol. I: Principles
of Kinematics and Dynamics . Oxford University Press, 431 pp.
Chen, T, et al, 1996: Interannual variation of global atmospheric angular momentum. J.
Atmos. Sci., 53, 2852-2857.
Crummet, W. P. and Western, A. B., University Physics (Modes and applications), (Wm.
C. Brown, Publishers, Dubuque, IA, 1994) pp 300-303.
Gutro, R., 2003: Changes in the Earth’s rotation are in the wind. Public Release
Hartmann, D.L., 1994: Global Physical Climatology. Academic Press, 411 pp.
Hense, A. and Stuck, J., 2003: Seasonal variability of simulated equatorial atmospheric
angular momentum and associated global pattern. Journal of Geophysical
Research, 5, 45-52.
Hewitt, P.G. Conceptual Physics, 7th Ed., (Harper Collins College Publishers, 1993) pp
131-134.
Holton, J.R., 1992: An Introduction to Dynamic Meteorology. Academic Press, 509 pp.
Lott, F. et al, 2001. Mountain torques and atmospheric oscillations. Geophysical research
letters, 0, 0-0.
Ostdick, V. S. and Bord, D. J., Inquiry Into Physics, 2nd Ed., (West Publishing Company,
St. Paul, MN, 1991) pp 133-137.
Riegel, C.A., 1992: Fundamentals of Atmospheric Dynamics and Thermodynamics.
World Scientific, 496 pp.
Schmintz-Hubsch, H. et al, 1999: The variability of length of day from seasonal to
subdiurnal time scales. German Geophysical Institution
Seitz, F. et al, 2002: Consistent atmospheric and oceanic excitation of the chandler
wobble. German Geophysical Institution
Viudez, A., and R.L. Haney, 1996: On the shear and curvature vorticity equations. J.
Atmos. Sci., 53, 3384-3394.
Warsi, Z.U.A., 1993: Fluid Dynamics. Theoretical and Computational Approaches. CRC
ress, 683 pp
Winkler, C. 2004: Studies of atmospheric angular momentum. NOAA-CIRES Climate
Diagnostics Center
Appendix I: Lecture and laboratory course outlines (for 15 week semester)
Week
Topics Covered
1
2
3
4
5
6
7
8
9
10
11
12
13
14
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Overview of earth's atmosphere, composition, vertical structure, air pressure and density.
Weather vs. Climate
Temp and heat transfer, Air temperature
Air Pressure winds, divergence, convergence, conservation of mass
Moisture: Humidity, dew point, vapor pressure, evaporation, condensation
Clouds and Precipitation
Circulation: divergence, conservation of mass, angular momentum
General circulation, local winds, wind patterns, jet stream
Air masses, fronts, extra-tropical cyclones
Climatology of world, topography, ocean currents, elevation
Synoptics: vorticity, divergence, convergence, COAM
Synoptics: severe weather and local trends
Local climate: in our case CA: west coast fog, sea breeze, rain shadow effect
Local conditions due to fronts, air masses, etc.
Air pollution
Current topics: El Nino, global warming