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```EESC V2100
The Climate System
spring 2004
Lecture 4:
Laws of Atmospheric
Motion and Weather
Yochanan Kushnir
Lamont Doherty Earth Observatory
of Columbia University
Palisades, NY 10964, USA
kushnir@ldeo.columbia.edu
Horizontal Motion in
the Atmosphere Geostrophic Balance
Horizontal Motion in the Atmosphere
•
•
•
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The motion of air on Earth is described in terms of a 3-dimensional vector with a
zonal (west-to-east) or x component, normally denoted u, a meridional (south-tonorth) or y component denoted v, and a vertical (upward) or z component w. In
most cases, the motion is described relative to the rotating Earth.
The more prominent component of atmospheric motion are in the horizontal
(parallel to the surface of the Earth) dimension. In the vertical dimension, on
scales larger than clouds and intense storms such as tornados, the motion is
much smaller than in the horizontal and the atmosphere is close to being in
hydrostatic balance.
The equations governing horizontal motion are based on Newton’s Second Law
of Motion applied to a fluid system in a spherical reference coordinate system
rotating with Earth.
The main driving forces in the atmosphere are the pressure gradient
produced either through thermal or dynamical effects, and gravity.
Retarding or balancing the pressure gradient force is the Coriolis force an
apparent force, which results from viewing the motion in reference to the
rotating Earth. Close to the surface, friction is also an important force retarding
force. Sometimes, when the motion is fast and circular, another apparent force,
the centrifugal force comes into play.
The Pressure Gradient Force
Just as we saw in the vertical dimension, the pressure gradient forces results from the
difference in pressure acting over a distance as described in the following diagram:
z - upward
•
r
y
o
-n
rd
a
thw
x - eastward
•
•
p+Δp
direction of motion
p
Δz
Δy
Δx
The pressure difference acting over a distance Δx create a force Fx= -- ΔpΔyΔz (negative
because eastward flow is produced by higher pressure to the west).
The eastward acceleration (= force per unit mass) ax, (in m/s) exerted by the pressure
gradient differece per unit mass is given by Newton’s second law of motion:
Fx= -- ΔpΔyΔz = (ρΔpΔyΔz)a, where ρ is the density of air.
•
After canceling identical terms on both side of the equation we obtain:
•
Similarly, in the y-direction:
ax = -- (1/ρ)(Δp/Δx).
ay = -- (1/ρ)(Δp/Δy).
Isobars
•
The mapping of pressure on a horizontal surface, such as the sea level, is the first step in
monitoring atmospheric motion
•
Pressure data are collected simultaneously by a network of stations, converted to sea
level and reported to weather centers four times daily, to create a synoptic
representation of sea level pressure.
•
The station data is interpolated to a regular grid and plotted as isobars - lines of equal
pressure.
Pressure Gradients
strong gradient
weak gradient
By viewing the patterns of the isobars, we can determine where the gradients are strong
and where they are weak, thus assessing the strength of the horizontal pressure gradient
and the direction of the pressure gradient force (PGF).
Units of pressure: the MKS unit for pressure is 1 Pascal = 1 Newton/m2 (the unit is named
after the French scientist Blaise Pascal - 1623-1662). The standard sea level pressure in
Pascal is 101325 and the order of magnitude of pressure variations is about 100 Pascals. A
more workable unit of pressure is a millibar (or one thousands of a bar) which equals 100
Pascals (also called hecto-pascal).
Non Inertial Frame of Reference and
Apparent Forces
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To study of body movement under force we use a frame of reference. A frame of
reference is called inertial if it is at rest or if it moves with constant velocity (that is,
constant speed and direction).
If the reference frame is moving under acceleration it is non-inertial. Examples for non
inertial frames of references are an accelerating car, a rotating platform (even if the angular
velocity is constant), and our planet Earth.
Non-inertial systems are under the influence of a force acting linearly or by exerting a
torque on the system to cause rotation. To use such system as a frame of reference we
need to introduce an apparent reaction force equal and opposite to the external one.
Consider for example a train moving at a fixed direction and constant speed in the
country side. If a passenger on that train decide to get out of her or his seat and move
forrard, they will feel and react just as if they are walking in their own back yard.
But if the trains decelerated suddenly the passenger would be jolted forward with the
same acceleration as that of the braking train.
That passenger may conclude that a force acted on her or him pushing them forward
while in reality the train they were riding was pushed back.
The train became a non-inertial frame of reference and in order to explain the effect on
the walking passenger in that frame, we will have to introduce and apparent accelerating
force equal to the force that slowed the entire train.
The Coriolis Force
•
•
•
•
•
d
Next consider an observer, standing in the center of a merry-goround, shaped like a circle with a radius R and rotating in a
counterclockwise direction (to the left) with an angular velocity Ω
(as in the figure to the right)
The observer sets a ball into motion towards the circumference of
the circle, in the direction of the radius and with a speed v.
Because the observer continues to rotate in the clockwise direction it will appear
to him that the ball is turning to the right as it moves away.
The faster the observer turns, the
faster the ball curves to the right.
The closer the ball gets to the
circumference of the circle, the
faster it will appear to move.
To explain the balls behavior, the
observer needs to introduce an
apparent force which causes the
apparent accelerated motion of
the ball to the right with respect
to the rotating merry-go-round.
This force is known as the
Coriolis force.
The Coriolis Force (continued)
•
•
•
•
•
•
•
To determine the magnitude of the Coriolis acceleration consider the following: after the
ball reached a distance of R from the observer, we can identify an arc that forms between
the point on the circumference of the circle at which the ball was originally aimed
(straight ahead of the observer) and the point where it actually crossed the circle.
The arc length, d is given by the time t elapsed from the beginning of the motion until
the arrival at the circumference of the circle R times the speed at which the
circumference is rotating (= angular velocity time the radius), namely: d = ΩRt
Under acceleration the distance covered by a moving body is proportional to the square
of time. Thus d is also equal: d = ac t2/ 2 where ac is the acceleration - in this case, the
Coriolis acceleration.
Equating d from the two equations results in an expression for the
Coriolis acceleration: ac= 2 ΩR / t
Now, the time t is given by the ratio between the radius and the
ball’s speed: t = R / v
Thus we obtain: ac= 2 Ωv
Ω
d
R
The Coriolis acceleration (or force per unit mass) acting on a particle viewed in a rotating
frame of reference is equal to twice the product of the angular velocity and the speed of
the particle. The direction of the force and acceleration is perpendicular to the motion,
acting in this case to the right when facing in the direction of the motion.
Coriolis Force on Earth
Ω
d
R
Ω
Ω
ΩsinΦ
Φ
How does the situation on Earth resemble the rotating disk?
The planet rotates around its axis (with the angular velocity
vector pointing to the north (see bottom figure on the left).
This is the direction of the vector at every point on the
Earth’s surface. At the North and South Poles Ω is vertical
to the surface and anywhere else it is slanted to the surface
with an angle equal to the local colatitude (90° minus the
latitude angle, Φ). The local angular velocity component
vertical to the surface is thus ΩsinΦ. It describes the rotation
of the local surface around the radius connecting it to the
center of the Earth with an angular velocity that decreases
from the North Pole (Ω) to the equator (0) and then
reversing sign in the Southern Hemisphere and continuing to
decrease to the South Pole (-Ω). The Coriolis acceleration
on Earth is thus a function of the latitude with:
ac= 2(ΩsinΦ)v,
On Earth:
Ω = 2π /84600
= 7.27 x 10-5 rad/sec
where v is the velocity of the the moving body with respect
to the Earth. Based on this relationship we define the
Coriolis factor f = 2(ΩsinΦ).The force is perpendicular to
the body’s motion acting to the right in the NH and to the
left in the SH.
Coriolis Effect on Earth
An aircraft heading out from the US West Coast (say San Francisco - latitude ~38°N),
flying towards the East Coast (New York - latitude ~40°N) with an almost direct eastward
heading, needs to continually correct its course to adjust for the Coriolis force otherwise it
will find itself drifting southward to a lower latitude.
Without correction, an aircraft flying at a speed of 900 km/hr (= 250 m/s) will drift
south at an initial acceleration of:
a = 2Ωsin(38)×250 = 2×7.27 x 10-5×0.62×250 = 0.0224 m/s2
c
This implies that for every hour of its flight the aircraft needs to correct its course by
heading north a distance of:
D = ac×(36002)/2 = 145152 m
or about 90 miles.
Geostrophic Balance I:
The primary horizontal balance of forces in atmospheric motion
is the geostrophic balance.
When an air parcel begins to move under a pressure gradient force (PGF), in a direction
perpendicular to the isobars, and towards the low pressure, the Colriolis force (CF) begins
to act, turning the parcel to the right (left in SH). Because CF depends of the particles
velocity, this turning intensifies as the particle accelerates until the turning angle is 90° and
CF is exactly equal to PGF and they both point perpendicularly to the motion, the
pressure gradient force pointing towards the low pressure and the Coriolis force in the
opposite direction.
Geostrophic Balance - Derivation
To find how to express the geostrophic
balance mathematically, examine the diagram
on the right.
The balance between CF and the PGF is
expressed by the equal and 180° apart
vectors P and C. In the NH the geostrophic
wind blows in a perpendicular direction such
that the low pressure is to its left (right in
the SH). Note that the CF depends on the
wind speed in the direction 90° to its left
(right in the SH).
N
PGF vector P = (px,py)
wind vector V = (u,v)
v
py
W
px
cx
u
cy
CF vector C = (cx,cy)
Northern Hemisphere (NH)
force diagram
The balance must also exist between the x
and y components of the forces, separately,
related the geostrophic wind components u
and v). In the component balance, cx depends
on v and cy on u, such that:
Balancing these forces with the PGF results
in the geostrophic balance:
cx = fv and cy = −fu
fu = −(1/ρ)(Δp/Δy)
where f is the Coriolis factor.
E
S
fv = (1/ρ)(Δp/Δx)
Note that the eqn in the x-direction
provides a solution for v and that in the ydirection, for u:
Geostrophic Flow with Friction
Friction slows down the wind, causing a weakening in the Coriolis force. A new balance is
achieved between the resultant of the Coriolis force (CF) and friction on one hand and the
pressure gradient force (PGF) on the other hand.
Friction, Convergence/Divergence,
and Mass Continuity
Friction leads to the convergence of air into the centers of low pressure and divergence out
of the centers of high pressure. An important principle of any fluid motion, mass continuity
(or mass balance) implies that there is rising motion in a low pressure system and sinking
motion in a high, leading to a reversal of the convergence/divergence patterns aloft.
The tendency of air to rise over a low pressure system creates favorable conditions for the
formation of rain clouds. In high pressure systems the sinking motion leads to clear and dry
conditions.
Mid-latitude Weather
Systems
Synoptic Map
This segment taken from a synoptic
(weather) map of surface pressure
shows isobars (contours of equal
pressure in mb) and small flags,
depicting the wind direction (the flags
fly in the direction of the wind) and
speed (each full flag bar is 10 knots
and half a bar is 5 knots with 1 knot =
1/2 m/s).
The flow is very close to geostrophic
balance everywhere with a small
tendency to flow across isobars
towards the low pressure center - a
result of the friction effect close to
the surface.
Horizontal Motion and Weather
The phenomena of weather are linked with the horizontal flow of air in other ways as well.
In the midlatitudes chains of lows (cyclones) and highs (anticyclones) migrate steadily
eastward mainly in winter. The overall tendency of air to rise over a low is combined with
the advection of air by the circulation around it. Northerly winds bring cold air from the
north southward to the west of the low-pressure center, and southerly winds bring warm
air from the south northward. When cold and warm air masses meet, the warm air tends to
move up creating favorable conditions for rain and severe weather. The bands along which
air masses meet are called fronts.
Midlatitude Weather Systems
Life Cycle of Midlatitude Cyclones
Life Cycle and Heat Transport
This three-dimensional
schematic of a midlatitude
cyclone life cycle
demonstrates how these
disturbances reduce the
north-south temperature
contrast in the
midlatitudes by mixing
cold air from the north
with warm air from the
south.
Tropical Weather
Systems
Tropical Cyclones
Tropical cyclones, also called
hurricanes and typhoons, are
intense low pressure
disturbances that forms and
migrates over the tropical ocean
regions and are associated with
intense winds and a very strong
convective activity, which brings
thunderstorms and large
amounts of rainfall. They have the
potential to cause major damage
and loss of life when they make
landfall.
Hurricane Vertical Cross Section
The massive disturbances that
sometimes grows in a time
frame of a week or so, need
specific and favorable conditions
to occur, such as high sea surface
temperatures (at lease 26°C)
and weak vertical wind shears.
Once they do, they spreads over
a radius of a few hundred
kilometers.
Hurricanes are surrounded by
rings of towering thunder clouds
spiraling up to a small circle at
the center of the storm, with a
radius of 30-40 km. Here the
winds can reach a speed of 100
km/hour and more and the most
intense rainfall occurs. Inside this
ring lies the eye of the storm,
where the air is still and the
convection is suppressed by slow
downward motion (subsidence).
Regions of Hurricane Activity
Hurricanes are active in the
"trade wind" belts - the regions
just north or south of the
equator where the winds blow
quite steadily from east to west
(easterlies).
Here tropical disturbances
generally form, initiated by weak
pressure perturbations that
exist all the time in the tropics.
They move west with the trade
winds in a steady, relatively slow
motion (10-20 km/hour).
During this phase they intensify
mainly through the release of
latent heat in the surrounding
clouds and a small percentage
reach full hurricane intensity.
Hurricanes tracks curve
eastward and they speed up
north of ~30°N
Intertropical Convergence Zone (ITCZ)
In the tropics, a belt of warmest surface temperatures, surrounds the Earth. Here there is
abundant moisture so that small vertical movements of air can lead to spontaneous
generation of deep convection. This convection then organizes itself in cells of massive
thunderstorms that tend to drift eastward carried in the prevailing winds and in weak
wave disturbances somewhat resembling midlatitude disturbances. This region is the
ITCZ.
```
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