#### Transcript Upper-Air Maps Newton`s Laws

```NATS 101
Lecture 12
Newton’s Laws of Motion
Upper-Air Maps and Winds
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Isobaric Maps
•
Weather maps at upper levels are analyzed on
isobaric (constant pressure) surfaces.
(Isobaric surfaces are used for mathematical reasons
that are too advanced to include in this course!)
•
Isobaric maps provide the same information
as constant height maps, such as:
Low heights on isobaric surfaces correspond to low
pressures on constant height surfaces!
Cold temps on isobaric surfaces correspond to cold
temperatures on constant height surfaces!
Isobaric Maps
(Constant height)
496 mb
504 mb
Some generalities:
1) The
2)
3)
Warm/Cold
High/Low
PGF on
heights
temps
an isobaric
on
onan
ansurface
isobar
isobaric
surface correspond
corresponds
to the downhill
to Warm/Cold
High/Low
direction
temps
pressures
on
a constant
on aheight
constant
surface
height surface
Ahrens, Fig. 2, p141
Contour Maps
How can we portray undulations of a 3D isobaric
surface on a 2D plane without loss of information?
Contour Maps
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Consider the terrain height of an island.
Contour Maps
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Denote the shoreline by a line. Labeled the line 0’
for zero feet above mean-sea-level.
Contour Maps
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Raise the water level by 500’. Denote the new
shoreline by another line. Labeled that line 500’
five-hundred feet above mean-sea-level.
Contour Maps
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Raise the water level by another 500’ to 1,000’.
Labeled that line 1000’ for one thousand feet
above mean-sea-level.
Contour Maps
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Labeled another line 1500’ above mean-sea-level.
Contour Maps
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2000’ above mean-sea-level.
Contour Maps
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2500’ above mean-sea-level.
Contour Maps
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3000’ above mean-sea-level.
Contour Maps
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3500’ above mean-sea-level.
Contour Maps
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A 4000’ line is not needed since the island is
completely submerged.
Contour Maps
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Lower the water level back to 0’. We are left with
lines every 500’ at 0’, 500’, 1000’,... above MSL
Contour Maps
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Rotate to a top-down perspective. We can see the entire island.
Contour Maps
If lines every 500’ is good, would lines every 250’ be better?
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Extra precision could be of value, but the map starts to get busy.
Contour Maps
Map can be clarified by accentuating every few contour lines
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Contours every 250’; labeled and thickened every 1250’
Note differences in the steepness of the mountain slopes.
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Contours every 500’.
Note that tight spacing of contours corresponds to steep slopes.
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Contours every 500’.
Topographic Map
Grand Canyon Village
570 dam contour
576 dam contour
570 and 576 dam contours
All contours at
6 dam spacing
All contours at
6 dam spacing
-20 C and –15 C
Temp contours
-20 C, –15 C, -10 C
Temp contours
All contours at
5o C spacing
Region of
Low Heights
TROUGH
and Cold
Region of
High Heights
RIDGE
and Warmth
Height contours
PGF
Wind
Do Rocks
BegsAlways
the Question….
Roll Downhill?
PGF
Gedzelman, p 247
The Empirical Evidence Shows
• Wind Direction and PGF Relationship
Winds more than 1 to 2 km above the ground
are perpendicular to PGF!
Analogous a marble rolling not downhill, but at
a constant elevation with lower altitudes to the
left of the marble’s direction
How can we explain observations?
Why does the wind blow?
To begin the answer to this question
we first have to revisit
Sir Isaac Newton
Newton’s Laws
of Motion
• Newton’s 1st Law
An object at rest will remain at rest and
an object in motion will remain at a
constant velocity (same speed and same
direction) if the net force exerted on it is
zero
An external force is required to speed up,
slow down, or change the direction of air
Newton’s Laws
of Motion
• Newton’s 2nd Law
The net force exerted on an object equals
its mass times its acceleration
Sum of All Forces = Mass  Acceleration
Acceleration = Velocity Change / Time
Acceleration = Change in Either Speed
or Direction
Velocity, Acceleration and
Force are Vectors
• Speed/Size Change
New
Velocity
Original New
Velocity Velocity
Original
Velocity
• Direction Change
New
Velocity
Original
Velocity
New
Velocity
Original
Velocity
Acceleration
and Force
Acceleration
and Force
Uniform, Circular Motion
Requires Acceleration
New
Velocity
Circular
Path
Original
Velocity
New
Velocity
Original
Velocity
Acceleration
directed toward
center of circle
Centripetal
Centripetal Force
CENTRIFUGAL
FORCE
CENTRIPETAL
FORCE
You experience acceleration
without a change in speed, for
example, on a tilt-a-whirl
carnival ride.
The force is directed toward
the center of the wheel.
An equal an opposite
(fictitious) centrifugal force is
exerted by the inertia of your
body on the wheel—so you
stay put and don’t fall off even
when upside down.
Important when considering
curved flows, as well see
later…
Newton’s 2nd law can be used to derive a
governing equation for atmospheric motion
The simplified form in the horizontal
that we’ll consider has four terms. By
understanding how each of these terms
works, we’ll be able to explain why the
wind blows.
Simplified equation of horizontal atmospheric
motion
1 p
V2
Total Force 
 2V sin  
 Fr
 d
r
(1)
(2)
(3)
(4)
Term
Force
Cause
1
Spatial differences in pressure
2
Coriolis force
Rotation of the Earth
3
Centripetal force
Curvature of the flow
4
Friction
force
FOCUS
Acts TODAY…
against direction of motion
ON FIRST TWO
due to interaction with surface
Force Balance
What we’re looking for in the equation of motion is the condition
where the forces exactly balance—or the sum of the forces is
equal to zero.
When this happens, there is no net acceleration and the wind
speed is constant, by Newton’s first law.
1 p
V2
0 
 2V sin 
 Fr
 d
r
0 = Pressure gradient force + Coriolis force + Centripetal Force + Friction
0 = Pressure gradient force + Coriolis force
Geostrophic
Balance
force
1 p1 p
 d d
Definition: Force to the difference in pressure (Δp) over
a distance (d). (In the equation ρ is the density of air)
force is directed
perpendicular to lines
of constant pressure
(isobars), toward lower
pressure.
Strength of the pressure gradient force
How strong the
depends on distance
between the areas of
high and low pressure,
or how close the lines
of constant pressure
are spaced.
STRONG
PRESSURE
WEAK
PRESSURE
Strong pressure
close together
Weak pressure
far apart.
The pressure gradient force is why the
wind blows, but you need the other
terms to complete the picture…
Upper Level
Chart for
Surface
Arctic Air
Outbreak
(300-mb)
Observations for upper level
winds:
Wind runs parallel to the lines of
constant height (i.e. isobars).
DENVER
105 knots
LOW
HIGH
PRESSURE
DENVER
ALBUQUERQUE
90 knots
Strength of the wind IS related to
the closeness, or packing, of the
isobars.
For example, compare the wind
speed at Denver (105 knots) to
some of the surrounding upper air
observations, like Albuquerque.
NEED AT LEAST ONE OF THE
OTHER THREE FACTORS TO
ACCOUNT FOR WIND MOTION
Coriolis Force

22V
Vsin
sin
Definition: Apparent force due to
rotation of the Earth (Ω). Depends on
the speed (V) and the latitude (Φ).
Gaspard Coriolis
Causes apparent deflection in reference
of an observer at a fixed point on Earth
Coriolis force on a merry-go-round
From perspective of person NOT on the merry-go-round,
path of ball is straight.
From perspective of person on merry-go-round, path of ball
deflects. It accelerates. This is an apparent (fictitious) force.
Life on a Rotating Platform
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World Weather Project 2010
Courtesy of M. Ramamurthy
U of Illinois, Urbana-Champaign
• From perspective of
person not on merrygo-round, path of ball
is straight.
• From perspective of
person on merry-goround, path of ball
deflects to left. There
is an apparent force.
Rotation of the Earth
(from the polar perspective)
NORTHERN HEMISPHERE
deflection
SOUTHERN HEMISPHERE
deflection
(Getzelman)
COUNTERCLOCKWISE ROTATION
Deflection to the right
CLOCKWISE ROTATION
Deflection to the left
SAME IDEA AS THE MERRY-GO-ROUND!
Coriolis Effect:
An Apparent Force
Cannonball follows a straight
path to an observer in space
Earth rotates counter-clockwise
underneath cannonball
(in Northern Hemisphere)
Cannonball appears to deflect
to the right to an observer on
earth
Shot misses Paris to the right
Coriolis Force and Latitude
All three airplanes travel in a straight line with respect to an
outside observer (from space).
The largest deviation, or deflection to the right, with respect to an
observer on Earth occurs for the one traveling closest to the pole.
The higher the latitude, the greater the Coriolis force. Accounted
for by the sine term in the mathematical expression.
Zero at equator (sin 0° = 0)
Maximum at poles (sin 90° = 1)
Coriolis force and speed
The Coriolis force is proportional
to the wind speed.
The faster the speed (or velocity),
the greater the amount of Coriolis
force.
Note also the dependence on
latitude here.
Coriolis Force vs. Wind Direction
NORTHERN HEMISPHERE
WIND
SOUTHERN HEMISPHERE
CORIOLIS FORCE
(TO LEFT)
WIND
CORIOLIS FORCE
(TO RIGHT)
Coriolis force acts perpendicular (at a right angle) to the wind
direction, to the right or left depending on which hemisphere.
Key Points: Coriolis Force
• Introduced to account for the earth’s rotation.
• Only deflects a moving object.
Never changes the speed of an object.
• Zero if the velocity of an object relative to
the earth’s surface is zero.
• Zero at the equator.
For given speed, it is maximum at the poles.
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World Weather Project 2010
Courtesy of M. Ramamurthy
U of Illinois, Urbana-Champaign
A. Parcel at rest initially
accelerates toward lower
pressure.
B. Coriolis Force rotates
parcel to right in NH.
C. As parcel speeds up,
Coriolis Force increases.
PGF equals CF and flow
is parallel to isobars.
Animate Picture
Geostrophic Wind
PARCEL
RELEASED
Positions 1 and 2:
Pressure gradient force accelerates the parcel towards the low
pressure.
Coriolis force acts to the right of the velocity of the parcel,
making it curve to the right.
Geostrophic Wind
Positions 3 and 4:
Pressure gradient force continues to accelerate the parcel
towards the low pressure.
As the velocity of the parcel increases, the Coriolis force
increases, making the parcel continue to curve to the right.
Geostrophic Wind
Position 5: FINAL STATE
Pressure gradient force is balanced by the Coriolis force.
Velocity of the parcel is constant (no acceleration). Direction is
parallel to the isobars.
FINAL STATE is called geostrophic balance.
Geostrophic Wind
FORCE
Isobar 2
WIND
Isobar 1
CORIOLIS FORCE
Pressure gradient force is equally balanced by the Coriolis force,
so net force is zero.
Wind speed and direction (velocity) is constant (no acceleration).
Direction of wind is parallel to the isobars, or lines of constant
pressure.
FORCE
Isobar 2
WIND
Isobar 1
WEAK
GEOSTROPHIC
WIND
Isobars far apart
CORIOLIS FORCE
FORCE
Isobar 2
WIND
STRONG
GEOSTROPHIC
WIND
Isobar 1
Isobars close
together
CORIOLIS FORCE
Geostrophic Wind and Upper Level Charts
CORIOLIS
FORCE
PRESSURE
FORCE
GEOSTROPHIC
WIND
Winds at upper
levels are pretty
close to being
geostrophic:
Wind is parallel to
isobars
Wind strength
dependents on
how close together
isobars are
Simplified equation of horizontal atmospheric
motion
1 p
V2
Total Force 
 2V sin  
 Fr
 d
r
GEOSTROPHY:(1)
No centripetal force or friction
X
(2)
(3)
(4)
Term
Force
Cause
1
Spatial differences in pressure
2
Coriolis force
Rotation of the Earth
3
Centripetal force
Curvature of the flow
4
Friction force
Acts against direction of motion
due to interaction with surface
DoNot
Rocks
if the
Always
Hill isRoll
Big Downhill?
Enough!
PGF
Gedzelman, p 247
Fundamental Concepts for Today
• Rotation of Earth
Geocentric =Accelerated Frame of Reference
• Introduce Coriolis “Force”
Apparent Force to Account for Deflection
Depends on Rotation, Latitude, Wind Speed
• Geostrophic Balance and Wind
Balance Between PGF and Coriolis Force
Geostrophic Wind Blows Parallel to Contours
About One Day Required to Reach Balance
Assignment Next Lecture
Surface Wind,Vertical Air Motions