NATS 101 Lecture 12 Newton’s Laws of Motion Upper-Air Maps and Winds

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NATS 101

Lecture 12Newton’s Laws of MotionNewton’s Laws of Motion

Upper-Air Maps and WindsUpper-Air Maps and Winds

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

Ahrens, Fig. 2, p141

504 mb504 mb

496 mb496 mb

PGF

Downhill(Constant height)

Some generalities:

1) High/Low heights on an isobar surface correspond to High/Low pressures on a constant height surface

2) Warm/Cold temps on an isobaric surface correspond to Warm/Cold temps on a constant height surface

3) The PGF on an isobaric surface corresponds to the downhill direction

Contour Maps

How can we portray undulations of a 3D isobaric surface on a 2D plane without loss of information?

Contour Maps

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/elevation.html



Consider the terrain height of an island.

Contour Maps




Denote the shoreline by a line. Labeled the line 0’ for zero feet above mean-sea-level.

Contour Maps




Raise the water level by 500’. Denote the new shoreline by another line. Labeled that line 500’

for five-hundred feet above mean-sea-level.

Contour Maps




Raise the water level by another 500’ to 1,000’. Labeled that line 1000’ for one thousand feet

above mean-sea-level.

Contour Maps




Labeled another line 1500’ above mean-sea-level.

Contour Maps




2000’ above mean-sea-level.

Contour Maps





Contour Maps




A 4000’ line is not needed since the island is completely submerged.

Contour Maps




Lower the water level back to 0’. We are left with lines every 500’ at 0’, 500’, 1000’,... above MSL

Contour Maps




Rotate to a top-down perspective. We can see the entire island.

Contour Maps

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/contourlabel.html



Extra precision could be of value, but the map starts to get busy.

If lines every 500’ is good, would lines every 250’ be better?

Contour Maps

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/contourlabel.html


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Contours every 250’; labeled and thickened every 1250’

Map can be clarified by accentuating every few contour lines

Contour Spacing and Gradient

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/contourspacing.html

Contours every 500’.

Note differences in the steepness of the mountain slopes.



Contour Spacing and Gradient

http://academic.brooklyn.cuny.edu/geology/leveson/core/linksa/contourspacing.html

Contours every 500’.

Note that tight spacing of contours corresponds to steep slopes.







Topographic MapGoogle maps

Grand Canyon Village

http://maps.google.com/maps?f=q&source=s_q&hl=en&geocode=&q=Grand+Canyon+Village,+Coconino+Arizona&sll=37.926868,-95.712891&sspn=29.454502,75.849609&ie=UTF8&hq=&hnear=Grand+Canyon+Village,+Coconino,+Arizona&ll=36.042992,-112.153931&spn=0.125476,0.296288&

570 dam contour570 dam contour

576 dam contour576 dam contour

570 and 576 dam contours570 and 576 dam contours

All contours at All contours at 6 dam spacing6 dam spacing

-20 C and –15 C -20 C and –15 C Temp contoursTemp contours

-20 C, –15 C, -10 C -20 C, –15 C, -10 C Temp contoursTemp contours

All contours at All contours at 55oo C spacing C spacing

Height contours Height contours Temps shadedTemps shaded

Region of Region of HighHigh Heights Heights RIDGERIDGE

and and WarmthWarmth

Region of Region of LowLow Heights Heights TROUGHTROUGH

and and ColdCold

PGF

Wind

Wind doesn’t blow downhill!Wind doesn’t blow downhill!

Do Rocks Always Roll Downhill?

Gedzelman, p 247

Upper-Level Winds

PGF

Begs the Question….

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 LawAn 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 zeroAn external force is required to speed up, slow down, or change the direction of air

Newton’s Laws of Motion

• Newton’s 2nd LawThe net force exerted on an object equals its mass times its accelerationSum of All Forces = Mass AccelerationAcceleration = Velocity Change / TimeAcceleration = Change in Either Speed

or Direction

Velocity, Acceleration and Force are Vectors

• Speed/Size Change

Original Velocity

New Velocity Original

Velocity

New Velocity

Acceleration and Force

Original Velocity

Original Velocity

New Velocity

New Velocity

Acceleration and Force

• Direction Change

Uniform, Circular Motion Requires Acceleration

Original Velocity

New Velocity

Acceleration directed toward center of circle Centripetal

Original Velocity

New Velocity

Circular Path

CENTRIPETAL CENTRIPETAL FORCEFORCE

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…

CENTRIFUGAL CENTRIFUGAL FORCEFORCE

Centripetal Force

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) (2) (3) (4)

Total Force = 1ρΔpd

+ 2ΩVsinφ+V2

r+ Fr

Term Force Cause

1 Pressure gradient force 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

FOCUS ON FIRST TWO TODAY…FOCUS ON FIRST TWO TODAY…

Force BalanceWhat 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.

0 = Pressure gradient force + Coriolis force + Centripetal Force + Friction

0 = Pressure gradient force + Coriolis force Geostrophic Balance

Pressure gradient force

Definition: Force to the difference in pressure (Δp) over a distance (d). (In the equation ρ is the density of air)

d

pΔρ1

The pressure gradient force is directed perpendicular to lines of constant pressure (isobars), toward lower pressure.

1

ρΔpd

Strength of the pressure gradient force

How strong the pressure gradient is depends on distance between the areas of high and low pressure, or how close the lines of constant pressure are spaced.

Strong pressure gradient: Isobars are close together

Weak pressure gradient: Isobars are far apart.

WEAK WEAK PRESSURE PRESSURE GRADIENTGRADIENT

STRONG STRONG PRESSUREPRESSUREGRADIENTGRADIENT

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)

PRESSURE PRESSURE GRADIENT AT GRADIENT AT

DENVERDENVER

LOWLOWHIGHHIGH

Observations for upper level winds:

Wind DOES NOT follow the pressure gradient.

Wind runs parallel to the lines of constant height (i.e. isobars).

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

DENVERDENVER105 knots105 knots

ALBUQUERQUEALBUQUERQUE90 knots90 knots

Coriolis Force φsinVΩ2

Definition: Apparent force due to rotation of the Earth (Ω). Depends on the speed (V) and the latitude (Φ).

Causes apparent deflection in reference of an observer at a fixed point on Earth

Gaspard Coriolis

2ΩVsinϕ

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

• From perspective of person not on merry-go-round, path of ball is straight.

• From perspective of person on merry-go-round, path of ball deflects to left. There is an apparent force.

(left click picture for animation) World Weather Project 2010 Courtesy of M. Ramamurthy U of Illinois, Urbana-Champaign Merry Go Round Link

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Rotation of the Earth (from the polar perspective)

NORTHERN HEMISPHERE SOUTHERN HEMISPHERE

COUNTERCLOCKWISE ROTATIONDeflection to the right

CLOCKWISE ROTATIONDeflection to the left

(Getzelman)

SAME IDEA AS THE MERRY-GO-ROUND!

deflectiondeflection

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 SOUTHERN HEMISPHERE

WINDWIND

CORIOLIS FORCECORIOLIS FORCE(TO RIGHT)(TO RIGHT)

WIND

CORIOLIS FORCECORIOLIS FORCE(TO LEFT)(TO LEFT)

Coriolis forceCoriolis force acts perpendicular (at a right angle) to the wind wind directiondirection, 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.

Geostrophic Adjustment

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.

D. Eventually (about a day), PGF equals CF and flow is parallel to isobars.

(left click picture to animate) World Weather Project 2010 Courtesy of M. Ramamurthy U of Illinois, Urbana-Champaign

Animate Picture

QuickTime™ and aYUV420 codec decompressor


https://www.e-education.psu.edu/gened/meteo101/Examples/Section2p03.html

Geostrophic Wind

Positions 1 and 2:

Pressure gradient forcePressure gradient force accelerates the parcel towards the low pressure.

Coriolis forceCoriolis force acts to the right of the velocityvelocity of the parcelof the parcel, making it curve to the right.

PARCEL RELEASED

Positions 3 and 4:

Pressure gradient force continues to accelerate the parcel towards the low pressure.

As the velocity of the parcelvelocity of the parcel increases, the Coriolis forceCoriolis force increases, making the parcel continue to curve to the right.

Geostrophic Wind

Position 5: FINAL STATE

Pressure gradient forcePressure gradient force is balanced by the Coriolis forceCoriolis force.

Velocity of the parcelVelocity of the parcel is constant (no acceleration). Direction is parallel to the isobars.

FINAL STATE is called geostrophic balance.

Geostrophic Wind

Geostrophic Wind

Pressure gradient forcePressure gradient force is equally balanced by the Coriolis forceCoriolis force, so net force is zero.

Wind speed and direction (velocity)Wind speed and direction (velocity) is constant (no acceleration).

Direction of wind is parallel to the isobars, or lines of constant pressure.

WINDWIND

CORIOLIS FORCECORIOLIS FORCE

PRESSURE GRADIENTPRESSURE GRADIENTFORCEFORCE

Isobar 1

Isobar 2

WINDWIND



Isobar 1

Isobar 2

WINDWIND


Isobar 2

Isobar 1

WEAKGEOSTROPHIC

WIND

Isobars far apart

STRONGGEOSTROPHIC

WIND

Isobars close together


PRESSURE PRESSURE GRADIENT GRADIENT

FORCEFORCE

Geostrophic Wind and Upper Level Charts

CORIOLIS CORIOLIS FORCE FORCE

GEOSTROPHIC GEOSTROPHIC WIND 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

GEOSTROPHY:No centripetal force or friction

Simplified equation of horizontal atmospheric motion

Term Force Cause

1 Pressure gradient force 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

(1) (2) (3) (4)

XTotal Force = 1ρΔpd

+ 2ΩVsinϕ +V2

r+ Fr

Do Rocks Always Roll Downhill?

Gedzelman, p 247

Upper-Level Winds

PGF

Not if the Hill is Big Enough!

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 LectureSurface Wind,Vertical Air Motions

• Reading - Ahrens3rd: 155-1624th: 157-1645th: 158-164

• Problems - HW05 D2L (Due Wed. Mar. 3)3rd-Pg 162: 6.23, 6.244th-Pg 164: 6.23, 6.245th-Pg 165: 6.24, 6.25

NATS 101 Lecture 12 Newton’s Laws of Motion Upper-Air Maps and Winds

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