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NATS 101NATS 101

Lecture 13Lecture 13Curved Flow and FrictionCurved Flow and Friction

Local Diurnal WindsLocal Diurnal Winds

Mullen’s 1Mullen’s 1stst Law of NATS 101 Law of NATS 101

Clicker=PointsClicker=Points

RUC surface analysis for 1800 UTC Mar 01, 2010

that shows winds blowing from high to low pressure

Last time we talked about two of the force terms in the simplified equation for

horizontal air motion

Geostrophic Balance:

PRESSURE GRADIENT = CORIOLIS

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

Total Force = 1ρΔpd

+ 2ΩVsinφ+V2

r+ Fr

Simplified equation of horizontal atmospheric motion

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

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 LAST TWO TODAY…FOCUS ON LAST TWO TODAY…

GEOSTROHIC BALANCE LAST TIME…GEOSTROHIC BALANCE LAST TIME…

The centripetal force and friction force are typically much smaller, but they are very

important for two reasons:

1. Cause mass divergence and convergence

2. Can be relatively large in special cases that are meteorologically important (i.e. cool)

MASS DIVERGENCE MASS CONVERGENCE

INITIALWIND

FASTERWIND

INITIALWIND

SLOWERWIND

MASS LOST MASS GAINED

AIR RISINGBELOW

AIR SINKING ABOVE

AIR RISINGABOVE

AIR SINKINGBELOW

To begin a discussion of centripetal force, let’s address the popular belief about how

water goes down the drain…

Popular belief: The way the toilet flushes or the sink drains depends on which hemisphere

you’re in.

Bart vs. Australia Simpson’s episode: Bart calls an Australian boy to see if his toilet really does flush clockwise…We’ll see what the surprising answer is later.

Centripetal Force =

t

vva 12 −=

Arises from a change in wind direction with a constant speed (v) due to the curvature of the flow around a radius (r)

Center of circle

V1

Initial velocity

V2

Final velocity

-V1

V2

Centripetal acceleration (a)(towards the center of circle)

The centripetal acceleration is always directed toward the center of the axis of rotation.

Note to be physically correct, the expression should have a negative sign, so +V2/r is actually the centrifugal acceleration.

r

V2

a

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.

CENTRIFUGAL FORCE

Centripetal Force

WINDS IN WINDS IN GEOSTROPIC GEOSTROPIC BALANCE FOR BALANCE FOR STRAIGHT FLOWSTRAIGHT FLOW

CENTRIPETAL CENTRIPETAL ACCELERATION ACCELERATION NEEDED ACCOUNT FOR NEEDED ACCOUNT FOR THE CURVATURE OF THE CURVATURE OF THE FLOWTHE FLOW

Recall: Uniform Circular Motion Requires Acceleration/Force

Initial Velocity

Final Velocity

Acceleration directed toward center of circleInitial

Velocity

Final Velocity

Circular Path

Circle Center

Centripetal (center seeking) acceleration is required for Centripetal (center seeking) acceleration is required for curved flow, i.e. to change the direction of the velocity vector! curved flow, i.e. to change the direction of the velocity vector!

Flow Around Curved Contours

5700 m5640 m

Centripetal Acceleration is Required for Air Parcel to Curve

Assume PGF constant size along entire channel

LL HHZer

Zer

oo

Height 2

Height 1

Flow Around Curved Contours

5700 m5640 m

Centripetal Acceleration

LL HHZer

Zer

oo

Assume PGF constant size along entire channel

How does atmosphere produce the necessary centripetal force? How does atmosphere produce the necessary centripetal force?

Height 2

Height 1

Forces for Curved Flow

5700 m5640 m

Centripetal = PGF + CF

Centripetal << PGF or CF

Gradient Wind Balance

Wind

Wind

Geo

Win

dPGF

PGF

PGF

CF

CF

CF

Assume PGF constant size along entire channel

Height 2

Height 1

Total Force = 1ρΔpd

+ 2ΩVsinφ+V2

r+ Fr

Simplified equation of atmospheric motion Gradient Wind Balance

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

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 surfaceGRADIENT WIND BALANCE…GRADIENT WIND BALANCE…

Gradient Wind Balance: End Result

5700 m5640 m

Wind speeds are Slower at trough Faster at ridge

Slower than Geo Wind

Faster than Geo Wind

Geo

Win

d

Win

d Spe

ed

Incr

ease

sWind Speed

Decreases

Assume PGF constant size along entire channel

Therefore, wind speeds Increase downwind of trough Decrease downwind of ridge

Height 2

Height 1

Gradient Wind Balance

Speeds and Areas: Increase downwind of trough Decrease downwind of ridge

Win

d Spe

ed

Incr

ease

sWind Speed

Decreases

Wind Speed

Decreases

Height 2

Height 1

Area

Incr

ease

s

1

2

Assume PGF constant size along entire channel

Area

Decreases

1

2

Divergence and Convergence

Parcel Shapes: Stretch Downwind of Trough so Area Increases Compress Downwind of Ridge so Area Decreases

Area I

ncrea

ses

Diver

gence

Area Decreases

Convergence

Assume PGF constant size along entire channel

Divergence: Horizontal Area Increases with Time Convergence: Horizontal Area Decreases with Time

Height 2

Height 1

Divergence and Convergence

Diver

gence

Net M

ass L

oss

Convergence

Net Mass G

ain

Mass transport across channel

Large

Small

Assume PGF constant size along entire channel

THERE MUST BE COMPENSATING VERTICAL MOTION DUE TO CHANGES IN WIND SPEED AHEAD OF THE TROUGH AND RIDGE.

Height 2

Height 1

MASS DIVERGENCE MASS CONVERGENCE

INITIALWIND

FASTERWIND

INITIALWIND

SLOWERWIND

MASS DIVERGENCE AND COVERGENCE AT UPPER LEVELS (DUE TO CURVATURE OF THE FLOW)

Stratosphere (acts as a lid)Stratosphere (acts as a lid)

AIR RISING

AIR SINKING

DOWNWIND OF A TROUGHUPWIND OF A RIDGE

UPWIND OF A TROUGHDOWNWIND OF A RIDGE

Gedzelman, p249

Relationship between upper-level troughs-ridges and vertical motion

Trough RidgeRidgeJET LEVEL

~300 mb

SURFACE

Surface Surface HighHigh Surface LowSurface Low

SINKING MOTION TYPICALLY STABLE

(clear skies likely)

RISING MOTIONMAY BE CONDITIONALLY UNSTABLE

(if clouds form and air is saturated)

Slower than geostrophicSlower than geostrophic

Faster than geostrophicFaster than geostrophic

TroughTrough

RidgeRidge

Converg

ence

Converg

ence D

iverg

enc

Div

erg

encee

DivergenceDivergence

ConvergenceConvergence

TroughTrough TroughTrough

Converg

ence

Converg

ence

Div

erg

ence

Div

erg

ence

ConvergenceConvergence

DivergenceDivergence

Gradient balance and flow around lows and highs (Northern Hemisphere)

Cent. forceCent. force

Cent. forceCent. force

Counterclockwise flow around lows

Clockwise flow Around highs

Flow around low pressure

Counterclockwise flow Clockwise flow(because Coriolis force reverses with respect to wind direction)

NORTHERN HEMISPHERE SOUTHERN HEMISPHERE

There is another force balance possible if the Coriolis

force is very small or zero, so it’s negligible.

In that case,

the pressure gradient force would

balance the centripetal force.

Total Force = 1ρΔpd

+ 2ΩVsinφ+V2

r+ Fr

Simplified equation of atmospheric motion Cyclostrophic Balance

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

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 surfaceCYCLOSTROPHIC BALANCE…CYCLOSTROPHIC BALANCE…

Cyclostrophic Balance

LLCentrifugal

Force

Pressure Gradient

Force

Pressure gradient balances the centrifugal force.

Occurs where flow is on a small enough scale where the Coriolis force becomes negligible.

PGF + Centripetal Force = 0

OR

PGF = Centrifugal Force

Important for understanding (really cool) meteorological phenomena that have extremely strong winds and tight pressure gradients!

Examples of Cyclostrophic Flow

HURRICANES

TORNADOES

And flushing toilets, too!!

The Unsolved Mystery of the

Flushing Toilet Solved!

PGFPGF

Centrifugal Centrifugal forceforce

To Bart and Lisa: “A swirling, flushing toilet is in cyclostrophic balance, so the way it flushes depends more on the shape of the drain—and nothing to do with whether you’re in Australia or not!”

One last force to consider…

FrictionFriction

Friction

Pressure Gradient Force

Coriolis Force

Geostrophic Wind

1004 mb

1008 mb

Once the wind speed becomes slower than the geostrophic value, geostrophic balance is destroyed because the Coriolis Force decreases.

Friction

Frictional Force is directed opposite to velocity. It acts to slow down (decelerate) the wind.

Friction

Pressure Gradient Force

Coriolis Force

Wind

1004 mb

1008 mb

Because PGF becomes larger than CF, air parcel will turn toward lower pressure.

Friction Turns Wind Toward Lower Pressure.

Friction

Friction

PGFCF

Wind1004 mb

1008 mb

Eventually, a balance among the PGF, Coriolis and Frictional Force is achieved.

PGF + CF + Friction = 0

Net force is zero, so parcel does not accelerate.

Fr

Friction

1004 mb

1008 mb

The decrease in wind speed and deviation to lower pressure depends on surface roughness. Smooth surfaces (water) show the least slowing and turning (typically 10o-30o from geostrophic).Rough surfaces (mtns) show the most slowing and turning (typically 30o-50o from geostrophic).

MtnsWater

10o-30o

30o-50o

Friction

1004 mb

1008 mb

Friction is important in the lowest km above surface.Its impact gradually decreases with height. By 1-2 km, the wind is close to geostrophic balance, gradient wind balance, or cyclostrophic.

SFC

~1 km0.6 km

0.3 km

www.met.tamu.edu

Gedzelman, p249

Flow in Surface LowsLows and HighsHighs

Spirals OutwardDivergence

Spirals InwardConvergence

MASS DIVERGENCE MASS CONVERGENCE

INITIALWIND

FASTERWIND

INITIALWIND

SLOWERWIND

MASS DIVERGENCE AND CONVERGENCE AT SURFACE (DUE TO THE FORCE OF FRICTION)

Ground is a solid barrier Ground is a solid barrier

AIR RISING

AIR SINKING

Flow into Lows Flow out of Highs

L H

Air curves outward away from surface high pressure

Mass divergence and sinking motion.

Ahrens, Fig 6.22

Friction Induced Vertical Motion

Air curves inward toward surface low pressure.

Mass convergence andrising motion

DivergenceDivergence

Conve

rgence

Conve

rgence

DivergenceDivergence

ConvergenceConvergence

Surface Convergence and Divergence

Summary of Force BalancesWhy the Wind BlowsWhy the Wind Blows

Force Balance Forces Involved Where it happens

Geostrophic Pressure gradient, CoriolisWinds at upper levels (with no curvature)

GradientPressure gradient, Coriolis, Centripetal (or Centrifugal)

Winds at upper levels with curvature

Cyclostrophic Pressure Gradient, CentrifugalSmaller-scale, tight rotations like tornadoes and hurricanes (sinks too)

Gradient + Friction

Pressure Gradient, Coriolis, Centripetal, Friction

Surface winds

Assignment for Next LectureLocal Winds, Monsoons

• Reading - Ahrens 3rd: Pg 165-1784th: Pg 167-1815th: Pg 169-184

• Homework06 - D2L (Due Monday Mar. 22)3rd-Pg 194: 7.3, 4, 54th-Pg 198: 7.3, 4, 55th-Pg 200: 7.3, 4, 5

Do Not Hand in 7.3

Assignment for Next Lecture

• QUIZ 2 this Thursday

D2L only

Similar format as before

7 am -12 pm time period for the exam

70 minutes to finish test