aoss 401, fall 2007 lecture 11 october 1, 2007 richard b. rood (room 2525, srb) 734-647-3530 derek...

AOSS 401, Fall 2007Lecture 11

October 1, 2007Richard B. Rood (Room 2525, SRB)

rbrood@umich.edu734-647-3530

Derek Posselt (Room 2517D, SRB)dposselt@umich.edu

734-936-0502

Class NewsOctober 1, 2007

• Ricky will be lecturing again starting Wednesday—I will lecture next on the 17th of October

• There is an exam next Wednesday, but you’re all probably well aware of that…

Material from Chapter 3(2)

• Balanced flow• Examples of flows in the atmosphere

Refresher from Friday…

Geostrophic & observed wind 300 mb

In order to understand the flow on maps that looked like this, we introduced “natural” coordinates.

The horizontal momentum equation

p

pp

pp

fDtD

fuydt

d

fxdt

du

uku

v

v

Assume no viscosity

Return to Geopotential (Φ) in upper troposphere

eastwestsouth

north

HIGH t t

tnn nLow

Do you see some notion of a radius of curvature? Sort of like a circle, but NOT a circle.

The horizontal momentum equation(in natural coordinates)

nfV

RV

sDtDV

nsfV

RV

DtDV

2

2

formcomponent in and

ntnnt

nfV

RV

2

Curved flow (Centrifugal Force)

Coriolis Pressure Gradient

One Diagnostic Equation

Natural Coordinates: Key Points• Velocity is defined to be positive• The n direction always points to the left of the

velocity (remember the right hand rule: k x t = n)• If n points toward the center of curvature, the

radius is positive• If n points away from the center of curvature, the

radius is negative• The pattern of isobars/height lines is assumed to

be fixed in space; no movement of weather systems

Uses of Natural Coordinates

• Geostrophic balance– Definition: coriolis and pressure gradient in

exact balance.– Parallel to contours straight line R is

infinite

nfV

RV

2

0

Geostrophic balance

nfV

xf

yfu

p

p

scoordinate p)n,(t, natural inor

v

scoordinate p)y,(x, In

Which actually tells us the geostrophic wind can only be equal to the real wind if the height contours are straight.

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

northn

fVg

Δn

How does curvature affect the wind?(cyclonic flow/low pressure)

nfV

RV

2

R

t

n

ΔnΦ0

Φ0+ΔΦ

Φ0-ΔΦ

HIGH

Low

From Holton

• If Vg/V < 1, geostrophic wind is an overestimate of the actual wind speed

• Since V is always positive, in the northern hemisphere (f > 0) this only happens for R positive

• For typical northern hemisphere large scale flow, R is positive for cyclonic flow; flow around low pressure systems

fRV

VVg 1

Geostrophic & observed wind 300 hPa

Geostrophic & observed wind 300 hPa

Observed:95 knots

Geostrophic:140 knots

How does curvature affect the wind?(anticyclonic flow/high pressure)

nfV

RV

2

R

t

n

Δn

Φ0

Φ0+ΔΦ

Φ0-ΔΦ

HIGH

Low

From Holton

• If Vg/V < 1, geostrophic wind is an underestimate of the actual wind speed

• Since V is always positive, in the northern hemisphere (f > 0) this only happens for R negative

• For typical northern hemisphere large scale flow, R is negative for anticyclonic flow; flow around high pressure systems

fRV

VVg 1

Geostrophic & observed wind 300 hPa

Geostrophic & observed wind 300 hPa

Observed:30 knots

Geostrophic:25 knots

Uses of Natural Coordinates:Balanced Flows

• Tornados• Hurricanes• General high and low pressure systems

Cyclostrophic Flow

nfV

RV

sDtDV

nsfV

RV

DtDV

2

2

formcomponent in and

ntnnt

Cyclostrophic Flow

• A balance in the normal, as opposed to tangential, component of the momentum equation.

• A balance of centrifugal force and the pressure gradient force.

• The following are needed– steady (time derivative = 0)– coriolis force is small relative to pressure

gradient and centrifugal force

Cyclostrophic Flow

nfV

RV

2

equation momentum ofcomponent normal

Get cyclostrophic flow with either large V small R

Cyclostrophic Flow

• Radical must be positive: two solutions

nRV

nRV

V:

2

for Solve

0 ,0 .2

0 ,0 .1

nR

nR

Cyclostrophic Flow

• Tornadoes: 102 meters, 0.1 km• Dust devils: 1 - 10 meters

– Small length scales– Strong winds

Low

Cyclostrophic Flow

Low

Pressure gradient force

Centrifugal force

0 ,0 .1

n

R 0 ,0 .2

n

R

Low

Cyclostrophic Flow

Low

0 ,0 .1

n

R 0 ,0 .2

n

R

Counterclockwise Rotation

Clockwise Rotation

http://www.youtube.com/watch?v=vgbzKF_pSXo

http://www.youtube.com/watch?v=k1dZpW5aFFk

http://www.youtube.com/watch?v=3jQoGm8JEPY

Anticyclonic Tornado (looking up)

Sunnyvale, CA 4 May 1998

In-Class Exercise: Compute Tornado Wind Speed

• Remember:

nRV

V:

for Solve

P=850 mb

P=750 mb

R = 100 m

np

n

1

(Assume ρ = 1 kg/m3)

In-Class Exercise: Compute Tornado Wind Speed

1222

2

3

100) 100(

) 100()100(

) 1(1) 100(

1

smsmV

mPa

mkgmV

npRV

nRV

P=850 mb

P=750 mb

R = 100 m

High

Cyclostrophic FlowAround a High Pressure System?

High

?0 ,0 .1

n

R ?0 ,0 .2

n

R

0

0

n

R

0

0

n

R

n

n

Gradient Flow(Momentum equation in natural coordinates)

• Balance in the normal, as opposed to tangential, component of the momentum equation

• Balance between pressure gradient, coriolis, and centrifugal force

nfV

RV

sDtDV

2

formcomponent In

Gradient Flow(Momentum equation in natural coordinates)

Vn

RfRVV

nfV

RV

speed for wind Solve

0

equation momentum ofcomponent normal

2

2

Gradient Flow(Momentum equation in natural coordinates)

nRfRfRV

nRfRfR

V

nRfRVV

4)(

2

2

4)(

0

2

2

2

Look for real and non-negative solutions

Gradient FlowSolution must be real

4

04

)(4

)(2

2

2

2

Rfn

nRfR

nRfRfRV

Low

Gradient Flow

0n

High

Definition of normal, n, direction

n

n

0n

R > 0 R < 0

Gradient FlowSolution must be real

4

2Rfn

Low∂Φ/∂n < 0

R > 0Always satisfied

High∂Φ/∂n < 0

R < 0Trouble!

pressure gradient MUST go to zero faster than R

Low

Gradient Flow(Solutions for Lows, remember that square root.)

Low

Pressure gradient force

Centrifugal forceCoriolis Force

V

V

Low

Gradient Flow(Solutions for Lows, remember that square root.)

Low

Pressure gradient force

Centrifugal forceCoriolis Force

NORMAL ANOMALOUS

V

V

High

Gradient Flow(Solutions for Highs, remember that square root.)

High

Pressure gradient force

Centrifugal forceCoriolis Force

V

V

NORMAL ANOMALOUS

Normal and Anomalous Flows

• Normal flows are observed all the time.– Highs tend to have slower magnitude winds

than lows.– Lows are storms; highs are fair weather

• Anomalous flows are not often observed.– Anomalous highs have been reported in the

tropics…– Anomalous lows are strange –Holton “clearly

not a useful approximation.”• But it is possible in tornadoes…

Compute Wind Speed Around a Hurricane

• R = 100 km• dP = -25 mb• f = 4 x 10-5

• V = 48 m/s = 107 mph = 93 kt• Category 2 hurricane…

nRfRfRV

4

)(2

2

We have covered a lot of material in a short time!

• Study and think about balances in the natural coordinate system from the point of view of

1. first, pressure gradient, 2. then coriolis force, 3. then the force due to curvature of the lines of

geopotential (or pressure)• Don’t confuse “curvature” in the natural

coordinate system with the curvature terms derived from use of a tangential coordinate system!

Next time

• Think about adding viscosity to the balance.

• And return to thermal wind balance…

Weather

• NCAR Research Applications Program– http://www.rap.ucar.edu/weather/

• http://www.aos.wisc.edu/weatherdata/eta_tempest/12UTC/eta_c850_h06.gif

• National Weather Service– http://www.nws.noaa.gov/dtx/