AOSS 401, Fall 2007 Lecture 12 October 3, 2007 Richard B. Rood (Room 2525, SRB) rbrood@umich.edu...

AOSS 401, Fall 2007Lecture 12

October 3, 2007

Richard B. Rood (Room 2525, SRB)rbrood@umich.edu

734-647-3530Derek Posselt (Room 2517D, SRB)

dposselt@umich.edu734-936-0502

Class News

• Homework– Homework and some review questions were posted last night.

• Homework due Monday• We will go over the review questions on Friday

– Think about them

• Exam next Wednesday– Today’s lecture is the last fundamentally new material that will

be on the exam • Friday we will talk about vertical velocity some more• Friday and Monday we will look at the material in different ways and

more thoroughly• Also have your questions

• Mid-term evaluation– “students will be notified soon thereafter that they can fill out the

midterm evaluations between October 8 and October 14”

Material from Chapter 3

• Balanced flow

• Examples of flows– Stratospheric Vortex

• Ozone hole

– Surface Flow• Friction

• Thermal wind

Picture of Earth

f=2Ωsin(Φ)

1.4X10-4 s-1

1.0X10-4 s-1

0.0 s-1

Picture of EarthΩ

k

k

k

Ω

Ω

Maximum rotation of vertical column.

No rotation of vertical column.

Rotation

• When a fluid is in rotation, the rotation comes to define the flow field; it provides structure.

• That structure aligns with the vector that defines the angular velocity.– So if the flow is quasi-horizontal, then how the flow

aligns in the vertical is strongly influenced by the rotation and its projection in the vertical.

– On a horizontal surface the curvature of the flow is important

And on the Earth.

• Tropics are more weakly influenced, defined by rotation than middle latitudes.– This also influences the vertical structure of

the dynamical features.

Length scales

• Planetary waves: 107 meters, 10,000 km– Have we seen one of these in our lectures?

• Synoptic waves: Our large-scale, middle-latitude, 106 meters, 1000 km– What’s a synoptic wave? What does synoptic mean?

• Hurricanes: 105 meters, 100 km• Fronts: 104 meters, 10 km• Cumulonimbus clouds: 103 meters, 1 km• Tornadoes: 102 meters, 0.1 km• Dust devils: 1 - 10 meters

Returning to our mid-latitude, large-scale flow.

• We saw last lecture that we could define natural coordinates that were (potentially) useful for determining the motion from maps of thermodynamic fields. That is, the pressure gradient or its analogue, geopotential height.

• We saw that, while a powerful constraint, geostrophy is formally true only when the lines of geopotential are straight.– It’s also a balance, steady state.

• Hence, while seductive, this is not adequate.

How do these natural coordinates relate to the tangential coordinates?

• They are still tangential, but the unit vectors do not point west to east and south to north.

• The coordinate system turns with the wind.

• And if it turns with the wind, what do we expect to happen to the forces?

Ω

Earth

Φ = latitude

a

Looking down from above

Looking down from above

Looking down from above

Looking down from above

Looking down from above

Balanced flows in natural coordinates(balanced, here, means steady)

gradient

hiccyclostrop

cgeostrophi

2

2

nfV

R

V

nR

V

nfV

Low

Cyclostrophic FlowHow do we get this kind of flow?

Low

Pressure gradient force

Centrifugal forceDo we have this balance

around a high?

Low

Gradient FlowWhat forces are being balanced?

0n

High

Definition of normal, n, direction

n

n

0n

Low

Gradient Flow

0n

High

Definition of normal, n, direction

n

n

0n

R>0 R<0

Gradient FlowSolution must be real

4

2Rf

n

Low∂Φ/∂n<0

R>0Always satisfied

High∂Φ/∂n<0

R<0Trouble!

pressure gradient MUST go to zero faster than R

What does this mean physically

• For a high, the pressure gradient weakens towards the center of the high. If pressure weakens, then wind speed weakens. Hence, highs associated with relatively weak winds.

• For a low, there is no similar constraint. Hence lows can spin up into strong storms.

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

Why do we call these flows anomalous?

• Where might these flows happen?

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.”

Balanced flow: an application of all that we know

Geopotential, 50 hPa surfacePressure units:

hPambar

inches of Hg

Length scale?

>1,000 km~10,000 km

What about the wind?

Pressure gradient

Coriolis forceWhat’s the latitude?

Centrifugal force

Wind

Wind

What would happen if I put dye in the low?

So we observe that what happens in this low stays in this low.

tinitialedyedye

dyedt

dyed

HH

dyeHdt

dyed

)()(

0. in, dye puttingQuit source. is

)()(

Ozone, October 23, 2006

Summary from ozone hole

• Ozone hole movie

• Cyclonic polar low isolates air from rest of Earth.

• Extreme cold temperature cause nitric acid and water clouds which changes basic chemical environment of atmosphere.

• Return of sun destroys ozone in isolated air with changed chemical environment.

Let’s move down to the surface.

• At 1000 mb

• How are things different?

• How would we have to modify the equations?

Geostrophic and observed wind 1000 mb (land)

Geostrophic and observed wind 1000 mb (ocean)

Think about this in terms of natural coordinates.

nfV

R

V

sDt

DV

nsfV

R

V

Dt

DV

2

2

erm?friction t some

formcomponent in and

ntnnt

Our geostrophic flow.

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

northn

fVg

Δn

We have said that what’s going on near the surface is related to viscosity.

positive is

- - Friction

draglinear a asfriction Model

motion ofdirection the toopposite actsFriction

2

k

kvku

forceFriction

ji

u

So what does it say if our wind crosses the height contours?

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

north nfVg

Δn ?

So what does it say if our wind crosses the height contours?(Staying in natural coordinates.)

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

north

ΔΦ

tn

So what does it say if our wind crosses the height contours?(Staying in natural coordinates.)

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

north

ΔΦ

tn

u

v

angle, α

Friction force

u

v

k

kvku

tan

o tangent tis

positive is

- - Friction

draglinear a asfriction Model

motion ofdirection the toopposite actsFriction

vt

ji

Friction force

kVvuk

kvku

22Friction

- - Friction

draglinear a asfriction Model

motion ofdirection the toopposite actsFriction

ji

Balance of forces (northern hemisphere)(Staying in natural coordinates.)

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

north

ΔΦ

tn angle, α

Balance of forces (northern hemisphere)(Staying in natural coordinates.)

eastwest

Φ0+ΔΦ

Φ0+3ΔΦ

Φ0

Φ0+2ΔΦ

south

north

ΔΦ

tn angle, α

angle, α, as well?

Angle in terms of forces

f

k

fV

kVvuk

force Coriolis

forceFriction tan

:figure From

force Coriolis

Friction

friction and force Coriolis of balance is angle

22

Can also be derived from

uuku

kfDt

D

kfuydt

d

kufxdt

du

p

pp

pp

v)()v

(

v)()(

Looks like a great homework problem!

Some basics of the atmosphere

Troposphere: depth ~ 1.0 x 104 m

Troposphere------------------ ~ 2Mountain

Troposphere------------------ ~ 1.6 x 10-3

Earth radius

This scale analysis tells us that the troposphere is thin relative to the size of the Earth and that mountains extend half way through the troposphere.

Structure of the atmospheric boundary layer

(Vertical length scales)

Viscous sublayerTransition layerInertial sublayer

Atmospheric Surface Layer (ASL)

Planetary (Convective) Boundary Layer (PBL)

Roughness sublayer

~ 10 1~2 m

~ 10 -1~1 m

~ 10 -3 m

~ 10 2-3 m

Free Atmosphere

Wind profile

Blending height

PBL height

Interfacial sublayer

from Bob Su ( www.itc.nl )

k

{

Let’s think about balance on a different scale

• Going back to our equations of motion in the tangential coordinate system.

Equations of motion in pressure coordinates(plus hydrostatic and equation of state)

pp

p

p

c

JS

y

Tv

x

Tu

t

T

py

v

x

u

fDt

D

0)(

uku

Linking thermal field with wind field.

• The Thermal Wind

Geostrophic wind

xfv

yfu gg

1

,1

Hydrostatic Balance

p

RT

p

Geostrophic wind

p

RT

xfp

v

p

RT

yfp

u

pxfp

v

pyfp

u

gg

gg

1 ,

1

1 ,

1

Take derivative wrt p.

Links horizontal temperature gradientwith vertical wind gradient.

Thermal wind

Tf

R

p

x

T

f

R

p

vp

y

T

f

R

p

up

p

RT

xfp

v

p

RT

yfp

u

pg

gg

gg

kU

ln

,

1 ,

1

p is an independent variable, a coordinate. Hence, x and y derivatives are taken with p constant.

A excursion to the atmosphere.Zonal mean temperature - Jan

north (winter)south (summer)

approximate tropopause

A excursion to the atmosphere.Zonal mean temperature - Jan

north (winter)south (summer)

∂T/∂y ?

A excursion to the atmosphere.Zonal mean temperature - Jan

north (winter)south (summer)

∂T/∂y ?

<0

<0

<0

>0

<0

<0

A excursion to the atmosphere.Zonal mean temperature - Jan

north (winter)south (summer)

∂T/∂y ?

<0

<0

<0

>0

<0

<0

> 0

<0

<0

>0

>0

>0

∂ug/∂p ?

A excursion to the atmosphere.Zonal mean wind - Jan

north (winter)south (summer)

Relation between zonal mean temperature and wind is strong

• This is a good diagnostic – an excellent check of consistency of temperature and winds observations.

• We see the presence of jet streams in the east-west direction, which are persistent on seasonal time scales.

• Is this true in the tropics?

Thermal wind

p

p

p

pU

pU

g

pg

pg

pTdf

Rd

pTdf

Rd

Tf

R

p

00

ln

ln

ln

@

@

kU

kU

kU

Thermal wind

p

pT

f

Rpp

pdTf

Rpp

T

pTdf

Rd

pgg

p

p

pgg

p

p

p

pU

pU

g

00

0

@

@

ln)()(

ln)()(

average andby drepresente

islayer ain T y)(x,any at that assume

ln

0

00

kUU

kUU

kU

Thermal wind

p

p

x

T

f

Rv

p

p

y

T

f

Ru

p

pT

f

Rpp

pT

p

T

pgg

0

0

00

ln

ln

ln)()(

kUU

Thermal wind

)(1

)(1

ln)()(

0

0

00

xfv

yfu

p

pT

f

Rpp

T

T

pgg kUU

?

From Previous LectureThickness

1

2

ln

)(

012

0

p

ppTd

g

RZZ

g

zZ

Z2-Z1 = ZT ≡ Thickness - is proportional to temperature is often used in weather forecasting to determine, for instance, the rain-snow transition. (We will return to this.)

Note link of thermodynamic variables, and similarity to scale heights calculated in idealized atmospheres above.

Similarity of the equations

p

p

pgg

p

p

pdTf

Rpp

pdTg

RZZ

0

1

2

ln)()(

ln

0

012

kUU

There is clearly a relationship between thermal wind and thickness.

Schematic of thermal wind.

from Brad Muller

Thickness of layers related to temperature. Causing a tilt of the pressure surfaces.

Another excursion into the atmosphere.

850 hPa surface 300 hPa surface

XX X

from Brad Muller

Another excursion into the atmosphere.

850 hPa surface 300 hPa surface

X

X X

from Brad Muller

Another excursion into the atmosphere.

850 hPa surface 300 hPa surface

from Brad Muller

Another excursion into the atmosphere.

850 hPa surface 300 hPa surface

from Brad Muller

A summary of ideas.

• In general, these large-scale, middle latitude dynamical features tilt westward with height.

• The way the wind changes direction with altitude is related to the advection of temperature, warming or cooling in the atmosphere below a level.– This is related to the growth and decay of these

disturbances. – Lifting and sinking of geopotential surfaces.

Balance and rotation

• We keep making a big deal of rotation and the balance of the coriolis force and the pressure gradient force, e.g. the geostrophic balance.

• We have all of these equations and scale analysis, and they keep leading use to these notions of geostrophic and hydrostatic balance.

• Let’s examine some of these ideas in a more visual way.

Rotation

• When a fluid is in rotation, the rotation comes to define the flow field; it provides structure.

• That structure aligns with the vector that defines the angular velocity.– So if the flow is quasi-horizontal, then how the

flow aligns in the vertical is strongly influenced by the rotation and its projection in the vertical.

And on the Earth.

• Tropics are more weakly influenced, defined by rotation than middle latitudes.– This also influences the vertical structure of

the dynamical features.

Some things that we learned (1)

• Organizing structure provided by rotation.• Rotation is less important in the tropics, which is

clearly observable in the atmosphere.• There is a theoretical limit on pressure gradients

associated with high pressure systems.– Highs tend to be smeared out; they tend to have

moderate wind speeds.

• There is not such a limit for low pressure systems.– Lows can be very intense; The highest wind speeds

are associated with lows.

Some things that we learned (2)

• There is the possibility of “anomalous” circulations.– Possibility of cyclonic highs– Possibility of anti-cyclonic lows

• We can estimate frictional dissipation based on the angle between lines of constant pressure, or height, and the observed wind.

Some things that learned (3)

• Dynamical features can isolate air and allow the evolution of extraordinary chemical processes.

Where do we need to go next?

• We need to understand the role of vertical motion in large-scale dynamics.

• We need to understand the role of thermodynamic variables in the dynamical balances.

Analysis of Hurricane

Let’s take a stab at a hurricane.(Northern hemisphere)

• What balance might we use?

Let’s take a stab at a hurricane.(Northern hemisphere)

L

Let’s take a stab at a hurricane.(Northern hemisphere)

L

r = radial coordinate

Gradient balance for hurricane

rfV

r

V

nfV

R

V

2

2

scoordinate lcylindricain formally

scoordinate naturalin balancegradient Our

Rewrite gradient wind for hurricane

r

rf

r

M

frVrM

4

well)asequation momentum the

sthat'(remember balance indgradient w

theintoput and conserved assume2

hurricane around momentumAngular

2

3

2

2

Define angular momentum(You’ve seen this before.)

r

T

H

R

z

M

r

H

RT

z

zrz

M

r

2

3

2

3

1

H,height scale with studied,

not have wesystem coordinate ain

1

z wrt toDerivative

Some analysis

then0, is you tellI If

1 2

3

z

Mr

T

H

R

z

M

r

Some analysis

center.at max is Hence, 0

then0, is you tellI If

1 2

3

Tr

Tz

Mr

T

H

R

z

M

r

Some analysis

K10:

10,50,100,7:

1

151

2

3

Tyields

sfmsUkmLkmHScales

r

T

H

R

z

M

r

Near ground we have friction.

-kV

or

kfuydt

d

kufxdt

du

pp

pp

ForceFriction

v)()v

(

v)()(

And hurricanes are observed to maintain themselves!

Latent heat from warm ocean water.

Hurricane Heat Engine

• Hurricanes are maintained by latent heat release from water that is evaporated from the ocean.– ~ 27o C is threshold.

• Bring in thermodynamic equation.

• Hurricanes are an efficient heat engines.