AOSS 401, Fall 2007 Lecture 15 October 17 , 2007

AOSS 401, Fall 2007Lecture 15

October 17, 2007

Richard B. Rood (Room 2525, SRB)[email protected]

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

[email protected]

mailto:[email protected]

mailto:[email protected]

October 17, 2007

• Exam results

• Roadmap for the next month

• Introduction to vorticity

Exam Results

• Class average: 21.7

• Class median: 21.0

Distribution of Scores, Exam 1

0

1

2

3

4

5

0.0-2.5

2.5-5.0

5.0-7.5

7.5-10.0

10.0-12.5

12.5-15.0

15.0-17.5

17.5-20.0

20.0-22.5

22.5-25.0

25.0-27.5

27.5-30.0

Exam Scores

Grades?

• We will not be assigning letter grades until the end of the semester

• Here is some guidance on how the scores might map to a letter grade on this exam

• (Remember that solid scores on homeworks will bump up your overall course grade and offset low exam scores…)

Grades?

• There were about 10-11 points on the test that we expected every one to get. Most everyone got these, and this is good.

• If you have 15 or less, we would like to make an appointment to talk with you.

• There were about 18-20 points on the test if you got the problems started. Average was 21.5. So most people got the problems started. This is better than good.

• At 22 points and higher, people have a very good grasp on the concepts and their application.

• At > 25 points, excellent grasp of material.

~A+

~C-

Exam Results

• Class averages for each question:1. 4.9 / 5.0

2. 5.1 / 6.0

3. 3.6 / 6.0

4. 3.1 / 6.0

5. 2.8 / 4.0

6. 2.2 / 3.0

1. Exam Question (5)Class Average: 4.9

1. In the equation below what is the physical meaning of each of the terms? If the term is a force, then state whether it is a surface force, body force, or apparent force.

)()cos(2)sin(v21)vtan( 2 uΩwΩ

x

p

a

uw

a

u

Dt

Du

Tendency, acceleration

Curvature or metric terms(accept apparent force)

Pressure gradient(surface force)

Coriolis force(apparent force)

Viscosity(surface force)

Remember how we derived PGF

x

y

z

≡ density = mass per unit volume (V)

V = xyz

m = xyz

-------------------------------------

p ≡ pressure = force per unit area acting on the particle of atmosphere

xy

z

ij

k

Force per unit area = surface force

x

y

z

.

x axis

FBx = (p0 - (∂p/∂x)x/2) (yz)

FAx = - (p0 + (∂p/∂x)x/2) (yz)

AB

(x0, y0, z0)

Fx = FBx + FAx

Fx/m = - 1/ (∂p/∂x)

Questions?


Above the equations of motion are provided using both z, height, as a vertical coordinate and p, pressure, as a vertical coordinate.

2) Write out the material derivative in both coordinate systems. Show explicitly whether or not z or p is held constant when partial derivatives are taken (4 points). What are the units of the vertical velocity in the two coordinate systems (2 points)?

Partial Derivatives…

• Important to remember that partial derivation implies we are holding everything else constant

• For a coordinate system that includes (x,y,z,t):

tyx

tzx

tzy

zyx

z

y

x

t

,,

,,

,,

,,

)

)

)

)

Partial Derivatives…

• Important to remember that partial derivation implies we are holding everything else constant

• For a coordinate system that includes (x,y,p,t):

tyx

tpx

tpy

pyx

p

y

x

t

,,

,,

,,

,,

)

)

)

)

2. Exam Question (6)

s

Pa units , ; )()

scoordinate pressurein derivative Material

s

m units , ; )()

scoordinateheight in derivative Material

Dt

Dp

ptDt

D( )

Dt

Dzw

zw

tDt

D( )

pp

zz

V

V

2. Exam Question (6)

s

Pa units , ; )()

scoordinate pressurein derivative Material

s

m units , ; )()

scoordinateheight in derivative Material

Dt

Dp

ptDt

D( )

Dt

Dzw

zw

tDt

D( )

pp

zz

V

V

Questions?


3) Write the mass conservation equation in pressure coordinates (1 point). Let the horizontal wind in the x and y direction, (u, v) = (ug+ua, vg+va), where subscript g represents a geostrophic wind and subscript a represents the ageostrophic wind. Using the definition of ω and of the geostrophic wind, with the assumption of f = f0 = constant, show that (2 points).

pa

V

Then, with the assumption that the divergence can be represented by a constant average value, use the definition of ω and rewrite the equation in terms of the time rate of change of surface pressure (3 points).

Answer to Question 3a

(8) Equation; 0)(

py

v

x

up

3) Write the mass conservation equation in pressure coordinates (1 point).

Answer to Question 3b3) Let the horizontal wind in the x and y direction, (u, v) = (ug+ua, vg+va), where subscript g represents a geostrophic wind and subscript a represents the ageostrophic wind. Using the definition of ω and of the geostrophic wind, with the assumption of f = f0 = constant, show that (2 points).

pa

V

Answer to Question 3b

p

py

v

x

u

xfv

yfu

aa

a

g

gg

agicageostrophcgeostrophi

is which

0 therefore

0 gives which

1

and 1

where wind,horizontal

v

v

vvvvv

0)( :Start with

py

v

x

up


p

py

v

x

u

xfv

yfu

aa

a

g

gg


is which

0 therefore

0 gives which

1

and 1


v

v

vvvvv

0)( :Start with

py

v

x

up


p

py

v

x

u

xfv

yfu

aa

a

g

gg


is which

0 therefore

0 gives which

1

and 1


v

v

vvvvv

0)( :Start with

py

v

x

up

Answer to Question 3cThen, with the assumption that the divergence can be represented by a constant average value, use the definition of ω and rewrite the equation in terms of the time rate of change of surface pressure (3 points).

Answer to Question 3c

column. theofout moving is mass

above; divergence withdecrease willpressure Surface

definitionby and )(

)0()()0(

constant is where assume

integrate and as rewrite

0

a

0

a

0@

@

a

a

DpDt

DpDpp

DpDpDdppp

D D

dpdp

p

sfcsfc

sfcsfc

sfcsfc

p

p

sfc

p

p

p

p

sfc

sfcsfc

v

vv

v




definitionby and )(

)0()()0(



0

a

0

a

0@

@

a

a

DpDt

DpDpp

DpDpDdppp

D D

dpdp

p

sfcsfc

sfcsfc

sfcsfc

p

p

sfc

p

p

p

p

sfc

sfcsfc

v

vv

v




definitionby and )(

)0()()0(



0

a

0

a

0@

@

a

a

DpDt

DpDpp

DpDpDdppp

D D

dpdp

p

sfcsfc

sfcsfc

sfcsfc

p

p

sfc

p

p

p

p

sfc

sfcsfc

v

vv

v




definitionby and )(

)0()()0(



0

a

0

a

0@

@

a

a

DpDt

DpDpp

DpDpDdppp

D D

dpdp

p

sfcsfc

sfcsfc

sfcsfc

p

p

sfc

p

p

p

p

sfc

sfcsfc

v

vv

v




definitionby and )(

)0()()0(



0

a

0

a

0@

@

a

a

DpDt

DpDpp

DpDpDdppp

D D

dpdp

p

sfcsfc

sfcsfc

sfcsfc

p

p

sfc

p

p

p

p

sfc

sfcsfc

v

vv

v

Questions?


4) Refer to the figure. This is geopotential height at a constant pressure level in the troposphere in the northern hemisphere far above the Earth’s surface. At points A, B, and C, draw the direction of the geostrophic wind and indicate whether the speed (magnitude) of the geostrophic wind is the same or different at these three points (1 point)? In class and text we derived the ratio of the geostrophic wind speed to the gradient wind speed:

curvature of radius where1 aa

g RfR

V

V

V

At points A, B, and C, draw the direction of the gradient wind and indicate whether the speed of the gradient wind is the same or different at these three points (3 points)? Using the definition of horizontal divergence, show regions where the gradient wind is divergent or convergent (= - divergence) (2 points).

Answer for Problem 4a

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

The geostrophic wind, red, is the same at A, B and C. It is parallel to the isolines.

Geostrophic wind speed only depends on gradient of pressure/height

Answer for Problem 4b

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

curvature of radius

1

a

a

g

R

fR

V

V

V

t

n

t

n

t

n

R > 0

R < 0

R > 0V < Vg

V > Vg

V < Vg

Gradient wind flows // to gradient of pressure/height


Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

The gradient wind, blue, is less than the geostrophic wind, red, at A, C and greater than the geostrophic wind at B. It is parallel to the isolines.

The geostrophic wind, red, is the same at A, B and C. It is parallel to the isolines.

Answer for Problem 4c

Φ0 - ΔΦ

Φ0 + ΔΦ

Φ0

ΔΦ > 0

A

B

C

٠

٠

٠

x, east

y, north

The divergence is ∂u/∂x + ∂v/∂y. Consider A and B. Δu is > 0, Δx > 0, Δv=0; hence, gradient is positive and there is divergence between the two points.

The divergence is ∂u/∂x + ∂v/∂y. Consider B and C. Δu is < 0, Δx > 0, Δv=0; hence, gradient is negative and there is convergence between the two points.

Questions?


5) Refer to the figure. This figure shows a jet stream in the northern hemisphere, upper troposphere. The direction is easterly, from the east. We saw that in a hydrostatic atmosphere the vertical gradient of the geostrophic wind, was related to the horizontal gradient of temperature. That is the thermal wind relationship. What is the sign of the vertical gradient of the wind below the jet stream? (1 point) With this information, is point A warmer or colder than point B (2 points)? Where are the temperature gradients strongest (1 points)? Be sure to justify your decisions.

Answer for Problem 5a

y, north

- p, vertical

-10 m/s

-20 m/s-30 m/s-5 m/s

Between lower and upper point Δu is < 0, Δp < 0, hence vertical gradient is positive.

5) What is the sign of the vertical gradient of the wind below the jet stream? (1 point)

A B

Answer for Problem 5b(Pressure Coordinates)

A. an warmer this Bhence,gradient; re temperatupositive

implies windofgradient verticalPositive

relation chydrostati thewith

1

derivative vertical taking1

y

T

pf

R

p

u

pyfp

u

yfu

g

g

g

5) With this information, is point A warmer or colder than point B (2 points)?

Answer for Problem 5b(Height Coordinates)

A. an warmer this Bhence,gradient; re temperatupositive

implies windofgradient verticalNegative

toleads relation chydrostati theusing

and derivative verticaltaking

1

y

T

fT

g

z

u

y

p

fu

g

g



y, north

- p, vertical

-10 m/s

-20 m/s-30 m/s-5 m/s

A, cooler B, warmer



y, north

- p, vertical

-10 m/s

-20 m/s-30 m/s-5 m/s

A, cooler

5) Where are the temperature gradients strongest (1 points)?

B, warmer

Strong shear.

y

T

pf

R

p

ug


y, north

- p, vertical

-10 m/s

-20 m/s-30 m/s-5 m/s

A, cooler

Strong temperature gradient.

5) Where are the temperature gradients strongest (1 points)?

B, warmer

Questions?

6. Exam Problem (3)Class Average: 2.2

• In several lectures we talked about the transport of trace “gases” such as ozone, smoke, or “dye.” – What is the conservation principle that governs the

behavior of such tracers? (1)– Write down the conservation equation for water vapor.

(1)

• As water vapor changes phases between liquid, gas, and ice, energy is absorbed and released from the atmosphere.– Specifically, what term in which of the equations of

motion represents this energy exchange? (1)

6a. Answer

– What is the conservation principle that governs the behavior of such tracers?

• Conservation of Mass (1)

4) (Eq. u Dt

D

6b. Answer

– Write down the conservation equation for water vapor. (1)

– PH2O: Production of water vapor (source term)

– LH2O: Loss of water vapor (sink term)

OHOH LPDt

OHD22

)( 2

6c. Answer

• As water vapor changes phases between liquid, gas, and ice, energy is absorbed and released from the atmosphere.– Specifically, what term in which of the equations of

motion represents this energy exchange? (1)• The diabatic heating term, J, in the Thermodynamic

equation

5) (Eq. or T

J

Dt

Dp

P

R

Dt

DT

T

c J

Dt

Dp

Dt

DTc p

v

Questions?

Roadmap to the Second Exam

• Exam 2 is scheduled for 16 November (Friday)• This exam will cover mostly chapter 4 in Holton,

specifically:– Holton Section 4.2: Vorticity– Holton Section 4.4: Vorticity equation

• tangential Cartesian coordinates• pressure coordinates• scale analysis in middle latitudes

– Holton Section 4.5: Vorticity in barotropic fluids– Holton Section 4.3: Potential vorticity

Roadmap to the Second Exam

• Exam 2 is scheduled for 16 November (Friday)

• If we have time, we may delve into chapter 6, section 2:– Quasi-geostrophic approximation– Quasi-geostrophic vorticity equation

Note some things we are NOT going to do in detail.

• Holton Section 4.1: Circulation

• Holton Section 4.6: The baroclinic (Ertel) potential vorticity equation

Introduction to Vorticity(From the Detroit NWS forecast office glossary of terms)

Vorticity: Simply put, the measure of rotation of an air parcel about a vertical axis. A parcel rotating clockwise is said to have negative vorticity, and a parcel rotating counterclockwise is said to have positive vorticity. There are two types of vorticity; shear vorticity, which arises from changes in wind speed over a horizontal distance, and curvature vorticity, which is due to turning of the wind flow.

There are parts of this definition that are not (strictly speaking) true…

We will be getting a bit more technical in the coming weeks…

Question: why is vorticity important?


• Vorticity Maximum: (VORT MAX) An area of maximum positive vorticity. The terms vort max and short wave are often used interchangeably. Areas downwind of a vort max experience positive vorticity advection (and rising motion), while areas upwind of a vort max experience negative vorticity advection (and sinking motion).


• Vorticity Maximum: (VORT MAX) An area of maximum positive vorticity. The terms vort max and short wave are often used interchangeably. Areas downwind of a vort max experience positive vorticity advection (and rising motion), while areas upwind of a vort max experience negative vorticity advection (and sinking motion).

Why is vorticity important?

• Positive vorticity is associated with cyclonic rotation in the northern hemisphere (low pressure systems)– Predict changes in vorticity = predict low and high pressure

systems– The first computer forecasts only predicted the changes in

vorticity—and did a decent job…

• Conservation of vorticity tells us how weather systems interact with mountains

• Can diagnose large scale vertical motion by looking at the horizontal advection of vorticity…(this is not exactly true—we will be getting a bit more technical in the coming weeks/months…)

Key questions:

• If vorticity is important, then– How is positive/negative vorticity generated?– How do we describe the time rate of change

of vorticity?– How do we describe conservation of vorticity

(is vorticity conserved following the motion?)– What is the role of the Earth’s rotation?

Documents

AOSS 401, Fall 2007 Lecture 15 October 17 , 2007