AOSS 401, Fall 2007Lecture 4

September 12, 2007

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

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

dposselt@umich.edu734-936-0502

Class News

• Posselt office hours: Tues/Thurs AM and after class– If you are coming from outside the building for office

hours (central or north campus), please email or call ahead

• Class cancelled Friday 14 September• No office hours Thursday 13 September

– I will be available during regular class time Friday• Homework 1 due today (Questions?)• Homework 2 posted by the end of the day

– Under “resources” in homework folder• Due Monday (September 17, 2007)

Weather

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

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

• Weather Underground– http://www.wunderground.com/cgi-bin/findweather/get

Forecast?query=ann+arbor– Model forecasts: http://

www.wunderground.com/modelmaps/maps.asp?model=NAM&domain=US

Outline

1. Review• Momentum equation(s)• Geopotential and atmospheric thickness• Transformation of vertical coordinates

2. Material Derivative• Lagrangian and Eulerian reference frames• Material (total, substantive) derivative• Mathematical tools needed for Homework 2

From last time

Our momentum equation

jikuu

fufvgpdt

d )(

1 2

Acceleration (change in

momentum)

Pressure Gradient Force: Initiates Motion

Friction/Viscosity: Opposes Motion

Gravity: Stratification

and buoyancy

Coriolis: Modifies Motion

Surface Body Apparent

Our momentum equation

jikuu

fufvgpdt

d )(

1 2

Surface Body Apparent

This equation is a statement of conservation of momentum.

We are more than half-way to forming a set of equations that can be used to describe and predict the motion of the atmosphere!

Once we add conservation of mass and energy, we will spend the rest of the course studying what we can learn from these equations.

Review: Vertical Structure and Pressure as Vertical Coordinate

Vertical Structure and Pressure as a Vertical Coordinate

• Remember, we defined the geopotential as

• And we were able to use hydrostatic balanceand the ideal gas law to show

zgdzzgdzd0

)(,

gdz

dp

pdRTp

RTdpdpdgdz ln

Vertical Structure and Pressure as a Vertical Coordinate

• Integrate from pressure p1 to p2 at heights z1 and z2

• From the definition of geopotential we get thickness and the fact that thickness is proportional to temperature

• So, hydrostatic balance and the ideal gas law form the basis for the relationship between and

2

1

ln12

p

p

pdTg

RZZ

2

1

ln)()(

ln

12

p

p

pTdRzz

pRTdd

p

Pressure Gradient in Pressure Coordinates

• Remember from Monday: horizontal pressure gradient force in pressure coordinates is the gradient of geopotential

• Remember, if we have hydrostatic balance:

pz

pz

yy

p

xx

p

1

1

dp

d

Pressure Coordinates: Why?

• From Holton, p2:“The general set of … equations governing the motion of the atmosphere is extremely complex; no general solutions are known to exist. …it is necessary to develop models based on systematic simplification of the fundamental governing equations.”

• Two goals of dynamic meteorology:

1. Understand atmospheric motions (diagnosis)2. Predict future atmospheric motions (prognosis)

• Use of pressure coordinates simplifies the equations of motion

Pressure Coordinates: Why?

fuy

p

dt

dv

fvx

p

dt

du

zz

zz

)1()(

)1()(

Horizontal momentum equations (u, v), no viscosity

Height (z) coordinates

fuydt

d

fxdt

du

pp

pp

)()v

(

v)()(

Pressure (p) coordinates

Density is no longer a part of the equations of motionHidden inside the geopotential… We will see that thissimplifies other relationships as well…

New Material: Holton Chapter 2

• Lagrangian and Eulerian Points of View

• Material (total) derivatives

• Review of key mathematical tools

Vector Momentum Equation(Conservation of Momentum)

jikuu

fufvgpdt

d )(

1 2

Vector Momentum Equation(Conservation of Momentum)

Coordinate system is defined as tangent to the Earth’s surface

jikuu

fufvgpdt

d )(

1 2

xiyj

z k

eastnorth

Local vertical

Velocity (u) = (ui + vj + wk)

Have entertained the possibility of several vertical coordinates z, p, …

Previously: Conservation of Momentum

Consider a fluid parcel moving along some trajectory.

Now we are going to think about fluids.

Consider a fluid parcel moving along some trajectory

(What is the primary force for moving the parcel around?)

CurvCorkuu

gpdt

d)(

1 2

Consider several trajectories

How would we quantify this?

Use a position vector that changes in time

Parcel position is a function of its starting point.The history of the parcel is known

Lagrangian Point of View

• This parcel-trajectory point of view, which follows a parcel, is known as the Lagrangian point of view.

• Benefits:– Useful for developing theory – Very powerful for visualizing fluid motion– The history of each fluid parcel is known

• Problems:– Requires considering a coordinate system for each parcel– How do you account for interactions of parcels with each other?– How do you know about the fluid where there are no parcels?– How do you know about the fluid if all of the parcels bunch

together?

Lagrangian Movie:Mt. Pinatubo, 1992

Consider a fluid parcel moving along some trajectory

Could sit in one place and watch parcels go by.

How would we quantify this?

• In this case:• Our coordinate system does not change• We keep track of information about the atmosphere at a

number of (usually regularly spaced) points that are fixed relative to the Earth’s surface

Eulerian Point of View

• This point of view, where is observer sits at a point and watches the fluid go by, is known as the Eulerian point of view.

• Benefits:– Useful for developing theory– Requires considering only one coordinate system for all parcels– Easy to represent interactions of parcels through surface forces– Looks at the fluid as a field.– A value for each point in the field – no gaps or bundles of

“information.”

• Problems– More difficult to keep track of parcel history—not as useful for

applications such as pollutant dispersion…

An Eulerian Map

Why Consider Two Frames of Reference?

• Goal: understanding. Will allow us to derive simpler forms of the governing equations

• Basic principles still hold: the fundamental laws of conservation– Momentum– Mass– Energy

• are true no matter which reference frame we use

Movies Eulerian vs. Lagrangian

Eulerian Lagrangian

Why Lagrangian?

• Lagrangian reference frame leads to the material (total, substantive) derivative

• Useful for understanding atmospheric motion and for deriving mass continuity…

On to the Material Derivative…

Material Derivative

Δy

Δx

Consider a parcel with some property of the atmosphere, like temperature (T), that moves some distance in time Δt

x

y

Material Derivative

zz

Ty

y

Tx

x

Tt

t

TT

HigherOrderTerms

Assume increments over Δt are small, andignore Higher Order Terms

We would like to calculate the change in temperature over time Δt, following the parcel.

Expand the change in temperature in a Taylor series around the temperature at the initial position.

Material Derivative

t

z

z

T

t

y

y

T

t

x

x

T

t

T

t

T

Divide through by Δt

dt

dz

z

T

dt

dy

y

T

dt

dx

x

T

t

T

dt

dT

Take the limit for small Δt

Material Derivative

Dt

Dz

z

T

Dt

Dy

y

T

Dt

Dx

x

T

t

T

Dt

DT

Introduce the convention of d( )/dt ≡ D( )/Dt

This is the material derivative: the rate of change of T following the motion

Material Derivative

Tt

T

Dt

DT

z

Tw

y

Tv

x

Tu

t

T

Dt

DT

U

Remember, by definition:

wDt

Dzv

Dt

Dyu

Dt

Dx ,,

and the material derivative becomes

LagrangianEulerian

Material Derivative (Lagrangian)

Dt

DTMaterial derivative, T change following the parcel

Local Time Derivative (Eulerian)

t

TT change at a fixed point

Change Due to Advection TU Advection

COLD

WARM

A Closer Look at Advection

z

Tw

y

Tv

x

TuT

U

Expanding advection into its components, we have

Change Due to Advection TU Advection

y

Tv

x

Tu

Class Exercise: Gradients and Advection

• The temperature at a point 50 km north of a station is three degrees C cooler than at the station.

• If the wind is blowing from the north at 50 km h-1 and the air is being heated by radiation at the rate of 1 degree C h-1, what is the local temperature change at the station?

• Hints:– You should not need a calculator– Use the definition of the material derivative and of

advection

Material Derivative

TDt

DT

t

T

z

Tw

y

Tv

x

Tu

Dt

DT

t

T

z

Tw

y

Tv

x

Tu

t

T

Dt

DT

U

We will use this again later…

Can be rewritten in terms of the local change

Advection: A Recent ExampleSix-hour time temperature change at St. Cloud, MN

1100 UTC 1200 UTC 1300 UTC

1400 UTC 1500 UTC 1600 UTC

Return to the Momentum Equation

CurvCokuu

gpdt

d)(

1 2

Remember, we derived from force balances

This is in the Lagrangian reference frame

CurvCokuuuu

gpt

)(1 2

In the Eulerian reference frame, we have

Non-linearThis comes from Eulerian point of view

Homework 2:Mathematical Tools

• Problem 2 in homework 2 asks you to expand various vector operators

• A quick review of these follows

Gradient: Three-Dimensional Partial Spatial Derivative

• A vector operator defined as

• The gradient of a scalar (f) is a vector

kjizyx

kjiz

f

y

f

x

ff

Dot Product

• The divergence is the dot product of the gradient with another vector

• The dot product of two vectors A and B is

zzyyxx

zyx

zyx

BABABA

BBB

AAA

BA

kjiB

kjiA

Laplacian: Divergence of a Gradient

• Three-dimensional partial spatial second derivative.

• Since it is a dot-product, it is NOT a vector itself…

• The Laplacian of a scalar (f) is

2

2

2

2

2

22

z

f

y

f

x

fff

Curl (Cross-Product)

• The curl will be closely related to rotation—we will use this extensively when we cover vorticity

• The result of taking the curl is a vector that is perpendicular (orthogonal) to both of the original vectors

• The direction of the resulting vector depends on the order of operations…

• We will return to this in more detail later…ABBA

Curl (Cross-Product)

For vectors A and B

the curl is

)(

)(

)(

xyyx

zxxz

yzzy

BABA

BABA

BABA

k

j

i

BA

zyx

zyx

BBB

AAA

kji

BA

Same as the determinant

Next time

• Conservation of mass (the continuity equation) (Holton, 2.5.1, 2.5.2)

• Scale analysis (Holton, 2.4, 2.5.3)

• Reversing these (compared to Holton) – derivation of the continuity equation uses the

distinction between Eulerian and Lagrangian reference frames

– Do this while the material is relatively fresh…