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Atmospheric Science 4310 / 7310

- Atmospheric Thermodynamics
- By
- Anthony R. Lupo

Syllabus

- Atmospheric Thermodynamics
- ATMS 4310
- MTWR 900 950 / 4 credit hrs.
- Location 1-120 Agruculture Building
- Class Ref 15505
- Instructor A.R. Lupo
- Address 302 E ABNR Building
- Phone 88-41638
- Fax 88-45070
- Email lupo_at_bergeron.snr.missouri.edu or

LupoA_at_missouri.edu - Homepage www.missouri.edu/lupoa/author.html
- Class Homepage www.missouri.edu/lupoa/atms4310.h

tml - Office hours MTWR 1000 1050
- 302 E ABNR Building

Syllabus

- Grading Policy Straight
- 97 100 A 77 79 C
- 92 97 A 72 77 C
- 89 92 A- 69 72 C-
- 87 89 B 67 69 D
- 82 87 B 62 67 D
- 79 82 B- 60 62 D-
- Grading Distribution
- Final Exam 20
- 2 Tests 40
- Homework/Labs 35
- Class participation 5 (Note, you WILL lose 1

point for each unexcused absence, up to 5 points.

This IS a half-letter grade, keep that in mind!) - Attendance Policy Shouldnt be an issue!

Syllabus

- Texts
- Holton, J.R., 2004 An Introduction to Dynamic

Meteorology, 4th Inter, 535 pp. (Required) - Bluestein, H.B., 1992 Synoptic-Dynamic

Meteorology in the Mid-latitudes Vol I Priciples

of Kinematics and Dynamics. Oxford University

Press, 431 pp. - Hess, S.L., 1959 An Introduction to Theoretical

Meteorology. Robert E. Kreiger Publishing Co.,

Inc., 362 pp. - Zdunkowski, W., and A. Bott, 2003 Dynamics of

the Atmosphere A course in Theoretical - Meteorology. Cambridge University Press, 719 pp.

(a good math review) - Zdunkowski, W., and A. Bott, 2004 Thermodynamics

of the Atmosphere A course in Theoretical

Meteorology. Cambridge University Press, 251 pp. - Various relevant articles from AMS and RMS

Journals. - Course Prerequisites
- Atmospheric Science 1050, Calculus through Math

1700, Physics 2750, or their equivalents. Senior

standing or the permission of the Instructor.

Syllabus

- Calendar Wednesday is Lab exercise day
- Week 1 21 22 23 24 August ? Introduction and

Friday makeup arrangements. Intro. To Atms. 4310.

Lab 1 The Thermodynamic diagram and upper air

information. - Week 2 28 29 30 31 August / September ? Fri.

makeup 1, 1 September. Lab 2 Adiabatic Motions

in the Atmosphere. - Week 3 hh 5 6 7 September ? Mon., Labour Day

Holiday / Fri makeup 2, 8 September. Lab 3 The

Thermodynamic Diagram Examining Moist Processes. - Week 4 11 12 13 14 September ? Fri. makeup 3,

15 September, Lab 5 Lab 4 The Thickness Equation

and its Uses in Operational Meteorology. (move

up other labs) - Week 5 18 19 20 21 September ? Friday make up

4, 22 September, Lab 5 the Lapse Rates of Special

Atmospheres. Test 1 22 Sept., covering material

to 19 Sept.? - Week 6 25 26 27 28 September ? Friday 29

September makeup 5. - Lab 6 Using Thermodynamic diagrams to Determine

Water Vapor Variables. - Week 7 2 3 4 5 October ? Friday makeup 6, 6

October. Lab 7 Estimating Vertical Motions Using

the First Law of Thermodynamics.

Syllabus

- Week 8 nn nn nn nn October ? No Class, UCAR-NCAR

member rep meetings and Heads and Chairs. Lab 8

Atmospheric Stability I Special Forecasting

Problems Fog Formation. - Week 9 nn nn nn nn October ? Gone to Cleveland,

OH NWA meet. - Lab 9 Atmospheric Stability II Special

Forecasting Problems Air Pollution. - Week 10 23 24 25 26 October ? Makeup 7, 27

October - Lab 10 Severe Weather The Synoptic-Scale sets

the table. - Week 11 30 31 1 2 October / November ? Makeup

8, 3 November Test covering material to 1

November. Lab 11 Using Thermodynamic diagrams in

forecasting Convective Outbreaks. - Week 12 nn nn nn nn November ? Severe and Local

Storms Conference in Saint, Louis, MO. Lab 12

Estimating Various Stability Indicies in

real-time. - Week 13 13 14 15 16 November ? Makeup number 9,

17 November Lab 13 Severe Weather I Using

thermodynamic diagrams Super Cell Formation and

Wind Gust Estimation. - Week 14 hh hh hh hh November ? No classes Turkey

day week! - Week 15 27 28 29 30 November / December ? Make

up number 10, 1 December. Lab 14 Severe Weather

II Using thermodynamic diagrams Hail Formation - Week 16 4 5 6 7 December ? Makeup 11, 8 Dec.,

Final 8 Dec.?

Syllabus

- ATMS 4310 Final Exam
- The Exam will be quasi-comprehensive. Most of the

material will come from the final third of the

course, however, important concepts (which I will

explicitly identify) will be tested. All tests

and the final exam will use materials from the

Lab excercises! Thus, all material is fair game!

The final date and time is - Friday, 15 December 2006 1030 am to 1230 pm

in ABNR 1-120 - University Important Dates Calendar
- August 14-18 FS2006 Regular Registration
- August 16 Residence Halls open 900 a.m.
- August 18 Easy Access registration - noon - 600

p.m. - August 21 Classwork begins 800 a.m.
- August 21 Late Registration and Add/Drop - Late

fee assessed - beginning August 21
- August 28 Last day to register, add, or change

sections - August 29-Sept. 25 Drop Only
- September 4 Labor day Holiday
- September 5 Last day to change grading option
- September 18 (Census Day) - Last day to register

for CDIS courses - for Fall

Syllabus

- TBA WS2007 Early Registration Appointments
- October 30 Last day to withdraw from a course -

FS2006 - November 15 Last day to change divisions
- November 18 Thanksgiving recess begins, close of

day - November 27 Classwork resumes, 800 a.m.
- December 8 Fall semester classwork ends
- December 8 Last day to withdraw from University
- December 9 Reading Day
- December 11 Final examinations begin
- December 15 Fall semester ends at close of day
- December 15-16 Commencement Weekend
- Please note This calendar is subject to change

Syllabus

- Syllabus
- Introductory and Background Material, including a

math review (Calculus III) - The Thermodynamics of Dry Air
- Hydrostatics
- The Thermodynamics of Moist Air
- Static Stability and Convection
- Vertical Stability, Instability, and Convection
- Cloud Microphysics
- The Thunderstorm and Non-hydrostatic Pressure
- These topics will be taught if there is time.

All Lecture schedules are tentative!

Syllabus

- Special Statements
- ADA Statement (reference MU sample statement)
- Please do not hesitate to talk to me!
- If you need accommodations because of a

disability, if you have emergency medical

information to share with me, or if you need

special arrangements in case the building must be

evacuated, please inform me immediately. Please

see me privately after class, or at my office. - Office location 302 E ABNR Building Office

hours ________________ - To request academic accommodations (for example,

a notetaker), students must also register with

Disability Services, AO38 Brady Commons,

882-4696. It is the campus office responsible for

reviewing documentation provided by students

requesting academic accommodations, and for

accommodations planning in cooperation with

students and instructors, as needed and

consistent with course requirements. Another

resource, MU's Adaptive Computing Technology

Center, 884-2828, is available to provide

computing assistance to students with

disabilities. - Academic Dishonesty (Reference MU sample

statement and policy guidelines) - Any student who commits an act of academic

dishonesty is subject to disciplinary action.

Syllabus

- The procedures for disciplinary action will be in

accordance with the rules and regulations of the

University governing disciplinary action. - Academic honesty is fundamental to the activities

and principles of a university. All members of

the academic community must be confident that

each person's work has been responsibly and

honorably required, developed, and presented. Any

effort to gain an advantage not given to all

students is dishonest whether or not the effort

is successful. The academic community regards

academic dishonesty as an extremely serious

matter, with serious consequences that range from

probation to expulsion. When in doubt about

plagiarism, paraphrasing, quoting, or

collaboration, consult the instructor. In cases

of suspected plagiarism, the instructor is

required to inform the provost. The instructor

does not have discretion in deciding whether to

do so. - It is the duty of any instructor who is aware of

an incident of academic dishonesty in his/her

course to report the incident to the provost and

to inform his/her own department chairperson of

the incident. Such report should be made as soon

as possible and should contain a detailed account

of the incident (with supporting evidence if

appropriate) and indicate any action taken by the

instructor with regard to the student's grade.

The instructor may include an opinion of the

seriousness of the incident and whether or not

he/she considers disciplinary action to be

appropriate. The decision as to whether

disciplinary proceedings are instituted is made

by the provost. It is the duty of the provost to

report the disposition of such cases to the

instructor concerned.

Syllabus

- Lab Exercise Write-up Format All lab write-ups

are due at the beginning of the next lab

Wednesday. Grading format also given. - Total of 100 pts Name
- Lab
- Atms 4310
- Neatness and Grammar 10 pts Date Due
- Title
- Introduction brief discussion of relevant

background material (5 pts) - Purpose brief discussion of why performed (5

pts) - Data used brief discussion of data used if

relevant (5 pts) - Procedure (15 pts)
- 1.
- 2.

Syllabus

- Results brief discussion of results (50 pts)
- observations
- discussion (answer all relevant questions here)
- Summary and Conclusions (10 pts)
- summary
- conclusions
- Write-ups need to be the appropriate length for

the exercise done. If one section does not apply,

just say so. However, one should never exceed 6

pages for a particular write up. Thats too

much! Finally, answer all questions given in the

assignment.

Day 1

- Thermodynamics ? the study of initial and final

equilibrium states of a "system" which has been

subjected to a specified energy process or

transformation. - System ? a specific sample of matter (air

parcels) - We will concentrate on the macroscale or parcel

properties only! We will not look at the

microscale (molecular level) thats atmospheric

physics (Dr. George, Dr. Fox).

Day 1

Day 1

- Variables of state (thermodynamic variables)
- pressure (hPa, mb) (Force Area-1)
- Temperature (oC, K, oF)
- Volume (typically m3 kg-1), but typically

assume unit mass)

Day 1

- Laws of thermodynamics
- Equation of State
- 1st law of thermodynamics (conservation of

energy) - 2nd law of thermodynamics (entropy) (direction of

heat flow) (warm to cold) - We will review Dimensions and Units, and

conventions.

Day 1

- Atmospheric science derives a set of standard

measurements or unit system, such that everyone

everywhere will be on the same page. - AMS endorsed the SI (Systeme International) or

International system of Units (BAMS, 1974, Aug.) - The basic units are Length, Mass, Time (meter,

m kilo, kg second, s)

Day 1

- A derived unit combines basic units example,

pressure - Force /Area kg m s-2 / m2 kg m-1 s-2

Pascals - 1000 Pa 1 kPa 10 hPa 10 mb
- Temperature (Kelvin, or absolute scale Celsius

(1742) Farenheit (1714)). - Coordinate System (Cartesian)

Day 1

- Coordinate system tangent to Earths surface

which is really a sphere (curvature for most

applications and approximations can be

neglected). - Cartesian coordinates
- x,y,z,t x,y,p,t, or x,y,q,t
- Could also use natural coordinates
- s(treamline),n(normal),z,t
- Spherical coordinates
- r(adius),q(longitude),f(latitude)

Day 2

- Wind
- Wind direction direction from which the wind

blows, and compass direction, not Cartesian! - West wind is blowing from 270o.
- Wind direction increasing with time or height

veering decreasing with time or height backing

Day 2

- Remember Direction of math 270 compass

(meteorology) direction. - Vector representation in Geophysical Fluid

Dynamics - Remember the atmosphere is a fluid, and a fluid

is liquid or gas. Thus, the primitive equations

will be valid in any atmosphere, terrestrial

(extraterrestrial).

Day 2

- Scalar quantity ? A quantity with magnitude only

(e.g., wind - speed has units m s-1) (zero order tensor)
- Vector ? (first order Tensor) A quantity with

magnitude and direction (e.g., wind velocity)

Day 2

- Wind

Day 2

- Dyadic? (2nd order Tensor) has a magnitude and

two directions! - Example stress (Force per unit area), where A is

the vector of some magnitude equal to the area

and in the direction of the normal. - In English Magnitude, direction (1) of the

force, and (2) on which surface applied

Day 2

- An example
- Example Stress Force Area times Stress (has

same units as pressure)

Day 2

- Vector Analysis
- Vector Notations
- A A
- A magnitude of A

Day 2

- Vectors are equal if they have equal magnitude

and directions!! - The unit vector any vector of unit length!
- where is a vector of unit

length - Cartesian Unit vectors vectors of

unit length in the positive x,y,z direction,

respectively.

Day 2

- Natural coordinates are
- Vector components (2 dimensions), but we can

extend to infinite number of directions - Magnitude of A ? A ? (Ax2 Ay2)

Day 2

- Vector addition and subtraction
- 1) A B C
- 2) AB B A C
- 3) A - B C

Day 2

- Addition Subtraction

Day 2

- Heres how
- Associative rule
- (AB) C A (BC)
- Negative Vector Is a vector of the same

magnitude, but opposite direction.

Day 2

- Vector multiplication
- Scalar x Vector
- In the atmospheric sciences The wind vector (2-D

3 D)

Day 2 /3

- Vector products
- The dot product (also the scalar or inner)

product - A dot B ABcos(q)
- Physically The dot product is the PROJECTION of
- vector B onto Vector A in direction of A!

Day 2/3

- Projection (mathworld.wolfram.com) (excellent

math site)

Day 3

- Properties of the Dot Product
- commutative
- A dot B B dot A
- associative
- A dot (B dot C) (A dot B) dot C
- distributive
- A dot (B C) A dot B A dot C

Day 3

- Dot product of perpendicular vector 0
- In order for the dot product to have a value, the

B vector must have a component parallel to vector

A! - Recall cos(0o) 1 and cos(90o) 0
- Thus, i dot j, and j dot k, etc 0, and i

dot i 1, etc

Day 3

- Orthogonal Vectors (Orthogonality property) When

the angle between two vectors is 90o, or the dot

product is zero, two vectors are said to be

orthogonal. - Other Dot Product Rules
- 1) A dot A AA cos(0) A2
- 2) A dot mA mA2

Day 3

- Dot product of two vectors (heres how)
- Remember Foil?
- AxBx (i dot i) Ax By ( i dot j) Ay Bx (j dot

i) Ay By (j dot j)

Day 3

- Ok, now you try ?
- Answer????? Ax
- Cross Product (or vector product)
- A x B ABsin(q)

Day 3

- What is it (again courtesy of Mathworld site)?

Day 3

- Einstien notation permutation
- Q "What do you get when you cross a

mountain-climber with a mosquito?" - A "Nothing you can't cross a scaler with a

vector," - Q "What do you get when you cross an elephant

and a grape?" - A "Elephant grape sine-of-theta."

Day 3

- The cross product of two vectors is a third

vector that is mutually perpendicular to the two

vectors and the plane containing these vectors. - Q Remember your Physics?
- The positive direction of A x B may be determined

by the right hand (or corkscrew) rule. Just

curl your fingers from A to B, and your right

thumb (vector C) is the result! (ayyyy)

Day 3

- Evaluating the cross product of A and B in the

cartesian coordinate system. - The vector or cross product is NOT commutative!

Day 3

- Try the right hand rule to show that we cannot

switch order - Also, remember sin(-90o) -1.0
- The cross product is distributive
- A X (B C) (A x B) (A x C)

Day 3/4

- The cross product of a vector with itself equals

0!! - q 0 so, sin(0) 0, or A x A

AAsin(0) 0 - Cross products of unit vectors and the cyclical

property of the cross product - 1) i x i 0, i x j k, i x k -j
- 2) j x i -k, j x j 0, j x k i
- 3) k x i j, k x j -i, k x k 0

Day 4

- A x B C , B x C A, C x A B
- Another trick or property
- Unit vector k x Ah Horizontal vector of length

A turned 90o to the left of A (try it with

right hand rule - horizontal vector!!!) - -k x Ah A vector of length A turned 90o to the

right of A

Day 4

- Multiple Vector Products
- Scalar Triple product
- A dot (B x C) a scalar value
- This is also cyclical
- A dot (Bx C) B dot (C x A) C dot (A x B)

Day 4

- Or dot and cross product may be interchanged
- A dot ( B x C) (A x B) dot C
- Triple vector product
- (A x (B xC)) Vector quantity
- A x (B xC) (A dot C ) B (A dot B) C
- The result is a third vector in the plane of B

and C!!!

Day 4

- But,
- (A x B) x C not equal to A x (B x C)
- since for former result is in the plane of A and

B!!

Day 4/5

- The mathematical description of the Atmosphere
- We must eventually develop from fundamental

physical laws and concepts (first principles),

the 5 vector (7 scalar) equations of geophysical

fluid dynamics, and describe and understand the

behavior of the atmosphere through the

manipulation of these equations. - Describing the atmosphere in terms of the

distribution in space and time of certain

properties of the atmosphere.

Day 5

- Independent variables ? these are the basis of

(or describe) our coordinate system. - Well use Cartesian system (x,y,z,t)
- and a right handed coordinate system

Day 5

- Dependent variables ? depend on your position in

space and time, and can be described as a

function of the independent variables. - In atmospheric science u,v,T,r,q,P,w,or w
- Example u(x,y,z,t)

Day 5

- Invariance ? A quantity that does not change if

measured in a different coordinate system - e.g., rotationally invariant ? quantities

that do not change even if coordinate system

rotates - Q Which variable might be rotationally

invariant? - A T(x,y,z,t) T(x,yz,t)
- Galilean invariant ? quantities that do not

change even if coordinate system is moving

horizontally

Day 5

- Important Definintion!
- Conserved ? a quantity that does not change with

time. (e.g., Potential Temperature and adiabatic

motions) - Conservation ? the change in some quantity with

time equals 0!

Day 5

- Conservation steady state balance (between

sources and sinks!) - where Q any quantity

Day 5

- The derivative (A review)
- Let take a quantity Q(x,y,z,t)
- Now one needs to take the Total Derivative. In

order to do this, we must use the Chain Rule!

Remember this?

Day 5

- Total derivative is (in x,y,z,t,)
- advective derivative Eulerian
- (in x,y,p,t)

Day 5

- in (x,y,q,t)
- in natural coordinates

Day 5

- now lets give our pens a break
- 1) u dx /dt,
- 2) v dy / dt,
- 3) w dz / dt
- In Mathematics
- The total derivative (heavy D) (substantial,

individual, material) is exact, thus the

derivative not path dependent!

Day 5

- Exactness!

Day 5

- But, if path dependent, total derivative has no

meaning, and we write with a small d. - If path dependent, then the process is sensitive

to the initial starting place and it corresponds

to (generally) one outcome. - But I, like most atmospheric scientists use the

notation d for a total derivative regardless.

Day 5

- The partial derivative is
- - the change in one variable or coordinate w/out

regard to the other components. - looks like..
- in the eyes of the partial derivative here, where

C z2y2t2.

Day 5/6

- Differentiation of Vectors
- The normal rules of differentiation apply, but

you must preserve the order when cross product is

applied. - Let vector A be time dependent, i.e., vector A is

changing size and/or direction with time and

space

Day 6

- If our coordinate system is not changing (i.e.,

i,j,k constant). - then Ax,Ay,Az change (of course!)
- (oh, you fill in the rest!)

Day 6

- Some more fun rules
- 1)
- 2)
- 3)

Day 6

- but. if i,j,k are changing.
- Position, Velocity, Acceleration
- The position vector R, the velocity vector V and

the Acceleration vector A

Day 6

- Position vector
- The velocity vector
- The acceleration vector

Day 6

- The del operator
- Also known as the Hamiltonian, gradient, or

nabla operator. - Let us define a differential operator with vector

properties

Day 6

- The del operator has no physical meaning until it

operates on another quantity such as a scalar or

another vector! (A ghost vector) - Operating on a scalar

Day 6

- Del Q is now a 3-D vector whose direction is in

the direction of the maximum increase of Q and

whose magnitude is equal to the rate of change of

Q per unit distance in that direction.

Day 6

- Del Q in normal (perpendicular to lines of Q). On

a 2 D surface is perpendicular to Q isolines. - Then in plane English delQ is simply the slope

of Q on some planar surface. The first derivative

in space (slope) is analogous to the first

derivative in time (velocity).

Day 6/7

- Now a proof! (show velocity vector is

perpendicular to gradient vector) - Step 1
- But, dQ 0 on a line of Q (correct?)

Day 7

- Now we know that
- a) each point of a surface of constant Q can be

defined by the position vector. - b) then on a Q surface dr (or V), must be on the

surface of Q, so dr must lie on the Q surface.

Day 7

- c) then dr dot del Q on Q surface
- So dr dot delQ dQ, this is the definition of

the total derivative. - but dQ 0 on Q surface as discussed above.

Day 7

- Therefore since dr and delQ separately are not 0,

but their dot product IS 0, they must be

perpendicular (or orthogonal)!!! - Furthermore, we know that delQ was perpendicular

to lines of Q, thus dr or Velocity, must be

parallel to lines of Q! - Point proved!

Day 7

- Advection of a scalar quantity
- 3-D transport of some quantity
- In Atms Sci, in (x,y,p,t) coordinates, advection

and flux are equivalent.

Day 7

- A 2-D example of advection
- Some other names for advective quantity
- Convective derivative or Lagrangian!

Day 7

- Two definitions
- Lagrangian measurement ? measurement that moves

with the flow (goes with the flow) - Eulerian ? measurement at a stationary point
- Important Concept!!
- ? If Q is conserved or a conserved property

(invariant with time) time derivative 0. We

also call this steady state.

Day 7

- Thus, if this is true either the source-sinks are

zero, or Equal and opposite. But it also implies

that advection equals the time rate of change.

Day 7

- - or
- where Q Any variable, vector or scalar!

Day 7

- The Laplacian operator
- ? Del dot Del Its a scalar operator! It

typically changes the sign of a function. A

measure of the curvature in a function. - ? as acceleration is to velocity, so is the

gradient operator (slope) to the Laplacian

(curvature)!

Day 7

- The divergence of a vector (del dot V) (a

scalar)! - Velocity divergence
- ? ?

??????? ???

- The curl of the vector V
- The curl of the velocity or vector vorticity

Day 7

- Vertical component of Vorticity (zeta)
- Magnitude in vertical is entirely dependent on

horizontal spatial variations or shears - Consider a case where all the terms contribute

positively

Day 7/8

- Then
- ? Then, the vertical component is POSITIVE for

cyclonic circulations (shear) and negative for

anticylonic circulations. It is the opposite in

the SH.

Day 8

- Rotational Vectors and vectors in Rotation
- Definition of the rotational vector (omega - w) ?

rotational vector has a direction along the axis,

positive in the sense of the Right hand rule, and

its magnitude, omega w, is porportional to

the angular velocity of the rotating system.

(ang. Vel. radians/sec) - The rate of change of a vector A of constant

magnitude due to its changing direction produced

by rotation (omega - W)

Day 8

- Now look down from above at the plane in which

the vector A is rotating.

Day 8

- The magn. Or length of DA DA and for small

angle Dq - Recall from Geometery
- DA A sin(f) Dq
- ? since for small angles tan q q opp. / adj.

Or opp. adj. tan q

Day 8

- So, (now include delta t)
- DA/ Dt A sin(f) (Dq / D t)
- And as Dt goes to 0 ?
- dA/dt A sin(f)(dq /dt)

Day 8

- But dq /dt omega w (the angular velocity) ? So
- dA/dt w A sin (f)
- Then we need to
- ? redefine w as W since were talking about

earth! - ? recall our definition of the cross product!!!

(W x A)

Day 7

- Thus the magnitude of DA/dt DA/dt equals the

mangitudes of W x A!! - dA/dt WxA!

Day 8

- Direction of dA/dt and W x A
- ? Since (W x A is mutually perpendicular to W and

A as is DA/dt, and is positive in the same

direction the two vectors DA/dt and W x A are

equal vectors! - Thus for A of constant magnitude dA/dt W x A

Day 8

- ? The rate of change of vector A in a fixed

(absolute) coordinate system vs. a rotating

coordinate system. (Fixed and absolute are

not good news we cant do this in practice (in

real atms.)). - Atms example coriolis force!!
- dV/dt W x V

Day 8

- OK, theres more than one way to skin a cat ..
- Consider (X,Y,Z) w/ unit vectors (I,J,K)
- Consider another (x,y,z) w/unit vectors (i,j,k).

Allow this one to rotate w/angluar velocity

(omega). - A AXI AYJ AZK Axi Ayj Azk

Day 8

- Now differentiate A w/r/t time (sytem 1 is

const.! System 2 is rotating) - DA/dt AX/dt I dAY/dt J dAZ/dt K dAx/dt i

Ax di/dt etc.. - i,j,k are unit vectors.
- Di/dt W x i and dj/dt W x j and.

Day 8

- (DA/dt)abs dAx/dt i dAy/dt j dAz/dt k (Ax

(W x i) Ay (W x j) Az (W x k) ) - DA/dt abs (dA/dt) (relative to rotating coord.

System) (W x (Axi Ayj Azk) - DA/dt dA/dt (relative to rot.) W x A (rot of

coord system w/r/t vertical)

Day 8/9

- The rate of change of a vector in an absolute

(inertial) frame of ref. Is equal to rate of

chage observed in the rotating system a term

to the cross product of the rotational vector and

the arbitrary vector A! - ? Thus, you have just derived the expression for

the coriolos force! ?

Day 9

- Dimensional Analysis (The rules!)
- 1. All terms of an equation must have the same

dimensions! e.g. potential temp relationship - 2) All exponents are non dimensional
- 3) All log and trig functions are also

non-dimentional.

Day 9

- 4. The dimensions of differentials are the

same as the dimensions of a differentiated

quantity. - e.g., we can say d (ln (T))
- ? Note specific as a prefix implies the

quantity is per unit mass and has the dimensions

of (Q x M-1) e.g., specific volume vol / unit

mass.

Day 9

- ? Valid physical relationships must be

dimensionally consistent with eachother, in other

words X Y must have the same units. Otherwise,

we say X proportional to Y. Or X AY where the

units of AY are the same as X. - Caveat Some proportionalities have consistent

units.

Day 9

- Non-dimensional analysis (Scale analysis)
- ? (Mathematical formalization) Theoreticians like

to look at equations in a non-dimensional sense,

that is we choose characteristic time and space

scales for some phenomena to be studied. - We essentially change coordinate systems
- (x,y) L(x,y) where L is 2000 km
- t Tt where T 100000 sec
- (u,v) U(u,v) U 10 m/s

Day 9

- ? In performing this type of analysis we can

determine what processes are important for du/dt,

for example, in the horizontal equation of motion

on the desired scale (cyclone), we can perform

this type of analysis for particular phenomena. - Informal Scale analysis
- ? Similar to that of non-dimensionalization.

Scale (size) analysis is also a powerful tool

often used in meteorological derivations the

analysis of physical processes. This is a more

informal method than non-dimensional analysis.

Day 9

- ? Suppose we have an equation based on a

physical law or principle (e.g. Newtons 2nd law,

3rd eqn. of motion). This equation is a

generalized equation, valid for many atmospheric

phenomena.

Day 9

- We choose the scale we are interested in, the

consider the order of magnitude, (e.g. the size

and space scale) each term in the equation would

have, for that particular scale of motion (again

typical values, or estimates). - Then, we can neglect the smaller terms and

simplify the equation. Well also know the error

introduced in doing this (ratio of neglected to

retained terms). - The equation is simplified, but now less general,

its the trade-off for a simpler relationship.

Day 9

- Hydrostatic balance
- This equation governs the movement of

synoptic-scale systems (highs/lows (waves in the

westerlies)) and fronts.

Day 9

- Hydrostatic balance the error ?
- Error Neglected Terms / Retained terms
- Error 10-3 / 10 10-4 0.0001 0.01
- ? Thus this is a darned good estimate for

synoptic and meso alpha scales. - ? Important Point! The equation is much simpler,

but its only valid for these scales and has now

lost its generality!

Day 9/10

- Scales of Atmospheric motions
- Scale Horiz Dimension Time
- Planetary 10,000 km weeks - 1 month
- Synoptic 2,000 6000 km 1 to 7 days
- Meso 10 km 2,000 km 1h 1 day
- Micro
- ? Can you think of examples of real phenomena

that fit each category?

Day 10

- Fundamental equations of geophysical

hydrodynamics - ? Seven dependent variables, four independent

variables, seven independent equations - p(x,y,z,t) Pressure (mass)
- r(x,y,z,t) density (mass, thermal)
- T(x,y,z,t) or q(x,y,z,t) (potential) Temperature

(thermal) - M(x,y,z,t) Mixing Ratio (mass, thermal)
- U(x,y,z,t) zonal wind (mass)
- V(x,y,z,t) meridional wind (mass)
- W(x,y,z,t) vertical wind (mass)

Day 10

- Concept Name
- Elemental Kinetic Theory Eqn. Of state

Day 10

- Cons. of Energy 1st Law of Thermo.
- Cons. of Mass Eq. of continuity

Day 10

- Cons. of mass Eq. of water mass

Day 10

- Cons. of momentum Eq. of motion
- A.K.A Navier Stokes Equation, Newtons 2nd

Law, etc.

Day 10

- Von Helmholtz 1858 In principle this is a

mathematically solvable system (closed) given

observed initial state and proper BCs, the

solution should yield all future states of the

system. (The rub ICs and BCs). - ? Thus, forecasting is an initial value problem

(Bjerknes, 1903) - ? These eqns. Are what will be studied in Atms

4310, 4320. These describe behavior of the

atmosphere. - ? Solving these equations (Numerical methods and

modeling classes Atms 4800)

Day 10

- The Thermodynamics of Dry Air (Holton Ch 2- 4)
- Reminder Dry air means DRY air (no moisture)!
- Moist air means water vapor present.
- Dry air (a homogeneous mixture of gasses from 0

80 km up Homosphere)

Day 10/11

- The Heterosphere is above that, gasses separate

by weight (mass) - The Atmosphere its makeup
- Gas (atomic weight) by Vol by mass
- Nitrogen (N2) 28.02 78.1 75.5
- Oxygen (O2) 32.00 20.9 23.1
- Argon (Ar) 39.94 0.93 1.3
- Carbon Dioxide (CO2) 44.01 0.036 0.05

Day 10/11

- ? Many other gases present in very small

quantities (Ne, He, H, O3) they are called Trace

Gases - ? Thus if we calculate the atomic weight of air
- 28.97 kg mol-1
- ? Three of these are very important because

despite the small quantities, they help determine

the temperature structure of the troposphere and

stratosphere H2O, CO2, and O3

Day 11

- H2O is important within the hydrologic cycle,

clouds, rain etc.. Water is the only substance in

earth atmosphere that exists in all three phase

at terrestrial pressures and temperatures. - ? Water Vapor and clouds are important in

determining atmospheric structure due to their

radiative properties (albedo, infrared). - ? Residence time 1 10 days.

Day 11

- ? It is the most important and potent greenhouse

gas, but its not homogeneously distributed! - CO2 has homogeneous concentration. It is

important because of its radiative properties in

infrared. Its residence time near 100 years, thus

important in longer term climate change. But, the

CO2 cycle is not well known yet. - O3 concentrated at 32km up (in stratosphere)

due to solar and chemical reactions. It absorbs

UV and emits infrared and responsible for the

Stratospheres inversion.

Day 11

- ? Near surface it is present in small amounts due

to pollution, but it is highly poisonous. - Moist air
- ? Water vapor extremely variable near 0 near 4

- Thats 0 - 40 g/kg!
- More typical 1 10 g/kg (Td 57F)

Day 11

- The variables of state
- Mass (M)
- ? Density (mass/unit vol) (r)
- ? Specific volume (vol / unit mass) a
- ? Pressure (Force/Area) is due to molecular

collisions and the associated momentum changes

independent of direction (a scalar). The force is

normal to gas container walls. N m-2 1 Pa

and 1mb 102 Pa - or - kg m-1s-1

Day 11

- ? Temperature (T) a measure of the average

internal energy of the molecules obtained during

a state of equilibrium. - ? To read temperature there needs to be

equilibrium established between the system and a

temp sensor. Temp. determines direction of heat

flow. (Kelvin, Celsius) 0o C 273.15 K. - Ideal Gas (Kinetic theory of gasses) is a

collection of molecules that are - completely elastic spheres
- with no attractive or repulsive forces, and
- occupying no volume.

Day 11

- Heat is a form of energy, which can be

transferred from a warmer to colder aubstance.

Heat transfer by (really kinetic energy

transfer) - Radiation ? Transfer of Electro- and Magnetic
- Conduction ? Transfer by molecular motions

(Contact) - Convection ? Transfer by turbulent mixing

(parcels, bulk transport, advection) - Latent (phase changes) ? "hidden heat"

Day 11

- Relationships between our state variables, r,

a, T, and P. - Boyles law
- Robert (Bob) Boyle (1600) said
- if T is constant, then

Day 11

- P1 x V1 P2 x V2 - or
- P x Volume Constant
- Recall, this is possible since Torricelli (1543)

invented the barometer! - So as a result of Boyles Law

Day 11

- As P increases, Volume decreases
- ? ?

Day 11

- Or as Volume increase, then P decreases
- ? ?

Day 11

- Charless Law
- Jaques Charles (1787) but stated formally J.

Gay-Lussac (1802) - (1787 Bonus question what else important

happened in Sept. 1787?) - Jack said if pressure is kept constant then
- Volume1/Volume2 Temperature1/Temperature2

Day 11

- Or
- Volume/Temperature Constant
- Thus, when gas is heated, volume goes up, when

gas is cooled, volume decreases!

Day 11/12

- Combined or Ideal gas law
- Constant Pressure x Volume / Temperature
- Derivation of Ideal Gas law for Atmosphere
- ? Nee Avagadros hypothesis (derived

experimentally, and derivable from kinetic theory

of gasses)

Day 12

- His hypothesis Different gasses, each containing

the same number of molecules, occupy the same

volume at the same temperature and pressure. - Kilogram molecular weight a kmol of material is

its molecular weight expressed in kg. Thus, one

kmol of water is 18.016 kg! - Number of molecules is Avagadros number (Av)
- 6.022x 1026

Day 12

- Universal gas constant
- ? if we have a mixture of gases, each with its

own ideal gas law (which is what Daltons Law

implies!) - Po Volume(gas) m(gas)R(gas)To where gas

1,2,3, etc. - where m is the number of moles of each gas!

Day 12

- Divide each equation by ng where ng is weight of

a kmol of gas. - Po Vg/ng (mg/ng) RgTo
- If each sample consists of 1 mol they have same

number of molecules. Avagadros hypotheses all

have same Volume. For each gas we can write - Po (V1/ng) / To Po (Vo/ng) / To

Day 12

- or Po(Vg/ng) / To ng mgRg
- Since mgRg will be the same by Avagadros

hypothesis - Then mgRg R (Universal Gas constant)
- R 8314.3 J K-1 kmol-1

Day 12

- Thus, we must find the apparent weight of air, or

take a weighted average of the gasses per mol. - Molecular weight 28.97 kg / kmol n
- So then Pa R/n T
- Pa RT
- Or
- P rRT (R is a constant depending

on individual Gas)

Day 12

- ? Or to include the effects of moisture
- P r Rd Tv (Rd 287.04 J K-1 kg-1)
- ? So lets go back to
- Po (Vg/ng) / To mg/ng R
- Can alternatively express RHS as R/ng R

Day 12

- Thus in ideal gas law we must express as
- PV R/n (air) T where air 28.97
- V is volume, where Volume could be anything. Well

well specify some specific volume. (a

vol/unit mass) which equals 1 kg. This is still

volume, we are not changing variables here or

playing fast and loose with the math. - Thus
- Pa R/ (n air) T RdT
- (- or - P r R/n T rRdT)

Day 12

- Thats for dry air. In accounting for moisture

ng 18.016. The amount of water is very

variable, and we could be specifying a different

R for air every time amount of moisture changes. - ?Thats where concept of virtual temperature

comes in. Thus, - P a Rd Tv -or- P r Rd Tv

Day 12

- Q Which is more dense, dry air or moist air?
- Ever heard a baseball announcer talk about the

ball carrying on a warm humid night? - Dry air weights 28.97, throw in 18.016 for moist

air.

Day 12

- In dry air, 1.00 28.97 for dry air 28.97
- Now, say the air was 4 vapor
- -age x mol. wt. Contribution
- 0.96 28.97 27.80

- 0.04 18.016 0.72
- ______________________________
- 28.53 (molecular weightof air vapor)

Day 12/13

- Application Lets derive expression for Virtual

temperature (Wallace and Hobbs p51 52) - Ideal Gas Laws, for water vapor and dry air

Day 13

- And use Daltons Law
- P Pd e
- r rd rv (1)
- so, substitute ideal gas laws into (1) to get
- Pd / (Rd T) (e / (Rv T))

Day 13

- And then
- (P e) / (Rd T) (e / (Rv T))
- use strategic multiplications of each term by

1 - 1st term multiply by P / P
- 2nd term multiply by P Rd / P Rd

Day 13

- then well have a common factor to pull out (P

/ (Rd T)) - and we get
- Then rearrange

Day 13

- And finally

Day 13

- Celsius (1742) Temperature scale (Centigrade)
- Define at P Po 1000 mb at a state of thermal

equilibruim - Pure Ice and water mixture ? temp 0o C
- Pure water and steam mixture at equilibrium ? 100

o C

Day 13

- If P Po alpha varies linearly with temp. Thus
- Y mx b
- T in oC is the slope
- T (Celsius) (at ao) / (a100 ao) 100
- This is the defining expression for Centigrade

scale!

Day 13

- The Absolute or Kelvin Temp Scale
- T oC 100 at / (a100 ao) 100ao / (a100

ao) - VRBL CONST
- ? Or we can use this relationship to re-define

the temperature scale by extrapolating to the

point where all molecular motion stops and

Specific Volume goes to 0! - T (Absolute) VRBL T oC CONST

Day 13

- CONST 273.16
- So,
- K C 273.16
- Show Absolute zero -273.16

Day 13

- Solve for (a) at at
- at ao T oC / (a100 ao) 100
- at ao (1 1 / 273.16ToC)
- so if at 0 (or all molecular motion ceases),

Day 13

- then 11/273.16 ToC 0
- ? Solve T oC -273.16
- and then 273.16 oC 0 Absolute
- or 0 K (Volume of Ideal gas goes to 0)!!

Day 14

- The work done by an expanding gas
- Lets draw a piston

Day 13

- ? Consider a mass of gas at Pressure P in a

cylinder of Cross section A - Now, Recall from Calc III or Physics
- Work force x distance or Work Force dot

distance - So only forces parallel to the distance travelled

do work!

Day 13

- Then,

Day 13

- But, we know that
- Pressure Force / Unit Area
- So then,
- Force P x Area

Day 13

- Total work increment now
- Well,
- Area x length Volume
- soo A ds dVol

Day 13

- Then we get the result
- ? Work
- Lets work with Work per unit mass
- ? Thus, we can start out with volume of only one

1 kg of gas!!!

Day 13

Day 13

Day 13