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Title: Atmospheric Science 4310 7310


1
Atmospheric Science 4310 / 7310
  • Atmospheric Thermodynamics
  • By
  • Anthony R. Lupo

2
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

3
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!

4
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.

5
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.

6
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.?

7
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

8
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

9
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!

10
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.

11
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.

12
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.

13
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.

14
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).

15
Day 1
16
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)

17
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.

18
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)

19
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)

20
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)

21
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

22
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).

23
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)

24
Day 2
  • Wind

25
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

26
Day 2
  • An example
  • Example Stress Force Area times Stress (has
    same units as pressure)

27
Day 2
  • Vector Analysis
  • Vector Notations
  • A A
  • A magnitude of A

28
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.

29
Day 2
  • Natural coordinates are
  • Vector components (2 dimensions), but we can
    extend to infinite number of directions
  • Magnitude of A ? A ? (Ax2 Ay2)

30
Day 2
  • Vector addition and subtraction
  • 1) A B C
  • 2) AB B A C
  • 3) A - B C

31
Day 2
  • Addition Subtraction

32
Day 2
  • Heres how
  • Associative rule
  • (AB) C A (BC)
  • Negative Vector Is a vector of the same
    magnitude, but opposite direction.

33
Day 2
  • Vector multiplication
  • Scalar x Vector
  • In the atmospheric sciences The wind vector (2-D
    3 D)

34
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!

35
Day 2/3
  • Projection (mathworld.wolfram.com) (excellent
    math site)

36
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

37
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

38
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

39
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)

40
Day 3
  • Ok, now you try ?
  • Answer????? Ax
  • Cross Product (or vector product)
  • A x B ABsin(q)

41
Day 3
  • What is it (again courtesy of Mathworld site)?

42
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."

43
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)

44
Day 3
  • Evaluating the cross product of A and B in the
    cartesian coordinate system.
  • The vector or cross product is NOT commutative!

45
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)

46
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

47
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

48
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)

49
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!!!

50
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!!

51
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.

52
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

53
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)

54
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

55
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!

56
Day 5
  • Conservation steady state balance (between
    sources and sinks!)
  • where Q any quantity

57
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?

58
Day 5
  • Total derivative is (in x,y,z,t,)
  • advective derivative Eulerian
  • (in x,y,p,t)

59
Day 5
  • in (x,y,q,t)
  • in natural coordinates

60
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!

61
Day 5
  • Exactness!

62
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.

63
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.

64
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

65
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!)

66
Day 6
  • Some more fun rules
  • 1)
  • 2)
  • 3)

67
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

68
Day 6
  • Position vector
  • The velocity vector
  • The acceleration vector

69
Day 6
  • The del operator
  • Also known as the Hamiltonian, gradient, or
    nabla operator.
  • Let us define a differential operator with vector
    properties

70
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

71
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.

72
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).

73
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?)

74
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.

75
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.

76
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!

77
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.

78
Day 7
  • A 2-D example of advection
  • Some other names for advective quantity
  • Convective derivative or Lagrangian!

79
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.

80
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.

81
Day 7
  • - or
  • where Q Any variable, vector or scalar!

82
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)!

83
Day 7
  • The divergence of a vector (del dot V) (a
    scalar)!
  • Velocity divergence
  • ? ?

84
??????? ???
  • The curl of the vector V
  • The curl of the velocity or vector vorticity

85
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

86
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.

87
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)

88
Day 8
  • Now look down from above at the plane in which
    the vector A is rotating.

89
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

90
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)

91
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)

92
Day 7
  • Thus the magnitude of DA/dt DA/dt equals the
    mangitudes of W x A!!
  • dA/dt WxA!

93
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

94
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

95
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

96
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.

97
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)

98
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! ?

99
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.

100
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.

101
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.

102
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

103
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.

104
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.

105
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.

106
Day 9
  • Hydrostatic balance
  • This equation governs the movement of
    synoptic-scale systems (highs/lows (waves in the
    westerlies)) and fronts.

107
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!

108
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?

109
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)

110
Day 10
  • Concept Name
  • Elemental Kinetic Theory Eqn. Of state

111
Day 10
  • Cons. of Energy 1st Law of Thermo.
  • Cons. of Mass Eq. of continuity

112
Day 10
  • Cons. of mass Eq. of water mass

113
Day 10
  • Cons. of momentum Eq. of motion
  • A.K.A Navier Stokes Equation, Newtons 2nd
    Law, etc.

114
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)

115
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)

116
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

117
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

118
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.

119
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.

120
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)

121
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

122
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.

123
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"

124
Day 11
  • Relationships between our state variables, r,
    a, T, and P.
  • Boyles law
  • Robert (Bob) Boyle (1600) said
  • if T is constant, then

125
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

126
Day 11
  • As P increases, Volume decreases
  • ? ?

127
Day 11
  • Or as Volume increase, then P decreases
  • ? ?

128
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

129
Day 11
  • Or
  • Volume/Temperature Constant
  • Thus, when gas is heated, volume goes up, when
    gas is cooled, volume decreases!

130
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)

131
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

132
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!

133
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

134
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

135
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)

136
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

137
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)

138
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

139
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.

140
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)

141
Day 12/13
  • Application Lets derive expression for Virtual
    temperature (Wallace and Hobbs p51 52)
  • Ideal Gas Laws, for water vapor and dry air

142
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))

143
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

144
Day 13
  • then well have a common factor to pull out (P
    / (Rd T))
  • and we get
  • Then rearrange

145
Day 13
  • And finally

146
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

147
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!

148
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

149
Day 13
  • CONST 273.16
  • So,
  • K C 273.16
  • Show Absolute zero -273.16

150
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),

151
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)!!

152
Day 14
  • The work done by an expanding gas
  • Lets draw a piston

153
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!

154
Day 13
  • Then,

155
Day 13
  • But, we know that
  • Pressure Force / Unit Area
  • So then,
  • Force P x Area

156
Day 13
  • Total work increment now
  • Well,
  • Area x length Volume
  • soo A ds dVol

157
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!!!

158
Day 13
159
Day 13
160
Day 13
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