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Atmospheric Science 4320 / 7320
Anthony R. Lupo
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Syllabus Atmospheric Dynamics ATMS 4320 / 7320 MTWR 9:00 – 9:50 / 4 credit hrs. Location: 123 Anheuser Busch Natural Resources Building Instructor: A.R. Lupo Address: 302 E ABNR Building Phone: 884-1638 Fax: 884-5070 Email: [email protected] or [email protected] Homepage: www.missouri.edu/~lupoa/author.html Class Home: www.missouri.edu/~lupoa/atms4320.html Office hours: MTWR 8:00 – 8:50 or by appointment 302 E ABNR Building 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- < 60 F Grading Distribution: Final Exam 20% 2 Tests 40% Homework/Labs 35% Class participation 5% Attendance Policy: “Shouldn’t be an issue!” (Each unexcused absence will be charged one participation point up to
5)
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Syllabus
Texts: Bluestein, H.B., 1992: Synoptic-Dynamic Meteorology in the Mid-
latitudes Vol I: Priciples of Kinematics and Dynamics. Oxford University Press, 431 pp. (Required)
Holton, J.R., 2004: An Introduction to Dynamic Meteorology, 4th Inter, 535 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)
Various relevant articles from AMS and RMS Journals. Course Prerequisites: Atmospheric Science 1050, 4320, Calculus through Calculus II, Physics
2750, or their equivalents. Senior standing or the permission of the Instructor.
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Syllabus Calendar: “Wednesday is Lab exercise day” Week 1: hh 17 18L 19 January Marty King day! Intro.
to ATMS 4320. Lab 1: Real and apparent forces: Coriolis Force. Week 2: 23 24 25L 26 January Lab 2: Methodologies
for calculating derivative in the fundamental equations of hydrodynamics. Week 3: 30 31 01L 02 February Lab 3: Estimating the
geostrophic wind Week 4: 06 07 08L 09 February Lab 4: The Nocturnal
boundary layer jet and severe weather. Week 5: 13 14 15L 16 February Test 1 – 16 February
covering materials up to 15 February. Lab 5: Using the kinematic method in estimating vertical motion.
Week 6: 20 21 22L 23 February Lab 6: The Use of isentropic coordinate maps in weather forecasting I.
Week 7: 27 28 01L 02 March Lab 7: The Use of isentropic coordinate maps in weather forecasting II.
Week 8: 06 07 08L 09 March Lab 8: The Thermal wind: Forecasting problems and the analysis of fronts.
Week 9: 13 14 15L 16 March Lab 9: Computing divergence using large data sets.
Week 10: 20 21 22L 23 March Test 2 – March 31 covering material up to 29 March. Lab 10: Vorticity and Cyclone Development.
Week 11: 27 28 29L 30 March No classes! Spring Break!!
Week 12: 03 04 05L 06 April Lab 11: Sutcliffe Methodology vs. Potential Vorticity Thinking.
Week 13: 10 11 12L 13 April Lab 12: The Rossby wave equation.
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Syllabus
Week 14: 17 18 19L 20 April Lab 13: The Omega Equation, a physical interpretation.
Week 15: 24 25 26L 27 April Lab 14: Convergent/Divergent patterns associated with jet maxima: Forecasting applications.
Week 16: 01 02 03L 04 May Lab 15: The practical uses of Q-G theory in daily analysis!
Finals Week: 8 - 12 May, 2006 ATMS 4320 / 7320 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:
Thursday 11 May 2006 – 8:00 am to 10:00 am in ABNR 123
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Syllabus REGULAR REGISTRATION INFORMATION - WINTER SEMESTER 2006 January 9-13, 2006 WS 2006 January 9-13 Regular Registration - WS2006 January 11 Residence halls open after 1:00 p.m. January 13 Easy Access Registration - 12 noon - 6:00 p.m. January 16 Martin Luther King Holiday-University closed *January 17 WS2006 Classwork begins (8:00 a.m.) January 17-24 Late registration - Late fee assessed beginning January 17 January 24 Last day to register, add or change sections Jan. 25-Feb. 21 Drop only January 30 Last day to change grading option February 21 Last day to drop a course without a grade March 6-24 SS2006/FS2006 Early Registration Appointment Times (currently enrolled students only)* April 15 Last day to transfer divisions March 26-April 2 Spring Break (begins at 12:00 a.m. March 26 and ends 8:00 a.m. April 2) April 3 Last day to withdraw from a course - WS2006 May 5 Winter semester classwork ends May 5 Last day to withdraw from the University - WS2006 May 6 Reading Day May 8 Final Examinations begin May 10-14 Registration - Add/Drop period for SS2006 and FS2006 (currently enrolled students) May 12 Semester ends at close of day - WS2006 **May 12-13-14 Commencement Weekend
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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. 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.
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Syllabus
Syllabus ** Introduction to the equations of motion The fundamental equations of geophysical hydrodynamics Horizontal flow and horizontal flow approximations The Isobaric coordinate system The isentropic coordinate system The vertical variation of the geostrophic wind (Thermal wind) Some fundamental kinematic concepts Vorticity and circulation the vorticity equation and vorticity theorems Inertial Instability Introduction to Quasi-geostrophic theory Fundamentals of numerical weather prediction* * These topics will be taught if there is time (also covered in
atms 4800 / 7800). All Lecture schedules are tentative! ** Students with special need are encouraged to schedule an
appointment with me as soon as possible!
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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 4320 (7320) 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. 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. That’s too much! Finally, answer all questions given in the assignment.
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Day One
The Equation of Motion - Newton’s Second Law
Dynamics will focus on these equations and their various incarnations!
The generalized form of Newton’s Second Law of Motion (sometimes called “Conservation of Momentum”) can be written as:
dtFVmd
)(
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Day One
(Recall Velocity - V and Force - F are VECTORS)
The differential change of momentum (following a particle):
“Impulse” )( VmddtF
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Day One
If there is no “force” acting, (F 0); then no change in momentum d(mV) = 0
“Conservation of Momentum”
If: 0dtVd
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Day One
Then the following MUST be true
Recall the basic form of a differential equation
0F
kssourcesdtdQ
sin
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Day One
The more “familiar” form of the Second Law:
dtVd
a
where
amF
,
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Day One
Thus, there are several forces which can contribute to the Left-hand-Side, and when applied to geophysical fluids (or F is defined for an oceanic or atmospheric environment) these would be called the Navier - Stokes equations (published in 1901).
These (N-S) equations are partial differential equations with an infinite number of solutions. The particular solution is dependent on the specification of boundary and initial conditions.
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Day One
Even though there are infinite number of solutions, we tend to see similar solutions over and over again.
Thus, while the atmosphere is a “chaotic” system, there are elements on different time and space scales which are quite predictable as solutions tend “cluster” on a time-space phase diagram.
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Day One
How did we get there?
Hold mass constant (unit mass - typical atmospheric assumption)
0
0
FdtVd
m
or
Vmdtd
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Day One
Q: Where is Newton’s second law valid?
A: In an Inertial frame of reference (or coordinate system) or one that is NOT being accelerated. This could be a “fixed” coordinate system, or:
This coordinate system can be moving though, it is just moving at constant V w/r/t another inertial frame.
0V
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Day One
The “other” frame sometimes is called an “absolute” frame or a “fixed frame”, but may really be neither!
Thus,
describes the motion of a particle or parcel of UNIT mass in an inertial frame of reference.
amF
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Day One / Two
Absolute and relative motion on a rotating system
We observe and refer motion to our position or coordinate system on the Earth’s surface which has a velocity and acceleration in “absolute” space due to the rotation of the Earth.
Thus,sysrelabs VVV
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Day One / Two
(Recall V is changing directions since the coordinate system we use is attached to the rotating Earth. Thus, it is NOT an inertial coordinate system.
Now Some Additions - The coriolis acceleration: is that portion of the total absolute
acceleration which is due to the change of orientation of the relative coordinate system as it moves relative to the absolute (rotating earth).
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Day Two
The coriolis acceleration exists whether you coordinate system defines the origin on the rotating earth or at the earth’s center.
The centrifugal acceleration:
the part of the total absolute acceleration which is due to a rotating earth. It is necessary that the centrifugal force keeps a surface based point of zero relative velocity (origin) in orbit about the earth’s axis in a plane normal to the axis at a constant distance normal to the axis.
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Day Two
In plain English: It only exists for a coordinate system defined on the earth’s surface, and whose origin is there. It does not exist in an absolute framework, or in a coordinate system defined at the center of the earth.
Varies as R = r cos where is the latitude. R is distance from the rotation axis, r is earth radius. Thus this force is maximum at the equator R is maximum. R is 0 at the poles.
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Day Two
Recall from last week we said that:
And yesterday we said that:
(1)Coriolis Centrifugal Acceleration Acceleration
sysrelabs VVV
crelrel
abs rrVdtVd
a ˆsin22
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Day Two
and we could substitute (1) in the equation of motion for dV/dt:
Now let’s put in the other real forces as we did yesterday (symbolically)
mF
dtVd
abs
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Day Two
Now Newton’s law with real and apparent forces:
viscgravityFricPGFm
rrVdtVd
crelrel
1
ˆsin22
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Day Two
Or
This form is similar to the inertial form, but is only valid on a rotating planet (Earth)! Note there is real and the apparent force, coriolis force, which appears as a “negative” since we included it as a forcing mechanism and not an acceleration per se!
viscgravityFricPGFm
rrVdtVd
crelrel
1
ˆsin22
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Day 2
Newtonian Gravitation
Newtonian gravitation (absolute gravity):
directed between the centres of mass, proportional to the product of masses and inversely proportional to the square of the distance between the centres of mass.
r
r
rGMem
gmF abs
2
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Day two
Thus for a particle of unit mass
r = re or radius of earth 6371 km.
M = Me or mass of earth 6.0 x 1024 kg
G is the universal gravitation constant G: 6.6 x 10-11 Nm2kg-2
2e
abs rGMe
g
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Day 2
thus: g approximately 9.8 ms-2 at the surface of the earth!!!
Apparent gravity: the centrifugal force:
Centrifugal Force The vector sum of the two is apparent gravity!
Rgg absapp
2
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Day Two
Net Force = Newtonian gravity + Centrifugal force
At equator:
at 45 N
at poles.
Rgg absapp
2
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Day Two
Apparent gravity only has a direction (which is always normal to or perpendicular to the surface) in k, now we can further simplify the equation of motion by stating:
gapp. = -g k = gabs. + centrifugal force
g does not appear in the horizontal equations of motion, only in the vertical!
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Day Two/Three
Thus, due to centrifugal forces, and the fact that earth is an oblate spheriod, not a sphere, the surface gravity:
at the poles: g = 9.83 ms-2
at the equator: g = 9.78 ms-2
in MO: g = 9.81 ms-2
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Day Two/Three
Geopotential Surfaces
geopotential surfaces are surfaces of constant gravitational geopotential energy (Potential energy).
Potential is defined as relative to some position, in this case, the earth’s surface: z = 0 (Mean Sea Level (MSL)).
z
gzgdz0
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Day Three
but, g = g(,z)
Thus, surfaces of constant geometric height (z) are not also surfaces of constant geopotential (due to the variation of g with latitude)
Surfaces of constant slope very slightly towards the poles forming oblate spheriods.
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Day Three
(Careful I: some atmospheric data sets give you GEOPOTENTIAL not height!)
(Careful II: MANY atmospheric scientists use the term height and geopotential interchangably! They are not!)
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Day Three
at z = 0 and = 45o g = go = 9.81 ms-2
we can express:
z z
o
oo
o
z
o
dzgdzgg
ggdz0
*
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Day Three
Where:
gravity is everywhere normal to surfaces of constant geopotential and is proportional in size to the vectoral gradient of geopotential!
dzgg
dzo
*
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Day Three
we have many ways of expressing apparent gravity in the equation of motion:
kz
g ˆ
Rgkgkz
g abs
2ˆ
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Day Three
The Pressure Gradient Force
Consider a unit volume of air with dimensions (xyz)
Consider the force on this volume due to pressure differences in the positive x direction (the x component of the total PGF) (This argument is also valid in the y and z direction!)
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Day Three
Pressure: Force / Area
Force = Pressure x Area
Total force on the LHS of the box:F1 = P1 Dy Dz
Total force on the right hand side (note sign convention!)
F2 = -P2 Dy Dz
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Day Three
But P2 can also be expressed as:
So F2 is:
xx
pPx
x
pP
11
zyxx
pzyP
1
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Day Three
Thus, the net force in the positive x direction: F1 + F2
--F1------ -------F2-----------------
zyxx
pzyPzyP
11
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Day Three
So the net force in the x direction is:
The net force per unit mass: Total Mass = Thus the x component of the Pressure
gradient force is:
dxdydzxp
zyxxp
dxdydzzyxVol
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Day Three
Net Force / Unit Mass
Then it follows that the y component is:
ixp
ixp
dxdydzdxdydz
xp ˆˆ1
jyp
jyp ˆˆ1
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Day Three
And finally the z component, (recall from thermo.)
Thus the three dimensional PGF is the sum of all three components:
kzp
kzp ˆˆ1
k
zp
jyp
ixp
PGF ˆˆˆ1
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Day Three
or in “Vector Notation”
where is the three dimensional Pressure gradient!
pPGF 3
1
P3
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Day Three/Four
We must realize that
1) the pressure gradient force is directed from high to low pressure, this means
2) air “flows” or is pushed from high to low pressure (source to sink)
We recall from thermodynamics, and our scale analyses that the vertical component dominates:
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Day Four
(1) (2) (3)
Term (1) ~ 100 hPa / km
Term (2),(3) ~ 1 hPa / 100 km
but, although the horizontal components are small, they are still very important to horizontal accelerations and motions.
yp
xp
zp
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Day Four
That is why it is customary to “scale” the “z” equation of motion as the hydrostatic approximation, while the horizontal equations are analyzed separately.
jyp
ixp
p
where
kzp
pp
h
h
ˆˆ
ˆ1113
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Day Four
Let’s now put in the Coriolis force, the centrifugal force and the PGF into the equations of motion
Grav. Fric Visc PGF Coriolis
This is the equation of motion! We have derived CF, gapp. and PGF.
VpFgdtVd
app
2
13
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Day Four
Frictional force + Viscous force (Piexoto and Oort, 1992, p. 36: The Physics of climate) in three dimensions:
divF
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Day Four
X-component
/
1
where
zU
zzyxFx zzyyxx
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Day Four
Kinematic Viscosity
where ( ) is a 2nd order “stress” tensor!
Recall from thermo. a tensor has magnitude, and 2 directions (Vector is 1st order tensor)!
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Day Four
Stress
zzzyzx
yzyyyx
xzxyxx
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Day Four
Or we can parameterize friction (e.g. Lupo et al., 1992, MWR, Aug.) (horizontal motions):
Now put in the Equations of Motion:
Fk
ˆ
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Day Four
PGF CO Grav. Friction
We now have the Navier-Stoke’s (1901) equations which represents the dynamics of a geophysical fluid (Atmosphere or Ocean).
12
13 gVp
dtVd
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Day Four
Again, it is common in atmospheric science to examine the horizontal and vertical components of the equation of motion separately. These equations in component form:
(Since your homework involves grinding out Coriolis force we won’t do here)
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Day Four
NS equations
Fzguz
p
dt
dw
Fyuy
p
dt
dv
Fxwvx
p
dt
du
cos21
sin21
cos2sin21
![Page 60: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Syllabus Atmospheric Dynamics ATMS 4320 / 7320 MTWR 9:00 – 9:50 / 4 credit hrs. Location: 123](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d7a5503460f94a5f007/html5/thumbnails/60.jpg)
Day Four
It is also VERY common to look at changes in momentum by scaling these relationships to fit the system we are trying to describe.
This was especially true in the early days of meteorology (Taylor, Richardson, Rossby, Eliassen, etc.), when there were no fancy computers to work with. These researchers had to find ways of simplifying the equations, and making approximations in order to make their life easier!
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Day Four
The “Z” component (we won’t re-derive from thermo.) is the hydrostatic relation:
which can be written as (recall from thermo.):
gzp
dzdp
gzp
,0
1
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Day Four
A scale analysis (using typical synoptic-scale values) of the hoizontal equations would show that:
01
gzp
oo
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Day Four
10-4 ms-2 10-3 ms-2 10-3 ms-2 10-5 ms-2 10-6 ms-2
Thus, the approximate form of the horizontal, inviscid N-S equations appopriate for synoptic and plaentary-scales (tropospheric and stratospheric flow):
FrictionCoriolisPGFdtVd
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Day Four
Scalar N-S equations
sin2
10
1
1
f
where
gzp
fuyp
dtdv
fvxp
dtdu
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Day Four
Vector N-S equations
kgkzp
kdzdw
jfujyp
jdtdv
ifvixp
idtdu
ˆˆ1ˆ
ˆˆ1ˆ
ˆˆ1ˆ
Vkfpdt
Vdh
h
ˆ1
![Page 66: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Syllabus Atmospheric Dynamics ATMS 4320 / 7320 MTWR 9:00 – 9:50 / 4 credit hrs. Location: 123](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d7a5503460f94a5f007/html5/thumbnails/66.jpg)
Day Five
The Equations of Motion (Navier Stokes Equations) in Spherical coordinates
Coordinate systems: Cartesian (x,y,z)
Polar (r,) Cylindrical (r,,z) Spherical (,,) Natural (n,s,z)
For scales of motion that are sufficently large, we must take into account changes in the orientation of earth relative to the coordinate system over the earth’s system (ie orientation of cartesian coordinates differs at poles and equator
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Day Five
The acceleration vector (N-S equations), where are treated as constants
From thermo. we said that they did not have to be so:
kji ˆ,ˆ,ˆ
kdtdw
jdtdv
idtdu
dtVd ˆˆˆ
wdtkd
vdtjd
udtid
kdtdw
jdtdv
idtdu
dtVd ˆˆˆˆˆˆ
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Day Five
Then the total derivative of the Unit vector is:
(An aside: can you write out the “j” and “k” equations?)
Now in spherical coords, meteorology defines:
zi
wyi
vxi
uti
dtid
ˆˆˆˆˆ
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Day Five
Math meaning meteorology
radius re latitude longitude
Visualize yourself on a sphere:
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Day Five
A Sphere The Earth
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Day Five
Angles
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Day Five
Thus, it follows that:
yk
vxk
uzk
wyk
vxk
utk
dtkd
yj
vxj
uzj
wyj
vxj
utj
dtjd
xi
uzi
wyi
vxi
uti
dtid
ˆˆˆˆˆˆˆ
ˆˆˆˆˆˆˆ
ˆˆˆˆˆˆ
![Page 73: Atmospheric Science 4320 / 7320 Anthony R. Lupo. Syllabus Atmospheric Dynamics ATMS 4320 / 7320 MTWR 9:00 – 9:50 / 4 credit hrs. Location: 123](https://reader036.vdocument.in/reader036/viewer/2022062421/56649d7a5503460f94a5f007/html5/thumbnails/73.jpg)
Day Five
Evaluate the partial derivatives of: “i” “j” and “k” terms, all nine!?
Let’s consider (partial “i” / partial x)
as: (i/x) the variation of i with increment of longitude
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Day Five
“look down” from North pole:
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Day Five
Thus, we can show (with ball) i must point inward! It must have components in the y and z components:
Magnitude in “y” dir:
cos11ˆ
limˆ
erRRx
i
xi
ee rrxi
xi
tan
sincos1
sinˆˆ
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Day Five
Magnitude in “z” dir:
Then:
ee rrxi
xi 1
coscos1
cosˆˆ
kru
jr
uxi
udtid
kr
jrx
i
ee
ee
ˆˆtanˆˆ
ˆ1ˆtanˆ
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Day Five
Now look at: (k/x)
Buuut,
xr
rxr
rxk
xk
rr
k e
e
e
ee
e
11ˆˆ
ˆ
xRre
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Day Five/Six
So,
and the direction of re is perpendicular to k in the positive x or i direction, so:
1
xRxR
xre
irx
k
e
ˆ1ˆ
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Day Six
Now consider:
We can do this using geometry, or use our cross product rules:
Since:
xj
ˆ
ikj ˆˆˆ
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Day Six
Then,
Then substitute our other expressions!
xi
kixk
ikxx
j
ˆˆˆ
ˆˆˆˆ
ijk
kk
ii
but
kr
jr
kiir eee
ˆˆˆ
,0ˆˆ
,0ˆˆ
ˆ1ˆtanˆˆˆ1
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Шестой днљй
Так (sooo….)
Then consider….
irx
j
e
ˆtanˆ
yj
yj
ˆˆ
ery
j ,ˆ
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Day Six
Then,
and the direction is in the –k
ee ryj
ryj 1ˆˆ
kry
j
e
ˆ1ˆ
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Day Six
Last, consider
We can consider this via geometry again, but let’s use our cross products!
yk
yk
ˆˆ
yj
ijyi
jiyy
k
ˆ
ˆˆˆ
ˆˆˆ
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Day Six
And;
substitute these in, and finally:
,0ˆ
yi
kry
j
e
ˆ1ˆ
jry
k
e
ˆ1ˆ
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Day Six
Then substitute back into the derivatives of the i,j,k vectors:
jruv
iruw
yk
vxk
udtkd
w
krv
iruv
yj
vxj
uvdtjd
v
kru
jru
xi
udtid
u
ee
ee
ee
ˆˆˆˆˆ
ˆˆtanˆˆˆ
ˆˆtanˆˆ
22
222
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Day Six
Now substitute back into the Navier Stoke’s equations:
FgV
pjruv
iruw
krv
iruv
kru
jru
kdtdw
jdtdv
idtdu
dtVd
eeee
ee
2
1ˆˆˆˆtan
ˆˆtanˆˆˆ
2
22
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Day Six
In component form:
Component – i:
10-4 10-5 10-7 10-3 10-3 10-5 10-6
xFFwfvxp
ruw
ruv
dtdu
ee
1
tan
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Day Six
In component form:
Component – j:
10-4 10-5 10-7 10-3 10-3 10-6
yFfuyp
rvw
ru
dtdv
ee
1
tan2
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Day Six
In component form:
Component – k:
10-6 10-5 10-5 10 10 10-3 10-6
zFFugzp
rv
ru
dtdw
ee
122
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Day Six
The Navier Stokes equations in the Natural Coordinate system
Natural Coordinates n,s,z
s is in the direction of horizontal flow (+ with flow, - against)
n is normal to the direction of flow (+ left, - right)
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Day Six
z is vertical coordinate
Thus,
The acceleration term in Natural coordinates:
sVVh ˆ
dtsd
Vsdt
Vd
dtsVd
dt
Vda hh ˆ
ˆˆ
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Day Six/Seven
Thus we must look at:
For , the direction is perpendicular to s and to it’s left in the n direction.
ts
dtsd
ˆˆ
xs
xs
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Day Seven
So;
thus, we know x = Rc, so;
ndtd
dtsd
or
ns
ˆˆ
ˆˆ
dtd
Rdtsd
or
Rs
c
c
ˆ
ˆ
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седЬмой днЬи
and in natural coordinates the definition of:
So;
dtsd
Vh
ˆ
nR
V
dtsd
R
V
dtd
c
h
c
h
ˆˆ
,
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Day Seven
сейчас(Now):
(1) (2) (1) Downstream or Tangential
Acceleration (speed)
(2) Centripetal acceleration (curvature)
n
R
Vs
dt
Vd
dt
sVd
dt
Vda
c
hh ˆˆˆ 2
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Day Seven
Natural Coordinates, the pressure gradient force
Natural Coordinates: the coriolis force
nnp
ssp
ph ˆ1
ˆ11
nVfVkf hh ˆˆ
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Day Seven
The horizontal invicid equations of motion in Natural Coordinates (e.g. Bell and Keyser, 1993, MWR, Jan)
The s-component, in the direction of motion:
sp
dt
Vd h
1
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Day Seven
The n-component
The energy equations (Bernoulli’s equations)
The 3-D Navier Stokes equation (x,y,z):
hc
Vfnp
RV f
12
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Day Seven
Dot N-S equations with V:
(For horizontal equations the result is a Kinetic Energy budget equation, Smith, Kung, Orlanski - e.g. Orlanski and Sheldon, 1993, MWR, November)
rFgVpdtVd
21
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Day Seven
OK, here’s the result
dt
d
dt
dzgwgkgV
VCOVV
pVpV
V
dt
dVV
dt
d
dt
VdV
ˆ
,02
11
22
2
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Day Seven
Recall dot product rules?
(Note Coriolis vector disappears, dot product rule, coriolis force contributes nothing to changes in Kinetic energy)
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Day Seven/Eight
so (following Orlanski and Sheldon),
(1) (2)
rFVpVgzV
dtd
rFVdtd
pVV
dtd
12
122
2
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Day Seven / Eight
where,
= geopotential
V2/2 = Kinetic energy per unit mass
(1) is the work done per unit mass by PGF
(2) is the work done per unit mass by FRIC
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Day Eight
Now Recall the First Law:
add KE equation and first law:
dtd
pdtdT
cQ v
rFVpVdtd
pQTcgzV
dtd
v
2
2
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Day Eight
Now take definition of change of pressure with time (multiply by ):
now substitute the above into the energy equation (where class?)
pV
tp
dtdp
rFVtp
dtdp
dtd
pQTcgzV
dtd
v
2
2
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Day Eight
we get (do you see the “product rule”?):
but since (using Eqn of state):
rFVtp
QpTcgzV
dtd
v
2
2
TcpTc pv
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Day Eight
Generalized Bernoulli Eq.
and if we assume, adiabatic, inviscid, and steady state flow, then what?
rFVtp
QTcgzV
dtd
p
2
2
02
2
Tcgz
Vdtd
p
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Day Eight
and then….
Moist Static Energy Equation
If the diabatic heating is assumed to involve condensation and evaporative processes within a parcel of air,
tConsTcgzV
p tan2
2
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Day Eight
Then;
where m is mixing ratio and L is the latent heat of condensation/evaporation.
dtLmd
dtdh
Q
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Day Eight
Then our generalized Bernoulli equation becomes:
Or;
rFV
tp
dtLmd
TcgzV
dtd
p
2
2
rFVtp
LmTcgzV
dtd
p
2
2
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Day Eight
and, of course we can assume again, steady state and inviscid flow:
or (total energy remains constant along the trajectory)
02
2
LmTcgz
Vdtd
p
tConsLmTcgzV
p tan2
2
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Day Eight
Consider these typical orders of magnitude:
V ~ 15 m/s V2/2~102 m2/s2 ~ 102 J/kg
g z ~ (3000m) g z ~3x104 m2/s2 ~ 3x104 J/kg
CpT ~ (270 K) CpT~2.7x105 m2/s2 ~ 2.7x105 J/kg
L m ~ (4 g/kg) L m~104 m2/s2 ~ 104 J/kg
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Day Eight
If the total energy E is:
102 104 105 104
Moist static energy!
LmTcgz
VE pT 2
2
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Day Eight
It is typical to neglect V2 /2 term, thus we could derive this quantity exclusively from the 1st law of thermodynamics for an inviscid flow.
Concept of moist static energy is important when talking about the energy balance of the general circulation, especially that of the tropics.
0
2
2
dtdE
constLmTcgzV
EE pmStatT
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Day Eight
Also, it should be obvious that if we examine an adiabatic atmosphere, we’re back to dry static energy, a concept we studied in thermo. (from which we can derive dry adiabatic lapse rate).
Geopotential temperature (dynamic potential temperature):
divide all terms by Cp, and viola!:
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Day Eight
This is the temperature an arbitrary sample of air would have after adiabatic decent to z=0 or sealevel!! (Similar concept to potential temperature)
Q: Is this the same as potential temperature (whose reference level is 1000 hPa)?
A: No! 1000 hPa not always at z=0 in the real atmosphere. But where 1000 hPa is at z =0, they are equal!
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Day Eight
You say: Prove it! 12Z 29 Friday, 1999 data near British Columbia-Alberta border
T700 = 264K SLP = 1000 hPa Z700 = 2820 m
Now consider case where latent energy is added or lost, the non-adiabatic case is (saturated):
And,
Now we can talk about geopotential equivalent potential temperature
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Day Eight
This is conserved for both saturated and unsaturated adiabatic process including air which is being evaporatively cooled by falling rain.
are similarly conservative and
porportional to their thermodynamic counter parts, but are simpler to calculate.
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Day Eight
The practical Applications of Geopotential Temperature
Storm-scale --> updrafts and downdrafts
Static Energy Index ( )
Layers in which are negative, possess potential convective instability, with respect to geopotential temperature.
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Day Eight
The Coupling Index (Bosart and Lackmann, 1995; Lupo et al; 2001)
Calculates a “lapse rate” using a moist low level, thus identifying regions of moist instability. Values lower than ‘10’ indicate very unstable air, and are associated with vigourous cyclogenesis. Can be identified using geopotential temperature also.
Subsynoptic and synoptic scale:
areas of maximum geopotential temperature at low levels indicate maximum updraft potential.
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Day Nine
The Equation of Continuity (Hess, p212 ch 13)!!
This is an expression of the principle of the conservation of mass.
In the cartesian system:
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Day Nine
A Cube:
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Day Nine
Consider the mass flux though the faces of a cube.
The flux of any quantity Q:
‘flux’ = QV
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Day Nine
mass flux:
the total mass flux into the left face
smkg
sm
mkg
V 23
dydzuorzyu 11
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Day Nine
the total mass flux out of the right face (recall convention)
but
dydzuorzyu 22
dxxu
uxxu
uu
112
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Day Nine
so the mass out of the right face:
the net inflow of mass due to the u-component of the flow equals:
mass into the left - mass out of the right
dydzdxxu
u
1
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Day Nine
The result?
the net mass increment per unit volume in the x direction (u - component)
dxdydzxu
dydzudydzu 11
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Day Nine
After cancellation:
where dV = dxdydz
xu
dVdxdydzxu
/)(
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Day Nine
the net mass increment per unit volume in the y direction (v - component)
the net mass increment per unit volume in the z direction (w component)
yv
zw
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Day Nine
the total increment in mass per unit volume per unit time:
so, t
Vzw
yv
xu
t
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Day Nine
thus,
which is the equation of continuity!
This represents the local rate of change in mass inside the cube equals the net divergence or convergence of mass.
Vt
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Day Nine
Mass increase mass convergence
(Hess way: more molecules stuffed in the box, which hasn’t changed size!)
Mass decrease mass divergence
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Day Nine
Alternative expressions for the equation of continuity
recall
ABBABA
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Day Nine
Then:
Thus,
VVVf
VVt
f
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Day Nine
Or
But since:
VVt
f
dt
dV
t
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Day Nine
the continuity equation may be rewritten as:
or
Vdtd
V
dtd
or
Vdtd
ln
1
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Day Nine
The fractional increment in the density of air following along a parcel trajectory is due to the 3- D convergence
Specific Volume form (Oceanography)
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Day Nine
Here it is!
Vdtd
V
dtd
or
Vdtd
ln
1
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Day Ten
The equation of continuity from Bluestein (p 190 - 193 Bluestein) another view.
Assume constant density = o:
Vdtd
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Day Ten
Then (“it’s the size of the box, stupid!”):
0,0 Vdtd
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Day Ten
Convergence:
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Day Ten
Divergence:
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Day Ten
Water is nearly incompressible, so oceanographers can use constant density form. The assumption is also reasonable for most atmospheric applications. Thus, let’s consider the continuity equation in constant pressure coordinates.
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Day Ten/11
In a hydrostatic atmosphere:
(x,y,z) (x,y,p)
0
gdV
dtd
dVdtd
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Day 11
Thus, by substituting for:
it follows that;
gp
zzw
Vhh
0
pVhh
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Day 11
This was derived by Sutcliffe and Godart (1942), thus making the continuity equation a Diagnostic equation.
Diagnostic No time derivatives appear in the equation! (or they approach 0 in a more formal definition)
0,0 dtdQ
dtdQ
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Day 11
Prognostic Time derivatives are explicit in the equation
This reduces the problem to two dimensions, since a hydrostatically balanced atmosphere must conserve the mass between two pressure levels. So our material behaves as an incompressible fluid!
SiSodtdQ
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Day 11
Continuity in isentropic coordinates:
(1) (2) (3)
(1) represents the local change in the inverse static stability. (Sz)
(2) represents the horizontal flux of inverse static stability
0
dtd
SVStS
zzz
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Day 11
(3) represents the vertical flux of inverse static stability
Thus we can think of this equation
as a static stability tendency equation as well.
p
S z
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Day 11
The equation of moisture Continuity
This is a statement of the conservation of water vapor mass per unit mass of air
A differential equation describing water vapor budget (coming from Geophysical Fluid dyn.):
Sdtdq
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Day 11
where:
S = Sources and Sinks
S1 evaporation and sublimation,
S2 condensation and precipitation
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Day 11
Where conservation of water vapor is:
0 Sdtdq
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Day 11
Which is commonly examined by examining horizontal and vertical transport (advection).
pq
qVtq
hh
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Day 11
Then we can rewrite in “flux” form (as is done in studies of the General Circulation budget)
pq
qVtq
hh
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Day 11
So, Let’s restate the fundamental equations of geophysical fluid dynamics. In this class we’ll look at:
Equation of State (Elemental Kinetic Theory or Gasses):
RTP
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Day 11
Conservation of Energy (1st Law of Thermodynamics):
dtdp
dtdT
cQ p
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Day 11
Conservation of mass (Continuity):
Dry
Moist
Vdtd
pq
qVtq
hh
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Day 11
Equation of Motion (also Navier Stokes, Newton’s 2nd Law, Conservation of Momentum:
Fzguz
p
dt
dw
Fyuy
p
dt
dv
Fxwvx
p
dt
du
cos21
sin21
cos2sin21
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Day 11
These equations are also called the “primitive equations and represent a closed set (7 variables [u,v,w(), p, T(), , q] , 7 equations. This is a mathematicallly solvable system which, given the proper initial and boundary conditions will yield all future states of the system.
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Day 11/12
Horizontal Flow
we have already derived a set of equations which represents the most rudimentary, but yet realistic, approximation of horizontal flow.
Geostrophic flow in cartesian coords (x,y,p!):
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Day 12
Geostropic wind!
kf
Vgˆ1
xxfv
yyfu
g
g
1
1
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Day 12
In natural coordinates:
(We need to discuss the properties of a geostrophically balanced system or geostrophic wind)
np
fV
nnp
p
VfpdtVd
g
h
geoh
1
ˆ
10
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Day 12
Geostrophic Wind
horizontal and non-accelerating and can be calculated where ever pressure gradient force and Coriolis force exist.
normal to the pressure gradient force, with low (high) pressure to the left (right)
0dtVd
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Day 12
magnitude is directly proportional to pressure gradient force (packing of isobars)
Virtually non-divergent (though there is a small amount of divergence and Helmholtz partitioning would demonstrate this). Simplest way to show:
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Day 12
non-divergent (f = fo):
y
v
x
uV gg
hh
0
xyfxfyy
v
yxfyfxx
u
oo
g
oo
g
2
2
11
11
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Day 12
And if we add this up, does it equal 0?
small divergences (f varies – div-ergence on the order of 10-4, 10-5 s-
1):
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Day 12
Here:
xy
f
fxyfxfyy
v
yx
f
fyxfyfxx
u
g
g
2
2
2
2
111
111
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Day 12
Adds up to be:
a term on order of 10-6 or 10-7 s-1.
f
v
xyf
fg
2
1
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Day 12
Balanced system
Geostrophic Balance (Is by strictest definition, a steady state system). Thus, disturbances (energy) cannot be generated in such an atmosphere, just moved around!)
In the strictest sense, this atmosphere (if f = fo) is barotropic! (Wind speeds are the same at all heights
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Day 12
Wind speed profile is constant, and there are no vertical motions!
Rossby Number (Ro)
Is a scaling parameter, or measure of geostrophic approximation validity:
Ratio of Accelerations to coriolis force.
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Day 12
The N-S equations in symbolic form:
Thus, the Rossby number is a measure of the departure from geostrophic flow:
1.
.
Co
Accel
CO
PGF
AccelCOPGF
Ro = U/fL
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Day 12
where U is the mean zonal wind
f is the Coriolis parameter
L is the characteristic length scale of an Atmospheric disturbance (synoptic-scale)
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Day 12
If Ro is << 1 flow is nearly geostrophic
If Ro = 0.1 flow is with 10% of geostrophic balance
If Ro = 1 then geostrophic balance begins to fail
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Day 12
Consequence of geostrophic balance (Pedlosky p43ff)
Taylor-Proudman theorem
If the atmosphere is in approximate geostrophic and hydrostatic balance, and flow is inviscid, and the baroclinic vector is zero!
02 p
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Day 12
Then velocity is perpendicular to rotation vector () and must always be so!
Flow is non-divergent!
Vorticity lines will always be parallel to rotation vector.
V0 hh V
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Day 12
Since flow is incompressible then:
Thus, the motions are COMPLETELY 2-dimensional!
0
zw
yv
xu
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Day 12
And
0,,
zv
zu
zw
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Day 12
Taylor’s experiment (1923):
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Day 12