1. Atmospheric Science 4320 / 7320 Anthony R. Lupo

2. 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:lupo@bergeron.snr.missouri.edu or LupoA@missouri.edu  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 – 100A+77 – 79C+  92 – 97A72 – 77C  89 – 92A-69 – 72C-  87 – 89B+67 – 69D+  82 – 87B62 – 67D  79 – 82 B-60 – 62D-  < 60 F  Grading Distribution:  Final Exam20%  2 Tests40%  Homework/Labs 35%  Class participation5%  Attendance Policy: “Shouldn’t be an issue!” (Each unexcused absence will be charged one participation point up to 5)

3. 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. 

4. Syllabus  Calendar: “Wednesday is Lab exercise day”  Week 1:hh1718L19January  Marty King day! Intro. to ATMS 4320.  Lab 1: Real and apparent forces: Coriolis Force.  Week 2:232425L26 January  Lab 2: Methodologies for calculating derivative in the fundamental equations of hydrodynamics.  Week 3:303101L02 February  Lab 3: Estimating the geostrophic wind  Week 4:060708L09 February  Lab 4: The Nocturnal boundary layer jet and severe weather.  Week 5:131415L16 February  Test 1 – 16 February covering materials up to 15 February. Lab 5:Using the kinematic method in estimating vertical motion.  Week 6:2021 22L23 February  Lab 6: The Use of isentropic coordinate maps in weather forecasting I.  Week 7:2728 01L 02 March  Lab 7: The Use of isentropic coordinate maps in weather forecasting II.  Week 8:060708L09March  Lab 8: The Thermal wind: Forecasting problems and the analysis of fronts.  Week 9:131415L16March  Lab 9: Computing divergence using large data sets.  Week 10:202122L23March  Test 2 – March 31 covering material up to 29 March. Lab 10: Vorticity and Cyclone Development.  Week 11:272829L30 March  No classes! Spring Break!!  Week 12:030405L06 April  Lab 11: Sutcliffe Methodology vs. Potential Vorticity Thinking.  Week 13:101112L13 April  Lab 12: The Rossby wave equation.

5. Syllabus  Week 14:171819L20April  Lab 13: The Omega Equation, a physical interpretation.  Week 15:242526L27April  Lab 14: Convergent/Divergent patterns associated with jet maxima: Forecasting applications.  Week 16:010203L04May  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

6. 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

7. 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.

8. 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!

9. 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. 

10. 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:

11. Day One  (Recall Velocity - V and Force - F are VECTORS)  The differential change of momentum (following a particle):  “Impulse” 

12. Day One  If there is no “force” acting, (F  0); then no change in momentum d(mV) = 0  “Conservation of Momentum”  If:

13. Day One  Then the following MUST be true  Recall the basic form of a differential equation

14. Day One  The more “familiar” form of the Second Law:

15. 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.

16. 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.

17. Day One  How did we get there?  Hold mass constant (unit mass - typical atmospheric assumption)

18. 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.

19. 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.

20. 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,

21. 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).

22. 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.

23. 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.

24. Day Two  Recall from last week we said that:  And yesterday we said that: (1) Coriolis CentrifugalAcceleration

25. 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)

26. Day Two  Now Newton’s law with real and apparent forces:

27. 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!

28. 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.

29. 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 10 24 kg  G is the universal gravitation constant G: 6.6 x 10 -11 Nm 2 kg -2

30. 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!

31. Day Two  Net Force = Newtonian gravity + Centrifugal force  At equator:  at 45 N  at poles.

32. 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: g app. = -g k = g abs. + centrifugal force  g does not appear in the horizontal equations of motion, only in the vertical!

33. 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

34. 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)).

35. 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.

36. 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!)

37. Day Three  at z = 0 and = 45 o g = g o = 9.81 ms- 2  we can express:

38. Day Three  Where:  gravity is everywhere normal to surfaces of constant geopotential and is proportional in size to the vectoral gradient of geopotential!

39. Day Three we have many ways of expressing apparent gravity in the equation of motion:

40. Day Three  The Pressure Gradient Force  Consider a unit volume of air with dimensions (xyz)  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!)

41. 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

42. Day Three  But P2 can also be expressed as: So F2 is:

43. Day Three  Thus, the net force in the positive x direction: F1 + F2  --F1------ -------F2-----------------

44. 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:

45. Day Three  Net Force / Unit Mass   Then it follows that the y component is:

46. Day Three  And finally the z component, (recall from thermo.)  Thus the three dimensional PGF is the sum of all three components:

47. Day Three  or in “Vector Notation”   where is the three dimensional Pressure gradient!

48. 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:

49. 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.

50. 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.

51. 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, g app. and PGF.

52. Day Four  Frictional force + Viscous force (Piexoto and Oort, 1992, p. 36: The Physics of climate) in three dimensions:

53. Day Four  X-component

54. 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)!

55. Day Four  Stress

56. Day Four  Or we can parameterize friction (e.g. Lupo et al., 1992, MWR, Aug.) (horizontal motions):  Now put in the Equations of Motion:

57. Day Four PGFCOGrav. Friction  We now have the Navier-Stoke’s (1901) equations which represents the dynamics of a geophysical fluid (Atmosphere or Ocean).

58. 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)

59. Day Four  NS equations

60. 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!

61. Day Four  The “Z” component (we won’t re- derive from thermo.) is the hydrostatic relation:  which can be written as (recall from thermo.):

62. Day Four A scale analysis (using typical synoptic-scale values) of the hoizontal equations would show that:

63. 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):

64. Day Four  Scalar N-S equations

65. Day Four  Vector N-S equations

66. 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

67. 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:

68. 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:

69. Day Five  Math meaning meteorology  radius re  latitude  longitude  Visualize yourself on a sphere:

70. Day Five  A SphereThe Earth

71. Day Five  Angles

72. Day Five  Thus, it follows that:

73. 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

74. Day Five  “look down” from North pole:

75. 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:

76. Day Five  Magnitude in “z” dir:  Then:

77. Day Five  Now look at: (k/x)  Buuut,

78. Day Five/Six  So,  and the direction of re is perpendicular to k in the positive x or i direction, so:

79. Day Six  Now consider:  We can do this using geometry, or use our cross product rules:  Since:

80. Day Six  Then, Then substitute our other expressions!

81. Шестой днљй  Так (sooo….)  Then consider….

82. Day Six  Then,  and the direction is in the –k

83. Day Six  Last, consider  We can consider this via geometry again, but let’s use our cross products!

84. Day Six  And;  substitute these in, and finally:

85. Day Six  Then substitute back into the derivatives of the i,j,k vectors:

86. Day Six  Now substitute back into the Navier Stoke’s equations:

87. Day Six  In component form:  Component – i: 10 -4 10 -5 10 -7 10 -3 10 -3 10 -5 10 -6

88. Day Six  In component form:  Component – j: 10 -4 10 -5 10 -7 10 -3 10 -3 10 -6

89. Day Six  In component form:  Component – k:  10 -6 10 -5 10 -5 1010 10 -3 10 -6

90. 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)

91. Day Six  z is vertical coordinate  Thus, The acceleration term in Natural coordinates:

92. Day Six/Seven  Thus we must look at:  For, the direction is perpendicular to s and to it’s left in the n direction.

93. Day Seven  So;  thus, we know x = Rc, so;

94. седЬмой днЬи  and in natural coordinates the definition of:  So;

95. Day Seven  сейчас(Now): (1)(2)  (1) Downstream or Tangential Acceleration (speed)  (2) Centripetal acceleration (curvature)

96. Day Seven Natural Coordinates, the pressure gradient force  Natural Coordinates: the coriolis force

97. 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:

98. Day Seven  The n-component  The energy equations (Bernoulli’s equations)  The 3-D Navier Stokes equation (x,y,z):

99. 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)

100. Day Seven  OK, here’s the result

101. Day Seven  Recall dot product rules?  (Note Coriolis vector disappears, dot product rule, coriolis force contributes nothing to changes in Kinetic energy)

102. Day Seven/Eight  so (following Orlanski and Sheldon), (1)(2)

103. Day Seven / Eight  where,   = geopotential  V 2 /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

104. Day Eight  Now Recall the First Law:  add KE equation and first law:

105. Day Eight  Now take definition of change of pressure with time (multiply by ):  now substitute the above into the energy equation (where class?)

106. Day Eight  we get (do you see the “product rule”?):  but since (using Eqn of state):

107. Day Eight  Generalized Bernoulli Eq.  and if we assume, adiabatic, inviscid, and steady state flow, then what?

108. 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,

109. Day Eight  Then;  where m is mixing ratio and L is the latent heat of condensation/evaporation.

110. Day Eight  Then our generalized Bernoulli equation becomes: Or;

111. Day Eight  and, of course we can assume again, steady state and inviscid flow:  or (total energy remains constant along the trajectory)

112. Day Eight  Consider these typical orders of magnitude:  V ~ 15 m/sV 2 /2~10 2 m 2 /s 2 ~ 10 2 J/kg  g z ~ (3000m)g z ~3x10 4 m 2 /s 2 ~ 3x10 4 J/kg  CpT ~ (270 K)CpT~2.7x10 5 m 2 /s 2 ~ 2.7x10 5 J/kg  L m ~ (4 g/kg)L m~10 4 m 2 /s 2 ~ 10 4 J/kg

113. Day Eight  If the total energy E is:  10 2 10 4 10 5 10 4  Moist static energy!

114. Day Eight  It is typical to neglect V 2 /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.

115. 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!:

116. 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!

117. 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

118. 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.

119. 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.

120. 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.

121. 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:

122. Day Nine  A Cube:

123. Day Nine  Consider the mass flux though the faces of a cube.  The flux of any quantity Q:  ‘flux’ = QV

124. Day Nine  mass flux:  the total mass flux into the left face

125. Day Nine  the total mass flux out of the right face (recall convention)  but

126. 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

127. Day Nine  The result?  the net mass increment per unit volume in the x direction (u - component)

128. Day Nine  After cancellation:  where dV = dxdydz

129. 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)

130. Day Nine  the total increment in mass per unit volume per unit time: so,

131. 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.

132. Day Nine  Mass increase  mass convergence (Hess way: more molecules stuffed in the box, which hasn’t changed size!)  Mass decrease  mass divergence

133. Day Nine  Alternative expressions for the equation of continuity recall

134. Day Nine  Then:  Thus,

135. Day Nine  Or  But since:

136. Day Nine  the continuity equation may be rewritten as:  or

137. 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)

138. Day Nine  Here it is!

139. Day Ten  The equation of continuity from Bluestein (p 190 - 193 Bluestein) another view.  Assume constant density  = o:

140. Day Ten  Then (“it’s the size of the box, stupid!”):

141. Day Ten  Convergence: 

142. Day Ten  Divergence: 

143. 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.

144. Day Ten/11  In a hydrostatic atmosphere:  (x,y,z)  (x,y,p)

145. Day 11  Thus, by substituting for:  it follows that;

146. 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)

147. 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!

148. Day 11  Continuity in isentropic coordinates:  (1)(2)(3)  (1) represents the local change in the inverse static stability. (S z )  (2) represents the horizontal flux of inverse static stability

149. 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.

150. 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.):

151. Day 11  where:  S = Sources and Sinks  S1 evaporation and sublimation,  S2 condensation and precipitation

152. Day 11  Where conservation of water vapor is:

153. Day 11  Which is commonly examined by examining horizontal and vertical transport (advection).

154. Day 11  Then we can rewrite in “flux” form (as is done in studies of the General Circulation budget)

155. 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):

156. Day 11  Conservation of Energy (1st Law of Thermodynamics):

157. Day 11  Conservation of mass (Continuity):  Dry  Moist

158. Day 11  Equation of Motion (also Navier Stokes, Newton’s 2nd Law, Conservation of Momentum:

159. 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.

160. 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!):

161. Day 12  Geostropic wind!

162. Day 12  In natural coordinates:  (We need to discuss the properties of a geostrophically balanced system or geostrophic wind)

163. 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)

164. 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:

165. Day 12  non-divergent (f = fo):

166. 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 ):

167. Day 12  Here:

168. Day 12  Adds up to be:  a term on order of 10 -6 or 10 -7 s -1.

169. 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

170. 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.

171. Day 12  The N-S equations in symbolic form:  Thus, the Rossby number is a measure of the departure from geostrophic flow: Ro = U/fL

172. 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)

173. 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

174. 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!

175. 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.

176. Day 12  Since flow is incompressible then:  Thus, the motions are COMPLETELY 2-dimensional!

177. Day 12  And

178. Day 12  Taylor’s experiment (1923):

179. Day 12