1. Richard B. Rood (Room 2525, SRB) rbrood@umich.edu 734-647-3530
AOSS 401 Geophysical Fluid Dynamics: Atmospheric Dynamics Prepared: Description / Analysis / Tropics
Richard B. Rood (Room 2525, SRB)
Cell:
Special thanks to Derek Posselt

2. Class News Ctools site (AOSS 401 001 F13)
Second Examination on December 10, 2013
Homework
Homework due November 26, 2013
Rest of lectures
Tropics
Reviews

3. Weather National Weather Service Weather Underground
Model forecasts:
Weather Underground
NCAR Research Applications Program

4. Equations of motion (z, height, as vertical coordinate)
Name __________________________
November 19, 2009
Equations of motion (z, height, as vertical coordinate)
tangential coordinate system on Earth’s surface
(x, y, z) = (+ east, + north, + local vertical)

5. Equations of motion (p, pressure, as a vertical coordinate)
Name __________________________
November 19, 2009
Equations of motion (p, pressure, as a vertical coordinate)
tangential coordinate system on Earth’s surface
(x, y, p) = (+ east, + north, pressure is vertical coordinate)

6. The quasi-geostrophic (QG) equations
momentum equation
geostrophic wind
continuity equation
thermodynamic equation

7. Geopotential tendency equation
Vorticity Advection
Thickness Advection
Linear partial differential equation for geopotential tendency.
Given a geopotential distribution at an initial time, can compute geopotential distribution at a later time.
The right hand side is like a forcing.

8. QG-omega equation Combine all QG equations
Link between vertical derivative of vorticity advection (divergence/stretching) and vertical motion

9. What is our strategy for exploring complex unsolvable equations?

10. The importance of rotation, the Coriolis parameter.
What are the differences between the tropics and the middle latitudes on Earth?
Tropics:
The area of the tropics – say + and – 30 degrees latitude is half the area of the Earth.
Might say the tropics is + and – 20 degrees of latitude, and subtropics are between 20 and 30 degrees of latitude.
The importance of rotation, the Coriolis parameter.
What else is different?

11. Picture of Earth: What can you say about this figure?

12. Differences between the tropics and middle latitudes
The contrast between summer and winter is not as large as at middle and high latitudes.
There is lot of solar heating.
There is a lot of water!
What is the “physical” difference between water and land?
Sea surface temperature is important to dynamics.
What happens to water when it is warm?

13. Coriolis force
The coriolis parameter decreases to zero at the equator.
Approximated by b X distance from the equator.
We see from our wave equation and the conservation of vorticity that b is a parameter of central importance to the dynamics
Advection of planetary vorticity.

14. Let’s think about the Coriolis parameter

15. Coriolis parameter in the tropics
What latitude is Coriolis parameter, say, 10% of mid-latitude value?

16. Perturbation equation: Barotropic Rossby wave at middle latitudes

17. Wave like solutions

18. Recalling our simple wave solution and comparing advection of planetary and relative vorticity. (mid-latitudes)

19. Coriolis force
Can we say that the advection of planetary vorticity is less important in the tropics?
We will come back to this.

20. Let’s think about waves some more
Some fundamental ideas.
Waves have some sort of restoring force
Buoyancy waves: gravity
Rossby waves: The gradient of planetary vorticity
Think about the conservation of potential vorticity
Waves tend to grow and decay at the expense of the “energy” in the mean state.
Waves tend to respond to out of balance situations.
Waves tend to move things towards equilibrium
Waves propagate
So they can communicate things happening in one part of the fluid to far away places.

21. In middle latitudes:
How do the Rossby waves that cause weather, the synoptic waves, get their energy?

22. Energetics of Mid-latitude Cyclone Development
The jet stream is commonly associated with strong temperature gradients in the middle/lower troposphere (thermal wind relationship)
Mid-latitude cyclones develop along waves in the jet stream
Mid-latitude cyclones are always associated with fronts (Norwegian cyclone model)
There is a link between temperature gradients and cyclone development…

23. Barotropic/baroclinic atmosphere
Energetics:
Baroclinic = temperature contrast = density contrast = available potential energy
Extratropical cyclones intensify through conversion of available potential energy to kinetic energy

24. Energetics in mid-latitude wave
Ability to convert potential energy to kinetic energy directly related to tilt with height (offset) of low/high pressure

25. Tropics and middle latitudes
In middle latitudes the waves grow from the energy available in the baroclinic atmosphere.
horizontal temperature gradients
scale is large
latent heat release is on scales small compared to baroclinic energy convergence.
In the tropics the horizontal temperature gradients are small.

26. An estimate of the January mean temperature
mesosphere
stratopause
note where the horizontal temperature gradients are large
stratosphere
tropopause
troposphere
south
summer
north
winter

27. Tropics and middle latitudes
Baroclinicity is less important in the tropics
Latent heat release is generally more important.
What does this mean?

28. Vertical Velocity
I emphasize the importance and the complications of the vertical velocity.
It is small, very small, but it is at the center of converting potential energy to kinetic energy. (thermodynamics to motion)
How do we calculate vertical velocity?
Something small in an environment full of big things.

29. Vertical velocity: Omega equation
Kinematic method
Horizontal advection
Diabatic method
Omega equation
What are the ways that we think about vertical motion? Diabatic, Horizontal Advection, Kinematic Method.

30. Characteristics of large-scale vertical velocity
In all of the estimates for vertical velocity what is missing?
The answer is _______________
The vertical velocity in this large-scale, mid-latitude description of dynamics is exactly what is needed to maintain what balances ____________ ?

31. Importance of latent heat release
Diabatic processes are more important in the tropics.
Hence, vertical velocity is more strongly related to diabatic heating than to temperature advection.
What about divergence?
The scale of the forcing of motions is small
Related to the phase change of water.

32. Estimating the vertical velocity: Diabatic Method
Start from thermodynamic equation in p-coordinates:
If you take an average over space and time, then the advection and time derivatives tend to cancel out.
Diabatic term

33. Picture of Earth: What can you say about this figure?

34. Inter-tropical Convergence Zone (ITCZ)

35. ITCZ Circulation
This is one place where heat is the direct driver of motion.

36. A couple of things to note
The winds at the surface in the tropics are, on average, easterly, from the east, towards the west.
Go back to our mid-latitude wave:
What does this say about waves in the tropics?
Well, it says, they are different!
(Not that they don’t exist!)

37. ITCZ: Seasonal differences

38. ITCZ: Seasonal differences
What is happening here and here?

39. South American Seasonal Cycle
CONVECTION GOES WILD

40. Cloud Liquid Water: Average
NOTE: Remarkable areas with no clouds! No rain!

41. Vertical circulation around the ITCZ
What is the direction of the zonal wind?

42. ITCZ: Seasonal differences
What is happening here and here?

43. Monsoonal Flow

44. Lets return to an old problem
DIVERGENCE
CONVERGENCE
PGF L
warm
core H
cold
core L H
PGF
CONVERGENCE
DIVERGENCE
Earth’s surface
OCEAN
LAND
SUMMER TIME

45. Thermal circulation H L H L PGF cooling warming PGF Earth’s surface
OCEAN
LAND
SUMMER TIME

46. Monsoonal Circulation
Driven by land-sea temperature contrast.
Reversal of flow from summer to winter.
Tremendously important to precipitation.
South and East Asian monsoon among most important of circulation features.

47. Circulation features of the tropics
Inter-tropical convergence zone
Hadley circulation
Monsoonal circulations
Madden-Julian Oscillation
African easterly waves
Walker circulation
El Nino and La Nina

48. Madden-Julian Oscillation
OLR = outgoing longwave radiation.
Cold is the top of the clouds.
Cold is the top of the hot towers.

49. African easterly waves

50. African easterly waves
That’s north Africa. It’s summer.
What is happening here and here?

51. African easterly waves
The Sahara gets SO HOT, that the meridional temperature gradient is important.
But it is reversed over our normal thinking!

52. So let’s think about these scales of motion.

53. Equations of motion in pressure coordinates (using Holton’s notation)

54. Scale factors for “large-scale” mid-latitude

55. Introduce another vertical coordinate.

56. Equations of motion in log pressure coordinates (using Holton’s notation)

57. Scale factors for “large-scale” tropics

58. Scaling: Momentum Equation

59. Scaling: Momentum equation
Geostrophic balance is not dominant.
How many km from the equator is geostrophic term no longer small?
What about b?
If the pressure gradient is balanced in the momentum equation, then ...

60. This means something
For a similar scale disturbances in the tropics and middle latitudes the geopotential perturbation is a smaller by an order of magnitude in the tropics.
What does this mean
for the scales of motion?
for the important physical terms?

61. Use the hydrostatic equation to say something about temperature
The temperature variability in tropical systems of scale H, is very small.

62. Go back to the scaling of the momentum equation
Vertical advection is very small.

63. What is the scale of divergence and vorticity?
So --- the divergence relative to the vorticity is even smaller than in middle latitudes.
The flow is also quasi-nondivergent.

64. Remember vorticity and divergence.
Remember that we earlier said that the flow could be defined as the sum of the rotational flow and the irrotational flow.
rotational  vorticity
irrotational  divergence

65. Momentum equation: approximately

66. Make a vorticity equation:
A VORTICITY EQUATION: Absolute vorticity conserved
The temperature variability in tropical systems of scale H, are very small.
The flow is quasi-nondivergent.
Vertical advection is very small.

67. Thinking about the tropics
These disturbances are nearly barotropic.
There is no mechanism for these disturbances to convert potential energy to kinetic energy.
Yet, we know there are lots of disturbances in the tropics.
What does this mean?

68. It means Tropical systems get their energy from either
Diabatic forcing from latent heat
Mid-latitude forcing

69. Some remembering How do we find wave-like solutions?
What is the strategy?

70. Linear perturbation theory
Assume: variable is equal to a mean state plus a perturbation
With these assumptions non-linear terms (like the one below) become linear:
These terms are zero if the mean is independent of x.
Terms with products of the perturbations are very small and will be ignored

71. Consider some equatorial waves
Kelvin waves (trapped waves):
coastal Kelvin waves (in the ocean!)
equatorial Kelvin waves (atmosphere and ocean)
Equatorial Rossby (ER) and Mixed Rossby-Gravity (MGR) wave
After end of lecture

72. Equatorial Kelvin waves
Amplitudes decay away from the ‘’boundary’’ (equator)

73. Coastal & Equatorial Kelvin waves
Connection between coastal and equatorial Kelvin waves:
Coastal Kelvin waves can turn the corner and circulate counterclockwise in northern hemisphere around a closed basin
Important for El Nino

74. Equatorial waves
Equatorial waves are important class of eastward and westward propagating disturbances
Present in atmosphere and ocean
Trapped about the equator (they decay away from the equator)
Types of waves:
Equatorial Kelvin waves
Equatorial Rossby (ER), Mixed Rossby-Gravity (MWR), inertia-gravity waves
Atmospheric equatorial waves excited by diabatic heating by organized tropical convection
Oceanic equatorial waves excited by wind stresses
Waves communicate effects of convective storms over large longitudinal distances

75. trapped wave?

76. Equatorial Kelvin waves: Derivation (1)
For equatorial Kelvin waves assume:
Flat bottom topography
Coriolis parameter at the equator is approximated by equatorial -plane (with f0=0, )
Meridional velocity vanishes: v = 0 (everywhere)
Shallow water equations become:
v=0
equator

77. Equatorial Kelvin waves: Derivation (2)
Linearize shallow water equations about a state at rest with mean height H:
Compute
Yields
c = phase speed

78. Equatorial Kelvin waves: Derivation (3)
Seek wave solutions of the form (allow amplitudes to vary in y):
Yields the system
Rearrange (1):
Plug (4) into (2):

79. Equatorial Kelvin waves: Derivation (4)
(5) can be integrated immediately: amplitude function (with u0: amplitude of the perturbation at the equator)
Solutions decaying away from the equator exist only for c > 0 with
Therefore: Atmospheric Kelvin waves always propagate eastward
Their zonal velocity and geopotential vary in latitude as Gaussian functions centered at the equator
e-folding decay width is
e.g. for c=30 m/s,
Yk=1600 km

80. Equatorial Kelvin waves: Derivation (5)
Amplitude of the height perturbations (use eq. 3):
The physical solutions are

81. Equatorial Kelvin waves: Velocity and height perturbations

82. That’s All Folks
Following slides introduce a number of other wave equations, from different scaling assumptions and a couple of new techniques.

83. One more wave equation I am showing this for the following reasons
1) To demonstrate that waves are a type of solution to the equations of motion.
They are not the solution.
2) That different types of waves exist with different balance of the terms in the equation.
3) How we explore those different balances

84. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (1)
Shallow water equations, no topography
Coriolis parameter at the equator is approximated by equatorial -plane

85. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (2)
Linearize shallow water equations about a state at rest with mean height H:
Seek wave solutions of the form (allow amplitudes to vary in y):

86. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (3)
Yields the system
Solve (1) for :

87. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (4)
Plug (4) into (2) and (3), rearrange terms
Solve (6) for :

88. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (5)
Plug (7) into (5) and rearrange terms. Yields second-order differential equation for amplitude function
More complicated equation than before since the coefficient in square brackets is not constant (depends on y)
Before discussing the solution in detail, let’s first look at two asymptotic limits when either H∞ or =0

89. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (6)
Asymptotic limit H∞
Means that the motion is non-divergent, consequence of the continuity equation (3):
Equation (8) becomes

90. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (7)
Asymptotic limit H∞
Solutions for the amplitude function in (9) exists of the form:
Provided that the frequency  satisfies the Rossby wave dispersion relationship:
It shows that for non-divergent barotropic flow, equatorial dynamics are in no way special
Earth’s rotation enters in form of , not f

91. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (8)
Asymptotic limit =0
All influence of rotation is eliminated
Solutions of (9) exists of the form:
Equation (8) reduces to the shallow water gravity model
Non-trivial solutions if frequency  satisfies
Pure gravity wave response

92. I have made my point. The general solution follows. It is
Pretty complex to talk about, requires thinking for it to make sense.
Introduces some new things that you can wait until AOSS 451 to get

93. There is a mix of waves that can be isolated under these conditions.
The positive root corresponds to an eastward moving equatorial inertia-gravity wave
The negative root corresponds to a westward moving wave, which resembles
an equatorial inertia-gravity (EIG) wave for long zonal scale (k  0)
An equatorial Rossby (ER) wave for zonal scales characteristic of synoptic-scale disturbances
This mode (n=0, westward moving) is therefore called Mixed Rossby-Gravity (MRG) wave.

94. Equatorial Mixed Rossby-Gravity wave
Plane view of horizontal velocity and height perturbations in the (n=0) Mixed Rossby-Gravity (MGR) wave, propagates westward H L
Equator L H

95. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (9)
In the general case with we seek solutions for the meridional distribution of
Boundary condition
For small y the coefficient in square brackets is positive and solutions oscillate in y
For large y the coefficient is negative, solutions either grow or decay
Only the decaying solution satisfies boundary condition

96. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (10)
Matsuno (1966) showed that decaying solutions only exist if the constant part of the coefficient in square brackets satisfies the relationship
Cubic dispersion relation for frequency 
Determines the allowed frequencies  of equatorially trapped waves for the zonal wavenumber k and meridional mode number n

97. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (11)
Replace y by the non-dimensional meridional coordinate
Then the solution has the form where v0 is a constant and Hn() designates the nth Hermite polynomial.
The first few Hermite polynomials have the form
Index n corresponds to the number of N-S nodes in v

98. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (12)
In general, the three  solutions of can be interpreted as an equatorially trapped
eastward moving inertia-gravity wave
westward moving inertia-gravity wave
westward moving Rossby wave
However, the case n = 0 must be treated separately
For n = 0 the meridional velocity perturbation has a Gaussian distribution centered at the equator

99. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (13)
For n = 0 (special case) the dispersion relation factors as
The root (westward propagating inertia-gravity wave) is not permitted, violates an earlier assumptions when eliminating
The two other roots are

100. Equatorial Rossby and Mixed Rossby-Gravity waves: Derivation (14)
The positive root corresponds to an eastward moving equatorial inertia-gravity wave
The negative root corresponds to a westward moving wave, which resembles
an equatorial inertia-gravity (EIG) wave for long zonal scale (k  0)
An equatorial Rossby (ER) wave for zonal scales characteristic of synoptic-scale disturbances
This mode (n=0, westward moving) is therefore called Mixed Rossby-Gravity (MRG) wave.

101. Equatorial Mixed Rossby-Gravity wave
Plane view of horizontal velocity and height perturbations in the (n=0) Mixed Rossby-Gravity (MGR) wave, propagates westward H L
Equator L H