Climate Modeling MEA 719 Lecture Set 7 PART3: Tropical Wave Dynamics

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Climate Modeling MEA 719Lecture Set 7 PART-3
Tropical Wave Dynamics
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Pressure and wind spatial distribution in the two
waves for n12
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Mixed Rossby-Gravity Waves

• this solution was obtained by Rosenthal (1965)
  before the publications of Matsuno (1966).
• this particular wave solution received
  considerable importance as it was actually
  observed in the real atmosphere in the lower
  stratosphere
• it is fundamental for the occurrence of the
  Quasi-Geostrophic Oscillation (QBO) in the lower
  stratosphere

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Mixed Rossby Gravity Waves
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Mixed Rossby-Gravity Waves continued

• ROSSBY WAVE CHARACTERISTICS
• Relative to basic state zonal current (in this
  case zero) waves moves westwards
• a little away from the equator, the flow in
  geostrophic with High to the right in the
  northern hemisphere and Low to the right in the
  southern hemisphere
• GRAVITY WAVE CHARACTERISTICS
• in parts of the solution the flow is
  cross-isobaric i.e., crossing from High to
  Low

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Kelvin Waves
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We have been examining the solutions for the
following equation
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the solution below is only valid for so long as v
tends to zero but does not identically vanish
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the nondimensional system becomes
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Dispersion Relationship
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Comments

• u increases very fast with y
• this solution is physically unacceptable

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Comments

• u decreases with y
• this solution is physically acceptable

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Kelvin Wave
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• Quite often in literature, this solution is
  referred to as Matsunos solution for n-1. Why
  this nomeclecture? The answer is as follows

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Westward propagating gravity waves
Westward propagating Mixed Rossby-gravity waves
Eastward propagating Kelvin waves
Eastward propagating gravity waves
SUMMARY DIAGRAM FOR ALL TYPES OF MODES
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Observed Properties
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The 40-50 day Oscillation

• In 1971 Roland Madden and Paul Julian stumbled
  upon a 40-50 day oscillation when analyzing zonal
  wind anomalies in the tropical Pacific
• They used ten years of pressure records at Canton
  (at 2.8 S in the Pacific) and upper level winds
  at Singapore.
• Until the early 1980's little attention was paid
  to this oscillation, which became known as the
  Madden and Julian Oscillation (MJO).

Madden, R. A., and P. R. Julian, 1971 Detection
of a 40-50 day oscillation in the zonal wind in
the tropical Pacific.J. Atmos. Sci., 28, 702-708
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Kelvin wave in the 40-50 day Oscillation and
Walker Circulation

• It is wave 1 in the zonal direction and moves
  slowly toward the east
• It is very prominent in the zonal velocity and
  surface pressure, but negligible in the
  meridional velocity component, thus consistent
  with the theory above
• This wave has since been identified as Kelvin wave

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Schematic for the 40-50 day Oscillation
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Relationship of 40-50 day Oscillation with ENSO

• Hypothesis - Individual or groups of MJO events
  can determine the timing and strength of a
  particular ENSO event. This is not to say the MJO
  alone causes ENSO. But the evolution of an ENSO
  event with and without the MJO can be
  considerably different.
• NOTE Other hypotheses exist

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ENSO Theory The Delayed Oscillator
http//iri.columbia.edu/climate/ENSO/theory/simpl
ified.html

• The Kelvin and Rossby wave signals propagate at
  different speeds
• The Kelvin wave travels eastward and in our
  idealized case has speed on the order of 2.9
  meters per second
• This means that a Kelvin wave will cross the
  Pacific Ocean, which extends from approximately
  120 East to 80 West (17,760 Kilometers in
  distance), in about 70 days
• The Rossby mode travels westward at one third the
  speed of the Kelvin wave, or about 0.93 meters
  per second. Thus a Rossby wave takes
  approximately 210 days to cross the Pacific.

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ENSO Theory The Delayed Oscillator
http//iri.columbia.edu/climate/ENSO/theory/simpl
ified.html
Ocean surface height anomaly along 140 West
Time-Zero
An eastward wind-stress forcing produces
equator-ward mass transport in both
hemispheres The mass surplus near the equator
then begins to disperse eastward as a so-called
(downwelling) Kelvin wave. The mass-deficit areas
begin to propagate westward as so-called
(upwelling) Rossby waves. Kelvin and Rossby waves
have different propagation speeds (and
directions).
Ocean surface height anomaly along 180 East
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Idealized El Nino Experiment(adapted from IRI
Website) http//iri.columbia.edu/climate/ENSO/the
ory/simplified.html

• After 25 days (see figures below) the Kelvin wave
  has moved from the central Pacific forcing region
  to the east
• At the same time, the Rossby wave has propagated
  to the west, but over a much shorter distance
• Over days 50 through 100 the Kelvin wave reaches
  the eastern boundary and reflects as a Rossby
  wave with positive sea surface height anomalies
• At the same time, the Rossby wave continues to
  propagate slowly to the west becoming visibly
  distorted by day 100 (associated with the
  interaction with the basin boundary).

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Idealized El Nino Experiment(adapted from IRI
Website)

• By day 125, the Rossby wave has reached the
  western boundary and is starting to reflect as a
  same-signed Kelvin wave
• We now see a time evolution similar to a Kelvin
  wave propagating eastward along the equator (this
  time starting from the western boundary) and a
  Rossby wave propagating westward from the eastern
  boundary. However, now the Kelvin wave has
  negative sea surface height anomalies, and is an
  upwelling wave. Over the period from day 125 to
  day 275 the Kelvin wave propagates from the
  western to the eastern boundary resulting in
  negative sea surface height anomalies along the
  equator in the east. During this same period, the
  reflected Rossby wave has traveled from near 120
  West to 170 West.

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Time-Zero
ENSO Theory The Delayed Oscillator
25 days
125 days
50 days
175 days
75 days
225 days
100 days
275 days
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Gills Dynamical Theory
Q is specified heating consistent with the
magnitude and spatial distribution of the SSTA
pattern
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Gills Dynamical Theory

• In the tropics, the so-called Gill model (Gill
  1980), a special case of the shallow water
  equations (Matsuno 1966), has proven quite
  successful at capturing the essential features of
  the tropical atmospheric response to diabatic
  heating
• The Gill model considers the linearized, resting
  basic state response to an equatorial heating
  anomaly that varies sinusoidally in the vertical,
  with a maximum at midlevels and zero values at
  the surface and "top" of the system
• The response is the first baroclinic mode that
  is, the atmospheric circulation at upper and
  lower levels are equal to each other, but of
  opposite sign
• Its horizontal structure is found using the
  shallow water equations. The relationship between
  winds and pressure show that the response is a
  packet of Kelvin waves to the east of the heating
  and two Rossby wave packets to the west of the
  heating. In the upper troposphere, the two Rossby
  wave packets correspond to a pair of anomalous
  anticyclones symmetric about the equator--quite
  similar to the observed ENSO response. In this
  model, by construction, the atmospheric response
  is equatorially trapped.

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The response of the Gill model (Gill 1980) to
equatorial heating of the form
cos(x)exp(-y2/4)sin(z). The surface wind field
and vertical motion field are shown in panel a,
where the upward motion roughly corresponds to
the forcing function the surface wind field,
repeated, and the pressure field are shown in
panel b and a longitude-height profile of the
vertical motion field is shown in c panel c.
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The 300hPa streamfunction response to a tropical
vorticity dipole source (located in shaded
region) in a barotropic model linearized about
solid body rotation. Panel b is as panel a, but
for the barotropic model linearized about time
mean January conditions (Branstator).
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Further reading

• Branstator, G., 1983 Horizontal energy
  propagation in a barotropic atmosphere with
  meridional and zonal structure. J. Atmos. Sci.,
  40, 1689-1708.
• Branstator, G., 1985 Analysis of general
  circulation model sea surface temperature anomaly
  simulations using a linear model, I, forced
  solutions. J. Atmos. Sci., 42, 2225-2241.
• Branstator, G., 1992 The maintenance of
  low-frequency atmospheric anomalies. J. Atmos.
  Sci., 49, 1924-1945.
• Branstator, G., 1995 Organization of storm track
  anomalies by recurring low-frequency circulation
  anomalies. J. Atmos. Sci., 52, 207-226.
• Gill, A., 1980 Some simple solutions for heat
  induced tropical circulation. Quart. J. Royal
  Meteorol. Soc., 106, 447-462.
• Held, I., and I.-S. Kang, 1987 Barotropic models
  of the extratropical response to El Niño. J.
  Atmos. Sci., 44, 3576-3586.
• Held, I., S. Lyons, and S. Nigam, 1989
  Transients and the extratropical response to El
  Niño. J. Atmos. Sci., 46, 163-174.
• Hsu, H.-H. and S.-H. Lin, 1992 Global
  teleconnections in the 250-mb streamfunction
  field during the northern hemisphere winter. Mon.
  Wea. Rev., 120, 1169-1190.
• James, I., 1995 Introduction to circulating
  atmospheres. Cambridge University Press (Great
  Britain), paperback ed., 422pp.
• Kasahara, A., and P. da Silva Dias, 1986
  Response of planetary waves to stationary
  tropical heating in a global atmosphere with
  meridional and vertical shear. J. Atmos. Sci.,
  43, 1893-1911.
• Matsuno, T., 1966 Quasi-geostrophic motions in
  the equatorial area. J. Meteorol. Soc. Japan, 44,
  25-42.
• Rosenthal S. L., (1965) Some preliminary
  theoretical considerations of tropospheric wave
  motions in equatorial latitudes Special case of
  Matsuno solutions Mon. Wea. Rev., 93, 606-612
• Rasmusson, E., and K.-C. Mo, 1993 Linkages
  between 200-mb tropical and extratropical
  anomalies during the 1986-1989 ENSO cycle. J.
  Clim., 6, 595-616.
• Sardeshmukh, P., and B. Hoskins, 1988 The
  generation of global rotational flow by steady
  idealized tropical divergence. J. Atmos. Sci.,
  45, 1228-1251.
• Schneider, E., and I. Watterson, 1984 Stationary
  Rossby wave propagation through easterly layers.
  J. Atmos. Sci., 41, 2069-2083.
• Simmons, A., J. Wallace, G. Branstator, 1983
  Barotropic wave propagation and instability, and
  atmospheric teleconnection patterns. J. Atmos.
  Sci., 40, 1363-1392.
• Trenberth, K., G. Branstator, D. Karoly, A.
  Kumar, N.-C. Lau, and C. Ropelewski, 1998
  Progress during TOGA in understanding and
  modeling global teleconnections associated with
  tropical sea surface temperatures. J. Geophys.
  Res., 103, 14291-14324.