Class Notes April 26
Review questions on the reflection of the long-waves at the boundaries
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1. Does the Wave transport mass?
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(related to the net mass transport by the Rossby Wave at tropics)
Yes! Because we are just looking at the certain period and the wave is
continuously forced on its way to the west.
2. Coastal Kelvin wave
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On the Eastern Boundary, the Kelvin wave comes in and the long Rossby
waves are reflected back. There's also coastal Kelvin wave traveling
poleward following the coast. The coastal Kelvin wave explains the
energy difference between the incoming Equatorial Kelvin wave and the
reflected Rossby wave, and starts from the critical latitude of each
meridional mode.
3. Reflected Long Rossby wave
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As the zonal scale is much greater than the meridional scale near the
Equator (k much less than l), the Rossby wave dispersion relation tells
us that the phase speed of the long Rossby wave is approximately
w/k=U-beta/l^2. So as you go to higher latitude the beta gets smaller
and the long Rossby wave propagates slower. This is why the reflected
long Rossby waves travels faster near the Equator.
Vertical Propagation of the reflected long-waves
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1. Justification for the use of 1.5 layer model on the ENSO problem We saw
how deep the 1 year period Rossby wave and 3 year Rossby wave dives as
they travel westward starting from the surface at the eastern boundary
at different latitudes. We also saw the same plot for the Kelvin wave
with the period of both 1 year and 3 year. These plots can give you the
sense about how far you can go wrong if you start to look at different
frequencies with 1.5 layer model. The diving angle increases with the
higher frequency and higher latitude. The l year Rossby wave dived down
to 500m when it got to the western boundary at 6'N and 1000m at 9'N. The
3 year Rossby wave dived to the 150m at 6'N and 300m at 9'N. So we can
say 1.5 layer model is still valid if you're interested in period longer
than 3 years, as the reflected long Rossby stay within the thermocline.
And the 1.5 layer model is only valid near the Equator, but fortunately
that's the area most of the energy is reflected. Remember that the
reflected energy is almost zero when it gets to 18N. So 1.5 layer model
is safe to use for the ENSO problem!
2. long-waves dive below the thermocline
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It was briefly mentioned that there are some studies showing the energy
dives below the thermocline feeds the deep equatorial jet.
Evaluation of the linear theory
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1. 'Obs. of the depth of the 20'C isotherm' vs '1.5 layer reduced gravity model'
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The comparison between the 20'C isotherm depth from the
observation and the interface depth from the 1.5 layer linear reduced
gravity model forced by observed wind at 13'N showed very good agreement
on the westward propagating features. Also the comparison was made
between the reduced gravity model runs with and without the eastern
boundary condition set to the observed values at the eastern boundary.
They showed no significant difference, but it was probably due to the
fact that the model was run at 13'N because the critical latitude for
the waves longer than 3 year period lies equatorward from 13'N. So if
it was test at 5'N, we would probably see influences of the Rossby waves
reflected from the eastern boundary. The response only to the local
Ekman pumping without propagation clearly shows the westward propagation
in the observed thermocline is due to the oceanic Rossby Wave rather
than the westward propagating component in the wind stress.
2. Hindcast of '82-83 Sea level using linear model
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The sum of 1st 4 modes matches very well with the observed sea level at
the islands in the tropical Pacific. Actually most of the contribution
was from the 1st and 2nd mode. Most of the rms difference between the
observation and model result were less than 1cm. As we are looking at
the signal of 30cm, this really simple physics captures most as long as
we are interested in the low frequency near the Equator.
3. 'Observed Sea level for 1961-1978' vs '1.5 layer reduced gravity model'
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Very Good!!!
4. Kelvin Wave reflected from the Island chains in the Western Pacific
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The eastern boundary in the tropical Pacific is almost solid wall, but we
have the island chains for the western boundary. So the linear theory
may fail at the western boundary. But the simulation shows 82% of the
perfect reflection is reflecting back, while the reflection efficiency
from the observation is 73%. So we don't have to worry about the island
chains.
5. A prognostic model for the SST in the Tropics
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The reduced gravity model whose dynamics is all adiabatic was forced by
observed wind plus an annual cycle of insolation. In addition to that,
the entrainment of the cold water below the mixed layer was added to
compute SST prognostically. SST calculation is purely prognostic in a
sense that any change in SST doesn't change dynamics. Temperature of the
thermocline water was related empirically to the depth of the
thermocline according to the observation. The model result showed very
good agreement with observation for both the annual cycle and 20 year
timeseries.
6. Conclusion
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Reduced gravity model and OGCMs governed by linear physics simulates
observed SST and Sea level incredibly!!!