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Thread: Geophysical Fluid Dynamics

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    Geophysical Fluid Dynamics

    This is a note on Shallow Water Wave theory
    and Planetary wave propogation in the Ocean
    and Atmosphere.

    note.pdf (281 KB)

    Courtesy: Prof. Dabasis Sengupta, CAOS, IISc, Banglore, India

    PS: Please post GFD related Notes in succession to this
    thread. Please feed back if the files or link does not work.
    ------------------------------------------------------------------------------------------------------

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    Quasi-Geostrophic Motions in the Equatorial Area, T. Matsuno, (1966)

    This classical paper is about shallow water wave solutions
    in the equatorial beta plane. This paper requires special respect as it
    dealt with the equatorial free oscillations for the first time.

    Quasi-Geostrophic Motions in the Equatorial Area, 1966: T. Matsuno,
    Journal of Meteorological Society of Japan, Vol. 44, No.1, 25-42.

    PS: Numerical Solutions of the equatorial osciallations will
    be available soon with this thread.

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    Forced Oscillations in the Equatorial Beta plane


    Equatorial Ocean (Atmosphere) dynamics is under serious discussion since early days. Several Classical works have
    been addressed in this area and solutions are well attested. Equatorial region acts as the 'wave guide' for planetary
    waves. These waves are of utmost importance in the climate of the Ocean and Atmosphere. For example the so called
    ENSO (ElNino and La Nina) is supposed as the coupled oscillation of these waves in the Ocean and the Atmosphere.
    Another example is the Madden-Julian Oscillations in the Atmosphere. In the Indian Ocean the well known Wyrtki jet
    is also explained by the equatorial waves driven by the seasonal switching of the Indian Ocean winds.

    The oscillation in the equatorial region is solved by using shallow water equations in the beta plane (Matsuno (1968), Gill (1980)
    and many other papers addressed as well. The dispersion relation of these waves in the k-w plane (wavenumber-frequency plane)
    is quite known to us. These waves cause typical 'circulations' in the ocean and atmosphere.

    The ocean (atmosphere) currents associated with these waves are interesting to notice and is shown here.

    The shallow-water equations in equatorial beta-plane is solved numerically in a 1&1/2 layer model.
    The basin is ideal (no realistic boundary) with a similar dimension of Pacific Ocean. In 1&1/2 layer model, only one
    layer (upper layer) is under motion (100 m thickness in this experiment). The density of upper layer and lower layer is
    chosen arbitrary values. The ocean is 'forced' using a 'zonal wind' in the equatorial region, symmetrical about
    the equator (Gaussian distribution -see Figure below). The meridional structure of the wind forcing along 175 E is also shown.
    The unit is in dynes/cm2.


    Figure: 1

    The wind is westerly and kept constant for 30 days to resonate the ocean and after that it is switched off.
    (Please note that this sudden switch-off of the wind-stress forcing is strictly an experimental case.
    In real ocean, wind blows continuously with seasonal changes).
    The ocean (which is in rest) starts to move due to this wind-forcing. The oscillations develop in the ocean and waves develop.
    Here two types of waves are discussed.

    1.Kelvin waves propagating to the east with maximum speed.

    2.Rossby waves propagating to the west with a speed of nearly 1/3 of the Kelvin wave.

    How these waves develop in our experiment ?

    Let us evaluate the process one by one. At first a wind suddenly blows from the west to east (please note
    that the wind in this context means the wind-stress). This will 'pile-up' the water to the east-side of the wind core.
    This piled-up water drives the Kelvin wave to the east. The 'word' pile-up means that the
    sea surface height is elevated about the equator (maximum at the Equator and is reduced to the either side of the
    Equator- a typical exponential wave height dampening of Kelvin wave).
    So the meridional structure of the Equatorial Kelvin wave will be something like shown in the figure below (black line).



    Figure: 2

    This 'piling-up' of water at the equator causes the down-welling along the way of this Kelvin Wave (because pressure is high
    as the water accumulate about the equator). This is called a 'down-welling' kelvin wave.

    Does it mean that the kelvin wave is always 'down-welling' ?

    No, the other way is possible as well. That is, at the equator the Sea surface height is minimum and increases either
    side away of the equator. This 'depression' in the surface height is obviously a 'low-pressure', and water to 'upwell'.
    Thus the 'upwelling' occurs along the path of this Kelvin Wave. This is called an 'up-welling Kelvin Wave.

    What happened to the west side of the 'wind-core' ?

    At the western side of the wind-core, since the water has shifted to the east, the sea level lowers. This causes a 'Rossby Wave'
    to develop. The phase speed of this Rossby wave is nearly 1/3rd of the Kelvin Waves and thus moves slower. The horizontal
    structure of these Rossby waves is different from the Kelvin Waves. It has two vortices on either side of the equator.
    The meridional structure of this Rossby wave is shown in Figure above (Red line). It has a maximum
    (negative SSH)of about '4 degrees on either side' of the equator and a minimum at the 'equator' (still negative, not zero).
    Since this Rossby wave has 'negative Sea Surface Height', its pressure is obviously low. Means water upwells along the path of
    this Rossby wave. So this is called 'up-welling Rossby Wave'. (A down-welling Rossby wave is possible as well.
    This will have positive Sea Surface Height instead).

    Why are these waves important to us?

    1. These waves cause the Sea-level to vary along its propagation path.

    2. This causes the 'Thermocline' (in the ocean) to move 'up and down'
    and affects the SST.

    3. Since these waves have large horizontal structure, the ocean currents
    associated with these waves are 'major' in the Oceanography.

    Now let us see how the 'ocean currents' vary due to these waves. The Figure below shows the evolution of the above
    mentioned waves in our experiment.





    Figure: 3 Both the Sea Surface Height anomaly (color shade )and currents (vector) are shown. Contour interval and vector
    scale are different for each plots, (only 'colors' are projected by 'changing the values')
    The plots are given for Day-25, 50, 75 ,100, 125, and 150 during the integration. (Remember that wind force is kept only for
    first 30 days, after that switched off).


    Days-25: The waves are well developed. The 'red' color is Positive Sea Surface Height anomaly and blue is negative. The 'red color'
    (positive SSH) shows the Kelvin Wave propagating to the east. Its horizontal structure
    is large. The ocean currents due to these waves (vector arrows) are at the east. The maximum current is at the center
    of the wave-core. Thus equatorial currents are like a 'jet' along the 'Kelvin Wave' paths. (Do you remember something
    similar 'jet' in the Indian Ocean? It is the Wyrtki Jet. So now it makes sense how these develop in the Indian Ocean.
    But don't believe it blindly. Wrytki jet needs much more explanation than this). Now let us look at the 'blue color'. It is negative
    SSH anomaly, and has 'two vortex' like structure about the equator. It is called the 'Rossby wave'. The currents are like
    'vortices on either side of the equator' (in this case cyclonic). It propagates much slower than the Kelvin Wave.

    Day-50: Here is not much to explain. The Rossby wave ('blue color')is reaching closer to the western boundary.

    Day-75: What happens to the Kelvin wave once it reaches the eastern boundary?
    It propagates along the coast towards the pole. It is called coastal Kelvin waves. Since it is the positive SSH, it is still the down-welling
    coastal Kelvin waves. (coastal Kelvin Wave moves pole-ward' in the eastern boundary' and 'equator-ward in the western boundary').

    Does all part of this Equatorial Kelvin Wave propagate coastaly?

    No, Some parts of its energy 'Reflect back' at the eastern boundary and propagate to the 'west' as 'Rossby Waves'.
    We know that Rossby wave has a 'vortices' like structure on either side of the equator. If you follow Day:50, 75, 100 and 125
    carefully, you can see how these Kelvin Wave reflect back as a Rossby Wave (the 'red' patch in the east) in the eastern
    boundary. Thus 'down-welling' kelvin wave reflects at the eastern boundary and becomes a 'down-welling Rossby Wave'. Since
    Rossby wave moves slower, it slowly advects to the west. If you look at the ocean currents over this Rossby Wave, we can
    see a 'anti-cyclonic' vortices on either side of the equator. (A similar anti-cyclonic vortices in the Equatorial Indian
    Ocean is recently noticed by Rahul et al(2004) called as 'Twin-Gyres'.

    Day-100: There is not much to tell about here.

    Day-125: What happened to the 'Rossby Waves' at 'western boundary' ?

    It 'can-not' propagate coastally pole-ward as coastal-kelvin wave, because Kelvin waves need land on the right hand
    side of its propagation in northern hemisphere. (land on the left hand side in the southern hemisphere).

    So it reflects at the western-boundary and propagates to the east as a 'Equatorial Kelvin' wave. But this
    time, look at the color of this equatorial kelvin wave. It is blue, means negative SSH, or a 'upwelling kelvin wave'.
    As we know, a Kelvin wave moves faster and suddenly propagates to the east very fast. Now, let us look at the ocean currents
    over it. Its westerly jet. (I don't know any example to show in the real ocean for this). If you follow Day:125 and 175 you can see this kelvin wave ('blue' color) more clearly.



    Day-175: Here you can see, the Kelvin Wave reflected Rossby wave (red) has reached up-to the center of the ocean.The Rossby Wave reflected Kelvin (blue), reaches the eastern side and almost propagated coastally toward the
    pole at the eastern boundary. (This Kelvin again reflects at the eastern boundary and above mechanism repeats.
    Its an oscillatory system. But these waves 'disperse' and thus the amplitude weakens gradually.)

    The upwelling/downwelling at the equatorial region associated with the wave propagation necessarily implies
    a divergence/convergence of the water at the equator. For example, if the wave patterns are occurring
    in the Atmosphere (not in the Ocean) the corresponding 'winds' will have the above patterns (just like ocean currents).
    In this case the ocean reacts to these winds by upwelling/downwelling at the equatorial region
    by the simple Ekman relation (water moves right to the wind in the northern hemisphere and to the left in southern hemisphere).
    However this is not the case at the 'exact Equator' where the Coriolis force vanishes and the balance is between
    inertial force and pressure-gradient.

    In the real case of the Atmospheric waves the moisture feedback to the waves are very important in determining
    the phase speed of the wave (in-additional to the stratification or forcing). For example, in the Pacific ocean,
    the waves which are generated in the Western parts are generally fed by the moisture, whereas it crosses the date line, the moisture supply is cut off (due to obvious reasons) and the waves become free.
    On the other hand, in the Indian Ocean it is not the case.

    Did we learn anything?

    1. Planetary waves in the Ocean can be detected by looking at the Sea Surface Height

    2. Ocean currents associated with the Planetary waves give unique pattern. This tool is useful to detect waves in the Ocean.

    3. Since planetary waves are large in horizontal structure, the associated currents advect water properties ( eg. SST) from place to place.

    4. It causes the thermocline movements as well as interaction of sub-surface variability with the surface.


    PS: If anyone noticed an error or miss-leading statements above PLEASE FEED BACK. Excuse is requested
    for any typographical errors seen above.

    Thanks: to Dr. Misuta for modeling instructions and Roxy for discussions. Francis helped to improve the writings.

    Some useful References:

    Quasi-Geostrophic Motions in the Equatorial Area, 1966: Matsuno, T,
    Journal of Met. Soc. of Japan, Vo.44, No.1

    Some simple solutions for heat-induced tropical circulation, 1980:
    Gill, A. E, Quart. Journal of Royal Met. Soc., 106, 447-462,

    Eastern Tropical Ocean Response to Changing Wind Systems: with
    Application to El Nino. 1976, McCreary J. P. Journal of Physical Oceanography, Vol.6, 632-645.

    Westward propagating Twin Gyres in the equatorial Indian Ocean, 2004:
    Rahul et al., Geophysical Res. Letters. Vol.31,


    __________________________________________________ ______
    [IMG]

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    Hi Praveen,
    Thanks for noticing me a mistake above, I corrected it.
    your feed back is appreciated
    -Vinu

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    Dear moderators,

    Would you please check why none of the figures are displayed in any of the posts?

    thanks and regards
    nisha

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    Thanks nisha for notifying. The figures were uploaded by vinu on the hokkaido university server before and they may have changed the location/access. Hopefully vinu can upload the same on the oceanographers.net upload folder and get the files displayed.

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    figures retrieved

    Thanks Nisha for noticing it.

    Hi Roxy,


    I got informed from Prescilla that they deleted files from old students,
    but they keep a backup. Akemi-san helped to retrieve some of them, but
    i am still trying to retrieve all figures.

    How can I upload these files to oceanographers.net

    -Vinu

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