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The Eliassen-Palm flux

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  • Location: The Netherlands
  • Location: The Netherlands

    One of the variables which has proven to be a challenge to interpret on the stratospheric charts on the FU-Berlin site is the Eliassen-Palm flux. With the help of some study, I am going to try to give an interpretation of what this flux actually indicates.


    Before we start, I want to emphasize that I am by no means an expert on this, so forgive any mistakes that could pop up in this post. If you see any, it would be very appreciated if you could tell, so that the explanation can be as complete and correct as possible!


    The Eliassen-Palm vector


    The Eliassen-Palm vector is the lowermost image on the figure given below:



    Zonal winds and fluxes of winter 2014-2015 as analysed by the ECMWF, as of 10-01-2015. The lowermost figure indicates the Eliassen-Palm vector.


    From now on the Eliassen-Palm vector will be abbreviated to EP-vector.


    Defining axes


    In order to avoid confusion during the explanation, first a brief explanation will be given of the different coordinates.

    • The x-coordinate is the so-called zonal coordinate or longitudinal coordinate.  The x-coordinate indicates movement toward the east or toward the west.
    • The y-coordinate is  the meridional coordinate or latitudinal coordinate. The y-coordinate indicates movement toward the pole (north) or equator (south), assuming one looks from the Northern Half (NH).
    • The z-coordinate is the altitudinal coordinate. This coordinate indicates movement upward or downward, or in other words, movement toward higher or lower altitudes (equivalent to pressure levels, or geopotential heights).

    Interpretation of the EP-vector


    Zonal averaging

    The Eliassen-Palm vector indicates the zonal average of a few quantities (which will be explained later on). Zonal averaging means one takes all points at a certain latitude (for example, 60N), and averages those to one mean quantity.


    An important result of this zonal averaging is that the x-coordinate does not matter anymore. In other words, it does not matter at which place you are going to stand as long as you are staying at 60N (for this example). The zonal averaged value will always be the same if one does not change the latitude. In layman’s terms, the zonal average will be the same whether you are standing in Vancouver or in London.


    Representation of axes


    Below is a representation of the EP-vector (as defined on the FU-Belin site) in terms of axes:




    As can be seen, there is no x-coordinate on this figure. This has to do with the fact that it could be ignored due to the zonal averaging.


    Taking the FU Berlin figure form above as a comparison, first taking the z-axis, upward arrows indicate movement of a certain quantity (which will be defined later) toward higher altitudes, while downward movement indicates movement of a certain quantity toward the Earth’s surface.


    Secondly, the y-axis is analysed. Movement of an arrow toward the right indicates movement of a certain quantity toward the pole, while movement of an arrow toward the left denotes equatorward movement.


    It is important to note that movement in the y- and z-axis can occur simultaneously.


    Interpretation of quantities


    After the crucial preparatory explanation, it is now time to go toward the most interesting part, being the interpretation of the different quantities. Because the mathematical description is rather complex, it will be left out of the explanation. Therefore, only a general description will be given. First, the y-axis will be discussed.





    The y-vector on the EP-flux denotes the momentum flux caused by atmospheric waves. Momentum is the tendency of a particle to move. Therefore, a particle that contains more momentum is in general moving faster. On a larger scale, one can think of pressure systems. When a gradient in pressure becomes larger toward the north (e.g. a low pressure area nearing Scotland when one lives in the UK) increases the transport of momentum. Relating this to the Impact of waves is a bridge too far for me at the moment, so it would be nice if somebody is able to add some or correct me.

    Going back to the figure from FU Berlin, it can be seen that the EP-flux can be positive as well as negative in the y-direction. This means that there could be transport of positive momentum toward the pole as well as to the equator.




    The z-vector is somewhat more difficult to interpret. The z-vector indicates the division between two terms. The nominator indicates the meridional flux of heat caused by waves. What this means is that if a wind toward the pole brings warmer air along with it, it is a positive flux of heat toward the pole. So basically one can interpret this as a flux of heat caused by waves.

    The denominator denotes the change of potential temperature with pressure level. As mentioned before, pressure level can be directly attributed to height, so it is the change of potential temperature with height.


    It is important to note that despite the fact that we are taking a meridional flux into account here, it is associated with the z-vector of the Eliassen-Palm flux.

    Physical interpretation

    For the z-vector, I’ll try to give a physical interpretation of both the nominator and the denominator.


    First, the meridional heat flux (belonging to the z-vector of the EP-flux). Usually, the temperature decreases when one moves toward the pole (in the stratosphere). This means that there is always a (positive) heat flux from the equator to the pole (i.e. heat is carried from the equator to the pole). However, the magnitude of this flux is influenced by the presence/absence of wave 1 or wave 2. The relation between this part of the flux and waves is beyond my knowledge, so I will not go into detail about this.


    Finally, the altitudinal potential temperature gradient (or change in potential temperature with height). In the stratosphere, the temperature does not change much with height. This means that the potential temperature increases with height. To illustrate this, below is a representation of the zonal mean potential temperature profile with height (see above for definition of zonal average).



    Zonal averaged potential temperature of February 2013. The x-axis indicates the latitude, while the y-axis denotes the height (in km). The colours show the potential temperature (in Kelvin).


    Note that there is a constant increase in potential temperature with height in the stratosphere (which starts from about 10 km).


    Linking this to the Fu-Berlin representation of the EP-flux, the Z-term appears to be positive at all times. This can be explained by the fact that the nominator and the denominator are always positive (due to the increase of potential temperature with height and the increase of temperature toward the equator).




    The Eliassen-Palm vector in itself is a very difficult term to digest. Hopefully, the explanation above has made it somewhat easier to understand the vector itself. However, the translation of this vector to the strength of atmospheric waves is not answered here, owing to my lack of knowledge on that subject. Therefore, the explanation is only a partial one, essentially missing the most important part.  Still, understanding the fundamentals of the vector is just as important, and perhaps this makes it somewhat easier to do so.


    Another aspect is that clarifying the first part of this vector only results in more questions, which is also the exciting part of science. There still remains much that can be learned.


    Finally, it would be great if anybody could notify me of any mistakes so that the explanation can be as good as possible. Also, contributions further explaining parts of the EP-vector that have not been explained yet are greatly appreciated!


    Further reading


    For anybody who wishes to explore the background or more details about the EP-vector, below are some articles about this flux:


    http://math.nyu.edu/~pauluis/TEM/TEM/Papers_files/Ellassen%26Palm_1961.pdf (original article of the founders of the EP-flux)


    http://www.sciencedirect.com/science/article/pii/S0997754614000120 (explanation of the EP-flux)









    Edited by Vorticity0123
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