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Atmosphere boundary layers influencing atmospheric refraction

Scope for this research

The following scope for the investigations of the atmospheric layers is chosen:

Layers in the atmosphere

For the determination of refraction the following atmospheric layers will be involved:
There is an aerodynamic roughness layer (characterized by the aerodynamic roughness length) which you don't see in the above model. This is because, IMHO, this layer is always hidden by the tallest object on the horizon (the aerodynamic roughness length/layer is much smaller then the highest object that makes the aerodynamic roughness length: Stull R., [1988], page 380) and thus not important for my scope. The viscous layer is still there: on top of the tallest object. Not convinced that the viscous layer is essential in the scope of this research?

What is needed for the height-temperature Profiler

For each layer in the above picture the following info is important:
  • the temperature and height of the bottom of the layer with a well defined zero height point
  • the dependency of this temperature and height on possible lower or higher layers or its internal structure.
  • the thickness/depth of the layer and its possible dependency on lower or higher layers or its internal structure.
  • the temperature changes within the layer; linear, exponential, constant, step, x-order Brezier/polynomial function, etc. and its possible dependency on lower or higher layers or its internal structure.
  • is more needed to get a better profile?
By the way the higher the layer the less important the accuracy is for determining the atmospheric refraction.

A model for (virtual) potential temperature emerges with two sections:
different parts in
              ABL 
  1. a bottom (blue) section;
    Which has three options: being a convective (surface temp>air temp), neutral (surface temp=air temp) or stable (surface temp<air temp) layer.
    The surface temperature and the air temperature at mixed/residual layer is in some way a given (on land it of course changes due to [lack of] solar radiation): so there is 'forcing'/'driving' from the surface.
  2. a top (red) section;
    This gives another three options: an inversion (entrainment/capping), direct FA (very stable/mature situation) or clouds. And above this the FA.
    The bottom FA temperature is seen as a more or less stable factor here (at least on a diurnal level): some 'forcing/driving' from the FA.
At this moment I assume one can combine one option of each section together to make the full land/marine ABL (not looking at the precise depth/temperature values of course), e.g.: Convective layer plus inversion layer (and FA).

The coordinates of height-temperature profile

Height of layer boundaries

Temperature at layer boundaries

The profiles are better defined in potential temperature.

Height-temperature profile of layers

Temperature profile seen by the light: the RPTP Profiler

The temperature Profiler is worked out in the below. The Profiler determines the temperature profile as seen by light path (Ray Path Temperature Profile: RPTP), with that in hand one can determine the temperature gradient the light sees as it travels its path and thus also the refraction.

A sample of elevation of the light path and the ground (derived from HeyWhatsThat) can be seen below (using the light path between The Hill and Pladda lighthouse). HeyWhatsThat can provide numeric output (instead of a picture) of the elevation of the light path (line of sight) and ground.
Elevation profile
          of a lightpath

By changing the scale, the lower elevations can be seen in more detail:
Elevation profile
          of light path

Number of height-temperature profiles needed

In principle the height-temperature profile should be in the path of the light ray (RPTP). If the RPTP is not available then one could use approximations and here are some general rules (exceptions are there of course):
The Standish/Auer methodology implemented by the author (using the algorithms of Hohenkerk&Sinclair) caters for these one or two profiles.

The height-temperature profile at several locations can be estimated (using e.g. balloon sondes or using a profile that is composed of [empirical derived] profiles as described in this link and this link).
The Profiler is now building the temperature seen by the path using likely boundary layer height-temperature profiles at certain points. So by breaking up the whole light path in segments, each starting and ending with (possible) different height-temperature profiles:

Within each segment the height-temperature profile for a certain distance is interpolated in such a way that the different sections of the height-temperature profile are respected.
Using this method one now has the height-temperature profile at each distance between the start and the end.
When knowing the height of the light path the Profiler determines the temperature at the location of the light for each distance (RPTP). The resulting temperature gradient can be used as input to the refraction module of Hohenkerk&Sinclair.

Remember that 'height' is seen from ground level and 'elevation' from sea level. For inversion/day time (CBL regime) the height-temperature profile might indeed relate to the height. But in stable (night) time (the SBL regime) height-temperature might related more to elevation-temperature profile or the height is per profile for a local minimum: a valley (not yet 100% sure about this!). MABL and CABL are most of the times elevation-temperature profiles (also the ground [height] in this case is the sea [elevation]).

An example of the RPTP

Lets take again The Hill as the viewing point (4.52727 West and 55.70659 North (WGS84)), the sea is some 15.7km away to the West. We look at (the blue line is the ground/sea elevation, the gray line the average ground/sea elevation and the orange line is the elevation of the Light of Sight):

Hilly and sloping terrain

It is important to realise that the landscape is hilly (Brown&Wood, 2003) and sloping (Garrett, 1994, page 180-182) towards the sea; thus the development of boundary layers in an horizontal equilibrium is complex. If wind speed is larger than 3.5 m/sec (windforce >2) (Lapworth 2003, page 1967 and Lapworth 2006, page 524) a more neutral environment is reached, lower wind speed has at this moment an unknown effect. More study is needed.
Some thoughts though around neutral/stable boundary layer (NBL/SBL):

Four cases around boundary layer and height/elevation

The RPTP (Ray Path Temperature Profile) for these two targets has been determined with the Profiler for four cases:
  1. The yellow dots give the RPTP if only the CBL/NBL/SBL height-temperature profile is used (in this case it is uses as the elevation-temperature profile).
  2. The blue dots give the RPTP if only the CMABL height-temperature profile is used (which is nor or less by definition: elevation-temperature profile).
  3. The dashed green line gives the RPTP if CBL/NBL/SBL and CMABL profiles are used and in both segments the elevation of the light path is used instead of the height.
  4. The orange line gives the RPTP if CBL/NBL/SBL and CMABL profiles are used in both segments the height-temperature profile is used. This is the most likely temperature profile the light sees.
In the below evaluation the average elevation of the ground has been used for case 4.
Case 1 and 2 can be evaluated using the Hohenkerk&Sinclair's algorithm (1985) as the height-temperature profile is independent of the location within the light path (because the atmosphere is seen as being in hydrostatic equilibrium and spherical symmetry).
Case 3 and 4 needs an algorithm where the height-temperature profile is depending on the location within the light path. I think van der Werf's algorithm (2017) is able to do this, as his temperature profile depends on the distance (in a sinus wave).

During daytime

For both examples we use a height-temperature profile over land so CBL (day profile) and a height-temperature profile over sea so CMABL (with sea temperature is higher than the air temperature). And the wind came from the West. A CABL is not explicitly included, as the light path is well above ground level near the coastal lines (at which height CBL and MABL profile suffice).

Height-Temperature
                profile MABL profile
CBL height-temperature profile
CMABL height-temperature profile
These profiles are derived (manipulated) from measurements (during strong inversion) made by tethered kite at the South Pole (Mahesh, 1997, Fig. 1).

The two targets show the below behavior for these three cases:
There are significant difference in these four case.
Although the resulting RPTPs depend of course on the actual height-temperature profiles: if the CBL and MABL are similar there would be less difference.
In Thom's situation [1958] the observation location is well in land and the target is well into the sea and thus two boundary layer profiles (like CBL and CMABL) are more obvious. 

Around Sunset

If the LABL is a near stable atmosphere (NBL) near sunset, the height-temperature profile is like below (using Lapworth, 2003, Fig. 19):
 Height-Temperature NBL
        profile


The RPTP of the two targets looks like:
With such a near stable atmosphere (NBL) at sunset, the CMABL is the one that determines the behavior mostly (case 2, 3 and 4 are quite similar). So, in these cases, the MABL will determine the refraction more than the NBL near the observer.

If one observes Sunsets over a vast plane surface from a height near the sea (such as Schaefer&Liller [1990] and Seidelmann [1968]) then using only the MABL would be ok. For observations from a high building towards to Western terrestrial horizon (such as Sampson [2003]), using only NBL would be ok.

Around Sunrise

If the LABL is a stable atmosphere (SBL) near sunrise, its height-temperature profile is like below (using Lapworth, 2003, Fig. 19):
 Height-Temperature
        NBL profile near SR


The RPTP of the two targets looks like:
With such a stable atmosphere (SBL) at sunrise, the CMABL and the SBL determine the behavior (case 4).
If one observes Sunrises over a vast plane surface from a height near the sea (such as Schaefer&Liller [1990] and Seidelmann [1968]) then using only the MABL would be ok. For observations from a high building towards to Eastern terrestrial horizon (such as Sampson [2003]), using only SBL would be ok.

Questions

When making the Profiler the following questions pop up, I hope you can help me with these (remember that my interest is mostly around sunset and sunrise events, so the questions below are related to that environment):

Open to feedback

Any ideas, errors, more/less layers, guideline values (heights and their reference point(s), temperature, lapse rate, function of the temp. change, etc.). In some way I want to be as practical as possible and thus also as usable as possible for the above stated scope. If you want to provide constructive feedback, let me know.

References

Brown, A.R., and N. Wood. 2003. 'Properties and parameterization of the Stable Boundary Layer over moderate topography', Journal of the atmospheric sciences, Vol November: pp. 2797-808.
Garrett, J.R. 1994. The atmospheric boundary layer (Cambridge: Cambridge University Press).
Hohenkerk, Catherine Y., and A.T. Sinclair. 1985. "The computation of angular atmospheric refraction at large zenith angles." ed. by HM nautical almanac office. Cambridge.
Lapworth, Alan J. 2003. 'Factors determining the decrease in surface windspeed following the evening transition', Quart. J. Roy. Meteorol. Soc., Vol 129: pp. 1945-68.
Lapworth, Alan J. 2006. 'The morning transition of the nocturnal boundary layer', Boundary-Layer Meteorology, Vol 119: pp. 501-26.
Mahesh, Ashwin, P. von Walden, and Stephen G. Warren. 1997. 'Radiosonde temperature measurements in strong inversions: Correction for thermal lag based on an experiment at the South Pole', Journal of Atmospheric and Oceanic Technology, Vol 2: pp. 45-53.
Thom, Alexander. 1958. 'An empirical investigation of atmospheric refraction', Empire Survey Review, Vol 14: pp. 248-62.
Werf, Siebren Y. van der. 2003. 'Ray tracing and refraction in the modified US1976 atmosphere', Applied optics, Vol 42: pp. 354-66.
Werf, Siebren Y. van der. 2017. 'Hafgeršingar and giant waves', Applied optics, Vol 56: pp. G51-58.

Acknowledgments

I would like to thank the following people for their help and constructive feedback: Wayne Angevine, Sam Barrett, Ian Brooks, Alan Lapworth, Barry Lesht, Brian Medeiros, Russ Sampson, Marcel Tschudin, Siebren van der Werf and Andrew Young and all unmentioned other people. Any remaining errors in methodology or results are my responsibility of course!!!
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Major content related changes: Sep. 5th, 2007