Atmosphere layers influencing atmospheric refraction

Scope for this research

• want to gather all relevant information around troposphere layers for determining terrestrial and astronomical (atmospheric) refraction
• the usage of this will be in archaeoastronomy environment, so it is difficult to do actual measurements at the moment the refraction was experienced;-)
  
• the apparent altitude that needs to be covered is between -4° to 90°.
  
• it should be able to look at a few specific atmospheric regimes; Convective (CBL), Nocturnal (NBL), Stable (SBL) boundary layers and any stability class. Also USA1976 (Standard Atmosphere) and MUSA76 profiles need to be incorporated.
  
• temperature and humidity ranges at observer should be normal for above mentioned atmospheric regimes.
• include the influences range that look at yearly, weekly and diurnal effects (inclusive Sun's set/rise events)
• vertical mixing is assumed to be the most important issues for this refraction scope.
  
• the observer is always on land. But several locations will have had observation towards the sea (many sunset/rise events are over a sea horizon). The land based Atmospheric Boundary Layer (LABL) will be quite different for marine/coastal environment (much more shallow and no real diurnal effects: Marine/Coastal atmospheric boundary layer [MABL/CABL]) then for land environment. An interesting introduction is here. The layer evaluation should include these three different environments: land, coastal and marine.
  
• My aim not about the theory of how (dynamic) atmospheres are formed, but about practical and realistic rules of thumb for height-temperature profile useable in the surface environment described in a specific steady state atmospheric regime and with an accuracy that is reasonable.
  So I am looking more at a phenomenological approach.
  Putting a figure to variability is difficult in a very dynamic weather system environment, but a predictability of say 68% (1 sigma) sounds ok. I am interested in the extremes (min./max.) and average/typical temperature profiles for each layer.
• priority will be given at the following conditions:
  
• sun set and rise moments,
  
• clear, isolated or scattered clouds, no real storms, no fog/mist/rain: fair-weather (otherwise no set or rise event),
• reasonable simple/uniform surfaces (like trees but no high houses: say non urban only neolithic settlements),  except when land/marine environments are nearby
  
• possibly small hills (they make the apparent horizon or they are used as observation point),
  
• latitudes between 20° and 70° (positive or negative),
  
• stable atmospheric regimes (mean values in the timescale between 10 and 60 minutes) ,
  
• land based observations looking at a sea or land horizon.
• temperature can be read in the below also as (virtual) potential temperature (Θ or Θ_v), because the mean virtual potential temperature might be easier to use for the proposed profiler.
  
• a vertical height resolution of (much) smaller then 10 m (certainly near the surface) is needed for determining the refraction accurately.
• for the astronomical refraction the temperature profile in the path of light ray is needed (Ray Path Temperature Profile: RPTP), and not the vertical profile at one location!
  
• hopefully (but not necessarily) it can also provide good profiles for special effect, like mirages, green flashes, etc.
• please provide feedback, ideas, hints etc. to improve the content of this page, let me know.

Layers in the atmosphere

• Mesosphere (between ~50 and ~80 km)
  
• mesopause (top of mesosphere)
  
• Stratosphere (between ~10 and ~50 km)
  
• stratopause (top of stratosphere)
• Troposphere (between 0 and 10 km)
  
• tropopause (top of troposphere)
  
• free atmosphere (FA; between ~2 and ~10 km)
  
• ABL
A nice picture of the LABL at homogenous land environment is copied  from Stull ([1988], page 11):

• for (homogenous) land environment (LABL)
  
• CBL regime (between 0 and ~2 km)
• entrainment zone (10-60% of ABL height)
• mixed layer or CBL (between 0 and ~2 km)
• surface layer (5-10% of ABL height)
  
• viscous layer (in region of cm's)
  
• NBL regime (between 0 and ~2 km)
• capping inversion (10% of ABL height)
• residual layer
• NBL (between 0 and ~500 m)
• viscous layer (in region of cm's)
• for coastal environment (CABL)
  
• TIBL (Temperature Inter Boundary Layer)
  
• wind from land (landwind)
  
• CTIBL (land temp < water temp)
• STIBL (land temp > water temp)
  
• wind from sea  (seawind)
  
• marine environment (MABL)
• for marine environment (MABL)
  
• water temp < air temp 
• SMABL (upto  ~500 m)
• viscous layer (in region of mm's)
• water temp > air temp
  
• CMABL (upto ~2000 m)
• viscous layer (in region of mm's)

What is needed for the height-temperature profile

• 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 refraction.

A model for (virtual) potential temperature emerges with two parts:

 

1. a bottom (blue) part;
   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) part;
   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 part 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

• a height-temperature profile is needed for atmospheric refraction (some people use a height-temperature gradient profile, but in essence that is the same), but the profiler can work in another coordinate system as long there is a one-to-one transformation.
  
• the temperature coordinate is well defined (being if °C, °F or K); it can (absolute) temperature (T), virtual temperature (T_v), potential temperature (Θ) or virtual potential temperature (Θ_v) and they can be translated into each other.
• the height coordinate: What is the zero height point of the height coordinate.
  
Defining the zero height point is essential, certainly as the work to be done is close to this zero height point!
So is the zero height point always the geoid or is it depending on the surface? If we have a vast flat plain surface (say more then 50 km in size), is that vast flat plain surface the zero height point for the temperature profile? The height axis of the temperature profile has to have a well defined zero height point (and may be different for different layers, as long as it is defined).

• some advantages/limitations of different vertical coordinate systems for the profiler; see here. The sigma coordinate system (s = (P₀-P)/P₀) might be handy for the profiler? Or would it be better to have a pressure-temperature profile (like Skew-T Log-P)? Let me know.
• geoid height: in this case the mean sea level (not the WGS84 height).
• sea height: is the actual sea height from the geoid height.
• surface height (the height of the ground/surface). This is surface height from the geoid height.
• observer height (location of the eyes/instrument). This is observer height from the geoid height.

Height of layer boundaries

• the tropopause depth is an distance from the geoid height (it ranges between 7000 and 20000 m, in USA1976/MUSA76 at 11000m). This height can vary with the underlying layers and the latitude.
  It is no problem to keep this one at 11000 [m] (as defined in Standard Atmosphere). Or should it be related to an average troposphere lapse rate (but not directly related to an other lapse rates of the surface layer) of 0.0065 [K/m] and the temperature at geoid height?
• The LABL consists of viscous layer or interfacial layer, surface layer, mixed layer and entrainment layer (and other layers in NBL). 
• The LABL depth is somewhere between 100-2000 m. Is this relative to the surface height or the geoid height? I am not sure myself... I think it is the surface height for vast flat plains.
  
The LABL depth is quite changeable during the day, diurnal effect (page 42).
And also during the year, and how?

• the surface layer depth, has a thickness of some 10% of LABL height. I assume this one is relative to the geoid or is it to the surface height? This layer is just above the viscous layer.
• the viscous layer depth (also called sometimes micro layer or interfacial layer). This is close to the surface and relative to the surface height. This is in the order of mm's. I don't think this is related to the aerodynamic roughness length.
  
This is the very bottom part of the LABL.

• LABL: to provide an educated guess of the layers involved at which time of the day (over homogenous land), a profiler has been made using several parameters:

• height of entrainment layer [m] at sunset
• height change [m] of top of entrainment layer between noon and sunset
  
• depth of entrainment layer [%] relative to height of top of entrainment layer
• height change [m] of top of capping layer between sunset and sunrise
  
• depth of capping layer [%] relative to height of top of capping layer
• height [m] of NBL layer at sunrise
  
• depth of surface layer [%] relative to height of entrainment layer
• period [hr] that sun is not strong enough before sunset
  


The profiler makes a heigth-potential temperature profiles as in the below section.

Temperature at layer boundaries

• 'tropopause temp': -56.5 degrees and quite stable for the above atmospheric regimes but depend on the latitude.
• at 'bottom LABL temp' (see below): close to 'sea/surface height temperature'
• at 'sea height temp': sea surface temperature (SST)
  

Height-temperature profile of layers

• balloon soundings
Balloon soundings provide input for a geopotential height-temperature profile. Normally the resolution is not that small (say every 100 to 400m a measurement), so near surface/geoid details (important for refraction calculations) are lost.
The ARM's Balloon-Borne Sounding System (SONDE) provides 4 times a day, every 2 seconds of its flight a measurements (a vertical resolution of around 10 meters). This comes closer to the needs for refraction calculations, but presumably not enough (will do some sensitivity analysis).

• above tropopause:
  For refraction calculations this profile is not really important, so taking the MASU76 sounds oke (table 1, van der Werf [2003]), which include upto mesopause.
• between ABL height and tropopause (free atmosphere)
  Is there is general behavior possible to describe that is close enough for all benchmarks, or do we need specific profiles here? Could we say it is a linear function between ABL depth and tropopause temperature.
• The viscous layer has a lineair temperature height profile, due to the molecular heat transport (Young A., pers. comm. [2007]).
• Variability of profiles:
  

  Environment	Event
  	Sunrise	Sunset	Diurnal	Seasonal	Remarks
  Land (LABL)	NBL	NBL/CBL
  Coast (CABL) landwind	NBL	NBL/CBL			more out to sea; MABL will take over
  Coast (CABL) seawind					well mixed layer
  Marine (MABL)					well mixed layer

  The above table provides an idea of the variability of the height-temperature profile in/around the mentioned event. Green means a relatively stable height temperature profile round that event. Yellow means that there are changes in the height-temperature profile and red means that significant changes are present (the variability is derived from Medeiros, B., [2005], Fig. 3 and 4).
  The colors say nothing if the profiles themselves are easy to define (normally also not the case); the color only says about the possible variation in the profiles.
• height - mean virtual potential temperature(Θ_v) idealized profiles during the day (from Stull, [1988], page 17):
  
  
  
  
  
• NBL: possible height-mean potential temperature(Θ) idealized profiles (from Stull, [1988], page 505):
  
  
  
  
  
• The temperature lapse rate (γ=δT/δz) related to the stability class is specifically designed for the surface layer, so it is used as a guideline (varying between -0.12 and 0.0342 K/m between zero height and surface level height). The potentional temperature lapse rate (δΘ/δz) is ~0.0098 K/m (dry adiabatic lapse rate) lower then temperature lapse rate (γ).
  
• 
  

Using the above mentioned profiler: Each of the layers at a certain time of the day (for land environment)  gets additional parameters to define the potential temperature (Θ) and its behavior inside that layer. The functions which at this moment implemented are: a step-ish function;  it can simulate (a), (b) and (c) in the above picture: the function is defined by the stepsize, the layer depth, the lapse rate or temperature strength a polynomial function; this can simulate (c) and (d) in the above picture: the function is defined by the order (from 0 to n; n can be any positive real number), the layer depth, the lapse rate or temperature strength. an exponential function, this can simulate (e) in the above picture: the function is defined by the size of the tail-off, the layer depth, the lapse rate or temperature strength. other functions or data sets can be added. The profiler incorporates a user defined temperature, pressure and lapse rate table (to provide for a diurnal effect on temperature related aspects). The left picture is generated by the profiler: it can be compared to the above development of the profiles from R. Stull.

The horizontal lines at the top means that above that the free atmosphere should start (not incorporated in the picture). The legend number [4 to 14] means the LST time of day. The temperature at Height=0 and the lapse rate are depending on time [LST].



• CABL: TIBL height-potential temperature (Θ) idealized profile for wind from warmer land to cooler sea (from I.M. Brooks, [1999], fig. 2):
  
  
  

  In this case the depth of the stable TIBL (STIBL) increased with the square root of the distance (Garratt [1987]). In case of convective TIBL (CTIBL) the mixed layer of the CTIBL also grows with the square root of the distance.
  
• MABL: Some different profiles can exist depening on the difference between surface sea temperature (SST) and air temperature. The stable MABL (SMABL) when the SST is lower then the air temperature and a convective MABL (CMANL) in case the air temperature is lower then the SST. A neutral MABL (NMABL) is also possible.
  
  
• Because the temperature profile in the light ray path (RPTP) is important, a composite is needed of the profiles in the path of the light. If the observer the profile is SBL and at horizon is CBL, the profile to be used for calculating the refraction is a composition of these two. Another example is the green line in the above picture
  

Questions

1. How important is the surface layer during night time (so at NBL)? Is this very surface layer a recognizable layer?
   At this moment I assume there is no need for the profiler to recognize a surface layer during NBL (night time).
   
2. If the daytime surface layer ends; when does it stop around sunset time; before the NBL starts to growth or is there some overlap?
   At this moment I assume there is an ovelap between NBL and daytime surface layer.
   
3. I think that the aerodynamic roughness length (ranging between 0.0001 m (calm sea), 0.001 m (off sea wind in coastal areas) to 0.1 m (farmland/neolithic environment; Stull R., [1988], page 380) has no relation to the viscous layer (a few mm/cm), correct? Aerodynamic roughness length is important for the wind profile, or does it influence the visible height temperature profile?
   At this moment I assume there is no significant relation between aerodynamic roughness length and height-temperature profile at the sunset/rise events. The aerodynamic roughness length is normally below the optical roughness of the horizon: tallest object; and thus not really important for the refraction.
4. Are there other functions or other layers important (in the scope of my research)?
   Some temp-height profiles in R. Stuff ([1988], Fig. 5.17, page 170) show that other functions/layers are present. In particular in Fig. 5.17 (g), (h), (j), (k) and perhaps (l) a potential temperature 'bump' is present near the top, this looks to be due to decoupled mixed layers (section 13.5.3, page 574) or stratocumulus-topped mixed layer (page 583). Thus not really important for regimes needed for sunset/rise events.
   I hope I have the normal occurring functions, but not sure! Certainly for the daytime surface layer I have no real concrete references to the functions!
   In Izumi [1971] (referred by Stull, R. [1988], page 439) a height-potential temperature profile is given for CBL regime (Kansas field experiment: 15:00-15:15 CST, July 29th, 1968):
   

   Height [m]	Potential temp Θ [K]
   2	303.88
   4	303.18
   8	302.67
   16	302.29
   22.63	302.13
   32	302

   This behaves like a 8.5^th order polynomial or exponential function (both with a very small H_ΔΘ of ~5 [m], much smaller then the expected related surface layer depth of around 150 [m]).
   I assume for now that the CBL behaves as a 9^th order polynomial function.
   
5. What is the best vertical coordinate for the profiler in this research? Height, pressure, sigma or other?
   At this moment I assume 'height' is ok.
6. The lapse rate defined in the stability class; is this the lapse rate of the temperature (T) or the potential temperature (Θ)?
   At this moment I assume it is for the temperature (T).
   
7. Is the Marine ABL comparable to day time surface layer or night time NBL? And how does it depend on the temperature difference of air and sea?
   From this reference (Medeiros, B., [2005], Fig. 2 and 3) it looks that around Ireland the yearly mean MABL is around 85 hPa (~750 m), a seasonal variation (between min. and max. monthly mean) of around 45 hPa (~400 m) and almost zero hPa diurnal variation.
8. The depth of the CTIBL is depending on potential temperature lapse rate γ=δΘ/δz of the air just above the TIBL (according to Stull, R., [1988], page 600). What value to take here?
   For now I assume it is the lapse rate in the capping layer of TIBL.
9. The layers above the TIBL will be comparable to the LABL layers p-hours ago at the shore (p related to distance from shore (x [m]) and windspeed (M [m/sec]): p = x/M/3600 [h])?
   I assume this is indeed the case.
10. Is a (T)IBL replacing or pushing up existing layers?
    As assumed in question 9 and seen in the diurnal behavoir over uniform land (of surface layer and NBL) it looks that new layers caused by IBL/TIBL effects replace existing layers and don't push them.
11. In case of the (T)IBL's in question 10; I am sure above the (T)IBL will not stay stable there for ever! What is the distance/time they say there?
    I would think they will be there not longer then say 60 minutes; so being in the region of turbulence scale/energy gap (as defined in Fig. 2.2 of Stull ([1988], page 32). Althought in land enviornment, the capping layer stays there for some 12 hours...
    

Number of height-temperature profiles needed

• apparent altitude is positive
  One height-temperature profile of as near as possible to the observer, looks ok
• apparent altitude is negative
  In case the height temperature profile changes over the distance the light is traveling (for refraction calculations), one needs to use the height-temperature profile as seen by the refracted light. For instance in case of an observer standing on the shore and watching a sunset over the sea with the wind in the back; there might be two height-temperature profiles involved:

• One from observer to horizon (dip) and this profile is not the profile at the observer or the horizon (dip), but is a composition of the whole path between these two locations (as the light travels): so using the height-temperature values as seen along the green line in the above picture. This is a profile following the CABL.
  
• Second from horizon (dip) to celestial object. Here one can hopefully take the height-temperature profile at the location of the horizon (dip). This is a profile MABL profile if the distance to the dipped horizon (distance [km] = 3.86*sqrt(h [m])) is greater then 50 km (a height of > 200 m).

Open to feedback

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