Atmosphere boundary 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 atmospheric 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/Stable/Neutral (NBL/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 from marine/coastal environment (much more shallow and no real diurnal effects: Marine/Coastal Atmospheric Boundary Layer [MABL/CABL]). An interesting introduction is here. The layer evaluation should include these three different environments: land, coastal and marine.
  
• My aim is not about the theory of how (dynamic) atmospheres are formed, but about practical and realistic rules of thumb for height-temperature-pressure-RH (HTPR) profile usable 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 30 minutes) ,
  
• land based observations looking at a vast plane sea or land surface.
• 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 Ray Path Temperature Profile: RPTP Profiler.
  
• a vertical height resolution of (much) smaller then 10 m (certainly near the surface) is needed for determining the refraction accurately.
• for the refraction the temperature profile in the path of light ray is needed (RPTP), and not the vertical HTPR profile at one location!
  
• hopefully (but not necessarily) it can also provide good HTPR profiles for moon observations (includes profiles during the whole day) and 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 diurnal picture of the LABL at homogeneous land environment is copied  from Stull ([1988], page 11):

• for (homogeneous) land environment (LABL: Land Atmospheric Boundary Layer)
  
• CBL: Convective Boundary Layer (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/SBL: Neutral/Stable Boundary Layer (between 0 and ~2 km)
• capping inversion (10% of ABL height)
• residual layer
• NBL/SBL (between 0 and ~500 m)
• viscous layer (in region of cm's)
• for coastal environment (CABL: Coastal Atmospheric Boundary Layer)
  
• TIBL (Temperature Inter Boundary Layer)
  
• wind from land (landwind)
  
• CTIBL: Convective Temperature Inter Boundary Layer (land temp < water temp)
• STIBL: Stable Temperature Inter Boundary Layer (land temp > water temp)
  
• wind from sea  (sea wind)
  
• marine environment (MABL)
• for marine environment (MABL: Marine Atmospheric Boundary Layer)
  
• water temp < air temp 
• SMABL: Stable Marine Atmospheric Boundary Layer (upto  ~500 m)
• viscous layer (in region of mm's)
• water temp ~ air temp 
• NMABL: Neutral Marine Atmospheric Boundary Layer (upto  ~500 m)
• viscous layer (in region of mm's)
• water temp > air temp
  
• CMABL: Convective Marine Atmospheric Boundary Layer (upto ~2000 m)
• viscous layer (in region of mm's)

What is needed for the height-temperature Profiler

• 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:

 

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

• 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 be (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 ground level or the geoid/sea level or is it depending on the surface? If we have a vast flat plane surface (say more then 50 km in size) is that vast flat plane 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).

• for 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 elevation: in this case referenced from the mean sea level (not the WGS84 height).
• surface elevation (the 'height' of the ground/surface). This is surface elevation from the sea.
• observer height (location of the eyes/instrument). This is observer height from ground level.

Height of layer boundaries

• the tropopause depth is an distance from the sea level or ground 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/SBL/CBL). 
• The LABL depth is somewhere between 100-2000 m. Is this relative to the ground height or the mean sea level? I am not sure myself... I think it is the ground height of vast plane surfaces.
  
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 ground height? This layer is just above the viscous layer.
• Lapworth (2003 and 2006) provides a good overview of dynamics of the boundary layer around sunset and sunrise.
  
• the viscous layer depth (also called sometimes micro layer or interfacial layer). This is close to the ground and relative to the ground elevation. 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 homogeneous 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 SBL (in below graph shown as 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 temperature'
• at 'sea 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 ok (table 1, van der Werf [2003]), which include up to mesopause.
• between ABL height and tropopause (FA)
  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 linear 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)	SBL	NBL	CBL
  Coast (CABL) land wind	SBL	NBL	CBL		more out to sea; MABL will take over
  Coast (CABL) sea wind					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):
  
  
  
  
  
• SBL: 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 potential 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 step size, 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 depending 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

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.

By changing the scale, the lower elevations can be seen in more detail:

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
  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 (vast plane surface), but is a composition of the whole path between these two locations (as the light travels): an example (here for CABL) is using the height-temperature values as seen along the green line in the above picture.
  
• Second from horizon (vast plane surface) to celestial object. Here one can hopefully take the height-temperature profile at the location of the horizon (vast plane surface). This is in many cases a MABL profile if the distance to the vast plane surface (distance [km] = 3.86*sqrt(h [m])) is greater then 50 km (a height of > 200 m).

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:

• Segment 1
  At distance 0 and around 15.6km one can use LABL height-temperature profiles (purple points). At point 0 we could measure the temperature gradient and at point 16km one could choice the same or a different LABL height-temperature profile
• Segment 2
  At the coast (15.65km) one can use an CABL (blue square) elevation-temperature profile.
  
• Segment 3
  From the coast (15.7km) to the far away point (48km: a small island) one can use an MABL profiles (orange square). Again the start and end could have different MABL elevation-temperature profiles.
• Segment n
  One could add other segment(s) where needed and necessary to include atmospheric chances: for instance for the last 100m on the island: like a LABL or CABL.
  

An example of the RPTP

• Pladda Lighthouse roof at around 48.5km.
  
  
• The sea horizon near the Pladda Ligthhouse, which would be at a distance of around 47.5km
  

Hilly and sloping terrain

• According to Brown&wood, 2003, page 2804) the effect of  a hilly terrain on a neutral boundary layer is not really distinguishable for a height difference between top and valley of lower than 50m (1 sigma variation of around 10m). In the above example a 1 sigma variation is seen from The Hill to the coast of around 15m.
  
• The effective roughness length can be used to parameterise a hilly terrain (at least upto 300m height difference between top and valley, at which the effective roughness length is around 10m) (Brown&Wood, 2003, page 2806)
  
• The average slope of the terrain from the The Hill to the sea coast s 150m over 15km, a terrain sloop angle of around 0.5deg. The influence of this might be small, but not sure. Drainage winds will happen (even on slopes of 0.5 deg: Stull, 1988, page 534-538).
• It is interesting/lucky that most hilly and sloping terrain simulations are done for neutral and stable situations, precisely the boundary layers relevant for Sunset and Sunrise.
• In the below evaluation the average elevation of the ground has been used (Brown&Wood, 2003, page 2820). This is expected to be better to determine the average height/elevation for the RPTP.

Four cases around boundary layer and height/elevation

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.

During daytime

CBL height-temperature profile	CMABL height-temperature profile

• From The Hill to Pladda Lighthouse roof (in below graph the lighthouse roof is in CBL, but one could argue it is in CABL or CMABL: for this example no third segment was added to the Profiler)
  
  
• From The Hill to the sea horizon (vast plane surface)
  

Around Sunset

• From The Hill to Pladda Lighthouse roof (in below graph the lighthouse roof is in NBL, but one could argue it is in CABL or CMABL: for this example no third segment was added to the Profiler)
  
  
• From The Hill to the sea horizon (vast plane surface)
  

Around Sunrise

• From The Hill to Pladda Lighthouse roof (in below graph the lighthouse roof is in SBL, but one could argue it is in CABL or CMABL: for this example no third segment was added to the Profiler)
  
  
• From The Hill to the sea horizon (vast plane surface)
  

Questions

1. How important is the surface layer during night time (so at SBL/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 SBL/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 overlap 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? How important is the viscous layer for this research?
   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. At this moment the viscous layer is not seen as essential for the scope of this research.
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 CBL or night time NBL/SBL? 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 wind speed (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 behavior over uniform land (of surface layer and CBL/NBL/SBL) 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). Although in land environment, the capping layer stays there for some 12 hours...
12. If one shifts the height-temperature towards another ground elevation: would one need to change the (potential) temperature in the height-temperature profile (as temperature changes with elevation/pressure)?
    At this moment it is expected that the height-temperature profile will have change (potential) temperature depending on the shift.
    
13. Still don't know how the SBL/NBL's height-temperature profile will behave at different elevations? As we have a stable atmosphere would this boundary layer have also been equalised over hills and valleys? As sunrise and sunset moments are close to these NBL events, this is important to know.
    Looking at mist banks at night time, it looks that the mist bank is almost everywhere in the valley at the same height. So at this moment at night time this behavior will be used. Still need to pinpoint the precise behavior at Sun rise/set moments. The article of Lapworth (2003 and 2006) might give an indication on models for the boundary layer during sunset and sunrise times.
    
14. In my earlier determinations of the atmospheric refraction for negative altitudes: the refraction from mirrored observer to space was added to twice the refraction from observer to a lower object/vast plane surface. That can only be done if everywhere there are the same height-temperature profiles, this is not the case in many circumstance (like shown above). Perhaps the refraction influence due to the vast plane surface is the same (seen by observer and mirrored observer), but the observer side's conditions (LABL) are not repeated at the mirrored observer side (more likely to be a MABL).
    This non-symmetry could pose some problems: one needs to calculate the near ground part two times for the differently involved height-temperature profiles.
    
15. When going from one height-temperature profile to another one, there is a transition over a certain distance. How this precisely works needs to be investigated.
    This transitional effect (square root) can be seen here when going from LABL to MABL through CABL.
    
16. The transition between LABL and MABL (aka CABL/TIBL) is depending on the wind direction, as explained here.
    So the Profiler has a wind direction parameter.

Open to feedback

References

Acknowledgments

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