1. Adaptive Optics in the VLT and ELT era Atmospheric Turbulence
François Wildi
Observatoire de Genève
Credit for most slides : Claire Max (UC Santa Cruz)

2. Atmospheric Turbulence Essentials
We, in astronomy, are essentially interested in the effect of the turbulence on the images that we take from the sky. This effect is to mix air masses of different index of refraction in a random fashion
The dominant locations for index of refraction fluctuations that affect astronomers are the atmospheric boundary layer , the tropopause and for most sites a layer in between (3-8km) where a shearing between layers occurs.
Atmospheric turbulence (mostly) obeys Kolmogorov statistics
Kolmogorov turbulence is derived from dimensional analysis (heat flux in = heat flux in turbulence)
Structure functions derived from Kolmogorov turbulence are  r2/3

3. Fluctuations in index of refraction are due to temperature fluctuations
Refractivity of air
where P = pressure in millibars, T = temp. in K,  in microns
n = index of refraction. Note VERY weak dependence on 
Temperature fluctuations  index fluctuations
(pressure is constant, because velocities are highly sub-sonic. Pressure differences are rapidly smoothed out by sound wave propagation)

4. Turbulence arises in several places
stratosphere
tropopause
10-12 km
wind flow around dome
boundary layer
~ 1 km
Heat sources w/in dome

5. Within dome: “mirror seeing”
When a mirror is warmer than dome air, convective equilibrium is reached.
Remedies: Cool mirror itself, or blow air over it, improve mount
credit: M. Sarazin
credit: M. Sarazin
convective cells are bad
To control mirror temperature: dome air conditioning (day), blow air on back (night)

6. Local “Seeing” - Flow pattern around a telescope dome
Cartoon (M. Sarazin): wind is from left, strongest turbulence on right side of dome
Computational fluid dynamics simulation (D. de Young) reproduces features of cartoon

8. Boundary layers: day and night
Wind speed must be zero at ground, must equal vwind several hundred meters up (in the “free” atmosphere)
Boundary layer is where the adjustment takes place, where the atmosphere feels strong influence of surface
Quite different between day and night
Daytime: boundary layer is thick (up to a km), dominated by convective plumes
Night-time: boundary layer collapses to a few hundred meters, is stably stratified. Perturbed if winds are high.
Night-time: Less total turbulence, but still the single largest contribution to “seeing”

9. Real shear generated turbulence (aka Kelvin-Helmholtz instability) measured by radar
Colors show intensity of radar return signal.
Radio waves are backscattered by the turbulence.

10. Kolmogorov turbulence, cartoonh
convection
solar
Wind shear
Outer scale L0
Inner scale l0
ground

11. Kolmogorov turbulence, in words
Assume energy is added to system at largest scales - “outer scale” L0
Then energy cascades from larger to smaller scales (turbulent eddies “break down” into smaller and smaller structures).
Size scales where this takes place: “Inertial range”.
Finally, eddy size becomes so small that it is subject to dissipation from viscosity. “Inner scale” l0
L0 ranges from 10’s to 100’s of meters; l0 is a few mm

12. Assumptions of Kolmogorov turbulence theory
Medium is incompressible
External energy is input on largest scales (only), dissipated on smallest scales (only)
Smooth cascade
Valid only in inertial range L0
Turbulence is
Homogeneous
Isotropic
In practice, Kolmogorov model works surprisingly well!
Questionable

13. Concept Question
What do you think really determines the outer scale in the boundary layer? At the tropopause?
Hints:

14. Outer Scale ~ 15 - 30 m, from Generalized Seeing Monitor measurements
F. Martin et al. , Astron. Astrophys. Supp. v.144, p.39, June 2000

15. Atmospheric structure functions
A structure function is measure of intensity of fluctuations of a random variable f (t) over a scale t :
Df(t) = < [ f (t + t) - f ( t) ]2 >
With the assumption that temperature fluctuations are carried around passively by the velocity field (for incompressible fluids), T and N have structure functions like
DT ( r ) = < [ T (x ) - v ( T + r ) ]2 > = CT2 r 2/3
DN ( r ) = < [ N (x ) - N ( x + r ) ]2 > = CN2 r 2/3
CN2 is a “constant” that characterizes the strength of the variability of N. It varies with time and location. In particular, for a static location (i.e. a telescope) CN2 will vary with time and altitude

16. Typical values of CN2 Index of refraction structure function
DN ( r ) = < [ N (x ) - N ( x + r ) ]2 > = CN2 r 2/3
Night-time boundary layer: CN2 ~ m-2/3
10-14
Paranal, Chile, VLT

17. Turbulence profiles from SCIDAR
Eight minute time period (C. Dainty, Imperial College)
Starfire Optical Range, Albuquerque NM
Siding Spring, Australia

18. Atmospheric Turbulence: Main Points
The dominant locations for index of refraction fluctuations that affect astronomers are the atmospheric boundary layer and the tropopause
Atmospheric turbulence (mostly) obeys Kolmogorov statistics
Kolmogorov turbulence is derived from dimensional analysis (heat flux in = heat flux in turbulence)
Structure functions derived from Kolmogorov turbulence are  r2/3
All else will follow from these points!

19. Phase structure function, spatial coherence and r0

20. Definitions - Structure Function and Correlation Function
Structure function: Mean square difference
Covariance function: Spatial correlation of a random variable with itself

21. Relation between structure function and covariance function
To derive this relationship, expand the product in the definition of D ( r ) and assume homogeneity to take the averages

22. Definitions - Spatial Coherence Function
Spatial coherence function of field is defined as
Covariance for complex fn’s
C (r) measures how “related” the field  is at one position x to its values at neighboring positions x + r .
Do not confuse  the complex field with its phase f

23. Now evaluate spatial coherence function C (r)
For a Gaussian random variable  with zero mean, So
So finding spatial coherence function C (r) amounts to evaluating the structure function for phase D ( r ) !

24. Next solve for D ( r ) in terms of the turbulence strength CN2
We want to evaluate
Remember that

25. Solve for D ( r ) in terms of the turbulence strength CN2, continued
But for a wave propagating
vertically (in z direction) from height h to height h + h. This means that the phase is the product of the wave vector k (k=2p/l [radian/m]) x the Optical path
Here n(x, z) is the index of refraction.
Hence

26. Solve for D ( r ) in terms of the turbulence strength CN2, continued
Change variables:
Then
Algebra…

27. Solve for D ( r ) in terms of the turbulence strength CN2, continued
Now we can evaluate D ( r )
Algebra…

28. Solve for D ( r ) in terms of the turbulence strength CN2, completed
But

29. Finally we can evaluate the spatial coherence function C (r)
For a slant path you can add factor ( sec  )5/3 to account for dependence on zenith angle 
Concept Question: Note the scaling of the coherence function with separation, wavelength, turbulence strength. Think of a physical reason for each.

30. Given the spatial coherence function, calculate effect on telescope resolution
Define optical transfer functions of telescope, atmosphere
Define r0 as the telescope diameter where the two optical transfer functions are equal
Calculate expression for r0

31. Define optical transfer function (OTF)
Imaging in the presence of imperfect optics (or aberrations in atmosphere): in intensity units
Image = Object  Point Spread Function
I = O  PSF   dx O(r - x) PSF ( x )
Take Fourier Transform: F ( I ) = F (O ) F ( PSF )
Optical Transfer Function is Fourier Transform of PSF:
OTF = F ( PSF )
convolved with

32. Examples of PSF’s and their Optical Transfer Functions
Seeing limited PSF
Seeing limited OTF
Intensity 
-1
l / D
l / r0
r0 / l
D / l
Diffraction limited PSF
Diffraction limited OTF
Intensity
-1 
l / D
l / r0
r0 / l
D / l

33. Now describe optical transfer function of the telescope in the presence of turbulence
OTF for the whole imaging system (telescope plus atmosphere)
S ( f ) = B ( f )  T ( f )
Here B ( f ) is the optical transfer fn. of the atmosphere and T ( f) is the optical transfer fn. of the telescope (units of f are cycles per meter).
f is often normalized to cycles per diffraction-limit angle (l / D).
Measure the resolving power of the imaging system by
R =  df S ( f ) =  df B ( f )  T ( f )

34. Derivation of r0  df B ( f ) =  df T ( f )  ( p / 4 ) ( r0 / l )2
R of a perfect telescope with a purely circular aperture of (small) diameter d is
R =  df T ( f ) = ( p / 4 ) ( d / l )2
(uses solution for diffraction from a circular aperture)
Define a circular aperture r0 such that the R of the telescope (without any turbulence) is equal to the R of the atmosphere alone:
 df B ( f ) =  df T ( f )  ( p / 4 ) ( r0 / l )2

35. Derivation of r0 , continued
Now we have to evaluate the contribution of the atmosphere’s OTF:  df B ( f )
B ( f ) = C ( l f ) (to go from cycles per meter to cycles per wavelength)
Algebra…

36. Derivation of r0 , continued
Now we need to do the integral in order to solve for r0 :
( p / 4 ) ( r0 / l )2 =  df B ( f ) =  df exp (- K f 5/3)
Now solve for K:
K = 3.44 (r0 / l )-5/3
B ( f ) = exp ( l f / r0 )5/3 = exp (  / r0 )5/3
Algebra…
(6p / 5) G(6/5) K-6/5
Replace by r

37. Derivation of r0 , concluded

38. Scaling of r0
r0 is size of subaperture, sets scale of all AO correction
r0 gets smaller when turbulence is strong (CN2 large)
r0 gets bigger at longer wavelengths: AO is easier in the IR than with visible light
r0 gets smaller quickly as telescope looks toward the horizon (larger zenith angles  )

39. Typical values of r0
Usually r0 is given at a 0.5 micron wavelength for reference purposes. It’s up to you to scale it by -1.2 to evaluate r0 at your favorite wavelength.
At excellent sites such as Paranal, r0 at 0.5 micron is cm. But there is a big range from night to night, and at times also within a night.
r0 changes its value with a typical time constant of minutes

40. Phase PSD, another important parameter
Using the Kolmogorov turbulence hypothesis, the atmospheric phase PSD can be derived and is
This expression can be used to compute the amount of phase error over an uncorrected pupil

41. Tip-tilt is single biggest contributor
Units: Radians of phase / (D / r0)5/6
Focus, astigmatism, coma also big
High-order terms go on and on….
Reference: Noll76