Download Structure functions derived from Kolmogorov

Adaptive Optics in the VLT and ELT era
Atmospheric Turbulence
François Wildi
Observatoire de Genève
Credit for most slides : Claire Max (UC Santa Cruz)
Page 1
Atmospheric Turbulence Main Points
• 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 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
Fluctuations in index of refraction are
due to temperature fluctuations
• Refractivity of air
3

  P
7.52
10
6
N  (n  1)  10  77.6  1 
 
2



 T
where P = pressure in millibars, T = temp. in K,  in microns
n = index of refraction. Note VERY weak dependence on 
• Temperature fluctuations  index fluctuations
 N  77.6  (P / T )T
2
(pressure is constant, because velocities are highly sub-sonic. Pressure
differences are rapidly smoothed out by sound wave propagation)
Turbulence arises in several places
stratosphere
tropopause
10-12 km
wind flow around dome
boundary layer
~ 1 km
Heat sources w/in dome
Within dome: “mirror seeing”
• When a mirror is warmer than
dome air, convective
equilibrium is reached.
credit: M. Sarazin
• Remedies: Cool mirror itself,
or blow air over it, improve
mount
credit: M. Sarazin
convective
cells are bad
To control mirror temperature: dome air conditioning (day), blow air on
back (night)
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
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”
Real shear generated turbulence (aka KelvinHelmholtz instability) measured by radar
• Colors show intensity of radar return signal.
• Radio waves are backscattered by the turbulence.
Kolmogorov turbulence, cartoon
solar
Outer scale L0
Inner scale l0
h
Wind shear
convection
h
ground
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
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
Questionable at best
• In practice, Kolmogorov model works surprisingly well!
Concept Question
• What do you think really determines the outer scale in
the boundary layer? At the tropopause?
• Hints:
Outer Scale ~ 15 - 30 m, from Generalized
Seeing Monitor measurements
• F. Martin et al. , Astron. Astrophys. Supp. v.144, p.39, June 2000
• http://www-astro.unice.fr/GSM/Missions.html
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
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 ~ 10-13 - 10-15 m-2/3
10-14
Paranal, Chile, VLT
Turbulence profiles from SCIDAR
Eight minute time period (C. Dainty, Imperial College)
Siding Spring, Australia
Starfire Optical Range,
Albuquerque NM
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!