Transcript Atmospheric Turbulence - IAG-Usp

Instrumentation Concepts
Ground-based Optical
Telescopes
Keith Taylor
(IAG/USP)
Aug-Nov, 2008
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Adaptive Optics
Atmospheric Turbulence
(appreciative thanks to USCS/CfAO)
Aug-Nov, 2008
IAG/USP (Keith Taylor)
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!
Kolmogorov Turbulence


Atmospheric refractive index inhomogeneities warp the
wave front incident on the telescope pupil.
These inhomogeneities appear to be associated with
atmospheric turbulence




Thermal gradients
Humidity fluctuations
Wind shear and associated hydrodynamic instabilities
Image quality is directly related to the statistics of the
perturbations of the incoming wave front

The theory of seeing combines



Theory of atmospheric turbulence
Optical physics
Predict the modifications to the image caused by refractive
index gradients.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Modeling the atmosphere

Random fluctuations in the motion of the atmosphere
occur predominantly due to:






Drag encountered by the air flow at the Earth's surface and
resultant wind shear
Differential heating of different portions of the Earth's
surface and the resultant thermal convection
Phase changes involving release of heat
(condensation/crystallization) and subsequent changes in
temperature and velocity fields
Convergence and interaction of air masses with various
atmospheric fronts;
Obstruction of air flows by mountain barriers that generate
wavelike disturbances and vortex motions on their leeside.
The atmosphere 6is difficult to study due to the high Reynolds
number (Re ~ 10 ), a dimensionless quantity, that
characterizes the ratio of inertial to viscous forces.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Cloud
formations
Indicative of
atmospheric
turbulence
profiles
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Atmospheric Turbulence: Outline
 What
determines the index of refraction in
air?
 Origins
of turbulence in Earth’s
atmosphere
 Kolmogorov
turbulence models
The Atmosphere
Kolmogorov model of turbulence


Kinetic energy in large scale turbulence cascades to smaller scales
Inertial interval





Inner scale l0 2mm. Outer scale L010 to 100 m
Turbulence distributed within discrete layers
The strength of these layers is described by a refractive index structure function:
Strength of turbulence can be described by a
single parameter, r0 , Fried’s parameter
Fried parameter is the diameter of a circular
aperture over which the wavefront phase
variance equals 1 rad2
Aug-Nov, 2008
IAG/USP (Keith Taylor)
J. Vernin, Universite de Nice.
Cerro Pachon for Gemini IGPO
Cn2 profiles
Median seeing conditions on Mauna
Kea are taken to be ro ~ 0.23 meters
at 0.55 microns.
The 10% best seeing conditions are
taken to be ro ~ 0.40 meters.
A set of SCIDAR data taken by Francois
Roddier et al. (1990 SPIE Vol 1236 485)
on Mauna Kea.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Wavefront variance
2 = 1.0039
  radians
DT
r0
5/3
2
• This gives the phase variance over a telescope of diameter DT
• A phase variance of less than ~ 0.2 gives diffraction limited performance
• There are 3 regimes
– DT < r0
Diffraction dominates
– DT ~ r0 - 4r0 Wavefront tilt (image motion) dominates
– DT >> r0
Speckle (multiple tilts across the telescope
aperture) dominates
4m telescope:
D/ r0(500nm)=20
D/ r0(2.2 m)=3.5
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Strehl ratio
Corrected
0.20” FWHM
Uncorrected
0.49” FWHM
MARTINI
WHT, K-band



There are two components of the PSF for 2 < 2 radians2
So ‘width’ of the image is not a useful parameter, use height of
PSF:
Peak intensity in a (un)corrected image
R
=
Strehl ratio:
Peak intensity in a diffraction limited image
2
2
 For small  : R ~ exp (- )
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Isoplanatic angle, temporal variation
• Angle over which wavefront distortions are essentially
the same:
3
2
5
8


2



 0   2.91  sec 3   Cn2 (h)h5 / 3 dh 
  


• It is possible to perform a similar turbulence weighted
integral of transverse wind speed in order to derive an
effective wind speed and approximate timescale of
seeing
• τ0 is the characteristic timescale of turbulence
• Note the importance of Cn2(h) in both cases
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Atmospheric Seeing - Summary
Dependence on Wavelength
r0  
6
5
6
0  
5
6
0  
5
=0.55m =1.6m =2.2m
r0
10cm
36
53
0
10ms
36ms
53ms
0
5’’
18’’
27’’
Aug-Nov, 2008
IAG/USP (Keith Taylor)
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.
credit: M. Sarazin
convective
cells are bad
To control mirror temperature: dome air conditioning (day), blow air on
back (night), send electric current through front Al surface-layer to
equalize temperature between front and back of mirror
Dome seeing
(TMT CFD calculations)
Wind direction
Wind velocities
Aug-Nov, 2008
IAG/USP (Keith Taylor)
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
Top View Flow pattern around a telescope dome
Computational fluid dynamics
simulation (D. de Young)
Thermal Emission Analysis: mid-IR
A good way to find heat sources that can cause turbulence
*>15.0¡C
14.0
12.0
10.0
8.0
VLT UT3, Chile

19 Feb. 1999

0h34 Local Time

Wind summit: 4m/s

Air Temp summit:
13.8C

Here ground is warm,
top of dome is cold.
Is this a setup for
convection inside the
dome?
6.0
4.0
2.0
*<1.8¡C
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 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 rising from hot ground. Quite turbulent.
 Night-time: boundary layer collapses to a few hundred meters,
is stably stratified. Perturbed if winds are high.

Boundary layer is much thinner at night
Surface layer: where
viscosity is largest effect
Credits: Stull (1988) and Haggagy (2003)
Convection takes place when
temperature gradient is steep
 Daytime:
ground is warm, air is cooler
 If temperature gradient between ground and ~ 1
km is steeper than the “adiabatic gradient,” a
warm volume of air that is raised upwards will
find itself still with cooler surroundings, so it
will keep rising
 These warm volumes of air carry thermal energy
upwards: convective heat transport
Boundary layer profiles from
acoustic sounders
Turbulence profile
Vertical wind profile
Boundary layer has strongest
effect on astronomical seeing

Daytime: Solar astronomers have to work with thick
and messy turbulent boundary layer

Big Bear Observatory
Night-time: Less total turbulence, but still the single
largest contribution to “seeing”
 Neutral times: near dawn and dusk


Smallest temperature difference between ground and air, so
wind shear causes smaller temperature fluctuations
Big Bear Solar
Observatory
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Wind shear mixes layers with
different temperatures
• Wind shear  Kelvin
Helmholtz instability
QuickTimeª and a
YUV420 codec decompressor
are needed to see this picture.
• If two regions have different
temperatures, temperature
fluctuations T will result
Computer
simulation
Real Kelvin-Helmholtz instability
measured by FM-CW radar
Colors show intensity of radar return signal.
 Radio waves are backscattered by the turbulence.

Temperature profile in atmosphere
Temperature gradient at low altitudes  wind shear
will produce index of refraction fluctuations
Turbulence strength vs. time of day
Day
Night
Day
Night
Night
Theory for different heights:
data Balance heat fluxes in surface layer.
Credit: Wesely and Alcaraz, JGR (1973)
Day
Night
Theory vs.
Leonardo da Vinci’s view of
turbulence
Reynolds number



When the average velocity (va) of
a viscous fluid of characteristic
size (l) is gradually increased, two
distinct states of fluid motion are
observed:
Laminar (regular and smooth in
space and time) at very low va
Unstable and random at va
greater than some critical value.
Aug-Nov, 2008
IAG/USP (Keith Taylor)
Reynolds number
Re = Reynolds number = v L / 
 Here L is a typical length scale, v is a typical velocity,  is
the viscosity
 Re ~ inertial stresses / viscous stresses

0-1
1 - 100
100 - 1,000
1,000 - 10,000
104 - 106
> 106
Creeping flow
Laminar flow, strong Re dependence
Boundary layer
Transition to slightly turbulent
Turbulent, moderate Re dependence
Strong turbulence, small Re dependence
Effects of variations in Re

When Re exceeds critical value a transition of
the flow from laminar to turbulent or chaotic
occurs.
Between these two extremes, the flow passes
through a series of unstable states.
 High Re turbulence is chaotic in both space and time
and exhibits considerable spatial structure

Aug-Nov, 2008
IAG/USP (Keith Taylor)
Velocity fluctuations


The velocity fluctuations occur on a wide range of
space and time scales formed in a turbulent cascade.
Kolmogorov turbulence model states that energy enters
the flow at low frequencies at scale length L0.


The size of large-scale fluctuations, referred to as large
eddies, can be characterized by their outer scale length L0.
(spatial frequency kL0 = 2/L0)
These eddies are not universal with respect to flow geometry;
they vary according to the local conditions.

Aug-Nov, 2008
A mean value L0 = 24m from the data obtained at Cerro Paranal,
Chile (Conan et al. 2000).
IAG/USP (Keith Taylor)
Energy transport
to smaller scales

The energy is transported to smaller and smaller
loss-less eddies until
At Re ~ 1 the KE of the flow is converted into heat
by viscous dissipation
 The small-scale fluctuations with sizes l0 < r < L0 is
the inertial subrange have scale-invariant behavior,
independent of the flow geometry.
 Energy is transported from large eddies to small
eddies without piling up at any scale.

Aug-Nov, 2008
IAG/USP (Keith Taylor)
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

Breakup of Kelvin-Helmholtz vortex
 Computer
simulation
 Start with large coherent
vortex structure, as is
formed in K-H instability
 Watch it develop smaller
and smaller substructure
 Analogous to Kolmogorov
cascade from large eddies to
small ones
 From
small k to large k
How large is the Outer Scale?

Dedicated instrument, the Generalized Seeing Monitor
(GSM), built by Dept. of Astrophysics, Nice Univ.)
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
Lab experiments agree
 Air
jet, 10 cm diameter (Champagne, 1978)
 Assumptions: turbulence is homogeneous,
isotropic, stationary in time
Power (arbitrary units)
Credit: Gary Chanan, UCI
Slope -5/3
l0
L0
 (cm-1)
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!
Typical values of CN

2
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!

Wavefronts
Zernike polynomials are normally used to
describe the actual shape of an incoming
wavefront
 Any wavefront can be described as a
superposition of zernike polynomials

Aug-Nov, 2008
IAG/USP (Keith Taylor)
Atmospheric Wavefront Variance after Removal of Zernike Polynomials
j
n
m
1
2
3
4
5
6
7
8
9
10
0
1
1
2
2
2
3
3
3
3
0
1
1
0
2
2
1
1
3
3
Aug-Nov, 2008
Zernike Polynomial
1
2  cos
2  sin 
3  2   1
2
6 sin 2
2
6 cos 2
2
8  3  2   sin 
2
8  3  2   cos
2
Name
Resid. Var. (rad2)
Constant
Tilt
Tilt
Defocus
Astigmatism
Astigmatism
Coma
Coma
1.030 (D/r0)5/3
8 3 sin 3
0.582 (D/r0)5/3
0.134 (D/r0)5/3
0.111 (D/r0)5/3
0.0880 (D/r0)5/3
0.0648 (D/r0)5/3
0.0587 (D/r0)5/3
0.0525 (D/r0)5/3
0.0463 (D/r0)5/3
0.0401 (D/r0)5/3
8 3 cos3
IAG/USP (Keith Taylor)
Gary Chanan, UCI
Piston
Tip-tilt
Astigmatism
(3rd order)
Defocus
Trefoil
Coma
“Ashtray”
Spherical
Astigmatism
(5th order)
Units: Radians of phase / (D / r0)5/6
Tip-tilt is single biggest contributor
Focus, astigmatism,
coma also big
High-order terms go
on and on….
Reference: Noll
What is Diffraction?
Slides based on those of James E. Harvey, Univ. of Central Florida, Fall ‘05
Details of diffraction from
circular aperture
1) Amplitude
First zero at
r = 1.22  / D
2) Intensity
FWHM
/D
Diffraction pattern from
hexagonal Keck telescope
Stars at Galactic Center
Ghez: Keck laser guide star AO
Considerations in the optical design of
AO systems: pupil relays
Pupil
Pupil
Pupil
Deformable mirror and Shack-Hartmann lenslet array
should be “optically conjugate to the telescope pupil.”
What does this mean?
Let’s define some terms

“Optically conjugate” = “image of....”
optical axis
object space

image space
“Aperture stop” = the aperture that limits the bundle of rays accepted by the
optical system
symbol for aperture stop

“Pupil” = image of aperture stop
So now we can translate:

“The deformable mirror should be optically conjugate to
the telescope pupil”
means

The surface of the deformable mirror is an image of the
telescope pupil
where

The pupil is an image of the aperture stop

In practice, the pupil is usually the primary mirror of
the telescope
Considerations in the optical design
of AO systems: “pupil relays”
Pupil
‘PRIMARY MIRROR
Pupil
Pupil
Typical optical design of AO
system
telescope
primary
mirror
Deformable
mirror
Pair of matched offaxis parabola mirrors
Wavefront
sensor
(plus
optics)
Science camera
Beamsplitter
More about off-axis parabolas

Circular cut-out of a parabola, off optical axis

Frequently used in matched pairs (each cancels out the
off-axis aberrations of the other) to first collimate light
and then refocus it
SORL
High order AO architecture
•
Wavefront controller
– Typically a deformable mirror (DM)
– May not be optically conjugate to an
image of the primary
•
Wavefront sensor (WFS)
– Shack Hartmann (WFS) or Curvature
Sensor (CS)
•
Beamsplitter
– Dichroic, multi-dichroic, intensity, spatial
or combination
•
Controller
– Typically multi-processor or multi-DSP
•
Interfaces
– Can be complex and include removal of
non-common path errors to science
instrumentation (hence an interface to
science data path)
•
Laser beacons
Aug-Nov, 2008
IAG/USP (Keith Taylor)