Turbulence and Seeing

AS4100 Astrofisika PengamatanProdi Astronomi 2007/2008

B. Dermawan

Introduction (1)• Ground-based astronomical observations are hobbled (the light

must pass through the Earth's atmosphere)• Since the atmosphere is layered by (or consists of varying

gradients in) temperature and pressure, it has refractive power• Worse than the presence of its net global refractive power is the

fact that atmospheric layering is not smooth

• Wind and convection and other currents create turbulence which mixes layers with differing indices of refraction in non-uniform and constantly changing ways

• The net result has a serious effect (e.g., tilting, bending and corrugating) on transiting, initially plane-parallel wavefronts

• This is the source of the "twinkling" phenomenon

Schroeder

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Introduction (2)

• The observed negative impacts of the turbulent atmosphere on astronomical images are encompassed globally under the expression "seeing"

• Understanding the physics of seeing allows us to: Improve site selection of telescopes for better image

quality/stability Improve the design of observatories to reduce the local

effects of seeing Improve the conditions at existing observatories by reducing

the local effects of seeing Design active/dynamic means for overcoming the image

degradation from atmospheric effects

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• The index of refraction of air as n =1• But in fact the index of refraction of air has a small variability,

depending on its physical state and composition• The variable part of the refraction index is given by Cauchy's

formula (extended by Lorenz to account for humidity):

Physics of Turbulence and Seeing

Index of Refraction of the Air

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where is the wavelength of light, p is the atmospheric pressure (mbars), T is the absolute temperature (K), and is the water vapor pressure (mbars)

Tp

Tn

48101052.71106.77

1 236

Index of Refraction of the Air (1)

• The dominant terms in this equation translate to

n - 1 = 77.6 X 10-6 p / T • To give some sense of the small variability in this index, for

500 nm light, n = 1.0003 at sea level and n = 1.0001 at 10 km altitude

• In terms of seeing, what we care about are changes in this quantity affecting the transiting wavefront, or, more insidiously, differentially affecting the wavefront on small scales (turbulence)

Majewski

Index of Refraction of the Air (2)• Fluctuations in water vapor have no significant effect on the refractive index at

optical wavelengths, except in extreme situations, such as in fog or just above sea surface

Water vapor can affect the radiative transfer properties of the air, and therefore alter the convective properties of air columns.

The effect of water vapor (latter term in equation above) is generally small for modern astronomical observatories, which are typically built in very dry sites, and which already typically have other weather-related observing problems when the humidity is high (e.g., condensation on mirror surfaces, etc.)

• If one takes derivatives of the previous equation with respect to temperature and pressure variations in temperature are far more important than variations in pressure, and, assuming adiabatic conditions and a perfect gas

dn / dT (p / T2) • Thus, the primary source of variations in the index of refraction are

attributable to thermal variations• In terms of seeing, what we care about is small scale variations in T, or

thermal turbulence

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Sources of Thermal TurbulenceAtmospheric turbulence is created on a variety of scales from several origins:

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• Convection: Air heated by conduction with the warm surface of the Earth becomes buoyant and rises into cooler air, while the cooler air descends

• Wind shear: High winds, particularly the very fast ones associated with the jet stream, generate wind shear and eddies at various scales, and create a turbulent interface between other layers that are in laminar (i.e., non-turbulent) flow

• Disturbances: Large landform variations can create turbulence, particularly on the lee side of mountains where the air flow becomes very non-laminar

Quirrenbach

Turbulence and the Eddy Cascade

• The properties of fluid flows are characterized by the Reynold's number, a dimensionless quantity relating inertial to viscous forces

Re = V L / where V is the fluid velocity, L is a characteristic length scale,

and is the kinematic viscosity (m2/s) of the fluid• Determines whether the flow will be dominated by viscosity

and be laminar (Re low) or be dominated by inertial forces and be turbulent (Re high)

• For air, = 1.5 X 10-5 m2/s• Thus, for typical wind speeds and length scales of meters to

kilometers, Re > 106 and the air is moving turbulently• One can think of turbulence as being made up of many

eddies of different sizes

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Turbulence and the Eddy CascadeThe nature of thermal turbulence is created by a process of Eddy transfer: • Kinetic energy is deposited into turbulence starting with the large scale air

flow processes of convection or wind shear The characteristic scale over which the energy is deposited, L0 is called

the external scale or outer scale L0 is generally larger than the aperture of a telescope, but there is

considerable debate over its typical value Somewhere between 1 to 100 meters• The large turbulent eddies created by the above processes create wind

shears on a smaller scale

http://www.lsw.uni-heidelberg.de/users/sbrinkma/thesis/node5.html

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Turbulence and the Eddy Cascade

• These still smaller eddies, in turn, spawn still smaller Eddies, and so on, in a cascade to smaller and smaller scales

This intermediate range of cascading turbulent scales is called the inertial range, and is where:

Inertial forces dominate and energy is neither created or destroyed but simply transferred from larger to smaller scales

All of the thermal fluctuations relevant to seeing occur • The cascade continues until the shears are so large relative to the eddy

scale (Re ~ 1) that the small viscosity of air takes over and the kinetic energy is "destroyed" (converted into heat)

Happens on scale, l0 , of a few millimeters Called the internal or inner scale The cascade stops The temperature fluctuations are smoothed out

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A Model for Turbulence:The Kolmogorov Spectrum (1)

• It is perhaps somewhat surprising that for the inertial range there is a universal description for the turbulence spectrum (the strength of the turbulences as a function of eddy size, usually expressed in terms of wave number )

• In 1941, Kolmogorov found that in the above process of an eddy cascade, the energy spectrum had a characteristic shape:

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A Model for Turbulence:The Kolmogorov Spectrum (2)

Léna et al. 1996

• The inertial range follows a spectrum of -5/3 (the Kolmogorov spectrum or Kolmogorov Law)

• When turbulence occurs in an atmospheric layer with a temperature gradient (differing from the adiabatic one) it mixes air of different temperatures at the same altitude and produces temperature fluctuations

Hence, the above spectrum also describes the expected variation of temp. in turbulent air

• Aside: The three-dimensional spectrum follows -11/3

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The Structure Function

A statistical measure of the fluctuation over a spatial span of r

2)()()( xTrxTrDT

For Kolmogorov spectrum: 322)( rCrD TT

2TC Temperature structure constant (coefficient)

Index of refraction structure constant

2

223622 1052.71106.77

T

pCC Tn

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Connection to Fried Parameter

• There is a characteristic transverse linear scale over which we can consider the atmospheric variations to flatten out, and in which plane parallel wavefronts are transmitted

• This scale is known as the Fried Parameter, r0, and is central to a description of the effects of turbulence on images of stellar sources

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53

2120 )()(cos67.1

dzzCr n

2nCIntegrating along a line of sight though the atmosphere

z is the altitude, and is the zenith distance

The Fried parameter is inversely proportional to the size of the PSF transmitted by the atmosphere

The FWHM of the observed PSF (in arcsecs)

Profile2nC

Bely

• Integrating the profile through the airmass Fried parameter• Large Fried parameter is better

53

21515 )()(cos1059.2

dzzCn

53

2 )(94.0

dzzCn For = 500 nm & typical mountain top observatory

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Time Dependence of The Turbulence• A simple model (the Taylor Hypothesis) is to consider the turbulence along a

line of sight as "frozen" with given spatial power spectrum and configuration, and assuming that a uniform wind translates the column of air laterally with a velocity V

• The physical basis: the timescales involved in the development of turbulence are much longer than the time for a turbulent element displaced by wind to cross the telescope aperture

• The temporal cut-off frequency (the quickest timescale for observed changes in image deformation):

fc = V/l0

• The turbulence at any particular site generally consists of a superposition of this "frozen" turbulence and a local turbulence which has a significant vertical component

Léna et al. 1996

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The Turbulent Layers of the Atmosphere Overall Vertical Structure of the Atmosphere

The pressure drops off exponentially: P(z) = P0 e-z/H; H scale height ~8 km

Léna et al. 1996

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The Turbulent Layers of the AtmosphereAtmospheric Layers in the Context of Seeing (1)

Basic atmospheric layers of relevance • The surface or ground layer (within a few to several tens of meters) Pays to have telescope, enclosure, primary mirror above this layer

• The planetary boundary layer (extending to of order 1 km) Still significant frictional effects from the Earth's surface, but also significant

vertical motion due to diurnal heating/cooling cycles of the ground and air in contact with it

Léna et al. 1996

Bely

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The Turbulent Layers of the AtmosphereAtmospheric Layers in the Context of Seeing (2)

• The atmospheric boundary layer (separated from the planetary boundary layer by the thermal inversion layer)

Abrupt changes in topography can create gravity waves, that can create turbulent flow

The atmospheric boundary layer above water generally exhibits smaller thermal fluctuations than over land

Bely

Léna et al. 1996

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The Turbulent Layers of the AtmosphereAtmospheric Layers in the Context of Seeing (3)

• The free atmosphere (the bulk of the atmosphere shown in the density/ pressure plot)

Mainly driven by very large-scale air flows (tropical trade winds, "the westerlies“, the jet stream)

The contribution to seeing from free atmosphere is about 0.4 arcsec on average

Bely

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The Turbulent Layers of the Atmosphere

Diurnal Effects in the Low Altitude Layers (1)

Significant daily changes in the thermal profile of the lower atmospheric layers, which drive the changing convective properties within these layers

Coulman 1985 Coulman 1985

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The Turbulent Layers of the Atmosphere

Diurnal Effects in the Low Altitude Layers (2) profile is generally given by the temperature variance through the convective layers (in principle can be estimated). In fact, the actual profile is more complicated

2nC

The existence of these discrete, high layers suggests that seeing compensation (e.g., with adaptive optics) might be best accomplished by optically complementing these discrete layers

2nC

Coulman 1985 Bely

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The Turbulent Layers of the Atmosphere

Diurnal Effects in the Low Altitude Layers (3)

Quirrenbach

Primarily free atmosphere effects above the VLT site at Paranal (which is above most of the other layers)

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Observed Seeing Effects

• There are various manifestations of the observed effects of turbulence (seeing)

• Scintillation changes the apparent brightness of a sort, whereas image wander and image blur degrade the long term image of a source

• Which of these effects comes into play depends on the telescope aperture size relative to the Fried parameter

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Scintillation (“twinkling”)

• Variations in the "shape" of the turbulent layer results in moments where it mimics a net concave lens that defocuses the light and other moments where it is like a net convex lens that focuses the light

• Scintillation only is obvious when the aperture/pupil diameter is of order r0 or less

• Since scintillation is ultimately an interference phenomenon, it is highly chromatic

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Léna et al. 1996

Image Wander

• Because the wavefront is locally plane parallel, the image can actually be diffraction-limited for each isoplanatic patch passes through the line of sight

• The center of that diffraction-limited Airy disk will wander around the image plane as different cells imposing different wavefront tilts pass over the aperture

• Need to ensure that the telescope aperture be smaller than r0 (by about a factor of 1.6)

Majewski

Léna et al. 1996

Bely

Image Blurring

• Things are more complicated for a large telescope (D > r0), since many isoplanatic patches will be in the beam of the telescope, and image blurring or image smearing dominates

• Each isoplanatic patch creates its own diffraction-limited Airy disk (FWHM ~ /D). These individual Airy spots are called speckles

• The ensemble of speckles will have an envelope given by FWHM ~ /r0

• A strategy for taking advantage of the speckles to recover the theoretical diffraction limit of a large aperture telescope is called speckle interferometry

Majewski

Léna et al. 1996

Léna et al. 1996

Short & Long Exposure Seeing (1)

• The image presented by the telescope with D > r0 is that presented by a telescope of diameter r0. It is seeing-limited

• Increasing the diameter of the telescope will not increase the resolution (decrease the size of the PSF)!

• The improvement in the cutoff frequency in the long exposure seeing is obvious

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Léna et al. 1996

Short exposure

Long exposure

Short & Long Exposure Seeing (2)

The improvement in the seeing as one moves away from the Earth surface is shown by the long and short exposure MTFs

Majewski Coulman 1985

Long exposure

Short exposure