Turbulence and Seeing AS4100 Astrofisika Pengamatan Prodi 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 Majewski

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 Majewski

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 Majewski 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)

Index of Refraction of the Air (1) The dominant terms in this equation translate to n - 1 = 77.6 X p / T To give some sense of the small variability in this index, for 500 nm light, n = at sea level and n = 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 / T 2 ) 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 Majewski

Sources of Thermal Turbulence Atmospheric turbulence is created on a variety of scales from several origins: Majewski 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 (m 2 /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 m 2 /s Thus, for typical wind speeds and length scales of meters to kilometers, Re > 10 6 and the air is moving turbulently One can think of turbulence as being made up of many eddies of different sizes Majewski

Turbulence and the Eddy Cascade The 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, L 0 is called the external scale or outer scale L 0 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 users/sbrinkma/thesis/node5.html Majewski

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, l 0, of a few millimeters Called the internal or inner scale The cascade stops The temperature fluctuations are smoothed out Majewski

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: Majewski

A Model for Turbulence: The Kolmogorov Spectrum (2) Léna et al 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 Majewski

The Structure Function A statistical measure of the fluctuation over a spatial span of r For Kolmogorov spectrum:  Temperature structure constant (coefficient) Index of refraction structure constant Majewski

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, r 0, and is central to a description of the effects of turbulence on images of stellar sources Majewski Integratingalong 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) Profile Bely Integrating the profile through the airmass  Fried parameter Large Fried parameter is better For = 500 nm & typical mountain top observatory Majewski

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): f c = V/l 0 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 Majewski

The Turbulent Layers of the Atmosphere Overall Vertical Structure of the Atmosphere The pressure drops off exponentially: P(z) = P 0 e -z/H ; H  scale height ~8 km Léna et al Majewski

The Turbulent Layers of the Atmosphere Atmospheric 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 Bely Majewski

The Turbulent Layers of the Atmosphere Atmospheric 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 Majewski

The Turbulent Layers of the Atmosphere Atmospheric 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 Majewski

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 Majewski

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 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 Coulman 1985 Bely Majewski

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) Majewski

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 Majewski

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 r 0 or less Since scintillation is ultimately an interference phenomenon, it is highly chromatic Majewski 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 r 0 (by about a factor of 1.6) Majewski Léna et al Bely

Image Blurring Things are more complicated for a large telescope ( D > r 0 ), 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 ~ / r 0 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

Short & Long Exposure Seeing (1) The image presented by the telescope with D > r 0 is that presented by a telescope of diameter r 0. 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 Majewski Léna et al 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