Scint Paper

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Title: Determining the refractive index structure constant using high-resolution

radiosonde data

Authors: M.G. Sterenborg1

J.P.V. Poiares Baptista2

S. Bhler3

Affiliations: 1 European Space Agency, ESTEC, Earth Observation Programme2 European Space Agency, ESTEC, Wave Interaction and Propagation3Institute for Environmental Physics, University of Bremen

Date of submission:

Author address: J.P.V. Poiares Baptista

ESTEC, European Space Agency

Keplerlaan 1 NL 2201 AZ

Noordwijk ZH, Netherlands

Email: [email protected]

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ABSTRACT

Within the framework of the Atmosphere and Climate Explorer (ACE+) radio-

occultation mission work has been carried out to determine the effects of scintillation

on its radio links. To that end a method to derive estimates of the refractive index

structure constant (Cn2) from high-resolution radiosonde data has been developed.

Data from four locations, from high to low latitudes, has been used, covering from

one up to four years of radiosonde measurements. From north to south the locations

are: Lerwick, Camborne, Gibraltar and St. Helena. A rigorous statistical analysis has

been performed, which seems to confirm the usefulness of these data to determine Cn2

with no assumptions regarding the statistics of turbulent layers.

1. INTRODUCTION

This work has been carried out within the context of the preparatory work of the

Atmosphere and Climate Explorer (ACE+), ESA [2004]. ACE+ proposed to use 3

radio links in occultation to determine atmospheric temperature and water vapour.

The frequencies proposed are 10, 17 and 23 GHz. The transmitter and receiver are

located on two different Low Earth Orbit (LEO) satellites.

From the attenuations measured at the receiving satellite, the water vapour and

temperature can be retrieved due to the different absorption at the three frequencies.

Since this technique uses the amplitude (or intensity) of the radio frequency

signal, measured in a finite period of time, scintillation may have an impact in the

accuracy of the estimation of the atmospheric attenuation. The time available for the

attenuation measurement is limited due to the velocity of the two satellites and the

required resolution of the temperature and water vapour profiles.

Scintillation is an incoherent radio propagation effect brought on by the passage

of the radio link through a random medium, such as Earths atmosphere. In essence

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the atmospheric turbulence introduces a stochastic component into the measured

amplitude and phase of the radio signal. An important aspect of this effect is that it

has no bias, meaning that, given an infinite observation time, the error due to

scintillation would be zero after averaging and the measurement would suffer only

from instrumental errors.

The impact of scintillation was evaluated using models based on Woo &

Ishimaru [1974] and Ishimaru [1973],that require, as input parameter the structure

constant of the index of refraction Cn2 .

This paper proposes a method to derive Cn2 profiles from currently available

high-resolution radiosonde data. To the authors knowledge this has never been

attempted, even if proposed or suggested by many in the field mainly because of the

lack of data, Warnock & VanZandt[1985], Vasseur[1999], VanZandt et al[1978].

1.1 Turbulence

Richardson [1922]first proposed a qualitative description of turbulence by imagining

it as a process of decay as it proceeds through an energy cascade, in which eddies

subdivide into ever smaller eddies until they disappear by means of heat dissipation

through molecular viscosity. This cascade begins at the outer scale wavenumber, with

an eddy size equal to the outer scale length L0, and continues on until the eddies are

equal the inner scale length l0.The main energy losses occur in the energy dissipation

region, which is separated from the energy input region by the inertial range. All the

energy is thus transmitted without any significant losses through the inertial range to

the viscous dissipation region. The energy transfer through the spectrum from small to

large wavenumbers, or from large scale eddies to small-scale ones, can be seen as a

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process of eddy division. If the Reynolds number, the dimensionless ratio of the

inertial to the viscous forces, of the initial flow is high, it becomes unstable and the

size of the resulting eddies is of the order of the initial scale of the flow L0. The

Reynolds number characterizing the motion of these eddies is smaller than that of the

initial flow, but still sufficiently high to make these eddies unstable and cause further

division into smaller eddies. During this process energy of a large decaying eddy is

transferred to smaller eddies, i.e., a flow of energy is established from small to large

wavenumbers.

Each division reduces the Reynolds number of the product eddies. This continues

on until the Reynolds number becomes sub-critical. At that point the eddies are stable

and have no tendency to decay any further. It is clear that the larger the Reynolds

number of the initial flow is, the greater the number of successive divisions. Thus the

inner scale length reduces with increasing Reynolds number corresponding to the

outer scale length. A finite inertial range is observed when the viscous range is

separated from the energy range. This occurs when Re >> Recr. In practice an inertial

range is observed for Re > 106 107, Tatarskii [1971] and can be described by a

universal theory based on dimensional analysis as advanced byKolmogorov [1941].

Assuming incompressible flow, Kolmogorov hypothesized that the velocity

fluctuations are both isotropic and homogeneous in the inertial range. For this range,

well removed from both the energy input and dissipation region, only the rate of

transfer of energy, , is of importance. The structure function for the velocity

component which is parallel to the separation vector depends on as well as on the

magnitude of the separation, Wheelon [2001]:

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( ) ( ) ( ) ( ) ,,,2

FtrvtrvDv =+=vvv

(1)

Employing dimensional analysis only one combination of and is found to generate

a squared velocity:

( ) LlCD vv


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Radar can measure the refractive index structure function because Cn2 is proportional

to the radar reflectivity, Ottersten [1969].

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38.0

=nC (4)

where is the radar reflectivity and is the radar wavelength. A review of radar-

based measurements ofCn2 is beyond the scope of this paper. Excellent reviews of the

technique may be found in VanZandt et al [1978], Rao et al [1997] and Rao et al

[2001].

The second advantage of the radar technique is its capability to acquire data in a

systematic and continuous manner. This capability has yielded the insight that the

conditional probability distribution of Cn2 (conditioned to the presence of turbulence)

is log-normal at all heights and times, Wheelon [2001] andNastrom et al[1986].

Unfortunately many authors, when publishing measurements ofCn2 have done so

in the form of averages (seasonal, monthly or diurnal means) with little information

on the probability distribution of the measurements, Ghosh et al [2001], Rao et al

[2001]. Often it is also not clear whether the means obtained refer only to turbulent

events, well above the minimum Cn2 detectable by the radar, or if these means are for

all measurements. The data in this form is difficult to use in any applications that

require statistical information such as those in radio-wave propagation where an

outage probability needs to be estimated.

Under an experimental point of view the radar has the disadvantage that it is an

expensive instrument to acquire and to operate. This is the reason why there are not

many around the globe when compared with for instance radiosondes.

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1.2.2 Thermosondes

Instrumented balloons carrying pairs of sensors at a given distance (e.g. 1 meter apart)

have provided very high quality profile data.

The technique derives the one-dimensional structure function directly from

differential measurement at the two sensors,Bufton et al[1972],Barlettti et al[1976]

and Coulman [1973]

>+=< 2)]()([)( rxTxTrDT (5)

where T is the variable measured (usually temperature), r the distance between

sensors. Local isotropy and homogeneity is assumed to derive Cn2.

This is a very powerful technique with vertical resolutions that, in principle, may be

better than those achievable by radar.

The disadvantage of this technique is that it requires specially purpose built

equipment and is limited to specific measurement campaigns. No systematic data is

available as it is more expensive than radar.

1.2.3 Radiosondes

Standard meteorological radiosondes have been used to derive Cn2 , Warnock &

VanZandt [1985], VanZandt et al [1978] and Vasseur [1999]. Radiosonde launches

are generally carried out at synoptic times (0, 6, 12 and 18 UTC) across the globe. In

more than 700 sites launches are carried out twice a day and in more than 300, four

times a day. These measurements are carried out as part of the global meteorological

network coordinated by the World Meteorological Organization.

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Radiosondes measure all atmospheric variables of interest (pressure, humidity

and temperature as well as wind speed and direction) across the full vertical profile

however only measurements at standard and significant pressure levels are stored and

archived.

These archived measurements have typical resolutions from 100 to 1000 meters,

which are much bigger than the typical outer scales of turbulence. These resolutions

are not sufficient to characterise turbulence, which in general occurs in relatively thin

layers, and as a consequence, assumptions on the occurrence of turbulent layers are

necessary to derive Cn2. Therefore probability distributions for wind shear, buoyancy

and the outer scale of turbulence have to be assumed.

The advantage of these data is that it is easily available and covering a wide

range of climates over long periods of time (some datasets cover more than 20 years).

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2. HIGH RESOLUTION RADIOSONDE DATA

High quality, high-resolution radiosonde data is increasingly becoming available for

scientific applications. Some research organisations have started to store and archive

the full resolution data (instead of only the standard and significant levels) from the

operational radiosonde launches. The acquisition of these data is justified when the

quality and response time of the equipment and sensors is adequate.

The British Atmospheric Data Centre has been archiving the high-resolution data

of the Vaisala RS80L radiosondes performed by the UK Met Office for around twenty

sites. This data is in the Vaisala PC-CORA binary format.

These data yield values for pressure, temperature, humidity, wind speed and

direction. Wind speed and direction are not directly measured by the radiosonde.

These are calculated from the position of the sonde at successive time intervals. The

equipment used to obtain the data is the Vaisala RS80L radiosonde. A short overview

of its technical specifications is shown below

The RS80L data employs Loran-C to determine wind speed and direction. The

estimated accuracies are 1-2 m/s and 5-10 degrees respectively. The resolution is 0.1

m/s and one degree respectively. With its sampling rate of 7 samples per 10 seconds

for each parameter it is ideally suited for this work, as this yields a data point

approximately every 8 meters of altitude, on average, given a typical radiosonde

ascent rate of about 5 meters per second. This resolution would allow in principle

allow the identification of turbulent layers, as thin as 8 meters, with no statistical

assumptions regarding the their occurrence.

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Data from four stations has been used, their locations spread out latitudinally. As

the aim was to characterise the Cn2 for different climate types, stations were chosen at

latitudes ranging from the most northern to the most southern available. The BADC

dataset covers a wider range of years that that used here however a subset was

selected based on the availability of a maximum number of launches per day with no

gaps throughout the years.

3. METHODOLOGY

To derive Cn2 we first identify, within the data for each individual radiosonde launch,

the turbulent layers. This is done through the calculation of the Reynolds and

Richardson (Ri) numbers for each high-resolution data point.

The Potential refractive Index Gradient is derived for all layers but is only used

to derive Cn2 in those identified as turbulent (i.e. where Ri is smaller than the critical

value).

In summary the following step-by-step approach was used for each high-

resolution radiosonde data point:

1. Calculate Reynolds number.

2. Calculate Richardson number.

3. Calculate Potential Refractive Index Gradient.

4. Create data subset based on (Ri < Ricr), this contains only measurements

where the layers are turbulent.

5. Calculate structure constant (Cn2) for all turbulent layers and set its value for

stable layers to 10-21 m-2/3.

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Figure 1 illustrates this methodology.

3.1 Reynolds number

The Reynolds number is used to determine whether a flow is laminar or turbulent.

Considered to be the most important dimensionless number in fluid dynamics it yields

the ratio between inertial and viscous forces. When Re exceeds a critical value a

transition of the flow from laminar to turbulent or chaotic occurs. For the atmosphere

the critical Reynolds number is around 106, 107.

lV=Re (6)

where V is the wind speed, l the characteristic length and = 1.1 10-5 m2/s is the

kinematic viscosity.

For the atmosphere the characteristic length has been taken to be equal to the

resolution of the radiosonde measurement. It has been included more as a measure of

completeness than as a conclusive method to determine whether or not a measurement

was turbulent. In fact, what the results show is that by definition of the Reynolds

number the entire atmosphere can be considered turbulent, which is of course the

basis for Kolmogorov's universal equilibrium theory.

3.2 Richardson number

Which radiosonde measurements are turbulent must first be established before the

structure constant can be calculated, or risk including non-turbulent measurements in

a theory that is specifically suited for turbulence only. To accomplish this the

Richardson number is calculated per measurement. Simply put the Richardson

number is a measure of how turbulent an atmospheric layer is. A stability criterion for

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the spontaneous growth of small-scale waves in a stably stratified atmosphere with

vertical wind shear, it yields the ratio between the work done against gravity by the

vertical motions in the waves to the kinetic energy available in the shear flow.

( )

( ) ( )[ ]22 VUTzzTg

Riv

av

+

+=

(7)

wheregis the gravitational acceleration, Tv the virtual temperature, a is the adiabatic

rate of decrease of temperature = 0.0098 K/m. With z the height and U, V the

components of the wind.

The smaller the value of the Richardson number, the less stable the flow is in

terms of shear instability. The most commonly used value for the start of shear-

induced turbulence is between 0.15 and 0.5, usually set at Ricr = 0.25. However, once

turbulence is established within a shear layer, it should be sustained as long as Ri x|

Turb(z) are represented in Figures 4, 5 and 6 respectively.

From figure 5, the probability of having a turbulent layer between 2 and 8

kilometres is always higher than 50%. This explains why the median values derived

from figure 4 are always smaller than those in figure 5, i.e. the median of all samples

underestimates the most probable case when there is turbulence. Thus the median of

the histogram in Figure 3 is Cn2=10-14, whereas the median for the overall histogram

for the same height is Cn2=10-16.

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The presence of the boundary layer (where the atmosphere interfaces with the

surface of the Earth) can be clearly identified below 2 km where there is an increased

probability of turbulence.

Note that in figure 6 below around 12 km the percentiles are almost symmetrical

around the median. This illustrates again the log-normal behaviour of Cn2 as already

discussed for figure 3.

Figure 7 shows the percentiles as a function of height derived from the cumulative

distribution of the turbulent layer thickness for each height. The thickness of the

turbulent layers was derived first from the Richardson analysis (see 3.2) and then by

creating a histogram from a thickness classification for each height.

The figure shows for all percentiles above 25%, as expected, thicker turbulent

layers in the boundary layer (below 2 km), an almost constant thickness up to the

tropopause and then a decrease above it. In the stratosphere (up to 20km) the figure

shows again a slowly decreasing value.

The values shown here are consistent with those for the outer scale of turbulence

derived for different seasons in Eaton & Nastrom [1998]. The trend however is

different. This may be due to the different techniques used, different climatology and

orography (Eaton & Nastrom measurements were carried out close to a mountain

range, 2700 m) and especially due to the different variables that are being compared

(turbulent layer thickness and outer scale of turbulence).

Figures 8, 9 and 10 show the results for Lerwick, Gibraltar and St. Helena in the form

of percentiles derived from the cumulative distributions of Cn2 for each height. Only

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the 10, 50 and 90 percentiles are shown, the 25 and 75 percentiles were omitted so

that the figures are not overcrowded.

For Lerwick the median value is very low as may be expected from a northern

site. The dashed line in the figure is the median-fit and was constructed between 2 and

8 km: Cn2 = 8.910-16 exp(-h/1054). With h the height in meters.

Gibraltar shows levels ofCn2 that are much higher than expected for a site at this

latitude. This may be due to orographic effects and the proximity of the radiosonde

launches to the Rock. The median was fitted between 3 and 10 km giving Cn2 =

2.3710-13 exp(-h/991).

St Helena shows values ofCn2 that higher than either Camborne or Lerwick as

may be expected for a site closer to the Equator. The median was fitted between 2 and

12 km: Cn2 = 6.8 10-15 e(-h/1284). The data shown in this figure is noisier than that

for all other sites, this is due to the smaller number of samples available (only one

launch per day). For the lowest altitudes a discontinuity in the data can be seen, this is

due to the altitude of the site from where the launches are performed (400 m). Some

influence of the local orography can also be observed in the median and 90%

percentile. The island has a peak at 832 meters.

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5. CONCLUSIONS

A method to determine the refractive index structure constant, Cn2, from high-

resolution radiosonde data has been developed. A full validation of this method was

not possible to carry out due to the lack of other datasets, e.g. radar measurements.

However, the results obtained present the values and behaviour similar and within the

range of those observed by other authors. The statistical behaviour of Cn2 also shows

the expected log-normality further confirming the general correctness of the approach.

The distributions of turbulent layer thickness are as well within the range of those

observed for the outer scale turbulence providing further reassurance on the taken

approach. Statistical results were obtained for 4 sites at different latitudes as well as

an exponential fit to the median for applications where simplified models for Cn2

suffice. These statistical results show the expected physical impact of the boundary

layer, orographic features and local climate.

The authors expect that further work will lead to the full validation of the method

and that high-resolution radiosonde data may become of widespread use, due to its

availability, to determine turbulence and its parameters.

Acknowledgements

The authors would like to thank the British Atmospheric Data Centre and the UK

MetOffice for providing access to its excellent database of high-resolution radiosonde

data. They would like to thank Danielle Vanhoenacker as well for providing the raw

data as used by Hugues Vasseur in his paper. Special mention has to be made of

Pierluigi Silvestrin for his unwaivering support to this activity and of Gottfried

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Kirchengast and Per Heg for the constructive criticism in the many discussions with

the authors.

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Figure 8: Mean of log Cn2 and 10, 50 and 90 percentiles as a function of height

derived from the cumulative distribution ofCn2 for Lerwick.


derived from the cumulative distribution ofCn2 for Gibraltar.


derived from the cumulative distribution ofCn2 for St. Helena.

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Measuring Range Accuracy Resolution

Pressure (hPa) 1060 to 3 0.5 0.1

Temperature (oC) +60 to -90 0.2 0.1

Humidity (%RH) 0 to 100 2 1

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Latitude

(o)

Longitude

(o)

AMSL

(m)

Launches

UTC

Years of

data

Lerwick 60.13 N 1.18 W 82 00, 06, 12, 18 99-02

Camborne 50.22 N 5.32 W 88 00, 06, 12, 18 94-96, 02

Gibraltar 36.14 N 5.35 W 10 00, 12 00-02St. Helena 15.23 S 5.18 W 400 12 00, 02

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DATACORRECT FOR

UNITS

REYNOLDS

NUMBER

RICHARDSON

NUMBER

POTENTIAL

REFRACTIVE

INDEX

GRADIENT

Ri < Ricr

CALCULATE Cn2 SET C

n2 TO 10-21

TURBULENT LAYERS STABLE LAYERS

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