## Optical measurement of rates of dissipation of temperature variance due to oceanic turbulence

Optics Express, Vol. 15, Issue 12, pp. 7224-7230 (2007)

http://dx.doi.org/10.1364/OE.15.007224

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### Abstract

Inhomogeneities in the refractive index induced by temperature fluctuations in turbulent flows have the effect of scattering light in near-forward angles. We have developed a method that extracts the rate of Temperature Variance Dissipation (TVD) and its spectrum from the properties of light scattering and have built an instrument - Optical Turbulence Sensor (OTS) - that implements the method. OTS uses a linear wavefront sensing Hartmann array and allows for nearly instantaneous measurements of temperature variance in turbulent flows. The instrument has been tested in an situ experiment carried out from a drifting vessel at a site off the coast of Newport, Oregon. Here we compare the temperature variance measured by OTS and its spectra with both theoretical predictions and with spectra obtained from a fast thermistor sensor.

© 2007 Optical Society of America

## 1. Introduction

*m*and 1

*mm*[1]. For some time [2], the optical oceanography community has theoretically postulated the effect of turbulence on light propagation in the ocean. Turbulent fluctuations of passive scalars (such as temperature and salinity) in water cause fluctuations in the density of the fluid, and hence in its refractive index, affecting light propagation. Inhomogeneities in the refractive index can be attributed mainly to the temperature field [3

3. D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Applied Optics **37**, 4669–4677 (1998). [CrossRef]

3. D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Applied Optics **37**, 4669–4677 (1998). [CrossRef]

^{-3}

*rad*), and for propagation distances of ≃ 0.3

*m*. However, experimental oceanic studies of the interaction between light and turbulence have been rare, in large part due to the complexity of appropriate experiments and to the difficulty of making angular measurements of scattered light at small angles in the presence of the unscattered light beam.

^{-7}to 10

^{-3}

*rad*) of the light beam on turbulent flow show that under energetic oceanographic conditions, the total scattering coefficient can be larger than that of particulates and that the turbulent inhomogeneities of fluid flow have the effect of scattering light in near-forward angles, thus providing an opportunity to use optics to quantify turbulence [4

4. D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. **43**, 5662–5676 (2004). [CrossRef] [PubMed]

4. D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. **43**, 5662–5676 (2004). [CrossRef] [PubMed]

6. R. G. Lueck, D. Huang, D. Newman, and J. Box, “Turbulence Measurement with a Moored Instrument,” Journal of Atmospheric and Oceanic Technology **14**, 143–161 (1997). [CrossRef]

^{-2}°

*C*

^{2}/

*s*a few meters below the surface to 10

^{-10}°

*C*

^{2}/

*s*in the deep ocean. TKED can vary from 10

^{-4}

*m*

^{2}/

*s*

^{3}in the fairly energetic upper layer to 10

^{-11}

*m*

^{2}/

*s*

^{3}in mid-water column [3

3. D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Applied Optics **37**, 4669–4677 (1998). [CrossRef]

^{-9}to 10

^{-1}°

*C*

^{2}/

*s*and TKED within 10

^{-10}to 10

^{-4}

*m*

^{2}/

*s*

^{3}range making it most suitable to the upper ocean measurements. Combining the optical measurement with appropriate processing allows nearly instantaneous estimation of spectra of temperature variance and thus of the rates of temperature variance dissipation.

## 2. Principles of the optical determination of the temperature variance spectrum

*L*= 0.3

*m*, can be expressed in terms of the function

*ψ*, which represents the complex amplitude of a 2-D light front propagating in the

*z*direction over the pathlength

*L*[8]:

*A*

_{0}is the initial value of

*ψ*(

*x*,

*y*) at

*z*= 0 (a constant, for a plane wave);

*λ*is the light wavelength, and

*n*(

*x*,

*y*,

*z*) is a small deviation of the local refractive index from its mean value. The exponent in Eq. (1), the function ∫

^{L}

_{0}

*n*(

*x*,

*y*,

*z*′)

*dz*′, depends on the refractive index of the medium where light propagates. This expression (for the given

*x*and

*y*) gives the phase of the light after it has propagated over the distance

*L*: Γ(

*x*,

*y*) = ∫

^{L}

_{0}

*n*(

*x*,

*y*,

*z*)

*dz*. The vector normal to Γ(

*x*,

*y*)

*α*(

*x*,

*y*) (

*α*is sometimes called angle-of-arrival) over the pathlength

*L*can thus be obtained as:

*ρ*is defined as:

*ρ*= (

*x*

^{2}+

*y*

^{2})

^{1/2}, and the domain

*D*is a plane which is perpendicular to the propagation axis

*Oz*. For measurements of light scattering on oceanic turbulence and propagation over pathlengths of

*L*= 0.3

*m*,

*D*is effectively truncated to a fraction of the entire plane because the majority of the turbulence scattered light is within a very narrow cone within 0.1

*°*(1.7·10

^{-3}

*rad*) of the original beam direction [3

**37**, 4669–4677 (1998). [CrossRef]

*E*(

_{T}*k*). This approach is based on the following relation between direct measurements of the correlation function of fluctuations in the angle-of-arrival,

*B*(

*ρ*), and the temperature variance spectra ([9], Eq. (42), p. 189):

*E*(

_{T}*k*) is that the temperature variance for length scales between

*k*and

*k*+

*dk*is proportional to

*E*. In general the spectrum

_{T}dk*E*(

_{T}*k*) depends parametrically on molecular properties of water, i.e. the kinematic viscosity

*ν*and the thermal diffusivity κ, as well as on the turbulent kinetic energy dissipation rate and the rate of temperature variance dissipation. The TVD rate can be directly found from its definition as a weighted integral of the temperature spectrum i.e.:

*E*(

_{T}*k*)

*k*

^{2}is the temperature dissipation spectrum [11

11. D. Bogucki, A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with Pr> 1 advected by turbulent flow,” J. Fluid Mech. **343**, 111–130 (1997). [CrossRef]

## 3. Optical Turbulence Sensor

*nm*laser diode. The beam was expanded, collimated and then propagated through the turbulent flow across the path length

*L*= 0.3

*m*exposed to the oceanic water. After interacting with the turbulent flow, the light beam passes through a linear wavefront sensing Hartmann array, consisting of 110 lenslets for a total length of 5

*cm*, and a line scan CCD. The lenslet spots are imaged onto the CCD linear 8k pixel array. The CCD light intensity data, consisting mostly of 110 bright spots, were then low-passed filtered to remove the effect of high frequency noise - oceanic particles. The ’cleaned up’ linear Hartmann array data was then transformed [4

4. D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. **43**, 5662–5676 (2004). [CrossRef] [PubMed]

*E*(

_{T}*k*). The Hartmann array was capable of measuring scattering angles from 0.3 μ rad to a few

*milirad*and operated at the speed of a 10000 CCD readings per second during a measuring interval.

## 4. Turbulence measurements using microstructure sensing package

*Hz*. The fast response thermistor, FP10, pressure sensor collected data with 256

*Hz*rate.

## 5. Results

12. D. Bogucki and J. Domaradzki, “Numerical study of light scattering by a boundary-layer flow,” Appl. Opt. **44**, 5286–5291 (2005). [CrossRef] [PubMed]

*m*initially till the ship drifted to a water depth of 38.5

*m*. The time series were collected during a period of 1.8 hours. In the following we compare the TVD rates from the OTS and the microstructure package by looking at the time series measurements at three different depths, ≃3

*m*(between 10.2 and 11.00 local time), ≃2

*m*(between 11 and 11.8 local time), and ≃6.5

*m*(between 11.8 and 12.2 local time), to capture variability in the dissipation rates.

*μsec*. The spectral ensemble typically became stable after approximately 50 realizations, corresponding to an averaging time of 5

*msec*(this time includes the overhead time for the camera-acquisition system). For this paper we averaged all OTS spectra down to 2

*sec*to enable comparison of TVD rates with those derived from the fast thermistor data, which yields a data point every 2 seconds. An example of 2-sec OTS temperature variance and dissipation spectra are presented in the Fig 2. Figure 3 shows the time series of

*χ*measured by OTS and compared to that obtained from the thermistor/conductivity measurements. The two measurements are consistent although OTS provides a much narrower

*χ*variance. We hypothesize that the major reason for the variance differences lies in the different physical nature of these two measurement. Traditional microstructure measurements use the ’Frozen Field’ assumption and thus require very accurate mean current speed,

*U*, to convert the time derivatives to spatial ones (i.e.

_{x}*∂*/

*∂x*= 1/

*U*∙

_{x}*∂*/

*∂t*). The OTS measurement does not rely on the existence of the current speed. The OTS measurement yields an instantaneous estimate of angular autocorrelation and resulting temperature spectra

*E*(

_{T}*k*) as long as they remain unchanged during the measurement time - 5 to 7

*μsec*.

*U*varied during the experiment between a few

_{x}*m*/

*s*down to few tens of

*cm*/

*s*and

*U*was highly variable over the integration time. We estimate that this factor alone increases the variance of

_{x}*χ*derived from the time series of thermistor/conductivity by at least a factor of 4, thus contributing to much larger variance of

*χ*in Fig. 3. In addition the OTS variance is more stable because each OTS data point (corresponding to the 2 sec long time series of thermistor/conductivity measurement) consists of 2000 values averaged to one number. The OTS needs more in situ testing in a variety of oceanographic conditions to fully address similarities and differences to existing oceanographic turbulence measurements methods.

## 6. Conclusions

*χ*from the OTS with those from a classical microstructure instrument, we conclude that the OTS provides consistent results. The main difference is the lower variance of the rate of temperature variance dissipation measured by the OTS. The highly variable mean flow affects the microstructure measurements but has no effect on the OTS results.

## Acknowledgments

## References and links

1. | R. V. Ozmidov, “On the turbulent exchange in a stably stratified ocean.” Izvestiya, Atmospheric and Oceanic Physics |

2. | W. H. Wells, “Theory of Small-angle Scattering,” (Advisory Group for Aerospace Research and Development, NATO, 92 Neuilly-Sur-Seine, France, 1973). |

3. | D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, “Comparison of nearforward scattering on turbulence and particles,” Applied Optics |

4. | D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, “Light scattering on oceanic turbulence,” Appl. Opt. |

5. | D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, C. R. Truman, and J. R. V. Zaneveld, “Near-forward light scattering on oceaqnic turbulence and particulates: an experimental comparison,” vol. SPIE, Ocean Optics XIV (1998). |

6. | R. G. Lueck, D. Huang, D. Newman, and J. Box, “Turbulence Measurement with a Moored Instrument,” Journal of Atmospheric and Oceanic Technology |

7. | M. C. Gregg, “Uncertainties and Limitations in Measuring |

8. | V. I. Tatarski, |

9. | V. I. Tatarski, |

10. | A. S. Monin and A. M. Yaglom, |

11. | D. Bogucki, A. Domaradzki, and P. K. Yeung, “Direct numerical simulations of passive scalars with Pr> 1 advected by turbulent flow,” J. Fluid Mech. |

12. | D. Bogucki and J. Domaradzki, “Numerical study of light scattering by a boundary-layer flow,” Appl. Opt. |

**OCIS Codes**

(010.7060) Atmospheric and oceanic optics : Turbulence

(010.7350) Atmospheric and oceanic optics : Wave-front sensing

**ToC Category:**

Atmospheric and ocean optics

**History**

Original Manuscript: April 4, 2007

Revised Manuscript: May 22, 2007

Manuscript Accepted: May 23, 2007

Published: May 29, 2007

**Virtual Issues**

Vol. 2, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

D. J. Bogucki, J. A. Domaradzki, C. Anderson, H. W. Wijesekera, R. V. Zaneveld, and C. Moore, "Optical measurement of rates of dissipation of temperature variance due to oceanic turbulence," Opt. Express **15**, 7224-7230 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-12-7224

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### References

- R. V. Ozmidov, "On the turbulent exchange in a stably stratified ocean," Izv., Acad. Sci. USSR, Atmos. Oceanic Phys. 1, 493-497 (1965).
- W. H. Wells, "Theory of small-angle scattering," (Advisory Group for Aerospace Research and Development, NATO, 92 Neuilly-Sur-Seine, France, 1973).
- D. Bogucki, J. A. Domaradzki, D. Stramski, and J. R. V. Zaneveld, "Comparison of nearforward scattering on turbulence and particles," Appl. Opt. 37, 4669-4677 (1998). [CrossRef]
- D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, and R. C. Truman, "Light scattering on oceanic turbulence," Appl. Opt. 43, 5662-5676 (2004). [CrossRef] [PubMed]
- D. J. Bogucki, J. A. Domaradzki, R. E. Ecke, C. R. Truman, and J. R. V. Zaneveld, "Near-forward light scattering on oceaqnic turbulence and particulates: an experimental comparison," SPIE, Ocean Optics XIV (1998).
- R. G. Lueck, D. Huang, D. Newman, and J. Box, "Turbulence measurement with a moored instrument," J. Atmos. Oceanic Technol. 14, 143-161 (1997). [CrossRef]
- M. C. Gregg, "Uncertainties and limitations in measuring ∑ and χ." J. Atmos. Oceanic Technol. 16, 1484-1490 (1998).
- V. I. Tatarski, Wave Propagation in Turbulent Media (McGraw-Hill, New York, 1961).
- V. I. Tatarski, The effects of the turbulent atmosphere on the wave propagation (Israel program for Scientific Translation, Jerusalem, 1971).
- A. S. Monin and A. M. Yaglom, Statistical Fluid Mechanics: Mechanics of Turbulence (The MIT press, 1981).
- D. Bogucki, A. Domaradzki, and P. K. Yeung, "Direct numerical simulations of passive scalars with Pr> 1 advected by turbulent flow," J. Fluid Mech. 343, 111-130 (1997). [CrossRef]
- D. Bogucki and J. Domaradzki, "Numerical study of light scattering by a boundary-layer flow," Appl. Opt. 44, 5286-5291 (2005). [CrossRef] [PubMed]

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