Simulation study on light propagation in an anisotropic turbulence field of entrainment zone Renmin Yuan,1,3,* Jianning Sun,2 Tao Luo,1 Xuping Wu,1,4 Chen Wang,1 and Yunfei Fu1 1
Key Laboratory of the Atmospheric Composition and Optical Radiation, CAS, School of Earth and Space Sciences, University of Science and Technology of China, Anhui, 230026, China 2 School of Atmospheric Sciences, Nanjing University, Jiangsu, 210093, China 3 Department of Atmospheric Science, University of Wyoming, Laramie, WY, 82070,USA 4 School of Mathematics and Physics, Anhui Jianzhu University, Anhui, 230601, China * [email protected]
Abstract: The convective atmospheric boundary layer was modeled in the water tank. In the entrainment zone (EZ), which is at the top of the convective boundary layer (CBL), the turbulence is anisotropic. An anisotropy coefficient was introduced in the presented anisotropic turbulence model. A laser beam was set to horizontally go through the EZ modeled in the water tank. The image of two-dimensional (2D) light intensity fluctuation was formed on the receiving plate perpendicular to the light path and was recorded by the CCD. The spatial spectra of both horizontal and vertical light intensity fluctuations were analyzed. Results indicate that the light intensity fluctuation in the EZ exhibits strong anisotropic characteristics. Numerical simulation shows there is a linear relationship between the anisotropy coefficients and the ratio of horizontal to vertical fluctuation spectra peak wavelength. By using the measured temperature fluctuations along the light path at different heights, together with the relationship between temperature and refractive index, the one-dimensional (1D) refractive index fluctuation spectra were derived. The anisotropy coefficients were estimated from the 2D light intensity fluctuation spectra modeled by the water tank. Then the turbulence parameters can be obtained using the 1D refractive index fluctuation spectra and the corresponding anisotropy coefficients. These parameters were used in numerical simulation of light propagation. The results of numerical simulations show this approach can reproduce the anisotropic features of light intensity fluctuations in the EZ modeled by the water tank experiment. ©2014 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence.
References and links 1. 2. 3. 4. 5. 6. 7.
A. Consortini, L. Ronchi, and L. Stefanutti, “Investigation of atmospheric turbulence by narrow laser beams,” Appl. Opt. 9(11), 2543–2547 (1970). M. Antonelli, A. Lanotte, and A. Mazzino, “Anisotropies and universality of buoyancy-dominated turbulent fluctuations: A large-eddy simulation study,” J. Atmos. Sci. 64(7), 2642–2656 (2007). A. S. Gurvich, and I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. 108, 4166 (2003). V. Kan, V. F. Sofieva, and F. Dalaudier, “Anisotropy of small-scale stratospheric irregularities retrieved from scintillations of a double star alpha-Cru observed by GOMOS/ENVISAT,” Atmos. Meas. Tech. 5(11), 2713–2722 (2012). V. F. Sofieva, F. Dalaudier, and J. Vernin, “Using stellar scintillation for studies of turbulence in the Earth's atmosphere,” Philos. Trans. R. Soc. A 371, 20120174 (2013). V. F. Sofieva, A. S. Gurvich, F. Dalaudier, and V. Kan, “Reconstruction of internal gravity wave and turbulence parameters in the stratosphere using GOMOS scintillation measurements,” J. Geophys. Res. 112, D12113 (2007). R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and C. Lu, “Simulation study on light propagation in an isotropic turbulence field of the mixed layer,” Opt. Express 22(6), 7194–7209 (2014).
#209326 - $15.00 USD Received 1 Apr 2014; revised 16 May 2014; accepted 17 May 2014; published 27 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013427 | OPTICS EXPRESS 13427
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
R. Yuan, J. Sun, K. Yao, Z. Zeng, and W. Jiang, “A laboratory simulation of the atmospheric boundary layer analyses of temperature structure in the entrainment zone (in Chinese),” Chin. J. Atmos. Sci. 26, 773–780 (2002). A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds number,” Dokl. Akad. Nauk SSSR 30, 299–303 (1941). V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961). J. P. L. C. Salazar and L. R. Collins, “Two-particle dispersion in isotropic turbulent flows,” Annu. Rev. Fluid Mech. 41(1), 405–432 (2009). J. L. Lumley, and A. M. Yaglom, “A century of turbulence,” Flow Turbul. Combust. 66, 241–286 (2001). R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988). A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. 15, 856–869 (1997). R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antennas Propag. 34(2), 258–261 (1986). A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media 11(3), 163–181 (2001). A. S. Gurvich and I. P. Chunchuzov, “Three-dimensional spectrum of temperature fluctuations in stably stratified atmosphere,” Ann. Geophys. 26(7), 2037–2042 (2008). E. S. Wheelon, I. Geometrical Optics (Cambridge University, 2001). E. S. Wheelon, II. Weak Scattering (Cambridge University, 2003). S. Kida and J. C. R. Hunt, “Interaction between different scales of turbulence over short times,” J. Fluid Mech. 201(1), 411–445 (1989). G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity,” J. Fluid Mech. 5(1), 113–133 (1959). R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. A 68(7), 892–899 (1978). L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005). H. M. Dobbins and E. R. Peck, “Change of refractive index of water as a function of temperature,” J. Opt. Soc. Am. A 63(3), 318–320 (1973). R. Yuan, X. Wu, T. Luo, H. Liu, and J. Sun, “A review of water tank modeling of the convective atmospheric boundary layer,” J. Wind Eng. Ind. Aerodyn. 99(10), 1099–1114 (2011). J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27(11), 2111–2126 (1988). J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, and F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21(6), 929–948 (1986). V. A. Kulikov and V. I. Shmalhausen, “Thread-shaped intensity field after light propagation through the convective cell,” Cornell University Library, Atmospheric and oceanic physics, arXiv:1310.5273 (2013).
1. Introduction So far, isotropic turbulence field is often assumed for study of light propagation in turbulent media. To the contrary, a large number of observations in the near-surface layer and at the top of convective boundary layer (CBL) reveal characteristics of anisotropy. In the near-surface layer, after the narrow beam covers some distance horizontally, the dancing of the narrow beam along horizontal and vertical direction appears to have different statistical characteristics [1]. Turbulent fluctuations of both velocity and temperature fields show anisotropy at the top of CBL [2]. The anisotropic situation can be often observed at the tropopause and stratosphere [3–6]. According to the light propagation experiments conducted in water tank, turbulent flows have obvious anisotropic characteristics at the top of the simulated atmospheric boundary layer [7, 8]. Since the theory of locally isotropic turbulence was introduced [9], the theory has been extensively applied to light propagating [10], pollution dispersion [11] and many other fields [12, 13]. However, a large number of observations indicate that turbulence field often exhibits anisotropic features. Therefore, we need to understand the impacts of anisotropy on light propagation in the turbulence field. Some models have been drafted, which all consider turbulence as isotropic in the horizontal plane and anisotropic in the vertical direction. These models can describe the observed anisotropic characteristics to a certain extent [3, 5], which are mainly divided into two categories: one focuses on the correlation moment for turbulent variables [1], another focuses on modified Kolmogorov's spectra [3, 5, 14, 15]. The measured temperature fluctuation was usually expressed as a sum of isotropic and anisotropic
#209326 - $15.00 USD Received 1 Apr 2014; revised 16 May 2014; accepted 17 May 2014; published 27 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013427 | OPTICS EXPRESS 13428
components. When the electromagnetic wave propagates in the turbulence field, if the anisotropic coefficient is large enough (for example, the anisotropy coefficient of 30 [3]), the influences of isotropy and anisotropy can be detected by judging the light intensity fluctuation spectra along the horizontal direction. All these researches mentioned above came from both theoretical studies and experimental observations. Usually the experimental observations are limited to 1D measurements based on Taylor’s hypothesis [13] and can hardly yield the 2D spectral distribution [16]. In this paper, we managed to generate an anisotropic turbulence field by means of simulation in water-tank. To be specific, a temperature gradient was generated along vertical direction so that different characteristics were revealed in horizontal and vertical directions. We measured the temperature in turbulence field, which enable us to obtain features of refractive index distribution in turbulence field. In the meantime, light propagation experiments were conducted to analyze the different features of light intensity fluctuations along horizontal and vertical directions in anisotropic turbulence field. Anisotropy was quantitatively described by using the anisotropy coefficient introduced in turbulence spectra model. The paper is organized as follows. In Section 2, we will introduce the anisotropic turbulence model by spectral method. Section 3 comes with settings of water-tank simulation and relevant experiment measurements. Numerical simulation method is also presented in Section 3. Section 4 gives the results of experiment. Section 5 shows conclusions and discussions. 2. Theory Temperature inversion often occurs at the top of CBL [13, 17–19]. The isotropic turbulence eddies (temperature fluctuation) are generated at the bottom of CBL [7]. Due to the effects of buoyancy, the turbulence eddies can rise through the CBL. When reaching the level above the CBL top (i.e. the entrainment zone, EZ), the turbulent eddies are compressed in the vertical direction and extended in the horizontal direction due to the temperature inversion in the EZ [8]. The turbulence can be considered isotropic along horizontal direction, but not the case in vertical direction [14]. In the vertical direction, turbulent eddies will have stronger fluctuations in shorter distance. Thus, horizontal correlation distance will be larger than the vertical one, which appears anisotropic to a certain degree [16]. Usually power spectrum was adopted to study the relationships between the temperature fluctuations and the turbulence effects [10] for the influence of temperature fluctuations on light propagations. For the three-dimensional (3D) isotropic spectra, spectral density value is the same on a sphere surface. As for the 3D spectra in anisotropic turbulence field, points with the same density can be regarded as being on an ellipsoid surface, whose two horizontal axes are of the same value and the length of vertical axis is not equal to those of horizontal ones, and which is called the spectrum density ellipsoid (SDE). In most cases, the length of vertical axis of the ellipsoid for the spectrum domain is larger than horizontal axes. The coefficient Caniso , defined as an anisotropy coefficient, can be used to denote the ratio of the length in vertical axis to the counterpart in horizontal axis. The turbulence exhibits isotropic features when Caniso = 1. Contrast to earlier researchers, for example Gurvich et al [3], we had different approaches to study anisotropic turbulence field: we do not separate the actual turbulence field into two parts, which are respectively isotropic and anisotropic components, but use a unified expression of spectrum to characterize the temperature fluctuation field (or refractive index fluctuation field). In fact, the two parts of turbulence fields may still has complicated interactions [20]. According to the experimental results, our theoretical method can reproduce the experimental results satisfactorily. The 3D turbulence spectral density of temperature fluctuation can be denoted as Φ T ( q ) , where q is the vector of wavenumber, which represents the point on the surface of the SDE. All points on the surface of the SDE have the same spectral density Φ T ( q ) . Thus, q has a relationship with conventionally-used wavenumber κ (κ x , κ y , κ z ) , shown as:
#209326 - $15.00 USD Received 1 Apr 2014; revised 16 May 2014; accepted 17 May 2014; published 27 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013427 | OPTICS EXPRESS 13429
q = (κ x * Caniso , κ y * Caniso , κ z / Caniso ) (1) We replace conventionally-used κ with q to express 2D spectral density of temperature
fluctuation Φ T (κ y , κ z ) to makes it easier to analyze 2D light intensity fluctuation, because the integration with wavenumber (κ y , κ z ) has nothing to do with anisotropy coefficient. Based on the consideration above, the 3D expression of anisotropic spectral density of temperature fluctuation in water can be written as the form of isotropic spectral density [21]: Φ T ( q) = K (α )CT2 q −α − 2φT ( q)
(2)
Γ(α + 1) π sin[(α − 1) ] and α is named as spectral power-law. Similar 4π 2 2 2 to the isotropic turbulent fields, CT is the temperature structure constant. In order to characterize the temperature fluctuations in dissipative range, we still adopt the method of isotropic turbulence and introduce the factor φT ( q) in Eq. (2). Besides, in this paper, the attenuation characteristics of anisotropic turbulence in the molecular dissipating range are considered the same as those of isotropic turbulence, which satisfies,
where K (α ) equals to
φT ( q) = 1
q
Key Laboratory of the Atmospheric Composition and Optical Radiation, CAS, School of Earth and Space Sciences, University of Science and Technology of China, Anhui, 230026, China 2 School of Atmospheric Sciences, Nanjing University, Jiangsu, 210093, China 3 Department of Atmospheric Science, University of Wyoming, Laramie, WY, 82070,USA 4 School of Mathematics and Physics, Anhui Jianzhu University, Anhui, 230601, China * [email protected]
Abstract: The convective atmospheric boundary layer was modeled in the water tank. In the entrainment zone (EZ), which is at the top of the convective boundary layer (CBL), the turbulence is anisotropic. An anisotropy coefficient was introduced in the presented anisotropic turbulence model. A laser beam was set to horizontally go through the EZ modeled in the water tank. The image of two-dimensional (2D) light intensity fluctuation was formed on the receiving plate perpendicular to the light path and was recorded by the CCD. The spatial spectra of both horizontal and vertical light intensity fluctuations were analyzed. Results indicate that the light intensity fluctuation in the EZ exhibits strong anisotropic characteristics. Numerical simulation shows there is a linear relationship between the anisotropy coefficients and the ratio of horizontal to vertical fluctuation spectra peak wavelength. By using the measured temperature fluctuations along the light path at different heights, together with the relationship between temperature and refractive index, the one-dimensional (1D) refractive index fluctuation spectra were derived. The anisotropy coefficients were estimated from the 2D light intensity fluctuation spectra modeled by the water tank. Then the turbulence parameters can be obtained using the 1D refractive index fluctuation spectra and the corresponding anisotropy coefficients. These parameters were used in numerical simulation of light propagation. The results of numerical simulations show this approach can reproduce the anisotropic features of light intensity fluctuations in the EZ modeled by the water tank experiment. ©2014 Optical Society of America OCIS codes: (010.1300) Atmospheric propagation; (010.1330) Atmospheric turbulence.
References and links 1. 2. 3. 4. 5. 6. 7.
A. Consortini, L. Ronchi, and L. Stefanutti, “Investigation of atmospheric turbulence by narrow laser beams,” Appl. Opt. 9(11), 2543–2547 (1970). M. Antonelli, A. Lanotte, and A. Mazzino, “Anisotropies and universality of buoyancy-dominated turbulent fluctuations: A large-eddy simulation study,” J. Atmos. Sci. 64(7), 2642–2656 (2007). A. S. Gurvich, and I. P. Chunchuzov, “Parameters of the fine density structure in the stratosphere obtained from spacecraft observations of stellar scintillations,” J. Geophys. Res. 108, 4166 (2003). V. Kan, V. F. Sofieva, and F. Dalaudier, “Anisotropy of small-scale stratospheric irregularities retrieved from scintillations of a double star alpha-Cru observed by GOMOS/ENVISAT,” Atmos. Meas. Tech. 5(11), 2713–2722 (2012). V. F. Sofieva, F. Dalaudier, and J. Vernin, “Using stellar scintillation for studies of turbulence in the Earth's atmosphere,” Philos. Trans. R. Soc. A 371, 20120174 (2013). V. F. Sofieva, A. S. Gurvich, F. Dalaudier, and V. Kan, “Reconstruction of internal gravity wave and turbulence parameters in the stratosphere using GOMOS scintillation measurements,” J. Geophys. Res. 112, D12113 (2007). R. Yuan, J. Sun, T. Luo, X. Wu, C. Wang, and C. Lu, “Simulation study on light propagation in an isotropic turbulence field of the mixed layer,” Opt. Express 22(6), 7194–7209 (2014).
#209326 - $15.00 USD Received 1 Apr 2014; revised 16 May 2014; accepted 17 May 2014; published 27 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013427 | OPTICS EXPRESS 13427
8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.
R. Yuan, J. Sun, K. Yao, Z. Zeng, and W. Jiang, “A laboratory simulation of the atmospheric boundary layer analyses of temperature structure in the entrainment zone (in Chinese),” Chin. J. Atmos. Sci. 26, 773–780 (2002). A. N. Kolmogorov, “The local structure of turbulence in incompressible viscous fluid for very large Reynolds number,” Dokl. Akad. Nauk SSSR 30, 299–303 (1941). V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, 1961). J. P. L. C. Salazar and L. R. Collins, “Two-particle dispersion in isotropic turbulent flows,” Annu. Rev. Fluid Mech. 41(1), 405–432 (2009). J. L. Lumley, and A. M. Yaglom, “A century of turbulence,” Flow Turbul. Combust. 66, 241–286 (2001). R. B. Stull, An Introduction to Boundary Layer Meteorology (Kluwer Academic, 1988). A. S. Gurvich, “A heuristic model of three-dimensional spectra of temperature inhomogeneities in the stably stratified atmosphere,” Ann. Geophys. 15, 856–869 (1997). R. M. Manning, “An anisotropic turbulence model for wave propagation near the surface of the Earth,” IEEE Trans. Antennas Propag. 34(2), 258–261 (1986). A. S. Gurvich and V. L. Brekhovskikh, “Study of the turbulence and inner waves in the stratosphere based on the observations of stellar scintillations from space: a model of scintillation spectra,” Waves Random Media 11(3), 163–181 (2001). A. S. Gurvich and I. P. Chunchuzov, “Three-dimensional spectrum of temperature fluctuations in stably stratified atmosphere,” Ann. Geophys. 26(7), 2037–2042 (2008). E. S. Wheelon, I. Geometrical Optics (Cambridge University, 2001). E. S. Wheelon, II. Weak Scattering (Cambridge University, 2003). S. Kida and J. C. R. Hunt, “Interaction between different scales of turbulence over short times,” J. Fluid Mech. 201(1), 411–445 (1989). G. K. Batchelor, “Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity,” J. Fluid Mech. 5(1), 113–133 (1959). R. J. Hill and S. F. Clifford, “Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,” J. Opt. Soc. Am. A 68(7), 892–899 (1978). L. C. Andrews and R. L. Phillips, Laser Beam Propagation Through Random Media (SPIE, 2005). H. M. Dobbins and E. R. Peck, “Change of refractive index of water as a function of temperature,” J. Opt. Soc. Am. A 63(3), 318–320 (1973). R. Yuan, X. Wu, T. Luo, H. Liu, and J. Sun, “A review of water tank modeling of the convective atmospheric boundary layer,” J. Wind Eng. Ind. Aerodyn. 99(10), 1099–1114 (2011). J. M. Martin and S. M. Flatté, “Intensity images and statistics from numerical simulation of wave propagation in 3-D random media,” Appl. Opt. 27(11), 2111–2126 (1988). J. L. Codona, D. B. Creamer, S. M. Flatte, R. G. Frehlich, and F. S. Henyey, “Solution for the fourth moment of waves propagating in random media,” Radio Sci. 21(6), 929–948 (1986). V. A. Kulikov and V. I. Shmalhausen, “Thread-shaped intensity field after light propagation through the convective cell,” Cornell University Library, Atmospheric and oceanic physics, arXiv:1310.5273 (2013).
1. Introduction So far, isotropic turbulence field is often assumed for study of light propagation in turbulent media. To the contrary, a large number of observations in the near-surface layer and at the top of convective boundary layer (CBL) reveal characteristics of anisotropy. In the near-surface layer, after the narrow beam covers some distance horizontally, the dancing of the narrow beam along horizontal and vertical direction appears to have different statistical characteristics [1]. Turbulent fluctuations of both velocity and temperature fields show anisotropy at the top of CBL [2]. The anisotropic situation can be often observed at the tropopause and stratosphere [3–6]. According to the light propagation experiments conducted in water tank, turbulent flows have obvious anisotropic characteristics at the top of the simulated atmospheric boundary layer [7, 8]. Since the theory of locally isotropic turbulence was introduced [9], the theory has been extensively applied to light propagating [10], pollution dispersion [11] and many other fields [12, 13]. However, a large number of observations indicate that turbulence field often exhibits anisotropic features. Therefore, we need to understand the impacts of anisotropy on light propagation in the turbulence field. Some models have been drafted, which all consider turbulence as isotropic in the horizontal plane and anisotropic in the vertical direction. These models can describe the observed anisotropic characteristics to a certain extent [3, 5], which are mainly divided into two categories: one focuses on the correlation moment for turbulent variables [1], another focuses on modified Kolmogorov's spectra [3, 5, 14, 15]. The measured temperature fluctuation was usually expressed as a sum of isotropic and anisotropic
#209326 - $15.00 USD Received 1 Apr 2014; revised 16 May 2014; accepted 17 May 2014; published 27 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013427 | OPTICS EXPRESS 13428
components. When the electromagnetic wave propagates in the turbulence field, if the anisotropic coefficient is large enough (for example, the anisotropy coefficient of 30 [3]), the influences of isotropy and anisotropy can be detected by judging the light intensity fluctuation spectra along the horizontal direction. All these researches mentioned above came from both theoretical studies and experimental observations. Usually the experimental observations are limited to 1D measurements based on Taylor’s hypothesis [13] and can hardly yield the 2D spectral distribution [16]. In this paper, we managed to generate an anisotropic turbulence field by means of simulation in water-tank. To be specific, a temperature gradient was generated along vertical direction so that different characteristics were revealed in horizontal and vertical directions. We measured the temperature in turbulence field, which enable us to obtain features of refractive index distribution in turbulence field. In the meantime, light propagation experiments were conducted to analyze the different features of light intensity fluctuations along horizontal and vertical directions in anisotropic turbulence field. Anisotropy was quantitatively described by using the anisotropy coefficient introduced in turbulence spectra model. The paper is organized as follows. In Section 2, we will introduce the anisotropic turbulence model by spectral method. Section 3 comes with settings of water-tank simulation and relevant experiment measurements. Numerical simulation method is also presented in Section 3. Section 4 gives the results of experiment. Section 5 shows conclusions and discussions. 2. Theory Temperature inversion often occurs at the top of CBL [13, 17–19]. The isotropic turbulence eddies (temperature fluctuation) are generated at the bottom of CBL [7]. Due to the effects of buoyancy, the turbulence eddies can rise through the CBL. When reaching the level above the CBL top (i.e. the entrainment zone, EZ), the turbulent eddies are compressed in the vertical direction and extended in the horizontal direction due to the temperature inversion in the EZ [8]. The turbulence can be considered isotropic along horizontal direction, but not the case in vertical direction [14]. In the vertical direction, turbulent eddies will have stronger fluctuations in shorter distance. Thus, horizontal correlation distance will be larger than the vertical one, which appears anisotropic to a certain degree [16]. Usually power spectrum was adopted to study the relationships between the temperature fluctuations and the turbulence effects [10] for the influence of temperature fluctuations on light propagations. For the three-dimensional (3D) isotropic spectra, spectral density value is the same on a sphere surface. As for the 3D spectra in anisotropic turbulence field, points with the same density can be regarded as being on an ellipsoid surface, whose two horizontal axes are of the same value and the length of vertical axis is not equal to those of horizontal ones, and which is called the spectrum density ellipsoid (SDE). In most cases, the length of vertical axis of the ellipsoid for the spectrum domain is larger than horizontal axes. The coefficient Caniso , defined as an anisotropy coefficient, can be used to denote the ratio of the length in vertical axis to the counterpart in horizontal axis. The turbulence exhibits isotropic features when Caniso = 1. Contrast to earlier researchers, for example Gurvich et al [3], we had different approaches to study anisotropic turbulence field: we do not separate the actual turbulence field into two parts, which are respectively isotropic and anisotropic components, but use a unified expression of spectrum to characterize the temperature fluctuation field (or refractive index fluctuation field). In fact, the two parts of turbulence fields may still has complicated interactions [20]. According to the experimental results, our theoretical method can reproduce the experimental results satisfactorily. The 3D turbulence spectral density of temperature fluctuation can be denoted as Φ T ( q ) , where q is the vector of wavenumber, which represents the point on the surface of the SDE. All points on the surface of the SDE have the same spectral density Φ T ( q ) . Thus, q has a relationship with conventionally-used wavenumber κ (κ x , κ y , κ z ) , shown as:
#209326 - $15.00 USD Received 1 Apr 2014; revised 16 May 2014; accepted 17 May 2014; published 27 May 2014 (C) 2014 OSA 19 May 2014 | Vol. 22, No. 11 | DOI:10.1364/OE.22.013427 | OPTICS EXPRESS 13429
q = (κ x * Caniso , κ y * Caniso , κ z / Caniso ) (1) We replace conventionally-used κ with q to express 2D spectral density of temperature
fluctuation Φ T (κ y , κ z ) to makes it easier to analyze 2D light intensity fluctuation, because the integration with wavenumber (κ y , κ z ) has nothing to do with anisotropy coefficient. Based on the consideration above, the 3D expression of anisotropic spectral density of temperature fluctuation in water can be written as the form of isotropic spectral density [21]: Φ T ( q) = K (α )CT2 q −α − 2φT ( q)
(2)
Γ(α + 1) π sin[(α − 1) ] and α is named as spectral power-law. Similar 4π 2 2 2 to the isotropic turbulent fields, CT is the temperature structure constant. In order to characterize the temperature fluctuations in dissipative range, we still adopt the method of isotropic turbulence and introduce the factor φT ( q) in Eq. (2). Besides, in this paper, the attenuation characteristics of anisotropic turbulence in the molecular dissipating range are considered the same as those of isotropic turbulence, which satisfies,
where K (α ) equals to
φT ( q) = 1
q
