Oceanic non-Kolmogorov optical turbulence and spherical wave propagation
Jinren Yao, Hantao Wang, Huajun Zhang, Jiandong Cai, Mingyuan Ren, Yu Zhang, Olga Korotkova
OOceanic non-Kolmogorov optical turbulence andspherical wave propagation J IN -R EN Y AO , H AN -T AO W ANG , H UA -J UN Z HANG , J IAN -D ONG C AI , M ING -Y UAN R EN , Y U Z HANG , AND O LGA K OROTKOVA , School of Physics, Harbin Institute of Technology, Harbin, 150001, China Department of Physics, University of Miami, Coral Gables, FL 33146, USA [email protected] * Corresponding author: [email protected]
Abstract:
Light propagation in turbulent media is conventionally studied with the help ofthe spatio-temporal power spectra of the refractive index fluctuations. In particular, for naturalwater turbulence several models for the spatial power spectra have been developed based onthe classic, Kolmogorov postulates. However, as currently widely accepted, non-Kolmogorovturbulent regime is also common in the stratified flow fields, as suggested by recent developmentsin atmospheric optics. Until now all the models developed for the non-Kolmogorov opticalturbulence were pertinent to atmospheric research and, hence, involved only one advectedscalar, e.g., temperature. We generalize the oceanic spatial power spectrum, based on twoadvected scalars, temperature and salinity concentration, to the non-Kolmogorov turbulenceregime, with the help of the so-called "Upper-Bound Limitation" and by adopting the concept ofspectral correlation of two advected scalars. The proposed power spectrum can handle generalnon-Kolmogorov, anisotropic turbulence but reduces to Kolmogorov, isotropic case if the powerlaw exponents of temperature and salinity are set to 11/3 and anisotropy coefficient is set to unity.To show the application of the new spectrum, we derive the expression for the second-order mutualcoherence function of a spherical wave and examine its coherence radius (in both scalar andvector forms) to characterize the turbulent disturbance. Our numerical calculations show that thestatistics of the spherical wave vary substantially with temperature and salinity non-Kolmogorovpower law exponents and temperature-salinity spectral correlation coefficient. The introducedspectrum is envisioned to become of significance for theoretical analysis and experimentalmeasurements of non-classic natural water double-diffusion turbulent regimes. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Oceanic optical turbulence is the phenomenon of the spatio-temporal water’s refractive-indexfluctuations caused by those in temperature and salinity concentration [1]. The Oceanic TurbulenceOptical Power Spectrum (OTOPS) being the Fourier transform of the spatial covariance functionof the refractive index provides an essential tool for characterizing the spatial statistics of anyorder for stationary light fields propagating through the natural waters. Within the last twodecades, the oceanic power spectrum model developed in [2] based on Kolmogorov turbulencetheory resulted, with the help of the Rytov and the extended Huygens-Fresnel methods, in anumber of theoretical predictions relating to light interaction with turbulent waters. In particular,evolution of the spectral density [3], the spectral shifts [4], the polarimetric [5] and coherence [6]changes and propagation of several other 2nd-order and 4th-order statistics [7–10] have beenrevealed. The theory has also benefited a number of underwater applications, such as the oceanicLight Detection and Randing (LiDaR) [11] systems, underwater optical communications [12–14],and underwater imaging [15].Since the oceanic optical turbulence is governed by two scalar fields, temperature and salinityconcentration, the OTOPS is approximately expressed as a linear combination of temperature a r X i v : . [ phy s i c s . a o - ph ] S e p ower spectrum, salinity power spectrum and their co-spectrum [2]. Therefore OTOPS containsmany parameters, such as the Kolmogorov scale η , the Prandtl number Pr , the Schmidt number Sc , as well as the dissipation rates of temperature, salinity, and kinetic energy, χ T , χ S , and ε , respectively, substantially complicating the predictions for the light - oceanic turbulenceinteractions.The OTOPS model of [2] and its derivatives [16, 17] were all based on the first of the fourmodels (called below H1) for a single-scalar turbulent advection developed by Hill [18]. Analternative model for the Kolmogorov oceanic optical turbulence has been recently obtainedin [19–22] by numerically fitting model 4 of the Hill’s paper (called below H4) [18]. The H4-basedmodels are more precise than the H1-based models in high spatial frequency region, and, hence,have advantages in oceanic cases with the wide-ranged Prandtl/Schmidt numbers [21, 23]. All theaforementioned OTOPS models are based on the Kolmogorov theory having a constant powerlaw − / , and the co-spectra in these models are obtained by analogy with a single scalar(temperature or salinity) spectrum. Kolmogorov theory rely on several assumptions including the homogeneous and isotropicnature of turbulent eddies. Such regime is clearly not universal, since it is not being able to accountfor several anomalous phenomena such as rampâĂŞcliff signature and unusual scaling exponent(e. g. [24]). Over the past 30 years, several experiments have revealed the presence of non-classicatmospheric optical turbulence [25–30]. The power spectrum model of the non-Kolmogorovturbulence advected by a single scalar and light interaction with such turbulence have beenwidely discussed in atmospheric optics literature [31–39].
However, it is our understanding thata comprehensive non-Kolmogorov model for oceanic waters does not exist .Non-Kolmogorov phenomena, as a result of inadequate rate of energy cascade, are common inunderdeveloped or vertically suppressed atmospheric turbulence, and do appear in stratified marineenvironment. In two oceanic experiments by Ichiye [40] and by Pochapsky and Malone [41] thenon-Kolmogorov fluctuations of temperature and salinity have been observed. In the Ichiye’smeasurement, the power law of temperature and salinity were between − / −
5, which wasinterpreted as the result of oceanic stratification. In Pochapsky and Malone measurement, a − we set the aim forthis paper to develop an OTOPS that extends the model suggested in [21, 22] to non-Kolmogorovregime. This requires (I) developing the non-Kolmogorov temperature/salinity spectrum whichis applicable for the marine environment with the wide-ranged Prandtl/Schmidt numbers, and (II)deriving the temperature-salinity co-spectrum which can not be directly obtained by analogywith a single-scalar spectrum, since the power law exponents of the two advected scalars can begenerally different.The paper is organized as follows: using a non-Kolmogorov structure function, we derivethe non-Kolmogorov temperature and salinity spectra based on an H4-based model (Section2.1); based on the Upper-Bound limitation, we develop a temperature-salinity co-spectrum(Section 2.2); on combining the results for the temperature spectrum, the salinity spectrum andthe co-spectrum, we introduce a non-Kolmogorov OTOPS (NK-OTOPS) model (Section 3); weapply the NK-OTOPS model for the analysis of the spherical wave propagation (Section 4); andwe summarize the obtained results (Section 5)
2. Temperature/salinity spectra and their co-spectrum in ocean
The OTOPS is composed of temperature spectrum, salinity spectrum, and temperature-salinityco-spectrum. In this section, we will derive the non-Kolmogorov temperature/salinity spectraand the temperature-salinity co-spectrum. .1. Non-Kolmogorov temperature/salinity spectra
We begin by recalling the H4-based temperature/salinity spectrum that has been develped forKolmogorov case in [21]. By comparing its structure function with the Kolmogorov structurefunction, we will first obtain its structure constant C i and its inner scale l i . Then, the H4-basedspectrum will be modified into a non-Kolmogorov spectrum. A. H4-based temperature/salinity spectrum
Here the H4-based temperature/salinity spectrum [21] is re-organized as Φ i ( κ ) = C k C i κ − / g i ( κη ) , with i ∈ { T , S } , (1)where κ is the wavenumber (cid:2) m − (cid:3) ; C i is the structure constant (dimensionless); C k is a dimen-sionless constant given by C k C i = βε − / χ i π , (2) β is the Obukhov-Corrsin constant (non-dimensional); ε is the dissipation rate of kineticenergy [ m s − ] ; χ i is the ensemble-averaged variance dissipation rate of temperature or salinity( i ∈ { T , S } ) with unit K s − or g s − ; the non-dimensional function g i ( x ) is g i ( x ) = (cid:213) j = a j x b j exp (cid:16) − . x c i . (cid:17) , (3)with (cid:8) a j (cid:9) = (cid:8) , . c i . , − . c i . (cid:9) , (4) (cid:8) b j (cid:9) = { , . , . } , (5) c i = a / β Pr − i , (6)where Pr T and Pr S are the temperature Prandtl number and salinity Schmidt number, respectively, a is constant and generally equals 0.072, and β is the Obukhov-Corrsin constant approximatingto 0.72 generally [18]. B. Structure constant C i and inner scale l i Structure constant C i and inner scale l i are the key parameters in the turbulence structurefunction, and they will be obtained by comparing the corresponding structure function in theKolmogorov case.The structure function of Eq.(1) is D i ( R ) = π ∫ ∞ κ Φ i ( κ ) (cid:18) − sin κ R κ R (cid:19) d κ = βε − / χ i η / (cid:213) j = a j (cid:40)(cid:16) . c i . (cid:17) − bj Γ (cid:18) − + b j (cid:19)(cid:20) − F (cid:18) − + b j , , − R × . c i . η (cid:19)(cid:21) (cid:27) , (7)where Γ (·) is a Gamma function, and F (· , · , ·) is a generalized hyper-geometric function. ForKolmogorov turbulence advected by single scalar (temperature or salinity), the structure functionis (Chapter 3 of [42]): D i ( R ) = C i l i − / R R (cid:28) l i , C i R / R (cid:29) l i , with i ∈ { T , S } . (8)y comparing Eqs. (7) and (8), and in combining with Eq. (2), we get the dimensionless constant C k = − Γ ( / ) / π Γ (− / ) Γ ( / ) ≈ . , (9) the structure constant C i = − βε − / χ i Γ (− / ) Γ ( / ) Γ ( / ) − / , (10)and the inner scale l i = η / T ( c i ) , (11)where T ( c i ) = π C k (cid:213) j = a j (cid:34)(cid:16) . c i . (cid:17) − bj − (cid:18) b j − (cid:19) Γ (cid:18) b j − (cid:19)(cid:35) / , (12)with a j , b j and c i defined in Eqs. (4)-(6), respectively. C. Non-Kolmogorov case
Now, we modify Eq. (1) to a non-Kolmogorov spectrum. Following the modification inatmospheric optics [31, 32, 43], we add two adaptive functions A ( α i ) and h ( α i , c i ) to Eq. (1), Φ i ( κ, α i ) = A ( α i ) C i κ − α i g i ( κη (cid:48) ) , (13)with η (cid:48) = η h ( α i , c i ) , (14)where A ( α i ) is a variable factor similar to the ‘ A ( α ) ’ in [31], h ( α i , c i ) is a scaling functionsimilar to the ‘ c ( α ) ’ in [31], it adjusts the location of viscous range on κ -axis. Expressions of A ( α i ) and h ( α i , c i ) are derived as follows .The structure function of Eq.(13) is D i ( R , α i ) = π ∫ ∞ κ Φ i ( κ, α i ) (cid:18) − sin κ R κ R (cid:19) d κ = π C i A ( α i ) η (cid:48) α − (cid:213) j = a j (cid:40)(cid:16) . c i . (cid:17) − − bj + α i Γ (cid:18) + b j − α i (cid:19)(cid:20) − F (cid:18) + b j − α i , , − R (cid:0) . c i . (cid:1) η (cid:48) (cid:19)(cid:21) (cid:27) , (15)and the non-Kolmogorov structure function [31, 32] is D i ( R , α i ) = C i l α i − i R R (cid:28) l i , C i R α i − R (cid:29) l i , (16)where C i and l i have been obtained in Eqs. (10) and (11), respectively. By comparing Eq. (15)with Eq. (16), we have A ( α i ) = Γ ( α i − ) π cos (cid:16) πα i (cid:17) , (17)and h ( α i , c i ) = G ( α i , c i ) T i ( c i ) , (18) log ( R ) F o r F F at = 11/3 F at = 12.5/3 F at = 14/3 F at = 11/3 F at = 12.5/3 F at = 14/3 Fig. 1. Functions F ( R ) and F ( R ) in Eq. (20). The solid curves represent F , and thedashed curves represent F . with G ( α i , c i ) = π A ( α i ) (cid:213) j = a j (cid:16) . c i . (cid:17) − − bj + α i (cid:18) + b j − α i (cid:19) Γ (cid:18) + b j − α i (cid:19) α i − , (19) T i ( c i ) has been given in Eq. (12). When α i = /
3, we have A ( / ) = C k ≈ .
033 and h ( / , c i ) =
1. Hence, the non-Kolmogorov spectrum Eq. (13) can degenerate to the traditionalKolmogorov model Eq. (1).To show the consistency between the proposed non-Kolmogorov spectrum Eq. (13) and thenon-Kolmogorov structure function Eq. (16), we plot the following two functions in Fig. 1, F ( R ) = (cid:16) C i l i α i − R (cid:17) − π ∫ ∞ κ Φ i ( κ ) (cid:0) − sin κ R κ R (cid:1) d κ, F ( R ) = (cid:0) C i R α i − (cid:1) − π ∫ ∞ κ Φ i ( κ ) (cid:0) − sin κ R κ R (cid:1) d κ. (20)It shows F ( R → ) = F ( R → ∞) =
1, which indicates that the modified non-Kolmogorovspectrum Eq.(13) agrees well with the asymptotic formula Eq. (16).
Equation (13) together with Eqs. (14), (17) and (18) constitute the main results of this section.They give the non-Kolmogorov spectrum of oceanic temperature/salinity turbulence, and theproposed spectrum agrees well with the widely accepted asymptotic structure function.
It mustbe noticed that parameter c i is related to Prandtl/Schmidt number in Kolmogorov case but thisdefinite relation is broken in non-Kolmogorov cases because of the presence of inhomogeneous,anisotropic and/or underdeveloped turbulence. In what follows, we consider c i as a directparameter, and set its range in Appendix I. In the Kolmogorov case the models of temperature-salinity co-spectrum have been obtained byanalogy with the single scalar (temperature/salinity) spectrum [2, 16, 19, 21] but such an analogyis unavailable if the exponents of temperature and salinity spectra are different. Hence, for thenon-Kolmogorov case, the temperature-salinity co-spectrum should be obtained by other means.In this section, we will derive the temperature-salinity co-spectrum based on the Upper-BoundLimitation [44–46] and the concept of spectral correlation [47].s proven in Section 5.2.5 of [44], the Upper-Bound Limitation gives the relation betweenscalar spectra φ T , φ S and their co-spectrum φ T S :0 ≤ φ T S ( κ ) ≤ [ φ T ( κ ) φ S ( κ )] / . (21)[48] extended the Upper-Bound Limitation to three-dimensional case:0 ≤ Φ T S ( κ, α T , α S ) ≤ [ Φ T ( κ, α T ) Φ S ( κ, α S )] / . (22)By adopting the concept of spectral correlation [47, 49], and combining Eq.(13) with Eq. (22),we obtain the temperature-salinity co-spectrum as Φ T S ( κ, α T , α S ) = γ ST ( κη ) [ Φ T ( κ, α T ) Φ S ( κ, α S )] / = γ ST ( κη ) A T S ( α T , α S ) C T S κ −( α T + α S )/ g T S ( κη ) , (23)with C T S = (cid:16) C T C S (cid:17) / , (24) A T S ( α T , α S ) = [ A ( α T ) A ( α S )] / , (25) g T S ( κη ) = (cid:20) g T (cid:18) κη h ( α T , c T ) (cid:19) g S (cid:18) κη h ( α S , c S ) (cid:19)(cid:21) / , (26)where γ ST ( κη ) is a correlation factor describing the degree of correlation between temperaturespectrum and salinity spectrum, and 0 ≤ γ ST ( κη ) ≤
1. When γ ST =
1, Eq. (23) refers to afully correlated co-spectrum; when γ ST =
0, Eq.(23) refers to a uncorrelated co-spectrum that Φ T S =
0; when 0 < γ ST <
1, the co-spectrum is partially correlated. Details about partiallycorrelated co-spectrum are given as follows.According to the concept of spectral correlation [47,48], temperature fluctuation T (cid:48) and salinityfluctuation S (cid:48) are highly correlated if they are both driven by eddy diffusion, but the correlationwill be broken down if T (cid:48) is driven by temperature molecular diffusion. Hence, the followingshould hold [50]:• When κ belongs to the inertial-convective range of Φ T (i.e. g T ∝ κ ), the salinityspectrum is generally in its inertial-convective range [51]. Thus, both T (cid:48) and S (cid:48) aregoverned by eddy diffusion, and they have a high correlation , i.e. γ ST = γ max ≤ κ belongs to the viscous-convective range of Φ T (i.e. g T ∝ κ / ), T (cid:48) is consumedby viscosity but S (cid:48) is still governed by eddy diffusion. The correlation begins to decreasein this range, and it has been observed in [52] that correlation decreases monotonically.Hence, we have d γ ST / d κ ≤ κ belongs to the viscous-diffusive range of Φ T (i.e. g T decreases fast with κ ), T (cid:48) is primarily depleted by temperature molecular diffusion, which leads to a very lowcorrelation between T (cid:48) and S (cid:48) , i.e. γ ST ≈ γ ST ( κη ) obey the following constraints: γ ST ( κη ) = γ max ≤ κ (cid:28) κ ,γ ST ( κη ) ∈ [ , γ max ] and d γ ST / d κ ≤ κ (cid:28) κ (cid:28) κ ,γ ST ( κη ) ≈ κ (cid:29) κ , (27) log ( ) g T g T (a) α T = / c T = . × − ; -3 -2 -1 0 1 log ( ) g T g T (b) α T = . / c T = . × − ; -3 -2 -1 0 1 log ( ) g T g T (c) α T = / c T = . × − ; -3 -2 -1 0 1 log ( ) g T g T (d) α T = / c T = . × − ; -3 -2 -1 0 1 log ( ) g T g T (e) α T = / c T = . × − ; -3 -2 -1 0 1 log ( ) g T g T (f) α T = / c T = . × − ; Fig. 2. The locations of κ η (‘|’) and κ η (‘|’) defined by Eqs. (29)-(30). where κ defines the transition between inertial-convective and viscous-convective ranges of Φ T , κ defines the transition between viscous-convective and viscous-diffusive ranges of Φ T .According to [18], we have following defining relations for κ and κ in H4-based non-Kolmogorovmodel: κ η h ( α T , c T ) = a , (28)and κ η h ( α T , c T ) = (cid:18) a / Qc T (cid:19) / , (29)where η is the Kolmogorov scale; h ( α T , c T ) is the non-Kolmogorov scaling function given in Eq.(18); a is a constant approximating to 0.072 [2]; Q is another constant about 2.35 [2]; and c T hasbeen given in Eq. (6). The locations of κ η and κ η are marked by ‘|’ and ‘|’ in Fig.2, respectively,and ‘—’ refers to g T . It shows that κ η and κ η mark the transitions between different rangesvery well.For mathematical simplicity of discussion, we suppose that the correlation factor in fullycorrelated case is γ ST = , (30)and in partially correlated case it takes form γ S T ( κη ) = − tanh {[ log ( κη ) − ( log ( κ η ) + log ( κ η )) / ] ρ } γ max , (31)with ρ = p log ( κ η ) − log ( κ η ) , γ max = , (32)where p controls the transition speed of γ ST from γ max to 0.In Figure 3 we compare Eq. (23) with the conventional co-spectrum [21] limiting ourselvesto Kolmogorov case ( α S = α T = / q ( κη ) = ( C T C S ) − / κ / Φ T S varying with log ( κη ) , where ‘---’ refers to the traditional co-spectrum [21];‘—’ refers to the proposed co-spectrum in Eq. (23) with a full correlation γ ST =
1; the curves‘---’ and ‘---’ refer to the partially correlated co-spectra with p = p =
2, respectively. The log( ) ( C T C S ) - / / T S (a) q ( κη ) ; (cid:51)(cid:68)(cid:85)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:79)(cid:92)(cid:3) correlated (cid:70)(cid:82)(cid:16)(cid:86)(cid:83)(cid:72)(cid:70)(cid:87)(cid:85)(cid:88)(cid:80)(cid:3) (cid:11)(cid:302)(cid:3)(cid:32)(cid:20)(cid:20)(cid:18)(cid:22)(cid:15) (cid:83)(cid:3)(cid:32)(cid:3)(cid:23) (cid:12) (cid:51)(cid:68)(cid:85)(cid:87)(cid:76)(cid:70)(cid:68)(cid:79)(cid:79)(cid:92)(cid:3) correlated (cid:3) (cid:3) (cid:70)(cid:82)(cid:16)(cid:86)(cid:83)(cid:72)(cid:70)(cid:87)(cid:85)(cid:88)(cid:80) (cid:11)(cid:302)(cid:3)(cid:32)(cid:20)(cid:20)(cid:18)(cid:22)(cid:15) (cid:83)(cid:3)(cid:32)(cid:3) (cid:21)(cid:12) (cid:55)(cid:85)(cid:68)(cid:71)(cid:76)(cid:87)(cid:76)(cid:82)(cid:81)(cid:68)(cid:79)(cid:3) (cid:43)(cid:23)(cid:16)(cid:69)(cid:68)(cid:86)(cid:72)(cid:71)(cid:3)(cid:80)(cid:82)(cid:71)(cid:72)(cid:79) Full y correlated co-spectrum((cid:127) =11/3) -2 -1 0 1 log( ) S T (b) γ ST ( κη ) ; Fig. 3. Comparing (a) the non-dimensional function q ( κη ) and (b) the correlationfactor γ ST ( κη ) corresponding to the proposed co-spectrum with these correspondingto conventional co-spectrum [21]. Values of parameters are listed in Appendix II. log ( ) f () (a) α T = / α S = / log ( ) f () (b) α T = / α S = . / log ( ) f () (c) α T = / α S = / log ( ) f () (d) α T = / α S = / log ( ) f () (e) α T = . / α S = / log ( ) f () (f) α T = / α S = / Partially correlated (cid:70)(cid:82)(cid:16)(cid:86)(cid:83)(cid:72)(cid:70)(cid:87)(cid:85)(cid:88)(cid:80)
Full y correlated co-spectrum
Fig. 4. The curves of the dimensionless function f ( κ ) = κ ( α T + α S )/ ( C T C S ) − / Φ T S with different values of α T and α S . vertical lines ‘|’ and ‘|’ mark the locations of κ η and κ η , respectively. With similar legends, Fig.3(b) shows correlation factor γ ST varying with log ( κη ) [53].Figure 3 shows that: for the Kolmogorov case and in comparison with the conventionalco-spectrum [21], the proposed partially correlated co-spectrum has a higher correlation in thetemperature inertial-convective range ( κ (cid:28) κ ), a lower correlation in the temperature viscous-convective range ( κ (cid:28) κ (cid:28) κ ), and also a low correlation in the temperature viscous-diffusiverange ( κ (cid:29) κ ). Furthermore, Fig.3 (a) indicates that the proposed fully correlated co-spectrumtends to the conventional co-spectrum when α T = α S = / in non-Kolmogorov case , and to verify its de-correlationwithin temperature viscous-convective range, we plot lo g of non-dimensional function f ( κ ) = κ ( α T + α S )/ ( C T C S ) − / Φ T S in Fig.4, and compare the fully correlated co-spectrum (‘—’) withthe partially correlated co-spectrum (‘---’) at p =
3. Same as before, κ η and κ η are markedby ‘|’ and ‘|’, respectively. It is shown that the proposed co-spectrum has low correlation in thetemperature inertial-convective range, as expected. This agrees with Eq. (27). hus we have obtained a temperature-salinity co-spectrum with a non-Kolmogorov powerlaw ( α T + α S )/ and a flexible correlation factor γ ST [see Eq. (23)]. If γ ST =
1, the proposedco-spectrum is fully correlated, and it approximately reduces to the conventional co-spectrumwhen α T = α S = /
3; if γ ST =
0, the proposed co-spectrum is uncorrelated, i.e. Φ T S =
0; if γ ST obeys Eq. (31), the proposed co-spectrum is partially correlated. As we expected, the newco-spectrum model has a power law between α T and α S ; if the temperature and salinity fields areboth Kolmogorov ( α T = α S = / α T S = /
3. OTOPS with anisotropy and non-Kolmogorov power law
In general, the oceanic refractive-index fluctuation n (cid:48) is approximately given by a linearcombination of temperature fluctuation T (cid:48) and salinity fluctuation S (cid:48) [2, 22, 54]: n (cid:48) ≈ n (cid:48) T T (cid:48) + n (cid:48) S S (cid:48) , (33)with n (cid:48) T = dn (cid:48) dT (cid:48) , n (cid:48) S = dn (cid:48) dS (cid:48) . (34)This implies that the spectrum of n (cid:48) is approximately given by linear combination Φ n ( κ ) = n (cid:48) T Φ T ( κ ) + n (cid:48) S Φ S ( κ ) + n (cid:48) T n (cid:48) S Φ T S ( κ ) , (35)where Φ T is the temperature spectrum, Φ S is the salinity spectrum, and Φ T S is the temperature-salinity co-spectrum. On combining Eqs. (13) and (23), we obtain the following expression forthe OTOPS: Φ n ( κ ) = n (cid:48) T Φ T ( κ ) + n (cid:48) S Φ S ( κ ) + n (cid:48) T n (cid:48) S γ ST ( κη ) (cid:112) Φ T ( κ ) Φ S ( κ ) = n (cid:48) T C T A ( α T ) κ − α T g T ( κη / h T ) + n (cid:48) S C S A ( α S ) κ − α S g S ( κη / h S ) + n (cid:48) T n (cid:48) S γ ST (cid:16) C T C S (cid:17) / [ A ( α T ) A ( α S )] / κ −( α T + α S )/ [ g T ( κη / h T ) g S ( κη / h S )] / . (36)with h T = h ( α T , c T ) , h S = h ( α S , c S ) . (37)To make the developed OTOPS more physical we now implement the finite outer-scale cut-offand extend it to the anisotropic case. To obtain the first extension we use the filter function withexponential form [20, 55]: Φ n ( κ ) = (cid:20) − exp (cid:18) − κ κ (cid:19)(cid:21) Φ n ( κ ) , (38)where κ is the outer-scale cut-off wavenumber defined by κ ≈ π / L with L ( m ) representingthe outer scale. Further the anisotropic OTOPS can be obtained on following [56] as: Φ n ( κ ) = µ Φ n ( κ iso ) , (39)where µ is the anisotropy parameter, κ is the three-dimensional wavenumber, and κ iso is aisotropisizing transformation of κ : κ = (cid:0) κ x , κ y , κ z (cid:1) T , κ iso = (cid:0) µκ x , µκ y , κ z (cid:1) T , κ iso = | κ iso | , (40) T is denoting vector transpose.Thus in this section, a non-Kolmogorov OTOPS (NK-OTOPS) is given in Eq. (36), whileits extended form for outer-scaled and anisotropic cases are presented by Eqs. (38) and (39),respectively . . Spherical wave propagation in oceanic optical turbulence As an example of applying the NK-OTOPS, and on taking into account the significance of thespherical wave statistics for the extended HuygensâĂŞFresnel principle, we will calculate andanalyze the 2nd-order statistics of a spherical wave. In particular, in Section 4.1, the wavestructure function (WSF) of a spherical wave in oceanic turbulence will be derived; in Section4.2, the vector and scalar versions of the coherence radius will be defined and examined; and inSection 4.3, the co-effect of temperature and salinity on spherical wave’s propagation will bediscussed by calculating the scalar coherence radius varying with α T , α S , c T and c S . A. 2nd-order statistical moments
We will first derive the 2nd-order statistical moment of a spherical wave propagating in thenon-Kolmogorov oceanic optical turbulence . According to Eq. (59) of chapter 5 in [42], forhorizontal channels (along y-axis) this quantity has form: E h ( r , r ) = π k n ∫ L d η ∫ ∫ + ∞−∞ d κ · Φ n ( κ ) exp (cid:20) i κ ( γ r − γ ∗ r ) − i κ k ( γ − γ ∗ ) ( L − η ) (cid:21) , (41)with κ = ( κ x , κ z ) T , r = ( r x , r z ) T , r = ( r x , r z ) T , (42)where L is the propagation distance from the source plane, k is the wavenumber defined as2 π n / λ , n being the average refractive-index, Φ n is the anisotropic NK-OTOPS as given by Eq.(39). For a spherical wave, γ = γ ∗ =
1. On assuming that κ t = ( µκ x , κ z ) T , r = ( r x / µ, r z ) T , r = ( r x / µ, r z ) T , (43)and combining Eq. (39) with Eq. (41), we get E h ( r , r ) = π k n µ ∫ L d η ∫ ∫ + ∞−∞ d κ t · µ Φ n ( κ t ) exp [ i κ t ( r − r )] = π k µ Ln ∫ + ∞ d κ t · κ t Φ n ( κ t ) J [ κ t | r − r |] , (44)where Φ n is the outer-scaled NK-OTOPS in Eq. (38). On setting ρ = r − r , we find that the2nd-order statistical moment of a spherical wave along a horizontal channel (along the y-axis)becomes E h ( ρ ) = π k µ Ln ∫ + ∞ d κ t · κ t Φ n ( κ t ) J (cid:20) κ t (cid:113) µ − ρ x + ρ z (cid:21) , (45)where κ t = (cid:12)(cid:12) ( µκ x , κ z ) T (cid:12)(cid:12) . (46)Similarly, the 2nd-order statistical moment of a spherical wave in a vertical channel (along thez-axis) becomes E v ( ρ ) = π k Ln ∫ + ∞ d κ t · κ t Φ n ( κ t ) J (cid:20) κ t (cid:113) µ − ρ x + µ − ρ y (cid:21) , (47)with κ t = (cid:12)(cid:12)(cid:12)(cid:0) µκ x , µκ y (cid:1) T (cid:12)(cid:12)(cid:12) . (48) . Wave structure function of spherical wave Next, based on the 2nd-order statistical moments given above, we derive the WSF of a sphericalwave in the non-Kolmogorov oceanic optical turbulence. According to the expressions in chapter6 of [42] the WSF of a spherical wave has form: D sp ( ρ , L ) = Re [ ∆ ( ρ , L )] = E ( r , r ) + E ( r , r ) − E ( r , r ) = E ( ) − E ( ρ ) , (49)where E is the 2nd-order statistical moment of a spherical wave. In combining with Eq. (45),we find that the WSF of a spherical wave in a horizontal channel ( ρ = ( ρ x , ρ z ) ) takes form D sp _ h ( ρ , L ) = π k µ Ln ∫ + ∞ d κ iso · κ iso Φ n ( κ iso ) (cid:20) − J (cid:18) κ iso (cid:113) µ − ρ x + ρ z (cid:19)(cid:21) . (50)Similarly, the WSF of a spherical wave in a vertical channel ( ρ = ( ρ x , ρ y ) ) becomes D sp _ v ( ρ , L ) = π k Ln ∫ + ∞ d κ iso · κ iso Φ n ( κ iso ) (cid:20) − J (cid:18) κ iso (cid:113) µ − ρ x + µ − ρ y (cid:19)(cid:21) . (51)When µ =
1, the WSFs in horizontal and vertical channels are equal and, hence, D sp _ h ( ρ , L ) = D sp _ v ( ρ , L ) = π k Ln ∫ + ∞ d κ · κ Φ n ( κ ) [ − J ( κ | ρ |)] . (52) Equations (50) - (52) are the main results of this section. They characterize the WSF of aspherical wave in an anisotropic, non-Kolmogorov turbulence by means of single integrals.
Wefirst plot the numerical results of the WSFs in an isotropic turbulence, with different values ofthe power law exponents in Fig. 5, and then compare isotropic and anisotropic cases in Fig. 6.Figure 5 shows that the power-law exponents α T and α S have significant effects on the WSF.Such power laws can result in a much higher or lower WSF in the non-Kolmogorov case than thatin the Kolmogorov case. Figure 6 shows that anisotropic turbulence leads to an anisotropic WSFwhich results in an elliptically shaped coherence radius, which we will further illustrate in thenext section. -5 0 5-505 r x (mm) r y ( mm ) D sp (a) -5 0 5-505 r x (mm) r y ( mm ) D sp (b) -5 0 5-505 r x (mm) r y ( mm ) D sp (c) -5 0 5-505 r x (mm) r y ( mm ) D sp (d) Fig. 5. The WSFs of spherical wave in isotropic turbulence ( µ =
1) with differentvalues of power laws. (a) ( α T , α S ) = ( / , / ) , (b) ( α T , α S ) = ( / , / ) , (c) ( α T , α S ) = ( / , / ) and (d) ( α T , α S ) = ( / , / ) . Values of other parametersare listed in Appendix II. The coherence radius of a spherical wave can be directly employed for assessing the opticalturbulence strength, and is also useful in calculating the statistics of various oprical beams(e.g. [57]). It is defined as a transverse separation distance between two points in the propagating
35 0 35-35035 r x (mm) r y ( mm ) D sp_h (a) -70 0 70-70070 r x (mm) r y ( mm ) D sp_v (b) -25 0 25-25025 r x (mm) r y ( mm ) D sp (c) Fig. 6. The WSFs of spherical wave (a) in horizontal channels with µ =
3, (b) invertical channels with µ =
3, and (c) in horizontal/vertical channels with µ =
1. Valuesof other parameters are listed in Appendix II. -35 0 35-35035 r x (mm) r y ( mm ) D sp_h (a) r D sp_h = 2 -70 0 70-70070 r x (mm) r y ( mm ) D sp_v (b) r D sp_v = 2 -25 0 25-25025 r x (mm) r y ( mm ) D sp (c) r D sp = 2 Fig. 7. The CRVs of the cases in fig. 6. spherical wave that corresponds to the WSF’s value of 2. As a rule, the coherence radius isconsidered to be a scalar quantity.However, as we have shown in Section 4.1, the WSF could be anisotropic. Hence, here the‘coherence radius’ is considered as a vector ρ and we define it as a coherence radius vector (CRV) ρ by setting D sp ( ρ , L ) = . (53)For horizontal and vertical channels, we rewrite Eq. (53) as D sp _ h ( ρ h , L ) = , D sp _ v ( ρ v , L ) = , (54)where ρ h = ( ρ x _ h , ρ z _ h ) T , ρ v = (cid:0) ρ x _ v , ρ y _ v (cid:1) T are the CRVs in horizontal channel andvertical channel, respectively.A coherence radius scalar (CRS) ρ is assumed as ρ = (cid:113) ρ x _ h + µ ρ z _ h in horizontal channel , (cid:113) ρ x _ v + ρ y _ v in vertical channel . (55) ρ equals the widely used coherence radius if µ = π k Ln ∫ + ∞ d κ · κ Φ n ( κ ) (cid:104) − J (cid:16) µ − κ ρ (cid:17)(cid:105) = µ − for horizontal channels , , (56)where Φ n is the outer-scaled NK-OTOPS. Eqs. (55) and (56) can be used to predict the CRS ρ and the CRV ρ in oceanic turbulence. For example, according to Eqs. (39) and (56), in the cases of Figs. 6 (a)-(c) are 25 . . . ρ into Eq.(55), we mark the CRVs by white arrows in Fig. 7. The derived coherence radius vector (CRV) and scalar (CRS) are the main results of thissection, which can be evaluated using Eqs. (55)-(56).
The CRS corresponds to the widelyused coherence radius if µ = In this section we will give a numerical example on the co-effect of temperature and salinity ofthe NK-OTOPS on the CRS by calculating it as a function of the power laws of temperature andsalinity spectra α T , α S , as well as parameters c T and c S , defined by Eq. (6).For brevity of discussion, we set the anisotropy constant µ = ρ in vertical channels as a measurement of turbulent disturbance. The ranges of related parametersare listed as follows (see more details in Appendix. I): · α T , α S ∈ [ / , / ) ; · c T ∈ (cid:2) . × − , . × − (cid:3) and c S ∈ (cid:2) . × − , . × − (cid:3) ; · C S / C T ≥ . × − ppt · deg − · m α T − α S .Figure 8 shows ρ ( α T , α S ) and ρ ( c T , c S ) for different spectral correlation of the powerspectrum (as above, fully correlated case refers to γ ST =
1, partially correlated case refers tothe γ ST obeying Eq. (31), and uncorrelated case refers to γ ST =
0, i.e. Φ T S = ρ ( α T , α S ) being very different from that in Figs. 8 (a)-(c), and Fig. 8(e) shows a distribution of ρ ( c T , c S ) being very different from that in Figs. 8 (f)-(h).Figure 9 shows ρ ( α T , α S ) and ρ ( c T , c S ) with different ratios of C S to C T . With theincrease of C S / C T , the variation of ρ with α S and c S becomes more pronounced.A comprehensive analysis of Figs. 8 and 9 reveals that• ρ substantially varies with α T and α S (can reach an order of magnitude).• γ ST ( κη ) , as a function describing the correlation between temperature and salinity spectra,has an obvious effect on ρ .• As expected, the structure constant C T or/and C S describes the contribution of temperatureor/and salinity fluctuation very well.
5. Summary and conclusion
The power spectrum of refractive-index fluctuations provides a rigorous physical descriptionof the 2nd-order statistics of natural random media, hence, bearing utmost significance forenvironmental optics. A number of non-Kolmogorov models have been recently developedfor ‘single-diffuser’ turbulence, i.e., based on a single advected scalar, as is temperature inatmsopheric case. However, to our knowledge, there was no model for non-Kolmogorov spectrumdescribing optical turbulence with two or more advected scalars, i.e., ‘double-diffuser turbulence’.The major obstacle for developing such a power spectrum was due to the fact that the co-spectrumof two scalar spectra in the non-Kolmogorov case could not be directly obtained by analogy witha method used for Kolmogorov case in which the power laws of the two scalar spectra are equal.In this paper, we have developed for the first time a non-Kolmogorov power spectrum ofoceanic refractive-index fluctuations, being an example of a double-diffuser, by deriving the T a S -1.735-1.155-0.575 (a) log (r ) (a) Full correlation: γ ST =
11 13 15111315 T a S -1.74-1.25-0.76 (b) log (r ) (b) Partial correlation: p =
11 13 15111315 T a S -1.735-1.246-0.756 (c) log (r ) (c) Partial correlation: p =
11 13 15111315 T a S -1.78-1.40-1.01 (d) log (r ) (d) Non-correlation: γ ST = c T (·10 ) c S ( · ) -1.27-1.26-1.25 log (r ) (e) (e) Full correlation: γ ST = c T (·10 ) c S ( · ) -1.287-1.283-1.279 log (r ) (f) (f) Partial correlation: p = c T (·10 ) c S ( · ) -1.283-1.279-1.275 log (r ) (g) (g) Partial correlation: p = c T (·10 ) c S ( · ) -1.512-1.509-1.506 log (r ) (h) (h) Non-correlation: γ ST = Fig. 8. The distributions ρ ( α T , α S ) and ρ ( c T , c S ) with different spectralcorrelation γ ST . Values of parameters are listed in Appendix II. temperature spectrum, the salinity spectrum, and their co-spectrum, based on the Upper-Boundlimitation and on the concept of spectral correlation. Our developed spectrum generallyhandles non-Kolmogorov turbulence with partially correlated temperature-salinity co-spectrum( α i ∈ [ / , / ) and γ ST ( κ ) ≤
1) which is common for the stratified flow fields, but reduces toconventional, Kolmogorov spectrum, with fully correlated co-spectrum ( α i = / γ ST = α T and α S ). Moreover, we have shown for the first time that the coherenceradius scalar ρ takes on very different values for different settings of spectral correlation.This also indicates the usefulness of developing the oceanic non-Kolmogorov power spectrumwith correlation factor γ ST .On finishing we mention that so far no literature of oceanic turbulence has provided modelsfor the correlation factor γ ST ( κ ) and other parameters such as c T , c S , α T and α S . But like inthe studies of atmospheric propagation, these parameters could be significant in characterizingoceanic optical turbulence, and any details about them are of importance for further experimentalcampaigns. Our model fills such a gap by providing a rather simple analytical model applicablein a variety of oceanic turbulence regimes. Appendix I. Ranges of parameters
For brevity of numerical calculation, we set the ranges of parameters as follows.
The ranges hereare based on references, and some of them are obtained in Kolmogorov case. The real rangescould be beyond what we set.
1. Constants
As given in [2, 18, 61], a = . β = .
72 and Q = . T a S -1.70-1.35-1.00 (a) log (r ) (a) C S / C T = . × − ;
11 13 15111315 T a S -1.7-1.2-0.7 (b) log (r ) (b) C S / C T = . × − ;
11 13 15111315 T a S -1.8-1.3-0.8 (c) log (r ) (c) C S / C T = . × − ; c T (·10 ) c S ( · ) -1.3740-1.3715-1.3690 log (r ) (d) (d) C S / C T = . × − ; c T (·10 ) c S ( · ) -1.326-1.323-1.320 log (r ) (e) (e) C S / C T = . × − ; c T (·10 ) c S ( · ) -1.808-1.799-1.790 log (r ) (f) (f) C S / C T = . × − ; Fig. 9. The distributions ρ ( α T , α S ) and ρ ( c T , c S ) with different values of C T / C S . All C S / C T has the unit ppt deg − m α T − α S . Values of parameters are listed inAppendix II.
2. The ranges of α T and α S According to the experimental data in [40] and the widely used range [32], non-Kolmogorovparameter α i ∈ [ / , / ) .
3. The range of C S / C T According to Eq. (2), C S / C T = χ S / χ T , (57)where the dispassion rate χ i of temperature and salinity are related through [17, 22] χ S / χ T = d r H − , (58)with d r ≈ (cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) + (cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) . (cid:0)(cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) − (cid:1) . , (cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) ≥ , . (cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) − . , . ≤ (cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) < , . (cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) , (cid:12)(cid:12) H θ T θ − S (cid:12)(cid:12) < . , (59)where d r is the eddy diffusivity ratio, θ T and θ S are the thermal expansion coefficient and thesaline contraction coefficient, respectively, and H is the temperature-salinity gradient ratio definedby H = d (cid:104) T (cid:105) / dzd (cid:104) S (cid:105) / dz . (60) d < T >/ dz (10 -3 ° C/m) z ( k m ) -4 -2 0 d < S >/ dz (10 -3 ppt/m) z ( k m ) -4 -2 0 T (10 -4 ° C -1 ) z ( k m ) -4 -2 0 S (10 -4 ppt -1 ) z ( k m ) Fig. 10. The distribution of temperature gradient d (cid:104) T (cid:105) / dz , salinity gradient d (cid:104) S (cid:105) / dz ,thermal expansion coefficient θ T and saline contraction coefficient θ S varying withdepth z in Pacific. -8 -4 0 4 log( C /C ) z ( k m ) C /C = 10 -4.498 -5 ppt deg -2 Fig. 11. The distribution of C S / C T varying with depth z in Pacific. Combining Eqs.(57)-(60), we have C S C T = (cid:12)(cid:12) H − θ T θ − S (cid:12)(cid:12) + (cid:12)(cid:12) H − θ T θ − S (cid:12)(cid:12) (cid:0) − (cid:12)(cid:12) H − θ S θ − T (cid:12)(cid:12)(cid:1) . , | H θ T θ − S | ≥ . (cid:12)(cid:12) H − θ T θ − S (cid:12)(cid:12) − . | H | − , . ≤ | H θ T θ − S | < . (cid:12)(cid:12) H − θ T θ − S (cid:12)(cid:12) , | H θ T θ − S | < . d (cid:104) T (cid:105) / dz , d (cid:104) S (cid:105) / dz , θ T and θ S of mid latitude Pacific in winter [62] (see alsoFig. 10), and based on Eq. (61), we plot C S / C T as a function of depth in Fig. 11. It shows that C S / C T ≥ . × − ppt · deg − . (62)For non-Kolmogorov cases, we assume C S / C T ≥ . × − ppt · deg − · m α T − α S . (63)
4. The ranges of c S and c T According to [22], Pr T varies from 5.4 to 13.4, and Pr S varies from 350.0 to 2210.0. Using therelation in Eq.(6) with constants a = .
072 and β = .
72, we have c T ∈ (cid:2) . × − , . × − (cid:3) and c S ∈ (cid:2) . × − , . × − (cid:3) . (64) Appendix II. The values of parameters in Figures
Here we list the values of parameters in figures. Figure 3: α T = α S = / c T = . × − , c S = . × − .• Figure 5: c T = . × − , c S = . × − , C T = . × − deg m − α T , C S = . × − ppt m − α S , η = . × − m, λ = n (cid:48) T = − . × − deg − n (cid:48) S = . × − g − L = L = α T = / α S = / c T = . × − , c S = . × − , C T = . × − deg m − α T , C S = . × − ppt m − α S , η = . × − m, λ = n (cid:48) T = − . × − deg − n (cid:48) S = . × − g − L = L = C T = . × − deg m − α T , C S = . × − ppt m − α S , λ = n (cid:48) T = − . × − deg − n (cid:48) S = . × − g −
1, , L = L = η = . × − m.(a)-(c) are plotted with ( c T , c S ) = ( . × − , . × − ) , and (d)-(e) are plotted with ( α T , α S ) = ( / , / ) .• Figure 9: C T = . × − deg m − α T , λ = n (cid:48) T = − . × − , n (cid:48) S = . × − , η = . × − , L = L = γ is given by Eq. (31) with p =
3. (a)-(c) are plotted with ( c T , c S ) = ( . × − , . × − ) , and (d)-(e) are plotted with ( α T , α S ) = ( / , / ) . Appendix III. Terminologies
Here we list a brief explanation about some terminology in this manuscript.•
Coherence radius vector (CRV) and coherence radius scalar (CRS):According to Section 4.1, the WSF D sp in anisotropic turbulence could be also anisotropic.Hence, the coherence radius | ρ | in D sp ( ρ ) = ρ .For brevity in discussion, we define ρ as CRV, and define a scalar — CRS — in Eq. (55).The CRS equals to coherence radius if µ = Hill’s model 1 (H1) and
Hill’s model 4 (H4):As widely accepted, the power spectrum of scalar fluctuations has two or three intervals [24].For the turbulence with large Pr or Sc, there are three intervals: inertial-convective, viscous-convective and viscous-diffusive intervals. For the turbulence with small Pr or Sc, there aretwo intervals: inertial and diffusive intervals. Hill’s models provide continuous transitionbetween different intervals. Hill’s model 1 is mathematically analytic but not as precise asHill’s model 4, and Hill’s model 4 is a non-linear differential equation that does not have aclosed-form solution. By numerical fitting, some approximate models for ocean [19, 21]and atmosphere [63] have been proposed based on Hill’s model 4.•
H1-based and
H4-based :They refer to the models based on Hill’s model 1 and 4, respectively.•
Upper-bound limitation :As proved in the Section 5.2.5 of [44], the co-spectrum φ ab of scalars a and b are limitedby | φ ab | ≤ φ a φ b , (65)where φ a and φ b are the spectrum of a and b , respectively. spectral correlation, fully correlated, partially correlated and uncorrelated :The ‘Correlation’ in this manuscript refers to the correlation between temperature fluctua-tions and salinity fluctuations. The spectral correlation factor is defined as γ ST = (cid:34) | Φ T S | Φ T Φ S (cid:35) / , (66)where Φ T and Φ S are the 3-D spectra of temperature and salinity, respectively. ‘fullycorrelated’ and ‘full correlation’ refer to the cases of γ ST =
1; ‘partially correlated’ and‘partial correlation’ refer to the cases of γ ST <
1; ‘uncorrelated’ and ‘non-correlation’ referto the cases of γ ST = Disclosures
The authors declare no conflicts of interest.
References
1. O. Korotkova, “Light Propagation in a Turbulent Ocean,” in Progress in Optics, Ed. T. D. Visser, , 1-43 (Elsevier,2018).2. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. FluidMech. Res. , 82–98 (2000).3. W. Lu, L. Liu, and J. Sun, “Influence of temperature and salinity fluctuations on propagation behaviour of partiallycoherent beams in oceanic turbulence,” J. Opt. A: Pure Appl. Opt. , 1052–1058 (2006).4. E. Shchepakina, N. Farwell, and O. Korotkova, “Spectral changes in stochastic light beams propagating in turbulentocean,” Appl. physics B , 415 (2011).5. O. Korotkova and N. Farwell, “Effect of oceanic turbulence on polarization of stochastic beams,” Opt. communications , 1740–1746 (2011).6. N. Farwell and O. Korotkova, “Intensity and coherence properties of light in oceanic turbulence,” Opt. Commun. , 872–875 (2012).7. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random ComplexMedia , 260–266 (2012).8. Y. Baykal, “Scintillation index in strong oceanic turbulence,” Opt. Commun. , 15–18 (2016).9. Y. Ata and Y. Baykal, “Structure functions for optical wave propagation in underwater medium,” Waves RandomComplex Media , 164–173 (2014).10. L. Lu, X. Ji, and Y. Baykal, “Wave structure function and spatial coherence radius of plane and spherical wavespropagating through oceanic turbulence,” Opt. express , 27112–27122 (2014).11. O. Korotkova, “Enhanced backscatter in lidar systems with retro-reflectors operating through a turbulent ocean,”JOSA A , 1797–1804 (2018).12. Y. Baykal, “Bit error rate of pulse position modulated optical wireless communication links in oceanic turbulence,”JOSA A , 1627–1632 (2018).13. X. Yi, Z. Li, and Z. Liu, “Underwater optical communication performance for laser beam propagation through weakoceanic turbulence,” Appl. Opt. , 1273–1278 (2015).14. Z. Cui, P. Yue, X. Yi, and J. Li, “Scintillation of a partially coherent beam with pointing errors resulting from aslightly skewed underwater platform in oceanic turbulence,” Appl. Opt. , 4443–4449 (2019).15. W. W. Hou, “A simple underwater imaging model,” Opt. Lett. , 2688–2690 (2009).16. J. R. Yao, Y. Zhang, R. N. Wang, Y. Y. Wang, and X. J. Wang, “Practical approximation of the oceanic refractiveindex spectrum,” Opt. Express , 23283–23292 (2017).17. M. Elamassie, M. Uysal, Y. Baykal, M. Abdallah, and K. Qaraqe, “Effect of eddy diffusivity ratio on underwateroptical scintillation index,” J. Opt. Soc. Am. A , 1969–1973 (2017).18. R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum inthe inertial and dissipation ranges,” Radio Sci. , 953–961 (1978).19. X. Yi and I. B. Djordjevic, “Power spectrum of refractive-index fluctuations in turbulent ocean and its effect onoptical scintillation,” Opt. Express , 10188–10202 (2018).20. Y. Li, Y. Zhang, and Y. Zhu, “Oceanic spectrum of unstable stratification turbulence with outer scale and scintillationindex of gaussian-beam wave,” Opt. Express , 7656–7672 (2019).21. J. R. Yao, H. J. Zhang, R. N. Wang, J. D. Cai, Y. Zhang, and O. Korotkova, “Wide-range Prandtl/Schmidt numberpower spectrum of optical turbulence and its application to oceanic light propagation,” Opt. Express , 27807–27819(2019).22. J. R. Yao, M. Elamassie, and O. Korotkova, “Spatial power spectrum of natural water turbulence with any average tem-perature, salinity concentration and light wavelength,” (Accepted by J. Opt. Soc. Am. A) doi: 10.1364/JOSAA.399150.3. A. Muschinski and S. M. de Bruyn Kops, “Investigation of hill’s optical turbulence model by means of directnumerical simulation,” J. Opt. Soc. Am. A , 2423–2430 (2015).24. K. R. Sreenivasan, “Turbulent mixing: A perspective,” Proc. Natl. Acad. Sci. , 18175–18183 (2019).25. M. S. Belen’kii, S. J. Karis, J. M. Brown II, and R. Q. Fugate, “Experimental study of the effect of non-kolmogorovstratospheric turbulence on star image motion,” in Adaptive Optics and Applications, vol. 3126 (International Societyfor Optics and Photonics, 1997), pp. 113–123.26. B. E. Stribling, B. M. Welsh, and M. C. Roggemann, “Optical propagation in non-kolmogorov atmospheric turbulence,”in Atmospheric Propagation and Remote Sensing IV, vol. 2471 (International Society for Optics and Photonics,1995), pp. 181–196.27. D. T. Kyrazis, J. B. Wissler, D. D. Keating, A. J. Preble, and K. P. Bishop, “Measurement of optical turbulence in theupper troposphere and lower stratosphere,” in Laser Beam Propagation and Control, vol. 2120 (International Societyfor Optics and Photonics, 1994), pp. 43–55.28. L. J. Otten III, M. C. Roggemann, B. Al Jones, J. Lane, and D. G. Black, “High-bandwidth atmospheric-turbulencedata collection platform,” in Optics in Atmospheric Propagation and Adaptive Systems III, vol. 3866 (InternationalSociety for Optics and Photonics, 1999), pp. 23–32.29. A. Zilberman, E. Golbraikh, and N. Kopeika, “Lidar studies of aerosols and non-kolmogorov turbulence in themediterranean troposphere,” in Electro-Optical and Infrared Systems: Technology and Applications II, vol. 5987(International Society for Optics and Photonics, 2005), p. 598702.30. A. Muschinski, “Non-kolmogorov turbulence,” in Imaging and Applied Optics 2017 (3D, AIO, COSI, IS, MATH,pcAOP), (Optical Society of America, 2017), p. PW2D.1.31. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Angle of arrival fluctuations for free space laser beampropagation through non kolmogorov turbulence,” in Atmospheric Propagation IV, vol. 6551 C. Y. Young and G. C.Gilbreath, eds., International Society for Optics and Photonics (SPIE, 2007), pp. 149 – 160.32. I. Toselli, L. C. Andrews, R. L. Phillips, and V. Ferrero, “Free-space optical system performance for laser beampropagation through non-kolmogorov turbulence,” Opt. Eng. , 026003 (2008).33. G. Wu, H. Guo, S. Yu, and B. Luo, “Spreading and direction of gaussian–schell model beam through a non-kolmogorovturbulence,” Opt. Lett. , 715–717 (2010).34. E. Shchepakina and O. Korotkova, “Second-order statistics of stochastic electromagnetic beams propagating throughnon-Kolmogorov turbulence,” Opt. Express , 10650–10658 (2010).35. O. Korotkova and E. Shchepakina, “Color changes in stochastic light fields propagating in non-Kolmogorov turbulence”Opt. Lett. , 3772–3774 (2010).36. A. Zilberman, E. Golbraikh, N. S. Kopeika, A. Virtser, I. Kupershmidt, and Y. Shtemler, “Lidar study of aerosolturbulence characteristics in the troposphere: Kolmogorov and non-kolmogorov turbulence,” Atmospheric Res. ,66–77 (2008).37. I. Toselli, O. Korotkova, X. Xiao and D. Voelz, “SLM-based laboratory simulations of Kolmogorov and non-Kolmogorov turbulence,” Appl. Opt. , 4740–4744 (2015).38. X. Xiao, D. Voelz, I. Toselli, and O. Korotkova, “Gaussian beam propagation in anisotropic turbulence alonghorizontal links: theory, simulation, and laboratory implementation” Appl. Opt. , 4079–4084 (2016).39. G. Funes, F. Olivares, C. G. Weinberger, Y. D. Carrasco, L. Nunez and D. G. Perez, “Synthesis of anisotropic opticalturbulence at the laboratory,” Opt. Lett. , 5696–5699 (2016).40. T. Ichiye, “Power spectra of temperature and salinity fluctuations in the slope water off cape hatteras,” Pure Appl.Geophys. , 205–216 (1972).41. T. E. Pochapsky and F. D. Malone, “Spectra of deep vertical temperature profiles,” J. Phys. Oceanogr. , 470–475(1972).42. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, Second Edition (SPIE PRESS,2005).43. B. Xue, L. Cui, W. Xue, X. Bai, and F. Zhou, “Generalized modified atmospheric spectral model for optical wavepropagating through non-kolmogorov turbulence,” J. Opt. Soc. A , 912–916 (2011).44. B. Harris, “Random data: Analysis and measurement procedures,” Technometrics , 271–271 (1975).45. L. Washburn, T. F. Duda, and D. C. Jacobs, “Interpreting conductivity microstructure: Estimating the temperaturevariance dissipation rate,” J. Atmospheric Ocean. Technol. , 1166–1188 (1996).46. H. E. Seim, “Acoustic backscatter from salinity microstructure,” J. Atmospheric Ocean. Technol. , 1491–1498(1999).47. J. D. Nash and J. N. Moum, “Estimating salinity variance dissipation rate from conductivity microstructuremeasurements,” J. Atmospheric Ocean. Technol. , 263–274 (1999).48. T. Ross, C. Garrett, and R. Lueck, “On the turbulent co-spectrum of two scalars and its effect on acoustic scatteringfrom oceanic turbulence,” J. Fluid Mech. , 107âĂŞ119 (2004).49. In some references, the Upper-Bound model (eq. (22)) was approximated as φ T S = ( φ T φ S ) / . In considering thatthe influence of this approximation on predicting light propagation is unknown, we adopt the concept of spectralcorrelation which provides elasticity under Upper-Bound limitation [47]. Examples of light propagation in fullycorrelated case, partially correlated case and uncorrelated case will be given in section 4.50. The Prandtl number of temperature is much less than the Schmidt number of salinity, which leads to an asynchronyof the sub-ranges of Φ T and Φ S , i.e., asynchronous consumption of T’ and S’. This asynchrony is responsible for theecrease of correlation which has been observed in numerical simulation [52].51. Generally, salinity Schmidt number is so larger than temperature Prandtl number that the inertial-convective range ofsalinity is wider than that of temperature. Hence, the salinity fluaction is in its inertial-convective range if κ locates inthe inertial-convective range of temperature spectrum.52. P. K. Yeung, M. C. Sykes, P. Vedula “Direct numerical simulation of differential diffusion with Schmidt numbersup to 4.0,” Phys. Fluids , 1601–1604 (2000).53. The concept of spectral correlation has not been introduced into the H4-based temperature-salinity co-spectrum [21].Here we calculate the γ ST of traditional co-spectrum by calculating Φ T S /( Φ T Φ S ) / .54. As shown in [22], n (cid:48) t and n (cid:48) s vary with environment (averaged temperature and salinity). Their variability also existsin non-Kolmogorov case. However, non-Kolmogorov case refers to inhomogeneous turbulence where the averagedtemperature and salinity could change temporally and spatially. Hence, the variability of n (cid:48) t and n (cid:48) s is not clear fornon-Kolmogorov case, and we will use fixed values in following numerical calculation for brevity.55. V. V. Voitsekhovich, “Outer scale of turbulence: comparison of different models,” J. Opt. Soc. Am. A , 1346–1353(1995).56. I. Toselli, “Introducing the concept of anisotropy at different scales for modeling optical turbulence,” J. Opt. Soc. Am.A , 1868–1875 (2014).57. O. Korotkova and J. R. Yao, “Bi-static lidar systems operating in the presence of oceanic turbulence,” Opt. Commun. , 125119 (2020).58. C. Wu, D. A. Paulson, J. R Rzasa and C.C. Davis, “Light field camera study of near ground turbulence anisotropyand observation of small outer-scales” Opt. Lett. , 1156–1159 (2020).59. F. Wang, I. Toselli, J. Li and O. Korotkova, “Measuring anisotropy ellipse of atmospheric turbulence by intensitycorrelations of laser light,” Opt. Lett. , 1129–1132 (2015).60. The anisotropic factor µ was firstly introduced into OTOPS in [64], where the range of µ is [ , ] . Here we assume µ = , 1849–1853 (1992).64. Y. Baykal “ Effect of anisotropy on intensity fluctuations in oceanic turbulence,” J. Mod. optics7