Spatial power spectrum of natural water turbulence with any average temperature, salinity concentration and light wavelength
SSpatial power spectrum of natural waterturbulence with any average temperature,salinity concentration and light wavelength J IN -R EN Y AO , M OHAMMED E LAMASSIE , O LGA K OROTKOVA , School of Physics, Harbin Institute of technology, Harbin, 150001, China Department of Electrical and Electronics Engineering, Özyeˇgin University, 34794 Istanbul, Turkey Department of Physics, University of Miami, Coral Gables, FL 33146, USA * Corresponding author: [email protected]
Abstract:
The power spectrum of water optical turbulence is shown to vary with its averagetemperature (cid:104) T (cid:105) and average salinity concentration (cid:104) S (cid:105) , as well as with light wavelength λ . Thisstudy explores such variations for (cid:104) T (cid:105) ∈ [ ◦ C , ◦ C ] , (cid:104) S (cid:105) ∈ [ ,
40 ppt ] covering most of thepossible natural water conditions within the Earth’s boundary layer and for visible electromagneticspectrum, λ ∈ [ nm , nm ] . For illustration of the effects of these parameters on propagatinglight we apply the developed power spectrum model for estimation of the scintillation index of aplane wave (the Rytov variance) and the threshold between weak and strong turbulence regimes. © 2020 Optical Society of America
1. Introduction
The demand for underwater applications has been steadily increasing in the recent years due tobroad expansion in human activities such as scientific data collection, environmental monitoring,oil field exploration, maritime archaeology, port and watercraft security. In their turn, theseapplications have boosted the demand for the underwater high speed wireless connectivity andhigh quality imaging. Optical signaling has the ability to achieve these goals but it is severelyaffected by water optical turbulence, i.e., rapid but spatially mild fluctuations in the water’srefractive index [1] (see also [2, 3]). Therefore it is crucial to establish accurate analytical modelsfor spatial power spectrum applicable to the wide range of Earth’s water conditions. It is wellknown that the two main factors affecting water optical turbulence are the temperature and thesalinity fluctuations. However, unlike in air turbulence, in the water not only the variations inthese quantities but also their average values may affect the optical signal transmission. Moreover,unlike in the air, on propagation in water light statistics may substantially depend on the sourcewavelength.Originally the spatial power spectra models of the fluctuating refractive index of the oceanicwaters resulting from temperature and salinity fluctuations have been developed in the seminalwork by Hill in 1978 [4], [5]. Not until 2000 have the two power spectra been combined into asingle power spectrum in a form of a linearized polynomial by Nikishovs [6], involving somedata from the previously developed model for double diffusers [7]. Due to its simplicity andversatility, over the last two decades the Nikishovs’ model became the standard for making thetheoretical predictions about light evolution in underwater turbulence [1]. Notwithstanding itslong-lasting impact on the field, several elaborations of the Nikishovs’ spectrum have been laterproposed [8–11].In particular, in recent work by the authors [11] (see also [12]) the attempt was made to use anumerical fit to one of the very accurate Hill’s models (model 4 of [4]) for the power spectrumwith Prandtl(Pr)/Schmidt(Sc) numbers varying in intervals sufficiently large to cover all possibileaverage temperatures occuring in the Earth’s ocean waters: (cid:104) T (cid:105) ∈ [ ◦ C , ◦ C ] . The effect of theaverage temperature on the evolution of the light waves was then revealed: with the increase a r X i v : . [ phy s i c s . a o - ph ] J u l n the average temperature the light statistics have been shown to be affected slightly less. Theproposed extension [11], while provided the insight into the average temperature dependence hasonly dealt with the ocean waters at the average salinity (NaCl) concentration of 35ppt.This study, building upon model in [11], considers turbulent waters with large ranges ofaverage temperature (cid:104) T (cid:105) ∈ [ ◦ C , ◦ C ] and average salinity concentration (cid:104) S (cid:105) ∈ [ ,
40 ppt ] covering practically all possible water basins on the planet. In particular, here we introduce twoimportant extensions: (I) we calculate the Prandtl/Schmidt numbers and the eddy diffusivityratio for arbitrary average temperature and salinity concentration and (II) we recalculate thelinear coefficients used in the Nikishovs’ linearized polynomial model, on the basis of a precisemodel of the refractive index of the ocean water developed by Quan and Fry [13]. The latterextension allows us to determine these linear coefficients having fine dependence on the averagetemperature, average salinity concentration and the wavelength of light. We point out that thedirect dependence of the underwater power spectrum on the light wavelength is a new type ofdependence and will be examined in detail in terms of its impact on light scintillation. In fact,this should not come as a surprise: water absorption has a very strong dependence on lightwavelength as well [14].
2. Calculation of temperature/salinity dependent Prandtl and Schmidt numbers
In this section, we analyze the dependence of the Prandtl and the Schmidt numbers on the water’saverage temperature and salinity concentration. In the Hill’s models [4, 5] these parameters arecalculated on the basis of the laminar flows and must not be confused with similar definitions forturbulent flows. We will assume here that the calculations of all derived quantities are made atthe atmospheric pressure.The
Prandtl number for the laminar flow is defined as Pr = ν / α T , (1)where ν (cid:2) m s − (cid:3) is the momentum diffusivity (also known as kinematic viscosity), α T [m /s] isthe molecular thermal diffusivity. Further, ν can be expressed as ν = µ / ρ, (2)with µ being dynamic viscosity (cid:2) N · s · m (cid:3) and ρ (cid:2) kg · m (cid:3) being the density of a fluid. Also inEq. (1) α T is α T = σ T /( ρ c p ) , (3)with σ T (cid:2) W · m − K − (cid:3) being the thermal conductivity and c p (cid:2) J · kg − · K − (cid:3) being the specificheat. Hence, Pr = c p µ / σ T . (4)As shown in Appendices I-III, three parameters: c p , µ , σ T may be directly related to the averagetemperature (cid:104) T (cid:105) and average salinity (cid:104) S (cid:105) . Thus, on combining Eq. (4) with Eqs. (27-33), wehave directly related Pr with (cid:104) T (cid:105) and (cid:104) S (cid:105) . The
Schmidt number for the laminar flow is defined as Sc = ν / α S , (5)where, as before, ν is kinematic viscosity and α S (cid:2) m s − (cid:3) is the molecular diffusivity of salt.According to the Stokes–Einstein law [15], α S µ /(cid:104) T (cid:105) = constant . (6)
10 20 30010203040 < S > ( pp t ) < T > ( o C) 5.4007.4009.40011.4013.40Prandtl number = 34.9ppt Pr = 7.16 (a) < S > ( pp t ) < T > ( o C) 350.0815.0128017452210Schmidt number = 34.9ppt Sc = 647.7 (b)
Fig. 1. (a) Schmidt number, and (b) Prandtl number varying with average temperatureand salinity.
On fitting the data in Ref. [15] with the least square method we concluded that α S ≈ . × − (cid:104) T (cid:105) µ . (7)Further, on combining Eq. (7) with Eqs. (2) and (5) we arrive at the formula Sc = µα S ρ ≈ µ . × − (cid:104) T (cid:105) ρ . (8)The details regarding variation of µ and ρ with (cid:104) T (cid:105) and (cid:104) S (cid:105) are given in Appendices III and IV,respectively. Thus, on combining Eq. (8) with Eqs. (31)-(36) we have directly related Sc with (cid:104) T (cid:105) and (cid:104) S (cid:105) .Figure 1 uses the results of this section by presenting the density plots of the Schmidt andthe Prandtl numbers varying with the average temperature and salinity concentration. The state ((cid:104) T (cid:105) , (cid:104) S (cid:105)) = ( ◦ C , . ) corresponding to Pr = .
16 and Sc = . Pr ≈ Sc ≈ Pr and Sc decrease with increasing (cid:104) T (cid:105) , and slowly increase with increasing (cid:104) S (cid:105) being inagreement with that of Ref. [11, 12].
3. Calculation of temperature/salinity dependent eddy diffusivity ratio
The aim of this section is to obtain the expression for the eddy diffusivity ratio d r varying withwater’s average temperature and average salinity [16]. This quantity is defined as [8, 11] d r = K S / K T ≈ R ρ + R . ρ ( R ρ − ) . , R ρ ≥ , . R ρ − . , . ≤ R ρ < , . R ρ , R ρ < . , (9)where K T and K S are the eddy diffusivity of temperature and salinity, respectively. Further, R ρ isa dimensionless quantity known as the density ratio, R ρ = α | H | β , (10)with H being the temperature-salinity gradient ratio ( d (cid:104) T (cid:105) / dz ) /( d (cid:104) S (cid:105) / dz ) , α and β are thethermal expansion coefficient and the saline contraction coefficient, respectively, which can be
15 300132639 < S > ( pp t ) < T > ( o C) 0.0001.7533.5075.260d r (a) < S > ( pp t ) < T > ( o C) 0.00016.1332.2748.40d r (b) < S > ( pp t ) < T > ( o C) 0.00064.00128.0192.0d r (c) Fig. 2. d r varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) at different values of H . (a) H = − ◦ C · ppt − ;(b) H = − ◦ C · ppt − ; (c) H = − ◦ C · ppt − . calculated from expressions α ( T , S ) = V ∂ V ∂ T (cid:12)(cid:12)(cid:12)(cid:12) S and β ( T , S ) = V ∂ V ∂ S (cid:12)(cid:12)(cid:12)(cid:12) T , (11)under the assumption of atmospheric pressure. The (specific) volume V has been described by a75-term polynomial expression in the new version of TEOS-10 standard. Related formulae andcalculations have been reported in [17], and developed in the TEOS-10 toolbox [18]. Based onEq. (9)-(10), and using the α ((cid:104) T (cid:105) , (cid:104) S (cid:105)) and β ((cid:104) T (cid:105) , (cid:104) S (cid:105)) in TEOS-10 toolbox, one can calculatethe eddy diffusivity ratio d r ((cid:104) T (cid:105) , (cid:104) S (cid:105) , H ) . Based on all the results of this section, Fig. 2 presents d r changing with (cid:104) T (cid:105) and (cid:104) S (cid:105) , forseveral fixed values of temperature-salinity gradient ratio H .
4. Calculation of the linear coefficients of temperature and salinity
In this section we derive, on the basis of the water refractive-index polynomial of Quan andFry [13], the expressions for linear coefficients to the power spectrum characterizing contributionsfrom temperature and salinity fluctuations which depend on average temperature (cid:104) T (cid:105) [ ◦ C ] , averagesalinity (cid:104) S (cid:105) [ ppt ] and light wavelength λ [ nm ] . The polynomial expression obtained in [13] for therefractive index n ( T , S , λ ) agrees well with Sager’s data [19] being less than 5 × −
1. This allowsus to assume the validity ranges for the average temperature and average salinity concentration as:0 ◦ C ≤ T ≤ ◦ C and 0 ppt ≤ S ≤
40 ppt, respectively. Besides, by comparing their model withAustin and Halikas’ experimental data [20], Quan and Fry have shown its occuracy in interval400 nm ≤ λ ≤
700 nm. The empirically fitted polynomial of [13] has form: n ( T , S , λ ) = a + ( a + a T + a T ) S + a T + a + a S + a T λ + a λ + a λ , (12)where constants a i , i = { , ..., } have the following values a = . , a = . × − , a = − . × − , a = . × − , a = − . × − , a = . , a = . , a = − . , a = − , a = . × . (13)This model was shown to be consistent with all the data taken up to then and to be a generalizationor correction for previously introduced models (see [13] and references wherein).Let us first represent the refractive index n as a sum of its average value n and relativefluctuation n (cid:48) : n = n ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) + n (cid:48) , (14)
10 20 30010203040 < S > ( pp t ) < T > ( o C) -13.30-10.16-7.03-3.89-0.75A (10 -5 deg -1 = 34.9ppt A = -10.31 10 -5 deg -1 (a) < S > ( pp t ) < T > ( o C) 1.831.871.911.952.00B (10 -4 g -1 = 34.9ppt B = 1.85 10 -4 g -1 (b) Fig. 3. The coefficients A and B varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) at λ = nm . where the latter portion can be approximately linearized as n (cid:48) ≈ A ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) T (cid:48) + B ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) S (cid:48) , (15)with T (cid:48) and S (cid:48) being the fluctuating components of the temperature and salinity concentrationdistributions, respectively, while A and B being the linear coefficients. Unlike in the previousoceanic refractive-index spectrum models, essentially all based on approach taken in [6], here A and B are not constants but functions of the water’s average temperature, average salinity andand the wavelength of light. In order to determine such functional dependence for A and B , weexamine the first-order Taylor approximation dn ( T , S , λ ) = ∂ n ( T , S , λ ) ∂ T dT + ∂ n ( T , S , λ ) ∂ S dS + ∂ n ( T , S , λ ) ∂λ d λ. (16)Setting d λ = A ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) = ∂ n ( T , S , λ ) ∂ T (cid:12)(cid:12)(cid:12)(cid:12) T = (cid:104) T (cid:105) , S = (cid:104) S (cid:105) = a (cid:104) S (cid:105) + a (cid:104) T (cid:105)(cid:104) S (cid:105) + a (cid:104) T (cid:105) + a λ , (17)and B ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) = ∂ n ( T , S , λ ) ∂ S (cid:12)(cid:12)(cid:12)(cid:12) T = (cid:104) T (cid:105) , S = (cid:104) S (cid:105) = a + a (cid:104) T (cid:105) + a (cid:104) T (cid:105) + a λ . (18)It can be deduced from Eqs. (17) and (18) that at fixed values (cid:104) T (cid:105) = ◦ C, (cid:104) S (cid:105) = . λ = A = − . × − deg − B = . × − g −
1. These values somewhat differ from the Nikishovs’ result developed with thehelp of formulas in Ref. [7]: A = − . × − deg − B = . × − g − A ≈ − − deg − B ≈ × − g − Equation (15) together with Eqs. (17) and (18) constitute the main result of this section. Theyprovide approximations to the linear coefficients of temperature and salinity contributions to thenatural water’s refractive-index fluctuations varying with (cid:104) T (cid:105) , (cid:104) S (cid:105) and λ . More details regarding of A and B varying with (cid:104) T (cid:105) , (cid:104) S (cid:105) and λ are given in Figs. 3 and 4.Figure 3 presents variation of A and B as functions of (cid:104) T (cid:105) and (cid:104) S (cid:105) when λ =
532 nm. Coefficient A decreases with increasing (cid:104) T (cid:105) and (cid:104) S (cid:105) but B only decreases with increasing (cid:104) T (cid:105) but does notvary with (cid:104) S (cid:105) . Such invariance can also be directly established from Eq.(18). Figure 4 shows thewavelength dependence of A and B at (cid:104) T (cid:105) = ◦ C and (cid:104) S (cid:105) = .
00 500 600 700 800-9.1-8.8-8.5 l (nm) 1.801.871.94B (10 -4 g -1 -5 deg -1
1) A B
Fig. 4. A and B varying with λ when (cid:104) T (cid:105) = ◦ C , (cid:104) S (cid:105) = . -3.0 -1.5 0.0 1.5 3.0-2.70.02.7 -1.6 -1.51.541.65 n ’ ( - l ) T’ (deg) (a) !" -3.0 -1.5 0.0 1.5 3.0-505
S ’ (ppt) n ’ ( - l ) (b) Fig. 5. (a) n (cid:48) varying with T (cid:48) , and (b) n (cid:48) varying with S (cid:48) in three models:Model 1: Nikishovs’ model;Model 2: Our linear model in Eq. (15) with Eqs. (16)-(17);Model 3: Quan and Fry’s formula containing full polynomial. Figure 5 compares linear approximation given in Eq. (15) with the Quan and Fry’s formula(see Eq. (12) of Ref. [13]) as well as with the Nikishovs’ model, by plotting n (cid:48) varying with T (cid:48) and S (cid:48) when (cid:104) T (cid:105) = ◦ C and (cid:104) S (cid:105) = .
5. Power spectrum for any average temperature and salinity
In this section we will incorporate the expressions obtained in the previous sections for thePrandtl/Schmidt numbers, the eddy diffusivity ratio and the linear coefficients A and B into thespatial power spectrum of the water refractive-index fluctuations. According to Ref. [6] the powerspectrum can be expressed as Φ n ( κ, (cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) = A ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) Φ T ( κ ) + B ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) Φ S ( κ ) + A ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) B ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) Φ TS ( κ ) , (19)where Φ T ( κ ) , Φ S ( κ ) , and Φ TS ( κ ) are the temperature spectrum, the salinity spectrum, and theco-spectrum, respectively, and A and B are the linear coefficients varying with < T > , < S > and λ obtained in Eqs. (17) and (18) [22].For each of these three spectra, we will apply the analytic fit [11] Φ i ( κ, (cid:104) T (cid:105) , (cid:104) S (cid:105)) = (cid:104) + . ( κη ) . c i . − . ( κη ) . c i . (cid:105) × π βε − κ − χ i exp (cid:104) − . ( κη ) c i . (cid:105) , i ∈ { T , S , TS } , (20)where the ensemble-averaged variance dissipation rates χ i ( i ∈ { T , S , T S }) are defined by [6, 8] χ T = K T (cid:18) d (cid:104) T (cid:105) dz (cid:19) , χ S = K S (cid:18) d (cid:104) S (cid:105) dz (cid:19) , χ T S = K T + K S (cid:18) d (cid:104) T (cid:105) dz (cid:19) (cid:18) d (cid:104) S (cid:105) dz (cid:19) , (21)
50 100 1500.00.51.0 s I , p l L (m) (a) s I , p l L (m)
0 C 10 C 20 C 30 C (b) s I , p l L (m) (c)
450 nm500 nm550 nm600 nm
Fig. 6. The scintillation index σ I , pl ( L ) with different values of (a) average salinity, (b)average temperature, and (c) wavelength. the Kolmogorov microscale η [ m − ] is η = ν / ε − / = (cid:20) µ ((cid:104) T (cid:105) , (cid:104) S (cid:105)) ρ ((cid:104) T (cid:105) , (cid:104) S (cid:105)) (cid:21) / ε − / , (22)where ε is the energy dissipation rate [ m / s ] . In Eq. (20) the non-dimensional parameters c i ( i ∈ { T , S , T S }) are c T = . / β Pr − ((cid:104) T (cid:105) , (cid:104) S (cid:105)) , c S = . / β Sc − ((cid:104) T (cid:105) , (cid:104) S (cid:105)) , c T S = . / β Pr ((cid:104) T (cid:105) , (cid:104) S (cid:105)) + Sc ((cid:104) T (cid:105) , (cid:104) S (cid:105)) Pr ((cid:104) T (cid:105) , (cid:104) S (cid:105)) Sc ((cid:104) T (cid:105) , (cid:104) S (cid:105)) , (23) c T S is based on the coupling between Pr and Sc [9, 10]; K T and K S , as before, are the eddydiffusivity of temperature and salinity, respectively. Combining Eq. (21) with Eqs. (9)-(11), weget χ S ((cid:104) T (cid:105) , (cid:104) S (cid:105) , H , χ T ) = d r ((cid:104) T (cid:105) , (cid:104) S (cid:105) , H ) H χ T ,χ T S ((cid:104) T (cid:105) , (cid:104) S (cid:105) , H , χ T ) = + d r ((cid:104) T (cid:105) , (cid:104) S (cid:105) , H ) H χ T . (24)where, as shown in Section 3, H is the temperature-salinity gradient ratio, and d r can be directlycalculated from (cid:104) T (cid:105) , (cid:104) S (cid:105) and H . The power spectrum model given by Eqs. (19) - (24) is the main result of our study. Incombination with the results of Sections 2, 3 and 4, it gives the 2nd-order analytic descriptionof the natural water optical turbulence with the wide-range average temperatures and salinityconcentrations occurring in the Earth’s oceans, seas, bays, rivers and lakes, at any geographicregion, under a variety of meteorological conditions, and for all visible wavelengths .
6. Light scintillation in natural waters
In this section we will explore the effects of the average temperature, the average salinityconcentration and the wavelength entering the developed power spectrum model Φ n ( κ, (cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) on the scintillation index of a plane wave, also known as the Rytov variance [23]. It is given byexpression σ I , pl ( L ) = π k Ln ∫ d ξ ∫ ∞ κ Φ n ( κ, (cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) (cid:20) − cos (cid:18) L κ ξ k (cid:19)(cid:21) d κ = π k Ln ∫ ∞ κ Φ n ( κ, (cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) (cid:34) − sin (cid:0) L κ / k (cid:1) L κ / k (cid:35) d κ, (25)
10 20 30 400.07.515.022.530.0 < T > ( o C ) < S > (ppt) 1.812.062.312.562.81 log ( L d ) Fig. 7. log ( L d ) varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) where L d has the unit [m]. where n is the average value equal to n ((cid:104) T (cid:105) , (cid:104) S (cid:105) , λ ) , k = π / λ is the wavenumber, and L is thepropagation distance. This quantity is one of the most crucial observables of optical turbulenceand is frequently used for separation of weak and strong turbulence regimes [23]. It was used inRef. [24] for the very first analysis of optical scintillation underwater.Figure 6 shows scintillation index σ I , pl ( L ) for several fixed values of (cid:104) S (cid:105) , (cid:104) T (cid:105) and λ and thefollowing fixed values of parameters: ε = − m s − , H = − ◦ C · ppt − and χ T = − K s − .We also set (cid:104) T (cid:105) = ◦ C and (cid:104) S (cid:105) = . (cid:104) T (cid:105) = ◦ C and λ =
532 nm in Fig.6(b); (cid:104) S (cid:105) = . λ =
532 nm in Fig. 6(c). It is shown that larger (cid:104) S (cid:105) and/or (cid:104) T (cid:105) lead tostronger effects of turbulence on the plane wave and result in a larger scintillation [25, 29]. Wealso conclude that the shorter the wavelength of light the stronger the scintillations are. In addition, on solving equation σ I , pl ( L d ) = , (26)we can find the threshold distance L d between weak ( L (cid:28) L d ) and strong ( L (cid:29) L d ) turbulenceregimes. The density plot of log [ L d ((cid:104) T (cid:105) , (cid:104) S (cid:105))] is illustrated in Fig. 7 with λ = ε = − m s − , H = − ◦ C · ppt − and χ T = − K s − . It is evident that larger values of L d correspond to lower (cid:104) T (cid:105) and (cid:104) S (cid:105) which is in agreement with Fig. 6(a)-(b).
7. Summary and conclusion
We have derived the expressions for the Prandtl/Schmidt numbers, the eddy diffusivity ratiovarying with average temperature (cid:104) T (cid:105) and average salinity (cid:104) S (cid:105) , as well as the coefficients A and B of the linear approximation of temperature and salinity contributions to the natural water powerspectrum as functions of (cid:104) T (cid:105) , (cid:104) S (cid:105) and wavelength λ . We have found the following:• Larger values of (cid:104) T (cid:105) or/and smaller values of (cid:104) S (cid:105) lead to smaller values of the Prandtl/Schmidtnumber;• Eddy diffusivity ratio d r increases with increasing (cid:104) T (cid:105) and/or (cid:104) S (cid:105) ;• Linear coefficient A decreases with increasing (cid:104) T (cid:105) and/or (cid:104) S (cid:105) while linear coefficient B decreases with increasing (cid:104) T (cid:105) but does not vary with (cid:104) S (cid:105) ;• A larger wavelength λ leads to a larger A and a smaller B .Using these results, we have obtained a model of the oceanic optical turbulence spectrumvarying with environmental parameters, and have used this model to calculate the scintillationindex of a plane wave. Based on the generic numerical calculations we concluded that a larger (cid:104) T (cid:105) , a larger (cid:104) S (cid:105) and a smaller λ would lead to stronger scintillations.The proposed power spectrum can be used in numerical calculations relating to light propaga-tion in natural turbulent waters with practically any average temperature and average salinityconcentration present in the Earth’s boundary layer and with any visible wavelength. ppendix: Related thermodynamic parameters varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) I. The specific heat varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) According to [30] and [31], at atmospheric pressure, the specific heat c p (cid:2) J · kg − · K − (cid:3) is c p = × ( a + a (cid:104) T (cid:105) + a (cid:104) T (cid:105) + a (cid:104) T (cid:105) ) , (27)where a = . − . × − (cid:104) S (cid:105) + . × − (cid:104) S (cid:105) , a = − . × − + . × − (cid:104) S (cid:105) − . × − (cid:104) S (cid:105) , a = . × − − . × − (cid:104) S (cid:105) + . × − (cid:104) S (cid:105) , a = . × − + . × − (cid:104) S (cid:105) − . × − (cid:104) S (cid:105) . (28) II. The thermal conductivity varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) Thermal conductivity σ T (cid:2) W · m − K − (cid:3) can be calculated using results of Ref. [32]:log ( σ T ) = . × (cid:18) . − . + . (cid:104) S h (cid:105)(cid:104) T h (cid:105) + . (cid:19) (cid:20) − (cid:104) T h (cid:105) + . . + . (cid:104) S h (cid:105) (cid:21) / + log ( + . (cid:104) S h (cid:105)) − , (29)where (cid:104) T h (cid:105) = . (cid:104) T (cid:105) , (cid:104) S h (cid:105) = (cid:104) S (cid:105)/ . . (30) III. The dynamic viscosity varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) At atmospheric pressure the dynamic viscosity µ (cid:2) N · s · m (cid:3) can be finely evaluated using amodel of Ref. [33], obtained as the analytic fit of data reported in [33]- [34], where: µ = µ ( a (cid:104) s (cid:105) + a (cid:104) s (cid:105) ) , (cid:104) s (cid:105) = (cid:104) S (cid:105) × − . (31)where a = . + . × − (cid:104) T (cid:105) − . × − (cid:104) T (cid:105) , a = . − . × − (cid:104) T (cid:105) + . × − (cid:104) T (cid:105) , (32)and µ = (cid:104) . × ((cid:104) T (cid:105) + . ) − . ] − + . × − . (33) IV. The density of water varying with (cid:104) T (cid:105) and (cid:104) S (cid:105) The density of water ρ (cid:2) kg · m (cid:3) at atmospheric pressure was fitted in Ref. [33] based on data ofRefs. [35] and [36]. ρ = ρ T + ρ S (34)where temperature-only contribution ρ T and its adjustment by salinity ρ T S are approximated bypolynomials: ρ T = . × + . × − (cid:104) T (cid:105) − . × − (cid:104) T (cid:105) + . × − (cid:104) T (cid:105) − . × − (cid:104) T (cid:105) , (35) T S = (cid:104) s (cid:105)[ . × − . (cid:104) T (cid:105) + . × − (cid:104) T (cid:105) − . × − (cid:104) T (cid:105) − . × − (cid:104) T (cid:105) (cid:104) s (cid:105)] , (36)where, as before, (cid:104) s (cid:105) = − × (cid:104) S (cid:105) . Acknowledgement
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. S. A. Thorpe, The Turbulent Ocean, (Cambridge: Cambridge University Press, 2007).3. O. Korotkova,
Random Beams: Theory and Applications, (CRC Press, 2013).4. R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. , 541âĂŞ562 (1978).5. R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. , 1067–1072 (1978).6. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. FluidMech. Res. , 82–98 (2000).7. B. Ruddick and T. Shirtcliffe, “Data for double diffusers: Physical properties of aqueous salt-sugar solutions,” Deep.Sea Res. Part A. Oceanogr. Res. Pap. , 775 – 787 (1979).8. 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).9. J. Yao, Y. Zhang, R. Wang, Y. Wang, and X. Wang, “Practical approximation of the oceanic refractive index spectrum,”Opt. Express , 23283–23292 (2017).10. 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).11. J. Yao, H. Zhang, R. Wang, J. Cai, Y. Zhang, and O. Korotkova, “Wide-range prandtl/schmidt number power spectrumof optical turbulence and its application to oceanic light propagation,” Opt. Express , 27807–27819 (2019).12. O. Korotkova and J.-R. Yao, “Bi-static lidar systems operating in the presence of oceanic turbulence,” Opt. Commun. , 125119 (2020).13. X. Quan and E. S. Fry, “Empirical equation for the index of refraction of seawater,” Appl. Opt. , 3477–3480 (1995).14. R. M. Pope and E. S. Fry, “Absorption spectrum (380âĂŞ700 nm) of pure water. II. Integrating cavity measurements,”Appl. Opt. , 8710–8723 (1997).15. A. Poisson and A. Papaud, “Diffusion coefficients of major ions in seawater,” Mar. Chem. , 265 – 280 (1983).16. As described in [8], eddy diffusivity ratio d r is not equal to 1, and it is a piecewise function of density ratio R ρ whichvaries with thermal expansion coefficient α and saline contraction coefficient β . Here we use TEOS-10 toolbox tocalculate α and β varying with < T > and < S > . In combining with d r ( R ρ ) , we will get details about d r varyingwith < T > and < S > .17. T. McDougall, R. Feistel, F. Millero, D. Jackett, D. Wright, B. King, G. Marion, C. Chen, P. Spitzer, and S. Seitz,“The international thermodynamic equation of seawater 2010 (teos-10): Calculation and use of thermodynamicproperties,” Glob. Ship-based Repeat Hydrogr. Manual, IOCCP Rep. No (2009).18. T. J. McDougall and P. M. Barker, “Getting started with teos-10 and the gibbs seawater (gsw) oceanographic toolbox,”SCOR/IAPSO WG , 1–28 (2011).19. G. Sager, “Zur refraktion von licht im meerwasser,” Beitr. Meeresk. , 63–72 (1974).20. R. W. Austin and G. Halikas, The index of refraction of seawater (UC San Diego: Library âĂŞ Scripps DigitalCollection, 1976).21. C. D. Mobley,
Light and Water: Radiative Transfer in Natural Waters (Academic press, 1994).22. The change of refractive index with temperature and salinity is nonlinear, which means the linear coefficients oftemperature and salinity should vary with < T > and < S > , and as we know, refractive index changes withwavelength. For above reasons, we consider the linear coefficients A and B environment-dependent. We have derivedthis in section 4. This result is a key point that different from traditional models like [9] and [11].23. L. C. Andrews and R. L. Phillips, Laser Beam Propagation in Random Media (SPIE Press, 2005).24. O. Korotkova, N. Farwell and E. Shchepakina, “Light scintillation in oceanic turbulence”, Waves Random ComplexMedia , 260–266 (2012).5. Note that due to natural water absorption most of the power will be absorbed at distances much shorter than thosegiven in Fig. 6. However, in the latest communication technologies some of the commercially available underwatertransmission links do operate at the ranges up to 200m (e.g. [26–28]).26. C. Pontbriand, N. Farr, J. Ware, J. Preisig, and H. Popenoe, “Diffuse high-bandwidth optical communications,” in OCEANS 2008, (2008), pp. 1–4.27. W. Liu, Z. Xu, and L. Yang, “SIMO detection schemes for underwater optical wireless communication underturbulence,” Photon. Res. , 48–53 (2015).28. N. Saeed, A. Celik, T. Y. Al-Naffouri, and M.-S. Alouini, “Underwater optical wireless communications, networking,and localization: A survey,” Ad Hoc Networks , 101935 (2019).29. The positive correlation between the plane wave scintillation and (cid:104) T (cid:105) is a new result different from previousreports [11, 12]. The difference comes from our extended consideration regarding A , B and d r varying with < T > and < S > .30. D. Jamieson, J. Tudhope, R. Morris, and G. Cartwright, “Physical properties of sea water solutions: heat capacity,”Desalination , 23 – 30 (1969).31. K. G. Nayar, M. H. Sharqawy, L. D. Banchik, and J. H. Lienghard, “Thermophysical properties of seawater: A reviewand new correlations that include pressure dependence,” Desalination , 1 – 24 (2016).32. D. Jamieson and J. Tudhope, “Physical properties of sea water solutions: thermal conductivity,” Desalination , 393 –401 (1970).33. M. H. Sharqawy, J. H. L. V, and S. M. Zubair, “Thermophysical properties of seawater: a review of existingcorrelations and data,” Desalination Water Treat. , 354–380 (2010).34. F. J. Millero, “Seawater as a multicomponent electrolyte solution,” The sea , 3–80 (1974).35. J. Isdale and R. Morris, “Physical properties of sea water solutions: density,” Desalination , 329 – 339 (1972).36. F. J. Millero and A. Poisson, “International one-atmosphere equation of state of seawater,” Deep. Sea Res. Part A.Oceanogr. Res. Pap.28