Oceanic non-Kolmogorov optical turbulence and spherical wave propagation

Light propagation in turbulent media is conventionally studied with the help of the spatio-temporal power spectra of the refractive index fluctuations. In particular, for natural water turbulence several models for the spatial power spectra have been developed based on the classic, Kolmogorov postulates. However, as currently widely accepted, non-Kolmogorov turbulent regime is also common in the stratified flow fields, as suggested by recent developments in atmospheric optics. Until now all the models developed for the non-Kolmogorov optical turbulence were pertinent to atmospheric research and, hence, involved only one advected scalar, e.g., temperature. We generalize the oceanic spatial power spectrum, based on two advected scalars, temperature and salinity concentration, to the non-Kolmogorov turbulence regime, with the help of the so-called"Upper-Bound Limitation"and by adopting the concept of spectral correlation of two advected scalars. The proposed power spectrum can handle general non-Kolmogorov, anisotropic turbulence but reduces to Kolmogorov, isotropic case if the power law 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 mutual coherence function of a spherical wave and examine its coherence radius (in both scalar and vector forms) to characterize the turbulent disturbance. Our numerical calculations show that the statistics of the spherical wave vary substantially with temperature and salinity non-Kolmogorov power law exponents and temperature-salinity spectral correlation coefficient. The introduced spectrum is envisioned to become of significance for theoretical analysis and experimental measurements of non-classic natural water double-diffusion turbulent regimes.


Non-Kolmogorov temperature/salinity spectra
We begin by recalling the H4-based temperature/salinity spectrum that has been develped for Kolmogorov case in [21]. By comparing its structure function with the Kolmogorov structure function, we will first obtain its structure constant C 2 i and its inner scale l i0 . Then, the H4-based spectrum 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 where κ is the wavenumber m −1 ; C 2 i is the structure constant (dimensionless); C k is a dimensionless constant given by β is the Obukhov-Corrsin constant (non-dimensional); ε is the dissipation rate of kinetic energy [m 2 s −3 ]; χ i is the ensemble-averaged variance dissipation rate of temperature or salinity (i ∈ {T, S}) with unit K 2 s −1 or g 2 s −1 ; the non-dimensional function g i (x) is with a j = 1, 21.61c i 0.02 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 approximating to 0.72 generally [18].
B. Structure constant C 2 i and inner scale l i0 Structure constant C 2 i and inner scale l i0 are the key parameters in the turbulence structure function, and they will be obtained by comparing the corresponding structure function in the Kolmogorov case.

C. Non-Kolmogorov case
Now, we modify Eq. (1) to a non-Kolmogorov spectrum. Following the modification in atmospheric optics [31,32,43], we add two adaptive functions A (α i ) and h (α i , c i ) to Eq. (1), where A (α i ) is a variable factor similar to the 'A (α)' in [31], h (α i , c i ) is a scaling function similar 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.
To show the consistency between the proposed non-Kolmogorov spectrum Eq. (13) and the non-Kolmogorov structure function Eq. (16), we plot the following two functions in Fig. 1, It shows F 1 (R → 0) = 1 and F 2 (R → ∞) = 1, which indicates that the modified non-Kolmogorov spectrum 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 the proposed spectrum agrees well with the widely accepted asymptotic structure function. It must be noticed that parameter c i is related to Prandtl/Schmidt number in Kolmogorov case but this definite 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 direct parameter, and set its range in Appendix I.

Non-Kolmogorov temperature-salinity co-spectrum
In the Kolmogorov case the models of temperature-salinity co-spectrum have been obtained by analogy with the single scalar (temperature/salinity) spectrum [2,16,19,21] but such an analogy is unavailable if the exponents of temperature and salinity spectra are different. Hence, for the non-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-Bound Limitation [44][45][46] and the concept of spectral correlation [47].
As proven in Section 5.2.5 of [44], the Upper-Bound Limitation gives the relation between scalar spectra φ T , φ S and their co-spectrum φ T S : [48] extended the Upper-Bound Limitation to three-dimensional case: By adopting the concept of spectral correlation [47,49], and combining Eq.(13) with Eq. (22), we obtain the temperature-salinity co-spectrum as with where γ ST (κη) is a correlation factor describing the degree of correlation between temperature spectrum and salinity spectrum, and 0 ≤ γ ST (κη) ≤ 1. When γ ST = 1, Eq. (23) refers to a fully 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 partially correlated co-spectrum are given as follows.
According to the concept of spectral correlation [47,48], temperature fluctuation T and salinity fluctuation S are highly correlated if they are both driven by eddy diffusion, but the correlation will be broken down if T is driven by temperature molecular diffusion. Hence, the following should hold [50]: • When κ belongs to the inertial-convective range of Φ T (i.e. g T ∝ κ 0 ), the salinity spectrum is generally in its inertial-convective range [51]. Thus, both T and S are governed by eddy diffusion, and they have a high correlation, i.e. γ ST = γ max ≤ 1.
• When κ belongs to the viscous-convective range of Φ T (i.e. g T ∝ κ 2/3 ), T is consumed by viscosity but S is still governed by eddy diffusion. The correlation begins to decrease in this range, and it has been observed in [52] that correlation decreases monotonically. Hence, we have dγ ST /dκ ≤ 0.
• When κ belongs to the viscous-diffusive range of Φ T (i.e. g T decreases fast with κ), T is primarily depleted by temperature molecular diffusion, which leads to a very low correlation between T and S , i.e. γ ST ≈ 0.
Thus the values of correlation parameter γ ST (κη) obey the following constraints: where κ 1 defines the transition between inertial-convective and viscous-convective ranges of Φ T , κ 2 defines the transition between viscous-convective and viscous-diffusive ranges of Φ T . According to [18], we have following defining relations for κ 1 and κ 2 in H4-based non-Kolmogorov model: and 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 has been given in Eq. (6). The locations of κ 1 η and κ 2 η are marked by '|' and '|' in Fig.2, respectively, and '-' refers to g T . It shows that κ 1 η and κ 2 η mark the transitions between different ranges very well. For mathematical simplicity of discussion, we suppose that the correlation factor in fully correlated case is and in partially correlated case it takes form 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 ourselves to Kolmogorov case (α S = α T = 11/3). Fig. 3(a) shows non-dimensional function q(κη) = (C 2 T C 2 S ) −1/2 κ 11/3 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 = 4 and p = 2, respectively. The Comparing (a) the non-dimensional function q(κη) and (b) the correlation factor γ ST (κη) corresponding to the proposed co-spectrum with these corresponding to conventional co-spectrum [21]. Values of parameters are listed in Appendix II.

Partially correlated
Fully correlated co-spectrum with different values of α T and α S . vertical lines '|' and '|' mark the locations of κ 1 η and κ 2 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 conventional co-spectrum [21], the proposed partially correlated co-spectrum has a higher correlation in the temperature inertial-convective range (κ κ 1 ), a lower correlation in the temperature viscousconvective range (κ 1 κ κ 2 ), and also a low correlation in the temperature viscous-diffusive range (κ κ 2 ). Furthermore, Fig.3 (a) indicates that the proposed fully correlated co-spectrum tends to the conventional co-spectrum when α T = α S = 11/3.
To examine the co-spectrum in non-Kolmogorov case, and to verify its de-correlation within temperature viscous-convective range, we plot log of non-dimensional function Fig.4, and compare the fully correlated co-spectrum ('-') with the partially correlated co-spectrum ('---') at p = 3. Same as before, κ 1 η and κ 2 η are marked by '|' and '|', respectively. It is shown that the proposed co-spectrum has low correlation in the temperature inertial-convective range, as expected. This agrees with Eq. (27).
Thus we have obtained a temperature-salinity co-spectrum with a non-Kolmogorov power law (α T + α S )/2 and a flexible correlation factor γ ST [see Eq. (23)]. If γ ST = 1, the proposed co-spectrum is fully correlated, and it approximately reduces to the conventional co-spectrum when α T = α S = 11/3; 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 new co-spectrum model has a power law between α T and α S ; if the temperature and salinity fields are both Kolmogorov (α T = α S = 11/3), the co-spectrum is also Kolmogorov (α T S = 11/3).

OTOPS with anisotropy and non-Kolmogorov power law
In general, the oceanic refractive-index fluctuation n is approximately given by a linear combination of temperature fluctuation T and salinity fluctuation S [2,22,54]: with This implies that the spectrum of n is approximately given by linear combination where Φ T is the temperature spectrum, Φ S is the salinity spectrum, and Φ T S is the temperaturesalinity co-spectrum. On combining Eqs. (13) and (23), we obtain the following expression for the OTOPS: with To make the developed OTOPS more physical we now implement the finite outer-scale cut-off and extend it to the anisotropic case. To obtain the first extension we use the filter function with exponential form [20,55]: where κ 0 is the outer-scale cut-off wavenumber defined by κ 0 ≈ 4π/L 0 with L 0 (m) representing the outer scale. Further the anisotropic OTOPS can be obtained on following [56] as: where µ is the anisotropy parameter, κ is the three-dimensional wavenumber, and κ iso is a isotropisizing transformation of κ: T is denoting vector transpose. Thus in this section, a non-Kolmogorov OTOPS (NK-OTOPS) is given in Eq. (36), while its 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 the spherical wave statistics for the extended HuygensâĂŞFresnel principle, we will calculate and analyze the 2nd-order statistics of a spherical wave. In particular, in Section 4.1, the wave structure function (WSF) of a spherical wave in oceanic turbulence will be derived; in Section 4.2, the vector and scalar versions of the coherence radius will be defined and examined; and in Section 4.3, the co-effect of temperature and salinity on spherical wave's propagation will be discussed by calculating the scalar coherence radius varying with α T , α S , c T and c S .

2nd-order statistical moments and wave structure function of spherical wave
A. 2nd-order statistical moments We will first derive the 2nd-order statistical moment of a spherical wave propagating in the non-Kolmogorov oceanic optical turbulence. According to Eq. (59) of chapter 5 in [42], for horizontal channels (along y-axis) this quantity has form: where L is the propagation distance from the source plane, k is the wavenumber defined as 2πn 0 /λ, n 0 being the average refractive-index, Φ n2 is the anisotropic NK-OTOPS as given by Eq. (39). For a spherical wave, γ = γ * = 1. On assuming that and combining Eq. (39) with Eq. (41), we get where Φ n1 is the outer-scaled NK-OTOPS in Eq. (38). On setting ρ = r 1 − r 2 , we find that the 2nd-order statistical moment of a spherical wave along a horizontal channel (along the y-axis) becomes where Similarly, the 2nd-order statistical moment of a spherical wave in a vertical channel (along the z-axis) becomes with B. Wave structure function of spherical wave Next, based on the 2nd-order statistical moments given above, we derive the WSF of a spherical wave in the non-Kolmogorov oceanic optical turbulence. According to the expressions in chapter 6 of [42] the WSF of a spherical wave has form: where E 2 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 Similarly, the WSF of a spherical wave in a vertical channel (ρ = (ρ x , ρ y )) becomes When µ = 1, the WSFs in horizontal and vertical channels are equal and, hence,

Equations (50) -(52) are the main results of this section. They characterize the WSF of a spherical wave in an anisotropic, non-Kolmogorov turbulence by means of single integrals.
We first plot the numerical results of the WSFs in an isotropic turbulence, with different values of the 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 that in the Kolmogorov case. Figure 6 shows that anisotropic turbulence leads to an anisotropic WSF which results in an elliptically shaped coherence radius, which we will further illustrate in the next section.

Coherence radius of a spherical wave
The coherence radius of a spherical wave can be directly employed for assessing the optical turbulence 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 spherical wave that corresponds to the WSF's value of 2. As a rule, the coherence radius is considered 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 0 and we define it as a coherence radius vector (CRV) ρ 0 by setting D sp (ρ 0 , L) = 2.
The derived coherence radius vector (CRV) and scalar (CRS) are the main results of this section, which can be evaluated using Eqs. (55)- (56). The CRS corresponds to the widely used coherence radius if µ = 1 or along a vertical channel, and it could measure the anisotropic turbulence strength along different directions. In fact, the atmospheric turbulence anisotropy has been recently directly assessed through a measurement of the elliptically shaped mutual coherence function of a laser beam [58] (see also a similar measurement via the elliptically shaped intensity-intensity correlation function [59]).

Co-effect of temperature and salinity on coherence radius scalar
In this section we will give a numerical example on the co-effect of temperature and salinity of the NK-OTOPS on the CRS by calculating it as a function of the power laws of temperature and salinity 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 = 3 [60], and choose the CRS ρ 0_iso in vertical channels as a measurement of turbulent disturbance. The ranges of related parameters are listed as follows (see more details in Appendix. I): · α T , α S ∈ [11/3, 15/3); · c T ∈ 1.61 × 10 −3 , 3.99 × 10 −3 and c S ∈ 9.76 × 10 −6 , 61.62 × 10 −6 ; Figure 8 shows ρ 0_iso (α T , α S ) and ρ 0_iso (c T , c S ) for different spectral correlation of the power spectrum (as above, fully correlated case refers to γ ST = 1, partially correlated case refers to the γ ST obeying Eq. (31), and uncorrelated case refers to γ ST = 0, i.e. Φ T S = 0.). Figure 8 (d) shows a distribution of ρ 0_iso (α T , α S ) being very different from that in Figs. 8 (a)-(c), and Fig. 8 (e) shows a distribution of ρ 0_iso (c T , c S ) being very different from that in Figs. 8 (f)-(h). Figure 9 shows ρ 0_iso (α T , α S ) and ρ 0_iso (c T , c S ) with different ratios of C 2 S to C 2 T . With the increase of C 2 S /C 2 T , the variation of ρ 0_iso with α S and c S becomes more pronounced. A comprehensive analysis of Figs. 8 and 9 reveals that • ρ 0_iso 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 0_iso .
• As expected, the structure constant C 2 T or/and C 2 S describes the contribution of temperature or/and salinity fluctuation very well.

Summary and conclusion
The power spectrum of refractive-index fluctuations provides a rigorous physical description of the 2nd-order statistics of natural random media, hence, bearing utmost significance for environmental optics. A number of non-Kolmogorov models have been recently developed for 'single-diffuser' turbulence, i.e., based on a single advected scalar, as is temperature in atmsopheric case. However, to our knowledge, there was no model for non-Kolmogorov spectrum describing 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-spectrum of two scalar spectra in the non-Kolmogorov case could not be directly obtained by analogy with a 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 of oceanic refractive-index fluctuations, being an example of a double-diffuser, by deriving the temperature spectrum, the salinity spectrum, and their co-spectrum, based on the Upper-Bound limitation and on the concept of spectral correlation. Our developed spectrum generally handles non-Kolmogorov turbulence with partially correlated temperature-salinity co-spectrum (α i ∈ [11/3, 15/3) and γ ST (κ) ≤ 1) which is common for the stratified flow fields, but reduces to conventional, Kolmogorov spectrum, with fully correlated co-spectrum (α i = 11/3 and γ ST = 1). We have also provided the extension to anisotropic non-Kolmogorov turbulence case. Besides, we have also illustarted how a non-Kolmogorov, isotropic and anisotropic oceanic turbulence affects the second-order statistics of a spherical wave. The numerical calculations have revealed that the turbulence's effect on a spherical wave substantially varies with the power laws exponents (α T and α S ). Moreover, we have shown for the first time that the coherence radius scalar ρ 0_iso takes on very different values for different settings of spectral correlation. This also indicates the usefulness of developing the oceanic non-Kolmogorov power spectrum with correlation factor γ ST .
On finishing we mention that so far no literature of oceanic turbulence has provided models for the correlation factor γ ST (κ) and other parameters such as c T , c S , α T and α S . But like in the studies of atmospheric propagation, these parameters could be significant in characterizing oceanic optical turbulence, and any details about them are of importance for further experimental campaigns. Our model fills such a gap by providing a rather simple analytical model applicable in 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 here are based on references, and some of them are obtained in Kolmogorov case. The real ranges could be beyond what we set.
3. The range of C 2 S /C 2 T According to Eq. (2), where the dispassion rate χ i of temperature and salinity are related through [17,22] with where d r is the eddy diffusivity ratio, θ T and θ S are the thermal expansion coefficient and the saline contraction coefficient, respectively, and H is the temperature-salinity gradient ratio defined by  Combining Eqs.(57)-(60), we have Using the data of d T /dz, d S /dz, θ T and θ S of mid latitude Pacific in winter [62] (see also Fig. 10), and based on Eq. (61), we plot C 2 S /C 2 T as a function of depth in Fig. 11. It shows that For non-Kolmogorov cases, we assume 4. The ranges of c S and c T According to [22], Pr T varies from 5.

Appendix II. The values of parameters in Figures
Here we list the values of parameters in figures.

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 0 | in D sp (ρ 0 ) = 2 could vary with the orientation of ρ 0 . For brevity in discussion, we define ρ 0 as CRV, and define a scalar -CRS -in Eq. (55). The CRS equals to coherence radius if µ = 1 or in vertical channels.
• 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, viscousconvective and viscous-diffusive intervals. For the turbulence with small Pr or Sc, there are two intervals: inertial and diffusive intervals. Hill's models provide continuous transition between different intervals. Hill's model 1 is mathematically analytic but not as precise as Hill's model 4, and Hill's model 4 is a non-linear differential equation that does not have a closed-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 limited by 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 fluctuations and salinity fluctuations. The spectral correlation factor is defined as where Φ T and Φ S are the 3-D spectra of temperature and salinity, respectively. 'fully correlated' 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' refer to the cases of γ ST = 0.

Disclosures
The authors declare no conflicts of interest.