## Abstract

Recent simulations and experiments have shown that the viscous-range temperature spectrum in water can be well described by the Kraichnan spectral model. Motivated by this, a tractable expression is developed for the underwater temperature spectrum that is consistent with both the Obukhov-Corrsin law in the inertial range and the Kraichnan model in the viscous range. In analogy with the temperature spectrum, the formula for the salinity spectrum and the temperature-salinity co-spectrum are also derived. The linear combination of these three scalar spectra constitutes a new form of the refractivity spectrum. This spectral model predicts a much stronger optical scintillation than the previous model.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

The study of propagation of optical waves through random media such as the atmosphere, the ocean, and the biological tissue, is very important in many applications including optical communications and imaging. In random media, the index of refractive fluctuations that is often referred to as optical turbulence, introduce random perturbations into the wave front phase, amplitude and angle of arrival. This leads to the serious distortion of the wave front quality. Knowledge of the spatial power spectra of refractive-index fluctuation in turbulent media is needed for assessing the reliability of optical communication links, the optical resolution of imaged objects, and other technical parameters of optical systems [1].

The refractivity fluctuation of the atmosphere is a function of the fluctuations of temperature and humidity [2], while the fluctuation of the refractive index in turbulent ocean is controlled by temperature and salinity fluctuations [3]. The spectra of such scalar fluctuations determine the spatial power spectrum of the refractivity fluctuations. In [4], Hill proposed four models of the scalar spectrum for an arbitrary Prandtl number $\mathrm{Pr}$, defined as the ratio of the kinematic viscosity to the molecular diffusivity. Typically, there are three wavenumber ranges on the scalar spectrum: an inertial-convective range, a viscous-convective range, and a viscous-diffusive range. The ${\kappa}^{-5/3}$ inertial-convective law of Obukhov [5] and Corrsin [6], and the ${\kappa}^{-1}$ viscous-convective law of Batchelor [7] and Kraichnan [8] have been experimentally validated with high precision for different Prandtl numbers. But for the viscous-diffusive range, the Kraichnan spectral model predicts a slower decrease in the dependences of spectrum level on wavenumber with respect to Batchelor spectrum. Based on modelling the spectral transfer function by the previously developed Corrsin [9], Pao [10], Leith [11], and Kraichnan’s models [8], Hill generalized the above characteristics of the scalar spectra and presented four models [4]. Hill’s Model 1 and 2 were derived from the Corrsin’s and Pao’s analysis, which introduced the velocity $s\left(\kappa \right)$ to set up the relation between the spectral flux function and the spectrum. Pao’s dimensional analysis result for $s\left(\kappa \right)$ in the viscous range is identical with the theory by Batchelor [10]. So these two models tend to Batchelor spectrum in the viscous range. As an improved version of Model 1, Model 2 uses the hyperbolic tangent to define the derivative of $s\left(\kappa \right)$, and thus making a smoother transition between the inertial- and viscous-convective ranges of the scalar spectrum. Hill’s Models 3 and 4 were deduced from the inertial-range form of Leith’s diffusion approximation [11] and from the viscous-range form of Kraichnan’s flux function [8]. Therefore, these two models represent the identical behavior to Kraichnan spectrum in the viscous range. In analogy with Model 2, Model 4 improves Model 3 by using the hyperbolic tangent function to smooth the transition from the inertial-convective range to the viscous-convective range. Out of these four models, Model 4 has the highest accuracy as compared to the experimental data that Champagne [12], and Williams and Paulson [13] observed in the compensated temperature spectrum in the atmospheric surface layer over land. Model 1 is the unique model that has the closed-form solution. However, Models 1 and 3 do not match well the measurements results in the inertial-convective range.

In the community of optical propagation in turbulent atmosphere, Model 4 of temperature (the effect of humidity is neglected) has become the standard model for the refractive-index power spectrum. Recently, its accuracy has been further verified by direct numerical simulation (DNS) [14]. Model 4, however, consists of a second-order differential equation that can be solved only numerically, and it seems that no closed-form solution has been published in the literature. Several theoretical approximations to Model 4 of temperature fluctuations in air have been proposed so far, such as the modified atmospheric [15], Churnside [16], and Frehlich’s spectra [17]. For the case of the turbulent ocean, in [3], the statistical properties of optical waves when the refractive index is controlled by a single component (temperature or salinity) was analyzed by numerically solving the Model 4. Then in [18], Nikishov expressed the power spectrum of refractivity fluctuation in terms of temperature spectrum, salinity spectrum, and temperature-salinity co-spectrum. Each scalar spectrum was obtained by using the Hill’s Model 1. To facilitate the analytical studies of optical wave propagation in ocean, very recently in [19], a mathematic approximation to Nikishov spectrum has been proposed. It employs the Gaussian function to replace the term $\mathrm{exp}\left(A{\kappa}^{4/3}+B{\kappa}^{2}\right)$ appeared in Nikishov spectrum. Here $A$ and $B$ are the spectrum parameters. In [20], the effect of the eddy diffusivity ratio has been added to Nikishov spectrum.

Since Nikishov spectrum derives from Hill’s Model 1, which has the exact mathematic form, it has been widely used in the analysis of the statistical properties of optical waves propagating in turbulent ocean [21–23]. Turbulence-induced irradiance fluctuations, also known as optical scintillation, is one of these statistical properties. The importance of optical scintillation lies in its influence on the performance of optical communication. All of the recent studies on optical scintillation in turbulent ocean were performed on basis of Nikishov spectrum [24–34]. In [24], the longitudinal (i.e., on-axis) as well as radial (i.e., off-axis) components of the scintillation index for plane, spherical and collimated Gaussian waves were numerically evaluated. The scintillation index of plane and spherical waves in weak turbulent ocean was studied in [25] in consideration of the effects of various system and channel parameters, e.g., wavelength, link length, balance of temperature and salinity fluctuations. In [26], the on-axis scintillation index of focused Gaussian beams in weak oceanic turbulence and the associated error performance of underwater optical communication were investigated. Scintillation and error rate analysis were further performed in [27,28], and [29], respectively, for partially coherent flat-topped beam (PCFT), annular beam, and phase-locked PCFT array beam. In [30, 31], the effects of increasing the size of the received aperture on the scintillation index and the associated error rate were investigated under weak turbulence condition. The effect of the eddy diffusivity ratio was included in the analysis of weak turbulence scintillation index for plane and spherical waves [20]. In [32], an equivalent refractive-index structure constant in ocean was derived by equating the spherical wave scintillation index of the Nikishov spectrum to that of atmospheric turbulence spectrum. This result was adopted in developing the scintillation index in strong oceanic turbulence [33, 34].

Although the Nikishov spectrum has been extensively used in the analysis of optical scintillation in ocean, it should be noted that this spectrum has some limitations. As mentioned above, Nikishov spectrum get deduced from Hill’s Model 1 which exhibits the deviation from the experimental data of atmospheric temperature spectrum in the inertial-convective range. Due to the similarity of atmospheric and oceanic turbulence, it is believed that such a deviation may also occur in comparing Nikishov spectrum with the underwater data. In addition, Hill’s Model 1 reduces to Batchelor spectrum at large wavenumber. However, recent results of the DNS with $\mathrm{Pr}=7$(temperature Prandtl number of water) [35] and the experimental observations of temperature fluctuation in water [36, 37] exhibit the behavior consistent with the Kraichnan spectrum. These limitations of Nikishov spectrum motivate our work here.

In this paper, we first review all the previous published scalar spectra, further compare and discuss their accuracy in modeling the temperature and salinity fluctuations of ocean. Then, we propose a new scalar spectrum to overcome the limitations of the present spectra. The result is used to generate the new spatial power spectrum of the refractive-index fluctuation in turbulent ocean. Finally, the plane- and spherical-wave scintillations in weak oceanic turbulence were studied by using the proposed spectrum.

## 2. Previous models of scalar spectrum

In this section, we review the previous spectral models of random, statistically homogeneous and isotropic, scalar fields. We consider the shell-averaged wavenumber spectrum $E\left(\kappa \right)$ [38], which is related to 3D spectrum $\Phi \left(\kappa \right)$ by $E\left(\kappa \right)=4\pi {\kappa}^{2}\Phi \left(\kappa \right)$. Here $\kappa $ is the magnitude of the 3D wavenumber vector $\kappa $, i.e., $\kappa \text{=}\left|\kappa \right|$. For the isotropic scalar field, the spatial power spectrum $E\left(\kappa \right)$ (or $\Phi \left(\kappa \right)$) is only the function of $\kappa $.

#### 2.1 Inertial and viscous range spectral models

There may exist several possible wavenumber ranges for the scalar spectrum determined by the Prandtl number $\mathrm{Pr}$ for the scalar. As mentioned in introduction,$\mathrm{Pr}$ is defined as the ratio of the kinematic viscosity $v$ to the molecular diffusivity $D$, i.e., $\mathrm{Pr}=v/D$. The terminology for these wavenumber ranges depends first on whether the wavenumber lies in the inertial or viscous range of the energy spectrum and second on whether the wavenumber lies in the convective or diffusive range of the scalar spectrum. For $\mathrm{Pr}\gg 1$, the relevant ranges are the inertial-convective, the viscous-convective, and the viscous-diffusive ranges. On the other hand, for $\mathrm{Pr}\ll 1$ the major wavenumber ranges are the inertial-convective and inertial-diffusive ranges [4]. In oceanographic fields temperature and salinity are important scalars. The typical values for Prandtl number are $\mathrm{Pr}\simeq 7$ and 700, respectively [3]. Therefore, the relevant cases here are ones for which $\mathrm{Pr}\gg 1$.

According to the Obukhov-Corrsin similarity theory [5,6], the spatial statistics of the scalar fluctuations in a statistically homogeneous and isotropic fluid characterized by a large Reynolds number depend on a set of similarity parameters, i.e., $\epsilon $, $\chi $, $v$, $D$, where $\epsilon $ is the dissipation rate of turbulent kinetic energy per unit mass of fluid, $\chi $ is the diffusive dissipation rate of the mean-square fluctuations of the scalar and given by [38, Eq. (33)]

Two useful length scales can be constructed from this set of parameters:In the inertial-convective range the parameters $v$ and $D$ play no role in the scalar spectrum, and for this case the scalar spectrum follows the well-known ${\kappa}^{-5/3}$ Oboukhov-Corrsin law [5, 6]:

Here $\beta $ is a non-dimensional constant which is called the Oboukhov-Corrsin constant.The common descriptions of both the viscous-convective and viscous-diffusive ranges were proposed by Batchelor [7] and Kraichnan [8]. The spectral forms of these two models are given, respectively, by [37, 40]

where ${q}_{B}$ and ${q}_{K}$ are the universal constants, and the subscript $B$ or $K$ signifies that the constant is used for Batchelor spectrum or Kraichnan spectrum. If $\kappa \ll 1/{\eta}_{B}$ that corresponds to the viscous-convective range, the factor ${\kappa}^{-1}$ dominates in Eq. (8) and Eq. (9). If the wavenumber lies in the viscous-diffusive range with $\kappa \gg 1/{\eta}_{B}$, Eq. (8) predicts a Gaussian damping whereas Eq. (9) suggests an exponential decrease that is slower than Eq. (8).There are various values for ${q}_{B}$ and ${q}_{K}$ have been reported in the literature [35, 37]. But throughout this paper the fixed values of ${q}_{B}=3.9$ and ${q}_{K}=5.26$ will be adopted. These values were obtained by the DNS method [35] and are in agreement with the most recent experimental results for temperature fluctuations in water [36, 37]. As shown in [37], at large wavenumber, the temperature spectra measured in a freshwater swimming pool exhibits the behavior consistent with the Kraichnan spectral form: $\sim \mathrm{exp}\left[-{\left(6{q}_{K}\right)}^{1/2}\kappa {\eta}_{B}\right]$ with constant ${q}_{K}=5.26$, significantly deviated from the Batchelor spectral form: $\sim \mathrm{exp}\left[-{q}_{B}{\left(\kappa {\eta}_{B}\right)}^{2}\right]$ with ${q}_{B}=3.9$.

#### 2.2 Hill’s scalar spectral models

Hill developed four models of the scalar spectrum for arbitrary Prandtl numbers by solving the scalar spectral equation in the steady state [4]:

where $T\left(\kappa \right)$ is the scalar spectral transfer function for the steady-state case. Hill considered several models of $T\left(\kappa \right)$ developed by Corrsin [9], Pao [10], Leith [11], and Kraichnan [8]. To facilitate the process of solving Eq. (10), Hill expressed $T\left(\kappa \right)$ in terms of the scalar spectral flux function $F\left(\kappa \right)$Then, the scalar spectral equation Eq. (10) becomesHill’s Model 1 and 2 derived on the basis of the Corrsin-Pao model for $T\left(\kappa \right)$ [9,10]. These two model consists of writing the spectral flux function in the form

where $s\left(\kappa \right)$ is the velocity at which the spectral element is transferred over $\kappa $. Hill’s Model 3 and 4 derived from the inertial-range form of Leith’s diffusion approximation [11] and from the viscous-range form of Kraichnan’s flux function [12]. These two diffusion models are based on a spectral flux function having the formIn Table 1, we list the details of Hill’s four models [4]. The first line for each model shows its form of $s\left(\kappa \right)$ (Models 1 and 2) or ${D}_{s}\left(\kappa \right)$ (Models 3 and 4). Model 2 (or 4) achieves the smooth transition between the inertial-convective and the viscous-convective ranges by using the hyperbolic tangent to define the derivative of $s\left(\kappa \right)$ (or ${D}_{s}\left(\kappa \right)$). Model 2 (or 4) includes Model 1 (or 3) as a special case with $a=1/3$(or $b=1/3$). The second line for each model gives its form for the scalar spectral equation, which is derived upon the substitutions of the corresponding $s\left(\kappa \right)$ (or ${D}_{s}\left(\kappa \right)$) into Eq. (13) (or Eq. (14)) and Eq. (12). Models 1 and 2 are described in terms of the first-order differential equation, while Models 3 and 4 are described by the second-order differential equation. The third line for each model shows the required free parameters to be determined from experimental data. Models 1 and 3 have one free parameter; whereas Models 2 or 4 have two free parameters, the first one (${\kappa}^{\text{*}}\eta $ or ${\kappa}^{\diamond}\eta $) representing the transitional point, and the second one ($a$ or $b$) defining the width of the transition range.

Only the Model 1 has the closed-form solution under the scalar variance dissipation constraint Eq. (1) [4,18]

These four models are applicable to arbitrary Prandtl number and all have the expected ${\kappa}^{-5/3}$ variation in the inertial-convective range and the ${\kappa}^{-1}$ variation in the viscous-convective range. The differences between these models lie in the viscous-diffusive range, Model 1 and its extended form, i.e., Model 2, tend to Batchelor spectrum, while Model 3 and the improved Model 4 exhibit the same behavior as Kraichnan spectrum. Furthermore, the extended models, Models 2 and 4 have two free parameters to control the shape of the transition region between the inertial- and viscous-convective ranges. In contrast, the basic models, Models 1 and 3 only have one free parameter, which limits their validity. As compared to the experiment data of temperature fluctuations in near-ground atmosphere [12, 13], Models 1 and 3 show significant deviations in the inertial-convective range. Model 4 gives the best fit to the data [4]. Under the assumption that the temperature fluctuations dominates the refractive-index fluctuations in air, Model 4 of temperature spectrum is used to express the refractive-index power spectrum of atmospheric turbulence. However, Model 4 does not have the tractable form and can only be numerically solved. To simply the theoretical analysis of optical beams propagation through atmospheric turbulence, several approximated spectra to Model 4 with $\mathrm{Pr}\simeq 0.72$(atmospheric temperature Prandtl number) were proposed, such as the modified atmospheric spectrum [15], Churnside spectrum [16], and Frehlich spectrum [17]. The modified atmospheric spectrum has been widely used in the studies of optical wave propagation through the turbulent atmosphere. However, recent research shows that both this model and Churnside spectrum violate the dissipation condition Eq. (1) [38]. Hill’s Model 4 still reveals high prediction accuracy that is demonstrated by the high-resolution DNS for $\mathrm{Pr}\text{=}0.7$ and $\mathrm{Pr}\text{=}1.0$ [14].

#### 2.3 Comparison of scalar spectra in the ocean

Unlike atmospheric turbulence mainly controlled by temperature fluctuations (when the effect of humidity fluctuations can be ignored in certain situations), oceanic turbulence is governed by both temperature and salinity fluctuations. Furthermore, temperature in air has $\mathrm{Pr}\simeq 0.72$, while temperature and salinity in water have $\mathrm{Pr}\simeq 7$and $\mathrm{Pr}\simeq 700$, respectively. The larger Prandtl number of the scalar has, the wider the viscous-convective range and the more remarkable the bump appears in the scalar spectrum. Since Hill’s four models are applicable for arbitrary Prandtl numbers, some of them have been employed for describing the temperature and salinity fluctuations in sea water. In the early investigation [3], the spectra of temperature and salinity fluctuations was obtained by numerically solving the Model 4 with $\mathrm{Pr}\simeq 7$and $\mathrm{Pr}\simeq 700$, respectively. The widely used spectrum for oceanic turbulence was proposed by Nikishov [18]. This spectrum was obtained from expressing temperature spectrum, salinity spectrum, and temperature-salinity co-spectrum by Hill’s Model 1 Eq. (15) with the free parameter ${Q}_{1}\approx 2.35$.

As an illustration, in Fig. 1, we plot the normalized spectrum of temperature fluctuations ${f}_{T}\left(\kappa \eta \right)={\chi}_{T}^{-1}{\epsilon}^{1/3}{\kappa}^{5/3}{E}_{T}\left(\kappa \right)$ [Fig. 1(a)] and the normalized spectrum of salinity fluctuations${f}_{S}\left(\kappa \eta \right)={\chi}_{S}^{-1}{\epsilon}^{1/3}{\kappa}^{5/3}{E}_{S}\left(\kappa \right)$ [Fig. 1(b)], where ${E}_{T}\left(\kappa \right)$ and ${E}_{S}\left(\kappa \right)$ are the wavenumber spectra of temperature and salinity fluctuations, respectively. We compare two previously proposed spectra: Hill numerical spectrum based on Hill’s Model 4 [3] and Nikishov spectrum, which is equivalent to the Hill’s Model 1 with ${Q}_{1}\approx 2.35$ [18]. To illustrate the scalar spectral behavior in the viscous range, we also plot Batchelor and Kraichnan spectrum in terms of $\kappa \eta $. With the spectral forms of Eqs. (8) and (9), and the relationship Eq. (3), we obtain $f\left(\kappa \eta \right)\text{=}{\chi}^{-1}{\epsilon}^{1/3}{\kappa}^{5/3}E\left(\kappa \right)$ for Batchelor and Kraichnan spectrum, respectively, as

In the numerical calculation, we choose $\mathrm{Pr}\text{=}7$ for ${f}_{T}\left(\kappa \eta \right)$; $\mathrm{Pr}\text{=}700$ for ${f}_{S}\left(\kappa \eta \right)$; and${q}_{B}=3.9$ and ${q}_{K}=5.26$ as [35–37].Clearly, the Hill spectrum obtained by numerically solving Model 4 exhibits the expected behavior: in the inertial-convective range where $E\left(\kappa \right)\sim {\kappa}^{-5/3}$, the functions $f\left(\kappa \eta \right)$ for temperature and salinity fluctuations are the horizontal lines and equal to the suggested value of the Obukhov-Corrsin constant $\beta \simeq 0.72$ [4]. In the viscous-convective range where $E\left(\kappa \right)\sim {\kappa}^{-1}$, $f\left(\kappa \eta \right)$ increases as ${\kappa}^{2/3}$ which produces a bump just prior to the onset of the viscous-diffusive range. Moreover, in the viscous range, Hill spectrum matches well with Kraichnan spectrum, which shows its better consistency with the simulation and experimental data [35–37]. This observation can be explained by the fact that Hill’s Model 4 derives from the viscous-range form of Kraichinan’s flux function. In the contrast, Nikishov spectrum shows a deviation from the horizontal line of the Obukhov-Corrsin constant in the inertial-convective range. This is an inherent problem of Hill’s Model 1 which also appears in modelling the atmospheric temperature spectrum [4]. Most importantly, Nikishov spectrum displays a remarkable deviation from Kraichnan spectrum in the viscous-convective range. It significantly underestimates the spectral values. It is known for the atmospheric turbulence case [1] that the spectral bump has important consequences on various aspects of optical beam propagation through turbulence, particularly in regards to scintillation effects. Through comparison, we conclude that Hill’s Model 4 can still be considered as the most accurate spectral model for the scalar fluctuations in ocean. However, due to its intractable form, Hill’s Model 4 cannot be used in the analytical studies.

## 3. The proposed oceanic turbulence spectrum

In this section, we propose a new spectral model for describing the refractive-index fluctuations in ocean. We also express the power spectrum of refractivity fluctuations in terms of the spectra of temperature and salinity and the temperature-salinity co-spectrum [18].

#### 3.1 Proposed spectra of oceanic scalar fluctuations

Based on previous studies, we concluded that the spectra of oceanic scalar fluctuations should satisfy two conditions: 1) they should have the tractable mathematic form that can facilitate the analytic studies as Nikishov spectrum does; 2) they should provide a good fit to the Obukhov-Corrsin law in the inertial range, while being consistent with Kraichinan spectrum for the constant ${q}_{K}=5.26$ in the viscous range just as the Hill’s Model 4 does. The starting point is to represent the scalar spectrum in a similar form as in [14, 38]

To show the accuracy of the proposed spectrum in the viscous range, we first compare the normalized dissipation forms ${\left(\kappa {\eta}_{B}\right)}^{2}E\left(\kappa \right)/\left(\chi {\left(v/\epsilon \right)}^{1/2}{\eta}_{B}\right)$ for the various proposed scalar spectral models as follows:

- • Batchelor spectrum [35, 37]:
- • Kraichnan spectrum [35, 37]:
- • Nikishov spectrum (from Eq. (15)):$$\begin{array}{c}\frac{{\left(\kappa {\eta}_{B}\right)}^{2}E\left(\kappa \right)}{\chi {\left(v/\epsilon \right)}^{1/2}{\eta}_{B}}=\beta {\mathrm{Pr}}^{-1/3}{\left(\kappa {\eta}_{B}\right)}^{1/3}\left[1+{Q}_{1}{\mathrm{Pr}}^{1/3}{\left(\kappa {\eta}_{B}\right)}^{2/3}\right].\\ \times \mathrm{exp}\left\{-1.5\beta {\mathrm{Pr}}^{-1/3}{\left(\kappa {\eta}_{B}\right)}^{4/3}-\beta {Q}_{1}{\left(\kappa {\eta}_{B}\right)}^{2}\right\}\end{array}$$
- • The proposed spectrum, derived from Eqs. (18) and (20), can be represented as:$$\frac{{\left(\kappa {\eta}_{B}\right)}^{2}E\left(\kappa \right)}{\chi {\left(v/\epsilon \right)}^{1/2}{\eta}_{B}}=\beta {\mathrm{Pr}}^{-1/3}{\left(\kappa {\eta}_{B}\right)}^{1/3}\left[1\text{+}{\displaystyle \sum _{n=1}^{N}{a}_{n}{\left({\mathrm{Pr}}^{1/2}\kappa {\eta}_{B}\right)}^{n}}\right]\mathrm{exp}\left(-\delta {\mathrm{Pr}}^{1/2}\kappa {\eta}_{B}\right).$$

In Fig. 2, we illustrate the normalized scalar dissipation spectrum for ${\mathrm{Pr}}_{T}\simeq 7$, i.e., Prandtl number of temperature in water. Figure 2(a) compares the results obtained from Kraichnan spectrum Eq. (23), Nikishov spectrum with ${Q}_{1}=2.35$ Eq. (24), the proposed spectrum Eq. (25) with the constant set Eq. (21), and DNS results of run a240 [35]. Clearly, the newly proposed spectrum is consistent with Kraichnan spectrum, which has been confirmed by DNS in the viscous range. But Nikishov spectrum with ${Q}_{1}=2.35$ deviates significantly from the DNS results all over the viscous range. Figure 2(b) illustrates the results from Batchelor spectrum Eq. (22), Nikishov spectrum with the increased ${Q}_{1}=4.6$ Eq. (24), the proposed spectrum Eq. (25) with the constant set Eq. (21), and DNS results of run a240 [35]. It is seen that with the increased free parameter, Nikishov spectrum can produce the values comparable to the DNS results in the range of small wavenumbers $\kappa {\eta}_{B}<0.25$. But as the wavenumber increases, it tends to Batchelor spectrum other than Kraichnan spectrum. The proposed spectrum shows a slightly faster decrease relative to Kraichnan spectrum and the DNS results at high $\kappa {\eta}_{B}$ near 1. This approximation error has less effect on the optical propagation studies as the absolute values of the scalar spectrum at such high wavenumbers are very small.

We also determine the constant set for salinity with ${\mathrm{Pr}}_{S}\simeq 700$ and for the coupled temperature-salinity with the equivalent ${\mathrm{Pr}}_{TS}=2{\mathrm{Pr}}_{T}\cdot {\mathrm{Pr}}_{S}/\left({\mathrm{Pr}}_{T}+{\mathrm{Pr}}_{S}\right)\simeq 13.86$ [19] as follows.

In Fig. 3, we plot the spectrum of temperature fluctuations ${f}_{T}\left(\kappa \eta \right)$ ([see Fig. 3(a)]) and spectrum of salinity fluctuations ${f}_{S}\left(\kappa \eta \right)$ ([see Fig. 3(b)]) versus scaled wavenumber $\kappa \eta $. In both cases, we can see that Nikishov spectrum with ${Q}_{1}=2.35$ significantly underestimates the spectral peak. Increasing free parameter (to ${Q}_{1}=4.6$ for temperature spectrum; to ${Q}_{1}=5.3$ for salinity spectrum) we can increase the spectral peak to the values comparable to that of Kraichnan spectrum. However, the deviation from the Obukhov-Corrsin law in the inertial-convective range increases correspondingly. The proposed new scalar spectrum, on the other hand, achieves a good fit to the Obukhov-Corrsin law and Kraichnan spectrum, simultaneously.

#### 3.2 Proposed spectrum of refractive-index fluctuations

Based on the developed scalar spectra, we derive the new form for the power spectrum of the refractivity in ocean. The dissipation rate of mean-squared refractive index ${\chi}_{n}$ is given by [18, 20]

where ${\rm A}$ is the thermal expansion coefficient and ${\rm B}$ is the saline contraction coefficient. ${\chi}_{T}$,${\chi}_{S}$, and ${\chi}_{TS}$ are respectively the dissipation rate of mean-squared temperature fluctuations, the dissipation rate of mean-squared salinity fluctuations and the covariance of temperature and salinity. These are used for setting the scalar variance dissipation constraint Eq. (1). ${\chi}_{S}$ and ${\chi}_{TS}$ are expressed in terms of ${\chi}_{T}$ [20]The shell-averaged spectrum of refractive-index is now expressed in terms of the spectra of temperature and salinity and the temperature-salinity co-spectrum as follows:

The calculated spectra of the refractive-index fluctuations are provided in Fig. 4. The vertical coordinate represents the values of the normalized spectrum of ${f}_{n}\left(\kappa \eta \right)={\chi}_{n}^{-1}{\epsilon}^{1/3}{\kappa}^{5/3}{E}_{n}\left(\kappa \eta \right)$, whereas the horizontal axis is the scaled wavenumber $\kappa \eta $. Both the proposed new spectrum and Nikishov spectrum with ${Q}_{1}=2.35$ are illustrated. To make a fair comparison, the modified Nikishov spectrum presented in [20] which includes the effect of the eddy diffusivity ratio is used. It is known that the spectrum of refractive-index fluctuations is a linear combination of the temperature spectrum, the salinity spectrum and the temperature-salinity co-spectrum Eq. (32), and each of them acquires a single peak located at different wavenumber. There should be three “bump” in Fig. 4. Evidently, in all cases, the spectra of ${f}_{n}\left(\kappa \eta \right)$ exhibit a double peak. This is because $\left|{\mathrm{Pr}}_{TS}-{\mathrm{Pr}}_{T}\right|\ll \left|{\mathrm{Pr}}_{S}-{\mathrm{Pr}}_{TS}\right|$which makes the peak of temperature spectrum overlap the co-spectrum’s peak. It is also observed that Nikishov spectrum with ${Q}_{1}=2.35$ obviously underestimates the refractivity fluctuations in the portion of the viscous-convective range as compared with the proposed spectrum. This may have obvious effects on optical scintillation.

## 4. Optical scintillation based on proposed oceanic turbulence spectrum

In this section, by using the proposed spectral model Eq. (33) in weak turbulence condition, we investigate the optical scintillation index of plane and spherical waves horizontally propagating through turbulent ocean. Based on Rytov theory, the scintillation index can be evaluated for plane and spherical waves, respectively, by [1]

In Fig. 5, we summarize the numerical results for the scintillation indices of both the plane and spherical waves with respect to the link length. We obtained the numerical results using the newly proposed spectrum Eq. (33), and substituting it into Eqs. (34) and (35), respectively. To show the difference between our results and the previous results [24, 25], we also present the scintillation index based on Nikishov spectrum with free parameter ${Q}_{1}=2.35$. Our results for the values of the scintillation index are limited to less than one so that they match with weak turbulence condition. In the numerical computation, we assume $A\text{=}2.56\times {10}^{-4}l/\mathrm{deg}$ and $\nu \text{=1}\text{.0576}\times {10}^{-6}{m}^{2}{s}^{-1}$when the temperature is 20°C and the salinity is 35% [20]. These values are for surface and near surface layer. To keep the consistency with the above parameter setting, we select ${\chi}_{T}={10}^{-5}{K}^{2}{s}^{-3}$ and $\epsilon \text{=}{10}^{-2}{m}^{2}{s}^{-3}$ that are the typical values for upper ocean [20]. The operating wavelength is $\lambda \text{=0}\text{.53}\mu m$. Here we choose $\omega =-1$ and ${d}_{r}=1$ for simplicity. Clearly, the scintillation index calculated based on Nikishov spectrum with ${Q}_{1}=2.35$ significantly underestimated the correct results for the scintillation index. This is due to the considerable underestimation of Nikishov spectrum in the viscous-convective range of the refractive-index spectrum.

## 5. Conclusion

In summary, we first study the previous scalar spectrum models used for oceanic temperature and salinity fluctuations. These spectra include the numerical solution based on Hill’s Model 4 and the exact Nikishov spectral form based on Hill’s Model 1. Hill’s Model 4 provides a good fit to the Obukhov-Corrsin law in the inertial range and Kraichinan spectrum with the constant ${q}_{K}=5.26$ in the viscous range, but has no closed-form solution. Nikishov spectrum with free parameter ${Q}_{1}=2.35$ has the closed-form expression, but has been found not to be consistent with the Obukhov-Corrsin law in the inertial-convective range. Additionally, the Nikishov spectrum underestimates the spectral values in the viscous-convective range, and reduces down to Batchelor spectrum in the viscous-diffusive range. To overcome these limitations, we propose a new scalar spectral form for temperature, salinity, and temperature-salinity co-spectrum. This spectrum matches well with the Obukhov-Corrsin law and Kraichinan spectrum with the constant ${q}_{K}=5.26$. Using this scalar spectral form, we develop a new power spectrum of refractive-index fluctuations. Then we investigate the plane- and spherical-wave scintillation index in weak turbulence regime. The results show that our spectrum predicts a remarkable reinforcement in the strength of optical scintillation in ocean.

## Funding

China Scholarship Council (201706965038); National Natural Science Foundation of China (NSFC) (61505155, 61571367); the Fundamental Research Funds for the Central Universities (JB160110, XJS16051); the 111 Project of China (B08038).

## Acknowledgements

We thank Peng Yue and Ruiqin Zhao for their invaluable advice, insight, and assistance. We also thank three anonymous reviewers for their detailed and conscientious comments.

## References and links

**1. **L. C. Andrews and R. L. Phillips, *Laser Beam Propagation through Random Media*, 2nd ed., (SPIE, 2005).

**2. **R. J. Hill, “Spectra of fluctuations in refractivity, temperature, humidity, and the temperature-humidity cospectrum in the inertial and dissipation ranges,” Radio Sci. **13**(6), 953–961 (1978). [CrossRef]

**3. **R. J. Hill, “Optical propagation in turbulent water,” J. Opt. Soc. Am. **68**(8), 1067–1072 (1978). [CrossRef]

**4. **R. J. Hill, “Models of the scalar spectrum for turbulent advection,” J. Fluid Mech. **88**(3), 541–562 (1978). [CrossRef]

**5. **A. M. Oboukhov, “Structure of the temperature field in turbulent flow,” Izv. Akad. Nauk SSSR, Ser. Geofr. I Geofiz. **13**(1), 58–69 (1949).

**6. **S. Corrsin, “On the spectrum of isotropic temperature fluctuations in isotropic turbulence,” J. Appl. Phys. **22**(4), 469–473 (1951). [CrossRef]

**7. **G. K. Batchelor, “Small scale variation of convected quantities like temperature in a turbulent fluid,” J. Fluid Mech. **5**(01), 113–133 (1959). [CrossRef]

**8. **R. Kraichnan, “Small-scale structure of a scalar field convected by turbulence,” Phys. Fluids **11**(5), 945–953 (1968). [CrossRef]

**9. **S. Corrsin, “Further generalization of Onsager’s model for turbulent spectra,” Phys. Fluids **7**(8), 1156–1159 (1964). [CrossRef]

**10. **Y.-H. Pao, “Structure of turbulent velocity and scalar fields a large wave-numbers,” Phys. Fluids **8**(6), 1063–1075 (1965). [CrossRef]

**11. **C. E. Leith, “Diffusion approximation for turbulent scalar fields,” Phys. Fluids **11**(8), 1612–1617 (1968). [CrossRef]

**12. **F. H. Champagne, C. A. Friehe, J. C. LaRue, and J. C. Wynagaard, “Flux measurements, flux estimation techniques, and fine-scale turbulence measurements in the unstable surface layer over land,” J. Atmos. Sci. **34**(3), 515–530 (1977). [CrossRef]

**13. **R. M. Williams and C. A. Paulson, “Microscale temperature and velocity spectra in the atmospheric boundary layer,” J. Fluid Mech. **83**(03), 547–567 (1977). [CrossRef]

**14. **A. Muschinski and S. M. de Bruyn Kops, “Investigation of Hill’s optical turbulence model by means of direct numerical simulation,” J. Opt. Soc. Am. A **32**(12), 2423–2430 (2015). [CrossRef] [PubMed]

**15. **L. C. Andrews, “An analytical model for the refractive index power spectrum and its application to optical scintillations in the atmosphere,” J. Mod. Opt. **39**(9), 1849–1853 (1992). [CrossRef]

**16. **J. H. Churnside, “A spectrum of refractive turbulence in the turbulent atmosphere,” J. Mod. Opt. **37**(1), 13–16 (1990). [CrossRef]

**17. **R. G. Frehlich, “Laser scintillation measurements of the temperature spectrum in the atmospheric surface layer,” J. Atmos. Sci. **49**(16), 1494–1509 (1992). [CrossRef]

**18. **V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. **27**(1), 82–98 (2000). [CrossRef]

**19. **J. Yao, Y. Zhang, R. Wang, Y. Wang, and X. Wang, “Practical approximation of the oceanic refractive index spectrum,” Opt. Express **25**(19), 23283–23292 (2017). [CrossRef] [PubMed]

**20. **M. Elamassie, M. Uysal, Y. Baykal, M. Abdallah, and K. Qaraqe, “Effect of eddy diffusivity ratio on underwater optical scintillation index,” J. Opt. Soc. Am. A **34**(11), 1969–1973 (2017). [CrossRef] [PubMed]

**21. **X. Peng, L. Liu, Y. Cai, and Y. Baykal, “Statistical properties of a radially polarized twisted Gaussian Schell-model beam in an underwater turbulent medium,” J. Opt. Soc. Am. A **34**(1), 133–139 (2017). [CrossRef] [PubMed]

**22. **Y. Li, L. Yu, and Y. Zhang, “Influence of anisotropic turbulence on the orbital angular momentum modes of Hermite-Gaussian vortex beam in the ocean,” Opt. Express **25**(11), 12203–12215 (2017). [CrossRef] [PubMed]

**23. **C. Lu and D. Zhao, “Statistical properties of rectangular cusped random beams propagating in oceanic turbulence,” Appl. Opt. **56**(23), 6572–6576 (2017). [CrossRef] [PubMed]

**24. **O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media **22**(2), 260–266 (2012). [CrossRef]

**25. **Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A **31**(7), 1552–1556 (2014). [CrossRef] [PubMed]

**26. **H. Gerçekcioğlu, “Bit error rate of focused Gaussian beams in weak oceanic turbulence,” J. Opt. Soc. Am. A **31**(9), 1963–1968 (2014). [CrossRef] [PubMed]

**27. **M. Yousefi, S. Golmohammady, A. Mashal, and F. D. Kashani, “Analyzing the propagation behavior of scintillation index and bit error rate of a partially coherent flat-topped laser beam in oceanic turbulence,” J. Opt. Soc. Am. A **32**(11), 1982–1992 (2015). [CrossRef] [PubMed]

**28. **H. Gerçekcioglu, “BER of annular beams in weak oceanic turbulence,” Selcuk University Journal of Engineering, Science and Technology **5**(3), 262–273 (2017). [CrossRef]

**29. **M. Yousefi, F. D. Kashani, S. Golmohammady, and A. Mashal, “Scintillation and bit error rate analysis of a phase-locked partially coherent flat-topped array laser beam in oceanic turbulence,” J. Opt. Soc. Am. A **34**(12), 2126–2137 (2017). [CrossRef] [PubMed]

**30. **X. Yi, Z. Li, and Z. Liu, “Underwater optical communication performance for laser beam propagation through weak oceanic turbulence,” Appl. Opt. **54**(6), 1273–1278 (2015). [CrossRef] [PubMed]

**31. **M. C. Gokce and Y. Baykal, “Aperture averaging and BER for Gaussian beam in underwater oceanic turbulence,” Opt. Commun. **410**, 830–835 (2018). [CrossRef]

**32. **Y. Baykal, “Expressing oceanic turbulence parameters by atmospheric turbulence structure constant,” Appl. Opt. **55**(6), 1228–1231 (2016). [CrossRef] [PubMed]

**33. **Y. Baykal, “Scintillation index in strong oceanic turbulence,” Opt. Commun. **375**, 15–18 (2016). [CrossRef]

**34. **M. C. Gokce and Y. Baykal, “Aperture averaging in strong oceanic turbulence,” Opt. Commun. **413**, 196–199 (2018). [CrossRef]

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

**36. **X. Sanchez, E. Roget, J. Planella, and F. Forcat, “Small-scale spectrum of a scalar field in water: the Batchelor and Kraichnan models,” J. Phys. Oceanogr. **41**(11), 2155–2167 (2011). [CrossRef]

**37. **D. J. Bogucki, H. Lou, and J. A. Domaradzki, “Experimental evidence of the Kraichnan scalar spectrum at high Reynolds numbers,” J. Phys. Oceanogr. **42**(10), 1717–1728 (2012). [CrossRef]

**38. **A. Muschinski, “Temperature variance dissipation equation and its relevance for optical turbulence modeling,” J. Opt. Soc. Am. A **32**(11), 2195–2200 (2015). [CrossRef] [PubMed]

**39. **V. I. Tatarskii, M. M. Dubovikov, A. A. Praskovsky, and M. Yu. Karyakin, “Temperature fluctuation spectrum in the dissipation range for statistically isotropic turbulent flow,” J. Fluid Mech. **238**(-1), 683–698 (1992). [CrossRef]

**40. **R. C. Mjolsness, “Diffusion of a passive scalar at large Prandtl number according to the abridged Lagrangian interaction theory,” Phys. Fluids **18**(10), 1393–1394 (1975). [CrossRef]