Polarimetric lidar measurements of aquatic turbulence - laboratory experiment

Darek J. Bogucki; Julian A. Domaradzki; Paul von Allmen

doi:10.1364/OE.26.006806

1. Introduction

Active optical remote sensing using ‘green’ wavelength lidar method is the only technique capable to sample a significant portion of the global upper ocean [1]. Shipboard and airborne lidars have long been used for a variety of ocean biology applications [2–5]. Currently deployed oceanographic lidar systems [6,7], are capable of measuring profiles of water optical properties such as the attenuation coefficients or the particulate backscattering coefficient as a function of the water depth z. Our approach goes one step further, by analyzing oceanic returns of polarimetric lidar system we can derive the depth-dependent oceanic turbulent parameters.

Most oceanic flows are turbulent and characterized by a wide range of coexisting scales of motion and temperature gradients which are as small as few millimeters in size. Turbulent fluctuations of the temperature in water cause fluctuations in the water refractive index. In the ocean, the nearforward part of the oceanic volume scattering function (VSF or the scattering differential cross section) is driven by the interaction of light with turbulent inhomogeneities of the water refractive index [8], arising mostly as the effect of water temperature fluctuations. These turbulence-induced inhomogeneities are very effective light scatterers at nearforward angles. It has been documented in aquatic VSF measurements [9] that in the range of θ₁ = 10⁻⁷ to θ₂ = 10⁻³ rad, in the nearforward direction, the scattering coefficient b_turb (defined as: $b_{turb} = 2 π \int_{θ_{1}}^{θ_{2}} VSF (θ) \sin (θ) d θ$ ) is exclusively due to turbulent inhomogeneities and can easily reach values of b_turb = 10 m⁻¹. Such a large value of the b_turb implies that the photon mean pathlength between scattering events l_phot on turbulent inhomogeneities is around a few centimeters, since l_phot ≈ 1/b_turb. That pathlength is much shorter than the typical mean pathlength due to scattering by particles, which is around a few meters to tens of meters [10]. Consequently, most of the lidar detected photons undergo many multiple forward scattering on turbulence events and a single backscattering event on a particle. However, little is known about the magnitude of the forward light depolarization on turbulent refractive index inhomogeneities.

Estimates carried out over the last 50 years by numerous researchers(e.g.: [11–14]) have established this depolarization to be negligible for the atmospheric turbulence. A quantity describing a medium’s ability to depolarize light is, for example, the depolarization rate γ, defined in [6] as γ ∼ dD/dz, where D is the ratio of the crosspolarized power to the copolarized incident power over distance z. For aquatic turbulence, the depolarization rate γ attains its largest value of γ ≈ 10⁻¹⁵ m⁻¹ for rather a specific type of turbulent flow - characterized by a low value of turbulent kinetic energy dissipation ∊ with simultaneously a large temperature variance dissipation rate χ, for example: ∊ = 10⁻¹⁰ Wkg⁻¹ and χ = 10⁻⁴ °C²s⁻¹. For turbulent flows, the γ value can be estimated [11–14] as $γ \approx 0.019 C_{n}^{2} k_{m}^{7 / 3} k^{- 2}$ , with k_m the turbulent flow Kolmogorov wavenumber, $C_{n}^{2}$ the flow refractive index structure parameter and k the light wavenumber. An estimate of $C_{n}^{2}$ can be found in [15] as $C_{n}^{2} \approx {(\frac{d n}{d T})}^{2} χ ∊^{- 1 / 3}$ , where n is the water refractive index and T is the water temperature. For the specific turbulent regime considered here, a light wavelength of 500 nm, and a water temperature around T = 20°C, the $C_{n}^{2}$ value is then $C_{n}^{2} ≃ 2.6 \cdot 10^{- 8} m^{- 2 / 3}$ .

For aquatic flows, the theoretically estimated upper bound of the depolarization rate, i.e., γ ≈ 10⁻¹⁵ m⁻¹ is in contrast to the laboratory results [16] where the authors measured γ = O (10⁻³) m⁻¹ in a turbulent convective flow. The work presented here aims to reconcile that difference. We present results of measurements collected in a 17 m long flume tank equipped with a grid such that the flow past the grid is turbulent. The flow was then sampled with a polarimetric lidar system to obtain the depolarization rate and the light attenuation coefficient for varying turbulent flow strength, particle concentration, and composition. The paper is organized as follows: we begin with an overview of the polarimetric theory then present our observations and finish with a hypothesis explaining our results.

2. Theory

Our approach follows the one presented in [6] and is based on a simplified version of the radiative transfer equation for the Stokes vector. The chief advantage of that approach stems from the availability of a large number of oceanographic observations [6], [7]. We assume a linearly polarized transmitter and consider the only light that is co-polarized with the transmitted light or cross-polarized with respect to it. We extend our approach to short propagation distances so as to describe our experimental results. Following [6], the power of an initially linearly polarized and perfectly collimated laser pulse propagating through the water column can be then described by a set of equations:

\frac{d P_{c}}{d z} = - α (z) P_{c} (z) + γ (z) P_{x} (z)

\frac{d P_{x}}{d z} = γ (z) P_{c} (z) - α (z) P_{x} (z),

where: P_c(z) is the power at a distance z in the initial linear polarization and P_x(z) is the power in the orthogonal polarization. The plane of polarization of the co-polarized component is assumed to be parallel to the plane of polarization of the laser pulses. The total power of the propagating pulse is P_T = P_c + P_x. The attenuation of a polarized beam is α(z) ≥ 0 and γ(z) ≥ 0 is the depolarization rate coefficient. The depolarization coefficient γ represents the rate at which light from one polarization is scattered into the other. The attenuation coefficient of an unpolarized lidar signal c(z) is then given by c(z) = α(z) − γ(z). Typically, in oceanic observations c(z) ≃ α(z) ≫ γ(z) - see [6].

The lidar observation is determined by three distinct physical processes: the propagation of the collimated light pulse away from the lidar transmitter, the scattering of the collimated pulse on a particle and the propagation of the scattered pulse back to the lidar receiver. Consequently, below, we will mathematically describe each of the processes and then combine the solutions to obtain the amount of power received by the lidar receiver.

For simplicity, we assume a lidar transmitter emitting a collimated (with vanishing divergence) short light pulse X̂_beam - Fig. 1. With the lidar transmitter located at z = 0 and the lidar pulse initially partitioned between co-polarized and cross-polarized pulse components as P_c(0) ≡ P_c₀ and P_x(0) ≡ P_x₀, the Eqs. (1) and (2) can be written more compactly in a matrix form as follows: $\frac{d {\hat{X}}_{beam}}{d z} = [\begin{matrix} - α (z) & γ (z) \\ γ (z) & - α (z) \end{matrix}] \cdot {\hat{X}}_{beam}$ ; ${\hat{X}}_{beam} \equiv [\begin{matrix} P_{c} (z) \\ P_{x} (z) \end{matrix}]$ . With initial conditions: ${\hat{X}}_{0} \equiv [\begin{matrix} P_{c} (0) \\ P_{x} (0) \end{matrix}] \equiv [\begin{matrix} P_{c_{0}} \\ P_{x_{0}} \end{matrix}]$ , the solution at a distance z₀ is given by:

\begin{array}{l} {\hat{X}}_{beam} (z_{0}) = e^{- A (z_{0})} \cdot T_{beam} \cdot [\begin{array}{l} P_{c_{0}} \\ P_{x_{0}} \end{array}]; & Where \end{array} : T_{beam} = [\begin{array}{l} cosh Γ (z_{0}) & sinh Γ (z_{0}) \\ sinh Γ (z_{0}) & cosh Γ (z_{0}) \end{array}]

The compact form of the solution is X̂_beam(z₀) = e^−A(z₀) · T_beam · X̂₀. Here, α(z), γ(z) are the spatially varying polarized beam attenuation and the depolarization rate while

A (z) \equiv \int_{0}^{z} α (z^{'}) d z^{'}

and

Γ (z) \equiv \int_{0}^{z} γ (z^{'}) d z^{'}

. Equation (3) describes the power at a distance z₀, in a pulse propagating away from the lidar and prior to the single backscattering event on a particle, see Fig. 1.

Fig. 1 A single backscattering event on a small spherical particle located at the distance z₀ from the lidar. X̂_beam represents the incoming and X̂_sc the scattered lidar pulse. The solid angle Ω corresponds to the lidar receiver field of view.

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For the lidar located at z = 0, to receive the signal from a distance z₀, the outgoing pulse power X̂_beam(z₀) has to be backscattered back to the detector by a particle located at z₀, Fig. 1. This single backscattering event redistributes the co- and cross-polarized outgoing collimated lidar pulse power X̂_beam(z₀) and scatters it into the solid angle Ω as X̂_sc(z₀), we then have [6]:

\begin{array}{l} {\hat{X}}_{sc} (z_{0}) = T_{part} (z_{0}) \cdot {\hat{X}}_{beam} (z_{0}); & where \end{array} T_{part} = [\begin{array}{l} β_{c} (π, z_{0}) & β_{x} (π, z_{0}) \\ β_{x} (π, z_{0}) & β_{c} (π, z_{0}) \end{array}]

where the vector X̂_beam(z₀) is the collimated power of the lidar pulse prior to the backscattering event on a particle. The elements of the vector

{\hat{X}}_{sc} (z_{0}) \equiv [\begin{matrix} P_{c_{sc}} (z_{0}) \\ P_{x_{sc}} (z_{0}) \end{matrix}]

represent the total backscattered power contained within the field of view of the lidar receiver Ω. The vector X̂_sc(z₀) can in principle be obtained by integrating an angle dependent scattered radiant intensity vector x̂_sc(z₀, Ω) such that

{\hat{X}}_{sc} (z_{0}) = \iint_{Ω} {\hat{x}}_{sc} (z_{0}, Ω) d Ω

. T_part is the backscattering matrix. The coefficients β_c(π, z₀) and β_x(π, z₀) represent the efficiency of power transfer between the same polarization or between two orthogonal polarizations, respectively, during the backscattering event and represent the average power in the backscattered direction over the solid angle Ω. The ratio δ(z₀) ≡ β_x(π, z₀)/β_c(π, z₀) is zero when the backscattering event preserves incident polarization. In general, the ratio δ is related to the linear polarization portion of the Mueller matrix M. In terms of the Muller matrix describing single particle backscattering event, we have:

M \equiv VSF (π) \cdot [\begin{matrix} 1 & 0 \\ 0 & 1 - d \end{matrix}]

, with the following relationship [3] between δ and d : δ = d/(2 − d). For the light scattered on a spherically symmetrical particle, δ = d ≡ 0. For an ensemble of particles, the deviation of d from zero indicates the presence of nonspherical particles. The range of the parameter d for non-spherical particles has been explored numerically. For aggregates of spherical particles with 10μm diameter, calculations in [17] demonstrated that d < 0.05.

The propagation process of the spherically symmetric, scattered pulse power X̂_sc, away from the particle center located at z₀ and towards the lidar receiver can be described as:

\frac{d {\hat{X}}_{sc}}{d z} = [\begin{matrix} - [α (z) + \frac{2}{z - z_{0}}] & γ (z) \\ γ (z) & - [α (z) + \frac{2}{z - z_{0}}] \end{matrix}] \cdot {\hat{X}}_{sc} (z) .

In Eq. (5), the factor −2/(z − z₀), is responsible for radial spreading of the scattered pulse power. Moreover, as for the collimated pulse, the unpolarized light attenuation is c = α − γ. This can be observed by solving Eq. (5) for P_T (z) with constant α and γ. The total scattered power within Ω can be found from Eq. (5) as:

P_{T} (z) = \frac{P_{T_{0}}}{4 π z^{2}} exp [- (α - γ) \cdot z]

. The general solution of the Eq. (5) in terms of the particle backscattered power X̂_sc(z₀) can be expressed as:

\begin{array}{l} \hat{X} (z) = \frac{e^{- A_{1} (z)}}{4 π {(z - z_{0})}^{2}} \cdot T_{sc} \cdot [\begin{array}{l} P_{c_{sc}} (z_{0}) \\ P_{x_{sc}} (z_{0}) \end{array}] . & Here \end{array} : T_{sc} \equiv [\begin{array}{l} cosh Γ_{1} (z) & sinh Γ_{1} (z) \\ sinh Γ_{1} (z) & cosh Γ_{1} (z) \end{array}]

and

A_{1} (z) \equiv \int_{z_{0}}^{z} α (z^{'}) d z^{'}

and

Γ_{1} (z) \equiv \int_{z_{0}}^{z} γ (z^{'}) d z^{'}

. After combining all results of Eq. (4), Eq. (3) and Eq. (6) we can find an expression for the power received by the lidar detector as a result of a single backscattering by a particle at a distance z₀ and as a function of the initial laser pulse power X̂₀:

\hat{X} (z_{0}) = \frac{π R_{t}^{2}}{4 π z_{0}^{2}} \cdot e^{- 2 A (z_{0})} \cdot T_{sc} \cdot T_{part} \cdot T_{beam} \cdot {\hat{X}}_{0},

where R_t is the aperture radius of the lidar receiving telescope. Equation (7) describes the power received by the lidar in the co- and cross polarized channels, P_x(z₀) and P_c(z₀), respectively. In order to obtain a closed-form expression for X̂(z₀), we have to determine the particle backscattering matrix T_part. Not much is known about the aquatic particles depolarization ratio δ defined as δ ≡ β_x(π)/β_c(π) [6]. Calculations in [17] for ensembles of 10 μm spherical particles constrained that value to δ < 0.025.

The relative strength of the cross-polarized to the co-polarized lidar return is typically analyzed in terms of easily measured quantity such as the depolarization ratio D(z₀), defined as D(z₀) ≡ P_x(z₀)/P_c(z₀). Physically, the depolarization D(z₀) represents a sum of contributions from multiple forward scattering events and a single depolarizing backscattering event.

In order to evaluate the backscattering contribution to the depolarization we initially assume that δ has a small and a constant value. To evaluate the δ effect on the depolarization D(z₀) we expand D(z₀), Eq. (7) in terms of a small value of the parameter δ as:

D (Z_{0}) = tanh {2 Γ (z_{0})} + [1 - {tanh}^{2} {2 Γ (z_{0})}] \cdot δ + \frac{sinh {2 Γ (z_{0})}}{{cosh}^{3} {2 Γ (z_{0})}} \cdot δ^{2} + O {(δ)}^{3}

In anticipation of the main result (i.e.: Eq. (13)) from Eq. (8) we find:

\frac{1}{2} \frac{d D (z_{0})}{d z_{0}} \frac{1}{[1 - {tanh}^{2} {2 Γ (z_{0})}]} \frac{1}{γ (z_{0})} = 1 - 2 tanh {2 Γ (z_{0})} \cdot δ + O {(δ)}^{2} .

Since

Γ (z_{0}) \equiv \int_{0}^{z_{0}} γ (z^{'}) d z^{'}

, for a constant γ we obtain Γ(z₀) = γz₀. Oceanic observations in [6], consistent with our measurements (Table 1), demonstrated that the aquatic depolarization rate γ is around γ ≃ 10⁻³ m⁻¹. Thus the effect of small and constant depolarization parameter δ would result in the LHS of Eq. (9) to differ from 1 by a factor of 4γz₀ · δ. Therefore, the relative uncertainty on γ when it is estimated by setting the LHS of Eq. (9) to 1, is better than 10⁻⁴, which results from assuming small values for δ (= 0.025) over distances of z₀ ≃ 1 m.

Table 1. Summary of all the experiment runs. The runs labeled with (*) - i.e.: (DEC31 runs) were carried out with the water containing polydisperse debris particles. Rest of the runs: the tank was filled with clean tap water and seeded with varying concentrations of monodisperse 10μm neutrally buoyant beads. Both dissipation parameters χ and ∊ were estimated at a distance of z₁ = 6.3 m downstream from the grid. The mean polarized beam attenuation coefficient α and the mean depolarization rate γ correspond to their values over the nominal interval. The relative errors on ∊ and χ were 2σ_∊/∊̄ = 0.6 and 2σ_χ/χ̄ = 1.2 respectively. All uncertainties are equal to two standard deviations - corresponding to a 95.5% confidence interval for the reported quantities.

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In a realistic turbulent flow, the value of the δ(z₀) likely varies with the distance z₀. Turbulent flow simulations in [18] or experiments in [19] have documented pronounced concentration fluctuations of small inertial particles resulting in hydrodynamic particle clustering. They observed the largest particles concentration gradient, over length scales comparable or smaller than the Kolmogorov lengthscale η, where: η ≡ (ν³/∊)^1/4 and ν is the kinematic viscosity, and typically η < 1 mm. To find out how hydrodynamic particle clustering, with the δ(z₀) varying in space, affects dD(z₀)/dz₀, we assume that α and γ are constant and, keeping the lowest order in γ²z₀δ(z₀), the solution of Eq. (7) in terms of dD(z₀)/dz₀ is:

\frac{d D (z_{0})}{d z_{0}} = - 2 δ^{2} (z_{0}) γ + \frac{d δ (z_{0})}{d z_{0}} + 2 γ + O (γ^{2} z_{0} δ (z_{0})) .

The ensemble average over a large number of realizations of Eq. (10) is:

\frac{\bar{d D (z_{0})}}{d z_{0}} = - 2 \bar{δ^{2} (z_{0}) γ} + \frac{d \bar{δ}}{d z_{0}} + 2 γ

where overline denotes the ensemble average. Note that the first term can not be simplified to dD̄/dz₀ since D(z) slowly varies over the considered interval of dz₀ ≃ 1 m - see Fig. 3. The situation is different for the term

\bar{d δ (z_{0}) / d z_{0}}

. As a result of the δ having a characteristic lengthscale of 1 mm which is much smaller than the lenghtscale over which we collect our data - dz₀ ≃ 1 m, we obtain:

\bar{\frac{d δ}{d z_{0}}} ≃ \frac{d \bar{δ}}{d z_{0}} = 0

. If δ̄ = 0.025 and γ ≃ 10⁻³ m⁻¹ we get:

\bar{d D (z_{0}) / d z_{0}} ≅ 2 \bar{γ}

with the uncertainty of 2δ̄²γ ≊ 6.4 · 10⁻⁷.

The above reasoning implies that the main source of lidar pulse depolarization originates from the set of nearforward scattering events that the propagating pulse experiences during its travel between the transmitter and the lidar receiver. Consequently, in our analysis, we assume δ → 0 and T_part becomes T_ij(z) = β_c(π, z)δ_ij where δ_ij is the Kronecker delta. From this result, we can derive the following expression for X̂(z₀):

\hat{X} (z_{0}) = \frac{R_{t}^{2} β_{c} (π, z_{0})}{4 z_{0}^{2}} \cdot e^{- 2 A (z_{0})} \cdot [\begin{matrix} cosh {2 Γ (z_{0})} & sinh {2 Γ (z_{0})} \\ sinh {2 Γ (z_{0})} & cosh {2 Γ (z_{0})} \end{matrix}] \cdot [\begin{matrix} P_{c_{0}} \\ P_{x_{0}} \end{matrix}] .

Assuming that initially all lidar power was contained in the co-polarized pulse, i.e. that initially, P_x₀ = 0 we then can solve Eq. (12) for the local value of γ as:

γ (z_{0}) = \frac{1}{2} {\frac{d D}{d z} |}_{z = z_{0}} \cdot {\frac{1}{[1 - D^{2} (z_{0})]} ≊ \frac{1}{2} \frac{d D}{d z} |}_{z = z_{0}} .

The last step in the Eq. (13) is justified since the term D²(z₀) < 4 · 10⁻⁵ - Fig. 3. The coefficient α can be found from the Eq. (12) as:

α (z_{0}) = - \frac{1}{2 P_{c} (z_{0})} {\frac{d P_{c} (z)}{d z} |}_{z = z_{0}} - \frac{1}{z_{0}} + γ (z_{0}) \cdot D (z_{0}) .

Since in our experiment: γ(z₀) · D(z₀) < 10⁻⁴ [1/m] ≪ α (see Table 1 and Fig. 3) we then get the following expression for α:

α (z_{0}) = \frac{d}{d z} {{- \frac{1}{2} log [z^{2} P_{c} (z)]} |}_{z = z_{0}}

Equations (12–15) were obtained with the assumption that the lidar transmitter beam is characterized by a vanishing divergence angle. If the lidar transmitter beam divergence can not be neglected, we can obtain modified expressions for α and γ by replacing in Eq. (2) the transmitter beam power P_c and P_x with a product of their respective irradiances: $P_{c}^{*}$ , $P_{x}^{*}$ and the transmitter beam cross section area S(z), such that $P_{c} (z) \equiv S (z) P_{c}^{*} (z)$ and $P_{x} (z) \equiv S (z) P_{x}^{*} (z)$ . (Irradiance is the power received by a normal surface per unit area). With these substitutions to the Eq. (3) and the subsequent equations, Eq. (12) becomes:

[\begin{array}{l} P_{c}^{*} (z_{0}) \\ P_{x}^{*} (z_{0}) \end{array}] = \frac{R_{t}^{2} β_{c} (π, z_{0})}{4 z_{0}^{2}} \cdot G (z_{0}) \cdot e^{- 2 A (z_{0})} \cdot [\begin{matrix} cosh {2 Γ (z_{0})} & sinh {2 Γ (z_{0})} \\ sinh {2 Γ (z_{0})} & cosh {2 Γ (z_{0})} \end{matrix}] \cdot [\begin{matrix} P_{c_{0}}^{*} \\ P_{x_{0}}^{*} \end{matrix}];

where

P_{c_{0}}^{*}

P_{x_{0}}^{*}

and

P_{c}^{*} (z_{0})

,

P_{x}^{*} (z_{0})

are the respective lidar emitted and the lidar measured copolarized or crosspolarized irradiances. G(z₀) is the geometrical factor reflecting outgoing beam divergence such that:

G (z_{0}) = r_{0}^{2} / {(r_{0} + β z_{0})}^{2}

; r₀ is the initial lidar beam radius and β is the lidar transmitter half beam divergence. With these modifications Eq. (13) remains unchanged while Eq. (15) becomes:

α (z_{0}) = {\frac{d}{d z} {- \frac{1}{2} log [z^{2} P_{c}^{*} (z) G (z)]} |}_{z = z_{0}} .

3. Experiment setup

Experiments were performed in the water channel facility at the University of Southern California which was modified to include a heating element and turbulence generating grid - Fig. 2. The 91.4 cm wide test section was filled to a depth of 38.1 cm during the experiments. A 5.50 cm thick acrylic window was installed at the end of the flow path such that a laser beam can be aligned with the axial center of the test section. Two grids were designed to introduce temperature fluctuations and generate homogeneous turbulence within the flow. The heating grid was integrated into an electrical circuit with 8.3, 2.7 and 0.78 kW power settings. A second grid was fabricated to generate turbulence in the water and is installed 0.48 m downstream of the heater. That grid consisted of a square mesh grid with the mesh size M = 0.1 m and solidity of 0.36.

Fig. 2 Left: The USC water tank with the lidar beam (green) location. 1. Lidar beam entrance, 2. Laser Beam, 3. Test Section 4. Draining Reservoir 5. Filling Reservoir 6. Turbulence Generating Grid 7. Heater Grid. Right: Photo of a lidar pulse propagating along the tank center.

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Fig. 3 Lidar measured irradiance $P_{c}^{*} (z)$ (left panel) and $D (z) \equiv P_{x}^{*} (z) / P_{c}^{*} (z)$ (right panel) over the nominal interval. The parameters $P_{c}^{*} (z)$ and D(z) were normalized to their respective values at a distance of 6.2m from the lidar. The least-square derived α and γ (Eq. (15) and Eq. (13)) are presented in Table 1.

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We used the Minilite II laser (Continuum Lasers) with the following operational parameters: the transmitted pulse energy was 0.025 J/pulse, the pulse length 4 ns (0.7 m length in water), the pulse repetition rate was 10 s⁻¹, the wavelength was 532 nm, with nominal linear polarization beam divergence < 3 mrad and a beam width < 3 mm. The laser beam was then expanded by a factor of 2 resulting in a lidar transmitter beam with a radius of r₀ ≃ 4.5 mm and a half divergence angle β ≃ 0.7 mrad (in water). The single receiver telescope with a diameter of 0.3 m was used to measure the reflected energy. The receiver telescope and the transmitted beam were aligned along the centerline of the tank to provide full overlap starting at the tank entrance. The light entering telescope was collimated. The telescope had an aperture at the focus of the primary lens. An interference filter (1 nm bandwidth) was used to limit the background light. A polarizer with extinction ratio 10⁵ (GL10A - Thorlabs) divided the incoming signal into two orthogonal polarizations. In each channel (cross- and co-polarized), a high sensitivity PIN photodetector (FPD510-FV, MenloSystems) was used to convert the collected light into an electrical signal. Each channel was connected to a 400 MSample/s and 14-bit digitizer (GaGe CS144002U-USB). The cross channel was additionally amplified by a factor of 18.7 with a Low Noise Amplifier ZFL-1000LN (Mini-Circuits). The collected data were offloaded via USB and stored on a laptop. Prior to some runs, the tank was seeded with varying concentration of monodisperse 10μm neutrally buoyant particles, with the refractive index of n = 1.5 (TSI PIV beads - model 10089).

The turbulence data were measured using a Rockland Int. microstructure profiler (VMP200), equipped with a fast thermistor (FP07) and two orthogonal shear probes. The VMP200 measured the turbulent kinetic energy dissipation rate ∊ and the temperature dissipation rate χ at selected spots along the tank centerline and for a selected range of current speeds and heater power settings. Measurements of the χ with the switched off heater provided an estimate of the background heat flux. We found an ambient heat flux of around 0.3 kW and likely varying between runs. This ambient heat flux contributed to a large scatter of the χ dissipation values at the lowest power settings i.e. 0.78 kW. Following [20] and based on VMP200 dissipation measurements along the tank centerline, we found the following set of equations for ∊ and χ as a function of the distance to the grid location z₁:

∊ = 0.90 \cdot U^{3} \cdot {(z_{1} / M)}^{- n - 1}, and : χ = 0.99 \cdot 10^{- 6} \cdot 1 / U \cdot P_{0}^{2} \cdot {(z_{1} / M)}^{- m - 1},

with the used grid mesh size M = 0.1 m and the respective exponents m = 1.19 and n = 1.12. (We used two sets of horizontal coordinates: z₁ associated with the grid such that z₁ = 0 corresponds to the grid location, and z associated with the lidar measurements such that z = 0 corresponds to the lidar location.) For the dissipation estimate to yield values of χ in [°C²s⁻¹], and ∊ in [m²/s³], the mean current speed U has to be expressed in m/s and the heater power P₀ in kW. Based on the tank geometry, the tank section free from sidewall reflection was located between z = 6.2 m to z = 6.8 m from the lidar detector (or z₁ = 6.3 m to z₁ = 6.9 m downstream from the grid). We labeled that range of distances as the ’nominal interval’. We thus have restricted our subsequent analysis to the nominal interval of length 0.6 m. The estimates of the turbulent dissipation parameters ∊ and χ were obtained using Eq. (18) for a distance, z₁ = 6.3m downstream from the grid and correspond to the turbulent dissipation at the origin of the nominal interval.

4. Results

We have carried out experiment runs over 5 consecutive days - results are summarized in Table 1. Each run was around 2.5 hours long and we have excluded the first 30 minutes worth of data to ensure that particles were thoroughly mixed and the laser attained its nominal condition. Each analyzed run consisted of 80000 horizontal lidar profiles. Each profile included $P_{x}^{*} (z)$ and $P_{c}^{*} (z)$ lidar returns over distances to the lidar detector in the range from z = 0 to 13 m, with a 0.28 m resolution.

In the first two runs (DEC31), the tank was filled with tap water without the addition of any particles. In these runs, we observed the presence of debris particles of varying sizes after the pump had established the constant water flow speed. To eliminate the debris in subsequent runs, we emptied the tank and re-filled it with clean tap water and then added 10 μm neutrally buoyant particles of varying concentration. In all experiment runs, either ∊ or the χ was varied by a factor up to 100 at a distance of z₁ = 6.3 m away from the grid. The turbulent dissipation rates in our experiment correspond to the ones observed within the upper ocean layers [21].

The depolarization D(z) was calculated from the lidar returns as $D (z) = P_{x}^{*} (z) / P_{c}^{*} (z)$ and is shown in the Fig. 3. In each run, D(z) was found to be approximately proportional to the distance z, consistent with the theoretical analysis of [11] or [14], where they found D(z) ∝ (χ∊^1/4) · z when considering depolarization on turbulent flow without particles. The mean attenuation of polarized light, α, was obtained via a least square fit to co-polarized intensities over the nominal interval- $P_{c}^{*} (z)$ - see Fig. 3 and then by applying Eq. (17).

The mean value of the depolarization rate γ was similarly obtained via a least square fit to D(z) over the nominal interval in Fig. 3 and then calculated from the Eq. (13) - see Table 1. Uncertainty estimates for all quantities reported in Table 1 and Fig. 4 were calculated from their sample variability and presented here as twice the standard deviation - thus corresponding to a 95.5% confidence interval.

Fig. 4 Left: Lidar measured: γ/α derived from Eq. (15) and Eq. (13) (averaged over nominal interval). Black circles denote runs seeded with 10μm particles and red dots - water laden with tank debris particles. Right: Comparison of lidar measured γ (black symbols with run names) in our grid turbulence flow with the data of [16] from the convective turbulent flow (blue dots). All errors are reported as two standard deviations.

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Previous measurements have documented the dependence of the depolarization rate γ on the turbulent flow strength [16]. Measurements in [16] were carried out with a continuous-wave laser, in a small 0.3 m long tank filled with purified water, where the authors generated a turbulent convective flow. Their setup permitted only to investigate the effects of nearforward depolarization with the implication that the measured rate γ did not include any contribution from particle backscattering - therefore their measured rate γ included only effects of the forward scattering. They also observed a variability of γ with the ambient particle concentrations.

In order to explore the effect of particles on the depolarization rate we note that, in general, at low particle concentrations (i.e.: when the single scattering assumption is valid [22]), we have c ≡ (α − γ) ∝ N_part · c^*, with N_part being the total number of particles in the scattering volume and c^* the single particle attenuation coefficient. Similarly, we expect the polarized beam attenuation coefficient α at low particle concentration to be additive i.e., to follow α ∝ N_part · α^*, where α^* is the single particle polarized beam attenuation. Consequently, at low particle concentrations, γ has to vary with particle concentration as γ ∝ N_part · γ^*, where γ^* is the equivalent single particle depolarization rate. As established earlier, the contribution to γ from particle backscattering events can be neglected when δ is small, as demonstrated by Eq. (9), Eq. (11) and the subsequent discussion. Thus the equation γ ∝ N_part · γ^* represents the effect of forward light scattering on turbulent inhomogeneities with γ^* being an effective coefficient that combines forward light depolarization by turbulence and single particle scattering.

Theoretical considerations in [11] or [14] for ’pure’ turbulent flows demonstrated that γ is a function of χ∊^1/4. Figure 4 shows our results for the ratio γ/α as a function of χ∊^1/4. A least square fit to our data gives the following relationship: γ/α = −0.017 log₁₀(χ∊^1/4) − 0.089, with R² = 0.81. R² is the ratio of the explained variance to the total variance. We can now evaluate the effect of varying particle sizes. In Fig. 4, the black symbols represent data collected when the tank was seeded with monodisperse 10μm particles. The red symbols denote runs when the tank was filled with polydisperse debris particles only - DEC31 runs. Debris particles in the DEC31 runs were likely distributed over a large interval of sizes from the smallest up to millimeter-sized pieces that were clearly visible in the water. This wide particle size distribution was likely a result of the water pump action breaking up larger debris particles. Values of γ/α in the DEC31 runs containing polydisperse particles, as well as runs seeded with monodisperse 10μm particles, follow γ/α = −0.017 log₁₀(χ∊^1/4) − 0.089. This result evidently suggests that the details of the particle size distribution were not important when the depolarization γ is normalized as the ratio γ/α and plotted as a function of (χ∊^1/4).

We thus posit the hypothesis that the observed variability of the nearforward depolarization rate γ is the result of a collective forward light scattering on the smallest particles and the refractive index inhomogeneities in the turbulent flow. Based on that assumption and in line with the earlier argument we expect the dependence of γ on the particle concentration to follow: γ = N_part · γ^* + γ_turb where γ_turb is the ’pure’ turbulent flow (i.e.: N_part = 0) forward depolarization rate. The depolarization rate γ^* is, as established earlier, an equivalent combined single particle and turbulent forward light depolarization coefficient.

The hypothesis that γ = N_part · γ^* + γ_turb is consistent with theoretical prediction in [11] [12] or [14]. This can be easily observed if we set N_part → 0, in which case γ reaches its smallest value γ = γ_turb. In the case of ’pure’ aquatic turbulent flow, the upper bound on that depolarization rate is γ_turb ≈ 10⁻¹⁵ m⁻¹. We can verify the above hypothesis by comparing our measurements of γ with results in [16], which were obtained for a convective turbulent flow of purified fresh water in an experiment that was only capable of measuring the contribution to γ arising from nearforward depolarization. For the sake of comparison, we have selected from our observations a set of runs with the smallest particle concentrations, i.e. with the lowest values of α (or unpolarized light attenuation c) - Table 1 - runs JAN2BX, JAN4TESTA, and JAN4AA. Figure 4 presents the results of this comparison in terms of the dependence of γ on χ· ∊^1/4. In Fig. 4, the blue dots represent the data from the convective turbulence measurements in [16] while the black circles with labels correspond to our lidar data. We observe an excellent agreement between our observations. This finding is consistent with our earlier argument that in our lidar observation, γ is mainly determined by nearforward light depolarization on turbulent inhomogeneities and particles. Furthermore, since the experiment in [16] was carried out in the purified water with a very low particle concentration γ in Fig. 4 represents a lower bound on the oceanic values of γ, such that any oceanic lidar measurements should yield γ ≥ 10⁻³ m ⁻¹. That level of depolarization rate γ is easily accessible to current lidar systems such as a plane based lidar system of [6] or to a space based observation such as the Cloud Aerosol Lidar with Orthogonal Polarization (CALIOP) launched as part of the CALIPSO instrument [23].

The relative insensitivity of our measured relationship γ/α = −0.017 log₁₀(χ∊^1/4) − 0.089 to the particle concentration or to their size distribution suggests that this type of relationship can become a tool in remote aquatic turbulence measurements. One should use some caution when applying our results to the oceanic data. Even though in our experiment, the turbulence variability corresponds to a significant subset of values expected in the oceanic mixed layer and covers roughly 2 orders of the magnitude of turbulent dissipation variability, ranges of particle concentration and their size distribution is a very small subset of the oceanic particle variability. We thus plan to conduct further research where we will use a more representative particle concentration and size distribution range in order to establish a more robust relationship between γ/α and (χ∊^1/4).

5. Conclusion

We have observed the nearforward light depolarization parameter γ to be dominated by turbulent processes and particle concentration. The presence of particles in the flow modifies the values of γ in such a way that γ/α becomes independent of the particle concentration and depends only on the turbulent flow parameter: χ· ∊^1/4. Our data indicate that in the limit of low particle concentration, γ likely reaches its lowest value, as derived for ’pure’ turbulent flow. We are unsure yet of the physics/optics details of nearforward depolarization by turbulent flow inhomogeneities laden with particles. The likely candidate for that optical process is the collective forward light scattering on particles embedded in the refractive index inhomogeneity. The independence of the parameter γ/α on the particle concentration combined with the likely insensitivity of γ/α to details of particle size distribution points to new applications of lidar systems to remotely measure turbulence strength in aquatic flows.

Funding

Caltech Director’s Research Discretionary Fund (DRDF) 2012 award; The Gulf of Mexico Research Initiative grant; National Science Foundation (NSF) (1434670).

Acknowledgments

We would like to thank Tyler Schlenker for help with preparation of the experiment and its execution. We thank NASA/JPL and Gary Spiers for lending the lidar system, Gary Spiers for help with lidar setup, its alignment, and the data acquisition and for numerous discussions and advice throughout the experiment. We thank Jim Silliman for proofreading and comments.

References and links

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16. S. Woods, J. Piskozub, W. Freda, M. Jonasz, and D. Bogucki, “Laboratory measurements of light beam depolarization on turbulent convective flow,” Appl. Opt. 49, 3545–3551 (2010). [CrossRef] [PubMed]

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Run name	U [ms⁻¹]	P [kW]	∊ [m²s⁻³]	χ [°C²s⁻¹]	(α ± 2 σ_α) × ×10²[m⁻¹]	(γ ± 2 σ_γ) × ×10³ [m⁻¹]

Dec31CC*	0.27	2.7	3.0e-06	3.5e-08	6.6 ± 1.6	3.9 ± 0.1
Dec31DD*	0.27	8.3	3.0e-06	3.3e-07	11.1 ±1.5	6.3 ± 0.1

Jan1CC	0.38	8.3	8.5e-06	2.3e-07	8.5 ± 1.2	3.5 ± 0.2
Jan1DD	0.14	8.3	4.2e-07	6.3e-07	8.0 ± 0.6	3.7 ± 0.1

Jan2AA	0.40	2.7	9.6e-06	2.4e-08	5.6 ± 2.1	2.5 ± 0.1
Jan2AB	0.40	2.7	9.6e-06	2.4e-08	5.7 ± 1.0	2.7 ± 0.04
Jan2BX	0.09	0.78	1.0e-07	1.7e-08	5.3 ± 1.9	2.8 ± 0.1

Jan3DD	0.41	3.6	1.1e-05	4.1e-08	7.0 ± 0.7	2.4 ± 0.2

Jan4TestA	0.09	0.78	1.3e-07	1.6e-08	4.5 ± 2.5	2.1 ± 0.1
Jan4TestB	0.09	2.7	1.3e-07	9.9e-08	5.8 ± 1.8	2.5 ± 0.1
Jan4TestC	0.09	3.6	1.3e-07	1.8e-07	7.2 ± 1.2	3.3 ± 0.1
Jan4GG	0.18	2.7	8.5e-07	5.3e-08	5.7 ± 1.0	2.3 ± 0.1
Jan4HH	0.18	3.6	8.5e-07	9.4e-08	6.5 ± 0.9	2.9 ± 0.04
Jan4II	0.18	8.3	8.5e-07	5.0e-07	7.9 ± 0.8	3.0 ± 0.1

Polarimetric lidar measurements of aquatic turbulence - laboratory experiment

Abstract

Corrections

1. Introduction

2. Theory

3. Experiment setup

4. Results

5. Conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (18)

Optics Express