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Remote Measurements of Horizontal Eddy Diffusivity DAREK J. BOGUCKI Division of Applied Marine Physics, Rosenstiel School of Marine and Atmospheric Science, University of Miami, Miami, Florida
BURTON H. JONES Department of Biological Sciences, University of Southern California, Los Angeles, California
MARY-ELENA CARR Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California (Manuscript received 12 August 2004, in final form 27 January 2005) ABSTRACT The rate of horizontal diffusivity or lateral dispersion is key to understanding the dispersion of tracers and contaminants in the ocean, and it is an elusive, yet crucial, parameter in numerical models of circulation. However, the difficulty of parameterizing horizontal mixing is exacerbated in the shallow coastal ocean, which points to the need for more direct measurements. Here, a novel and inexpensive approach to remotely measure the rate of horizontal diffusivity is proposed. Current shipboard measurement techniques require repeated surveys and are thus time consuming and labor intensive. Furthermore, intensive in situ sampling is generally impractical for routine coastal management or for rapid assessment in the case of emergencies. A remote approach is particularly useful in shallow coastal regions or those with complex bathymetry. A time series of images from a dye-release experiment was obtained with a standard three-megapixel digital camera from a helicopter that hovered over the study area. The red–green–blue (RGB) images were then 1) analyzed to distinguish the dye from the ambient color of the water and adjacent land features, 2) orthorectified, and 3) analyzed to obtain advection and diffusion rates of the thin subsurface dye layer. A horizontal current of the order of 6 cm s⫺1 was found. The estimated horizontal eddy diffusivity rate for scales of O(10 m) in the harbor was 0.1 m2 s⫺1. The dye diffusivity and advection rate that are calculated from the images are consistent with independent calculations based on in situ measurements of current speed fluctuations.
1. Introduction The accurate assessment of mixing is an important yet challenging task for oceanography, especially in the dynamic and complex coastal ocean. Mixing rates are necessary to quantify the dispersive processes that affect the distribution of naturally occurring dissolved and particulate matter, as well as contaminants released into the sea. Because environmental flows are characterized by large Reynolds numbers, the exact solution
Corresponding author address: Dr. Darek J. Bogucki, Division of Applied Marine Physics, Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Cswy. Miami, FL 33149-1098. E-mail:
[email protected]
© 2005 American Meteorological Society
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of the equation of motion is beyond the scope of the present numerical capabilities. In practical applications, we use the averaged equation of motion and the parameter known as eddy diffusivity, which contains information about the energetics of small, unresolved scales of fluid motion. The horizontal eddy diffusivity that is associated with turbulent horizontal dispersion can vary over a few orders in magnitude in the coastal environment, that is, from 0.1 m2 s⫺1 up to O(10) m2 s⫺1 (Okubo 1971; Csanady 1980; Sundermeyer and Ledwell 2001). The relative roles of horizontal and vertical diffusivity are illustrated in the following example. Imagine a contaminant that is initially bounded by a surface area of 1 m2 and extends 1 m in depth to form a nearly neutrally buoyant plume. The initial pollutant volume is V0 ⫽ x0 ⫻ y0 ⫻
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z0, where x0 ⫽ y0 ⫽ z0 ⫽ 1 m and z is the vertical coordinate. Assuming horizontal homogeneity and the absence of horizontal and vertical shear, we can express the normalized pollutant volume change dV/V0 as dV/V0 ⫽ 2dx/x0 ⫹ dz/z0 ⫹ O([dx/x0]2), where changes dx, dy, and dz correspond to an increase of the pollutant volume dV. If the volume change is the result of Fickian diffusivity (Sundermeyer and Ledwell 2001), then the corresponding linear change of the pollutant volume is related to the associated diffusion coefficients (x, y, z), such that x ⫽ 1/2 ⫻ dx2/dt for the x direction. This results in the following expression for the pollutant dispersion rate: (1/V0) ⫻ dV/dt ⫽ (2x /x02) ⫹ (z/z02). Given an assumed vertical eddy diffusivity of 10⫺3 m2 s⫺1 for an energetic near-surface layer (Rehmann and Duda 2000) and a horizontal diffusivity of ⫺1 10 m2 s⫺1 corresponding to the lower bound observed for near surface horizontal diffusivity rates (Sundermeyer and Ledwell 2001), then horizontal mixing is responsible for at least 95% of the pollutant dispersion in this scenario. This underlines the importance of horizontal eddy diffusivity for environmental applications. Current shipboard approaches to measure mixing are difficult, being both time consuming and labor intensive (Tennekes and Lumley 1972). Horizontal diffusivity is usually measured by releasing dye and recording its concentration along an isopycnal (density surface) at intervals ranging between a few minutes and few years (Ledwell and Watson 1991; Sundermeyer and Ledwell 2001). After correcting for lateral shear, this measurement yields the “true” horizontal diffusivity value, that is, the irreversible lateral dispersion. In an infinitesimally small initial patch, the lateral shear correction is negligible. For a review of tracer-release experiments, see Watson and Ledwell (2000). In some dye experiments (Sundermeyer and Ledwell 2001), the vertically integrated dye concentration is measured instead of the isopycnal concentration. The data from this type of measurement have to be corrected for the both horizontal and vertical shear to give the true horizontal diffusivity. Therefore, measurements of isopycnal dye concentrations are generally easier to interpret in terms of horizontal diffusivity. Although in situ dye measurements are the most accurate approach (Watson and Ledwell 2000), they are frequently of limited practical application. Dye injections of a reduced spatial extent may be preferable in small regions or shallow environments (especially where water depth is of a few meters). When the dye occupies a small area (which is necessary if the study region is small), the frequent in situ traverses that are required for precise mapping of the dye location can
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themselves disturb the dye distribution. Furthermore, the cost may be prohibitive for routine coastal management, and repeated surveys may be impractical in the case of emergencies, when a rapid estimate of diffusivity is required or intensive in situ sampling is not feasible. Under such circumstances, remote means to map the dye location may prove to be more appropriate. In this paper we present a noninvasive, inexpensive approach to remotely map the dye location along isopycnals to quantify the true turbulent horizontal eddy diffusivity that is associated with horizontal irreversible dispersion (diffusivity) in the near-surface layer of the coastal ocean. Our approach is based on taking photographs with a standard digital camera from a hovering helicopter during dye-release experiments. Our test experiment was carried out in Avalon Harbor in Catalina Island (California) in October 2001. We show that our optical approach can be used to analyze the Lagrangian behavior of the water parcels, such as in the studies of Richardson (1926) and Sundermeyer and Ledwell (2001). Our objective thus is to validate the use of standard digital photographs to map the time series of dye spreading for short-term horizontal mixing studies. First, we address the optical properties of the dye, deriving two optical limits corresponding to the regimes of either vertically integrated or isopycnal-marking dye. Then, we discuss the method to identify the dye regimes and to discriminate between them. We show that the isopycnal-marking regime presents a unique optical property that renders it suitable for the study of Lagrangian isopycnal processes. Because the relation between the observed signal and the concentration is strongly nonlinear, it is possible to identify and follow the dye isoconcentration. A measure of horizontal eddy diffusivity is obtained by following the isopycnal dye isoconcentration contour. This remotely based measurement of diffusivity is compared with an in situ estimate. Throughout the paper we use the term dye and tracer interchangeably because the optical properties of the dye allow us to trace it in time.
2. The dye experiment The experiment was carried out in Avalon Harbor of Catalina Island, California (33.34°N, 118.2°E) (Fig. 1). Average water depth is 10–15 m, and the maximum depth is ⬃30 m at the entrance to the harbor. The stratification is two-layer-like, with a density jump of ⬃25 m from the surface. Three acoustic Doppler current profilers (ADCPs) were deployed across the harbor mouth from 25 September 2001 through 6 November 2001 for a total of 42 days. Two of the strongest
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FIG. 1. Aerial view from the helicopter of Avalon Harbor. The pink-shaded area represents the surface covered with the dye Rhodamine WT.
components of the flow through the mouth are thesemidiurnal tide (M2, period 12.42 h) and the mixed luni– solar diurnal tide (K1, 23.93 h). The major axis of the dominant M2 tidal ellipse corresponds to a speed of a few centimeters per second at the surface. A series of dye experiments with Rhodamine were performed to examine dispersive processes within the harbor. Rhodamine WT was chosen because it has a relatively slow light-induced decay rate and does not readily absorb onto particles, making it a conservative tracer in surface waters Suijlen and Buyse (1994). Here we use the data from the dye-release experiment of 10 October 2001. The dye was premixed with ambient subsurface water (⬃10 cm below the surface) and was released to provide an initial in situ concentration of 10–50 g L⫺1; the corresponding optical absorption coefficient exceeds 10 m⫺1. The dye was deployed at the surface of the central harbor in a cross pattern (Fig. 1). A time series of images of the spreading dye at ⬃1 min intervals was obtained with a standard threemegapixel digital camera (Canon G-1) from a helicopter that hovered over the harbor for approximately 1 h.
a. Optical properties of the dye in the water and dye identification The optical quantity that the camera detects is the remotely sensed reflectance, that is, the radiance reflectance (R), which has been normalized by the anisotropy of the near-surface upwelling light field, known as the Q factor Walker (1994). The Q factor is typically around 1. In this study, we assume that Q ⫽ 1 and is constant, and that the measured radiance reflectance is
a good approximation for the remotely sensed reflectance (Walker 1994; Barnard et al. 1999). An analytical solution to the radiative transfer (RT) equations in terms of down-/upwelling radiance is not generally possible for a realistic ocean because the optical properties of the varying water constituents cannot be explicitly determined. However, fewer parameters are required in the integral version of the RT equation. Here we describe light propagation in terms of angularintegrated down-/upwelling radiance, that is, the twostream model version of the RT equations. At the water surface the radiance reflectance R is defined as the ratio of the upwelling irradiance Eu(z ⫽ 0) to downwelling irradiance Ed(z ⫽ 0) at the surface (z ⫽ 0), that is, R ⫽ Eu(z ⫽ 0)/Ed(z ⫽ 0) (Walker 1994). The coefficient R always depends on wavelength, but here for brevity, it is not expressed explicitly. The starting point for our description is the twostream model; in the absence of in situ light sources (Mobley 1994), the two-flow RT equations can be written as d关Ed共z兲兴 ⫽ ⫺kduEd ⫹ BudEu, dz
共1兲
d关Eu共z兲兴 ⫽ kudEu ⫺ BduEd, dz
共2兲
where kdu and kud are the attenuation of light on its way down or up, respectively, resulting from absorption and scattering (the two are equal to first order). Likewise, Bud and Bdu are the backscattering coefficients, assumed as being equal, and hereafter shown as B. Assuming that Ed(z) Ⰷ Eu(z) with appropriate boundary conditions, Ed(z ⫽ 0) ⫽ Ed0, and Eu(z ⫽ ⬁) ⫽ 0, we obtain
R共z ⫽ 0兲 ⫽
冕
z
0
冋 冕
z⬘
B共z⬘兲exp ⫺2
0
册
K共z⬙兲 dz⬙ dz⬘, 共3兲
where K(z⬘) is the depth-dependent attenuation coefficient. For our observations we assume that 1) the dye is fully dissolved in a second layer of constant thickness, 2) the dye contributes to absorption in this layer only, 3) the dye is separated from the surface by an optically thin layer of constant thickness, and 4) the layer underlying the layer of dye is infinitely deep optically. If we additionally assume that all the optical coefficients are a piecewise constant function of depth in three layers, then the solution is
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R共z ⫽ 0兲 ⫽
兺 2
再 冋 冕
1 bi exp ⫺2 2 ki
冋 冕
⫺ exp ⫺2
Zi
0
Zi⫹1
0
K共z⬘兲
K共z⬘兲 dz⬘
册冎
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dz⬘.
共4兲
We use the following notation for each layer: the attenuation coefficient is ki ⫽ ai ⫹ bi, where ai and bi are, respectively, the absorption and scattering coefficients of the ith layer. We assume that the dye has fully dissolved in the second layer, and after dissolution it only affects the absorption coefficient of the layer, such that in the presence of the dye it becomes equal to a2 ⫹ a⬘2 where a2 is the background absorption coefficient and a⬘2 is its increase resulting from the dye. The physical quantity that we measure directly from the photographs is not the reflectance R itself, but rather the spectral contrast in R between dye-laden and -free water ␦ R(x0, x1):
␦R共x0,x1兲 ⫽ R共x0兲 ⫺ R共x1兲,
共5兲
where x0 denotes the water with dye and x1 denotes values of R for water in locations free from dye. In our approach we use ␦ R(x0, x1) to mark the dye location instead of R, because we cannot follow the dye-marked water parcel indefinitely; at a given point its optical signature weakens resulting from diffusion, photodegradation, or changes in the ambient optical conditions. Defining a threshold spectral contrast level ␦ R(x0, x1) ⫽ ␦0, below which the dye becomes indistinguishable from the background, provides an efficient criterion to follow the dye location. Guided by our observations we distinguish two important optical limits: the optically thin and optically thick dye layers. As we demonstrate below, these two optical scenarios, which are two distinct limiting cases in the mathematical sense, are associated with different physical processes in the water column. In the optically thin dye-layer limit, we assume that the product of absorption resulting from the dye and the layer thickness a⬘2 ⫻ 䉭z2 Ⰶ 1 is small and the dye absorption is larger than the layer attenuation coefficient a⬘2 ⬎ k2. This yields
␦R共x0,x1兲 ⫽ ⫺3T 21共1 ⫺ T 23兲共a⬘2⌬z2兲 ⫹ O共a⬘2⌬z2兲2, 共6兲 where the transmission through layers 1 and 2, that is, T1, is given by T1 ⫽ exp[⫺(a1 ⫹ b1) ⫻ ⌬z1], T3 ⫽ exp[⫺(a3 ⫹ b3) ⫻ ⌬z2], and 3 is the single scattering albedo of layer 3; the calculation is carried out to the second order in (a⬘2䉭z2)2 The minus sign results from
FIG. 2. The value of a⬘2 /(a ⫹ b ⫹ a⬘2) as a function of a⬘2, and assuming a fixed value of a ⫹ b ⫽ O(0.1) m⫺1.
the fact that the layer with the dye is strongly absorbent. The strength of the optical contrast then depends on the scattering albedo and transmission of the underlying layer (layer 3) and the integrated amount of the dye in the layer, that is, the product (a⬘2䉭z2). In this limit the dye optical signature ␦R(x0, x1) is proportional to the dye absorption a⬘2. From a remote sensing perspective, the situation is dramatically different when the dye creates an optically thick layer. In this case, the dye concentration (proportional to the dye absorption a⬘2) is large, such that a⬘2 Ⰷ (a2 ⫹ b2). a⬘2 Additionally, the dye is contained in the layer of thickness ⌬z2, such that exp[⫺2(a⬘2 ⫹ a2 ⫹ b2) ⫻ 䉭z2] Ⰶ 1. Assuming that absorption a and backscattering b are similar in layers 1 and 2, the optical contrast can then be calculated as
冉
1 a⬘2 ␦R共x0,x1兲 ⫽ ⫺ 2T 21 ⫻ 2 a ⫹ b ⫹ a⬘2 ⫻ 关1 ⫹ O共T 21T 22T 2a兲兴,
冊 共7兲
where 2 is the single scattering albedo of layer 2; T1 and T2 are the transmission through layers 1 and 2 all in the absence of the dye, and Ta ⫽ exp(⫺a⬘2䉭z2) is the fraction of energy absorbed by the dye. For a large concentration of dye, the value of the first bracketed expression is constant (⯝ 1) and independent of the dye concentration. The ␦ R(x0, x1) is then independent of both dye concentration and layer thickness; it attains a value of ⫺(1/2) ⫻ w2T 21. That is, the photons reaching the camera have traveled across the thin upper layer and reflected once from the thin dye sublayer. The thickness of this sublayer is ⬇ (1/2) ⫻ 1/a⬘2. In our experiment, the initial dye concentration and corresponding dye absorption was a⬘2 ⬇ 10 m⫺1, and
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the coefficient of light attenuation in the harbor a ⫹ b ⬇ 0.1 m⫺1 The dye was contained in a 20-cm-thick layer such that exp[⫺2(a⬘2 ⫹ a2 ⫹ b2) ⫻ 䉭z2] ⬇ 0.2. Figure 2 shows that the value of a⬘2/(a ⫹ b ⫹ a⬘2) is practically independent of the dye concentration a⬘2 when a⬘2 exceeds 1 m⫺1. The steplike dependence of ␦ R(x0, x1) on a⬘2 [Eq. (7) and Fig. 2] allows the isoconcentration contour of a⬘2 ⬇ 2 to be identified unequivocally and makes it possible to use the dye in the optically thick regime as an optical isopycnal tracer. Initially (up to t ⫽ 4000 s), the dye is in the optically thick regime and the photographic time series reflects movement and diffusion of the 5-cm- [⫽ (1/2) ⫻ 1/ a⬘2 ] thick dye sublayer. We can further assume, following Sundermeyer and Ledwell (2001), that the movement of this thin iospycnal layer provides an estimate of horizontal diffusivity after correcting for lateral shear. To locate the water parcels with dye, the spectral contrast must exceed the threshold value of ␦0. Therefore, ␦ R(x0, x1) ⬎ ␦0 determines the spatial extent of the dye cloud in the water for any given image. The red–green–blue (RGB) images were analyzed first to distinguish the dye from the ambient color of the water and adjacent land features. We also arbitrarily set the threshold value of the contrast ␦0, which was determined by how well we could distinguish the dye cloud. Then, we divide the visible dye into “new” and “old” dye where the former corresponds to the optically thick regime. Specifically, the properties of new and old dye were obtained in the following way: the properties of wavelength versus intensity of the dye and of the surrounding water (Mobley 1994) were obtained from a series of test photos. These properties provided the defining range for the dye presence ␦0. The earliest photos, with the highest dye concentration, were used to optically characterize the new dye. These experimentally determined ranges were then used to process the remaining photographs, and the dye-covered area was covered with a binary mask. The result of this operation is shown in Fig. 1, where the pink streak is the location of the new dye, overlaying the original image. This stage of analysis is complete when the extent of the new dye has been mapped on the time series of images.
b. Image mapping, orthorectification The next step is to correct the individual photographs to a common geographic location, which, in this case, is equivalent to transforming the time series of images to a fixed orthocorrected image, also referred to here as the initial image. Any given image can be mapped onto another image of the same two-dimensional area using
a projective transformation of the form (Tsai and Huang 1984) ˆ⬘⫽ X
ˆX ˆ ⫹ bˆ A , ˆ ⫹1 cˆ X
共8兲
ˆ ⬘ ⫽ [x⬘, y⬘] are the coordinates of the point where X in the mapped image, and X ⫽ [x, y] are the coordinates ˆ , bˆ, and cˆ are maof the point on the initial image; A trices of 2 ⫻ 2, 2 ⫻ 1, and 1 ⫻ 2 dimensions, with unknown coefficients, yielding a total of eight unknown coefficients. The coefficients of the transformation for each image are found using at least four pairs of points (four on the initial image and four on the mapped image), such that the chosen parameters of the projective transformation were optimal in the least squares sense. Here we have used six pairs of points for each image. The resultant pixel size of the resulting orthorectified image was at least 0.25 m or better, usually around 0.1 m. The orthorectification procedure leads to a series of binary masks (a total of 76), which provide the dye location at a given time. The ensemble view of the distribution of the optically thick dye layer for successive time steps and after orthorectification is shown in Fig. 3.
3. Results a. Horizontal diffusivity from the time series of images The ensemble of binary masks from Fig. 3 was used to estimate mean advection and the rate of shear. In the shallower part of the harbor (10-m water depth), the estimated advection ranges from 0 to 7 cm s⫺1 in the center of the figure. This sets the Reynolds number of the flow at ⬃106. In addition, we estimate (based on Fig. 3) the horizontal shear rate ␥ ⫽ u/y ⫽ 4 ⫻ 10⫺4 s⫺1 and /x to be 1 ⫻ 10⫺4 s⫺1. We use the approach of Okubo and Karweit (1969) or Csanady (1980) and Sundermeyer and Ledwell (2001) to obtain an expression describing the horizontal rate of dye dispersion. We assume that the initial dyepatch linelike structure collapses into a set of ellipsoids, each characterized by the length of major X(t) and minor Y(t) axes. The major axis X(t) of the largest ellipsoid is approximately aligned with the x axis, shown on Fig. 3. The ellipsoids are embedded in a flow with the mean velocity U( y) ⫽ U0 ⫹ ␥ y, where ␥ ⫽ u/y is the horizontal shear and U0 constant background current. Following Okubo and Karweit (1969) we assume that the dye patch was a point source initially located at x0 and
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FIG. 3. Ensemble of orthorectified dye images observed during the initial 11 min of the experiment. Different colors correspond to different time steps.
y0 ⫽ 0. In this situation, the horizontal diffusion of a patch is subject to the equation ⭸C ⭸C ⫹ 共U0 ⫹ ␥ y兲 ⫽ ⵜ2C. ⭸t ⭸x
共9兲
major X(t) and minor Y(t) axes of the dye isoconcentration C[X(t), Y(t), t] ⫽ C0. To the first order we obtain, in nondimensional time,
⫽ X共t兲2Ⲑt ⫹ Y共t兲2Ⲑt ⫺
The solution is
冋
册
q y2 ⫺共x ⫺ U0t ⫺ 1Ⲑ2␥ yt兲2 exp ⫺ , C共x,y,t兲 ⫽ 2xy 2 2x 2 2y 共10兲
2y
2x
2y
⫽ t, ⫽ [1 ⫹ (1/12)␥ t ], and is the where constant and uniform horizontal diffusion coefficient. The quantity ␥ t is a nondimensional time such that shear dispersion becomes relevant when |␥ t| ⬎ 1. From the solution of Eq. (10), we find the value of the diffusivity as a function of the location of the 2
2
⫹
冋
冋
1 dX共t兲2 dY共t兲2 ⫹ 4 dt dt
册
册
1 dX共t兲 dY共t兲 ⫹ X共t兲 Y共t兲 共␥t兲 ⫹ O共␥t兲2. 4 dt dt 共11兲
For a short time after release, that is, |␥ t| ⬍ 1, and for elongated ellipsoids, that is, Y(t) ⫽ X(t), the expression for becomes
⫽ X共t兲2Ⲑt ⫺
冋 册
1 dX共t兲2 . 4 dt
共12兲
The value of the mapped dye concentration is set by the
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FIG. 4. (top) Time series of the length of the largest dye patch and of the calculated value of (1⁄2)␥Y(t)t; (bottom) horizontal diffusivity , calculated from the Eq. (12) is shown.
choice of the threshold parameter ␦0, which is also used to delineate the new from the old dye. The value of the diffusivity coefficient is obtained by analyzing the time series of the size of the largest new dye patch. This dye patch is very elongated in the x axis, and we used Eq. (12) to calculate the diffusivity. The time series of the length of the largest patch and the corresponding horizontal eddy diffusivity (horizontal rate of increase in size of the major axis) is shown in Fig. 4. Following Sundermeyer and Ledwell (2001), the value of true diffusivity for an infinitesimally thin dye patch is given solely by the rate of dye spreading. Hence, by accurately measuring the spreading of an isopycnal dye patch, we obtain the time series of diffusivity without correcting for vertical shear. Such calculated horizontal diffusivity is a true (irreversible) measure of diffusivity, because it corresponds to measurements of dye concentration along thin isopycnal layers. Our measured value of horizontal diffusivity is O(0.1 m2 s⫺1). The length of the experiment is limited by the initial concentration of the dye (for a higher initial dye concentration, the optically thick regime can be measured longer) and by the contribution of the shear term, which, in the case of our experiment, would become important after time t ⫽ 104 s.
ing arguments from Csanady (1980). We assume that the extent of the major axis of the dye patch is marked by a separated set of fluid particles and that their separation distance is related to their rms velocity u⬘ as
2共t兲 ⫽ 2u⬘2
冕
t
共t ⫺ s兲共s兲 ds,
共13兲
0
where (s) is the Lagrangian autocorrelation function of particle horizontal position. We can then use the relation ⫽ 1/2d(t)2/dt (Csanady 1980) to estimate . For a dye patch a few meters in thickness, and for a time period of O(L/u⬘), where u⬘ is the rms of velocity at energy containing scales, and L is the large-eddy size, such that (L/u⬘) represents a large-eddy time scale, we find
⫽ 共u⬘兲2
L . u⬘
共14兲
Our ADCP measurements of velocity fluctuations yield u⬘ ⫽ 2 cm s⫺1 at scales of the order of L ⬃10 m, leading to estimates of of ⬃0.2 m2 s⫺1. This value is consistent with the previous photographic-based estimate of ⬃0.1 m2 s⫺1.
b. Independent estimate of horizontal diffusivity
4. Conclusions and impact
For the case of vanishing large-scale shear ␥ ⫽ 0, we carried out an independent estimate of diffusivity us-
• We show that remote photographs of dye spreading
can be used to obtain an estimate of horizontal
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diffusivity for scales on the order of kilometers and a few hours. Because it is necessary to use the optically thick limit to obtain a measure of the isopycnal, the diffusivity can be obtained with relative ease if the dye patch is sufficiently small (i.e., without needing to estimate shear later). Consequently, this approach is most indicated for short-term studies of mixing rates. The present experimental approach can be rapidly deployed with reasonably short notice and is not very expensive, requiring only a small vessel to release the dye and an observing platform, which enables imaging of the entire scene. It is thus well suited for routine management studies or for emergency situations in either coastal or inland water sites. The diffusivity rates that are obtained from the measurement can be used to validate numerical models or to guide the selection of an appropriate model parameterization. They can also be used directly to estimate the time scale of dilution for contaminants that are released into a body of water to predict the impact on protected locations or those that are heavily used (such as beaches). In our study, we obtained a horizontal diffusivity value that is close to the lower values observed in a shelf environment using longer-term in situ sampling (Sundermeyer and Ledwell 2001). Our value, 0.1 m2 s⫺1, is consistent with low-energy conditions, that is, the prevailing low wind speeds and stratified water column.
Acknowledgments. We thank Rob Clark and the city of Avalon (California) for their support of this project through the California Clean Beaches Initiative. Logistical support was provided by Brian Bray, Harbormaster, City of Avalon. John Moore and Island Express provided the helicopter for the aerial photography. Technical support was provided by Matthew Ragan and Zhihong Zheng (USC). We also thank anonymous reviewers for their comments, which led to a muchimproved paper. Funding for M.E.C. was provided by
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the NASA Ocean Biogeochemistry Program within the Carbon Cycle Science Program. A portion of the research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. REFERENCES Barnard, A. H., J. V. Zaneveld, and W. S. Pegau, 1999: In situ determination of the remotely sensed reflectance and the absorption coefficient: Closure and inversion. Appl. Opt., 38, 5106–5117. Csanady, G. T., 1980: Turbulent Diffusion in the Environment. D. Reidel Publishing, 250 pp. Ledwell, J. R., and A. J. Watson, 1991: The Santa Monica tracer experiment: A study of diapycnal and isopycnal mixing. J. Geophys. Res., 96, 16 709–16 719. Mobley, C. D., 1994: Light and Water: Radiative Transfer in Natural Waters. Academic Press, 592 pp. Okubo, A., 1971: Oceanic diffusion diagrams. Deep-Sea Res., 18, 789–802. ——, and M. J. Karweit, 1969: Diffusion from a continous source in a uniform shear flow. Limnol. Oceanogr., 14, 514–520. Rehmann, C. R., and T. F. Duda, 2000: Diapycnal diffusivity inferred from scalar microstructure measurements near the New England shelf/slope front. J. Phys. Oceanogr., 30, 1354– 1371. Richardson, L. F., 1926: Atmospheric diffusion shown on a distance-neighbour diagrams. Proc. Roy. Soc. London, 110A, 709–737. Suijlen, J., and J. Buyse, 1994: Potentials of photolytic rhodamine WT as a large-scale water tracer assessed in a long-term experiment in the Loosdrecht lakes. Limnol. Oceanogr., 39, 1411–1423. Sundermeyer, M. A., and J. R. Ledwell, 2001: Lateral dispersion over the continental shelf: Analysis of dye release experiments. J. Geophys. Res., 106C, 9603–9621. Tennekes, H., and J. L. Lumley, 1972: A First Course in Turbulence. MIT Press, 301 pp. Tsai, R. Y., and T. S. Huang, 1984: Multiframe image restoration and registration. Trans. J. ACM, 31, 317–339. Walker, R. E., 1994: Marine Light Field Statistics. Wiley Interscience, 660 pp. Watson, A. J., and J. R. Ledwell, 2000: Oceanographic tracer release experiments using sulphur hexafluoride. J. Geophys. Res., 105 (C6), 14 325–14 337.
