Taylor's single-particle dispersion model is revisited and applied to unstratified and density stratified flows using observationally based and theoretical models of the Lagrangian velocity and density spectra, which are compared with existing parameterizations of diapycnal diffusion in these flows. For unstratified homogeneous turbulence, the vertical particle dispersion coefficient Kz computed from model Lagrangian velocity spectra agrees well with contemporary estimates of the diffusivity. For internal waves with no mixing, a large apparent dispersion occurs for times somewhat larger than the inverse buoyancy frequency 1/N. No dispersion occurs at long times. For stratified homogeneous turbulence with energy dissipation rate ɛ, Kz = ΓdɛN−2, the same form as Osborn, but with Γd of about 2.5. This high value is attributed to apparent dispersion due to internal waves and an improper form of the model spectra that allows internal waves to exist at low frequencies. A diapycnal dispersion coefficient K∗ is formulated based on a white spectrum of Lagrangian density change Dρ/Dt with level βρχ, where χ is the rate of dissipation of density variance. This yields K∗ = (π/2)βρχ/ρ−2z, where ρz is the mean vertical density gradient. This has the same form as the Osborn and Cox model for diapycnal diffusivity if βρ = 1/π.
Measurements of ocean turbulence using neutrally buoyant floats have produced empirical forms for the spectra of velocity (Lien et al. 1998) and density (Lien et al. 2002) following a Lagrangian trajectory in stratified or unstratified turbulence. These spectra predict the dispersion of particles and thus imply mixing rates. In this paper, we compare these mixing rates to those derived by other methods and thus attempt to clarify the relationship between Lagrangian and Eulerian turbulence measurements.
Taylor (1921) pioneered the study of turbulent mixing through the examination of the statistics of single particle dispersion. In a stationary and shear-free flow, the dispersion coefficient Kz is defined as
where Rw(τ) = 〈w(t)w(t + τ)〉 is the Lagrangian velocity covariance function at a time lag τ, z is the displacement of a particle from its initial position, and angle brackets denote an average over an ensemble of particles. Two fundamental results were obtained by Taylor without invoking explicit properties of turbulence. First, at time scales much smaller than the integral time scale of velocity T = ∫∞0 Rw(τ) dτ/Rw(0), Kz grows linearly with time. Second, for t ≫ T, Kz converges to a constant value. The temporal evolution of Kz depends on the details of Rw(τ).
The turbulent transport of a scalar c—for example, temperature or density—is often parameterized by an eddy diffusivity defined as the ratio of the flux Fc to its mean gradient ∂zc; that is, Ke = −Fc/∂zc. Dispersion predicts the flux of particles, and so, if c is the particle concentration and Kz is the eddy diffusivity for particles, then Ke should equal Kz. Here, we will compare estimates of Kz from Lagrangian models with estimates of Ke from Eulerian models.
For stratified turbulence, two parameterizations of Ke are commonly used, particularly in oceanic measurements. The Osborn and Cox (1972) method is based on measuring χc, the dissipation rate of scalar variance, and a mean scalar gradient (∂zc); that is,
This expression was originally derived from the scalar variance equation by assuming a balance between the turbulence flux and dissipation. In this derivation, Kh is the vertical eddy diffusivity and (∂zc) is the average vertical gradient. Winters and D'Asaro (1996) derive (2) exactly, but with χc describing the diascalar flux, that is, the flux of scalar across isoscalar surfaces, and (∂zc) is the gradient of the profile of c after it has been monotonically sorted. For a well-stratified fluid, the difference between these definitions does not create large differences in the computed eddy coefficient (D'Asaro et al. 2004). For the rest of this paper, (2) will be used for density flux, that is, c = ρ.
The Osborn (1980) method is based on ɛ, the dissipation rate of turbulent kinetic energy,
Here Γ is the mixing efficiency, closely related to the flux Richardson number, and N is the buoyancy frequency. The value of Γ is typically 0.2 or less in the stratified ocean (Osborn 1980). The values of Kρ and Kh, when sufficiently averaged, generally agree with each other and with other estimates of diapycnal diffusivity (Gregg 1998).
This paper will 1) revisit the dispersion coefficient Kz using up-to-date Lagrangian velocity spectra in unstratified turbulence (section 2), internal waves (section 3), and stratified turbulence (sections 4 and 5); 2) investigate diapycnal dispersion by defining a diapycnal dispersion coefficient K∗ based on Lagrangian scalar spectra (section 5); and 3) reconcile the dispersion coefficients Kz and K∗ with the diapycnal eddy diffusivity coefficients Kh and Kρ.
2. Unstratified turbulence
Taylor's dispersion coefficient Kz (1) can be expressed in terms of Lagrangian velocity spectrum Φw(ω) or structure function D(τ) = 〈[w(t + τ) − w(t)]2〉; for example,
In the inertial subrange, Kolmogorov scaling suggests forms for both the structure function D(τ) = C0ɛτ and the spectrum Φw(ω) = βɛω−2. Monin and Yaglom (1975) show that the two constants are related as C0 = βπ. Lien and D'Asaro (2002) find that the value C0 = 6 ± 0.5 and β = 2 ± 0.2 are consistent with existing data at high Reynolds number.
The second form (Mordant et al. 2001) assumes an exponential form e−τ/T for the Lagrangian velocity covariance function; the velocity spectrum has a Lorentzian form,
The third form is constructed empirically using observations taken in the oceanic surface mixed layer (Lien et al. 1998):
The three spectral forms are plotted in Fig. 1a. In each, the spectrum transitions from a flat slope at low frequencies to an inertial subrange with a −2 slope. The transition occurs at a large-eddy frequency ωTLo, ωMo, or ωo. The large- eddy frequency is defined as ωo = ɛ1/3k2/3o, where ko is the large-eddy wavenumber (Lien et al. 1998). The transition is most abrupt for ΦTL and the smoothest for ΦL. The inertial subrange Kolmogorov constants are βTL = 1.2, βM = 1.8 and βL = 1.8. Given these values and requiring all three forms to have the same velocity variance, σ2w = 1.6ɛ/ωo, requires ωTLo = 1.6ωo, and ωMo = 1.9ωo.
Correlation and structure functions are computed from the spectrum and plotted in Figs. 1b and 1c, respectively. For these plots, the inertial subrange has been extended to a Nyquist frequency 106ωo; our results are insensitive to this factor as long as the inertial subrange is sufficiently wide. The three correlation functions have a similar exponential form. The three structure functions normalized by ɛτ have a plateau at τωo ≪ O(1). The level of the plateaus gives C0(=πβ), yielding 3.8 for ΦTL and 5.7 for ΦM and ΦL, consistent with the results of previous work (Yeung 2002). Figure 2 shows the particle dispersion coefficients Kz(t) computed from the three Lagrangian spectra using (4). The curves are consistent with Taylor's results. Kz(τ) increases linearly as σ2wτ at τ ≪ ω−1o and reaches a constant level Kz (∞) = (π/2)α at τ > ω−1o, where α is the spectral level at ω ≪ ωo. Thus we scale Kz(∞) with ɛω−2o, the spectral level at low frequencies; Kz(∞)ω2oɛ−1 equals 0.73, 0.78, and 0.58 for ΦTL, ΦM, and ΦL, respectively.
The model of Mellor and Yamada (1974) will be used as a summary of the rates of mixing in unstratified turbulence. They parameterized the vertical eddy diffusivity KM = 0.4ql, where q2 = 3σ2ω and l is the master length scale. Assuming l = k−1o and using σ2w = 1.6ɛω−1o and ωo = ɛ1/3k2/3o, predicts KM = 0.88ɛω−2o. Thus for unstratified homogeneous turbulence the coefficients for dispersion Kz and diffusion KM are very close.
3. Oceanic internal waves—Vertical dispersion
Internal gravity waves are the dominant cause of vertical motion in the ocean for frequencies greater than f, the inertial frequency. A large body of literature starting with Fofonoff (1969) and reviewed by Garrett and Munk (1979) finds the vertical displacements due to these motions to be well represented by linear wave theory with a “Garrett–Munk” (henceforth GM) vertical velocity spectrum, which can be approximated as
where ϕ0 sets the internal wave energy level. Figures 3a and 3b show ΦGM and the associated dispersion function KGM(t) for a typical value of N/f = 100. Since internal waves are adiabatic, they cannot cause mixing and KGM(∞) = 0. However, they do cause a net displacement of water from its initial position, resulting in an apparent vertical dispersion with a maximum value of about (π/2)ϕ0 near Nt = 2. The value decreases to zero for Nt = 200 and then oscillates around zero with decreasing amplitude. Smoothing ΦGM near f and N results in a decreased amplitude of oscillations and a more rapid decay of KGM(t) for Nt ≫ 100.
4. Stratified turbulence—Vertical dispersion
D'Asaro and Lien (2000a) argue that in stratified turbulence the large-eddy frequency ωo should be approximately N. They support this by oceanic observations in which Lagrangian vertical velocity spectra exhibit a form nearly indistinguishable from (7) and N = 2(±0.3)ωo. The spectra are isotropic for ω ≫ N, supporting the idea that these are turbulent motions. The spectra are highly anisotropic in the same sense as internal waves for ω ≪ N, supporting the idea that these motions are due to internal waves. These results should apply for flows with gradient Richardson numbers of order 1; for sufficiently larger Richardson numbers the spectral form near N changes (D'Asaro and Lien 2000b); for sufficiently smaller Richardson numbers the flow geometry determines ωo.
These results can be used to compute the dispersion coefficient for stratified turbulence. Applying N = 2(±0.3)ωo to the result in the previous section, that is, Kz(∞)ω2oɛ−1 is equal to 0.73, 0.78, and 0.58 for ΦTL, ΦM, and ΦL, respectively, we find
where Γd = 3.0(±0.9), 3.2(±0.9), and 2.4(±0.7) for ΦTL, ΦM, and ΦL, respectively. Expression (10) is identical to (3) except that Γd is an order of magnitude greater than Γ and thus does not properly model diapycnal mixing.
The large value of Kz(∞) is explained by adiabatic internal waves being responsible for most of the vertical excursions. As shown in Fig. 3 internal waves produce a large apparent vertical dispersion for 1 < Nt < N/f. However, if the vertical motions of stratified turbulence are dominated by internal waves for ω < N, then the spectrum must also reflect the low frequency limit of internal waves. In the open ocean, this is due to f. In a confined sea or a laboratory vessel it might be set by the lowest mode of oscillation. In either case, the vertical velocity spectrum cannot remain white at low frequency as in (5), (6), and (7); instead, it must decrease by a factor Γ/Γd so as to bring Kz(∞) into agreement with Kρ.
An example of a Lagrangian spectrum with a low frequency cutoff is shown in Fig. 4. The dispersion function Kz(t) is similar to those in Fig. 2 for Nt ≤ 10 but decreases as in Fig. 3 for large Nt, ultimately reaching a value of Γɛ/N2 at long times. The initial large apparent dispersion is due to the wave part of the flow; the final smaller value is the true dispersion due to diapycnal mixing.
5. Stratified turbulence—Diapycnal dispersion
A more direct connection between dispersion and diffusion is obtained by examining dispersion in an isopycnal coordinate system (Pearson et al. 1983). Define an isopycnal displacement z∗ as the displacement of a particle away from its initial isopycnal. For a flow with a well-defined vertical gradient in density ρz, z∗ can be approximated by
where ρ′ is the deviation of the density from its initial value.
Following Taylor's dispersion theory, as in (1), we define a diapycnal dispersion coefficient as
Here, w∗ is the diapycnal velocity, Rw* is the diapycnal velocity covariance function, Dw* is the corresponding structure function, Φw* is the diapycnal velocity spectrum, and Φρ̇ is the spectrum of Dρ/Dt, the Lagrangian rate of change of density.
To evaluate (12), Lagrangian spectra of density change are needed. There is only scant theoretical or measurement guidance available. Kolmogorov scaling suggests Φρ̇(ω) = βρχ in an inertial subrange extending from ω > ωo ≈ N/2 to ω < ων = ɛ1/2ν−1/2. Here βρ is a Kolmogorov constant, ων is the Kolmogorov frequency at which viscous effects become important, and ν is the kinematic viscosity. Lagrangian observations in an oceanic shear-stratified flow provide vague evidence of white Φρ̇ spectrum in the inertial subrange (Lien et al. 2002). Yeung (2001) performed direct numerical simulations but did not observe a white Φρ̇ in the inertial subrange.
The true diapycnal dispersion rate is given by K∗(∞), which depends on the spectral level of Φρ̇ at low frequency. All contributions to Dρ/Dt and thus Φρ̇ must occur at diffusive scales since adiabatic motions cannot affect Dρ/Dt. Diffusive motions have frequencies near ων and thus contribute to the increase in K∗ for small times. Motions within the inertial subrange are adiabatic and do not contribute to K∗. Motions at lower frequencies should also be adiabatic and will also not contribute to K∗. Thus we expect (12) to be accurate at all frequencies and K∗(∞) to be given by (13).
Figure 7 helps to illustrate the above ideas. Consider the motion of an isopycnal surface within a given volume in a turbulent stratified fluid. The volume is chosen to be somewhat larger than the overturning scale of the turbulence. The average position of the surface (black) moves vertically in response to internal waves. If there is no mixing, the internal waves move dye vertically from its initial position, causing apparent vertical dispersion at short times. The motion is bounded by the wave amplitude so that the dispersion decreases to zero for long times (Fig. 3).
Suppose dye is placed on the surface at time t = 0 and spreads away from the isopycnal by mixing. The gray shading indicates the spread of the dye. At short times, waves are still the dominant cause of vertical motion of the dye. At longer times, the spread of the dye exceeds the amplitude of the waves, resulting in a finite dispersion that is smaller than the initial wave- induced apparent dispersion. This sequence is described by Fig. 4. The transition time between these two regimes depends on the amplitude and frequency of the internal waves, even if these play no role in the mixing.
The Lagrangian vertical velocity spectrum, or equivalently the dispersion function Kz(t), contains information on both diabatic and adiabatic motions. The diabatic processes can be isolated by taking a sufficiently long time average, that is, finding Kz(∞). In some systems this may not be possible. For example, we have several measurements of Φw in stratified fluids, but none extend to low enough frequency to see the expected decrease in level. This issue is not unique to Lagrangian spectra, but is common to many mixing diagnoses that use an Eulerian coordinate system (Winters and D'Asaro 1996).
These problems can be mostly overcome by working in an isopycnal coordinate system. Obviously, the mixing rate is more easily determined by measuring the diapycnal spreading of the dye around the isopycnal than by measuring its vertical spreading. Thus it is not surprising that the Lagrangian spectrum of density, which measures the motion of particles across isopycnals, gives a measurement of diapycnal diffusivity (13) that is completely insensitive to adiabatic motions of the isopycnal and therefore does not need to be averaged over long time scales to provide useful information.
Measurements of Φw and Φρ̇ can, in principle, be made using water-following floats (D'Asaro 2003). Practically, however, these floats only follow water at high frequencies. At low frequencies they are usually isopycnal; that is, they return to a given density surface. Isopycnal floats would follow the black line in Fig. 7, while truly Lagrangian floats, like dye, would undergo random walks filling the shaded region. This distinction is critical for the estimation of Kz(∞) from Φw since it is the low frequency value of Φw that determines Kz(∞). As discussed in D'Asaro (2003), it becomes increasingly difficult to design floats that are accurately Lagrangian at frequencies very much smaller than N. Float technology may be a better match for estimation of K∗ from Φρ̇. If, as hypothesized here, the spectrum is indeed white for all frequencies below the Kolmogorov frequency, then measurements of Φρ̇ within the inertial subrange will be sufficient. Existing floats can be accurately Lagrangian for these frequencies. However, a better understanding of the form and scaling of Φρ̇, gained by both theory and observations, will be required before diapycnal mixing rates can be reliably computed using Lagrangian measurements of high frequency density fluctuations.
This study has examined the relationship between diapycnal mixing and Lagrangian spectra of vertical velocity and temperature change through the concept of single particle dispersion [(1)]. This is motivated by recent measurements that define empirical forms for these spectra in turbulent flows.
In unstratified homogeneous turbulence, Lagrangian velocity spectra predict dispersion coefficients that closely match known diffusion coefficients.
Adiabatic internal gravity waves in density stratified flows produce large apparent dispersions at time lags somewhat larger than 1/N, but no average dispersion at long times.
In density stratified turbulence with Richardson numbers of O(1), the existing empirical Lagrangian velocity spectra predict vertical dispersion coefficients with the Osborn (1980) [(3)] form but with a mixing efficiency Γd much larger than observed. This is explained by the presence of a large adiabatic gravity wave component in these flows that leads to a large apparent dispersion at short times. If gravity waves and other adiabatic vertical motions become sufficiently weak at low frequency, then the Lagrangian velocity spectrum can be used to measure diapycnal mixing. The existing spectral forms must be modified to represent these low frequency changes. Similarly, measurements of the Lagrangian velocity spectrum must extend to sufficiently low frequency to be useful for the estimation of diapycnal mixing.
Inertial subrange theory and limited observations predict a white Lagrangian inertial subrange spectrum for density change with a level βρχ, where χ is the dissipation rate for density. This form gives a diapycnal dispersion coefficient with the Osborn and Cox (1972) [(2)] form and thus predicts a value βρ = 1/π for the Kolomogorov constant. This formulation is insensitive to adiabatic motions such as gravity waves and is thus a promising approach for estimation of diapycnal mixing from neutrally buoyant floats.
This work was supported by NSF Grants OCE9617671 and OCE0117411.
Corresponding author address: Dr. Ren-Chieh Lien, Applied Physics Laboratory, University of Washington, 1013 NE 40th Street, Seattle, WA 98105-6698. Email: firstname.lastname@example.org