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On the influence of a β-effect on Lagrangian diffusion

Authors


Abstract

[1] Geophysical turbulence is strongly influenced by the variation of the Coriolis force with latitude (β-effect): when this effect is significant an anisotropic inverse cascade is developed since energy is preferentially transferred towards zonal modes. The consequent emergence of flow structures along the zonal direction can strongly impact turbulent transport and modify meridional scalar diffusion. In this letter we investigate zonal and meridional diffusion in laboratory experiments of turbulence affected by aβ-effect. The degree of anisotropy and the flow characteristic scales have been quantified according to a Lagrangian approach based on the reconstruction of tracer trajectories, i.e., in analogy with oceanic drifters data, and on the Finite-Scale Lyapunov Exponent (FSLE) analysis. An experimental confirmation of a recently introduced scaling law for the meridional diffusion in the marginally zonostrophic regime is also presented.

1. Turbulence and Diffusion in Flows With a β-Effect

[2] Large scale planetary and terrestrial circulation are studied under the paradigm of the geostrophic turbulence introduced by Charney [1971]to model a turbulent, rotating, stably stratified fluid in near geostrophic balance. This model, based on the three-dimensional inviscid conservation equation for the potential vorticity, presents several analogies with purely 2D turbulence, e.g., the conservation of two quadratic invariants. Nevertheless, important differences exist [Read, 2001]. Here, among these differences, we will focus on turbulence modification by the meridional variation of the Coriolis parameter, i.e., the so-calledβ-effect, defined approximately asf = f0 + βy, where f0 is twice the rotation angular velocity Ω and y is the latitude coordinate. Rhines considered this problem in the context of unforced barotropic flows [Rhines, 1975] demonstrating that the β-effect induces the anisotropization of the upward energy transfer preferentially directing it towards zonal modes. The emergence of the flow anisotropy was explained in terms of a competition between nonlinear andβ terms and anisotropy of the dispersion relation. The associated characteristic scale separating the regions of wave vector space where either β or nonlinear effects respectively dominate is known as the Rhines wavenumber, nR, and can be expressed in terms of the root mean square of the velocity and β.Similar ideas can be applied to geostrophic turbulence where the baroclinic instability takes the role of the small-scale forcing and the non-linear interaction between eddies and the mean flow plays an important role in flow anisotropization; nevertheless, for small Burger numbers, the inverse cascade develops in the barotropic mode due to the barotropization mechanism [Salmon, 1980; Rhines, 1994; Read, 2001; Galperin et al., 2010].

[3] The concept of the cascade arrest was introduced by Rhines [1975] for unforced flows and recently revisited by Sukoriansky et al. [2007]in the context of small-scale forced, large-scale damped, barotropic vorticity equation on the surface of a rotating sphere. They show how, in the presence of aβ-effect, 2D turbulence features a complex interplay with waves at large scales and develops an anisotropic inverse energy cascade. As a consequence, several steady-state regimes characterized by different degree of anisotropy and energy spectra may be attained. These flow regimes can be classified in terms of relationship between the wavenumbers associated with the forcing,nf, the small-scale dissipation,nd, the β-effect,nβ, and the large-scale drag,nfr. A prominent role in characterization of flow regimes plays the non-dimensional parameterRβ = nβ/nR, termed the zonostrophy index. Sukoriansky et al. [2007] and Galperin et al. [2010] studied two regimes: friction dominated and zonostrophic. In the former, Rβ < 1.5 while in the latter, Rβ > 2. The intermediate range refers to a transitional regime [Sukoriansky et al., 2009]. In the friction dominated regime, the flow presents features similar to classical 2D turbulence, i.e., a Kolmogorov n−5/3 scaling for the total spectrum and a nearly isotropic inverse cascade for n > nβ. The zonostrophic regime is characterized by a strong spectral anisotropy, e.g., the zonal spectrum follows a n−5 scaling while the residual spectrum follows the classical n−5/3 scaling. The most distinguished feature of zonostrophic turbulence is the formation of a nearly stable system of eastward/westward zonal jets. Jets related to the zonostrophic regime were studied in laboratory experiments [Whitehead, 1975; Armi, 1989; Sommeria et al., 1991; Aubert et al., 2002; Rasmussen et al., 2006; Read et al., 2007; Slavin and Afanasyev, 2012], as well as in numerical simulations [Galperin et al., 2010]. The outputs of Oceanic General Circulation Models [Nakano and Hasumi, 2005; Berloff et al., 2009] and the analysis of satellite altimetry observations [Maximenko et al., 2005; Sokolov and Rintoul, 2007] also yield zonally elongated structures resembling zonal jets [Galperin et al., 2004; Nadiga, 2006].

[4] Turbulent meridional diffusion is strongly influenced by the presence of a β-effect. An analogy with the modification imposed on the diapycnal diffusion by the presence of a strong stable stratification has been recently proposed bySukoriansky et al. [2009]. In particular, in the friction dominated regime, for Rβ ≤ 1, the meridional (or, equivalently, radial) diffusivity, Drad, follows the Richardson law behavior Drad ≃ 2ε1/3nE−4/3, where ε is the rate of the inverse energy cascade and nE is the wavenumber corresponding to the most energetic scale. In the range 1 < Rβ < 1.5, the threshold of the transitional regime, the meridional diffusion coefficient follows the law Drad ≃ 0.3ε1/3nβ−4/3. This law holds for values of Rβ belonging to either the transitional or the zonostrophic regime, indicating that scales of order of magnitude larger than 1/nβ do not contribute to meridional transport [Galperin et al., 2010]. Sukoriansky et al. [2009]found some confirmation for these predictions analyzing field measurements of diffusion in deep large-scale oceanic circulation.

2. Laboratory Experiments

[5] Three experiments have been performed using a circular tank whose radius is RT = 18 cm, placed on a rotating table. Turbulence is generated in an electromagnetic cell [Espa et al., 2010] by imposing a continuous electromagnetic forcing on a thin layer of saline solution. In order to simulate flows in the northern hemisphere the rotating table rotation is counter-clockwise. The parabolic free surface assumed by the fluid under rotation is used to model theβ-effect in laboratory [Afanasyev and Wells, 2005; Espa et al., 2008], in our model the point of maximum depression of the fluid surface represents the pole while the peripheral areas of the domain correspond to lower latitudes. The corresponding expression for β can be found in Espa et al. [2009]. The experiments are characterized by different values of the fluid depth, the rotation rate of the table, and the forcing intensity in terms of the input current I, see Table 1. In performing flow measurements, the fluid surface is seeded with styrene particles (mean size ∼ 5 10−5m) supposed to be passively advected by the flow, the tank is covered with a transparent lid to prevent interaction with air and the free surface is lit with two lateral lamps to gain a high contrast between the white particles and the black bottom. A video camera (spatial resolution of 1023 × 1240 pixels) co-rotating with the system, perpendicular to the tank and with the optical axis parallel to the rotation axis, acquire flow images at 25 Hz directly into the memory of a computer also placed on the rotating table. The horizontal velocity components are measured in a Lagrangian framework by means of a non intrusive image analysis technique called Feature Tracking, described in the cited literature. Basically, this method allows one to reconstruct particle trajectories, while the ratio between the estimated displacements in subsequent frames and the time step between two frames (0.04 s here) gives the instantaneous Lagrangian velocity fields. In this case we are able to reconstruct a large number of long (i.e., lasting for time intervals of the order of the characteristic time scale associated to the investigated phenomena) trajectories suitable for evaluating Lagrangian statistics. After this procedure, data sets of order ∼O(104) trajectories are used to analyze zonal and meridional relative dispersion.

Table 1. Experimental Parametersa
EXPΩ (1/s)I (A)H0 (cm)β (1/cm/s)
  • a

    Ω is the rotation rate; I is the input current generating the electromagnetic forcing; H0 is the depth of the fluid at rest; β is the meridional gradient of the Coriolis parameter.

E12.5239.6 10−2
E23.4732.2 10−1
E33.43.532.2 10−1

3. Finite-Scale Lyapunov Exponents

[6] From the Lagrangian point of view, two-particle relative dispersion is the key process that reveals much of the physical properties of the underlying dynamics. Normally, one expects that the mean square relative displacement between two (passive) particles, denoted as 〈R2(t)〉, may follow a certain scaling law with the time 〈R2(t)〉 ∼ tν, depending on the particular dispersion regime. This classical description can suffer from some drawbacks [Boffetta et al., 2000] that will not be discussed here. A more objective evaluation of the relative dispersion properties is obtained by measuring the mean dispersion rates at fixed separation scale, adopting the consolidated technique based on the computation of the Finite-Scale (Finite-Size) Lyapunov Exponents, or FSLE [Aurell et al., 1996, 1997], widely applied and discussed in a variety of Geophysical contexts [LaCasce, 2008; Károlyi et al., 2010; Hernández-Carrasco et al., 2011].

[7] If δ is the separation between two particles, and 〈τ(δ)〉 is the average time that δ takes to be amplified by a factor ϱ > 1, then the FSLE is defined as:

display math

The quantity τ(δ) is defined as the time interval that a separation of size δ takes to attain the size ϱδ. Some useful scaling exponent relations between 〈R2(t)〉 and λ(δ) can be assessed on the basis of dimensional arguments. If 〈R2(t)〉 ∼ t2/μ then the FSLE is expected to scale as λ(δ) ∼ δμ. Notable cases are: a) μ → 0, i.e., λ(δ) = constant, corresponding to exponential separation between trajectories; b) μ = 2/3, corresponding to Richardson's t3 law [Richardson, 1926] and to Kolmogorov's n−5/3 scaling for the energy density spectrum [Frisch, 1995]; c) μ= 2, corresponding to standard diffusion with constant, scale-independent diffusion coefficient [Taylor, 1921]. It is worth noting that, in the limit of small separations, the FSLE coincides with the (Lagrangian) Maximum Lyapunov Exponent, or MLE, i.e., λ(δ) → λL for δ → 0 [Boffetta et al., 2000]. In this case, the exponential divergence between trajectories is due to chaotic advection by structures of size much larger than the particle separation scale. Richardson law, as well as the Kolmogorov's scaling, occur either in the inertial range of 3D fully developed turbulence or in the 2D inverse cascade turbulence regime. By standard diffusion regime we mean the asymptotic large scale particle scattering (i.e., random walk) by the most energetic eddies, i.e., the largest coherent structures of the flow. As we will see in the next section, the FSLE analysis not only recovers the variety of possible dispersion rates, as primary information, but it also provides a number of other interesting quantities useful for characterization of the Lagrangian dynamics.

4. Results and Conclusion

[8] The mean radial velocity profiles, as well as the time evolution of the mean zonal wind as a function of the radius, i.e., the distance from the centre of the tank, are characterized by an alternation of opposite signed jet-like structures surviving the long time averaging (not shown here). These Eulerian features are reported and discussed inEspa et al. [2010]. As far as Lagrangian dispersion is concerned, the FSLE is measured on a range of scales for both radial (meridional) and zonal components of the particle dispersion in each experiment. In all cases, the small scale dispersion is compatible with isotropic exponential separation, whose mean growth rate is given by the Lagrangian MLE, λL. The FSLE curves, grouped by component and normalized by λL, are reported in Figures 1a and 1b. Anisotropy in the dispersion process is reflected by the different scaling behaviors fitting radial and zonal components, as the particle separation leaves the isotropic exponential regime: radial FSLE are close to the standard diffusion scaling, δ−2, while zonal FSLE are close to inverse cascade Richardson's turbulent scaling, δ−2/3. Let us observe that these scaling laws have multiplicative factors which must be adjusted in order to fit the data. From the classic Richardson's law 〈R(t)2〉 = CRεt3, where CR, roughly ∼O(10−1), is the Richardson's constant and ε is the mean rate of the inverse cascade, we deduce that the FSLE in the Richardson's regime must scale like

display math

where ϱ = 21/4 is the constant amplification factor between first neighbor scales, throughout the FSLE analysis in all experiments. From formula (2) we see that the fitting parameter is substantially equal to ε1/3, except for a ∼O(1) factor. We estimate the most energetic wavenumber, nE, as the inverse of the scale δE corresponding to the maximum radial eddy kinetic energy distribution defined, from the radial FSLE, as EKE(rad)(δ) = 0.5[δ · λ(δ)(rad)/lnϱ]2, i.e., δE is such that EKE(rad)(δE) > EKE(rad)(δ) for any δδE. Rhines' wavenumber is estimated directly from the relation nR = (β/2Vrms)1/2 [Rhines, 1975]. At last, the wavenumber nβ is derived from the formula nβ = 0.5(β3/ε)1/5 [Sukoriansky et al., 2007], where the values of ε are those obtained from (2). All these parameters are listed in Table 2. As far as radial (meridional) diffusion is concerned, let us observe that, in a standard diffusion regime two particles separate, on average, as 〈R(t)2〉 = 4Kt, where R(t) is the distance between two particles at time t and K is the eddy diffusion coefficient. By dimensional arguments, we obtain the corresponding scaling law for the FSLE:

display math

where, in our case, K = Drad is the radial (meridional) diffusion coefficient and 4 · lnϱ/(ϱ2 − 1) is a numerical factor of order ∼O(1). The estimated values of the zonostrophy index, Rβ, of the meridional diffusion coefficients, Drad, as well as the reference quantities D(fric) = ε1/3nE−4/3 and D(zonos) = ε1/3nβ−4/3 are reported in Table 3. The experimental flows we are considering explore the edge of the friction dominated range, Rβ ≃ 1.5, a regime relevant, e.g., to oceanic current systems. Since nEnβ we have that Drad/D(fric) ∼ Drad/D(zonos) and that, notably, the normalized diffusion coefficients (Figure 2) provide a first, even though partial, experimental confirmation, at least as order of magnitude, of values found around Rβ ≃ 1.5 in recent numerical simulations [Sukoriansky et al., 2009]. At this regard, we remark that our results have been obtained with a relatively large range of variation of Ω and ε in the experiments, and that, consequently, this implies that the prediction by Sukoriansky et al. [2009]turns out to be quite robust. We observe that the behavior of the meridional diffusion coefficient “anticipates” the onset of the full zonostrophic regime and that, more in general, the whole relative dispersion process displays an evident anisotropy between zonal and radial directions even for zonostrophy index values still in the friction-dominated/transition range. In particular, while the meridional diffusion coefficient tends to become constant and scale-independent, as the meridional component of the separation between two particles becomes larger thanδE, the zonal diffusivity tends to maintain a Richardson-like behavior, indicating that the jet-like zonal currents are not yet developed enough to dominate the dispersion with strong advection effects. At last, we would like to remark that the dynamical systems approach, based on the FSLE analysis, allows one to estimate all the needed quantities characterizing the flow regimes from a Lagrangian point of view, and provides, in this way, an independent diagnostics against which Eulerian estimates can be eventually tested.

Figure 1.

(a) Radial FSLE and (b) zonal FSLE for all experiments, normalized to the MLE λL, displaying tendency to standard diffusion scaling δ−2, and to inverse cascade Richardson's scaling δ−2/3, respectively. Statistical error bars are of order of the point size.

Table 2. λL Is the Lagrangian Maximum Lyapunov Exponent; Vrms Is the Root Mean Square Lagrangian Velocity; ε Is the Mean Turbulent Energy Flux; nE, nβ and nR Are the Most Energetic Wavenumber, the Transitional Wavenumber and the Rhines' Wavenumber, Respectively
EXPλL (1/s)Vrms (cm/s)ε (cm2/s3)nE (1/cm)nβ (1/cm)nR (1/cm)
E10.200.826 10−30.300.340.24
E20.401.881.25 10−10.360.310.24
E30.220.926 10−30.560.570.35
Table 3. Rβ = nβ/nR Is the Zonostrophy Index; Drad Is the Experimental Radial Diffusion Coefficient Estimated From the FSLE; D(fric) = ε1/3nE−4/3 and D(zonos) = ’ε1/3nβ−4/3
EXPRβDrad (cm2/s)D*(fric) (cm2/s)D*(zonos) (cm2/s)
E11.411.7 10−10.90.8
E21.274.0 10−12.02.4
E31.629.0 10−20.40.4
Figure 2.

Radial diffusion coefficients, normalized to D∗ = D(fric) = ε1/3nE−4/3 (open triangles) and to D∗ = D(zonos) = ε1/3nβ−4/3 (open squares), as function of the zonostrophy index Rβ. Estimated relative error ∼20%. The dotted line roughly represents the zonostrophic asymptotic limit [see Sukoriansky et al., 2009, Figure 2].

Acknowledgments

[9] The authors would like to thank Gabriella Di Nitto for her contribution in the acquisition of the experimental data-set, and Volfango Rupolo for inspiring discussions.

[10] The Editor thanks two anonymous reviewers for assisting in the evaluation of this paper.

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