Mixing properties of a shallow basin due to wind-induced chaotic flow

https://doi.org/10.1016/j.advwatres.2007.11.001Get rights and content

Abstract

As important environmental features, mixing properties of inland water bodies in unsteady flow conditions are investigated. Time-dependent motion, often resulting in chaotic behavior, requires the Lagrangian description of the transport. As a simple example, unsteady hydrodynamics driven by periodical wind forcing in a simplified shallow lake geometry is considered to explore the main chaotic properties. In the modelled flow field methods identifying strong and weak shearing sub-regions are proposed and applied as mixing indicators. These include the determination and inter-comparison of the finite size Lyapunov exponents (FSLE), the residence time, and the implementation of the so-called leaking method. Coherent structures as stable and unstable manifolds are also identified, playing the role of Lagrangian barriers that hinder local transversal material transfer, and avenues that significantly channel transport. The primary effect of turbulent diffusion on the FSLE fields is also demonstrated.

Introduction

In shallow environmental flows horizontal mixing is of particular importance. Its understanding and accurate modeling are essential for applied sciences, such as estimating water exchange mechanisms, interpreting plankton movement or planning and operating pollutant outfalls. Mixing in water takes place due to two main processes: diffusion and advection. The role of diffusion alone is usually minor in the efficiency of mixing, but its combined effect with advection or even more, the complexity of advection in itself can also result in large-scale spreading. In time-dependent velocity fields the basic mechanism is chaotic advection, which is best handled as inherently Lagrangian transport as shown by Aref [2]. With methods originating in chaos theory we are able to locate spatial structures which govern the flow and areas where the most effective mixing occurs. Such structures as hyperbolic (also called saddle) points and manifolds (see Fig. 1) have long been used for classifying the evolution of trajectories in abstract dynamical systems (for an introduction see e.g. [20]). Their application in the context of fluid dynamics, in turn offers a new tool with clear physical meaning: identifying vortex boundaries, barriers and avenues of transport, or lines of strong stretching [5], [8], [16]. Especially, chaotic dynamics are characterized by complex intersection of stretching and contracting manifolds around the hyperbolic points. Mixing is typically strong in these regions: trajectories of initially close particles are quickly separated along the stretching directions.

Hyperbolic points are the Lagrangian analogs of Eulerian stagnation points. As is well known, a stagnation point is the intersection of streamlines at a certain instant of time (Fig. 1A). If the flow field were frozen, the fluid would be motionless at such points. As the definition implies, the stagnation point is an instantaneous property of the flow. A hyperbolic point is, on the contrary, a point moving with the fluid, along a periodic orbit in temporally periodic flows. At any instant of time there is a curve running towards the hyperbolic point. This curve is the set of all points which, when followed at integer multiples of the period of the flow, will hit the hyperbolic point in the future. This curve can be called the curve of contraction, or, in terms of dynamical system theory, the stable manifold [20] (see Fig. 1B). Similarly, another curve can be defined, the stretching curve or unstable manifold, along which points leave an infinitesimally small neighborhood of the hyperbolic point. The unstable manifold can also be considered as the stable manifold in the time-reversed Lagrangian dynamics. Near hyperbolic points, the rate of separation of nearby particles is exponential. The manifolds of the hyperbolic point reflect the entire history of the fluid around the hyperbolic point. Stagnation and hyperbolic points are thus basically different, although both are surrounded by a cross-like pattern, as seen in Fig. 1. (They coincide in stationary flows only.) Often they happen to be close to each other, as we shall illustrate in the present paper.

In our work we compare different Lagrangian methods, with special emphasis on the finite size Lyapunov-exponent (FSLE) method, the residence time distribution and the leaking method, which have not been applied earlier in inland water context. We take a simple wind-forced shallow basin model, analyze the mixing properties and, in particular, the locations with strong chaotic behavior responsible for efficient mixing. All this reveals new aspects in understanding and interpreting mixing processes in shallow wind-forced lakes, providing additional tools to the water resources management in such an environment.

Section snippets

The hydrodynamical model

In order to investigate the chaotic features in a simple, but realistic case, a shallow, wind-forced sample lake was chosen with dimensions close to the ones typical for shallow inland waters. Representing, e.g., small reed-enclosed inner ponds of shallow lakes (as can be found in the Everglades, or in Lake Neusiedl in Central Europe) while keeping the shape as simple as possible, a horizontally 2 km × 2 km square-shaped lake was set up. Apart from the nearshore zone, the bottom of such bays

Lagrangian methods and traditional approaches for chaos analysis

The Eulerian velocity components are provided on the aforementioned grid at discrete time steps of 600 s. Particle positions are calculated using velocity fields with East and North components u(x, y, t), v(x, y, t) linearly interpolated between time steps ti and ti + 1 at intermediate time level t to obtain the Lagrangian particle paths. The advection equationsdxdt=u(x,y,t),dydt=v(x,y,t)were solved by a fourth-order Runge–Kutta scheme using bilinear spatial interpolation. The solution of these

Leaking advection dynamics

A novel way of visualizing mixing properties is provided by the method of “leaking“ introduced by Schneider et al. [17] (see also [18], [19], [22]). This method is based on monitoring particles which do not enter a pre-selected area, the leak, over very long times. If the advection dynamics is chaotic, the monitored particles trace out fractal patterns (see Fig. 5). The starting position of these particles indicates the so-called stable manifold, from where they reach hyperbolic points. Their

Finite size Lyapunov exponents

The leaking method is unable to provide a quantitative measure of the strength of particle separation. The traditional way to characterize this feature is the determination of the standard Lyapunov exponent of chaotic advection (see e.g. [20]). This is defined as the average exponential rate of separation of initially nearby fluid parcels, averaged over long times. The concept of finite size Lyapunov exponent (FSLE) has been developed as a generalisation of the average Lyapunov exponent by

The effect of diffusion on FSLE

We devote this section to estimating the effect of turbulent diffusion on the filamentary patterns of the advection dynamics. Assume that a dye is distributed in a band along an unstable manifold. The width of this band will slowly increase due to diffusion. The growth would go on without any limitation if the flow were not present. The permanent stretching along the unstable manifold is, however, accompanied with a contraction across the unstable manifold. This contraction will slow down the

Conclusions

Mixing properties of inland water bodies were studied in an essentially Lagrangian framework. As chaos-induced properties, coherent structures such as manifolds and hyperbolic points were determined. In a simplified shallow lake geometry unsteady hydrodynamics driven by periodical wind forcing was considered and the focus was on exploring the main chaotic mixing properties. In the numerically modeled depth-averaged flow field methods identifying strong and weak shearing sub-regions and

Acknowledgement

This research was supported by the OTKA Grants T047233 and TS044839.

References (23)

  • G. Haller

    Finding finite-time invariant manifolds in two-dimensional velocity fields

    Chaos

    (2000)
  • Cited by (22)

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