No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
The role of self-consistent Lagrangian chaos in Benard convection in an annulus
Abstract
The nonlinear behavior of the two-dimensional Benard problem with periodic boundary conditions in the horizontal direction is studied with particular emphasis on the role of self-consistent chaotic advection. The results show a complex interplay between vortices driven by the Benard (Rayleigh--Taylor) instability and shear flow, which is driven by the vortices [J. Drake [ital et] [ital al]., Phys. Fluids B [bold 4], 4881 (1992)] and which causes their decay. Chaotic advection occurs in the transition from the low Rayleigh number (Ra) regime to the high Ra regime [J. Finn, Phys. Fluids B [bold 5], 415 (1993)]. For the former, vortex flow and shear flow coexist, possibly with slow relaxation oscillations. In the high Ra regime there are vortices localized near the upper and lower boundaries with a shear flow in between. As Ra is decreased from the high Ra regime, these vortices broaden, eventually overlapping, causing self-consistent Lagrangian chaos. This onset of chaos is responsible for several properties of the transition state between the low Ra and the high Ra regimes, most notably the damping of the relaxation oscillations involving vortex and shear flow. It is also observed that the Nusselt number Nu has a peak with respect to Ra in this transition regime characterized by Lagrangian chaos. In the low Ra regime, on the other hand, the relaxation oscillations are on a much slower time scale than the eddy turnover time and the Lagrangian behavior is described by separatrix crossing.
Turbulent motions due to flux-driven thermal convection are investigated by numerical simulations and stochastic modeling. Tilting of convection cells leads to the formation of sheared flows and quasi-periodic relaxation oscillations for the energy integrals far from the threshold for linear instability. The probability density function for the temperature and radial velocity fluctuations in the fluid layer changes from a normal distribution at the onset of turbulence to a distribution with an exponential tail for large fluctuation amplitudes for strongly driven systems. The frequency power spectral density has an exponential shape, which is a signature of deterministic chaos. By use of a novel deconvolution method, this is shown to result from the presence of Lorentzian pulses in the underlying time series, demonstrating that exponential frequency spectra can also persist in turbulent flow regimes.
In this work the nature of confinement improvement implied by the self-consistent generation of mean flows in two-dimensional convective turbulence is studied. The confinement variations are linked to two distinct regulation mechanisms which are also shown to be at the origin of low-frequency bursting in the fluctuation level and the convective heat flux integral, both resulting in a state of large-scale intermittency. The first one involves the control of convective transport by sheared mean flows. This regulation relies on the conservative transfer of kinetic energy from tilted fluctuations to the mean component of the flow. Bursting can also result from the quasi-linear modification of the linear instability drive which is the mean pressure gradient. For each bursting process the relevant zero-dimensional model equations are given. These are finally coupled in a minimal model of convection in fluids and plasmas. The results of the modeling are used to discuss confinement scaling and intermittency, and in a heuristic way, more complex issues such as criticality and transport avalanches.
Nonlinear evolution of interchange mode produces both vortex flow and shear flow in a scrape-off layer (SOL) plasma with unfavorable magnetic curvature. When the Rayleigh number becomes the order of 105, the shear flow exceeds the vortex flow and the zonal counterstreaming flow appears in the perpendicular direction to the ambient magnetic field. Simultaneously, the fluctuation level decreases and the associated cross-field transport becomes almost classical. However, since reduction of the saturation level weakens the shear flow generation, an intermittent oscillatory behavior appears and continues. Transport due to the vortex flow measured with a Nusselt number may depend on the Prandtl number for a given Rayleigh number.
Available from http://www3.interscience.wiley.com/cgi-bin/abstract/62004587/START
Nonlinear evolution of two dimensional pressure-driven interchange mode produces relaxation oscillations in the presence of poloidal shear flow for Reyleigh number Ra˜104. Here the flow velocity has a symmetric profile with respect to the center of slab plasma where the velocity shear disappears. There is a tendency that the velocity shear becomes weak after the saturatioin. When the velocity shear almost disappears, fluctuations grow rapidly. Then the velocity shear is again generated to suppress the growth of fluctuations. This kind of relaxation oscillation may be related to ELM [Edge Localized Mode] in the H-mode of tokamak.
Two mechanisms at the origin of profile consistency in models of electrostatic turbulence in magnetized plasmas are considered. One involves turbulent diffusion in collisionless plasmas and the subsequent turbulent equipartition of Lagrangian invariants. By the very nature of its definition, this state can only be reached in the absence of imposed fluxes of the transported quantities. As such, the concept of turbulent equipartition cannot be used to interpret profiles in numerical simulations of interchange modes, as it has nevertheless been done in the past. It is shown in this article that for interchange modes, profile consistency is in fact due to mixing by persistent large-scale convective cells. This mechanism is not a turbulent diffusion, cannot occur in collisionless systems, and is the analog of the well-known laminar “magnetic flux expulsion” in magnetohydrodynamics. This expulsion process involves a “pinch” across closed streamlines and further results in the formation of pressure fingers along the separatrix of the convective cells. By nature, these coherent structures are dissipative because the mixing process that leads to their formation relies on a finite amount of collisional diffusion. Numerical simulations of two-dimensional interchange modes confirm the role of laminar expulsion by convective cells for profile consistency and structure formation. They also show that the fingerlike pressure structures ultimately control the rate of heat transport across the plasma layer and thus the transport scaling at large Rayleigh numbers. This contradicts mixing-length arguments which do not account for collisional processes. For interchange modes, the problem of coherent structure formation, profile consistency, and transport scaling are thus intimately linked.
In this contribution some recent investigations of two-dimensional thermal convection relevant to ordinary fluids as well as magnetized plasmas are reviewed. An introductory discussion is given of the physical mechanism for baroclinic vorticity generation and convective motions in stratified fluids, emphasizing its relation to interchange motions of non-uniformly magnetized plasmas. This is followed by a review of the theories for the onset of convection and quasi-linear saturation in driven-dissipative systems. Non-linear numerical simulations which result in stationary convective states reveal the process of laminar scalar gradient expulsion, leading to the formation of temperature plumes and vorticity sheets. These dissipative structures are demonstrated to result in temperature profile consistency and power law transport scaling far from the threshold. The last part of this paper deals with the generation of differential rotation by fluctuating motions through tilting of the convective structures. The role of kinetic energy transfer and shearing due to differential advection is pointed out. Numerical simulations for strongly driven systems reveal turbulent states with a bursty behaviour of the fluctuation level which is associated with relaxation oscillations in the kinetic energy of the azimuthally mean flows. This leads to a state of large-scale intermittency manifested by exponential tails in the single-point probability distribution function of the dependent variables. The global bursting is interpreted in terms of a predator–prey regulation from the point of view of energetics. Finally, a discussion is given of the relevance of these phenomena to a variety of magnetized plasma experiments.
Transport and confinement within the resistive-g paradigm are investigated by means of two-dimensional numerical simulations. The system is driven by a constant incoming heat flux at the inner radial boundary. Different confinement and transport states are identified, involving self-sustained sheared poloidal flows. At the onset of turbulent convection the probability distribution functions of pressure and radial velocity fluctuations measured in the centre of the plasma layer have a nearly Gaussian form. Further increasing the heat flux drive these distributions become increasingly non-Gaussian, developing exponential tails. This large-scale intermittency is ascribed to the presence of bursting in the domain averaged convective transport and the fluctuation energy integrals. The quasi-periodic bursts are separated by shear-dominated quiescent periods in which the mean flow energy decreases and the confined heat increases on diffusive timescales. The time-averaged thermal energy confined within the plasma layer shows a power law dependence and significant increase with the injected power over the range of turbulent convection investigated.
In the present contribution, the quasilinear dynamics of convective turbulence is studied. In essence, and contrary to the ''frozen gradient'' assumption, the quasilinear approach takes into account the back-reaction of the convective flux on the mean gradient driving the instability. The dynamical regulation of convective transport by a sheared mean flows is also included. Close to the instability threshold it naturally gives rise to a transition from low to high confinement modes. Further away, regular relaxation oscillations are sustained. In this time-dependent state, each transient maximum of the convective flux activity triggers a ballistic transport event observed on the mean profile. The period of the oscillations is not controlled by the nonlinearity but by the dissipation on the mean flow. A ''Dimits-shift'' regime is thus identified in the limit of zero damping on the mean flow. This infinite period cycle corresponds to a single ballistic transport event triggered before the system settles into its diffusive state. Far away from the threshold, relaxation oscillations are still sustained in the presence of mean flow dissipation, but are superimposed on high-frequency fluctuations. This particular behavior makes the convective transport to follow exponential statistics when measured at a local probe. © 2003 American Institute of Physics.
The length scales dominating physical phenomena in nanotechnology are at the boundary of the macroscopic and microscopic world. Nanoobject formation is governed by a competition between deterministic and random forces. Therefore, classical physical noise phenomena play an important role in determining structural formations and noise measurements can be used as important tools of nanotechnology.
Self-consistent chaotic transport is the transport of a field F by a velocity field v according to an advection-diffusion equation in which there is a dynamical constrain between the two fields, i.e., O(F,v)=0 where O is an integral or differential operator, and the Lagrangian trajectories of fluid particles exhibit sensitive dependence on initial conditions. In this paper we study self-consistent chaotic transport in two-dimensional incompressible shear flows. In this problem F is the vorticity zeta, the corresponding advection-diffusion equation is the vorticity equation, and the self-consistent constrain is the vorticity-velocity coupling z nabla xv=zeta. To study this problem we consider three self-consistent models of intermediate complexity between the simple but limited kinematic chaotic advection models and the approach based on the direct numerical simulation of the Navier-Stokes equation. The first two models, the vorticity defect model and the single wave model, are constructed by successive simplifications of the vorticity-velocity coupling. The third model is an area preserving self-consistent map obtained from a space-time discretization of the single wave model. From the dynamical systems perspective these models are useful because they provide relatively simple self-consistent Hamiltonians (streamfunctions) for the Lagrangian advection problem. Numerical simulations show that the models capture the basic phenomenology of shear flow instability, vortex formation and relaxation typically observed in direct numerical simulations of the Navier-Stokes equation. Self-consistent chaotic transport in electron plasmas in the context of kinetic theory is also discussed. In this case F is the electron distribution function in phase space, the corresponding advection equation is the Vlasov equation and the self-consistent constrain is the Poisson equation. This problem is closely related to the vorticity problem. In particular, the vorticity defect model is analogous to the Vlasov-Poisson model and the single wave model and the self-consistent map apply equally to both plasmas and fluids. Also, the single wave model is analogous to models used in the study of globally coupled oscillator systems. (c) 2000 American Institute of Physics.
The effect of an externally imposed perturbation on an unstable or weakly stable shear flow is investigated, with a focus on the role of Lagrangian chaos in the bifurcations that occur. The external perturbation is at rest in the laboratory frame and can form a chain of resonances or cat's eyes where the initial velocity v(x0)(y) vanishes. If in addition the shear profile is unstable or weakly stable to a Kelvin-Helmholtz instability, for a certain amplitude of the external perturbation there can be an unlocking bifurcation to a nonlinear wave resonant around a different value of y, with nonzero phase velocity. The interaction of the propagating nonlinear wave with the external perturbation leads to Lagrangian chaos. We discuss results based on numerical simulations for different amplitudes of the external perturbation. The response to the external perturbation is strong, apparently because of non-normality of the linear operator, and the unlocking bifurcation is hysteretic. The results indicate that the observed Lagrangian chaos is responsible for a second bifurcation occurring at larger external perturbation, locking the wave to the wall. This bifurcation is nonhysteretic. The mechanism by which the chaos leads to locking in this second bifurcation is by means of chaotic advective transport of momentum from one chain of resonances to the other (Reynolds stress) and momentum transport to the vicinity of the wall via chaotic scattering. These results suggest that locking of waves in rotating tank experiments in the presence of two unstable modes is due to a similar process. (c) 2002 American Institute of Physics.
Shear flow dynamics described by the two-dimensional incompressible Navier-Stokes equations is studied for a one-dimensional equilibrium vorticity profile having two minima. These lead to two linear Kelvin-Helmholtz instabilities; the resulting nonlinear waves corresponding to the two minima have different phase velocities. The nonlinear behavior is studied as a function of two parameters, the Reynolds number and a parameter lambda specifying the width of the minima in the vorticity profile. For parameters such that the instabilities grow to a sufficient level, there is Lagrangian chaos, leading to mixing of vorticity, i.e., momentum transport, between the chains of vortices or cat's eyes. Lagrangian chaos is quantified by plotting the finite time Lyapunov exponents on a grid of initial points, and by the probability distribution of these exponents. For moderate values of lambda, there is Lagrangian chaos everywhere except near the centers of the vortices and near the boundaries, and there are competing effects of homogenization of vorticity and formation of structures associated with secondary resonances. For smaller values of lambda Lagrangian chaos occurs in the regions in the centers of the vortices, and the Eulerian behavior of the flow undergoes bifurcations leading to Eulerian chaos, as measured by the time series of several Galilean invariant quantities. A discussion of Lagrangian chaos and its relation to Eulerian chaos is given.(c) 2001 American Institute of Physics.
The tilting mechanism, which generates differential rotation in two-dimensional turbulent convection, is shown to produce relaxation oscillations in the mean flow energy integral and bursts in the global fluctuation level, akin to Lotka-Volterra oscillations. The basic reason for such behavior is the unidirectional and conservative transfer of kinetic energy from the fluctuating motions to the mean component of the flows, and its dissipation at large scales. Results from numerical simulations further demonstrate the intimate relation between these low-frequency modulations and the large-scale intermittency of convective turbulence, as manifested by exponential tails in single-point probability distribution functions. Moreover, the spatio-temporal evolution of convective structures illustrates the mechanism triggering avalanche events in the transport process. The latter involves the overlap of delocalized mixing regions when the barrier to transport, produced by the mean component of the flow, transiently disappears.
Two-dimensional convection in a Boussinesq fluid with infinite Prandtl number, confined between rigid horizontal boundaries and stress-free lateral boundaries, has been investigated in a series of numerical experiments. In a layer heated from below steady convection becomes unstable to oscillatory modes caused by the formation of hot or cold blobs in thermal boundary layers. Convection driven by internal heating shows a transition from steady motion through periodic oscillations to a chaotic regime, owing to the formation of cold blobs which plunge downwards and eventually split the roll. The interesting feature of this idealized problem is the interaction between constraints imposed by nonlinear dynamics and the obvious spatial structures associated with the sinking sheets and changes in the preferred cell size. These spatial structures modify the bifurcation patterns that are familiar from transitions to chaos in low-order systems. On the other hand, even large-amplitude disturbances are constrained to show periodic or quasi-periodic behaviour, and the bifurcation sequences can be followed in considerable detail. There are examples of quasi-periodic behaviour followed by intermittency, of period-doubling cascades and of transitions from quasi-periodicity to chaos, associated with a preference for narrower rolls as the Rayleigh number is increased.
The reduced nonlinear low ?? tokamak magnetohydrodynamic equations are extended to the case of high ??. A large aspect ratio ordering is used. The dynamics can be described using only three variables: the stream functions for the poloidal magnetic field and velocity, and the pressure. The equations are solved numerically to find two???dimensional equilibria. These equilibria can be unstable to ballooning modes, which tend to grow to large amplitude in nonlinear, three???dimensional computations.
Two-dimensional convection in a Boussinesq fluid confined between free boundaries is studied in a series of numerical experiments. Earlier calculations by Fromm and Veronis were limited to a maximum Rayleigh number R 50 times the critical value R, for linear instability. This range is extended to 1000Rc. Convection in water, with a Prandtl number p = 6·8, is systematically investigated, together with other models for Prandtl numbers between 0·01 and infinity. Two different modes of nonlinear behaviour are distinguished. For Prandtl numbers greater than unity there is a viscous regime in which the Nusselt number , independently of p. The heat flux is a maximum for cells whose width is between 1·2 and 1·4 times the layer depth. This regime is found when . At higher Rayleigh numbers advection of vorticity becomes important and N [infty infinity] R0·365. When p = 6·8 the heat flux is a maximum for square cells; steady convection is impossible for wider cells and finite amplitude oscillations appear instead, with periodic fluctuations of temperature and velocity in the layer. For p < 1 it is also found that N [infty infinity] R0·365, with a constant of proportionality equal to 1·90 when p [double less-than sign] 1 and decreasing slowly as p is increased. The physical behaviour in these regimes is analysed and related to astrophysical convection.
A periodic array of convection cells is subject to a ‘‘shear flow’’ instability. The generation of the sheared flow is a consequence of ‘‘peeling’’ of the convection cells. Fluid simulations demonstrate that the efficiency of shear flow generation is high. Implications for understanding poloidal rotation in tokamaks are discussed.
A new operational regime has been observed in neutral-injection-heated ASDEX divertor discharges. This regime is characterized by high βp values comparable to the aspect ratio A(βp<~0.65A) and by confinement times close to those of Ohmic discharges. The high-βp regime develops at an injection power ≥1.9 MW, a mean density n¯e>~3×1013 cm−3, and a q(a) value ≥2.6. Beyond these limits or in discharges with material limiter, low βp values and reduced particle and energy confinement times are obtained compared to the Ohmic heating phase.
A slowly varying Hamiltonian with one degree of freedom and nearly closed orbits has an adiabatic invariant. This adiabatic invariant is conserved to all orders in, the slowness parameter, except for orbits that cross a separatrix. The present work discusses the change of the adiabatic invariant during this crossing process through order . First, a calculation of the change of the adiabatic invariant is presented. This calculation is general and, hence, encompasses previous results for specific cases. The change in the adiabatic invariant is shown to depend on a maximum of five parameters, which are functions of the Hamiltonian of interest. Second, the statistics of this process are derived. Finally, these results are applied to the motion of a particle in a wave with changing amplitude and phase velocity.
Results on the nonlinear behavior of the Rayleigh–Taylor instability and consequent development of shear flow by the shear instability [Phys. Fluids B 4, 488 (1992)] are presented. It is found that the shear flow is generated at sufficient amplitude to reduce greatly the convective transport. For high viscosity, the time-asymptotic state consists of an equilibrium with shear flow and vortex flow (with islands, or ‘‘cat’s eyes’’), or a relaxation oscillation involving an interplay between the shear instability and the Rayleigh–Taylor instability in the presence of shear. For low viscosity, the dominant feature is a high-frequency nonlinear standing wave consisting of convective vortices localized near the top and bottom boundaries. The localization of these vortices is due to the smaller shear near the boundary regions. The convective transport is largest around these convective vortices near the boundary and there is a region of good confinement near the center. The possible relevance of this behavior to the H mode and edge-localized modes (ELM’s) in the tokamak edge region is discussed.
A fluid equilibrium consisting of a periodic array of counter‐rotating vortices is found to be unstable to the generation of one‐dimensional sheared flow along the direction of periodicity. This instability is inviscid (exists for zero viscosity μ) or viscous (with growth rate γ∼μ3/4) depending on the elongation of the vortices. Nonlinearly, the instability goes through a vortex reconnection or ‘‘peeling’’ phase in which one of the vortices per period is destroyed, leading to a state with a chain of islands. Without a source, the flow evolves to pure one‐dimensional shear flow, which decays because of viscosity on a much longer time scale. In the presence of a source driving the initial vortices, the flow evolves to an equilibrium having vortex flow plus shear flow and, for sufficiently high Reynolds number, having only one vortex per periodicity length rather than two, i.e., with islands.
Time-dependent magnetohydrodynamic (MHD) convection is studied by
numerical modeling on several levels. An intuitive model, the mixing
length theory of convection (MLT), is reviewed; from this, a scaling law
for convective structures is derived and compared with observations of
the solar convection zone. MLT lays the foundation for the
magneto-anelastic equations--a model for MHD convection at low Mach and
Alfven Mach numbers--which is rederived with particular attention to its
relation to the familiar Boussinesq equations. A variant of the
anelastic approximation is offered which obviates the need for a
first-order pressure equation. The magneto-anelastic model is used as
the basis for SUMO, a computer model that generates numerical solutions
on a two-dimensional Cartesian mesh by a finite-difference,
predictor-corrector algorithm. Thermodynamic properties of the fluid are
held fixed and constant at the rigid, stress-free top and bottom
boundaries of the computational box, while lateral boundaries are
treated as periodic. A numerical study of the equations is made for a
zero-order polytrope, at a Rayleigh number 5.44 times the critical
value, Prandtl number 1.8, and aspect ratio unity. The dynamical
behavior is highly influenced by a horizontal magnetic field which is
imposed at the bottom boundary (the top of the domain is attached to a
nonconducting region). As the magnetic parameter is increased, the
initially tilted character of convection rolls changes. A single-roll,
highly sheared, time-dependent state is replaced by a steady
"traveling-wave" tilted state; then an oscillatory or "sloshing" state;
then a steady two-roll state with no tilting; and finally, a stationary
state. The concentrated nature of magnetic fields at the top of these
flows is explained by a simple advection-diffusion balance. The complex
dynamical behavior is shown to be well-represented by a nine-mode
truncation of the Fourier expansion of the Boussinesq equations in a
uniform, horizontal magnetic field. The key mode is a "tilting" one,
which breaks the up-down mirror symmetry. Qualitative and quantitative
comparisons are made between the simulations and the nine -mode model,
which demonstrate that the latter captures all the essential physics of
the former.
A mathematical model of convection, obtained by truncation from the two-dimensional Boussinesq equations, is shown to exhibit a bifurcation from symmetrical cells to tilted non-symmetrical ones. A subsequent bifurcation leads to time-dependent flow with similarly tilted transient plumes and a large-scale Lagrangian mean flow. This change of symmetry is similar to that occurring with the advent of a large-scale flow and transient tilted plumes seen in laboratory experiments on turbulent convection at high Rayleigh number. Though not intended as a description of turbulent convection, the model does bring out in a theoretically tractable context the possibility of the spontaneous change of symmetry suggested by the experiments.
Further bifurcations of the model lead to stable chaotic phenomena as well. These are numerically found to occur in association with heteroclinic orbits. Some mathematical results clarifying this association are also presented.
An experimental study of the onset of chaos in a linear array of forced vortices is presented. The vortices are driven by electromagnetic forces in a layer of electrolyte. The system is found to behave like a chain of nonlinearly coupled oscillators, each oscillator being sustained by a pair of vortices. Systems with a small number of vortices exhibit scenarios characterized by a small number of degrees of freedom. Increasing the number of vortices leads to a rapid increase of the complexity of the regimes of transition to chaos. For moderately long systems, quasi-periodicity preceding the onset of chaos and intermittent behaviour is observed.
The purpose of this review article is to demonstrate via a few simple models the mechanism for a very general, universal instability - the Arnold diffusion—which occurs in the oscillating systems having more than two degrees of freedom. A peculiar feature of this instability results in an irregular, or stochastic, motion of the system as if the latter were influenced by a random perturbation even though, in fact, the motion is governed by purely dynamical equations. The instability takes place generally for very special initial conditions (inside the so-called stochastic layers) which are, however, everywhere dense in the phase space of the systsm.The basic and simplest one of the models considered is that of a pendulum under an external periodic perturbation. This model represents the behavior of nonlinear oscillations near a resonance, including the phenomenon of the stochastic instability within the stochastic layer of resonance. All models are treated both analytically and numerically. Some general regulations concerning the stochastic instability are presented, including a general, semi-quantitative method-the overlap criterion—to estimate the conditions for this stochastic instability as well as its main characteristics.
Results on the transition from soft to hard turbulence in simulations of two-dimensional Boussinesq convection are reported. The computed probability densities for temperature fluctuations are exponential in form in both soft and hard turbulence, unlike what is observed in experiments. In contrast, a change is obtained in the Nusselt number scaling on Rayleigh number in good agreement with the three-dimensional experiments.
In DIII-D H-mode plasmas, large amplitude modulation of steep edge electron pressure gradients are observed during the occurrences of giant edge-localized modes (ELM's). For a considerable range of plasma currents (1 to 2 MA), the experimental pressure gradient before a giant ELM is close to the theoretically predicted marginal stability limit for the first regime of the ideal ballooning mode. These results, together with observations showing that ELM's originate in the region of bad curvature, strongly suggest that ELM's are triggered by ballooning mode instabilities.
The temporal instabilities of a linear array of vortices are studied experimentally; a short-wavelength instability is observed. The results are compared to a one-dimensional model of coupled nonlinear oscillators, leading to excellent agreement.
The experimental study of a linear array of vortices shows the existence of spatially modulated states of oscillation above onset. A model of coupled oscillators is proposed, and the measurements of the corresponding first- and second-neighbor couplings show that these regimes are the result of frustration.
Using numerical simulations of the anelastic partial differential equations, convective flow in a two-dimensional polytropic atmosphere is investigated. Fixing the aspect ratio at unity and choosing the viscosity as the order parameter, the numerical solutions have indicated a variety of dynamic behavior including (1) oscillatory motion with a period of order the eddy turnover time, (2) quasi-periodic motion with a long period much longer than an eddy turnover time, and (3) at least two branches of solutions in certain regimes.