PHY1460H Course Project: Mixing and Chaotic Advection in Atmospheric Dynamics

Home , Chaotic mixing, Mixing (mathematics)

Andre R. Erler

April 20, 2009

1 Andre R. Erler PHY1460H April 20, 2009

Contents

1 Introduction 3

2 The Theory of Kinematic Mixing 3 2.1 Kinematic Foundations ...... 3 2.1.1 Measures of Mixing ...... 6 2.2 Chaotic Advection and Hamiltonian Chaos ...... 7 2.3 Mixing in Discrete Maps ...... 11

3 Chaotic Advection in the Atmosphere 11 3.1 Idealized Examples of Kinematic Mixing ...... 12 3.1.1 The Barotropic Rossby Wave ...... 12 3.1.2 A Simple Shear–Flow Map ...... 16 3.2 Realistic Mixing in the Atmosphere ...... 19 3.3 Dynamically Consistent Tracer Advection ...... 21 3.3.1 Dynamically Active and Passive Tracers ...... 25

4 Summary and Conclusion 27

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1 Introduction

Like many problems in fluid dynamics mixing is both, theoretically challenging and math- ematically involved, and at the same time of very high practical importance. Applications range from the chemical industry over life–sciences to geophysical fluid dynamics and as- trophysics. Most often mixing is associated with diffusion, however, in most fluids advection is a much faster process, so that kinematic mixing becomes dominant. Furthermore kinematic mixing is often understood to be synonymous with turbulence, which in the proper sense only exists in three dimensions. Yet, kinematic mixing does occur in two–dimensional flow. In this essay I will review our current understanding of kinematic mixing in two– dimensional flow and its application to atmospheric dynamics. I will further discuss the mixing of dynamically active tracers and the formation of mixing barriers in the form of jets ant large vortices in the atmosphere.

2 The Theory of Kinematic Mixing

Mixing as discussed in this paper is a purely kinematic phenomenon. The Eulerian flow field v(x, t) is assumed to be know; the theory of mixing is then concerned with the stretching and folding of material fluid elements along their trajectories (streaklines) as seen in Fig. 1. In stationary flow streaklines and streamlines generally coincide, but in time–dependent flow trajectories (streaklines) typically exhibit a much more complicated structure.

2.1 Kinematic Foundations

Most of the discussion in this section follows Ottino (1989). The treatment here will be focused on two–dimensional ﬂows, in three dimensions “line elements” will typically be

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Figure 1: Mixing in a journal–bearing flow (a)-(d) and a time–periodic cavity flow (e)-(f). Panels (b), (e), and (f) show stretching and folding of line elements; this process is what effects kinematic mixing. (a) shows typical initial conditions, (c) illustrates the initial stretching of a line element, and (d) gives an idea of the final mixed state. In panels (a), (c), and (d) line elements break up due to interfacial tension between the fluids. Note that the area around the upper blob does not participate in the mixing at all. (Ottino, 1990)

4 Andre R. Erler PHY1460H April 20, 2009 replaces by “surface elements” and the relations below change accordingly.

The inﬁnitesimal deformation of a line–element dx0 by the ﬂow is given by

dx = F · dx0 , (1) where F = ∇v denotes the deformation tensor. Since we are not primarily interested in absolute length or orientation, we introduce the stretch parameter λ as a local measure of line stretching:

|dx| 1 λ = lim = eT C e 2 . (2) |dx|→0 |dx0|

The tensor C = FT F is called the Cauchy–Green strain tensor and is a measure for local (and directional) stretching;1 e is a unit vector which deﬁnes the direction of strain under consideration. The rate of deformation of a line element is given by

D dx = dx · ∇v , (3) Dt so that he rate of stretching can be written as

D ln λ = eT D e , (4) Dt

1 T with the symmetric deformation tensor D = 2 (∇v) + ∇v . Given the measure of stretching λ, an approach to quantify mixing would be to estimate the magnitude of the deformation D along a typical trajectory in the Eulerian flow field (the stretching function), and integrate the rate of deformation (4) forward in time. If fluid trajectories are sufficiently chaotic, the deformation a line element is subject to is essentially random and can be approximated by a mean value (cf. random strain theory). A constant

1Deformation can be decomposed into rotation and stretching: F = RU; with RT R = 1, we obtain C = FT F = UT U, a measure for stretching.

5 Andre R. Erler PHY1460H April 20, 2009 stretch rate will result in exponential stretching of line elements but can usually not be achieved in a finite volume; for the same reason, non–vanishing stretch rates in a finite volume also imply folding. Only stretching and folding together achieve efficient mixing. Note that stationary area preserving 2D–flow can not produce efficient mixing: streamline must close inside the domain (and can not cross), so that the amount of folding is bounded and absolute stretching due to shear is at most linear in time.

2.1.1 Measures of Mixing

There are numerous ways to quantify mixing, and which one is used largely depends on the specific purpose; here I will give some examples to illustrate some common approaches. Inspired by the concept of stretching of line elements, a measure of mixing efficiency often used in atmospheric science is contour stretching: the stretching of iso–lines of some materially conserved tracer (Haynes et al., 2001, and references therein). Contour lines are either obtained from the Eulerian grid point values, or from Lagrangian contour advection schemes (Dritschel, 1989). The latter usually resolve finer scales of mixing. Another way of quantifying mixing is via Lagrangian decorrelation times: essentially flow characteristics of a material fluid element are correlated with the characteristics of the same fluid element at a previous time–step (lag–auto–correlation). The time–scale at which the auto–correlation becomes small, is the decorrelation time (Ngan and Shepherd, 1999, and references therein). Pierrehumbert (1990) takes a slightly different approach, where he considers the separation of fluid parcels which are initially close together. The evolution of the distribution (as a function of separation distance) permits inferences upon the characteristics of the mixing process. Pierrehumbert and Yang (1993) also give estimates for striation spacing and folding length and uses the characteristic length– and time–scales to characterize kinematic mixing (as opposed to diffusive mixing for instance).

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2.2 Chaotic Advection and Hamiltonian Chaos

Intuitively it is obvious that chaotic motion will likely lead to good mixing. Furthermore, we might expect that chaotic dynamics will thermalize a system and equilibrate gradients. While the former is genrally true the latter is only partially true: we will see that chaotic systems can maintain a degree of order which can lead to the steepening of gradients. A straight forward connection between mixing and chaotic dynamics arises, when Lya- punov exponents are related to the rate deformation: Lyapunov exponents are the eigenval- ues of the Lagrangian long–time average of the deformation tensor D(x, t), and are thus an estimate for the stretching rate: if dx is taken to be a line element between two infinitely close end points, the divergence of these two end points is proportional to the stretch of the line element (sensitive dependence on initial conditions). But in non–divergent 2D–flow the relevance of chaos theory for kinematic mixing goes much beyond that: if one exploits the formal analogy between the stream–function formulation and Hamiltonian dynamics, results from Hamiltonian chaos theory can be directly transferred to vortical (i.e. non–divergent) flow in two dimensions. Identifying the (area–preserving) stream–function with the Hamiltonian of a dynamical system

Ψ ⇐⇒ H , (5) the components of the ﬂow vector v = (u, v) correspond exactly to the generalized coordinate q and momentum p of a one–dimensional Hamiltonian system (two degrees of freedom, excluding time):

∂ dq ∂ u = − Ψ ⇐⇒ = − H (6) ∂y dt ∂p ∂ dp ∂ v = Ψ ⇐⇒ = H . (7) ∂x dt ∂q

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Critical points correspond to extrema of the stream–function (i.e. v = 0). They are clas- sified into hyperbolic, elliptic, and parabolic points. Elliptic points correspond to true extrema of the stream–function and are associated with vortices. Hyperbolic points are actually saddle points of the stream–function and are called stagnation points in the fluid dynamics literature; hyperbolic points have stable and unstable manifolds (directions of contraction and stretching, respectively). Depending on the orientation, orbits (or streamlines) crossing a hyperbolic point can either belong to the stable or the unstable manifold. Orbits which belong to both, the stable and the unstable manifold, are called homoclinic; if the stable and unstable manifolds do not belong to the same hyperbolic point, they are called heteroclinic (cf. Fig. 2a). The most important result of chaos theory concerns the behaviour of orbits in the vicinity of critical points under perturbation: if G(x, t) is a time–dependent perturbation the Hamiltonian/stream–function of the system H0(x, t) = H(x) + G(x, t)( being a small parameter), attains more than two degrees of freedom and looses its integrability;2 it can exhibit chaotic behaviour. Closed orbits are structurally unstable and break up under perturbation; this means that homoclinic and heteroclinic connections disconnect, as illustrated in Fig. 2b. Orbits on the stable manifold are slightly perturbed and begin to fold up and stretch under increasingly divergent transverse flow as they approach the hyperbolic point. Because the (Hamiltonian) flow preserves area, the striation of the folds becomes increasingly thinner. Thus, under time–dependent perturbation of the stream–function we expect to see chaotic behaviour (and strong mixing) along homoclinic/heteroclinic streamlines near hyperbolic/stagnation point. This behaviour can be visualized in Poincarésections, using Lagrangian tracer trajectories: the position of a particular fluid element is traced and plot-

2In the case of non–divergent 2D–ﬂow, the two integral invariants constraining the motion are enstrophy and energy.

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Figure 2: A critical points of a Hamiltonian system and associated stable and unstable manifolds. (a) a homoclinic orbit associated with a hyperbolic point (left) circles around an elliptic point (right), (b) under a small time–dependent perturbation the stable and unstable manifold disconnect and the homoclinic orbit becomes a region of chaotic dynamics, (c) illustrates stretching and folding as a consequence of ﬂow deformation near hyperbolic points.(Ottino, 1990)

9 Andre R. Erler PHY1460H April 20, 2009 ted onto the Poincar´esection at time intervals matching the period of the perturbation. If the motion is regular, the resulting points will trace out a streamline, while point clouds are indicative of chaotic mixing (cf. Fig. 3). Kinematic mixing in regions of chaotic dynamics is often referred to as chaotic advection. Hyperbolic points and heteroclinic cycles/streamlines under time–dependent perturbation are characteristic of chaotic advection.

Heuristic Explanation The results of Hamiltonian chaos theory described above can also be interpreted heuristically from a fluid dynamics point of view. Consider a hyperbolic or stagnation point in a 2D–fluid. The heteroclinic streamline approaching the hyperbolic point on the stable manifold coincides with the separation line between streamlines which pass the stagnation point to the left and to the right (hence the name separatrix); trajectories following this streamline are highly unstable since small perturbations will cause the fluid element to diverge either to the left or the right (extreme sensitivity to initial conditions). If the flow is subject to small perturbations, the separatrix– streamline will shift and wiggle slightly so that fluid elements end up either to the left or the right. If we imagine a continuous strand of fluid along the heteroclinic streamline, a small perturbation will cause parts of this strand to diverge to the right, while others diverge to the left. Extreme stretching and folding of fluid lines is the consequence. Fig. 2c illustrates this behaviour. The preservation of area immediately follows from the non–divergence condition; in this picture it is also clear why the number of heteroclinic points is infinite: the folded line elements only stretch and get thinner as they approach the hyperbolic point, but the fluid strand as a whole can never pass the hyperbolic point: layer on layer of thinning striation accumulate in a limiting process.

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2.3 Mixing in Discrete Maps

Kinematic mixing can be treated in a conceptually (and numerically) simplified form, if the continuous displacement flow (trajectories of fluid elements) is replaced by a discrete displacement map. Note that it is necessary to know the Eulerian flow field a priori, so that the study of discrete maps is restricted to kinematic effects. The map Φ can easily be constructed from a given time–periodic Eulerian flow field by integrating trajectories (streaklines) over the period of the flow T .

Z tk+1 xk+1 = Φ(xk) Φ(xk) = v(x(t)) dt (8) tk The map Φ defines a discrete dynamical system. Note that this formulation directly re- lates to Poincarésections of the continuous system with period T . But of course discrete dynamical systems do not necessarily have a continuous analog. This perspective is conceptually attractive because certain prototypical effects can be isolated in continuous flows. The most efficient mixing is effected by the so–called “baker’s map”: stretching by a factor of two and subsequent folding in the middle (Ottino, 1989).3 An approximation to the baker’s map to continuous flow systems is the “horseshoe map” (so called for its shape).

3 Chaotic Advection in the Atmosphere

Mixing processes are of fundamental importance to the atmosphere. Not only for the transport of trace gases and pollutants, but also from a dynamical perspective: dynamical quantities like entropy (heat) and (potential) vorticity are materially conserved and are thus mixed in a similar as tracers. The governing ﬂow structure which eﬀects chaotic advection

3This process of mixing is akin to the kneading of a dough, hence the name, but also, for example, to forge–welding in metalworks.

11 Andre R. Erler PHY1460H April 20, 2009 under perturbation is called organizing structure; the Rossby wave in the following example is such a structure. Furthermore we will see that the relationship between organizing structures and dynamically active tracers is an important constraint for the dynamical structure of the atmosphere.

3.1 Idealized Examples of Kinematic Mixing

In the following I will give two examples to illustrate some additional points concerning chaotic advection. The examples are chosen in a way that they correspond to observed phenomena in the atmosphere and are directly relevant for an understanding of mixing in the atmosphere. Nevertheless many of the points made here are also applicable outside the realm of atmospheric science.

3.1.1 The Barotropic Rossby Wave

Pierrehumbert (1990) discusses chaotic advection caused by a periodically perturbed barotropic Rossby wave — a structure commonly found in the atmosphere. The barotropic Rossby wave is a solution of the nonlinear barotropic vorticity equation on the β–plane:4

∂ ∂ ∇2Ψ + J(Ψ, ∇2Ψ) + β Ψ = 0 . (9) ∂t ∂x

A plane–wave Ansatz with zonal (longitudinal) and meridional (latitudinal) wave numbers

4The β–plane is a linear approximation to the sphere where the coordinates are treated as locally Cartesian and the planetary vorticity varies only linearly with latitude. β is the ﬁrst order coeﬃcient of the planetary vorticity expansion. See Pedlosky (1987) for more information.

12 Andre R. Erler PHY1460H April 20, 2009 k and l leads to a solution with a dispersion relation for ω:

Ψ(x, y, t) = < (Ψ0exp [i (k x + l y − ωt)]) (10) ω = −βk / k2 + l2 . (11)

The zonal phase speed cx = ω/k is negative, so that the wave propagates to the west. For the following analysis we will use a co-moving frame, so that the wave appears stationary. Linearizing about the solution (10), we can add another Rossby wave of small amplitude and shorter wave–length (and smaller frequency). This introduces a time–dependent periodic perturbation to the dominant stationary wave (in the co-moving frame). Choosing a meridional wave number of l = 1/2 for the dominant quasi–stationary mode, we can write

Ψ(x, y, t) = sin(y) [sin k(x − ct) + sin k(x − ct + φ)] , (12) where << 1 is the amplitude of the perturbation, and k and c are the wavenumber and phase speed associated with the perturbation. Fig. 4 (left) illustrates the process which leads to kinematic mixing: a line element is first stretched and distributed along the heteroclinic stream–line and subsequently folded, when the filament length exceeds the size of the area enclosed by the heteroclinic streamline. The process continues and eventually disperses the line element over almost the entire area enclosed by the heteroclinic streamline. Fig. 4 (right) shows the final state of mixing for different values of . Evidently the degree to which the vortex centre is included in the mixing process depends on the amplitude of the perturbation (this point will become clearer in the next section). Furthermore in none of the examples shown in Fig. 4 (right) mixing occurs between the two vortices: the domain is divided into an upper and a lower part by the meandering streamlines in the middle between the vortices; these continuous streamlines are invariant sets and represent mixing barriers. However, if the amplitude of the perturbation is further increased, even the central barrier will eventually break and mixing will occur over the whole domain.

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Figure 3: The stream–function of the unperturbed quasi–stationary Rossby wave (top panel); also indicated is a heteroclinic connection between two hyperbolic/stagnation point. In the Poincar´esection (bottom panel) exactly these areas exhibit chaotic advection and mixing (indicated by the cloud of red points); the meandering streamlines in the middle (black lines between the vortices) remain connected and thus form a barrier to chaotic advection. Also note that areas near the vortex centre remain unmixed (green lines); these areas shrink and eventually disappear when the perturbation is increased (with suﬃciently strong perturbation the barrier in the middle also breaks). (Pierrehum- bert, 1990)

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Figure 4: Left: the dispersion of an initial line element into a well mixed point cloud around the vortex centre. Note the initial stretching and subsequent folding, as the line elements become too long for the area contained within the heteroclinic streamline. Right: the ﬁnal stage of mixing for successively increasing perturbation amplitude (the number indicated on the panels is the value of used in (12)). (Pierrehumbert, 1990)

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The main conclusions which will also generalize to more complex ﬂow pattern are:

1. The distribution of tracers ﬁrst occurs by stretching and ﬁlamentation over large length scales. The stretching is exponentially fast near hyperbolic points (Ottino, 1989).

2. When the ﬁlament length reaches the domain size, folding occurs. Chaotic advection can mix tracers over an area (as opposed to simple shear–stretching of ﬂuid strands).

3. Chaotic advection is conﬁned to areas near heteroclinic streamlines. Mixing barriers exist, which can maintain large gradients (e.g. jets in atmospheric dynamics).

Also note that the above characteristics clearly separate chaotic advection (kinematic mixing) from diﬀusive mixing. In principle kinematic mixing is reversible since the equations of motion are strictly time–reversible. However, because of the extreme sensitivity to initial conditions this reversibility is of no practical consequence and does not occur after only a few iterations of stretching and folding. The reversibility of chaotic mixing is to be understood in exactly the same way, as the in–principle reversibility of gas kinematics (due to the reversibility of Newtonian Mechanics).

3.1.2 A Simple Shear–Flow Map

In atmospheric dynamics discrete displacement maps of the form (8) have been used to study chaotic advection in the vicinity of jet structures. A recent example studied by Haynes et al. (2007) is displayed in Fig. 5. The map is given by the relation5

2 xk+1 = xk + exp − (yk − π) + 0.65 (yk − π) , (13)

yk+1 = yk + (xk − π) . (14)

5 The relation given by Haynes et al. (2007) for the y–component is yk+1 = yk + xk; I believe this to be in error, because it would imply monotonous growth in y (∀ x ∈ (0, 2π]).

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Figure 5: Poincar´esection of the discrete map (8) representing ﬂuid displacement near a jet stream ( = 1.5). The left panel shows the displacement (solid line) at each iteration and the corresponding shear (dashed line) as a function of y. The continuous region of almost no mixing (low point density) in the upper half is a mixing barrier; it can be interpreted as a meandering jet in the atmosphere. Note that the location of zero shear is close to the centre of the perturbed jet. (Haynes et al., 2007)

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As before, is a measure for the perturbation amplitude. Haynes et al. (2007) used the discrete map (8) in extensive numerical experiments to show that jets form a central barrier to chaotic advection of the form shown in Fig. 5 (right), which is persistent over a large range of perturbations. The value for in Fig. 5 is chosen just below the value where the major mixing barrier dividing the domain breaks up. They further argue that jets in the atmosphere (which exhibit a similar behaviour) are associated with invariant tori, and that these are typically also the last tori to break up, when the perturbation is increased. According to the KAM–theorem, tori of sufficiently irrational winding number survive small perturbations, while others break up and form self–similar sub–tori, termed cantori. These cantori break up again under ever increasing perturbation amplitude and form an infinite self–similar sequence (akin to the Cantor Set, hence the name). The important point here is that these cantori can still significantly impede mixing between larger regions of chaos, although they can not prevent it entirely (Ottino, 1989). This behaviour can already be observed in Fig. 5 (right): the centre of the domain is a region of reduced mixing, reminiscent of the vortex centre in the previous example (cf. Fig. 3). The break up of invariant tori and the formation of self–similar cantori in-between surviving tori can be beautifully observed here. Haynes et al. (2007) further argue that the formation of cantori after the break–up of the invariant tori is rather unrealistic and is not consistent with the dynamical constraints of a fluid dynamical jet. They conclude that for this reason purely kinematic mixing models do not faithfully represent the break up of mixing barriers.

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3.2 Realistic Mixing in the Atmosphere

The motion in the atmosphere in mid–latitudes is to good approximation adiabatic, so that the flow is confined to isentropic surfaces.6 Therefore the dynamics are quasi–two– dimensional and we can hope that the theory developed in Chapter 2 and the examples above can provide insights into the mixing processes that occur in the atmosphere. The main difference to the examples presented above is of course the simple fact that the flow of the atmosphere is constantly changing. But Rossby waves do occur in the atmosphere and their life–time is of the order of weeks, so that they can still serve as organizing structures. If a structure exhibits hyperbolic/stagnation points and heteroclinic streamlines, we can expect chaotic advection (for waves it will be some topological equivalent of the well know “Cat’s Eye”–phase portrait). Suitable time–dependent perturbations will almost always be available in the form of shorter waves. Only they may become too strong, so that the organizing structure breaks up. Pierrehumbert and Yang (1993) used wind fields from a GCM (General Circulation Model) simulation to compute tracer transport and mixing (off–line). The results of one such tracer transport calculation is shown in Fig. 6: the tracer was initialized with a steep meridional gradient centred around a sinusoid of wavenumber 3 in the zonal direction;7 the advecting wind fields were obtained from a GCM. The resulting tracer distribution exhibits the characteristics generally associated with chaotic advection, namely: filamentation, advection on large scales, before smaller scales are mixed, and a self–similar cascade of ever finer strands and folds, until the striation is not distinguishable anymore and the fluid can be considered mixed (on sufficiently large scales). Pierrehumbert and Yang (1993) conclude that chaotic advection does explain a great deal of the large–scale mixing observed in the

6In atmospheric sciences isentropic surfaces are commonly denoted by their potential temperature (θ), that is, the temperature the air would have under sea–level pressure (after adiabatic compression). 7One may imagine a pattern of three Rossby waves along a mid–latitudes small–circle; cf. Fig. 3.

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Figure 6: Global tracer transport and mixing on the 315K isentrope; advection was com- puted oﬀ–line using wind ﬁelds from a GCM. The vertical axis is latitude (with the equator in the middle), the horizontal is longitude, and time progresses from left to right and top to bottom. The tracer was initialized with a sinusoidal pattern of wavenumber 3 in the zonal direction. Evidently tracers become well mixed in the atmosphere within only a few weeks. The mixing patterns are characteristic of chaotic advection. (Pierrehumbert and Yang, 1993)

20 Andre R. Erler PHY1460H April 20, 2009 earth’ atmosphere, and their results were confirmed by other authors (Ngan and Shepherd, 1999, to name only one). However Pierrehumbert and Yang (1993) give a caveat to their results, which we will explore in more detail in the next section: if the tracer field is taken to be the vorticity field of a wavenumber 3 Rossby wave (not an unreasonable scenario), the implication would be that the wave would decay within approximately 10 to 15 days (since its characteristic vorticity gradient would be dispersed). By the same token, jet structures in the atmosphere should not persist much longer than a couple of weeks. On the other hand, Haynes et al. (2001) (and others before) showed that jets are in fact barriers to mixing, in very much the same sense as the the continuous meandering streamlines in Fig. 3 and the mixing barrier in Fig. 5 (right). Fig. 7 shows the strength of mixing8 in isentropic coordinates, derived from an idealized GCM. It is immediately evident that mixing in the atmosphere exhibits a much richer structure than one would expect from Fig. 6. Moreover, structures in mixing properties are also associated with dynamical structures: regions of very low mixing correspond to strong and persistent jets, while regions of high mixing generally exhibit no persistent large–scale dynamical structures.

3.3 Dynamically Consistent Tracer Advection

Vorticity/PV and entropy are materially conserved tracers and at the same time dynamically relevant quantities, which have a direct eﬀect on the evolution of the governing ﬂow; such quantities are called “dynamically active” tracers. Tracers which have no feedback on the dynamics are called “passive”. From the preceding section we are left with two seemingly contradictory results: one might expect that mixing of dynamically active tracers should lead to a state of the at-

8Measured by the stretching of PV–contours. Potential vorticity PV is the isentropic analog to vorticity in 2D; it is a materially conserved scalar in adiabatic 3D–ﬂow.

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Figure 7: Mixing in an idealized GCM simulation; displayed is the stretching of potential vorticity–contours, a measure for mixing. The vertical coordinate is potential temperature (∼ entropy) and the horizontal coordinate is latitude (northern hemisphere only). Note that mixing is very high in layers intersecting the ground (the troposphere); in the mid– latitude stratosphere a mixing barrier is evident. (Haynes et al., 2001)

22 Andre R. Erler PHY1460H April 20, 2009 mosphere which exhibits no dynamical structure; on the other hand there appears to be distinct structure in the mixing process itself. The resolution is that the organizing structure of the mixing process are of course them- selves associated with characteristic patterns of dynamically active tracers (usually potential vorticity on isentropic surfaces). Only structures which are stable solutions of the equations of motion (like the Rossby wave) emerge as organizing structures, since they must preserve their own characteristic vorticity gradient, in order to persist. Of course a perturbation to the organizing structure is required, but as long as it is sufficiently small, chaotic advection in the proper sense is constrained by the preservation of the organizing structure. Mixing barriers then form along dynamically relevant gradients: in the case of the Rossby wave along the meandering central streamlines (which traverse the characteristic vorticity gradient of the barotropic Rossby wave). Haynes et al. (2007) expanded on the Rossby wave example originally given by Pierre- humbert (1990) and increased the perturbation amplitude in a fully nonlinear simulation9, and compared the results to a purely kinematic superposition of the advection fields obtained from the organizing structure (the long wavelength Rossby wave) and the initial perturbation (a shorter Rossby wave of half the wavelength). Final quasi–stationary states for different perturbation amplitudes (defined as before) are shown in Fig. 8. Panel (a) shows the vorticity pattern of the initial Rossby wave (without perturbation), panels (b) through (e) show how successively stronger perturbations lead to homogenization within the regions bounded by the central mixing barrier, leading to ever stronger gradients across the mixing barrier; panel (f) shows a state of complete homogenization after break–up of the central mixing barrier. The point Haynes et al. (2007) make here is that the tran-

9Pierrehumbert (1990) also integrated the nonlinear vorticity equation to obtain the advection ﬁelds, however his perturbation amplitude was suﬃciently small in all cases. Also note that Haynes et al. (2007) applied an additional barotropic shear, which was found to facilitate barrier break–up.

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Figure 8: Quasi–stationary ﬁnal states for diﬀerent values of the perturbation amplitude . The shading indicates vorticity; initial streamlines of the “organizing structure” are superimposed (black dashed lines). –values for panels (a)-(f) are = {0, 0.05, 0.1, 0.3, 0.4, 0.5}. Note that only the largest perturbation (f) results in a complete homogenization of vorticity; (b)-(e) show successively stronger homogenization in the two areas separated by the central continuous mixing barrier. Haynes et al. (2007)

24 Andre R. Erler PHY1460H April 20, 2009 sition from the regime dominated by the mixing barrier (e, = 0.4), to the completely homogenized state (f, = 0.5) is very fast — much faster than might be anticipated from purely kinematic calculations. To illustrate this point they produced Fig. 9, which shows the average time it takes for trajectories to cross the central mixing barrier. Evidently the transition (break–up of the barrier) is almost instantaneous in the dynamically consistent calculation. The kinematic calculation significantly underestimates the mixing efficiency after the break–up of the barrier; the reason for this is, Haynes et al. (2007) conjecture, that cantori which survive to a significant extend in the kinematic case, can severely impede mixing by maintaining a partial (but open) boundary in areas where the original mixing barrier was destroyed. In the dynamically consistent case the dynamics change radically once vorticity becomes significantly mixed and the organizing structure breaks down (and all tori and cantori with it), resulting in a regime transition. From a dynamical point of view the organizing structure breaks down because the Rossby waves start to interact nonlinearly once the perturbation is sufficiently strong, so that the resulting streamfunction is no longer a stable solution of the equations of motion and decays.

3.3.1 Dynamically Active and Passive Tracers

Dynamically active tracers place severe constraints on the dynamics and the structure of mixing. However, this does not mean that the presence of dynamically active tracers prevents efficient kinematic mixing altogether (cf. Fig. 6). Wirth et al. (2004) showed in simulations using a surface–quasi–geostrophic model that only dynamically active tracers (surface potential temperature in this case) suffer from low mixing efficiency due to dynamical constraints; tracers which are uncorrelated with dynamically active tracers are mixed with high efficiency. Snapshots of typical tracer distribution are shown in Fig. 10. The active tracer indicates the location of two vortex patches at

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Figure 9: Mixing across the central jet as a function of perturbation amplitude. The measure of mixing here is the average time it takes for a trajectory to cross the central mixing barrier. The dashed line corresponds to a purely kinematic calculations while the solid line refers to the dynamically consistent calculations (some of which are shown in Fig. 8). The break–up of the mixing barrier in the dynamically consistent case is almost discontinuous, in contrast to the purely kinematic calculation. Haynes et al. (2007)

26 Andre R. Erler PHY1460H April 20, 2009 the surface which govern the dynamics. The passive tracer has the same distribution but is located at a higher level (above the surface anomaly), so that the tracer is only partially correlated with the vortex patches. Evidently the dynamically active tracer does not disperse but rather forms an even larger vortex, which is in accord with the existence of an inverse energy cascade in two–dimensional flow. The passive tracer on the other hand gets mixed and disperses over the entire domain (although the homogenization is less than perfect). Wirth et al. (2004) conclude that, when mixing rates are of interest, the dynamical role of a materially conserved quantity has to be taken into account. That is, whether the tracer is part of a stable coherent structure or takes part in the nonlinear inverse energy cascade (in two dimensions), or is simply a passive tracer.10 In the case of vorticity, the most important active tracer, the matter is further complicated by McIntyre’s “scale effect”: due to the structure of the inversion operator (LΨ = ζ) small patches of vorticity can be mixed without dynamical constraint, while large–scale patches are dynamically active. (This effect may explain the initial success of linear superposition in Fig. 3 and the persistence of the mixing barrier in Fig. 8 (b)-(e).)

4 Summary and Conclusion

The dynamics of the atmosphere on large scales are quasi–two–dimensional, and the two processes commonly associated with mixing, diﬀusion and turbulence, are weak. Transport is dominated by advection, but the patterns appear highly irregular. Signiﬁcant theoretical progress was made by exploiting the formal analogy between two–

10Note that tracers which are inherently not dynamically relevant but highly correlated with an active tracer, are of course subject to the same dynamical constraints as the active tracer (to the extent that they are correlated).

27 Andre R. Erler PHY1460H April 20, 2009

Figure 10: Advection of dynamically active and passive tracers. The upper panel shows how two vortex patches merge and form a larger coherent structure (inverse energy cascade). The lower panel shows a situation with slightly diﬀerent dynamics, where the tracer distribution is only weakly correlated with dynamical quantities. The passive tracer forms ﬁlaments and is distributed over the entire domain, in sharp contrast to the active tracer. (Wirth et al., 2004)

28 Andre R. Erler PHY1460H April 20, 2009 dimensional non–divergent (vortical) flow and Hamiltonian dynamical systems: the streamfunction of a non–divergent 2D–flow can be identified with the Hamiltonian of a dynamical system with two degrees of freedom. Under time–dependent perturbation these dynamical systems can exhibit chaotic behaviour near so–called heteroclinic connections between hyperbolic points (stagnation points in fluid dynamics). Chaotic trajectories are taken as indicative of strong mixing. Chaotic advection is characterized by transport over large scales before mixing at small scales takes place. Due to chaotic dynamics fluid filaments are not only stretched but eventually distributed over an area (as opposed to a streamline). Furthermore kinematic mixing is restricted to said regions of chaotic dynamics, so that mixing can exhibit a rich spatial structure. Mixing barriers exist and can even lead to enhanced gradients. Transport in large–scale atmospheric flow exhibit similar characteristics, so that chaotic advection appears to be a useful concept in atmospheric science. In particular, the paradigm of chaotic advection put the concept of wave breaking and irreversible mixing in baroclinic life–cycles on solid theoretical grounds Pierrehumbert (1990). However, a purely kinematic approach to tracer advection in the atmosphere is somewhat problematic. Dynamically active tracers (such as vorticity) change the governing flow (the organizing structure), but are at the same time also subject to severe dynamical constraints: active tracers have to preserve the integrals of motion (of which there are two: energy and enstrophy) in addition to mass conservation, while passive tracer are merely subject to the latter. In order to study the dynamical implications of active tracer advection, it is of paramount importance to ensure the dynamical consistency of tracer advection and the advecting flow. Conceptually we imagine chaotic advection to be governed by organizing structures (such as Rossby waves or large vortices). These structures form mixing barriers, which prevent advection of active tracers that would cause their own break–up.

29 Andre R. Erler PHY1460H April 20, 2009

Of course there is no new dynamics here: this is just another way of saying that these structures are stable solutions of the equations of motion (and in order to persist, they must). However looking at atmospheric dynamics from a mixing point of view still has its use; not only is it possible to infere mixing constraints from dynamical structures (such as jets, or, most prominently, the polar vortex), it can also provide dynamical insights: for example (a) based on correlations between dynamically active tracers and external transport constraints (such as the global heat budget, cf. Schneider, 200411), or (b) the local homogenization and simultaneous steepening of gradients at mixing barriers (“PV– staircase”).

11Schneider (2004) uses a diffusive closure for PV–mixing, which appears arkward in the light of this discussion, however, given the crudeness of his approximations and the scales on which it is applied, this simplification appears defensible. In fact I am not aware of any accepted closure scheme for chaotic advection or eddy–fluxes on planetary scales.

30 Andre R. Erler PHY1460H April 20, 2009

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