Vertical shear plus horizontal stretching as a route to mixing

Transcription

1 Vertical shear plus horizontal stretching as a route to mixing Peter H. Haynes Department of Applie Mathematics an Theoretical Physics, University of Cambrige, UK Abstract. The combine effect of vertical shear an horizontal stretching leas to thin, sloping structures in tracer fiels, whose vertical length scale is much smaller than their horizontal length scale. These structures are then vulnerable to vertical mixing processes. This effect nees to be taken into account when interpreting the horizontal structure of oceanic (an atmospheric) tracers an may explain recent oceanographic observations. 1. Introuction In both atmosphere an ocean it is useful to ivie velocity fiels into isentropic or isopycnal parts, which move flui parcels along potential temperature or potential ensity surfaces, an iabatic or iapycnal parts, which move flui parcels across such surfaces. This is because iabatic or iapycnal motion can be accomplishe only by non-conservative physical processes such as raiative heating (in the atmosphere) an molecular iffusion of temperature enhance by three-imensional turbulence (in atmosphere an ocean). In large parts of the atmosphere an ocean, such processes are weak in some average sense, so that iabatic or iapycnal velocities must be much smaller than isentropic or isopycnal velocities. The process of stirring may therefore be regare as layerwise two-imensional, in the sense that ifferential avection by the quasi-horizontal isentropic/isopycnal velocity fiel acts inepenently on each isentropic or isopycnal surface to istort tracer fiels into complex geometrical configurations. If atmospheric an oceanic flows were purely two-imensional, similar to many numerical simulations or laboratory experiments, then stirring woul simply strengthen horizontal spatial graients of tracer fiels until horizontal iffusion became competitive with avection an mixing occure. But in layerwise two-imensional atmospheric an oceanic flows, horizontal avection that varies in the vertical may also act to increase vertical graients, thereby enhancing the effects of vertical iffusion, or more complicate vertical mixing processes. The fact that vertical scales are almost always observe to be consierably smaller than horizontal scales suggests that mixing iffusion rather than horizontal mixing is the ominant process. Most previous theoretical investigations of stirring an mixing of tracers have focusse on moels of quasiisotropic flows, either in two or three imensions. The consierations above suggest that the most relevant moel for the atmosphere an the ocean is an anisotropic moel in which the vertical velocity is zero, but the horizontal velocities epen on the vertical coorinate. Haynes an Anglae (1997, hereafter HA97) use such an anisotropic moel to stuy the enhancement of vertical graients by horizontal stirring plus vertical shear. This is a relatively simple extension to previous work, but essential if theoretical results are to be relevant to real flows. The HA97 work was motivate primarily by the stratosphere. This article reviews an expans on some of the arguments in that paper, focussing in particular on the implications for stirring an mixing in the ocean. The moel to be presente is base on the paraigm of chaotic avection, that flows that are smoothly varying in space an time may lea to patterns of avecte tracer that are spatially highly complex. One of the requirements on a chaotic avection flow is that there is a useful ivision of scales between the smallest active scale in the avecting velocity fiel an the iffusive scale (or more generally scale at which genuine mixing processes act). The case of three-imensional turbulence, for example, oes not fall into this category, since the velocity fiel has complex spatial structure at scales own to the iffusive scale an it may be argue that the stirring of tracer features at any scale larger than the iffusive scale is ominate by the flow at the same scale (rather than at the large scale). In fact the chaotic avection paraigm is relevant in three-imensionally turbulent flows for tracers with iffusivity much less than the momentum iffusivity (large Schmit number). The tracer fiel may then have non-trivial structure on a spatial scale smaller than the Kolmogorov scale, with the flow at the Kolmogorov 73

2 74 HAYNES scale proviing the large-scale stirring. This is often calle the Batchelor regime of turbulence an has been stuie as an important special case in recent theoretical work on tracer fiels in turbulent flows (e.g. see Shraiman an Siggia, 2000 an references therein). In the stratosphere it seems to be useful to consier tracer fiels to be etermine by the stirring effect of the large-scale flow (varying on spatial scales of hunres to thousans of kilometers) an by the mixing effect of localise patches of three-imensional turbulence, with vertical scales of a hunre meters or so an horizontal scales of a few kilometers. There is no strong evience of active stirring by eies on scales from a few hunre own to tens of kilometers, an in this range of scales the chaotic avection paraigm is therefore relevant. Similarly in the ocean one might argue that mesoscale eies with scales of tens of kilometers are the ominant part of the flow in stirring the tracer fiel at scales from a few tens of kilometers own to the scale of small-scale mixing events. The structure of the paper is as follows. Section 2 escribes a simple moel problem of steay flow incluing both horizontal stretching an vertical shear, focussing on the spreaing of a tracer from a point release in such a flow. Section 3 iscusses some aspects of tracer evolution in more general ranom flows. Section 4 consiers application of these ieas to the observations of Lewell et al. (1993) (hereafter L93). Section 5 gives a brief iscussion of the possible competing role of mixing by ouble-iffusive intrusions. Section 6 escribes some possible future lines of research. 2. Steay flow moel The simplification allowe by the chaotic avection paraigm is that there is a finite scale below which the flow may be consiere as a linear function of the space coorinates, i.e. the velocity fiel may be expane as a (multiimensional) Taylor series about some reference location an only the linear term in the series retaine. The effects of small-scale mixing events may represente by a iffusion term (but more sophisticate representations are also possible). We follow HA97 in consiering first a steay linear flow that has the two ingreients of horizontal strain, Γ, plus vertical shear, Λ. In much of the atmosphere an ocean it is relevant to consier the parameter regime Λ Γ, i.e. vertical shear much larger than horizontal strain. This may be argue from the observe fact that horizontal length scales ten to be larger than vertical scales or by appealing to theoretical arguments that the ratio of horizontal to vertical scales is of orer f/n (Prantl s ratio), where f is the Coriolis parameter an N the buoyancy frequency. f/n is O(100) in the atmosphere an in the oceanic thermocline an perhaps reuces to O(10) in the eep ocean. The flow is taken to have components (Γx, Γy + Λz,0) in the x, y an z irections respectively, where (x, y, z) are Cartesian coorinates, with z vertical. The steainess assumption implies that the horizontal strain Γ an the vertical shear Λ are constants. The equation for the evolution in this flow of a tracer with concentration χ(x, y, z, t) is therefore χ t +Γx χ χ +(Λz Γy) x y = κ( 2 χ x + 2 χ 2 y + 2 χ ), (1) 2 z2 where κ is the iffusivity, assume constant. HA97 consiere sinusoial solutions of this equation. Another useful solution of this equation, an inee of the equation for tracer concentration in any flow where velocity components are linear functions of space coorinates (incluing time-epenent flows), is an ellipsoial Gaussian solution, i.e. an exponential function of a negative efinite quaratic function of x, y an z. This is conveniently obtaine by consiering the secon-orer spatial moments of the solution, such as m xx = x 2 χv/ χv, m xy etc. The six moments are sufficient to etermine the quaratic function an the constraint that the total amount of tracer remains constant then etermines the coefficient of the exponential function. In the above flow, the equations for the moments may be shown to be t m xx =2Γm xx + κ (2) t m xy =Λm xz (3) t m xz =Γm xz (4) t m yy = 2Γm yy +2Λm yz +2κ (5) t m yz = Γm yz +Λm zz (6) t m zz = κ (7) These equations may be solve straightforwarly. For brevity we consier the case where all moments ten to zero as t 0 +, corresponing to a point release. Then the solutions are m xx = κ Γ e2γt (8) m xy = m xz = 0 (9)

3 VERTICAL SHEAR AS A ROUTE TO MIXING 75 m yy = κλ2 Γ 3 { 2Γt +4e Γt e 2Γt 3 } + κ Γ { 1 e 2Γt} (10) m yz =2 κλ Γ 2 { Γt 1+e Γt} (11) m zz =2κt (12) (κt) (I) 1/2 (κt) 1/2 Λt (κt) 1/2 (II) (κt) 1/2 First consier the case where Λ = 0, i.e. the case of purely two-imensional flow. Then m xx = κ(e 2Γt 1)/Γ an m yy = κ(1 e 2Γt )/Γ. For small times, t Γ 1, iffusion ominates an both m xx an m yy increase linearly with time. When t Γ 1 avection by the horizontal strain flow becomes competitive with iffusion. The tracer elongates in the irection of the stretching axis, so that for t Γ 1, m xx κe 2Γt /Γ, whilst in the irection of the compression axis a steay-state balance between strain an iffusion is achieve, with m yy κ/γ. At all times the only vertical transport is through iffusion, so m zz increases linearly with t at all times. What is the effect of aing vertical shear, i.e. taking Λ 0 (with Λ Γ)? The elongation of the tracer along the stretching axis of the horizontal strain axis is unchange. However, the spreaing of the tracer in the irection of the compression axis is rather ifferent. Examination of the expression for m yy above shows that there are three regimes. For t < Λ 1 (regime I) there is purely iffusive spreaing as above. For Λ 1 < t< Γ 1 (regime II) the iffusive spreaing of the tracer in the vertical allows the vertical shear to augment the spreaing in the horizontal, so that m yy κλ 2 t 3. Finally when t Γ 1 horizontal avection begins to inhibit the horizontal spreaing, so that for t > Γ 1 (regime III), m yy κtλ 2 /Γ 2. These three regimes are epicte graphically in Figure 1. Note that m yy continues to increase with time, in contrast to the case with Λ = 0. In regime III m yy Λm yz /Γ Λ 2 m zz /Γ 2 suggesting that the y an z scales of the tracer patch are in the ratio Λ/Γ. Inee what happens is that the tracer patch forms a sheet that slopes at angle Γ/Λ (from the horizontal) in the (y, z) plane. The extent of the sheet in the vertical irection is (κt) 1/2, therefore the extent of the sheet in the horizontal (y) irection is Λ(κt) 1/2 /Γ. This explains the ominant behaviour of m yy, m yz an m zz. In orer to euce the sprea of the tracer on a given horizontal surface it is useful to consier m yy 2Λm yz /Γ+m zz Λ 2 /Γ 2, i.e. the moment of (y Λz/Γ) 2. It is straightforwar to show that m yy 2Λm yz /Γ+m zz Λ 2 /Γ 2 = κ Λ2 (1 + Γ Γ )(1 2 e 2Γt ) κ Λ 2 Γ Γ (1 2 e 2Γt ). (13) Figure 1. (Λ/Γ) 2 (κt) 1/2 (III) (κt) 1/2 The sprea of tracer in the (y, z) in each of regimes I (t < Λ 1 ), regime II (Λ 1 < t < Γ 1 ) an regime III (Γ 1 < t). Thus the sprea of the tracer on a given horizontal surface is (κ/γ) 1/2 Λ/Γ. The effect of the vertical shear is therefore to increase the equilibrium with of the tracer patch by a factor Λ/Γ. In summary, the effect of the vertical shear is to lea to sloping structures in the tracer fiel with aspect ratio (ratio of horizontal length scale to vertical length scale) α = Λ/Γ. The equlibrium with of these structures is larger by a factor α than woul be the case without vertical shear. One might say that it is as if a horizontal iffusivity κα 2 were acting. 3. Time-epenent flow moels In a flow that varies in space an time, the velocity graient encountere by a flui parcel changes with time as the particle moves through the flow. The evolution of small-scale tracer features is therefore governe by the equation for tracer evolution in a time-epenent linear flow. In the stuy of turbulence there is a long traition of consiering the effect of a linear flow that varies ranomly in time (using so-calle ranom-straining moels). This approach has been use most recently by various authors to give a rather complete escription of the statistics of tracer fiels in Batchelor-regime turbulence. [See Balkovsky an Fouxon, 1999 an Falkovich et al., 2001 for further etails of this work.] The ranom-straining moel most relevant to the atmosphere an ocean is non-stanar in that the statistics of the velocity graient tensor is not isotropic (since the vertical velocity is zero, but the vertical graient of the horizontal velocity is not). HA97 consiere such ranom-straining moels an showe that the important preictions of the steay flow moel presente in

4 76 HAYNES Section 2 carrie over, in the sense that the tracer fiel was preicte to form sloping sheets, with an aspect ratio α that epene on the statistics of the horizontal strain an vertical shear fiels. We review some of the HA97 results below, emphasising the epenence of α on the ifferent flow parameters. One particular goal is to ientify conitions uner which α is not simply equal to the aspect ratio Λ/Γ of the flow itself. It is convenient to consier the normalise graient of the tracer fiel, efine by k(t) =χ 1 χ, evaluate at the position x = X(t) following a flui parcel. k might be consiere as a local wavenumber of the tracer fiel in the case where the tracer fiel varies rapily in space compare to the velocity fiel. It may be shown from the graient of the tracer avection equation that k = ( u).k (14) t where the scalar prouct on the right-han sie applies to the secon inex of the tensor u. The evolution of k therefore epens on the time series of u (encountere following the point X(t)). In a ranom-straining moel the effect of turbulent flow on graients of tracer (or line elements) is consiere as a kinematical problem, in which the tensor u is taken to be given by a ranom time series, without aressing questions of the flow ynamics. For the layerwise two-imensional flows of interest here, the matrix W of components of the velocity graient tensor, with W ij = u i / x j, has W 31 = W 32 = W 33 = 0. (The inex 3 is taken to correspon to the z coorinate.) It is useful to write k =(k (h),m), with k (h) the horizontal wavenumber an m the vertical wavenumber, an efine W (h) to be the 2n rank tensor with components W 11,W 21,W 12 an W 22, an Λ to be the vector with components (W 13,W 23 ). It then follows that k (h) i t = W (h) ji k (h) j (15) an m t = W j3k (h) j = Λ j k (h) j (16) In what follows the summation convention is use for repeate suffices an the suffices run through the values {1, 2}. Equation (15) is just that for the tracer wavenumber in two-imensional (horizontal) flow. Note that it may be solve inepenently of (16). Thus variation in the z-irection makes no ifference to the evolution of the horizontal wavenumber vector. However (16) shows that the vertical wavenumber evolves through the vertical shear (in the horizontal flow) acting on the horizontal wavenumber vector. To formulate a suitable ranom-straining moel we assume that each of the six non-zero components of the matrix W is a realisation of a ranom function of time, with average (over all realisations) zero. We also assume that the wavenumber at t = 0 is ranomly chosen an is statistically inepenent of the W ij. Equation (16) may be integrate to give m(t) =m(0) + t 0 Λ j (t )k (h) j (t )t (17) an then squaring, an taking the ensemble average, it follows that m(t) 2 = m(0) 2 + t 0 t t 0 t Λ j (t )k (h) j (t )Λ k (t )k (h) k (t ), (18) where. enotes the ensemble average, over all realisations of the flow fiel an all realisations of the initial wavenumber vector. Useful quantitative estimates are possible if a number of further assumptions are mae concerning the statistics of the velocity graient tensor. The first is that the components of Λ are inepenent of all components of W (h), i.e. that the vertical shear is statistically inepenent of the horizontal eformation. It follows that Λ is also inepenent of k (h) so that averages of terms involving Λ an terms involving k (h) may be taken separately in (18). The secon important assumption is stationarity, from which it follows Λ j (t )Λ k (t ) Λ 2 g(σ Λ t t ), for some function g, where σ Λ is an inverse correlation time for the vertical shear an Λ 2 = Λ 2. In orer to estimate the integral appearing in (18) it is also necessary to have information on the evolution of the horizontal wavenumber. This follows exactly as in the ranom straining moels of isotropic two-imensional flow consiere by Kraichnan (1974) an others. The basic preictions of these theories are that the horizontal wavenumber increases exponentially in time, at a rate, S say, governe by the statistics of the horizontal strain fiel an epening in particular on the root mean square rate of strain, Γ say, an on the inverse correlation time σ h for the horizontal strain fiel. Detaile analysis of explicit moels (Kraichnan, 1974; Chertkov et al., 1995; HA97) suggests that S Γ min(γ/σ h, 1). Again, making suitable stationarity assumptions, this suggests that k (h) i (t )k (h) j (t ) k0e 2 S(t +t ) h(µ t t ), where k 2 0 = k (h) (0) 2. The inverse time scale µ an the precise form of the function h(.), epen on the statistics of the horizontal strain fiel, but explicit calculation in various moels suggests that µ S.

5 VERTICAL SHEAR AS A ROUTE TO MIXING 77 Substituting these estimates into (18) it follows that, for large times, when the secon term on the right-han sie ominates the first, σ Λ m(t) 2 Λ2 k0e 2 2St min{1, S }. (19) S 2 σ Λ Note that the ominant contribution to the first integral, over t, performe in (18) comes from a neighbourhoo of t = t of size min{s 1,σ 1 Λ } an that to the secon integral, over t, comes from a neighbourhoo of t = t of size S 1. The ratio α of horizontal length scale to vertical length scale, or, equivalently, the ratio of vertical wavenumber to horizontal wavenumber may therefore be estimate as α 2 m(t) 2 k (h) (t) 2 Λ2 S 2 min{1, S σ Λ } Λ2 Γ max(1, σ2 h 2 Γ ) min{1, Γ min(1, Γ )}.(20) 2 σ Λ σ h with the last estimate following from the estimate for S. This semi-quantitative analysis suggests that the vertical wavenumber increases exponentially at the same rate as the horizontal wavenumber, an that the ratio α is equal to the ratio of vertical shear to horizontal strain, Λ/Γ, multiplie by a number that epens on σ h /Γ an σ Λ /Γ. These preictions agree with explicit calculations base on suitably formulate ranom-straining moels. (See HA97 for more etails.) Figure 2 summarises the variation of the aspect ratio α with the parameters σ h an σ Λ. Note that α is anomalously large (i.e. greater than Λ/Γ) when the horizontal strain varies on a timescale that is shorter than both the inverse horizontal strain rate an the timescale of variation of the vertical shear. On the other han, α is anomalously small (i.e. less than Λ/Γ), when the vertical shear varies on a timescale that is shorter than both the inverse horizontal strain rate an the timescale of variation of the horizontal strain. As might be expecte, in the limit where σ h Γ an σ Λ Γ the estimate for α agrees with that obtaine for the steay flow consiere in the previous section. Note that in this section we have not consiere the effects of iffusion explicitly. Nonetheless, it is plausible that iffusion acts in a similar way to that euce from the steay flow moel in Section 2, i.e. iffusion κ acts on structures in the tracer fiel that slope with aspect ratio α an the result, e.g. in achieving a balance between iffusion an horizontal straining, is that it is as if there is a horizontal iffusivity with magnitue κα 2. This is supporte (with certain limitations) by explicit calculations in Vanneste an Haynes (2001). Γ Λ 2 / Γσ Λ Λ 2 / Γ 2 Λ σ h / Γ Γ Λ 2 σ h / Γ 2 σ Λ Figure 2. Scaling of α 2 (where α is aspect ratio of horizontal to vertical scale) with parameters Γ (horizontal strain rate), Λ (vertical shear), σh (inverse correlation time for horizontal strain) an σ Λ (inverse correlation time for vertical shear). The thin ashe lines elimit the regions where the ifferent estimates hol. The thick ashe lines elimit the regions where α scales as the aspect ratio of the velocity fiel (Λ/Γ) (lower left-han region), is larger than that aspect ratio (right-han region) an is smaller than that aspect ratio (upper region). 4. Application to the Lewell et al. (1993) observations In the tracer release experiment reporte by L93 the ispersion of a tracer in the ocean thermocline was followe over a perio of several months. The vertical (cross-isopycnal) iffusion coul be estimate irectly an a value of vertical iffusivity κ v of about 10 5 m 2 s 1 was inferre. In the horizontal the tracer was, in the later stages of the experiment, observe to be confine to thin streaks, whose with apparently reache an equilibrium value of about 3 km. This was interprete as the stretching out of the tracer patch by the mesoscale ey fiel, resulting in filaments of tracer whose with was the equilibrium value etermine by a balance between horizontal stretching (which tens to reuce the with of the filament) an horizontal ispersive effects, perhaps associate with small-scale mixing processes (which ten to increase the with of the filament). From the observe length of the tracer filament it was estimate that the average stretching rate experience was about s 1. It followe that if the horizontal ispersive effects coul be represente by an effective horizontal iffusivity κ h, it must have a value of about 3 m 2 s 1 ( (3 km) s 1 ). Thus κ h /κ v Previous work by Young et al. (1982), taking account of the combine effect of vertical mixing an horizontal avection by inertio-gravity waves ha σ h

6 78 HAYNES suggeste κ h /κ v (N/f) , i.e. 200 times too small to account for the L93 observations. HA97 note that the combine effect of horizontal stretching an vertical shear coul account for the value of κ h /κ v if the aspect ratio α was about 500. Accoring to the analysis presente in Section 3 this requires Λ s 1, base on the estimate α Λ/S (which hols if σ Λ < S, i.e. the correlation time for the vertical shear must be larger than S 1 ). This value of vertical shear (equivalent to 15 cm s 1 per km) oes not seem inefensible as realistic. If the moels escribe in this article are to be relevant to the L93 observations, one might ask further whether the observe morphology of the tracer is consistent with the moel preictions. First it is clear that in a flow with finite length scales the exponential stretching in a single irection preicte by the moel in Section 2 must break own when the largest length scale of the tracer patch (or, at this stage, tracer filaments) becomes comparable to the length scale of the flow. What happens is simply that the filaments meaner on the length scale of the flow (as observe in countless laboratory an numerical experiments). Perhaps more crucial is the shape of tracer filaments in cross-section. The moel of Section 2 preicts that this will ultimately be highly elongate, so that the tracer is actually concentrate within sloping sheets. However, this applies on time scales much greater than Γ 1, where Γ is the strain rate an therefore, in the steay moel, the stretching rate. For times comparable to Γ 1 (see Figure 1), the tracer istribution is not sheet-like, but it is simply the case that the filament cross section has much larger horizontal extent than vertical extent (by a factor α). The observing perio reporte by L93 is such that Γt < 6, so that sheet-like features are not necessarily expecte. [Note that the extent of the sheets is proportional to (Γt) 1/2.] 5. Possible effects of ouble iffusion Garrett (1982) (hereafter G82) note that the stirring by mesoscale eies of temperature an salinity along isopycnal surfaces woul lea to large local graients in these quantities manifeste as thermohaline fronts. These in turn might lea to ouble-iffusive intrusions. G82 argue that the sharpening of graients of temperature an salinity by stirring woul be halte by the mixing effects of the intrusions when the growth rate of the intrusions was equal to the convergence acting on the fronts, i.e. to the stretching rate associate with the stirring process. He gave the formula (base on previous theoretical an experimental work) λ max =0.075 gβs x N (21) for the maximum growth rate λ max of the intrusions, where g is the gravitational acceleration, β = ρ 1 ( ρ/ S) T is the rate of change, at constant temperature, of ensity with salinity, an S x is the salinity graient in the neighbourhoo of the front. He argue that if the front ha with L f then the approximate value of S x woul be S x = Sx L ey /L f where Sx was the large-scale salinity graient an L ey was the typical size of the mesoscale eies. If s is the stretching rate then it follows that L f is given by the formula L f =0.075 gβs x N For the values of s an L f L ey. (22) s observe by L93 (taking L f to be the with of the filaments) an assuming L ey 100 km, this woul require gβs x /N 10 7 s 1. This is, in fact, consierably weaker than the values suggeste by G82 as examples. This therefore appears to leave open the possibility that it is the mixing effects, along isopycnal surfaces, of ouble iffusive intrusions, rather than any process irectly relate to backgroun vertical mixing, that etermines the with of the filaments observe by L93. On the other han it is not at all clear that the filaments of a tracer injecte at an arbitrary location will match the regions of enhance graients in temperature an salinity whose istribution is set on the large scale an therefore feel the same mixing effects. 6. Discussion The work reporte in HA97 may be extene in various ways, some of which are relevant to oceanographic consierations. For example, Vanneste an Haynes (2001) have consiere the effect of vertical shear on the horizontal wavenumber spectrum of passive tracers, in particular on the range of scales where the effects of iffusive mixing become important. In particular this work highlights the limitations of estimating the effective horizontal iffusivity as κα 2. The recent theoretical work on Batchelor-regime turbulence has mae explicit preictions about probability ensity functions for tracer concentrations, tracer concentration ifferences (over a finite istance) an tracer concentration graients an this work nees to be extene to the layerwise two-imensional case if it is to be applicable to real atmospheric an oceanic flows. Much insight into atmospheric tracer istributions has been obtaine by using moels riven by observe velocity fiels, or velocity fiels extracte from quasirealistic global moels. Approaches have inclue full numerical integrations of the tracer evolution equation (e.g. on an isentropic surface), non-iffusive reconstructions of tracer fiels by following back trajectories to an

7 VERTICAL SHEAR AS A ROUTE TO MIXING 79 initial conition an calculations of the statistical properties of large numbers of trajectories, e.g. to give the istribution of finite-time stretching rates. See, for example, the papers by Schoeberl an Newman (1995), Ngan an Shepher (1999) an Hu an Pierrehumbert (2001). All of these methos have substantial savings over integrations of a full moel incluing tracer equations an ynamical equations an some may be worth applying in the oceanic context. Velocity fiels might be extracte from ey-resolving moels for this purpose, but as a first step it might be worth consiering synthetic velocity fiels that are generate quite artificially to have a plausibly realistic spatial an temporal structure in space an time. Acknowlegments. I am grateful to Chris Garrett for bringing possible ouble-iffusive effects to my attention. References Balkovsky, E. an A. Fouxon, Universal long-time properties of Lagrangian statistics in the Batchelor regime an their application to the passive scalar problem. Phys. Rev. E, 60, , Chertkov, M., Falkovich, G., Kolokolov, I. an V. Lebeev, V., Statistics of a passive scalar avecte by a large-scale two-imensional velocity fiel: analytic solution. Phys. Rev. E, 51, , Falkovich, G., I. Kolokolov, V. Lebeev, an S. Turitsyn, Statistics of soliton-bearing systems with aitive noise. Phys. Rev. E, 63, (R), Garrett, C., On the parametrization of iapycnal fluxes ue to ouble-iffusive intrusions. J. Phys. Oceanogr., 12, , Haynes, P. H. an J. Anglae, The vertical-scale cascae of atmospheric tracers ue to large-scale ifferential avection. J. Atmos. Sci., 54, , Hu, Y. an R.T. Pierrehumbert, The avection-iffusion problem for stratospheric flow. Part I: Concentration probability. J. Atmos. Sci., 58, , Kraichnan, R. H., Convection of a passive scalar by a quasiuniform ranom straining fiel. J. Flui Mech., 64, , Lewell, J. R., Watson, A. J. an C. S. Law, Evience for slow mixing across the pycnocline from an open-ocean tracer-release experiment, Nature, 364, , Ngan, K. an T.G. Shepher, A closer look at chaotic avection in the stratosphere. Part I: Geometric structure. J. Atmos. Sci., 56, , Schoeberl, M. R. an P. A. Newman, A multiple-level trajectory analysis of vortex filaments. J. Geophys. Res., 100, , Shraiman, B. I. an E. D. Siggia, Scalar turbulence. Nature, 405, , Vanneste, J. an P. H. Haynes, The role of iffusion an vertical shear in etermining stratospheric tracer spectra, J. Atmos. Sci. (submitte), Young, W. R., Rhines, P. B., Garrett, C. J. R., Shear-flow ispersion, internal waves an horizontal mixing in the ocean. J. Phys. Oceanogr., 12, , This preprint was prepare with AGU s LATEX macros v4, with the extension package AGU ++ by P. W. Daly, version 1.6a from 1999/05/21.