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10.21.10: Notes on the literature

Marshall and Shutts (1981)

I do not understand this paper as it is based mostly on the equation for the temperature variance and enstrophy.

Holland and Rhines (1980)

The authors use a two-layer quasi-geostrophic model of a flat-bottom basin forced by a steady sinusoidal zonal wind stress. The model has bottom friction, interfacial stress between layers and biharmonic lateral friction. As in our case, the Eulerian-mean flow in the bottom layer is entirely driven by eddies. The mean potential vorticity field in that layer is dominated by the beta effect; in particular, the contours are block by the meridional boundaries so that nonconservative processes are necessary to explain the presence of the mean flow. As they write, it is the dissipation of enstrophy that allows Lagrangian mean flow to cross the geostrophic contours (Rhines (1979). The analysis of the simulation reveals that the interfacial stress is the main driver of the circulation in the bottom layer which is slown down by lateral and bottom friction.

The eddy PV flux in the bottom layer is directed southward (downgradient) outside the domain and is mainly due to the eddy advection of thickness anomaly. Within the western boundary, the eddy PV flux is southward and is mainly due to eddy advection of relative vorticity (The numbers 1 and 3 in the label of their Fig. 7 should be swap). The turbulent Sverdrup equation is proposed to explain the mean flow, in which the eddy PV flux, rotated by 90 deg. clockwise, can be seen as a wind stress. See Rhines and Holland (1979) for an interpretation of the eddy PV flux in terms of a Lagrangian diffusivity.

This type of simulation could be our next simulation. The simulation would be more realistic as the eddies would be produced by the instabilities of the mean flow in the upper layer and not by the actual surface forcing.

Rhines and Holland (1979)

The authors review the induction by eddies of a Eulerian mean circulation in terms of the turbulent Sverdrup equation, the eddy PV flux and Lagrangian diffusivity.

In their Eq. 8, the authors re-write the eddy PV flux in terms of Lagrangian diffusivity and a term due to forcing and dissipation. Their Eq. 9a is then the Eulerian-mean PV equation for which there is no forcing and dissipation and in which the eddy PV flux has been re-written in terms of diffusive term across Eulerian mean PV. For me, the equation is qualitatively misleading: it suggests that parcels will cross Eulerian-mean PV contours which is not possible as there is no forcing and dissipation (except if there is a subtle difference that I need to understand between Lagrangian-mean and Eulerian-mean PV even in the case where the eddies are stationary?). Their Eq. 9c will then explain the component of the Eulerian-mean flow that cancels the Stokes component, but it does not address the actual transport across mean PV contours (Lagrangian-mean velocity) because forcing and dissipation are not included. The introduction of the term “Lagrangian diffusivity” (in their Eq. 8) is again misleading as you can observe large values of that term even if there is actually no diffusion across mean PV contours.

Their enstropy Eq. 12, however, does include dissipation but I do not understand how they deduce their corollary from this equation.

There is a discussion on p. 302 on the scaling of the mean circulation induced when mean PV contours are blocked or not. In the first case, the authors argue that it is so small that it has no practical interest. I do not understand that part nor the difference with a similar discussion on the previous page.

About example (i): again no dissipation is explicitly included in Eq. 17 but in the following paragraph, the authors speak of PV transport down its mean gradient. I do not understand how this is possible. Furthermore, if there is no dissipation, then the Eulerian-mean flow deduced from averaging the inviscid wave field should be zero. The fact that there is a mean flow implies that dissipation plays a role albeit not an explicit one in the equations. I do not understand as well the discussion about the Kelvin circulation.

The paper does acknowledge that dissipation is necessary to get Lagrangian-mean flow as p. 315, first paragraph. What I am afraid is that the diffusivity term, as defined in the paper and calculated from float trajectories, is dominated mostly by the Stokes component (for weak dissipation) and cannot, alone, indicate the strength and pattern of the Lagrangian-mean flow.

The last example, example (iv), is a summary of Holland and Rhines; in this case, again, the Eulerian-mean flow is explained but the Lagrangian-mean flow is only slightly mentioned. If it is argued that dissipation is very weak, then the Lagrangian-mean flow is quasi null and the Eulerian-mean flow can be considered an artifact of both the eddies and the time averaging but does not represent any actual circulation of tracers and water masses. As we are interested on how eddies modify the transport –especially the transport of southward North Atlantic Deep Water–, an analysis that explicitly takes into account dissipation and computes the Lagrangian-mean flow is needed.

The authors finish the paper (p. 322) with a discussion on the GLM theory. They argue that the theory is valid only for wave-like processes only. Although Eric suspects this is correct, I still do not see any reason in the theory for this to be true.

Rhines (1977)

I do not understand the discussion on p. 279 about the Eulerian-mean flow and the Stokes drift.

The author discusses the inviscid theory (section 8C) and the time-dependent viscous theory (section 8D) but not the forced-dissipative stationary theory. However, section 9E (p. 309) discusses the trajectories of SOFAR floats at 1500 m in the mode water region. The relevant comment is the observation that the trajectories are not “slight oscillations about latitude lines”, which is the regime I am studying at the present time. If the Lagrangian-mean flow is, over long time scales, stationary, then I argue that it has to be linked to the dissipation. If, however, the Lagrangian-mean flow varies with time, the theory I study does not apply and transient transport across mean PV contours (themselves varying with time) can occur without dissipation. I, thus, need to know if the hypothesis of stationarity is valid or not. Notice, however, that on p. 315, the author notices that dissipation has to occur to explain reasonably the meridional excursions of the SOFAR floats. See the paper on floats and PV (LaCasce 2000).

Plumb and Mahlman (1987)

The authors’ goal is to parameterize the zonally-averaged eddy PV flux by a flux gradient relationship with diffusivity K. Plumb (1979) justified this form for small eddy amplitude (see also Matsuno 1980). K is composed of an antisymmetric L and symmetric D part. L is the flux along mean contours of tracer (it is advective) while D is the flux across them (it is diffusive). L and (for a conserved tracer) D are expressed in term of parcel displacements; in this case, each represent a purely kinematic dispersion of tracers. If the tracer is not conserved –which is usually the case most of the time–, other terms need to be added to D; say differently, D contains the source/sink term of the tracer equation. In case the tracer is PV, D thus contains the dissipative term. Because sources and sinks vary with each type of tracer, there is no universal way to represent the dispersion D.

The L term can be absorbed into the zonally-averaged advective term to form a new zonally-averaged advective term with an effective transport velocity. This transport velocity equals the Lagrangian-mean flow only when the diffusion D is isotropic (their Fig. 1 and their discussion is crystal clear). Unlike the Lagrangian-mean flow, this transport velocity satisfies all the time the continuity equation.

On p. 301-302, there is an interesting discussion on the simplification of the K tensor. Especially, can the K tensor be purely advective, relegating D to subgrid processes? The answer is no as scaling of the transport equation shows that L and D have globally –but not necessarily locally– the same scale.

Using the GFDL model and two synthetic conserved tracers, for which the gradients of their respective initial distribution are perpendicular, each coefficient of each kinematic dispersion term is computed. As said above, the non-kinematic part of the dispersion is directly dependent on the type of source/sink for the tracer and cannot be universalized. The difficulty is that, if the tracers are indeed perfectly conserved, then the calculation becomes singular –anytime the gradient of one of the tracer vanishes or the two gradients become parallel. For this reason, the calculation is performed with quasi-conserved tracers relaxed to their initial distribution. The relaxation is strong enough that the singularities never appear, weak enough that D stays dominated by its kinematic component.

From this paper, I learn that there is a component of the eddy flux that is diffusive (across mean tracers contours), yet does not actually involve the source/sinks of the tracers. This component involves instead time derivatives of parcel displacement statistics. We could think as this term playing a role when the flow is non-stationary (similar to the time-derivative of the Lagrangian-mean PV), but the authors add that the term is non-zero even when Eulerian statistics are stationary (p. 300). More importantly, it is this term that seems to be considered the dominant one in the rest of the paper. Understand why even when Eulerian statistics are stationary, the term is non-zero. Tung (1984; see below) discusses, for instance, the atmospheric case where this diffusion is due to a series of sudden wave growth followed by radiative damping.

Vallis (2006)

Section 10.4 (p. 419): the flux-gradient relation with diffusivity K is a way to parameterize the dispersive effects of the eddies alone (without respect of the presence or not of real molecular diffusion). But, as pointed in section 10.4.1, molecular diffusion has to occur for the relation to be valid, even if such diffusion is not taken into account quantitatively.

A first problem is that by choosing the parameterization, we may impose the way the tracer is being transported. It is thus important to know how the transport is dependent on the structure of the parameterization itself (Laplacian, biharmonic, etc) versus its dependency to the flow: Is the transport the same given the same eddy statistics but different parameterization?

A second problem is the fact that the actual diffusion is not taken into account quantitatively whatsoever. This bothers me a lot. If the PV equation was valid at all scales, then it would be the integrated effect of diffusion along parcel trajectories that matters (the Lagrangian-mean diffusion), the effect of eddies playing an indirect role via the re-organization of the flow and the gradients. One way to study this problem would be to use two numerical simulations: one high-resolution such as the explicit dissipation is considered as the true one *versus* one coarse-resolution for which the explicit dissipation is considered as a parameterization. Are the parcel trajectories and thus the transport different between the two? If not then the parameterization is justified with respect to Lagrangian-mean transport. If no, is there a dissipation scheme that can reproduce the results of the high-resolution model? Plumb and Mahlman (1987) impose the scheme (flux-gradient relation) and find the diffusive coefficient K for a zonally-averaged model (coarse-resolution) from the 3D model (high-resolution). This is a way to assure that the transport will be similar in both simulations. Is there anybody who did a similar study for the ocean?

Tung (1984)

If the off-diagonal coefficients in K are zero, K is purely diffusive and the diffusion is downgradient. If the off-diagonal coefficients are non-zero but K is still symmetric, there is a system of axis in which the diffusion is purely diffusive and still downgradient –but it opens the door that there is counter-gradient diffusion in the original system of axis. In reality, K has an asymmetric component. See for instance Matsuno (1980) in the case of planetary waves. And in some cases (Clark and Rogers 1989; Plumb 1979; Matuson 1980), K is completely asymmetric. In this case, the eddies are purely advective.

Interestingly (p. 419), if dynamical coordinates are used (such as potential density in the vertical), then K becomes in all cases purely diffusive, as long as the processes stay adiabatic. If, furthermore, the eddy field is steady, then K vanishes and “no eddy transport is accomplished by an adiabatic, steady eddy field” (see Tung 1982).

The author discusses (p. 430-431) the nature of the symmetric part of K that does not involve the source/sink of the tracer. In the context of the paper (meteorology), only Kyy is non-zero and Kyy is the time-derivative of the mean squared of eddy horizontal displacements, 1/2.d(<n’.n’>)/dt, where <.> stands for an average over the eddy scales. Thus, steady periodic wave cannot give rise to net transport. The waves need to be dissipated, either directly or after a cascade of its energy toward the small scales. Here, we see that, even for a tracer, actual wave dissipation or wave transience are needed for the diffusive part of K to be non-zero. The author discusses, for instance, the case of sudden warming for which there is a wave growth followed by radiative damping (Is this similar to Eric’s example where an eddy would first advect inviscidly northward before being damped in some localized band of latitudes?) and computes the resulting Kyy due to a series of such event. Thus, here, although each event is *irreversible* (*via* radiative damping), the diffusion coefficient is given by the *reversible* component of the process (the wave growth).

This explains the nature of the symmetric part of K that does not involve the source/sink of tracer (in Plumb and Mahlman 1987). Wave dissipation is still needed and implicit but the diffusivity is given by the transient but nonetheless reversible part of the eddy advection. Thus, as long as the eddy process is irreversible, the time derivative in K will be non-zero even if the eddy statistics are stationary. What is the valid paradigm in the ocean? Such sudden reversible advection followed by dissipation or rather a constant dissipation acting at all time? Is there any paper in oceanography that uses the paradigm as described by Tung?

Holloway and Kristmannsson (1984)

  • “parapycnal” == along isopycnal
  • see Armi and Haidvogel (1982, JPO, 787-794) for a study of eddy mixing where the eddy diffusion coefficient is non-homogeneous.

Plumb (1979)

This paper is summarizing the role of eddies on tracer advection and diffusion when the eddies have small amplitude and dissipation is present.

  • When the wave field is steady and there is no tracer source/sink, the eddy flux has to be along mean tracer contours. Note that, in this case, the flux can be up or downgradient in any direction.
  • When the wave field is transient and there is no tracer source/sink, the eddy flux is both advective and diffusive. It is downgradient for a growing disturbance and upfradient for a decaying one. Note, however, that this is not true for every direction.
  • When the wave field is steady but the tracer is not conserved, the eddy flux will be both advective and diffusive. The direction of the diffusive component depends on the form of the source/sink term.

In section 4, the form of the eddy flux is discussed for conservative tracer. It can be written in different forms. The first form is as in Plumb and Mahlman (1987) in which the eddy flux is composed of the symmetric (diffusive, due to non-steady waves) and asymmetric (advective) part.

The second form (Eq. 25) rewrites the advective component to explicitly introduce the Stokes drift velocity (the Stokes drift is the divergence of the diffusivity –see Eq. 24); when the advection by the mean flow is added, the Lagrangian mean velocity appears instead. The total advective velocity is then the Lagrangian mean velocity plus a velocity associated to the changing wave field and it satisfies the continuity equation. This formulation adds a nondivergent term in the flux that has no effect on the tracer but has to be taken into account nonetheless when interpretating flux data. When the waves are steady, the diffusive component vanishes, the divergent part of the eddy flux reduces to the advection by the Stokes velocity and the total advective velocity reduces to the Lagrangian mean velocity. When the waves are growing, the diffusion is downgradient and when they are decaying, the diffusion is upgradient.

I realize that the Lagrangian-mean PV is not the Eulerian-mean PV (It should have been more than obvious at this time!). Thus, in Eric’s example of reversible eddy advection followed by irreversible eddy dissipation, the departure between the Lagrangian-mean PV and the Eulerian-mean PV may be large so that the GLM theory stays valid albeit difficult to interpret in terms of mean transport across the (zonal) Eulerian-mean PV contours.

Can we estimate dissipation and the tracer-independent component of diffusion (and thus the diffusivity tensor) directly from float trajectories?

Haynes (2004)

  • 3D turbulence, type I flow: “the time taken to stir a chemical field from some finite scale to the the scale at which molecular diffusion is important is independent of diffusivity (when the latter is small)”
  • 2D turbulenece, type II flow (also called the Batchelor regime): “the deformation at given scale is dominated by the flow at the large scale” and “the time taken to stir a chemical field from some finite scale to the the scale at which molecular diffusion is important increases logarithmically with the inverse of diffusivity (when the latter is small)”

Transport and mixing can be different if the model is purely kinematic or contains dynamics such as the PV conservation (see Poet 2004).

Cessi et al. (2006)

Using idealized numerical simulations and scaling theory, the authors show that the meridional heat transport is directly linked to small-scale diapycnal mixing and dissipation (bottom drag in their case). In the simulations, PV is homogenized so that the PV constraint on meridional transport does not exist in this case.

Gent et al. (1995)

“most obvious and simplest parameterization, downgradient Fickian diffusion, has been demonstrated to be inadequate. A major reason for this inadequacy is that a purely diffusive parameterization means that advection in the large-scale model is by the large-scale velocity”, which is not the case in general.

The tracer equation is written in term of the tracer concentration times the thickness of its isopycnal layer. This equation and that for the thickness are then decomposed into mean and eddy components. The equation for the tracer concentration alone is then written as a classic advective-diffusive equation. The diffusion represents mixing along the isopycnal layer. The advective velocity is the large-scale scale velocity plus the eddy bolus velocity. The large-scale equation for the thickness is also re-written in a similar fashion. As in Plumb and Mahlman (1987), they view the eddy effect as a diffusive tensor acting on the large-scale tracer gradient, with its symmetric part which is diffusive and its anti-symmetric part which is advective. They mention Middleton and Loder (1989) as well as Davis (1994) as two previous studies that view the eddy effects similarly.

The advective velocity thus defined has the advantage to satisfy the continuity equation, unlike the Lagrangian-mean velocity. It is subsequently parameterized as a Fickian diffusion using the thickness diffusivity but the exact form is still an open question.

The overall result is that the mixing scheme written for a z-level stays adiabatic.

Bryan et al. (1999)

“As Gent and McWilliams point out, the traditional representation of mesoscale eddies in level-coordinate ocean general circulation models as a purely diffusive p ocess in the horizontal plane does not conserve density and so is clearly an inadequate representation. In this respect ocean circulation models of low horizontal resolution using an isopycnal vertical coordinate have a clear advantage.”

Griffies (1998)

The author describes the GM mixing scheme as non-dissipative (purely advective), reversible and could be considered as pure stirring because it involves only the asymmetric part of the total diffusive tensor. He adds that “a symmetric component to the mixing tensor should be present in order to represent irreversible downgradient diffusive effects of various subgrid-scale processes. This process is called “isoneutral diffusion” (also called Redi diffusion tensor following Redi 1982) as it is assumed to apply also along isopycnal surfaces (see Griffies et al. 1998 for kinematical behavior of such diffusivity).

The author explains well that, because of the properties of the antisymmetric tensor, the antisymmetric part can be written as the divergence of an advective tracer flux involving a non-divergent advective velocity. The antisymmetric tensor can also be written as the divergence of a skew-diffusive flux. The latter flux is always perpendicular to the gradient. The advective and skew fluxes have different amplitude and orientation; in some circumstances, they are even opposite (see Middleton and Loder). Because their convergence are identical, they differ by the curl of some function.

I stopped to read at section 3.

Killworth (1997)

  • Two different approaches in oceanography: parameterization of the effect of eddies on tracers (Gent and McWilliams 1990, hereafter GM) and parameterization of effect of eddies on momentum (Eby and Holloway 1994). See Lee and Leach (1996) for a link between the two approaches.
  • The GM approach is based on the effect of eddies in baroclinic instability but does not address the eddy transport over long distance or associated with other instabilities.
  • See Treguier et al. (1997) for a discussion on the effect of boundaries and the use of the potential vorticity rather than thickness in the GM parameterization.
  • “This paper has set of to achieve two parallel goals: to use linear perturbation theory to suggest a structure for a parameterization of baroclinic instability which is, to leading order at least, a solution of the equations of motion; and to deduce conditions on parameterizations which must be satisfied in general because of potential vorticity conservation.”
  • The eddy parameterization is mean flows that are wide compared with a deformation radius and the associated scheme is mostly downgradient flux of PV and thickness. The fluxes are computed from the linearized equations of motion around the slowly-varying (in time and space) background flow and the pertubation being represented by a single plane wave. The lack of skewed flux (anti-symmetric term) is due to the slowly varying assumption.
  • His Eq. 29 relates, for small perturbations, the eddy flux of potential vorticity to that of thickness flux. This is because, when geostrophy holds, the perturbation velocity is out of phase with the relative vorticity so that their correlation is zero. In regime where quasi-geostrophy becomes invalid, this is not the case [Dukowicz and Greatbatch (1999) says, however, that this is still true even if there is a linear combination of waves –of same frequency, I suppose– or in the general case as long as the average is a spatial average].
  • “Although the scheme evaluates thickness mixing, it differs from Gent and McWilliams (1990) in many respects. First, the diffusion coefficient is predicted to vary with position, including a strong vertical structure to the mixing, similar to that deduced by Treguier (1997) from an eddy-resolving model. Second, the diffusion coefficient is outside the vertical derivative. Third, the mixing is turned by the matrix A, to align itself along the pathways of the mixing of the fastest growing linear normal mode. Fourth, thickness itself is not mixed, but potential vorticity is.”

Treguier (1999)

  • “A numerical model of a baroclinically unstable jet in a zonally periodic channel is used to analyze mesoscale eddy fluxes and their relationship with the gradients of the mean flow. The quasigeostrophic approximation proves the best way to calculate potential vorticity fluxes in the primitive equation model. Away from the surface layers, eddy fluxes of potential density are consistent with advection by eddy-induced velocities v* and w* as suggested by Gent et al. (1995). Eddies mix potential vorticity along isopycnals, so that v* is related to the gradients of potential vorticity rather than potential density as implicitly assumed by Gent et al. The mixing coefficient for potential vorticity, associated with the advective component of the eddy fluxes, is found to be similar to the mixing coefficient of tracer anomalies on isopycnals.Both show a maximumat mid-depth below the jet core.”
  • “the flux-gradient relationships that exist in an unstable baroclinic jet are not compatible with some of Gent et al.’s assumptions.”
  • zonal average is used for the mean,
  • The term associated with the bolus velocity in Gent et al‘s Eq. 6 is equivalent in z-coordinate to the advection by a 3D nondivergent velocity field. The meridional and vertical velocities are approximately the vertical gradient and horizontal divergence of the meridional eddy flux of density anomalies, respectively.
  • Because of the similarity between the bolus velocity in isopycnal coordinate and its equivalent quasi-geostrophic version in z-coordinate, the author argue that “[t]he quasi-geostrophic form of potential vorticity must be preferred for an analysis in z coordinates, even in a primitive equation model”.
  • The bolus velocity in isopycnal coordinate and its equivalent quasi-geostrophic version in z-coordinate differ, however, a lot near the surface. Not really understood why.
  • It is found that the advection of the mean density field by the z-coordinate version of the bolus velocity is similar to the divergence of the eddy flux of density anomalies (their Fig. 4).
  • As in Killworth (1997), the bolus velocity is proportional to the eddy flux of potential vorticity which is everywhere downgradient in the simulation. Thus a parameterization of the latter is also a parameterization of the former.
  • The mixing coefficent for potential vorticity (computed as the ratio of the eddy potential vorticity flux and the mean potential vorticity gradient) has a maximum below the core of the eastward flow. Such profile confirms the analytical expression for the mixing coefficient developped by Killworth (1997).
  • The author was not able to recover the bolus velocity using the GM parameterization with a depth-independent diffusivity coefficient. Furthermore, the GM diffusivity is also different from the potential vorticity diffusivity which has to be considered, at least in this regime, as the “true” diffusivity in the parameterization of the bolus velocity.
  • An explanation is given on why Rix and Willebrand (1996) failed to find a statistically significant structure of the diffusivity coefficient in their eddy-resolving model.
  • The author reminds correctly that, according to GM, thickness is only advected by eddies while tracers are both advected and diffused. Thus, the diffusivity coefficient for a passive tracer is not equal to the diffusivity coefficient for thickness.
  • The diffusivity coefficient computed for a tracer from its eddy flux and mean gradient should contain both the advective and diffusive effect of the eddies. As long as two tracers are conserved –that is no molecular or sub-grid diffusion–, their diffusivity coefficient should be the same. Thus, if friction is negligigle, potential vorticity is conserved and the diffusivity coefficient for potential vorticity should equal the diffusivity coefficient for any passive conserved tracer. This is, indeed, qualitatively and to some extent quantitatively true.
  • The diffusivity coefficient for a tracer is computed as in Plumb in Mahlman (1987): either as an initial value problame and over a short time (before the mean gradient disappears) or using a weak restoring force to maintain the mean gradient. As in Plumb and Mahlman (1987), both procedures give the same result.
  • Any study that furthers this work?
    • Eden (OM, 2010): Parameterising meso-scale eddy momentum fluxes based on potential vorticity mixing and a gauge term
    • Ferrari and Nikurashin (JPO, 2010): Suppression of Eddy Diffusivity across Jets in the Southern Ocean
    • Griesel et al. (JGR, 2010): Isopycnal diffusivities in the Antarctic Circumpolar Current inferred from Lagrangian floats in an eddying model
    • Eden and Greatbatch (OM, 2009): A diagnosis of isopycnal mixing by mesoscale eddies
    • Lee et al. (JPO, 2007): Eddy advective and diffusive transports of heat and salt in the Southern Ocean
    • Eden et al. (JPO, 2007): A diagnosis of thickness fluxes in an eddy-resolving model
    • Berloff (DAO, 2005): On dynamically consistent eddy fluxes
    • Drijfhout and Hazeleger (2001): Eddy mixing of potential vorticity versus thickness in an isopycnic ocean model
    • Roberts and Marshall (2000): On the validity of downgradient eddy closures in ocean models
    • Marshall et al. (1999): The relation between eddy-induced transport and isopycnic gradients of potential vorticity

Greatbatch (1998)

  • “By writing the momentum equations in terms of the isopycnal flux of potential vorticity, the author shows that any parameterization of the eddy-induced transport velocity must be consistent with the conservation equation for potential vorticity. This places a constraint on possible parameterizations, a constraint that is satisfied by the Gent and McWilliams parameterization only if restrictions are placed on the diffusivity coefficient. A new parameterization is suggested that is the simplest extension of Gent and McWilliams based on the potential vorticity formulation. The new parameterization parameterizes part of the time-mean flow driven by the Reynolds stress terms in addition to the eddy-induced transport velocity.”
  • The author argues that the bolus velocity is not exactly the eddy-induced transport velocity. Smith and Dukowicz (unpublished manuscript) have shown that, in general, the eddy-induced transport velocity need only be a part of u*, the remaining part being associated with a purely rotational eddy thickness flux. [...] Smith and Dukowicz argue that as a consequence, the eddy-induced transport velocity may differ from the bolus velocity by a “rotational” component (see his Eq. 40 for the exact property of that component). The author does not give, however, what is the definition of the proper eddy-induced transport velocity.
  • The author writes the Eulerian-mean equation for potential vorticity using the thickness-weighted potential vorticity. The advective velocity is still the Eulerian mean plus the bolus velocity but, unlike the similat equation developped in Gent et al. (1995), no assumption has been made to develop it. Notice that a similar form can be found for any tracer.
  • Similarly as in Killworth (1997) and Treguier (1999), the author notices that if the flow is geostrophic, the bolus velocity can be written directly in term of the eddy flux of potential vorticity.
  • As in other papers, however, the eddy flux is parameterize by a purely diffusive tensor, excluding the part of the eddy flux that is purely advective.
  • What are the other papers that reference this paper:
    • Wilson and Williams (2006)
    • Wilson and Williams (2004)
    • Dukowicz and Greatbatch (1999)
    • Marshall et al. (1999)

Dukowicz and Greatbatch (1999)

  • The authors re-iterate Greatbatch (1998)’s claim that only the component of the bolus velocity associated with the divergent part of the the eddy thickness flux is effective in tracer transport.
  • The authors apply a compressible stochastic theory of turbulence developped by Dukowicz and Smith (1997) to the mean continuity equation and the mean tracer equation. Assuming some sort of ergotic hypothesis and comparing these equations with the mean continuity equation and the equation of the mean thickness-weighted tracer, the authors find new formula for the bolus velocity and the eddy tracer flux.
  • The formula for the bolus velocity invokes a Lagrangian-mean velocity. This velocity is postulated following Monin and Yaglom (1971)” the Stokes component can be written as the divergence of a diffusive tensor. This results in the bolus velocity being equivalent to a thickness diffusion, the form guessed in Gent et al. (1995).
  • Thus, the authors justify, based on their theory, the choice of parameterization for the bolus velocity and the eddy flux made by Gent et al. (1995).
  • Using potential vorticity as tracer, however, they find that the bolus velocity is composed of two components: one component associated with the eddy flux of potential vorticity, written as a diffusive term and a component that involves correlation between eddy relative vorticity and velocity.
  • This avoids to make the Monin and Yaglom postulate and gives, instead, a formula for the Lagrangian-mean velocity.
  • The paper then studies the case of quasi-geostrophy and shows that, even in this case in which the bolus velocity can be written uniquely in term of an eddy flux of potential vorticity, there is a contribution to the bolus velocity involving the beta term (the so-called beta velocity) that needs to be taken into account and that is absent in the form used by Gent et al. (1995). This beta velocity appears in the middle layer of the simulation performed by Lee et al. (1997) [the authors notice, however, that this velocity is caused in part by the diabatic heating present in the simulation, which is outside the adiabatic environment assumed in the rest of the paper].
  • Even if the contribution of the beta velocity will be in general small, the work supports Treguier et al. (1997)’s claim that the parameterization of the bolus velocity should be a parameterization of the eddy flux of potential vorticity.

Marshall et al. (1999)

  • “When there is an active eddy field, the eddy-induced transport is found to correlate with isopycnic gradients of potential vorticity, rather than gradients of layer thickness. For any unforced layers, the eddy transfer leads to a homogenization of potential vorticity and a vanishing of the eddy-induced transport in the final steady state.”
  • “An important issue that remains unresolved is whether the eddy-induced transport is related to gradients of isopycnic layer thickness, as advocated by Gent et al. (1995), or isopycnic gradients of potential vorticity, as advocated by Treguier et al. (1997).”
  • They give the formula for the correct version of the eddy-induced velocity. This version has been developed by Greatbatch (1998) but I have yet understood correctly the rationale. The definition adds to the bolus velocity a Reynolds component that has two terms: a term associated with the eddy correlation between relative vorticity and velocity and a term linked to the eddy kinetic energy.
  • The first of these terms is exactly opposite to one component of the bolus velocity as described by Dukowicz and Greatbatch (1999). Thus, the formula given in this paper for the eddy-induced velocity can be written differently, involving instead the eddy flux of thickness-weighted potential vorticity.

Lee et al. (1997)

  • The authors study the meridional transport across a baroclinically unstable zonal jet. As found by Marshall et al. (1999), the bolus velocity is correlated to gradient of potential vorticity rather than thickness.
  • The eddy tracer flux is also parameterized by a purely diffusive process.
  • The initial spreading of the tracer will be dominated by diffusion. At long time scale, however, the spreading is controlled by advection.
  • The term left between the eddy flux of potential vorticity and the eddy flux of thickness is defined as the Reynolds velocity (see their Eq. 20).
  • In region where the potential vorticity is mixed, the eddy flux is zero and the Reynolds velocity exactly cancels the bolus velocity. Is this equivalent to the Stokes drift exactly canceling the Eulerian mean flow?
  • The diffusivity of potential vorticity is found to be close to that of a passive tracer
  • The diffusivity of the eddy tracer flux is found positive everywhere (thus downgradient) although it is far from being uniform in latitude. It is, however, similar to the diffusivity of the eddy flux of potential vorticity. This is expected if the tracers are conserved; in this case, the diffusivity of one tracer is a universal feature of the flow (see Plumb and Mahlman 1987).
  • Because they do not consider a 2D tensor for the diffusivity but a scalar, we do not know if that tensor would have a non-zero anti-symmetric component.
  • What I am realizing is that when the tracer is potential vorticity, the eddy flux on the r.h.s. appears also in the definition of the bolus velocity in the quasi-geostrophic case. So that, the same parameterization is used for both the advective and diffusive part.
  • In this paper, they find similarly that the diffusivity for the eddy flux of tracer is the same as the diffusivity for the eddy flux of potential vorticity which happens to be also in the quasi-geostrophic framework, the diffusivity for the bolus velocity.
  • They conclude by noticing that the total transport velocity could be deduced if we follow the migration of tracer contours over long time scale (at which time the advection is dominant). The time scale needs, however, to be not too long, before the system reaches the quasi-steady state.

Middleton and Loder (1989)

  • The skew flux is perpendicular to the mean scalar gradient (it is parallel to mean contours of tracer).
  • small wave amplitude
  • The skew flux can affect the evolution of the mean tracer if it is divergent
  • “Wave fields that give rise to the skew flux are characterized by a preferred sense of rotation”
  • The skew flux is composed of a non-divergent component that has no effect on the tracer and an component that can be written as an advective flux and can have a component up or downgradient. The velocity in that flux is one component of the Stokes drift. It is exactly the Stokes drift when the waves are steady. mv

Lee et al. (2007)

  • They write the conservation equation for the tracer concentration per unit area in an isopycnal layer.
  • As in Gent et al. (1995), the eddy flux is considered purely diffusive and the advective velocity is composed of a mean velocity and an eddy-induced velocity. The latter decomposition is, however, different from that in Gent et al. (1995): although the advective velocities are considered parapycnal velocities, the mean velocity is from the time-average taken in z-coordinates. I do not understand the reason why. See McDougall and McIntosh (2001).

Matsuno (1980)

  • upward propagating steady planetary waves embedded in a uniform zonal flow.
  • trajectories are computed up to second-order in wave amplitude
  • trajectories at first order are closed elliptic trajectories
  • trajectories at the second order adds the Stokes drift component, as well as other second-order correction to the mean flow
  • as long as the wave is steady, the correction to the mean flow is exactly cancelled by the Stokes correction so that the trajectories stay elliptic even at second order.
  • even if individual fluid particles have no mean motion, eddy transport is still possible
  • eddy flux of a conservative tracer by steady waves are perpendicular to the mean tracer gradient (and is entirely the skew flux). In that case, the skew flux is non-divergent and does not take part in the tracer. This explains why a parameterization of the eddy flux needs to have an anti-symmetric component.
  • when the eddy transport cancels the transport by the mean flow, the eddy parameterization needs to be advective.
  • the author develops a new eddy diffusivity on the basis of the parcel trajectories and applying a mixing-length type assumption.
  • according to the author, the velocity associated with the eddy flux (including both the symmetric and anti-symmetric components) is the Stokes drift. Note, however, that this is the case of zonal average so that there is term isolated as the bolus velocity.

Eden (2010)

  • “Meso-scale fluctuations are known to drive large-scale zonal flows in the ocean, a mechanism which is currently missing in non-eddy-resolving ocean models. A closure for meso-scale eddy momentum fluxes is evaluated in a suite of idealised eddying channel models, featuring eddy-driven zonal jets. It is shown how the appearance of zonal jets, which act as mixing barriers for turbulent exchange, and reduced lateral diffusivities are linked in a natural way by implementing mixing of potential vorticity and using a gauge term to insure that no spurious forces are introduced. It appears, therefore, possible to parameterise the appearance of zonal jets and its effect on the ventilation of interior ocean basins in non-eddyresolving, realistic ocean models.”
  • “the existence and statistical persistence of up-gradient eddy momentum fluxes in eddy-driven jets – indicating negative turbulent viscosities – has been known, but left aside in ocean modelling”
  • When parameterizing eddy momentum flux as diffusion of potential vorticity, “care has to be taken in the momentum budget, such that no forces are introduced by the parameterisation which would lead to spurious integral acceleration.” This is achieved by using a gauge term, the divergence of which is zero.
  • “zonally averaged case in quasi-geostrophic approximation”
  • They tried to reproduce zonal jets that alternate latitudinally in a zonally-averaged version of the model. But it would be interesting also to be able to reproduce in a coarse-resolution model, the meridional transport of mass –but this will be imposed anyway by the boundary conditions at the domain (except if we consider now the ocean-atmosphere system, in which case, the oceanic Lagrangian mean flow is not directly constrained by the boundary conditions).

Eden and Greatbatch (2009)

a parallel to Plumb (1979) and a discussion on the symmetric part of the tensor K of Plumb and Mahlman (1987).

Ferrari and Nikurashin (2010)

  • “The rate of dispersion of floats and drifters can also be related to an eddy diffusivity. However, questions remain as to how such Lagrangian diffusivities are related to the Eulerian eddy diffusivities employed in large-scale ocean models.” Can I resolve this question?
  • “Satellite altimetry provided the first global estimates of eddy statistics at the ocean surface. Holloway (1986), Keffer and Holloway (1988), and Stammer (1998) relied on mixing length theory (Prandtl 1925) to compute maps of eddy diffusivity from sea surface height variability. [...] The inferred K peaked in the core of strong currents, such as the western boundary currents and the Antarctic Circumpolar Current (ACC) of the Southern Ocean, where eddy velocities are largest.”
  • “Marshall et al. (2006) took a different approach to estimating K from altimetric measurements. They followed the technique pioneered by Nakamura (1996) to study eddy transport in the stratosphere. The idea is to numerically advect an idealized tracer using the geostrophic flow measured with the altimeter.”
  • “Marshall et al. (2006) used the technique to estimate K across the ACC in the Southern Ocean (the technique cannot be used to estimate eddy mixing along mean currents). They found that K is largest on the equatorward flank of the ACC in the subtropics, whereas it is somewhat smaller in the core of the current. This pattern is opposite to that obtained in the studies reviewed above.”
  • “We show that the cross-current effective diffusivity does indeed scale as K~v.l, consistent with mixing length arguments. However, the calculation of the mixing length ‘ must be modified to account for the propagation of eddies.”
  • “Using altimetric data,we find that themixing length suppression is so large that K is often smaller within the ACC than on its flanks, despite the rms eddy velocity y being largest in the ACC.”
  • “Papanicolaou and Pironneau (1981) showed that the analogy [between molecular diffusion and eddy mixing] holds as long as there is a clear separation between the spatial and temporal scales of the eddies and the large-scale circulation.”
  • “Even in this limit, however, the analogy is not perfect. Eddy diffusivities, unlike molecular diffusivities, can be strongly modulated by variations in the large-scale currents. Such modulations are often ignored in the oceanographic literature, but they are a crucial feature of eddy mixing (e.g., Andrews et al. 1987).”
  • “We will herein use the surfaceQGmodel to illustrate mixing across the ACC”
  • “To simplify the problem, the nonlinear term is represented with a fluctuation–dissipation stochastic model.”
  • “The stochastic model used to describe eddy mixing across a jet relies on three drastic assumptions: 1) the eddy motions can be represented with a linear stochastic model, 2) the eddy forcing consists of a single wave with specific wavenumber, and 3) the eddy scale is much smaller than the mean current width.”
  • “We show that the speed of propagation of eddies modulates the eddy diffusivity and must be included in theories of eddy mixing.”
  • “Nakamura (1996) showed that it is possible to estimate eddy diffusivities by numerically advecting a passive tracer with an observed velocity field.”
  • The geostrophic velocity field is made nondivergent as described above, a requirement for Nakamura’s approach to work. The basic idea is that a nondivergent velocity field is area preserving. Therefore, only molecular (or numerical) diffusion can change the area enclosed by tracer contours.”
  • “Geostrophic eddies, however, increase themolecular fluxes by twisting and folding tracer contours so that the interface available for molecular diffusion is enhanced.”
  • Nakamura (1996)’s theory also shows that the eddy diffusivity is directly linked to the molecular diffusivity, a result of the Generalized Lagrangian-Mean theory. However, the authors say that “The approach does not depend on details of the numerical dissipation because K [the eddy diffusivity] becomes independent of mu [the molecular diffusivity] for mu sufficiently small.” Where is this proven? See Marshall et al. (2006).
  • “The technique cannot be used to estimate the eddy diffusivity along mean currents because tracers are quickly homogenized along mean streamlines.”
  • The across-current diffusivity is suppressed by the mean flow. The theoretical diffusivity is close to the diffusivity computed from Nakamura (1996)’s technique.
  • It is not clear how the expression works in regimes other than eddies superimposed on a mean uniform flow.
  • “We expect our formula to break locally in regions where the mean currents meander strongly due to barotropic instabilities or topographic steering. In these regions both the assumption of scale separation between the mean and eddies and the relationship between mean current speed and eddy phase speed break.”
  • Eddies with a finite lifetime are not fully reversible and are efficient mixers. In this scenario, mixing occurs everywhere, but mixing is suppressed in regions where the mean flow is sufficiently fast to advect tracer out of the eddy faster than the eddy lifetime and hence reduce the time for which eddies can stir the tracer.
  • “Our results are consistent with the idea that mixing is suppressed in regions with strong potential vorticity gradients (e.g., Dritschel and McIntyre 2008).”
  • “Our results, instead, depart from previous literature that finds suppression of mixing where the horizontal shear in the mean currents is large (e.g., d’Ovidio et al. 2009).”

Marshall et al. (2006)

  • “if Pe >> 1, then one might expect Keff to be independent of the magnitude of k [the molecular diffusivity], because it is the stirring of tracers by the large-scale velocity field that controls the gradients of q on which the microscale diffusion acts. This has been shown to be true for the case of simple chaotic advection flows (Shuckburgh and Haynes 2003), and we will study whether it is also true for the oceanographic flows under consideration here.”
  • They use very high resolution so that they can reduced the dissipation to low level and explore a high range of Peclet (Pe) number.
  • When Pe is small, diffusion dominates over advection and the effective diffusivity scales as the squared root of the molecular diffusion. When Pe is large, advection dominates over diffusion and the effective diffusivity is independent on molecular diffusivity.
  • In the regime where the effective diffusivity is independent of the molecular one, what are the consequences with respect to the Lagrangian-mean tracer equation?
  • Plotting the Nussel (Nu) number versus Pe, different slopes indicates the type of regime (advection or diffusion dominates).
  • “A balance between advection and diffusion occurs on the “Batchelor scale.”
  • “From Fig. 4b it can be seen that the effective diffusivity is largely independent of the diffusivity when the total numerical diffusivity is used in the calculation.”

Shuckburgh and Haynes (2003)

  • “Calculating the transport and mixing properties of a flow by solving the advection-diffusion equation for a tracer rather than by direct analysis of particle trajectories, might seem unnecessarily complicated and expensive. However, the fact is that identification of partial barriers is difficult using particle or trajectory based methods—it is easy enough to calculate the transport across a specified control surface, but identification of a partial barrier requires identification of a control surface, probably a moving control surface, across which the transport is minimized.”

    Which methods of using parcel trajectories they are referring to (method #2 of inferring diffusivity described by Marshall et al. 2006 in their Introduction?)? Is the method that I am using (the Generalized Lagrangian Mean theory) a way to use parcel trajectories and avoiding the caveat of a moving surface as described here?

  • “At present, there is no precise mathematical theory relating effective diffusivity to transport and mixing properties of a flow, only heuristic arguments.”

  • The “effective diffusivity” has a priori a heuristic interpretation: “The mixing ability of a flow is essentially the ability to produce complex tracer geometry. This motivates the heuristic idea, exploited in previous studies, that the equivalent length or area may be used as a diagnostic of the flow, being relatively large in mixing regions and relatively small in barrier. It is this heuristic idea that we aim to make more quantitative in this paper.”

  • They use an artificial tracer that diffuses weakly with a diffusivity coefficient k to deduce the “effective diffusivity” Keff and they say that as long as the Peclet number is larger (k is small), then Keff is independent of the choice of k and becomes a characteristics of the flow alone (and not on the way the artificial tracer diffuses). This is different from my approach which is to quantify where particles cross on average a tracer contour, which, in the case where tacer is potential vorticity, indicates where PV is changed by diffusion of momentum (friction) and/or of density. The latter is thus a measure of where actual dissipation occurs. If the tracer is something else, it should also show (I think) where actual diffusion of that tracer is occurring (because diffusionof atracer is the process during which a parcel crosses a tracer contour).

  • Keff is a function of an area or volume which is itself a monotonic function of contour concentration. Keff cannot give information on change in mixing along tracer contours. It is a one-dimensional or two-dimension map of mixing in the case of two-dimensional and three-dimensional flow, respectively: “In practice, the mixing properties might vary significantly along a tracer contour, however the effective diffusivity, which is an average quantity, will not represent this.”

  • Look in Rhines and Young (1983) (and maybe Nakamura and Ma 1997) to see if any of the following has any relevance to my GLM work: “For a steady two-dimensional flow, with small values of the diffusivity, there is a straightforward relationship between the effective diffusivity and the cross-streamline diffusivity calculated by Rhines and Young (1983; see also Nakamura and Ma 1997). Since the Rhines and Young diffusivity is defined without reference to any tracer field, it follows that for a steady flow, the effective diffusivity can indeed be regarded as a well-defined diagnostic of a steady flow.”

  • About the comparison of effective diffusivity to particle transport: “Note that we have no a priori reason to believe that the advective transport of particles should correspond in any quantitative sense to the transport of tracer subjected to both advection and small diffusion. We adopt this approach simply as a modeling hypothesis—we have no theoretical basis for suggesting that Eq. (11) should precisely capture the transport of advected particles, nevertheless one might expect some similarity for small diffusion.” Then: “the distribution of advective particles seems to be well modeled in a quantitative sense by the effective diffusivity derived from an advected and diffused tracer. We emphasize again that we have no precise theoretical understanding of why this should be the case.”

Lee (2002)

For a discussion about the link between the Lagrangian-mean flow and the bolus velocity in her appendix.

Griesel et al. (2009)

  • “The spatial structure of eddy fluxes and their relation to mean quantities is complex: eddy fluxes can be directed up, down, and along mean gradients of the tracer. They act on different spatial scales and have rotational components that bias analyses. When diffusivities are diagnosed from eddying models, correlations are typically low and the distributions are unphysical and noisy (Rix and Willebrand, 1996; Roberts and Marshall, 2000; Nakamura and Chao, 2000; Eden et al., 2007a; Eden et al., 2007b).”
  • “Here we assess whether correlations of eddy fluxes with their parameterizations are improved when the eddy fluxes are averaged spatially.”

Moffatt (1983)

Moffatt argues that in the presence of the Coriolis force, the Lagrangian correlation tensor is not symmetric. In this case, the turbulence does not satisfy anymore the time-reversibility condition and particles follow a random path that nonetheless exhibits a preferred sense of rotation (see his Fig. 2) and it gets, instead, helicity, that is the velocity is correlated with the relative vorticity. This effect introduces a component to the eddy flux that is perpendicular to the tracer contours. This is the skew diffusion term associated with the asymmetric part of the diffusive tensor.

In Section 4, the author gives an account of the Kolmogorov theory of turbulence for the tracer equation, and describes the spectrum of the tracer depending on the Prandtl number (the ratio of molecular viscosity over molecular diffusivity).

Welander (1970 or 1973)

lateral friction in the ocean as an effect of potential vorticity mixings, Geophys. Fluid Dyn., 5 101–120.

Other papers to read

  • Nakamura (1996)

  • Smith et al. 02

  • McDougall and McIntosh 2001

  • See Visbeck, M., J. Marshall, T. Haine, and M. Spall, 1997: On the specification of eddy transfer coefficients in coarse-resolution ocean circulation models. J. Phys. Oceanogr., 27, 381

  • See Wardle, R., and J. Marshall, 2000: Representation of eddies in primitive equation models by a PV flux. J. Phys. Oceanogr., 30, 2481

  • See Where do ocean eddy heat fluxes matter? C Wunsch - Journal of Geophysical Research, 1999 -

  • See Green, J. S. A., 1970: Transfer properties of the large-scale eddies

    and the general circulation of the atmosphere. Quart. J. Roy. Meteor. Soc., 96, 157

  • See Rix and Willebrand (1996)

  • Griesel et al. (JGR, 2010): Isopycnal diffusivities in the Antarctic Circumpolar Current inferred from Lagrangian floats in an eddying model

  • Berloff (DAO, 2005): On dynamically consistent eddy fluxes

  • Kraichman (1976)

  • Ringler and Gent (submitted)

  • Wilson and Williams (2004)

  • Wilson and Williams (2006)

  • Eby and Holloway (1994): parameterization of the effect of eddies on momentum (referred by Killworth 1997).

Thiebaux (1975)

“Determination of one-particle effective eddy diffusivity tensor in linearly inhomogeneous turbulent flows”, JAS, 2136. Referenced by Matsuno (1980) for the case where the eddy flux is completely skewed albeit without having no effect on the tracer evolution.

Rhines (1977)

Referenced by Matsuno (1980) about a discussion between the Eulerian and Lagrangian-mean flow.