Abstract

Three alternative methods of averaging the general conservation equation of a fluid property in a turbulent flow in the Boussinesq approximation are compared: Lagrangian, residual, and isopycnal (or semi-Lagrangian) mean. All methods differentiate consistently but in different ways between effects of advection and irreversible changes of the average property. Because the three average properties differ, the mean transport velocities and the mean irreversible changes in the mean conservation equation differ in general.

The Lagrangian and the semi-Lagrangian (or isopycnal) mean frameworks are shown to be approximately equivalent only for weak irreversible changes, small amplitudes of the turbulent fluctuations, and particle excursion predominantly along the mean property gradient. In that case, the divergent Stokes velocity of the Lagrangian mean framework can be replaced in the Lagrangian mean conservation equation by a nondivergent, three-dimensional version of the quasi-Stokes velocity of T. J. McDougall and P. C. McIntosh, for which a closed analytical form for the streamfunction in terms of Eulerian mean quantities is given.

1. Introduction

The general conservation equation, /Dt = Q, describes the changes Q of a property α of a fluid particle along its path in a given flow in the Boussinesq approximation. If this flow is turbulent, a description of the average evolution of the property is often more useful than the instantaneous one. Using the Eulerian framework for such a mean description (i.e., applying some kind of average at a fixed location), however, mixes effects of advection and irreversible changes of the particle’s properties given by Q, which then complicates or even sometimes inhibits a useful physical interpretation of the average conservation equation. Therefore, some effort has to be made to decipher effects of turbulent advection and irreversible changes in a mean conservation equation. In this study, three alternative methods for such a consistent differentiation between both effects are discussed: the generalized Lagrangian mean (Andrews and McIntyre 1978a), the semi-Lagrangian or isopycnal mean (McDougall and McIntosh 2001; Nurser and Lee 2004), and the residual mean (e.g., Eden 2010).

It is the aim of this study to compare the different methods, in particular with respect to the similarities of the Lagrangian and the semi-Lagrangian mean framework. While in the Lagrangian mean the average of the fluid property is taken following instantaneous three-dimensional particle trajectories, in the semi-Lagrangian mean excursions of isosurfaces of the property under consideration are followed in a specified spatial direction. This conceptual similarity motivates to look for the connection between both frameworks. It will turn out that both become approximately equivalent under circumstances, which are stated below. Note that, specifying the property α as density, the semi-Lagrangian mean becomes the isopycnal mean, which is in particular popular in the oceanographic community. However, the difference of the widely used isopycnal mean framework to the generalized Lagrangian mean theory of Andrews and McIntyre (1978a) has not received much attention so far.

A further motivation for the comparison of Lagrangian and semi-Lagrangian mean is the missing closed analytical expression for the mean property and the mean transport velocity in the Lagrangian mean framework. Such a closed expression is available in terms of Eulerian mean quantities for the residual mean, and an approximate form is available for the semi-Lagrangian mean for the case of strong stratification. The discussion in this study yields an approximate form of the Stokes velocity in terms of a generalized, three-dimensional quasi-Stokes streamfunction given by Eulerian mean quantities, which can be used in the Lagrangian mean framework.

To start the discussion, the alternative frameworks for consistent averages of the conservation equation and their most important aspect are reviewed, followed by a comparison of the resulting eddy-related velocities valid to second order. The last section summarizes and discusses the results.

2. The Lagrangian mean

The Eulerian mean of an Eulerian quantity α(x, t) denotes a statistical mean with the fundamental property taken at the fixed position x. The Eulerian mean might be a time (space) average in a system with separated time (space) scales or an ensemble average (for which the requirement would be exactly met). In contrast to the Eulerian mean, the Lagrangian mean denotes an average of the property α following the fluid particle path. To make the connection of Lagrangian and Eulerian variables, it is useful to introduce the displacement vector ξ with

 
formula

Note that the displacement ξ is a vector function of space and time and that it is evaluated at the point x. The latter requirement allows the assumption that the Eulerian mean of ξ(x, t) will vanish: that is, . The mean position of the particle is then given by , and its actual position is given by xξ. The relation Eq. (1) can also be considered as a transformation rule, translating the mean position x to the actual position xξ. It is a necessary assumption for the generalized Lagrangian mean framework that this transformation is invertible. It is important to note that this might not always be the case, indicating a possible break down of the theory (Andrews and McIntyre 1978a).

The Lagrangian mean operator can now be defined by

 
formula

where denotes the Eulerian mean operator applied to the property α of a moving particle at its position xξ. The notation α(xξ, t) = αξ is introduced to indicate that α is evaluated at the actual position xξ = x + ξ during the averaging. Applying the Lagrangian mean operator to the velocity u(x, t) yields the Lagrangian mean velocity , which is the velocity of the mean point x such that x denotes the mean particle trajectory. The point x moves with on mean trajectories while xξ moves with uξ on the actual trajectories. More details about the formal derivation of the Lagrangian mean and its physical interpretation can be found in Dunkerton (1980), Plumb and Mahlman (1987), Bühler (2009), and Olbers et al. (2012).

It is the main result of the generalized Lagrangian mean theory of Andrews and McIntyre (1978a) that it is possible to define the Lagrangian mean material derivative for which

 
formula

holds for any property α. If Q = 0,1 the Lagrangian mean property is conserved along trajectories given by the Lagrangian mean velocity , which may be called quasi-material trajectories.2 The Lagrangian mean property is only changed by the Lagrangian mean forcing on the quasi-material trajectories. Note that the transport velocity is the same for any property; it only depends on the kinematics of the turbulent flow and not on the specific property under consideration.

3. The residual mean

Applying the Eulerian mean directly to the conservation equation yields

 
formula

with the velocity and property perturbations and , respectively. In Eq. (4), the familiar divergence of the eddy fluxes shows up, which cannot be related in an obvious way to either the effect of irreversible changes or advection in the Eulerian mean conservation equation. Further, it becomes clear that, setting Q = 0, the Eulerian mean property is not conserved along trajectories given by .

In fact, it is the residual velocity u* (e.g., Eden 2010) that defines the quasi-material trajectories for the Eulerian mean property: By adding nondivergent components to the Eulerian mean eddy flux , which do not figure in the Eulerian mean conservation equation but in the Eulerian mean variance equation, and also adding nondivergent components to fluxes of higher-order moments , it is possible to rewrite Eq. (4) as the residual mean conservation equation

 
formula

The turbulent diffusivity K is only related to irreversible changes of α (and changes in time) through correlations with Q and the properties of α—its variance and higher-order moments of α (or changes in times of those moments). It follows for vanishing Q (and steady state) that K = 0. In that limit, the residual velocity , given by the sum of the Eulerian mean velocity and the eddy-driven velocity , defines the quasi-material trajectories for the Eulerian mean property .

The eddy-driven velocity is given by a vector streamfunction, which can, in principle, be calculated from the Eulerian mean eddy fluxes and an infinite series of fluxes of higher-order moments of α of increasing order (e.g., Eden 2010),

 
formula

where denotes the (Eulerian mean) variance and with the unit vector . The leading-order term (with respect to α′) in the expression for Ψe equals the definition for the eddy-driven streamfunction in the transformed residual mean (TEM) framework, which was formulated first by Andrews and McIntyre (1976) for the zonal mean case and is also discussed in, for example, Andrews and McIntyre (1978b), Plumb (1979), Andrews et al. (1987), and Held and Schneider (1999). Note that the TEM for the zonal mean case can be extended to the general three-dimensional case considered here without complication (e.g., Plumb and Ferrari 2005). The TEM was extended by McDougall and McIntosh (1996) (for the three-dimensional case within the limit of strong stratification) by including the along-isopycnal flux of variance as the definition for the rotational flux potential θ [here a second-order term in Eq. (6)] and was formulated for the full hierarchy of moments and the general case by Eden et al. (2007), which then finally yields the physically consistent property that K = 0 if Q ≡ 0 in steady state. Note that it follows from Eq. (6) that quasi-material trajectories of are different for different properties, in contrast to the Lagrangian mean, where only a single advection velocity (i.e., the Lagrangian mean velocity) is transporting any property of the fluid particles. More details about the residual mean can be found in Olbers et al. (2012).

4. The semi-Lagrangian (isopycnal) mean

A further alternative averaging framework, which is often discussed in the oceanographic community, is given by applying the average to a variable evaluated at the depth of a material surface instead of constant geopotential height (McDougall and McIntosh 2001).3 In analogy to the Lagrangian mean Eq. (1), a semi-Lagrangian mean of a variable α can be defined as

 
formula

where the vertical displacement ξ3 is the deviation of the instantaneous depth of a material surface z(xh, t) from its mean: that is, . When compared to the Lagrangian mean Eq. (1), the semi-Lagrangian mean resembles a Lagrangian mean in which only one space coordinate is varied during averaging.

It is convenient to use an isosurface of the property α(x, t) = constant as the material surface z(xh, α, t). If potential density (i.e., isopycnals) is used as material surfaces, the mean defined by Eq. (7) might be called isopycnal mean (McDougall and McIntosh 2001) and is equivalent to averaging in isopycnal coordinates. However, note that, following Nurser and Lee (2004), the general case is discussed here, where isosurfaces of α define the material surfaces in Eq. (7) and where α is a general property of the fluid subject to the general conservation equation /Dt = Q.

The choice of α as semi-Lagrangian coordinate is motivated by the fact that α is conserved for vanishing forcing Q, such that particles stay on the material surfaces during their movements; only effects included in Q lead to excursions from the material surfaces. Note that it is assumed here for simplicity that the function ρ(z) or α(z) is monotonic, such that its inverse exists. A generalization of the concept for nonmonotonic functions can be found in Nurser and Lee (2004). In contrast to the Lagrangian mean, the semi-Lagrangian mean is therefore always well defined. Note that depth for the definition for ξ3 can also be exchanged with any other spatial direction, which is, however, not considered here.

Using isosurfaces of α itself as material surfaces in Eq. (7), yields trivially . However, it is important to note that the semi-Lagrangian mean is defined at the mean isosurface height . In other words, yields the value of the property α, whose mean isosurface height at the position xh equals the actual depth z at which is given. Note that differs in general from the Eulerian mean and the Lagrangian mean . It can be shown that for the semi-Lagrangian mean

 
formula

holds (Nurser and Lee 2004; Olbers et al. 2012). Note that, in Eq. (8), isosurfaces of α are used as material surfaces in the definition of the semi-Lagrangian mean Eq. (7). The transport velocity is the sum of the Eulerian mean and the quasi-Stokes velocity and σ denotes the infinitesimal layer thickness ∂z(xh, α, t)/∂α. For Q = 0, the velocity mark the quasi-material trajectories in this framework and only the layer thickness-weighted forcing changes the semi-Lagrangian mean property .

The quasi-Stokes velocity is given by a vector streamfunction (McDougall and McIntosh 2001)

 
formula

where the vertical integral is taken from the mean isosurface depth to the instantaneous depth . It can be shown that the sum of the horizontal Eulerian mean and quasi-Stokes velocity equals the thickness-weighted velocity, . The latter can be decomposed into the isopycnally averaged velocity and the Bolus velocity (Rhines 1982), but note that the quasi-Stokes velocity differs in general from the Bolus velocity. Compare also Bühler (2009) for the relation between the Bolus and Stokes velocity. A parameterization for the Bolus velocity in ocean models was first proposed by Gent and McWilliams (1990), and a physical interpretation was first given by Gent et al. (1995). Note that the differences between Bolus, quasi-Stokes, and eddy-driven velocity of the residual mean framework, as well as between the Eulerian mean and semi-Lagrangian mean property, complicate the exact formal formulation of the parameterization of Gent and McWilliams (1990) (for a detailed discussion, see, e.g., Olbers et al. 2012).

Equation (8) applies to only; for different properties or, equivalently, using different properties to define the material surfaces in the semi-Lagrangian mean Eq. (7), additional eddy fluxes would show up in the mean conservation equation for α, which are usually interpreted as “isopycnal mixing” (McDougall and McIntosh 2001) [using isopycnals as material surfaces in Eq. (7)]. Similar to the eddy fluxes in the Eulerian mean conservation equation, these isopycnal eddy fluxes cannot be related in an obvious way to advection or irreversible changes of the property under consideration (in fact, it is likely that a dominant part of the isopycnal eddy fluxes is related to turbulent advection). On the other hand, using isosurfaces of the specific property in the semi-Lagrangian mean Eq. (7) yields a consistent semi-Lagrangian mean conservation equation but consequently different quasi-Stokes velocities for each property, similar to the residual mean but unlike the Lagrangian mean framework.

5. The relation between the averaging frameworks

The steady conservation equations for vanishing Q for the different averaging frameworks are given by

 
formula

with the Stokes velocity . Each averaging framework defines different eddy-related velocities, different mean properties, and thus different quasi-material trajectories. It is only the Lagrangian mean in which the quasi-material trajectories are independent of the property under consideration. This feature is unique to the Lagrangian mean. On the other hand, no closed analytical form exists for the Lagrangian mean velocity. It is therefore useful to connect to the other velocities for which (approximate) closed forms exist.

Subtracting the Lagrangian mean Eq. (3) and the residual mean conservation equation Eq. (5) in the steady and adiabatic limit yields

 
formula

with the Stokes correction . The last step in Eq. (11) approximates the Lagrangian advection by Eulerian mean advection, which is valid to second order in perturbation quantities, as detailed below. Note that because of this approximation small irreversible change Q and deviations from steady state become possible in Eq. (11). The difference between Stokes and eddy-driven velocity is related to the advection of the Stokes correction . A similar relation as Eq. (11) holds for the difference between Stokes and quasi-Stokes velocities, which is related to the advection of the difference in Lagrangian and semi-Lagrangian mean property. One might argue that this difference is smaller than the difference between Eulerian and Lagrangian mean property: that is, Stokes correction in Eq. (11). However, because holds in general, Stokes and quasi-Stokes velocities are not identical in general. This is also reflected by the fact that the Stokes velocity may be divergent, whereas the quasi-Stokes velocity is not.

To relate Stokes velocity , eddy-driven velocity , and quasi-Stokes velocity , a truncated expansion of the Stokes correction is considered. The results will therefore only be valid for small amplitudes of the fluctuating quantities. Note that, although Eq. (11) applies only to the adiabatic and steady limit, small deviations from that limit are possible, as long as the effect of these deviations do not exceed the order of truncation of the expansion. A Taylor expansion at the mean position x up to second order in perturbation quantities of the instantaneous value of the particle property αξ is given by

 
formula

Taking the mean of Eq. (12) yields because the first-order term vanishes. It follows that the Stokes correction is of second order. It will turn out below that the Stokes velocity (as the eddy-driven velocity and the quasi-Stokes velocity ) is also of second order in perturbation quantities.

Small irreversible changes Q in the conservation equation for the property α are assumed now, such that and thus . Further, it is assumed that the component of the displacement vector ξ normal to the gradient of the mean property is of second order, whereas the component in the direction of is of first order,

 
formula

This assumption is necessary to connect the displacement vector with the Eulerian mean property and will lead to a connection between the Stokes and the quasi-Stokes velocity as shown below. For a two-dimensional flow and small wavelike perturbation the assumption Eq. (13) appears valid. On the other hand, in most three-dimensional geophysical flows, the displacements along isosurfaces of the mean property will normally be much greater than those across. It follows that, if those particle excursions along mean isosurfaces will introduce local and instantaneous perturbations of the property α larger than the particle excursions across the mean isosurfaces, the connection between the Stokes and the quasi-Stokes velocity cannot be established.

Using Eq. (13) to calculate the second-order Stokes correction in Eq. (12) yields after some manipulations, which are detailed in the  appendix,

 
formula

with the Eulerian mean variance φ = α2/2. The second-order Stokes correction Eq. (14) is now used in Eq. (11) to calculate the difference between Stokes and eddy-driven velocity. To do so, the Eulerian mean velocity on the right-hand side of Eq. (11) is first decomposed using the three-dimensional unit vector into a component in the direction of n and a component perpendicular to n: that is, as with and . This decomposition yields, in Eq. (11),

 
formula

Because small irreversible changes Q in the Eulerian mean conservation equation were assumed, holds and it follows that un = O(a2). Because , this means that the term related to un in Eq. (15) can be neglected. The term related to in Eq. (15) can be decomposed into a nondivergent and a divergent part,

 
formula

for which it is shown in the  appendix that the latter vanishes in Eq. (15) to O(a3). Using now the vector streamfunction for , Eq. (15) simplifies to

 
formula

which defines the generalized three-dimensional version of the quasi-Stokes streamfunction . It can also be rewritten as

 
formula

using the Stokes correction , and the definition for the streamfunction given by Eq. (6). In the small slope limit, , the streamfunction for the horizontal quasi-Stokes velocity becomes

 
formula

This form for Ψ+, valid for strong stratification only, is identical to Eq. (4a) of McDougall and McIntosh (2001). A two-dimensional version of the quasi-Stokes velocity is given by

 
formula

with and .

6. Discussion

Applying different averaging operators to the general conservation equation /Dt = Q implies different quasi-material trajectories: that is, different trajectories along which the mean property is conserved for Q = 0 and on which only the mean of Q acts to irreversibly change the mean property. Three alternative averaging frameworks are discussed and compared in this study: the Lagrangian, residual, and semi-Lagrangian mean framework, with quasi-material trajectories given by the sum of Eulerian mean velocity plus either Stokes, eddy-driven, or quasi-Stokes velocity, respectively. Although the Stokes velocity is independent of the property under consideration and depends on the kinematics of the flow only, both eddy-driven and quasi-Stokes velocity are special to a certain property, which means that, for different properties under consideration, different quasi-material trajectories result. On the other hand, the Stokes velocity may in general be divergent, whereas both eddy-driven and quasi-Stokes velocity are given by a streamfunction. Further, no closed analytical form is available for the Stokes velocity, whereas the eddy-driven streamfunction can be expressed by an infinite series of fluxes of (Eulerian mean) property moments and for the quasi-Stokes streamfunction an approximate form in (Eulerian mean) perturbation quantities can be given.

Because each of the three alternative averaging frameworks are related to a different averaged property, all three resulting mean transport velocities are advecting different quantities and are thus inherently different. Note that consequently the implied turbulent mixing also becomes different in the different frameworks. The difference is most obvious for the Eulerian mean and the Lagrangian mean property. On the other hand, the definition of Lagrangian mean and semi-Lagrangian mean differs only by the choice of the displacement vector x + ξ: it follows instantaneous three-dimensional particle trajectories in the Lagrangian mean and excursions of isosurfaces of the property under consideration in a single spatial direction in the semi-Lagrangian mean framework. This similarity motivates looking for the differences between both frameworks and in particular for the differences between Stokes and quasi-Stokes velocities.

The comparison of the eddy-driven velocities in the alternative averaging frameworks yields that for

  • small fluctuations;

  • weak irreversible changes Q [i.e., for ]; and

  • local fluctuations of α generated to leading order by particle excursions directed parallel to the mean property gradient [Eq. (13)]

the generalized quasi-Stokes velocity given by the streamfunction Eq. (18) approximates the Stokes velocity to second order in perturbation quantities in the Lagrangian mean conservation equation. In contrast to the Stokes velocity, the generalized quasi-Stokes streamfunction Eq. (18) is given in a closed analytical form in terms of Eulerian mean quantities, which are often easier to evaluate than the respective Lagrangian quantities.

Specifying α as density, the equivalence between Lagrangian and semi-Lagrangian framework is due to identical [to O(a3)] diapycnal components of Stokes and quasi-Stokes velocities. In other words, the use of the quasi-Stokes instead of the divergent Stokes velocity in the Lagrangian mean conservation equation of the property under consideration introduces no error larger than third order, provided that the assumptions (i)–(iii) hold. Note that this applies only to the specific property under consideration, because the along-isopycnal components of the Stokes velocity, important for properties with gradients on isopycnals, might differ from the quasi-Stokes velocity.

The equivalence between the diapycnal components of Stokes and quasi-Stokes velocities only holds when particle excursions along mean isosurfaces do not yield local perturbations of the property larger than the excursions directed parallel to the gradient of the mean property. For a two-dimensional flow and small wavelike perturbation, this assumption appears valid. On the other hand, in most three-dimensional geophysical flows the displacements along isosurfaces of the mean property will normally be much greater than those across. It follows that if those particle excursions along mean isosurfaces will introduce perturbations of the property α larger than the particle excursions parallel to the gradient of the mean property, the connection between the Stokes and the quasi-Stokes velocity cannot be established.

The quasi-Stokes streamfunction of McDougall and McIntosh (2001) was formulated for density as a monotonic function of depth. Nurser and Lee (2004) generalized the concept to different properties, nonmonotonic functions, and different spatial coordinates. In this study, the quasi-Stokes streamfunction Ψ+ was formulated without reference to a special coordinate and thus resembles a generalized, three-dimensional version of the quasi-Stokes streamfunction by McDougall and McIntosh (2001), without the need to specify a particular spatial direction. This generalized streamfunction can be used in the Lagrangian mean conservation equation together with the approximate Stokes correction Eq. (14). Here, Ψ+ becomes equivalent to the quasi-Stokes streamfunction of McDougall and McIntosh (2001) or Nurser and Lee (2004) for strong stratification in a single spatial direction.

A weakness of the present derivation is that Ψ+ is only given by the leading-order terms in the expansion and for small forcing Q. In contrast, it is possible to evaluate also higher-order terms for the expansion of the quasi-Stokes velocity by McDougall and McIntosh (2001) (e.g., Olbers et al. 2012). Further, no closed (semi-Lagrangian) definition for Ψ+ was given, which is available for the quasi-Stokes streamfunction of McDougall and McIntosh (2001) given by the lateral transport below an instantaneous isopycnal [Eq. (9)]. It would be desirable—but left for future work—to formulate a corresponding definition for Ψ+ as well and to specify the averaging operation leading to a consistent conservation equation involving Ψ+ as transport velocity.

Acknowledgments

This study was supported by the Deutsche Forschungsgemeinschaft within the Cluster of Excellence CliSAP. Discussions with Trevor McDougall, Dirk Olbers, Jürgen Willebrand, Jan Viebahn, and Stefan Riha and comments by David Marshall and another, anonymous reviewer have been helpful and are appreciated.

APPENDIX

Derivation of the Stokes Correction

The derivation of the Stokes correction Eq. (14) is detailed here,

 
formula

using Eq. (13) [i.e., ] and with the Eulerian mean variance φ = α2/2. It is also shown that the second term on the right-hand side of Eq. (16), , vanishes to O(a3) in Eq. (15),

 
formula

using because un = O(a2), and thus . Using and , it follows that .

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Footnotes

1

Note that the case of vanishing nonconservative forcing Q (locally and for all times) in the instantaneous conservation equation for α is only hypothetical. In fact, a nonzero Q related to molecular diffusion is needed to cascade variance to small scales in a turbulent flow. Here, however, the hypothetical case Q → 0 is used to test whether the implied irreversible effects of the averaging framework on the mean property are vanishing as well. For the Eulerian mean, they do not in general, such that this framework is considered to be inconsistent.

2

Because actual trajectories are different from the mean, quasi-material trajectories are hypothetical, not material trajectories.

3

Note that the concept discussed in this section is called temporal residual mean (TRM) by McDougall and McIntosh (2001). However, the name TRM was first introduced by McDougall and McIntosh (1996), where in fact a framework for the Eulerian mean density is considered. Sometimes, the later concept by McDougall and McIntosh (2001) is called TRM-II. The generalization of the concept of McDougall and McIntosh (1996) by Eden et al. (2007) was also called (generalized) TRM. Here, the concept by McDougall and McIntosh (2001) is called the semi-Lagrangian or isopycnal mean and the concept by Eden et al. (2007) is called the residual mean to avoid any confusion.