Ocean Dynamics (2010) 60:1319–1337DOI 10.1007/s10236-010-0325-z

Geophysical flows with anisotropic turbulence and dispersivewaves: flows with stable stratification

Boris Galperin · Semion Sukoriansky

Received: 31 March 2010 / Accepted: 29 July 2010 / Published online: 19 August 2010© Springer-Verlag 2010

Abstract The quasi-normal scale elimination (QNSE)is an analytical spectral theory of turbulence basedupon a successive ensemble averaging of the velocityand temperature modes over the smallest scales ofmotion and calculating corresponding eddy viscosityand eddy diffusivity. By extending the process of suc-cessive ensemble averaging to the turbulence macro-scale one eliminates all fluctuating scales and arrivesat models analogous to the conventional Reynoldsstress closures. The scale dependency embedded inthe QNSE method reflects contributions from differentprocesses on different scales. Two of the most impor-tant processes in stably stratified turbulence, internalwave propagation and flow anisotropization, are explic-itly accounted for in the QNSE formalism. For rela-tively weak stratification, the theory becomes amenableto analytical processing revealing just how increasing

Responsible Editor: Leo Oey

B. Galperin (B)College of Marine Science, University of South Florida,St. Petersburg, FL 33701, USAe-mail: bgalperin@marine.usf.edu

S. SukorianskyDepartment of Mechanical Engineering,Ben-Gurion University of the Negev,Beer-Sheva 84105, Israele-mail: semion@bgu.ac.il

S. SukorianskyPerlstone Center for Aeronautical Engineering Studies,Ben-Gurion University of the Negev,Beer-Sheva 84105, Israel

stratification modifies the flow field via growing aniso-tropy and gravity wave radiation. The QNSE theoryyields the dispersion relation for internal waves inthe presence of turbulence and provides a theoreti-cal reasoning for the Gargett et al. (J Phys Oceanogr11:1258–1271, 1981) scaling of the vertical shear spec-trum. In addition, it shows that the internal wavebreaking and flow anisotropization void the notion ofthe critical Richardson number at which turbulence isfully suppressed. The isopycnal and diapycnal viscosi-ties and diffusivities can be expressed in the form ofthe Richardson diffusion laws thus providing a theo-retical framework for the Okubo dispersion diagrams.Transitions in the spectral slopes can be associatedwith the turbulence- and wave-dominated ranges andhave direct implications for the transport processes. Weshow that only quasi-isotropic, turbulence-dominatedscales contribute to the diapycnal diffusivity. On larger,buoyancy dominated scales, the diapycnal diffusivitybecomes scale independent. This result underscores thewell-known fact that waves can only transfer momen-tum but not a scalar and sheds a new light upon theEllison–Britter–Osborn mixing model. It also providesa general framework for separation of the effects ofturbulence and waves even if they act on the samespatial and temporal scales. The QNSE theory-basedturbulence models have been tested in various appli-cations and demonstrated reliable performance. It issuggested that these models present a viable alternativeto conventional Reynolds stress closures.

Keywords Anisotropic turbulence · Dispersive waves ·Stable stratification · Turbulence spectra ·Okubo diagrams

1320 Ocean Dynamics (2010) 60:1319–1337

1 Introduction

Oceanic flows entwine anisotropic turbulence and avariety of waves that populate a broad range of spa-tial and temporal scales. Turbulence-wave interactionsare stochastic and highly irregular. Their intrinsic non-linearity precludes the existence of stable analyticalsolutions. Stochastic description is more feasible yet itscharacteristics are well known only for the simplest,locally isotropic turbulent flows that depend on a singlenon-dimensional parameter, the Reynolds number.

In the linearized representation, the ocean dynam-ics features three general classes of dispersive wavesacting on different scales. Internal gravity waves arecaused by stable stratification and populate relativelysmall scales. Flows on larger scales are affected bythe planetary rotation that introduces inertial waves.The largest, planetary scales are dominated by Rossbywaves brought about by the latitudinal variation of theCoriolis parameter.

The ‘extra strains’ (Bradshaw 1973) that supportthese waves evoke fundamental changes in turbulenceregimes and associated transport processes. As eluci-dated by Smith and Waleffe (2002), each type of the ex-tra strains elicits the large-scale flow anisotropization.In addition, flows develop slow manifolds in lower-dimensional subspaces normal to the wave vectorsaligned in the directions of the respective zero frequen-cies. In the physical space, the resulting flows wouldappear either as a system of vertically decorrelatedhorizontal layers (stable stratification, e.g. Lilly 1983;Fernando 2000; Fritts et al. 2003) or a system of large-scale cyclonic vortical columns (rotation, e.g. Smith andWaleffe 1999) or a system of alternating zonal jets(a β-effect, e.g. Rhines 1975; Huang et al. 2001). Inturn, turbulence and non-linear interactions modify thewaves’ dynamics. For instance, a small-scale, random,white-noise forcing is capable of exciting a complete setof large-scale quasi-linear Rossby waves (Sukorianskyet al. 2007).

Small- and large-scale geophysical flows differ bytheir mechanisms of the energy transfer. The formerare three-dimensional (3D), maintained by the large-scale instabilities and feature the direct energy cascadedown to the scales of the molecular dissipation. Thelatter are quasi two-dimensional (2D) due to the strongeffect of rotation and ensuing dynamics restrictionsimposed by the Taylor–Proudman theorem. In the pres-ence of a relatively small-scale forcing these flows maydevelop the inverse energy cascade that maintains thelarge-scale eddies and zonal jets (Vallis 2006; Galperinet al. 2006). The understanding of the dynamics andtransport processes in flows combining waves and tur-

bulence is essential for successful prediction of geo-physical and planetary systems.

Polzin (2004) describes the literature on oceanic in-ternal waves as a ‘disordered melánge’ dealing with di-verse processes on many scales. These processes rangefrom wave generation and interactions with geostrophiccurrents to non-linear wave interactions and dissipationby instabilities and wave breaking. He mentions thatdespite extensive research, there is still ‘an intrinsicdisorder in that the various elements are usually notconnected to each other in a systematic way. The mostintrinsically difficult piece of a synthetic treatment is toaccount for non-linearity and dissipation in the wavefield. The key...is developing a sufficiently simple un-derstanding of non-linearity and wave breaking suchthat their effects can be more easily recognized andappreciated.’

The oceanographic research could greatly benefitfrom the significant progress in studies of the effectof gravity waves in the middle atmosphere (Fritts andAlexander 2003; Fritts et al. 2009b, c). These studiescover source characteristics, spectral properties andenergy transfer, instabilities, wave–wave and wave-flow interactions and implications of all these vari-ous processes for the atmospheric circulation. As inthe ocean, interactions of atmospheric gravity wavesamongst themselves and with other flow structures pro-vide a vast reservoir of turbulence energy on small andlarge scales (Fritts et al. 2009a). However, one of themain difficulties that has hampered direct numericalsimulations spanning the entire range of scales is com-mon to the two fields and stems from the large disparitybetween the inertia-gravity and dissipation scales.

Turbulence parameterization in many oceanicand atmospheric models is accomplished using theReynolds averaging at the level of the second andhigher order turbulence moments (Mellor and Yamada1982). This approach will be referred to as a Reynoldsstress modeling (RSM). By its very nature, RSM doesnot discern different processes on different scalesand simply lumps all of them together. To relatethe unknown correlations to the known ones, RSMrelies upon the closure hypotheses which utilize theprinciple of the ‘invariant modeling’ (Lewellen 1977).According to this principle, the constants in the closureassumptions are assumed flow independent and thusamenable to calibration in simple, well controlledneutral flows with no waves. More flexible but far morecomplicated models allow for the closure coefficientsto be functions of the Reynolds stress tensor invariants(e.g., Ristorcelli 1997). This approach may help toaccount for turbulence anisotropy but would still havedifficulties with waves.

Ocean Dynamics (2010) 60:1319–1337 1321

The necessity to account for the effect of wavesin addition to turbulence has long been recognized(Stewart 1969). Early attempts to include waves inRSM go back to the works by Einaudi and Finnigan(Finnigan and Einaudi 1981; Finnigan et al. 1984;Einaudi et al. 1984). Numerous efforts in the samedirection have been undertaken since then, see e.g.,Finnigan (1999), Jin et al. (2003), Zilitinkevich et al.(2009). Significant progress has been achieved in thedescription of flows featuring time and/or scale sepa-ration between turbulence and waves. When scale sep-aration does not exist, the progress has been slow. Infact, Jacobitz et al. (2005) conclude that ‘there is proba-bly no really clear-cut distinction between turbulenceand waves, at least when both are coexisting at thesame scales.’

An alternative to RSM is a spectral approachproviding a time- and scale-dependent description(Wyld 1961; Orszag 1977; McComb 1991; Canutoand Dubovikov 1996; Cambon and Scott 1999;Cambon 2001). This approach naturally accommo-dates turbulence-wave interaction and flow anisotropy.The main drawback of most spectral theories is theirimmense mathematical complexity when applied toanisotropic flows with waves. This complexity has ham-pered theoretical developments and practical applica-tions of spectral models.

Recently, we have developed a new spectral the-ory based upon the successive scale elimination proce-dure that explicitly deals with processes on all scales(Sukoriansky et al. 2005b). This theory, known as thequasi-normal scale elimination (QNSE), provides acomprehensive analysis of turbulence and waves whilelending itself to complete analytical treatment. Eventhough the intermediate calculations are quite compli-cated, the final results are concise and easy to imple-ment. The QNSE accurately predicts many importantflow characteristics unavailable in RSM, e.g., turbu-lence spectra and flow anisotropization. Of particularsignificance is the derivation of the dispersion relationfor internal waves in the presence of turbulence whichdemonstrates that the theory is capable of accountingfor the combined effect of turbulence and waves evenwhen there is no space and time scale separation be-tween them. Thus, the theory fills an important gap inour understanding of the turbulence-wave interaction.As will be shown later, the theory suggests an objectiveway to discern turbulence- and wave-dominated scalesand identify their effects. This identification will berelated to the well known fact that unlike turbulence,waves can only transfer momentum but not a scalar. Onthe practical side, the QNSE theory produces expres-sions for the isopycnal and diapycnal eddy viscosities

and eddy diffusivities that can be utilized in variousapplications.

This paper is a companion to the paper published inPart 1 of the special issue that dealt with anisotropicturbulence and Rossby waves (Galperin et al. 2010).Together, they present a comparative study of thewave-turbulence interaction and transport processes insystems affected by internal or Rossby waves. Thispaper concentrates on flows with stable stratification.Along with the analysis of the flow, it discusses similar-ities in their dynamics and transport characteristics.

The organization of the paper is as follows. Section 2briefly presents the foundations of the QNSE theoryand provides the description of the scale-dependentisopycnal and diapycnal viscosities and diffusivities.Technical details are given in the Appendix. Section 3explores the vertical spectrum of the kinetic energy,elaborates the physical origin of the Gargett et al.(1981) scaling and compares the results with atmo-spheric and oceanic data. Section 4 considers thelimit of strong stable stratification, revisits the Ellison–Britter–Osborn (EBO thereon) model, applies the re-sults to the diapycnal and isopycnal diffusion in theocean and discusses the Okubo dispersion diagrams.Section 5 analyzes the vertical spectrum of the tem-perature fluctuations, presents it in the format of theGargett et al. scaling and compares the results withvarious data. Section 6 elaborates the absence of thecritical Richardson number. Finally, Section 7 providesdiscussion and conclusions and emphasizes that theQNSE model offers a viable alternative to the Reynoldsstress closures.

2 Basics of the QNSE theory of stablystratified turbulence

The theory is formulated for a fully three-dimensionalturbulent flow field with imposed vertical, stabilizingtemperature gradient. The flow is governed by themomentum, temperature and continuity equations inthe Boussinesq approximation,

∂u∂t

+ (u · ∇)u − αgTe3 = ν0∇2u − 1ρ

∇ P + f0, (1)

∂T∂t

+ (u · ∇)T + d�

dzu3 = κ0∇2T, (2)

∇ · u = 0, (3)

where the unit vertical vector e3 is directed upward,P is the pressure, ρ is the constant reference density,ν0 and κ0 are the molecular viscosity and diffusivity,respectively, α is the thermal expansion coefficient,

1322 Ocean Dynamics (2010) 60:1319–1337

g is the acceleration due to gravity directed downwards,� is the mean potential temperature, and T is thefluctuation of �. The force f0 represents the effect ofexternal large-scale stirring that maintains turbulencein a statistically steady state. The details of the forcingare immaterial for flow properties on scales smallerthan the forcing scale.

The space and time Fourier representation of thevelocity and temperature yields respective Fouriermodes, ui(k) and T(k), where k ≡ (ω, k) is a four-dimensional vector in Fourier space. The spectraldomain is bounded by the viscous dissipation orKolmogorov wave number, kd = (ε/ν3

0)1/4, where ε isthe rate of the viscous dissipation. Fourier-transformedEqs. 1 and 2 involve strong non-linearity of the velocitymodes and coupling between the velocity and tempera-ture modes. These prominent features of the governingequations make the problem very difficult for analyti-cal treatment. The non-linear terms in the momentumequation exceed the linear ones by the factor O(Re),Re being the Reynolds number. On large scales, whereRe is very large, the analytical tools become practi-cally useless. The situation is different on the smallestscales, of the order of the Kolmogorov scale, kd, whereRe = O(1) (Tennekes and Lumley 1972). The small-ness of Re prompts the exploration of the methodologyof the renormalized perturbation theory which oper-ates with ‘dressed,’ or effective, or eddy viscosity andeddy diffusivity rather than their ‘bare’ (i.e., molecular)values (Forster et al. 1977; Yakhot and Orszag 1986;McComb 1991, 1995). This methodology has been usedto develop the quasi-normal scale elimination tech-nique used in this paper. The technical details of thismethodology are given in the Appendix. For simplicity,this section provides only a schematic description of theQNSE technique.

We begin with derivation of the self-contained equa-tion for ui(k) whose domain of definition extends from0 to kd. We then subdivide this domain onto two sub-domains, (0, kd − ��] and (kd − ��, kd], where �� isinfinitesimal. For modes inside the latter subdomain,Re = O(1). The smallness of Re allows one to designa procedure of ensemble averaging of the fluctuatingmodes over the shell �� that coarsens the domain from(0, kd] to (0, kc]. The wave number kc = kd − �� willbe referred to as the dynamic dissipation cutoff. Thisaveraging involves several assumptions the most impor-tant of which is the hypothesis of the quasi-normality.According to this hypothesis, the third moments of thefluctuating variables are small and can be discardedwithin the shell �� (Sukoriansky et al. 2005b). Theaveraging generates corrections to the viscosity anddiffusivity which account for the transport processes on

now unresolved scales. Emerging from this procedure‘dressed’ viscosity and diffusivity are flow-dependent,in contrast to their ‘bare’ constant molecular valuesobtained when all scales are resolved.

At the next step, the new domain of definition isagain subdivided onto two subdomains, (0, kc − ��]and (kc − ��, kc]. The procedure of elimination ofthe shell �� is repeated yielding new corrections tothe viscosity and diffusivity. Successive repetition ofthis procedure results in a cyclic process each cycle ofwhich modifies ‘dressed’ viscosity and diffusivity andshrinks the domain of definition by ��. The decreaseof kc and increase of the eddy viscosities facilitatethe Reynolds number in the vicinity of kc to remainO(1) thus indicating that the scale elimination proce-dure theory is mathematically sound. The use of the‘dressed’ viscosities and diffusivities at every step of thescale elimination procedure is equivalent to introducingeddy damping and, thus, places QNSE in the class ofthe quasi-normal eddy-damped theories of turbulence(Orszag 1977; McComb 1991; Chasnov 1991).

Due to the anisotropization introduced by stablestratification, the corrections to the viscosity and diffu-sivity are different in the vertical and horizontal direc-tions such that it is convenient to represent the diffusiveterms in Fourier transformed Eqs. 1 and 2 as

νk2 = νhk2h + νzk2

3, (4)

κk2 = κhk2h + κzk2

3, (5)

where k2 = k21 + k2

2 + k23 = k2

h + k2z; kx = k1 and ky =

k2 are the zonal and the meridional wave numbers,respectively; kh = (k2

1 + k22)

1/2 and kz = k3 are the hor-izontal and the vertical wave numbers, respectively; νh

and νz are the horizontal and vertical eddy viscosities,and κh and κz are the horizontal and vertical eddydiffusivities. Clearly, at the very first iteration, ν0 =νh = νz and κ0 = κh = κz. Each step of the scale elim-ination procedure generates corrections, �νh, �νz, �κh

and �κz, to νh, νz, κh and κz. All these corrections areof the order O(��). We now take a limit �� → 0and obtain a system of four coupled integro-differentialequations for νh, νz, κh and κz as functions of k.Sukoriansky et al. (2005b) show that this systemof equations can be non-dimensionalized to produceequations for νh/νn, νz/νn, κh/νn and κz/νn as functionsof k/k0, where νn = 0.46ε1/3k−4/3 (Yakhot and Orszag1986; Sukoriansky et al. 2003) and k0 = (N3/ε)1/2 isthe Ozmidov wave number. Here, νn is the value ofthe isotropic eddy viscosity which the flow would haveacquired in the absence of stratification. The Ozmidovwave number defines a scale at which the turbulenceeddy turnover time scale becomes equal to the period

Ocean Dynamics (2010) 60:1319–1337 1323

of a linear internal wave, N−1, where N ≡ (αg d�

dz

)1/2

is the buoyancy, or Brunt-Väisälä frequency. Turbu-lence prevails in the range k/k0 � 1 while internalwaves dominate the flow for k/k0 � 1. Keeping inmind the application of the QNSE theory in physicaloceanography, we identify the horizontal and verticaleddy viscosities and eddy diffusivities with their respec-tive isopycnal and diapycnal values. While the scalingwith the Ozmidov wave number has often been usedin oceanographic literature, the QNSE theory demon-strates its fundamental role from nearly first principles.

In the limit of weak stable stratification, k/k0 � 1,the QNSE theory yields analytical expressions for theisopycnal and diapycnal viscosities and diffusivities(Sukoriansky et al. 2005b),

νh/νn = 1 + 0.38 (k/k0)−4/3, (6)

νz/νn = 1 − 1.24 (k/k0)−4/3, (7)

κh/νn = α + 0.22 (k/k0)−4/3, (8)

κz/νn = α − 1.6 (k/k0)−4/3, (9)

where α−1 = Prt0 � 0.72 is the turbulent Prandtl num-ber for neutral flows. These equations show that sta-ble stratification affects the isopycnal and diapyc-nal mixing: the diapycnal viscosity and diffusivity aresuppressed compared to their values in the neutralcase while their isopycnal counterparts are enhanced.When the stratification effect becomes strong, k/k0 →1, these equations cannot be used and full solutionsare necessary. These solutions are shown in Fig. 1.Evidently, the tendencies exhibited by the simplifiedEqs. 6–9 are preserved in the full solution. Due to theeffect of the breaking internal waves fully accounted forin the QNSE formalism, the diapycnal viscosity remainsfinite even in very strong stratification while the diapyc-nal diffusivity may become very small. As shown in theAppendix, the vertical dashed line in Fig. 1 correspondsto the length scale of the threshold of internal waveradiation which turns out to closely coincide with thebeginning of flow anisotropization.

The integration of the system of equations for eddyviscosities and eddy diffusivities can be extended to anarbitrary wave number �. If π�−1 is smaller than theturbulence macroscale, πk−1

L , then the eddy viscositiesand eddy diffusivities can be used as subgridscale pa-rameters in large-eddy simulations (LES) where thegrid resolution is π�−1. By extending the integra-tion to kL, one eliminates all fluctuating modes fromthe governing equations and arrives at the Reynolds-averaged Navier–Stokes (RANS) description of thesystem (Sukoriansky et al. 2005b, 2006).

0.01 0.1 1 10 1000

0.5

1

1.5

2

2.5

Fig. 1 Normalized isopycnal and diapycnal viscosities and diffu-sivities as functions of k/k0. Dashed vertical line shows the thresh-old of internal wave generation in the presence of turbulence; itcoincides with the onset of anisotropization of the viscosity anddiffusivity

3 Kinetic energy spectra

The analytical results obtained for the case of weakstratification can be utilized to derive various spectra.Due to the anisotropy, the traditional 3D kinetic en-ergy spectrum provides only limited information aboutthe flow energetics. More detailed information canbe gained from various one-dimensional (1D) spectra.Here, we shall consider one of them, probably the mostoften discussed, vertical spectrum of the horizontalvelocity (Sukoriansky et al. 2005b),

E1(k3) = 4π

∫ ∞

0

∫ ∞

0

dk1dk2

(2π)2

∫ ∞

−∞dω

2πU11(ω, k)

= 0.626 ε2/3k−5/33 + CB N2k−3

3 , (10)

where U11(ω, k) is the horizontal component of thespectral tensor, Eq. 47, and CB = 0.214. Equation 10reveals the transition from the Kolmogorov spectralslope of −5/3 to the slope of −3 stipulated by stablestratification. This transition is associated with the hor-izontal layering of the flow and formation of the slowmanifold (Bretherton 1969; Smith and Waleffe 2002;Fritts et al. 2003)). This transition takes place on scalesof the order of the Ozmidov scale. The numerical valueof the coefficient, CB = 0.214, is in agreement withthe direct numerical simulations (Bouruet-Aubertotet al. 1996) (CB � 0.1), LES (Carnevale et al. 2001)

1324 Ocean Dynamics (2010) 60:1319–1337

(CB � 0.2), and laboratory experiment (Benielli andSommeria 1996) (CB � 0.2); see also a review (Staquet2007).

Equation 10 agrees with and quantifies the pioneer-ing measurements by Shur (1962) in the atmosphericboundary layer. These measurements pointed to the−5/3 to −3 transition in the kinetic energy spectrumthat takes place around the Ozmidov scale. Shur’s workstimulated the rise of early theories of stably strati-fied turbulence (Lumley 1964, 1965; Phillips 1965;Weinstock 1985; Holloway 1986a).

The spectrum Eq. 10 has important implicationsnot only for the boundary layer but for the entireatmosphere. VanZandt (1982) noticed that the ob-served vertical spectra of the horizontal velocity in theatmosphere have a tendency to develop a universaldistribution for all seasons, meteorological conditions,and geographical locations throughout the atmosphere.This distribution has been referred to as a canonicalgravity wave spectrum (Fritts and Alexander 2003) asits origin has often been attributed to interacting in-ternal gravity waves (see e.g. Dewan 1979; Dewan andGood 1986; Fritts and Alexander 2003, and referencestherein).

Figure 2 compares the QNSE-based spectrum Eq. 10with the spectral data throughout the troposphere,stratosphere, mesosphere and thermosphere along withthe canonical spectrum E1(k3) = 0.5N2k−3

3 as pre-sented in Smith et al. (1987). Some papers (Tsuda et al.1989; Fritts and Alexander 2003; Janches et al. 2006)and references therein provide extensive observationalsupport for the vertical spectrum in the form of Eq. 10with CB = 1/6 throughout the atmosphere.

Other studies related the canonical spectrum to thebuoyancy range turbulence (Weinstock 1985; Sidi et al.1988). Among the existing theories, the QNSE is theonly one capable of representing turbulence, internalwaves and the transitional range in a unified frame-work. The observed universality of the vertical energyspectrum and its quantitative agreement with theQNSE expression 10 point to the fundamental charac-ter of this result.

Diverse views of the medium combining turbulenceand waves should not be surprising. As will be elab-orated later, it is indeed difficult to classify wave-turbulence composite as either solely turbulence orsolely waves on any scale. The peculiar properties ofthe wave-turbulence interaction will be discussed inSection 4.1 by comparing their effects on the transportof momentum and tracers.

The universality of the vertical spectrum Eq. 10 canalso be explored for oceanic flows where the verticalscales dominated by turbulence or waves can be re-

Fig. 2 Vertical spectra of the horizontal velocity for differentregions of the Earth atmosphere (after Smith et al. 1987). Thethin dashed line represents the equation, 0.5N2k−3

3 , suggested inSmith et al. (1987). The thick line is the QNSE Eq. 10. Sincethe spectrum uses dimensional variables, it needs to be adjustedto accommodate different values of N in different regions. Thetheoretical lines use N = 0.0033 cps typical of the stratosphereand the mesosphere. As suggested in Smith et al. (1987), the datalines need to be elevated by a factor of 4 for the troposphere andlowered by a factor of 2 for the thermosphere

solved. In the oceanographic literature, the attention isusually focused on the spectrum of the vertical shear,ES(k3) = k2

3 E1(k3). In a pioneering paper, Gargettet al. (1981) suggested a universal or canonical (Polzinet al. 1995) scaling of the vertical shear in the oceanicmixed layer,

ES(x)

EB= F(x), x ≡ k3/k0, (11)

where EB = (εN)1/2 and F(x) is a universal function.A fundamental difference between the Gargett et al.(1981) spectrum and the one obtained for the atmo-sphere is that the former includes waves, small-scaleturbulence and the transitional range while the latterdeals solely with waves. Gargett et al. (1981) foundthat ‘in the intermediate-wavenumber range...the col-lapse of the observations to a single curve [F(x)] over

Ocean Dynamics (2010) 60:1319–1337 1325

the range 0.1 < k3/k0 < 1 is remarkable...’ Later, thesame scaling was verified in Gregg et al. (1993) usingPATCHEX and PATCHEX north data sets. Theseauthors noticed that ‘Gargett et al. (1981) are uncer-tain whether the spectral collapse effected by theirbuoyancy scaling demonstrates the existence of theShur–Lumley buoyancy subrange or whether it is sim-ply fortuitous...The buoyancy scaling used by Gargettet al. (1981) does collapse their observations...thesame scaling is applied to PATCHEX and PATCHEXnorth...The results are dramatic, achieving a bettercollapse than obtained by Gargett et al. The collapseextends across the internal wave range...If this collapseis not also fortuitous, the scaling must represent in-ternal wave dynamics rather than buoyancy-modifiedturbulence...’

Using Eq. 10, the QNSE theory provides the follow-ing expression for the vertical shear spectrum,

ES(k3/k0)

EB= 2k2

3 E1(k3)

(εN)1/2 = F(k3/k0)

= 1.252(k3/k0)1/3 [

1 + 0.34(k3/k0)−4/3] .

(12)

The coefficient 2 accounts for the contribution fromtwo horizontal components. By confirming and quanti-fying the empirical scaling (11) via Eq. 12, the QNSEtheory demonstrates that Gargett et al. scaling isnot fortuitous; it faithfully reflects the physics of theturbulence—internal wave interaction.

Figure 3 compares Eq. 12 with the observationaldata by Gregg et al. (1993). Although all data pointscollapse well on one line for k3/k0 < 1, there is adiscrepancy between the observations for k3/k0 > 1.The theoretical expression 12 is in very good agree-ment with the data from PATCHEX north while forPATCHEX, the agreement is good only for k3/k0 ≤ 1.The difference can be understood if the strength ofturbulence, measured by the magnitude of the buoy-ancy Reynolds number, Reb = ε/(ν0 N2), is taken intoaccount (Phillips 1991). Note that Reb , also knownas the isotropy index (Thorpe 2005), characterizesthe distance between the Kolmogorov viscous dissi-pation and Ozmidov wave numbers, Reb = (kd/k0)

4/3.Using the data from Gregg et al. (1993), one findsthat for PATCHEX, Reb � 14 and kd/k0 � 7, whilefor PATCHEX north, Reb � 125 and kd/k0 � 37. It isself-evident that for the PATCHEX data, the inertialsubrange of isotropic 3D turbulence was practicallyabsent while for the data from PATCHEX north thissubrange was well developed.

The remarkable universality of the vertical energyspectrum throughout the atmosphere and the ocean

ES

/EB

10-2

10-1

100

101

100

101

k3/ko

PATCHEX north 2.5-5.5 MPa

PATCHEX 2.5-5.5 MPa

10-2

10-1

100

101

100

101

ES

/EB

k3/ko

PATCHEX north 5.75-9.25 MPa

PATCHEX 5.75-9.25 MPa

Fig. 3 Comparison of the measured spectrum of the verticalshear in the ocean with the theoretical QNSE expression 12(grey thick line). The data for the shallow (top) and deep (bot-tom) PATCHEX and PATCHEX north is from Gregg et al.(1993). Note that for k3/k0 > 1 the PATCHEX data was stronglyaffected by the molecular viscosity

suggests that both media are governed by the samephysical laws. Figure 3 recast for the vertical energyspectrum demonstrates that the transition from the−5/3 to −3 spectral slope is not sharp but takes placeover some range of scales. For k/k0 < 0.5, the −3 slopebecomes well established. Echoing the meteorologicalliterature, Gregg et al. (1993) attributed this spectrumto the internal wave dynamics. This assumption needsto be reconciled with the fact that the vertical spec-trum scales with N2 although the internal gravity wavescannot propagate in the vertical direction. Indeed, ac-cording to Eq. 49 the wave frequency vanishes in thatdirection and so N does not appear to be an appropriatescaling parameter for the 1D vertical spectrum. Thissomewhat counter-intuitive result resembles similar be-havior in a β-plane turbulence where the spectrumsteepens and scales with β2 in the zonal direction along

1326 Ocean Dynamics (2010) 60:1319–1337

which the β-term vanishes thus precluding Rossby wavepropagation (Chekhlov et al. 1996) (recall that β is themeridional gradient of the Coriolis parameter). Bothresults are direct products of turbulence anisotropiza-tion by the extra strains that support respective linearwaves. This anisotropization can be explained withinthe theory of triad interactions, e.g., Holloway (1979,1986b) for internal waves and Rhines (1975), Hollowayand Hendershott (1977), Huang et al. (2001), Lee andSmith (2007) for Rossby waves. The necessity to satisfythe frequency resonance condition in addition to thewave number resonance hampers the efficiency of thetriad interactions and reduces their ability to transferenergy across the spectrum. As a result, the energy pilesup and 1D spectra steepen in the directions of zerofrequencies. Those directions are kx = 0 and kh = 0 fora β-plane and stably stratified turbulence, respectively.Even though the waves do not propagate in thesedirections, the changes in the spectral behavior arefacilitated by the interaction between waves and turbu-lence. It is now clear that as β2k−5

y is not a spectrumof Rossby waves in β-plane turbulence (Galperin et al.2010), N2k−3

3 is not a spectrum of gravity waves instably stratified turbulence. These spectra correspondto the slow manifolds developing in the zero frequencydirections. They reveal themselves as a system of alter-nating zonal jets or of vertically decorrelated horizontallayers in a β-plane or stably stratified turbulence, re-spectively. The effect of turbulence-wave interactionsand changes in the spectral characteristics upon thetransport processes in the case of strong stratificationare considered in the next section.

4 The limit of strong stable stratification

4.1 The Ellison–Britter–Osborn model

Figure 1 can be used to establish the asymptotic expres-sions for νh, νz and κh in the case of strong stratification,i.e., k/k0 � 1,

νh � 1.3νn � 0.6ε1/3k−4/3, (13)

νz � 0.2νn � 0.1ε1/3k−4/3, (14)

κh � 2.3νn � 1.1ε1/3k−4/3. (15)

The asymptotic behavior of the diapycnal diffusivityis different from Eqs. 13–15 and can be deter-mined using dimensional analysis. From the governingEqs. 1–3 one deduces that in horizontally homogeneousstratified flows, κz can only depend on k, ε and N. In

the case of strong stratification, the QNSE asymptoticsindicates that κz becomes k-independent (Sukorianskyet al. 2005b) yielding

κz = �εN−2, (16)

where the ‘mixing’ coefficient � needs to be deter-mined. Equation 16 recovers the EBO model (Osborn1980; McIntyre 1989; Davis 1994 and Lindborg andBrethouwer 2008).

4.2 The Richardson diffusion law format

The Richardson diffusion laws (Lesieur 1997; Salazarand Collins 2009) offer a powerful tool to char-acterize the transport of momentum and scalar instrongly stratified turbulence. We recall that in neu-tral, isotropic, homogeneous 3D turbulence, the QNSEexpressions for the eddy viscosity and eddy diffusivityat a scale k−1 are (Sukoriansky et al. 2005b)

νn = 0.46ε1/3k−4/3, κn = 0.64ε1/3k−4/3. (17)

These equations represent the classical Richardsondiffusion laws for momentum and heat and suggest thatνn and κn are the effective viscosity and diffusivity thataccumulated at a scale k−1 as a result of contributionsfrom all eddies smaller than k−1.

In strongly stratified, anisotropic turbulence, theKolmogorov −5/3 scaling is preserved for the hor-izontal spectra of the kinetic and potential energies(Riley and Lindborg 2008). This scaling leads to theRichardson diffusion laws for the isopycnal mixingwhich is indeed reflected by Eqs. 13 and 15. The diapy-cnal viscosity is significantly reduced compared to theneutral case but preserves its Richardson law nature.The asymptotic expression for the diapycnal diffusivity(16) can be rearranged in a form congruent with theRichardson diffusion law,

κz = �ε1/3k−4/30 . (18)

By comparing Eq. 14 with Eq. 18 one infers that whileall scales contribute to the diapycnal viscosity, only thescales up to a fraction of k−1

0 contribute to the diapycnaldiffusivity. This result provides a new interpretation ofthe EBO model. The consequence of the Eqs. 14 and 18is the growth of the vertical turbulent Prandtl number,Prt = νz/κz, with the increasing stratification. Observa-tional, laboratory and numerical data provide qualita-tive and quantitative support for this result (Deardorff1973; Mauritsen and Svensson 2007; Zilitinkevich et al.2008; Huq and Stewart 2008; Yagüe et al. 2001, 2006).

Ocean Dynamics (2010) 60:1319–1337 1327

Equations 14 and 18 quantify the rate of growth of Prt

in terms of k, Prt ∝ (k0/k)4/3, as k → 0.To quantify the mixing coefficient � in Eqs. 16 and

18, we assume the balance between the turbulenceenergy production, P = νzS2, S being the mean shear;buoyancy destruction, B = κz N2, and the dissipation,ε, P = B + ε, and invoke the flux Richardson num-ber, R f = B/P (Pardyjak et al. 2002). It follows fromEq. 16 that B = �ε giving R f = �/(� + 1). At strongstratification the QNSE theory predicts R f in the rangebetween 0.5 and 0.3 (Sukoriansky et al. 2006) whichyields � between 1 and 0.4. These values are somewhatlarger than the widely accepted in the oceanographicliterature estimate of 0.2 (Osborn 1980; Gregg et al.2003; Wunsch and Ferrari 2004; Palmer et al. 2008).However, they are well in line with the atmosphericdata where � was found in the range between 0.33 and1 (Lilly et al. 1974; Pardyjak et al. 2002; Hermawan andTsuda 1999; Bishop et al. 2004; Clayson and Kantha2008). Note that � derived from observations may dif-fer from these values due to, e.g., contribution ofdouble diffusion processes (Canuto et al. 2008a) orlateral stirring by mesoscale fluctuations (Smith andFerrari 2009). Some studies even suggest that � is nota constant at all and its value may vary by an order ofmagnitude (e.g., McIntyre 1989).

Comparison of the expressions 17 and 18 suggeststhat the upper limit of scales that contribute to κz

nearly coincides with the scale (0.5k0)−1 at which the

vertical energy spectrum attains the slope N2k−33 as-

sociated with the buoyancy dominated spectral range.As was elaborated in the previous section, this spec-trum cannot be directly attributed to internal wavesbecause waves do not propagate in the vertical direc-tion. Fluid particles, on the other hand, can travel inthe vertical, and they are engaged in ascending anddescending excursions under the action of the buoy-ancy force. On scales smaller than (0.5k0)

−1 turbulentoverturn is possible such that those scales can con-tribute to irreversible mixing of both momentum andscalar. Larger excursions are wave-like and thus theycannot lead to irreversible mixing of the scalar. Onecould think of these excursions as of the product ofthe internal wave elasticity, the notion similar to theRossby wave elasticity (Baldwin et al. 2007) for theRossby waves. Fluid particles carry momentum and canstill cause irreversible momentum exchange on scalesexceeding (0.5k0)

−1. Note that as long as the Ozmidovwave number remains smaller than the Kolmogorovdissipation wave number kd, there will be vertical tur-bulent mixing and the vertical layers do not becomemixing barriers. Only when k0 → kd, or the buoyancyReynolds number, Reb → 1, vertical turbulent scalar

mixing degenerates. The horizontal mixing, however, isnot affected by the vertical particle excursions and canstill be substantial.

4.3 Diapycnal and isopycnal viscositiesand diffusivities and the Okubo diagrams

The scale-independence of κz in strongly stratifiedflows with nearly constant ε is an important result whichcan be verified against data. Studies of the mixing inthe oceanic pycnocline by Ledwell et al. (1998) indicatethat the diapycnal diffusivity remains approximatelyconstant and largely scale independent on the timescales from 6 to 24 months even though the verticalspread of the tracer during that time was considerable.From the physical viewpoint, since the vertical diffusionis carried out within turbulent patches dominated by3D overturning turbulence, the scale of these patchesrather than the vertical spread of the tracer is a per-tinent scale for the eddy diffusivity. Taking the typi-cal value of the rate of the viscous dissipation in thepycnocline at 10−9 m2s−3, typical N2 at 10−4 s−2 andusing Eq. 18, one estimates κz in the pycnocline atO(10−5) m2s−1. The estimate of κz in Ledwell et al.(1998) was close, about 0.15 × 10−4 m2s−1.

The observations by Ledwell et al. (1998) can alsobe used to quantify the QNSE results for the isopycnaldiffusivity. Note that unlike the scale-independentdiapycnal diffusivity, its isopycnal counterpart was ob-served to be substantially scale-dependent. This factis consistent with Eq. 15. In the observations, κh wasfound to increase from 2 to 103 m2s−1 on scales fromabout 1 to about 300 km. These numbers are in goodquantitative agreement with Eq. 15 assuming, again,that ε � 10−9 m2s−3. As mentioned earlier, however,the results on very large scales need to be taken withcaution because the processes there are affected by theplanetary rotation and a β-effect, the factors excludedin the present analysis.

Extensive studies of the relative dispersion in theocean were conducted by Okubo (Okubo 1971; Ikawaet al. 1998). The well-known Okubo diagrams show thedependence of the horizontal diffusivity coefficient onthe diffusion length scale and demonstrate that it obeysthe Richardson diffusion law over several decadesalbeit with variable amplitude. Okubo noticed thatsince the dissipation rate, ε, enters the Richardson lawwith the power of 1/3, large variations in ε would yieldonly moderate changes in the horizontal diffusivitycoefficient.

Regarding the background physics of the Okubo dia-grams, Garrett (2006) remarked, ‘Okubo (1971) foundthat much oceanographic data on relative dispersion

1328 Ocean Dynamics (2010) 60:1319–1337

does seem to obey (Richardson) law, but this mustbe for other (than 3D turbulence) reasons, as theassumption of homogeneous 3D turbulence is hardlyvalid over the large scales analyzed.’ Equation 15partially resolves this conundrum as it demonstratesthat the isopycnal diffusion in strongly anisotropic sta-bly stratified turbulence does obey the Richardson law.The quantification of this agreement is more difficultbecause ε is unknown. Another limitation of the pre-sented here analysis is in that it does not account forthe effect of rotation and so it only applies to a limitedrange of scales. Note, however, that, firstly, a study byRiley and Lindborg (2008) reports on the Kolmogorov−5/3 spectrum in the horizontal direction (which is aprerequisite for the Okubo diagrams) on scales affectedby both stratification and rotation and, secondly, asdiscussed in Galperin et al. (2010), on scales affectedby a β-effect, the meridional diffusivity also obeys theRichardson law.

It is of considerable practical interest to test theexpressions for the horizontal viscosity and diffusivity,Eqs. 13 and 15, in numerical models of oceanic cir-culation. For instance, in a study by Wallcraft et al.(2005) it was determined that a factor of 2 increasein the horizontal resolution warrants decrease in thehorizontal eddy viscosity by a factor of 3 while inHogan and Hurlburt (2000) this factor was appraisedbetween 2 and 3. According to Eq. 13, the horizontaleddy viscosity can be described by the Richardson lawsuch that the decrease of a scale by a factor of 2 yieldsdecrease in the eddy viscosity by a factor of 24/3 � 2.5,in good agreement with the findings in both Hogan andHurlburt (2000) and Wallcraft et al. (2005). A possibleimpact of the value of ε on these results is reduced dueto the relatively weak sensitivity of Eqs. 13 and 15 tothat parameter.

5 Potential energy spectra

The QNSE theory can also be used to derive spectra ofthe turbulent potential energy defined as

Ep(k) = (αg)2

N2 ET(k), (19)

where ET(k) is the spectrum of the temperature vari-ance, 1

2 T2. Using the Langevin equation for the tem-perature, Eq. 36, it can be shown (Yakhot and Orszag1986) that when the temperature is a passive scalar,the temperature variance obeys the Corrsin–Obukhovspectrum,

ET(k) = CCOε−1/3εθk−5/3, (20)

where εθ is the dissipation rate of 12 T2, and CCO = 1.03

is the Corrsin–Obukhov constant. The value of thisconstant is in fair agreement with laboratory measure-ments and direct numerical simulations which showsome spread but point to CCO being about 1 (Watanabeand Gotoh 2004; Yeung et al. 2002).

In the case of weak stratification, Eqs. 6–9 can beused to estimate the vertical temperature spectrumdefined similarly to Eq. 10 but with the appropriatecorrelation function,

ET(k3) = CTεθε−1/3k−5/3

3

[1 + CN1

(k3

k0

)−4/3]

+ CN2

(d�

dz

)2

k−33 , (21)

where CT = 0.62, CN1 = 0.406 and CN2 = 0.134. Fullderivation of this result is beyond the scope of thispaper and will be presented elsewhere.

Even though the derivation of Eq. 21 implies k/k0 >

1, it can be extrapolated to the range of strongstratification, k/k0 < 1, where the Corrsin–Obukhovterm is small compared with the other two terms. Notethat the exploration of theories beyond their range ofapplicability is customary in physics and was useful inearlier studies of the vertical shear spectrum.

One can introduce the potential energy dissipationrate, εp = (αg/N)2εθ , and, using Eqs. 19–21, derive ascaling for the spectrum of the ‘vertical temperatureshear’ in the same format as the Gargett et al. (1981)spectrum for the vertical shear of the horizontal veloc-ity, Eq. 12,

ESp(k3/k0)

EB= k2

3 Ep(k3)

(εN)1/2

εp

ε= G(k3/k0)

= CTεp

ε

(k3

k0

)1/3 [1 + CN1

(k3

k0

)−4/3]

+ CN2

(k3

k0

)−1

. (22)

In the case of strong stratification, using the expres-sion for εθ ,

κz

(d�

dz

)2

= εθ , (23)

valid for horizontally homogeneous fluids (Lilly et al.1974; Galperin et al. 1989; Wilson 2004), and the EBOmodel, Eq. 16, one derives

εp

ε= �, (24)

where the mixing coefficient � and its range of variationare described in the previous section. It can be shown

Ocean Dynamics (2010) 60:1319–1337 1329

that this result is equivalent to the linear dependencyof the vertical turbulent Prandtl number, Prt = νz/κz,on the gradient Richardson number, Ri, at large Ri. Asusual, Ri is defined as Ri = N2/S2. As a corollary, theanalogue of the Ozmidov wave number for tempera-ture fluctuations, kθ = (N3/εp)

1/2, is not independentof k0 in this limit as according to Eq. 24, kθ = �−1/2k0.Furthermore, for � = 0.4 the function G(x) in Eq. 22becomes

G(x) = 0.25x1/3(1 + 0.95x−4/3), (25)

and kθ = 1.58k0. Note that the value � = 0.4 in Eq. 24is consistent with that found in direct numerical simu-lations by Riley and deBruynKops (2003). Various ex-pressions for the temperature spectra were developedpreviously (e.g., Weinstock 1985; Dalaudier and Sidi1987) but they ignored the anisotropy. Present deriva-tions mark significant step forward over these earlystudies.

The transition from the Corrsin–Obukhov to thebuoyancy dominated, N2k−3

3 spectral range takes placeat about the same scales as in the case of the kineticenergy spectrum (10). In strong stratification, the k−3

3branch of the spectrum (21) prevails and one obtains

Ep(k3) = γ N2k−33 , (26)

where γ = CTCN1� + CN2 is equal to 0.2 for � = 0.4.Numerical simulations reported in Bouruet-Aubertotet al. (1996) gave Ep(k3) = 0.2N2k−3

3 , in good agree-ment with (26). Stratospheric observations generallyconfirm the scaling (26) but suggest that the propor-tionality coefficient is about 0.1 (Gurvich and Kan 2003;Sofieva et al. 2007, 2009). This scaling indicates thatthe vertical spectra of the horizontal velocity and ofthe temperature variance are proportional on scaleswhere stratification is strong. Despite observationaldifficulties, various atmospheric data generally sup-port this proportionality. Cot (2001) mentions largefluctuations of Ep(k3) but nevertheless gives an esti-mate of the ratio of the 1D kinetic to potential energyspectra to be about 2.5–3 in the troposphere and thestratosphere. The QNSE Eqs. 10 and 26 yield a ratio of2 which is consistent with this range (note that E1(k3)

needs to be doubled to account for the contributionfrom both horizontal components).

6 The absence of the critical Richardson number

The dispersion relation (49) indicates that the internalwaves populating large scales become practically in-

distinguishable from the linear waves. The flow itself,however, remains far from laminar if judged by theisopycnal and diapycnal transfers of momentum andheat which are strongly anisotropic. In addition, thedifferent asymptotic behaviors of νz and κz reflect thefact that the waves can mix momentum but not a scalar.

In terms of the RSM, these results provide anevidence of the absence of the critical gradientRichardson number, Ricr, associated with completesuppression of turbulence and flow laminarization instrong stratification (Galperin et al. 2007). This con-clusion agrees with abundant data. To name only afew examples, Izakov (2007) mentions that ‘direct mea-surements of turbulent pulsations of temperature andwind speed...show that turbulent pulsations are alwayspresent in the troposphere and lower stratosphere...’By analyzing turbulence variability in very stable at-mospheric boundary layers, Mahrt (2010) concludesthat ‘the turbulence never vanishes at large Richardsonnumber partly because of shear generation by the sub-meso motions.’ Mack and Schoeberlein (2004) were un-able to identify Ricr in the oceanic data. In simulationsby Fritts et al. (2009a) it was found that ‘...an initialRi ≥ 1/4 is not a reliable predictor of the absence ofturbulence in flows exhibiting significant scale inter-actions.’ Models with no critical Richardson numberwere discussed by Savijärvi (2009). The absence of Ricr

has been adopted in recent RSM (Zilitinkevich et al.2007; Canuto et al. 2008b; Violeau 2009; Alexakis 2009;Kantha and Carniel 2009). Note that in order to elim-inate Ricr, one of the constant coefficients in RSMwas made ad hoc Ri-dependent (Canuto et al. 2008b;Kantha and Carniel 2009). This modification was in-troduced a posteriori, in order to comply with the dataand other results, while in QNSE, the absence of Ricr isone of the intrinsic features of the theory stipulated byexplicit accounting for the effect of internal waves andflow anisotropy (Galperin et al. 2007).

Note also that Ri is a characteristic of the verticalgradients of the mean fields which carries no infor-mation on the molecular processes. As discussed inSection 3, the buoyancy Reynolds number, Reb , pro-vides a suitable characterization of the molecularprocesses and should be used in flows where moleculareffects are important. Both numbers complement eachother and their combination provides more completecharacterization of stably stratified turbulence than itcould be achieved by using either of them separately.Consistent with this approach, some recent researchalso indicates that turbulent mixing is never com-pletely quashed, even under very strong stratification,for as long as Reb � 1 (Billant and Chomaz 2001;Brethouwer et al. 2007).

1330 Ocean Dynamics (2010) 60:1319–1337

7 Conclusions

This paper provides a succinct presentation of a quasi-normal scale elimination theory of turbulence anddiscusses its application to stably stratified flows. QNSEis a spectral theory maximally proximate to first princi-ples and empirical constants free. The main tool of theQNSE theory, the algorithm of successive eliminationof small scales, makes it possible to properly account forvarious factors affecting turbulence on different scales.In the case of stable stratification, among such factorsare internal gravity waves.

How does the QNSE theory compare with RSM?The latter has been traditionally used to account forthe small-scale turbulent mixing in the vertical inmost oceanic and atmospheric circulation models. Thismethod employs the concept of eddy viscosity and eddydiffusivity denoted as KM and KH and given by theexpressions KM = qlSM and KH = qlSH , respectively.Here, q2 is twice the turbulence kinetic energy (TKE), lis the turbulence macroscale, and SM and SH are thenon-dimensional stability functions. These functionsare calculated using certain closure assumptions for tur-bulence correlations. Most popular sets of such assump-tions belong either in the Mellor and Yamada (Mellorand Yamada 1982; Galperin et al. 1988; Kantha andClayson 1994) or Launder, Reece and Rodi (Launderet al. 1975; Cheng et al. 2002) frameworks or their com-bination (Kurbatskiy and Kurbatskaya 2009). Eitherframework employs the Reynolds decomposition of thegoverning equations and their ensemble averaging thatlead to equations for the Reynolds stresses, turbulentheat fluxes and turbulent heat variance decay, uiu j, uiθ

and θ2, respectively. The full set of the Reynolds stressequations includes ten prognostic equations which inturn include many more unknown correlations. Theprocedure of the ensemble averaging is scale indepen-dent and lumps together all processes on all scalesthus making it impossible for RSM to discern betweenturbulence and waves unless additional assumptionsare introduced. The most important—and most difficultto model—are the correlations involving the pressurebecause it is a non-local variable. Finally, the closure as-sumptions involve empirical constants that are assumedinvariant in different types of flows (Lewellen 1977)although in some recent models one of the constantswas allowed to become a function of Ri in order toaccommodate the absence of Ricr (Canuto et al. 2008b;Kantha and Carniel 2009). The resulting equations arestill complicated and usually simplified in such a way asto obtain algebraic equations for SM and SH . A popu-lar scheme for such simplification is the Mellor and

Yamada hierarchy of models in terms of the departureof the Reynolds stress tensor, uiu j, and the turbulentheat flux vector, uiT, from the isotropy (Mellor andYamada 1974, 1977, 1982). The scheme is comple-mented by two additional prognostic or diagnosticequations, for TKE or q and l, q2l (Mellor and Yamada1982) or ε. Due to the absence of a simple conservationlaw, the equation for the length scale, l, or the dissipa-tion rate, ε, is usually empirical.

The QNSE theory, on the other hand, operates withthe governing equations which are simpler than theReynolds stress equations. As shown in the Appendix,the pressure term in the spectral representation can besolved for and excluded from the equations exactly. Infact, the main results pertaining to the effect of internalwaves on turbulent flow field can be obtained alreadyfrom the consideration of the pressure term. Further-more, to achieve a coarse-grain description of the sys-tem, the QNSE theory uses an algorithm of successiveensemble averaging over small shells of high wave num-ber modes. This is a scale-dependent procedure thataccounts for the physical processes (such as waves) onthe eliminated scales. Strictly speaking, QNSE providesan algorithm for subgridscale parameterization in LESas it produces eddy viscosities and eddy diffusivitiesthat depend on the subgridscale dissipation and gridresolution. By extending the scale elimination to the in-tegral macroscale one derives the QNSE equivalent ofan RSM for the eddy viscosity and eddy diffusivity. Inthis format, the QNSE theory provides expressions forthe vertical and horizontal stability functions while stillrequiring additional equations or closure assumptionsfor l and ε, similarly to a practice in RSM. Along withthe scale-dependent physics of the vertical mixing, theQNSE model accounts for turbulence anisotropizationby producing, in addition to the vertical eddy viscosityand eddy diffusivity, their horizontal counterparts. De-spite the complexity of the intermediate QNSE deriva-tions, the final equations for the stability functions canbe approximated by simple fraction-polynomial equa-tions and used in practical applications. Such expres-sions have already been implemented and tested in var-ious planetary boundary layer and weather predictionmodels (Sukoriansky and Galperin 2008; Sukorianskyet al. 2005a, 2006). The QNSE-based RANS modelrecently became a part of the standard physics packageof the Weather Research & Forecasting (WRF) model(http://www.wrf-model.org/index.php).

Being a spectral theory, QNSE provides an accessto the spectral structure of the flow field. The theoryyields the dispersion relation for internal waves inthe presence of turbulence and generates expressions

Ocean Dynamics (2010) 60:1319–1337 1331

for various 3D and 1D spectra. It supports the earlyShur–Lumley model of stably stratified turbulenceand details the transition from isotropic to buoyancy-dominated regimes of turbulence. In addition, it gen-erates analytical expressions for the vertical spectraof the kinetic energy, vertical shear and temperaturevariance, and reveals the fundamental nature of theGargett et al. universal scaling of the composite verticalshear spectrum. It also provides a similar scaling forthe ‘vertical shear’ of the potential temperature. Thevertical spectra obtained with QNSE are in good agree-ment with oceanic and atmospheric data and othertheories. The QNSE results suggest that the theoreticalexplanation of the universality of the large-scale spectrastems from interactions between anisotropic turbulenceand internal waves rather than the wave interactionsalone.

The QNSE theory is not free of limitations. As aspectral theory, it assumes spatial homogeneity whichseldom occurs in the natural environment. In practicalsimulations, the theory is applied locally assuming localhomogeneity. The direct dynamic effect of shear onturbulent fluctuations is not included in this theory;rather, it is introduced via the TKE balance equation.As a result, the theory may become somewhat inaccu-rate in flows with a very strong shear. In principle, thederivations could include the shear directly, similarly tothe way they included the temperature gradient. This isan interesting task which is left to the future research.Among other promising directions of research wouldbe the inclusion of the Coriolis parameter and studies ofthe combined effect of stable stratification and rotation.So far, among the most significant achievements of thetheory were the analytical representation of the transi-tion from neutral to stably stratified turbulence, explicitdemonstration of the turbulent mixing anisotropizationand its quantification under strong stratification, andthe evidence of the absence of Ricr.

Finally, the QNSE theory provides a new look atdiapycnal and isopycnal mixing in geophysical systemsthrough the prism of the Richardson diffusion. CitingSalazar and Collins (2009), ‘the classical frameworkcreated by Richardson (1926), placed in the contextof (Kolmogorov theory) K41 by Obukhov (1941), andextended and clarified by Batchelor (1950) has heldup remarkably well over time. Nothing has effectivelychallenged the existence of the Richardson–Obukhovregime. Similarly, there has not been an experi-ment that has unequivocally confirmed Richardson–Obukhov scaling over a broad enough range of timeand with sufficient accuracy.’ In this paper we demon-strate that the QNSE theory is not only consistent with

the Richardson diffusion laws in neutral stratificationbut also allows one to extend the Richardson frame-work to anisotropic turbulent flows with internal waveson scales exceeding the Ozmidov scale. The analysisof theoretical, observational, laboratory and numeri-cal data reveals that only quasi-isotropic, turbulencedominated scales contribute to the diapycnal diffusivitywhile on larger scales it becomes scale-independent.The Richardson diffusion format (18) or, equivalently,the EBO model provide an elegant quantification ofthis result. On the other hand, the isopycnal diffusivity,as well as the isopycnal and diapycnal viscosities, whilebeing affected by internal waves on large scales, pre-serve, nevertheless, their Richardson law-like charac-ter, albeit with different amplitudes. The underlyingphysical explanation for different behaviors of the di-apycnal viscosity and diffusivity stems from the abilityof waves to transport momentum but not a scalar.

It is well possible that the persistence of theRichardson–Obukhov diffusion laws in various naturalenvironments with anisotropic turbulence and disper-sive waves stands behind the Salazar and Collins’s(2009) assertion that nothing has challenged the ex-istence of these laws over the years. On the otherhand, the changing values of the coefficients in theselaws in anisotropic flows with different types of extrastrains could possibly explain why there have not beenexperiments or observations that could unequivocallyconfirm the Richardson–Obukhov scaling over a broadrange of temporal and spatial scales with sufficientaccuracy.

As a concluding remark we note that the QNSEis a relatively new theory which is being extensivelytested in various oceanic and atmospheric applications.It has produced encouraging theoretical and practicalresults. Testing of new models is an elaborate andlong process which is only in its midst for the QNSEtheory. We expect that one of the outcomes of thisprocess will be a new generation of reliable and use-ful turbulence models that present a viable alternativeto RSM.

Acknowledgements The authors thank Michael McIntyre forilluminating discussions on the roles of waves and turbulencein mixing processes in the atmosphere and the ocean. He alsobrought to our attention that the expression 16 most often knownas the Osborn model was established earlier by Ellison andBritter and should in fact be referred to as the Ellison–Britter–Osborn model.

Partial support of this research by the ARO grants W911NF-05-1-0055 and W911NF-09-1-0018, ONR grant N00014-07-1-1065, and the Israel Science Foundation grant No. 134/03 isgratefully appreciated.

1332 Ocean Dynamics (2010) 60:1319–1337

Appendix: Basics of the QNSE theory of stablystratified turbulence

The space and time Fourier decompositions of thevelocity and temperature are

ui(x, t) = 1(2π)4

∫

k≤kd

dk∫

dω ui(k) exp[i(kx − ωt)], (27)

T(x, t) = 1(2π)4

∫

k≤kd

dk∫

dω T(k) exp[i(kx − ωt)]. (28)

Fourier-transformed momentum and temperatureequations, Eqs. 1 and 2, are

(−iω + ν0k2)uβ(k) = f 0β (k) − ikβ

P(k)

ρ+ αgT(k)δβ3

− ikμ

∫uβ(q)uμ(k − q)

dq(2π)4 ,

(29)

(−iω + κ0k2)T(k) = −d�

dzu3(k)

− ikα

∫uα(q)T(k − q)

dq(2π)4 ,

(30)

while the continuity Eq. 3 becomes

uα(k)kα = 0, (31)

where q ≡ (�, q) and k ≡ (ω, k) are four-dimensionalvectors in Fourier space and the Einstein’s summationrule is implied.

In the physical space representation, the pressure is anon-local variable whose correlations with the velocityand temperature are unknown and difficult to evaluate.In RSM, these correlations are usually parameterizedusing the Rotta hypothesis which is known to be prob-lematic in anisotropic flows with waves (Mellor andHerring 1973; Cheng et al. 2002; Canuto et al. 2008a).In the spectral representation, the pressure, P(k), canbe rigorously evaluated using the momentum and con-tinuity Eqs. 29 and 31 and then excluded from Eq. 29to yield

G−10 (k)uβ(k) = f 0

β (k) + αgT(k)P3β(k)

− i2

Pβμν(k)

∫uμ(q)uν(k − q)

dq(2π)4 ,

(32)

where

Pij(k) = δij − kik j/k2, Plmn(k) = km Pln(k) + kn Plm(k)

are projection operators,

G0(k) = (−iω + ν0k2)−1 (33)

is the ‘bare’ auxiliary Green function, and δij is theKronecker δ-symbol.

Using Eq. 30 we derive an expression for thetemperature,

T(k) = GT0(k)

[−d�

dzu3(k)

− ikα

∫uα(q)T(k − q)

dq(2π)4

], (34)

where GT0(k) is the ‘bare’ temperature Green function,

GT0(k) = (−iω + κ0k2)−1. (35)

Anticipating further developments consisting of thesuccessive scale elimination and calculating the ensuingcorrections to the viscosity and diffusivity that willmodify the Green functions, we now introduce a for-mal solution to Eq. 34 in the form of a temperatureLangevin equation,

T(k) = GT(k)

[fT(k) − d�

dzu3(k)

], (36)

where GT(k) involves an ef fective or ‘dressed’ diffu-sivity. The term d�

dz u3(k) describes buoyancy generatedforcing at the scale k, and

fT(k) = ikα

∫uα(q)T(k − q)

dq(2π)4 (37)

is the forcing due to non-linear interactions between allnon-eliminated scales which would be present even inneutrally stratified flows.

By substituting (36) in (32) we obtain a self-contained equation for the velocity,

uα(k)

= G0αβ(k)

[�T

β (k)− i2

Pβμν(k)

∫uμ(q)uν(k − q)

dq(2π)4

],

(38)

where �Tβ (k) is a ‘thermal stirring’ evoked by fT(k), and

G0αβ(k) is the ‘bare’ velocity Green function,

G0αβ(k) = G0(k)

[δαβ + N2GT(k)G0(k)Pα3(k)δβ3

]−1.

(39)

Ocean Dynamics (2010) 60:1319–1337 1333

The next important step is the inversion of the matrixon the right side of (39) which can be accomplishedanalytically,

G0αβ(k) = G0(k)

[δαβ + A(k)Pα3(k)δβ3

], (40)

A(k) = − N2

G−10 (k)G−1

T (k) + N2 sin2 φ, (41)

where φ is the angle between k and the vertical.The results (38–41) illuminate various aspects of

the effect of stable stratification on turbulence. Themost noticeable are the anisotropization of the velocityGreen function; its acquisition of the tensorial nature isreflected by Eq. 40, and the appearance of the complexpoles in (41) points to the presence of internal waves inthe turbulent flow field. The spectral approach is crucialfor derivation of these results.

Further advance is hampered by strong non-linearityof the momentum Eq. 38 where the non-linear termsexceed the linear ones by the factor O(Re). To over-come this difficulty, we apply the successive scale elimi-nation technique briefly outlined in Section 2. Fulldetails of this technique are given in Sukoriansky et al.(2005b). The procedure is based upon Eqs. 34–41. Asmentioned in Section 2, the eddy viscosity and eddydiffusivity become anisotropic in the course of the suc-cessive coarsening. As a result, the ‘dressed’ auxiliaryGreen function, G(k), and the ‘dressed’ temperatureGreen function, GT(k), assume the form

G(k) = [−iω + νhk2h + νzk2

3

]−1, (42)

GT(k) = [−iω + κhk2h + κzk2

3

]−1, (43)

where νh and νz are the horizontal and vertical‘dressed,’ or eddy viscosities, and κh and κz are thehorizontal and vertical eddy diffusivities, respectively.At each step of the process of scale elimination, thevelocity and temperature modes subject to ensembleaveraging are recast in the form of the Langevin equa-tions,

ui(k) = Gij(k) f j(k) (44)

and Eq. 36, which are then used to calculate correctionsto the viscosity and diffusivity. Equation 44 has trans-parent physical interpretation: the stochastic force f j(k)

represents the non-linear stirring of a given velocitymode, ui(k), by all other modes. Equation 36 indicatesthat temperature fluctuations are excited by velocityfluctuations.

Along with the stochastic forcing, Eqs. 44 and 36also imply that the modes ui(k) and T(k) are dampedvia non-linear interactions with all other modes. This

damping is represented by the eddy viscosities andeddy diffusivities in the Green functions. Thus, Eqs. 44and 36 enforce the statistical balance between the non-linear forcing and non-linear damping of every velocityand temperature mode.

Equations 38 and 34 are used to calculate correctionsto the inverse Green functions, G−1

αβ and G−1T , resulting

from the ensemble averaging of a small shell �� of thehigh wavenumber modes adjacent to the dynamic dissi-pation cutoff kc. Since the modes ui(k) and T(k) in thatshell are governed by the Langevin equations (44) and(36), one needs to know statistical characteristics of theforcing f j(k). As shown in Sukoriansky et al. (2005b),an important demand to f j(k) is that its triple momentsbe zero in the shell ��. This property alone sufficesto develop a rigorous and self-contained mathematicalprocedure of successive averaging. Thus, f j(k) does notneed to be Gaussian although a Gaussian field wouldmeet the above requirement. The forcing f j(k) will thusbe characterized as quasi-normal. As explained earlier,the eddy viscosities and eddy diffusivities provide eddydamping. The combination of the quasi-normal forcingand eddy damping places the QNSE in the class ofthe quasi-normal eddy-damped theories of turbulence(Orszag 1977; McComb 1991; Chasnov 1991).

The corrections to G−1αβ (ω, k, k3) and G−1

T (ω, k, k3)

are

�[G−1

αβ (ω, k, k3)]

= Pαμθ (k)

∫

��

Pνσβ(k−q)Gθν(ω−�, |k−q|, k3−q3)

× Uμσ (�, q, q3)dq d�

(2π)4 , (45)

�[G−1

T (ω, k, k3)]

= kαkβ

∫

��

Uαβ(�, q, q3)GT(ω − �, q, q3)

× dq d�

(2π)4 . (46)

The integrals are evaluated in the assumption thatk/q � 1 (the distant interaction approximation, seee.g., Yakhot and Orszag 1986; Sukoriansky et al. 2005b)and yield corrections to the eddy viscosities and eddydiffusivities.

The spectral tensor, Uαβ(q), is given by

Uαβ(q) = Cεq−3Gαμ(q)G∗βσ (q)Pμσ (q). (47)

As elaborated in Sukoriansky et al. (2005b), the projec-tion operator in (47) is due to the incompressibility. Theproportionality to ε relates the velocity correlator to the

1334 Ocean Dynamics (2010) 60:1319–1337

forcing of the mode q, and the factor q−3 is dictated bydimensional considerations. The coefficient C could, inprinciple, depend on the non-dimensional combinationof k, ε and N (the spectral Froude number), and the an-gle φ but we assume, for simplicity, that C is a constantand thus that the dependences on N and φ are fullyabsorbed in the Green functions. Note that C is notan adjustable constant; it is calculated from the energybalance considerations after the Green functions aredetermined.

Equations 45, 46 and 47 provide corrections to theisopycnal and diapycnal viscosities and diffusivities re-sulting from the elimination of the shell ��. At theend of this step, the dissipation cutoff moves from kd

to kc = kd − ��. This procedure can then be repeatedresulting in elimination of the next shell �� adjacent tothe new dissipation cutoff, etc. Finally, taking the limit�� → 0, we obtain a coupled system of four ordinarydifferential equations for νh, νz, κh and κz as functionsof the dynamic dissipation cutoff.

The eigenfrequencies of the Langevin equation (44)are given by the secular equation

det[G−1

αβ (k, ω)]

= 0. (48)

The real parts of these eigenfrequencies provide thedispersion relation for internal waves in the presenceof turbulence (Sukoriansky et al. 2005b),

ω=ω0

⎧⎪⎨

⎪⎩1−

(kk0

)4/3⎡

⎣

(κzνn

− νzνn

)cos2 φ+

(κhνn

− νhνn

)sin2 φ

4 sin φ

⎤

⎦

2⎫⎪⎬

⎪⎭

1/2

,

(49)

where ω0 = N sin φ is the classical dispersion relationfor linear internal waves. Equation 49 specifies theinternal wave frequency shift due to turbulence. Inthe limit of k/k0 → 0, the classical dispersion relationfor linear waves is recovered. Consistently with ob-servations of flows with internal waves (D’Asaro andLien 2000a, b) ω ≤ ω0 ≤ N for any stratification. As kincreases, waves undergo attenuation due to turbulencewhich leads to the decrease of ω. At large k, ω → 0and the radiation of internal waves is completely over-whelmed by the turbulence scrambling. The thresholdwave number of the internal wave radiation can beestimated, approximately, at 32k0 (Sukoriansky et al.2005b). It is shown on Fig. 1 by a vertical dashed line.Remarkably, this threshold closely coincides with thebeginning of flow anisotropization caused by stablestratification.

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