geophysical flows with anisotropic turbulence and dispersive waves: flows with a β-effect
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ISSN 1616-7341, Volume 60, Number 2
Ocean Dynamics (2010) 60:427–441DOI 10.1007/s10236-010-0278-2
Geophysical flows with anisotropic turbulenceand dispersive waves: flows with a β-effect
Boris Galperin · Semion Sukoriansky ·Nadejda Dikovskaya
Received: 24 September 2009 / Accepted: 16 February 2010 / Published online: 16 March 2010© Springer-Verlag 2010
Abstract Geostrophic turbulence is a key paradigm inthe current understanding of the large-scale planetarycirculations. It implies that a flow is turbulent, rotat-ing, stably stratified, and in near-geostrophic balance.When a small-scale forcing is present, geostrophic tur-bulence features an inverse energy cascade. When themeridional variation of the Coriolis parameter (or aβ-effect) is included, the horizontal flow symmetrybreaks down giving rise to the emergence of jet flows.The presence of a large-scale drag ensures that theflow attains a steady state. Dependent on the governingparameters, four steady-state flow regimes are possible,two of which are considered in this study. In one ofthese regimes, a flow is dominated by the drag whilein the other one, the recently discovered regime ofzonostrophic turbulence, a flow becomes strongly
Responsible Editor: Leo Oey
B. Galperin (B)College of Marine Science, University of South Florida,St. Petersburg, FL 33701, USAe-mail: [email protected]
S. Sukoriansky · N. DikovskayaDepartment of Mechanical Engineering,Ben-Gurion University of the Negev,Beer-Sheva 84105, Israel
S. Sukorianskye-mail: [email protected]
N. Dikovskayae-mail: [email protected]
S. SukorianskyPerlstone Center for Aeronautical Engineering Studies,Ben-Gurion University of the Negev,Beer-Sheva 84105, Israel
anisotropic and features slowly evolving systems ofalternating zonal jets. Zonostrophic turbulence is dis-tinguished by anisotropic inverse energy cascade andemergence of a new class of nonlinear waves knownas zonons. In addition, meridional scalar diffusion isstrongly modified in this regime. This paper providesan overview of various regimes of turbulence witha β-effect, elaborates main characteristics of friction-dominated and zonostrophic turbulence, elucidates thephysical nature of the zonons, discusses the meridionaldiffusion processes in different regimes, and relatesthese results to oceanic observations.
Keywords Geostrophic turbulence ·Turbulent diffusion · Nonlinear waves
1 Introduction
The notion of geostrophic turbulence has been intro-duced by Charney (1971) as a paradigm of large-scale planetary and oceanic macroturbulence (Rhines1979; Salmon 1998; Held 1999; Read 2001; Vallis2006). Geostrophic turbulence is stipulated by theTaylor–Proudman theorem in rapidly rotating systems(Pedlosky 1987) and pertains to a highly anisotropicstably stratified turbulence in near-geostrophic bal-ance. Among the most productive applications of thisparadigm have been two-layer studies of baroclinicallyunstable systems with an imposed environmental verti-cal shear which develops mesoscale eddy activity. Thebasic theory of such systems has been developed inRhines (1977), Salmon (1978, 1980, 1998), Hoyer andSadourny (1982), Larichev and Held (1995), and Heldand Larichev (1996). These studies analyzed barotropic
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428 Ocean Dynamics (2010) 60:427–441
and baroclinic energy cascades, scaling relationships,energy and potential vorticity fluxes, and meridionaleddy diffusivity.
The dominant energy transfers in a two-layer modelwere succinctly characterized by Salmon (1998; seeFig. 6.6 in Salmon 1998). The large-scale baroclinicenergy cascades down the spectrum toward the internalRossby radius of deformation, LD. Around that scale,due to the action of baroclinic instability, the energyis transferred to the barotropic mode. This energybecomes a small-scale agitator of the inverse cascadein the barotropic mode. If LD is much smaller thana scale L at which the inverse cascade is dissipatedor, equivalently, the Burger number is small, Bu ≡(LD/L)2 � 1, then the barotropic mode may becomethe most energetic component of the flow (Larichevand Held 1995). It has become widely accepted thatwhen the inverse cascade reaches the Rhines scale,LR = (2U/β)1/2, where U is the rms of the barotropicvelocity and β is the meridional gradient of the Coriolisparameter, the inverse cascade becomes arrested andthe flow evolves into a system of alternating zonal jets.
Sukoriansky et al. (2007) studied the dynamics ofbarotropic flows with inverse cascade in detail anddemonstrated that rather than halting the inverse cas-cade, a β-effect only causes its anisotropization. Whena β-effect is strong, the flow may attain a regimeof zonostrophic turbulence (from the Greek wordsζωνη—band, belt, and στρoφη—turn, thus referring tozonal structures in a rotating environment; Galperinet al. 2006; Galperin and Sukoriansky 2008). Thisregime is distinguished by a strongly anisotropic kineticenergy spectrum whose zonal mode alone may containmore energy than all other modes combined (Chekhlovet al. 1996; Huang et al. 2001; Galperin et al. 2006).The most profound visual feature of zonostrophic tur-bulence is the formation of a slowly changing systemof alternating zonal jets spanning the entire flow do-main. The emergence of such jet systems is sometimesreferred to as zonation. The regime of zonostrophicturbulence has been hypothesized to be a basic mecha-nism of generation and maintenance of zonal jets inthe atmospheres of giant planets and in the terres-trial oceans (Galperin et al. 2001, 2004; Sukorianskyet al. 2002).
Recent investigations have revealed two more attri-butes of the zonostrophic regime—nonlinear waves,or zonons (Sukoriansky et al. 2008), and the scaleindependency of the coefficient of lateral, or cross-jeteddy diffusivity (Sukoriansky et al. 2009). Figure 1 pro-vides schematic representation of the transition fromgeostrophic to zonostrophic turbulence as well as themain features of the latter regime.
Fig. 1 Schematic representation of physical processes leadingfrom geostrophic to zonostrophic turbulence and the basic fea-tures of the latter regime. The zonostrophy index, Rβ , is definedin Section 2. The red arrows point to the three main features ofthis regime elaborated in Sections 4, 5, and 6
This paper presents a brief review of possible flowregimes in the barotropic mode using the vorticityequation on the surface of a rotating sphere. Thenext section will provide a succinct survey of thequasi-2D turbulence with a β-effect and outlines fourpossible steady-state regimes in it. Sections 3 and 4detail the two of most interesting of these regimes:friction-dominated and zonostrophic, respectively.Section 5 elucidates a new class of nonlinear wavesin zonostrophic turbulence, zonons. Section 6 elabo-rates diffusion processes in various regimes. Section 7provides discussion and conclusions.
2 Quasi-2D turbulence with a β-effect
Some basic features of large-scale planetary and ter-restrial circulations can be studied in the frameworkof an unforced quasi-2D turbulence on a β-plane oron the surface of a rotating sphere. Such models retainthe physics of turbulence–Rossby wave interaction andturbulence anisotropization (Newell 1969; Rhines 1975;Holloway and Hendershott 1977). Studies that followedup extended this framework by including the small-scale forcing necessary to maintain the inverse cascade(Vallis and Maltrud 1993; Chekhlov et al. 1996; Smithand Waleffe 1999; Huang et al. 2001; Sukoriansky et al.2002, 2007; Galperin et al. 2006; Nadiga 2006; Scott andPolvani 2007).
When a large-scale dissipation is also present, thebalance between the forcing and the dissipation leadsto the establishment of a steady-state flow regime. Thisregime is not unique. Sukoriansky et al. (2007) provided
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classification of the multitude of steady-state regimesusing long-term simulations utilizing the barotropicvorticity equation on the surface of a rotating sphere.We briefly recap these results.
Consider small-scale forced, linear-friction damped,barotropic, 2D vorticity equation on the surface of arotating sphere,
∂ζ
∂t+ J(ψ, ζ + f ) = ν∇2pζ − λζ + ξ, (1)
where ζ is the vorticity, ψ is the stream function,∇2ψ = ζ , f = 2� sin θ is the Coriolis parameter, � isthe angular velocity of the sphere’s rotation, θ is thelatitude, φ is the longitude, ν is the hyperviscositycoefficient, p is the power of the hyperviscous opera-tor (p = 4 in this study), and λ is the linear frictioncoefficient. The scale at which the upscale energyflux is absorbed by the large-scale friction is associ-ated with the friction wavenumber, nfr. The small-scaleforcing, ξ , acting on the scales around n−1
ξ , pumpsenergy into the system at a constant rate. Part of thisenergy becomes available for the inverse cascade ata rate ε. The Jacobian, J(ψ, ζ + f ), where J(A, B) =(R2 cos θ)−1(Aφ Bθ − Aθ Bφ) and R is the radius of thesphere, represents the nonlinear term.
The stream function in Eq. 1 can be represented viaspherical harmonics decomposition,
ψ(μ, φ, t) =N∑
n=1
n∑
m=−n
ψmn (t)Ym
n (μ, φ), (2)
where Ymn (μ, φ) are the spherical harmonics (the asso-
ciated Legendre polynomials), μ = sin θ , φ is the longi-tude, θ is the latitude, n and m are the total and zonalwavenumbers, respectively, and N is the total trun-cation wavenumber. Conventionally, the indices n, mand N are nondimensional. However, when appear inequations below, the wavenumbers n and m have the di-mension of the inverse length. Setting R = 1 eliminatesthe difference between the indices and wavenumbers.
Equation 1 admits a class of linear Rossby–Haurwitzwaves (RHW; Holton 2004) with the dispersion relation
ωR(n, m) = −2 βm
n(n + 1), (3)
where β = �/R. Equation 3 is consistent with a β-planeapproximation format (Huang et al. 2001).
The presence of RHWs in the nonlinear dynamicssubstantially modifies the energy transfer. It was foundthat the wave vector triads that satisfy the frequencyresonance condition provide the dominant contributionto the energy transfer. Combined with the RHW dis-persion relation (Eq. 3), this condition results in theanisotropization of the energy flux and its redirection
to modes with small n and m, i.e., large-scale, energeticzonal jets (Rhines 1975; Holloway and Hendershott1977; Holloway 1986; Chekhlov et al. 1996; Huang et al.2001; Galperin et al. 2006).
To investigate various aspects of the nonlinear dy-namics, we conducted a series of long-term simulationswith Eq. 1 using the decomposition (Eq. 2). A Gaussiangrid was employed with resolutions of 400 × 200 and720 × 360 nodes while enforcing the 2/3 dealiasing rule(rhomboidal truncations R133 and R240, respectively).The simulations were performed with low resolution ifpossible. The high resolution was only used when thecompliance with the chain inequality (Eq. 14) belowrequired that the forcing be set at high wavenumbers.We checked that the results did not depend on the reso-lution. The hyperviscosity coefficient, ν, was chosen insuch a manner as to effectively suppress the enstrophyrange. The Gaussian random forcing had a constantvariance, was uncorrelated in time and between themodes, and was distributed among all modes in smallshells centered around nξ = 84, 100, or 160 dependenton the resolution. Most of the presented results em-anate from these simulations.
The energy spectrum for flows in spherical geometryis given by
E(n) =n∑
m=−n
E (n, m) = n(n + 1)
4
n∑
m=−n
⟨∣∣ψmn
∣∣2⟩, (4)
where the modal spectrum, E (n, m), is the spectral en-ergy density per mode (n, m) and the angular bracketsindicate an ensemble or time average (Boer 1983; Boerand Shepherd 1983). The spectrum E(n) can be dis-sected into a sum of the zonal and nonzonal, or residualcomponents, E(n) = EZ(n) + ER(n), where the zonalspectrum is EZ(n) = E (n, 0).
As elaborated in Sukoriansky et al. (2007), themodes dominated by the large-scale drag absorb theupscale propagation of the inverse energy cascade.The scale where this process takes place is denotedas the friction wavenumber, nfr, and its magnitudedepends on λ, β, and ε, i.e.,
nfr = f (λ, β, ε). (5)
Another important scale characterizing these flowsis associated with the transitional wavenumber nβ ∝(β3/ε)1/5. This wavenumber was introduced by Vallisand Maltrud (1993) for flows on a β-plane by equat-ing the Rossby wave period with the turbulent eddyturnover time in the regime of isotropic 2D turbu-lence (Kolmogorov–Kraichnan (KK) regime). In thisrespect, nβ is analogous to the Ozmidov wavenum-ber, kO, in stably stratified turbulence discussed in
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430 Ocean Dynamics (2010) 60:427–441
Galperin and Sukoriansky (2010). Sukoriansky et al.(2007) elucidated that nβ = 0.5(β3/ε)1/5 characterizesthe threshold of the inverse cascade anisotropization.Similarly, the wavenumber 0.5kO can be associatedwith the anisotropization of the direct cascade by stablestratification (Galperin and Sukoriansky 2010). Finally,recall that the Rhines’s wavenumber (Rhines 1975),nR = L−1
R = (β/2U)1/2, is an additional characteristicof the large-scale flow (Sukoriansky et al. 2007). Sincethe kinetic energy, E = U2/2, in the forced flows in asteady state is related to the forcing, ε, and the lineardrag coefficient, λ, by
ε = 2λE, (6)
the Rhines wavenumber depends on λ, β, and ε. Thesethree dimensional quantities quantities allow to forma single nondimensional parameter, γ = β1/2ε1/4λ−5/4,that fully characterizes the system. This parameter wasintroduced in Danilov and Gryanik (2004) and Danilovand Gurarie (2004). Using the dimensional analysis,Eq. 5 can be rewritten in the form
nR
nfr= f (γ ). (7)
When a β-effect is weak, β � (λ5/ε)1/2, the flowshould remain nearly isotropic on all scales not sup-pressed by the large-scale friction. In this case, nfr
can be found by equating λ−1 to the turnover time ofisotropic turbulence in inverse cascade regime (i.e., KKregime) which gives nfr = (λ3/ε)1/2. This relationship,combined with the definition of nR, yields
nR
nfr∝ γ, γ → 0, (8)
which, in fact, is the first term of the Taylor seriesexpansion of f (γ ). Assuming that f (γ ) is continuousand monotonic function and taking note that f (γ ) isbounded, one obtains the γ → ∞ asymptotics, f (γ ) →const, or
nR
nfr→ const, γ → ∞. (9)
The cases of small and large γ were considered byDanilov and Gryanik (2004) and Danilov and Gurarie(2004) where the ensuing regimes were characterizedas “weak zonal” and “strong zonal”, respectively. Theyshowed that in the weak zonal regime, the large-scalefriction damps the modes where a β-effect is significant.The resulting flow appeared nearly isotropic and exhib-ited features of the classical KK turbulence. The strongzonal regime, on the other hand, was distinguishedby profound anisotropy and slowly varying alternatingzonal jets.
Galperin et al. (2006) and Sukoriansky et al. (2007)present the analysis of 2D turbulence on the surfaceof a rotating sphere based upon over 300 numericalsimulations with the described earlier settings. A widerange of parameters was explored and four possibleflow regimes were identified. These regimes differedfrom each other by the degree of anisotropy, spec-tral and transport properties, and the nature of thewave–turbulence interaction. For classification of theseregimes, they employed a nondimensional parameterRβ = nβ/nR � 0.7γ 1/5 which will be referred to as azonostrophy index. As will be shown in Section 4,Rβ has a distinct and important physical meaning.Sukoriansky et al. (2007) evaluated the asymptoticbehavior of Eq. 7 and confirmed both expressions 8 and9 with the constant in Eq. 9 being equal to 0.83. It turnsout that Rβ provides a more convenient parameterthan γ for quantitative characterization of boundariesbetween different regimes. The classification of the fourregimes in terms of nβ and nR is shown in Fig. 2 whereevery symbol denotes a result of a single simulationperformed for many tens of thousands of �−1 after asteady state was attained. The regimes can be catego-rized as friction-dominated, transitional, zonostrophic,and large-scale condensation.
The friction-dominated regime is similar to the weakzonal regime of Danilov and Gryanik (2004) andDanilov and Gurarie (2004). In the transitional regime,the large-scale drag is smaller and the flow begins toundergo anisotropization. In the zonostrophic regime,the flow becomes strongly anisotropic and its zonal
0 10 20 300
5
10
15
nβ
nR
Friction- dominated
Transitional
Zonostrophicturbulence
Condensation
Fig. 2 The four possible flow regimes in 2D turbulence on thesurface of a rotating sphere in the space of parameters nR andnβ . Open squares pertain to the friction-dominated regime, opentriangles correspond to the transitional regime, f illed circles showthe zonostrophic regime, and the asterisks denote the regime ofthe large-scale condensation
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Ocean Dynamics (2010) 60:427–441 431
spectrum is fully determined by β and the local wave-number n thereby pointing to the regime’s universality.The condensation regime mostly occurs in flows withvery small λ whereas the energy accumulates in a fewlowest modes unable to carry out the inverse cascade.They facilitate the “bottleneck” phenomenon that leadsto the large-scale energy condensation (Smith andYakhot 1993; Sukoriansky et al. 1999; Galperin andSukoriansky 2005; Bos and Bertoglio 2009).
In the present study, we are concerned with theintercomparison of the two universal regimes, friction-dominated and zonostrophic, while the other tworegimes are of lesser interest.
3 The friction-dominated regime
As evident from Fig. 2, the friction-dominated regimeis approximately delineated by the inequality Rβ � 1.5.Figure 3 shows the energy spectra typical of this regime.The total spectrum, E(n), closely follows the classicalKK scaling (Huang et al. 2001; Sukoriansky et al. 2002),
E(n) = CKε2/3n−5/3, CK � 6. (10)
For n > nβ , the modal spectrum is
E (m, n) � (1/2)CKε2/3n−8/3. (11)
100
101
10210
10
108
10_
_
_
_
_6
E
n
m=1
m=2m=6
n 8/3 E
n 5/3
m=0
Fig. 3 The kinetic energy spectra for a friction-dominated regime(nR = 9.2, nβ = 14.3). The thick solid line shows the total spec-trum, E(n), while the thin lines show the modal spectra, E (n, m).The values of m are shown near the lines. The dashed straightlines correspond to the spectra 10 and 11
It is important to understand how turbulence inter-acts with the Rossby–Haurwitz waves. Are these wavespresent in the fully nonlinear system and, if theanswer is yes, how do they effect and are affected byturbulence? To answer these questions, we consider thespace–time Fourier transform of the two-point velocitycorrelation function,
U(ω, n, m) = n(n + 1)
4
⟨∣∣ψmn (ω)
∣∣2⟩, (12)
where ψmn (ω) is the time Fourier transform of the spec-
tral coefficient ψmn (t) in Eq. 2. The methodology of the
use of the two-point correlation function is elucidatedin the “Appendix”. In pure 2D turbulence with nowaves, U(ω, n, m) has symmetric bell shape aroundthe zero frequency. In the case at hand, waves mani-fest themselves as peaks in U(ω, n, m) at frequenciesωR(n, m).
Figure 4 shows U(ω, n, m) in the friction-dominatedregime. The large-scale modes are populated by nar-row spikes corresponding to linear RHWs which arepresent at n far exceeding both nR and nβ where thespectral peaks become increasingly broadened by tur-bulence. This result suggests that on the large scales,the flow is dominated by RHWs and that there is nosharp scale separation between waves and turbulence;both phenomena coexist on virtually all scales. Eventhough the effect of rotation is significant and the flowdynamics is dominated by strong nonlinearity punc-tuated by the inverse cascade and the energy spec-trum of well-developed turbulence, the flow featureslinear waves with no frequency shift. This strikingly
0
0.2
0
5
0
5
0
0.2
0
0.2
0
0.02
0
5
0
1
0 −0.10
5
0.20 −0.2 0
0.2
0
1
0
5
0
0.2
0
5
0
0.1
0
2
0
5
0
0.2
0−0.40
0.01
U
0 −0.20
0.5
ω
n/nβ
m/n 0.41 0.82 1.64 3.28
0.2
0.4
0.6
0.8
1.0
n 5 10 20 40
0
1
30
2.45
0
1
0
1
0
1
0.20−0.20
1
Fig. 4 The velocity correlation function, U(ω, n, m) × 107, forthe friction-dominated regime; nR = 9.2, nβ = 12.3. The f illedtriangles correspond to the Rossby–Haurwitz wave dispersionrelation (Eq. 3)
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432 Ocean Dynamics (2010) 60:427–441
counter-intuitive phenomenon is quite common inthe terrestrial atmosphere which exhibits a nonlinear,friction-dominated regime (Sukoriansky et al. 2007)while featuring linear RHWs (Holton 2004). Note thatin a truly linear system, the excitation of every RHWwould require a specific source. In the nonlinear frame-work, however, all possible waves are excited by therandom noise of diverse nature.
We now return to Fig. 3 and take a note that itreflects a β-effect caused anisotropization of the inverseenergy cascade at the modes with n < nβ . As explainedearlier, these modes are dominated by RHWs. Theanisotropic inverse cascade pumps energy to the modeswith low m and n. The march of energy to the largescales is arrested by the large-scale friction. Equatingthe RHW frequency to the inverse friction time scale,2λ, we find the approximate boundary of the energycontaining region,
m ∼ ±λ
βn(n + 1). (13)
This relationship provides a quantitative explanationof the spectral behavior shown in Fig. 3 for n < nβ . Aβ-plane analog of Eq. 13 is the dumbbell shape (Vallisand Maltrud 1993; Vallis 2006). It is important to accen-tuate that the large-scale friction rather than a β-effectcauses inverse cascade termination. Sukoriansky et al.(2007) emphasized that since a β-effect is nondissipa-tive, it cannot arrest the inverse cascade in principle;it can only cause its anisotropization. They designedsimulations with no large-scale drag at all and demon-strated that the energy containing region expanded towavenumbers much smaller than nβ . When the dragwas present, as in Fig. 3, the inverse cascade terminuswas determined by nfr.
4 The regime of zonostrophic turbulence
Let us now define a zonostrophic inertial range as theinterval (nR, nβ) whose width is measured by the zono-strophy index, Rβ = nβ/nR. As evident from Fig. 2,if Rβ � 2.5, i.e., this range is wide enough, the flowdevelops a regime of zonostrophic turbulence. In thisrange, processes become independent of the large-scalefriction and the flow is governed solely by the inverseenergy cascade and a β-effect. Galperin et al. (2006)and Sukoriansky et al. (2007) emphasized the importantrole of the forcing wavenumber, nξ , in the formation ofthe zonostrophic regime. Using Fig. 2, one can establisha more concise condition for the development of this
regime and express it in a form of an approximate chaininequality,
nξ � 4nβ � 10nR � 40. (14)
Generally, this inequality instructs us that the zono-strophic regime can be attained when (1) the forcingacts on scales only weakly impacted by a β-effect, (2)there exists a sufficiently wide zonostrophic inertialrange (nR, nβ), and (3) there exists a minimum numberof the lowest modes necessary to resolve the large-scalefriction processes and avoid a condensation of energyon the largest scales (in the condensation regime, thesmall number of modes involved in the large-scale en-ergy removal are unable to fully absorb the energy fluxassociated with the inverse cascade without distortingit (Smith and Yakhot 1993, 1994; Sukoriansky et al.1999)). Let us consider the requirement that nξ � 40. Ifthe small-scale forcing is associated with the baroclinicinstability, then the forcing scale can be identified withthe first internal deformation radius, LD. Recall thataccording to the linear theory, the maximum growthrate of the baroclinic instability corresponds to nξ =0.64nD where nD = π R/LD such that the inequalitynξ � 40 yields (LD/R)2 = Bu � 2.5 × 10−3.
The zonal and residual kinetic energy spectra inthe zonostrophic regime are distinguished by stronganisotropy,
EZ(n) = CZβ2n−5, CZ ∼ 0.5, (15a)
ER(n) = CKε2/3n−5/3, CK ∼ 5 to 6. (15b)
E(n)n-5
n-5/3
n-8/3
nfr nznβ n
zonostrophicinertialrange
Fig. 5 Schematic representation of the regime of zonostrophicturbulence. The red and blue lines show the zonal and residualspectra, EZ and ER, respectively. They intersect at the transi-tional wavenumber nβ . Note that the steepening of the zonalmode, E (0, n), extends beyond nβ up to the intersection withthe spectrum 11 at nz (Sukoriansky et al. 2002). On the largescales, n < nfr, both zonal and residual spectra flatten out underthe action of the large-scale drag
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Ocean Dynamics (2010) 60:427–441 433
These spectra were established and extensively inves-tigated in computer simulations (Huang et al. 2001;Sukoriansky et al. 2002, 2007). Figure 5 provides aschematic representation of this regime. The frictionscale, nfr, is now determined by the shape of the zonalenergy spectrum and defined as the wavenumber atwhich the low wavenumber plateau intersects with thezonostrophic n−5 spectrum (Sukoriansky et al. 2007).For n > nfr, as expected, the spectra 15a and 15b do notdepend on the large-scale drag. For n < nfr, the effect ofthe drag becomes important while nfr and nR are barelydistinguishable from each other, nfr � 1.2nR (Galperinet al. 2001; Sukoriansky et al. 2007). The zonal andresidual spectra in the zonostrophic regime intersect atthe introduced above transitional wavenumber,
nβ =(
CZ
CK
)3/10 (β3
ε
)1/5
� 0.5
(β3
ε
)1/5
. (16)
In studies by Danilov and Gryanik (2004) andDanilov and Gurarie (2004) it was stated that the spec-trum 15a is not universal. Most of their simulations,however, did not comply with one or more of the in-equalities (Eq. 14). We replicated the simulations fromDanilov and Gryanik (2004) and Danilov and Gurarie(2004) in spherical geometry and found out that whenthe inequality (Eq. 14) was fulfilled, the ensuing spec-trum agreed well with Eq. 15a.
−4
−8
0 1 2
E
E
1 2 m=3
n n
z
−5/3
−8/3
−5
R
R β
nn
n
Fig. 6 The kinetic energy spectra for a zonostrophic regime(nR = 5.5, nβ = 16.2). The thick gray and black solid lines showthe zonal and nonzonal spectra, EZ(n) and ER(n), respectively,while the thin lines show the modal spectra, E (n, m). The valuesof m are shown near the lines. The dashed straight lines corre-spond to the spectra 15a, 15b, and 11
Figure 6 shows the kinetic energy spectra for theregime of zonostrophic turbulence. Evidently, the zonaland residual spectra approach distributions 15a and15b, respectively. For relatively small n belonging inthe zonostrophic inertial range, the anisotropizationbecomes so profound that the zonal modes containmore energy than all other modes combined.
From the theoretical standpoint, the regime ofzonostrophic turbulence is distinguished not only bystrongly anisotropic spectra but also by a new class ofnonlinear waves and specific transport properties whichwill be considered in the next two sections.
5 Nonlinear waves—zonons
An important feature of the nonzonal spectral modesevident in Fig. 6 is their steepening above the KKspectrum (Eq. 11). The most energetic are the modeswith small n which are still unaffected by the large-scale drag. Unlike the zonal modes, their nonzonalcounterparts harbor waves that are highly energetic atlow n. It is important to understand wave behavior inthis strongly nonlinear anisotropic system. To addressthis issue, we analyze the velocity correlation function,U(ω, n, m), shown in Fig. 7. Along with the linearRHWs, one now discerns a new class of waves whichwere not present in the linear system. Clearly, thesewaves emerge from nonlinear interactions. We refer tothem as zonons and denote their frequency as ωz(n, m)
(Sukoriansky et al. 2008). Most intriguingly, the zonalmodes harbor zonal zonons (zonons with m = 0) whilethe RHWs in these modes do not exist. With increasing
Fig. 7 The velocity correlation function, U(ω, n, m), for theregime of zonostrophic turbulence; nR = 5.5, nβ = 16.2. Thef illed triangles correspond to the RHW dispersion relation(Eq. 3) while the f illed circles refer to the zonons
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434 Ocean Dynamics (2010) 60:427–441
Fig. 8 The velocity correlation function, U(ω, n, 1), (solid lines)for a zonostrophic regime (nR = 5.5, nβ = 16.2). The f illed tri-angles at the top of the panel correspond to the RHW dispersionrelation (Eq. 3) with values of n shown near the triangles. To easetracing the lines at different n, they are shown at dif ferent grayscales
n, both zonal and nonzonal modes may acquire multiplezonons.
What is the physical nature of the zonons? To answerthis question, we present in Fig. 8 the correlation func-tion U(ω, n, 1) viewed through a “magnifying glass” ofthe logarithmic scale. For a given n, each line U(ω, n, 1)
exhibits a peak at its own RHW frequency (Eq. 3).The additional secondary peaks are associated withthe zonons. It is apparent that the zonons are forcedoscillations excited by energetic RHWs in modes withfixed m and practically all other n. The frequencies ofthese forced waves are equal to the frequencies of thecorresponding “master” RHWs and thus are indepen-dent of n.
Fig. 9 Frequencies of RHW and zonons as functions of m fordifferent n; nR = 5.5, nβ = 16.2. The empty circles and dashedlines correspond to the RHW dispersion relation (Eq. 3); thef illed circles and the solid black and gray lines show the dispersionrelations for the zonons excited by RHW with n = 4 and 5,respectively
Z
4to
t
Fig. 10 The total energy of the zonal modes as a function of time
Figure 9 compares zonons with RHWs by presentingω(m) for different n. The RHWs are clearly evidentfor all modes including those with n > nβ . Along withRHWs, one discerns two groups of zonons identifiablewith RHWs with n = 4 and 5. Figure 6 indicates thatthese RHWs are most energetic. Furthermore, ωz(n, m)
is proportional to m and virtually independent of nfor both groups of zonons such that they form non-dispersive wave packets whose zonal phase speeds, cz =ωz(n, m)/m, are equal to the zonal phase speeds of thecorresponding “master” RHWs. Note the presence ofthe zonal zonons, i.e., the waves with m = 0.
Figure 10 clarifies the nature of the zonal zonons bydemonstrating that the total energy of the zonal modes,Etot
Z , oscillates in time and is in fact a wave comprisedof two modes with the frequencies of the zonal zonons.This behavior is stipulated by nonlinear interactions be-tween the zonal and nonzonal modes that cause energyoscillations around the saturated value which can becalculated from the spectrum 15a (Huang et al. 2001).
We hypothesize that all zonons are “slave” wavesexcited by RHWs. Since their dispersion relations differfrom Eq. 3 they should be recognized as a class of wavescompletely different from RHWs.
Can the zonons be detected in the physical space?Since the RHWs with n = 4 (denoted as nE) are themost energetic, their respective packets of zonons
Fig. 11 The Hovmöller diagrams of the stream function at threedifferent latitudes θ in the zonostrophic regime. The white linesshow the slope of the diagrams used to calculate the angularvelocities of the eddies
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Ocean Dynamics (2010) 60:427–441 435
0
0c
cz
RE
−4
−4
−0.4−0.4
0
0
Fig. 12 Comparison of the phase speed of the most energeticRHWs, cRE, computed from the dispersion relation (Eq. 3) andthe zonal velocities of the packets of zonons, cz, computed fromthe Hovmöller diagrams for a number of different simulations
would be dominant in the physical space and the eas-iest to observe. The phase speed of these packets isωR(nE, m)/m ≡ cRE. In the physical space, these pack-ets are expected to form westward propagating eddiesdetectable in the Hovmöller diagrams. The slope ofthe demeaned diagrams (i.e., the stream function iscomputed via the inverse transform with the m = 0modes discarded) yields the phase speed of the zonallypropagating eddies relative to the local zonal flows.If the eddies are really comprised of zonons, theirzonal phase speed should be equal to cz = cRE. TheHovmöller diagrams shown in Fig. 11 reveal westwardpropagating eddies at three different off-equatorial lati-tudes at which the zonal jets have their maximum, mini-mum, and zero velocity. Figure 12 further demonstratesthat cz = cRE at all three latitudes. Thus, the westwardpropagating eddies can indeed be identified with theenergetic zonon packets moving largely independentlyof the zonal flows.
6 Meridional diffusion in turbulence with a β-effect
Small-scale forced, homogeneous, and isotropic 2Dturbulence features the upscale energy cascade in theenergy subrange, i.e., on scales larger than the forc-ing scale. Flows of this type are said to exhibit thenegative viscosity, or antidiffusion phenomenon (Starr1968). The antidiffusion is only a part of the picture,however. The simultaneous conservation of the energyand the enstrophy (Kraichnan 1971; Vallis 2006) im-plies concurrent inverse energy cascade with the con-stant rate ε and zero flux of enstrophy in the energysubrange (Kraichnan 1971). A single viscosity term,either negative or positive, cannot fulfill this require-
ment. Analytical theories of turbulence as well asnumerical simulations inform us that along with thenegative Laplacian viscosity, an infinite series of dis-sipative hyperviscosities is needed (Sukoriansky et al.1996; Sukoriansky and Galperin 2005; Chekhlov et al.1994). The first term of this series already providesan approximation sufficient for practical applications.In such two-term formulation, the biharmonic, dis-sipative hyperviscosity compensates the destabilizingeffect of the negative Laplacian viscosity. An equa-tion combining the negative Laplacian viscosity anddissipative biharmonic hyperviscosity was named afterKuramoto and Sivashinsky (Kuramoto and Tsuzuki1976; Kuramoto 1984; Sivashinsky 1977, 1985) whoderived it for flows with chemical reactions and com-bustion. The original Kuramoto–Sivashinsky equationemployed constant viscosity coefficients. In the case ofhydrodynamic 2D turbulence, the Laplacian viscositycoefficient must be flow-dependent in order to ensurenumerical stability. The Kuramoto–Sivashinsky typeformulation with a flow-dependent negative Laplacianviscosity was developed in Sukoriansky et al. (1996) andSukoriansky and Galperin (2005) and used for success-ful simulation of 2D turbulence with unresolved small-scale forcing. That formulation was termed stabilizednegative viscosity.
Unlike the antidiffusion of the momentum, the scalardiffusivity in the energy subrange of 2D flows is posi-tive and, similarly to 3D flows, obeys the Richardsondiffusion law (Ollitrault et al. 2005; LaCasce 2008).Clearly, the Reynolds analogy (Tennekes and Lumley1972) does not apply to 2D turbulence.
Planetary rotation substantially complicates 2D tur-bulence by imposing an extra strain (Bradshaw 1973)that supports the Rossby–Haurwitz waves and rendersturbulence anisotropic. One can inquire whether ornot the diffusion processes are affected by the flowanisotropy and the action of waves. This query wasaddressed in a series of simulations with Eq. 1 coupledwith the diffusion equation for a passive scalar. Theflow was in a steady state with Rβ varying from 0to about 4 such as to include all flow regimes dis-cussed in Section 2. The meridional diffusivity, Dy, wascalculated using three different methods. In the firstmethod, the diffusion equation for a passive scalar cwas solved as the initial value problem with the initialconcentration, c0(θ), agglomerated in thin zonal ringsor in the polar caps with a Gaussian distribution inthe meridional direction. The meridional diffusivity wasdetermined from the expression, σ 2
y = 2Dyt, where σy
is the Gaussian spread and t is the time elapsed fromthe beginning of the diffusion experiment. The secondmethod was based upon the Reynolds averaging,
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436 Ocean Dynamics (2010) 60:427–441
whereas Dy was estimated by dividing the meridionalturbulent scalar flux, −v′c′, by the mean meridionalscalar gradient, ∂C/∂y. In the third method, trac-ers were released from circles at different latitudes.Such release eliminated complications due to the zonaladvection while allowing for testing of the effect of thezonal jets on the meridional diffusion. Tracer’s evolu-tion was followed until it reached high latitudes fromwhich it could escape via polar regions. The meridionaldiffusivity, Dy, was found based upon the growth intime of the standard deviations of the tracers fromtheir initial latitude. The results obtained from all threemethods were close to each other; they are summarizedin Fig. 13.
In the friction-dominated regime, for Rβ � 1, thediffusion is nearly isotropic and obeys the Richardsonlaw,
Dy � 2ε1/3n−4/3E , (17)
where nE ∝ nfr is the wavenumber of the mode withthe maximum kinetic energy. While still in the frictionregime, 1 � Rβ � 1.5, Dy undergoes the transition to
Dy � 0.3ε1/3n−4/3β . (18)
Afterward, Dy keeps this value throughout the tran-sitional and zonostrophic ranges, Rβ � 1.5, thus sug-gesting that scales larger than O(n−1
β ) do not contributeto Dy.
The laws 17 and 18 can be given transparent inter-pretation in terms of the mixing length theory applied
0 1 2 3 4
0.5
1
1.5
2
2.5
Dy /
D*
zonostrophy index, Rβ
Fig. 13 The meridional diffusivity, Dy, normalized with D∗ =ε1/3n−4/3
E in the friction-dominated regime (open triangles) and
with D∗ = ε1/3n−4/3β in the transitional and zonostrophic regimes
(open diamonds and asterisks), respectively. In the former re-gime, nE ∝ nfr is the wavenumber of the scale with the maximumkinetic energy
to the meridional diffusion. In the framework of thistheory, the meridional diffusion coefficient at a scale lis given by
Dy ∝ E1/2l, (19)
where E is the turbulence energy contained in all scalesfrom l and smaller. In the friction-dominated regime,the scale l ∼ n−1
E ∝ n−1fr belongs in the KK regime such
that E can be estimated by integrating the spectrum 10from ∞ to nE giving Dy ∼ ε1/3n−4/3
E , in agreement withEq. 17. In the zonostrophic regime, if the maximumscale contributing to Dy is O(n−1
β ), then again the KKspectrum can be used for the estimation of E and Eq. 19becomes Dy ∼ ε1/3n−4/3
β in agreement with Eq. 18.In order to understand the transition from the laws
17 to 18 for Rβ > 1 in terms of the Richardson diffusionlaw, we recall that this law,
κ ∝ ε1/3k−4/3, (20)
is basically equivalent to Eq. 19. More specifically, κ
is the ef fective diffusivity that reflects the cumulativecontribution from eddies of all scales smaller thank−1 (Galperin and Sukoriansky 2010). In the friction-dominated regime, the flow preserves its isotropy andthe KK spectra over all energy containing scales, theeffect of the RHWs is relatively small, and the com-pliance of Eq. 17 with the Richardson diffusion lawcomes at no surprise. The lateral diffusion law in thetransitional and zonostrophic regimes, on the otherhand, is rather remarkable as it demonstrates thatthe scales where a β-effect is important and whichwere shown in Section 4 to be dominated by RHWs,make no contribution to the meridional diffusion. Theeffective transport on these scales is determined byquasi-isotropic turbulent eddies with scales not exceed-ing a fraction of n−1
β . This result conforms with the well-known property of waves to transfer momentum butnot scalar. An extensive discussion of this property wasgiven in Galperin and Sukoriansky (2010) in the contextof stably stratified flows.
Recasting Eq. 18 in terms of β and ε yields a scale-independent expression,
Dy � 0.76ε3/5β−4/5. (21)
A scaling similar to Eq. 21 was obtained in Held andLarichev (1996), Smith et al. (2002), and Lapeyre andHeld (2003) based upon a theory of the baroclinic eddyheat fluxes and confirmed in numerical simulations(Smith 2005). This scaling emerged from the dimen-sional considerations in the assumption that the inverse
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Ocean Dynamics (2010) 60:427–441 437
energy cascade is halted by a β-effect and, thus, that Dy
depends on ε and β only. As discussed earlier, a β-effectdoes not halt the inverse cascade. The physics behindEq. 18 is that the scales dominated by RHWs do notcontribute to the scalar diffusion.
Tracer dispersion in the deep large-scale oceanic cir-culation may provide an example of the transition fromthe scale-dependent law 20 to its scale-independentcounterpart Eq. 21. As estimated in Galperin et al.(2004), this circulation is transitional to marginallyzonostrophic with Rβ � 1.5 such that one could ex-pect to detect a transition in the diffusion laws. Suchtransition was indeed observed in some experimentson relative dispersion of subsurface floats in the NorthAtlantic (LaCasce 2008). The scale of this transition canbe estimated as n−1
β . With ε ∼ 10−10 m2 s−3 (Ollitraultet al. 2005; Galperin et al. 2006) and β ∼ 10−11 m−1 s−1,n−1
β is approximately 100 km, in good agreement withthe observations.
7 Discussion and conclusions
This paper describes recently discovered new featuresof 2D turbulence with a β-effect: various steady stateregimes, nonlinear waves, and the effect of waves onmeridional diffusion of a scalar. From the viewpointof general fluid dynamics, the new results demonstratethat the commingling of anisotropic turbulence anddispersive waves may produce counter-intuitive effectswhich, nevertheless, appear to be common to flows withdifferent types of extra strains, different types of waves,and even different types of dynamics (2D flows withinverse and 3D flows with direct cascade of energy).
External forcing plays a fundamental role in flowdynamics via establishing turbulence energy cascade.It facilitates the development of the Kolmogorovkinetic energy spectrum in both 2D and 3D turbu-lence. In flows with extra strains that support linearwaves, the rate of the energy cascade, ε, combineswith the extra strain parameters to form transi-tional wavenumbers that separate regions of turbu-lence and wave domination. In turbulence with aβ-effect, such a wavenumber is nβ while in turbulencewith stable stratification, the role of the separatorplays the Ozmidov wavenumber, kO (Galperin andSukoriansky 2010).
For turbulence with a β-effect as well as for tur-bulence with stable stratification, directions with zerowave frequency play a special. In flows with a β-effect, the zero-frequency direction is that with m = 0.Recalling that the energy flux is guided in this direc-
tion and following conventional scaling arguments, onewould expect that the energy spectrum along m = 0would scale with a turbulence characteristic such as ε
rather than β. This is not so in the nonlinear dynamicswith anisotropic waves, however. The steepening of thezonal spectrum in zonostrophic turbulence, its scalingwith β2, and independence of ε, Eq. 15a, are observedprecisely along m = 0.
Flow anisotropization entwined with inverse energycascade renders modes with low n and m very energetic.These modes include Rossby–Haurwitz waves whosehigh energy facilitates strong direct interactions withother modes and the excitation of secondary nonlinearwaves, zonons. Zonons can be viewed as forced oscilla-tions slaved to the energetic RHWs. They are an inte-gral part of the zonostrophic regime. The mechanism ofzonon generation is accommodated neither in conven-tional second-moment closure theories (Holloway andHendershott 1977; Ishihara and Kaneda 2001) nor intheories of wave turbulence and weakly nonlinear waveinteractions (Zakharov et al. 1992).
The sharpening of the spectral peaks of the velocitycorrelation functions shown in Figs. 4 and 7 reveals anincreasing effect of waves on large scales. In Galperinand Sukoriansky (2010), we discussed the effect ofwaves on turbulent transport in stably stratified flowsand showed that scales dominated by internal wavesdo not contribute to diapycnal diffusion while thediapycnal diffusivity becomes scale-independent and isrepresented by the Osborn model. Those scales exceed(0.5kO)−1, the transitional scale at which the 1D verticalenergy spectrum steepened up above the Kolmogorovslope and acquires a buoyancy-dominated distribu-tion, N2k−3
3 , where N is the Brunt–Väisälä frequencyand k3 is the vertical coordinate. Section 6 demon-strates that turbulence with a β-effect reveals a similarfeature, namely scales exceeding the transitional scale,(0.5nβ)−1, are dominated by waves and do notcontribute to the meridional diffusivity. Note thatmeridional and diapycnal directions coincide with thedirections of the respective zero frequencies.
In summary, zonostrophic turbulence is similar toflows with strong stable stratification in three as-pects: its 1D, zonal spectrum steepens along the zero-frequency (m = 0) direction; its anisotropy increaseswith scale, and waves dominate the dynamics on scaleslarger than n−1
β . The spectral thresholds of these phe-nomena can be expressed, respectively, in terms of thetransitional and Ozmidov (Galperin and Sukoriansky2010) wavenumbers, nβ and kO. These similarities inturbulent transport complement similarities betweenlinear internal gravity waves and RHWs studied inBühler (2009).
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438 Ocean Dynamics (2010) 60:427–441
These findings underscore fundamental physical dif-ferences in transport of momentum and scalar in flowsthat combine turbulence and waves and establishscales of the crossover between turbulence- and wave-dominated regimes. The analysis based upon the spec-tral space representation of the two-point, two-timevelocity or vorticity correlation functions helps to de-tect waves in turbulent environment. These resultsshould be useful for quantitative understanding ofphysical processes in a wide variety of flows that com-bine turbulence and waves. They largely alleviate asomewhat pessimistic conclusion reached in Stewart(1969) and Jacobitz et al. (2005) that “there is probablyno really clear-cut distinction between turbulence andwaves, at least when both are coexisting at the samescales.”
As discussed in Sukoriansky et al. (2007), Rβ � 1 forthe terrestrial troposphere so that its macroturbulenceis strongly affected by large-scale drag. For the solargiant planets, Rβ may well exceed 2.5 leading to aslowly variable, distinct zonal band structure of theirweather layer circulations and well observable zonalspectra which are close to Eq. 15a (Galperin et al. 2001,2006). Recent studies explore both the zonostrophicregime and the presence of the zonons in planetarycirculations (Del Genio et al. 2009; Guillot et al. 2009).For the oceanic flows, Rβ � 1.5 and so they can beclassified as transitional or marginally zonostrophic.Visually, these flows are highly variable and usuallydefy description in the framework of a single theory.Averaging in time reveals anisotropization (Zang andWunsch 2001; Huang et al. 2007) and zonation althoughthe zonal structures may have meridional components(Maximenko et al. 2005, 2008; Ollitrault et al. 2006).Maximenko et al. (2008) express doubts regarding thenonlinear jet-like nature of the zonal structures andrefer to them as “striations”. They suggest that thestriations are manifestations of linear stationary Rossbywaves. On the other hand, Schlax and Chelton (2008)suggest that the striations are caused by nearly west-ward propagating eddies. Remarkably, Scott and Wang(2005) determined the existence of the inverse energycascade from the satellite altimetry data which pointsto the nonlinear character of the zonal structures. Theanisotropization of the material transport is anothersign of the nonlinear dynamics (Kamenkovich et al.2009a). The existing data resolution is probably in-sufficient to determine the exact physical mechanism ofthe zonal structures (Schlax and Chelton 2008), and itremains an area of active research (Berloff et al. 2009;Kamenkovich et al. 2009b). Whatever this mechanismis, these quasi-zonal structures play an important rolein the general oceanic circulation and a recent study
proposes a new paradigm of global circulation whichaccords these structures a prominent role (Maximenkoet al. 2009).
With regard to the deep circulation, some numer-ical models show a build-up of the barotropic zonalspectrum that agrees with Eq. 15a (Galperin et al.2004). Other studies reveal that the alternating jets inthe ocean extend throughout the water column up tothe surface (Berloff et al. 2009; Kamenkovich et al.2009b). Another recent study reveals an evidence ofdeep, vertically extending zonal jets, but the data areinsufficient for spectral analysis (Herbei et al. 2008).Deep circulation data of appropriate geographical cov-erage, duration, and resolution is difficult to obtain, andour best hope to have a glance at it would be throughthe use of oceanic drifters (LaCasce and Bower 2000;Rossby 2007).
Large westward propagating oceanic eddies weredetected recently in satellite altimetry data (Cheltonet al. 2007). Their speed corresponds to that of theenergetic baroclinic Rossby waves. The nature of theeddies propagation, their nondispersive behavior, andthe magnitude of their speed suggest that they may be amanifestation of zonons in the near-surface circulation.It would be of great interest to test this hypothesis in thefuture studies that would combine data and numericalmodels.
Finally, we note that nondivergent, 2D flows areoften used as simple models of geophysical flows.Nevertheless, as shown in Section 2, in the presenceof strong nonlinearity and a β-effect, these flows arecharacterized by three independent wavenumbers andmay harbor four different regimes and a new class ofnonlinear waves in the energy subrange alone where nξ
is the largest wavenumber of the three. Furthermore,the importance of forcing and related to it wavenumbernβ still has not been fully appreciated (Sukorianskyet al. 2007). More complicated multilayer baroclinicmodels are expected to be characterized by a largernumber of critical scales and feature a larger varietyof different flow regimes. Many studies assume thatthe main nondimensional parameter characterizing thedynamics of such systems is a criticality, ξ = U(βL2
D)−1,U being a baroclinic shear. A recent study by Zurita-Gotor (2007) mentions difficulties of characterizing atwo-layer baroclinic system in terms of the criticalityonly. In terms of a barotropic model, the criticalitywould be equivalent to (nξ /nR)2 if the wavenumbercorresponding to the Rossby radius of deformation isassociated with the wavenumber of the forcing scale,nξ . Inequality (Eq. 14) shows that this parameter onits own is insufficient to characterize even a barotropicsystem where it needs to be complemented by Rβ . This
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Ocean Dynamics (2010) 60:427–441 439
example underscores the utility of studies of barotropicsystems that help to establish better guidelines for char-acterization of various physical processes in their morecomplicated baroclinic counterparts.
Acknowledgements Discussions with Peter Read, DavidMarshall, Michael McIntyre, Rob Scott, Andy Thompson, PavelBerloff, Igor Kamenkovich, Sergey Nazarenko, and ShaferSmith as well as reviews by Sergei Danilov and anonymousreviewer helped us to improve and clarify the manuscript. Partialsupport of this research by the ARO grants W911NF-05-1-0055and W911NF-09-1-0018, ONR grant N00014-07-1-1065, andthe Israel Science Foundation grant No. 134/03 is gratefullyappreciated.
Appendix: Identification of waves using the two-point,two-time correlation functions
In this appendix, we shall elucidate the method used inSections 3, 4, and 5 to identify dispersion relations ofRossby–Haurwitz waves and zonons in flows featuringturbulence and waves. At first, consider linear forcedvorticity equation on the surface of a rotating spherewith Laplacian viscosity and linear large-scale damping,
∂ζ
∂t+ 2�
R2
∂ψ
∂φ= ν∇2ζ − λζ + ξ, (22)
where ξ is a homogeneous, stochastically steady whiterandom force with the correlation function
〈ξ(ω, n, m)ξ ∗(ω′, n′, m′)〉= Cδ(m + m′)δ(n + n′)δ(ω + ω′). (23)
Here, the angular brackets denote ensemble averaging,the δ-functions reflect the homogeneity of space andtime, and C is a constant of appropriate dimension-ality. We perform spherical harmonics decompositionin space and Fourier transform in time of Eq. 22. Thesolution to this equation in spectral space can be writtenin the form
ζ = G−1(ω, n, m)ξ(ω, n, m), (24)
where G(ω, n, m) = −i{ω + 2βm[n(n + 1)]−1} + Dn isthe Green function and Dn = νn(n + 1) + λ is thedissipation.
The two-point, two-time vorticity correlation func-tion can be written as
〈ζ(ω, n, m)ζ ∗(ω′, n′, m′)〉= Uζ (ω, n, m)δ(m + m′)δ(n + n′)δ(ω + ω′). (25)
Using Eqs. 23, 24, and 25 we find
Uζ (ω, n, m) = C|G(ω, n, m)|−2 (26)
where
|G(ω, n, m)|2 =[ω + 2β
mn(n + 1)
]2
+ D2n. (27)
Equations 26 and 27 show that in inviscid flows withno large-scale drag, the poles of Uζ (ω, n, m) returnthe dispersion relation for the linear Rossby–Haurwitzwaves, Eq. 3. The dissipation term in Eq. 27 causesthe broadening of the peaks which increases on smallerscales where n is larger. Conversely, if a linear flow fieldwith waves is sampled for a long time with sufficientspatial and temporal resolution, the analysis employingthe vorticity (or velocity) correlation function wouldrecover the linear dispersion relations directly from thedata. This dispersion relation is given by the peaks ofthe plots of Uζ as a function of ω for every pair (n, m).Since velocity and vorticity are linearly dependent, thepeaks of their respective correlation functions coincide.Thus, one can utilize the peaks of U(ω) given by Eq. 12for every n and m to establish the dispersion relationω = ω(n, m).
One can apply this method to nonlinear flows thatcomprise turbulence and waves. The effect of the non-linear terms in such flows can be written as a scale-dependent viscosity which may turn out to be complex.Then, the real part of this viscosity can be associatedwith the eddy viscosity (Kraichnan 1976; McComb1991) while the imaginary part can be viewed as theRossby–Haurwitz wave frequency shift (Legras 1980;Kaneda and Holloway 1994; Ishihara and Kaneda2001).
In Section 3, the analysis based upon the velocitycorrelation function 12 is applied to the friction-dominated regime of 2D turbulence on the surface of arotating sphere. This analysis reveals a family of waveswith frequencies exactly corresponding to the linearRossby–Haurwitz waves with the dispersion relation(Eq. 3; see, e.g., Fig. 4). The narrow wave-generatedpeaks are increasingly broadened by turbulence ondecreasing scales. Contrary to Legras (1980), Kanedaand Holloway (1994), and Ishihara and Kaneda (2001),these results demonstrate the absence of the RHWsfrequency shift by turbulence. In Sections 4 and 5, thisanalysis is applied to the zonostrophic regime where itreveals, along with the traditional RHWs, a new classof nonlinear waves, zonons.
Note that the utility of the method based upon theanalysis of the correlation functions is quite general.The method can be applied to either simulated ormeasured 2D and 3D turbulent and nonturbulent flowswith different types of waves with the purpose of iden-tification of these waves and finding their dispersionrelations.
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