Role of Third-Order Structure Function in Studying Two-Dimensionalisation of Turbulence

# Role of Third-Order Structure Function in Studying Two-Dimensionalisation of Turbulence

Sagar Chakraborty S.N. Bose National Centre for Basic Sciences
Saltlake, Kolkata 700098, India
July 3, 2019
###### Abstract

We look at various correlation functions, which include those that involve both the velocity and the vorticity fields, in two-dimensional (2D) isotropic homogeneous unforced turbulence. We adopt the more intuitive approach due to Kolmogorov (and subsequently, Landau in his text on fluid dynamics) and show that how the 2D turbulence’s results, obtainable using other methods, may be established in a simpler way. Same method is used to calculate some third-order structure functions for quasi-geostrophic (QG) turbulence for the forward cascade of pseudo-potential enstrophy and the inverse energy cascade in quasi-geostrophic turbulence. These results motivate us to study the two-point third order structure function in the context of the two-dimensionalisation effect. Consequent studies enable us to give a reason for the inverse energy cascade in the two-dimensionalised rapidly rotating three dimensional (3D) incompressible turbulence. For such a system, literature shows a possibility of the exponent of wavenumber in the energy spectrum’s relation to lie between -2 and -3. We argue the existence of a stricter range of -2 to -7/3 for the exponent in the case of rapidly rotating turbulence which is in accordance with the recent experiments. Also, a derivation for the two point third order structure function has been provided helping one to argue that even with slow rotation one gets, although dominated, a spectrum with the exponent -2.87, thereby hinting at the initiation of the two-dimensionalisation effect with rotation. Moreover, using the Gledzer-Ohkitani-Yamada (GOY) shell model, modified for rotation, these signatures of two-dimensionalisation effect have been verified.

###### pacs:
47.27.âi, 47.27.Jv, 47.32.Ef, 92.60.hk, 92.10.Lq

## I Introduction

Rotation, in the face of the discovery of two-dimensionalisation effect, has emerged as a parameter that can progressively make a 3D turbulent flow look like a quasi-2D or a 2D turbulent flow. The phrase ‘look like’ basically means that certain properties of 3D turbulence, such as wavenumber dependence of energy spectrum, direction of energy cascade etc., become such that they give impression that the flow is getting two-dimensionalised. In view of the fact that the dynamics of oceans, atmospheres, liquid planetary cores, fluid envelopes of stars and, other bodies of astrophysical and geophysical interest do require an understanding of inherent properties of turbulence in the rotating frame of reference, the problem of two-dimensionalisation is of central interest to any serious scientist; turbulence in rotating bodies is even of some industrial and engineering interest.
In the steady non-turbulent flow, for low Rossby number () and high Reynolds number (), Taylor-Proudman theoremBatchelor () argues that rotation two-dimensionalises the flow. This argument is often carelessly extended to turbulent flows to explain the rotation induced two-dimensionalisation arising therein. The two-dimensionalisation of the 3D turbulent flow in presence of rotation has begun to be understood as a subtle non-linear effect which is distinctly different from Taylor-Proudman effect.
Cambon et al.Cambon1 () have showed that in the presence of rotation, the transfer of energy from small to high wavenumbers is inhibited; at the same time, the strong angular dependence of this effect leads to a draining of the spectral energy from the parallel to the normal wave vectors (w.r.t. the rotation axis) showing a trend towards two-dimensionalisation. WaleffeWaleffe () has used helical decomposition of the velocity field to study the nature of triad interactions in homogeneous turbulence and coupling it with the instability assumption predicted a transfer of energy toward wave vectors perpendicular to the rotation axis under rapid rotation. The helical decomposition turns out to be very handy to deal with rapidly rotating turbulent flow. In that case the linear eigensolutions of the problem, the so-called inertial waves, have the structure of helical modes. The assumption about the triadic transfers, coupled with resonance condition for non-linear interaction between inertial waves, show that there will be a tendency toward non-linear two-dimensionalisation of the flow.
Simulations by Smith et al.Smith () speak volumes for the two-dimensionalisation effect. They show the coexistence of inverse cascade (a typical feature of 2D turbulence) and forward cascade in forced rotating turbulence within a periodic box of small aspect ratio. In the simulations, the ratio of the mean rates of energy dissipated to the energy injected decreased almost linearly, for less than a critical value, with decrease in (increase in angular velocity ). By the way, a very recent numerical studyWaite () shows similar transition from stratified to quasi-geostrophic turbulence, manifested by the emergence of an inverse cascade – a conclusion that agrees with that of LindborgLindborgQG ().
Although recent experiments by Baroud et al.Baroud1 (); Baroud2 () and Morize et al.Morize1 (); Morize2 () have shed some light on the two-dimensionalisation effect, the scaling of two-point statistics and energy spectrum in rotating turbulence remains a controversial topic. ZhouZhou () in analogy with MHD turbulence has proposed an energy spectrum for rapidly rotating 3D turbulent fluid (also see Canuto ()) and this does seem to be validated by some experimentsBaroud1 (); Baroud2 () and numerical simulationsYeung (); Hattori (); Reshetnyak (); Muller (). But some experimentsMorize1 () do not tally with this proposed spectrum. They predict steeper than spectrum and this again seem to be drawing some support from numerical resultsYang (); Bellet () and analytical results found using wave turbulence theoryGaltier (); Cambon2 ().
Unbiasedly speaking, if one wishes angular velocity to become a relevant parameter in constructing the energy spectrum , simple dimensional analysis would lead one to:

 E(k)∝Ω3m−52ε3−m2k−m (1)

where is a real number. should be restricted within the range 5/3 to 3 to keep the exponents of and in relation (1) positive. The two limits and corresponds to isotropic homogeneous 3D turbulence and 2D turbulence respectively. The spectrum due to Zhou — — is due an intermediate value of . So, as far as the present state of the literature on rotating turbulence goes, two-dimensionalisation of 3D turbulence would mean the dominance of a spectrum which goes towards and which may choose to settle at — an issue yet to be fully resolved.
As a turbulent flow can be treated as the manifestation of a random velocity field, one actually hopes to unveil the statistical properties of the flow rather than every other detail of the flow. This means that we basically are after some probability distribution for the flow: Even today, this remains a tough nut to crack. However, the knowledge of the structure functions assists one to take the first step towards finding the distribution. The structure functions are experimentally measurable and hence are of extreme practical importance. The scaling relations of the structure functions, thus, are the lynchpins of turbulence theory although uncertainty lingers as to their general validity and the details of the derivation as far as the present status of research in turbulence is concerned. Therefore, naturally a lot of time and effort are spent by the scientists working in the field of turbulence to determine the exact forms for these functions and to study various phenomena in their light.
In this article, we shall deal with the phenomenon of two-dimensionalisation of 3D incompressible high Reynold’s number fluid turbulence and try to see what can be said about it from the study of structure functions, especially (to be defined below). Basically, herein we shall comprehensively review the worksSagarE0 (); SagarE5 (); SagarE4 (); SagarE1 (); SagarE2 (); SagarE3 () done in this direction by the author and observe how the method of calculating structure functions, developed by Kolmogorov and subsequently Landau, serves as the cornerstone for studying the two-dimensionalisation effect from the angle adopted by the author.

## Ii 2D turbulence

It may be unanimously accepted that Kolmogorov’s four-fifths lawKolmogorov () is a landmark in the theory of turbulence because it is an exact non-trivial result. In three spatial dimensions, this law says that the two-point third order velocity correlation function behaves as:

 S3≡⟨[{→v(→x+→l)−→v(→x)}.→l|→l|]3⟩=−45εl

where is the rate per unit mass at which energy is being transferred through the inertial range. is the velocity field. The inertial range is the intermediate spatial region postulated by Kolmogorov where the large scale disturbances (flow maintaining mechanisms) and the molecular scale viscous dissipation play no part. This result is of such central significance that attempts are regularly made to understand it afresh and to extend it in other situations involving turbulence. There appears to be following two primary methodsBhattacharjee () of obtaining this result:

1. The original Kolmogorov method put forward in details in the fluid dynamics text due to Landau and LifshitzLandau (). There is no external forcing in this approach, and the equality of dissipation rate and forcing rate for the energy is never enforced.

2. A field-theoretic technique that invokes the so-called ‘dissipation anomaly’ in the high Reynold’s number fluid turbulence. In this approach, there is an external forcing that maintains a steady state turbulence.

The two approaches yield the same important results as they should.
If one goes by the standard procedure given in the book by FrischFrisch () to derive the form of the correlation function in -D turbulence with the assumption of forward energy cascade, one would land up onGawedzki ():

 S3=−12d(d+2)εl (2)

where is the mean rate of dissipation of energy per unit mass. This result is not quite true for the two-dimensional case since it gives for : and not . This is so because the calculation doesn’t take into account the conservation of enstrophy in 2D turbulence which causes the reverse cascade of energyKraichnann (). It might be noted that for is for the regime of scales larger than the forcing scaleSagarE0 (); Bernard ().
Actually, if we consider the two-dimensional turbulence, then in the inviscid limit, we have two conserved quantities – energy and enstrophy. This gives rise to two fluxes with the enstrophy flux occurring from the larger to the smaller spatial scales. The energy flux goes in the reverse direction. Recently, BernardBernard () and LindborgLindborg2D () have used the above mentioned techniques to obtain the third order structure function for both the energy and the enstrophy cascade regions in forced 2D turbulence. We believe that the issue is important enough that a derivation of these results using the Kolmogorov-Landau approach should be useful. This is what we have attempted here and our results do come out in agreement with them. Besides, we also have derived some other correlation functions which deal with vorticity fields in the inertial region and also some two-point second order correlation functions in the dissipative region following the arguments of Landau, thereby consolidating the equivalence between the two approaches mentioned in the beginning.

### ii.1 Second order velocity correlation function

It is a well-established fact that there exists a direct-cascade of enstrophy in 2D turbulence. One defines total enstrophy as where is the vorticity in the Cartesian coordinates; being the velocity field. As we shall consider incompressible fluids only (), we shall take density to be unity and let take over the task of representing position vector in 2D plane. The enstrophy flows through the inertial range and gets dissipated near dissipation scale. Using the antisymmetric symbol that has four components, viz. and , one may define the mean rate of dissipation of enstrophy per unit mass as:

 η≡ν⟨→∇ω.→∇ω⟩ (3) ⇒ η=νεταεθβ⟨(∂τ∂γvα)(∂θ∂γvβ)⟩ (4)

Here, angular brackets denote an averaging procedure which averages over all possible positions of points and at a given instant of time and a given separation. Now, if and represent the fluid velocities at the two neighbouring points at and respectively, one may define rank two correlation tensor:

 Bαβ≡⟨(v2α−v1α)(v2β−v1β)⟩ (5)

For simplicity, we shall take a rather idealised situation of turbulence flow which is homogeneous and isotropic on every scale — a case achievable in practice in a vigorously-shaken-fluid left to itself. The component of the correlation tensor will obviously, then, be dependent on time, a fact which won’t be shown explicitly in what follows. As the features of local turbulence is independent of averaged flow, the result derived below is applicable also to the local turbulence at a distance much smaller than the fundamental scale. Isotropy and homogeneity suggests following general form for

 Bαβ=A1(ρ)δαβ+A2(ρ)ρoαρoβ (6)

where and are functions of time and . The Greek subscripts can take two values and which respectively mean the component along the radial vector and the component in the transverse direction. Einstein’s summation convention will be used extensively. Also,

 →ρ=→ρ2−→ρ1,xxxρoα≡ρα/|→ρ|,xxxρoρ=1,xxxρo⊥=0 (7)

using which in the relation (6), one gets:

 Bαβ=B⊥⊥(δαβ−ρoαρoβ)+Bρρρoαρoβ (8)

One may break the relation (5) as

 Bαβ=⟨v1αv1β⟩+⟨v2αv2β⟩−⟨v1αv2β⟩−⟨v2αv1β⟩ (9)

and defining , one may proceed, keeping in mind the isotropy and the homogeneity, to write

 Bαβ=⟨v2⟩δαβ−2bαβ (10)

Again, having assumed incompressibility, one may write:

 ∂βBαβ=0 (11) ⇒ B⊥⊥=ρB′ρρ+Bρρ (12)

where the equation (8) has been used and prime () is allowed to denote derivative w.r.t. . Near the dissipation region the flow is regular and its velocity varies smoothly which allows to expand in a series of power of . One must take neglecting the higher powers ( is not taken because it leads to the contradictory result that as can be seen from the relation (4)). So, treating as a proportionality constant, let , which means (using equation (12)) and hence,

 ⟨v1αv2β⟩=12⟨v2⟩δαβ−52aρ4δαβ+2aρ2ραρβ (13) ⇒ ⟨(∂1τ∂1γv1α)(∂2θ∂2γv2β)⟩=−72aδθτδαβ+24aδβθδατ+24aδαθδβτ (14) ⇒ εταεθβ⟨(∂τ∂γvα)(∂θ∂γvβ)⟩=−192a (15) ⇒ Bρρ=−ηρ4192ν (16)

In the equation (16), we have put , for these relations are assumed to be valid for arbitrarily small . While writing the relation (16), equation (4) has been recalled. This is the two-point second order correlation function for enstrophy cascade in dissipation range.

### ii.2 Third order velocity correlation function

Now, we shall focus thoroughly on the inertial range. Let’s again define:

 bαβ,γ≡⟨v1αv1βv2γ⟩

Invoking homogeneity and isotropy once again along with the symmetry in the first pair of indices, one may write the most general form of the third rank Cartesian tensor for as

 bαβ,γ = C(ρ)δαβρoγ+D(ρ)(δγβρoα+δαγρoβ)+F(ρ)ρoαρoβρoγ (17)

where, , and are functions of . Yet again, incompressibility dictates:

 ∂∂ρ2γbαβ,γ=∂∂ργbαβ,γ=0 (18) ⇒C′δαβ+Cρδαβ+2Dρδαβ+2D′ρ2ραρβ−2Dρ3ραρβ+F′ρ2ραρβ+Fρ3ραρβ=0 (19)

Putting in equation (19) one gets:

 2C+2D+F=constantρ=0 (20)

where, it as been imposed that should remain finite for . Again, using equation (19), putting and manipulating a bit one gets:

 D=−12(ρC′+C) (21)

using which in relation (20), one arrives at the following expression for :

 F=ρC′−C (22)

Defining

 Bαβγ ≡ ⟨(v2α−v1α)(v2β−v1β)(v2γ−v1γ)⟩ (23) = 2(bαβ,γ+bγβ,α+bαγ,β)

and putting relations (21) and (22) in the equation (23) and using relation (17), one gets:

 Bαβγ=−2ρC′(δαβρoγ+δγβρoα+δαγρoβ)+6(ρC′−C)ρoαρoβρoγ (24) ⇒ S3≡Bρρρ=−6C (25)

which along with relations (21), (22) and (17) yields the following expression:

 bαβ,γ=−S36δαβρoγ+112(ρS′3+S3)(δγβρoα+δαγρoβ)−16(ρS′3−S3)ρoαρoβρoγ (26)

Navier-Stokes equation suggests:

 ∂∂tv1α=−v1γ∂1γv1α−∂1αp1+ν∂1γ∂1γv1α (27) ∂∂tv2β=−v2γ∂2γv2β−∂2βp2+ν∂2γ∂2γv2β (28)

multiplying equations (27) and (28) with and respectively and adding subsequently, one gets the following after averaging the consequent result:

 ∂∂t⟨v1αv2β⟩ = −∂1γ⟨v1γv1αv2β⟩−∂2γ⟨v2γv1αv2β⟩ (29) −∂1α⟨p1v2β⟩−∂2β⟨p2v1α⟩⟩ +ν∂1γ∂1γ⟨v1αv2β+ν∂2γ∂2γ⟨v1αv2β⟩

Due to isotropy, the correlation function for the pressure and velocity, (), should have the form . But since due to solenoidal velocity field, must have the form that in turn must vanish to keep correlation functions finite even at . Thus, equation (29) can be written as:

 ∂∂tbαβ=∂γ(bαγ,β+bβγ,α)+2ν∂γ∂γbαβ (30)

For isotropic and homogeneous turbulence, the condition of incompressibility gives the easily derivable well-known result:

 4∂γbαγ,α=∂γBααγ (31)

Defining and noting that , we get from relations (30) and (31):

 −∂W∂t=12∂δ∂δ(∂γBααγ)−2ν∂δ∂δW (32)

Again, if one defines (which is not to be confused with the rotation rate discussed earlier), for homogeneous isotropic turbulence one may write . So, equation (32) can be manipulated into the following:

 12∂Ω∂t−∂⟨ω2⟩∂t=12∂δ∂δ∂γBααγ+ν∂δ∂δΩ−2ν∂δ∂δ⟨ω2⟩ (33) ⇒ ∂δ∂δ∂γBααγ=4η (34) ⇒ Bααρ=14ηρ3 (35)

Here, we have assumed to be relatively negligible and let so that the terms proportional to vanish. Also, we have recalled that . From equations (17) and (23), and the condition of incompressibility, it readily follows that putting which in expression (35) and integrating subsequently (keeping in mind that shouldn’t blow up at ), we arrive at:

 Bρρρ=+18ηρ3 (36)

This is the one-eighth law for the unforced 2D incompressible turbulence proved using the Kolmogorov-Landau approach.
Let us go back to equation (30). Using expressions (10) and (26), one can rewrite the equation as:

 12∂∂t⟨v2⟩−12∂∂tBρρ=ν∂γ∂γ⟨v2⟩+16ρ3∂∂ρ(ρ3Bρρρ)−νρ∂∂ρ(ρ∂Bρρ∂ρ) (37)

As we are interested in the enstrophy cascade, the first term in the R.H.S. is zero due to homogeneity and the first term in the L.H.S. is zero because of energy remains conserved in 2D turbulence in the inviscid limit (and of course, it is the high Reynolds number regime that we are interested in); it cannot be dissipated at smaller scales. Also, as we are interested in the forward cascade which is dominated by enstrophy cascade, on the dimensional grounds in the inertial region (as it may depend only on and ) may be written as:

 ∂∂tBρρ=Aηρ2 (38)

where is a numerical proportionality constant. Hence, using the relation (38), the equation (37) reduces to the following differential equation:

 16ρ3∂∂ρ(ρ3Bρρρ)=νρ∂∂ρ(ρ∂Bρρ∂ρ)−A2ηρ2 (39)

which when solved using expression (25) in the limit of infinite Reynolds number (), one gets

 Bρρρ=−Aη2ρ3 (40)

Comparing it with the expression (36) for the two-point third order velocity correlation function for the isotropic and homogeneous 2D unforced turbulence in the inertial range of the forward enstrophy cascade, we determine the value of to be -1/4. Therefore, relation (38) yields

 ∂∂tBρρ=−14ηρ2 (41)

This, to the best of our knowledge, is yet another exact new result that has to be verified experimentally and numerically to test its validity.
Suppose in the homogeneous isotropic fully-developed turbulence in two-dimensional space, energy is being supplied and the mean rate of injection of energy per unit mass is denoted by . Let us concentrate on the inverse energy cascade. Then technically we have to proceed as before and on doing so one would re-arrive at the differential equation (37); only that now the arguments would differ. In the larger scales viscosity is not as significant and anyway we shall be interested in the infinite Reynolds number case which would mean that the last term in the R.H.S. of equation (37) would go to zero. One obviously would also set and lets assume in the inverse cascade regime, justification of which can be sought from the fact that the ultimate result that is obtained has been experimentally and numerically verified. So, we are left with the following differential equation:

 16ρ3∂∂ρ(ρ3Bρρρ)=ε (42) ⇒ Bρρρ=+32ερ (43)

where in the last step the integration constant has been set to zero to prevent from blowing up at . This equation (43) is the expression for two-point third order velocity correlation function of the energy cascade in inertial the range.

### ii.3 Second order vorticity correlation function

Recall that:

 W≡⟨ω1ω2⟩ (44) and, Ω≡⟨(ω2−ω1)(ω2−ω1)⟩ (45)

and that due to homogeneity, may be expressed as:

 Ω=2⟨ω2⟩−2W (46)

In the dissipation range: so, and hence we may, choosing a proportionality constant (say), assume:

 Ω=bρ2 (47)

Using relations (44), (46) and (47), one gets:

 ⟨ω1ω2⟩=⟨ω2⟩−b2ρ2 (48) ⇒ ⟨(∂1αω)(∂2αω)⟩=2b (49)

But we know,

 η=ν⟨(∂αω)(∂αω)⟩ (50)

So, relation (49) would yield:

 η=2νb (51)

where, we have put in the relation (49), for, being in the dissipation range, these relations are assumed to be valid for arbitrarily small . Combining relations (47) and (51), one arrives at a experimentally verifiable result for two-point second order vorticity correlation function in the dissipation range:

 Ω=η2νρ2 (52)

### ii.4 Third order mixed correlation function

Now, we wish to find two-point third order mixed correlation function in the inertial range of enstrophy cascade. We start by defining a two-point third order mixed correlation tensor in inertial range:

 Ωβ≡⟨(v2β−v1β)(ω2−ω1)(ω2−ω1)⟩ (53) ⇒ Ωβ=2Mβ+4Wβ (54)

where, and . Due due to isotropy and homogeneity, we can write following form for :

 Mβ=M(ρ)ρoβ (55) ⇒ ∂∂ρ2βMβ=⟨ω1ω1∂2βv2β⟩=0 (56) ⇒ ∂∂ρM(ρ)+M(ρ)ρ=0 (57) ⇒ M(ρ)=constantρ=0 (58)

In the relation (56), we are assuming incompressibility and in writing the relation (58) we have taken into account the fact that should remain finite when . Relations (55) and (58) imply that:

 Mβ=0 (59)

using which in the relation (54), we get:

 Ωβ=4Wβ (60)

From the equations (27) and (28), we may write respectively:

 ∂∂tω1=−v1γ∂1γω1+ν∂1γ∂1γω1 (61) ∂∂tω2=−v2γ∂2γω2+ν∂2γ∂2γω2 (62)

Multiplying equations (61) and (62) by and respectively and adding subsequently, we get the following differential equation after averaging:

 ∂∂tW=2∂βWβ+2ν∂β∂βW (63)

where we have used the fact . Using relations (46) and (60) in the equation (63), one gets for the inertial range for the enstrophy cascade in homogeneous, isotropic and fully-developed freely decaying turbulence in two-dimensional space in the infinite Reynolds number limit (i.e., ) following differential equation:

 ∂∂t⟨ω2⟩−12∂∂tΩ=12ρ∂∂ρ(ρΩρ) (64) ⇒ Ωρ=−2ηρ (65)

In getting relation (65) from the equation (64), we have used the facts: and as it may be supposed that the value of varies considerably with time only over an interval corresponding to the fundamental scale of turbulence and in relation to local turbulence the unperturbed flow may be regarded as steady which mean that for local turbulence one can afford to neglect in comparison with the enstrophy dissipation rate . This result (relation (65)) has gained importance by serving as the starting point in deriving various rigorous inequalities for short-distance scaling exponents in 2D incompressible turbulenceEyink ().

## Iii QG turbulence

Quasi-geostrophic (QG) turbulence is a rather more realistic class of turbulent flow than the isotropic homogeneous 3D turbulence. It can be seen in the large scale flows on oceans and atmosphere; thus having profound geophysical and astrophysical significance. QG turbulenceCharney () stands somewhere in between 2D and 3D turbulences. Thus, naturally it is very appealing candidate that deserves study if one is interested in the two-dimensionalisation effect. In the inviscid limit, besides total energy, QG flows enjoy the possession of yet another conserved quantity which is conserved at the horizontal projection of the particle motion. We shall call this pseudo-potential vorticity to distinguish it from the potential vorticity that is conserved at a particle in a homentropic fluid. Defining pseudo-potential enstrophy as half the square of the pseudo-potential vorticity, one would say that like 2D turbulence there are two cascades — forward cascade of pseudo-potential vorticity and inverse cascade of energy — in QG turbulence which, however, is inherently three dimensional in nature.
Recently, a paperLindborg () has calculated some structure functions in QG turbulence and has made illuminating revelation that isotropy in the sense of CharneyCharney () is useless in deriving the structure functions for QG turbulence. It has gone on to show that formulation of QG turbulence under the constraint of axisymmetry is productive. However, it criticized (though somewhat rightly) the ineffectiveness of use of tensorial quantities in the case of QG turbulence in deriving the results. Now, manipulating the tensorial quantities are at the heart of the derivation of many important two-point velocity correlation functions and other onesLandau (). The technique is very intuitive and straightforward. It has, recently, also been thoroughly used to find out various correlation functions for 2D turbulenceSagarE0 (). In this article, we shall closely (and trickily) follow the original Kolmogorov method put forward in details in the fluid dynamics text due to Landau and LifshitzLandau (); and repeated in the ref.-(SagarE0 ()), to derive structure functions in QG turbulence. The method has the extra advantage to being able to probe into the form for the two-point third order velocity correlation function in the forward pseudo-potential enstrophy cascade regime — this has remained uninvestigated earlier in ref.-(Lindborg ()).

### iii.1 Third order mixed correlation function

First of all, we shall briefly introduce the necessary equations (see ref.-(Salmon ()) for details). Let be the three dimensional velocity field of the fluid in a frame rotating with constant angular velocity . The fluid body (such as ocean) is assumed to be of uniform density with free surface at . Suppose the bottom is rigid. The shallow-water equations, then, are:

 ∂h∂t+→∇.(→vh)=0 (66) and, D→vDt+→f×→v=−g→∇ξ (67)

Here, , , , and . is Coriolis parameter that is Taylor-expanded to write . Using the equations (66) and (67), one gets the relation:

 DDt[^z.({\bf curl}→u)+fh]=0 (68)

Let us assume: a) Rossby number , b) Fractional changes in are small, and c) where is the horizontal scale of the flow. Imposing these three assumptions on the shallow-water equations one can modify the relation (68) to yield

 ∂q∂t+→v.→∇q=0 (69)

where ( being ) may be called pseudo-potential vorticity. Under the same assumptions, for QG flow, one also has the condition:

 →∇.→v=0 (70)

Now, the trick is to select an arbitrary two-dimensional plane in the QG turbulent flow such that the plane’s normal is along and impose the properties of homogeneity and isotropy in the plane only. By the way, one must keep in mind that the so-called fundamental scale of 3D turbulence has its analogy as the horizontal length scale for the case of QG turbulence. The correlation functions to be derived for the forward cascade in this article are valid in the range (which we shall call inertial range) that is much smaller than but quite larger than the scale at which the dissipation is effective. Whereas, the structure function to be derived for the inverse cascade is valid in the range whose scale is larger than the scale at which energy had been fed in. As we shall consider fluid bodies of uniform density only, we shall take density to be unity and let , as usual in this article, take over the task of representing position vector in the 2D plane. The Greek subscripts used herein can take two values and which respectively mean the component along the radial vector and the component in the transverse direction. Whenever we shall use the Latin subscript (e.g., ), it should mean that it can take one more value apart from the ones mentioned above: the third value ‘’ would signify the vertical direction. As before, Einstein’s summation convention will be used extensively. Also,

 →ρ=→ρ2−→ρ1,xxxρoα≡ρα/|→ρ|,xxxρoρ=1,xxxρo⊥=0 (71)

Now, if and represent the horizontal fluid velocities at the two neighbouring points at and respectively then with similar meaning for and , one may define, just for the sake of notational convenience:

 K ≡ ⟨q1q2⟩ (72) and,xxxQ ≡ ⟨(q2−q1)(q2−q1)⟩ (73)

The angular brackets denote an averaging procedure which averages over all possible positions of points and at a given instant of time and a given separation. Due to homogeneity, may be re-expressed as:

 Q=2⟨q2⟩−2K (74)

For simplicity, we shall take a rather idealised situation of QG turbulence which is homogeneous and isotropic on every scale in the plane. For the unforced case, the component of the correlation tensor will obviously be dependent on time, a fact which we won’t be showing explicitly in what follows. As the features of local QG turbulence should be independent of averaged flow, the result derived below is applicable also to the local turbulence in the plane at scale much smaller than the fundamental scale.
Again, we define a two-point third order mixed correlation tensor in inertial range:

 Qβ≡⟨(v2β−v1β)(q2−q1)(q2−q1)⟩ (75) ⇒ Qβ=4Kβ+2Lβ (76)

where just to reduce the effort of writing, we have defined:

 Kβ ≡ ⟨v1βq1q2⟩ (77) and,xxxLβ ≡ ⟨q1q1v2β⟩ (78)

Obviously, isotropy, homogeneity and the condition (70) compels to vanish. Hence, equation (76) reduces to:

 Qβ=4Kβ (79)

From the equation (69), we may write for the points 1 and 2 respectively:

 ∂∂tq1=−v1γ∂1γq1 (80) ∂∂tq2=−v2γ∂2γq2 (81)

Multiplying equations (80) and (81) by and respectively and averaging subsequently after adding, we get the following differential equation:

 ∂K∂t=2∂βKβ (82)

Using relations (74) and (79) in the equation (82), one gets for the inertial range for the pseudo-potential enstrophy cascade in homogeneous and isotropic QG turbulence (forced at an intermediate scale or unforced) in inviscid limit the following differential equation:

 ∂∂t⟨q2⟩−12∂∂tQ=12ρ∂∂ρ(ρQρ) (83) ⇒ Qρ=−2εqρ (84)

In getting relation (84) from the equation (83), we have assumed the following:

1. does not blow up at . This sets the integration constant as zero.

2. , i.e., there exists a pseudo-potential enstrophy sink at small scales due to some dissipative force such as viscosity and is the finite and constant dissipation rate of the mean pseudo-potential enstrophy.

3. due to quasi-stationarity. It may be supposed that the value of varies considerably with time only over an interval corresponding to the fundamental scale of turbulence and in relation to local turbulence the unperturbed flow may be regarded as steady which mean that for local turbulence one can afford to neglect in comparison with the pseudo-potential enstrophy dissipation rate .

### iii.2 Third order velocity correlation function

Having explored the form for two-point third order mixed correlation function in the preceding discussion, we now proceed to find the scaling for the two-point third order velocity correlation function. For this motive, one may define a rank two correlation tensor:

 Bαβ≡⟨(v2α−v1α)(v2β−v1β)⟩ (85)

Isotropy and homogeneity in the plane suggests following general form for

 Bαβ=A1(ρ)δαβ+A2(ρ)ρoαρoβ (86)

where and are functions of time and . Making use of the relations (71) in the equation (86), one gets:

 Bαβ=B⊥⊥(δαβ−ρoαρoβ)+Bρρρoαρoβ (87)

One may expand the R.H.S. of the relation (85) and defining , one may proceed, keeping in mind the isotropy and the homogeneity, to arrive at:

 Bαβ=⟨v2⟩δαβ−2bαβ (88)

Let’s concentrate on the following statistically averaged quantity that will prove to be of crucial importance for deriving the desired results:

 bαβ,γ≡⟨v1αv1βv2γ⟩

Invoking homogeneity and isotropy in the plane once again along with the symmetry in the first pair of indices, one may write the most general form of the third rank Cartesian tensor for this case as

 bαβ,γ = C(ρ)δαβρoγ+D(ρ)(δγβρoα+δαγρoβ)+F(ρ)ρoαρoβρoγ (89)

where, , and are functions of . Imposing the condition (70) on the expression (89), one can get (in the same way as done earlier for the 2D case) the following relations:

 D=−12(ρC′+C) (90) and, F=ρC′−C (91)

Here, prime () denotes derivative w.r.t. . Defining

 Bαβγ ≡ ⟨(v2α−v1α)(v2β−v1β)(v2γ−v1γ)⟩ (92) = 2(bαβ,γ+bγβ,α+bαγ,β)

and putting relations (90) and (91) in the equation (92) and using relation (89), one gets:

 Bαβγ=−2ρC′(δαβρoγ+δγβρoα+δαγρoβ)+6(ρC′−C)ρoαρoβρoγ (93)

which along with relations (89), (90) and (91) yields the following expression:

 bαβ,γ=−Bρρρ6δαβρoγ+112(ρB′ρρρ+Bρρρ)(δγβρoα+δαγρoβ)−16(ρB′ρρρ−Bρρρ)ρoαρoβρoγ (94)

The equation (67) suggests:

 ∂∂tv1α=−v1γ∂1γv1α+f1aϵaαγv1γ−g∂1αξ1 (95) ∂∂tv2β=−v2γ∂2γv2β+f1aϵaβγv2γ−g∂2βξ2 (96)

multiplying equations (95) and (96) with and respectively and adding subsequently, one gets the following:

 ∂∂t⟨v1αv2β⟩ = −∂1γ⟨v1γv1αv2β⟩−∂2γ⟨v2γv1αv2β⟩ (97) +ϵaαγ⟨f1av1γv2β⟩+ϵaβγ⟨f2av2γv1α⟩ −g∂1α⟨ξ1v2β⟩−g∂2β⟨ξ2v1α⟩

Due to isotropy, the correlation function should have the form . This should not be confused with the Coriolis parameter. But since, owing to the relation (70), must have the form , where is a constant. Now, k must vanish to keep correlation functions finite even at . Thus, equation (97) can be written as:

 ∂∂tbαβ=∂γ(bαγ,β+bβγ,α)+f0ϵzαγbγβ+f0ϵzβγbαγ (98)

Here we have used the approximation: . Using equations (88) and (94), one can rewrite equation (98) as:

 12∂∂t⟨v2⟩−12∂∂tBρρ=16ρ3∂∂ρ(ρ3Bρρρ) (99)

Note that the terms containing the Levi-Civita symbol vanish by virtue of the joint effect of the expressions (87) and (88), and the antisymmetry property of Levi-Civita symbol. As we are interested in the pseudo-potential enstrophy cascade, the first term in the L.H.S. is zero because of energy remains conserved in QG turbulence in the inviscid limit: it cannot be dissipated at smaller scales. Also, as we are interested in the forward cascade which is dominated by pseudo-potential enstrophy cascade, on the dimensional grounds in the inertial range (if it is assumed to depend only on and ) may be written as:

 ∂∂tBρρ=Γεqρ2 (100)

where is a numerical proportionality constant. Hence, using the relation (100), the equation (99) reduces to the following differential equation:

 16ρ3∂∂ρ(ρ3Bρρρ)=−Γ2εqρ2 (101)

which when solved, imposing finiteness of for , gives:

 Bρρρ=−Γεq2ρ3 (102)

The relation (102) is the expression for the two-point third order correlation function in the isotropic and homogeneous plane of QG turbulence (forced or unforced) in the range of the forward cascade where there is no overlapping with energy cascade. Since has not been determined, one must confess that the equation (102) is just a scaling law at this stage.
Now suppose the fluid body is being forced at small scales i.e., energy is being supplied and the mean rate of injection of energy per unit mass is denoted by (assumed finite and constant). Let us focus on the inverse energy cascade. Then technically we have to proceed just as before to finally arrive at the differential equation (99). One obviously would set invoking the hypothesisCharney () that there should be equipartition of energy between potential energy and the energy content in each of the two horizontal velocity components in the plane. This equipartition had been proposed in view of the assumption that at sufficiently small scales the interaction of the mean flow with the eddies (and thus the eddy-energies) diminishes; as a result, for increasingly smaller vertical and horizontal scales the energies will tend to become homogeneous and equally distributed among the perturbations. By the way, the concept of equipartition of energy is very old and wide-spread in the literature of statistical mechanics. Historically, equilibrium statistical mechanics had been used to justify many aspects of turbulence e.g., the dual cascades in 2D turbulence etc. A detailed discussion may be found in the books by ChorinChorin () and Lim et al.Lim (). Now, lets also assume that in the inverse cascade regime supposing the forced QG turbulence to be in the state of quasi-stationarity. So we are left with the following differential equation:

 16ρ3∂∂ρ(ρ3Bρρρ)=23εu (103) ⇒ Bρρρ=+εuρ (104)

where in the last step the integration constant has been set to zero to prevent from blowing up at . The expression (104) is the expression for the two-point third order velocity correlation function in the isotropic and homogeneous plane of forced QG turbulence for the inverse energy cascade.
The fact that the structure functions for the inherently three-dimensional QG turbulence are more like that for the 2D turbulence than that for the 3D turbulence speaks volumes for the importance of study of third order structure functions for demystifying the two-dimensionalisation effect of the 3D turbulent fluid due to rapid rotation. This serves as the motivation for jumping into the subject of rotating flows and to attempt finding the form of therein.

## Iv Rotating turbulence

All the studies on the two-dimensionalisation effect are mainly for low high limit while the high and high limit has been rather less ventured in relation to the two-dimensionalisation effect of turbulence, although the second case, we believe, should be analytically more tractable. If, using calculations of structure functions, in the limit of high and high , one wishes to see whether a trend towards two-dimensionalisation of 3D homogeneous isotropic turbulence occurs or not, then basically one would have to check (a) if at small scales for 3D turbulence shows a tilt towards at large scales for the 2D turbulence and (b) if the forward energy cascade is depleted at the smaller scales. As we shall show, in the lowest order calculation this is what one may get, hinting at the initiation of the effect of two-dimensionalisation of 3D turbulence owing to the small anisotropy induced by slow rotation.

### iv.1 Relevant scales

Let us look in to the various length scales that have to be taken into consideration while talking about a homogeneous rotating turbulence which basically satisfies following version of Navier-Stoke’s equation:

 ∂→v∂t+(→v.→∇)→v = −1ρ→∇P−→Ω×(→Ω×→x)−2→Ω×→v+ν∇2→v+→f (105)

In this context is external force and is angular velocity. Various parameters to be considered are: (kinematic viscosity), (finite mean rate of dissipation of energy per unit mass), (angular velocity) and (integral scale which typically is the system-size). The three important time-scales involved in the system are: