The third-order law for increments in magnetohydrodynamic turbulence with constant shear
aa r X i v : . [ phy s i c s . p l a s m - ph ] J u l The third-order law for increments in magnetohydrodynamic turbulence withconstant shear
M. Wan , S. Servidio , S. Oughton , and W. H. Matthaeus Bartol Research Institute and Department of Physics and Astronomy,University of Delaware, Newark, Delaware 19716 Department of Mathematics, University of Waikato, Hamilton, New Zealand
We extend the theory for third-order structure functions in homogeneous incompressible magne-tohydrodynamic (MHD) turbulence to the case in which a constant velocity shear is present. Ageneralization is found of the usual relation [Politano and Pouquet, Phys. Rev. E,
21 (1998)]between third-order structure functions and the dissipation rate in steady inertial range turbulence,in which the shear plays a crucial role. In particular, the presence of shear leads to a third-orderlaw which is not simply proportional to the relative separation. Possible implications for laboratoryand space plasmas are discussed.
PACS numbers: XXX
I. INTRODUCTION
A well known result in hydrodynamic turbulence the-ory is the Kolmogorov–Yaglom (“4/5”) law that relatesthe third-order structure function to the energy dissipa-tion rate [1–3]. Often regarded as a rigorous result of thefluid equations, this law requires assumptions of isotropy,homogeneity, and time stationarity of the statistics of ve-locity increments δ u = u ( x + r ) − u ( x ) (velocity u , spa-tial positions x + r and x ). In addition, and crucially, italso requires adoption of the von K´arm´an hypothesis [4]that the rate of energy dissipation ǫ approaches a con-stant nonzero value as Reynolds number tends to infinity.Without the need for assuming isotropy, one finds ∂∂r i h δu i | δ u | i = − ǫ, (1)where h· · · i indicates an ensemble average and a sum onrepeated indices is implied. If isotropy is further assumedthen, h δu L | δ u | i = − d ǫ | r | , (2)where d is the number of spatial dimensions and δu L =ˆ r · δ u is the increment component measured in the direc-tion of the unit vector ˆ r parallel to the relative separation r . Extension of the third-order law to the case of incom-pressible MHD was reported by Politano and Pouquet[5], who remained close to the approximations made inthe hydrodynamic case. Without assuming isotropy, theyfound ∂∂r k h δz ∓ k | δ z ± | i = − ǫ ± , (3)which, after adoption of isotropy, reduces to, h δz ∓ L | δ z ± | i = − d ǫ ± r, (4)where δ z ± = z ± ( x + r ) − z ± ( x ) are the increments of theEls¨asser variables and δz ± L = ˆ r · δ z ± . The constants ǫ ± are the mean energy dissipation rates of the correspond-ing variables z ± = u ± b , where b is the magnetic fieldfluctuation in Alfv´en speed units.Here we extend the third-order law in MHD turbu-lence to cases in which the isotropy assumption is re-laxed. This is accomplished by introducing homogeneousshear in the velocity field, a simplified and well-studiedapproach in hydrodynamics [6–9]. In particular, it sup-ports departures from strict isotropy and introduction ofshear without consideration of rigid boundaries. MHDthird-order laws have been applied to systems that mayalso admit departures from strict uniformity, due to co-herent large-scale gradients; e.g., plasma confinement de-vices [10, 11] and the solar wind [12–15]. For systems likethese, the homogeneous shear approximation may be areasonable step towards including such large-scale effectsin the relevant MHD turbulence scaling laws. To this end,our derivation of the MHD third-order law will includethe effect of homogeneous shear, leading to a necessarilyanisotropic form for the law.More specifically, we find that a uniform shear intro-duces new terms in the third-order law, so that one canno longer conclude that a particular third-order structurefunction, or even a particular integral of a third-orderstructure function, is proportional to the dissipation ratetimes the relative separation length r . This is in markedcontrast to the situation for the fully isotropic hydrody-namic and MHD cases, given here as Eqs. (2) and (4). Itis, however, entirely consistent with the work of Lindborg[16] and Casciola et al. [17], who derived modifications tothe form of the third-order law for hydrodynamics withshear.The principle theoretical result given below is that auniform shear indeed is responsible for changing the formof the third-order law, whereas a mean magnetic fielddoes not produce such structural changes. Implicationsfor solar wind, laboratory, and astrophysical measure-ments of turbulence are suggested, and in particular theprimacy of the third-order law in unambiguously definingan inertial range is challenged. II. ENERGY DECAY WITH LARGE-SCALEFIELDS
The third-order law is often derived from the steady-state version of an equation related to energy decay.To obtain the version of the law appropriate for MHDwith uniform velocity shear, we follow the same proce-dure used previously for MHD [5, 13], combined with themethod of Casciola et al. [17] for extending Eq. (2) toinclude shear. A uniform magnetic field is also retained,although only the simplest of its consequences will enterthe discussion.First, let us employ a Reynolds decomposition of thevelocity field v = U + u into a mean velocity U ( x ) anda fluctuating component u ( x , t ), where h v i = U and h u i = 0. Similarly we write the total magnetic field, con-veniently expressed in Alfv´en speed units, as B = b + B .We assume B is constant and uniform, but that U ( x )varies in space. However this variation will be taken asnon-random and slowly-varying, so that the turbulenceproperties can be treated as locally homogeneous.Now we write the incompressible MHD equations attwo positions, x and x ′ = x + r : ∂ t z ± i = − ( z ∓ k + U k ∓ B k ) ∂ k ( U i + z ± i ) − ∂ i P + ν∂ k ∂ k z ± i , (5) ∂ t z ±′ i = − ( z ∓′ k + U ′ k ∓ B k ) ∂ ′ k ( U ′ i + z ±′ i ) − ∂ ′ i P ′ + ν∂ ′ k ∂ ′ k z ±′ i . (6)Here the prime denotes quantities at position x ′ , P is thepressure, and ν is the kinematic viscosity, taken equal tothe resistivity hereafter. Subtracting Eq. (5) from Eq. (6)yields the following equation for the Els¨asser increments δ z ± = z ± ( x ′ ) − z ± ( x ): ∂ t δz ± i = − ( δU k + δz ∓ k ) ∂ ′ k δz ± i − ( z ∓ k + U k ∓ B k )( ∂ ′ k + ∂ k ) δz ± i − ( δz ∓ k + δU k ) ∂ k U i − ( z ∓′ k + U ′ k ∓ B k ) δ ( ∂ k U i ) − ( ∂ ′ i + ∂ i ) δP + ν ( ∂ ′ k ∂ ′ k + ∂ k ∂ k ) δz ± i , (7)where we use the property that the primed and unprimedcoordinates are independent, so that ∂ k z ±′ i = 0 and ∂ ′ k z ± i = 0.As noted above, we seek an equation related to energydecay. Multiplying the previous equation by 2 δz ± i andaveraging yields ∂ t h| δz ± i | i = − ∂∂r k h ( δU k + δz ∓ k ) | δz ± i | i + h| δz ± i | ( ∂ k U k + ∂ ′ k U ′ k ) i− h ∂ k U i δz ± i ( δz ∓ k + δU k ) i− h ( z ∓′ k + U ′ k ∓ B k ) δ ( ∂ k U i ) δz ± i i +2 ν ∂ ∂r k h| δz ± i | i − ν h| ∂ k z ± i | i . (8)In arriving at this expression we make use of ∂ k h•i = − ∂∂r k h•i and ∂ ′ k h•i = ∂∂r k h•i . These latter relations fol- low from spatial homogeneity (i.e., translation invarianceof the statistical properties), which can be considered forsome systems to be an exact property (see following sec-tion) or an approximation, e.g., in the case of a weaklyinhomogeneous system. The main results here will be forstrict homogeneity.The last term of Eq. (8) can be identified with thedissipation rates ǫ ± = ν h| ∂ k z ± i | i , (9)which for steady state are also the mean energy trans-fer rates. Following the usual arguments [4], in thelimit of vanishing viscosity ν →
0, it is assumed—notproven—that the ǫ ± remain nonzero, and in effect areexternally prescribed by the rate of supply of turbu-lence energy (and cross helicity). Although this non-trivial assertion is physically plausible [18], it nonethe-less prevents the subsequent developments, including theclassical 4/5-law, from being considered an exact conse-quence of the fluid equations themselves. Furthermore,the penultimate term in Eq. (8), also involving the vis-cosity, is assumed to vanish at high Reynolds numberwhen we are examining the inertial range of separations.For the above-stated set of approximations, the incre-ments r are restricted to lie in the inertial range, that isseparations smaller than the correlation length (energy-containing scale) and bigger than the dissipation scale(scale at which fluctuations are critically damped). Forvariations of the set of assumptions that lead to a third-order law, see e.g., Hill [19]. III. MHD WITH HOMOGENEOUS SHEAR
The above relations need not be strictly homogeneous,as variations in U over the slowly-varying large scalesmay be present. To rectify this and arrive at a generallaw that is translation invariant, we now specialize to thecase of a homogenous shear flow, alluded to in the intro-ductory section. With this choice the tensor ∂U i /∂x j is aconstant matrix independent of position. The turbulenceis then homogeneous and all terms in Eq. (8)—both co-efficients and averaged terms—are only a function of theseparation vector r .Under the hypothesis of steady-state turbulence, theleft-hand side of Eq. (8) vanishes. Integrating in r , overa volume V that is enclosed by a surface S , the equationbecomes: I S (cid:2) ˆ n k h ( δz ∓ k + δU k ) | δz ± i | i (cid:3) d S r + 2 ∂U i ∂x k Z V h δz ± i δz ∓ k i d V r = − V ǫ ± , (10)where V is the volume of the region V and ˆ n k is a unitvector normal to S .If the region of integration is a three dimensionalsphere of radius r , volume V r and surface S r , the in-tegration yields I (cid:2) ˆ r k h ( δz ∓ k + δU k ) | δz ± i | i (cid:3) d S r + 2 ∂U i ∂x k Z h δz ± i δz ∓ k i d V r = − πr ǫ ± , (11)where, in the first term of the equation, ˆ r k is the unit vec-tor normal to the surface of the sphere, and now in spher-ical ( r, θ, φ ) coordinates d S r = r d(cos θ )d φ ≡ r dΩ.Equation (11) may be interpreted as the integral formof the third-order law for incompressible homogeneousMHD turbulence with an external velocity field that isconstant in time but which can vary linearly in space.By setting U = 0 and assuming isotropic turbulence,Eq. (11) will recover the standard third-order law forisotropic MHD turbulence [5], given here as Eq. (4).In standard derivations for isotropic turbulence [5, 13,14], shear is necessarily lacking, and it is assumed thatthe structure functions are rotationally symmetric. Inthat case the above relation is simplified by carrying outthe integrals explicitly. (For a more general case, seebelow.) Here we allow for anisotropy induced by eitherthe large-scale magnetic field, or by the imposed homo-geneous shear. Note that the large-scale magnetic field B does not appear explicitly in the third-order relation,even though it is well documented that such a field in-duces spectral anisotropy in MHD turbulence [20].We now further specialize to the large-scale homoge-neous shear flow U = U x ( y )ˆ x = αy ˆ x in a cartesian( x, y, z ) system, with α = const. The integral form ofthe third-order relation becomes r I (cid:2) ˆ r · h ( δ z ∓ | δ z ± | i (cid:3) dΩ+ αr I (ˆ r · ˆ x )(ˆ r · ˆ y ) h| δ z ± | i dΩ+ 2 α Z h δz ± x δz ∓ y i d V r = − πr ǫ ± . (12)Denoting an angular average over a shell of radius r as h . . . i Ω and a volume average over a sphere of radius r as h . . . i V we may rearrange the above equation as hh δz ∓ L | δ z ± | ii Ω = − αr hh (ˆ r · ˆ x )(ˆ r · ˆ y ) | δ z ± | ii Ω − αr hh δz ± x δz ∓ y ii V − rǫ ± , (13)where, again, δz ± L = ˆ r · δ z ± . This form, based on a spher-ical region of radius r , indicates that all three terms onthe right hand side of the equation have an explicit pro-portionality to r ; moreover, the first and second of thesealso admit an implicit dependence on r . The quantityon the left side of Eq. (13) is the MHD analog of theusual third-order structure function that appears in theYaglom and Kolmogorov laws [1, 2], and we see that inthe presence of homogeneous shear it is not simply pro-portional to the dissipation ǫ ± .At this point we remark on an alternative form thatthe third-order law can assume that may be revealing in anisotropic cases. Recall that Eq. (10) is valid foran arbitrary volume V and its associated bounding sur-face S . The advantage of employing a spherical vol-ume V is that when the flux is isotropic, the integrandin the surface integral will be independent of the direc-tion of r , making the integration trivial. Unfortunately,this property is lost when the turbulence is anisotropic[20, 21]. However, provided that the (energy-like) vec-tor flux F + = h ( δ z − + δ U ) | δ z + | i is smoothly varying in r , it is in principle possible to find a set of nested sur-faces S ( V ) [labeled by their enclosing volume V and withunit normal vectors ˆ n S ], such that the normal compo-nent of the vector flux F + is uniform across S ( V ). Then H S d S ˆ n S · F + = F + n ( V ) S , where the constant normal flux F n is labeled by the volume V bounded by the surface,and S is the value of the surface area. The partner quan-tity F − is defined analogously. Provided these nested sur-faces can be found, the homogeneous shear case, Eq. (10),can then be reduced to F ± n ( V ± ) = h ˆ n ±S ± · ( δ z ∓ + δ U ) | δ z ± | i = − αV ± S ± hh δz ± x δz ∓ y ii V − V ± S ± ǫ ± , (14)where V ± and S ± are the volumes and associated surfaceareas that admit constant normal fluxes F ± n ( V ± ). Notethat in general the constant flux surfaces S + and S − areexpected to be different, e.g., due to cross helicity effects.When homogeneous shear is absent the result inEq. (14) reduces to the formal anisotropic third-orderlaw F ± n ( V ± ) = − V ± S ± ǫ ± . (15)The latter can have application in the cases in whichanisotropy is present due to a mean magnetic field B =0. IV. SUMMARY AND DISCUSSION
We examined the mixed third-order Els¨asser structurefunctions for MHD turbulence, incorporating a constantsheared velocity (homogeneous shear) field in additionto homogeneous fluctuations, under a set of assumptionsthat parallels those used in standard turbulence theoryto derive the Kolmogorov 4/5-law. In analogy to thefindings of Casciola et al. [17] and Lindborg [16] for hy-drodynamics, we find that a law can be obtained for sta-tionary homogeneous turbulence that relates third-orderstructure functions and dissipation, but which also in-volves additional terms. For MHD with a constant im-posed shear, there are shear-related terms that appearin this modified third-order law, as in the hydrodynamiccase. On the other hand, a uniform magnetic field doesnot appear explicitly in this relation.On the basis of a very simple estimate we expect thenew terms in the third-order equation to be of signifi-cance when the large-scale velocity increments are of thesame order or larger than the fluctuation increments atthe same separation r , that is, when δU ∼ δz . In someapplications this condition may be realized, and conse-quently the classical third-order law is modified by thesenew terms. We suspect that for solar wind turbulence,as well as for laboratory devices, the present generalizedform of the third-order law will be relevant. In particu-lar, the modified MHD third-order law no longer admitsan interpretation purely in terms of energy transfer anddissipation, and therefore differs from the isotropic casewithout shear.Further extensions of the third-order law can also beundertaken. For example, by including a large-scale butnon-uniform magnetic field. This will induce additionalterms in the generalization of Eqs. (11)–(13).As a final remark, we note that the modifications ofthe third-order law for energy decay that we describehere can be anticipated in the structure of scale-separatedtransport equations derived for MHD in a weakly inho-mogeneous medium [22, 23]. These two-scale transportequations provide a formalism for evolution of second-order correlation functions, and include nonlinear decay, analogous to our third-order structure functions, alongwith advection and shear terms. On this basis, one couldhave already concluded that the third-order law requiresmodification in the presence of large-scale shear. Thepresent study concentrated only on the special case of ho-mogeneous shear, and generalizations of the third-orderlaw have been found.We expect that future studies based on numerical sim-ulations may provide explicit verification and examplesof the relationships we propose here. Taking into accounteffects like shear, observational studies may prove usefulin a variety of systems with large-scale shear flows, suchas astrophysical and laboratory plasmas. Acknowledgments
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