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Are the energy and magnetic potential cascades direct or inverse in 2D MHD turbulence?
Physica D 165 (2002) 213–227
Are the energy and magnetic potential cascades direct or inverse in 2D MHD turbulence? Eun-jin Kim a,b,∗ , Béreng‘ere Dubrulle a,c b c
a NCAR, P.O. Box 3000, Boulder, CO 80307-3000, USA Department of Physics, University of California, San Diego, La Jolla, CA 92093-0319, USA CNRS, UMR 5572, Observatoire Midi-Pyrénées, 14 Avenue Belin, F-31400 Toulouse, France
Received 29 February 2000; received in revised form 1 February 2002; accepted 23 February 2002 Communicated by U. Frisch
Abstract Transfer of energy and magnetic potential is studied in 2D MHD turbulence in the presence of a background shear flow and magnetic field. The energy is injected into both fluid and magnetic field by external forcing. By using a two-scale analysis and the Gabor transform, the direction of the cascade of energy and magnetic vector potential is found to depend on the properties of forcings. In particular, in the case of a magnetic forcing that is isotropic, the magnetic vector potential cascades from small to large-scales whereas the energy from large to small-scales. © 2002 Elsevier Science B.V. All rights reserved. PACS: 47.65.+a; 05.45.−a; 02.50.−r Keywords: Magnetohydrodynamics; Stochastic processes and statistics; Nonlinear dynamics
1. Introduction The enstrophy (the mean square vorticity), as well as the energy, is an ideal invariant of 2D hydrodynamics (HD) due to the lack of vortex stretching in 2D. As a result, the energy is transferred from small to large-scales (inverse cascade). In 2D magnetohydrodynamics (MHD), a magnetic field allows the possibility of generation of vorticity through Lorentz force, breaking the conservation of enstrophy; in this case, the conserved quantities are total energy, mean square magnetic vector potential, and cross helicity. Therefore, it is not immediately clear in which direction the energy will be transferred in 2D MHD. Previous study [1], through EDQNM closure, suggested direct cascade of energy and finite enstrophy dissipation rate as viscosity/diffusivity approaches zero, thereby implying the similarity of 2D MHD to 3D HD, rather than to 2D HD. Orszag and Tang [2] confirmed the formation of singularity in a finite time, by direct numerical simulation of a decaying MHD turbulence; they however also found that the singularity is not so severe as in 3D HD and invoked a concept of 2.5 HD. To reconcile these results, Pouquet [1] has noted the possibility of inverse cascade of ∗
Corresponding author. Present address: Department of Physics, University of California–San Diego, La Jolla, CA 92093-0319, USA. E-mail address: [email protected] (E. Kim). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 4 2 5 - 6
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energy, despite the formation of small-scale structures, since the vorticity conservation is only weakly broken in 2D MHD. On the other hand, Fyfe and Montgomery [3] conjectured the existence of the inverse cascade of magnetic vector potential in 2D MHD due to an ideal invariant of the mean square magnetic vector potential. This conjecture was later confirmed by the EDQNM computations of [1] as well as by a direct simulation of [4]. In this paper, we present analytic results on the transfer of energy and magnetic vector potential in 2D MHD, including its direction. Like in [5], we shall exploit an important property of MHD turbulence, linked with the Alfvén effect. This effect accounts for the strong influence of large-scale magnetic fields on smaller eddies, and is intrinsically non-local. We thus adopt a formalism in which non-local effects due to large-scale magnetic fields as well as large-scale background flow are exactly accounted for, while local effects, like the seeding of small-scales through local cascades, are parameterized via stochastic small-scale forces, with prescribed statistics. Hence, we assume that both the fluid and magnetic field are stirred by small-scale random forcings that are homogeneous in space and delta-correlated in time; without an external forcing to the magnetic field, the magnetic vector potential, being a conserved scalar field in 2D, always decays in the presence of Ohmic diffusion (anti-dynamo in 2D). We note that due to the assumption of strong large-scale fields, our results cannot be compared with [4] where large-scale fields are much weaker than small-scale fields (see Section 4 for more discussion). The structure of the paper is as follows. Section 2 contains the formulation of the problem. Analytical solutions for fluctuations are provided in terms of the Gabor transform in Section 3. We compute turbulent viscosity and diffusivity by evaluating momentum and magnetic potential fluxes and provide the numerical computations obtained by EDQNM closure in Section 4. Section 5 discusses the small-scale formation in the presence of a magnetic field. Conclusion is provided in Section 6. 2. Formulation of the problem The governing equations for incompressible fluid and magnetic field in two spatial dimensions (x, z) are: ∂ + u · ∇ ω = −(b · ∇)∇ 2 a + ν∇ 2 ω + Fω , ∂t ∂ + u · ∇ a = η∇ 2 a + Fa . ∂t
(1) (2)
Here ω and a are the vorticity and magnetic vector potential. These are related to the velocity u and magnetic field b = (bx , 0, bz ) by ωyˆ = ∇ × u = (∂z ux − ∂x uz )yˆ and b = ∇ × a yˆ = (−∂z a, 0, ∂x a). Fω and Fa are the (random) forcings; ν and η are the viscosity and Ohmic diffusivity, respectively, which shall be taken to be equal (ν = η) to simplify the analysis. We consider a situation where initially a strong mean large-scale shear flow and a large-scale magnetic field are present at scale L and where small-scales are built via the forcings concentrated over some small characteristic scale l. To account for the non-local interactions between large-scale and small-scale quantities, we introduce a two-scale analysis by assuming scale separation with = l/L 1 and decompose the fields as u = u + u = U + u ,
ω = ω + ω = Ω + ω ,
b = b + b = B + b ,
a = a + a = A + a ,
(3)
where the angular brackets denote an average over the statistics of the random forcings Fω and Fa ; U, . . . , B are large-scale fields and u , . . . , b are small-scale fields. By definition, u = ω = b = a = Fω = Fa = 0. Furthermore, the large-scale fields are assumed to be much stronger than the fluctuations, leading to another small parameter ˜ 1, the ratio of the amplitude of fluctuation to that of mean field. Then, we can neglect in the
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evolution equations for fluctuations the interaction between two fluctuations compared to that between mean field and fluctuation. Thus, non-local interaction such as the Alfvén effect is exactly taken into account. For simplicity, we shall assume that both the large-scale velocity and magnetic field are stationary and directed in x, depending solely on z, i.e., U = U (z)xˆ and A = A(z) (Ω = −∂z U and B = B xˆ with B = −∂z A) and that Ω > 0. This configuration with strong U and B may be found in the solar tachocline where a strong toroidal large-scale magnetic field is thought to be present in the direction parallel to a mean shear flow provided by radial differential rotation. Then, to first order in and ˜ , we obtain the following linear equations for fluctuations: ∂ 2 (4) + U ∂x − ν∇ ω = −B∂x ∇ 2 a + Fω , ∂t ∂ + U ∂x − ν∇ 2 a = −u z ∂z A + Fa , (5) ∂t where the derivative of mean fields A and U are kept only to first order in , ignoring second derivatives of A and U , which are O( 2 ). We note that the above Eqs. (4) and (5) are similar to main governing equations used in [6]. The latter, however, adopted periodic boundary conditions for large-scale fields and treated the effects of background shear to be smaller than that of viscosity and Ohmic diffusion via their perturbation scheme (see Eqs. (3.5)–(3.8) in [6]). In contrast, allowing possible non-periodic boundary conditions on z for U and A, we shall exactly solve full Eqs. (4) and (5) in the Gabor space without perturbation (see Section 3). We are interested in the effect of fluctuations on the evolution of mean fields through the interaction between two fluctuations, which is second-order in ˜ . Thus, the mean field equations are treated to second-order in ˜ as well as , with the resulting equations: [∂t − ν∂zz ]U = −∂x Π − ∂z u x u z − bx bz , [∂t − ν∂zz ]A =
−∂z u z a ,
(6) (7)
where Π ≡ p + b 2 /2. The non-linear effect of fluctuations, reflected in the correlation functions on the right hand side of Eqs. (6) and (7), can be represented by turbulent viscosity νT and turbulent diffusivity ηT , respectively:
u x u z − bx bz = −νT ∂z U = νT Ω,
u z a = −ηT ∂z A = ηT B.
(8)
When νT (ηT ) is negative, fluctuations act as a source for a large-scale velocity (magnetic field), while the opposite holds for positive νT (ηT ). In order to obtain these transport coefficients, first, we analytically solve equations for fluctuations in terms of the Gabor transform in the next section by assuming that U and A are stationary in time. In 2D, a stationary solution for A (or B) is possible only when B is uniform regardless of the sign of νT (see also [5]). Thus, in our formulation, B is implicitly taken to be uniform. Next, by using obtained analytic solutions for fluctuations, we then calculate the total momentum flux u x u z − bx bz and magnetic vector potential flux u z a . Note that the total momentum flux (total stress) consists of Reynolds (fluid) stress u x u z and Maxwell (magnetic) stress bx bz . 3. Fluctuations 3.1. Gabor transform The variation of U and A in z in the equations for fluctuations can be incorporated to first order in via the Gabor transform: the Gabor transform of a given function ψ being ∞ ˆ ψ(k, x, t) ≡ d2 x f (|x − x |) eik·(x−x ) ψ(x , t), (9) −∞
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where f (x) is a filter function with a characteristic scale λ (where l λ L) which decreases rapidly for large x. Since the fluctuation has a characteristic length scale l which is smaller than λ, k in Eq. (9) should satisfy |k| 1/λ. An example of the filter function is the Gaussian f (x) = exp(−x 2 /λ2 ). The Gabor transform can be viewed as a localized Fourier transform around x with a compact support λ. It clearly commutes with time derivative ∂t . It also commutes with space derivative, subject to surface terms which vanish farther from the boundaries than the distance λ. A few key properties of the Gabor transform are as follows. First, the derivative of the Gabor transform can be shown to be 1 ∂i uˆ ≈ iki uˆ + O , (10) kλ where ∗ ≡ 1/(kλ) 1 is a small parameter: ∗ 1. Secondly, the Gabor transform of the product of a function varying over large-scales (e.g. U and A) and a function varying over small-scales (e.g. u and b ) can be expressed to first order in ≡ l/L: ∞ Uj u (k, x, t) = f (x − x ) eik·(x−x ) Uj (x , t)u (x , t) dx ≈ Uj (x, t)uˆ + i∇l (Uj (x, t))∇kl uˆ + O( 2 ), −∞
(11) where we used a Taylor expansion of U around the point x to first order in and an integration by parts. The Taylor expansion converges rapidly since the kernel f varies over scales of the order λ, while U varies over scales of the order L. Note that ∇kl denotes the derivative with respect to l-th component of k, and that the summation over index l is implied. Thirdly, the inverse transform of the Gabor transform is just an integration over all wavenumbers with a proper normalization factor: ∞ 1 ˆ ψ(x, t) = d2 k ψ(x, k, t). (12) f (0)(2π)2 −∞ By using the above properties, the Gabor transform of the equations for fluctuations (4) and (5) can easily be obtained as follows: [Dt + ν(k 2 + p 2 )]ωˆ = iBk(k 2 + p 2 )aˆ + Fˆω ,
(13)
[Dt + ν(k 2 + p 2 )]aˆ = uˆ z B + Fˆa ,
(14)
ˆ a, ˆ ω, ˆ ˆ b, ˆ Fˆω and Fˆa are the Gabor transforms of u , ω , b , a , where k = (k, 0, p) and u(x, k, t) = (uˆ x , 0, uˆ z ); u, Fω and Fa . Here, the total derivative Dt is defined as Dt ≡ ∂t + U ∂x − k∂z U ∂p = ∂t + U ∂x + kΩ∂p .
(15)
This motivates us to introduce the ray equations Dt x = U,
Dt z = 0,
Dt k = 0,
Dt p = kΩ,
which imply x = x0 + U (t − t0 ),
z = z0 ,
k = k0 ,
p = p0 + kΩ(t − t0 ),
ˆ where subscript “0” denotes the initial value at t = t0 . In addition, the following relations hold between ωˆ and u: uˆ x =
−ip ω, ˆ k 2 + p2
uˆ z =
ik ω, ˆ k 2 + p2
(16)
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to leading order in . Note that in deriving Eqs. (13) and (14), we have neglected the surface terms. Thus, the former are valid at a distance large than λ from the boundaries. Due to the neglect of surface terms (boundary terms), our treatment amounts to a local analysis. When the surface terms are not negligible, they should be considered to be lumped into the forcings Fω and Fa . In the sequel, these forcings influence our results via their correlations, which in turn alter some constants in the determination of the viscosity and diffusivity. Physically, this means that the boundary conditions can influence the mean profiles or the energy exchanges even above the surface. This effect is well known, and has been observed e.g. in flows over topography, or over wavy surfaces, where both velocity profiles of heat exchanges were observed to differ from the flat case [7]. In the present case, we shall see that the boundary conditions can influence the direction of the cascades, and the sign of the transport coefficients. 3.2. Solutions Following a similar procedure as in [5], solutions to Eqs. (13) and (14) are sought in terms of expansion parameter 1/γ in the regime 1/γ 1, where γ ≡ |Bk/Ω| ∼ (B/U )kL ∼ (B/L) −1 —this constraint is reasonable since it only requires that the Alfvén frequency of k mode is larger than the inverse of the characteristic time-scale of a large-scale flow. In this limit, one can obtain solutions valid up to second-order in 1/γ as (see Appendix A for intermediate steps): a(x, ˆ k, t) = α
−∞
×
uˆ z (x, k, t) =
∞
Ω B
2
d x
∞ −∞
2
∞
t
d k
dt g(x, k, t : x , k , t)
0
iα ψ sin ζ Fˆω + k|k|
∞
1 α β ψ cos ζ + sin ζ Fˆa , α γ
t
(17)
iα αψ k|k|
γ cos ζ ψ −∞ −∞ 0
1 α γ 3 2 αβ ψ ˆ × Fω + −α βψ sin ζ + cos ζ + α β ψ cos ζ − sin ζ Fˆa . γ α ψ α d2 x
d2 k
dt g(x, k, t : x , k , t)
−α 2 βψ 2 sin ζ +
(18)
Here α≡
|k|
|k |
p β≡ , k
p β ≡ , k
α4 ψ ≡ 1− 2γ 2
−1
, α ≡ , k 2 + p2 k 2 + p 2 −1 α 4 1 , ζ ≡ γ Ω(t − t ) − ( tan −1 β − tan −1 β − α 2 β − α 2 β ). ψ ≡ 1 − 2 4γ 2γ
, (19)
Fˆω ≡ Fˆω (x , k , t), Fˆa ≡ Fˆa (x , k , t ), and g(x, k, t : x , k , t ) is a Green’s function along the particle trajectory, modified by the viscosity/diffusivity and the magnetic field:
α 4 − α 4 g(x, k, t : x , k , t ) = δ(x − x − U (t − t ))δ(z − z )δ(k − k )δ(p − p − kΩ(t − t )) exp 4γ 2 p3 p 3 ×exp −ν k 2 t + . exp ν k 2 t + 3Ωk 3Ωk
(20)
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4. Turbulent transport coefficients 4.1. Analytical results The total momentum flux and magnetic vector potential flux can readily be computed from Eqs. (8), (17)–(20): (a) by using the inverse Gabor transform (12); (b) by approximating correlation functions of forcings in Gabor space in terms of those in Fourier space as follows (see [8]):
Fˆj (x1 , k1 , t1 )Fˆ+ (x2 , k2 , t2 ) ∼ (2π)2 δ(k1 + k2 )f 2 (| 21 (x1 − x2 )|) ei(x1 −x2 )·k2 δjl φ˜ j (k2 , t2 − t1 ),
(21)
where j , l = ω, a; φ˜ j is the Fourier transform of φj (r, t) ≡ Fj (x, t1 )Fj (x + r, t2 ) which is assumed to be ˜ 2 , t2 − t1 ) homogeneous. Note that there is no summation over the index j , and that Fˆa Fˆω = 0. We shall take φ(k to be delta-correlated in time, i.e ˜ 2 , t2 − t1 ) = φ(k ˆ 2 )δ(t2 − t1 ). φ(k
(22)
The total momentum flux and magnetic vector potential obtained this way are provided in Eqs. (B.1) and (B.3), together with some of intermediate steps, in Appendix B. The τ -integrals in Eqs. (B.1) and (B.3) can easily be obtained in the limit of strong shear such that ξ = νk 2 /Ω 1 or in the limit of weak shear such that ξ = νk 2 /Ω 1. In the weak shear case, the shearing (or, tilting) of convective cells (or, fluid motion) by a shear flow can easily be destroyed by the viscosity. This is very different from what happens in the limit of strong shear where the shearing can continue without being inhibited by viscosity. Therefore, the effect of the viscosity is more important than that of the shear in the weak shear limit, while the opposite holds in the strong shear limit. Since we are mainly interested in the effect of shear, we shall limit ourselves to a strong shear case (ξ = νk 2 /Ω 1) for the remainder of the paper. Note that the limit ξ 1 is an additional assumption to that of ˜ ∼ |b |/B ∼ |u |/U 1, on which our analysis is based. In such a strong shear case, the results on turbulent viscosity and diffusivity are as follows:
2/3 ∞ 2ξ 1 1 1 2 4 ηT = d k −φˆ a (k) + 4 Γ (23) [φˆ ω (k) + k φˆ a (k)] , 3 3 2B 2 (2π)2 −∞ 2k νT =
1 2 4B (2π)2
∞ −∞
d2 k
k2 2p 2 − k 2 ˆ ω (k) + ˆ a (k) , φ φ (k 2 + p 2 )3 k 2 + p2
(24)
in the limit γ 1 and t → ∞: here Γ (x) is the Gamma function, and ξ(p/k)3 1 is used to obtain (23). It is interesting to notice that the sign of turbulent diffusivity ηT and that of turbulent viscosity νT in Eqs. (23) and (24) depend on the dominant contribution, between φˆ ω and φˆ a as well as the properties of forcings. In the extreme case where only the fluid is stirred (φˆ a = 0) 1 , ηT is positive 2 , indicating the decay of A. It is because a magnetic field always decays in 2D in the absence of an external magnetic forcing φˆ a . In the same limit, νT is also positive regardless of the property of φˆ ω ; energy is consequently transferred from large to small-scales due to the effect of a magnetic field (see Section 5 for more discussion). This finding supports the direct energy cascade suggested by [1]. In contrast, the sign of turbulent viscosity in 2D HD is negative (in the limit of ξ 1) as can be shown through a similar analysis (cf. [5]). In the opposite extreme case where the magnetic field is forced externally (φˆ ω = 0), ηT becomes negative (recall ξ 1), leading to an inverse cascade of the magnetic vector potential. This result agrees with the previous 1
This limit was considered in [5]. For a real function Fj (x, t), where j = ω, a, the integrals d2 k k 2n φˆ i (k), d2 k k −8/3 φˆ ω (k), and d2 k k 4/3 φˆ a (k) are positive (n is integer), as shown in appendix A in [5]. 2
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studies [1,3,4]. However, unlike our results obtained under the assumption of strong large-scale fields compared to small-scale fields (˜ 1), the numerical simulation reported in [4] was performed in the opposite limit (˜ 1), which implies an inverse cascade of the magnetic vector potential even in the absence of large-scale fields (or Alfvén effect). Therefore, the inverse cascade of magnetic vector potential may just be a consequence of the conservation of mean square magnetic vector potential of 2D MHD in the ideal limit, as mentioned in Section 1. Furthermore, Eq. (24) indicates that the sign of νT hinges on the property of the forcing φˆ a . Let us look at this interesting dependence in ˆ ˆ ˆ detail. First, in the case of isotropic forcing φa (k) = φa (k) = φa ( k 2 + p 2 ), we use polar coordinates (r, θ ) such that d2 k = dr r dθ , k = r cos θ, and p = r sin θ . Then the k-integral becomes ∞ ∞ 2π ∞ 2p 2 − k 2 2 2 ˆ a (r) d2 k φˆ a ( k 2 + p 2 ) 2 = dr φ dθ(2 sin θ − cos θ) = π dr φˆ a (r) > 0. k + p2 −∞ 0 0 0 Therefore, the turbulent viscosity is positive for an isotropic magnetic forcing. However, in the limit where the forcing 2 2 2 2 φˆ a mainly consists of components ∞with2 p/k 1, the integrand of k-integral becomes (2p − k )/(k + p ) ∼ −1 to leading order. Thus, νT ∼ − −∞ d k φˆ a (k) < 0 so that the turbulent viscosity becomes negative. These results reveal an important fact that the negative viscosity can be favored with a highly anisotropic forcing, which was also observed in [9] in a different context. It is worth noting that [4] found no indication of build-up of a large-scale velocity in the case of a magnetic forcing (implying a positive turbulent viscosity), but, as mentioned above, in a situation where the large-scale fields are much weaker than small-scale fields (˜ 1 opposite to our basic assumption ˜ 1). Therefore, strictly speaking, our results cannot be compared with theirs. Nevertheless, our results are in agreement with [4] in the case of an isotropic magnetic forcing (as shown immediately above). It thus suggests that the relaxation of vorticity conservation in 2D MHD may be intrinsically responsible for the positive turbulent viscosity. In between the two extreme cases discussed above, the direction of cascade may hinge on the properties of forcings, such as isotropy/anisotropy, spectrum, amplitude and cutoff. First, the borderline between direct and inverse cascade of magnetic vector potential (equivalent to ηT > 0 or < 0) is obtained for d2 k φˆ a (k) = d2 k ξ 2/3 φˆ ω (k)/k 4 . However, recalling again ξ 1, an inverse cascade of magnetic potential (ηT < 0) can easily established as long as φˆ a does not vanish. This result is numerically confirmed by EDQNM closure in the next subsection. Secondly, the turbulent viscosity is always positive in the case of an isotropic forcing whereas it becomes negative for anisotropic forcing with p/k 1 when d2 k φˆ a (k) > d2 k φˆ ω (k)/k 4 . In real MHD turbulence, these forcings are not externally chosen but derived internally from local cascade which provides the continuous seeding of small-scales. A shortcoming of the present approach, reminiscent of that of the traditional RNG approach [10], is that these local properties cannot be determined recursively. We have however identified an interesting clue in the understanding of cascade direction. In numerical simulations like [1] or RNG computations like [10], external forcings are added to the local dynamics, and may well supersede the influence of the local cascade, therefore externally driving the direction of the cascade through the input forcing. This possible pitfall of numerical simulations or RNG approach is thus one important result of our calculations. Another interesting result of this paper is that the magnitude of ηT and νT decreases as 1/B 2 , becoming very small as B → ∞. In particular, a small amplitude of νT , being of second-order in 1/B, is most likely a consequence of fluctuations turning into Alfvén waves (u i = ±bi ) as B → ∞, which leads to the cancellation between Reynolds and Maxwell stresses (see Appendix B). The sign of this small turbulent viscosity hinges on the relative sign of total stress uz ux − bz bx to Ω. This relative sign turned out to be always positive in the case of fluid forcing while it can be both positive and negative in the case of magnetic forcing depending on the property of the latter. In comparison, in 2D HD, the relative sign of uz ux to Ω is always negative (for ξ 1), rendering a negative turbulent viscosity. Unfortunately, detailed dynamics which controls this relative sign remains elusive at present time. Nevertheless, the existence/lack of vorticity conservation law seems to provide us a guideline for
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the robustness of the negative viscosity for different forcings. In other words, due to the lack of vorticity conservation, the negative turbulent viscosity found for an anisotropic magnetic forcing turns into the positive turbulent viscosity when the property forcing changes to being isotropic. In comparison, in 2D HD, the vorticity is conserved in the ideal limit, thereby leading to a negative turbulent viscosity in the case of either isotropic or anisotropic forcing. Finally, it is important to note that total diffusivity (viscosity) is given by a sum of molecular and turbulent diffusivities (viscosities). Therefore, in the presence of a sufficiently strong large-scale magnetic field, the sign of total turbulent diffusivity (viscosity) may become positive for large enough molecular diffusivity (viscosity) as the amplitude of turbulent diffusivity (viscosity) may become smaller than the molecular value. 4.2. Numerical results The detailed check of our predictions requires direct numerical simulations of the system of Eqs. (1) and (2) which are beyond the scope of the present paper. An interesting confirmation of one of our prediction can however easily been obtained using a second-order closure eddy-damped quasi normal Markovian (EDQNM) approximation, which
Fig. 1. Spectrum of magnetic potential at some selected temporal stages. In the top panel c = M/I = 1, in the bottom panel c = 0.01. Also shown are the initial conditions (the straight lines with power law k −2 spanning the wavenumbers up to 40). On the top panel is also shown the straight line corresponding to the law k −7/3 . Energy injection is at wavenumber k = 23/4 1.682.
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has proven to effectively reproduce the main features of 2D magnetic turbulence [1]. In this limit, the equations for the evolution of the magnetic and kinetic energies are the following: ∂t EkV + 2
k2 V E = NLVl (E V , E M ) + FkV , R k
∂t EkM + 2
k2 M V M M E = NLM l (E , E ) + Fk . RM k
(25)
V M Here, NLVl (E V , E M ) and NLM l (E , E ) are the non-linear terms, responsible for the non-linear cascades and for the energy transfers between the magnetic and the velocity field (their complete expression is given in [1]). R and RM are respectively the Reynolds number and the magnetic Reynolds number. In our case, R = RM . In the following, most of the runs were performed at R = 4500. Some of the runs were repeated up to R = 105 , to make sure the qualitative picture is independent of the Reynolds number. FkV and FkM are the kinetic and magnetic forcing. In our study, they are taken as constant over a narrow band of wavenumber, centered at k = kV , and with amplitude I and M for the kinetic and magnetic part respectively. The EDQNM equations are “one-dimensional”, in the sense that they only describe scalar quantities. They cannot therefore be used to check our predictions related to anisotropic forcings. They can however be used to check some cascade predictions. Indeed, our prediction is that, independently of the isotropy properties of the forcing, the magnetic potential and energy cascades should become direct, as the ratio c = M/I (see Eq. (25))
Fig. 2. Kinetic energy spectrum (top) and magnetic energy spectrum (bottom) at some selected temporal stages. The injection ratio is c = 1. Also shown are the initial conditions (the horizontal line spanning the wavenumbers up to 40), and the straight line corresponding to the law k −1/3 .
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Fig. 3. Kinetic energy spectrum (top) and magnetic energy spectrum (bottom) for c = 0.1.
tends to zero. To check this point, we have undertaken a series of simulation taking as initial condition magnetic and energy spectra concentrated at the large-scale EkM = EkV = 2.41 × 10−2 for k < 40, and sharply decreasing at small-scales. Both kinetic and magnetic energies are injected at wavenumbers k = 23/4 1.682. In the different simulations we vary the injection ratio c = M/I = 1 while keeping the total injection rate M + I constant (equal to one). In Fig. 1 we show the spectra of the magnetic potential at some selected temporal stages for two simulations having injection ratio c = M/I = 1 (top) and c = 0.01 (bottom). When this ratio is 1, one clearly sees the build-up of large-scale magnetic potential, i.e. an inverse cascade. However, as c decreases, the rate of inverse cascade slows down, and in the limit when c = 0 (no magnetic forcing), the inverse cascade is switched off, exactly as we predicted. The other figures (Figs. 2–4) show the spectra of kinetic and magnetic energy at selected temporal stages. When c is the largest, one observes the build-up of large-scale magnetic and kinetic energy, resulting from an inverse cascade of energy. However, in the limit when c is small, one observes almost no build-up of energy at large-scale. This is in agreement with our prediction, that in the limit φa → 0, the viscosity should become positive, irrespectively of anisotropic effects. Note also the very strong coupling between the evolution of the magnetic and kinetic energy. This is a consequence of the existence of the Alfvén effect, a basic ingredient of our theory.
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Fig. 4. Kinetic energy spectrum (top) and magnetic energy spectrum (bottom) for c = 0.1.
5. Energy balance By energy consideration, we now discuss how the small amplitude of turbulent viscosity is related to the formation of singularities in small-scale total energy u 2 + b 2 and enstrophy ω 2 + j 2 (j yˆ = ∇ × b ). In our model, the evolution of small-scale total energy can easily be shown to satisfy the following relation: 1 d 2
u + b 2 = −ν ω 2 + j 2 + Ω 2 νT + Fu · u + Fb · b , 2 dt
(26)
there is no work done by pressure. In order for small-scale total energy to be stationary, the balance needs be obtained among: (i) the work done by forcing Fu · u + Fb · b , where Fω yˆ = ∇ × Fu and Fb = ∇ × Fa y; ˆ (ii) the total energy dissipation ν ω 2 + j 2 ; (iii) the energy carried from (or to) large-scales by the total momentum flux (or total stress) uz ux − bz bz Ω = νT Ω 2 . Note that when νT > 0, the energy (total energy) is carried from large to small-scales (direct cascade) due to total stress, whereas when νT < 0, the energy is carried from small to large-scales (inverse cascade). We note that in the presence of non-uniform large-scale magnetic field, extra terms involving ∂z B are expected to appear in Eq. (26).
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In the following, for the sake of simplicity, we shall consider the case of anisotropic forcing with p/k 1 only since the same analysis can easily be carried over to a more general case. First of all, the work done by forcing (i) can be shown to be ∞ 1 2 1 ˆ ω (k) + k 4 φˆ a (k) ,
Fu · u + Fb · b = d k φ (27) 2(2π)2 −∞ k2 which is of zeroth order in 1/γ and independent of ν. This term can be balanced only by the total energy dissipation (ii) since the total stress term (iii) is of second-order in 1/γ . Therefore, ω 2 + j 2 ∼ O(1/ν) to zeroth order, implying the formation of a singularity in the total small-scale enstrophy as ν → 0. This can formally be checked by calculating the latter as follows: ∞ 1 1 1 2 1 4ˆ ˆ ω (k) + 1 −
ω 2 + j 2 = φ 1 + φ k d k (k) , (28) a 2ν(2π)2 −∞ k2 2γ 2 2γ 2 which diverges as ν → 0. In other words, no matter how the system is forced, either by Fω or Fa , small-scale structures must develop to destroy (or balance) the energy being injected by the forcing. The next order term in the total energy dissipation exactly balances the energy transfer by the total stress (iii), as can be seen from Eqs. (24), (26) and (28). The formation of small-scale structure may also be inferred by calculating a typical scale χ of u and b , defined by
u 2
b 2 ∼ χ≡ . 2
ω
j 2 To leading order in 1/γ , ω 2 j 2 ∝ ν −1 , whereas u 2 b 2 ∝ ν 1/3 since: ∞ 1 1 3Ω 1/3 2 2 2 1
u b d k 2Γ [φˆ ω (k) + k 4 φˆ a (k)]. 3 6(2π)2 Ω −∞ k 2νk 2
(29)
Thus, χ ∝ ν 1/3 , becoming small as ν → 0. It is important to note that in 2D HD, the major balance is between (i) and (iii) to leading order (see, for example, [5]), with energy dissipation ν ω 2 ∝ ν 2/3 vanishing as ν → 0. In other words, the work done by a forcing is always carried to large-scales by Reynolds stress (inverse cascade). Therefore, we have confirmed the results [1,2] on the formation of small-scale structures in 2D MHD in comparison to 2D HD. Nevertheless, it is worth noting that this result may be a consequence of having an arbitrary external forcing. In a decaying turbulence without forcing, the leading order balance will be between (ii) and (iii).
6. Conclusion The direction of transfer of energy and magnetic vector potential has been demonstrated to crucially depend on the properties of forcings. In particular, in the case of a fluid forcing, we illustrated the direct cascades of magnetic vector potential and energy for both isotropic and anisotropic forcings. In the case of a magnetic forcing, we have confirmed the inverse cascade of magnetic vector potential, which is rather well known through various numerical computations (cf. [1,4]). We have also demonstrated that the direction of energy cascade may depend on the property of the forcing, being direct for an isotropic forcing and possibly inverse for a highly anisotropic forcing. These results, showing the robustness of inverse cascade of magnetic vector potential compared to that of energy, may not be surprising considering that mean square magnetic vector potential is an ideal invariant of 2D MHD whereas the vorticity is not. As an aside, we comment that the interesting dependence of the direction of
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transfer of energy and magnetic vector potential is reminiscent of the case of the passive scalar in a compressible smooth flow, where the cascade of the scalar is direct or inverse depending on the dimension and on the degree of compressibility [11]. Finally, we note that in our problem, the forcing may be physically interpreted to represent the effect of boundary conditions on the inertial range (see also [10]). Therefore, we may choose a spectrum for both forcings to ensure Komogorov −3/2 spectrum for the kinetic and magnetic energies; then φˆ ω (k)/k 4 and φˆ a (k) have the same power spectrum. Moreover, if two forcings have the same cutoff for the power spectra, the only free parameter that can be controlled would be their amplitudes.
Acknowledgements We are grateful to L. Valdettaro for the numerical simulations and thank A. Pouquet for interesting discussions and references and B.C. Low for a careful reading of and helpful comments on the manuscript. NCAR is supported by the National Science Foundation. EK was partially funded by US DOE FG03-88ER 53275 and US Department of Energy Contract no. DE-AC02-76-CHO-3073, and BD by a NATO fellowship.
Appendix A In this Appendix, we provide intermediate steps leading to Eqs. (17) and (18). First, to solve coupled Eqs. (13) and (14), we introduce a variable p p0 R= = + Ωt, k k where k = k0 is used. Note that DR = (Dt/DR)Dt = Dt /Ω and R > 0 for t > |p0 /k|/Ω. Then, Eqs. (13) and (14) can be rewritten in terms of R as B 3 1 k (1 + R 2 )a˜ + F˜ω , Ω Ω
(A.1)
B 1 ω˜ + F˜a . kΩ 1 + R 2 Ω
(A.2)
DR ω˜ = i DR a˜ = i
Here P˜ is related to Pˆ as R3 Pˆ , P˜ ≡ exp −ξ R + 3
(A.3)
for P = ω, a, Fω , and Fa . From Eqs. (A.1) and (A.2), we can form a single equation for a˜ as follows: DR [(1 + R 2 )DR a] ˜ + γ 2 (1 + R 2 )a˜ =
iB ˜ 1 Fω + DR [(1 + R 2 )F˜a ], 2 Ω Ω k
(A.4)
where γ ≡ |kB/Ω|. The above equation can be simplified in terms of θ = tan −1 R as: [Dθθ − Q]a˜ =
iB 1 sec 2 θ F˜ω + Dθ [ sec 2 θ F˜a ], 2 Ω Ω k
(A.5)
where Q ≡ −γ 2 sec 4 θ . We shall present here a WKB solution to the above equation, which is a good approximation when γ (1 + R 2 ) 1.
(A.6)
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The above condition is satisfied for all R when γ 1. To solve the inhomogeneous Eq. (A.5) in the WKB approximation, we construct a Green’s function G(θ, θ ) from WKB solutions to the homogeneous equation that are correct to third order in 1/γ : θ + sin 2θ/2 cos 4 θ a˜ ∼ cos θ exp ±iγ tan θ − + , 4γ 2 4γ 2 as cos θ cos θ 1 4 4 G(θ, θ ) = Θ(θ − θ ) sin ϕ exp [ cos θ − cos θ ] . (A.7) γ (1 − cos 4 θ /2γ 2 ) 4γ 2 Here ϕ ≡ γ ( tan θ − tan θ ) − [θ − θ + ( sin 2θ − sin 2θ )/2]/4γ , and Θ(x) is a step function. By assuming aˆ = DR aˆ = ωˆ = 0 at R = R0 (or t = 0), the WKB solution to Eq. (A.5), in terms of R, is R ˜ ) 1 1 dR ψ(R a(R, ˜ R0 ) = sin {ϕ(R; ˜ R )}exp{χ˜ (R; R )} √ √ γ Ω 1 + R 2 R0 1 + R 2 iB ˜ × Fω (R ) + DR [(1 + R 2 )F˜a (R )] , (A.8) kΩ where ˜ ψ(R ) ≡ 1 −
−1 1 , 2γ 2 (1 + R 2 )2 1 R R −1 −1 ϕ(R; ˜ R ) ≡ γ (R − R ) − , − tan R − tan R + 4γ 1 + R2 1 + R 2 1 1 1 . χ˜ (R; R ) ≡ − 2 2 2 4γ (1 + R ) (1 + R 2 )2 After integrating the term involving F˜a by part, we express a solution for a(x, ˆ k, t) in terms of k, p = kR = k(R0 + Ωt), and t to obtain Eq. (17). Next, by virtue of (A.2), a solution for uˆ z in Eq. (18) can be derived in a similar way. We note that the solution for uˆ z is correct to second-order in 1/γ although the solution for aˆ is correct to third order.
Appendix B In this Appendix, we show some of steps to get to Eqs. (23) and (24). We use the inverse Gabor transform Eq. (12) to express magnetic flux a u z as follows: 1 2 d k d2 k a(x, ˆ k , t)uˆ z (x, k , t) .
a u z = f (0)2 (2π)4 |k |≥0 |k |≥0 Then, by plugging Eqs. (17) and (18) in the above equation, by using Eqs. (19)–(22), and then by a change of variable τ ≡ p/k + (t − t2 )Ω, we can obtain the following equation:
∞ 1 ψ 2 φˆ ω (k) 2 2χ¯ ˆ ˆ
a uz = d k −φa (k) + ξ φa (k)[1 + cos (2ϕ)]+ dτ e ¯ [1− cos (2ϕ)] ¯ 2B(2π)2 |k|≥k0 k 2 (k 2 +p 2 ) p/k p 1 p 3 1 3 × exp 2ξ + exp −2ξ τ + τ + O(γ −3 ), (B.1) k 3 k 3
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in the limit as t → ∞. Here
p
νk 2 1 τ pk −1 −1 p , ϕ¯ ≡ γ τ − ξ≡ − tan τ + − tan , − 2 Ω k 4γ k 1 + τ2 k + p2 −1 k4 1 1 1 k4 , χ ¯ ≡ . ψ ≡ 1− − 2γ 2 (k 2 + p 2 )2 4γ 2 (1 + τ 2 )2 (k 2 + p 2 )2
227
(B.2)
In the limit of ξ 1 and γ 1 and for ξ(p/k)3 1, the integral over τ can be performed to leading order in 1/γ , yielding Eq. (23) in the text. In a similar way, the total momentum flux u x u z − bx bz can be evaluated by using uˆ x = −p uˆ z /k, and Eqs. (12), (17), (19)–(22). Keeping terms up to order 1/γ 2 , the result is: ∞ 1 3 Ω p 1 p 3 2 2χ¯
ux uz − bx bz = − 2 exp −2ξ τ + τ d k dτ e exp 2ξ + k 3 k 3 2B (2π)2 |k|≥k0 β φˆ ω (k) 2γ 2 τ 2γ τ 2 τ3 − τ × cos (2 ϕ) ¯ − sin (2 ϕ) ¯ + k 2 (k 2 + p 2 ) 1 + τ 2 (1 + τ 2 )2 (1 + τ 2 )3 φˆ a (k) −2γ 2 τ τ2 τ 3 −τ 2τβ + cos (2ϕ) ¯ + 2γ sin (2ϕ) ¯ + . (B.3) − α2 (1 + τ 2 ) (1+τ 2 )2 1+τ 2 (1+τ 2 )3 Here β = p/k, α = 1/(1 + β 2 ) and ϕ¯ is given by Eq. (B.2). Approximate values for the τ -integrals in Eq. (B.3) can easily be obtained in the case of γ 1 and ξ 1, leading to Eq. (24) in the main text. We note that to leading order, both Reynolds and Maxwell stresses involve a term that is of order γ 0 with the same sign. Therefore, these two leading order terms are exactly cancelled each other for the total stress u x u z − bx bz . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
A. Pouquet, On two-dimensional magnetohydrodynamic turbulence, J. Fluid Mech. 88 (1978) 1–16. S.A. Orszag, C. Tang, Small-scale structure of two-dimensional magnetohydrodynamics, J. Fluid Mech. 90 (1979) 129–143. D. Fyfe, D. Montgomery, High-beta turbulence in two-dimensional magnetohydrodynamics, J. Plasma Phys. 16 (1976) 181–191. D. Biskamp, U. Bremer, Dynamics and statistics of inverse cascade processes in 2D magnetohydrodynamic turbulence, Phys. Rev. Lett. 72 (1993) 3819–3822. E. Kim, B. Dubrulle, Turbulent transport and equilibrium profiles in 2D MHD with background shear, Phys. Plasmas 8 (2001) 813–824. D. Montgomery, T. Hatori, Analytical estimates of turbulent MHD transport coefficients, Plasma Phys. Contr. Fusion 26 (1984) 717–730. S. Ciliberto, C. Laroche, Random roughness of boundary increases the turbulent convection scaling exponent, Phys. Rev. Lett. 82 (1999) 3998–4001. S. Nazarenko, N. Kevlahan, B. Dubrulle, Non-linear RDT of near wall turbulence, Physica D 139 (1999) 158–176. B. Dubrulle, U. Frisch, The eddy viscosity of parity invariant flow, Phys. Rev. A 43 (1991) 5355–5364. W.Z. Liang, P.H. Diamond, A renormalization group analysis of two-dimensional magnetohydrodynamic turbulence, Phys. Fluid B 5 (1993) 63–73. M. Chertkov, I. Kokolov, M. Vergassola, Inverse versus direct cascades in turbulent advection, Phys. Rev. Lett. 80 (1998) 512–515.
Are the energy and magnetic potential cascades direct or inverse in 2D MHD turbulence? Eun-jin Kim a,b,∗ , Béreng‘ere Dubrulle a,c b c
a NCAR, P.O. Box 3000, Boulder, CO 80307-3000, USA Department of Physics, University of California, San Diego, La Jolla, CA 92093-0319, USA CNRS, UMR 5572, Observatoire Midi-Pyrénées, 14 Avenue Belin, F-31400 Toulouse, France
Received 29 February 2000; received in revised form 1 February 2002; accepted 23 February 2002 Communicated by U. Frisch
Abstract Transfer of energy and magnetic potential is studied in 2D MHD turbulence in the presence of a background shear flow and magnetic field. The energy is injected into both fluid and magnetic field by external forcing. By using a two-scale analysis and the Gabor transform, the direction of the cascade of energy and magnetic vector potential is found to depend on the properties of forcings. In particular, in the case of a magnetic forcing that is isotropic, the magnetic vector potential cascades from small to large-scales whereas the energy from large to small-scales. © 2002 Elsevier Science B.V. All rights reserved. PACS: 47.65.+a; 05.45.−a; 02.50.−r Keywords: Magnetohydrodynamics; Stochastic processes and statistics; Nonlinear dynamics
1. Introduction The enstrophy (the mean square vorticity), as well as the energy, is an ideal invariant of 2D hydrodynamics (HD) due to the lack of vortex stretching in 2D. As a result, the energy is transferred from small to large-scales (inverse cascade). In 2D magnetohydrodynamics (MHD), a magnetic field allows the possibility of generation of vorticity through Lorentz force, breaking the conservation of enstrophy; in this case, the conserved quantities are total energy, mean square magnetic vector potential, and cross helicity. Therefore, it is not immediately clear in which direction the energy will be transferred in 2D MHD. Previous study [1], through EDQNM closure, suggested direct cascade of energy and finite enstrophy dissipation rate as viscosity/diffusivity approaches zero, thereby implying the similarity of 2D MHD to 3D HD, rather than to 2D HD. Orszag and Tang [2] confirmed the formation of singularity in a finite time, by direct numerical simulation of a decaying MHD turbulence; they however also found that the singularity is not so severe as in 3D HD and invoked a concept of 2.5 HD. To reconcile these results, Pouquet [1] has noted the possibility of inverse cascade of ∗
Corresponding author. Present address: Department of Physics, University of California–San Diego, La Jolla, CA 92093-0319, USA. E-mail address: [email protected] (E. Kim). 0167-2789/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 7 - 2 7 8 9 ( 0 2 ) 0 0 4 2 5 - 6
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energy, despite the formation of small-scale structures, since the vorticity conservation is only weakly broken in 2D MHD. On the other hand, Fyfe and Montgomery [3] conjectured the existence of the inverse cascade of magnetic vector potential in 2D MHD due to an ideal invariant of the mean square magnetic vector potential. This conjecture was later confirmed by the EDQNM computations of [1] as well as by a direct simulation of [4]. In this paper, we present analytic results on the transfer of energy and magnetic vector potential in 2D MHD, including its direction. Like in [5], we shall exploit an important property of MHD turbulence, linked with the Alfvén effect. This effect accounts for the strong influence of large-scale magnetic fields on smaller eddies, and is intrinsically non-local. We thus adopt a formalism in which non-local effects due to large-scale magnetic fields as well as large-scale background flow are exactly accounted for, while local effects, like the seeding of small-scales through local cascades, are parameterized via stochastic small-scale forces, with prescribed statistics. Hence, we assume that both the fluid and magnetic field are stirred by small-scale random forcings that are homogeneous in space and delta-correlated in time; without an external forcing to the magnetic field, the magnetic vector potential, being a conserved scalar field in 2D, always decays in the presence of Ohmic diffusion (anti-dynamo in 2D). We note that due to the assumption of strong large-scale fields, our results cannot be compared with [4] where large-scale fields are much weaker than small-scale fields (see Section 4 for more discussion). The structure of the paper is as follows. Section 2 contains the formulation of the problem. Analytical solutions for fluctuations are provided in terms of the Gabor transform in Section 3. We compute turbulent viscosity and diffusivity by evaluating momentum and magnetic potential fluxes and provide the numerical computations obtained by EDQNM closure in Section 4. Section 5 discusses the small-scale formation in the presence of a magnetic field. Conclusion is provided in Section 6. 2. Formulation of the problem The governing equations for incompressible fluid and magnetic field in two spatial dimensions (x, z) are: ∂ + u · ∇ ω = −(b · ∇)∇ 2 a + ν∇ 2 ω + Fω , ∂t ∂ + u · ∇ a = η∇ 2 a + Fa . ∂t
(1) (2)
Here ω and a are the vorticity and magnetic vector potential. These are related to the velocity u and magnetic field b = (bx , 0, bz ) by ωyˆ = ∇ × u = (∂z ux − ∂x uz )yˆ and b = ∇ × a yˆ = (−∂z a, 0, ∂x a). Fω and Fa are the (random) forcings; ν and η are the viscosity and Ohmic diffusivity, respectively, which shall be taken to be equal (ν = η) to simplify the analysis. We consider a situation where initially a strong mean large-scale shear flow and a large-scale magnetic field are present at scale L and where small-scales are built via the forcings concentrated over some small characteristic scale l. To account for the non-local interactions between large-scale and small-scale quantities, we introduce a two-scale analysis by assuming scale separation with = l/L 1 and decompose the fields as u = u + u = U + u ,
ω = ω + ω = Ω + ω ,
b = b + b = B + b ,
a = a + a = A + a ,
(3)
where the angular brackets denote an average over the statistics of the random forcings Fω and Fa ; U, . . . , B are large-scale fields and u , . . . , b are small-scale fields. By definition, u = ω = b = a = Fω = Fa = 0. Furthermore, the large-scale fields are assumed to be much stronger than the fluctuations, leading to another small parameter ˜ 1, the ratio of the amplitude of fluctuation to that of mean field. Then, we can neglect in the
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evolution equations for fluctuations the interaction between two fluctuations compared to that between mean field and fluctuation. Thus, non-local interaction such as the Alfvén effect is exactly taken into account. For simplicity, we shall assume that both the large-scale velocity and magnetic field are stationary and directed in x, depending solely on z, i.e., U = U (z)xˆ and A = A(z) (Ω = −∂z U and B = B xˆ with B = −∂z A) and that Ω > 0. This configuration with strong U and B may be found in the solar tachocline where a strong toroidal large-scale magnetic field is thought to be present in the direction parallel to a mean shear flow provided by radial differential rotation. Then, to first order in and ˜ , we obtain the following linear equations for fluctuations: ∂ 2 (4) + U ∂x − ν∇ ω = −B∂x ∇ 2 a + Fω , ∂t ∂ + U ∂x − ν∇ 2 a = −u z ∂z A + Fa , (5) ∂t where the derivative of mean fields A and U are kept only to first order in , ignoring second derivatives of A and U , which are O( 2 ). We note that the above Eqs. (4) and (5) are similar to main governing equations used in [6]. The latter, however, adopted periodic boundary conditions for large-scale fields and treated the effects of background shear to be smaller than that of viscosity and Ohmic diffusion via their perturbation scheme (see Eqs. (3.5)–(3.8) in [6]). In contrast, allowing possible non-periodic boundary conditions on z for U and A, we shall exactly solve full Eqs. (4) and (5) in the Gabor space without perturbation (see Section 3). We are interested in the effect of fluctuations on the evolution of mean fields through the interaction between two fluctuations, which is second-order in ˜ . Thus, the mean field equations are treated to second-order in ˜ as well as , with the resulting equations: [∂t − ν∂zz ]U = −∂x Π − ∂z u x u z − bx bz , [∂t − ν∂zz ]A =
−∂z u z a ,
(6) (7)
where Π ≡ p + b 2 /2. The non-linear effect of fluctuations, reflected in the correlation functions on the right hand side of Eqs. (6) and (7), can be represented by turbulent viscosity νT and turbulent diffusivity ηT , respectively:
u x u z − bx bz = −νT ∂z U = νT Ω,
u z a = −ηT ∂z A = ηT B.
(8)
When νT (ηT ) is negative, fluctuations act as a source for a large-scale velocity (magnetic field), while the opposite holds for positive νT (ηT ). In order to obtain these transport coefficients, first, we analytically solve equations for fluctuations in terms of the Gabor transform in the next section by assuming that U and A are stationary in time. In 2D, a stationary solution for A (or B) is possible only when B is uniform regardless of the sign of νT (see also [5]). Thus, in our formulation, B is implicitly taken to be uniform. Next, by using obtained analytic solutions for fluctuations, we then calculate the total momentum flux u x u z − bx bz and magnetic vector potential flux u z a . Note that the total momentum flux (total stress) consists of Reynolds (fluid) stress u x u z and Maxwell (magnetic) stress bx bz . 3. Fluctuations 3.1. Gabor transform The variation of U and A in z in the equations for fluctuations can be incorporated to first order in via the Gabor transform: the Gabor transform of a given function ψ being ∞ ˆ ψ(k, x, t) ≡ d2 x f (|x − x |) eik·(x−x ) ψ(x , t), (9) −∞
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where f (x) is a filter function with a characteristic scale λ (where l λ L) which decreases rapidly for large x. Since the fluctuation has a characteristic length scale l which is smaller than λ, k in Eq. (9) should satisfy |k| 1/λ. An example of the filter function is the Gaussian f (x) = exp(−x 2 /λ2 ). The Gabor transform can be viewed as a localized Fourier transform around x with a compact support λ. It clearly commutes with time derivative ∂t . It also commutes with space derivative, subject to surface terms which vanish farther from the boundaries than the distance λ. A few key properties of the Gabor transform are as follows. First, the derivative of the Gabor transform can be shown to be 1 ∂i uˆ ≈ iki uˆ + O , (10) kλ where ∗ ≡ 1/(kλ) 1 is a small parameter: ∗ 1. Secondly, the Gabor transform of the product of a function varying over large-scales (e.g. U and A) and a function varying over small-scales (e.g. u and b ) can be expressed to first order in ≡ l/L: ∞ Uj u (k, x, t) = f (x − x ) eik·(x−x ) Uj (x , t)u (x , t) dx ≈ Uj (x, t)uˆ + i∇l (Uj (x, t))∇kl uˆ + O( 2 ), −∞
(11) where we used a Taylor expansion of U around the point x to first order in and an integration by parts. The Taylor expansion converges rapidly since the kernel f varies over scales of the order λ, while U varies over scales of the order L. Note that ∇kl denotes the derivative with respect to l-th component of k, and that the summation over index l is implied. Thirdly, the inverse transform of the Gabor transform is just an integration over all wavenumbers with a proper normalization factor: ∞ 1 ˆ ψ(x, t) = d2 k ψ(x, k, t). (12) f (0)(2π)2 −∞ By using the above properties, the Gabor transform of the equations for fluctuations (4) and (5) can easily be obtained as follows: [Dt + ν(k 2 + p 2 )]ωˆ = iBk(k 2 + p 2 )aˆ + Fˆω ,
(13)
[Dt + ν(k 2 + p 2 )]aˆ = uˆ z B + Fˆa ,
(14)
ˆ a, ˆ ω, ˆ ˆ b, ˆ Fˆω and Fˆa are the Gabor transforms of u , ω , b , a , where k = (k, 0, p) and u(x, k, t) = (uˆ x , 0, uˆ z ); u, Fω and Fa . Here, the total derivative Dt is defined as Dt ≡ ∂t + U ∂x − k∂z U ∂p = ∂t + U ∂x + kΩ∂p .
(15)
This motivates us to introduce the ray equations Dt x = U,
Dt z = 0,
Dt k = 0,
Dt p = kΩ,
which imply x = x0 + U (t − t0 ),
z = z0 ,
k = k0 ,
p = p0 + kΩ(t − t0 ),
ˆ where subscript “0” denotes the initial value at t = t0 . In addition, the following relations hold between ωˆ and u: uˆ x =
−ip ω, ˆ k 2 + p2
uˆ z =
ik ω, ˆ k 2 + p2
(16)
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to leading order in . Note that in deriving Eqs. (13) and (14), we have neglected the surface terms. Thus, the former are valid at a distance large than λ from the boundaries. Due to the neglect of surface terms (boundary terms), our treatment amounts to a local analysis. When the surface terms are not negligible, they should be considered to be lumped into the forcings Fω and Fa . In the sequel, these forcings influence our results via their correlations, which in turn alter some constants in the determination of the viscosity and diffusivity. Physically, this means that the boundary conditions can influence the mean profiles or the energy exchanges even above the surface. This effect is well known, and has been observed e.g. in flows over topography, or over wavy surfaces, where both velocity profiles of heat exchanges were observed to differ from the flat case [7]. In the present case, we shall see that the boundary conditions can influence the direction of the cascades, and the sign of the transport coefficients. 3.2. Solutions Following a similar procedure as in [5], solutions to Eqs. (13) and (14) are sought in terms of expansion parameter 1/γ in the regime 1/γ 1, where γ ≡ |Bk/Ω| ∼ (B/U )kL ∼ (B/L) −1 —this constraint is reasonable since it only requires that the Alfvén frequency of k mode is larger than the inverse of the characteristic time-scale of a large-scale flow. In this limit, one can obtain solutions valid up to second-order in 1/γ as (see Appendix A for intermediate steps): a(x, ˆ k, t) = α
−∞
×
uˆ z (x, k, t) =
∞
Ω B
2
d x
∞ −∞
2
∞
t
d k
dt g(x, k, t : x , k , t)
0
iα ψ sin ζ Fˆω + k|k|
∞
1 α β ψ cos ζ + sin ζ Fˆa , α γ
t
(17)
iα αψ k|k|
γ cos ζ ψ −∞ −∞ 0
1 α γ 3 2 αβ ψ ˆ × Fω + −α βψ sin ζ + cos ζ + α β ψ cos ζ − sin ζ Fˆa . γ α ψ α d2 x
d2 k
dt g(x, k, t : x , k , t)
−α 2 βψ 2 sin ζ +
(18)
Here α≡
|k|
|k |
p β≡ , k
p β ≡ , k
α4 ψ ≡ 1− 2γ 2
−1
, α ≡ , k 2 + p2 k 2 + p 2 −1 α 4 1 , ζ ≡ γ Ω(t − t ) − ( tan −1 β − tan −1 β − α 2 β − α 2 β ). ψ ≡ 1 − 2 4γ 2γ
, (19)
Fˆω ≡ Fˆω (x , k , t), Fˆa ≡ Fˆa (x , k , t ), and g(x, k, t : x , k , t ) is a Green’s function along the particle trajectory, modified by the viscosity/diffusivity and the magnetic field:
α 4 − α 4 g(x, k, t : x , k , t ) = δ(x − x − U (t − t ))δ(z − z )δ(k − k )δ(p − p − kΩ(t − t )) exp 4γ 2 p3 p 3 ×exp −ν k 2 t + . exp ν k 2 t + 3Ωk 3Ωk
(20)
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4. Turbulent transport coefficients 4.1. Analytical results The total momentum flux and magnetic vector potential flux can readily be computed from Eqs. (8), (17)–(20): (a) by using the inverse Gabor transform (12); (b) by approximating correlation functions of forcings in Gabor space in terms of those in Fourier space as follows (see [8]):
Fˆj (x1 , k1 , t1 )Fˆ+ (x2 , k2 , t2 ) ∼ (2π)2 δ(k1 + k2 )f 2 (| 21 (x1 − x2 )|) ei(x1 −x2 )·k2 δjl φ˜ j (k2 , t2 − t1 ),
(21)
where j , l = ω, a; φ˜ j is the Fourier transform of φj (r, t) ≡ Fj (x, t1 )Fj (x + r, t2 ) which is assumed to be ˜ 2 , t2 − t1 ) homogeneous. Note that there is no summation over the index j , and that Fˆa Fˆω = 0. We shall take φ(k to be delta-correlated in time, i.e ˜ 2 , t2 − t1 ) = φ(k ˆ 2 )δ(t2 − t1 ). φ(k
(22)
The total momentum flux and magnetic vector potential obtained this way are provided in Eqs. (B.1) and (B.3), together with some of intermediate steps, in Appendix B. The τ -integrals in Eqs. (B.1) and (B.3) can easily be obtained in the limit of strong shear such that ξ = νk 2 /Ω 1 or in the limit of weak shear such that ξ = νk 2 /Ω 1. In the weak shear case, the shearing (or, tilting) of convective cells (or, fluid motion) by a shear flow can easily be destroyed by the viscosity. This is very different from what happens in the limit of strong shear where the shearing can continue without being inhibited by viscosity. Therefore, the effect of the viscosity is more important than that of the shear in the weak shear limit, while the opposite holds in the strong shear limit. Since we are mainly interested in the effect of shear, we shall limit ourselves to a strong shear case (ξ = νk 2 /Ω 1) for the remainder of the paper. Note that the limit ξ 1 is an additional assumption to that of ˜ ∼ |b |/B ∼ |u |/U 1, on which our analysis is based. In such a strong shear case, the results on turbulent viscosity and diffusivity are as follows:
2/3 ∞ 2ξ 1 1 1 2 4 ηT = d k −φˆ a (k) + 4 Γ (23) [φˆ ω (k) + k φˆ a (k)] , 3 3 2B 2 (2π)2 −∞ 2k νT =
1 2 4B (2π)2
∞ −∞
d2 k
k2 2p 2 − k 2 ˆ ω (k) + ˆ a (k) , φ φ (k 2 + p 2 )3 k 2 + p2
(24)
in the limit γ 1 and t → ∞: here Γ (x) is the Gamma function, and ξ(p/k)3 1 is used to obtain (23). It is interesting to notice that the sign of turbulent diffusivity ηT and that of turbulent viscosity νT in Eqs. (23) and (24) depend on the dominant contribution, between φˆ ω and φˆ a as well as the properties of forcings. In the extreme case where only the fluid is stirred (φˆ a = 0) 1 , ηT is positive 2 , indicating the decay of A. It is because a magnetic field always decays in 2D in the absence of an external magnetic forcing φˆ a . In the same limit, νT is also positive regardless of the property of φˆ ω ; energy is consequently transferred from large to small-scales due to the effect of a magnetic field (see Section 5 for more discussion). This finding supports the direct energy cascade suggested by [1]. In contrast, the sign of turbulent viscosity in 2D HD is negative (in the limit of ξ 1) as can be shown through a similar analysis (cf. [5]). In the opposite extreme case where the magnetic field is forced externally (φˆ ω = 0), ηT becomes negative (recall ξ 1), leading to an inverse cascade of the magnetic vector potential. This result agrees with the previous 1
This limit was considered in [5]. For a real function Fj (x, t), where j = ω, a, the integrals d2 k k 2n φˆ i (k), d2 k k −8/3 φˆ ω (k), and d2 k k 4/3 φˆ a (k) are positive (n is integer), as shown in appendix A in [5]. 2
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studies [1,3,4]. However, unlike our results obtained under the assumption of strong large-scale fields compared to small-scale fields (˜ 1), the numerical simulation reported in [4] was performed in the opposite limit (˜ 1), which implies an inverse cascade of the magnetic vector potential even in the absence of large-scale fields (or Alfvén effect). Therefore, the inverse cascade of magnetic vector potential may just be a consequence of the conservation of mean square magnetic vector potential of 2D MHD in the ideal limit, as mentioned in Section 1. Furthermore, Eq. (24) indicates that the sign of νT hinges on the property of the forcing φˆ a . Let us look at this interesting dependence in ˆ ˆ ˆ detail. First, in the case of isotropic forcing φa (k) = φa (k) = φa ( k 2 + p 2 ), we use polar coordinates (r, θ ) such that d2 k = dr r dθ , k = r cos θ, and p = r sin θ . Then the k-integral becomes ∞ ∞ 2π ∞ 2p 2 − k 2 2 2 ˆ a (r) d2 k φˆ a ( k 2 + p 2 ) 2 = dr φ dθ(2 sin θ − cos θ) = π dr φˆ a (r) > 0. k + p2 −∞ 0 0 0 Therefore, the turbulent viscosity is positive for an isotropic magnetic forcing. However, in the limit where the forcing 2 2 2 2 φˆ a mainly consists of components ∞with2 p/k 1, the integrand of k-integral becomes (2p − k )/(k + p ) ∼ −1 to leading order. Thus, νT ∼ − −∞ d k φˆ a (k) < 0 so that the turbulent viscosity becomes negative. These results reveal an important fact that the negative viscosity can be favored with a highly anisotropic forcing, which was also observed in [9] in a different context. It is worth noting that [4] found no indication of build-up of a large-scale velocity in the case of a magnetic forcing (implying a positive turbulent viscosity), but, as mentioned above, in a situation where the large-scale fields are much weaker than small-scale fields (˜ 1 opposite to our basic assumption ˜ 1). Therefore, strictly speaking, our results cannot be compared with theirs. Nevertheless, our results are in agreement with [4] in the case of an isotropic magnetic forcing (as shown immediately above). It thus suggests that the relaxation of vorticity conservation in 2D MHD may be intrinsically responsible for the positive turbulent viscosity. In between the two extreme cases discussed above, the direction of cascade may hinge on the properties of forcings, such as isotropy/anisotropy, spectrum, amplitude and cutoff. First, the borderline between direct and inverse cascade of magnetic vector potential (equivalent to ηT > 0 or < 0) is obtained for d2 k φˆ a (k) = d2 k ξ 2/3 φˆ ω (k)/k 4 . However, recalling again ξ 1, an inverse cascade of magnetic potential (ηT < 0) can easily established as long as φˆ a does not vanish. This result is numerically confirmed by EDQNM closure in the next subsection. Secondly, the turbulent viscosity is always positive in the case of an isotropic forcing whereas it becomes negative for anisotropic forcing with p/k 1 when d2 k φˆ a (k) > d2 k φˆ ω (k)/k 4 . In real MHD turbulence, these forcings are not externally chosen but derived internally from local cascade which provides the continuous seeding of small-scales. A shortcoming of the present approach, reminiscent of that of the traditional RNG approach [10], is that these local properties cannot be determined recursively. We have however identified an interesting clue in the understanding of cascade direction. In numerical simulations like [1] or RNG computations like [10], external forcings are added to the local dynamics, and may well supersede the influence of the local cascade, therefore externally driving the direction of the cascade through the input forcing. This possible pitfall of numerical simulations or RNG approach is thus one important result of our calculations. Another interesting result of this paper is that the magnitude of ηT and νT decreases as 1/B 2 , becoming very small as B → ∞. In particular, a small amplitude of νT , being of second-order in 1/B, is most likely a consequence of fluctuations turning into Alfvén waves (u i = ±bi ) as B → ∞, which leads to the cancellation between Reynolds and Maxwell stresses (see Appendix B). The sign of this small turbulent viscosity hinges on the relative sign of total stress uz ux − bz bx to Ω. This relative sign turned out to be always positive in the case of fluid forcing while it can be both positive and negative in the case of magnetic forcing depending on the property of the latter. In comparison, in 2D HD, the relative sign of uz ux to Ω is always negative (for ξ 1), rendering a negative turbulent viscosity. Unfortunately, detailed dynamics which controls this relative sign remains elusive at present time. Nevertheless, the existence/lack of vorticity conservation law seems to provide us a guideline for
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the robustness of the negative viscosity for different forcings. In other words, due to the lack of vorticity conservation, the negative turbulent viscosity found for an anisotropic magnetic forcing turns into the positive turbulent viscosity when the property forcing changes to being isotropic. In comparison, in 2D HD, the vorticity is conserved in the ideal limit, thereby leading to a negative turbulent viscosity in the case of either isotropic or anisotropic forcing. Finally, it is important to note that total diffusivity (viscosity) is given by a sum of molecular and turbulent diffusivities (viscosities). Therefore, in the presence of a sufficiently strong large-scale magnetic field, the sign of total turbulent diffusivity (viscosity) may become positive for large enough molecular diffusivity (viscosity) as the amplitude of turbulent diffusivity (viscosity) may become smaller than the molecular value. 4.2. Numerical results The detailed check of our predictions requires direct numerical simulations of the system of Eqs. (1) and (2) which are beyond the scope of the present paper. An interesting confirmation of one of our prediction can however easily been obtained using a second-order closure eddy-damped quasi normal Markovian (EDQNM) approximation, which
Fig. 1. Spectrum of magnetic potential at some selected temporal stages. In the top panel c = M/I = 1, in the bottom panel c = 0.01. Also shown are the initial conditions (the straight lines with power law k −2 spanning the wavenumbers up to 40). On the top panel is also shown the straight line corresponding to the law k −7/3 . Energy injection is at wavenumber k = 23/4 1.682.
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has proven to effectively reproduce the main features of 2D magnetic turbulence [1]. In this limit, the equations for the evolution of the magnetic and kinetic energies are the following: ∂t EkV + 2
k2 V E = NLVl (E V , E M ) + FkV , R k
∂t EkM + 2
k2 M V M M E = NLM l (E , E ) + Fk . RM k
(25)
V M Here, NLVl (E V , E M ) and NLM l (E , E ) are the non-linear terms, responsible for the non-linear cascades and for the energy transfers between the magnetic and the velocity field (their complete expression is given in [1]). R and RM are respectively the Reynolds number and the magnetic Reynolds number. In our case, R = RM . In the following, most of the runs were performed at R = 4500. Some of the runs were repeated up to R = 105 , to make sure the qualitative picture is independent of the Reynolds number. FkV and FkM are the kinetic and magnetic forcing. In our study, they are taken as constant over a narrow band of wavenumber, centered at k = kV , and with amplitude I and M for the kinetic and magnetic part respectively. The EDQNM equations are “one-dimensional”, in the sense that they only describe scalar quantities. They cannot therefore be used to check our predictions related to anisotropic forcings. They can however be used to check some cascade predictions. Indeed, our prediction is that, independently of the isotropy properties of the forcing, the magnetic potential and energy cascades should become direct, as the ratio c = M/I (see Eq. (25))
Fig. 2. Kinetic energy spectrum (top) and magnetic energy spectrum (bottom) at some selected temporal stages. The injection ratio is c = 1. Also shown are the initial conditions (the horizontal line spanning the wavenumbers up to 40), and the straight line corresponding to the law k −1/3 .
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Fig. 3. Kinetic energy spectrum (top) and magnetic energy spectrum (bottom) for c = 0.1.
tends to zero. To check this point, we have undertaken a series of simulation taking as initial condition magnetic and energy spectra concentrated at the large-scale EkM = EkV = 2.41 × 10−2 for k < 40, and sharply decreasing at small-scales. Both kinetic and magnetic energies are injected at wavenumbers k = 23/4 1.682. In the different simulations we vary the injection ratio c = M/I = 1 while keeping the total injection rate M + I constant (equal to one). In Fig. 1 we show the spectra of the magnetic potential at some selected temporal stages for two simulations having injection ratio c = M/I = 1 (top) and c = 0.01 (bottom). When this ratio is 1, one clearly sees the build-up of large-scale magnetic potential, i.e. an inverse cascade. However, as c decreases, the rate of inverse cascade slows down, and in the limit when c = 0 (no magnetic forcing), the inverse cascade is switched off, exactly as we predicted. The other figures (Figs. 2–4) show the spectra of kinetic and magnetic energy at selected temporal stages. When c is the largest, one observes the build-up of large-scale magnetic and kinetic energy, resulting from an inverse cascade of energy. However, in the limit when c is small, one observes almost no build-up of energy at large-scale. This is in agreement with our prediction, that in the limit φa → 0, the viscosity should become positive, irrespectively of anisotropic effects. Note also the very strong coupling between the evolution of the magnetic and kinetic energy. This is a consequence of the existence of the Alfvén effect, a basic ingredient of our theory.
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Fig. 4. Kinetic energy spectrum (top) and magnetic energy spectrum (bottom) for c = 0.1.
5. Energy balance By energy consideration, we now discuss how the small amplitude of turbulent viscosity is related to the formation of singularities in small-scale total energy u 2 + b 2 and enstrophy ω 2 + j 2 (j yˆ = ∇ × b ). In our model, the evolution of small-scale total energy can easily be shown to satisfy the following relation: 1 d 2
u + b 2 = −ν ω 2 + j 2 + Ω 2 νT + Fu · u + Fb · b , 2 dt
(26)
there is no work done by pressure. In order for small-scale total energy to be stationary, the balance needs be obtained among: (i) the work done by forcing Fu · u + Fb · b , where Fω yˆ = ∇ × Fu and Fb = ∇ × Fa y; ˆ (ii) the total energy dissipation ν ω 2 + j 2 ; (iii) the energy carried from (or to) large-scales by the total momentum flux (or total stress) uz ux − bz bz Ω = νT Ω 2 . Note that when νT > 0, the energy (total energy) is carried from large to small-scales (direct cascade) due to total stress, whereas when νT < 0, the energy is carried from small to large-scales (inverse cascade). We note that in the presence of non-uniform large-scale magnetic field, extra terms involving ∂z B are expected to appear in Eq. (26).
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In the following, for the sake of simplicity, we shall consider the case of anisotropic forcing with p/k 1 only since the same analysis can easily be carried over to a more general case. First of all, the work done by forcing (i) can be shown to be ∞ 1 2 1 ˆ ω (k) + k 4 φˆ a (k) ,
Fu · u + Fb · b = d k φ (27) 2(2π)2 −∞ k2 which is of zeroth order in 1/γ and independent of ν. This term can be balanced only by the total energy dissipation (ii) since the total stress term (iii) is of second-order in 1/γ . Therefore, ω 2 + j 2 ∼ O(1/ν) to zeroth order, implying the formation of a singularity in the total small-scale enstrophy as ν → 0. This can formally be checked by calculating the latter as follows: ∞ 1 1 1 2 1 4ˆ ˆ ω (k) + 1 −
ω 2 + j 2 = φ 1 + φ k d k (k) , (28) a 2ν(2π)2 −∞ k2 2γ 2 2γ 2 which diverges as ν → 0. In other words, no matter how the system is forced, either by Fω or Fa , small-scale structures must develop to destroy (or balance) the energy being injected by the forcing. The next order term in the total energy dissipation exactly balances the energy transfer by the total stress (iii), as can be seen from Eqs. (24), (26) and (28). The formation of small-scale structure may also be inferred by calculating a typical scale χ of u and b , defined by
u 2
b 2 ∼ χ≡ . 2
ω
j 2 To leading order in 1/γ , ω 2 j 2 ∝ ν −1 , whereas u 2 b 2 ∝ ν 1/3 since: ∞ 1 1 3Ω 1/3 2 2 2 1
u b d k 2Γ [φˆ ω (k) + k 4 φˆ a (k)]. 3 6(2π)2 Ω −∞ k 2νk 2
(29)
Thus, χ ∝ ν 1/3 , becoming small as ν → 0. It is important to note that in 2D HD, the major balance is between (i) and (iii) to leading order (see, for example, [5]), with energy dissipation ν ω 2 ∝ ν 2/3 vanishing as ν → 0. In other words, the work done by a forcing is always carried to large-scales by Reynolds stress (inverse cascade). Therefore, we have confirmed the results [1,2] on the formation of small-scale structures in 2D MHD in comparison to 2D HD. Nevertheless, it is worth noting that this result may be a consequence of having an arbitrary external forcing. In a decaying turbulence without forcing, the leading order balance will be between (ii) and (iii).
6. Conclusion The direction of transfer of energy and magnetic vector potential has been demonstrated to crucially depend on the properties of forcings. In particular, in the case of a fluid forcing, we illustrated the direct cascades of magnetic vector potential and energy for both isotropic and anisotropic forcings. In the case of a magnetic forcing, we have confirmed the inverse cascade of magnetic vector potential, which is rather well known through various numerical computations (cf. [1,4]). We have also demonstrated that the direction of energy cascade may depend on the property of the forcing, being direct for an isotropic forcing and possibly inverse for a highly anisotropic forcing. These results, showing the robustness of inverse cascade of magnetic vector potential compared to that of energy, may not be surprising considering that mean square magnetic vector potential is an ideal invariant of 2D MHD whereas the vorticity is not. As an aside, we comment that the interesting dependence of the direction of
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transfer of energy and magnetic vector potential is reminiscent of the case of the passive scalar in a compressible smooth flow, where the cascade of the scalar is direct or inverse depending on the dimension and on the degree of compressibility [11]. Finally, we note that in our problem, the forcing may be physically interpreted to represent the effect of boundary conditions on the inertial range (see also [10]). Therefore, we may choose a spectrum for both forcings to ensure Komogorov −3/2 spectrum for the kinetic and magnetic energies; then φˆ ω (k)/k 4 and φˆ a (k) have the same power spectrum. Moreover, if two forcings have the same cutoff for the power spectra, the only free parameter that can be controlled would be their amplitudes.
Acknowledgements We are grateful to L. Valdettaro for the numerical simulations and thank A. Pouquet for interesting discussions and references and B.C. Low for a careful reading of and helpful comments on the manuscript. NCAR is supported by the National Science Foundation. EK was partially funded by US DOE FG03-88ER 53275 and US Department of Energy Contract no. DE-AC02-76-CHO-3073, and BD by a NATO fellowship.
Appendix A In this Appendix, we provide intermediate steps leading to Eqs. (17) and (18). First, to solve coupled Eqs. (13) and (14), we introduce a variable p p0 R= = + Ωt, k k where k = k0 is used. Note that DR = (Dt/DR)Dt = Dt /Ω and R > 0 for t > |p0 /k|/Ω. Then, Eqs. (13) and (14) can be rewritten in terms of R as B 3 1 k (1 + R 2 )a˜ + F˜ω , Ω Ω
(A.1)
B 1 ω˜ + F˜a . kΩ 1 + R 2 Ω
(A.2)
DR ω˜ = i DR a˜ = i
Here P˜ is related to Pˆ as R3 Pˆ , P˜ ≡ exp −ξ R + 3
(A.3)
for P = ω, a, Fω , and Fa . From Eqs. (A.1) and (A.2), we can form a single equation for a˜ as follows: DR [(1 + R 2 )DR a] ˜ + γ 2 (1 + R 2 )a˜ =
iB ˜ 1 Fω + DR [(1 + R 2 )F˜a ], 2 Ω Ω k
(A.4)
where γ ≡ |kB/Ω|. The above equation can be simplified in terms of θ = tan −1 R as: [Dθθ − Q]a˜ =
iB 1 sec 2 θ F˜ω + Dθ [ sec 2 θ F˜a ], 2 Ω Ω k
(A.5)
where Q ≡ −γ 2 sec 4 θ . We shall present here a WKB solution to the above equation, which is a good approximation when γ (1 + R 2 ) 1.
(A.6)
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The above condition is satisfied for all R when γ 1. To solve the inhomogeneous Eq. (A.5) in the WKB approximation, we construct a Green’s function G(θ, θ ) from WKB solutions to the homogeneous equation that are correct to third order in 1/γ : θ + sin 2θ/2 cos 4 θ a˜ ∼ cos θ exp ±iγ tan θ − + , 4γ 2 4γ 2 as cos θ cos θ 1 4 4 G(θ, θ ) = Θ(θ − θ ) sin ϕ exp [ cos θ − cos θ ] . (A.7) γ (1 − cos 4 θ /2γ 2 ) 4γ 2 Here ϕ ≡ γ ( tan θ − tan θ ) − [θ − θ + ( sin 2θ − sin 2θ )/2]/4γ , and Θ(x) is a step function. By assuming aˆ = DR aˆ = ωˆ = 0 at R = R0 (or t = 0), the WKB solution to Eq. (A.5), in terms of R, is R ˜ ) 1 1 dR ψ(R a(R, ˜ R0 ) = sin {ϕ(R; ˜ R )}exp{χ˜ (R; R )} √ √ γ Ω 1 + R 2 R0 1 + R 2 iB ˜ × Fω (R ) + DR [(1 + R 2 )F˜a (R )] , (A.8) kΩ where ˜ ψ(R ) ≡ 1 −
−1 1 , 2γ 2 (1 + R 2 )2 1 R R −1 −1 ϕ(R; ˜ R ) ≡ γ (R − R ) − , − tan R − tan R + 4γ 1 + R2 1 + R 2 1 1 1 . χ˜ (R; R ) ≡ − 2 2 2 4γ (1 + R ) (1 + R 2 )2 After integrating the term involving F˜a by part, we express a solution for a(x, ˆ k, t) in terms of k, p = kR = k(R0 + Ωt), and t to obtain Eq. (17). Next, by virtue of (A.2), a solution for uˆ z in Eq. (18) can be derived in a similar way. We note that the solution for uˆ z is correct to second-order in 1/γ although the solution for aˆ is correct to third order.
Appendix B In this Appendix, we show some of steps to get to Eqs. (23) and (24). We use the inverse Gabor transform Eq. (12) to express magnetic flux a u z as follows: 1 2 d k d2 k a(x, ˆ k , t)uˆ z (x, k , t) .
a u z = f (0)2 (2π)4 |k |≥0 |k |≥0 Then, by plugging Eqs. (17) and (18) in the above equation, by using Eqs. (19)–(22), and then by a change of variable τ ≡ p/k + (t − t2 )Ω, we can obtain the following equation:
∞ 1 ψ 2 φˆ ω (k) 2 2χ¯ ˆ ˆ
a uz = d k −φa (k) + ξ φa (k)[1 + cos (2ϕ)]+ dτ e ¯ [1− cos (2ϕ)] ¯ 2B(2π)2 |k|≥k0 k 2 (k 2 +p 2 ) p/k p 1 p 3 1 3 × exp 2ξ + exp −2ξ τ + τ + O(γ −3 ), (B.1) k 3 k 3
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in the limit as t → ∞. Here
p
νk 2 1 τ pk −1 −1 p , ϕ¯ ≡ γ τ − ξ≡ − tan τ + − tan , − 2 Ω k 4γ k 1 + τ2 k + p2 −1 k4 1 1 1 k4 , χ ¯ ≡ . ψ ≡ 1− − 2γ 2 (k 2 + p 2 )2 4γ 2 (1 + τ 2 )2 (k 2 + p 2 )2
227
(B.2)
In the limit of ξ 1 and γ 1 and for ξ(p/k)3 1, the integral over τ can be performed to leading order in 1/γ , yielding Eq. (23) in the text. In a similar way, the total momentum flux u x u z − bx bz can be evaluated by using uˆ x = −p uˆ z /k, and Eqs. (12), (17), (19)–(22). Keeping terms up to order 1/γ 2 , the result is: ∞ 1 3 Ω p 1 p 3 2 2χ¯
ux uz − bx bz = − 2 exp −2ξ τ + τ d k dτ e exp 2ξ + k 3 k 3 2B (2π)2 |k|≥k0 β φˆ ω (k) 2γ 2 τ 2γ τ 2 τ3 − τ × cos (2 ϕ) ¯ − sin (2 ϕ) ¯ + k 2 (k 2 + p 2 ) 1 + τ 2 (1 + τ 2 )2 (1 + τ 2 )3 φˆ a (k) −2γ 2 τ τ2 τ 3 −τ 2τβ + cos (2ϕ) ¯ + 2γ sin (2ϕ) ¯ + . (B.3) − α2 (1 + τ 2 ) (1+τ 2 )2 1+τ 2 (1+τ 2 )3 Here β = p/k, α = 1/(1 + β 2 ) and ϕ¯ is given by Eq. (B.2). Approximate values for the τ -integrals in Eq. (B.3) can easily be obtained in the case of γ 1 and ξ 1, leading to Eq. (24) in the main text. We note that to leading order, both Reynolds and Maxwell stresses involve a term that is of order γ 0 with the same sign. Therefore, these two leading order terms are exactly cancelled each other for the total stress u x u z − bx bz . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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