ASTR/ATOC-5410: Fluid Instabilities, Waves, and Turbulence November 18, 2016, Axel Brandenburg

Handout 19: Inverse Cascades and Inverse Transfer

Inverse cascades are not that uncommon, but they are usually always related to some extra conservation laws in turbulence. Magnetic helicity in 3-D MHD is perhaps the most famous one, but in 2-D MHD we have the conservation of ⟨A^{2}⟩ and in 2-D HD we have the conservation of ⟨ω^{2}⟩. In closed or periodic domains we have

\frac{d}{dt}⟨ω^{2}⟩ = −2ν⟨(∇ × ω)^{2}⟩ (in 2-D HD),  (1)
\frac{d}{dt}⟨A^{2}⟩ = −2η⟨B^{2}⟩ (in 2-D MHD),  (2)
\frac{d}{dt}⟨A · B⟩ = −2η⟨J · B⟩ (in 3-D MHD),  (3)

The consequences can be dramatic. Below some examples.

1 2-D Hydrodynamic Turbulence

The possibility of an inverse cascase in 2-D hydrodynamic turbulence was demonstrated conclusively by Frisch & Sulem (1984); see Figure 1 the spectra from their paper. In this case, energy was injected at intermediate lenth scales. This is important, because otherwise there is no room for the inverse cascase to move energy to larger length scales.

Figure 1: Forced 2-D hydrodynamic turbulence (Frisch & Sulem, 1984).

2 3-D MHD Turbulence

The case of decaying turbulence is perhaps most dramatic, because we actually see a real increase of spectral energy at small wavenumbers; see Figure 2. To understand what happens, we first need to explain the realizability condition, which means that magnetic helicity can only be realized in a field up to a certain value.

Figure 2: Comparsion of spectra of decaying MHD turbulence with helicity (left) and without (right).

3 Realizability Condition

Magnetic energy and helicity spectra are usually calculated as

M_{k} = \frac{1}{2} ∫k-shell B_{k}^{∗} · B_{k} k^{2} dΩ_{k},  (4)

H_{k} = \frac{1}{2} ∫k-shell(A_{k}^{∗} · B_{k} + A_{k} · B_{k}^{∗}) k^{2} dΩ_{k},  (5)

where dΩ_{k} is the solid angle element in Fourier space, B_{k} = ik × A_{k} is the Fourier transform of the magnetic field, and A_{k} is the Fourier transform of the vectors potential. These spectra are normalized such that

∫_{0}^{∞} H_{k} dk = ⟨A · B⟩V ≡ H,  (6)

∫_{0}^{∞} M_{k} dk = ⟨\frac{1}{2}B^{2}⟩V ≡ M,  (7)

where H and M are magnetic helicity and magnetic energy, respectively, and angular brackets denote volume averages.

It is convenient to decompose the Fourier transformed magnetic vector potential, A_{k}, into a longitudinal component, h^{∥}, and eigenfunctions h^{±} of the curl operator. This decomposition has been used in studies of turbulence Waleffe (1993). We have seen such functions as Beltrami waves, but they can be oriented in arbitrary directions.

h(k) = R · h(k)^{(nohel)} with R_{ij} = \frac{δ_{ij} − iσϵ_{ijk}k̂}{\sqrt{1 + σ^{2}}} ,  (8)

where the parameter σ characterizes the fractional helicity of f, and

h(k)^{(nohel)} = (k × ê) /\sqrt{k^{2} − (k · ê)^{2}},  (9)

is a non-helical forcing function. Here ê is an arbitrary unit vector not aligned with k, k̂ is the unit vector along k, and |h|^{2} = 1.

Using this decomposition we can write the Fourier transformed magnetic vector potential as

A_{k} = a_{k}^{+}h_{k}^{+} + a_{k}^{−}h_{k}^{−} + a_{k}^{∥}h_{k}^{∥},  (10)

with

ik × h_{k}^{±} = ±kh_{k}^{±},  k = |k|,  (11)

and

⟨h_{k}^{+}^{∗} · h_{k}^{+}⟩ = ⟨h_{k}^{−}^{∗} · h_{k}^{−}⟩ = ⟨h_{k}^{∥}^{∗} · h_{k}^{∥}⟩ = 1,  (12)

where asterisks denote the complex conjugate, and angular brackets denote, as usual, volume averages. The longitudinal part a_{k}^{∥}h_{k}^{∥} is parallel to k and vanishes after taking the curl to calculate the magnetic field. In the Coulomb gauge, ∇ · A = 0, the longitudinal component vanishes altogether.

The (complex) coefficients a_{k}^{±}(t) depend on k and t, while the eigenfunctions h_{k}^{±}, which form an orthonormal set, depend only on k and are given by

h_{k}^{±} = \frac{1}{\sqrt{2}} \frac{k × (k × e) ∓ ik(k × e)}{k^{2}\sqrt{1 − (k · e)^{2}/k^{2}}},  (13)

where e is an arbitrary unit vector that is not parallel to k. With these preparations we can write the magnetic helicity and energy spectra in the form

H_{k} = k(|a_{k}^{+}|^{2} − |a_{k}^{−}|^{2})V,  (14)
M_{k} = \frac{1}{2}k^{2}(|a_{k}^{+}|^{2} + |a_{k}^{−}|^{2})V,  (15)

where V is the volume of integration. (Here again the factor μ_{0}^{−1} is ignored in the definition of the magnetic energy.) From Equations (14) and (15) one sees immediately that

\frac{1}{2}k|H_{k}| ≤ M_{k},  (16)

which is also known as the realizability condition. A fully helical field has therefore M_{k} = ±\frac{1}{2}kH_{k}.

For further reference we now define power spectra of those components of the field that are either right or left handed, i.e.

H_{k}^{±} = ±k|a_{k}^{±}|^{2}V, M_{k}^{±} = \frac{1}{2}k^{2}|a_{k}^{±}|^{2}V.  (17)

Thus, we have H_{k} = H_{k}^{+} + H_{k}^{−} and M_{k} = M_{k}^{+} + M_{k}^{−}. Note that H_{k}^{±} and M_{k}^{±} can be calculated without explicit decomposition into right and left handed field components using

H_{k}^{±} = \frac{1}{2}(H_{k} ± 2k^{−1}M_{k}), M_{k}^{±} = \frac{1}{2}(M_{k} ± \frac{1}{2}kH_{k}).  (18)

This method is significantly simpler than invoking explicitly the decomposition in terms of a_{k}^{±}h_{k}^{±}.

In Section ?? plots of M_{k}^{±} will be shown and discussed in connection with turbulence simulations. Here the turbulence is driven with a helical forcing function proportional to h_{k}^{+}; see Equation (13).

4 Argument for inverse cascade

The occurrence of an inverse cascade can be understood as the result of two waves (wavenumbers p and q) interacting with each other to produce a wave of wavenumber k. The following argument is due to Frisch et al. (1975). Assuming that during this process magnetic energy is conserved together with magnetic helicity, we have

M_{p} + M_{q} = M_{k},  (19)
|H_{p}| + |H_{q}| = |H_{k}|,  (20)

where we are assuming that only helicity of one sign is involved. Suppose the initial field is fully helical and has the same sign of magnetic helicity at all scales, we have

2M_{p} = p|H_{p}| and 2M_{q} = q|H_{q}|,  (21)

and so Equation (19) yields

p|H_{p}| + q|H_{q}| = 2M_{k} ≥ k|H_{k}|,  (22)

where the last inequality is just the realizability condition (16) applied to the target wavenumber k after the interaction. Using Equation (20) in Equation (22) we have

p|H_{p}| + q|H_{q}| ≥ k(|H_{p}| + |H_{q}|).  (23)

In other words, the target wavevector k after the interaction of wavenumbers p and q satisfies

k ≤ \frac{p|H_{p}| + q|H_{q}|}{|H_{p}| + |H_{q}|}.  (24)

The expression on the right hand side of Equation (24) is a weighted mean of p and q and thus satisfies

min(p, q) ≤ \frac{p|H_{p}| + q|H_{q}|}{|H_{p}| + |H_{q}|} ≤ max(p, q),  (25)

and therefore

k ≤ max(p, q).  (26)

In the special case where p = q, we have k ≤ p = q, so the target wavenumber after interaction is always less or equal to the initial wavenumbers. In other words, wave interactions tend to transfer magnetic energy to smaller wavenumbers, i.e. to larger scale. This corresponds to an inverse cascade. The realizability condition, \frac{1}{2}k|H_{k}| ≤ M_{k}, was the most important ingredient in this argument.

References

Frisch, U. & Sulem, P. L., “Numerical simulation of the inverse cascade in two dimensional turbulence,” Phys. Fluids 27, 1921-1923 (1984).

U. Frisch, A. Pouquet, J. Léorat, and A. Mazure, “r,” J. Fluid Mech. 68, 769-F (1975).isch, U., Pouquet, A., Léorat, J., & Mazure, A., “Possibility of an inverse cascade of magnetic helicity in hydrodynamic turbulence,” J. Fluid Mech. 68, 769-778 (1975).

Waleffe, F., “Inertial transfers in the helical decomposition,” Phys. Fluids A5, 677-685 (1993).

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