Reports of the Academy of Sciences of the USSR

1. Volume 195, No. 4

UDC 538.4

HYDROMECHANICS

S. I. VAINSTEIN

THE PROBLEM OF GENERATION OF A MAGNETIC FIELD IN THE PRESENCE OF ACOUSTIC TURBULENCE

(Presented by Academician R. Z. Sagdeev on 3 III 1970)

One of the basic problems of magnetohydrodynamic turbulence consists in clarifying the question of whether a turbulent medium is stable with respect to weak perturbations of the magnetic field, i.e., whether weak magnetic fields will grow with time. The main difficulty is the absence of a small parameter in ordinary hydrodynamic turbulence \((^1)\). Such a small parameter can be found in acoustic turbulence, namely \(\tau v/\lambda\), where \(\tau\) is the correlation time at one point, \(\lambda\) is the characteristic wavelength, and \(v\) is the characteristic velocity.

In problems of generation of the magnetic field \(\mathbf{H}\), the velocity field \(\mathbf{v}\) is assumed to be prescribed, while \(\mathbf{H}\) satisfies the equation

\[ \partial \mathbf{H}/\partial t=\operatorname{rot}[\mathbf{v},\mathbf{H}]+v_m\Delta\mathbf{H}, \tag{1} \]

where \(v_m\) is the magnetic viscosity.

I. We shall assume the velocity field to be stationary and distributed homogeneously and isotropically in space. It is convenient to pass to \(k-\omega\)-space

\[ \mathbf{v}=\int \frac{\mathbf{k}}{k}\,\varphi(\mathbf{k},\omega)e^{i[(\mathbf{k}\mathbf{r})-\omega t]}\,d\mathbf{k}\,d\omega . \]

Next, averaging over the ensemble, we obtain

\[ \langle u_i(\mathbf{k},\omega)u_j^*(\mathbf{k}',\omega')\rangle = \frac{k_i k_j}{k^2}\,I(k,\omega)\delta(\mathbf{k}-\mathbf{k}')\delta(\omega-\omega'); \tag{2} \]

\[ \langle v_i(\mathbf{x},t)v_j(\mathbf{x}+\mathbf{r},t+s)\rangle = \int \frac{k_i k_j}{k^2}\,f(k,s)e^{i(\mathbf{k}\mathbf{r})}\,d\mathbf{k}; \]

\[ f(k,s)=\int I(k,\omega)e^{-i\omega s}\,d\omega . \tag{3} \]

Let \(E(k)\) be the power spectrum of the acoustic oscillations; then \(E(k)=4\pi k^2 f(k,0)\). The correlation time may be introduced in the following way:

\[ \tau=\int dk\int_0^\infty ds\,f(k,s)\Big/\int dk\,f(k,0) = \pi\int dk\,I(k,0)\Big/\int dk\,f(k,0). \tag{4} \]

We now determine \(I(k,0)\). For this purpose we represent the density \(\rho\) and the velocity as follows:
\(\rho=\rho_0+\rho_1+\rho_2;\ \mathbf{v}=\mathbf{v}_1+\mathbf{v}_2\)
\((\rho_0\gg\rho_1\gg\rho_2,\ \mathbf{v}_1\gg\mathbf{v}_2)\).
The Fourier images in \(k-\omega\)-space of the quantities \(\rho_1,\rho_2,\mathbf{v}_1,\mathbf{v}_2\) will be denoted by
\(\rho_1(\mathbf{k},\omega),\rho_2(\mathbf{k},\omega),\frac{\mathbf{k}}{k}\varphi_1(\mathbf{k},\omega),\frac{\mathbf{k}}{k}\varphi_2(\mathbf{k},\omega)\).
Linearization of the equations of gas dynamics gives, as is known, a solution in the form of acoustic waves, with

\[ \rho_1(\mathbf{k},\omega)=\frac{1}{c}\rho_0\varphi_1(\mathbf{k},\omega); \]

\[ \langle \varphi_1(\mathbf{k},\omega)\varphi_1^*(\mathbf{k}',\omega')\rangle = \delta(\mathbf{k}-\mathbf{k}')\delta(\omega-\omega')f(k,0)\frac{1}{2} [\delta(\omega-ck)+\delta(\omega+ck)] \tag{5} \]

(\(c\) is the speed of sound). To obtain \(I(k,0)\) we write the equation for the correction of second approximation (it is sufficient to use only the equation—

continuity equation):

\[ \rho_0 k \varphi_2(\mathbf{k},\omega)=\omega \rho_2(\mathbf{k},\omega) -\int \rho_1(\mathbf{k}-\mathbf{k}_1,\omega-\omega_1) \frac{(\mathbf{k}\mathbf{k}_1)}{k_1}\, \varphi_1(\mathbf{k}_1,\omega_1)\,d\mathbf{k}_1\,d\omega_1 . \tag{6} \]

Multiplying (6) by \(\varphi_2^*(\mathbf{k}',\omega')\), using (5), and averaging, the expression on the right-hand side of (6), when multiplied by the complex conjugate, then gives a fourth-order moment. Here one may use the random-phase approximation and integrate over \(d\mathbf{k}'\) and \(d\omega'\); then we obtain \((\mathbf{p}=\mathbf{k}-\mathbf{q})\):

\[ I(\mathbf{k},0)=\frac{1}{2c^2k^2} \int\left[ \frac{(\mathbf{k}\mathbf{q})^2}{q^2} + \frac{(\mathbf{k}\mathbf{q})(\mathbf{k}\mathbf{p})}{qp} \right] f(p,0)f(q,0)\delta(cp-cq)\,d\mathbf{q} = \]

\[ =\frac{\pi k}{2c^3}\int_{k/2}^{\infty} f^2(q,0)\,dq; \]

\[ \pi=\frac{\pi^2}{16c^3} \int k^2 f^2\!\left(\frac{k}{2},0\right)\,dk \bigg/ \int f(k,0)\,dk . \tag{7} \]

In order of magnitude \(\tau \approx (v^3/c^3)\lambda/v\), so that \(\tau \ll \lambda/v\).

II. To derive the equation for the spectral function of the magnetic field \(B(k,t)\), we shall use the Fourier representation (1) in \(k\)-space

\[ \mathbf{H}(\mathbf{k},t)=\mathbf{H}(\mathbf{k},0)e^{-k^2\nu_m t} +i\int_0^t dt_1 e^{-k^2\nu_m(t-t_1)} \int d\mathbf{q} \left[ \mathbf{k}\left[ \frac{\mathbf{p}}{p}\psi(\mathbf{p},t_1)\mathbf{H}(\mathbf{q},t_1) \right] \right]. \tag{8} \]

\(\mathbf{H}(\mathbf{k},t)\) and \(\psi(\mathbf{k},t)\mathbf{k}/k\) are the Fourier transforms of \(\mathbf{H}(\mathbf{r},t)\) and \(\mathbf{v}(\mathbf{r},t)\).

We use a perturbation-theory expansion in the velocity:

\[ \mathbf{H}(\mathbf{k},t)=\sum_{n=0}^{\infty}\mathbf{H}^{(n)}; \qquad \mathbf{H}^{(0)}=\mathbf{H}(\mathbf{k},0)e^{-\nu_m k^2 t}, \]

\[ \mathbf{H}^{(n)}= i\int_0^t dt_1 e^{-\nu_m k^2(t-t_1)} \int d\mathbf{q} \left[ \mathbf{k}\left[ \frac{\mathbf{p}}{p}\psi(\mathbf{p},t_1)\mathbf{H}^{(n-1)}(\mathbf{q},t_1) \right] \right]. \tag{9} \]

We restrict ourselves to terms of second order in \(\psi\) (which is justified when \(\tau \ll \lambda/v\)). Multiplying (9) by \(H_j^*(\mathbf{k}',t)\), we shall assume: 1) \(\mathbf{H}(\mathbf{k},0)\) is statistically independent of \(\psi(\mathbf{k},t)\); 2) \(\nu_m\) is small, so that in (9) one may put \(\exp(-\nu_m k^2 t)=1-\nu_m k^2t\); 3) \(\langle H_i(\mathbf{k},0)H_j^*(\mathbf{k}',0)\rangle=B(k,0)\delta(\mathbf{k}-\mathbf{k}')\times(\delta_{ij}-k_i k_j/k^2)\).

Averaging the resulting expression and integrating over \(d\mathbf{k}'\), we obtain

\[ B(k,t)=B(k,0) -2B(k,0)\left[ \int_0^t ds\,(t-s)\int f(q,s)\frac{(\mathbf{k}\mathbf{q})^2}{q^2}\,d\mathbf{q} +\nu_m k^2t \right] + \]

\[ +\int_0^t ds\,(t-s) \int f(p,s)B(q,0) \frac{(\mathbf{k}\mathbf{p})^2q^2+k^2(\mathbf{q}\mathbf{p})^2}{p^2q^2} \,d\mathbf{p}. \tag{10} \]

From the asymptotic behavior of (10) at large \(t\) (\(t\gg\tau\)), one can construct an equation for \(B(k,t)\):

\[ \frac{\partial B(k,t)}{\partial t} +2(\chi+\nu_m)k^2B(k,t) = \pi\int I(p,0)B(q,t) \frac{(\mathbf{k}\mathbf{p})^2q^2+k^2(\mathbf{q}\mathbf{p})^2}{p^2q^2}\,d\mathbf{p}; \tag{11} \]

\[ \chi=\frac{\pi}{3}\int I(q,0)\,d\mathbf{q} = \frac{\pi^2}{48c^3}\int k^2 f^2\!\left(\frac{k}{2},0\right)\,d\mathbf{k}. \]

(11) can be transformed to \(r\)-space:

\[ \frac{\partial B_1}{\partial t} = 2\left\{ AB_1'' + \frac{A_1}{r}B_1' - 2\frac{v'}{r}B_1 \right\}; \]

\[ B_1=r^{-3}\int_0^r \rho^2 B(\rho,t)\,d\rho; \qquad B(r,t)=\int B(k,t)e^{i(\mathbf{k},\mathbf{r})}\,d\mathbf{k}; \]

\[ v(r)=\pi\int I(k,0)e^{i(\mathbf{k},\mathbf{r})}\,d\mathbf{k}; \qquad A=\chi+\nu_m-(rv_1'+v_1); \tag{12} \]

\[ A_1=4(\chi+\nu_m)-(\nu' r+2\nu'_1 r+4\nu_1);\qquad \nu_1=r^{-3}\int_0^r \nu(\rho)\rho^2\,d\rho . \]

III. We shall seek the eigenfunctions (12) by the substitution \(B_1=h e^{-2Et}\); then, by the substitution

\[ z=h_1 \exp \left[\frac12 \int_0^r \frac{d\rho\, A_1(\rho)}{\rho A(\rho)}\right] \]

we pass to the Schrödinger equation

\[ z''+m(E-U)z=0, \tag{13} \]

where the “mass” is \(m=1/A\), and the potential is

\[ U=2\nu'/r+\frac12 A(A_1/rA)'+A_1^2/4r^2A . \]

As \(r\to 0\), \(U\to 2\nu_m/r^2\); as \(r\to\infty\), \(U\to 2(\chi+\nu_m)/r^2\). The wave function of a bound state in the potential corresponds to an unstable (growing) solution in the present problem, the eigenenergy of the state being the growth increment with the opposite sign. A bound state appears if \(U\) has a potential well. The well arises if \(\chi\gg \nu_m\); indeed, in this case, for \(r\) such that

\[ \nu(0)/\nu_2(0)\gg r^2\gg \nu_m/\nu_2(0),\qquad \text{where}\quad \nu_2(r)=\int \pi I(q,0)q^2\,dq, \]

\[ U\simeq \nu_2(0)/15 . \]

The presence of a potential well is not a sufficient condition for the existence of eigenfunctions with \(E<0\)—the well may prove insufficiently deep; moreover, the eigenfunction must have the properties of a correlation function (its Fourier image \(B(k,t)\) must be positive). To determine a sufficient condition, let us turn to equation (11). By the substitution \(B=h e^{-2Et}\) we pass to the eigenvalue problem for an integral equation. To find the minimal \(E\), we formulate the variational principle

\[ E=\left[(\chi+\nu_m)\int k^2h^2\,dk -\frac{\pi}{2}\int dk\,dp\,h(k)h(q)I(p,0) \frac{({\bf kp})^2q^2+k^2({\bf qp})^2}{q^2p^2}\right]\Big/\int h^2\,dk; \tag{14} \]

\[ \delta E=0 . \]

Assign to \(h(k)\) the following form: for \(k\le k_1\),

\[ h(k)=k_0^{-1/2}k_1^{-3}k^2\exp(-k_1/2k_0), \]

where

\[ \nu_2(0)/\nu(0)\ll k_1^2\ll \nu_2(0)/\nu_m,\qquad k_0^2\simeq \nu_2(0)/\nu_m; \]

for \(k\ge k_1\),

\[ h(k)=k^{-1}k^{-1/2}\exp(-k/2k_0). \]

For \(k^2<\nu_2(0)/\nu(0)\) only a negligible part of the energy \(h(k)\) is concentrated; therefore the right-hand side of (14) can be simplified:

\[ E=\left[\nu_m\int k^2h^2\,dk -\nu_2(0)/5\int \left\{\frac23 h^2+2hh'k+h''hk^2\right\}\,dk\right]\Big/\int h^2\,dk = \]

\[ =-\frac{7}{30}\nu_2(0)+2k_0^2\nu_m+\nu_2\alpha(k_1/k_0), \]

where \(\alpha(k_1/k_0)\) is a small quantity of order \(k_1/k_0\). It is clear that if \(k_0^2\lesssim 7\nu_2(0)/60\nu_m\), then our \(h(k)\) gives the functional (14) a negative value. Consequently, the eigenfunctions of problem (11) with \(E<0\) exist. Thus, acoustic turbulence is unstable with respect to perturbations of the magnetic field. It is not difficult to show that \(\min U(r)=-\frac23\nu_2(0)\); therefore the growth increment of the field is

\[ \gamma\simeq \nu_2(0)=\frac{\pi}{24c^3}\int k^4 f^2\left(\frac{k}{2},0\right)\,dk . \tag{15} \]

We see that the most essential quantity determining the possibility of field generation is

\[ \nu(0)/\nu_m=3\chi/\nu_m\simeq (v^3/c^3)v\lambda/\nu_m=S={\rm M}^3{\rm Re}_m, \tag{16} \]

(\({\rm M}\) is the Mach number, \({\rm Re}_m\) is the magnetic Reynolds number). For instability it is necessary that \(S\gg 1\).

IV. The velocity field of acoustic turbulence is potential. The vortical component is generated in the same way as the magnetic one, since the equations for \(\mathbf H\) and \(\operatorname{rot}\mathbf v\) are analogous and \(\operatorname{rot}\mathbf v(\mathbf r,0)\) may be taken to be statistically independent of the potential component. Here the quantity most essential for generation is \(S_\omega=\mathrm{M}^3\mathrm{Re}\). If \(S_\omega\gg 1\), the generation of vortices is analogous to the instability described above; if \(S_\omega\ll 1\), there occurs growth of the vortical component \(v_\omega\) as a nonlinear effect—a phenomenon known as acoustic streaming \((^2)\).

The growth of \(v_\omega\) for \(S_\omega\gg 1\) is limited by turbulent viscosity: the point is that the vortex–vortex interaction has the usual character of energy transfer into the region of large \(k\). For a rough estimate of the energy of the established level \(v_\omega\), we note that, since the width of the potential pit is \(\sim\lambda\), the energy \(v_\omega^2\) is concentrated at \(k\sim 1/\lambda\). Let us write the equation for \(v_\omega\):

\[ \frac{d}{dt}v_\omega^2=\gamma v_\omega^2-v_\omega^2/\lambda . \]

In the stationary state \(v_\omega\sim\lambda\gamma\). Let us calculate \(v_\omega\) for the following form of \(E\): for \(k<1/\lambda\), \(E=0\); for \(1/\lambda\le k\le k_\nu\), \(E=v^2\lambda^{1-\alpha}k^{-\alpha}\), where \(k_\nu\) is the spectral cutoff due to viscosity; for \(k>k_\nu\), \(E=0\). Here there are 3 cases: 1) \(\alpha>3/2\), then \(\gamma\sim \mathrm{M}^3v/\lambda\) and \(v_\omega\sim \mathrm{M}^3v\); 2) \(1<\alpha<3/2\), then \(\gamma\sim \mathrm{M}^3(\lambda k_\nu)^{2-2\alpha}v k_\nu\), \(k_\nu=\mathrm{Re}^{2/(1+\alpha)}\lambda^{-1}\), \(v_\omega\sim v\mathrm{M}^3\mathrm{Re}^{2(3-2\alpha)/(1+\alpha)}\le vS_\omega\) (for \(\alpha<3/2\), \(v_\omega\sim \mathrm{M}^3v\)); 3) \(\alpha=3/2\), then \(\gamma\sim \mathrm{M}^3\ln(k_\nu\lambda)v/\lambda\), \(v_\omega\sim \mathrm{M}^3v\ln\mathrm{Re}\).

In conclusion I express my gratitude to Academician R. Z. Sagdeev for discussions.

Siberian Institute of Terrestrial Magnetism,
Ionosphere and Radio-Wave Propagation
of the Siberian Branch of the Academy of Sciences of the USSR
Irkutsk

Received
9 II 1970

REFERENCES

\(^1\) R. Kraichnan, S. Nagarajan, Phys. Fluids, 10, No. 4, 859 (1967).
\(^2\) L. K. Zarembo, V. A. Krasil’nikov, Introduction to Nonlinear Acoustics, “Nauka,” 1966.