One-Dimensional Analytical Solutions of the Transport Equations for Incompressible Magnetohydrodynamic (MHD) Turbulence

Abstract

We derive one-dimensional (1D) analytical solutions for the transport equations of incompressible magnetohydrodynamic (MHD) turbulence, including the Elsässer energies and the correlation lengths. The solutions are suitable for an arbitrary given background convection speed and Alfvén speed profiles but require near equipartition of turbulent kinetic energy and magnetic field energy. These analytical solutions provide a simple tool to investigate the evolution of turbulence and resulting energetic particle diffusion coefficients in various space and astrophysical environments that possess simple geometry.

1. Introduction

Turbulence is important to a variety of astrophysical phenomena from diffusive shock acceleration to cosmic ray propagation, but there are few analytic solutions to turbulence transport models. The traditional Wentzel–Kramer–Brillouin (WKB) turbulence transport model describes the evolution of small amplitude incompressible fluctuations in a large-scale inhomogeneous flow [1,2]. Although the analytical solution is accessible, the classic WKB model does not include the interaction between anti-propagating waves that initiates the dissipation of turbulence energy, nor can it describe the evolution of the correlation length. In an important paper, Zank et al. [3] clarified the relation between the WKB description of the fluctuating magnetic field variance and a simplified turbulence transport model that included the physics of dissipation and the corresponding correlation length. Unlike the WKB model, Zank et al. [4], Adhikari et al. [5] derived much more complex transport equations based on the incompressible MHD equations that represent the leading-order description of nearly incompressible MHD in the limit of large plasma beta. As the model consists of six coupled equations, it is far less tractable than the WKB model and is only solved numerically [6,7]. In this work, by recognizing that the system of equations can be simplified to four equations by assuming near equipartition between the turbulent kinetic and magnetic field energy, we can derive 1D analytical solutions for the turbulence Elässer energies and the corresponding correlation lengths. The solutions work for arbitrary convection speed and Alfvén speed profiles. A set of approximate solutions is also developed for turbulence with extremely weak dissipation or when the dissipation is balanced by some form of driving often manifested as plasma instabilities (examples already considered include turbulence generated by large-scale shear of different speed flows, shock waves, pick-up ion creation [3,4,8], and cosmic ray streaming ([7])). These analytical solutions provide a simple and practical tool for studying the evolution of turbulence in various space and astrophysical environments, and are useful extensions of earlier analytical solutions with different simplifications by Zank et al. [3], Adhikari et al. [9].

For the purposes of illustration, we apply the new analytic solutions of the turbulence transport equations to calculate the cosmic ray diffusion coefficient in the galactic halo. With the commonly adopted convection speed profile in cosmic ray studies [10,11], we find a somewhat self-consistent cosmic ray diffusion coefficient, although the reality is likely to be much more complicated. We derive the analytical solutions of the turbulence transport equations in Section 2, and as an application, apply the analytical solutions to investigate the cosmic ray diffusion coefficient in Section 3. Finally, we summarize our results in Section 4.

2. 1D Analytical Solutions of the Turbulence Transport Equations

With the assumption that the turbulent kinetic energy nearly equals the turbulent magnetic field energy, the simplified 1D transport equations for leading order nearly compressible turbulence in a high plasma beta regime can be written as [4,5]

$$\begin{array}{c}(U-V_A){\displaystyle \frac{d{z}_{+}^2}{dx}}+\left({\displaystyle \frac{1}{2}}\nabla ·\mathbf{U}+\nabla ·{\mathbf{V}}_{\mathbf{A}}\right)z_{+}^2=-2\alpha {\displaystyle \frac{z_{+}^2}{{\lambda }_{+}}}{\left(z_{-}^2\right)}^{1/2};\end{array}$$

(1)

$$\begin{array}{c}(U+V_A){\displaystyle \frac{d{z}_{-}^2}{dx}}+\left({\displaystyle \frac{1}{2}}\nabla ·\mathbf{U}-\nabla ·{\mathbf{V}}_{\mathbf{A}}\right)z_{-}^2=-2\alpha {\displaystyle \frac{z_{-}^2}{{\lambda }_{-}}}{\left(z_{+}^2\right)}^{1/2};\end{array}$$

(2)

$$\begin{array}{c}(U-V_A){\displaystyle \frac{d{\lambda }_{+}}{dx}}=2\beta {\left(z_{-}^2\right)}^{1/2};\end{array}$$

(3)

$$\begin{array}{c}(U+V_A){\displaystyle \frac{d{\lambda }_{-}}{dx}}=2\beta {\left(z_{+}^2\right)}^{1/2},\end{array}$$

(4)

where the fluctuating Elsässer variables ${\mathbf{z}}_{\pm }=\mathbf{u}\pm \mathbf{b}/\sqrt{\mu \rho }$, and $\mathbf{u}$, $\mathbf{b}$, $\rho$, and $\mu$ are the turbulent velocity and magnetic fields (not assumed small but instead are the fluctuating components of a mean field decomposition), the mass density of the background plasma, and the vacuum magnetic permeability, respectively, and $V_A$ is the Alfvén speed. The first two terms in Equations (1) and (2) describe convection and “heating/cooling” of the Elsässer energy associated with the expansion/compression of the background flow [4]. And the right hand terms represent the turbulence dissipation where ${\lambda }_{\pm }$ is the corresponding correlation length for $z_{\pm }$. $\alpha$ and $\beta$ are the von Kármán–Taylor constants. $z_{\pm }^2$ are Elsässer energies. This set of equations is valid for 1D Cartesian coordinates or for a spherical coordinate system where $\mathbf{U}$ and ${\mathbf{V}}_{\mathbf{A}}$ are along the radial direction.

By writing $z_{\pm }^2$ as the function of ${\lambda }_{\pm }$ through Equations (3) and (4) and substituting them into Equations (1) and (2), we obtain

$$\begin{array}{c}{\displaystyle \frac{d^2{\lambda }_{\pm }}{d{x}^2}}=-\left[a_{\pm }+{\displaystyle \frac{1}{{\lambda }_{\mp }}}{\displaystyle \frac{d{\lambda }_{\mp }}{dx}}\right]{\displaystyle \frac{d{\lambda }_{\pm }}{dx}},\end{array}$$

(5)

after using the common assumption $\alpha =2\beta$. Here, $a_{\pm }$ has the form

$$a_{\pm }={\displaystyle \frac{\nabla (U\mp V_A)}{U\mp V_A}}+{\displaystyle \frac{\frac{1}{2}\nabla ·\mathbf{U}\mp \nabla ·{\mathbf{V}}_{\mathbf{A}}}{2(U\pm V_A)}}.$$

(6)

Obviously, Equation (5) represents the full derivation of ${\lambda }_{\pm }{\displaystyle \frac{d{\lambda }_{\mp }}{dx}}$ and the solutions are given by

$$\begin{array}{c}{\lambda }_{+}{\displaystyle \frac{d{\lambda }_{-}}{dx}}={\lambda }_{0+}{\left.{\displaystyle \frac{d{\lambda }_{-}}{dx}}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right);\end{array}$$

(7)

$$\begin{array}{c}{\lambda }_{-}{\displaystyle \frac{d{\lambda }_{+}}{dx}}={\lambda }_{0-}{\left.{\displaystyle \frac{d{\lambda }_{+}}{dx}}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right),\end{array}$$

(8)

where the ${\left.d{\lambda }_{\pm }/dx\right|}_{x_0}$ are given by Equations (3) and (4) at the initial position $x_0$. Since ${\lambda }_{+}{\displaystyle \frac{d{\lambda }_{-}}{dx}}+{\lambda }_{-}{\displaystyle \frac{d{\lambda }_{+}}{dx}}={\displaystyle \frac{d\left({\lambda }_{+}{\lambda }_{-}\right)}{dx}}$, adding Equations (7) and (8) yields the expression for ${\lambda }_{+}{\lambda }_{-}$ as

$${\lambda }_{+}{\lambda }_{-}={\lambda }_{0+}{\lambda }_{0-}+{\lambda }_{0+}{\left.{\displaystyle \frac{d{\lambda }_{-}}{dx}}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right)dx+{\lambda }_{0-}{\left.{\displaystyle \frac{d{\lambda }_{+}}{dx}}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right)dx.$$

(9)

By writing ${\lambda }_{-}=\left({\lambda }_{+}{\lambda }_{-}\right)/{\lambda }_{+}$, using Equation (9) and substituting it into Equation (8), we obtain

$$\begin{array}{c}{\displaystyle \frac{1}{{\lambda }_{+}}}{\displaystyle \frac{d{\lambda }_{+}}{dx}}={\displaystyle \frac{{\lambda }_{0-}{\left.\frac{d{\lambda }_{+}}{dx}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right)}{{\lambda }_{+}{\lambda }_{-}}};\end{array}$$

(10)

$$\begin{array}{c}{\lambda }_{+}={\lambda }_{0+}exp\left({\int }_{x_0}^x{\displaystyle \frac{{\lambda }_{0-}{\left.\frac{d{\lambda }_{+}}{dx}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right)dx}{{\lambda }_{0+}{\lambda }_{0-}+{\lambda }_{0+}{\left.\frac{d{\lambda }_{-}}{dx}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right)dx+{\lambda }_{0-}{\left.\frac{d{\lambda }_{+}}{dx}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right)dx}}\right).\end{array}$$

(11)

Following the same method, we derive expressions for ${\lambda }_{-}$ as

$$\begin{array}{c}{\displaystyle \frac{1}{{\lambda }_{-}}}{\displaystyle \frac{d{\lambda }_{-}}{dx}}={\displaystyle \frac{{\lambda }_{0+}{\left.\frac{d{\lambda }_{-}}{dx}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right)}{{\lambda }_{+}{\lambda }_{-}}};\end{array}$$

(12)

$$\begin{array}{c}{\lambda }_{-}={\lambda }_{0-}exp\left({\int }_{x_0}^x{\displaystyle \frac{{\lambda }_{0+}{\left.\frac{d{\lambda }_{-}}{dx}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right)dx}{{\lambda }_{0+}{\lambda }_{0-}+{\lambda }_{0+}{\left.\frac{d{\lambda }_{-}}{dx}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right)dx+{\lambda }_{0-}{\left.\frac{d{\lambda }_{+}}{dx}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right)dx}}\right).\end{array}$$

(13)

The Elsässer energy $z_{\pm }^2$ can be derived from Equations (3) and (4) through

$$\begin{array}{c}{z}_{\pm }^2={\displaystyle \frac{{(U\pm V_A)}^2}{4{\beta }^2}}{\left({\displaystyle \frac{d{\lambda }_{\mp }}{dx}}\right)}^2.\end{array}$$

(14)

We have now obtained four exact analytical expressions for the Elsässer energy and associated correlation lengths.

For a special case in which the generation of the turbulence is balanced by the dissipation, the turbulence equations take the form $\alpha \to 0$. Equation (11) then becomes

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\lambda }_{+}& \approx {\lambda }_{0+}exp\left({\int }_{x_0}^x{\displaystyle \frac{{\lambda }_{0-}{\left.\frac{d{\lambda }_{+}}{dx}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right)}{{\lambda }_{0+}{\lambda }_{0-}}}\right)\hfill \\ \hfill & \approx {\lambda }_{0+}+{\left.{\displaystyle \frac{d{\lambda }_{+}}{dx}}\right|}_{x_0}{\int }_{x_0}^{x}exp\left({\int }_{x_0}^x-a_{-}dx\right)dx,\hfill \end{array}\end{array}$$

(15)

and $z_{-}^2$ is given by Equation (14) as

$$z_{-}^2={\displaystyle \frac{{(U-V_A)}^2}{4{\beta }^2}}{\left({\displaystyle \frac{d{\lambda }_{+}}{dx}}\right)}^2\approx z_{-0}^2{\left({\displaystyle \frac{U-V_A}{U_0-V_{A0}}}\right)}^{2}exp\left(-2{\int }_{x_0}^x{a}_{-}dx\right).$$

(16)

According to Equation (7), $d{\lambda }_{-}/dx$ can be written as

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\displaystyle \frac{d{\lambda }_{-}}{dx}}& \approx {\displaystyle \frac{{\lambda }_{0+}{\left.\frac{d{\lambda }_{-}}{dx}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right)}{{\lambda }_{0+}+{\left.\frac{d{\lambda }_{+}}{dx}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{-}dx\right)dx}}\hfill \\ \hfill & \approx {\left.{\displaystyle \frac{d{\lambda }_{-}}{dx}}\right|}_{x_0}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right).\hfill \end{array}\end{array}$$

(17)

After integration over x, we obtain

$$\begin{array}{c}{\lambda }_{-}\approx {\lambda }_{0-}+{\left.{\displaystyle \frac{d{\lambda }_{-}}{dx}}\right|}_{x_0}{\int }_{x_0}^{x}exp\left(-{\int }_{x_0}^x{a}_{+}dx\right)dx.\end{array}$$

(18)

Like Equation (16), the solution for $z_{+}^2$ is

$$z_{+}^2\approx z_{+0}^2{\left({\displaystyle \frac{U+V_A}{U_0+V_{A0}}}\right)}^{2}exp\left(-2{\int }_{x_0}^x{a}_{+}dx\right).$$

(19)

3. A Toy Model for Cosmic Ray Diffusion Coefficient in the Halo

Based on the analytic solutions, we build a toy model to illustrate the cosmic ray diffusion coefficient vertical to the galactic disk in the galactic halo. Assuming that the turbulent magnetic field in the interstellar medium follows a Kolmogorov power spectrum, the spatial diffusion coefficient (D) of galactic cosmic ray particles is related to the strength of the turbulent magnetic field (b) and its correlation length (${\lambda }_b$) as well as the large-scale magnetic field (B) (b and B should be interpreted as the perpendicular and parallel component of the field with respect to the outward galactic wind) through [7,12]

$$D\propto {\beta }_v\left({\displaystyle \frac{B^{5/3}}{b^2}}\right){\lambda }_b^{2/3}{p}^{1/3},$$

(20)

where ${\beta }_v$ is the particle speed in units of speed of light, p is the particle momentum, and $b^2$ and ${\lambda }_b$ are given by [8],

$$b^2=\mu \rho {\displaystyle \frac{z_{+}^2+z_{-}^2}{4}};{\lambda }_b={\displaystyle \frac{z_{+}^2{\lambda }_{+}+z_{-}^2{\lambda }_{-}}{z_{+}^2+z_{-}^2}}={\lambda }_{+}+({\lambda }_{-}-{\lambda }_{+}){\displaystyle \frac{1}{1+z_{+}^2/z_{-}^2}},$$

(21)

for the case that turbulent kinetic energy equals to turbulent magnetic field energy. Obviously, the correlation length of the turbulent magnetic field ${\lambda }_b$ is in the range of ${\lambda }_{-}$ and ${\lambda }_{+}$. Once the speed profiles for convection and Alfvén speed are determined, we can calculate the essential turbulence quantities and the resulting diffusion coefficient with Equations (11), (13) and (14). Following cosmic ray propagation models—see e.g., [13]—the convection speed is assumed to increase linearly from the galactic disk to the halo, $U\left(x\right)=xdU/dx$, where x is the distance from the galactic disk.The Alfvén speed is defined as $V_A=B/\sqrt{\mu \rho }$. The mass density profile $\rho \left(x\right)$ can be obtained from the mass continuity equation

$$\rho \left(x\right)={\displaystyle \frac{{\rho }_0{U}_0}{U\left(x\right)}}.$$

(22)

Thus, the Alfvén speed can be parametrized as $V_A=V_{A0}\sqrt{x/x_0}$.

We set the boundary at $x_0=0.1\mathrm{kpc}$ which is the height of the galaxy disk. For the typical values of $B=1.5$$\mathsf{\mu }$G, ${\rho }_0/m_p=n_0=0.003{\mathrm{cm}}^{-3}$ ($m_p$ is the proton mass), and $b_0^2/B^2=0.7$, we obtain $V_{A0}=60\mathrm{km}{\mathrm{s}}^{-1}$. This should not be confused with the effective Alfvén speed in cosmic ray reacceleration models not only because it only represents the Alfvén speed in x direction, but also because the effective Alfvén speed is scaled by some factors related to height of the reacceleration zone and the turbulence level [14]. Assuming that the diffusion coefficient for particle with momentum $1\mathrm{GeV}/c$ at $x_0$ is $4\times {10}^{28}{\mathrm{cm}}^2$/s, the correlation lengths ${\lambda }_{0\pm }$ are modeled as $40\mathrm{pc}$ and $z_{+}^2=1/3z_{-}^2=2500{\left(\mathrm{km}{\mathrm{s}}^{-1}\right)}^2$ for the purpose of illustration. The appropriate von Kármán–Taylor constant ($\alpha$) for the interstellar medium is also unknown and can only be obtained by comparing the predicted and measured gas temperature, which is beyond the scope of this toy model. We consider a simple model where the damping of turbulence is balanced by the cosmic ray streaming instability [15,16] or other possible coexisting instabilities; this formally corresponds to $\alpha \to 0$. Accordingly, in practice, we set $\alpha =1\times {10}^{-4}$ in this work.

According to Equation (20) and with Equations (21) and (22), we can define the spatial dependence of the diffusion coefficient as

$$k_x=x{(z_{+}^2+z_{-}^2)}^{-5/3}{(z_{+}^2{\lambda }_{+}+z_{-}^2{\lambda }_{-})}^{2/3}.$$

(23)

According to Equations (3) and (4), correlation lengths ${\lambda }_{\pm }$ can be appropriated as constants when $\alpha \to 0$. If ${\lambda }_{+}/{\lambda }_{-}\sim \mathcal{O}\left(1\right)$, the evolution of diffusion coefficient is basically determined by the turbulent magnetic field magnitude and nearly follows $1/b^2$.

We illustrate the variation in $k_x$ with distance from the galactic disk (x) for three different choices of convection speed gradient ($dU/dx$) in Figure 1. It can be seen that the diffusion coefficients near the galactic disk within $1\mathrm{kpc}$ increase rapidly at about the same rate. Thereafter, the diffusion coefficients continue to increase at slower rates until about $6\mathrm{kpc}$. The diffusion coefficient with the smaller speed gradient increases faster than that with the larger gradient. Above $6\mathrm{kpc}$, the diffusion coefficient with a large speed gradient $dU/dx=35\mathrm{km}{\mathrm{s}}^{-1}{\mathrm{kpc}}^{-1}$ (green dash–dotted line) decreases with x. This contradicts the standard cosmic ray propagation model, where the diffusion coefficient is expected to be larger outside the cosmic ray diffusion halo to allow for the escape of cosmic rays.

4. Summary

The six coupled transport equations for incompressible MHD turbulence can be simplified to four under the assumption of near equipartition of the turbulent kinetic and magnetic field energies. Exact 1D analytic solutions were derived and valid for any specified convection and Alfvén speed. A set of approximate solutions was presented for the case where the generation and dissipation of turbulence are balanced. These analytic solutions have wide applications in space and astrophysical environments that possess simple geometry. Based on the analytic solutions, a toy model is built to study the variation in the cosmic ray diffusion coefficient with distance from the galactic disk under the assumption that the turbulence dissipation is balanced by the cosmic ray streaming instability, which may work for sub-TeV cosmic rays in the self-confinement transport regime. Our aim is neither to fix the diffusion coefficient unambiguously nor to explore the possible parameter spaces. Rather, we considered a simple phenomenological scenario in order to illustrate a not completely unreasonable and self-consistent diffusion coefficient based on the turbulence properties. This is crucial for developing a realistic spatially dependent cosmic ray propagation model. Using the current toy model with the assumptions that the turbulent kinetic energy equals to turbulent magnetic field and a constant magnetic field pointing the galactic wind direction as well as the estimated initial condition at the boundary, we showed that the convection speed gradient should be sufficiently small to allow the diffusion coefficient to continue to increase with distance from the galactic disk. We address that in order to accurately model the diffusion coefficient, a complete galactic wind model should be developed in the future with the basis of the gas dynamic equations, the turbulence transport model, and the comic ray transport equation.