2 Governing equations

We consider full three-dimensional incompressible, resistive MHD equations with primitive variables of velocity, pressure and magnetic field, $q=\left\{\boldsymbol{U},p,\boldsymbol{B}\right\}$ . The governing equations are (Goossens Reference Goossens2012)

(2.1) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{U}=0, & \displaystyle\end{eqnarray}$$

(2.2) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{U}}{\unicode[STIX]{x2202}t}+(\boldsymbol{U}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{U}=-\frac{1}{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D735}p+\frac{1}{\unicode[STIX]{x1D70C}}\boldsymbol{j}\times \boldsymbol{B}+\unicode[STIX]{x1D708}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{U}, & \displaystyle\end{eqnarray}$$

(2.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{B}=0, & \displaystyle\end{eqnarray}$$

(2.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}\boldsymbol{B}}{\unicode[STIX]{x2202}t}=\unicode[STIX]{x1D735}\times (\boldsymbol{U}\times \boldsymbol{B})+\frac{1}{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D6FB}^{2}\boldsymbol{B}, & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}$ is the density, $\boldsymbol{j}$ is the current density, $\unicode[STIX]{x1D708}$ is the kinematic viscosity, $\unicode[STIX]{x1D707}_{0}$ is the permeability of free space and $\unicode[STIX]{x1D70E}$ is the conductivity of the MHD fluid. The current density, $\boldsymbol{j}$ , can be expressed in terms of the magnetic field, $\boldsymbol{B}$ , using Ampere’s law:

(2.5) $$\begin{eqnarray}\boldsymbol{j}=\frac{1}{\unicode[STIX]{x1D707}_{0}}(\unicode[STIX]{x1D735}\times \boldsymbol{B}).\end{eqnarray}$$

The Lorentz force term, $\boldsymbol{j}\times \boldsymbol{B}$ , on the right-hand side of (2.2) then reduces to

(2.6) $$\begin{eqnarray}\boldsymbol{j}\times \boldsymbol{B}=-\unicode[STIX]{x1D735}\left(\frac{B^{2}}{2\unicode[STIX]{x1D707}_{0}}\right)+(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\frac{\boldsymbol{B}}{\unicode[STIX]{x1D707}_{0}}.\end{eqnarray}$$

The effect of Lorentz force manifests as a magnetic pressure term, $-\unicode[STIX]{x1D735}(B^{2}/2\unicode[STIX]{x1D707}_{0})$ , and a magnetic tension term, $(\boldsymbol{B}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{B}/\unicode[STIX]{x1D707}_{0}$ . The magnetic pressure supplements the fluid pressure, resulting in a total pressure, $p_{t}$ , given by

(2.7) $$\begin{eqnarray}p_{t}=p+\frac{B^{2}}{2\unicode[STIX]{x1D707}_{0}}.\end{eqnarray}$$

For the purpose of analysis, the velocity, pressure and magnetic fields are decomposed into mean (ensemble) and perturbation fields:

(2.8a-c ) $$\begin{eqnarray}u_{i}=\overline{u}_{i}+u_{i}^{\prime },\quad p=\overline{p}+p^{\prime },\quad B_{i}=\overline{B}_{i}+B_{i}^{\prime }.\end{eqnarray}$$

The mean can represent ensemble, time or space average as appropriate to the specific flow under consideration. The equations are derived for the most general ensemble average first. The governing equations for the mean field are given by

(2.9a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{i}}=0,\quad \frac{\unicode[STIX]{x2202}\overline{B}_{i}}{\unicode[STIX]{x2202}x_{i}}=0,\end{eqnarray}$$

(2.10) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}t}+\overline{u}_{j}\frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{j}} & = & \displaystyle -\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\overline{p}}{\unicode[STIX]{x2202}x_{i}}-\overline{\frac{\unicode[STIX]{x2202}u_{i}^{\prime }u_{j}^{\prime }}{\unicode[STIX]{x2202}x_{j}}}-\frac{1}{2\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}(\overline{B}_{j}\overline{B}_{j})-\frac{1}{2\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}}\overline{\frac{\unicode[STIX]{x2202}B_{j}^{\prime }B_{j}^{\prime }}{\unicode[STIX]{x2202}x_{i}}}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\overline{B}_{j}}{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}\overline{B}_{i}}{\unicode[STIX]{x2202}x_{j}}+\frac{1}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\overline{\frac{\unicode[STIX]{x2202}B_{i}^{\prime }B_{j}^{\prime }}{\unicode[STIX]{x2202}x_{j}}}+\unicode[STIX]{x1D708}\frac{\unicode[STIX]{x2202}^{2}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{j}^{2}},\end{eqnarray}$$

(2.11) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}\overline{B}_{i}}{\unicode[STIX]{x2202}t}+\overline{u}_{j}\frac{\unicode[STIX]{x2202}\overline{B}_{i}}{\unicode[STIX]{x2202}x_{j}} & = & \displaystyle \overline{B}_{j}\frac{\unicode[STIX]{x2202}\overline{u}_{i}}{\unicode[STIX]{x2202}x_{j}}+\overline{\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{j}}\left(u_{i}^{\prime }B_{j}^{\prime }-u_{j}^{\prime }B_{i}^{\prime }\right)}+\unicode[STIX]{x1D702}\frac{\unicode[STIX]{x2202}^{2}\overline{B}_{i}}{\unicode[STIX]{x2202}x_{j}\unicode[STIX]{x2202}x_{j}},\end{eqnarray}$$

where $i,j$ vary from 1 to 3, and $\unicode[STIX]{x1D702}=1/\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}$ is the magnetic diffusivity of the MHD fluid. Einstein summation convention is used. Overbars indicate ensemble averaging. For the case of homogeneous shear flows, the ensemble average is equivalent to the spatial average on the streamwise–spanwise plane.

Figure 1.

(a) Problem setup. (b) Wavevector orientation with respect to the streamwise direction.

3 Linear analysis

The setup for linear analysis and numerical simulations is shown in figure 1(a). The mean velocity is oriented along the 1 direction and it varies along the 2 direction. The magnetic field is applied in the 1 direction. The colinear alignment of velocity and magnetic fields is seen in plasma propulsion jets such as the variable specific impulse magnetoplasma rocket (VASIMR) engine (Diaz Reference Diaz2000). Other magnetic field orientations will be considered in the future. In subsequent discussions, the 1, 2 and 3 directions will be referred as streamwise, normal and spanwise directions. For this study we consider

(3.1a,b ) $$\begin{eqnarray}\bar{\boldsymbol{u}}\equiv (U_{1}(x_{2}),0,0),\quad \bar{\boldsymbol{B}}\equiv (B_{0},0,0).\end{eqnarray}$$

Mean density is taken to be uniform. For linear analysis, we consider an ideal MHD fluid. In DNS, the full nonlinear and viscous/resistive effects are included. Under the assumptions of idealized MHD linear analysis, mean pressure is uniform throughout the flow field:

(3.2) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}\overline{p}}{\unicode[STIX]{x2202}x_{i}}=0.\end{eqnarray}$$

The evolution of fluctuating fields is isolated by subtracting the mean field evolution terms from the total field evolution equations upon neglecting nonlinear terms. The resulting equations are (Chagelishvili et al. Reference Chagelishvili, Hristov, Chanishvili and Lominadze1993)

(3.3a,b ) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{i}}=0,\quad \frac{\unicode[STIX]{x2202}B_{i}^{\prime }}{\unicode[STIX]{x2202}x_{i}}=0,\end{eqnarray}$$

(3.4) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\overline{\text{D}}u_{i}^{\prime }}{\overline{\text{D}}t}\equiv \frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}t}+U_{1}(x_{2})\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{1}}=-u_{2}^{\prime }S\unicode[STIX]{x1D6FF}_{i1}-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x_{i}}\left(p^{\prime }+\frac{B_{0}B_{1}^{\prime }}{\unicode[STIX]{x1D707}_{0}}\right)+\frac{B_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\frac{\unicode[STIX]{x2202}B_{i}^{\prime }}{\unicode[STIX]{x2202}x_{1}} & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\overline{\text{D}}B_{i}^{\prime }}{\overline{\text{D}}t}\equiv \frac{\unicode[STIX]{x2202}B_{i}^{\prime }}{\unicode[STIX]{x2202}t}+U_{1}(x_{2})\frac{\unicode[STIX]{x2202}B_{i}^{\prime }}{\unicode[STIX]{x2202}x_{1}}=B_{2}^{\prime }S\unicode[STIX]{x1D6FF}_{i1}+B_{0}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}x_{1}}, & \displaystyle\end{eqnarray}$$

where the uniform flow shear is given by

(3.6) $$\begin{eqnarray}S\equiv \frac{\text{d}U_{1}}{\text{d}x_{2}}.\end{eqnarray}$$

A coordinate frame ( $\boldsymbol{X}$ ) that convects and deforms with the mean flow is utilized to examine the flow evolution (Rogallo Reference Rogallo1981):

(3.7a-c ) $$\begin{eqnarray}X_{1}=x_{1}-\int _{0}^{t}U_{1}(x_{2})\,\text{d}\unicode[STIX]{x1D709},\quad X_{2}=x_{2},\quad X_{3}=x_{3},\quad t=t.\end{eqnarray}$$

Equations in the convective coordinate frame ( $\boldsymbol{X}$ ) then become

(3.8) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}X_{i}}-St\frac{\unicode[STIX]{x2202}u_{2}^{\prime }}{\unicode[STIX]{x2202}X_{1}}=0, & \displaystyle\end{eqnarray}$$

(3.9) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{i}^{\prime }}{\unicode[STIX]{x2202}X_{i}}-St\frac{\unicode[STIX]{x2202}B_{2}^{\prime }}{\unicode[STIX]{x2202}X_{1}}=0, & \displaystyle\end{eqnarray}$$

(3.10) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}t}=-u_{2}^{\prime }S\unicode[STIX]{x1D6FF}_{i1}-\frac{1}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}X_{i}}-\frac{B_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\frac{\unicode[STIX]{x2202}B_{1}^{\prime }}{\unicode[STIX]{x2202}X_{i}}+\frac{St}{\unicode[STIX]{x1D70C}}\frac{\unicode[STIX]{x2202}p^{\prime }}{\unicode[STIX]{x2202}X_{1}}\unicode[STIX]{x1D6FF}_{i2}+\frac{StB_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\frac{\unicode[STIX]{x2202}B_{1}^{\prime }}{\unicode[STIX]{x2202}X_{1}}\unicode[STIX]{x1D6FF}_{i2}+\frac{B_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\frac{\unicode[STIX]{x2202}B_{i}^{\prime }}{\unicode[STIX]{x2202}X_{1}}, & \displaystyle\end{eqnarray}$$

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x2202}B_{i}^{\prime }}{\unicode[STIX]{x2202}t}=B_{0}\frac{\unicode[STIX]{x2202}u_{i}^{\prime }}{\unicode[STIX]{x2202}X_{1}}+B_{2}^{\prime }S\unicode[STIX]{x1D6FF}_{i1}. & \displaystyle\end{eqnarray}$$

Following RDT methodology (Taylor & Batchelor Reference Taylor and Batchelor1949; Batchelor & Proudman Reference Batchelor and Proudman1954; Cambon, Coleman & Mansour Reference Cambon, Coleman and Mansour1993; Pope Reference Pope2001), we now consider perturbations that are periodic in all $X_{i}$ directions:

(3.12) $$\begin{eqnarray}q^{\prime }(\boldsymbol{X},t)=\hat{q}(t)\text{e}^{\text{i}\unicode[STIX]{x1D73F}(\mathbf{0})\boldsymbol{\cdot }\boldsymbol{X}}=\hat{q}(t)\text{e}^{\text{i}\unicode[STIX]{x1D73F}(\boldsymbol{t})\boldsymbol{\cdot }\boldsymbol{x}}.\end{eqnarray}$$

Owing to the coordinate transformation, the wavenumbers evolve as (Pope Reference Pope2001)

(3.13) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D705}_{i}}{\text{d}t}=-\unicode[STIX]{x1D705}_{j}\frac{\unicode[STIX]{x2202}U_{j}}{\unicode[STIX]{x2202}x_{i}}.\end{eqnarray}$$

For the mean velocity profile ( $U_{1}(x_{2})$ ) considered in this study, the above equation reduces to

(3.14a-c ) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D705}_{1}}{\text{d}t}=0,\quad \frac{\text{d}\unicode[STIX]{x1D705}_{2}}{\text{d}t}=-\unicode[STIX]{x1D705}_{1}S,\quad \frac{\text{d}\unicode[STIX]{x1D705}_{3}}{\text{d}t}=0,\end{eqnarray}$$

leading to

(3.14d-f ) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{1}(t)=\unicode[STIX]{x1D705}_{1}(0),\quad \unicode[STIX]{x1D705}_{2}(t)=\unicode[STIX]{x1D705}_{2}(0)-St\unicode[STIX]{x1D705}_{1}(0),\quad \unicode[STIX]{x1D705}_{3}(t)=\unicode[STIX]{x1D705}_{3}(0).\end{eqnarray}$$

The implication is that the wavevector magnitude and direction in the streamwise–spanwise plane remain constant, and the wavenumber in the normal direction grows linearly. The perturbation amplitudes evolve as (using (3.14d–f ))

(3.15) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\hat{u} _{i}}{\text{d}t}=-\hat{u} _{2}S\unicode[STIX]{x1D6FF}_{i1}-\text{i}\unicode[STIX]{x1D705}_{i}\left(\frac{\hat{p}}{\unicode[STIX]{x1D70C}}+\frac{B_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\hat{B}_{1}\right)+\frac{B_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\text{i}\unicode[STIX]{x1D705}_{1}\hat{B}_{i}, & \displaystyle\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\hat{B}_{i}}{\text{d}t}=\hat{B}_{2}S\unicode[STIX]{x1D6FF}_{i1}+\text{i}\unicode[STIX]{x1D705}_{1}B_{0}\hat{u} _{i} & \displaystyle\end{eqnarray}$$

subject to the divergence-free constraints

(3.17a,b ) $$\begin{eqnarray}\text{i}\unicode[STIX]{x1D705}_{i}\hat{u} _{i}=0,\quad \text{i}\unicode[STIX]{x1D705}_{i}\hat{B}_{i}=0.\end{eqnarray}$$

It is evident from the above equations that the stability characteristics of the linear system are contingent on the initial wavenumber of the perturbations. The equations demonstrate that $\unicode[STIX]{x1D705}_{2}$ plays a critical role in the perturbation evolution as it is the only wavenumber component that changes with time. We consider two distinct classes of initial conditions: $\unicode[STIX]{x1D705}_{2}\neq 0$ and $\unicode[STIX]{x1D705}_{2}=0$ .

Initial perturbations with $\unicode[STIX]{x1D705}_{2}(0)\neq 0$ . In the early stages, the initial value of $\unicode[STIX]{x1D705}_{2}$ plays a critical role in determining the evolution of the perturbations. This transient stage can have interesting implications on perturbation growth. Chagelishvili et al. (Reference Chagelishvili, Hristov, Chanishvili and Lominadze1993) have investigated the early-time energy interactions for two-dimensional perturbations on the streamwise–normal ( $\unicode[STIX]{x1D705}_{1}$ – $\unicode[STIX]{x1D705}_{2}$ ) plane. Further analytical examination of this parameter regime will performed in future works.

Initial perturbations with $\unicode[STIX]{x1D705}_{2}(0)=0$ . In this study, we focus on perturbations that are initially located on the $\unicode[STIX]{x1D705}_{1}$ – $\unicode[STIX]{x1D705}_{3}$ plane for several reasons. First and foremost, these perturbations provide crucial insight into the long-term stability characteristics of general shear flow instabilities subject to streamwise magnetic field. Second, owing to the nature of $\unicode[STIX]{x1D705}_{2}$ growth, the asymptotic stability of the linear equations can be completely captured by this initial condition. Finally, this system is amenable to rigorous mathematical analysis, which leads to profound insight into the stabilizing influence of magnetic field. We express these initial wavevectors in terms of their magnitude $\unicode[STIX]{x1D705}_{0}$ and orientation $\unicode[STIX]{x1D6FD}$ with respect to the streamwise direction (see figure 1 b). The wavenumber equation (3.14d–f ) can be written as

(3.18a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{1}(t)=\unicode[STIX]{x1D705}_{0}\cos \unicode[STIX]{x1D6FD},\quad \unicode[STIX]{x1D705}_{2}(t)=-\unicode[STIX]{x1D705}_{0}St\cos \unicode[STIX]{x1D6FD},\quad \unicode[STIX]{x1D705}_{3}(t)=\unicode[STIX]{x1D705}_{0}\sin \unicode[STIX]{x1D6FD}.\end{eqnarray}$$

The Poisson equation for total pressure amplitude ( $\hat{p}+B_{0}\hat{B}_{1}/\unicode[STIX]{x1D707}_{0}$ ) can be obtained using (3.14b ), (3.15) and (3.17):

(3.19) $$\begin{eqnarray}\frac{\hat{p}}{\unicode[STIX]{x1D70C}}+\frac{B_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\hat{B}_{1}=\frac{2\text{i}\unicode[STIX]{x1D705}_{1}}{\unicode[STIX]{x1D705}^{2}}\hat{u} _{2}S.\end{eqnarray}$$

Equation (3.15) then becomes

(3.20) $$\begin{eqnarray}\frac{\text{d}\hat{u} _{i}}{\text{d}t}=\hat{u} _{2}S\left(-\unicode[STIX]{x1D6FF}_{i1}+\frac{2\unicode[STIX]{x1D705}_{1}\unicode[STIX]{x1D705}_{i}}{\unicode[STIX]{x1D705}^{2}}\right)+\text{i}\frac{B_{0}}{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}\unicode[STIX]{x1D705}_{1}\hat{B}_{i}.\end{eqnarray}$$

The perturbation equations (3.16) and (3.20) are non-dimensionalized using the Alfvén wave speed ( $V_{A}$ ), mean flow shear, initial wavenumber and imposed magnetic field:

(3.21a-e ) $$\begin{eqnarray}u_{i}^{\ast }=u_{i}/V_{A},\quad V_{A}=\frac{B_{0}}{\sqrt{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D707}_{0}}},\quad \unicode[STIX]{x1D70F}=St,\quad \unicode[STIX]{x1D705}_{i}^{\ast }=\unicode[STIX]{x1D705}_{i}/\unicode[STIX]{x1D705}_{0},\quad B_{i}^{\ast }=B_{i}/B_{0}.\end{eqnarray}$$

Finally, the governing dimensionless equations are

(3.22) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\hat{u} _{i}^{\ast }}{\text{d}\unicode[STIX]{x1D70F}}=\hat{u} _{2}^{\ast }\left(-\unicode[STIX]{x1D6FF}_{i1}+\frac{2\cos (\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D705}_{i}^{\ast }}{1+\cos ^{2}(\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D70F}^{2}}\right)+\text{i}R_{A}\cos (\unicode[STIX]{x1D6FD})\hat{B}_{i}^{\ast }, & \displaystyle\end{eqnarray}$$

(3.23) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\hat{B}_{i}^{\ast }}{\text{d}\unicode[STIX]{x1D70F}}=\hat{B}_{2}^{\ast }\unicode[STIX]{x1D6FF}_{i1}+\text{i}R_{A}\cos (\unicode[STIX]{x1D6FD})\hat{u} _{i}^{\ast }, & \displaystyle\end{eqnarray}$$

where

(3.24) $$\begin{eqnarray}R_{A}\equiv \frac{\unicode[STIX]{x1D705}_{0}V_{A}}{S}=\frac{\text{shear timescale}}{\text{magnetic timescale}}.\end{eqnarray}$$

From the above equations, it can be seen that the flow behaviour can be completely characterized in terms of the parameters $R_{A}$ and $\unicode[STIX]{x1D6FD}$ . The role of $R_{A}$ has been recognized to some extent in the other studies (Chagelishvili et al. Reference Chagelishvili, Hristov, Chanishvili and Lominadze1993). The influence of $\unicode[STIX]{x1D6FD}$ is examined in detail for the first time in this study. Equations (3.22) and (3.23) clearly indicate that the velocity and magnetic field perturbations are coupled via the last terms on their right-hand sides. Here $R_{A}\cos (\unicode[STIX]{x1D6FD})$ is the coupling parameter in the problem. Therefore, for a given value of $R_{A}$ , streamwise perturbations experience the highest degree of coupling. The coupling becomes weaker with increasing $\unicode[STIX]{x1D6FD}$ . For spanwise orientations, velocity and magnetic field perturbations are decoupled.

We now investigate the evolution of the magnitudes of velocity and magnetic perturbation amplitudes. For ease of notation, we denote the dimensionless variables $q^{\ast }$ as $q$ . To obtain the evolution equation for perturbation kinetic energy, we multiply equation (3.22) by $\bar{\hat{u} }_{i}$ , the complex conjugate of $\hat{u} _{i}$ ,

(3.25) $$\begin{eqnarray}\bar{\hat{u} }_{i}\frac{\text{d}\hat{u} _{i}}{\text{d}\unicode[STIX]{x1D70F}}=\bar{\hat{u} }_{i}\hat{u} _{2}\left(-\unicode[STIX]{x1D6FF}_{i1}+\frac{2\cos (\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D705}_{i}}{1+\cos ^{2}(\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D70F}^{2}}\right)+\text{i}R_{A}\cos (\unicode[STIX]{x1D6FD})\hat{B}_{i}\bar{\hat{u} }_{i}.\end{eqnarray}$$

Utilizing the complex conjugate of the continuity equation (3.17a ), the above kinetic energy equation (3.25) reduces to

(3.26) $$\begin{eqnarray}\bar{\hat{u} }_{i}\frac{\text{d}\hat{u} _{i}}{\text{d}\unicode[STIX]{x1D70F}}=-\bar{\hat{u} }_{i}\hat{u} _{2}\unicode[STIX]{x1D6FF}_{i1}+\text{i}R_{A}\cos (\unicode[STIX]{x1D6FD})\hat{B}_{i}\bar{\hat{u} }_{i}.\end{eqnarray}$$

Similarly, equation (3.23) is multiplied by $\bar{\hat{B}}_{i}$ (the complex conjugate of $\hat{B}_{i}$ ) to obtain the magnetic energy equation:

(3.27) $$\begin{eqnarray}\bar{\hat{B}}_{i}\frac{\text{d}\hat{B}_{i}}{\text{d}\unicode[STIX]{x1D70F}}=\bar{\hat{B}}_{i}\hat{B}_{2}\unicode[STIX]{x1D6FF}_{i1}+\text{i}R_{A}\cos (\unicode[STIX]{x1D6FD})\bar{\hat{B}}_{i}\hat{u} _{i}.\end{eqnarray}$$

Summing (3.26) and (3.27) with their complex conjugate counterparts yields

(3.28) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\frac{\text{d}\hat{u} _{i}\bar{\hat{u} }_{i}}{\text{d}\unicode[STIX]{x1D70F}}=-\text{Re}(\bar{\hat{u} }_{i}\hat{u} _{2})\unicode[STIX]{x1D6FF}_{i1}-R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{i}\bar{\hat{u} }_{i}), & \displaystyle\end{eqnarray}$$

(3.29) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{1}{2}\frac{\text{d}\hat{B}_{i}\bar{\hat{B}}_{i}}{\text{d}\unicode[STIX]{x1D70F}}=\text{Re}(\hat{B}_{2}\bar{\hat{B}}_{i})\unicode[STIX]{x1D6FF}_{i1}+R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{i}\bar{\hat{u} }_{i}), & \displaystyle\end{eqnarray}$$

where $\text{Re}(q),\text{Im}(q)$ indicate the real and imaginary parts of a complex quantity $q$ . The first term on the right-hand side of (3.28) represents production, the process in which perturbation kinetic energy is transferred from the mean flow. Similarly, the first term on the right-hand side of (3.29) represents magnetic stretching, the process in which perturbation magnetic energy is produced by the stretching action of the mean flow. Production and magnetic stretching occur at shear ( $S$ ) timescale. It can be seen that the last terms on the right-hand side of (3.28), (3.29) are identical, but with opposite signs. This implies that they result in a harmonic exchange between magnetic and kinetic energies. The harmonic exchange process takes place at the timescale, $V_{A}\unicode[STIX]{x1D705}_{0}\cos (\unicode[STIX]{x1D6FD})$ . Therefore, $R_{A}\cos \unicode[STIX]{x1D6FD}$ determines the relative timescales of production and exchange processes.

We next investigate the energy transfer among individual components of kinetic and magnetic energies. We utilize the following notation for various energies for the sake of simplicity:

(3.30a-d ) $$\begin{eqnarray}k\equiv \hat{u} _{i}\bar{\hat{u} }_{i},\quad k_{1}\equiv \hat{u} _{1}\bar{\hat{u} }_{1},\quad k_{2}\equiv \hat{u} _{2}\bar{\hat{u} }_{2},\quad k_{3}\equiv \hat{u} _{3}\bar{\hat{u} }_{3},\end{eqnarray}$$

(3.30e-h ) $$\begin{eqnarray}b\equiv \hat{B}_{i}\bar{\hat{B}}_{i},\quad b_{1}\equiv \hat{B}_{1}\bar{\hat{B}}_{1},\quad b_{2}\equiv \hat{B}_{2}\bar{\hat{B}}_{2},\quad b_{3}\equiv \hat{B}_{3}\bar{\hat{B}}_{3}.\end{eqnarray}$$

The governing equations for the energies in velocity components are

(3.31a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}k_{1}}{\text{d}\unicode[STIX]{x1D70F}}\equiv \frac{\text{d}(\hat{u} _{1}\bar{\hat{u} }_{1}\!)}{\text{d}\unicode[STIX]{x1D70F}}=2\text{Re}(\hat{u} _{2}\bar{\hat{u} }_{1}\!)\frac{2\cos ^{2}(\unicode[STIX]{x1D6FD})}{1+\cos ^{2}(\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D70F}^{2}}-\!2\text{Re}(\hat{u} _{2}\bar{\hat{u} }_{1})\!-2R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{1}\bar{\hat{u} }_{1}),\qquad & \displaystyle\end{eqnarray}$$

(3.31b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}k_{2}}{\text{d}\unicode[STIX]{x1D70F}}\equiv \frac{\text{d}(\hat{u} _{2}\bar{\hat{u} }_{2})}{\text{d}\unicode[STIX]{x1D70F}}=-2\text{Re}(\hat{u} _{2}\bar{\hat{u} }_{2})\frac{2\cos ^{2}(\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D70F}}{1+\cos ^{2}(\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D70F}^{2}}-2R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{2}\bar{\hat{u} }_{2}), & \displaystyle\end{eqnarray}$$

(3.31c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}k_{3}}{\text{d}\unicode[STIX]{x1D70F}}\equiv \frac{\text{d}(\hat{u} _{3}\bar{\hat{u} }_{3})}{\text{d}\unicode[STIX]{x1D70F}}=2\text{Re}(\hat{u} _{2}\bar{\hat{u} }_{3})\frac{2\cos (\unicode[STIX]{x1D6FD})\sin (\unicode[STIX]{x1D6FD})}{1+\cos ^{2}(\unicode[STIX]{x1D6FD})\unicode[STIX]{x1D70F}^{2}}-2R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{3}\bar{\hat{u} }_{3}). & \displaystyle\end{eqnarray}$$

$\hat{u} _{i}$

$\hat{u} _{2}\bar{\hat{u} }_{1}$

$(-1)$

$\bar{\hat{u} }_{1}\hat{u} _{2}$

(3.32) $$\begin{eqnarray}\hat{u} _{1}\unicode[STIX]{x1D705}_{1}+\hat{u} _{2}\unicode[STIX]{x1D705}_{2}+\hat{u} _{3}\unicode[STIX]{x1D705}_{3}=0.\end{eqnarray}$$

Substituting for the values of $\unicode[STIX]{x1D705}_{i}$ using (3.18),

(3.33) $$\begin{eqnarray}\hat{u} _{1}\cos \unicode[STIX]{x1D6FD}-\unicode[STIX]{x1D70F}\hat{u} _{2}\cos \unicode[STIX]{x1D6FD}+\hat{u} _{3}\sin \unicode[STIX]{x1D6FD}=0.\end{eqnarray}$$

Multiplying the above equation by the complex conjugate of $\hat{u} _{1}$ ,

(3.34) $$\begin{eqnarray}\cos \unicode[STIX]{x1D6FD}|\hat{u} _{1}|^{2}+\bar{\hat{u} }_{1}\hat{u} _{3}\sin \unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D70F}\bar{\hat{u} }_{1}\hat{u} _{2}\cos \unicode[STIX]{x1D6FD}.\end{eqnarray}$$

Therefore, the term $\bar{\hat{u} }_{1}\hat{u} _{2}$ can be written as

(3.35) $$\begin{eqnarray}\bar{\hat{u} }_{1}\hat{u} _{2}=\frac{1}{\unicode[STIX]{x1D70F}}(|\hat{u} _{1}|^{2}+\bar{\hat{u} }_{1}\hat{u} _{3}\tan \unicode[STIX]{x1D6FD}).\end{eqnarray}$$

For $\unicode[STIX]{x1D6FD}=0^{\circ }$ , the continuity equation stipulates $\bar{\hat{u} }_{1}\hat{u} _{2}$ to be always positive. For other orientations, $\bar{\hat{u} }_{1}\hat{u} _{2}$ could be positive or negative depending upon the initial conditions. However, the value of $\bar{\hat{u} }_{1}\hat{u} _{2}$ for all orientations decreases with time as $\unicode[STIX]{x1D70F}^{-1}$ .

The evolution equations for energies in the magnetic field components are given by

(3.36a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}b_{1}}{\text{d}\unicode[STIX]{x1D70F}}\equiv \frac{\text{d}(\hat{B}_{1}\bar{\hat{B}}_{1})}{\text{d}\unicode[STIX]{x1D70F}}=2\text{Re}(\hat{B}_{2}\bar{\hat{B}}_{1})+2R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{1}\bar{\hat{u} }_{1}), & \displaystyle\end{eqnarray}$$

(3.36b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}b_{2}}{\text{d}\unicode[STIX]{x1D70F}}\equiv \frac{\text{d}(\hat{B}_{2}\bar{\hat{B}}_{2})}{\text{d}\unicode[STIX]{x1D70F}}=2R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{2}\bar{\hat{u} }_{2}), & \displaystyle\end{eqnarray}$$

(3.36c ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}b_{3}}{\text{d}\unicode[STIX]{x1D70F}}\equiv \frac{\text{d}(\hat{B}_{3}\bar{\hat{B}}_{3})}{\text{d}\unicode[STIX]{x1D70F}}=2R_{A}\cos (\unicode[STIX]{x1D6FD})\text{Im}(\hat{B}_{3}\bar{\hat{u} }_{3}). & \displaystyle\end{eqnarray}$$

$\hat{u} _{2}\bar{\hat{u} }_{1}$

$\hat{B}_{2}\bar{\hat{B}}_{1}$

(3.37) $$\begin{eqnarray}\bar{\hat{B}}_{1}\hat{B}_{2}=\frac{1}{\unicode[STIX]{x1D70F}}(|\hat{B}_{1}|^{2}+\bar{\hat{B}}_{1}\hat{B}_{3}\tan \unicode[STIX]{x1D6FD}).\end{eqnarray}$$

Overall, the energy interactions involve four important processes: production, magnetic stretching, pressure redistribution and harmonic kinetic–magnetic exchange. It can be observed that production and magnetic stretching depend on the behaviour of normal components of velocity and magnetic fields, $u_{2}^{\prime },B_{2}^{\prime }$ . Therefore, it is crucial to examine the evolution of $u_{2}^{\prime },B_{2}^{\prime }$ at different $R_{A},\unicode[STIX]{x1D6FD}$ .

We examine the evolution of spanwise perturbations ( $\unicode[STIX]{x1D6FD}=90^{\circ }$ ) first, followed by an investigation of other orientations ( $\unicode[STIX]{x1D6FD}\in [\!0^{\circ },90^{\circ }\!)$ ).

3.1 Regimes of evolution

3.1.1 Spanwise perturbations, $\unicode[STIX]{x1D6FD}=90^{\circ }$

As seen in previous sections, this represents a special limit because the velocity and magnetic fields are decoupled. The wavevectors remain at their initial state:

(3.38a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D705}_{1}(t)=0,\quad \unicode[STIX]{x1D705}_{2}(t)=0,\quad \unicode[STIX]{x1D705}_{3}(t)=\unicode[STIX]{x1D705}_{3}(0).\end{eqnarray}$$

The total pressure Poisson equation becomes

(3.39) $$\begin{eqnarray}\frac{1}{\unicode[STIX]{x1D70C}}\hat{p}+\frac{B_{0}}{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70C}}\hat{B}_{1}=0.\end{eqnarray}$$

The individual velocity components evolve as

(3.40a-c ) $$\begin{eqnarray}\frac{\text{d}\hat{u} _{1}}{\text{d}\unicode[STIX]{x1D70F}}=-\hat{u} _{2},\quad \frac{\text{d}\hat{u} _{2}}{\text{d}\unicode[STIX]{x1D70F}}=0,\quad \frac{\text{d}\hat{u} _{3}}{\text{d}\unicode[STIX]{x1D70F}}=0.\end{eqnarray}$$

The velocity equations are unaffected by pressure similar to spanwise perturbations in hydrodynamic homogeneous shear flow (Kumar et al. Reference Kumar, Bertsch and Girimaji2014). This is called pressure-released behaviour (Simone, Coleman & Cambon Reference Simone, Coleman and Cambon1997). Integrating the above equations,

(3.41a-c ) $$\begin{eqnarray}\hat{u} _{1}(\unicode[STIX]{x1D70F})=-\hat{u} _{2}(0)\unicode[STIX]{x1D70F}+\hat{u} _{1}(0),\quad \hat{u} _{2}(\unicode[STIX]{x1D70F})=\hat{u} _{2}(0),\quad \hat{u} _{3}(\unicode[STIX]{x1D70F})=\hat{u} _{3}(0).\end{eqnarray}$$

If $\hat{u} _{1},\hat{u} _{3}$ are zero initially, the kinetic energy ( $k$ ) in the flow will evolve with respect to the initial kinetic energy ( $k_{0}$ ) as (Simone et al. Reference Simone, Coleman and Cambon1997)

(3.42) $$\begin{eqnarray}\frac{k}{k_{0}}=1+\unicode[STIX]{x1D70F}^{2}.\end{eqnarray}$$

The individual magnetic components evolve as

(3.43a-c ) $$\begin{eqnarray}\frac{\text{d}\hat{B}_{1}}{\text{d}\unicode[STIX]{x1D70F}}=\hat{B}_{2},\quad \frac{\text{d}\hat{B}_{2}}{\text{d}\unicode[STIX]{x1D70F}}=0,\quad \frac{\text{d}\hat{B}_{3}}{\text{d}\unicode[STIX]{x1D70F}}=0.\end{eqnarray}$$

This implies

(3.44a-c ) $$\begin{eqnarray}\hat{B}_{1}(\unicode[STIX]{x1D70F})=\hat{B}_{2}(0)\unicode[STIX]{x1D70F}+\hat{B}_{1}(0),\quad \hat{B}_{2}(\unicode[STIX]{x1D70F})=\hat{B}_{2}(0),\quad \hat{B}_{3}(\unicode[STIX]{x1D70F})=\hat{B}_{3}(0).\end{eqnarray}$$

Magnetic perturbations remain unchanged when the initial magnetic energy is zero.

3.1.2 Perturbation orientations with $\unicode[STIX]{x1D6FD}\in [0^{\circ },90^{\circ })$

Recall that the evolution of kinetic and magnetic energies critically depends on the behaviour of $u_{2}^{\prime }$ and $B_{2}^{\prime }$ . To eliminate the explicit dependence of $\unicode[STIX]{x1D6FD}$ on $\hat{u} _{2},\hat{B}_{2}$ , we define a new timescale ( $\unicode[STIX]{x1D70F}^{\ast }$ ), similar to the transformation performed by Dimitrov et al. (Reference Dimitrov, Maneva, Hristov and Mishonov2011):

(3.45) $$\begin{eqnarray}\unicode[STIX]{x1D70F}^{\ast }=\unicode[STIX]{x1D70F}\cos \unicode[STIX]{x1D6FD}.\end{eqnarray}$$

The resultant equations are given by

(3.46a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\hat{u} _{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast }}=-\frac{2\unicode[STIX]{x1D70F}^{\ast }\hat{u} _{2}}{1+\unicode[STIX]{x1D70F}^{\ast 2}}+\text{i}R_{A}\hat{B}_{2}, & \displaystyle\end{eqnarray}$$

(3.46b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}\hat{B}_{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast }}=\text{i}R_{A}\hat{u} _{2}. & \displaystyle\end{eqnarray}$$

This is a coupled system of homogeneous linear ordinary differential equations with $R_{A}$ as the coupling parameter. To isolate the evolution of each field, a second derivative with respect to $\unicode[STIX]{x1D70F}^{\ast }$ is calculated. The resulting equations are

(3.47a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}\hat{u} _{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}=-\frac{2\unicode[STIX]{x1D70F}^{\ast }}{1+\unicode[STIX]{x1D70F}^{\ast 2}}\frac{\text{d}\hat{u} _{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast }}-2\hat{u} _{2}\frac{1-\unicode[STIX]{x1D70F}^{\ast 2}}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}+\text{i}R_{A}\frac{\text{d}\hat{B}_{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast }}, & \displaystyle\end{eqnarray}$$

(3.47b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}\hat{B}_{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}=\text{i}R_{A}\frac{\text{d}\hat{u} _{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast }}. & \displaystyle\end{eqnarray}$$

Using (3.46a ), (3.46b ), the decoupled equations for $\hat{u} _{2}$ and $\hat{B}_{2}$ are given by

(3.48a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}\hat{u} _{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}+\frac{2\unicode[STIX]{x1D70F}^{\ast }}{1+\unicode[STIX]{x1D70F}^{\ast 2}}\frac{\text{d}\hat{u} _{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast }}+2\hat{u} _{2}\frac{1-\unicode[STIX]{x1D70F}^{\ast 2}}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}+R_{A}^{2}\hat{u} _{2}=0, & \displaystyle\end{eqnarray}$$

(3.48b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}\hat{B}_{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}+\frac{2\unicode[STIX]{x1D70F}^{\ast }}{1+\unicode[STIX]{x1D70F}^{\ast 2}}\frac{\text{d}\hat{B}_{2}}{\text{d}\unicode[STIX]{x1D70F}^{\ast }}+R_{A}^{2}\hat{B}_{2}=0. & \displaystyle\end{eqnarray}$$

To facilitate an analytical solution, the following transformation of variables is performed (Lerner & Knobloch Reference Lerner and Knobloch1985):

(3.49a ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{u} _{2}=g(\unicode[STIX]{x1D70F}^{\ast })\exp \left(-\!\int \frac{\unicode[STIX]{x1D70F}^{\ast }\,\text{d}\unicode[STIX]{x1D70F}^{\ast }}{1+\unicode[STIX]{x1D70F}^{\ast 2}}\right)=\frac{g(\unicode[STIX]{x1D70F}^{\ast })}{\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}}, & \displaystyle\end{eqnarray}$$

(3.49b ) $$\begin{eqnarray}\displaystyle & \displaystyle \hat{B}_{2}=h(\unicode[STIX]{x1D70F}^{\ast })\exp \left(-\!\int \frac{\unicode[STIX]{x1D70F}^{\ast }\,\text{d}\unicode[STIX]{x1D70F}^{\ast }}{1+\unicode[STIX]{x1D70F}^{\ast 2}}\right)=\frac{h(\unicode[STIX]{x1D70F}^{\ast })}{\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}}. & \displaystyle\end{eqnarray}$$

The new variables $g$ and $h$ represent scaled values of velocity and magnetic field perturbation amplitudes. The resulting equations for the evolution of $g(\unicode[STIX]{x1D70F}^{\ast }),h(\unicode[STIX]{x1D70F}^{\ast })$ are

(3.50a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}g}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}=-\!\left(\frac{1-2\unicode[STIX]{x1D70F}^{\ast 2}}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}+R_{A}^{2}\right)g, & \displaystyle\end{eqnarray}$$

(3.50b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}h}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}=-\!\left(-\frac{1}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}+R_{A}^{2}\right)h. & \displaystyle\end{eqnarray}$$

A similar magnetic field evolution (3.50b ) was derived by Dimitrov et al. (Reference Dimitrov, Maneva, Hristov and Mishonov2011). Those authors note that (3.50b ) resembles the effective Schrödinger equation, and derived an analytical solution in terms of confluent Heun functions. The solution is not amenable to simple physical analysis. In this study, we extend the work of Dimitrov et al. (Reference Dimitrov, Maneva, Hristov and Mishonov2011) to identify different parameter regimes of self-similar behaviour and characterize the precise nature of energy exchange between velocity and magnetic fields. Following the analysis, we perform DNS to verify the analytical findings.

It is evident from (3.50a ), (3.50b ) that, for a given $R_{A}$ , the $u_{2}^{\prime }$ and $B_{2}^{\prime }$ perturbations are self-similar in scaled time $\unicode[STIX]{x1D70F}^{\ast }$ for all $\unicode[STIX]{x1D6FD}$ . The value of $\unicode[STIX]{x1D6FD}$ merely determines the timescale of evolution.

The first term on the right-hand side of (3.50a ) represents the pressure reorientation mechanism, which manifests as the pressure redistribution process in the energy equation discussed previously. Similarly, the first term on the right-hand side of (3.50b ) represents a higher-order implication of the effect of total pressure on $B_{2}^{\prime }$ . For any $\unicode[STIX]{x1D70F}^{\ast }$ , the maximum values of the coefficients of these terms are of the order unity. The last terms on the right-hand sides of (3.50a ), (3.50b ) represent the effect of harmonic exchange on the evolution of $\hat{u} _{2}$ and $\hat{B}_{2}$ . If the value of $R_{A}$ under consideration is larger than unity, the second terms influence the physical processes from the beginning. Taking these considerations into account, we examine different regimes of $R_{A}$ to identify dominant behaviour: $R_{A}\ll 1$ and $R_{A}\gg 1$ .

$R_{A}\gg 1$ or high magnetic field limit. In this limit, equations (3.50a ) and (3.50b ) simplify to

(3.51a ) $$\begin{eqnarray}\frac{\text{d}^{2}g}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}\approx -R_{A}^{2}g,\end{eqnarray}$$

(3.51b ) $$\begin{eqnarray}\frac{\text{d}^{2}h}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}\approx -R_{A}^{2}h.\end{eqnarray}$$

The solutions to these equations are

(3.52a ) $$\begin{eqnarray}\displaystyle & \displaystyle g(\unicode[STIX]{x1D70F}^{\ast })\approx A_{1}\exp (\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast })+A_{2}\exp (-\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast }), & \displaystyle\end{eqnarray}$$

(3.52b ) $$\begin{eqnarray}\displaystyle & \displaystyle h(\unicode[STIX]{x1D70F}^{\ast })\approx B_{1}\exp (\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast })+B_{2}\exp (-\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast }), & \displaystyle\end{eqnarray}$$

where the values of $A_{1}$ , $A_{2}$ , $B_{1}$ and $B_{2}$ can be obtained from appropriate initial conditions. The solution for the amplitudes of $u_{2}^{\prime }$ , $B_{2}^{\prime }$ can then be written as

(3.53) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}\displaystyle \hat{u} _{2}(\unicode[STIX]{x1D70F}^{\ast })\approx A_{1}\frac{\exp (\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast })}{\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}}+A_{2}\frac{\exp (-\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast })}{\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}},\\ \displaystyle \hat{B}_{2}(\unicode[STIX]{x1D70F}^{\ast })\approx B_{1}\frac{\exp (\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast })}{\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}}+B_{2}\frac{\exp (-\text{i}R_{A}\unicode[STIX]{x1D70F}^{\ast })}{\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}}.\end{array}\right\}\end{eqnarray}$$

The behaviour of $u_{2}^{\prime },B_{2}^{\prime }$ is oscillatory, as a result of the dominating influence of harmonic exchange. Owing to pressure reorientation, the magnitude of $u_{2}^{\prime }$ and consequently $B_{2}^{\prime }$ decrease. Therefore, using Parseval’s theorem, the evolution of normal kinetic energy ( $k_{2}$ ) in physical space is given by

(3.54) $$\begin{eqnarray}k_{2}=\frac{\cos ^{2}(R_{A}\unicode[STIX]{x1D70F}^{\ast })}{1+\unicode[STIX]{x1D70F}^{\ast 2}}.\end{eqnarray}$$

For the case of no initial magnetic field perturbations, the evolution of normal magnetic energy ( $b_{2}$ ) is given by

(3.55) $$\begin{eqnarray}b_{2}=\frac{\sin ^{2}(R_{A}\unicode[STIX]{x1D70F}^{\ast })}{1+\unicode[STIX]{x1D70F}^{\ast 2}}.\end{eqnarray}$$

A solution to (3.50b ), similar to (3.55), was derived by Lau & Liu (Reference Lau and Liu1980). In this limit, $u_{2}^{\prime }$ and $B_{2}^{\prime }$ are oscillatory with a phase difference of $\unicode[STIX]{x03C0}/2$ and a time period of $T=2\unicode[STIX]{x03C0}/R_{A}$ . This oscillatory evolution results in negligible production and magnetic stretching when averaged over a full cycle. This implies that harmonic exchange between magnetic and kinetic energy dominates at all times. As a result, equipartition of energy between perturbation velocity and magnetic fields is indicated.

$R_{A}\ll 1$ or low magnetic field limit. We first consider the evolution of the coefficients of the first terms on the right-hand sides of (3.50a ) and (3.50b ). At initial times,

(3.56) $$\begin{eqnarray}\frac{1-2\unicode[STIX]{x1D70F}^{\ast 2}}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}\sim O(1)\end{eqnarray}$$

and asymptotes to zero at late times. Similarly, at early times,

(3.57) $$\begin{eqnarray}\frac{1}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}\sim O(1)\end{eqnarray}$$

and asymptotes to zero at subsequent times. Thus, the first terms dominate during the early time in the low $R_{A}$ regime. At late times, however, the first terms have negligible contributions and the last terms dominate, i.e. harmonic exchange prevails. We now investigate the behaviour at early times.

Early-time behaviour. In this limit, the last terms in (3.50a ), (3.50b ) can be omitted. This implies that exchange is negligible and pressure reorientation is the crucial process at this stage. As a result, the evolution equations for $g,h$ can be written as follows:

(3.58a ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}g}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}\approx -\frac{1-2\unicode[STIX]{x1D70F}^{\ast 2}}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}g, & \displaystyle\end{eqnarray}$$

(3.58b ) $$\begin{eqnarray}\displaystyle & \displaystyle \frac{\text{d}^{2}h}{\text{d}\unicode[STIX]{x1D70F}^{\ast 2}}\approx \frac{1}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}h. & \displaystyle\end{eqnarray}$$

The $g$ equation (3.58a ) is identical to the pure hydrodynamic case, implying that the magnetic field does not affect the initial behaviour. The solution for such flows is given by

(3.59) $$\begin{eqnarray}g(\unicode[STIX]{x1D70F}^{\ast })=\frac{c_{1}}{\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}}.\end{eqnarray}$$

The evolution of $k_{2}$ is given by

(3.60) $$\begin{eqnarray}k_{2}=\frac{1}{(1+\unicode[STIX]{x1D70F}^{\ast 2})^{2}}.\end{eqnarray}$$

The following is the analytical solution for $h,\hat{B}_{2}$ :

(3.61) $$\begin{eqnarray}h=\sqrt{1+\unicode[STIX]{x1D70F}^{\ast 2}}(c_{1}+c_{2}\arctan (\unicode[STIX]{x1D70F}^{\ast }))\Rightarrow \hat{B}_{2}=c_{1}+c_{2}\arctan (\unicode[STIX]{x1D70F}^{\ast }).\end{eqnarray}$$

If there is no initial magnetic field perturbation, then $c_{1}=0$ . $c_{2}$ can be determined from (3.46b ). Thus, the final solution for $b_{2}$ at early times is

(3.62) $$\begin{eqnarray}b_{2}=R_{A}^{2}\arctan ^{2}(\unicode[STIX]{x1D70F}^{\ast }).\end{eqnarray}$$

Late-time behaviour. For late times, harmonic exchange dominates as seen from the order of magnitude analysis performed above. As a result, the evolution of $\hat{u} _{2}$ and $\hat{B}_{2}$ becomes oscillatory, similar to the behaviour of cases with $R_{A}\gg 1$ .

Therefore, $u_{2}^{\prime }$ and $B_{2}^{\prime }$ for flows with $R_{A}\ll 1$ experience a three-stage evolution. At early times, pressure reorientation determines the dynamics. However, for later stages, harmonic exchange is the dominant mechanism. At late times, the equipartition between magnetic and kinetic energies is indicated once again. The behaviour at intermediate times where pressure reorientation and exchange are equally dominant will be deduced from numerical simulations.

4 Numerical setup

DNS performed in this study employ the magneto-gas kinetic scheme (MGKM) developed by Araya et al. (Reference Araya, Ebersohn, Anderson and Girimaji2015). MGKM utilizes the kinetic-based gas kinetic method (GKM) for solving the flow field equations. The magnetic field is computed from Ohm’s law using a finite-difference solver. Lorentz force and Joule heating terms are added as source terms in the momentum and energy equations, respectively. The GKM portion of the scheme has been exhaustively validated in Kumar et al. (Reference Kumar, Bertsch and Girimaji2014) for homogeneous shear turbulence, Xie & Girimaji (Reference Xie and Girimaji2014) for wall-bounded flows and Karimi & Girimaji (Reference Karimi and Girimaji2016) for free shear flows. The details of the numerical scheme and validation for the case of shock tube and channel flow simulations are described in Araya et al. (Reference Araya, Ebersohn, Anderson and Girimaji2015).

The schematic of the background velocity and magnetic fields with perturbations is shown in figure 1(a). Periodic boundary conditions are applied in streamwise and spanwise directions. For boundaries in normal direction, shear periodic boundary conditions are applied (Baron Reference Baron1982):

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D719}(t,x+m_{1}L_{1},y+m_{2}L_{2},z+m_{3}L_{3})=\unicode[STIX]{x1D719}(t,x-Sm_{2}L_{2}t,y,z),\end{eqnarray}$$

where $L_{2}$ is the length of the domain in the normal direction and $m_{1},m_{2},m_{3}$ are integers.

Table 1.

Simulation parameters.

Simulations are performed for different shear-to-magnetic timescales ( $R_{A}$ ). Table 1 lists the details of various simulations. As $u_{2}^{\prime }$ is the most critical for the development of fluid instabilities, only these perturbations are considered in the initial conditions:

(4.2a-c ) $$\begin{eqnarray}u_{1}^{\prime }(0)=0,\quad u_{2}^{\prime }(0)=\sin (\unicode[STIX]{x1D705}_{1}x_{1}+\unicode[STIX]{x1D705}_{3}x_{3}),\quad u_{3}^{\prime }(0)=0,\end{eqnarray}$$

(4.2d-f ) $$\begin{eqnarray}B_{1}^{\prime }(0)=0,\quad B_{2}^{\prime }(0)=0,\quad B_{3}^{\prime }(0)=0,\end{eqnarray}$$

where $\unicode[STIX]{x1D705}_{1}$ and $\unicode[STIX]{x1D705}_{3}$ are integers and $\unicode[STIX]{x1D705}_{1}$ and $\unicode[STIX]{x1D705}_{3}$ are modified such that the perturbation wavevector orientations vary from streamwise to spanwise directions, i.e.  $\unicode[STIX]{x1D6FD}\in [0^{\circ },90^{\circ }]$ . Reynolds numbers, fluid and magnetic ( $\mathit{Re}=S/\unicode[STIX]{x1D708}\unicode[STIX]{x1D705}^{2},\mathit{Re}_{m}=S/\unicode[STIX]{x1D702}\unicode[STIX]{x1D705}^{2}$ ) and $\unicode[STIX]{x1D6FD}$ values considered in different simulations are listed in table 1. Since MGKM is a compressible flow solver, we choose low gradient Mach numbers, $M_{g}\equiv S/\unicode[STIX]{x1D705}a\leqslant 0.03$ , to emulate the incompressible limit. A grid sensitivity study is performed for the case of $\unicode[STIX]{x1D6FD}=0^{\circ }$ in the low $R_{A}~(=0.39)$ limit. Figure 2 shows that $128\times 128\times 4$ and $256\times 256\times 4$ give nearly identical results except for a higher numerical dissipation in the low-resolution simulation. As the evolution of long-wavelength perturbations is of current interest, the grid resolution requirements are much less stringent than turbulent flows. The smallest energetic wavelength encountered at the end of the run is of the order of the domain size. Thus, the grid resolution employed in the simulation is quite adequate. Nonlinear breakdown and transition to turbulence would require much finer grids and such investigation is currently underway. The analytical results derived in the previous sections would validate the implementation of shear periodic boundary conditions into MGKM for future nonlinear studies.

Figure 2.

Grid independence for $R_{A}=0.39$ and $\unicode[STIX]{x1D6FD}=0^{\circ }$ .

In addition to DNS, the RDT equations (3.22) and (3.23) are evolved temporally using a fourth-order Runge–Kutta method. To contrast the MHD flow evolution against hydrodynamic flows, cases with $R_{A}=0$ are also considered. Analytical results and numerical solutions from RDT and DNS are compared in the next section.

6 Summary and conclusions

The goal of this paper was to investigate the fundamental flow and flow–magnetic field interaction mechanisms that exert a stabilizing influence on perturbations in sheared MHD flows. Four constituent mechanisms have been identified: pressure redistribution, kinetic energy production, kinetic–magnetic harmonic exchange and magnetic stretching. Linear analysis and DNS are performed to characterize the role of different mechanisms in different parameter regimes of magnetic field strength ( $B_{0}$ ) and wavevector orientation ( $\unicode[STIX]{x1D6FD}$ ).

To establish the fundamental character of these mechanisms, velocity and magnetic field perturbations are investigated in homogeneously sheared MHD flows. Linear analysis identifies the ratio of shear and magnetic timescales, $R_{A}\equiv V_{A}\unicode[STIX]{x1D705}/S$ , as the parameter that characterizes the effect of relative timescales of magnetic and fluid processes. Velocity and magnetic field perturbations in the normal direction are identified as crucial, as they influence the production and magnetic stretching terms. For a given value of $R_{A}$ , these perturbations have identical evolution at all $\unicode[STIX]{x1D6FD}$ with respect to $\unicode[STIX]{x1D70F}^{\ast }=\unicode[STIX]{x1D70F}\cos \unicode[STIX]{x1D6FD}$ . This implies that as $\unicode[STIX]{x1D6FD}$ increases, the time elapse becomes slower. The primary findings of this paper are now summarized.

Spanwise perturbations. For this orientation, the velocity and magnetic field equations are decoupled. In addition the velocity evolution is impervious to pressure (pressure-released). This results in a quadratic growth of kinetic energy.

Low magnetic field regime for $\unicode[STIX]{x1D6FD}\in [\!0^{\circ },90^{\circ }\!)$ . In this limit, a three-stage perturbation evolution is seen. At initial times, the kinetic energy follows the hydrodynamic evolution. A slight transfer to magnetic perturbations is also seen in this stage. During the second stage, magnetic perturbations are amplified owing to magnetic stretching. At late times, harmonic exchange dominates. This results in equipartition of energy at late times.

Intermediate magnetic field regime for $\unicode[STIX]{x1D6FD}\in [\!0^{\circ },90^{\circ }\!)$ . For this shear-to-magnetic timescale ratio, the first and second stages seen in the low magnetic field regime occur simultaneously. As a result, magnetic stretching, production and pressure redistribution occur in the first stage. The second stage is characterized by harmonic kinetic–magnetic exchange. This again results in equipartition of kinetic and magnetic energies.

High magnetic field regime for $\unicode[STIX]{x1D6FD}\in [0^{\circ },90^{\circ })$ . In this limit, all three stages occur simultaneously. This results in negligible net production and magnetic stretching. Pressure redistribution and harmonic exchange are the only dominant processes. Equipartition is prevalent at all times owing to harmonic exchange.

This work is the first of a three-part study. The findings of this paper establish the fundamental nature of perturbation evolution in homogeneous shear flows at various values of shear-to-magnetic timescale ratios ( $R_{A}$ ). The second study extends the non-modal linear analysis to inhomogeneous shear flows, specifically mixing layers. For inhomogeneous shear flows the flow domain at a particular moment experiences various $R_{A}$ regimes owing to the inhomogeneity in shear. Since the transient behaviour of perturbations is understood for various $R_{A}$ values, the inferences from the current study will help establish the locations of stabilization of magnetic field in both linear and nonlinear stages of flow field evolution. This is unique to non-modal analysis as modal analysis would only indicate whether there is linear stabilization in the asymptotic limit. The final (third) study of this sequence involves comparing and contrasting the effects of compressibility and magnetic field on shear flow instabilities.