2 Formulation of the problem

Consider a channel of width $L$ and depth $\unicode[STIX]{x1D6FF}$ between two solid vertical parallel walls filled with an inviscid liquid of density $\unicode[STIX]{x1D70C}$ and electrical conductivity $\unicode[STIX]{x1D70E}$ . The liquid carries electric current of a uniform density $\boldsymbol{j}_{0}$ directed upwards along the rigid edge of the channel, which coincides with the $y$ -axis of the Cartesian system of coordinates shown in figure 1. The layer is assumed to be thin with $\unicode[STIX]{x1D6FF}\ll L$ and, thus, treated as a planar undeformable sheet with the areal density $\bar{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6FF}$ and the respective electrical conductivity $\bar{\unicode[STIX]{x1D70E}}=\unicode[STIX]{x1D70E}\unicode[STIX]{x1D6FF}$ . The electric current with the linear density $\boldsymbol{J}_{0}=\boldsymbol{e}_{y}j_{0}\unicode[STIX]{x1D6FF}$ that flows along the sheet generates a magnetic field whose normal $(z)$ component through the sheet can be found using the Biot–Savart law as

(2.1) $$\begin{eqnarray}B_{0}(x)=\frac{\unicode[STIX]{x1D707}_{0}J_{0}}{2\unicode[STIX]{x03C0}}\int _{0}^{L}\frac{\text{d}\unicode[STIX]{x1D709}}{\unicode[STIX]{x1D709}-x}=\frac{\unicode[STIX]{x1D707}_{0}J_{0}}{2\unicode[STIX]{x03C0}}\ln \left(\frac{L}{x}-1\right)\approx \frac{\unicode[STIX]{x1D707}_{0}J_{0}}{2\unicode[STIX]{x03C0}}\ln \frac{L}{x},\end{eqnarray}$$

where the last approximate expression holds in the vicinity of edge for $x\ll L$ , which will be the main focus of this study. On the other hand, the sheet approximation implies $x\gg \unicode[STIX]{x1D6FF}$ . Note that expression (2.1) represents a superposition of the fields of straight wires which may be thought to form the sheet. As only the gradient of the field $B_{0}^{\prime }(x)\approx -(\unicode[STIX]{x1D707}_{0}J_{0}/2\unicode[STIX]{x03C0}x)$ is required in the following, the width $L$ drops out of the model. It means that the layer is treated as semi-infinite. The interaction of the electric current with its own magnetic field gives rise to the pinch force with the areal density $\boldsymbol{F}_{0}=\boldsymbol{J}_{0}\times \boldsymbol{B}_{0}=\boldsymbol{e}_{x}J_{0}B_{0}(x)$ , which is balanced by the $x$ -component of the effective pressure gradient

(2.2) $$\begin{eqnarray}P_{0}^{\prime }(x)=-J_{0}B_{0}(x).\end{eqnarray}$$

Note that for sheet, the effective pressure $P$ is defined as force per unit length.

Disturbance of this quiescent equilibrium state by an arbitrary slow liquid flow, which is confined to the plane of the sheet by the solid bounding walls, is governed by the linearized two-dimensional (2-D) Euler’s equation

(2.3) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D70C}}\unicode[STIX]{x2202}_{t}\boldsymbol{v}_{1}=-\unicode[STIX]{x1D735}P_{1}+\boldsymbol{F}_{1},\end{eqnarray}$$

with the velocity distribution $\boldsymbol{v}_{1}$ subject to the incompressibility constraint $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{v}_{1}=0$ . The linearized perturbation of the electromagnetic force $\boldsymbol{F}_{1}=(\boldsymbol{J}_{1}B_{0}+\boldsymbol{J}_{0}B_{1})\times \boldsymbol{e}_{z}$ produced by the flow results from the interaction of the current perturbation $\boldsymbol{J}_{1}$ with the base magnetic field $B_{0}$ as well as from the interaction of the base current $\boldsymbol{J}_{0}$ with the perturbation of the normal magnetic field $B_{1}$ . Note that the quadratic terms in the perturbation amplitudes are neglected as usual in the linear stability analysis because all perturbations are assumed to be infinitesimal. The perturbation of the pressure gradient $\unicode[STIX]{x1D735}P_{1}$ is eliminated from (2.3) by applying the curl operator. Then the flow is governed by the $z$ -component of the resulting equation

(2.4) $$\begin{eqnarray}\bar{\unicode[STIX]{x1D70C}}\unicode[STIX]{x2202}_{t}\unicode[STIX]{x1D714}_{1}=-(\boldsymbol{J}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{0}+\boldsymbol{J}_{0}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{1}),\end{eqnarray}$$

where $\unicode[STIX]{x1D714}_{1}=\boldsymbol{e}_{z}\boldsymbol{\cdot }\unicode[STIX]{x1D735}\times \boldsymbol{v}_{1}$ is the normal component of vorticity. The associated current perturbation is governed by Ohm’s law for a moving medium

(2.5) $$\begin{eqnarray}\boldsymbol{J}_{1}=\bar{\unicode[STIX]{x1D70E}}(\boldsymbol{E}_{1}+\boldsymbol{v}_{1}\times \boldsymbol{e}_{z}B_{0}).\end{eqnarray}$$

A time-dependent perturbation of the magnetic field induces a rotational electric field according to the first Maxwell equation $\unicode[STIX]{x1D735}\times \boldsymbol{E}_{1}=-\unicode[STIX]{x2202}_{t}\boldsymbol{B}_{1}$ . Note that in highly resistive media this induction effect is usually negligible, which corresponds to the so-called quasi-stationary or inductionless approximation commonly used in the liquid-metal MHD (Roberts Reference Roberts1967). In order to keep our model general and applicable also to well conducting fluids, we forgo this approximation here.

The charge conservation $\unicode[STIX]{x1D735}\boldsymbol{\cdot }\boldsymbol{J}_{1}=0$ is satisfied by introducing the electric streamfunction $h_{1}$ , which, as shown below, is directly related to the scalar magnetic potential at the current sheet. It allows us to represent the electric current in the plane of sheet as

(2.6) $$\begin{eqnarray}\boldsymbol{J}_{1}=-\boldsymbol{e}_{z}\times \unicode[STIX]{x1D735}h_{1}.\end{eqnarray}$$

Note that a truly 2-D ( $z$ -independent) current distribution would generate a 2-D magnetic field with $z$ -component only which would then coincide with the current streamfunction introduced above. The current sheet considered here, however, produces a 3-D magnetic field whose normal component at the sheet is related with the electric stream function by the induction-type equation which is obtained as follows. Substituting (2.6) into (2.5) and taking the $z$ -component of the curl of the resulting equation, we obtain

(2.7) $$\begin{eqnarray}\unicode[STIX]{x2202}_{t}B_{1}+\boldsymbol{v}_{1}\boldsymbol{\cdot }\unicode[STIX]{x1D735}B_{0}=\bar{\unicode[STIX]{x1D70E}}^{-1}\unicode[STIX]{x1D6FB}^{2}h_{1}.\end{eqnarray}$$

The magnetic field perturbation in the surrounding space is sought as

(2.8) $$\begin{eqnarray}\boldsymbol{B}_{1}=-\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D735}\unicode[STIX]{x1D6F9}_{1}\end{eqnarray}$$

using the scalar magnetic potential $\unicode[STIX]{x1D6F9}_{1}$ , which is governed by the Laplace equation

(2.9) $$\begin{eqnarray}\unicode[STIX]{x1D6FB}^{2}\unicode[STIX]{x1D6F9}_{1}=0.\end{eqnarray}$$

The surface current density is related to the jump of the tangential magnetic field over the current sheet by Ampère’s circuital law: $\boldsymbol{J}_{1}=(1/\unicode[STIX]{x1D707}_{0})\boldsymbol{e}_{z}\times [\boldsymbol{B}_{1}]_{S}$ , where $\unicode[STIX]{x1D707}_{0}=4\unicode[STIX]{x03C0}\times 10^{-7}~\text{H}~\text{m}^{-1}$ is the vacuum permeability. Combining this current sheet condition with (2.6) and (2.8), we obtain the boundary condition

(2.10) $$\begin{eqnarray}\unicode[STIX]{x1D6F9}_{1}|_{z\rightarrow +0}=-\unicode[STIX]{x1D6F9}_{1}|_{z\rightarrow -0}={\textstyle \frac{1}{2}}h_{1},\end{eqnarray}$$

which relates the scalar magnetic potential with the electric stream function. It is important to note that the problem is defined in terms of the linear current density $J_{0}$ , which is the relevant electromagnetic parameter in the current sheet approximation.

3 Linear stability analysis

Owing to the $y$ -invariance of the base state, perturbations can be sought in the normal mode form

(3.1) $$\begin{eqnarray}\{\boldsymbol{v}_{1},p_{1},h_{1}\}(\boldsymbol{x},t)=\{\hat{\boldsymbol{v}},\hat{p},{\hat{h}}\}(x)\text{e}^{\unicode[STIX]{x1D6FE}t+\text{i}ky},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}$ is a generally complex growth rate and $k$ is a real wavenumber. Then (2.4) and (2.7) for the flow and current perturbations take the form

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}\boldsymbol{D}_{k}^{2}\hat{v}=-\bar{\unicode[STIX]{x1D70C}}^{-1}J_{0}k^{2}\left(\hat{B}+\frac{\unicode[STIX]{x1D707}_{0}}{2\unicode[STIX]{x03C0}}\frac{{\hat{h}}}{x}\right), & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}\hat{B}=\bar{\unicode[STIX]{x1D70E}}^{-1}\boldsymbol{D}_{k}^{2}{\hat{h}}+J_{0}\frac{\unicode[STIX]{x1D707}_{0}}{2\unicode[STIX]{x03C0}}\frac{\hat{v}}{x}, & \displaystyle\end{eqnarray}$$

where $\hat{v}$ is the $x$ -component of the velocity perturbation, which is further referred to as the transverse velocity, $\hat{B}$ is the normal $(z)$ component of the induced magnetic field at the sheet and $\boldsymbol{D}_{k}\equiv \boldsymbol{e}_{x}(\text{d}/\text{d}x)+\text{i}k\boldsymbol{e}_{y}$ is a spectral counterpart of the nabla operator acting on the mode with the wavenumber $k$ . The $y$ -component of velocity perturbation $\hat{u}$ has been eliminated from (3.2) by using the incompressibility constraint $\boldsymbol{D}_{k}\boldsymbol{\cdot }\hat{\boldsymbol{v}}=\hat{v}^{\prime }+\text{i}k\hat{u} =0$ , while the $z$ -component of velocity is absent owing to the undeformability of the sheet.

The boundary conditions which follow from the vanishing of the $x$ -component of velocity and electric current at the fixed insulating edge $(x=0)$ and far from the edge $(x\rightarrow \infty )$ are

(3.4) $$\begin{eqnarray}\hat{v}={\hat{h}}=0.\end{eqnarray}$$

These boundary conditions imply $(\hat{v},{\hat{h}})\sim x$ at the edge. It means that the terms ${\sim}1/x,$ which result from the logarithmic singularity of the base magnetic field (2.1) in (3.2) and (3.3), are expected to be regular as $x\rightarrow 0:{\hat{h}}/x\rightarrow {\hat{h}}^{\prime }(0)$ and $\hat{v}/x\rightarrow \hat{v}^{\prime }(0)$ . Then the finite longitudinal current perturbation at the edge ${\hat{h}}^{\prime }(0)={\hat{J}}_{y}(0)$ would produce a perturbation of the magnetic field with a logarithmic singularity $\hat{B}\sim \ln x$ similar to that of the base magnetic field (2.1). According to (3.2) and (3.3), which reduce to $(\hat{v},{\hat{h}})^{\prime \prime }\sim \ln x$ for $x\rightarrow 0$ , logarithmic singularity produces a higher-order small perturbation $(\hat{v},{\hat{h}})\sim x^{2}\ln x$ . This confirms the regularity of ${\hat{h}}^{\prime }$ and $\hat{v}^{\prime }$ at the edge.

To find $\hat{B}$ required in (3.2) we need to solve (2.9) in the space surrounding the sheet. This can conveniently be done in the cylindrical coordinates with the axis aligned along the edge and the poloidal angle $\unicode[STIX]{x1D703}$ measured from the plane of the sheet so that $x=r\cos \unicode[STIX]{x1D703}$ , $z=r\sin \unicode[STIX]{x1D703}$ and $\unicode[STIX]{x1D6F9}_{1}(\boldsymbol{x},t)=\hat{\unicode[STIX]{x1D6F9}}(r,\unicode[STIX]{x1D703})\text{e}^{\unicode[STIX]{x1D6FE}t+\text{i}ky}$ . Then (2.9) takes the form

(3.5) $$\begin{eqnarray}r\unicode[STIX]{x2202}_{r}(r\unicode[STIX]{x2202}_{r}\hat{\unicode[STIX]{x1D6F9}})+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}^{2}\hat{\unicode[STIX]{x1D6F9}}-(rk)^{2}\hat{\unicode[STIX]{x1D6F9}}=0.\end{eqnarray}$$

Boundary condition (2.10), which now reads as $\hat{\unicode[STIX]{x1D6F9}}|_{\unicode[STIX]{x1D703}=0}=-\hat{\unicode[STIX]{x1D6F9}}|_{\unicode[STIX]{x1D703}=2\unicode[STIX]{x03C0}}={\hat{h}}(x)/2$ , and the reflection symmetry of the problem $\hat{\unicode[STIX]{x1D6F9}}(r,\unicode[STIX]{x1D703})=-\hat{\unicode[STIX]{x1D6F9}}(r,2\unicode[STIX]{x03C0}-\unicode[STIX]{x1D703})$ , suggests a solution in the form

(3.6) $$\begin{eqnarray}\hat{\unicode[STIX]{x1D6F9}}(r,\unicode[STIX]{x1D703})=\frac{1}{2}{\hat{h}}(r)\cos \frac{\unicode[STIX]{x1D703}}{2}-\frac{1}{2\unicode[STIX]{x03C0}}\mathop{\sum }_{n=1}^{\infty }c_{n}\hat{\unicode[STIX]{x1D6F9}}_{n}(r)\sin (n\unicode[STIX]{x1D703}),\end{eqnarray}$$

where $c_{n}=\int _{0}^{2\unicode[STIX]{x03C0}}\sin (n\unicode[STIX]{x1D703})\cos (\unicode[STIX]{x1D703}/2)\,\text{d}\unicode[STIX]{x1D703}=8n/(4n^{2}-1)$ . Substituting this expression into (3.5), we obtain a sequence of ordinary differential equations

(3.7) $$\begin{eqnarray}r(r\hat{\unicode[STIX]{x1D6F9}}_{n}^{\prime })^{\prime }-(n^{2}+(rk)^{2})\hat{\unicode[STIX]{x1D6F9}}_{n}=r(r{\hat{h}}^{\prime })^{\prime }-(2^{-2}+(rk)^{2}){\hat{h}},\end{eqnarray}$$

which define the harmonics of the scalar magnetic potential $\hat{\unicode[STIX]{x1D6F9}}_{n}$ in terms of the electric streamfunction ${\hat{h}}$ . For (3.6) to be single valued at $r=0$ , $\hat{\unicode[STIX]{x1D6F9}}_{n}(0)=0$ is required, which serves as a boundary condition for (3.7). Then the normal magnetic field perturbation in (3.2) is given by

(3.8) $$\begin{eqnarray}\hat{B}(x)=\left.-\frac{\unicode[STIX]{x1D707}_{0}}{x}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D703}}\hat{\unicode[STIX]{x1D6F9}}\right|_{\unicode[STIX]{x1D703}=0}=\frac{\unicode[STIX]{x1D707}_{0}}{2\unicode[STIX]{x03C0}x}\mathop{\sum }_{n=1}^{\infty }nc_{n}\hat{\unicode[STIX]{x1D6F9}}_{n}(x).\end{eqnarray}$$

The problem is converted into the dimensionless form by using $v_{m}=(\unicode[STIX]{x1D707}_{0}\bar{\unicode[STIX]{x1D70E}})^{-1}$ and

(3.9) $$\begin{eqnarray}\unicode[STIX]{x1D70F}_{m}=\frac{\bar{\unicode[STIX]{x1D70C}}}{\bar{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D707}_{0}^{2}J_{0}^{2}}\end{eqnarray}$$

as the characteristic velocity and time scales. The former corresponds to the magnetic diffusion speed, whereas the latter, when represented in terms of the characteristic magnetic flux density $B_{0}=\unicode[STIX]{x1D707}_{0}J_{0},$ coincides with the so-called magnetic response time (Roberts Reference Roberts1967), also known as the magnetic damping time (Davidson Reference Davidson2001), which typically appears in the inductionless $(\mathit{Pm}=0)$ limit. The length scale can conveniently be chosen as

(3.10) $$\begin{eqnarray}k^{-1}=\frac{\unicode[STIX]{x1D706}}{2\unicode[STIX]{x03C0}},\end{eqnarray}$$

where $k$ and $\unicode[STIX]{x1D706}$ are the wavenumber and wavelength of perturbation, respectively. The electric streamfunction and the scalar magnetic potential are both scaled with $J_{0}/k$ . Then (3.3) and (3.2) take the following dimensionless form

(3.11) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}\boldsymbol{D}_{k}^{2}\hat{v}=-\left(\hat{B}+\frac{1}{2\unicode[STIX]{x03C0}}\frac{{\hat{h}}}{x}\right), & \displaystyle\end{eqnarray}$$

(3.12) $$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6FE}S_{\unicode[STIX]{x1D706}}^{2}\hat{B}=\boldsymbol{D}_{k}^{2}{\hat{h}}+\frac{1}{2\unicode[STIX]{x03C0}}\frac{\hat{v}}{x}, & \displaystyle\end{eqnarray}$$

where the dimensionless wavenumber $k=1$ and $S_{\unicode[STIX]{x1D706}}=\bar{\unicode[STIX]{x1D70E}}\unicode[STIX]{x1D707}_{0}J_{0}\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D706}/2\unicode[STIX]{x03C0}\bar{\unicode[STIX]{x1D70C}}}=v_{A}/v_{m}$ is the Lundquist number based on the wavelength $\unicode[STIX]{x1D706}$ . This is the only parameter of the problem, which defines the Alfvén speed $v_{A}=J_{0}\sqrt{\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D706}/2\unicode[STIX]{x03C0}\bar{\unicode[STIX]{x1D70C}}}$ relative to that of the magnetic diffusion $v_{m}$ introduced above. A small $S_{\unicode[STIX]{x1D706}}$ corresponds to the quasi-stationary limit in which the induction effect represented by the left-hand-side term of (3.12) becomes negligible. In the limit of vanishing Lundquist number $(S_{\unicode[STIX]{x1D706}}=0),$ the growth rate $\unicode[STIX]{x1D6FE}$ becomes independent of $S_{\unicode[STIX]{x1D706}}$ . In the opposite limit $(S_{\unicode[STIX]{x1D706}}\gg 1)$ , the magnetic diffusion represented by the first term on the right-hand side of (3.12) is expected to become negligible. In this case, which corresponds to an ideally conducting liquid, $\hat{v}$ can be substituted from (3.12) into (3.11). Then $\unicode[STIX]{x1D6FE}^{2}S_{\unicode[STIX]{x1D706}}^{2}$ emerges as the only parameter (eigenvalue) of the reduced problem. Consequently, in this limit, we expect $\unicode[STIX]{x1D6FE}\sim S_{\unicode[STIX]{x1D706}}^{-1}$ . The respective physical instability development time is

(3.13) $$\begin{eqnarray}\frac{\unicode[STIX]{x1D70F}_{m}}{\unicode[STIX]{x1D6FE}}\sim J_{0}^{-1}\sqrt{\frac{\bar{\unicode[STIX]{x1D70C}}\unicode[STIX]{x1D706}}{2\unicode[STIX]{x03C0}\unicode[STIX]{x1D707}_{0}}}=\frac{1}{2\unicode[STIX]{x03C0}}\frac{\unicode[STIX]{x1D706}}{v_{A}},\end{eqnarray}$$

which is the characteristic Alfvén time for the length scale (3.10).

4 Numerical method

The eigenvalue problem posed by (3.11), (3.12) and (3.7) was solved numerically using a Chebyshev collocation method with Chebyshev–Lobatto nodes $\unicode[STIX]{x1D702}_{m}=\cos (m\unicode[STIX]{x03C0}/(M+1))$ , $m=0,\ldots ,M+1$ , and the coordinate transform $\unicode[STIX]{x1D702}=(\unicode[STIX]{x1D6FC}x-1)/(\unicode[STIX]{x1D6FC}x+1)$ that maps the semi-infinite domain $x=r\in [0,\infty )$ onto $\unicode[STIX]{x1D702}=[-1,1]$ using parameter $\unicode[STIX]{x1D6FC}$ and transforms the differentiation operator $\text{d}/\text{d}x$ into $(1/2)\unicode[STIX]{x1D6FC}(1-\unicode[STIX]{x1D702})^{2}(\text{d}/\text{d}\unicode[STIX]{x1D702})$ . Equations were approximated at the internal collocation points $x_{m}$ corresponding to $m=1,\ldots ,M$ and the boundary conditions were applied at $x_{M+1}=0$ and $x_{0}=\infty$ (Boyd Reference Boyd2013). This eliminates the potential edge singularity of the normal magnetic field from the discretizied problem. The problem was reduced to a standard matrix eigenvalue problem as follows. First, the Fourier series (3.6) was truncated at the length $N$ . Second, the normal magnetic field $\hat{B}$ defined by (3.8) was expressed in terms of the electric streamfunction ${\hat{h}}$ by inverting $N$ matrices of size $M\times M$ representing the left-hand-side operators in (3.7) for $n=1,\ldots ,N$ Fourier modes, and then substituted into (3.11) and (3.12). Eventually, inverting the matrix representations of the left-hand-side operators of the resulting equations, we obtained a standard matrix eigenvalue problem of size $2M\times 2M$ for the growth rate $\unicode[STIX]{x1D6FE}$ and the unknown vector $(\hat{\boldsymbol{v}},\hat{\boldsymbol{h}})$ at the internal collocation points. In the quasi-stationary limit, which corresponds to $S_{\unicode[STIX]{x1D706}}=0$ , the problem can be simplified further by expressing ${\hat{h}}$ from (3.12) in terms of $\hat{v}$ and then substituting it into (3.11). This results in a standard matrix eigenvalue problem of size $M\times M$ for the growth rate $\unicode[STIX]{x1D6FE}$ and the vector of  $\hat{\boldsymbol{v}}$ . In the ideally conducting limit, which corresponds to $S_{\unicode[STIX]{x1D706}}\gg 1$ , the problem is reduced to a standard matrix eigenvalue problem of the same size $M\times M$ for the eigenvalue $\unicode[STIX]{x1D6FE}^{2}S_{\unicode[STIX]{x1D706}}^{2}$ and the vector $\hat{\mathbf{h}}$ as outlined in the previous section. Matrices were inverted and eigenvalue problem solved using the $LU$ and $QR$ factorization algorithms from the standard linear algebra software library LAPACK.

Figure 2.

Growth rate of the leading quasi-stationary $(S_{\unicode[STIX]{x1D706}}=0)$ mode computed with various number of collocation points $M$ and Fourier modes $N$ versus the mapping parameter $\unicode[STIX]{x1D6FC}$  (a); growth rates of the three leading modes versus the Lundquist number computed with $M=64$ , $N=4096$ , $\unicode[STIX]{x1D6FC}=0.3$  (b).

Table 1.

Growth rates of the three leading quasi-stationary $(S_{\unicode[STIX]{x1D706}}=0)$ modes computed with various number collocation points $M$ , Fourier modes $N$ and $\unicode[STIX]{x1D6FC}=0.3$ .

6 Summary and conclusions

In this study, we considered a pinch-type instability in a semi-infinite planar sheet of an inviscid incompressible liquid with a straight rigid edge carrying a uniform tangential electric current. The electromagnetic pinch force resulting from the interaction of the electric current with its own magnetic in this model is balanced by the pressure gradient in quiescent liquid. This equilibrium state was found to be inherently unstable with respect to the internal flow perturbation caused by a row of counter-rotating vortices along the edge. The vortices compress the electric current lines where the flow is directed towards the edge and disperse them in the opposite case. The compression and dispersion of the current lines occurs as a weaker magnetic field is advected by the flow towards the edge and a stronger field is carried away. The interaction of the current perturbation with the base magnetic field produces a pinch force that amplifies initial flow perturbation. In an ideally conducting liquid, this part of the pinch force dominates over the opposing one which results from the interaction of the magnetic flux perturbation with the base current.

The role of the latter changes in the resistive limit, where it becomes destabilizing and dominant. The difference in the instability mechanism is due to the current and the magnetic field distributions at the edge. In an ideally conducting liquid, perturbation of the magnetic field remains finite at the edge, whereas the associated current density increases as ${\sim}x^{-1/2}$ becoming unbounded at the edge. In a resistive liquid, this current singularity is smoothed out by the magnetic diffusion, which thus produces a logarithmic singularity in the magnetic field. The latter enhances the base magnetic field at the edge. This gives rise to a destabilizing perturbation of the pinch force which dominates in the highly resistive liquid over the interaction of the current perturbation with the base magnetic field.

Different instability mechanisms lead to disparate instability development times in the highly resistive and well conducting fluids. In the former, the instability develops on the magnetic response time scale, which depends on the conductivity. In the latter, the instability develops on the much shorter Alfvén time scale. This is because the perturbation of the magnetic field in poorly conducting fluid depends on the product of conductivity and velocity, whereas in well conducting fluid the magnetic field is effectively frozen in and, thus, its perturbation depends directly on the displacement. Therefore these two instability development times are generic and, thus, expected to hold also in other geometries with fixed boundaries regardless of singularities in the magnetic field or the electric current distributions.

It follows from our model that the relevant electromagnetic parameter of the problem is the linear current density. In the eutectic alloy of GaInSn with $\unicode[STIX]{x1D70C}=6.4\times 10^{3}~\text{kg}~\text{m}^{-3}$ and $\unicode[STIX]{x1D70E}=3.3\times 10^{6}~\text{S}~\text{m}^{-1}$ (Weber et al. Reference Weber, Galindo, Stefani, Weier and Wondrak2013) carrying linear current density of $J_{0}=1~\text{kA}~\text{cm}^{-1}$ , which is comparable to the respective quantity for the $m=1$ instability mode in the cylinder with the radius of 5 cm and the critical current of 2.9 kA (Rüdiger, Schultz & Gellert Reference Rüdiger, Schultz and Gellert2011; Seilmayer et al. Reference Seilmayer, Stefani, Gundrum, Weier, Gerbeth, Gellert and Rüdiger2012), the characteristic instability development time is $\unicode[STIX]{x1D70F}_{m}/\unicode[STIX]{x1D6FE}_{1}\approx 30\unicode[STIX]{x1D70C}/(\unicode[STIX]{x1D70E}\unicode[STIX]{x1D707}_{0}^{2}J_{0}^{2})\approx 4~\text{s}$ , where $\unicode[STIX]{x1D6FE}_{1}\approx 0.0335$ is the dimensionless growth rate of the most unstable resistive mode. Note that in a Li–Te liquid-metal battery, where the areal current density can reach up to $j_{0}=7~\text{A}~\text{cm}^{-2}$ (Kim et al. Reference Kim, Boysen, Newhouse, Spatocco, Chung, Burke, Bradwell, Jiang, Tomaszowska and Wang2013), the linear current density $J_{0}=1~\text{kA}~\text{cm}^{-1}$ corresponds to a layer of thickness $\unicode[STIX]{x1D6FF}\sim 1~\text{m}$ . The respective instability development time in Li ( $\unicode[STIX]{x1D70E}=3.3\times 10^{6}~\text{S}~\text{m}^{-1}$ , $\unicode[STIX]{x1D70C}=0.5\times 10^{3}~\text{kg}~\text{m}^{-3}$ ) (Müller & Bühler Reference Müller and Bühler2001) is then by an order of magnitude shorter than that in GaInSn. Comparing the instability development time in GaInSn with the viscous damping time $\unicode[STIX]{x1D70F}_{\unicode[STIX]{x1D708}}=\unicode[STIX]{x1D6FF}^{2}/12\unicode[STIX]{x1D708}$ due to the kinematic viscosity $\unicode[STIX]{x1D708}$ in the respective Hele-Shaw flow (Batchelor Reference Batchelor1967), we can estimate the critical Hartmann number $\mathit{Ha}_{c}=\unicode[STIX]{x1D707}_{0}J_{0}\unicode[STIX]{x1D6FF}\sqrt{\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D70C}\unicode[STIX]{x1D708}}=\sqrt{12/\unicode[STIX]{x1D6FE}_{1}}\approx 20$ . This value is comparable with $\mathit{Ha}_{c}\approx 25$ for the non-axisymmetric $(m=1)$ instability mode in the cylindrical geometry (Rüdiger et al. Reference Rüdiger, Schultz and Gellert2011), and it is also not far from $\mathit{Ha}_{c}\approx 40$ for the axisymmetric mode in the annular geometry (Priede Reference Priede2015). For $\mathit{Ha}\gtrsim 10$ , a more adequate estimate may be provided by the so-called Hartmann damping time $\unicode[STIX]{x1D70F}_{H}=(\unicode[STIX]{x1D6FF}^{2}/2\unicode[STIX]{x1D708})\,\mathit{Ha}^{-1}$ (Sommeria & Moreau Reference Sommeria and Moreau1982), which is based on the Hartmann rather than Poiseuille velocity profile, as in the Hele-Show flow, and yields $\mathit{Ha}_{c}=2/\unicode[STIX]{x1D6FE}_{1}\approx 60$ . For liquid metals, which are relatively poor conductors characterized by a low magnetic Prandtl number $\mathit{Pm}=\unicode[STIX]{x1D707}_{0}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D708}\sim 10^{-5}{-}10^{-6}$ , all these critical Hartmann numbers ${\sim}10$ correspond to the Lundquist number $S_{\unicode[STIX]{x1D706}}=\mathit{Ha}\mathit{Pm}^{1/2}\lesssim 0.1$ , which means a highly resistive mode of instability.

In conclusion, note that in a viscous fluid, the threshold of instability, which is defined by $\unicode[STIX]{x1D6FE}=0$ , depends only on the Hartmann number (Rüdiger et al. Reference Rüdiger, Schultz and Gellert2011). This is because the magnetic Prandtl number vanishes from (3.11) and (3.12) together with the Lundquist number similarly to the thermal Prandtl number in the case of Rayleigh–Bénard instability (Chandrasekhar Reference Chandrasekhar1961). But it is important to stress that the actual mechanism of the instability and, in particular, its growth rate depend on the Lundquist number. Also nonlinear evolution of the instability depends not on only on the Hartmann number but also on the Lundquist (magnetic Prandtl) number as recently demonstrated by Herreman et al. (Reference Herreman, Nore, Cappanera and Guermond2015).