Magnetic skin layer ($\delta =\sqrt{2\eta /\omega }$)

A magnetic skin layer is the thin region penetrated by an oscillating magnetic field in a conductor. For locally tangential boundary amplitude $C(x)$, negligible material motion and variations along the boundary much slower than those across it, the resistive induction equation gives $B_x=C(x)e^{-(1+i)y/\delta }$ with time convention $e^{i\omega t}$. The solenoidal magnetic-field constraint requires the smaller component $B_y=C^{\prime }(x)e^{-(1+i)y/\delta }\delta /(1+i)$ to leading order. The tangential-field approximation therefore does not mean that the normal component is exactly zero at finite magnetic diffusivity.

In a thin magnetic skin layer next to a rigid no-slip boundary condition, vertical pressure balance gives $p=P-C^2{e}^{-2y/\delta }/(4{\mu }_0)$. With negligible inertia in the layer, the tangential Stokes flow balance is $\rho \nu u_{yy}=p_x$. Matching zero leading shear outside the layer gives $u=U(1-e^{-2y/\delta })$ and the displayed effective tangential slip. The physical wall still has zero velocity; the slip is a boundary condition for the outer flow extrapolated through the unresolved layer. It points toward larger squared boundary-field amplitude.

For real boundary amplitude $C$ and $q=(1+i)/\delta$, the leading electric current density is $j_z=qC{e}^{-qy}/{\mu }_0$. Averaging products of real harmonic fields gives $\overline{\mathbf{F}}=\mathrm{Re}[(\nabla \times \mathbf{B})\times {\mathbf{B}}^{\ast }]/(2{\mu }_0)$. Its leading normal component is the displayed positive force. The first candidate tangential contribution is proportional to $\mathrm{Re}(q/q^{\ast })C{C}^{\prime }=0$, so retaining the small normal field does not introduce a tangential force of that order.