\Lambda\nu)\right],\end{aligned}$$ where $\beta$, $\beta'$ and $\beta''$ are phenomenological coefficients. On substituting the mutual friction force in the Euler equation for the superfluid, Eq. (\[eq:AS:EULER\_S\]), we see that the vorticity propagates with a velocity ${\bf v}_L$, that is $$\begin{aligned}
\frac{\partial {\omega}}{\partial t}
={\nabla}\times
\left({\bf v}_L\times{\omega}\right),\end{aligned}$$ which is defined, assuming $\beta''\ll \beta\, , \beta'$, as $$\begin{aligned}
{\bf v}_L &=&{\bf v}_S +\nabla\times\Lambda\nu  
+\beta'({\bf v}_N -{\bf v}_S- \nabla \times\Lambda\nu) \nonumber\\ 
&+& \beta\left[\,\omega\times({\bf v}_N-{\bf v}_S-
\nabla\times\Lambda\nu)\,\right].\end{aligned}$$ The latter equation can be put in a form reflecting the balance of forces acting on a vortex $$\begin{aligned}
\label{eq:AS2.4.2}
\rho_S\left[\left({\bf v}_S
+\nabla\times\Lambda\nu-{\bf v}_L\right)\times
\omega\right] - \eta\left({\bf v}_L-{\bf v}_N\right)
+\eta'\left[\left({\bf v}_L-{\bf v}_N\right)\times\nu\right] = 0,\end{aligned}$$ with the new phenomenological coefficients $\eta$ and $\eta'$ defined as $$\begin{aligned}
\label{eq:AS:etas}
\beta = \frac{\eta\rho_S\omega}{\eta^2+\left(\rho_S\omega-\eta'\right)^2},\quad
\beta' = 1- 
\frac{\rho_S\omega\,(\rho_S\omega-\eta')}{\eta^2+\left(\rho_S\omega-\eta'\right)^2}.\end{aligned}$$ The first term in Eq. (\[eq:AS2.4.2\]) is a non-dissipative lifting force due to a superflow past the vortex (the Magnus force). The remaining terms reflect the friction between the vortex and the normal fluid. The coefficients $\eta$ and $\eta'$, therefore, measure the friction parallel and orthogonal to the vortex motion in the plane orthogonal to the average direction of the vorticity. A nonzero $\beta''$ implies friction along the [*average*]{} direction of the vorticity, which is possible 