It is well established that the magnetically structured solar atmosphere supports the propagation of MHD waves along various kind of jets including also the solar wind. It is well-known as well that under some conditions, namely high enough jet speeds, the propagating MHD modes can become unstable against to the most common Kelvin–Helmholtz instability (KHI). In this article, we explore how the propagation and instability characteristics of running along a slow solar wind MHD modes are affected when they are investigated in the framework of the ideal Hall-magnetohydrodynamics. Hall-MHD is applicable if the jet width is shorter than or comparable to the so called Hall parameter lHall=c/ωpi\documentclass[12pt]{minimal}
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\begin{document}$l_{ \mathrm{Hall}} = c/\omega _{\mathrm{pi}}$\end{document} (where c\documentclass[12pt]{minimal}
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\begin{document}$c$\end{document} is the speed of light and ωpi\documentclass[12pt]{minimal}
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\begin{document}$\omega _{\mathrm{pi}}$\end{document} is the ion plasma frequency). We model the solar wind as a moving with velocity v0\documentclass[12pt]{minimal}
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\begin{document}$\boldsymbol {v}_{0}$\end{document} cylindrical flux tube of radius a\documentclass[12pt]{minimal}
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\begin{document}$a$\end{document}, containing incompressible plasma with density ρi\documentclass[12pt]{minimal}
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\begin{document}$\rho _{\mathrm{i}}$\end{document} permeated by a constant magnetic field Bi\documentclass[12pt]{minimal}
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\begin{document}$\boldsymbol {B} _{\mathrm{i}}$\end{document}. The surrounding plasma is characterized with its density ρe\documentclass[12pt]{minimal}
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\begin{document}$\rho _{\mathrm{e}}$\end{document} and magnetic field Be\documentclass[12pt]{minimal}
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\begin{document}$\boldsymbol {B}_{\mathrm{e}}$\end{document}. The dispersion relation of MHD waves is derived in the framework of both standard and Hall-MHD and is numerically solved with input parameters: the density contrast η=ρe/ρi\documentclass[12pt]{minimal}
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\begin{document}$\eta = \rho _{\mathrm{e}}/\rho _{\mathrm{i}}$\end{document}, the magnetic fields ratio b=Be/Bi\documentclass[12pt]{minimal}
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\begin{document}$b = {B}_{\mathrm{e}}/{B}_{\mathrm{i}}$\end{document}, and the Hall scale parameter lHall/a\documentclass[12pt]{minimal}
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\begin{document}$l_{\mathrm{Hall}}/a$\end{document}. It is found that the Hall current, at moderate values of lHall/a\documentclass[12pt]{minimal}
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\begin{document}$l_{\mathrm{Hall}}/a$\end{document}, stimulates the emerging of KHI of the kink (m=1\documentclass[12pt]{minimal}
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\begin{document}$m = 1$\end{document}) and high-mode (m⩾2\documentclass[12pt]{minimal}
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\begin{document}$m \geqslant 2$\end{document}) MHD waves, while for the sausage wave (m=0\documentclass[12pt]{minimal}
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\begin{document}$m = 0$\end{document}) the trend is just the opposite—the KHI is suppressed.
