We analyse the kinematics and dynamics of a homogeneous sample of red clump stars selected from the second Gaia data release catalogue in the direction of the Galactic poles. The level of completeness of the sample at heights between 0.6 and 3.5 kpc is asserted by comparison with the 2 Micron All Sky Survey catalogue. We show that both the density distribution and velocity dispersion are significantly more perturbed in the North than in the South, in all analysed regions of our Galactic neighbourhoods. We provide a detailed assessment of these North-South asymmetries at large heights. We then proceed to evaluate how such asymmetries could affect determinations of the dynamical matter density under equilibrium assumptions. We find that a Jeans analysis delivers relatively similar vertical forces and integrated dynamical surface densities at large heights above the plane in both hemispheres. At these heights, the densities of stars and gas are very low and the surface density is largely dominated by dark matter, which allows to estimate, separately in the North and South, the local dark matter density derived under equilibrium assumptions. In the presence of vertical perturbations, such values should be considered as an upper limit. This Jeans analysis yields values of the local dark matter density above 2~kpc, $rho_{rm DM} sim 0.013 , {rm M}_odot/{rm pc}^3$ ($ sim 0.509 , {rm GeV/cm}^3$) in the perturbed Northern hemisphere, and $rho_{rm DM} sim 0.010 , {rm M}_odot/{rm pc}^3$ ($ sim 0.374 , {rm GeV/cm}^3$) in the much less perturbed South. As a comparison, we determine the local dark matter density by fitting a global phase-space distribution to the data. We end up with a value in the range of $rho_{rm DM} sim 0.011 - 0.014 , {rm M}_odot/{rm pc}^3$ in global agreement with Jeans analysis.
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