# Contour lines on a torus not joining up

I have created a set of contour lines on a torus, but am unable to make some of them join up, even when increasing the number of points dramatically. Is there a sensible way to do this?

Show[ContourPlot3D[y^2 == 1 - (Sqrt[x^2 + z^2] - 3)^2, {x, -4, 4}, {y, -4, 4}, {z, -4, 0}, Boxed -> False, Axes -> False, Mesh -> False, ContourStyle -> Opacity[0.5]],     {ParametricPlot3D[{{-Sqrt[10 - #^2 - y^2 - 6 Sqrt[1 - y^2]], y, #}, {-Sqrt[10 - #^2 - y^2 + 6 Sqrt[1 - y^2]], y, #}, {Sqrt[10 - #^2 - y^2 - 6 Sqrt[1 - y^2]], y, #}, {Sqrt[10 - #^2 - y^2 + 6 Sqrt[1 - y^2]], y, #}}, {y, -4, 4}, PlotStyle -> {{Thick, Black}}, PlotPoints -> 100]}&/@Range[-4, 0, 0.5], PlotRange -> {{-5, 5}, {-5, 5}, {-4, 0}},BoxRatios -> {1, 1, 0.5}, ImageSize -> 1000]


The output is:

where it's clear that the contours aren't quite joining up as they should be. Any help on this would be great.

Try MeshFunctions inside ContourPlot3D:
ContourPlot3D[y^2 == 1 - (Sqrt[x^2 + z^2] - 3)^2, {x, -4, 4}, {y, -4, 4}, {z, -4, 0}, Boxed -> False, Axes -> False, MeshFunctions -> {#3 &}, ContourStyle -> Opacity[0.5]]