A solenoidal tangent field is one whose divergence vanishes. They are also called incompressible. I understand why they are called incompressible - a fluid flow is called incompressible when a small fluid parcel retains constant density when it moves along along a streak line. This means that its material derivative vanishes and this in turn means that the divergence of its velocity field vanishes.

But why the description of such a field as solenoidal? I expect that this name had historical origins but its unlikely that it was so named without some link to some aspect of what is generally meant by a solenoid. However, checking a few sources online hasn't resolved this mystery. Generally, a solenoidal field is defined mathematically as above without any discussion why it was named as such.

Now, a solenoid is a helix which suggests that a fluid particle whose velocity field is solenoidal should be moving helically. Is this because such a field would normally be rotational and so has a non-zero curl and so fluid particles in small tubes around a streakline will move helically and because the motion is also incompressible, the cross-section of this tube is constant and hence we have described a solenoid, of a kind?