Let $Q$ be an n-dimensional Riemanian manifold and $TQ$ its tangent boundle. In coordinates, the elements of $TQ$ can be written as $(x^i,v^i)$, where $x^i$ are the coordinates on $Q$ and $v^i=\dot x^i$. Now in this case it is easy to see that these coordinates have an intuitive meaning because if we think of a 2d manifold then $TQ$ is a plane and $x^i$ tells you the point, $v^i$ the direction so that a vector in $TQ$ can be identified.
I´m struggling to do the same, even for a 2d manifold, for the double tangent boundle. More specifically, a vector $z$ in $T_vTQ$ can be written as, if $v=(x^i,v^i)$, $z=(x^i,v^i,w^i,r^i)$, but I don´t understand what $w^i,r^i$ are.
Why is the horizontal part of $z$ then $z_{hor}=(x^i,w^i)$? I also don´t see what $\text{shape}$ $T_vTQ$ has, and therefore what $\text{shape}$ $TTQ$ has. is there a way to see it, even just for 2d manifolds, that continues the intuition for $TQ$?