Jon,
In a spark gap type Tesla coil, critical coupling is undesirable.
Critical
coupling is that coupling which occurs when the coupling is reduced
just to the point where there's no splitting of
frequencies. This concept of critical coupling is more suited for
continuous wave applications such as radio circuits. The degree
of coupling which results from critical coupling is a very low k
value,
maybe around k = 0.01 or so (I don't remember exactly). If such a
low coupling value is used, the coil is unlikely to produce any sparks
at all, or extremely weak sparks at best.
For a spark gap Tesla coil, we want a coupling value which is much
higher than critical coupling. We want the highest possible coupling
we can get without getting racing sparks on our secondary. This is
usually around k = 0.12 or so.