C-connected frame congruences

We discuss the congruences θ that are connected as elements of the (totally disconnected) congruence frame \CFL, and show that they are in a one-to-one correspondence with the completely prime elements of L, giving an explicit formula. Then we investigate those frames L with enough connected congruences to cover the whole of \CFL. They are, among others, shown to be TD -spatial; characteristics for some special cases (Boolean, linear, scattered and Noetherian) are presented.