We know that superconductors generate a powerful magnetic field when carrying a current. We also know that they expel all external magnetic fields, which is called the Meisner effect, which will cause a superconductor to levitate if placed over a magnet. However, we also know from classical physics that 2 parallel current carrying conductors will experience a force (the Lorentz force). If both currents are going in the same direction, an attraction effect will result, and if they are going in opposite directions, they will repel. I do know that superconductors have a critical current density limit, which when passed will no longer result in its superconductivity.
So, my question is: If we have 2 parallel current carrying superconductor wires who's currents are travelling in the same direction, will they attract each other? If so, this would seem to go against the Meisner effect.