MD88 CREW AND A DASH 8 HAD LESS THAN LEGAL SEPARATION IN ZJX CLASS E.

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ACR X FLT XXX LEFT JAX; FL; ON THE FRI OF SEP AROUND XA10 LCL. WE WERE HEADING APPROX 330 DEGS FOR DUBLIN; GA. WE LEVELED AT 15000 FT FOR TFC AT 1 TO 2 O'CLOCK 16000 FT HEADING SBOUND. I ESTABLISHED A VISUAL AND CLRNC WAS GIVEN TO CLIMB THROUGH 16000 FT WHILE MAINTAINING A VISUAL. MY FIRST OFFICER WAS FLYING. HE SELECTED VNAV AND THE ACFT BEGAN A SLOW CLIMB. I WAS WATCHING THE OTHER ACFT WHICH DID NOT APPEAR TO BE MOVING ON OUR WINDSHIELD. I INSTRUCTED MY FO TO INCREASE THE CLIMB RATE. SHORTLY AFTER THAT THE ACFT INCREASED IN SIZE RAPIDLY. I DISCONNECTED THE AUTOPLT AND CLIMBED QUICKLY THROUGH 16000 FT. THE ACR Y DASH 8 DID A SLIGHT LEFT TURN TO PASS BEHIND US ABOUT 4000 FT TO 5000 FT. OUR INITIAL PERCEPTION WAS THE ACFT WAS PARALLELING OUR COURSE AND WOULD NOT BE A FACTOR. MY EXPECTATIONS WERE TO CLIMB RAPIDLY THROUGH 16000 FT. I DID NOT RELAY THIS INITIALLY TO THE FO; BY THE TIME I DID WE WERE BEHIND. AT THE POINT OF TAKING OVER MANUALLY I DECIDED IT WAS BETTER TO CONTINUE/INCREASE THE CLIMB RATHER THAN TRY TO LEVEL AND THEN DESCEND. IF I HAD TO DO IT AGAIN I WOULD HAVE REMAINED BELOW 16000 FT UNTIL AFTER PASSING BEHIND THE OTHER ACFT. IF GIVEN THE OTHER ACFT'S HEADING RATHER THAN GENERAL SBOUND TFC; SITUATIONAL AWARENESS WOULD BE INCREASED.