B737 WITH PCT TRACON; ON ARR TO BWI; EXPERIENCED TCAS RA WITH CLBING CIVIL JET THAT WAS MAINTAINING VISUAL SEPARATION.

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MY FLT WAS ON A STEPDOWN ARR INTO BWI. I WAS LEVEL AT 210 KTS AND 14000 FT MSL. I HAD JUST TURNED N TO REJOIN THE ARR AFTER DEVIATING FOR TSTMS. FO WAS PF AND I WAS THE PLT MONITORING. A GULFSTREAM JET WAS APCHING US FROM OUR R SIDE AT 13000 FT IN MVFR CONDITIONS. THE GULFSTREAM RPTED US IN SIGHT TO POTOMAC APCH AND REQUESTED A CLB. POTOMAC APCH CLRED THE GULFSTREAM TO CLB VFR REF COMPANY B737. IMMEDIATELY OUR TCAS SYS INSTRUCTED US TO CLB AT 2200 FPM. I INSTRUCTED THE FO TO FOLLOW THE TCAS AS HE ADDED MAX THRUST. THE TCAS SYS INSTRUCTED US TO 'INCREASE CLB RATE;' WHICH WE DID. I QUICKLY SCALED DOWN THE TCAS DISTANCE TO 3 MI AND NOTICED THE GULFSTREAM WAS NEARLY DIRECTLY UNDER US AND WITHIN 200 FT VERTICALLY. FO HAD PICKED HIM UP IN SIGHT WHILE THE GULFSTREAM BANKED INTO A TURN UNDER US TO HIS R NEARLY PARALLEL TO OUR DIRECTION OF FLT. HE CONTINUED HIS CLB THROUGH OUR ALT IN FRONT OF US AND TO OUR L. BECAUSE OF OUR STEEP CLB AND HIS APCH ANGLE; I HAD DIFFICULTY EVEN SEEING THE OTHER ACFT. FO SAID HE HAD THE JET IN SIGHT WHEN IT BANKED AWAY FROM US AND ESTIMATED IT WITHIN 300-500 FT HORIZONTALLY. WE HAD CLBED 2000 FT TO 16000 FT MSL AS DIRECTED BY OUR TCAS SYS. IT WAS UNLIKELY THAT THE OTHER PLT COULD HAVE SEEN US WHEN THEY ENTERED THE TURN AWAY FROM US. I THINK THE TURN WAS A RESULT OF THE HIGH CLOSURE RATE AND FOR COLLISION AVOIDANCE. THE DECISION TO CLB IN MARGINAL CONDITIONS BY THE OTHER CREW WAS UNFORTUNATE. I THINK HIS MISJUDGEMENT OF PROPER SEPARATION WAS DIRECTLY RELATED TO THE CONDITIONS. PERHAPS HE BELIEVED HE COULD CLB ABOVE US WITHOUT A PROB. HOWEVER; MISJUDGING THE DISTANCE AND OUR TCAS RA INSTRUCTING US TO CLB HAD COMPLICATED THE CLB HE INTENDED TO MAKE.