Medium Transport; Low Wing; 2 Turbojet Eng
Flight Crew Air Transport Pilot (ATP)
Communication Breakdown; Confusion; Workload; Situational Awareness
ATC Issue All Types; Conflict Airborne Conflict
Air Traffic Control Issued New Clearance; Flight Crew FLC complied w / Automation / Advisory; Flight Crew Took Evasive Action; Flight Crew Became Reoriented
Airport; Environment - Non Weather Related; Procedure; Weather
We were cleared for takeoff from XXR at ZZZ.  Shortly after liftoff the Tower Controller advised us a B737 was going around and to report the traffic in sight (this happened at less than 400 FT AGL.)  The Pilot Monitoring and I both spotted the aircraft; which apparently was performing a go-around from either XYL or XYR and called the traffic to the Tower. We were told to maintain visual with the 737. At this point the traffic was approximately 2-2.5 NM laterally and 200 FT vertically separated looking at the TCAS. We were climbing out at Rated Takeoff power and at V2 + 10 (142 KIAS) and quickly were higher than the 737. The Tower Controller said a couple times 'stay low ' which honestly confused me for a second; and I was not positive if he wanted me to push the nose over and attempt to fly under the crossing traffic (we were still less than 1000 FT AGL.)  I elected to continue climbing out at V2 + 10 and Rated Takeoff power as I could see the 737 out my side window and ascertain that we were out climbing it. When the TCAS engaged we got an RA to 'Monitor Vertical Speed' while in the climb attitude. By the time we crossed paths with the B737 we were 800 FT above them. Once we received the 'Clear of Conflict' message is when I lowered the nose to a more normal climb attitude and reduced to climb power to prepare for flap retraction. Due to the fact that we were on different Tower frequencies; we did not hear any of the communications between the B737 and ZZZ Tower. I believe they ended up on the same departure frequency as us (a B737 flight stated they went around due to a 20 KTS tailwind at 100 FT but I did not catch the flight number.)  Should this happen again I would suggest issuing a turn to the south for the aircraft coming off Runway XX and a turn west for the aircraft coming off XY in VMC to try and get some lateral separation.
A corporate aircraft departed ZZZ Runway XXR and took evasive action visually and using TCAS from a B737 after that aircraft's crew executed a go around from Runway XY because of winds.