B737 Undifferentiated or Other Model
Flight Crew Air Transport Pilot (ATP)
Situational Awareness
ATC Issue All Types; Deviation / Discrepancy - Procedural Published Material / Policy; Inflight Event / Encounter Weather / Turbulence; Inflight Event / Encounter CFTT / CFIT
Air Traffic Control Issued Advisory / Alert; Flight Crew Executed Go Around / Missed Approach; Flight Crew Became Reoriented
Human Factors; Weather
Conditions on arrival in SAN were 300 OVC; 1 1/4 vis; light winds. The FO was flying. At minimums we did not see the airport and the FO executed a go around.  Procedures and call outs were perfect.  He called 'check power' and I did. It was approximately 85% N1.  Prior to 400 ft; ATC gave us instructions and I read them back with a incorrect altitude.  As I was with ATC getting the instructions; the FO called for 'Level change; Set clean maneuvering and flaps 5.'  At this time; ATC asked us our altitude and said he received a low altitude warning.  We replied 1100 ft.  We were VFR above the cloud layer and clear of any obstacles when they asked about our altitude.  We realized at this time we had not re-engaged the autothrottles.  We engaged the auto throttle and the power setting was a little low.  ATC instructed us for holding and a subsequent approach with a successful landing as the fog was lifting.
Here is the meat of it...because of our forgetting to rearm the auto throttle upon missed approach; we received a low altitude alert from ATC; due to a shallow rate of climb. The flight was uneventful until the missed approach phase. Our briefings were standard. Because the ceiling was 300 overcast and the visibility 1 3/4 miles; we did not expect to have to go missed approach. [Another carrier] went missed ahead of us; so at that point it was on our mind. When the autopilot captured the glide slope; it aggressively pitched over; so I disconnected the autopilot and auto throttles. It was a smooth ride down the approach; however; upon reaching minimums we executed a go around (runway not in sight). The go around went very smoothly and all of our call outs were timely and correct. I pressed TOGA and advanced the thrust levers at arms length (roughly 78-80% N1 was set). I hand flew the go around to around 2000 feet. The captain called 'thrust set' and I did not realize the auto throttle was not engaged. The airplane was accelerating and climbing at what felt like a normal rate. At 400 feet ATC issued us the instruction 'climb to 5000 and after 1200 feet fly heading 090.' We armed heading select and were pretty much already on a 090 heading; as runway heading was like 095. At 800 AGL I called 'level change; set clean man speed; flaps 5.' At this point; as we sped up and cleaned up; I noticed our rate of climb was shallowing to the point of near zero climb. I attributed this to lvl change wanting to accelerate to clean man speed before it would pick up the climb rate. I hit TOGA a second time; to try and get more power (still believing the auto throttles were armed). When nothing happened; I realized they were not armed. At the same time; ATC issued us the low altitude alert. We had just reached the cloud tops and could see visually at this point. There was some radio confusion between the Captain and ATC. It took 2 calls for him to acknowledge the alert. At this point we began climbing over 2500 feet per minute with no other issues. It took me so long to realize the AT were not armed; because I believed 1. We were in reduced thrust go around mode; and have only done a couple of go arounds in the actual airplane and 2. I felt the airplane would pick up its climb rate after reaching clean man speed. It is common practice for me to hand fly approaches in visual conditions; and this one was hand flown because the autopilot was being so aggressive with the glide path intercept. It was an eye opening experience for me; and is certainly a mistake that I do not plan on repeating.
B737 NG flight crew reported receiving a low altitude alert from ATC after executing a missed approach from SAN. Lack of autoflight mode awareness was cited as contributing.