Date: 202009
Local time of day: 0601-1200
Location: RIV.Airport, CA
Weather elements / visibility: Fog; Haze / Smoke; 3
Aircraft: Commercial Fixed Wing
Operator: Air Carrier
Operating under: Part 121
Flight plan: IFR
Flight phase: Other Go Around
Airspace: Class C RIV
ATC / advisory: Tower RIV
Reporter organization: Air Carrier
Reporter function: Pilot Not Flying; Captain
Reporter qualification: Flight Crew Air Transport Pilot (ATP); Flight Crew Instrument; Flight Crew Multiengine
Human factors: Workload; Confusion; Situational Awareness
Second reporter function: First Officer; Pilot Flying
Anomaly: ATC Issue All Types; Deviation - Altitude Overshoot; Deviation / Discrepancy - Procedural Published Material / Policy; Deviation / Discrepancy - Procedural Clearance; Inflight Event / Encounter Unstabilized Approach
Detector: Person Air Traffic Control; Person Flight Crew
When detected: In-flight
Result: Air Traffic Control Issued Advisory / Alert; Flight Crew Became Reoriented
Contributing factors: Human Factors; Procedure
Primary problem: Human Factors

Narrative: My FO (First Officer) was PF (Pilot Flying) and ATC kept us high for ILS 32 RIV.  After all attempts to go down and slow down; performing S turns and seeing the airport; knowing we could not possibly be stable by 1000 ft.; we elected to go around at about 1300 ft. AGL.When I told tower we were going around; she said 'yes; you're cleared to land'.  When I repeated that we were 'going around; we are too high to land'; she said 'oh; I misunderstood' and then proceeded to give us these instructions:  'Aircraft X; roger go around and at the end of the runway; turn left heading 140; climb and maintain 4000.My FO had initiated the go around before these instructions were given; and had started flying the MAP with an altitude of 5300 ft.; and as the PM (Pilot Monitoring) I was trying desperately to keep the airport in sight.  Even the field was reporting better than 'five and five'; there was fog everywhere else; and inflight visibility was about 3 miles in haze. We selected 4000 ft.; and verified it; but when my scan came back inside; we were at 4500 ft.  I instructed the PF to descend back down to 4000 ft. about the time tower said 'Aircraft X maintain 4000' and I responded that we were descending back down.When I changed back to approach control; we were level or about level and we were sequenced for a proper ILS and landed without incident.  We were not given a phone number; nor was there any known traffic conflict.I don't believe ATC has any idea what a go around is like.  We brief MAP procedures prior to our descent check; but we really don't know what ATC instructions will include on a go around because there are so many variables to consider.  I hate go arounds; but I hate writing ASAPs more.  A bit ironic because during cruise I was reading the latest Cleared Direct! Streamlining SOCAL approach into RIV would be helpful; 3 frequency changes in a few minutes is difficult; and they are so busy with GA it's easy to forget us.  By the time we checked on with Approach; he asked us to 'report established;' and when he cleared us for the ILS we were several thousand feet high.  I feel as the Captain; I should have been more focused inside; but our instructions preoccupied my scan outside for the go around.


Second narrative: While on the ARKOE 1 Arrival ATC had us maintain 12;000 ft. until shortly after passing the ARKOE Fix. When handed off to Tower we were given the clearance to descend and maintain 4;000 ft. and subsequently cleared for the ILS Y 32 Approach. I was unable to descend in the allotted time to achieve the Stable Approach Criteria. I excited a Go Around. Tower was informed and was given vectors back to re-sequence for the approach. I then executed another approach and landed without incident.


Synopsis: Air Carrier flight crew reported deviating from clearance on go-around.