Narrative:

Red cas message and aural annunciation of 'wind shear' required us to go-around. When we practice this maneuver in the simulator; it is associated with a convective microburst at a lower altitude and survival requires an urgent climb.in this case; there was no convective activity; but moderate continuous chop and gusting winds; both getting worse as we made our approach. I made an immediate PIREP of our lost 20 knots and our go-around. ATC replied 'fly the published miss.' by this time - only seconds had passed - we were well above (at about 2500 feet) the published missed approach altitude of 2000 feet. Noticing our failure to follow his simple instructions; tower gave us a heading; altitude; and a frequency for approach control to re-sequence us for another approach.the second approach; a visual backed up by the ILS; also resulted in a missed approach; because the winds were gusting above our crosswind limitation. Once again; we climbed above the published missed approach altitude. There was no traffic conflict reported to us.the turbulence on approach was continuous moderate. Everything was shaking so violently that i could not easily read my approach plate; much less the instruments. Even taking hold of the frequency selector knob and changing to the correct numbers was a struggle. I was getting nauseated and my head - even my eyes - hurt.initially; I meant to chastise my copilot after his first altitude violation. But; I decided not to because this was the correct response in more serious conditions. Judging from his second excessive climb on missed approach; I realized that he did need a reminder from me; which I failed to provide.we subsequently bungled an approach at ZZZ1. Tower issued updated wind report every few minutes. We were close to our crosswind limitation; so every report required another calculation of crosswind. This simple task was complicated by turbulence. On four mile final approach; winds exceeded our limits and we announced our intention to go-around. Tower instructed us to 'fly the published miss'. By then; we had climbed to about 2000 feet; not the required 1500 feet; and were still heading 240; not the required 260. Tower handed us back to approach. We made an uneventful approach and landing at ZZZ2.

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Original NASA ASRS Text

Title: C750 Captain reported severe weather resulted in several missed approaches and a diversion.

Narrative: Red CAS message and aural annunciation of 'wind shear' required us to go-around. When we practice this maneuver in the simulator; it is associated with a convective microburst at a lower altitude and survival requires an urgent climb.In this case; there was no convective activity; but moderate continuous chop and gusting winds; both getting worse as we made our approach. I made an immediate PIREP of our lost 20 knots and our go-around. ATC replied 'fly the published miss.' By this time - only seconds had passed - we were well above (at about 2500 feet) the published missed approach altitude of 2000 feet. Noticing our failure to follow his simple instructions; tower gave us a heading; altitude; and a frequency for approach control to re-sequence us for another approach.The second approach; a visual backed up by the ILS; also resulted in a missed approach; because the winds were gusting above our crosswind limitation. Once again; we climbed above the published missed approach altitude. There was no TRAFFIC conflict reported to us.The turbulence on approach was continuous moderate. Everything was shaking so violently that i could not easily read my approach plate; much less the instruments. Even taking hold of the frequency selector knob and changing to the correct numbers was a struggle. I was getting nauseated and my head - even my eyes - hurt.Initially; I meant to chastise my copilot after his first altitude violation. But; I decided not to because this was the correct response in more serious conditions. Judging from his second excessive climb on missed approach; I realized that he did need a reminder from me; which I failed to provide.We subsequently bungled an approach at ZZZ1. Tower issued updated wind report every few minutes. We were close to our crosswind limitation; so every report required another calculation of crosswind. This simple task was complicated by turbulence. On four mile final approach; winds exceeded our limits and we announced our intention to go-around. Tower instructed us to 'fly the published miss'. By then; we had climbed to about 2000 feet; not the required 1500 feet; and were still heading 240; not the required 260. Tower handed us back to approach. We made an uneventful approach and landing at ZZZ2.

Data retrieved from NASA's ASRS site and automatically converted to unabbreviated mixed upper/lowercase text. This report is for informational purposes with no guarantee of accuracy. See NASA's ASRS site for official report.