A Tower Ground Controller reported the Low Altitude Alarm activated three times for an arriving aircraft but the Local Controller did not issue the Low Altitude Alert.

Narrative

I was working Ground Control / Flight Data. There was a supervisor on CIC (Controller in Charge); and another supervisor on Local Control. There was an IFR turboprop; on a visual approach; approximately 3.5 miles southwest of the airfield. The Low Altitude Alarm went off. At approximately the same time; a VFR aircraft was calling inbound on Local Controls frequency. I expected the Local Controller to issue the low altitude alert to the turboprop; once the inbound aircraft transmission was complete. Instead; as the Low Altitude Alarm is sounding for the second time; the Local Controller chose to issue instruction to the VFR aircraft inbound and did NOT issue the Low Altitude Alert. As soon as I heard him issuing instruction to the VFR inbound; and not to the turboprop; I said loudly in the cab; 'Duty Priorities! Low altitude alert to the southwest! The Local Controller said nothing.My statement alerted the CIC; who then asked the local controller; 'Did you issue it?'. The Local Controller didn't respond to him either and instead just looked out the window for the turboprop. Since the Local Controller didn't respond to the CIC; I replied; 'He did not'. As the Local Controller continue to just look out the window for the turboprop to get him in sight; the low altitude alarm sound for the third time. I said 'just issue it'. Shortly thereafter he said that he had them in sight. Duty priorities were not followed. Furthermore; a turboprop in close proximity to an antenna farm was never issued a low altitude alert. After the alarm sounded 3 times; and even though the controller was informed about the situation; he disregarded the need to issue the alert more than once. Looking out the window is what we do preventatively. The alert should have been given. Supervisors need more proficiency time ON POSITION; not just 'in the tower cab'. His decision making showed an extreme lack of judgement."