Scotland, Northern Ireland minimum distance.

In Northern Ireland the closest High Water Mark point to Scotland, is in my opinion at Irish Grid 323445 440782 (D 23445 40782) = ETRS89 55° 11’ 55.310897” North, 6° 03’ 43.570090” West, on Torr Head, Antrim. See first screenshot below this text: The closest Mean High Water Springs point on the Scottish mainland to Northern Ireland, is in my opinion on Rubha na Lice, Mull of Kintyre, Argyll & Bute, at British National Grid 158937.51 607194.41 (NR 58937.51 07194.41) = ETRS89 55° 17’ 58.746111” N, 5° 47’ 57.029350” W, to the closest High Water Mark point of Northern Ireland, a distance to the nearest whole number metre of 20.148 km or 12.519 miles or 10.879 nautical miles. This was calculated using the GRS80 ellipsoidal distance of 20147.479720921 metres and dividing by an elevation factor results in a more accurate distance. Though it only very slightly increases the distance by 17.4877062 centimetres to 20147.654597983 metres. However, the extra 17.4877062 cm results in the distance being rounded from 20.147 km to 20.148 km, when rounded the nearest whole number metre. An elevation factor is used, because at the closest Mean High Water point of Torr Head to Scotland, the GRS80 ellipsoid is 55.789 metres below Belfast mean sea level, and at the closest Mean High Water Springs point to Northern Ireland on Rubha na Lice, Mull of Kintyre, the GRS80 ellipsoid is 55.262 metres below Newlyn mean sea level. At the midpoint of the minimum distance between Northern Ireland mainland MHW, and Scotland mainland MHWS, the GRS80 ellipsoid is 55.539 metres below Belfast mean sea level, and 55.282 metres below Newlyn mean sea level. Adding the two heights, and dividing the result, gives a mean midpoint GRS80 ellipsoid height of 55.411 metres. Using Simpson's rule, results in a mean GRS80 ellipsoid height of 55.449 metres, for this GRS80 ellipsoid path of the minimum distance between Northern Ireland mainland MHW and Scotland mainland MHWS. The most accurate method of calculating an elevation factor, is by calculating the Earth's radius of curvature at a specific azimuth, which results in 6388245.484364689832935118207859 metres, rather than simply using the Earth's mean radius of 6371000 metres, for the midpoint of the minimum distance between Northern Ireland mainland MHW, and Scotland mainland MHWS, the closest point of the Northern Ireland mainland MHW, and the closest point of the Scottish mainland MHWS.  The WGS84 ellipsoid path midpoint between the closest points MHW mainland Northern Ireland - MHWS mainland Scotland, is at 55.24925° N, 5.93080° W, calculated using the Geo.Javawa website. See fourth, fifth, and sixth screenshots below this text:
Calculation in GRS80: Earth's radius of curvature at a specific azimuth 6388245.484364689832935118207859 metres ÷ (Earth's radius of curvature at a specific azimuth 6388245.484364689832935118207859 metres + GRS80 ellipsoid mean height below Belfast and Newlyn mean sea level 55.449 metres + height above Belfast and Newlyn mean sea level 0 metres = 6388300.933364689832935118207859 metres) 6388300.933364689832935118207859 metres, results in an elevation factor of 0.99999132022730638719972360269727. Dividing the GRS80 ellipsoidal distance of 20147.479720921 metres by this elevation factor, results in a "true" distance of 20147.654597983 metres. Raising the height of the GRS80 ellipsoid from 55.449 metres below mean sea level to mean sea level, does not add any more distance.
The Airy 1830 (OSGB36) ellipsoid distance is 20148.130033291 metres, and dividing an elevation scale factor, calculated by me, results in a "true" distance of 20148.142649043 metres. The Airy 1830 ellipsoid being about 4 metres below the "British Geoid" at the Mull of Kintyre. Though one should not really use the Airy 1830 ellipsoid (OSGB36) for Ireland.
In my opinion, according to Ordnance Survey Northern Ireland 1:2,500 mapping closest mainland Mean High Water, and Ordnance Survey Great Britain 1:2,500 mapping closest mainland Mean High Water Springs, furthermore personally calculated both in the GRS80 ellipsoid and Airy 1830 ellipsoid, with elevation factors personally calculated using the Earth's radius of curvature at a specific azimuth, the distance to the nearest whole number metre is 20.148 km or 12.519 miles or 10.879 nautical miles.
Incidentally the Scottish Gaelic Rubha na Lice, loosely translates as Flat stone or Slab headland. See second screenshot below this text:

Though the closest Scottish Mean High Water Springs point to Northern Ireland, is in my opinion, at British National Grid 158873.50 607284.92 (NR 58873.50 07284.92) = ETRS89 55° 18’ 01.554646” N, 5° 48’ 00.930264” W, a distance of 20.139 km or 12.514 miles or 10.874 nautical miles, to the closest High Water Mark point of Northern Ireland. The distance rounded to the nearest whole number metre is the same, whether or not an elevation factor is added to the GRS80 ellipsoidal distance. BNG 158873.50 607284.92 (NR 58873.50 07284.92) is on a part of the Scottish mainland, that becomes a skerry / rock at Mean High Water Springs, i.e. at least at Mean High Water Springs this point is no longer a part of the Scottish mainland. This skerry / rock at MHWS, is from north-east to south-west about 18 metres in length, and from north to south about 8 metres in length. See third screenshot below this text:


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The closest Low Water Mark point on the Northern Irish mainland to Scotland, is in my opinion at 323438 440800 (D 23438 40800) = ETRS89 55° 11’ 55.898896” N, 6° 03’ 43.937341” W, on Torr Head. See first screenshot below this text: Furthermore the closest Mean Low Water Springs point on the Scottish mainland to Northern Ireland, is in my opinion on Rubha na Lice, Mull of Kintyre, at British National Grid 158888.99 607188.92 (NR 58888.99 07188.92) = ETRS89 55° 17’ 58.483399” N, 5° 47’ 59.757338” W, a distance of 20.098 km or 12.488 miles or 10.852 nautical miles to the closest Low Water Mark point of the Northern Irish mainland. The distance rounded to the nearest whole number metre is the same, whether or not an elevation factor is added to the GRS80 ellipsoidal distance. See second screenshot below this text:
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The closest Low Water Mark point of Northern Ireland in my opinion, at 323478 440774 (D 23478 40774) = ETRS89 55° 11’ 55.022726” N, 6° 03’ 41.7181667” W, on a "Flat Rock", it is just off Torr Head. This "Flat Rock" is at Low Water about 26 metres south/east to north/west, and 7.5 metres south/west to north/east. According to the mapping, at least at High Water, it is under water. See first screenshot below this text: The closest Scottish Mean Low Water Springs point to Northern Ireland, is in my opinion at BNG 158878.01 607113.91 (NR 58878.01 07113.91) = ETRS89 55° 17’ 56.042978” N, 5° 48’ 00.146778” W, to the closest Low Water Mark point of Northern Ireland a distance of 20.033 km or 12.448 miles or 10.817 nautical miles. The distance rounded to the nearest whole number metre is the same, whether or not an elevation factor is added to the GRS80 ellipsoidal distance. BNG 158878.01 607113.91 (NR 58878.01 07113.91) is at Mean Low Water Springs on a skerry / rock that is above water, but at least at Mean High Water Springs this skerry / rock is under water. This skerry / rock at MLWS, is from south-west to north-east about 23 metres in length, and from  south-east to north-west about 23 metres in length. See second screenshot below this text:
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For a larger 1:25,000 mapping view of the Rubha na Lice, Mull of Kintyre area, see the screenshot below this text: On this 1:25,000 mapping extract, one can see that Torr as in Torr Head, also occurs on the Scottish side, in Torr na lice and Torr Môr. Torr in Scottish Gaelic translates as eminence or mound or hill.
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From Torr Head to Rubha na Lice, Mull of Kintyre is a GRS80 and WGS84 geodetic azimuth of 55.98561538° True North, and from Rubha na Lice, Mull of Kintyre to Torr Head is a GRS80 and WGS84 geodetic azimuth of 236.20164740° True North.

Northern Irish grid references personally determined from the nidirect government services website, Ordnance Survey Northern Ireland Online Map Shop, Large scale vector mapping, Torr Head at 1:2,500 scale. Scottish grid references personally determined using Ordnance Survey Great Britain 1:2,500 Detail MasterMap mapping, that is on the Promap website.
Grid references personally transformed to ETRS89 latitude-longitudes, using Ordnance Survey's Grid InQuest II, and Ordnance Survey's Coordinate transformation tool.
GRS80 ellipsoidal distances and azimuths, personally calculated using Charles Karney's Online geodesic calculations using the GeodSolve utility: "GeodSolve is accurate to about 15 nanometers (for the WGS84 ellipsoid)" or 0.000015 of a millimetre or 0.00000059 of an inch. In emails to me from Charles Karney: "The accuracy of 15 nanometers that I quote is for paths up to half-way round the earth." "The accuracy for the Airy" [1830] "ellipsoid will be (very nearly) the same as for the WGS84 ellipsoid (because the parameters are roughly the same)."
GRS80 ellipsoid heights from Ordnance Survey's Grid InQuest II, and Ordnance Survey's Coordinate transformation tool.
Earth's radius of curvature at a specific azimuth, and the resulting elevation factors, calculated using the formulas, that are in the PDF document GPS, Geodesy, and a Ghost in the Machine, A Workshop for for Surveyors and GIS Professionals, Workbook state edition: Arizona, version 8 March 2010.