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From Iceberg - Wikipedia, the closest mountain-shaped iceberg seem to be dome iceberg:

Different shapes of icebergs. 1: Tabular; 2: Wedge; 3: Dome; 4: Drydock; 5: Pinnacled; 6: Blocky.

However, dome icebergs seem to be like two mountains attached to their bases, not a single mountain one:

Is there an iceberg that is simply gradually larger to the bottom, even when passing the water surface?

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    $\begingroup$ Some shapes are impossible because they will tip. $\endgroup$
    – Keith McClary
    Commented Jan 24, 2021 at 22:38
  • $\begingroup$ Even mountains have their negative called the crustal root, for the sake of equilibrium (i.e., isostasy). $\endgroup$
    – Jean-Marie Prival
    Commented Jan 25, 2021 at 9:50
  • $\begingroup$ @KeithMcClary I'm not sure how a mountain-shaped iceberg would tip? It seems to be stable to me. $\endgroup$
    – Ooker
    Commented Jan 26, 2021 at 18:43
  • $\begingroup$ All the research seems to be for cones with vertex down (ship shape), but we could use the same method for vertex up. $\endgroup$
    – Keith McClary
    Commented Jan 26, 2021 at 19:07

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Ice being less dense than water, icebergs float to the degree they displace water, with about 1/8 of their mass above the surface. Icebergs are also dynamic, melting from exposure to sunlight, warm air and water, eroding, and fracturing, and pushed by both wind and waves. (Notably the descriptions used in the Wikipedia article generally apply to the shape of that small portion of an iceberg visible above the surface, not the gross geometry of the whole berg.)

Geophysicist Henry Pollack explored the potential stable positions of icebergs in a Physics Today article titled "Tip of the Iceberg":

When a totally submerged lower-density body is released, the buoyancy force causes it to rise until it reaches a floating equilibrium. The tip then rests above the surface and the root below it, with the mass of each determined by the density contrast between the floating solid and the surrounding fluid.

A floating object is only stable to the extent that its center of gravity is aligned above its center of buoyancy, Pollack writes, "...ice cylinders floating in water will stabilize in only two orientations—with the cylindrical axis either perpendicular or parallel to the water surface;" With an idealized ice-like cylinder 9/10 the density of water, the mass can't be stable if it's taller than it is wide.

The geometry won't be much different using an idealized cone shape -- a cylinder widening along its length -- for your mountain. It can only be short and squat, and if it's at all unbalanced at any point in its life, it'll flip to a new stable equilibrium, on its side or upside-down.

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  • $\begingroup$ "With an idealized ice-like cylinder 9/10 the density of water, the mass can't be stable if it's taller than it is wide." Why is that? I thought the dimension ratio (length over width) is not a factor here? Isn't that Henry Pollack says that the only factor is density ratio between the floating solid and the surrounding fluid? Assuming the mass is uniformly distributed $\endgroup$
    – Ooker
    Commented Jan 26, 2021 at 8:20
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    $\begingroup$ @Ooker According to the sources Pollack cites, the cylinder can only be stable with its axis perpendicular to the surface of the fluid or parallel to it -- if it's proportionately too tall, any torque will destabilize it and force it into another state (like a sailing vessel with a tall mast listing far enough to capsize). "An ice cylinder will float with its rotational axis perpendicular to the water surface when H/D<0.7266, and with its rotational axis parallel to the surface when H/D>1.1785." $\endgroup$
    – jeffronicus
    Commented Jan 26, 2021 at 17:01
  • $\begingroup$ ah I see. You are talking about permanently stable (one and only stable equilibrium), while I'm talking about temporarily stable (two or more stable equilibriums). So when H/D is over than 0.7266 and lower than 1.1785, then the dimensional ratio is a factor here, alongside with the density ratio. Can you confirm this? $\endgroup$
    – Ooker
    Commented Jan 26, 2021 at 17:44
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    $\begingroup$ That's probably another question. A large proportion of bergs are created where glaciers meet the ocean and fracture; glaciers are formed from compressed snow, so the density of their ice varies along the depth of the glacier, for one thing. (Bergs also come from cracking off floating ice sheets, vast slabs of ice protruding onto the ocean but still attached to shore.) $\endgroup$
    – jeffronicus
    Commented Jan 26, 2021 at 19:00
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    $\begingroup$ After this question was asked and answered, Joshua Tauberer created a web app that lets you draw an iceberg shape and see how his algorithm thinks it will float: joshdata.me/iceberger.html $\endgroup$
    – jeffronicus
    Commented Feb 22, 2021 at 15:52

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