Inverted-Barometer Sea Level Change from Air Pressure

Air has weight, and that weight presses on the sea. Where the air pressure over the ocean is low, less weight sits on the water and the surface rises; where it is high, the surface is pushed down. The sea does this because it adjusts until the pressure at depth is the same everywhere, so a difference in the pressure of the air above must be made up by a difference in the height of the water. In the formula, $p_{ref}$ is the normal pressure for the place, $p$ is the sea-level pressure now, and $p_{ref} - p$ is how far below normal the air has fallen. Dividing by $\rho\,g$ turns that pressure difference into a depth of water, as in a tank, where pressure equals $\rho\,g$ times the depth.

Reading the answer. A positive $\Delta\eta$ is a rise of the sea surface and a negative one is a fall. The handy summary is about one centimetre of sea level for each hectopascal of pressure. With sea water at 1025 kg/m³, a pressure 10 hPa below 1013.25 hPa gives 9.95 cm on the page, close to the rule's 10 cm. A fall of 100 hPa raises the sea about a metre, so a 1.00 m rise from a normal of 1013.25 hPa corresponds to 913.25 hPa by the rule and 912.7 hPa by the formula with sea water at 1025 kg/m³. The small gap is the rounding in the rule.

A worked example. The page's example uses inches of mercury, which shows how the density term works. Mercury is 13.6 times as dense as water, so a pressure difference of one inch of mercury holds up 13.6 inches of fresh water, which is 0.34544 m or 34.544 cm. Enter 29 for $p$ and 30 for $p_{ref}$ (both in inches of mercury) with 1000 kg/m³ for $\rho$, and the page gives about 34.5 cm, since 13.6 is a rounded ratio. Sea water is denser than fresh, so the same pressure difference gives a slightly smaller rise.

Where it comes from, and its limits. The relation is the inverted barometer effect, treated by Wunsch and Stammer (1997) in their paper on atmospheric loading and the oceanic inverted barometer effect. It is a balance, so it gives the sea level once the water has adjusted to the air pressure. It is also only the pressure part of a storm surge, which adds to the tide and to the pile-up of water that the wind causes, so on a tide gauge it is one contribution among others. The reference pressure must be the normal for the place; change it and every answer shifts with it.