# How pressure tide gauges work

Pressure tide gauges measure the water pressure above the sensor. Whether pneumatic, strain gauge or quartz crystals, they do not required a stilling well. They must be immersed deep enough so they remain submerged during equinox low tides. Placed on the seabed, they measure the ambient pressure, which reflects the height of the water column and the atmospheric pressure at the surface. It is important to know the atmospheric pressure and the water density to determine water depths.

## A little physics

Consider the pressure sensor positioned on the seabed.

Where:

- H: depth of the measurement location (average immersion of the sensor)
- h (t): the change in sea level, a function of time t and the zero mean h(t) = 0
- p (t): the pressure measured by the sensor
- Pa (t): the atmospheric pressure at sea level,
- : the average density of the sea water (a function of temperature, salinity, the effect of the pressure being neglected for immersions less than a few hundred meters) on the height H h (t),
- G: the acceleration of gravity

The pressure given by the sensor is equal to the sum of the atmospheric pressure and the hydrostatic pressure, namely:

The height of water above the bottom sensor is:

Some systems directly measure the differential pressure p(t) - p_{a }(t) which is equal to the hydrostatic pressure of the water column. But this means that atmospheric pressure measurement must be available at the differential sensor through an air intake, which is not feasible when the unit is submerged far from the shore. Moreover, in some applications, especially in the field of physical oceanography, the pressure p(t) is the useful information. Thus, to determine changes in sea level h(t) at sea based on pressure on the seabed p(t) three variables must be known:

- g
- p
_{a }(t)

As regards the acceleration of gravity g, it varies with latitude L according to the formula (in m/s²)

g = 9.7803185 (1 + 0.005302357 sin²L – 0.0000059 sin² 2L) m/s²

From the equator to the pole, its intensity increases by approximately 0.5%. This corresponds, for the same hydrostatic pressure amplitude variations, to a relative decrease of tidal range of 5mm/m. If we adopt a value of g corresponding to a mid-latitude, the maximum possible error is 2.5mm/m of tidal range. This error is not negligible. In particular, it affects the determination of depth H which is the basis of studies on climate change (subject to corrections induced by tectonic movements).

The average density** **of seawater is around 1.028 . 10^{3} kg/m^{3}. It is a function of temperature, salinity and pressure. Because of the high pressures involved, it is necessary to take into account the compressibility of sea water at great depths, but the effect of the pressure on p is negligible in the surface layers.

The salinity curves were drawn without taking into account the adiabatic compression of the sea water and with gravity g = 9.81 m/s². Thus, for example, a hydrostatic pressure of 10^{4 }Pa for a water column at 4° C corresponds to a height of 101.94 cm freshwater (S = 0) and 99.34 cm for sea water whose salinity is equal to S = 37, a difference of 2.6 cm. Recall that this is the same type of error that occurs when determining the sea level by measuring the level in a stilling well.

**To find out more:**

**Reference**

- Simon B. (2007). La Marée - La marée océanique et côtière. Edition Institut océanographique, 434pp.

*Last updated: 12/12/2012*