Wind setup

Wind setup, also known as wind effect or storm effect, refers to the rise in water level in seas or lakes caused by winds pushing the water in a specific direction. As the wind moves across the water's surface, it applies a shear stress to the water, prompting the formation of a wind-driven current. When this current encounters a shoreline, the water level along the shore increases, generating a hydrostatic counterforce in equilibrium with the shear force.

During a storm, wind setup is a component of the overall storm surge. For instance, in the Netherlands, the wind setup during a storm surge can elevate water levels by approximately 3 metres above the normal tide. In the case of cyclones, the wind setup can reach up to 5 metres. This can result in a significant rise in water levels, particularly when the water is forced into a shallow, funnel-shaped area.

Observation
In lakes, water level fluctuations are typically attributed to wind setup. This effect is particularly noticeable in lakes with well-regulated water levels, where the wind setup can be clearly observed. By comparing this with the wind over the lake, the relationship between wind speed, water depth, and fetch length can be accurately determined. This is especially feasible in lakes where water depth remains fairly consistent, such as the IJsselmeer.

At sea, wind setup is usually not directly observable, as the observed water level is a combination of both the tide and the wind setup. To isolate the wind setup, the (calculated) astronomical tide must be subtracted from the observed water level. For example, during the North Sea flood of 1953 at the Vlissingen tidal station (see image), the highest water level along the Dutch coast was recorded at 2.79 metres, but this was not the location of the highest wind setup, which was observed at Scheveningen with a measurement of 3.52 metres.

Notably, the highest wind setup ever recorded in the Netherlands (3.63 metres) was in Dintelsas, Steenbergen in 1953, a location approximately 40 km from the sea along the Haringvliet estuary.

Calculation of wind setup
Based on the equilibrium between the shear stress due to the wind on the water and the hydrostatic back pressure, the following equation is used:
 * $$\frac{dh}{dx}=\frac{\kappa u^2 \cos \phi}{gh}$$

in which:
 * h = water depth
 * x = distance
 * u= wind speed
 * $$\kappa = c_w \frac{\rho_{air}}{\rho_{water}}$$, Ippen suggests $$\kappa$$ = 3.3*10−6
 * $$\phi$$ = angle of the wind relative to the coast
 * g  = acceleration of gravity
 * cw has a value between 0.8*10−3 and 3.0*10−3

Application at open coasts
For an open coast, the equation becomes:
 * $$ \Delta h = \sqrt{2 \kappa \frac{u^2}{g} F \cos \phi + h^2} - h$$

in which
 * Δh = wind setup
 * F = fetch length, this is the distance the wind blows over the water

However, this formula is not always applicable, particularly when dealing with open coasts or varying water depths. In such cases, a more complex approach is needed, which involves solving the differential equation using a one- or two-dimensional grid. This method, combined with real-world data, is used in countries like the Netherlands to predict wind setup along the coast during potential storms.

Application at (shallow) lakes
To calculate the wind setup in a lake, the following solution for the differential equation is used:
 * $$\Delta h= 0.5 \kappa \frac{u^2}{gh} F \cos \phi $$

In 1966 the Delta Works Committee recommended using a value of 3.8*10−6 for $$\kappa$$ under Dutch conditions. However, an analysis of measurement data from the IJsselmeer between 2002 and 2013 led to a more reliable value for $$\kappa$$, specifically $$\kappa$$ = 2.2*10−6.

This study also found that the formula underestimated wind setup at higher wind speeds. As a result, it has been suggested to increase the exponent of the wind speed from 2 to 3 and to further adjust $$\kappa$$ to $$\kappa$$=1.7*10−7. This modified formula can predict the wind setup on the IJsselmeer with an accuracy of approximately 15 centimetres.

Note
Wind setup should not be mistaken for wave run-up, which refers to the height which a wave reaches on a slope, or wave setup which is the increase in water level caused by breaking waves.