Ekman spiral
The oceanic, wind driven Ekman spiral is the result of a force balance created by a shear stress force, Coriolis force and the water drag. This force balance gives a resulting current of the water different from the winds. In the ocean, there are two places where the Ekman spiral can be observed. At the surface of the ocean, where the shear stress force needed corresponds with the wind stress force and at the bottom of the ocean, where the shear stress force is created by friction with the ocean floor. This phenomenon was first observed at the  surface by the Norwegian oceanographer Fridtjof Nansen during his Fram expedition. He noticed that icebergs did not drift in the same direction as the wind. His student, the Swedish oceanographer Vagn Walfrid Ekman, was the first person to physically explain this process. Bottom Ekman Spiral. In order to derive the properties of an Ekman spiral a look is taken at a uniform, horizontal geostrophic interior flow in a homogeneous fluid. This flow will by denoted by formula_1, where the two components are constant because of uniformity. Another result of this property is that the horizontal gradients will equal zero. As  a result, the continuity equation will yield, formula_2. Note that the concerning interior flow is horizontal, so formula_3 at all depths, even in the boundary layers. In this case, the Navier-Stokes momentum equations, governing geophysical motion can now be reduced to: Where formula_5 is the Coriolis parameter, formula_6 the fluid density and formula_7 the eddy viscosity, which are all taken as a constant here for simplicity. These parameters have a small variance on the scale of an Ekman spiral, thus this approximation will hold. A uniform flow requires a uniformly varying pressure gradient. When substituting the flow components of  the interior flow, formula_8 and formula_9, in the equations above, the following is obtained: Using the last of the three equations at the top of this section, yields that the pressure is independent of depth. formula_12 and formula_13 will suffice as a solution to the differential equations above. After substitution of these possible solutions in the same equations, formula_14 will follow. Now, formula_15 has the following possible outcomes: Because of the no-slip condition at the bottom and the constant interior flow for formula_17, coefficients formula_18 and formula_19 can be determined. In the end, this will lead to the following  solution for formula_20: Here, formula_22. Note that the velocity vector will approach the values of the interior flow, when the formula_23 takes the order of formula_24. This is the reason why formula_24 is defined as the thickness of the Ekman layer. A number of important properties of the Ekman spiral will follow from this solution: Surface Ekman Spiral. The solution for the flow forming the bottom Ekman spiral was a result of the shear stress exerted on the flow by the bottom. Logically, wherever shear stress can be exerted on a flow, Ekman spirals will form. This is the case  at the air-water interface, because of wind. A situation is considered where a wind stress formula_36 is exerted along a water-surface with an interior flow formula_37 beneath. Again, the flow is uniform, has a geostrophic interior and is homogeneous fluid. The equations of motion for a geostrophic flow, which are the same as stated in the bottom spiral section, can be reduced to: The boundary conditions for this case are as follows: With these conditions, the solution can be determined: Some differences with respect to the bottom Ekman spiral emerge. The deviation from the interior flow is exclusively  dependent on the wind stress and not on the interior flow. Whereas in the case of the bottom Ekman spiral, the deviation is determined by the interior flow. The wind-driven component of the flow is inversely proportional with respect to the Ekman-layer thickness formula_24. So if the layer thickness is small, because of a small viscosity of the fluid for example, this component could be very large. At last, the flow at the surface is 45 degrees to the right on the northern hemisphere and 45 degrees to the left on the southern hemisphere with respect to the  wind-direction. In case of the bottom Ekman spiral, this is the other way around. Observations. The equations and assumptions above are not representative for the actual observations of the Ekman spiral. The differences between the theory and the observations are that the angle is between 5-20 degrees instead of the 45 degrees as expected and that the Ekman layer depth and thus the Ekman spiral is less deep than expected. There are three main factors which contribute to the reason why this is, stratification, turbulence and horizontal gradients. Other less important factors which play a role in this  are the Stokes drift, waves and the Stokes-Coriolis force.