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In the open sea the speed of the current in the developed sea can be estimated as 3% the wind velocity at 10 meters.

Why is ocean surface velocity around 3% of wind velocity?

Is there a similar estimation for the depth of such a current?

Intuitively, a surface current can't continue to move towards the shoreline, as there is not enough space underneath for a bottom current to compensate the surface current, and those currents would interfere. Also, there should be some mixing of the water layer in the breaking waves near the shore, that would contribute to canceling the surface current too.

Having a wind-induced surface current directed towards the shoreline, how close to the shore can such current continue, depending on the steepness of the seabed, the wind and the wave conditions?

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First, a bit of background, following the Navier-Stokes equations and assuming steady state conditions, we can obtain a simplified balance known as Ekman balance for the boundary layer of the upper ocean (order hundred meters). In this part of the ocean horizontal gradients are small compared to vertical gradients. Under these conditions, there is a balance between Coriolis, and friction (sometimes the pressure gradient force needs to also be considered in the balance).

The resulting theoretical flow has a vertical structure with decreasing magnitude with depth and flow direction progressively rotating to the right (left in the southern hemisphere) as depth is increasing. This is known as the Ekman spiral. Ekman spiral Source: http://offshoreengineering.com

Under these conditions, in the open ocean (in fact assuming an infinitely deep ocean), the transport in the Ekman layer is perpendicular to the wind direction. The depth of the Ekman layer, or Ekman depth, is simply dependent on Coriolis and viscosity: $$D_e = \sqrt{2K_m\over f}$$ where $K_m$ is the kinematic turbulent viscosity (assumed constant) and $f$ is the Coriolis parameter (a typical value for mid-latitudes is $10^{-4}$ $s^{-1}$). Typical Ekman depths are of order 10 to 30 meters. Typical values for $K_m$ are around $0.1$ $m^2/s$ (not to be confused with molecular viscosity that is around $10^{-6}$ $m^2/s$) but can vary by several orders of magnitude, resulting in a range of $D_e$ of $10-150m$. A good explanation of Ekman balance and much more information can be found in Cushman-Roisin & Beckers (Chapter 8).

While all that is useful in a theoretical setting, the depths obtained from observations of the Ekman depth can be approximated by the expression: $$D_e = 0.4\sqrt{\tau\over f^2 \rho_0 }$$ where $\tau$ is the wind stress magnitude and $\rho_0$ is water density. The factor $0.4$ has been derived empirically and can varied slightly under different oceanic conditions.

All that was describing the surface boundary layer, but in reality a bottom boundary layer also forms. The solution to the bottom layer can also be approximated as an Ekman layer for which the forcing is not the wind, but the bottom friction. The resulting boundary layer also has an Ekman depth approximated with the expression above. In an sloping bottom, the boundary layer follows the bottom slope and the thickness (depth above bottom) of the layer (measured perpendicular to the bottom) is scaled by the cosine of the bottom slope.

Taking 10 meters (one of the smallest values in mid-latitudes) for the surface and bottom boundary layer thicknesses, that implies that in water depths shallower than 20 m the two boundary layers overlap. In water depths shallower than that, the transport in the surface layer is no longer perpendicular to the wind direction. The momentum input into the water by the wind (wind stress) is affected by the presence of the bottom and it is directly dissipated by bottom friction. The mixing by the wind (and wave breaking near the surf zone) results in additional mixing through the water column and facilitating well-mixed water columns. Under these conditions the wind-induced currents extend all the way to the bottom with the direction of the flow being a function of bottom depth, wind direction, and bottom slope. The presence of wind-induced currents does not preclude the occurrence of flow in the opposite direction of the wind.

A fantastic article summarizing the different flow conditions under different wind and wave fields in shallow water depths is given in Lentz and Fewings (2012). (Reprint)

Additional factors to consider are:

  • stratification by temperature and salinity that reduces mixing and helps maintain a separation between surface and bottom layers (some effects described here).
  • what happens when the wind is changing speed and direction (non-steady state)?
  • what happens at the Equator where the Coriolis parameter is zero?
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  • $\begingroup$ According to Lentz and Fewings, "standard deviations of subtidal depth-averaged cross-shelf currents are much smaller (0.01– 0.05 m s−1) and increase with the distance offshore from near-zero values near the coast." So basically my assumption was correct - there is no cross-shelf current in the surf zone. $\endgroup$ – George Polevoy Aug 8 '17 at 15:13
  • $\begingroup$ The answer is no! They are talking about depth-averaged cross-shelf current. By definition the depth average velocity has to be zero at the coast. The issue is that you still can have currents near the surface and bottom that integrate to "near-zero" close to the coast. Also there is a three-dimensional component to the problem with convergences and divergences near the coast. A good example is a rip current. $\endgroup$ – arkaia Aug 8 '17 at 15:35

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