Ekman layer

Ekman developed the theory of the Ekman layer after Fridtjof Nansen observed that ice drifts at an angle of 20°–40° to the right of the prevailing wind direction while on an Arctic expedition aboard the Fram. Nansen asked his colleague, Vilhelm Bjerknes to set one of his students upon study of the problem. Bjerknes tapped Ekman, who presented his results in 1902 as his doctoral thesis.{{cite book |last=Cushman-Roisin |first=Benoit |title=Introduction to Geophysical Fluid Dynamics |edition=1st |year=1994 |publisher=Prentice Hall |pages=76–77 |chapter=Chapter 5 – The Ekman Layer |isbn=978-0-13-353301-9 }}

The mathematical formulation of the Ekman layer begins by assuming a neutrally stratified fluid, a balance between the forces of pressure gradient, Coriolis and turbulent drag.

:$$

\begin{align}

-fv &= -\frac{1}{\rho_o} \frac{\partial p}{\partial x}+K_m \frac{\partial^2 u}{\partial z^2}, \\[5pt]

fu &= -\frac{1}{\rho_o} \frac{\partial p}{\partial y}+K_m \frac{\partial^2 v}{\partial z^2}, \\[5pt]

0 &= -\frac{1}{\rho_o} \frac{\partial p}{\partial z},

\end{align}

where $\backslash \; u$ and $\backslash \; v$ are the velocities in the $\backslash \; x$ and $\backslash \; y$ directions, respectively, $\backslash \; f$ is the local Coriolis parameter, and $\backslash \; K\_m$ is the diffusive eddy viscosity, which can be derived using mixing length theory. Note that $p$ is a modified pressure: we have incorporated the hydrostatic of the pressure, to take account of gravity.

There are many regions where an Ekman layer is theoretically plausible; they include the bottom of the atmosphere, near the surface of the earth and ocean, the bottom of the ocean, near the sea floor and at the top of the ocean, near the air-water interface. Different boundary conditions are appropriate for each of these different situations. Each of these situations can be accounted for through the boundary conditions applied to the resulting system of ordinary differential equations. The separate cases of top and bottom boundary layers are shown below.

We will consider boundary conditions of the Ekman layer in the upper ocean:{{cite book |last=Vallis |first=Geoffrey K. |title=Atmospheric and Oceanic Fluid Dynamics |edition=1st |year=2006 |publisher=Cambridge University Press |location=Cambridge, UK |pages=112–113 |chapter=Chapter 2 – Effects of Rotation and Stratification |isbn=978-0-521-84969-2 }}

:$$

\text{at } z = 0 :

\quad A \frac{\partial u}{\partial z} = \tau^x \quad \text{and}

\quad A \frac{\partial v}{\partial z} = \tau^y,

where $\backslash \; \backslash tau^x$ and $\backslash \; \backslash tau^y$ are the components of the surface stress, $\backslash \; \backslash tau$, of the wind field or ice layer at the top of the ocean, and $\backslash \; A\; \backslash equiv\; \backslash rho\; K\_m$ is the dynamic viscosity.

For the boundary condition on the other side, as $\backslash \; z\; \backslash to\; -\backslash infty\; :\; u\; \backslash to\; u\_g,\; v\; \backslash to\; v\_g$, where $\backslash \; u\_g$ and $\backslash \; v\_g$ are the geostrophic flows in the $\backslash \; x$ and $\backslash \; y$ directions.

Image:EkmanSpiral.svg

These differential equations can be solved to find:

:$$

\begin{align}

u &= u_g + \frac{\sqrt{2}}{\rho_ofd}e^{z/d}\left [\tau^x \cos(z/d - \pi/4) - \tau^y \sin(z/d - \pi/4)\right ],

\\[5pt]

v &= v_g + \frac{\sqrt{2}}{\rho_ofd}e^{z/d}\left [\tau^x \sin(z/d - \pi/4) + \tau^y \cos(z/d - \pi/4)\right ],

\\[5pt]

d &= \sqrt{2 K_m/|f|}.

\end{align}

The value $d$ is called the Ekman layer depth, and gives an indication of the penetration depth of wind-induced turbulent mixing in the ocean. Note that it varies on two parameters: the turbulent diffusivity $K\_m$, and the latitude, as encapsulated by $f$. For a typical $K\_m=0.1$ m$^2$/s, and at 45° latitude ($f=10^\{-4\}$ s$^\{-1\}$), then $d$ is approximately 45 meters. This Ekman depth prediction does not always agree precisely with observations.

This variation of horizontal velocity with depth ($-z$) is referred to as the Ekman spiral, diagrammed above and at right.

By applying the continuity equation we can have the vertical velocity as following

:$w\; =\; \backslash frac\{1\}\{f\backslash rho\_o\}\backslash left\; [-\backslash left\; (\backslash frac\{\backslash partial\; \backslash tau^x\}\{\backslash partial\; x\}\; +\; \backslash frac\{\backslash partial\; \backslash tau^y\}\{\backslash partial\; y\}\; \backslash right\; )e^\{z/d\}\backslash sin(z/d)\; +\; \backslash left\; (\backslash frac\{\backslash partial\; \backslash tau^y\}\{\backslash partial\; x\}\; -\; \backslash frac\{\backslash partial\; \backslash tau^x\}\{\backslash partial\; y\}\; \backslash right\; )(1-e^\{z/d\}\backslash cos(z/d))\backslash right\; ].$

Note that when vertically-integrated, the volume transport associated with the Ekman spiral is to the right of the wind direction in the Northern Hemisphere.

The traditional development of Ekman layers bounded below by a surface utilizes two boundary conditions:

• A no-slip condition at the surface;
• The Ekman velocities approaching the geostrophic velocities as $z$ goes to infinity.

There is much difficulty associated with observing the Ekman layer for two main reasons: the theory is too simplistic as it assumes a constant eddy viscosity, which Ekman himself anticipated,{{cite journal | last = Ekman | first = V.W. | title = On the influence of the earth's rotation on ocean currents | journal = Ark. Mat. Astron. Fys. | volume = 2 | issue = 11 | pages = 1–52 | year = 1905}} saying

{{cquote|It is obvious that $\backslash \; \backslash left[\backslash nu\; \backslash right]$ cannot generally be regarded as a constant when the density of water is not uniform within the region considered}}

and because it is difficult to design instruments with great enough sensitivity to observe the velocity profile in the ocean.

The bottom Ekman layer can readily be observed in a rotating cylindrical tank of water by dropping in dye and changing the rotation rate slightly.[http://paoc.mit.edu/labguide/ekman.html] {{Webarchive|url=https://web.archive.org/web/20131022034132/http://paoc.mit.edu/labguide/ekman.html |date=2013-10-22 }} Surface Ekman layers can also be observed in rotating tanks.[http://dennou-k.gaia.h.kyoto-u.ac.jp/library/gfd_exp/exp_e/exp/ek/1/app.htm]

In the atmosphere, the Ekman solution generally overstates the magnitude of the horizontal wind field because it does not account for the velocity shear in the surface layer. Splitting the planetary boundary layer into the surface layer and the Ekman layer generally yields more accurate results.{{cite book |last=Holton |first=James R. |title=Dynamic Meteorology |edition=4th |series=International Geophysics Series |volume=88 |year=2004 |publisher=Elsevier Academic Press |location=Burlington, MA |pages=129–130 |chapter=Chapter 5 – The Planetary Boundary Layer |isbn=978-0-12-354015-7 }}

The Ekman layer, with its distinguishing feature the Ekman spiral, is rarely observed in the ocean. The Ekman layer near the surface of the ocean extends only about 10 – 20 meters deep, and instrumentation sensitive enough to observe a velocity profile in such a shallow depth has only been available since around 1980. Also, wind waves modify the flow near the surface, and make observations close to the surface rather difficult.{{cite journal | last1 = Santala | first1 = M. J.| first2 = E. A. | last2 = Terray| title = A technique for making unbiased estimates of current shear from a wave-follower | journal = Deep-Sea Research | volume = 39 | pages = 607–622 | year = 1992 | doi = 10.1016/0198-0149(92)90091-7 | issue = 3–4 |bibcode = 1992DSRA...39..607S }}

Observations of the Ekman layer have only been possible since the development of robust surface moorings and sensitive current meters. Ekman himself developed a current meter to observe the spiral that bears his name, but was not successful.{{cite journal | last = Rudnick | first = Daniel | title = Observations of Momentum Transfer in the Upper Ocean: Did Ekman Get It Right? | journal = Near-Boundary Processes and Their Parameterization | publisher = School of Ocean and Earth Science and Technology | location = Manoa, Hawaii | year = 2003}}

The Vector Measuring Current Meter {{cite journal | last1 = Weller | first1 = R.A. | last2 = Davis | first2 = R.E. | title = A vector-measuring current meter | journal = Deep-Sea Research | volume = 27 | pages = 565–582 | year = 1980 | doi = 10.1016/0198-0149(80)90041-2 | issue = 7|bibcode = 1980DSRA...27..565W }} and the Acoustic Doppler Current Profiler are both used to measure current.

The first documented observations of an Ekman-like spiral in the ocean were made in the Arctic Ocean from a drifting ice floe in 1958.{{cite journal | last1 = Hunkins | first1 = K. | title = Ekman drift currents in the Arctic Ocean | journal = Deep-Sea Research | volume = 13 | issue = 4 | pages = 607–620 | year = 1966 | doi=10.1016/0011-7471(66)90592-4|bibcode = 1966DSRA...13..607H }} More recent observations include (not an exhaustive list):

• The 1980 mixed layer experiment{{cite journal | last1 = Davis | first1 = R.E. | first2 = R. | last2 = de Szoeke | first3 = P. | last3 = Niiler. | title = Part II: Modelling the mixed layer response | journal = Deep-Sea Research | volume = 28 | pages = 1453–1475 | year = 1981 | doi = 10.1016/0198-0149(81)90092-3 | issue = 12|bibcode = 1981DSRA...28.1453D }}
• Within the Sargasso Sea during the 1982 Long Term Upper Ocean Study {{cite journal | last1 = Price | first1 = J.F. | last2= Weller | first2= R.A.| last3= Schudlich|first3=R.R.| s2cid = 45511024 | title = Wind-Driven Ocean Currents and Ekman Transport | journal = Science | volume = 238 | issue = 4833 | pages = 1534–1538 | year = 1987 |bibcode = 1987Sci...238.1534P |doi = 10.1126/science.238.4833.1534 | pmid=17784291}}
• Within the California Current during the 1993 Eastern Boundary Current experiment {{cite journal|last=Chereskin|first=T.K.|title=Direct evidence for an Ekman balance in the California Current|journal=Journal of Geophysical Research|year=1995|volume=100|issue=C9|pages=18261–18269|bibcode = 1995JGR...10018261C |doi = 10.1029/95JC02182 }}
• Within the Drake Passage region of the Southern Ocean {{cite journal|last1=Lenn|first1=Y|last2=Chereskin|first2=T.K.|title=Observation of Ekman Currents in the Southern Ocean|journal=Journal of Physical Oceanography|year=2009|volume=39|issue=3|pages=768–779|doi=10.1175/2008jpo3943.1|bibcode = 2009JPO....39..768L |s2cid=129107187|doi-access=free}}
• In the eastern tropical Pacific, at 2°N, 140°W, using 5 current meters between 5 and 25 meters depth.{{cite journal|last1=Cronin|first1=M.F.|last2=Kessler|first2=W.S.|title=Near-Surface Shear Flow in the Tropical Pacific Cold Tongue Front|journal=Journal of Physical Oceanography |year=2009|volume=39|issue=5|pages=1200–1215|doi=10.1175/2008JPO4064.1|bibcode=2009JPO....39.1200C|citeseerx=10.1.1.517.8028}} This study noted that the geostrophic shear associated with tropical stability waves modified the Ekman spiral relative to what is expected with horizontally uniform density.
• North of the Kerguelen Plateau during the 2008 SOFINE experiment {{cite journal|last1=Roach|first1=C.J.|last2=Phillips|first2=H.E.|last3=Bindoff|first3=N.L.|last4=Rintoul|first4=S.R.|title=Detecting and Characterizing Ekman Currents in the Southern Ocean|journal=Journal of Physical Oceanography |year=2015|volume=45|issue=5|pages=1205–1223|doi=10.1175/JPO-D-14-0115.1|bibcode = 2015JPO....45.1205R |url=http://ecite.utas.edu.au/99058|doi-access=free}}

Common to several of these observations spirals were found to be "compressed", displaying larger estimates of eddy viscosity when considering the rate of rotation with depth than the eddy viscosity derived from considering the rate of decay of speed.

{{portal|Oceans}}

• {{annotated link|Ekman number}}
• {{annotated link|Ekman spiral}}
• {{annotated link|Ekman transport}}
• {{annotated link|Ekman velocity}}
• {{annotated link|Tea leaf paradox}}
• {{annotated link|Stewartson layer}}
• {{annotated link|Vagn Walfrid Ekman}}

{{reflist}}

• [http://paoc.mit.edu/labguide/ekman.html Bottom Ekman layer lab demonstration] {{Webarchive|url=https://web.archive.org/web/20131022034132/http://paoc.mit.edu/labguide/ekman.html |date=2013-10-22 }}
• [http://dennou-k.gaia.h.kyoto-u.ac.jp/library/gfd_exp/exp_e/exp/ek/1/app.htm Surface Ekman layer lab demonstration]

{{physical oceanography}}

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Category:Boundary layer meteorology

Category:Oceanography

Category:Vertical distributions

fr:Spirale d'Ekman#Couche d'Ekman