The Ekman spiral is an arrangement of ocean currents: the directions of horizontal current appear to twist as the depth changes. 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, the shear stress force corresponds with the wind stress force. At the bottom of the ocean, 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.
32-452: 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 be denoted by u → = ( u ¯ , v ¯ ) {\displaystyle {\vec {u}}=({\bar {u}},{\bar {v}})} , where the two components are constant because of uniformity. Another result of this property
64-485: A high fat content, such as mayonnaise or melted cheese. The no-slip condition is an empirical assumption that has been useful in modelling many macroscopic experiments. It was one of three alternatives that were the subject of contention in the 19th century, with the other two being the stagnant-layer (a thin layer of stationary fluid on which the rest of the fluid flows) and the partial slip (a finite relative velocity between solid and fluid) boundary conditions. However, by
96-428: A nonzero but nanoscale slip length. While the no-slip condition is used almost universally in modeling of viscous flows, it is sometimes neglected in favor of the 'no-penetration condition' (where the fluid velocity normal to the wall is set to the wall velocity in this direction, but the fluid velocity parallel to the wall is unrestricted) in elementary analyses of inviscid flow , where the effect of boundary layers
128-416: A region of high pressure (or high sea level) to a region of low pressure (or low sea level). The force pushing the water towards the low pressure region is called the pressure gradient force. In a geostrophic flow, instead of water moving from a region of high pressure (or high sea level) to a region of low pressure (or low sea level), it moves along the lines of equal pressure ( isobars ). This occurs because
160-623: A role in this are the Stokes drift , waves and the Stokes-Coriolis force . Geostrophic A geostrophic current is an oceanic current in which the pressure gradient force is balanced by the Coriolis effect . The direction of geostrophic flow is parallel to the isobars , with the high pressure to the right of the flow in the Northern Hemisphere , and the high pressure to the left in
192-401: 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, u = u ¯ {\displaystyle u={\bar {u}}} and v = v ¯ {\displaystyle v={\bar {v}}} , in
224-588: A solid boundary, a viscous fluid attains zero bulk velocity. This boundary condition was first proposed by Osborne Reynolds , who observed this behaviour while performing his influential pipe flow experiments. The form of this boundary condition is an example of a Dirichlet boundary condition . In the majority of fluid flows relevant to fluids engineering, the no-slip condition is generally utilised at solid boundaries. This condition often fails for systems which exhibit non-Newtonian behaviour . Fluids which this condition fails includes common food-stuffs which contain
256-414: A solution to the differential equations above. After substitution of these possible solutions in the same equations, ν E 2 λ 4 + f 2 = 0 {\displaystyle \nu _{E}^{2}\lambda ^{4}+f^{2}=0} will follow. Now, λ {\displaystyle \lambda } has the following possible outcomes: Because of
288-419: A wave-like, periodic, dependence in time: In this case, if we set ω = 0 {\displaystyle \omega =0} , we have reverted to the geostrophic equations above. Thus a geostrophic current can be thought of as a rotating shallow water wave with a frequency of zero. No-slip condition In fluid dynamics , the no-slip condition is a boundary condition which enforces that at
320-467: Is exerted along a water-surface with an interior flow u → = ( u , v ) {\displaystyle {\vec {u}}=(u,v)} 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,
352-400: Is greater than that between the fluid particles (cohesive forces). This force imbalance causes the fluid velocity to be zero adjacent to the solid surface, with the velocity approaching that of the stream as distance from the surface increases. When a fluid is at rest, its molecules move constantly with a random velocity. When the fluid begins to flow, an average flow velocity, sometimes called
SECTION 10
#1732854685964384-442: Is neglected. The no-slip condition poses a problem in viscous flow theory at contact lines : places where an interface between two fluids meets a solid boundary. Here, the no-slip boundary condition implies that the position of the contact line does not move, which is not observed in reality. Analysis of a moving contact line with the no slip condition results in infinite stresses that can't be integrated over. The rate of movement of
416-401: Is no viscosity, and that the pressure is hydrostatic . The resulting balance is (Gill, 1982): where f {\displaystyle f} is the Coriolis parameter , ρ {\displaystyle \rho } is the density, p {\displaystyle p} is the pressure and u , v {\displaystyle u,v} are the velocities in
448-410: Is that the horizontal gradients will equal zero. As a result, the continuity equation will yield, ∂ w ∂ z = 0 {\displaystyle {\frac {\partial w}{\partial z}}=0} . Note that the concerning interior flow is horizontal, so w = 0 {\displaystyle w=0} at all depths, even in the boundary layers. In this case,
480-511: Is the coordinate normal to the wall, ℓ {\displaystyle \ell } is the mean free path and C {\displaystyle C} is some constant known as the slip coefficient, which is approximately of order 1. Alternatively, one may introduce β = C ℓ {\displaystyle \beta =C\ell } as the slip length. Some highly hydrophobic surfaces, such as carbon nanotubes with added radicals, have also been observed to have
512-502: The x , y {\displaystyle x,y} -directions respectively. One special property of the geostrophic equations, is that they satisfy the incompressible version of the continuity equation. That is: ∇ ⋅ u = 0 {\displaystyle \nabla \cdot \mathbf {u} =0} The equations governing a linear, rotating shallow water wave are: The assumption of steady-state ( no net acceleration ) is: Alternatively, we can assume
544-446: The Earth is rotating. The rotation of the earth results in a "force" being felt by the water moving from the high to the low, known as Coriolis force . The Coriolis force acts at right angles to the flow, and when it balances the pressure gradient force, the resulting flow is known as geostrophic. As stated above, the direction of flow is with the high pressure to the right of the flow in
576-501: The Navier-Stokes momentum equations , governing geophysical motion can now be reduced to: Where f {\displaystyle f} is the Coriolis parameter , ρ 0 {\displaystyle \rho _{0}} the fluid density and ν E {\displaystyle \nu _{E}} the eddy viscosity , which are all taken as a constant here for simplicity. These parameters have
608-596: The Northern Hemisphere , and the high pressure to the left in the Southern Hemisphere . The direction of the flow depends on the hemisphere, because the direction of the Coriolis force is opposite in the different hemispheres. The geostrophic equations are a simplified form of the Navier–Stokes equations in a rotating reference frame. In particular, it is assumed that there is no acceleration (steady-state), that there
640-452: The Southern Hemisphere . This concept is familiar from weather maps , whose isobars show the direction of geostrophic winds . Geostrophic flow may be either barotropic or baroclinic . A geostrophic current may also be thought of as a rotating shallow water wave with a frequency of zero. The principle of geostrophy or geostrophic balance is useful to oceanographers because it allows them to infer ocean currents from measurements of
672-579: The no-slip condition at the bottom and the constant interior flow for z ≫ d {\displaystyle z\gg d} , coefficients A {\displaystyle A} and B {\displaystyle B} can be determined. In the end, this will lead to the following solution for u → ( z ) {\displaystyle {\vec {u}}(z)} : Here, d = 2 ν E f {\displaystyle d={\sqrt {\frac {2\nu _{E}}{f}}}} . Note that
SECTION 20
#1732854685964704-689: The sea surface height (by combined satellite altimetry and gravimetry ) or from vertical profiles of seawater density taken by ships or autonomous buoys. The major currents of the world's oceans , such as the Gulf Stream , the Kuroshio Current , the Agulhas Current , and the Antarctic Circumpolar Current , are all approximately in geostrophic balance and are examples of geostrophic currents. Sea water naturally tends to move from
736-459: The velocity vector will approach the values of the interior flow, when the z {\displaystyle z} takes the order of d {\displaystyle d} . This is the reason why d {\displaystyle d} is defined as the thickness of the Ekman layer. A number of important properties of the Ekman spiral will follow from this solution: The solution for
768-436: 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
800-556: The bulk velocity, is added to the random motion. At the boundary between the fluid and a solid surface, the attraction between the fluid molecules and the surface atoms is strong enough to slow the bulk velocity to zero. Consequently, the bulk velocity of the fluid decreases from its value away from the wall to zero at the wall. As the no-slip condition was an empirical observation, there are physical scenarios in which it fails. For sufficiently rarefied flows , including flows of high altitude atmospheric gases and for microscale flows,
832-484: 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. u = u ¯ + A e λ z {\displaystyle u={\bar {u}}+Ae^{\lambda z}} and v = v ¯ + B e λ z {\displaystyle v={\bar {v}}+Be^{\lambda z}} will suffice as
864-484: 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 τ → = ( τ x , τ y ) {\displaystyle {\vec {\tau }}=(\tau _{x},\tau _{y})}
896-432: 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. The equations and assumptions above are not representative for
928-649: The no-slip condition is inaccurate. For such examples, this change is driven by an increasing Knudsen number , which implies increasing rarefaction, and gradual failure of the continuum approximation . The first-order expression, which is often used to model fluid slip, is expressed as (also known as the Navier slip boundary condition) u − u Wall = C ℓ ∂ u ∂ n , {\displaystyle u-u_{\text{Wall}}=C\ell {\frac {\partial u}{\partial n}},} where n {\displaystyle n}
960-522: 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 d {\displaystyle d} . So if
992-454: The start of the 20th century it became generally accepted that slip, if it did exist, was too small to be measured. The stagnant layer was deemed too thin, and the partial slip was considered to have negligible effect on the macroscopic scale. While not derived from first principles, two possible mechanisms have been offered to explain the no-slip behaviour, with one or the other being dominant under different conditions. The first contends that
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1024-427: The surface roughness is responsible for bringing the fluid to rest through viscous dissipation past the surface irregularities. The second is related to the attraction of fluid molecules to the surface. Particles close to a surface do not move along with a flow when adhesion is stronger than cohesion . At the fluid-solid interface, the force of attraction between the fluid particles and solid particles (adhesive forces)
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