How does the Coriolis effect explain Ekman transport?

Question

Before answering the question, keep in mind that I am a second year Biology student, with no experience in studying Physics and a very basic understanding of Mathematics. Thus, I would prefer a qualitative explanation if possible.

---

I do not understand why water drifts at an angle to the wind. I understand that,

Ekman transport occurs when ocean surface waters are influenced by the friction force acting on them via the wind. Source: Wikipedia.

I also understand that,

As the wind blows it casts a friction force on the ocean surface that drags the upper 10-100m of the water column with it. Source: Wikipedia.

However, I do not understand this point,

However, due to the influence of the Coriolis effect, the ocean water moves at a 90° angle from the direction of the surface wind. Source: Wikipedia.

I do not understand this point, because the wind itself is also veering off to the right when travelling from the equator to the poles and vice versa. What causes the differential in 'veering' to the right or the left, between the wind and the water travelling in a Northerly or Southerly direction, if both are affected by the Coriolis effect? In other words, simply stating Coriolis effect does not explain why water drifts at an angle to the wind since they are both affected by the Coriolis effect.

I know this phenomenon is explained by the Ekman spiral, which I partly understand. I understand that,

If the ocean is divided vertically into thin layers, the magnitude of the velocity (the speed) decreases from a maximum at the surface until it dissipates. Source: Wikipedia.

This makes sense to me, because pressure increases with water depth, meaning more wind energy is required to move 'heavier' water.

Albeit, I do not understand why,

The direction also shifts slightly across each subsequent layer (right in the northern hemisphere and left in the southern hemisphere). Source: Wikipedia.

My only idea is that it may have something to do with this,

The impact of the Coriolis effect is most significant with high speeds or long distances. Source.

Explained a little bit here,

Because it largely depends on how large the difference is between an object's velocity is and the ground's velocity, the effect is really only significant at high speeds (either type) or long distances (north/south especially). Source.

So the direction of water shifting slightly across each subsequent layer might have to do with the different velocities of the water layers compared to each other and the wind. Furthermore, I get why the Coriolis effect is greater over longer distances on Earth, because points on the equator are moving faster then points on the poles. However, what do these sources mean by high speeds exactly? Are they referring to the object's speed due to the Earth's rotation (pushing it in an Easterly direction) or due to the force pushing an object in a Northerly or Southerly direction for example? How do higher speeds make the Coriolis effect greater? The answers to these sub-questions will help me understand why water drifts at an angle to the wind.

Answer 1

In this answer I will first generally address the properties of the rotation-of-Earth-effect that is taken into account in Geophysical Fluid Dynamics. (Geophysical Fluid Dynamics is the umbrella name for the fluid dynamics that Meteorology and Oceanography have in common.) Understanding the properties of the rotation-of-Earth-effect will provide the basis for physical understanding.

Next I will discuss the 'wind & water' question: the assertion that when prevailing winds set water into motion: the motion of the water does not necessarily remain parallel with the motion of the wind.
You raise the question: if air mass and water mass are equally subject to the Coriolis effect, shouldn't their motions remain co-linear? I will address that.

The rotation-of-Earth-effect

The following is a condensed version of a discussion of the rotation-of-Earth-effect that is available on my own website. (To navigate to my website: there is a link to it on my stackexchange profile page.)

In this answer here I will - for the sake of brevity - discuss only the case of circumnavigating motion along the surface of a plane. For the generalization to motion of fluid over the surface of a celestial body: check out the discussion on my own website.

The case I will discuss is circumnavigating motion over a plane surface, with the circumnavigating motion sustained by a centripetal force that increases proportional to the radial distance.

In physics, when there is a restoring force that is proportional to displacement the case is referred to as an instance of Hooke's Law. Hooke's law is named after John Hooke, who first discussed it in the context of coil springs being extended.

When the restoring force is exactly according to Hooke's law the resulting motion is called 'harmonic oscillation'. It is called harmonic oscillation because the position as a function of time is given by a sine function

Among the properties of harmonic oscillation is that the period of oscillation is independent of the amplitude. With larger amplitude the object covers more distance, but the the object is also subjct to more acceleration, so the object moves over the midpoint faster. It turns out the factors all drop away against each other. In the end: for small oscillation and large oscillation: the period of oscillation is in all cases the same.

Circumnavigating motion over the surface of a plane can be thought of as a superposition of two perpendicular oscillations. When the centripetal force is according to Hooke's law then the circumnavigating motion will be cyclic, and along an ellipse.

Diagram 1

The following parametric equation gives the planar coordinates $x$ and $y$ as a function of the time parameter $t$:

With:
$\Omega$ = angular velocity in units of angle per unit of time

$$ \begin{matrix} x = a \cos(\Omega t) \\ y = b \sin(\Omega t) \end{matrix} \tag{1} $$

Of course, when $a$ and $b$ are the same value the motion is along a circle. But it's when $a$ and $b$ are non-equal that things get especially interesting.

The following animation demonstrates that the two perpendicular harmonic oscillations can be restated as a superposition of two circular motions:

Animation 1

The parametric equation correspondingly rearranged:

$$ \begin{matrix} x = \left(\frac{a+b}{2}\right)\cos(\Omega t) + \left(\frac{a-b}{2}\right)\cos(\Omega t) \\ y =\left(\frac{a+b}{2}\right)\sin(\Omega t) - \left(\frac{a-b}{2}\right)\sin(\Omega t) \end{matrix} \tag{2} $$

The following animation represents transformation to a rotating coordinate system:

Animation 2

The angular velocity of the rotating coordinate system is of course matched to the period of oscillation of the two perpendicular harmonic oscillations.

With the angular velociity matched in that way: after the transformation to the rotating coordinate system only the motion along the epi-circle remains.

Note especially how fast the point goes around the epi-circle. Relative to the rotating coordinate system the point goes twice around the epi-circle for every cycle of the true circumnavigating motion.

The corresponding parametric equation that features that factor '2'

$$ \begin{matrix} x & = & \ \ \ \left(\frac{a-b}{2}\right)\cos (2 \Omega t) \\ y & = & - \left(\frac{a-b}{2}\right)\sin (2 \Omega t) \end{matrix} \tag{3} $$

Animation 3

In the above animation the object is stationary relative to the rotating system. As stated earlier: this is the case were at every distance to the central axis there is a centripetal force that is proportional to the distance to the central axis. With a centripetal force like that: at any distance to the central axis: a test object that is co-moving with the rotating system will remain co-moving with the rotating sytem.

The left hand side show the true motion: circumnavigating motion. The right hand side displays the motion as seen from a co-rotating point of view. Think of it as a recording by a video camera that is suspended above a platform, co-rotating with it.

Things get more interesting when you give the test object a nudge:

Animation 4

If the test object is initially co-moving with the rotating system and you give it a nudge: that change is a change from circular circumnavigating motion to circumnavigating motion along an ellipse.

In the animation the arrow represents the centripetal force. When the test object has received a nudge the motion relative to the rotating coordinate system is along the epi-circle as depicted in animations 1 and 2.

What the rotation effect is proportional to:
-The faster the overall system is rotating the faster the corresponding motion along the epi-circle. The faster the motion along the epi-circle, the larger the corresponding acceleration relative to the rotating coordinate system. Using $\Omega$ for the angular velocity of the system: the acceleration relative to the rotating system will be proportional to $2\Omega$
-faster velocity relative to the rotating system corresponds to a more flattened ellipse (ellipse as depicted in the left hand panel of animation 4.) A more flattened ellipse corresponds to a larger epi-circle for the motion relative to the rotating system. For circular motion we have: for a given angular velocity: the larger the radius of that circle the larger the velocity along the perimeter of that circle. So the acceleration relative to the rotating system will be proportional to the velocity relative to the rotating system.

Combining: we expect for the magnitude of the acceleration $a$ due to the coriolis effect:

$$ a = 2\Omega v \tag {4} $$

Some general remarks:
In all GFD textbooks it is emphasized that the coriolis effect is in all directions the same. No matter what your direction relative to the rotating coordinate system is, the coriolis effect is the same.
In the animations offered here a centripetal force is present, such that the rotation effect is the same for all directions of motion. If that centripetal force is not be present then the motion relative to the non-rotating coordinate system is along a straight line, and the object will fly off into the distance. That would result in different effect depending on the direction of motion relative to the rotating coordinate system.
That is why in the animations a centripetal force is present. Additional constraint: to result in the same effect for any direction of motion: that centripetal force must be according to Hooke's law. Only a force according to Hooke's law has that symmetry property.

Formation of cyclonic flow

The diagram above presents formation of cyclonic flow at a very, very schematic level.

The blue arrows represent pressure gradient.
The red arrows represent rotation-of-Earth-effect

The initial response of air mass to pressure gradient is always to move down the pressure gradient.

In the case of formation of cyclonic flow:
Initially the flows are down the pressure gradient. Due to the rotation-of-Earth-effect all the incoming flows acquire an additional velocity component, perpendicular to that initial velocity. The total effect of all the flows and pressure gradients is that the flows shepherd each other into cyclonic flow. Once the wind flow has settled to a pattern of cyclonic flow the coriolis effect (red arrows) acts in opposition ot the pressure gradient

Because of the pattern of cyclonic flow the low pressure area in the center tends to persist. It think it is quite common to have a situation where a direction of wind flow persists for days.

Overall:
The direction of prevailing wind is determined by the distribution of pressure gradient.

The 'wind & water' question

In general a body of water will be subject to different constraints than the winds above it.

In general the body of water that is set in motion by wind flow above it will not have opportunity to go into the same cyclonic flow as the wind flow above it.

The water mass is set in motion by the wind flow, and the initial velocity will be co-linear with the wind flow. Over time that water flow will acquire a velocity component perpendicular to the initial flow.

The rotation-of-Earth-effect is perpendicular to the instantaneous direction of flow. So we can expect that over time the velocity component perpendicular to the wind flow above will keep increasing, while the velocity component co-linear with the wind flow will not increase. So: over time I can see an accumulation of angle between the direction of wind flow and direction of water flow, approaching an angle of 90 degrees.

Answer 2

I'm using a second answer to adress some misunderstandings.

You write:
" [...] pressure increases with water depth, meaning more wind energy is required to move 'heavier' water."

Actually, the density of water is pretty much the same at every depth.

From an online book 'Introduction to oceanography the chapter density

Water without other substances dissolved in it, at 4 degrees centrigrade, defines the standard density of 1 gram per cubic centimeter. Compared to that the largest density that occurs in the oceans is a density of 1.03

By contrast: it is the atmosphere where we do see a large gradient in density. Near the top of mount Everest, which is just 8 kilometers above sea level, the density of the atmosphere is only a third of the density at sea level.

Let's simplify to a stratification of just two layers, that tend to hardly mix. The deeper layer is oblivious to the direction of the wind above the top layer. There is some dragging of the deeper layer by the upper layer, just as there is some dragging of the upper layer by the wind above it.

Over time the upper layer accumulates coriolis effect, so it is moving at an angle to the wind above it. In turn the deeper layer is dragged by the upper layer; the motion of the deeper layer accumulates coriolis effect as well, so over time the deeper layer will be moving at an angle to the motion of the upper layer.