Why are the waves so much higher with 'wind against tide'?

Or more precisely, how can wave size be increased so much by even a weak counter-current?

The effect can clearly be seen in river mouths and straits where an ocean swell encounters a counter current. It also happens in deep water, such as when the Gulf Stream encounters head seas. There need not be any wind at all.

The often given reasons (increased apparent wind speed, shallow waters, steepening of the waves, etc.) cannot account for the strength of the effect.

Wave height increases from reduced wave speed (celerity) when encountering an evenly spread counter current, and there is a formula that calculates this increase exactly (conservation of energy). The increased wave height remains as long as the new speed is maintained and re-increasing the speed (of the waves) would reduce the height. It's similar to waves approaching shallow waters. However this modest effect doesn't fully account for many real ocean scenarios.

My mind is more visual and in 1992 my eureka moment came one day while heading east against the current through the Current Rock passage in the Virgin Islands where I had a birds-eye view of whole effect clearly displayed in front of me.

My explanation of this additional stronger effect is the difference in speed of forward motion of the wave-train inside the area of counter-current versus outside (a current of 1kt is an appreciable difference for a wave speed of say 10kt). The speed change can arguably have two effects: increase (wave focusing) or decrease of wave height.

For clarity, the illustrations show waves in a regular and parallel pattern (unlikely in the messy real world).

 

Figure 1 A counter-current will cause a distortion in the wave-train, as the waves in the center of the current move slightly slower than those on either side. This bending of the wave-train causes the waves on the sides to slowly converge towards the center, increasing the energy and height near the center at C. Note: Shallow water also slows down the wave velocity causing a bend towards the shallower side (wave refraction).
Figure 2 Even a very weak counter-current could have a noticeable effect as a modest convergence will eventually concentrate the wave energy given enough distance.
Figure 3 The opposite effect is also true (wave diffraction): if the current were to run in the direction of the waves, the wave height at the center would be reduced by divergence of the wave-train (see river example below). When waves curve towards the shore in shallower water it's mostly by refraction.
Figure 4 When wind blows parallel to a river that has no current, the waves are fairly uniform in size across the width (disregarding a slight reduction near the shore from reduced wind speed and refraction in shallow depth).
Figure 5 The current (black/white arrows) in a river is usually strongest in the deeper center. When the wind is blowing with the current, the wave-train is bent by the stronger current in the center, causing divergence (red arrows) and a reduction in wave height.
Figure 6 When the current flows against the wind, the wave-train also bends but now the waves converge (red arrows) towards the center where their energy piles up into noticeably higher waves. Near the river banks, the waves are reduced.
Figure 7 Wave diffraction is sometimes used to great effect in harbours exposed to large ocean seas. After a narrow entrance at A, the jetties curve away from each other, causing waves to spread. This reduces the height of the waves that reach the harbour proper at B.

 

 

Leo, 10/ 2015 (updated 6/2020)

Comments are appreciated.

 

© Leo Lindstrand. All rights reserved.