Circles in a Circle

May 12, 2023
CIRCLES WITHIN A CIRCLE: a wheel within a wheel. Photo: NASA GOES

Circles are everywhere. On a cosmic level, the earth is an orb in a collection of orbs that rotate on their axes as they spin around the sun, which also turns on its axis (every 25 days). This plural of revolving planets (and a sun) make up a packet - a solo solar system - that orbits around the centre of the Milky Way Galaxy.

The only reason I speak of things that twirl while whirling around other rotating things in an ever repeating spiral - think Matroyshka dolls or illusory windows within windows that disappear into eternity - is based on a confounding fact of physics that one finds in the ocean.

Propagated into existence over long distances over time by strong winds that blow around circles of high winds called storms, swells move through the ocean in individual groups, or “sets”.

The speed each swell travels, bizarrely, is double the speed of the group it moves within. To understand this, you must see the group of waves as a solo packet like our solar system, which is also made up of parts (planets) constantly moving within it.

Swells move forward through the group at double its speed, constantly fading from the front and reforming at the back.

This alone is enough to make your head spin. But wait, there’s more. The way that energy oscillates and rotates pervades the universe, so perhaps it is worth pondering this inter-connectedness.

To explain existence, humans developed mythology based on metaphor. The ubiquitous circle - or variations thereof - is key to decrypting this code. The Ouroboros - the ancient Greek symbol of a serpent eating its own tail - is one such cosmic emblem of the cycle, or circle if you will, of life. Wikipedia defines the Ouroboros as "something constantly re-creating itself” just like waves in a set.

The Ouroboros represents “the infinite cycle of nature's endless creation and destruction, life and death”. In his 1979 book, The Eternal Golden Braid, Douglas Hofstadter explores similar concepts by examining common themes in the lives and works of logician Kurt Gödel, artist Maurits Escher, and composer Johann Bach.

Musicians speak of recursive musical notes that meet at the top and bottom of the scale like a quantum bending loop. Artists try to replicate the illusory glitches you find in the etchings of Escher that lead somewhere but nowhere. Even in mathematics, there are impossibly circular routes where meaning is lost and algorithms fail tests of logic, certainly in the Incompleteness Theorems of Gödel.

English professors pontificate on the meaning of the “widening gyre” of poet Yeats that ends when “things fall apart” after “the centre cannot hold”. In the song Windmills of the Mind by Noel Harrison (also covered by Sting) we hear: “Round like a circle in a spiral; Like a wheel within a wheel; Never ending nor beginning; On an ever spinning reel.”

There are countless examples of how these mystic shapes manifest on just one planet in our cosmos - a spinning orb we cling too with whitened knuckles. In the Maluti Mountains of Lesotho, look out for the insanely beautiful and rare Spiral Aloe (Aloe Polyphylla) found only in the mountain kingdom. The fleshy leaves of this hypnotic succulent - don't worry, it's not psychotropic - are arranged in spirals.

Spirals are found in the smallest of particles, such as a DNA molecule, where you will find a double helix of organic matter. In other words, two intertwined helices. Yes, the plural of helix is helices.

DOUBLE HELIX: The double spiral staircase, designed by Leonardo da Vinci, brings fame to the stunning Château de Chambord in France.

You also find the helix in human endeavour, particularly engineering. Spiral staircases are common, but have you seen the spectacular open double-spiral staircase at the centre of the Château de Chambord in France? This double helix edifice ascends three floors but never meets. People going up or down each staircase will never make eye contact. It is believed to have been designed by that engineering savant and polymorph Leornado da Vinci himself.

And on we go trying to make sense of it all with loops and twirls and curls and coils and whorls, just some of many synonyms.

The last step before we broker the science behind the breaker lies with the snake that entwines the Orphic Egg in Greek Mythology. This snake wraps itself in a spiral around the “cosmic egg” at the centre of many creation myths. The Orphic Egg hatched primordial deity Zeus, and Poseidon was his brother.

If we were to believe in deity’s like Poseidon or Neptune, the genesis of waves and ocean storms would be easy - they come from that puffed-cheek god depicted on an ancient marine chart who blows on the sea. Easy!

The truth is not dissimilar. Waves are not formed by a Greek God, but wind does indeed create waves, and wind is formed by weather systems driven by variations in atmospheric pressure.

As you may know from school geography, cyclones are Low Pressure pockets of warm air rising off the ocean and interfacing with cool air sucking in from areas of High Pressure. Cyclones rotate clockwise in the southern hemisphere and anti clockwise in the northern hemisphere.

When they’re into their game, cyclones are spinning masses of bad weather and high winds pumping around a motionless area in the middle called the eye, a form of rotational inertia you might find in a spinning top. The outside is whipping around at high speed. The middle is not moving at all. The spinning part of the storm is expressed as wind, which imparts energy into the surface of the ocean by scouring it with unevenness - pushed from behind.

The ruffled surface begins to move forward as micro ripples, then ripples, then wavelets, or wind chop.

Were the wind to drop, the chop would die out quickly. This is because, due to the law of physics as per our friend Sir Isaac Newton, the opposing force of the wavelets is surface tension, or viscosity, which restores the ocean to calm should the wind die.

However, wind with the right velocity that blows for long enough over the right distance will create much larger waves that grow into swells. As the sea surface becomes scoured and dented, so its surface area grows. This gives the wind more area to grip, and the energy transfer goes into a sort of exponential hyper drive.

Given enough time, the opposing force transitions from viscosity to gravity, and it is the latter that enables swells to roll through the ocean for hundreds, sometimes thousands, of nautical miles. Their physical height may gradually wane as they move through the sea, but their relative energy dissipates much slower.

The mystic three: Time, Distance and Wind Velocity

The result can be modelled with mathematical certainty based on the key ingredient of wind speed.

As an example, assuming the minimum time frame and distance the wind blows over (fetch) have been reached, a 20 kt wind will create relatively weak 1m waves with intervals of 7.5 seconds. In other words a succession of small waves occurring frequently, but lacking any power at all.

However, boost the wind to 40kts, and an exponential boost takes place. Wind speed has only doubled, but total energy of the sea state has grown 16 times, and maximum wave heights of almost 15 metres have formed with intervals of 15 seconds. The intervals of waves, otherwise known as period, is a key indicator of energy.

Beware big waves with long period. A 20 second wave may only occur every 20 seconds, but it travels at 31.2 m/s in the open ocean (112km/h), and possesses a wavelength of 624 metres, with energy that extends to a depth of half its wavelength (312m) - a mathematical given in the physics of wave dynamics.

A 10 second swell occurs more frequently, one every 10 seconds. It travels at half the speed (15.6m/s in the open ocean), but possesses a wavelength four times less (156m), with 16 times less energy that extends to a paltry depth of 78m.

This long period energy is prized by surfers for powerful breaking waves, but treated warily by ship captains, who can lose their vessels, as we have seen in Table Bay when moored ships are quietly lifted and dragged by these long wavelength behemoths.  

After its long journey, the swell starts to change as its energy begins to scrape along the sea floor. Shoaling takes place as it approaches the surf spot. The swell slows down (but keeps its group formation), and eventually bunches up into a breaking wave as the increasing shallows push the volume of water upwards, and the wave curves into a curl.

The surfer watches for the one where the energy is best bunched up to catch at the moment the wave rears over sand, rock or coral but one that will continue curling to create a ride.

The surfer turns, and within a few paddle strokes, quickly stands and drops down the face, looking along the wall of the unbroken wave to see where best to maximise the glide.

As the side rail of the surfboard splices the wave face and its fins steer the board straight and true, the top of the wave begins to topple. The surfer pulls into the womb they call the tube, completely encased in a rotating cascade of crystalline water, yet untouched by it.

As the surfer comes spitting out of the chamber, she looks back and sees another mystical round shape that brought her to this moment.