Rogue wave formation in the Agulhas current

Rogue wave formation in the Agulhas current | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Rogue wave formation in the Agulhas current D. J. PONS This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-4906129/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Context: Harmonic summation and amplification by winds blowing contrary to currents are known contributions to rogue waves, but the causes of the observed wave steepness, asymmetric form, and non-breaking are poorly understood. The potential effect of bathymetric and meteorological features has not been addressed. Method: Vortex theory was applied qualitatively to the weather and ocean conditions of the Agulhas region. Results : Rogue wave formation is attributed to: (1) Wind lee vortices cause steepening of wave leeward face, and suppresses wave breaking. (2) Boundary layer vortices from the meteorological cold front transfer energy to the wind lee vortices thereby enhancing their wave sharpening effect. (3) Agulhas current boundary layer vortices interact with water lee vortices to accelerate a jet of water between them, thereby steepening the wave and enhancing the preceding trough. (4) Bathymetric topology, especially a canyon on the continental slope, generates a vortex in the Agulhas current. This vortex is detached from the canyon by prising of the coastal downwelling current (induced by the meteorological cold front), and combines with the water lee vortex to heighten the wave. (5) Jetting arises when the canyon vortex and the Agulhas current boundary layer vortices pass each other, thereby accentuating wave height, steepness, and asymmetry. Conclusions: The novel contribution is the provision of a mechanism for rogue wave formation, using vortex theory, that conceptually integrates wave formation, Agulhas sea currents, bathymetric features including submarine canyons, and meteorological cold front weather systems. oceanography turbulence rogue South Africa Figures Figure 1 Figure 2 Figure 3 Figure 4 1 Introduction There have been numerous abrupt sinkings of ships in mysterious situations in peacetime, and rogue waves may have been involved [ 1 ]. Rogue waves have been reported by mariners through the ages, but were rejected by physicists on the assumption of the impossibility for stochastic superposition of multiple wave trains to cause the claimed heights or steepness, as breaking would dissipate the energy. Acceptance only arose after an exceptionally large wave was recorded in 1995 at the Draupner oil rig in the North Sea [ 2 ], but still the physics of the phenomenon are poorly understood. A wave is named rogue by its height relative to the prevailing wave height. It must be at least twice the significant wave height , which is the mean of the third highest waves. Rogue waves are much steeper than they ought to be, generally appear in stormy conditions, can have a height of 18m or even more, move quickly but are highly localised and do not travel far, disappear quickly, may be in a train of up to about three waves, and may be preceded by a depression (‘hole’) in the water [ 3 ] [ 4 ]. They occur in all the world’s oceans, more often than might be thought [ 4 ], and satellite photography suggests there is at least one wave of 25m somewhere in the world every two days [ 5 ]. They can also appear on the coast, where they are a hazard to beach-goers and people fishing from rocks [ 6 ]. They also depend on bathymetric features, though the relationships are unclear. How they might depend on meteorological features is an unexplored area. This paper presents a conceptual theory for the formation of rogue waves in the Agulhas current, including bathymetric features of the continental slope, and the passage of meteorological cold fronts. 2 Rogue wave literature Principles of wave growth In the first instance wave height is determined by the strength and fetch of the wind. Fully developed waves are those for which the wind has blown over a long enough fetch that the waves have grown as high as they can, and the crests of the waves start to break, dissipating energy. Wave height is proportional to wind speed squared for fully developed waves. The formation of conventional waves in real seas are complex and the multiple mechanics are still imperfectly understood. There are several phases in the growth of a wave. The first stage is from initiation through to formation of a small growing wave, represented by the Phillips and Miles mechanisms. The Phillips mechanism [ 7 ] requires a turbulent wind to start with, in which random small pressure distributions occur on the interface, and some of these move at the same speed as the phase velocity for the medium, and hence form waves by resonance. The Miles mechanism is based on a velocity profile for the wind, which causes shear on the surface, and the effect is related to the curvature of that profile [ 8 ]. In practice both theories successfully predict waves based on surface tension (capillary) and gravity, and both rely on a disturbance or turbulence for initiation. Experiments in a wind-wave tunnel show that numerous vortices, smaller than the size of the wave, and with different spin, arise at the water surface, and migrate down into the water [ 9 ]. Water vortices of the same spin have also been observed merging [ 9 ]. Both Phillips and Miles theories include viscosity, e.g. as a sheltering coefficient, to explain why wave growth only occurs once air velocity reaches a critical value. Phillips has pressure distributions moving across the surface, and Miles has shear – they are different aspects of the same phenomenon, and can as readily be represented by vortices. The above theories ignore the possibility of correlated motions between the air and water compartments. The above theories inadequately explain ocean wave growth at the extreme end, or real three dimensional seas. For increased height, say 20m, single waves would need unrealistically fast propagation velocities. Even then they would be long swells, without much steepness, and hence low hazard to shipping. Rogue waves do not show this characteristic: they are both high and steep. Another contribution to wave height comes from wave train frequency divergence. A wave train (wave packet) is a set of waves of similar wavelength travelling in the same direction. Waves in deep water do not keep their frequency but diverge into a variety of frequencies, some of which are longer wavelength and hence higher and faster. The faster waves enter the train from the rear, build up height in the middle, and then decrease as they move to the front of the train. The speed of an individual wave is the phase velocity (wavelength divided by the period), and is twice as fast as the wave train as a whole ( group velocity). For fully developed conditions, the fastest waves move at or slightly faster than the wind speed [ 10 ]. In general for deep (as opposed to shallow) water waves, longer wavelength waves have greater vertical height, and propagate faster, than shorter wavelength waves, but are not particularly steep. Hence this mechanism also fails to explain the rogue phenomenon. A third height contribution is stochastic or harmonic superposition of multiple wave trains. These trains may come from different directions since waves continue to propagate after the winds that cause them have died away. Thus lateral wave trains contribute towards constructive interference [ 3 ]. However a linear superposition still does not explain rogue waves. They are higher, deeper, sharper, and more common than they should be. Second-order summations have been attempted, and these do give steeper waves [ 11 ] but do not fully account for them. The underlying phenomena that might give rise to non-linear summations are unclear. The fourth known contribution is that winds blowing contrary to water currents dramatically increase wave height, i.e. wave-current coupling. A wave that meets an opposing current reduces its wavelength and speed, and increases its height. Waves are highly sensitive to current, and can be stopped by an opposing current flowing at ¼ of the wave speed [ 10 ]. Generally currents flow much slower than winds & waves, so stopping is not the issue, but the point is that even a small current can cause waves to gain height. This is believed to be the primary amplification in the Agulhas current, and has been known since at least the early 1900s [ 12 ]. Also, currents may refract waves [ 13 ] whereby the waves are slowed more at the faster centre of the current than at the edges. This bends the edges of waves, and can focus them on an area. There is some evidence, though not conclusive, that the counter-clockwise eddies in the Agulhas retroflection may increase the wave height [ 14 ]. Contributions to wave steepening Even all the above do not fully account for the shape of rogue waves. This has been a challenge to explain. The rogue waves are not symmetrical nor even sinusoidal in shape. Rather they have a steep wall of water. Also, rogue waves –at least in the Agulhas region - have ‘always been associated with a correspondingly long deep trough – which occurs in advance of the wave’, i.e. a hole-in-the-sea on the NE side of the wave [ 3 ]. It is this combination of hole and wave that is particularly hazardous to ships travelling towards the wave. The depth of the trough before the wave adds to the overall height that the ship has to survive. The cause of the trough is unknown. There are theoretical and empirical models [ 15 ] of wave shape that give waves with greater height (not merely twice the amplitude as with sinusoidal), sharper crests and wider troughs. However these models are one dimensional sea surfaces with clean starting assumptions, and do not transfer readily to real sea conditions. Another part of the difficulty of explaining rogue waves is that waves should break at their crest – hence loose energy and height - before getting so large [ 16 ], but apparently this does not occur. Mechanisms for suppressing wave breaking are poorly understood. Predictive approaches Another approach to rogue waves is stochastic, seeking to predict the occurrence probability of a rogue wave arising somewhere in a given storm, and this typically uses the Rayleigh-Haring-Tayfun distribution, which is a geometrical composite of several approximation models [ 15 ]. Predicting the crest height and trough depth is not possible to do in a satisfactory manner with the stochastic models, but there are other approaches that separately predict wave heights [ 17 ]. Other approaches focus on trying to model the sea state with a view to giving a precise prediction of a rogue wave within the timeframe of several seconds, i.e. to provide warning for conditions that might spawn rogue waves [ 10 , 15 ]. Rogue waves in the Agulhas region One of the earliest papers to address rogue waves, well before the Draupner incident, was written in 1974 by Mallory [ 3 ], a Master Mariner, Captain in the SA Navy, and Professor of Oceanography. Mallory surveyed several ships that had experienced and survived a rogue wave, and noted that rogue waves in the Agulhas current tended to occur on the 100 fathom line and near submarine canyons, with an approaching cold front [ 3 ]. Thus there were three factors that he identified: the continental shelf, underwater topology (canyons) and frontal weather. Mallory’s explanatory work is still unsurpassed for its descriptive accuracy and its practical implications for the only known advice for shipping to reduce the risk to rogue waves in the Agulhas region: ‘ keep away from the vicinity of the outer edge of the continental shelf or 100 fathom (200m) line … when steaming to the southwest with a falling barometer, a fresh northeasterly wind blowing, and a change to fresh to strong southwesterly winds forecast in the next twelve hours ’ [ 3 ]. However Mallory did not identify the underlying physics, nor has the modern literature. There does not appear to be any further refinement of the link between weather systems and rogue waves. The existing literature on rogue waves focusses on the physics of deep water waves, and typically takes a quantitative analytical approach based on idealised water conditions, but still does not explain the mechanisms whereby these waves occur in real seas. In summary, the literature on rogue waves identifies harmonic summation and amplification by winds blowing contrary to currents as the main contributions, but the causes of the observed wave steepness and asymmetric form are poorly understood, and they remain difficult to predict. The combined effect of bathymetric and meteorological features on rogue wave formation has not been addressed. 3 Method Research objective The current objective was to explain rogue wave formation in way that integrates wave formation, sea currents, bathymetric features, and meteorological weather systems. This has not previously been shown in the literature. The area under examination is the Agulhas region, specifically the South African east coast between Durban and Gqeberha (Port Elizabeth), which has a high prevalence of rogue waves [ 3 ]. Approach The approach used the vortex perspective of fluid flow. This is justified on several grounds. First, the mechanics of wave trains and packets are a purely analytical perspective, but have not been successful in explaining rogue waves, and hence it is worth trying different approaches. Second, the fluid dynamics of real seas, with their winds and waves, is fundamentally turbulent. While there is no explicit mechanics available to describe turbulence – this being one of the unsolved problems of fluid mechanics – it is nonetheless instructive to consider the flow as comprising chaotic vortices. In addition there is some evidence – albeit limited – that wind vortices may have some unknown role in heightening of waves. This is because larger waves are commonly explained as being raised by a shear instability, whereby the air flow locally reverses downwind of a wave. This is a type of vortex, though often not specifically identified as such. A more explicit vortex consideration is provided by considering a wave to be a Kelvin-Helmholtz (KH) instability in the interface between two fluid streams, whereby reduced pressure over the crest lifts the weight of the wave [ 18 ]. This causes vortices in counter-rotating pairs, and these vortices cause further rotations of the interface [ 19 ]. This interpretation is usually applied to fluids of the same phase, and has only rarely been applied to ocean waves [ 18 ], and not to rogue waves. Having selected the vortex perspective, the next stage was to examine the meteorological and oceanographic literatures for established evidence of vortices. In order of greatest to least importance were field studies, experimental studies (e.g. wave tanks), finite element analyses, and explicit analyses of ideal geometry. A considerable literature was discovered, almost none of it directly connected to rogue waves, but nonetheless interesting in the way it identified how and where vortices formed. Little to no mathematical treatment was discovered for real sea states, but this was unsurprising because, as already mentioned, there is no mathematics available to describe turbulence other than in the aggregate. Finally a conceptual process was applied to identify candidate mechanisms whereby the various types of vortices could combine to create an extreme sea state. This part of the work is purely conceptual – no calculations or simulations were performed. This is justified as quantitative methods are unavailable for modelling vortex combination, except by numerical simulation of the fluid dynamics. The combination of meteorological, oceanographic, and bathymetric induced vortices makes the problem excessively complex and to create a simulation would be a major programme of work. 4 Results 4.1 Vortices Vortices naturally arise in a fluid in two principal ways. One is when the flow changes direction due to an obstruction or the shape of its duct. The other is due to viscosity and friction with the duct forming a boundary layer velocity profile. In this context the duct is whatever physically bounds the flow, which can be submarine terrain, an adjacent counter flow, or the interface between two immiscible fluids (water and air in this case). The duct is not necessarily rigid, as evident in Kelvin-Helmholtz instability at the interface. The vortices created by viscous friction are approximated by idealised irrotational (or free) vortices, i.e. the tangential velocity \(\:\overrightarrow{v}\) about the vortex axis decreases with radius per \(\:\overrightarrow{v}=\frac{{\Gamma\:}}{2\pi\:r}\) where \(\:{\Gamma\:}\) is the circulation about the vortex axis, with \(\:{\Gamma\:}=\oint\:v\bullet\:\:dl\) where \(\:dl\) is perimeter length around a closed curve enclosing the vortex axis, and \(\:r\) is the distance from that axis. The vorticity is a measure of the rotation rate of a fluid particle, given by \(\:\overrightarrow{\omega\:}=\nabla\:\times\:\overrightarrow{v}\) where \(\:\nabla\:\times\:\) is the curl (rotation) of the velocity field. For an ideal irrotational vortex \(\:\overrightarrow{\omega\:}=0\) . This relationship is idealised because in practice the velocity cannot be infinite at the axis, but instead is zero. As each outward layer in the vortex moves faster, there is a viscous shear stress generated through the body of the vortex, and this dissipates energy. Or conversely, an ongoing supply of energy is required to sustain the vortex. The other extreme of the idealised vortex is a rotational vortex (or forced vortex, or rigid body rotation) where the angular velocity \(\:{\Omega\:}\) is constant such that tangential velocity is proportional to radius, \(\:v={\Omega\:}r\) and in which case vorticity is \(\:\overrightarrow{\omega\:}=2{\Omega\:}\) . These vortices have a pressure applied at their outer boundary. Real vortices combine elements of both above idealised types, and in addition they need not be circular. They can also change shape, move with the fluid, bend and extend on their axis, merge, split, and form tubes (including toroidal). The inner regions of vortices tend to retain fluid particles, and hence can transfer fluid and momentum as the vortex migrates. Vortices therefore can contain substantial energy, and transport it to new locations. 4.2 Proposed mechanisms for rogue waves We propose that Agulhas rogue waves form by the combination three effects: Harmonic superposition of multiple wave trains; Amplitude modulation of waves by opposing water currents; and Fluid-air interface mediated by multi-scale multi-medium chaotic vorticity. The first two have well-established physics, but the third is a novel proposal, and suggests that the missing ingredient to explain Agulhas rogue waves is vorticity. (1) Harmonic superposition of multiple wave trains During winter, there are waves coming up from deep in the Southern ocean. The winds have a long fetch of about 800-1,200km, hence fully developed waves are created (typical height 6m, wavelength 120m). They travel from the SW. For example, for 12 July 2023 the forecast for Marion Island – which is deep in the Southern Ocean - was: ‘MARION FORTIES EAST (40S/50S, 35E/50E): WIND : W to SW 25 to 35 reaching 40 in places in the south. VIS : Moderate in showers and rain. SEA STATE: 5.5 to 6.5m reaching 7.0 to 8.0m in the south, SW to W swell’ [ 20 ]. Note the wind strength of up to 40 knots, and the wave height of up to 8m. However those waves would be long swells, not steep sided. These waves propagate up towards Gbergha (Port Elizabeth). In addition, the SW winds created by cold meteorological fronts create waves > 0.5m height (e.g. height 3m, wavelength 60m, period 7 s) that travel north eastwards on the continental shelf. These travel in the same direction as the Southern Ocean waves, and periodically reinforce them to greater height. They are also able to reverse the surface Agulhas current [ 21 ]. Modelling shows them to have the greatest effect at the shelf break itself [ 22 ]. Meteorological conditions in South Africa are affected by the terrain which consists of an elevated plateau of 1,400m in the interior. This has a marked effect on the evolution of weather systems including the formation of the coastal low NE of the escarpment [ 23 ] and localised strong NE or NW winds (up to 30 knots) with clear skies [ 24 ]. These preceding winds create additional local waves of high frequency [ 3 ]. These NW winds and waves converge onto SW winds and waves at the head of the front. These waves of different wavelength and direction are constantly interfering and summing into higher waves with deeper troughs. However this superposition on its own is insufficient to account for the size and steepness of rogue waves. (2) Amplitude magnification of waves by opposing water currents. Waves from the SW wave are magnified on meeting the Agulhas current. This has well established physics and is familiar to mariners. These waves meet the strongly flowing Agulhas current which is flowing towards the SW direction. The intersection causes the waves to rise in height, typically by 20–40% and up to 60% in extreme situations [ 25 ]. These SW winds and their waves are opposite in direction to the Agulhas current, and strongest where the current is most concentrated. When meeting the current the waves become shorter in wavelength, and steeper. The effect is ‘ more pronounced where the opposing current is strongest, i.e. just outside the 100 fathom (200 m) line’ , and ship logs show much heavier sea conditions here [ 3 ]. This interaction on its own is sufficient to generate large waves of about 10m, and the South Africa Weather Services forecasts this interaction [ 20 ] based on a model developed by [ 25 ]. There is also refraction occurring, whereby SW waves that meet the current are slowed, but those on the slower edges of the current are not. This effect is likely to be more pronounced where the current speed drops off most sharply, which is the edge of the continental shelf. These waves interfere from the side (crossing sea). Even so, these mechanics are insufficient to describe the occurrence of rogue waves, because these conditions arise frequently. (3) Fluid-air interface mediated by multi-scale multi-medium chaotic vorticity. We propose that superimposed and current-amplified waves receive an energy boost from vortices. There are numerous vortices in the Agulhas case, air and water, which appear not to have been previously identified as such. These are at different physical scales and frequencies. Our proposal is that the chaotic combination of these vortices, on top of the preceding two phenomena, is what creates rogue waves. There are a number of fluid mechanics principles that apply to vortices: A. Vortices arises from three mechanisms: (a) wherever there is fluid shear on a boundary layer, (b) moving fluid that changes its direction of flow due to obstruction from another fluid or solid body, or (c) Coriolis force acting on a fluid moving across latitudes. The Coriolis effect operates at the large scale and is apparent in the direction of rotation of the atmospheric fronts and the Agulhas current as a whole, but is less important for present considerations. B. Vortices in a 3D medium are chaotic – they have a regularity of sorts, but not a fixed periodicity. Consequently they do not engage in harmonic superposition like the wave trains of classical physics. Instead their interactions are irregular and unpredictable. There is no analytical physics that can predict their behaviour accurately. C. Vortices – either in the air or the water - with sufficient scale cause waves when they act on the water-air interface. D. Adjacent vortices with the same spin combine to produce a stronger vortex. This is an amplification mechanism. E. The effect of vortices on an air-water interface is complex and poorly understood [ 26 ]. We propose that vortices in the liquid at a liquid-gaseous fluid interface will preferentially transfer their kinetic energy into waves on that interface, rather than take the Kolmogorov breakdown route to progressively smaller vortices. We could not find explicit confirmation of this in the literature, nonetheless it is consistent with the theoretical finding that “the surface deforms to satisfy the conditions that the tangential stress be equal to zero and the normal stress be equal to a constant at all times” and that the evolution of the movements on the air side are negligible [ 27 ]. This provides a mechanism for additional amplification. F. Furthermore we propose that contrary rotation water vortices can, if converged (impacted together appropriately) cause vertical jets that will accentuate wave height (potentially at an angle) and create a trough alongside. This formation of jets for converged counter-rotation vortices pairs is experimentally observed [ 28 ]. It is also consistent with the observation (in slurries) that for contrary vortices the “impact energy is conveyed through the tangential component” [ 29 ]. In addition analytical studies on an idealised shelf break show that contrary vertical spin vortices will usually repel each other, but there are conditions whereby they can move in parallel in jet-like flows [ 30 ]. 4.3 Elaboration of the vortex proposal We propose an explanation for rogue wave formation under the conditions encountered of: strong Agulhas current; cold front from the SW; and bathymetry of the continental slope. Using the above principles it is possible use the morphology of the vortices to make inferences about which vortices contribute to heightening waves, and how different vortices could affect each other, even if the mathematics are not yet ready to represent the interactions with precision. The vortex axis is denoted with the right-hand rule: \(\:⨀\) out the page is anticlockwise, \(\:⨂\) is clockwise. Effect of lee vortices on wave height We assume that a set of waves have already been generated by winds some distance away (e.g. Marion Island) by Phillips/Miles [ 7 , 8 ] or other starting mechanisms, and these waves have propagated into a region above which a SW cold front is moving and winds are blowing (e.g. SW 35 knots), see Fig. 1 . The wind experiences friction on the water, due to the combination of the small scale roughness of the water surface, and the topology of the waves. The drag results in a boundary layer. Initially, this drag force increases with the wind speed [ 31 ]. A high degree of microscale turbulence arises in this air boundary layer – theorised by [ 7 ] and evident experimentally by its effects on the water [ 9 ]. The effect of turbulence is to inject faster moving air into the boundary layer, i.e. causes mixing, and makes the velocity profile more abrupt. Turbulence also involves vorticity. At some point the size and topology of the wave is such that air flow separation occurs, and a lee vortex appears (downwind of the wave crest), see λ1 ( \(\:⨂\) ) in the figure. This drives surface water (v L1 ) towards the wavecrest, which grows the wave. In other perspectives this localised region of backwards air flow is called a shear instability. At the same time a windward vortex arises by impact with the back of the wave, see ω1, which likewise grows the wave from the rear (v W1 ). However as the wave is also moving forward, this is expected to be reduced in efficacy. Hence the lee vortex is expected to make the greater contribution to wave growth. The asymmetry of the effects is expected to steepen the leeward face of the wave. Both air vortices move forward with the wave. Vortices typically evolve topographically over time [ 32 ], and in this case we propose that the lee vortex grows over several consecutive waves, see λ1 to λ2 in the figure, at the expense of the windward vortex. They are then extinguished by turbulent inflows from above or laterally, as the bulk air stream recolonises the lacuna, and the cycle recommences. As the sea state worsens, so its frictional resistance to the SW winds increases. This will thicken the boundary layer and create stronger wind vortices. The air vortices induce movement of water, and hence also corollary vortices in the water, denoted L (lee) and W (windward) in the diagram. These have opposite spin to their air counterparts. Vortices arising from meteorological cold front The cold front also contains many additional vortices [ 33 ]. We propose that these interact with the air vortices shaping the waves, see Fig. 2 . Ahead of the cold front is a warm air uplift vortex (U \(\:⨂\) ), in response to the cold front pushing forward. There is considerable vorticity in this region [ 33 ]. The axis of vorticity is the same as the lee λ vortices ( \(\:⨂\) ), hence there is the potential for these vortices to combine, i.e. for the uplift vortex U to transfer energy to lee vortex λ, thereby strengthening wave growth. This depends on these two vortices being in close proximity, which might occur if a SW gust was to extend beyond the front to create a lee vortex under the uplift vortex. Behind the front is a large main vortex (H \(\:⨀\) ) with high wind speeds [ 33 ]. This has spin like the air windward ( \(\:{\omega\:}⨀\) ) vortex, and hence has the potential to combine and transfer energy into it. It may be that this flattens the sea, rather than builds waves. This vortex spins off a trail of secondary head vortices (H2 \(\:⨀\) ) that are pushed forward by the wind [ 33 ]. Close to the terrain (which in this case is the sea surface) and about 100m up into the atmosphere are a series of boundary layer vortices (B \(\:⨂\) ) [ 33 ]. These originate in the friction over the surface. They are at a larger scale than the flow separation lee vortices λ \(\:⨂\) . As they have the same spin, the boundary layer vortices have the potential to combine with or transfer energy to the lee vortices, thereby heightening the wave. These vortices are moving in the same direction, at approximately the same speed, so there is an extended period of time for this interaction. We propose this is an appreciable contribution to wave heightening. The air boundary layer vortices (B \(\:⨂\) ) and the secondary head vortices (H2 \(\:⨀\) ) have opposite spin, so will not merge but will rather tend to create jets between them (v HB ), resulting in wind gusts at terrain-level. These gusts will tend to enhance flow separation and strengthen the lee vortices λ, hence further raising wave height. This is consistent with modelling work that shows gustiness allows the waves to develop to greater heights than theory would otherwise allow, especially for following swell (no great difference for opposing swells) [ 34 ], which is exactly the situation under consideration. Those authors proposed the underlying mechanics might be smaller waves absorbing wind energy and transferring it to larger waves, but we proposed instead the mechanisms are strengthening of the lee vortex by interactions with air boundary layer air vortex, and front secondary vortex. In addition, the Agulhas water that moves across the leading edge of the weather front will experience a reversal of vorticity. Consequently there are likely to be new waves generated at the front, which will add energy to other waves. The front itself would raise sea level ahead of it [ 35 ], by winds pushing water ahead. This is a forcing effect imposed on the coastal waters. This would pump water onto the continental shelf, which is relatively wider in the location of Algoa bay. Storm surges regularly raise the level of coastal waters in this way. Data for the Beaufort sea show surges of 1m are common (mean 0.4m, maximum 2m) with regression analysis identifying the magnitude correlated primarily with wind speed and direction, and then percentage open water (the location has ice which does not apply in the Agulhas case), with air pressure not being a significant variable [ 36 ]. Their winds speeds were a maximum of 16 knots, whereas much higher speeds are observed for cold fronts in the Agulhas region. Another study of the Beaufort sea identified peak surges of about 3m [ 37 ], but this was based on driftwood on beaches so probably includes wave runup. So a sea level rise of about 2m at the edge of the continental shelf at Algoa bay is not unreasonable to expect for a severe storm. We propose that this head of water periodically overcomes the trapping forces, by localised escaping surges. Both the pumping of water onto the continental shelf, and its escape, are facilitated by bathymetric features, especially submarine canyons (see below). Water escapes may locally lower the water level, contributing to formation of depressions in the water surface, which accentuate wave height. Vortices arising from the Agulhas current The Agulhas current flows against the SW wind, and hence wave amplification occurs [ 10 ]. However water vortices will also be generated by the boundary layer (against the air) and velocity profile of the current. As the surface layer of water is slowed by storm winds – or even driven backwards as has been reported by mariners [ 12 ], a train of substantial current boundary layer vortices (C \(\:⨂\) ) will arise, carried along in the current, see Fig. 3 . These vortices (C \(\:⨂\) ) move in the opposite linear direction to the lee (L \(\:⨀\) ) and windward (W \(\:⨂\) ) vortices within the volume of the wave. Nonetheless they can be expected to interact in passing. As the spin is the same for the C and W vortices, these could combine or strengthen each other. However as identified above, a component of this interaction is in the same direction as the wave travel hence probably accelerates the wave more than heightening it. The C and L vortices have opposite spin hence would not combine, but they could accelerate a jet of water between them. This would remove water from the foot of the wave hence steepening it (rather than heightening it), and enhancing the preceding trough. This is an observed feature of rogue waves, as is the high forward speed. Submarine canyons on the continental slope Then there is bathymetric topology to consider. Historical data showed that canyons on the South African continental slope were associated with greater incidence of rogue waves [ 3 ], for reasons not clear at the time. The reasons are still not fully elucidated. Several ideas emerge from the modern literature. A current will divert a filament into such a feature [ 38 ], creating a vortex at that location. Analytical flow analysis of the theoretical situation of a vortex passing along a gap in a vertical wall shows that the vortex strays into the gap, or can be severed into two vortices, one continuing with the flow and the other passing into and through the gap [ 39 ]. Other work has shown by numerical methods that a current passing over an idealised 2D square trench in shallow liquid will generate upstream-propagating solitary waves [ 32 ]. Hence it is reasonable to expect that a canyon presents a negative forcing function onto the Agulhas current, which creates a topology vortex. This would be a long rope-like vortex laid out in the canyon, hence extending from deep water to the edge of the continental shelf. It is conceivable that part of the vortex could be detached and come close to the water surface. Possible detachment mechanisms include prising by the Agulhas deep countercurrent, and coastal upwelling and downwelling (see below). In particular, a cold front comes with SW winds which cause coastal downwelling, and we hypothesise that this causes accentuated water flows down the canyon thereby detaching the vortex. In the case of the Agulhas current, the topology vortex T will have \(\:⨀\) spin, see Fig. 4 . Consequently when it comes to the surface it could combine with the lee (L \(\:⨀\) ) vortex to drive heightening of the wave and scavenging of the preceding trough. In addition, jetting would arise where the topology (T \(\:⨀\) ) and boundary layer (C \(\:⨂\) ) vortices were approximated. This would accentuate wave height. The continental slope is the interface between the coastal and pelagic bodies of water, and the bathymetry imposes physical constraints on the flow. Analytical methods show that the upwelling flow is most intense over the upstream part of a bank [ 40 ]. These high water velocities are accentuated by the presence of a canyon. Slope vortices arising from movement of the Agulhas current along the continental slope Any strong current that impacts steep underwater topography will generate vortices [ 41 ]. Currents like the Agulhas that flow along a submarine surface experience a frictional boundary layer (shear) along the slope, which can be expected to result in the formation of cylindrical vortices of considerable depth, with axis of spin parallel to the slope gradient (pointing downwards towards the SE). Analytical modelling of idealised submarine ridges identifies the vortices to be about 100m-10km in size [ 42 ]. Such vortices have the capability to generate waves in multiple locations, as they are carried along in the current. In the SW-NE transect plane, the slope vortices are denoted S \(\:⨀\) . These have the same spin as the canyon vortex T \(\:⨀\) , and hence the potential to combine. In addition empirical measurements show that off Algoa Bay there is a strong difference in velocities (magnitude and direction) between the Agulhas current and the coastal counter current, forming a vertical interface above the 120m depth contour [ 21 ] (see Fig. 23 therein). This boundary layer will create columnar vortices reaching the surface (axis pointing downwards for the vortices in the Agulhas current, opposite direction for the countercurrent). These vortices would intersect with the other water vortices, producing irregular waves. Vortices arising from upwelling water movement between the continental shelf and the deeper ocean Coastal upwelling of cold water is a regular feature of continental shelves like the Algoa Bay. In general, upwelling arises from winds transporting, via friction, the warm surface layers of water offshore, with cold deeper water flowing in at depth to compensate. The upwelling and downwelling occurs via vortices at the water front, of about 4 km wide and 40m deep, with strong currents of about 0.5m/s [ 43 ] (for California coast). This also results in a heightened sea level offshore. Numerical modelling of the Heceta Bank (Oregon) show the upwelling occurs over the 100-m isobath [ 44 ] and is strongest over the edge of the continental shelf [ 40 ]. Similar effects exist for the Agulhas Bank and Algoa Bay shelves. Here the forcing winds are the W, NW and NE, and the upwelling of cold water occurs 19–60 hours later [ 45 ]. The upwelling results in changes in sea level > 0.5m, and is associated with trapping of coastal waves [ 45 ]. Upwelling also occurs on the shallow Agulhas Bank due to wind forcing [ 46 ]. Upwelling off Algoa Bay may be represented as a large scale vortex with horizontal spin axis pointing to the NE. Another forcing mechanism for upwelling are meanders (also called pulses or rings) that arise off Natal in the Agulhas current. They are of the order 30km in diameter, and when these arrive at Algoa Bay the leading edge thereof initiates upwelling near Bird Island, and strong SW currents of 0.8m/s [ 47 ]. The meanders themselves travel at a typical speed of 10–20 km/day, up to 65 km/day, and the bulk of the Agulhas current travels around them on the seaward side [ 47 ]. For the Algoa Bay, the more intense thermal gradients occur in summer, whereas more isothermal conditions tend to apply in winter due to the prevalence of the SW wind [ 21 ]. Nonetheless there is a front between warm and cold waters at about 60m depth in winter [ 21 ]. Upwelling flows are dramatically changed by the presence of an underwater canyon on the continental slope. Finite element methods on an idealised canyon with water flowing along the continental margin show greatly intensified upwelling at the head of the canyon, followed downstream (in the wind direction) by downwelling [ 38 ]. Each of these features was about 10km in in its shortest dimension. Canyons that cut down to deeper levels show enhanced upwelling [ 48 ]. Upwelling flows in the Wilmington canyon (recorded with a submarine glider) originated at the 150-215m depth in the canyon, were 10–30 m thick and 5–20 km wide, and flowed at a maximum of 0.5 m/s [ 49 ]. However the literature on canyons is focussed exclusively on upwelling and how this affects transfer of nutrients and coastal ecological systems. There appears to be nothing on the implications for the sea state for navigation. In addition, the canyon upwelling works are decoupled from meteorological effects such as cold front weather systems – the fronts mentioned in the upwelling papers are exclusively fronts between warm and cold bodies of water, not air. Hence it is difficult to determine exactly what effect a cold front would have on the sea state above or near a canyon. Existing models of upwelling are necessarily simplistic [ 47 ] and in reality there are likely to peculiarities and progressively finer level flows, as is universally typical for turbulence. In the Eastern Cape location under examination, prior to the cold front there are westerly winds that drive upwelling. There is strong evidence analytically and experimentally that canyons accentuate high flow and vortices in the wind-forced upwelling situation, and that the most intense flows occur at the edge of the shelf, at 100-200m depth. Once the cold front passes, its winds cause downwelling, and this can be expected to be likewise concentrated around canyons. Reversing the direction of flow for these large bodies of water will generate additional detached turbulent filaments and vortices. Hence the interaction of winds, currents, and bathymetric features result in a complex movement of water, even a chaotic flow regime: “There is considerable spatial variability in the currents, densities, and dynamics over the Heceta Bank complex” [ 44 ]. It is difficult to anticipate exactly how this might contribute to rogue wave formation, but it seems likely that chaotic wave generation must arise, and amplification of existing waves by vortices. 5 Discussion 5.1 A proposed mechanism for rogue wave formation In summary, we propose that rogue waves off the South African east coast arise from a progressive development. The underlying swells originate in the Southern Ocean, and are initially wind forced. They travel up the coast, and are amplified on meeting the contrary Agulhas current. This mechanism is well-established [ 3 ] but on its own does not generate particularly steep waves. The critical sharpening is proposed to occur by lee vortices, of which there are aerial and aquatic types. The air vortices arise from flow separation off the crest of the wave under high wind conditions. They are reinforced by vortices in the meteorological cold front, particularly the air boundary layer vortices. The water lee vortices are initially merely corollaries to the air lee vortices, but subsequently energised by other water vortices to further heighten the wave. Of these other vortices, the topology or canyon vortices appear to have the characteristics to energise the water lee vortices. Considerations of vortex spin suggests there should also be occasional upward jetting between coincident canyon and water boundary layer vortices. The reported preceding trough before the rogue wave [ 3 ] is, per this analysis, a consequence of depletion of the trough by the lee vortices, and by the water boundary layer vortices. The canyon vortex protrudes to the surface presumably as a consequence prising by the return current of coastal downwelling. On the landward (NW) margin of the Agulhas current, waves are refracted in to create a crossing sea, and several water vortices are expected to arise from the Agulhas current interacting with the continental slope and the inshore counter current, with additional vortices arising at the head of canyons due to upwelling-downwelling and the reversals thereof due to movement of the meteorological front. These phenomena make additional waves available for superposition, or the vortices energise selectively waves in a chaotic process. We anticipate that a storm arena may occur where a meteorological front with strong winds meets a locally intensified part of the Agulhas current. The various vortices may be non-linearly strengthened by stronger winds and currents, as the rougher sea state would increase the friction experienced by both fluids. Fronts on land are impeded by terrain, and it is reasonable to suppose that the same occurs over sea. In which case there would also be more waves refracted in from the side, adding further energy, and creating a concentrated localised storm arena. A component of air in the cold front could be retarded and eventually overtaken by the bulk of the frontal air mass forcing its way forward, but not without intense aerial turbulence and gusts. Even then it could remain as a cyst of intense sea conditions within the rear of the cold front. Hence conceptually there is no difficulty providing plausible vortex mechanisms for heightening a wave and creating a trough, in a chaotic manner that would present as a random rogue wave. Many of the mechanisms have already been identified in the literature, though not specifically put together in this way nor associated with rogue waves. Mallory identified the 100 fathom (200 depth) isobar as the risk area for rogue waves [ 3 ]. A partial explanation is now available, as the literature shows there are a variety of flows and vortices generated at 100-200m depth. Qualitatively the vertical section at the 200m isobar may be understood as the flow aperture for water movements on and off the continental shelf. The shelf being shallow cannot supply or accept a large volume of water quickly, without appreciable change in water level, whereas the deep ocean can do this. This makes the edge the location that witnesses the dynamic lateral flow interactions between these two large bodies of water, each of which responds in its own way to the currents, tides, atmospheric pressure, and winds acting on it. Any discrepancies in how those bodies of water behave will generate waves at the interface. 5.2 Limitations and implications The limitation of this explanation is that it is theoretical. There are reasonable grounds for each of the identified vortices, but there are uncertainties: the magnitudes are unknown and the interactions are hypothesised rather than proved. Testing the theory by modelling is not straightforward either. The fluid mechanics is complex because of the multiple vortices and the interactions at the water-air interface. Explicit solutions do not exist and are unlikely to exist until they are able to represent turbulence. Hence realistically, computational fluid dynamics (CFD) appears to be only analytical method available to explore the theory presented here. Even so this is expected to be challenging because of the large range of scales that need to be accommodated. A rogue wave is of the order of 20m tall, hence would need elements approximately 0.5m or smaller in size, whereas the bathymetric terrain and meteorological features are tens of km in size, thus a large model arises. The current literature does not report on any models of this resolution and size. Most of the CFD models are of simple idealised geometry and even these are computationally effortful. Nonetheless it is reasonable to believe that computational models will eventually be possible. We recommend focussing such models on actual bathymetric features. The specific area under examination was the Agulhas region, specifically the South African east coast between Durban and Gqeberha (Port Elizabeth). It is possible that the mechanisms may generalise to other locations where frontal weather interacts with ocean currents, and those currents interact with bathymetric features. Tentative implications for mariners in this region, building on Mallory’s work [ 3 ] are that the risk factors for rogue waves may be: ship heading towards the SW; high sea state with long waves from the SW; navigating along the 100 fathom isobar, i.e. the upper side of the continental shelf; approach of a SW cold front; high winds approximately Beaufort 9 or stronger; severe wind gusts; meeting the cold front near a submarine canyon that excises the continental shelf break; strong Agulhas current (approximately 3 knots). For mariners already in risky circumstances, possible remedies may be changing course to avoid the continental shelf break, or reducing forward speed. The continental shelf break is the most risky area because cold fronts cause reversal of the coastal upwelling-downwelling currents, and these show most turbulent behaviour at the shelf break and especially near heads of canyons on the continental downslope. 6 Conclusions The novel contribution of this paper is the provision of a mechanism for rogue wave formation, using vortex theory. This has not previously been shown in the literature. It is proposed that the first stage in wave growth is harmonic superposition of multiple wave trains, and the second stage being amplitude magnification of waves by the opposing Agulhas water current. However such waves are not rogue and additional mechanisms are necessary to develop the rogue wave characteristics of height, steepness, non-breaking, asymmetrical form and a preceding trough. This is attributed to several vortex interactions, as follow. (1) Wind lee vortices cause steepening, especially on the leeward face of the wave. This suppresses wave breaking. The vortices depend on the wind speeds, hence also on the strength of the meteorological cold front. (2) Boundary layer vortices from the meteorological cold front transfer energy to the wind lee vortices thereby enhance their wave sharpening effect. (3) Agulhas current boundary layer vortices (at the water-air surface) interact with water lee vortices to accelerate a jet of water between them, thereby steepen the wave forward by removing water from the foot of the wave, and also enhance the preceding trough. (4) Bathymetric topology, especially a canyon on the shelf break (continental slope), generates a vortex in the flow of the Agulhas current. The upper end of this rope-like vortex is detached from the canyon by prising of the coastal downwelling current (induced by the meteorological cold front). The vortex moves upwards and combines with the water lee vortex to drive heightening of the wave and scavenging of the preceding trough. (5) Jetting arises when the canyon vortex and the Agulhas current boundary layer vortices pass each other, thereby accentuating wave height, steepness, and asymmetry. In addition slope vortices arise from frictional movement of the Agulhas current along the continental slope, which reinforce the canyon vortices. Thus the vortex theory is able to qualitatively explain all the main features of rogue waves: their height, steepness, non-breaking crest, asymmetrically steep front face, and preceding trough. It also partly why the edge of the continental shelf, e.g. the 100 fathom line, is so prone to forming rogue waves, the reason being that this region is the flow aperture between the continental shelf waters and the deep ocean, and has numerous vortices and flow reversals at up and down-dwelling cycles during the passage of meteorological cold fronts. 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Continental Shelf Research, 2015. 101 : p. 34-46,DOI: https://doi.org/10.1016/j.csr.2015.04.004. Allen, S.E., On subinertial flow in submarine canyons: Effect of geometry. J. Geophys. Res., 2000. 105 (C1): p. 1285-1297,DOI: https://doi.org/10.1029/1999JC900240. Wang, H., Gong, D., Friedrichs, M.A.M., Harris, C.K., Miles, T., Yu, H.-C., and Zhang, Y., A Cycle of Wind-Driven Canyon Upwelling and Downwelling at Wilmington Canyon and the Evolution of Canyon-Upwelled Dense Water on the MAB Shelf. 2022. 9 ,DOI: http://dx.doi.org/10.3389/fmars.2022.866075. Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-4906129","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":348299091,"identity":"97650eb7-7685-4372-82a4-f25f2e130378","order_by":0,"name":"D. J. PONS","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABEElEQVRIiWNgGAWjYDCCwwzMEAZ7A4Q2IF4LzwEgkUCMlgMwLRIJRGrhO8772IBxh02e/MzHDz8X/rDJN5fIfSbBUGPHYM5/AKsWycPsxgmMZ9KKDW6nGUvPSEiz3Dkj3UyC4Vgyg2UDdi0Gh9mYDzC2HU7cIJ3DIM2TcNjA4EYamwQD2wEGg4MN+LXMn3mG+TdPwn+oln9ALYex+wWkJQGkpeEGDxvQlgMQLYxtQC3HsGuRBGoxSDyTlrjhTJqZNU9asoFlzzNmi8S+ZB6DMzhC7PwxZomPO2wS57cffnybx8bOwJw9jfHGh292cgbnsXsfDBLR/MkCiiMe3OqBgBFNC/MHvMpHwSgYBaNgpAEATMpYBGuVGt0AAAAASUVORK5CYII=","orcid":"","institution":"University of Canterbury","correspondingAuthor":true,"prefix":"","firstName":"D.","middleName":"J.","lastName":"PONS","suffix":""}],"badges":[],"createdAt":"2024-08-13 09:59:46","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-4906129/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-4906129/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":64205634,"identity":"985211f0-8ec3-4e19-98d5-fbc9b8aafa3a","added_by":"auto","created_at":"2024-09-10 04:58:14","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":86445,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eProposed vortex architecture for wind to water interactions. Wind vortices \u003c/em\u003eλ\u003cem\u003e arise in the lee of the wave, and via frication drag, push water towards the crest, heightening the wave. Windward vortices do likewise but with less efficacy as the wave is moving forward. Corollary vortices are induced in the water. The diagram represents the location and spin of the vortices, and is not to scale. The ocean surface is represented by the green line. The explanation is constructed for the South African east coast, with the view being from offshore facing NW towards the land, i.e. a longitudinal section on the SW-NE axis.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-4906129/v1/384a4896492e5b26edb24a0c.png"},{"id":64205638,"identity":"603c29fa-a844-439c-bea4-1f53b5dc268b","added_by":"auto","created_at":"2024-09-10 04:58:15","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":86445,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eProposed vortex architecture for cold front. Vortices in the upper structure of the meteorological interact with the surface winds. Not to scale.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-4906129/v1/76bb6fd404694e8474617ea8.png"},{"id":64205636,"identity":"a8044554-a89d-4de1-b73f-824c2c4d2f90","added_by":"auto","created_at":"2024-09-10 04:58:14","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":56374,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eProposed vortex architecture for water current. Vortices arise in the Agulhas current due to the effect of the wind overhead, and the interaction with the water current. Not to scale.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-4906129/v1/3d7545d34c5ede1c68d77e70.png"},{"id":64205635,"identity":"6a9b989b-8507-44f5-af99-d7b06caa02bb","added_by":"auto","created_at":"2024-09-10 04:58:14","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":59941,"visible":true,"origin":"","legend":"\u003cp\u003e\u003cem\u003eProposed vortex architecture for bathymetric features. The Agulhas current interacts with the ground topology to create vortices. These include an irregular train of current boundary current vortices (C), a topology (canyon) vortex (T1, T2), and a series of slope (or ground contact) vortices (S). These interact each other to cause jetting (T2-C). They also interact with and reinforce the wind-induced lee vortices (T2-L). Hence the bathymetry topology generated vortices provide a mechanism whereby energy may be transferred to generate wave steepening and asymmetry of surface waves. Not to scale.\u003c/em\u003e\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-4906129/v1/712a863f6791c1aa289ab53c.png"},{"id":83442493,"identity":"6f8aee3e-6302-461b-98ba-22926e024ecd","added_by":"auto","created_at":"2025-05-26 10:01:54","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":892085,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-4906129/v1/195604bf-6fb0-4212-8895-c02783b66de3.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Rogue wave formation in the Agulhas current","fulltext":[{"header":"1 Introduction","content":"\u003cp\u003eThere have been numerous abrupt sinkings of ships in mysterious situations in peacetime, and rogue waves may have been involved [\u003cspan citationid=\"CR1\" class=\"CitationRef\"\u003e1\u003c/span\u003e]. Rogue waves have been reported by mariners through the ages, but were rejected by physicists on the assumption of the impossibility for stochastic superposition of multiple wave trains to cause the claimed heights or steepness, as breaking would dissipate the energy. Acceptance only arose after an exceptionally large wave was recorded in 1995 at the Draupner oil rig in the North Sea [\u003cspan citationid=\"CR2\" class=\"CitationRef\"\u003e2\u003c/span\u003e], but still the physics of the phenomenon are poorly understood.\u003c/p\u003e \u003cp\u003eA wave is named \u003cem\u003erogue\u003c/em\u003e by its height relative to the prevailing wave height. It must be at least twice the \u003cem\u003esignificant wave height\u003c/em\u003e, which is the mean of the third highest waves. Rogue waves are much steeper than they ought to be, generally appear in stormy conditions, can have a height of 18m or even more, move quickly but are highly localised and do not travel far, disappear quickly, may be in a train of up to about three waves, and may be preceded by a depression (\u0026lsquo;hole\u0026rsquo;) in the water [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e]. They occur in all the world\u0026rsquo;s oceans, more often than might be thought [\u003cspan citationid=\"CR4\" class=\"CitationRef\"\u003e4\u003c/span\u003e], and satellite photography suggests there is at least one wave of 25m somewhere in the world every two days [\u003cspan citationid=\"CR5\" class=\"CitationRef\"\u003e5\u003c/span\u003e]. They can also appear on the coast, where they are a hazard to beach-goers and people fishing from rocks [\u003cspan citationid=\"CR6\" class=\"CitationRef\"\u003e6\u003c/span\u003e]. They also depend on bathymetric features, though the relationships are unclear. How they might depend on meteorological features is an unexplored area. This paper presents a conceptual theory for the formation of rogue waves in the Agulhas current, including bathymetric features of the continental slope, and the passage of meteorological cold fronts.\u003c/p\u003e"},{"header":"2 Rogue wave literature","content":"\u003cp\u003ePrinciples of wave growth\u003c/p\u003e \u003cp\u003eIn the first instance wave height is determined by the strength and fetch of the wind. Fully developed waves are those for which the wind has blown over a long enough fetch that the waves have grown as high as they can, and the crests of the waves start to break, dissipating energy. Wave height is proportional to wind speed squared for fully developed waves. The formation of conventional waves in real seas are complex and the multiple mechanics are still imperfectly understood.\u003c/p\u003e \u003cp\u003eThere are several phases in the growth of a wave. The first stage is from initiation through to formation of a small growing wave, represented by the Phillips and Miles mechanisms. The Phillips mechanism [\u003cspan citationid=\"CR7\" class=\"CitationRef\"\u003e7\u003c/span\u003e] requires a turbulent wind to start with, in which random small pressure distributions occur on the interface, and some of these move at the same speed as the phase velocity for the medium, and hence form waves by resonance. The Miles mechanism is based on a velocity profile for the wind, which causes shear on the surface, and the effect is related to the curvature of that profile [\u003cspan citationid=\"CR8\" class=\"CitationRef\"\u003e8\u003c/span\u003e]. In practice both theories successfully predict waves based on surface tension (capillary) and gravity, and both rely on a disturbance or turbulence for initiation. Experiments in a wind-wave tunnel show that numerous vortices, smaller than the size of the wave, and with different spin, arise at the water surface, and migrate down into the water [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. Water vortices of the same spin have also been observed merging [\u003cspan citationid=\"CR9\" class=\"CitationRef\"\u003e9\u003c/span\u003e]. Both Phillips and Miles theories include viscosity, e.g. as a sheltering coefficient, to explain why wave growth only occurs once air velocity reaches a critical value. Phillips has pressure distributions moving across the surface, and Miles has shear \u0026ndash; they are different aspects of the same phenomenon, and can as readily be represented by vortices. The above theories ignore the possibility of correlated motions between the air and water compartments.\u003c/p\u003e \u003cp\u003eThe above theories inadequately explain ocean wave growth at the extreme end, or real three dimensional seas. For increased height, say 20m, single waves would need unrealistically fast propagation velocities. Even then they would be long swells, without much steepness, and hence low hazard to shipping. Rogue waves do not show this characteristic: they are both high and steep.\u003c/p\u003e \u003cp\u003eAnother contribution to wave height comes from wave train frequency divergence. A \u003cem\u003ewave train\u003c/em\u003e (wave packet) is a set of waves of similar wavelength travelling in the same direction. Waves in deep water do not keep their frequency but diverge into a variety of frequencies, some of which are longer wavelength and hence higher and faster. The faster waves enter the train from the rear, build up height in the middle, and then decrease as they move to the front of the train. The speed of an individual wave is the \u003cem\u003ephase velocity\u003c/em\u003e (wavelength divided by the period), and is twice as fast as the wave train as a whole (\u003cem\u003egroup velocity).\u003c/em\u003e For fully developed conditions, the fastest waves move at or slightly faster than the wind speed [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. In general for deep (as opposed to shallow) water waves, longer wavelength waves have greater vertical height, and propagate faster, than shorter wavelength waves, but are not particularly steep. Hence this mechanism also fails to explain the rogue phenomenon.\u003c/p\u003e \u003cp\u003eA third height contribution is stochastic or harmonic superposition of multiple wave trains. These trains may come from different directions since waves continue to propagate after the winds that cause them have died away. Thus lateral wave trains contribute towards constructive interference [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. However a linear superposition still does not explain rogue waves. They are higher, deeper, sharper, and more common than they should be. Second-order summations have been attempted, and these do give steeper waves [\u003cspan citationid=\"CR11\" class=\"CitationRef\"\u003e11\u003c/span\u003e] but do not fully account for them. The underlying phenomena that might give rise to non-linear summations are unclear.\u003c/p\u003e \u003cp\u003eThe fourth known contribution is that winds blowing contrary to water currents dramatically increase wave height, i.e. wave-current coupling. A wave that meets an opposing current reduces its wavelength and speed, and increases its height. Waves are highly sensitive to current, and can be stopped by an opposing current flowing at \u0026frac14; of the wave speed [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e]. Generally currents flow much slower than winds \u0026amp; waves, so stopping is not the issue, but the point is that even a small current can cause waves to gain height. This is believed to be the primary amplification in the Agulhas current, and has been known since at least the early 1900s [\u003cspan citationid=\"CR12\" class=\"CitationRef\"\u003e12\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eAlso, currents may refract waves [\u003cspan citationid=\"CR13\" class=\"CitationRef\"\u003e13\u003c/span\u003e] whereby the waves are slowed more at the faster centre of the current than at the edges. This bends the edges of waves, and can focus them on an area. There is some evidence, though not conclusive, that the counter-clockwise eddies in the Agulhas retroflection may increase the wave height [\u003cspan citationid=\"CR14\" class=\"CitationRef\"\u003e14\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eContributions to wave steepening\u003c/p\u003e \u003cp\u003eEven all the above do not fully account for the shape of rogue waves. This has been a challenge to explain. The rogue waves are not symmetrical nor even sinusoidal in shape. Rather they have a steep wall of water. Also, rogue waves \u0026ndash;at least in the Agulhas region - have \u0026lsquo;always been associated with a correspondingly long deep trough \u0026ndash; which occurs in advance of the wave\u0026rsquo;, i.e. a hole-in-the-sea on the NE side of the wave [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. It is this combination of hole and wave that is particularly hazardous to ships travelling towards the wave. The depth of the trough before the wave adds to the overall height that the ship has to survive. The cause of the trough is unknown.\u003c/p\u003e \u003cp\u003eThere are theoretical and empirical models [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e] of wave shape that give waves with greater height (not merely twice the amplitude as with sinusoidal), sharper crests and wider troughs. However these models are one dimensional sea surfaces with clean starting assumptions, and do not transfer readily to real sea conditions. Another part of the difficulty of explaining rogue waves is that waves should break at their crest \u0026ndash; hence loose energy and height - before getting so large [\u003cspan citationid=\"CR16\" class=\"CitationRef\"\u003e16\u003c/span\u003e], but apparently this does not occur. Mechanisms for suppressing wave breaking are poorly understood.\u003c/p\u003e \u003cp\u003ePredictive approaches\u003c/p\u003e \u003cp\u003eAnother approach to rogue waves is stochastic, seeking to predict the occurrence probability of a rogue wave arising somewhere in a given storm, and this typically uses the Rayleigh-Haring-Tayfun distribution, which is a geometrical composite of several approximation models [\u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e]. Predicting the crest height and trough depth is not possible to do in a satisfactory manner with the stochastic models, but there are other approaches that separately predict wave heights [\u003cspan citationid=\"CR17\" class=\"CitationRef\"\u003e17\u003c/span\u003e]. Other approaches focus on trying to model the sea state with a view to giving a precise prediction of a rogue wave within the timeframe of several seconds, i.e. to provide warning for conditions that might spawn rogue waves [\u003cspan citationid=\"CR10\" class=\"CitationRef\"\u003e10\u003c/span\u003e, \u003cspan citationid=\"CR15\" class=\"CitationRef\"\u003e15\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eRogue waves in the Agulhas region\u003c/p\u003e \u003cp\u003eOne of the earliest papers to address rogue waves, well before the Draupner incident, was written in 1974 by Mallory [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e], a Master Mariner, Captain in the SA Navy, and Professor of Oceanography. Mallory surveyed several ships that had experienced and survived a rogue wave, and noted that rogue waves in the Agulhas current tended to occur on the 100 fathom line and near submarine canyons, with an approaching cold front [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. Thus there were three factors that he identified: the continental shelf, underwater topology (canyons) and frontal weather. Mallory\u0026rsquo;s explanatory work is still unsurpassed for its descriptive accuracy and its practical implications for the only known advice for shipping to reduce the risk to rogue waves in the Agulhas region: \u0026lsquo;\u003cem\u003ekeep away from the vicinity of the outer edge of the continental shelf or 100 fathom (200m) line \u0026hellip; when steaming to the southwest with a falling barometer, a fresh northeasterly wind blowing, and a change to fresh to strong southwesterly winds forecast in the next twelve hours\u003c/em\u003e\u0026rsquo; [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eHowever Mallory did not identify the underlying physics, nor has the modern literature. There does not appear to be any further refinement of the link between weather systems and rogue waves.\u003c/p\u003e \u003cp\u003eThe existing literature on rogue waves focusses on the physics of deep water waves, and typically takes a quantitative analytical approach based on idealised water conditions, but still does not explain the mechanisms whereby these waves occur in real seas. In summary, the literature on rogue waves identifies harmonic summation and amplification by winds blowing contrary to currents as the main contributions, but the causes of the observed wave steepness and asymmetric form are poorly understood, and they remain difficult to predict. The combined effect of bathymetric and meteorological features on rogue wave formation has not been addressed.\u003c/p\u003e"},{"header":"3 Method","content":"\u003cp\u003eResearch objective\u003c/p\u003e \u003cp\u003eThe current objective was to explain rogue wave formation in way that integrates wave formation, sea currents, bathymetric features, and meteorological weather systems. This has not previously been shown in the literature. The area under examination is the Agulhas region, specifically the South African east coast between Durban and Gqeberha (Port Elizabeth), which has a high prevalence of rogue waves [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e].\u003c/p\u003e \u003cp\u003eApproach\u003c/p\u003e \u003cp\u003eThe approach used the vortex perspective of fluid flow. This is justified on several grounds. First, the mechanics of wave trains and packets are a purely analytical perspective, but have not been successful in explaining rogue waves, and hence it is worth trying different approaches. Second, the fluid dynamics of real seas, with their winds and waves, is fundamentally turbulent. While there is no explicit mechanics available to describe turbulence \u0026ndash; this being one of the unsolved problems of fluid mechanics \u0026ndash; it is nonetheless instructive to consider the flow as comprising chaotic vortices. In addition there is some evidence \u0026ndash; albeit limited \u0026ndash; that wind vortices may have some unknown role in heightening of waves. This is because larger waves are commonly explained as being raised by a shear instability, whereby the air flow locally reverses downwind of a wave. This is a type of vortex, though often not specifically identified as such. A more explicit vortex consideration is provided by considering a wave to be a Kelvin-Helmholtz (KH) instability in the interface between two fluid streams, whereby reduced pressure over the crest lifts the weight of the wave [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e]. This causes vortices in counter-rotating pairs, and these vortices cause further rotations of the interface [\u003cspan citationid=\"CR19\" class=\"CitationRef\"\u003e19\u003c/span\u003e]. This interpretation is usually applied to fluids of the same phase, and has only rarely been applied to ocean waves [\u003cspan citationid=\"CR18\" class=\"CitationRef\"\u003e18\u003c/span\u003e], and not to rogue waves.\u003c/p\u003e \u003cp\u003eHaving selected the vortex perspective, the next stage was to examine the meteorological and oceanographic literatures for established evidence of vortices. In order of greatest to least importance were field studies, experimental studies (e.g. wave tanks), finite element analyses, and explicit analyses of ideal geometry. A considerable literature was discovered, almost none of it directly connected to rogue waves, but nonetheless interesting in the way it identified how and where vortices formed. Little to no mathematical treatment was discovered for real sea states, but this was unsurprising because, as already mentioned, there is no mathematics available to describe turbulence other than in the aggregate.\u003c/p\u003e \u003cp\u003eFinally a conceptual process was applied to identify candidate mechanisms whereby the various types of vortices could combine to create an extreme sea state. This part of the work is purely conceptual \u0026ndash; no calculations or simulations were performed. This is justified as quantitative methods are unavailable for modelling vortex combination, except by numerical simulation of the fluid dynamics. The combination of meteorological, oceanographic, and bathymetric induced vortices makes the problem excessively complex and to create a simulation would be a major programme of work.\u003c/p\u003e"},{"header":"4 Results","content":"\u003cdiv id=\"Sec5\" class=\"Section2\"\u003e\n \u003ch2\u003e4.1 Vortices\u003c/h2\u003e\n \u003cp\u003eVortices naturally arise in a fluid in two principal ways. One is when the flow changes direction due to an obstruction or the shape of its duct. The other is due to viscosity and friction with the duct forming a boundary layer velocity profile. In this context the duct is whatever physically bounds the flow, which can be submarine terrain, an adjacent counter flow, or the interface between two immiscible fluids (water and air in this case). The duct is not necessarily rigid, as evident in Kelvin-Helmholtz instability at the interface.\u003c/p\u003e\n \u003cp\u003eThe vortices created by viscous friction are approximated by idealised irrotational (or free) vortices, i.e. the tangential velocity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{v}\\)\u003c/span\u003e\u003c/span\u003e about the vortex axis decreases with radius per \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{v}=\\frac{{\\Gamma\\:}}{2\\pi\\:r}\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Gamma\\:}\\)\u003c/span\u003e\u003c/span\u003e is the circulation about the vortex axis, with \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Gamma\\:}=\\oint\\:v\\bullet\\:\\:dl\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:dl\\)\u003c/span\u003e\u003c/span\u003e is perimeter length around a closed curve enclosing the vortex axis, and \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:r\\)\u003c/span\u003e\u003c/span\u003e is the distance from that axis. The vorticity is a measure of the rotation rate of a fluid particle, given by \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{\\omega\\:}=\\nabla\\:\\times\\:\\overrightarrow{v}\\)\u003c/span\u003e\u003c/span\u003e where \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\nabla\\:\\times\\:\\)\u003c/span\u003e\u003c/span\u003e is the curl (rotation) of the velocity field. For an ideal irrotational vortex \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{\\omega\\:}=0\\)\u003c/span\u003e\u003c/span\u003e. This relationship is idealised because in practice the velocity cannot be infinite at the axis, but instead is zero. As each outward layer in the vortex moves faster, there is a viscous shear stress generated through the body of the vortex, and this dissipates energy. Or conversely, an ongoing supply of energy is required to sustain the vortex.\u003c/p\u003e\n \u003cp\u003eThe other extreme of the idealised vortex is a rotational vortex (or forced vortex, or rigid body rotation) where the angular velocity \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\Omega\\:}\\)\u003c/span\u003e\u003c/span\u003e is constant such that tangential velocity is proportional to radius, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:v={\\Omega\\:}r\\)\u003c/span\u003e\u003c/span\u003e and in which case vorticity is \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:\\overrightarrow{\\omega\\:}=2{\\Omega\\:}\\)\u003c/span\u003e\u003c/span\u003e. These vortices have a pressure applied at their outer boundary.\u003c/p\u003e\n \u003cp\u003eReal vortices combine elements of both above idealised types, and in addition they need not be circular. They can also change shape, move with the fluid, bend and extend on their axis, merge, split, and form tubes (including toroidal). The inner regions of vortices tend to retain fluid particles, and hence can transfer fluid and momentum as the vortex migrates. Vortices therefore can contain substantial energy, and transport it to new locations.\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec6\" class=\"Section2\"\u003e\n \u003ch2\u003e4.2 Proposed mechanisms for rogue waves\u003c/h2\u003e\n \u003cp\u003eWe propose that Agulhas rogue waves form by the combination three effects: Harmonic superposition of multiple wave trains; Amplitude modulation of waves by opposing water currents; and Fluid-air interface mediated by multi-scale multi-medium chaotic vorticity. The first two have well-established physics, but the third is a novel proposal, and suggests that the missing ingredient to explain Agulhas rogue waves is vorticity.\u003c/p\u003e\n \u003cp\u003e(1) Harmonic superposition of multiple wave trains\u003c/p\u003e\n \u003cp\u003eDuring winter, there are waves coming up from deep in the Southern ocean. The winds have a long fetch of about 800-1,200km, hence fully developed waves are created (typical height 6m, wavelength 120m). They travel from the SW. For example, for 12 July 2023 the forecast for Marion Island \u0026ndash; which is deep in the Southern Ocean - was: \u0026lsquo;MARION FORTIES EAST (40S/50S, 35E/50E): WIND : W to SW 25 to 35 reaching 40 in places in the south. VIS : Moderate in showers and rain. SEA STATE: 5.5 to 6.5m reaching 7.0 to 8.0m in the south, SW to W swell\u0026rsquo; [\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e]. Note the wind strength of up to 40 knots, and the wave height of up to 8m. However those waves would be long swells, not steep sided. These waves propagate up towards Gbergha (Port Elizabeth).\u003c/p\u003e\n \u003cp\u003eIn addition, the SW winds created by cold meteorological fronts create waves\u0026thinsp;\u0026gt;\u0026thinsp;0.5m height (e.g. height 3m, wavelength 60m, period 7 s) that travel north eastwards on the continental shelf. These travel in the same direction as the Southern Ocean waves, and periodically reinforce them to greater height. They are also able to reverse the surface Agulhas current [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e]. Modelling shows them to have the greatest effect at the shelf break itself [\u003cspan class=\"CitationRef\"\u003e22\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003eMeteorological conditions in South Africa are affected by the terrain which consists of an elevated plateau of 1,400m in the interior. This has a marked effect on the evolution of weather systems including the formation of the coastal low NE of the escarpment [\u003cspan class=\"CitationRef\"\u003e23\u003c/span\u003e] and localised strong NE or NW winds (up to 30 knots) with clear skies [\u003cspan class=\"CitationRef\"\u003e24\u003c/span\u003e]. These preceding winds create additional local waves of high frequency [\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e]. These NW winds and waves converge onto SW winds and waves at the head of the front.\u003c/p\u003e\n \u003cp\u003eThese waves of different wavelength and direction are constantly interfering and summing into higher waves with deeper troughs. However this superposition on its own is insufficient to account for the size and steepness of rogue waves.\u003c/p\u003e\n \u003cp\u003e(2) Amplitude magnification of waves by opposing water currents.\u003c/p\u003e\n \u003cp\u003eWaves from the SW wave are magnified on meeting the Agulhas current. This has well established physics and is familiar to mariners. These waves meet the strongly flowing Agulhas current which is flowing towards the SW direction. The intersection causes the waves to rise in height, typically by 20\u0026ndash;40% and up to 60% in extreme situations [\u003cspan class=\"CitationRef\"\u003e25\u003c/span\u003e]. These SW winds and their waves are opposite in direction to the Agulhas current, and strongest where the current is most concentrated. When meeting the current the waves become shorter in wavelength, and steeper. The effect is \u0026lsquo;\u003cem\u003emore pronounced where the opposing current is strongest, i.e. just outside the 100 fathom (200 m) line\u0026rsquo;\u003c/em\u003e, and ship logs show much heavier sea conditions here [\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e]. This interaction on its own is sufficient to generate large waves of about 10m, and the South Africa Weather Services forecasts this interaction [\u003cspan class=\"CitationRef\"\u003e20\u003c/span\u003e] based on a model developed by [\u003cspan class=\"CitationRef\"\u003e25\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003eThere is also refraction occurring, whereby SW waves that meet the current are slowed, but those on the slower edges of the current are not. This effect is likely to be more pronounced where the current speed drops off most sharply, which is the edge of the continental shelf. These waves interfere from the side (crossing sea). Even so, these mechanics are insufficient to describe the occurrence of rogue waves, because these conditions arise frequently.\u003c/p\u003e\n \u003cp\u003e(3) Fluid-air interface mediated by multi-scale multi-medium chaotic vorticity.\u003c/p\u003e\n \u003cp\u003eWe propose that superimposed and current-amplified waves receive an energy boost from vortices. There are numerous vortices in the Agulhas case, air and water, which appear not to have been previously identified as such. These are at different physical scales and frequencies. Our proposal is that the chaotic combination of these vortices, on top of the preceding two phenomena, is what creates rogue waves.\u003c/p\u003e\n \u003cp\u003eThere are a number of fluid mechanics principles that apply to vortices:\u003c/p\u003e\n \u003cp\u003eA. Vortices arises from three mechanisms: (a) wherever there is fluid shear on a boundary layer, (b) moving fluid that changes its direction of flow due to obstruction from another fluid or solid body, or (c) Coriolis force acting on a fluid moving across latitudes. The Coriolis effect operates at the large scale and is apparent in the direction of rotation of the atmospheric fronts and the Agulhas current as a whole, but is less important for present considerations.\u003c/p\u003e\n \u003cp\u003eB. Vortices in a 3D medium are chaotic \u0026ndash; they have a regularity of sorts, but not a fixed periodicity. Consequently they do not engage in harmonic superposition like the wave trains of classical physics. Instead their interactions are irregular and unpredictable. There is no analytical physics that can predict their behaviour accurately.\u003c/p\u003e\n \u003cp\u003eC. Vortices \u0026ndash; either in the air or the water - with sufficient scale cause waves when they act on the water-air interface.\u003c/p\u003e\n \u003cp\u003eD. Adjacent vortices with the same spin combine to produce a stronger vortex. This is an amplification mechanism.\u003c/p\u003e\n \u003cp\u003eE. The effect of vortices on an air-water interface is complex and poorly understood [\u003cspan class=\"CitationRef\"\u003e26\u003c/span\u003e]. We propose that vortices in the liquid at a liquid-gaseous fluid interface will preferentially transfer their kinetic energy into waves on that interface, rather than take the Kolmogorov breakdown route to progressively smaller vortices. We could not find explicit confirmation of this in the literature, nonetheless it is consistent with the theoretical finding that \u0026ldquo;the surface deforms to satisfy the conditions that the tangential stress be equal to zero and the normal stress be equal to a constant at all times\u0026rdquo; and that the evolution of the movements on the air side are negligible [\u003cspan class=\"CitationRef\"\u003e27\u003c/span\u003e]. This provides a mechanism for additional amplification.\u003c/p\u003e\n \u003cp\u003eF. Furthermore we propose that contrary rotation water vortices can, if converged (impacted together appropriately) cause vertical jets that will accentuate wave height (potentially at an angle) and create a trough alongside. This formation of jets for converged counter-rotation vortices pairs is experimentally observed [\u003cspan class=\"CitationRef\"\u003e28\u003c/span\u003e]. It is also consistent with the observation (in slurries) that for contrary vortices the \u0026ldquo;impact energy is conveyed through the tangential component\u0026rdquo; [\u003cspan class=\"CitationRef\"\u003e29\u003c/span\u003e]. In addition analytical studies on an idealised shelf break show that contrary vertical spin vortices will usually repel each other, but there are conditions whereby they can move in parallel in jet-like flows [\u003cspan class=\"CitationRef\"\u003e30\u003c/span\u003e].\u003c/p\u003e\n\u003c/div\u003e\n\u003cdiv id=\"Sec7\" class=\"Section2\"\u003e\n \u003ch2\u003e4.3 Elaboration of the vortex proposal\u003c/h2\u003e\n \u003cp\u003eWe propose an explanation for rogue wave formation under the conditions encountered of: strong Agulhas current; cold front from the SW; and bathymetry of the continental slope. Using the above principles it is possible use the morphology of the vortices to make inferences about which vortices contribute to heightening waves, and how different vortices could affect each other, even if the mathematics are not yet ready to represent the interactions with precision. The vortex axis is denoted with the right-hand rule: \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e out the page is anticlockwise, \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e is clockwise.\u003c/p\u003e\n \u003cp\u003eEffect of lee vortices on wave height\u003c/p\u003e\n \u003cp\u003eWe assume that a set of waves have already been generated by winds some distance away (e.g. Marion Island) by Phillips/Miles [\u003cspan class=\"CitationRef\"\u003e7\u003c/span\u003e, \u003cspan class=\"CitationRef\"\u003e8\u003c/span\u003e] or other starting mechanisms, and these waves have propagated into a region above which a SW cold front is moving and winds are blowing (e.g. SW 35 knots), see Fig.\u0026nbsp;\u003cspan class=\"InternalRef\"\u003e1\u003c/span\u003e. The wind experiences friction on the water, due to the combination of the small scale roughness of the water surface, and the topology of the waves. The drag results in a boundary layer. Initially, this drag force increases with the wind speed [\u003cspan class=\"CitationRef\"\u003e31\u003c/span\u003e]. A high degree of microscale turbulence arises in this air boundary layer \u0026ndash; theorised by [\u003cspan class=\"CitationRef\"\u003e7\u003c/span\u003e] and evident experimentally by its effects on the water [\u003cspan class=\"CitationRef\"\u003e9\u003c/span\u003e]. The effect of turbulence is to inject faster moving air into the boundary layer, i.e. causes mixing, and makes the velocity profile more abrupt. Turbulence also involves vorticity.\u003c/p\u003e\n \u003cp\u003eAt some point the size and topology of the wave is such that air flow separation occurs, and a lee vortex appears (downwind of the wave crest), see \u0026lambda;1 (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e) in the figure. This drives surface water (v\u003csub\u003eL1\u003c/sub\u003e) towards the wavecrest, which grows the wave. In other perspectives this localised region of backwards air flow is called a shear instability.\u003c/p\u003e\n \u003cp\u003eAt the same time a windward vortex arises by impact with the back of the wave, see \u0026omega;1, which likewise grows the wave from the rear (v\u003csub\u003eW1\u003c/sub\u003e). However as the wave is also moving forward, this is expected to be reduced in efficacy. Hence the lee vortex is expected to make the greater contribution to wave growth. The asymmetry of the effects is expected to steepen the leeward face of the wave. Both air vortices move forward with the wave. Vortices typically evolve topographically over time [\u003cspan class=\"CitationRef\"\u003e32\u003c/span\u003e], and in this case we propose that the lee vortex grows over several consecutive waves, see \u0026lambda;1 to \u0026lambda;2 in the figure, at the expense of the windward vortex. They are then extinguished by turbulent inflows from above or laterally, as the bulk air stream recolonises the lacuna, and the cycle recommences. As the sea state worsens, so its frictional resistance to the SW winds increases. This will thicken the boundary layer and create stronger wind vortices.\u003c/p\u003e\n \u003cp\u003eThe air vortices induce movement of water, and hence also corollary vortices in the water, denoted L (lee) and W (windward) in the diagram. These have opposite spin to their air counterparts.\u003c/p\u003e\n \u003cp\u003eVortices arising from meteorological cold front\u003c/p\u003e\n \u003cp\u003eThe cold front also contains many additional vortices [\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e]. We propose that these interact with the air vortices shaping the waves, see Fig. \u003cspan class=\"InternalRef\"\u003e2\u003c/span\u003e. Ahead of the cold front is a warm air uplift vortex (U\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e), in response to the cold front pushing forward. There is considerable vorticity in this region [\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e]. The axis of vorticity is the same as the lee \u0026lambda; vortices (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e), hence there is the potential for these vortices to combine, i.e. for the uplift vortex U to transfer energy to lee vortex \u0026lambda;, thereby strengthening wave growth. This depends on these two vortices being in close proximity, which might occur if a SW gust was to extend beyond the front to create a lee vortex under the uplift vortex.\u003c/p\u003e\n \u003cp\u003eBehind the front is a large main vortex (H\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e) with high wind speeds [\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e]. This has spin like the air windward (\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:{\\omega\\:}⨀\\)\u003c/span\u003e\u003c/span\u003e) vortex, and hence has the potential to combine and transfer energy into it. It may be that this flattens the sea, rather than builds waves. This vortex spins off a trail of secondary head vortices (H2\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e) that are pushed forward by the wind [\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003eClose to the terrain (which in this case is the sea surface) and about 100m up into the atmosphere are a series of boundary layer vortices (B\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e) [\u003cspan class=\"CitationRef\"\u003e33\u003c/span\u003e]. These originate in the friction over the surface. They are at a larger scale than the flow separation lee vortices \u0026lambda;\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e. As they have the same spin, the boundary layer vortices have the potential to combine with or transfer energy to the lee vortices, thereby heightening the wave. These vortices are moving in the same direction, at approximately the same speed, so there is an extended period of time for this interaction. We propose this is an appreciable contribution to wave heightening.\u003c/p\u003e\n \u003cp\u003eThe air boundary layer vortices (B\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e) and the secondary head vortices (H2\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e) have opposite spin, so will not merge but will rather tend to create jets between them (v\u003csub\u003eHB\u003c/sub\u003e), resulting in wind gusts at terrain-level. These gusts will tend to enhance flow separation and strengthen the lee vortices \u0026lambda;, hence further raising wave height. This is consistent with modelling work that shows gustiness allows the waves to develop to greater heights than theory would otherwise allow, especially for following swell (no great difference for opposing swells) [\u003cspan class=\"CitationRef\"\u003e34\u003c/span\u003e], which is exactly the situation under consideration. Those authors proposed the underlying mechanics might be smaller waves absorbing wind energy and transferring it to larger waves, but we proposed instead the mechanisms are strengthening of the lee vortex by interactions with air boundary layer air vortex, and front secondary vortex.\u003c/p\u003e\n \u003cp\u003eIn addition, the Agulhas water that moves across the leading edge of the weather front will experience a reversal of vorticity. Consequently there are likely to be new waves generated at the front, which will add energy to other waves.\u003c/p\u003e\n \u003cp\u003eThe front itself would raise sea level ahead of it [\u003cspan class=\"CitationRef\"\u003e35\u003c/span\u003e], by winds pushing water ahead. This is a forcing effect imposed on the coastal waters. This would pump water onto the continental shelf, which is relatively wider in the location of Algoa bay. Storm surges regularly raise the level of coastal waters in this way. Data for the Beaufort sea show surges of 1m are common (mean 0.4m, maximum 2m) with regression analysis identifying the magnitude correlated primarily with wind speed and direction, and then percentage open water (the location has ice which does not apply in the Agulhas case), with air pressure not being a significant variable [\u003cspan class=\"CitationRef\"\u003e36\u003c/span\u003e]. Their winds speeds were a maximum of 16 knots, whereas much higher speeds are observed for cold fronts in the Agulhas region. Another study of the Beaufort sea identified peak surges of about 3m [\u003cspan class=\"CitationRef\"\u003e37\u003c/span\u003e], but this was based on driftwood on beaches so probably includes wave runup.\u003c/p\u003e\n \u003cp\u003eSo a sea level rise of about 2m at the edge of the continental shelf at Algoa bay is not unreasonable to expect for a severe storm. We propose that this head of water periodically overcomes the trapping forces, by localised escaping surges. Both the pumping of water onto the continental shelf, and its escape, are facilitated by bathymetric features, especially submarine canyons (see below). Water escapes may locally lower the water level, contributing to formation of depressions in the water surface, which accentuate wave height.\u003c/p\u003e\n \u003cp\u003eVortices arising from the Agulhas current\u003c/p\u003e\n \u003cp\u003eThe Agulhas current flows against the SW wind, and hence wave amplification occurs [\u003cspan class=\"CitationRef\"\u003e10\u003c/span\u003e]. However water vortices will also be generated by the boundary layer (against the air) and velocity profile of the current. As the surface layer of water is slowed by storm winds \u0026ndash; or even driven backwards as has been reported by mariners [\u003cspan class=\"CitationRef\"\u003e12\u003c/span\u003e], a train of substantial current boundary layer vortices (C\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e) will arise, carried along in the current, see Fig. \u003cspan class=\"InternalRef\"\u003e3\u003c/span\u003e. These vortices (C\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e) move in the opposite linear direction to the lee (L\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e) and windward (W\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e) vortices within the volume of the wave. Nonetheless they can be expected to interact in passing. As the spin is the same for the C and W vortices, these could combine or strengthen each other. However as identified above, a component of this interaction is in the same direction as the wave travel hence probably accelerates the wave more than heightening it. The C and L vortices have opposite spin hence would not combine, but they could accelerate a jet of water between them. This would remove water from the foot of the wave hence steepening it (rather than heightening it), and enhancing the preceding trough. This is an observed feature of rogue waves, as is the high forward speed.\u003c/p\u003e\n \u003cp\u003eSubmarine canyons on the continental slope\u003c/p\u003e\n \u003cp\u003eThen there is bathymetric topology to consider. Historical data showed that canyons on the South African continental slope were associated with greater incidence of rogue waves [\u003cspan class=\"CitationRef\"\u003e3\u003c/span\u003e], for reasons not clear at the time. The reasons are still not fully elucidated. Several ideas emerge from the modern literature. A current will divert a filament into such a feature [\u003cspan class=\"CitationRef\"\u003e38\u003c/span\u003e], creating a vortex at that location. Analytical flow analysis of the theoretical situation of a vortex passing along a gap in a vertical wall shows that the vortex strays into the gap, or can be severed into two vortices, one continuing with the flow and the other passing into and through the gap [\u003cspan class=\"CitationRef\"\u003e39\u003c/span\u003e]. Other work has shown by numerical methods that a current passing over an idealised 2D square trench in shallow liquid will generate upstream-propagating solitary waves [\u003cspan class=\"CitationRef\"\u003e32\u003c/span\u003e]. Hence it is reasonable to expect that a canyon presents a negative forcing function onto the Agulhas current, which creates a topology vortex. This would be a long rope-like vortex laid out in the canyon, hence extending from deep water to the edge of the continental shelf. It is conceivable that part of the vortex could be detached and come close to the water surface. Possible detachment mechanisms include prising by the Agulhas deep countercurrent, and coastal upwelling and downwelling (see below). In particular, a cold front comes with SW winds which cause coastal downwelling, and we hypothesise that this causes accentuated water flows down the canyon thereby detaching the vortex.\u003c/p\u003e\n \u003cp\u003eIn the case of the Agulhas current, the topology vortex T will have \u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e spin, see Fig. \u003cspan class=\"InternalRef\"\u003e4\u003c/span\u003e. Consequently when it comes to the surface it could combine with the lee (L\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e) vortex to drive heightening of the wave and scavenging of the preceding trough. In addition, jetting would arise where the topology (T\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e) and boundary layer (C\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨂\\)\u003c/span\u003e\u003c/span\u003e) vortices were approximated. This would accentuate wave height.\u003c/p\u003e\n \u003cp\u003eThe continental slope is the interface between the coastal and pelagic bodies of water, and the bathymetry imposes physical constraints on the flow. Analytical methods show that the upwelling flow is most intense over the upstream part of a bank [\u003cspan class=\"CitationRef\"\u003e40\u003c/span\u003e]. These high water velocities are accentuated by the presence of a canyon.\u003c/p\u003e\n \u003cp\u003eSlope vortices arising from movement of the Agulhas current along the continental slope\u003c/p\u003e\n \u003cp\u003eAny strong current that impacts steep underwater topography will generate vortices [\u003cspan class=\"CitationRef\"\u003e41\u003c/span\u003e]. Currents like the Agulhas that flow along a submarine surface experience a frictional boundary layer (shear) along the slope, which can be expected to result in the formation of cylindrical vortices of considerable depth, with axis of spin parallel to the slope gradient (pointing downwards towards the SE). Analytical modelling of idealised submarine ridges identifies the vortices to be about 100m-10km in size [\u003cspan class=\"CitationRef\"\u003e42\u003c/span\u003e]. Such vortices have the capability to generate waves in multiple locations, as they are carried along in the current. In the SW-NE transect plane, the slope vortices are denoted S\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e. These have the same spin as the canyon vortex T\u003cspan class=\"InlineEquation\"\u003e\u003cspan class=\"mathinline\"\u003e\\(\\:⨀\\)\u003c/span\u003e\u003c/span\u003e, and hence the potential to combine.\u003c/p\u003e\n \u003cp\u003eIn addition empirical measurements show that off Algoa Bay there is a strong difference in velocities (magnitude and direction) between the Agulhas current and the coastal counter current, forming a vertical interface above the 120m depth contour [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e] (see Fig.\u0026nbsp;23 therein). This boundary layer will create columnar vortices reaching the surface (axis pointing downwards for the vortices in the Agulhas current, opposite direction for the countercurrent). These vortices would intersect with the other water vortices, producing irregular waves.\u003c/p\u003e\n \u003cp\u003eVortices arising from upwelling water movement between the continental shelf and the deeper ocean\u003c/p\u003e\n \u003cp\u003eCoastal upwelling of cold water is a regular feature of continental shelves like the Algoa Bay. In general, upwelling arises from winds transporting, via friction, the warm surface layers of water offshore, with cold deeper water flowing in at depth to compensate. The upwelling and downwelling occurs via vortices at the water front, of about 4 km wide and 40m deep, with strong currents of about 0.5m/s [\u003cspan class=\"CitationRef\"\u003e43\u003c/span\u003e] (for California coast). This also results in a heightened sea level offshore. Numerical modelling of the Heceta Bank (Oregon) show the upwelling occurs over the 100-m isobath [\u003cspan class=\"CitationRef\"\u003e44\u003c/span\u003e] and is strongest over the edge of the continental shelf [\u003cspan class=\"CitationRef\"\u003e40\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003eSimilar effects exist for the Agulhas Bank and Algoa Bay shelves. Here the forcing winds are the W, NW and NE, and the upwelling of cold water occurs 19\u0026ndash;60 hours later [\u003cspan class=\"CitationRef\"\u003e45\u003c/span\u003e]. The upwelling results in changes in sea level\u0026thinsp;\u0026gt;\u0026thinsp;0.5m, and is associated with trapping of coastal waves [\u003cspan class=\"CitationRef\"\u003e45\u003c/span\u003e]. Upwelling also occurs on the shallow Agulhas Bank due to wind forcing [\u003cspan class=\"CitationRef\"\u003e46\u003c/span\u003e]. Upwelling off Algoa Bay may be represented as a large scale vortex with horizontal spin axis pointing to the NE.\u003c/p\u003e\n \u003cp\u003eAnother forcing mechanism for upwelling are meanders (also called pulses or rings) that arise off Natal in the Agulhas current. They are of the order 30km in diameter, and when these arrive at Algoa Bay the leading edge thereof initiates upwelling near Bird Island, and strong SW currents of 0.8m/s [\u003cspan class=\"CitationRef\"\u003e47\u003c/span\u003e]. The meanders themselves travel at a typical speed of 10\u0026ndash;20 km/day, up to 65 km/day, and the bulk of the Agulhas current travels around them on the seaward side [\u003cspan class=\"CitationRef\"\u003e47\u003c/span\u003e]. For the Algoa Bay, the more intense thermal gradients occur in summer, whereas more isothermal conditions tend to apply in winter due to the prevalence of the SW wind [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e]. Nonetheless there is a front between warm and cold waters at about 60m depth in winter [\u003cspan class=\"CitationRef\"\u003e21\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003eUpwelling flows are dramatically changed by the presence of an underwater canyon on the continental slope. Finite element methods on an idealised canyon with water flowing along the continental margin show greatly intensified upwelling at the head of the canyon, followed downstream (in the wind direction) by downwelling [\u003cspan class=\"CitationRef\"\u003e38\u003c/span\u003e]. Each of these features was about 10km in in its shortest dimension. Canyons that cut down to deeper levels show enhanced upwelling [\u003cspan class=\"CitationRef\"\u003e48\u003c/span\u003e]. Upwelling flows in the Wilmington canyon (recorded with a submarine glider) originated at the 150-215m depth in the canyon, were 10\u0026ndash;30 m thick and 5\u0026ndash;20 km wide, and flowed at a maximum of 0.5 m/s [\u003cspan class=\"CitationRef\"\u003e49\u003c/span\u003e].\u003c/p\u003e\n \u003cp\u003eHowever the literature on canyons is focussed exclusively on upwelling and how this affects transfer of nutrients and coastal ecological systems. There appears to be nothing on the implications for the sea state for navigation. In addition, the canyon upwelling works are decoupled from meteorological effects such as cold front weather systems \u0026ndash; the fronts mentioned in the upwelling papers are exclusively fronts between warm and cold bodies of water, not air. Hence it is difficult to determine exactly what effect a cold front would have on the sea state above or near a canyon.\u003c/p\u003e\n \u003cp\u003eExisting models of upwelling are necessarily simplistic [\u003cspan class=\"CitationRef\"\u003e47\u003c/span\u003e] and in reality there are likely to peculiarities and progressively finer level flows, as is universally typical for turbulence. In the Eastern Cape location under examination, prior to the cold front there are westerly winds that drive upwelling. There is strong evidence analytically and experimentally that canyons accentuate high flow and vortices in the wind-forced upwelling situation, and that the most intense flows occur at the edge of the shelf, at 100-200m depth. Once the cold front passes, its winds cause downwelling, and this can be expected to be likewise concentrated around canyons. Reversing the direction of flow for these large bodies of water will generate additional detached turbulent filaments and vortices. Hence the interaction of winds, currents, and bathymetric features result in a complex movement of water, even a chaotic flow regime: \u0026ldquo;There is considerable spatial variability in the currents, densities, and dynamics over the Heceta Bank complex\u0026rdquo; [\u003cspan class=\"CitationRef\"\u003e44\u003c/span\u003e]. It is difficult to anticipate exactly how this might contribute to rogue wave formation, but it seems likely that chaotic wave generation must arise, and amplification of existing waves by vortices.\u003c/p\u003e\n\u003c/div\u003e"},{"header":"5 Discussion","content":"\u003cdiv id=\"Sec9\" class=\"Section2\"\u003e \u003ch2\u003e5.1 A proposed mechanism for rogue wave formation\u003c/h2\u003e \u003cp\u003eIn summary, we propose that rogue waves off the South African east coast arise from a progressive development. The underlying swells originate in the Southern Ocean, and are initially wind forced. They travel up the coast, and are amplified on meeting the contrary Agulhas current. This mechanism is well-established [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] but on its own does not generate particularly steep waves. The critical sharpening is proposed to occur by lee vortices, of which there are aerial and aquatic types. The air vortices arise from flow separation off the crest of the wave under high wind conditions. They are reinforced by vortices in the meteorological cold front, particularly the air boundary layer vortices. The water lee vortices are initially merely corollaries to the air lee vortices, but subsequently energised by other water vortices to further heighten the wave. Of these other vortices, the topology or canyon vortices appear to have the characteristics to energise the water lee vortices. Considerations of vortex spin suggests there should also be occasional upward jetting between coincident canyon and water boundary layer vortices. The reported preceding trough before the rogue wave [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] is, per this analysis, a consequence of depletion of the trough by the lee vortices, and by the water boundary layer vortices. The canyon vortex protrudes to the surface presumably as a consequence prising by the return current of coastal downwelling. On the landward (NW) margin of the Agulhas current, waves are refracted in to create a crossing sea, and several water vortices are expected to arise from the Agulhas current interacting with the continental slope and the inshore counter current, with additional vortices arising at the head of canyons due to upwelling-downwelling and the reversals thereof due to movement of the meteorological front. These phenomena make additional waves available for superposition, or the vortices energise selectively waves in a chaotic process.\u003c/p\u003e \u003cp\u003eWe anticipate that a storm arena may occur where a meteorological front with strong winds meets a locally intensified part of the Agulhas current. The various vortices may be non-linearly strengthened by stronger winds and currents, as the rougher sea state would increase the friction experienced by both fluids. Fronts on land are impeded by terrain, and it is reasonable to suppose that the same occurs over sea. In which case there would also be more waves refracted in from the side, adding further energy, and creating a concentrated localised storm arena. A component of air in the cold front could be retarded and eventually overtaken by the bulk of the frontal air mass forcing its way forward, but not without intense aerial turbulence and gusts. Even then it could remain as a cyst of intense sea conditions within the rear of the cold front.\u003c/p\u003e \u003cp\u003eHence conceptually there is no difficulty providing plausible vortex mechanisms for heightening a wave and creating a trough, in a chaotic manner that would present as a random rogue wave. Many of the mechanisms have already been identified in the literature, though not specifically put together in this way nor associated with rogue waves.\u003c/p\u003e \u003cp\u003eMallory identified the 100 fathom (200 depth) isobar as the risk area for rogue waves [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e]. A partial explanation is now available, as the literature shows there are a variety of flows and vortices generated at 100-200m depth. Qualitatively the vertical section at the 200m isobar may be understood as the flow aperture for water movements on and off the continental shelf. The shelf being shallow cannot supply or accept a large volume of water quickly, without appreciable change in water level, whereas the deep ocean can do this. This makes the edge the location that witnesses the dynamic lateral flow interactions between these two large bodies of water, each of which responds in its own way to the currents, tides, atmospheric pressure, and winds acting on it. Any discrepancies in how those bodies of water behave will generate waves at the interface.\u003c/p\u003e \u003c/div\u003e \u003cdiv id=\"Sec10\" class=\"Section2\"\u003e \u003ch2\u003e5.2 Limitations and implications\u003c/h2\u003e \u003cp\u003eThe limitation of this explanation is that it is theoretical. There are reasonable grounds for each of the identified vortices, but there are uncertainties: the magnitudes are unknown and the interactions are hypothesised rather than proved. Testing the theory by modelling is not straightforward either. The fluid mechanics is complex because of the multiple vortices and the interactions at the water-air interface. Explicit solutions do not exist and are unlikely to exist until they are able to represent turbulence. Hence realistically, computational fluid dynamics (CFD) appears to be only analytical method available to explore the theory presented here. Even so this is expected to be challenging because of the large range of scales that need to be accommodated. A rogue wave is of the order of 20m tall, hence would need elements approximately 0.5m or smaller in size, whereas the bathymetric terrain and meteorological features are tens of km in size, thus a large model arises. The current literature does not report on any models of this resolution and size. Most of the CFD models are of simple idealised geometry and even these are computationally effortful. Nonetheless it is reasonable to believe that computational models will eventually be possible. We recommend focussing such models on actual bathymetric features.\u003c/p\u003e \u003cp\u003eThe specific area under examination was the Agulhas region, specifically the South African east coast between Durban and Gqeberha (Port Elizabeth). It is possible that the mechanisms may generalise to other locations where frontal weather interacts with ocean currents, and those currents interact with bathymetric features.\u003c/p\u003e \u003cp\u003eTentative implications for mariners in this region, building on Mallory\u0026rsquo;s work [\u003cspan citationid=\"CR3\" class=\"CitationRef\"\u003e3\u003c/span\u003e] are that the risk factors for rogue waves may be: ship heading towards the SW; high sea state with long waves from the SW; navigating along the 100 fathom isobar, i.e. the upper side of the continental shelf; approach of a SW cold front; high winds approximately Beaufort 9 or stronger; severe wind gusts; meeting the cold front near a submarine canyon that excises the continental shelf break; strong Agulhas current (approximately 3 knots). For mariners already in risky circumstances, possible remedies may be changing course to avoid the continental shelf break, or reducing forward speed.\u003c/p\u003e \u003cp\u003eThe continental shelf break is the most risky area because cold fronts cause reversal of the coastal upwelling-downwelling currents, and these show most turbulent behaviour at the shelf break and especially near heads of canyons on the continental downslope.\u003c/p\u003e \u003c/div\u003e"},{"header":"6 Conclusions","content":"\u003cp\u003eThe novel contribution of this paper is the provision of a mechanism for rogue wave formation, using vortex theory. This has not previously been shown in the literature. It is proposed that the first stage in wave growth is harmonic superposition of multiple wave trains, and the second stage being amplitude magnification of waves by the opposing Agulhas water current. However such waves are not rogue and additional mechanisms are necessary to develop the rogue wave characteristics of height, steepness, non-breaking, asymmetrical form and a preceding trough. This is attributed to several vortex interactions, as follow.\u003c/p\u003e \u003cp\u003e(1) Wind lee vortices cause steepening, especially on the leeward face of the wave. This suppresses wave breaking. The vortices depend on the wind speeds, hence also on the strength of the meteorological cold front.\u003c/p\u003e \u003cp\u003e(2) Boundary layer vortices from the meteorological cold front transfer energy to the wind lee vortices thereby enhance their wave sharpening effect.\u003c/p\u003e \u003cp\u003e(3) Agulhas current boundary layer vortices (at the water-air surface) interact with water lee vortices to accelerate a jet of water between them, thereby steepen the wave forward by removing water from the foot of the wave, and also enhance the preceding trough.\u003c/p\u003e \u003cp\u003e(4) Bathymetric topology, especially a canyon on the shelf break (continental slope), generates a vortex in the flow of the Agulhas current. The upper end of this rope-like vortex is detached from the canyon by prising of the coastal downwelling current (induced by the meteorological cold front). The vortex moves upwards and combines with the water lee vortex to drive heightening of the wave and scavenging of the preceding trough.\u003c/p\u003e \u003cp\u003e(5) Jetting arises when the canyon vortex and the Agulhas current boundary layer vortices pass each other, thereby accentuating wave height, steepness, and asymmetry. In addition slope vortices arise from frictional movement of the Agulhas current along the continental slope, which reinforce the canyon vortices.\u003c/p\u003e \u003cp\u003eThus the vortex theory is able to qualitatively explain all the main features of rogue waves: their height, steepness, non-breaking crest, asymmetrically steep front face, and preceding trough. It also partly why the edge of the continental shelf, e.g. the 100 fathom line, is so prone to forming rogue waves, the reason being that this region is the flow aperture between the continental shelf waters and the deep ocean, and has numerous vortices and flow reversals at up and down-dwelling cycles during the passage of meteorological cold fronts.\u003c/p\u003e \u003cp\u003eHence a conceptual explanation is provided for rogue wave formation, that integrates wave formation, Agulhas sea currents, bathymetric features including submarine canyons, and meteorological cold front weather systems.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e \u003ch2\u003eConflict of interest\u003c/h2\u003e \u003cp\u003eThe authors declare no conflict of interest.\u003c/p\u003e \u003c/p\u003e\u003ch2\u003eFunding\u003c/h2\u003e \u003cp\u003eThe authors declare no funding.\u003c/p\u003e\u003ch2\u003eAuthor Contribution\u003c/h2\u003e\u003cp\u003eD.P. conducted the research and wrote the manuscript\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eKotze, P., \u003cem\u003eMonsters of the seas.\u003c/em\u003e New Scientist, 2021. \u003cstrong\u003e249\u003c/strong\u003e(3327): p. 41-45,DOI: https://doi.org/10.1016/S0262-4079(21)00523-6.\u003c/li\u003e\n \u003cli\u003eCavaleri, L., Benetazzo, A., Barbariol, F., Bidlot, J.-R., and Janssen, P.A.E.M., \u003cem\u003eThe Draupner Event: The Large Wave and the Emerging View %J Bulletin of the American Meteorological Society.\u003c/em\u003e 2017. \u003cstrong\u003e98\u003c/strong\u003e(4): p. 729-735,DOI: https://doi.org/10.1175/BAMS-D-15-00300.1.\u003c/li\u003e\n \u003cli\u003eMallory, J.K., \u003cem\u003eAbnormal Waves on the South East Coast of South Africa.\u003c/em\u003e The International Hydrographic Review, 1974. \u003cstrong\u003e51\u003c/strong\u003e(2),DOI: https://www.swimhistory.co.za/files/Big%20Waves%20Transkei%20coast.pdf.\u003c/li\u003e\n \u003cli\u003eDidenkulova, E., \u003cem\u003eCatalogue of rogue waves occurred in the World Ocean from 2011 to 2018 reported by mass media sources.\u003c/em\u003e Ocean \u0026amp; 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https://doi.org/10.3390/jmse9030326.\u003c/li\u003e\n \u003cli\u003eSald\u0026iacute;as, G.S. and Allen, S.E., \u003cem\u003eThe Influence of a Submarine Canyon on the Circulation and Cross-Shore Exchanges around an Upwelling Front\u0026nbsp;\u003c/em\u003eJournal of Physical Oceanography, 2020. \u003cstrong\u003e50\u003c/strong\u003e(6): p. 1677-1698,DOI: https://doi.org/10.1175/JPO-D-19-0130.1.\u003c/li\u003e\n \u003cli\u003eJohnson, E.R. and McDonald, N.R., \u003cem\u003eThe motion of a vortex near a gap in a wall.\u003c/em\u003e Physics of Fluids, 2004. \u003cstrong\u003e16\u003c/strong\u003e(2): p. 462-469,DOI: http://dx.doi.org/10.1063/1.1637603\u003c/li\u003e\n \u003cli\u003eWhitney, M.M. and Allen, J.S., \u003cem\u003eCoastal Wind-Driven Circulation in the Vicinity of a Bank. Part I: Modeling Flow over Idealized Symmetric Banks.\u003c/em\u003e Journal of Physical Oceanography, 2009. \u003cstrong\u003e39\u003c/strong\u003e(6): p. 1273-1297,DOI: https://doi.org/10.1175/2008JPO3966.1.\u003c/li\u003e\n \u003cli\u003evan der Boog, C.G., Molemaker, M.J., Dijkstra, H.A., Pietrzak, J.D., and Katsman, C.A., \u003cem\u003eGeneration of Vorticity Near Topography: Anticyclones in the Caribbean Sea.\u003c/em\u003e JGR Oceans, 2022. \u003cstrong\u003e127\u003c/strong\u003e(8): p. e2021JC017987,DOI: https://doi.org/10.1029/2021JC017987.\u003c/li\u003e\n \u003cli\u003eJagannathan, A., Srinivasan, K., McWilliams, J.C., Molemaker, M.J., and Stewart, A.L., \u003cem\u003eBoundary-Layer-Mediated Vorticity Generation in Currents over Sloping Bathymetry\u0026nbsp;\u003c/em\u003eJournal of Physical Oceanography, 2021. \u003cstrong\u003e51\u003c/strong\u003e(6): p. 1757-1778,DOI: https://doi.org/10.1175/JPO-D-20-0253.1.\u003c/li\u003e\n \u003cli\u003eWashburn, L. and Armi, L., \u003cem\u003eObservations of Frontal Instabilities on an Upwelling Filament\u0026nbsp;\u003c/em\u003eJournal of Physical Oceanography, 1988. \u003cstrong\u003e18\u003c/strong\u003e(8): p. 1075-1092,DOI: https://doi.org/10.1175/1520-0485(1988)018\u0026lt;1075:OOFIOA\u0026gt;2.0.CO;2.\u003c/li\u003e\n \u003cli\u003eWhitney, M.M. and Allen, J.S., \u003cem\u003eCoastal Wind-Driven Circulation in the Vicinity of a Bank. Part II: Modeling Flow over the Heceta Bank Complex on the Oregon Coast\u0026nbsp;\u003c/em\u003eJournal of Physical Oceanography, 2009. \u003cstrong\u003e39\u003c/strong\u003e(6): p. 1298-1316,DOI: https://doi.org/10.1175/2008JPO3967.1.\u003c/li\u003e\n \u003cli\u003eGoschen, W.S., Schumann, E.H., Bernard, K.S., Bailey, S.E., and Deyzel, S.H.P., \u003cem\u003eUpwelling and ocean structures off Algoa Bay and the south-east coast of South Africa.\u003c/em\u003e African Journal of Marine Science, 2012. \u003cstrong\u003e34\u003c/strong\u003e(4): p. 525-536,DOI: http://dx.doi.org/10.2989/1814232X.2012.749810.\u003c/li\u003e\n \u003cli\u003eHancke, L., Smeed, D., Roberts, M., Russo, C., Rayner, D., and Jebri, F., \u003cem\u003eAtmospheric and advective forcing of upwelling on South Africa\u0026apos;s central Agulhas Bank.\u003c/em\u003e Deep Sea Research Part II: Topical Studies in Oceanography, 2023. \u003cstrong\u003e209\u003c/strong\u003e: p. 105293,DOI: https://doi.org/10.1016/j.dsr2.2023.105293.\u003c/li\u003e\n \u003cli\u003eGoschen, W.S., Bornman, T.G., Deyzel, S.H.P., and Schumann, E.H., \u003cem\u003eCoastal upwelling on the far eastern Agulhas Bank associated with large meanders in the Agulhas Current.\u003c/em\u003e Continental Shelf Research, 2015. \u003cstrong\u003e101\u003c/strong\u003e: p. 34-46,DOI: https://doi.org/10.1016/j.csr.2015.04.004.\u003c/li\u003e\n \u003cli\u003eAllen, S.E., \u003cem\u003eOn subinertial flow in submarine canyons: Effect of geometry.\u003c/em\u003e J. Geophys. Res., 2000. \u003cstrong\u003e105\u003c/strong\u003e(C1): p. 1285-1297,DOI: https://doi.org/10.1029/1999JC900240.\u003c/li\u003e\n \u003cli\u003eWang, H., Gong, D., Friedrichs, M.A.M., Harris, C.K., Miles, T., Yu, H.-C., and Zhang, Y., \u003cem\u003eA Cycle of Wind-Driven Canyon Upwelling and Downwelling at Wilmington Canyon and the Evolution of Canyon-Upwelled Dense Water on the MAB Shelf.\u003c/em\u003e 2022. \u003cstrong\u003e9\u003c/strong\u003e,DOI: http://dx.doi.org/10.3389/fmars.2022.866075.\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"oceanography, turbulence, rogue, South Africa","lastPublishedDoi":"10.21203/rs.3.rs-4906129/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-4906129/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003e\u003cstrong\u003eContext:\u003c/strong\u003e Harmonic summation and amplification by winds blowing contrary to currents are known contributions to rogue waves, but the causes of the observed wave steepness, asymmetric form, and non-breaking are poorly understood. The potential effect of bathymetric and meteorological features has not been addressed.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eMethod:\u003c/strong\u003e Vortex theory was applied qualitatively to the weather and ocean conditions of the Agulhas region.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eResults\u003c/strong\u003e: Rogue wave formation is attributed to: (1) Wind lee vortices cause steepening of wave leeward face, and suppresses wave breaking. (2) Boundary layer vortices from the meteorological cold front transfer energy to the wind lee vortices thereby enhancing their wave sharpening effect. (3) Agulhas current boundary layer vortices interact with water lee vortices to accelerate a jet of water between them, thereby steepening the wave and enhancing the preceding trough. (4) Bathymetric topology, especially a canyon on the continental slope, generates a vortex in the Agulhas current. This vortex is detached from the canyon by prising of the coastal downwelling current (induced by the meteorological cold front), and combines with the water lee vortex to heighten the wave. (5) Jetting arises when the canyon vortex and the Agulhas current boundary layer vortices pass each other, thereby accentuating wave height, steepness, and asymmetry.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eConclusions:\u003c/strong\u003e The novel contribution is the provision of a mechanism for rogue wave formation, using vortex theory, that conceptually integrates wave formation, Agulhas sea currents, bathymetric features including submarine canyons, and meteorological cold front weather systems.\u003c/p\u003e","manuscriptTitle":"Rogue wave formation in the Agulhas current","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-09-10 04:58:10","doi":"10.21203/rs.3.rs-4906129/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"897878b3-3e6c-4fb3-860b-30f0e4793a04","owner":[],"postedDate":"September 10th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2025-05-26T09:53:43+00:00","versionOfRecord":[],"versionCreatedAt":"2024-09-10 04:58:10","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-4906129","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-4906129","identity":"rs-4906129","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}