Conceptual Analysis of Vortex Contributions to Rogue Wave Formation in the Agulhas Current

Abstract

Harmonic summation and amplification by winds blowing contrary to currents are known contributions to rogue waves in the region of the Agulhas current, 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. Vortex theory was applied to develop a theory of wave formation, based on conceptual reasoning. Rogue wave formation is attributed to the following: (1) wind lee vortices causing steepening of a wave’s leeward face, and suppressing wave breaking; (2) boundary layer vortices from the meteorological cold front transferring energy to the wind lee vortices thereby sharpening the wave; (3) Agulhas current boundary layer vortices interacting 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, generating 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, and (5) jetting, which arises when the canyon vortex and the Agulhas current boundary layer vortices pass each other, thereby accentuating wave height, steepness, and asymmetry.

1. Introduction

Rogue waves have been reported by mariners through the ages and present an appreciable if rare hazard to shipping, passengers, and engineering structures [1]. In the open ocean they are primarily associated with storm weather. Hence, if climate change increases the severity or frequency of storms, then rogue wave encounters may become more hazardous. Examples of rogue wave heights experienced by ships in the Atlantic ocean are Royal Mail Ship (RMS) Lusitania (1910) ~23 m, RMS Homeric ~24 m (1924), QE2 ~29 m (1995) [2]. The number of ships positively known to have been sunk by a rogue wave is relatively small, though there have been numerous abrupt sinkings of ships in mysterious situations in peacetime, and rogue waves may have been involved [3]. Satellite photography suggests there is at least one wave of 25 m somewhere in the world every two days [4].

Until relatively recently these waves were considered physically impossible, since stochastic superposition of multiple wave trains cannot cause the claimed steepness, and 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 [5]. The evidence is compelling for extremely large and destructive waves, albeit as a rare and highly localised phenomenon. Nonetheless, the physics of the phenomenon are incompletely understood.

A wave is defined as rogue when it is at least twice the significant wave height, which itself is the mean of the third highest waves. Rogue waves are much steeper and sharper than they ought to be, generally appear in stormy conditions, can have a height of 18 m or 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, are not necessarily symmetrical nor even sinusoidal in shape, and may be preceded by a depression in the water [6,7]. The depth of the trough before the wave adds to the overall height that the ship has to survive. They occur in all the world’s oceans, more often than might be thought [4,7]. Multiple mechanisms have been proposed for rogue waves, but none individually adequately explains their formation for real sea states. There are at least three manifestations of rogue wave. Oceanic rogue waves occur in deep ocean [8], even over abyssal plains. This oceanic type tend to occur within storms [9], and a possible explanation is that solitons and breathers build large waves [10], which then experience non-linear superposition when meeting crossing seas or contrary currents. Coastal rogue waves are a hazard to beach-goers and people fishing from rocks [11] and appear to be caused by different mechanisms, specifically focusing the effects of large waves [12] by inshore submarine topology [11,13,14], variable depth [15], and reflection [16]. In between is the outer continental shelf area, which is the topic of this paper. Here, the literature is sparse [17]. There appears to be a complex but unknown involvement of bathymetry and meteorology, along with the deep ocean mechanisms.

The area under examination is the outer continental shelf region of the Agulhas current off the coast of South Africa, which is a known risk area for rogue waves [6] for reasons unknown. This paper analyses this region and seeks to develop a theory that includes bathymetric features of the continental slope, and the passage of meteorological cold fronts.

2. Rogue Wave Literature

The primary focus of the literature is to derive an explicit mathematical representation of the underlying physics based on wave mechanics. Multiple approaches are evident but there is no certainty as to which, if any, is correct. Some of the works include validation using artificial wave channels. A weak area is the problem of applying mathematical models based on simplistic starting conditions, to real sea states. There is also literature on statistical prediction of likelihood or size of rogue waves for a given sea state.

The formation of conventional non-rogue waves in real seas is complex. The first stage is from initiation through to formation of a small growing wave, represented by the Phillips or Miles mechanisms. The Phillips mechanism [18] requires a turbulent wind in which random small pressure distributions move at the phase velocity, and hence form waves by resonance. The Miles mechanism requires that the velocity profile causes shear on the surface [19].

In ocean conditions, fully developed waves are those for which the wind has blown over sufficient fetch that the waves have grown as high as they can for that wind strength. The fastest waves in a group move at or slightly faster than the wind speed [20] and wave height is proportional to wind speed squared. For deep water (as opposed to shallow) waves, longer wavelength gives greater vertical height and faster propagation, but they are not particularly steep. Large waves, of the order of 20 m, are precluded by this mechanism as they would need unrealistically fast propagation velocities. 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 [21]. Mechanisms for suppressing wave breaking are poorly understood.

Waves may be heightened by linear first order summation of random (stochastic) waves, crossing seas, or harmonic superposition of multiple wave trains. In addition, for waves in deep ocean there are wave–current interactions and modulational instability effects, and for coastal waves, bottom topography.

Wave trains may come from different directions since waves continue to propagate after the winds that cause them have died away, these waves being called swells. In the Agulhas case, the main swells propagate towards the north-east [22], and lateral wave trains contribute towards constructive interference [6]. Also, non-linear wave interactions involve high frequency components increasing the height of the crests and creating a broader trough [23].

A known mechanism for heightening waves is that winds blowing contrary to water currents dramatically increase wave height (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 [20]. Generally, currents flow much slower than winds and 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 off the east coast of South Africa, where the Agulhas current flows towards the south-west, directly opposing the waves propagated up from the Southern Ocean. This creates heavy seas, and the cause has been known by mariners since at least the early 1900s [24]. This region is also particularly hazardous for rogue waves [6].

Also, currents may refract waves [1] 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 through such mechanisms [25].

The above mechanisms are well established theoretically and validated in real sea states. Nonetheless, they inadequately account for rogue wave features of height, steepness, sharpness, and trough. Additional mechanisms are speculated to assist, particularly second-order wave theory, and Peregrine breather theory.

Waves may interact with each other, by transferring energy from one wave to another, hence second-order wave theory [26]. The phenomenon is real, and can be materialised in physical wave generators [27]. The mathematical representation is an approximation of complex and incompletely understood mechanisms; nonetheless, the theory does give steeper and sharper waves with flattening of the trough [23]. However, it shows discrepancies when compared to wave-tanks and numerical simulations [8].

Oceanic waves comprise a wave train of individual waves of similar wavelength travelling in the same direction. These diverge into a variety of frequencies, some of which are longer wavelength and hence higher and faster. The faster waves appear in the rear of the train, build up height in the middle, and then decrease as they move to the front. Wave packets of a few localised waves (solitons) arise naturally in the ocean as a consequence of the slow dispersion of waves, countered by non-linear localisation of the energy. These have been physically observed, notably a packet of three waves measured by a buoy in the Gulf of Biscay, with a height of 27.8 m [10].

It is possible for a modulation to occur within a wave packet, creating a temporary localised standing wave, and the mathematical physics thereof is the Peregrine breather [28], a type of soliton based on the non-linear Schrödinger equation. Other related breathers also exist [29]. The physics represents a carrier wave with a group velocity, the modulations of which at some moment in time summate to peak at a specific location, and then rapidly decline. It is a localised and temporary concentration of energy of the background carrier wave, with the peak being three times higher than the surrounding wave amplitude. This creates a steep wall of water, preceded by a deep trough. The physics has been replicated empirically in a linear wave tank [30] (see Figure 5 therein) [31]. The breather is consistent with the observed spatio-temporal presentation of rogue waves, their steepness, and the hole. However, the laboratory tests had a number of artificialities. The measured wave was at the centimetre scale where surface tension effects could be appreciable, and the absence of wind required artificial generation of the wave by forcing a carrier amplitude and frequency. Also, the non-linear Schrödinger equation, and the wave tank experiments, only represent one-dimensional propagation, and this is known not to properly represent the mechanics of real sea states [13]. Furthermore, analysis of ocean buoy data for deep water showed that the largest breather does not always correspond to the rogue wave [32]. Hence, it is not established that breathers really are causal in real seas, even if they have attractive attributes in theory.

Stochastic properties of waves have been studied by Fourier analysis, including non-linear versions, a method which can represent the breathers and other non-linear behaviours. For a theoretical perspective, see [33,34]. However, when applied to real data the method is sensitive to the sampling window [32]. A case study in shallow seas showed that solitons, present in the rogue waves and single large solitons but not clusters thereof, were associated with the observed heights [12]. However, that study was limited by being conducted in shallow water (10 m), with a sloping seabed, a shoal offshore, a buoy that could move laterally with the waves, and a tidal range of 2.5 m. In this context, the reported rogue waves of up to 7.45 m total height (approximately 4 m above and below the nominal waterline, see Figure 4 therein) might not be what they seem. Depending on the tide, a feature of those dimensions might be caused by a shore breaker, with appreciable horizontal displacement of the surface-following buoy. If so, this would weaken the causal role attributed to solitons.

Analysis using non-linear Fourier transforms on data from a deep-water (5000 m) ocean buoy implied that the rogue waves were caused mainly by non-linear superposition of conventional sine waves, rather than breathers [32]. They also found that breathers and solitons were present in all large ocean waves, rogue or not. It appears that large waves were formed by these unstable modes, with breathers contributing up to 20% of wave crest height and solitons up to 50%, which non-linear superposition worked on to produce rogue waves.

There are other empirical models [35] of wave shape that represent the shape of the wave surface, rather than the physics. A typical approach uses the Rayleigh–Haring–Tayfun distribution, which is a geometrical composite of several approximation models [35]. This does give waves with greater height (not merely twice the amplitude as with sinusoidal), sharper crests and wider troughs. Other approaches predict wave heights [36]. Likelihood of rogue waves has been correlated with the crest–trough parameters, i.e., the non-linear wave shape [37]. This line of research seeks to model the sea state to predict a rogue wave within the timeframe of several seconds, i.e., to provide warning for conditions that might spawn rogue waves [20,35].

Two other factors have been implicated in rogue wave formation: the effect of bathymetry, and meteorology. The literature on bathymetric considerations is focussed on shallow coastal water, where they can be particularly destructive to property [38]. The main factor is variable water depth and the interaction of the wave therewith [15]. Waves are amplified by run up on a sloping seabed [13], though simulations show that, if there is rogue wave, it is more likely to occur at the deeper location [39]. Likewise, breathers and solitons from the deep water can persist into shallower waters, where they interact with shoals and a sloping seabed [40], although, in contrast, others claim that modulational instabilities are not present in the coastal region [16]. Raised submarine features (sea mounts) enhance the asymmetry of incoming waves [37]. They also refract wave energy with the effect being dependent on the wave period [11]. Wave tank experiments suggest that waves that break offshore may recombine to produce rogue waves closer in [14].

However, the effect of deep water bathymetry on rogue wave formation has not been explored in the literature. The effect is known to be real, with the edges of continental shelves (200 m depth) being known risk areas [6]. These depths are too great for any conceivable interaction between the wave and the seabed, as occurs for shallow water, yet the mechanisms are unknown.

The other factor of meteorology is known in the simplistic sense of rogue waves being associated with stormy weather. The premise is that wind strength generally increases the energy of the sea state, thereby providing more energy to be combined in wave-to-wave interactions. However there is a deeper problem in that rogue waves are sometimes particularly associated with the timing of passage of frontal systems, not merely the strength of the winds [6].

In the area under examination, which is the Agulhas current off South Africa, both bathymetry and meteorology appear to play a part. Mallory surveyed 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 or recently passed cold front [6]. 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 (200 m) 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’ [6]. However, Mallory did not identify the underlying physics, and this is still an open question in the modern literature. There does not appear to be any further refinement of his association of waves with weather systems and bathymetry.

In summary, multiple mechanisms have been proposed for deep-water rogue wave formation. These are, in order of the most established mechanisms first:

• Wind strength and fetch build wave height (and wavelength, but not steepness).
• Wave height is increased by linear summation (first order) of random waves or crossing seas.
• Winds blowing contrary to water currents dramatically increase wave height (wave–current coupling).
• Adjacent waves can transfer energy between themselves (second-order wave theory).
• Wave trains can create a temporary localised standing wave (breathers and solitons).

3. Materials and Methods

3.1. Research Objective

The objective was to explain rogue wave formation in a way that integrates wave formation, sea currents, bathymetric features, and meteorological weather systems, for the Agulhas current. 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 [6]. This area was selected because of the quality of the available data. This limits the overly complex phenomenon to a sub-category that is more tractable. In contrast, the open ocean rogue waves, e.g., in the North Atlantic (see examples above) in deep water, may have a different causation.

3.2. Approach

The approach comprised several phases: an initial exploration of available data; the drawing of inferences as to possible underlying mechanisms for rogue wave formation; and the elaboration of a theory for the causal effects of continental shelf bathymetry and cold front meteorology.

3.2.1. Spatial Analysis Approach

The first phase explored for patterns in known data about ships that had encountered rogue waves in the region under examination. A spatial analysis was applied using geographic information systems (GIS) with software QGIS (version 3.38.3-Grenoble). The geospatial analysis was used as a preliminary conceptual screening of variables that might be associated with rogue wave prevalence. The data were sparse. There is a list of rogue waves [41] but it does not identify any cases off the South African coast, neither does it provide precise locations. The only data available were from Mallory [6]. Bathymetric data were obtained from NOAA (USA) [42] in the form of the digital elevation model “DEM global mosaic”. Contours were extracted using QGIS, and colour mapped. The associated hillshades were also obtained to provide a visualisation of the terrain. Open street maps (OSM) were used for the place names. Velocity of the Agulhas current was obtained from [43] (EU) in the form of the dataset “MULTIOBS_GLO_PHY_MYNRT_015_003” mean over year 2022.

The results of the first phase indicated that rogue waves were indeed associated with a complex combination of (i) the Agulhas water current, (ii) bathymetric terrain features, and (iii) meteorological air fronts, as Mallory suggested.

3.2.2. Vortex Conceptualisation

The next phase sought a common mechanism that could account for the interaction of these three different domains. The magnitude of fluid velocity is an obvious candidate and is well represented in the literature. However, considerations of fluid velocity alone have been unsuccessful in explaining rogue waves. The results of the present analysis confirmed that current velocity had some, but imperfect, relationship with rogue waves. This was evident in the rogue wave encounters not being closely correlated to where the Agulhas current was strongest. Instead, the data showed a better fit to bathymetric terrain.

This gave rise to the idea that turbulent flow, i.e., vortices, may be involved. However, there is no literature on this idea, so a theory had to be constructed from scratch. Also, turbulence is an unsolved problem in fluid mechanics, ruling out a theoretical approach at this scale.

Having identified the various types of vortices that exist in the three compartments, the remaining task was to develop a conceptual model of how the 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 motivated on the grounds that 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 development of a simulation could be future work once the conceptual basis is established.

4. Results

The results draw from meteorological, oceanographic, and bathymetric literatures, and a brief explanation is necessary of the term “front”, which has variable meanings in the literature. In the meteorological field, it refers to the boundary between hot and cold air masses (e.g., in weather systems), and in the oceanographic field it is water masses (e.g., up and down welling at continental shelves). In what follows, an attempt is made to be specific about this term.

4.1. Analysis of Mallory Data

The objective of this part was to explore trends in location of rogue waves. The Mallory data [6], summarised in

Table 1. Summary of Mallory data.

Ship	Latitude	Longitude	Wind Direction	Wind Strength [Beaufort Scale]	Time After Front Was at Wave Location [h]	Month
Gaastekerk	−31.783333	29.53333	SSW	4	0	4
Oranjefontein	−32.3	29.03333	SSW	4	11	4
Jagersfontein	−31.75	29.48333	SW	7	11	12
Edinburgh Castle	−31.65	29.76667	SW	6	8	8
World Glory	−29.633333	32.25	SW	9	25	6
Esso Lancashire	−29.333333	32	SW	8	0	8
Clan Maclay	−30.583333	30.73333	SW	3	22	10
Southern Cross	−32.033333	29.28333	SW	8	5	10
Moreton Bay	−33.6	27.5	WSW	11	0	8
Bencruachan	−31.166667	31.16667	SW	7	38	5
Svealand	−33	28.35	SW	10	21	9

, were imported into QGIS. For each ship, the known coordinates allowed the location of its meeting a rogue wave to be represented. The size of the rogue wave was unavailable, but other data were available: the wind strength and direction, the time after a meteorological front had passed the location of the wave, and the date (time of year). Note wind directions: north (N), north-east (NE), east (E), south-east (SE), south (S), south of south-west (SSW), south-west (SW), west of south-west (WSW), west (W), and north-west (NW).

The wind strength and the time after a cold front were represented in the point for each ship: diameter for wind velocity, and colour for time. The location of these encounters relative to the bathymetry is shown in Figure 1. This confirms the observation of Mallory that the 200 m isobar is a particularly hazardous location. There is also an association with the presence of submarine canyons in the vicinity of the 200 m isobar. In all cases the rogue wave was encountered at or after the cold front, not before. For the two ships in deep water, the encounter occurred long after the front. It is inferred that there is an interaction of the weather front with the continental shelf in some, as yet to be determined, manner.

The locations are also shown relative to the surface water current from [43], replotted to Figure 2. This was represented for (a) absolute speed, (b) eastward velocity components (uo), and (c) northward velocity component (vo). The results show that rogue waves are better correlated with the north/south component of current velocity (Figure 2c), not the absolute velocity, or the east/west component. All the data points fit this observation (N = 11).

This is unexpected because the current travels towards the SW, directly against the prevailing SW wind [6] and the NE direction of propagation of wave power [22]. Hence, by the current-amplification theory, the waves ought to be most prevalent where the SW flowing current was strongest. This finding might be due to current data being from a different year to the shipping data, but this seems unlikely as the Agulhas current has long been known to have a SW flow tendency. Instead, our interpretation is that the southerly current represents the component of current moving off the continental shelf, i.e., there is some hidden interaction between the current and the continental shelf that contributes to rogue waves.

Building on the literature review above, a plausible mechanism for rogue wave formation in the Agulhas current, sans bathymetric or meteorological effects, would be as follows:

• Wind strength and fetch build wave height, and wavelength (but not steepness). During winter, there are waves coming up from deep in the Southern ocean. The winds have a long fetch of about 800–1200 km, hence fully developed waves are created (typical height 6 m, wavelength 120 m), travelling 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 6.5 m reaching 7.0 to 8.0 m in the south, SW to W swell’ [44]. Note the wind strength of up to 40 knots, and the wave height of up to 8 m. However, those waves would be long swells, not steep sided. These waves propagate up towards Gqeberha (Port Elizabeth).
• Wave height is increased by linear summation (first order) of random waves or crossing seas. Meteorological conditions in South Africa are affected by the terrain which consists of an elevated plateau of 1400 m 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 [45] and localised strong NE or NW winds (up to 30 knots) with clear skies [46]. These preceding winds create additional local waves of high frequency [6]. These NW winds and waves converge onto SW winds and waves at the head of the front to create a crossing sea with wave superposition. There is also refraction occurring, whereby SW waves (of any origin) that meet the Agulhas 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).
• Winds blowing contrary to water currents dramatically increase wave height (wave–current coupling). Waves from the SW wave are magnified on meeting the Agulhas current [6,47]. This has well-established physics. These SW winds and their waves are opposite in direction to the Agulhas current, and strongest where the current is most concentrated. The intersection causes the waves to rise in height, typically by 20–40% and up to 60% in extreme situations [47]. When meeting the current, the waves become shorter in wavelength, and steeper. Empirical observation are that 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 [6]. This interaction on its own is sufficient to generate large waves of about 10 m, and the South Africa Weather Services forecasts this interaction [44] based on a model developed by [47]. Even so, there is much that is not known about this interaction at smaller scales, including specifically in the Agulhas current [48]. Waves also interact frictionally with the sea bottom at sufficiently shallow depths. Theoretical modelling shows this, coupled with the opposing current, ought to cause an increase in wave height and induced current on the outside of the continental sedge [49], though the size of the effects is difficult to predict.
• In addition, two other theoretical mechanisms may be at work, though there is no empirical evidence to confirm or deny their operation in the Agulhas: adjacent waves may transfer energy between themselves (second-order wave theory); and wave trains may contain temporary localised standing waves (breathers and solitons).

However, that still leaves open the question of how bathymetric and meteorological frontal features, which are observationally known factors in the Agulhas region, affect wave formation.

In the Mallory data there appear to be two sub-populations of rogue waves: (1) deep ocean; and (2) continental edge.

The deep ocean rogue occurred far from the continental shelf (N = 2: World Glory; Bencruachan). Both cases are associated with high wind strengths. For these cases, the rogue wave was encountered a considerable period of time (24 h or longer) after the meteorological cold front. Such fronts are often followed by a strong and spatially deep wind field, hence the potential for a long fetch. Bathymetric features do exist at depth and some horizontal distance away in both cases, but they are not as acute as on the continental edge. Hypothetically, this type of rogue wave may have similar causality to those in other oceans such as the mid-Atlantic ocean.

Continental edge rogue waves formed within 5 km of the 200 m depth contour (N = 9). All these waves occurred within the high southerly velocity region of the Agulhas current. High wind strength was not a primary factor as there are several cases where the winds were only moderate. Proximity to bathymetric irregularities, especially canyons, appears to be a risk factor (N = 8). The canyon proximity has a range of 20 km, which is rather larger than the canyon arm widths (length 2 km to 5 km by inspection from GIS model), though less than the canyon lengths (about 60 km). The mechanisms whereby canyons may affect rogue wave generation are not provided in [6] nor elsewhere in the literature. These rogue waves also appear to have a weather dependency: they occur at the location of a meteorological cold front or within 12 h afterwards (N = 7). No cases are reported pre-frontal.

The attention of this research focusses on the Continental edge rogue waves. They have a complex causality, being dependent on (i) proximity to the edge of the continental shelf, (ii) proximity to submarine canyons, (iii) passage of a weather cold front, (iv) wind velocity, and (v) southward water velocity. The question, then, is what type of hidden variables could link these multiple factors? Ideally, the number of hidden variables should be small (parsimonious), and the physics of their mechanisms should be plausible. It is here proposed that vortex generation could be the latent variable, as it can link all these variables.

4.2. The Vortex Theory of Rogue Wave Formation

It was unexpected to find that rogue waves were associated with the southern vector component of the Agulhas flow, not the flow magnitude (as would be expected for the wave amplification mechanism). This, plus the association of the waves with the continental margin, suggests the possibility of a causal contribution by vortices. There are numerous vortices in the Agulhas case which appear not to have been previously identified as such. These are in different media, at different physical scales and frequencies. The chaotic combination of these vortices, on top of the preceding phenomena, is proposed as the mechanism for converting large waves into rogue waves.

There are three compartments that need consideration: air movements in the form of meteorological cold fronts and the interactions between cold and warm air masses; water movement with the Agulhas current interacting with bathymetric features; and the wind–water interaction where the waves form on the interface.

A considerable vortex literature was discovered, unconnected to rogue waves, but nonetheless interesting in the way it identified how and where vortices formed. The literature included field studies, experimental studies (e.g., wave tanks), finite element analyses, and explicit analyses of ideal geometry. 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.

The data on turbulence in meteorological cold fronts was found to be extremely limited, with one exception that provided quantitative data on the internal structure of a cold front and vortex spin rates [50]. For the ocean compartment, the most relevant literature was found to be the modelling of water flow along and across continental shelves. This was mostly from the perspective of ecological consequences for fisheries and benthic species. There was also literature in the fluid mechanics area, typically using numerical methods to simulate flow over simple geometry.

4.2.1. Motivation

Additional justification for taking a vortex perspective is that the fluid dynamics of real seas, with their winds and waves, is fundamentally turbulent. While there is no explicit mechanics available to describe turbulence, it is nonetheless instructive to consider waves as comprising 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 in the boundary layer [19], whereby the air flow locally reverses downwind of a wave. Both Phillips [18] and Miles [19] 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 alternatively be represented by vortices. However, these theories ignore the possibility of correlated motions between the air and water compartments, which is central to the current work. 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 [51]. This causes vortices in counter-rotating pairs, and these vortices cause further rotations of the interface [52]. This interpretation is usually applied to fluids of the same phase, and has only rarely been applied to ocean waves [51]. 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 [53]. Water vortices of the same spin have also been observed merging [53]. Hence the experimental evidence supports the idea of a turbulent coupling between air and water.

4.2.2. Mechanisms Whereby Vortices Are Involved in Wave Formation

Vortices naturally arise in a fluid from three mechanisms:

(a) Fluid shear on a boundary layer. This 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

$$\overrightarrow{v}={\displaystyle \frac{\Gamma }{2\pi r}}$$

(1)

where $\mathsf{\Gamma }$ is the circulation about the vortex axis, with

$$\mathsf{\Gamma }=\mathit{\oint }v· dl$$

(2)

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

$$\overrightarrow{\omega }=\nabla \times \overrightarrow{v}$$

(3)

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.

(b) Changes in fluid angular momentum. Vortices form in moving fluid that changes its direction of flow due to obstruction from another fluid, solid body, or the shape of its duct. Hence the other extreme of the idealised vortex is a rotational vortex (or forced vortex, or rigid body rotation) where the angular velocity $\mathsf{\Omega }$ is constant such that tangential velocity is proportional to radius,

$$v=\mathsf{\Omega }r$$

(4)

and in which case vorticity is

$$\overrightarrow{\omega }=2\mathsf{\Omega }$$

(5)

These vortices have a pressure applied at their outer boundary.

(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.

Real vortices combine elements of the 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.

4.2.3. Established and Proposed Vortex Principles

There are a number of fluid mechanics principles that apply to vortices and their effects on waves. The following principles are uncontroversial:

A.

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.

B.

Vortices, either in the air or the water, cause waves when they act on the water–air interface.

C.

Same spin (para) vortices can combine. Adjacent vortices with the same spin combine to produce a stronger vortex. This is an aggregation mechanism for vortex energy.

We propose that additional principles apply to vortices and waves. These principles were inferred from the experimental literature and are somewhat tentative because of the sparseness of that literature and the lack of explicit attestation for these principles.

D.

Premise: Liquid vortices preferentially transfer energy to wave formation. The effect of vortices on an air–water interface is complex and poorly understood [54]. 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 [55]. This principle provides a mechanism for additional amplification of waves by water vortices of extrinsic origin to the wave.

E.

Premise: Contrary (ortho) rotation water vortices cause jetting and scavenging. We propose that contrary rotation water vortices can, if converged (impacted together appropriately) cause vertical jets that will accentuate wave height. The direction of this jet depends on the relative positions of the vortices. Hence, the jet may be at a skew angle (not vertical), and its evolution may scavenge water from elsewhere to create a trough alongside. This formation of jets for converged counter-rotation vortices pairs is experimentally observed [56]. It is also consistent with the observation (in slurries) that for contrary vortices the “impact energy is conveyed through the tangential component” [57]. In addition, analytical studies on an idealised continental 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 [58].

We propose that the above vortex principles are the conceptual mechanism for acute wave sharpening and trough widening, especially the combination of para-spin vortices (C), water vortices transferring energy to waves (D), and jetting from ortho-spin water vortices (E). The next sections describe how these vortices may arise, from the interaction between air, water, and bathymetric terrain. The morphology of the vortices is used 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, see Figure 3.

4.2.4. Wind Lee Vortices

We assume that a set of waves have already been generated by winds some distance away (e.g., Marion Island) by Phillips [18] or Miles [19] 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). 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 [59]. A high degree of microscale turbulence arises in this air boundary layer, theorised by [18] and evident experimentally [53]. 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, via the fluid shear gradient.

At some point, the size and topology of the wave is such that air flow separation occurs, and a lee vortex λ1 $⨂$ appears downwind of the wave crest, see in Figure 4. This drives surface water (v_L1) towards the wave crest, 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 [60], 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 wind increases. This thickens the boundary layer and creates 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.

4.2.5. Air Vortices Arising from Meteorological Cold Front

A cold front contains many vortices [50]. We propose that these interact with the air vortices shaping the waves, see Figure 5. Ahead of the cold front is a warm air uplift vortex (U$⨂$), in response to the cold front pushing forward. There is considerable magnitude of vorticity in this region [50]. The axis of vorticity would be the same as the lee λ$⨂$ vortex, 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 at the front. 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 [50]. This has spin like the air windward $\mathsf{\omega }⨀$ 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 [50].

Close to the terrain (which in this case is the sea surface) and about 100 m up into the atmosphere would be a series of boundary layer vortices B$⨂$ [50]. 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 air jets between them (v_HB), resulting in wind gusts impacted down onto the terrain. These gusts may have been observed [61]. 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) [9], 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. This is because the winds of the cold meteorological fronts are able to reverse the surface Agulhas current [62]. Consequently, there are likely to be new waves generated at the front, which will add energy to other waves.

The front itself would raise the sea level ahead of it [17], as a result of 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, narrower elsewhere. Storm surges regularly raise the level of coastal waters in this way. Data for the Beaufort sea show that surges of 1 m are common (mean 0.4 m, maximum 2 m) with regression analysis identifying the magnitude correlated primarily with wind speed and direction, and then the percentage of open water (the location has ice which does not apply in the Agulhas case), with air pressure not being a significant variable [63]. 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 3 m [64], but this was based on driftwood on beaches so probably included wave runup.

So, a sea level rise of about 2 m 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, causing localised escaping surges back to deeper ocean. Furthermore, it is proposed that 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 the formation of depressions in the water surface, which accentuate wave height.

4.2.6. Water Vortices Arising from the Agulhas Current

The Agulhas current flows against the SW wind, and hence wave amplification occurs [20]. 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 [24,62], a train of substantial current boundary layer vortices C$⨂$ will arise, carried along in the current, see Figure 6. These C$⨂$ vortices 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 it probably accelerates the wave more than heightening it. The C$⨂$ and L$⨀$ vortices have opposite spin and therefore 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 enhance the preceding trough. This is an observed feature of rogue waves, as is the high forward speed.

Satellite data using synthetic aperture radar (SAR) has identified that winds increase in velocity when encountering the Agulhas current [61]. These increases exist irrespective of the wind direction relative to the Agulhas current but are greatest when blowing contrary to the current. The effect appears to be a thermal wind driven by differences in the sea surface temperature (SST), itself caused by the warm Agulhas current. The magnitude of the thermal wind was maximally about 1 m/s (which augments or opposes the background wind) and was located 10 km offshore of the SST thermal front. The SAR data showed that windspeeds over the centre of the Agulhas current were up to 10 m/s faster than at the coast. They also included Jason-2 data which showed that the significant wave height increased abruptly at the SST front. Their analysis was conducted on two transects bracketing the Algoa Bay region. They also expressed an opinion that the observed wind speeds were unrealistic because they did not fit existing models. They instead proposed that the increase in ocean roughness caused overestimation of wind velocity by wave motion being aligned with the radar emission, and that the actual increase in wind speed was only up to 2 m/s. They did not identify the location of the SST front, but inspection of their maps suggests it was approximately over the −1000 m contour. The resolution of their wind data was 1 km to 5 km, hence they would only be able to see macroscopic features of the wind field, and not vortices at the air–water interaction at wave scale. Hence, it is conceivable that the wind spikes they rejected as artefacts might instead be detection of finer scale turbulence. This was not a possibility they considered.

4.2.7. Water Vortices Arising from Submarine Canyons on the Continental Slope

Then, there is bathymetric topology to consider. Several ideas emerge from the modern literature. A current will divert a filament into such a feature [65], 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 [66]. Other work has shown by numerical methods that a current passing over an idealised two dimensional square trench in shallow liquid will generate upstream-propagating solitary waves [60]. However, the literature is highly idealised and for simplistic geometry, so there is a large gap in understanding of how vortices form at real bathymetric features. Nonetheless, 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 counter-current, and coastal upwelling and downwelling (see below).

In the case of the Agulhas current, the topology vortex T will have $⨀$ spin, see Figure 7. 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.

4.2.8. Water Vortices Arising from Upwelling Water Movement Between the Continental Shelf and the Deeper Ocean

A key idea in this theory is that the bathymetric canyon vortices T$⨀$ are prised out their canyons, towards the surface, where they strengthen the wave lee vortices L$⨀$. The proposed mechanism is water flow off the continental shelf. This is termed downwelling, for its effect at the coast, and there is a strong body of literature about down- and upwelling, but this is focussed on transfer of nutrients, fisheries, and coastal ecological systems, rather than wave formation.

4.2.9. Up- and Downwelling

Coastal upwelling of cold water is a regular feature of continental shelves like 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 continental slope is the interface between the coastal and pelagic bodies of water, and the bathymetry imposes physical constraints on the flow. The upwelling and downwelling occurs via vortices at the water front (between the cold and warm water), of about 4 km wide and 40 m deep, with strong currents of about 0.5 m/s [67] (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 [68] and is strongest over the upstream edge of the continental shelf rather than the deeper water [69].

Similar effects exist for the Agulhas Bank and Algoa Bay shelves. The upwelling results in changes in sea level > 0.5 m, and is associated with trapping of coastal waves [70]. Upwelling also occurs on the shallow Agulhas Bank due to wind forcing [71].

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 30 km in diameter, and when these arrive at Algoa Bay the leading edge thereof initiates upwelling near Bird Island, and strong SW currents of 0.8 m/s [72]. 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 [72]. For 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 [62]. Nonetheless there is a front between warm and cold waters at about 60 m depth in winter [62].

4.2.10. Effect of Canyons on Up- and Downwelling

Upwelling flows are dramatically changed by the presence of an underwater canyon on the continental slope. Finite element methods on an idealised canyon with wind-forced water flowing along the continental margin show greatly intensified upwelling at the head of the canyon, followed downstream (in the wind direction) by downwelling [65]. Each of these features was about 10 km in in its shortest dimension. Canyons that cut down to deeper levels show enhanced upwelling [73]. Upwelling flows in the Wilmington canyon (recorded with a submarine glider) originating at the 150–215 m depth in the canyon, were 10–30 m thick and 5–20 km wide, and flowed at a maximum of 0.5 m/s [74].

The dynamics of real upwelling flows show high spatial variability [68] (Heceta Bank), and existing models of upwelling are necessarily simplistic [72] (Algoa Bay). There are likely to be peculiarities and progressively finer level flows, as is typical for turbulence.

4.2.11. Inferences for Rogue Wave Formation

The canyon upwelling literature 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. Nonetheless, the connection may be made with reasonable confidence. For the South African east coast, a weather cold front is preceded by W, NW, and NE winds [45] which cause upwelling of cold water at the coast 19–60 h later [70]. The passage of the weather front reverses the flow to coastal downwelling, hence warmer waters inshore.

Upwelling on this continental shelf may be represented as a large-scale vortex with horizontal spin axis pointing to the NE, downwelling to the SW, see Figure 8. The passage of a weather front causes reversal of the vortices. The literature does not describe what effects this would have on the sea state above a canyon. Nonetheless, there is strong evidence analytically and experimentally that canyons accentuate high flow and generate vortices in the wind-forced upwelling situation, and that the most intense flows occur at the edge of the shelf, at 100–200 m depth. The literature is less specific about downwelling, but 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, with significant momentum involved.

Note that in all cases the rogue wave was encountered at or after the cold front, not before. Therefore, we propose that passage of the weather front causes reversal from upwelling to downwelling (this is not controversial), hence complex fluid flows comprising vortices and filaments. This results in water vortices at multiple length scales. These vortices occasionally and chaotically couple with the existing waves generated by the mechanisms identified above, either by vortex combination to amplify existing waves, or vortex opposition to produce jets. These interactions occur at the edge of the continental shelf, and preferentially at the heads of canyons.

We propose that the downwelling flow at the passage of a weather front, prises the bathymetric vortex T$⨀$ out the canyon, toward the surface. This vortex is forced by the shape of the canyon onto the Agulhas current and will therefore have a large kinetic energy. It is proposed that this vortex, or the breakdown filaments thereof, interacts with the vortices of the surface waves, see the T2$⨀$-L$⨀$ and T2$⨀$-C$⨂$ in Figure 7.

Historical rogue wave encounters appear to be better associated with the southward velocity component of the Agulhas current (see Figure 2c) rather than its SW magnitude or eastern component. As the current is orthogonal to the layout of the canyons, there is nominally no along-canyon flow of the Agulhas current. However, the canyon is inclined, and this suggests a possible explanation: that at the 200 m isobar, the rope-like canyon vortex is longer in length towards the deeper waters than it is towards the head of the canyon (being limited there by the water surface). Hence, to the extent that the vortex is irrotational, it will have a lower central pressure and the longer arm of the vortex would have greater capacity to move water—the down-canyon flow predominates and strengthens the downwelling. A complementary explanation would be that the westward component of the Agulhas velocity is obstructed by the continental rise and shallowness of the shelf (smaller cross-sectional area or aperture), whereas the southward component towards deep waters is less impeded, and hence strands of the Agulhas current are diverted down-canyon. These explanations are tentative, and do not fully accommodate the fact that the data in Figure 2c are for surface rather than deep currents.

4.2.12. Water Vortices from Movement of the Agulhas Current Along the Continental Slope

Any strong current that impacts steep underwater topography will generate vortices [75]. 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). These are represented in Figure 7 as slope S$⨀$ vortices, though that figure only shows the component in the vertical plane. There is also a component Sh$⨂$ in the horizontal plane viewed from above. Analytical modelling of idealised submarine ridges identifies the vortices to be about 100 m–10 km in size [76].

Empirical measurements show that off Algoa Bay there is a strong difference in velocities horizontally (magnitude and direction) between the Agulhas current and the coastal counter current, forming a vertical interface above the 120 m depth contour [62] (see Figure 23 therein). We expect this boundary layer will create vertical columnar vortices reaching the surface. In the horizontal plane, the Agulhas current vortices Ah$⨂$ and counter-current ACh$⨂$ vortices have axes pointing downwards. Such vortices have the capability to generate waves in multiple locations, as they are carried along in the current. These vortices would primarily affect the lateral edges of the Agulhas current, which is typically in the 100 m to 200 m isobar region, creating a Kelvin–Helmholtz instability at the interface between the currents. In the horizontal plane, the slope vortices Sh$⨂$ and Agulhas current vortices Ah$⨂$ can be expected to combine and drive lateral movements in surface water.

Waves from the SW are also refracted into this region, creating crossing seas. In addition, there will be interactions with the canyon T$⨀$ and slope S$⨀$ vortices (vertical plane), hence chaotic interactions will arise. These interactions are difficult to predict, but it seems reasonably certain that significant chaotic energy is available in the region of the continental edge (200 m). This is supported by evidence from ships logs, showing much heavier sea conditions exactly here [6]. Therefore, there are opportunities for irregular waves in this location.

5. Discussion

5.1. A Proposed Mechanism for Rogue Wave Formation

In summary, it is proposed that rogue waves off the South African east coast arise from the combination of multiple mechanisms at the macroscopic and mesoscopic scales.

Long fetch, fully developed waves arise. The underlying swells originate in the Southern Ocean and are initially wind forced. The waves travel up the coast and are amplified on meeting the contrary Agulhas current. This mechanism is well established [6] but on its own does not generate particularly steep waves.

Crossing seas arise, which add waves laterally, and contribute to superposition. These crossing seas arise from two mechanisms: prefrontal winds blowing from the NE; and waves from the SW being refracted at the landward margin of the Agulhas current, which also corresponds to the edge of the continental shelf. An additional source of chaotic wave energy over the continental edge is the combination, in the horizontal plane, of the slope vortices Sh$⨂$ and Agulhas current vortices Ah$⨂$.

Wave sharpening and heightening occurs by wind lee vortices. 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. Thus, the SW winds of a cold weather front create a wind lee vortex λ$⨂$, itself strengthened by air jetting from the weather H/H2$⨀$ and B$⨂$ vortices.

The wind vortex λ frictionally induces a water lee vortex L$⨀$ which heightens and steepens the wave. The water lee vortex L$⨀$ is further strengthened by occasional meeting of a continental slope S$⨀$ cylindrical vortex.

Large scale water vortices arise at the heads of canyons due to upwelling–downwelling and the reversals thereof due to movement of the meteorological front. The canyons on the continental slope create, via negative forcing function on the Agulhas current, a topology rope vortex T$⨀$. Downwelling water movements, caused by the weather front and augmented by distraction of some of the Agulhas current down-canyon, then prise vortex T$⨀$ out of the canyon, toward the surface. The canyon vortex T$⨀$ then strengthens the water lee vortex L$⨀$ and causes heightening and steepening of the face, and scavenging of the trough ahead. Considerations of vortex spin suggests there should also be occasional upward jetting between coincident canyon T$⨀$ and water boundary layer vortices C$⨂$. The canyon vortex T$⨀$ is fixed to the topology, but the wind λ$⨂$ and water L$⨀$ lee vortices move forward with the waves (towards the NE), and the slope S$⨀$ vortex moves with the Agulhas current (towards the SW).

The vortices are at very different length scales. The lee vortices (wind and wave) are of similar size to the wave (20 m), the Agulhas boundary layer vortices perhaps the same, the upwelling–downwelling vortices are geometrically constrained to the depth of the continental shelf (200 m), and the canyon vortices limited to the cross section of the canyon (2 km). The larger vortices can be expected to form a smaller one, per the Kolmogorov breakdown route. Hence, it is feasible to conceive of large vortex structures such as the canyon vortex, developing smaller vortices at the 20 m scale that can interact with the lee vortex in the wave. The result is chaotic interactions, within which wave superposition arises. There is complex timing for all these vortices to interact, which is consistent with the rogue wave presenting as an occasional rather than regular process.

Hence, conceptually, a vortex mechanism has been presented 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.

5.1.1. Why Is the Edge of the Continental Shelf So Dangerous?

Mallory identified the 100-fathom (200 m) isobar as the risk area for rogue waves [6]. An explanation is now available. First, there are a variety of additional vortices generated in this region that are not present in coastal or deep waters. These include the prised canyon vortex T$⨀$, Agulhas current edge vortex Ah$⨂$, and downwelling vortex, in addition to the air and water lee vortices that exist irrespective of ocean depth. The continental shelf break experiences reversal of the coastal upwelling–downwelling currents, which are especially strong near heads of canyons on the continental downslope.

The vertical section at the 200 m 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 an 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.

5.1.2. Comparison with the Literature

In comparison, the literature does not show vortex theory being applied to explain rogue waves. Possibly, this maybe because existing approaches are mathematical, whereas there is no explicit fluid mechanics for vortices. First and second order wave theory are more amenable to a mathematical approach. Even so, there may be correspondences. The second order theory attempts to represent energy transfers between laterally adjacent waves [26]. Conceptually, this has some similarities to the present conjecture of interaction between vortices. However, the present theory proposes that the really important interaction is between the surface and sub-surface water vortices. This vertical interaction appears not to be represented in second order theory. Nor is the weather interaction addressed in the extant literature on rogue waves.

5.2. Limitations and Implications

The limitation of this explanation is that it is speculative. The proposed theory is based on conceptual reasoning. 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. Potential methods for validation include explicit mathematical modelling, numerical simulation and finite element methods, or experimental validation, e.g., via wave tank. However, each of these are large endeavours in their own right. The fluid mechanics is complex because of the multiple vortices and the interactions at the water–air interface. Explicit mathematical solutions do not exist and are unlikely to exist until they are able to represent turbulence; testing the theory by simulation is not straightforward (more on this below), and experimental methods would require a complex water flume that included wind and topology.

Realistically, computational fluid dynamics (CFD) appears to be the 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 20 m high, hence would need elements approximately 0.5 m 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 elsewhere in the world, where frontal weather interacts with ocean currents, and those currents interact with bathymetric features.

Rogue waves in deep ocean (Type 1) are not explained here, and it seems unlikely that the complex set of vortices proposed to exist near the continental shelf would be relevant. What is needed is better data on the location of other deep-water encounters, such as the North Atlantic ocean. Such data do not appear to exist. As the present study shows, there is value in knowing the location of the wave (hence bathymetric topology can be inferred), wind speed and direction, and weather conditions.

Implications for Navigation

Tentative implications for mariners in this region, building on Mallory’s work [6], 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; onset of a SW cold front and up to 24 h after; high winds approximately Beaufort 9 or stronger; severe wind gusts; meeting the cold front near a submarine canyon; and a strong Agulhas current (approximately 3 knots). For mariners already in risky circumstances, possible remedies may be changing course inshore or offshore to avoid the continental shelf break (200 m depth contour) or reducing forward speed.

6. Conclusions

This work suggests that the missing ingredient to explain Agulhas rogue waves is vorticity. 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 superposition of multiple wave trains, the second stage being amplitude magnification of waves by the opposing Agulhas water current. However, such waves are not necessarily rogue [44,47,48] 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. Wind lee vortices cause wave 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. Air boundary layer vortices from the meteorological cold front transfer energy to the wind lee vortices thereby enhancing their wave-sharpening effect. Agulhas current boundary layer vortices (at the water–air surface) interact with water lee vortices to accelerate a jet of water between them, thereby steepening the wave forward by removing water from the foot of the wave, and also enhancing the preceding trough. 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. 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.

The unpredictability of the rogue wave is attributed to the complex timing of the vortices. The canyon vortex is fixed to the topology, but the wind and water lee vortices move forward with the waves (towards the NE), and the slope vortex moves with the Agulhas current (towards the SW). Hence, the interaction of these vortices is irregular.

The vortex theory is able to qualitatively explain the main features of rogue waves: their height, steepness, non-breaking crest, asymmetrically steep front face, and preceding trough. It also explains why the continental shelf (200 m isobar) and canyons are prone to experiencing rogue waves in the Agulhas region.

Hence, a conceptual explanation is provided for rogue wave formation that integrates wave formation, Agulhas sea currents, bathymetric features including continental edge and submarine canyons, and meteorological cold front weather systems.