Significance to Climate Change and Forcing Abstract Ther

An overview of the swell dynamics and their implications over Africa: significance to climate change and forcing

Abstract There is a growing interest in climate dynamics as the quantity and quality of new observational and theoretical applications are increasing. The ideas involved in understanding large-scale atmosphere-land-ocean dynamics and their interactions continue to hold special fascination because of their central importance for both theoretical and practical applications. This paper presents a theoretical assessment of the African swell dynamics imploring atmospheric formulation. Africa exhibits substantial inter-annual and inter-decadal climatic variability due to cyclone activity, storm surges and sea waves. Most of these surges and corresponding swell trains form over the tropical environment as easterly waves propagate westward across the Indian Ocean primarily between 10° and 20° S, termed source region. Localized sea surface temperatures (SSTs) and ocean upwelling play a vital role to provide moist enthalpy to power the surges. Also, multi-decadal variations in major wave activity are associated with SST changes in the Atlantic because tropical North Atlantic correlates positively with major hurricane activity. A key remote factor is temperature variability in the central and eastern equatorial Pacific associated with El Niño Southern Oscillation. The continuous erosion, perennial ocean surges, coastal swells and associated flooding due to the wave energy and its pounding effect are of great concern. Just like in most parts of the world where development of these systems is critical, they need to be closely watched particularly over southern Africa. Key Words: climate dynamics; swell train; source region; wave energy; ocean surge.

1. Introduction temperatures (SSTs), sea level rise and Africa experiences a wide variety of ocean upwelling. It is clear that any climate regimes and is one continent substantial changes in numbers or strongly affected by the subtropical fatalities of these events have significant anticyclonic belts and has extensive consequences in the active and/or areas of tropical-subtropical climates. vulnerable regions during periods of Changes in climate dynamics and wind disturbance, storm surge, cyclone variability are likely to exacerbate the activity and/or damaging floods. For existing problems due to episodic wave example, southern east Africa –including events that occur with little advanced the island of Madagascar appears to be warning [1]. These impacts are likely to influenced by SST changes over much affect many different aspects of the larger regions [3], and references world’s environments. Year-to-year therein]. The region is frequently variations in tropical cyclone (TC) affected by cyclonic and other numbers are great, and in many regions significant weather events [4] that strongly linked to El Niño Southern unleash large wave events over land [1] Oscillation [2]. Also, regions of wave between December and March. Majority activity may increase poleward as a of these TCs track south-westerly away result of increasing sea surface from the mainland and back towards the Indian Ocean. According to Mather and 1

Stretch [1], these cyclones sometimes climate regime spanning to seasons, remain semi-stationary south of decades or centuries. Understanding the Madagascar that cause the biggest swells root causes of these changes and the in the region. The southern part of perceived value of seasonal prediction is Madagascar also appears to be well- vital, given the overwhelming positioned to receive strong south- importance of the timescales of climate westerly swells from these systems as information and adaptive responses [9]. they move over the ocean [5]. Although advances to understand these The northwest coast often experience complex mechanisms and how they may mid-latitude depressions that track influence future climate variability are across the North Atlantic [6] –the region made, this is a critical area requiring with a wave activity subject to a high further research [10]. For example, seasonal variation and enhances during projecting future climate evolution needs the northern-hemisphere winter. some kind of mathematical- or physical Similarly, West-Africa receive north- approach to understand and/or model. As westerly swells from the North Atlantic such, climate scientists try to tackle during the northern winter and the these issues by understanding the link occasionally long-range swells from the between various factors that relate air South Pacific during the southern winter. movement and clouds, heat and On the other hand, east Africa is often temperature, rainfall and cloud affected by locally-generated wind seas, formation, pressure changes and wind together with the south (south-easterly) field. One application is the use of swells from mid-latitude depressions that existing mathematical laws (the so- would have otherwise occurred or called atmospheric equations) to propagate over the South West Indian understand the underlying physical Ocean (SWIO). One significant problems. Fundamental laws of fluid difference between the South Indian dynamics and thermodynamics Ocean and the South Atlantic Ocean is governing atmospheric behavior are that the former is much larger and allows expressed in terms of partial differential the nature of the Indian Ocean High equations involving the field variables as (IOH) to be more modified [7]. dependent or independent variables in According to the author, the IOH is not a space and time. These are basic laws of single steady synoptic entity, and tends motion derived from the same equations to slowly move eastwards across the governing all classical fluid dynamics. South Indian Ocean and often splits into This paper aims to understand the status two waves, particularly with a strong and evolution of the swells and their depression poleward and the transition influence on the African climate from one cell to the next that provides variability. We address this on the issues the trickiest navigation dilemma. of geophysical fluid dynamics applying The interplay between the atmosphere, some fundamental atmospheric land and oceans driving these systems is equations. The main objective is to extremely complex. Indeed interactions analyze the swell trains as a possible aid between the waves and currents induce to theoretical and/or operational change in the wave direction [8], and forecasting. The paper also highlights on references therein. Interactions can result interannual and intraseasonal variability in large variations in weather and/or in the wave activity, including links to 2

large-scale phenomena such as El Niño areas of influence by these systems. The Southern Oscillation (ENSO), SSTs and five broad wave environments have been the quasi-binneal oscillation (QBO). recognized as: Also described are some characteristics i) Storm wave environments: occurs demonstrated about African storms and where gales generate short high-energy global wave environments. waves of varying direction. ii) East coast swell: consist of weaker 2. Wave climates and swell generation swell, amplified by local wave 2.1 Major global wave environments generation There are significant patterns of wind iii) West coast swell: occurs on coasts belts and oceans identified globally, that experiencing low long-swell waves of cause the major wave climates. The consistent direction with rare storms largest of which are associated with gale iv) Monsoon influence environments: force winds of the sub-polar and experience a reversal of principal wind temperate latitude regions, characterized wave directions, and by the occurrence of westerly winds and v) Protected sea environments: large frontal activity [11]. According to the water bodies in which wave generation author, these occur in latitudes around is localized rather than swells from the 60S in the southern hemisphere (SH), large ocean basins. whereas in the northern hemisphere As shown in the figure, the east and west (NH) they occur in the “roaring forties”, coasts are often affected by the east and 40–60N. The trade wind belts are west swells, with the southeast also characterized by strong persistent winds subject to the TCs from the SWIO and maintain the swells, although they tracking westwards. Moving further east are rare in the doldrums of equatorial towards south Asia, the region is subject latitudes. Tropical cyclones (also to monsoon influence which sometimes described in different nomenclature in reach Africa via the so-called Great different places) significantly influence Horn of Africa. waves on many tropical coasts, but do not occur within 5N–5S region of the equator. Figure 1 illustrates the major

Fig. 1. Major wave climates [11]

disturbance is one necessary condition 2.2 Wave generation and swell trains for a TC development, a vast majority of over Africa which form over tropical environments Africa is generally characterized by stark [14]. climatic contrasts, significant inter- Almost all swell trains reaching Africa annual and inter-decadal climate are generated from the southern oceans’ variability [12]. An upper anticyclonic storms with less frequency coming from circulation continues aloft in both the northern oceanic areas. Occasionally, hemispheres all year round so that a high storms approach from the east driven by zonal index prevails. The continent is frontal systems over the SWIO (Fig. 2). surrounded by the Atlantic and Indian The figure shows regions of active oceans that play an important role in its swells across the globe, indicating that climate. Generally, the ocean surface is oceanic areas are the most active source either ‘wind sea’ or swell dominated. regions (SRs). Most of these occur over The surface displays a combination of the warm waters of the Indian Ocean and wide gravity waves with varying propagate westwards, where they amplitudes and frequencies. These are sometimes result in devastating impacts swell waves which can propagate out of over land. Adverse conditions are often their generation zones and travel longer expected offshore, especially during the distances without significant attenuation summer season. Short waves comprising [13]. Because they can propagate to local seas and swells are generated distant areas without much disturbance, mainly by passing frontal systems (low they contain a significant nature and pressure systems) from the Atlantic, cut- wind intensity along, making them off low-pressure systems and occasional precursors to various atmospheric TCs over Mozambique Channel [15]. phenomena such as storms, TCs and According to Gray et al. [16], about 87% other large-scale sea breeze systems. A of TCs develop between latitudes 20 preexisting low-level cyclonic north and south, two thirds of which

develop in the NH, and twice as many in the eastern than western hemisphere.

Fig. 2. Global Swell model, showing swell generation areas in the main oceanic areas. (http://www.surftrip.com/surfresources/swellmodels/swellmodels.html).

influence of local wind conditions [13]. Consistent with figure 2, figure 3 also Mid-latitude waves are refracted shows global cyclone tracks, again equatorward, due to large index values indicating the Indian and Pacific Oceans resulting from diminishing westerly as active SRs of cyclone activity. winds [18]. Unlike in the NH where the Normally, the surges tend to move Coriolis force deflects winds to the right, westwards, curving from the equator, winds are deflected to the left in the SH. sometimes comprising of large rotating Coriolis force helps to spin the thunderstorm masses [17]. Quite often, it aggregation of thunderstorms into a has been observed that the swells arrive characteristic spiral circulating winds. from the oceanic region south of 35 S – The basic horizontal balance between a a region defined as the Southern Ocean, TC above the boundary layer is due to and grow significantly under the the sum of the Coriolis ‘acceleration’

and the centripetal acceleration balanced significantly influenced by the swells by the pressure gradient force [19]. which are predominant during southwest Coriolis acceleration is the product of monsoon [13,20]. For example, the horizontal air parcel velocity and the World Meteorological Organization Coriolis parameter f ( , (WMO) report indicate that there was where  is earth angular velocity and  near-average tropical storm activity in is latitude), which becomes zero at the the SWIO during the 2001-2010 decade; equator and 2 at the poles. the most active season was observed in 2009 with a total of 16 tropical storms Recent studies indicate that sea state and 7 cyclones (Fig. 4). along the coastal Indian Ocean is

Fig. 3. Global cyclone tracks (Figure curtsey of: https://geozoneblog.files.wordpress.com/2014/05/globalwrappedclipped.png)

Fig. 4. Statistical storm number over the SWIO in the 2001-2010 decade. (Source: WMO, [2])

Generally, waves travel from the same number and group speed and iv) local direction in groups as a collection of influence of wind vector relative to sinusoids with different periods. The current vector. These can lead to a generation and growth of the waves is decrease in the wave steepness and primarily dependent on wind duration, eventually become long-crested swells speed and fetch [13]. This is an [23]. The principal fate of a wave important concept that determines the depends on the large-scale environment rate at which energy propagates both in which it is embedded, its synoptic spatially and temporally [21]. However, structure, mesoscale- and convective- even in the absence of significant wind scale processes associated with its disturbance, there can still exist waves of evolution [24]. Wind-waves generated varying wavelengths over the water by intense storms become swells as they surface. A swell generation is largely leave their generation zone [25,26] and governed travel very long distances across the ocean basin without much disturbance by the deep-water dispersion [22] given by: [13,27]. Wave components with faster wind speed are considered swells, (1) whereas slower ones are taken as wind- seas [28]. where cg is group velocity, T is wave period and g is gravitational field The wave train evolution can be strength. Nevertheless, even after expressed [6] as: generation, some processes can still (2a) operate in the wave evolution [6], such as: i) refraction of waves related to ratio (2b) of current gradient to wave group velocity; ii) advection of wave action by the current vector; iii) change in wave (2c)

where is experienced the most powerful cyclone the dispersion function and of recent time [31]. is the wave action, k The WMO also reported that the SWIO turned out to be an active region during is wave vector along the trajectory path the 1999–2000 season due to a high and u is surface current vector. As the number of cyclone activity and will be waves move towards the ocean remembered for a long time to have the boundaries, wind speeds diminish and most intense depressions, high winds waves move faster. Eventually they and severe flooding. For example, would reach a point when the peak cyclone Eline-Leon earned accolade as waves have phase speeds larger than the the longest lived and farthest traveled wind and then propagate out of their cyclone on record in the SH [30,32]. generation zones as the wave energy Remnants of Eline traversed the E(k) grow and the peak period increases. subcontinent with torrential rains Tropical cyclone generated waves are causing floods further inland, including more energetic and usually have ephemeral desert rivers [33]. Other catastrophic consequences in many parts typical swells include a series of high of the globe. waves that broke over La Reunion island in May 2007, resulting in extreme 2.3. Cyclonic events and associated weather events off the southeast coast of swell vagaries over Africa the mainland [20]. The waves resulted in Africa is susceptible to extreme weather numerous damages to property and events that sometimes cause damages to many lives lost within the neighbouring infrastructure, livelihoods and islands. Just recently, another ecosystems. These events account for the devastating cyclone Idai developed over large percentage of natural disaster the SWIO between Mozambique deaths [29]. For example, during the Channel and Madagascar in March 2019 1991–2000 period, a number of Africans and caused very catastrophic effect in lost their lives as a consequence of Mozambique, Malawi and Zimbabwe. A severe meteorological and hydrological lot of infrastructure and dams were events almost doubled [30]. The destroyed between March and April, occurrence of these severe weather with an estimated $1 billion of events and their resulting disasters is infrastructure damage and more than gradually increasing and noticeable signs 100,000 homes destroyed according to of the vagaries resulting from global the United Nations report warming and climate changes are (https://www.worldvision.org/disaster- evident. Some of the most devastating relief-news-stories/2019-cyclone-idai- TCs over the Indian Ocean include facts). Eline-Leon (February 2000), Gloria (February-March) and Hudah (March– 3. Convective storms April) 2000 which struck Mozambique, Storms develop over the oceans when a Madagascar and other parts of southern large part of the ocean or sea attains a Africa, causing severe flooding and certain temperature gradient (at least about 700 lives lost in Mozambique and 27oC) from the vertical air motions [34], millions being displaced. Mozambique due to the buoyant or mechanical forces that provide efficient vertical heat

transport, mass and momentum to the heavy rain, and/or damaging tornadoes. rising air. Once this temperature is Initiation requires the presence of reached, vast quantities of water are specific ingredients such as moisture and evaporated and the warm air is carried instability –a source of upward motion upwards by spiral winds that condense and strong vertical wind shear [39]. upwards to form huge clouds and However, a split may occur so that the develop into a convective system. storm develops into 2 forms; one moving Convection initiation is linked to left and the other moving right. Usually mechanisms by which the air is lifted the former dies rapidly while the latter and accelerates upwards, driven slowly evolves into a rotating circulation primarily by buoyancy [35]. The air with a single updraft core and trailing experience atmospheric instabilities as it downdrafts. is heated from below while cooling at 3.1. The right-moving storm the top, or when it becomes saturated in the atmosphere. Latent heat plays an Much of the atmosphere-ocean motion is important role in the storm growth and governed by a set of equations known as energy dissipation. The size of the storm the Navier-Stokes equations. The is important because it determines the equations are used in making weather fetch conditions, together with wind forecasts, computational analysis as well energy and speed. Maximum winds as ocean current studies. Although there constrain the wave field and its main are uncertain details about the equations, characteristics [8]. Depending on the they are for most parts accepted as fact amount of energy released and [40]. When describing a system, we atmospheric instability, the overall express simple relationships between process can lead to thunderstorm relevant variables; and in the present formation. case, express continuity of mass, momentum, potential temperature etc. Convective storms can take a large then eliminate variables to discover variety of forms and scale ranges –from wave and diffusion equations as special isolated thunderstorms involving a single cases. For example, pressure can be convective cloud (or cell) to mesoscale eliminated by differentiating the convective complexes consisting of momentum equation ensembles of multicelled systems. Their behaviour is inherently dependent on the (3) environment in which they develop, including thermodynamic stability, which consists essentially of integrating vertical wind profiles and mesoscale round a closed circuit so that the effect forcing [36]. Moisture and instability, of the large potential force due to together with a triggering mechanism are pressure field vanishes identically. This key parameters of deep convection and integration round a circuit produces a evolution, followed by a lifting process vector equation for the line integral of [37]. Deep convective cells are parent velocity, known as ‘vorticity’ (a storms for most severe weather events microscopic measure of fluid rotation often found in regions such as equatorial and a vector field defined as the curl of Africa, Amazon southeast Asia, velocity). Vertical buoyancy gradient Indonesia and northern Australia [38]. can influence the perturbation pressure Such super cell storms often produce p’ field [41], which can be obtained by

taking the divergence of the horizontal In order to understand the generation of momentum equation (3) above: updrafts in the vortices on the flanks of the storm, perturbations are examined on (3*) pressure field [42]. For example, the Euler momentum- and continuity where . A positive buoyancy equations may be expressed (using anomaly in the upper troposphere and Boussineq approximation) as: negative anomaly in the lower (7) troposphere produce a positive B/z in the mid-troposphere, thus creating a meso-low cyclonic circulation (2p’ 0). When the environmental wind shear (a where is the 3-D velocity, ρ0 small change in the horizontal wind over is deviation pressure from the horizontal some vertical distance between two mean and is the total altitudes) is unidirectional, the anticyclonic (left-moving) and cyclonic buoyancy, while is as defined in (right-moving) updraft cores are equally equation (4) above. It is therefore, forced. convenient to rewrite the momentum equation using the vector identity; More often, it is convenient to deal with vorticity than circulation. For example, to we use absolute vorticity ω and relative a obtain the relation: vorticity ωr defined by: (8) (4a) (4b) Taking the curl of equation (7), recalling where  is the 3-D del operator while Ua that the curl of the gradient vanishes, we and Ur denote the absolute and relative obtain the 3-D vorticity equation: velocities respectively. In Cartesian (9) coordinates; where (5) Letting to be the vertical In most severe storms hitting Africa, the component of vorticity and taking mean flow turns cyclonically with derivative of k from equation (9), we height. The dominance of the right- obtain the tendency equation of in a moving storm can be understood non-rotating reference frame: qualitatively by considering the (10) dynamical pressure perturbations. We can define the basic state wind shear from which we note that buoyancy only vector , (assumed to turn affects the horizontal vorticity components. clockwise with height) noting that the basic state horizontal vorticity as: Taking the divergence of (8), we obtain a diagnostic equation for the disturbance (6) pressure:

(11)

From equation (6), we see that there is a cause [18]; typhoon (North Pacific); contribution to the dynamic pressure of cyclone (North Indian Ocean) and SH equation (10) of the form: (clockwise rotation): cyclone (South Pacific and South Indian Ocean). (12) Using equation (11) the sign of the Typically, hurricanes have radial scales pressure perturbation due to this effect of several hundred kilometers similar to may be determined noting that: those of some midlatitude synoptic systems. They are well-organized spiral (13) rain bands where the radial shear of swirling winds are large with high wave This indicates that there is a positive number are often dumped [43]. The dynamical pressure perturbation up- triggering action is typically a tropical shear of the cell and a negative wave, its motion controlled by a deep perturbation down-shear. In the case of layer of mean environmental flow [43]. unidirectional shear, the induced The chief initiating mechanisms in the pressure pattern favors updraft growth of Atlantic is easterly waves [38]. Energy is the leading edge of the storm. However, derived from horizontal shearing motion when the shear vector rotates clockwise in the easterlies and its core is cold with height, equation (13) shows a because of dynamically induced lifting dynamical pressure disturbance pattern in the presence of weak horizontal in which there is an upward directed temperature gradients [44]. vertical pressure gradient force on the flank of the cyclonically rotating cell and Horizontal shear instability must be a downward directed pressure gradient produced in the equatorial convergence force on the flank of the anticyclonic zone; the locus of most easterly waves of cell. this zone is located some distance off the equator and the converging air masses 4. Hurricanes carry their angular momentum with Hurricanes develop from pre-existing them. A mature hurricane has different tropical depressions that have somehow dynamical features (Fig. 5). Its center acquired a degree of circular symmetry (called the eye) is surrounded by and a warm/cold core, although such outwardly deep cloud known as the ‘eye disturbances are rare. Also referred to as wall.’ This is the region of increasing TCs and typhoons in some parts of the severity with heavy precipitation and world, hurricanes are intense vertical strong swirling winds. Swirling in the storms developing over tropical oceans core becomes strongly axisymmetric as of very warm upwelling surface water the hurricane matures [19]. The eye is an and move around subtropical high. area of relatively clear calm/fair weather Although they are described in various sinking air, but warmest temperatures terminologies in different parts of the aloft, separated by an inversion [45]. world, they are basically similar types of Outside the eye wall are regions of storms. For example, some nomenclature loosely organized rainbands with a wide exist such as in the NH scale, often with convective cells. (counterclockwise rotation): hurricane (Atlantic and East Pacific) –where they are classified by the damage they can

Fig. 5. Schematic-cross section of air circulation in a mature hurricane. Air spirals in towards the eye within the boundary layer, ascends along constant surfaces in the eye wall (red arrows), and subsides in the drier regions away from the eye wall (blue arrows). (https://www.hko.gov.hk/informtc/nature.htm).

by contributing to: (a) suppressed Many factors affect tropical storm and vertical wind shear across the hurricane activity over the North subtropical North Atlantic between 10– Atlantic; some of these show 20°N, and (b) a tendency for stronger surprisingly strong long-range predictive easterly low-pressure waves (at signals for Atlantic basin seasonal TC approximately 15°N) which typically activity up to 11 months in advance move across west Africa and propagate [46,47]. Another study by Gray and westward over the subtropical North colleagues (Reference) identified three Atlantic. prominent factors associated with this 4.1. Balance in the pre-hurricane and predictive skill: (1) the stratospheric tropical depression QBO; (2) the ENSO cycle; and (3) West African rainfall. QBO describes a layer In this context, the incipient hurricane is of winds that encircle the earth in the taken as a forced circulation driven by lower stratosphere, roughly between 20 the heat released in the organized and 40 km altitudes and 15 N and 15 S cumulus convection, not a free latitudes. The air masses alternate both circulation driven by unbalanced easterly and westerly, reversing every buoyant forces [44]. Thus, the large- 12-13th month. ENSO refers to the scale flow could be assumed as quasi- interaction between the atmosphere and hydrostatic and discount unbalanced ocean in the tropical Pacific, resulting in Coriolis and centrifugal forces, taking a periodic variation between below- the horizontal forces to be approximately normal and above-normal SSTs and in a gradient state or geostrophic dry/wet conditions over a few years. balance. The concept of balance is often According to Gray et al. [48], the factors used extensively in the analysis of large- appear to be in a phase favorable for scale gravitationally stable, extra- above normal Atlantic tropical activity tropical motions. For example, applying the perturbation technique to small- 12

amplitude wave motions of a From equation (15), the divergence conditionally unstable saturated forms of the laws of conservation of atmosphere, it is found that the small- angular momentum and energy may be scale modes grow at a greater rate and written as: ultimately predominate. We can assume a pre-hurricane depression as sufficiently (16a) small horizontal scale so that we ignore the curvature of the earth and the (16b) variation of the Coriolis parameter. where  is potential temperature, cp is The conditions for balance in the radial specific heat capacity at constant and vertical directions [44,49] may be pressure and is the rate of external written as: heating per unit mass. (14a) From equations (14a), (14c) and (16a), one notices that no frictional forces appear. This is because of the (14b) assumption that the entire effect of where p is pressure, ρ is air density, r is friction is confined to a surface boundary radial coordinate, z is counterclockwise layer [44]. Although the turbulent moist- tangential velocity component, f is the convective process which transports Coriolis parameter and g is gravitational latent heat must also transport field strength. Introducing perturbation momentum and sensible heat both analysis to equation (14a) reduces it to: vertically and radially, these transports unlike friction in the surface layer, are (14c) essentially dissipative and can cut down Under weak assumption that the the rate of swell development. characteristic phase and orbital Although the wind and convective frequencies are small compared to the clouds observed in a hurricane are not frequency of sound, it may be shown very axisymmetric about the vortex that the local and horizontal fluctuations center, the fundamental aspects of of pressure, density and temperature are hurricane dynamics can be modeled by fractionally small. Taking horizontal idealizing the hurricane as an density in space-time averages, equation axisymmetric vortex [42]. The thermal (14a) may be approximated by: wind in an axisymmetric hurricane can (14d) be derived from the gradient wind balance cylindrical coordinates [43], where and m is the expressed as: angular momentum per unit mass given (17) by where r is the radial distance from the centre (positive outward) and vλ is the The continuity equation may also be tangential velocity (positive for approximated by: anticlockwise flow). It is sometimes useful to use the absolute angular (15) momentum in place of vλ where u is the radial velocity.

because above the boundary layer Mλ is [42]. Conversely, potential energy is nearly conserved following the air carried out by a transverse secondary motion. In terms of Mλ the gradient wind circulation associated with the hurricane. balance can be expressed as: Circulation consists of boundary layer inflow into a region of enhanced (18) convection surrounding the eye wall

Eliminating Φ in equation (18) with the ascent within convective cloud towers hydrostatic equation in log-pressure that tend to be concentrated in the coordinates narrow outward-sloping direction, radially out flow in a thin layer near the we obtain the relationship tropopause and gentle subsidence at a between the radial temperature gradient large radius. The water that evaporates and the vertical shear of the absolute from the sea surface into the inward angular momentum flowing air within the boundary layer causes a large increase in the equivalent (19) potential temperature , as the air where z is altitude, R is gas constant, H approaches the aye wall [42]. Within the is the pressure height scale and T is air eye wall, potential temperature and temperature. The cyclonic flow in a absolute angular momentum surfaces hurricane becomes maximum near the coincide so that a parcel ascent in the top of the boundary layer. However, the eye wall neutralizes with respect to inflow picks up latent heat through conditional instability, and does not evaporation and exchanges sensible heat require external forcing [42]. with the underlying ocean. Near the 5. Monsoon circulations vortex center, the inflow turns upward The dynamical theories of thermally and draws in latent heat from the direct, zonally symmetrical circulations boundary layer into the free atmosphere in the tropical atmosphere are reported in [19]. Dynamically, strong cyclonic literature [50–52]. A summary from the swirling flow indicates strong inertial findings is that below a certain threshold stability [43] given by: value of thermal forcing, the atmosphere (20) maintains a steady state of thermal equilibrium. Beyond this threshold where VT is tangential wind velocity. value, equilibrium breaks down and a Because at the top of the boundary layer meridional circulation is initiated. , equation (19) implies that Effects of forcing on monsoon strength are due to changes in the amplitude of . Thus temperature maximum seasonal insolation and the resulting land-sea thermal and pressure contrasts must occur at the center of the [53], and references therein. The storm/hurricane. This is consistent with subtropical thermal forcing can be observation that hurricanes are ‘warm distinctively described for dry and moist core’ systems. atmospheres. For example, the moist The hurricane kinetic energy is atmosphere is more appropriate over maintained in the presence of boundary tropical Africa relative to Sahel Africa. layer dissipation by conversion of latent One important characteristic of moisture heat required from the underlying ocean 14

transport in the atmosphere is residence latitude and the meridional distribution time [54] -defined as the time spent by of boundary-layer entropy. Hence, the moisture in the atmosphere between onset of a monsoon circulation depends evaporation and precipitation. This is in a nonlinear fashion on these two also a fundamental balance in water factors. But the monsoon is not cycle. Depending on the distribution of controlled by simple land-sea thermal entropy, two possible regimes may contrast only as zonal asymmetric dominate the dynamics in the tropics; a diabatic heating and large-scale radiative-convective equilibrium regime orography can also significantly affect it or an angular momentum conserving [56]. According to the authors, theory regime [55]. The thermal wind can be predicts that a flat distribution of entropy expressed (for a zonally symmetrical does not drive any circulation and that a atmosphere) as: relatively large gradient of entropy should strongly drive the monsoon (21) circulation. A zero absolute vorticity in where u is zonal wind, α is specific the upper troposphere is indicative of the volume (= 1/) and y is meridional presence of monsoon circulation [57]. distance. A monsoon circulation is not simply driven by the east-west differential Using Maxwell’s theory, we can write: heating, but rather associated with the (22) differential heating profile and maintained dynamically by the balance where is saturation entropy given by between a vorticity source and advection. The balance is reflected by a and is equivalent spatial quadrature relationship between potential temperature. the monsoon divergent circulation and Besides the monsoon, there can exist the monsoon high/low at upper/lower other intraseasonal oscillations related to levels [56], and references therein. westerly bursts that modulate stability 6. Long westerly waves and wind shear near the equator [38]. For example the Madden-Julian The atmosphere-land-ocean is a complex Oscillation (MJO) is a large synoptic system supporting many different types scale region that favours deep of motions on a wide range, both convection originating near east Africa spatially and temporally. For example, and migrates eastwards, typically the huge cyclones and anticyclones have reaching its maxima over the Indian horizontal length scales, wave motion Ocean. Meridional monsoon circulation have periodic time scales in seconds, can develop over any tropical region (off days or months [49]. In order to attempt the equator) when the absolute vorticity a theoretical study of the atmosphere- near the tropopause reaches a threshold ocean motions, we begin by identifying value of zero. However, for a moist scales of motion interest to us and try to atmosphere satisfying a quasi- isolate important mechanisms because equilibrium balance between moist equations of motion are a very rich convection and the radiative forcing, matrix from which a wide range of absolute vorticity at the upper- physical phenomena can occur. Among tropospheric levels is a function of both the phenomena permitted are the wave

motion and vorticity. Using the vorticity latitude are parallel to those at any other equation, and making additional latitude. Thus ζ = ∂v/∂x, from which: assumptions of zero horizontal (24) divergence purely horizontal flow autobarotropy, and neglecting additional forces, the vorticity equation may be We seek as a solution, waves which expressed as: move in the x-direction with a constant speed c, without changing shape. Thus, (23a) if one follows a parcel moving in the east-west direction with speed c, no or (23b) changes will be observed in any of the variables. Therefore, the operator where is relative , where vorticity, V = (u,v) is velocity vector means individual derivative following an (assuming a 2-D fluid flow) and u and v air parcel moving with speed c. Here are zonal and meridional velocity components.  is the vertical component is different from the operation since of the 3-D vorticity vector , sometimes the latter represents an individual written as  and is positive for counter- z derivative following an air parcel. With clockwise fluid rotation [58]. However, this assumption the vorticity theorem since  is usually smaller than f but becomes: greatest at the edge of fast currents, we normally use absolute vorticity (a =  + (25) f) which is the sum of planetary and Equation (24) is nonlinear because it relative vorticity. involves a product of dependent A planetary wave may be studied within variables and is therefore not easy to the framework of the Cartesian equation solve. We may apply the perturbation by making the so-called ‘beta’ or - method in order to reduce it to a linear plane approximation [48].  is the differential equation. For example, we change in Coriolis force with latitude as assume the horizontal flow to be made a result of wave amplitude. The up of a large basic current U (the zonal expression for  may be obtained from wind component), with a west-east flow the Coriolis parameter f, noting that, upon which are superimposed small f=2Ωsinφ, whereupon. variable perturbations and which df/dy=2Ωcosφ×dφ/dy are functions of distance x and time t. Defining (where a is Thus; the earth radius) and assuming a broad westerly current of infinite lateral extent Here U is considered to be a quantity of which is undulating north and south in a zero-order magnitude, while and wave-like fashion, we may take the (which are one step smaller) are of the dependent variable to be independent of first order magnitude. If one adopts a log y. In this case, we deal with a system of scale to base ten, then and will be waves in which the streamlines at any about one-tenth as large as U, so that equation (25) becomes:

(26) (30)

The first two terms are first order in size where U is speed from west to east of while the last term consists of the the component of the upper-level wind product of two first-order quantities and due to uniform flow, β is the meridional hence the second order in magnitude. or north-south gradient of the Coriolis Within this degree of precision, the parameter. Since the magnitude of f second-order term can be neglected so increases towards the poles, β is positive that equation (26) reduces to a linear and hence waves with shorter equation of the form: wavelengths have a relatively small (27) component due to propagation. Under this condition, advection overwhelms the Since x is the only independent variable, effect of propagation and waves move this corresponds to an ordinary second- downstream. On the other hand, if in the order differential equation with constant midlatitudes the wavelength is long, coefficients. Solution for is known to effects of propagation may cancel those be sinusoidal in x. To show this we can of advection and waves may become use a trial solution of the form stationary or retrograde. The wavelength of stationary waves occurs when c = 0, (28) from which equation (30) yields: where is a constant amplitude and L (31) is the wavelength. Substituting this into equation (26) yields: 7. Jet streams and wave tracks (29) When the longitudinally asymmetric Equation (29) is a relationship existing geopotential anomalies associated with between the wave speed and wavelength stationary waves are added to the zonal- of sinusoidal waves if equation (27) is to mean geopotential distribution, the be satisfied. Such an expression of wave resulting time mean field includes local phenomena is called the frequency regions of enhanced meridional potential equation. One can see that when c is gradient that are manifested in the wind positive (waves progress from west to field of the NH by the Asian and north east) U – c is smaller than U and the American jet streams. In addition to the wavelengths are relatively short. When c two jet cores in West Pacific and West is negative (waves retrogress from east Atlantic, there is also a third jet core to west), U – c is larger than U and the centered over North Africa, although it wavelengths are relatively long. Upper- is weaker than the two [59]. To level waves in the westerlies in understand the momentum budget in the midlatitudes usually move from west to jet streams and its relationship to east as a result of advection (a process in weather distribution, we consider the which the airflow transports a property zonal component of the momentum of the atmosphere downstream) and in equation part as a result of propagation, which acts in the opposite direction towards the (32) west. From equation (29) the phase where va is the meridional component of speed of the waves can be expressed as: the ageostrophic wind. This equation

indicates that the westerly acceleration equatorward ageostrophic motion. This (Dgug/Dt >0) that air parcels experience meridional flow with accompanying as they enter the jet can only be provided vertical circulation is illustrated in figure by a poleward ageostrophic wind below. component (va >0) and conversely acceleration that air parcels experience as they leave the jet requires an

Fig. 6. Meridional cross sections showing the relationship between the time mean secondary meridional circulation (continuous thin lines with arrows) and the jet stream (denoted by J) at locations upstream & downstream from the jet stream cores [42]. It would be nice if you generate your own figure, rather than just copy, (anyway reference is missing).

storm development. This jet mode Downstream of the jet core however, the appears to be forced by anomalous secondary circulation is thermally tropical heating associated with the indirect, but much stronger than the ENSO cycle and intra-seasonal zonally averaged Farrell cell. Because variability such the MJO [60], and the growth rate of baroclinically unstable references therein. Synoptic waves travel synoptic scale disturbances is along the subtropical jet over the South proportional to the strength of the basic Indian Ocean in the region of strongest state thermal wind, it would not be surface westerly winds [60] and south of surprising that the Pacific and Atlantic Australia [61]. When the waves jet streams can be important SRs for propagate equatorward into the westerly

dust in the upper troposphere over summer (May–September). Equatorial equatorial Pacific they trigger deep West Africa generally has light winds convection [62]. Typically transient with diurnal sea breezes. However, there baroclinic waves develop in the jet is a weak monsoon effect in the Gulf of entrance region and grow as they are Guinea, which means continuous advected downstream and decay in the onshore winds between June and jet exit region. The African Easterly Jet August. Relatively cool moist air from (AEJ), also popularly called the Somali the Gulf of Guinea is often advected Jet can be viewed as a western boundary onto the warm dry continental landmass current of the East African highlands. where the resulting rainband can travel This is normally observed over North from the coast to Sahel and back again, Africa [63] and occurs due to the following the movement of the ITCZ reversed surface temperature gradient [69]. when the Sahara becomes hotter than the 8. Conclusion (It seems to me that equatorial regions [38,64]. The jet flow conclusion should start from here, so develops from instabilities over sub- the discussion of the jet was finished, Saharan Africa [34]; its origin lies over and you start doing some summary) the Indian Ocean in the form of southeast trades. However, trades tend to Several other studies have been be more from an east-southeast direction conducted on tropical African waves and than their name implies. A steady cyclogenesis [14,70–73]. Most of these zonally symmetric solution indicates that studies outline the climatology of the combination of inertial forces, African easterly waves (AEWs) and surface friction and weak subsidence their propagation over African and provide adequate ingredients of the global environment. AEWs are synoptic- southeast trades over the South Indian scale tropical disturbances that form Ocean [65]. over northern Africa [74] and propagate westwards as unbroken progressions The AEJ plays a crucial role than the across Africa and the Atlantic Ocean intertropical convergence zone (ITCZ) during summer [71–73]. They move in and generally linked to the well- trade wind belt and occur most organized mesoscale features [66] in the frequently in August–September rainfall production by triggering between 5 and 20 N and appear to be atmospheric disturbances at its southern breeding grounds for hurricanes. They and northern boundaries. Some are a fundamental component of the west emerging theory also relates the African climate [75] and commonly formation of such atmospheric known as the main precursors of TCs in disturbances in the vicinity of the AEJ to the Atlantic [72,76]. It is estimated that a barotropic-baroclinic energy they account for 60% of the Atlantic conversion process [67,68]. The wind tropical storms and about 85% of the regime in northwest Africa is dominated major hurricane activity [14,63]. The by the north to north-easterly trades that north wave primarily grows in blow on the eastern flank of the quasi- association with dry baroclinic energy permanent anticyclone in the North conversions at the southern margin of Atlantic (the ‘Azores’ high). These the Sahara, while southern-track mainly winds are stronger and tend to move result from barotropic instability and towards the northwest during the NH interactions with deep conversion [73], 19

and references therein. The southward cold, equatorward-flowing currents track is driven by barotropic-baroclinic means that the water can be energy conversion and can exhibit uncharacteristically cold for the latitude. stronger circulation over West Africa For example, the western part of South [72]. Although it has been hypothesized, Africa is renowned for episodic up for decades that AEW generation was welling events associated with the Cape due to a mixed baroclinic-barotropic Doctor [78]. instability of the AEJ, recent studies now 8. Conclusion indicate that baroclinic and barotropic conversions from these instabilities The global distribution of continental alone do not account for the AEW landmasses is a major factor determining growth. The triggering mechanisms are the nature of swells on the global wave the mesoscale convection and upstream climate. However, the relative development from the previous AEW importance of remote forcings on the [74,77]. African climate remains a key area of uncertainty [12]. For example, there is Along the southwest coast, predominant considerable uncertainty at the regional winds are south or south-easterly trades, scale whether TC numbers may decrease and strongest during southern summer or increase, or whether SRs will remain (October–April). The more southerly the same. Despite the presence of regions are often influenced by the mid- massive land barriers, extratropical latitude depressions during southern swells from the NH can also contribute winter, and so winds are often stormy significantly to the wave activity in and appear to come from a westerly remote locations, in addition to tropical quarter. However, during summer, the environment-generated waves. This south-easterly trades predominate. In the demonstrates that the earth’s climate was Cape Peninsular, with virtually no land and will possibly never be in equilibrium topography, these winds can reach [79]. Nevertheless, climate science has strong to gale force [78]. Over east been active in taking advantage of Africa, winds are predominantly from advances in basic mathematical and north or northeast, whereas over physical approach to provide simulations southern Africa, the influence of mid- to address new fundamental problems latitude weather systems means the [80]. In the present study, we use the so- possibility of winds from a south- called Navier-Stokes equations to westerly quarter during winter. Much of understand the swell dynamics around the east-coast which is out of the Africa. The study gives motivation to the influence of cold currents and up- climatological pattern of wave activity welling is generally warm, with greater and swell propagation characteristics as seasonal variation further south. In the primarily related to the main sea state trade-wind belts of the northwest and and wind patterns and may also help southwest Africa there is up welling. improve extreme weather predictions Oceanic upwelling occurs mainly as a and monitoring. result of surface water divergence due to wind stress so that adjacent surface does There are key concepts to note: tropical not flow back, obstructed by the waves play an important role in the landmass in the upwind direction [60]. formation of large scale sea breezes This, together with the proximity of which frequently take shape over Africa;

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