INTERACTIONS WITH EQUATORIAL WAVES AND MODES OF VARIABILITY

Interest of equatorial waves and modes

Even with the use of multi-model forecast ensembles (TIGGE), rainfall forecasting over West Africa (Vogel et al. 2018) remains poor and slightly better than climatology after post-processing of the ensembles.

The forecaster is thus well placed to improve these forecasts on the basis of his expertise and experience, relying on other variables better predicted than rain. It is therefore necessary to look for other indirect sources of predictability. We propose here to use equatorial waves and other modes of intraseasonal variability as a source of predictability.

According to the study of Schlueter et al. (2019) using GPCP observations, equatorial waves and modes contribute strongly to the rainfall variability over the Sahel for a total of ~20 mm²d^-2. The first contribution is due to easterly waves (AEW for ~45%) and is documented in the AEW object. Next in decreasing order of importance are the contributions of,

Kelvin waves (K for ~25 %),
Equatorial Rossby waves (ER for ~16 %),
Mixed Rossby-Gravity waves (MRG for ~8 %)
and the MJO (less than 4 %).

The opposite figure shows maps of the contributions of the different equatorial waves to the variance of rainfall during the full monsoon (JAS).

The Sahel zonal channel corresponds to the largest AEW (or TD) contribution increasing from the Ethiopian plateau towards West Africa to reach its maximum over the near Atlantic.
The Sahel zonal channel corresponds to the largest AEW (or TD) contribution increasing from the Ethiopian plateau towards West Africa to reach its maximum over the near Atlantic.
Although ER and MRG waves have lesser contributions, they cannot be neglected.

. Standard deviation of TRMM precipitation (1998–2016) within the specific wave domains for (a)–(f) the transition season and (g)–(l) the full monsoon. Source : modified Fig. 4 of Schuelter *et al.* (2019).

These mean maps can be misleading because for a given situation the contribution of a wave or mode of variability can be much higher than that reported for the season. Furthermore, strong and extreme events are often associated with the combination of the favourable influence of several waves or modes, as for example the crossing of a K wave with an ER one.

Convection coupled equatorial waves (CCEW) theoretical framework

These waves coupled to convection (dynamically and thermodynamically), constitute coherent structures, named convection coupled equatorial waves (CCEW). They result from “trapping” along the equator due to the Earth’s rotation.

The structural and dispersion characteristics of these CCEWs are predicted from the shallow-water model equations projected onto an equatorial “β-plane”. Matsuno (1966) was the first to derive the complete set of zonally propagating wave solutions from these equations, isolating the normal modes of this system and their dependence on the zonal wavenumber, thus introducing the zonal wavenumber (k) and frequency (ω).

The opposite figure shows the dispersion relation k-ω of the brightness temperature (characteristic of the convective activity, in colour) superimposed on the theoretical Matsuno relation (family of curves) for the symmetric (a) and antisymmetric (b) components.

The y-axis of the frequency ω is also graduated in periods from 20 to 1.25 days and the zonal wavenumber k (x-axis) can be negative or positive depending on whether the waves are propagating westwards or eastwards respectively. The vector from the origin to a point on the diagram corresponds to the propagation speed, which is faster the more vertical.

The different packets shown in this diagram (framed windows) correspond to different types of CCEWs, in particular :

(a) Modes with a symmetric structure
- Kelvin propagating eastwards,
- MJO (paquet proche de l’origine k entre 1 et 5, période de 30 à 60 j
- ER propagating westwards
(b) Modes with an antisymmetric structure,
- MRG propagating rapidly westwards but propagating its energy more slowly eastwards

Wave number–frequency power spectrum for symmetric (a) and antisymmetric (b) spectrum of brightness temperature Tb from the European CLAUS for July 1983–2005, averaged over 15°N, 15°S. Source : Figure 7.4 of the Handbook extracted from Kiladis et
al. (2009).

Structure and impact of CCEWs

1. Kelvin Waves

Symmetric /Equateur
Eastwards Propagation : 15-25 ms^-1 (can circumnavigate the earth in 20-30 days)
Period : 4-15 d
Wave length : ~4 000 to 20 000 km
Signature : zonal wind, divergence in upper levels (VP200)
Frequent over Africa
Convection favoured in the low layer convergence zone
Explains ~25% of precipitation variance
Strengthens the AEJ at its passage and thus favours AEWs in its wake

The opposite composite of the wet and dry phases of Kelvin waves shows the modulation of the associated rainfall. Relative to the rainfall median, the modulation by Kelvin waves is very strong and extends over much of the Sahel, not just the equatorial band.

The opposite figure shows the spatial structure of the Kelvin wave over West Africa.

The circulation fields are roughly symmetric with respect to the Equator and are characterised by a wavelength of about 5000 km and an eastward phase velocity of about 15 m/s. It thus crosses Africa in 5-6 days, with a large area of convective enhancement (green) to the east of a zonal 850 hPa westerly wind anomaly according to the conceptual scheme.

Conceptual scheme of the Kelvin wave structure.

CHIRPS rainfall composite of the wet and dry phases of Kelvin waves Composite phase humide/ sèche de l’onde de Kelvin des pluies. Source: Schulter et al AMS 2018.

Composite maps of May to October anomalous brightness temperature Tb (shading), geopotential height (contours) and wind (vectors) associated with a −10 K perturbation in Kelvin wave Tb at the base point 2°N, 10°E, for (a) day −2 at 850 hPa, (b) day 0 at 850 hPa and (c) day +2 at 850 hPa. The contour interval is 12 mgp, with negative contours dashed. Cold shading is for negative and warm shading is for positive Tb perturbations. Tb and wind vectors are locally significant at the 95% level, with the largest vectors around 2 m s⁻¹. Source: Figure 7.6 of the Handbook.

2. Equatorial Rossby waves (ER)

Symmetric /Equator
Westward propagation : -5 ms^-1
Period : 15-20 days
Wavelength : ~10 000 km
Signature : vorticity, meridional wind, PW
Modulation of the monsson flow
Possible strengthening of a trough for a few days
Explain ~16 % of the precipitation variance
Moistens the lower layers, resulting in a PW signature
The indian monsoon is an important source of ER waves

The opposite figure shows the typical structure of equatorial Rossby waves ER coupled with convection over West Africa, computed over the months of May to October.

The dipole of weakening/strengthening of the convection associated with a specific circulation propagates westward between days – 5 and + 5. Note at 850 hPa a strengthening of the south-west monsoon flow feeding the convection strengthening zone (cold colours). On the other hand, the zone of weakening convection (warm colours) corresponds to a reinforcement of the north-east Harmattan flow. The ER contribute partly to the Sahelian mode.

Conceptual scheme of the ER wave structure.

Cartes composites sur mai-octobre pour l’onde ER identiques à celle précédente pour l’onde de Kelvin, pour les jours J-5, J et J+5. Source : Figure 7.8 du Handbook.

3. Mixed Rossby-Gravity waves (MRG)

Antisymmetric /Equator
Westward propagation : 15-25 ms^-1
Group velocity (energy propagation) towards East : ~5 ms^-1 (as AEWs)
Period : 4-5 days
Wavelength : ~10 000 km
Signature : PW, meridional wind maximum at the Equator in the low levels, corresponding to interhemispheric transport with a thickening and strengthening of the monsoon flow
Convection favoured in phase with the intensification of the monsoon flow
Explains ~8 % of the rainfall variance
MRGs are similar to AEWs, but with a longer zonal wavelength
Faster than AEWs, MRGs when they join an AEW and are in phase, a sudden amplification is possible contributing to extreme events like the one in Ouagadougou on 1^st September in 2009 (Lafore et al. 2017).

Conceptual scheme of the structure of a MRG

4. The Madden–Julian Oscillation over Africa (MJO)

Eastward propagation : ~5 ms^-1 on average (between 3 and 9 ms^-1) and circles the globe in 30-40 days
As illustrated in the diagram opposite, its wind and convection modulation structure is a mixture of equatorial waves
- of Rossby behind the MJO (West) with 2 symmetric cyclonic gyres / Equator, and a westerly acceleration of the zonal wind at the Equator
- and of Kelvin in the front the MJO (East) with an easterly zonal wind anomaly
Convection is reinforced between the 2 opposite zonal accelerations
The MJO envelope includes Kelvin waves propagating faster than it but in the same direction
Signature : meridional wind, upper levels divergence (VP200)
The MJO has global impacts: modulation of monsoons and cyclone activity, impacts on El Niño, rainfall in North America, Europe…

MJO detection and tracking :

Real-time monitoring and forecasting of the MJO is done using the MJO index (Figure opposite), proposed by Wheeler and Hendon (2004). It is based on the EOF pair of the combined zonal wind fields at 850 hPa and 200 hPa averaged over the equatorial band, and the OLR. The projection of daily observed data onto these multivariate EOFs with the annual cycle and interannual variability components removed, constructs time series of principal components that vary mainly on the intra-seasonal scale of the MJO.

The RMM (Real-time Multivariate MJO) index visualises the MJO in the phase space defined by the time series of the first 2 principal components: RMM1 and RMM2.
In this diagram 8 phases of the MJO are defined corresponding to 8 geographical areas
Within the circle of radius 1, the MJO is low or even non-existent.

Conceptual scheme of the structure of the MJO

Phase‐space representation of the two‐ component MJO index, for the period 1 May–16 June 2012 (thin lines) and the forecast period valid 17 June–1 July 2012. Each dot represents the value of the index on a particular day, with the starting and ending days labelled. Also shown are the eight defined phases of the MJO and the region used to signify weak MJO activity. Also labelled (with words) are the approximate locations of the near‐equatorial enhanced convective signal of the MJO for each quadrant of the phase space (e.g. the ‘Indian Ocean’ for phases 2 and 3). After Wheeler and Hendon (2004). Source: Figure 7.18 of the Handbook. NOAA’s Climate Prediction Center MJO web portal: http://www.cpc.ncep.noaa.gov/products/precip/CWlink/MJO/mjo.shtml

The composite analysis below highlights the impact of the MJO over West Africa (Wheeler et al. 2009). The probability of rainfall is enhanced during phase 1 in conjunction with low-level westerly wind anomalies. On the other hand, rainfall is decreased from phases 3 to 7 in conjunction with easterly wind anomalies. The MJO-based wind and rainfall anomaly composites are used as a guide for the intraseasonal forecast.

MJO composites of weekly rainfall probabilities (contours and shading) and 850 hPa wind anomalies (vectors) for the MJJ
season for phases 1 and 5. Rainfall probabilities refer to the chance of weekly rainfall exceeding the upper decile, expressed as a ratio with the
mean probability (nominally 33%). Source : Extract from Figure 7.14 of the Handbook, adapted from Wheeler et al. (2009).

The figure opposite summarises the impact of the MJO over Africa and the tropical Atlantic using the RMM diagram (after Ventrice et al. 2011) :

AEJ acceleration during phases 8, 1 and 2
Enhanced convection over tropical Africa during phases 8, 1 and 2
Enhanced activity of AEWs during phases 1-3
Enhanced cyclogenesis during phases 1-3

Schematic RMM phase diagram of the MJO representing the positive impact of the MJO over Africa for JEA (orange), convection (blue), AEW activity (red), and cyclonic activity over the Atlantic (green). Source: Ventrice et al. (2011).

5. The quasi‐biweekly Zonal Dipole Mode (QBZB)

The quasi-biweekly zonal dipole (QBZD) is the dominant mode of variability over West Africa in the 10-25 day range. The “dipole” aspect refers to a quasi-stationary oscillation with a phase opposition between the African rains and the rains over the western Atlantic and Central America. The figure opposite shows its composite structure.

When convection is minimal over West and Central Africa, solar radiation reaching the surface is strong, which increases surface temperatures and decreases surface pressures. This creates an east-west pressure gradient at the latitudes of the ITCZ and the Saharan thermal low, leading to increased moisture advection over the continent.

Composite pattern of the subseasonal QBZD mode in terms of non‐filtered OLR modulation (W m⁻²). Source: extract from Figure 7.11 of the Handbook.

The arrival of a positive pressure signal from the Atlantic, associated with the structure of a Kelvin wave, amplifies the zonal component of the low-level winds and the moisture advection over the continent, leading to an increase in convective activity over West and Central Africa. Then the opposite phase of the dipole develops (Mounier et al., 2008). This mode is also present in spring.

6. The Sahel Mode

The “Sahel mode” is the second mode of 10-25 day variability (Figure opposite) and was detected by Sultan et al. (2003).

The convection enhancement in the African ITCZ is associated with a propagative mode first appearing over central Africa, moving northwards towards the Sahelian latitudes, and then propagating westward towards the eastern tropical Atlantic. This structure is associated with a cyclonic circulation located northwest of the convective enhancement pole (the convective enhancement zone in the dipole structure), increasing the moisture advection towards this pole.

Composite pattern of the subseasonal Sahel mode in terms of non‐filtered OLR modulation (W m⁻²). Source: extract from Figure 7.11 of the Handbook.

The Sahel mode can be partly explained by the intra-seasonal variability of the mid-latitudes via the Saharan thermal low (cf. HL mode) for about 1/3 of the cases and by the dynamics of equatorial Rossby waves for another 1/3 of the cases.

Adapted products

MISVA

Convection coupled equatorial waves (CCEW) are rarely visible on the raw fields and even on their anomalies. Filtering techniques in the occurrence windows of these modes are needed to extract and visualise them. This is the strategy adopted by the MISVA site, as well as other sites, to provide forecasters with analyses and forecasts of CCEW. These can be maps, Hovmoller (longitude – time) diagrams in appropriate latitude bands, or appropriate indices. The variables used are multiple and depend on the CCEW mode considered, summarised in the table below.

	Zonal wind	Meridional wind	Velocity Potential VP @200 hPa	Vorticity @850 hPa	Stream Function	PW
MJO	X		X
Kelvin	X		X
Rossby		X		X		X
MRG		X		X		X
AEW	X	X		X		X
OLR*<0 Enhanced Convection			VP<0 Divergent		SF>0 Anticyclonic
MISVA Variables		925-600 hPa layer	VP200		SF	PW

Relevant variables for detecting the different CCEW modes.

Wave filtered Maps:
- Synoptic scale: Wave_filtered_Maps – Misva (aeris-data.fr)
  - Parameters: Parametre – Niveau-hPa – Ondes – Domaine
Hovmöllers d’ondes filtrées:
- Sub-seasonal scale: ECMWF analysis over the last 3 months and ensemble forecasts (11 members) for the next 41 days Hovmollers_Monthly – Misva (aeris-data.fr)
  - For 8 variables Parameter
  - and for 7 Latitude bands
- Synoptic scale: ECMWF analysis over the last 3 weeks and deterministic forecasts for the next 10 days Prévision Synoptique Hovmollers – Misva (aeris-data.fr)
  - Parameters: Domain – Parameter – Latitude
QBZD INDEX QBZD index – Misva (aeris-data.fr)
Intra-seasonal activity Index :
- Not currently available
SAHELIAN MODE INDEX Sahelian mode index – Misva (aeris-data.fr)
Phase‐space representation: HL AND_SAHELIAN MODES HL and_Sahelian modes – Misva (aeris-data.fr)

Products of others web-sites

Site of M. Ventrice http://mikeventrice.weebly.com/tropical-waves.html
Site of K. Schrek https://ncics.org/portfolio/monitor/mjo/
Australian Site of Wheller: http://cawcr.gov.au/staff/mwheeler/
- RMM index: Indice en temps réel de la MJO real-time index http://cawcr.gov.au/staff/mwheeler/maproom/RMM/
Site of the NOAA’s Climate Prediction Center:
- RMM index: http://www.cpc.ncep.noaa.gov/products/precip/CWlink/MJO/

Références

Handbook

Chapitre 7 Subseasonal Forecasting

Section 7.1.3 Detection of the Main Modes of Subseasonal Variability of Convection pages 258-260
Section 7.1.4 Ondes équatoriales couplées à la convection pages 260-264
- 7.1.4.1 Kelvin waves
- 7.1.4.2 Equatorial Rossby waves (ER)
- 7.1.4.3 Mixed Rossby–Gravity Waves (MRG)
Section 7.1.5 Other Convectively Coupled Signals and Links with Equatorial Waves pages 265-275

Articles

Lafore, J.-P., Beucher, F., Peyrillé, P., Diongue-Niang, A., Chapelon, N., Bouniol, D., Caniaux, G., Favot, F., Ferry, F., Guichard, F., Poan, E., Roehrig, R. and Vischel., T. (2017a) A multi-scale analysis of the extreme rain event of Ouagadougou in 2009. Quarterly Journal of the Royal Meteorological Society, 143(709), 3094–3109. https://doi.org/10.1002/qj.3165.

Mounier F, Janicot S, Kiladis GN. 2008. The West African Monsoon Dynamics. Part III: The Quasi-Biweekly Zonal Dipole. J. Climate. 21: 1911-1928.

Schlueter, A., A. H. Fink, and P. Knippertz, 2019a: A systematic comparison of tropical waves over northern Africa. Part II: Dynamics and thermodynamics. J. Climate, 32, 2605–2625, https://doi.org/10.1175/JCLI-D-18-0651.1.

Schlueter, A., A. H. Fink,Seloobse P. Knippertz, and P. Vogel, 2019b: A systematic comparison of tropical waves over northern Africa. Part I: Influence on rainfall. J. Climate, 32, 1501–1523, https://doi.org/10.1175/ JCLI-D-18-0173.1.

Skinner, C. B., and N. S. Diffenbaugh, 2013: The contribution of African easterly waves to monsoon precipitation in the CMIP3 ensemble. J. Geophys. Res. Atmos., 118, 3590–3609, https:// doi.org/10.1002/jgrd.50363.

Sultan B, Janicot S, Diedhiou A. 2003. The West African Monsoon Dynamics. Part I: Documentation of Intraseasonal Variability. J. Climate. 16: 3389-3406.

Ventrice, M. J., C. D. Thorncroft, and P. E. Roundy, 2011: The Madden–Julian oscillation’s influence on African easterly waves and downstream tropical cyclogenesis. Mon. Wea. Rev., 139, 2704–2722, https://doi.org/10.1175/MWR-D-10-05028.1.

Wheeler MC, Hendon HH. 2004. An all-season real-time multivariate MJO index: development of an index for monitoring and prediction. Mon. Weather Rev. 132: 1917-1932.

Wheeler MC, Hendon HH, Cleland S, et al. 2009. Impacts of the Madden–Julian oscillation on Australian rainfall and circulation. J. Climate 22: 1482–1498.