A Study of the Energetics of African Easterly waves
Using A Regional Climate Model
Jen-Shan Hsieh* and Kerry H. Cook
Department of Earth and Atmospheric Sciences, Cornell University
Ithaca, NY 14850
Submitted as an Article to the Journal of Atmospheric Sciences
February 12, 2006
*
Corresponding author: Jen-Shan Hsieh,
Present affiliation: 618B Department of Oceanography, Texas A & M University
College Station, TX 7784. Email address: [email protected]
Abstract
The evolution and spatial distribution of the energetics of African waves are studied.
Complete eddy energy equations for an open system are derived for the computation of energy
transformations during wave generation and dissipation. It is found that baroclinic overturning is
the dominant energy source although barotropic conversions can be equally important when there is
concentrated moist convection south of the jet or shallow cumulus convection beneath the jet. The
generation of active waves usually results from the nearly in-phase evolution of baroclinic and
barotrpic conversions, which are accompanied by significant rainfall over Africa.
Significant barotropic instability associated with the horizontal shear is usually induced by
concentrated deep convection on the southern flank of the jet. Barotropic conversions associated
with the vertical wind shear may attain even greater magnitudes than that associated with the
horizontal shear when the shallow cumulus convection beneath the jet is strong. The eddy available
potential energy consumed by the baroclinic overturning is compensated directly by the conversion
of zonal to eddy available potential energy and the generation of eddy potential energy by diabatic
heating. These direct conversions of latent heat and zonal available potential energy suggest that
interactions across space scales, from convective space scales to the large scales, are important for
generating African waves.
The convectively induced barotropic instability may feedback to
enhance baroclinic overturning, leading to the formation of organized precipiation migrating with
the waves through the nonlinear interaction of the waves with convection.
A space-time spectral analysis shows that the dispersion characteristics of African easterly
waves with wavelengths between 2650 km and 4000 km do not follow the dispersion relation of the
shallow water waves, indicating that these waves, similar to other easterly waves in the tropics,
possess significant nonlinearity and cannot be fully explained by linear wave theory.
1
1. Introduction
Westward propagating wave disturbances over West Africa and the Atlantic Ocean have
been studied in many previous investigations. They typically are categorized into two groups. One
has periods between 3 and 5 days, and the other has periods between 6 and 9 days (e.g., Viltard et
al. 1997, Diedhiou et al. 1998). Diedhiou et al. (1998, 2001) identified two main tracks of the 3-5day wave regime over West Africa at 5°N and 15°N, which merge into one track over the tropical
Atlantic along 17.5°N. Reed et al. (1977) analyzed observations during Phase III of GATE to form
a composite wave structure and found strong convection ahead of wave troughs over West Africa.
Duvel (1990) had similar findings using Metosat data and European Center for Medium-range
wather Forecast (ECMWF). However, Rowell and Milford (1993) studied the generation of African
squall lines and found that African waves have no significant impact on squall line development
near Central Africa. It is not clear if this convection is a cause of the waves, or a result of the
waves, or if the waves simply organize the large-scale convection.
Numerous studies associate the generation of 3-5-day waves with an instability of the
African easterly jet (Burpee 1972, Rennick 1976, Simmons 1977, Mass 1979, Kwon 1989,
Thorncroft and Hoskins 1994a and b, Thorncroft 1995), while Schubert et al. (1991) suggested that
easterly waves are a result of the zonal flow instability associated with the intertropical convergence
zone (ITCZ). Ferreira and Schubert (1997) showed that unstable mean flow can be produced by
ITCZ convection in just a couple of days though their study does not focus on African waves.
Hsieh and Cook (2005, 2006) analyzed the realistically simulated African easterly waves in a
regional model and found that the generation of African waves are more closely associated with the
ITCZ convection than the strength of the jet and the potential vorticity (PV) gradient reversals on
southern flank of the jet is mainly caused by convective heating.
2
A connection between the waves and the ITCZ implies that baroclinic conversions provide
the primary wave energy, while a connection with the jet implies barotropic conversions dominate.
Therefore, a thorough energetics analysis of the waves will help resolve confusion about why these
waves occur. However, a complete and realistic energetics analysis for African easterly waves has
not yet been performed.
Previous energetics analyses using observational data and modern
reanalyses were not able not include realistic diabatic heating and frictional dissipation. Analyses
based on numerical studies, meanwhile, have not included the effects of longitudinal variations in a
realistic background state. It is especially important to include the longitudinal variations in the
model to examine how African waves are initiated over East Africa and growing over West Africa.
In addition, the inclusion of zonal variation reflects the reality of the background state gradually
changing from East Africa to West Africa. These studies are reviewed below.
The aim of this paper is to provide a complete energetics analysis using a realistic regional
model simulation.
In the next section, the energy equations are briefly reviewed along with
previous studies of African easterly wave energetics. The regional climate model simulation is
described in section 3a, and the horizontal and vertical wave structures in the model are compared
with previous observational studies in section 3b. In section 4, the results of the energetics analysis
are presented. Section 5 is the summary and conclusions.
2. Background
a. Energy equations. Synoptic-scale disturbances, such as African easterly waves, can be
considered to be eddies propagating on a zonal current.
One way to understand how these
disturbances are generated and maintained is through a diagnosis of their energy processes. Lorenz
(1955) first partitioned the potential and kinetic energy equations into zonal and eddy forms in his
investigation of the general circulation of the atmosphere. He defined the concept of available
3
potential energy, which is the primary source for driving eddies and the general circulation of the
atmosphere.
Oort (1964) presented three different methods of separating the total kinetic and
available potential energy into their mean and eddy parts in space and time domains, defining
energy types and identifying the various energy conversion processes.
Considering the zonal propagation of African easterly waves, the associated energy
transformations are best partitioned into the zonal-mean state and the eddies. The energy equations
for the energetics analysis in the present study are presented in similar forms by Norquist et al
(1977), except that a more complete set of energy equations are derived here to include diabatic
generation, dissipation, and boundary flux terms for an analysis in a limited region.
The governing equations for eddy kinetic and available potential energy in an open system
are
#K E
= C k + C pk " D E + BK E + B! E
#t
(1)
"AE
= C A ! C pk + G E + BAE .
"t
(2)
and
The approximate mathematical expression of each component for the computation of energy
conversions in the averaging domain is given in the Appendix.
KE is the average eddy kinetic
energy (Eq. A4) in the average domain, and AE is the eddy available potential energy (Eq. A5). The
barotropic energy conversion, Ck, represents the sum of four barotropic conversion terms (Ck1 - Ck4)
(see Eq. A6), which capture the conversion of zonal to eddy kinetic energy through the zonal ([u])
and meridional wind ([v]) shears. Cpk is the baroclinic energy conversion (Eq. A7). This term is
associated with vertical overturning, and denotes a conversion of eddy available potential energy to
eddy kinetic energy. The reappearance of this term with opposite sign in Eq. 2 indicates that the
4
consumed eddy available potential energy is converted to eddy kinetic energy, or vice versa. DE is
frictional dissipation, which always acts as a sink of eddy kinetic energy (Eq. A10).
CA in Eq. 2 represents the conversion of zonal available potential energy to eddy available
potential energy due to eddy heat flux along the zonal mean temperature gradient (Eq. A8). GE is
the generation of eddy available potential energy by diabatic heating (Eq. A9). A positive value of
GE (generation of AE) results from heating in warmer regions or cooling in colder regions at the
same latitude (Michadelids 1992). In contrast, a negative value of GE (destruction of AE) is due to
cooling in warmer regions or heating in colder regions at the same latitude.
BKE and BAE represent boundary fluxes of eddy kinetic and available potential energy,
respectively, advecting into and out of the region of interest (Eqs. A11 and A13). BΦE denotes
boundary pressure work done by the eddies, which produces eddy kinetic energy at the boundary
(Eq. A12). From the viewpoint of the energy transport, these boundary flux terms would appear
with identical magnitudes but opposite sign in the set of energy equations for the remainder of the
atmosphere. Higher order terms involving triple products of perturbations (see Eqs. A14 and A15)
are neglected in the computation.
b. Previous applications to African easterly waves. Observational analyses of African wave
energetics are relatively rare because of the scarcity of observations in this region. Norquist et al.
(1977) estimated the related composite energy conversions using GATE data from Phase III and
found that baroclinic conversions are stronger than barotropic conversions over land, but over the
ocean the situation is reversed. They did not estimate the effects of latent heat release, frictional
dissipation, boundary fluxes and eddy pressure work on the energy transformations of the waves.
More recently, Diedhiou et al. (2002) estimated the energetics of 3-5-day and 6-9-day
easterly waves using the 1979-97 NCEP/NCAR reanalyses. They suggested that the 3-5-day waves
over land located south of the African easterly jet core mainly result from the barotropic instability
5
of the jet. In contrast, their analysis indicated that waves propagating north of the jet core, in the
Sahel-Sahara region, grow through both barotropic and baroclinic conversions below the jet level.
They did not estimate GE, DE, BKE and BAE.
A number of modeling studies have focused on the energetics of the African waves.
Rennick (1976) used a linear pseudospectral primitive equation model, and found that the primary
energy source for the waves in the model is barotropic conversions associated with the jet’s
horizontal and vertical wind shears. The underestimation of baroclinic conversions may be due to
the fact that a crude latent heat parameterization and the hydrostatic approximation are invoked.
Dissipation, which may play a role in the balance of wave energetics, was also not included in her
model.
Estoque and Lin (1977) studied the energetics of easterly waves forced by a prescribed
heating source moving westward in the tropical Atlantic in a quasi-geostrophic model. They found
that diabatic heating is the ultimate source of energy for the generation of kinetic energy in the
upper troposphere, and it is exported by wave potential energy fluxes to other levels to counteract
dissipation. However, they used a initial background state from the tropical Atlantic with smaller
meridional temperature gradients than are observed over Africa.
Thus, they may be
underestimating CA (Eq. A8) for the generation of the easterly waves over Africa. Moreover,
prescribing the heating source in their model eliminates interactions between convection and the
large-scale flow.
Thorncroft and Hoskins (1994a and b) used a spectral model governed by the primitive
equations on a sphere to investigate the linear and nonlinear behavior of the easterly-wave normal
modes. They found that the easterly waves are initially dominated by barotropic conversions in this
model, but later grow through baroclinic conversions in a nonlinear stage.
6
From these previous studies and others, it is interesting to note that simplified models with
prescribed basic state of the African easterly jet tend to find that barotropic conversions are the
main energy source of African waves (e.g. Mass 1979, Kwon 1989, Thorncroft and Hoskins 1994
ab), while the models with prescribed cumulus convective heating of ITCZ usually suggest that
diabatic heating is the ultimate energy sources for the easterly waves (e.g. Estoque and Lin 1977).
Either way of prescribing basic sates in their simple models may inherently underestimate the
instability from the other though they are all able to simulate similar wave characteristics and
structures.
Therefore, a regional climate model, which can realistically simulate these two
important background states without prescribing them in the model, is used to further explore the
energetics processes for the generation of African waves
3. Numerical simulation and Model Validation
a. Numerical simulation.
The results presented in this study are based on a realistic
simulation conducted by Hsieh and Cook (2005; hereafter, HC05). The integration analyzed here is
referred to as the realistic “climate-mode” simulation in that paper. As detailed in HC05, the
simulation was performed using a regional climate model that realistically captures the climatology
of northern Africa and the tropical Atlantic. The domain extends from 40˚E to 85˚W and 6˚S to
45˚N with horizontal resolution of 80 km (see Fig. 1 in HC05). The model uses 24 vertical ! - level
resolution from the surface to 25 hPa. A seasonal simulation from May 15th to September 14th was
carried out by updating the lateral boundary conditions for winds, temperature, and moisture every
12 hours from the climatology of the NCEP/NCAR reanalysis. Similar treatments for surface
boundary conditions of SST and soil moisture distribution were also performed.
In another study with this model, Hsieh and Cook (2006; hereafter HC06) calculated the
potential vorticity budget from the model output, and showed that the PV produced by the
7
stretching and tilting effect of the air column is well balanced by the advection and diffusion of PV.
This suggests that the model can realistically simulate the production and frictional dissipation of
the eddies. The model was run with the Blackadar high-resolution PBL scheme (Blackadar 1979),
which provides a sophisticated treatment of vertical diffusion of momentum and heat to include
convective mixing effect in the model. In addition to this PBL scheme, parameterizations include
the Kain-Fritsch cumulus convection parameterization (Kain and Fritsch 1993) with the simple ice
scheme for moisture predication and the RRTM radiation scheme (Mlawer et al. 1997).
b. Model validation. HC05 presented a detailed validation of the simulated background
climatology over Africa and the tropical Atlantic. For example, they showed that the African
easterly jet (at 13˚N near 600 hPa) and the ITCZ are simulated well in the model. In addition, the
horizontal characteristics of the wave disturbances, such as wavelength and phase speed, are also
realistic. Here, the validation emphasizes wave characteristics and structure.
To isolate the 3-5 day wave activity, model output is filtered with a 3-5-day bandpass filter
and composites are computed based on a reference point at 725 hPa near 15°N and 0°W (see HC05
for details). Fig. 1 shows the modeled 725 hPa composite winds and geopotential height deviations
for 3-5-day waves.
Darker shading, which occurs slightly ahead of the troughs over Africa,
indicates strong (i.e., magnitude greater than 0.36 hPa/hr) ascending motion averaged between 325
hPa and 725 hPa in the filtered p-velocity ( ! ). The lighter shading ahead of the ridges marks
regions of descent, with ! values greater than 0.36 hPa/hr. This suggests the organized convection
in the African easterly waves. The northeast-southwest and southeast-northwest tilt of the wave
axes on either side of the jet is consistent with the composite waves (Fig. 4c) of Reed et al. (1977).
The 3-hourly time series of the filtered meridional wind at the reference point (15°N-0°W) is
shown in Fig. 2. Values greater than 1 ms-1 are marked with open circles and are selected to
compute the composites. Based on this selection criterion, there are totally about 18 significant
8
waves to make up the composite waves. Evidently, the selected wave disturbances in August are
weaker and thus have less contribution to the composite waves.
Fig. 3a displays the longitude-pressure cross-section of the composite 3-5-day meridional
winds and geopotential heights along 12°N. The level of the maxima of the 3-5-day meridional
winds increases from nearly below 700 hPa to 625 hPa in propagating from east to west. Fyfe
(1999) also studied the vertical wave structure of African waves using the ECMWF and the NCEP
reanalyses. He also found that the elevation of the perturbation maxima of the waves increases
from the lower to mid-troposphere as the waves propagate from Central Africa to the Atlantic.
Secondary maxima near 200 hPa with reversed meridional wind perturbations are due to
anticyclonic (cyclonic) divergent (convergent) flows near the tropopause, in agreement with Reed et
al.’s finding (1977). The maxima of the 3-5-day meridional wind perturbations and geopotential
height oscillations are over West Africa (5°E~15°W).
Note that the meridional wind contours (Fig. 3a) tilt eastward with height beneath 700 hPa
over Africa and the eastern Atlantic (20°W), and tilt westward with height above, in agreement with
Burpee’s observational analysis (1974, 1975). This vertical tilt, combined with the horizontal tilt
shown in Fig. 1, suggests that both Ck and CA are positive. The weaker vertical tilt of the wave axes
over the ocean (west of 30˚W) suggests that CA is weaker as the waves propagate off Africa.
The composite contours of the vertical velocity perturbation are shown in Fig. 3b. A
comparison with Fig. 3a shows that, below the level of the jet where meridional temperature
gradients are positive, the southward flow west (or ahead of) of the trough is rising, and the
northward flow east of the trough (ahead of the ridge) is sinking. The maxima rising motion
between 5˚W and 15˚W is located around 700 - 800 hPa, consistent with the observations of Reed et
al. (1977).
9
Fig. 3c shows composite geopotential height deviations from the zonal mean. A comparison
with Fig.3a and b shows that the geopotential height perturbations are out of phase with the
merdional wind perturbations but in-phase with the vertical velocity, in agreement with
observations (Reed et al. 1977). The maximum geopotential height perturbations also move upward
somewhat as the waves propagate to the west. According to the geopotential tendency equation, the
local change of geopotential basically is determined by the distribution of vorticity and diabatic
heating. The deep moist convection associated with condensational heating south of the jet induces
negative pressure perturbation. In addition, the cyclonic vorticity on the southern side of the jet,
enhanced by the vertically differential heating due to convection, also causes the negative
geopotential perturbations near mid-troposphere.
Temperature deviations are contoured in Fig. 3d. Below the level of the jet (about 600 hPa),
the basic state temperature gradient in positive, and above the jet it is negative. Due to this reversal
in the meridional temperature gradient, the southward wind perturbation ahead of the trough near
5°W-10˚W (Fig. 3a) advects warm air (Fig. 3a) below the jet and cold air above. Thus, the
composite wave has a warm core ahead of the trough below the jet and a cold core at 500-600 hPa
between 20˚W and 15˚W, as shown in Fig. 3d. Likewise, the northward wind perturbation ahead of
the ridge at 20°W causes a cold core below the jet and warm core at 500-600 hPa between 25˚W
and 35˚W. Reed et al. (1977) showed similar distributions of vertical velocity and temperature
deviations.
The cold advection (10˚W-0˚W) in the trough deepens the trough at the jet level (600 hPa) at
5˚W, and the warm advection in the ridge builds the ridge at the same level near 20˚W. Thus the
eastward tilt of the wave troughs below the jet level, in the presence of the westward zonal mean
flow, releases zonal available potential energy to the developing waves, corresponding to a positive
CA. The temperature advection above the jet has the opposite effect to that below the jet, and this
10
offsets the oscillations of the geopotential heights above the jet. Temperature deviations above 500
hPa are weak, suggesting weak thermal advection and zonal mean temperature gradients above this
level.
Therefore, thermal advection in the lower troposphere is mainly responsible for the
baroclinic processes that convert zonal to eddy available potential energy.
We conclude from this analysis that the simulated wave structure is consistent with the
observations. As discussed above, the model must correctly simulate the details of the wave
structure, including the vertical tilt of trough and ridges, to be suitable for the energetics analysis
since these are related to the wave energetics.
4. Results
The evolution of wave events in the model is examined using wavelet analysis in the model
output to better understand how the developing waves change with time. Fig. 4a and b shows the
time series of the daily mean meridional winds at 725 hPa near 12˚N-0°W and the corresponding
wavelet transform (Hauf et al. 1996, Torrence and Compo 1998, Nappo 2002), respectively.
Wavelet transform is computed using the complex Morlet wavelet with smallest time dilation of 1
day.
The Morlet wavelet represents a sine wave with amplitudes modulated by a Gaussian
envelope. The convolution of the signal with the Morlet wavelet with scales of time dilation from 1
to 10 days acts as a bandpass filter for the meridional wind signal.
The wavelet modulus, shown in Fig. 4b, represents the amplitudes of the meridional wind at
725 hPa near 12°N-0°W at different periods (2-10 days) and time. Two primary groups of waves
are simulated in the model, consistent with the previous findings (Vitard et al. 1997, Diedhiou et al.
1998). One has periods between 3 and 5 days; the other has periods between 6 and 10 days. The 610-day waves in the model are more active in June and the beginning of July and then decay from
mid-July to the beginning of September. The 3-5-day waves are more persistent in late June and
July and recurrent in September. Near July 20, significant modes with periods from 2.5 - 7 days are
11
in phase and lead to a large fluctuation of the meridional wind, as shown in Fig. 3a. The 3-5-day
waves are weaker during August in comparison with that in late June and July and reinvigorate in
September.
To examine if the wave activity is associated with moist convection over Africa, the time
series of the modeled rainfall averaged between 10°W and 20°E in longitude and 9°N and 15°N in
latitude is shown in Fig. 5. The area average rainfall is calculated to reflect the intensity of ITCZ
convection over the region near the African easterly jet though the modeled rainfall near 0°E-12°N
shows a similar trend. It can be observed that intense rainfall events are closely associated with the
occurrence of active waves, as shown in Fig. 4b. This is consistent with the observation conducted
in the jet2000 project (Thorncroft et al. 2003). They found practical difficulty diagnosing the
African waves over Africa even though they observed a strong African easterly jet during their
experiment, which happened to be in an anomalous dry period.
Fig. 6 displays the geographical distributions of daily mean precipitations for major rainfall
events occurring at 30 June, 17-21 July and 6 September shown in Fig. 5. As shown in Fig. 6a, an
undulating strip of the modeled precipitation at 30 June can be clearly observed from Central to
West Africa, a prominent signature of African waves. Starting from 16 July, more precipitation
falls south of 12°N with a concentration near 5°E-9°N and then spreads along 9°N at 17 July,
suggesting a continuous concentrated moist convection on the southern side of the jet though the
overall average precipitation is not high. The intense precipitation falls near 10°E-10°N at 18 July
and then migrates westward in about 5 degree per day. The westward moving precipitation attains
its maximum at 20 July, which brings a high precipitation rate up to 43 mmday-1 near 0°E-12°N, as
shown in Fig.6f. The heavy rainfall starts to decrease and continue moving westward to 5°W
around 21 July (Fig. 6g). The precipitation intensifies and is organized between 10°N and 15°N
12
when moving westward, suggesting the enhanced precipitation coinciding with the wave trough.
Near the end of the rainfall event in early September shown in Fig. 6h, a wavy precipitation strip is
again observed over West Africa. To understand the processes for the generation of African waves
in association with these heavy rainfall events, a temporal energetics analysis is performed in what
follows.
Computations of the 3-5-day wave energetics are carried out in the region from 5°N to 25°N
in latitude and 10°W and 20°E in longitude over Africa because the waves originate near 20°E and
are strongest near 10°W in this region (Fig. 1). Fig. 7a displays the time series of Cpk and Ck (see
Eqs. A6 and A7) averaged over this analysis region. Fig. 7b shows the evolutions of Ck1 and Ck2,
the first two leading terms of Ck. Cpk is the dominant energy source for these waves in general. Ck
usually tends to increase when Cpk is increasing. Around 23 June, Ck rapidly increases to the same
magnitude of Cpk, which leads to the generation of significant waves though there is no very intense
precipitation during this period. The in-phase oscillation between Cpk and Ck does not lead to
stronger Cpk or Ck, suggesting no resonant effect for the generation of these waves. As shown in
Fig. 7b, the major part of Ck is contributed by a large production of Ck2, which is due to the strong
shallow cumulus convection beneath the jet from late June to early July. This indicates that Ck2 can
be even greater than Ck1 when the shallow cumulus convection at lower levels is strong. The
African easterly jet becomes meridionally broader from 21 to 27 June and attains its maximum
zonal wind speed of 22 ms-1 with significant vertical shear above and below the jet core around 27
June, indicating that the strength of the jet core is not weakened by the waves. With this enhanced
jet, barotropic conversions Ck continues decreasing until 1 July, suggesting that a strong jet in late
June is not the main cause of large production of Ck.
Note that during late June Ck1 on the northern side of the jet is stronger than that on the
southern side of the jet. A potential vorticity gradient reversal on the northern flank of the jet is also
13
observed during this period (not shown). This may be caused by a strong dry convection at low
levels from the Sahara or the formation of squall lines near north of the jet (Diongue et al. 2002).
After early July, the barotropic conversion Ck1 is mainly located on the southern flank of the jet.
This is in agreement with Burpee’s findings (1972), suggesting that at 700 hPa the direction of eddy
momentum transport changes from a poleward transport of easterly momentum in June to an
equatorward transport of easterly momentum in July, August and September.
Cpk continues to grow until 1 July due to a rainfall event with area average precipitation rate
over 6 mmday-1 around 30 June (see Fig. 5). The baroclinic overturning Cpk in association with this
rainfall event is the main energy source that sustains the waves near the beginning of July. This
suggests that African waves without barotropic conversions can still organize rainfall in a wavy
strip over West Africa, as shown in Fig. 6a. Note that Ck1 is slightly negative near 30 June and 1
July, indicating that horizontal eddy momentum flux is consumed to strengthen the zonal mean
flow.
Near mid-July, moderate waves are primarily maintained by Cpk since Ck is weak, which
corresponds to a relatively low precipitation during this period. As shown in Fig. 7b, Ck is mainly
dominated by Ck1 after mid-July, suggesting that shallow cumulus convection becomes weaker.
Around 16 and 17 July, Ck starts to increase significantly on the equatorward side of the jet when
the precipitation is more concentrated near 9°N, as shown in Fig.6b and c. An analysis of the
Eliassen-Palm flux indicates that the rapid growth of Ck1 is associated with the strengthening of
potential vorticity gradient reversals around 17 July (not shown), corresponding to an increase of
the cyclonic shear near on the southern side of the jet. The increase of the cyclonic shear is due to
the eddy forcing in association with the concentrated convective heating (see Fig. 6c, d) on the
zonal mean flow. As noted by Schubert et al. (1991), an unstable zonal flow can be expected when
potential vorticity gradient is significantly reversed by typical convective heating in just a couple of
14
days, which enhances the easterly flow and westerly flow on the northern and southern sides of the
potential vorticity maximum (near 9°N in this case), respectively.
The rapid increasing Ck1 also feeds back to organize and enhance convection south of the jet
when there is enough moisture supply from ITCZ. Cpk becomes in phase with Ck around 18 July,
which leads to even stronger Cpk at 20 July through resonance, corresponding to very intense
rainfall between 12°N and 15°N shown in Fig. 6f. The resonance between Cpk and Ck indicates a
strong coupling between the waves and deep convection south of the jet. Ck does not further
increase and is mainly maintained at its peak values with small oscillations with Cpk until 22 July.
The cyclonic shear of the jet is slightly weakened by Ck at 19 July but the strength of the jet core,
migrating northward to 15°N, mainly remains at 14 ms-1. This is due to the strong deep organized
convection south of 15°N, which contributes to maintaining part of the jet’s strength and thus its
horizontal and vertical shears. The waves decay rapidly when the deep convection weakens. Note
that near 22 July the zonal wind of the jet core near 600 hPa drops from14 ms-1 to12 ms-1 and thus
the shears of the jet further relax. This is mainly due to the diminishment of the contribution from
intense convection and the barotropic energy conversion from the jet.
In general, Cpk and Ck both are weaker in August than in June and July but still show similar
trend with the average precipitation rate south of the jet. Note that baroclinic overturning and
barotropic conversions slightly increase near 6 August, corresponding to a period of low
precipitation (see Fig. 5). This is mainly due to the energy conversions from Ck2, Ck1 (slightly
excited by the rainfall near 4 August), CA2 and CA1 (see Fig. 8b), which suggests the baroclinic
instability north of the jet at low levels. This can explain why around 6 August a significant wave
signal near 15°N appears (see Fig. 2) while the overall wave activity is not so strong as that in late
June and July. From mid-August to late August, Cpk and Ck are both weak although average
15
precipitation rate is increasing. ITCZ convection is not concentrated to significantly strengthen the
potential vorticity gradient south of the jet though the overall precipitation rate is significantly high.
Near the end of August, Ck starts to increase and oscillate with Cpk again when Cpk is
increasing, which is associated with the increasing rainfall (convection), shown in Fig. 5.
Barotropic conversions Ck1 rapidly increase over 0.03 Wm-2 near 4 September as the potential
vorticity gradient is strengthened due to a concentrated convective heating near 11°N. A clear wavy
strip of rainfall starts to form over West Africa at 5 September and migrate westward as Ck1 attains
its maximum around 6 September (see Fig. 6h). Both of Cpk and Ck start to decrease as the moist
convection is weakening. Note that the significant precipitation rate from late August to early
September does not correspond to a production of Cpk as large as that in June and July. This is
partly due to less eddy available potential energy provided by baroclinic instabilities (CA1 and CA2,
see Fig.8b) in August and September. The ITCZ convection also does not contribute to the
production of GE as much as that in June and July, suggesting that ITCZ convection in late August
has weaker interaction with the waves.
Fig. 7c shows the time series of DE, BKE and B"E . (Eqs. 1 and A10, A11 and A12). Large
dissipation (thick solid line) due to friction and small-scale turbulence slightly lags behind the
!
occurrence of large barotropic and baroclinic
conversions (thick solid line in Fig. 7a).
The
dissipation peak in September, corresponding to the significant waves shown in Fig. 4b, shows a
more significant time lag from the peak time of energy production in Fig. 7a because their
generation and dissipation rates are both smaller in comparison with the waves in June and July.
The evolution of BKE and B! E are shown in thin solid line and dash line, respectively. The
oscillation of BKE in negative range indicates that more eddy kinetic energy propagates westward
out of the averaging domain than propagates into the domain. The oscillation of B! E around zero
16
suggests that boundary pressure work can act as a minor source or sink of eddy kinetic energy. In
general, the net effect of these two boundary terms transports the eddy kinetic energy out of the
averaging domain.
Fig. 7d displays the balance between the time rate of change of eddy kinetic energy (lhs of
Eq. 1) and the net energy production rate (rhs of Eq. 1). The agreement between the two sides of
the energy equation evidently shows how these modeled African waves are generated, transformed
and dissipated temporally through the above energy processes. The major deviations around 16
June and 20 July suggests strong nonlinear effect of the waves (Eq. A14). The one near 16 June
indicates that nonlinear interaction acts as the main energy source for the waves at decaying stage,
which also causes large dissipation shown in Fig.7c. Another significant nonlinear effect occurs
near 20 July when the waves are strongly coupled with convection.
Fig. 8a displays the time series of the generation of eddy available potential energy by
diabatic heating (GE, defined in Eq. A9). The evolution of GE with time is similar to that of Cpk,
suggesting that the eddy available potential energy (AE) consumed by Cpk is partly compensated
immediately by the generation of AE by diabatic heating. GE can sustain the waves to continually
trigger other instabilities to occur (e.g. Ck1 and CA). Other sources of AE are CA1 and CA2 (Eq. A8),
i.e., the conversions from zonal available potential energy. These are displayed in Fig. 8b. CA1
shows strong diurnal oscillations, which is due to the strong diurnal response of the low-level static
stability (see Eq. A8). CA2 has a weaker diurnal response since most of this energy conversion
occurs in the upper troposphere where the diurnal effect is weaker.
Fig. 8c displays the time series of BAE (Eq. A13). The nearly zero values of BAE indicate no
contribution to the domain’s average eddy available potential energy. Fig. 8d shows that there is a
good correspondence between the time rate of change of AE and the net production rate of AE (the
rhs of Eq. 2). The net production of AE appears with significant diurnal oscillations within the
17
domain, suggesting that most of the residual AE is generated by CA1 in the lower troposphere. GE
and CA, the sources of AE, are balanced by Cpk as the sink of AE for the 3-5-day waves over Africa.
Fig. 9 shows the seasonal mean energy cycle averaged from June 15th to September 14th.
The seasonal mean rates of change of eddy kinetic and available potential energy in Eqs. (1) and (2)
are close to zero. Both of the mean eddy energies (KE, AE) are small in comparison with the
corresponding mean zonal mean energies (KZ, AZ). The positive CZ (Eq. A3) suggests that the
conversion of AZ to KZ due to the meridional overturning is an important energy source for the
maintenance of zonal mean flow. The mean kinetic energy KZ is about triple the mean available
potential energy AZ. On the other hand, the seasonal mean KE is more than triple the seasonal mean
AE. This is different from typical mid-latitude baroclinic systems where the available potential
energy greatly exceeds the kinetic energy (Holton 1972). Note that the energy diagram is not fully
complete due to the incomplete part of the energetics of zonal mean flow. However, this would not
affect the results about the energetics of African easterly waves since the part of eddy energy is
complete.
The eddy kinetic energy is mainly maintained by Cpk. Barotropic conversions Ck, which is a
minor source of KE, is about two thirds of the magnitude of Cpk. Dividing KE by its net production
rate (Ck + Cpk - DE ) gives a doubling time of about 4 days for KE, which is approximately the time
for the waves to propagate westward with phase speed of 6~8 m/s (5~6° longitude day-1) from
Central to West Africa. The increase of eddy kinetic energy during the doubling time is removed
from the averaging domain by the boundary flux of eddy kinetic energy and the boundary eddy
pressure work (0.008 W/m2 and -0.0002 W/m2, respectively). The slight imbalance of the seasonal
mean eddy kinetic energy budget is due to the higher order nonlinear terms (Eq. A14). Without
frictional dissipation, the total production rate of Ck and Cpk would double the eddy kinetic energy
in about 1 day, faster than the estimate (1.7 days) of Norquist et al. (1977).
18
AE is maintained by CA and GE. The average magnitude of CA is approximately twice that of
GE. The doubling time for total eddy energy, estimated by dividing AE+KE by CA+GE+CK-DE, is
approximately 5.2 days. This is longer than doubling time for KE since most of the produced AE is
converted into eddy kinetic energy directly. In summary, the seasonal mean energy cycle shows
that African easterly waves are maintained by a balance between the main energy sources Cpk (about
3
2
KE), the secondary source Ck ( KE) and dissipation DE. AE is replenished by CA and GE .
5
5
To further examine whether the mechanism that initiates African waves is different from
!
that over our analysis !
domain covering West Africa, a similar analysis is performed for Cpk and Ck
over a smaller domain between 5°N and 20°N in latitude and 15°E and 30°E in longitude, the origin
of African waves as noted by Burpee (1972). As shown in Fig. 10, Cpk is the dominant energy
source throughout the analysis period for the domain over East Africa since Ck is even weaker than
that over Central and West Africa. Over East Africa, Ck2 generally appears more important than
Ck1. This suggests the importance of shallow cumulus convection for the initiation of African
waves and the lack of deep convection over Central and East Africa. This is in agreement with
Fig.3a, showing the maximum perturbations at lower levels near Central Africa. This indicates that
these modeled African waves are not initiated by barotropic conversions of the jet. For the domain
between 15°E and 30°E, the seasonal mean of Cpk is about twice the seasonal mean of Ck.
The spatial distributions of the waves’ energy sources and sinks may offer insights into how
the energy conversions occur to energize the waves. Figs. 11a and b display the vertical and
meridional distribution of the two major terms of the barotropic conversions, Ck1 and Ck2 (see the
integrands in Eq. A6). The maximum of Ck1 distribution is located at 8.5˚N between 600 hPa and
700 hPa. This major production of Ck1 on the southern flank of the jet, together with a smaller
region of production at 700 hPa near 17.5˚N, leads to the northeast-southwest and slight southeast-
19
northwest tilts of the wave axes on either side of the jet, as shown in Fig.1. The downgradient
(
![u ]
)
!y
eddy momentum flux ( [u !v !] ) converts zonal kinetic energy to eddy kinetic energy in the mid-
troposphere. Two other minor production regions of Ck1 appear near the surface (20˚N) and the
tropopause (6˚N). The one near the tropopause is associated with anticyclonic divergent flow of the
ITCZ and the large shear of the tropical easterly jet, and the one near the surface is associated with
the surface confluent zone of the ITCZ between 18˚N and 21˚N. The smaller Ck1 near 18°N at 700
hPa mainly results from the contribution in late June, as discussed previously in Fig. 7a.
The distribution of Ck2 (Fig. 11b) is primarily confined below 700 hPa where there are large
vertical shears below the jet. Ck2 is a result of the downgradient transport of easterly momentum,
[u "! "] , interacting with the vertical shear
![u ]
.
!p
Under the quasi-geostrophic condition, Ck2 is rather
small. The secondary maximum of Ck2 near 700 hPa at 10°N suggests its close association with
convection since the vertical wind shear near 10°N is relatively weak as compared with that beneath
the jet. Near the tropics over Africa, however, Ck2 can be more important than Ck1 especially when
there is strong cumulus shallow convection beneath the jet.
Another energy conversion associated with the vertical zonal wind shear is CA1 (Eq. A8), the
leading term in the conversion of zonal to eddy available potential energy. The distribution of CA1,
as shown in Fig. 11c, is mainly located in the lower troposphere with maximum centered near
16.5˚N at 850 hPa. Thermal advection, in association with correlations between southward heat
fluxes [v !T !] and large temperature gradients, converts AZ to AE. Note that negative values of CA1
distribution are observed above 800 hPa with locations varying from 10-12°N to 15-18°N when the
waves are decaying, suggesting some feedback to the zonal mean flow. However, the overall
conversion rate is still positive.
20
Fig. 11d displays the vertical and meridional distribution of CA2 (the integrand in Eq. A8),
which is a measure of the correlation between the vertical heat flux [" !T !] and the vertical gradient
of zonal mean temperature deviations (
![T ]"
!p
). Below 800 hPa over the Sahara, the occurrence of
negative values of [" !T !] is in association with the ascent of relatively warm, dry air from the north
or the descent of cool, moist air from the south (depending on the phase of the wave). This
contributes to the negative values of CA2 near the surface over the Sahara by correlating "[# !T !] > 0
with negative values of
![T ]"
!p
. These increasing zonal mean temperature deviations ( [T ]! ) with
increasing height over the Sahara are associated with a more uniform surface temperature
distribution north of 15°N.
In the upper troposphere between 200 hPa and 300 hPa, the condition is reversed because
"# !T ! > 0
correlates with positive values of
available potential energy. Positive
![T ]"
!p
![T ]"
!p
, leading to the positive conversion of zonal to eddy
at these levels results from a decrease of [T ]! with
increasing height, which is associated with a more uniform condensational heating at these levels.
As noted by Norquist et al. (1977), satellite-derived IR brightness decreases less rapidly with
latitude than precipitation observed at the ground, indicating the presence of clouds in upper levels
north of the ITCZ.
As discussed above and from Eq. A8, CA2 is closely associated with overturning baroclinic
conversions, Cpk. The distribution of Cpk averaged between 10°W and 20°E, shown in Fig. 12a,
indicates that there is positive conversion (i.e., the generation of eddy kinetic energy by baroclinic
overturning) in the lower and upper troposphere, and a region of negative conversion (the
destruction of eddy kinetic energy) centered near 700 hPa at 10°N.
21
The region of negative
conversion slopes upward to the north.
This distribution is in agreement with observational
analyses (Norquist et al. 1977, Pytharoulis and Thorncroft 1999).
The positive conversions near 15˚N in the lower troposphere convert most of the AE
generated from CA1 into eddy kinetic energy directly. As noted by Reed et al. (1977), northerly
(southerly) winds are associated with relatively warm and dry ascending (cool and moist
descending) air. This suggests positive integrands of CA1 ( "v !T ! > 0 ) and Cpk ( "# !T ! > 0 ) at the same
time and same region. Large baroclinic overturning occurs from 500 hPa to 200 hPa between 9°N
and 12°N and above 300 hPa between 9˚N and 24˚N. This is mostly the result of warm ascending
(cool descending) air associated with latent heat release (evaporative cooling) due to convection
above the mid-troposphere in the wave troughs (ridges). The release of latent heat in convective
precipitation regions has been diagnosed as a major source of wave energy at lower latitudes (Nitta
1972, Holton 1972).
Negative baroclinic conversions result from perturbed ascending cold (descending warm)
air, which converts the eddy kinetic energy to eddy available potential energy. Yanai (1961)
suggested that ascending motion in easterly waves is accompanied by cold air, and that dynamical
forcing induces this cold ascent. Norquist et al. (1977) suggested that evaporative cooling in
convective downdrafts may be responsible for the association of negative temperature perturbations
with large-scale ascending motion. Diedhiou et al. (2002) also showed negative values of the Cpk
integrand from 800 hPa to 600 hPa between 5˚N and 10˚N over Africa and further west at 30˚W.
Fig. 12b displays the distribution of GE (the integrand of Eq. A9), which is another source of
AE. The similarity between the spatial distributions of GE and Cpk (Fig. 8a) suggests that the eddy
available potential energy consumed by baroclinic overturning in the upper troposphere is
supplemented by diabatic heating. Negative values of the GE integrand centered at 10˚N between
800 hPa and 600 hPa denote the destruction of eddy available potential energy, implying ascending
22
cold air ( T ! < 0 ) with the release of latent heat ( Q ! > 0 ) or descending warm air ( T ! > 0 ) with
evaporative cooling ( Q ! < 0 ). In the wave troughs, latent heat liberation below 600 hPa and the
advection of relatively warm air from the north are not strong enough to warm this cold region with
ascending motion. Above 600 hPa, warm ascending air due to large amounts of condensational
heating leads to the generation of eddy available potential energy in the upper troposphere. The
positive values of Cpk distribution between 200 hPa and 300 hPa are approximately balanced by the
sum of the positive GE and CA2 distributions at these levels.
In the upper region of negative GE integrand near 15-20ºN, evaporative cooling ( Q ! < 0 ) is
associated with descending warm air ( T ! > 0 ), opposite to the release of latent heat ( Q ! > 0 ) in
association with ascending cold air ( T ! < 0 ) near 700 hPa at 10˚N. The negative values of GE
integrand below 800 hPa at 18˚N are a consequence of ascending warm air in the southward flow
( T ! > 0 ) associated with diabatic cooling ( Q ! < 0 ), or descending cold air ( T ! < 0 ) in northward flow
with diabatic heating ( Q ! > 0 ). Both of these lead to the destruction of eddy available potential
energy.
In summary, the evolution and spatial distributions of Cpk, CA1, CA2, and GE suggest that most
of the AE generated by CA1, CA2 and GE is directly converted to KE. This is why the boundary flux
of eddy available potential energy is nearly zero all the time. There is not much storage of AE
during the energy transformation processes, an important characteristic of energy conversions for
equatorial wave disturbances.
Fig. 12c displays the seasonal mean distribution of frictional dissipation, DE. The maximum
dissipation is located in the lower troposphere over the Sahara, in association with a thicker
boundary layer and stronger vertical mixing of momentum and heat. The dissipation maximum
near 700 hPa between 9˚N and 12˚N is associated with the greater depth and strength of convection
23
in that region. The larger values of DE near 200 hPa occur because of dissipation by eddies in the
anticyclonic divergent outflow at the top of the deep convection. The lower troposphere near the
ground acts as the major sink of the wave kinetic energy.
Fig. 12d shows the net zonal flux of eddy kinetic energy and boundary eddy pressure work
on the zonal boundaries. Negative values indicate a net westward flux of eddy kinetic energy and
eddy pressure work out of the averaging domain. The maximum net westward flux is located at 600
hPa, indicating strong zonal advection by the African easterly jet at that level. Similarly, the
negative flux near 200 hPa is associated with strong zonal advection by the tropical easterly jet.
The positive flux at upper levels near 23°N suggests more westward flux of eddy kinetic energy
entering the domain than leaving the domain. The net westward flux of eddy kinetic energy and
eddy pressure work indicate the westward propagation of wave energy, suggesting a westward
group speed.
Figure 13 displays the seasonal mean eddy kinetic energy distribution averaged between
10°W and 20°E. African waves are active at lower levels below 800 hPa north of the jet and near
700 hPa south of the jet, in agreement with Pytharoulis and Thorncroft’s observational analysis
(1999). In addition, active waves are also observed between 700 and 800 hPa beneath the jet. A
significant amount of eddy kinetic energy also appears near 200 hPa, consistent with the secondary
meridional wind maxima near 200 hPa shown in Fig. 2a and Reed et al. (1977). The eddy kinetic
energy generated by Cpk and the Ck is redistributed by advection and eddy pressure work in
association with baroclinic overturning (- " !T ! > 0 ), e.g. " #! # and " !v ! (see A12) (in the meridional
cross section), leading to maximum wave amplitudes at upper levels above 300 hPa and below 500
hPa.
An analysis of the Eliassen-Palm cross-section (not shown) suggests that wave energy
generated in the upper troposphere propagates downward through eddy pressure work.
24
The
distribution of wave kinetic energy is also in agreement with the finding of Holton (1971), who
applied a prescribed oscillatory heating with a period of 4 days and a wavelength of 4000 km in a
linear model to initiate forced Rossby waves driven by the release of latent heat. He found that the
maximum wave amplitudes are located above and below the prescribed heating. The meridional
distribution of the eddy kinetic energy, which shows different characteristics from that (see Fig. 3)
of Thorncroft (1995), suggests that these waves are not caused by the shear instabilities of the jet as
shown in his idealized simulations.
To explore the dispersion characteristics of African easterly waves, a space-time spectral
analysis (Wheeler and Kiladis 1999) is conducted. Figure 14 displays the latitudinal mean spectrum
of the meridional wind at 725 hPa between 5˚N and 20˚N on the wavenumber-frequency domain.
Major spectral peaks are located between frequencies of 0.33 day-1 (period of 3 days) and 0.2 day-1
(period of 5 days) and between wavenumbers of 15 (wavelength of 2650 km) and 10 (4000 km).
The dispersion relation curves for mixed Rossby-gravity waves (MRG) and equatorial Rossby
waves (ER) from the linear shallow water equation are superimposed. The equivalent depths of the
dispersion curves range from 25 m to 400 m, which are typical equivalent depths for the tropical
troposphere (James 1994, Wheeler and Kiladis 1999). The spectral peaks between 3 days and 5
days are not located on these linear dispersion curves, indicating that African waves cannot be
explained by linear wave theory. As shown in our energetics analysis, African waves demonstrate
significant nonlinearity when the waves are strongly coupled with convection.
From the
wavenumber-frequency spectra, these waves can be categorized as tropical depression (TD)-type
disturbances (Wheeler and Kiladis 1999), which usually prevail in the tropical western north Pacific
(Takayabu and Nitta 1993).
Some spectral power, between wavenumbers 10 and 5 (4000~8000 km wavelength) and
frequencies 0.17 and 0.25 CPD (6~4-day period), is close to the region where the dispersion curves
25
of MRG waves are located. This suggests that some disturbances over Africa and the tropical
Atlantic may possess hybrid features of TD-type disturbances and MRG waves.
Such
characteristics of these hybrid modes can also be seen in Fig. 4a of Diedhiou et al. (1998). For
example, the cyclonic/anticyclonic circulations on the equatorward side of the jet over Africa,
which usually have larger wavelengths to the south, may extend to 10˚S and exhibit the
antisymmetric features of MRG waves with respect to the equator. As the waves propagate off the
equator to the Atlantic, the TD-type disturbances with shorter wavelengths (about 3000 km) and
higher frequencies (near 0.3 CPD) usually dominate.
The disturbance usually has a hybrid
structure of the two types in the process of transformation (Takayabu and Nitta 1993).
The fact that there is some spectral power near 8 and 10 days (Fig. 14) indicates that 6-9-day
waves (Diedhiou et al. 1998, 1999) may be classified as symmetric equatorial Rossby waves (ER)
with equivalent depths of about 200-400 m. This suggests that 6-9-day waves can be explained by
linear wave theory. This may be due to their less effective convergence, leading to weak nonlinear
effect in the planetary boundary layers. Such waves are symmetric about the equator, and they can
be seen in the 6-9-day composite waves in Fig. 4b of Diedhiou et al. (1998).
As noted by Takayabu and Nitta (1993), off-equatorial TD-type disturbances (easterly
waves) usually posses organized convection in correspondence with low pressure cyclonic vortexes,
in agreement with the African waves shown in this study. This suggests that African easterly waves
possess very similar characteristics of other easterly waves in western Pacific, which are mainly
maintained by baroclinic conversion through convection.
5. Summary and conclusions
The evolution and spatial distribution of 3-5-day wave energetics over Africa are
investigated using a regional climate model. The complete eddy energy equations for an open
26
system are derived for the computation of energy transformations and transport for the wave
generation and dissipation. The inclusion of the energetics terms that have been previously ignored
provides the way to examine the balance of eddy energy equations, which makes a complete
energetics analysis possible. It is found that baroclinic overturning is the dominant energy source
for the wave generation although barotropic conversions can be equally important when there is
concentrated ITCZ convection nearby or strong shallow cumulus convection beneath the jet. The
generation of active waves in the model is usually associated with the occurrence of the intense
rainfall events over Africa, suggesting a close relation with the moist convection of ITCZ. Based
on our energetics analysis, the principal processes responsible for the generation of African waves
over Africa and their significances are summarized as follows.
(1) The modeled waves generated in late June are caused by baroclinic overturning through
convection and about the same magnitude of barotropic conversions. From late June to early July,
barotropic conversions associated with the vertical wind shear is even greater than that associated
with the horizontal shear. This is due to the active shallow cumulus convection beneath the jet,
which is usually underestimated by the simplified models based on quasi-geostrophic or hydrostatic
assumptions. Near the beginning of July, the waves are excited by baroclinic overturning through
the moist convection and sustained mainly by barotropic conversions associated with the vertical
zonal wind shear due to active shallow cumulus convection.
(2) Near the mid-July, the waves are initially excited by overturning baroclinic conversions with
weak barotropic conversions. As the moist convection concentrates near 9°N, barotropic
conversions rapidly increase due to the strengthening of the potential vorticity gradient (the increase
of the horizontal wind shear) on the southern side of the jet. The large barotropic conversions feed
back to enhance baroclinic overturning and oscillate in phase with it. This leads to the occurrence
of an even larger baroclinic overturning, corresponding to the formation of organized deep
27
convection migrating with the waves through the resonance between baroclinic overturning and
barotropic conversions. The occurrence of such strong organized convection may depend on the
strengths of barotropic conversions and moisture convergence from ITCZ.
(3) Near the end of August and early September, barotropic conversions on the southern side of the
jet gradually increase and oscillate nearly in-phase with baroclinic overturning due to the increase
of rainfall. However, the induced barotropic conversions do not feedback to enhance the ITCZ
convection.
(4) About three fifths of the seasonal mean eddy kinetic energy is produced from baroclinic
conversions through convection and two fifths is obtained through barotropic conversions for the
region between 10°W and 20°E.
Over East Africa, the origin of African waves, barotopic
instability is even weaker and always smaller than baroclinic overturning throughout the summer.
This suggests that in our model simulation African easterly waves over East Africa are not initiated
by the barotropic instability of the jet.
(5) The spatial distribution of the braoclinic overturning shows great similarity with that of the
diabatic generation of eddy available potential energy, suggesting that the eddy available potential
energy consumed by baroclinic overturning is compensated directly by the generation of eddy
potential energy due to the convective heating. The convective heating acts as an important role to
sustain the baroclinic overturning and continually trigger other instabilities to occur.
(6) A space-time spectral analysis indicates that African waves mainly posses westward
wavenumbers between 15 and 10 (wavelengths of 2650-4000 km) and periods from 3 to 5 days, in
agreement with the spectral distribution of tropical depression (TD)-type disturbances (Wheeler and
Kiladis 1999). In wavenumber-frequency space, 3-5-day waves are not located along the dispersion
relation curves of the linear wave theory with realistic equivalent depths, suggesting strong
nonlinearity of these African waves.
28
The results in this study show that active waves are mainly generated by convectively
induced unstable zonal flow, different from many previous investigations based on the simplified or
idealized models. The occurrence of significant barotropic conversions is closely associated with
convection, i.e. deep moist convection or shallow cumulus convection. Note that many previous
studies (e.g. Simmons 1977, Kwon 1989, Thorncroft and Hoskins 1994ab, Thorncroft 1995) have
been able to simulate African waves with similar characteristics (e.g. wavelength, tilt, propagation
speed, doubling time). This may partly be due to the mean zonal wind profile of the African
easterly jet prescribed in their models, which reflects an important background climatology over
Africa. However, the African easterly jet may play a role in setting the scale of the waves but may
not be directly involved in their generation (Hsieh and Cook 2005).
As noted by Thorncroft (1995), the role of diabatic heating is not well understood and more
realistic basic states including both their time and zonal variations should be included in future
work.
With these improvements, the processes that initiate and maintain African waves are
investigated through a detailed energetics study. Though the results are based on numerical
modeling, the energetics analysis of these simulated waves offers more insights into how the energy
of these waves is produced and dissipates as a function of time and space. Such an analysis is not
possible with existing observational systems and reanalyses.
Acknowledgements: The authors would like to thank Dr. Peter Gierasch for helpful discussions and
suggestions for this study, and Dr. Christopher Thorncroft for his valuable comments on this paper.
The assistance of Dr. Edward Vizy in the modeling of West Africa climatology is also appreciated.
This paper is also greatly improved by two anonymous reviewers’ detailed comments.
research was supported by NOAA’s CLIVAR Atlantic Program (Award NA16GP1568).
29
This
APPENDIX
The definitions of variables used in the energy equations are as follows:
u : zonal wind velocity, positive to the east
v : meridional wind velocity, positive to the north
! : vertical pressure velocity (= dp dt )
T : temperature
p : pressure
Fx : friction in the zonal direction
Fy : friction in the meridional direction
Q:
diabatic heating
! :
geopotential
! :
mean static stability (= gc [T ] " gpR ![T ] / !p )
"1
"1
P
cp : specific heat at constant pressure
R:
specific gas constant for dry air
g :
gravitational acceleration
x :
zonal coordinate, positive to the east
y :
meridional coordinate, positive to the north
Lx :
the zonal distance for the averaging domain
Ly :
the latitudinal distance for the averaging domain
The mathematical expressions for the components in the energy equations are as follows:
p2
KZ =
!
[u ] 2 + [v] 2
dp
2g
(A1)
p1
30
p2
AZ =
!
p1
[T ]*2
dp
2"
P2
CZ= #
R
! p [$ ] [T ]
"
(A2)
dp
g
"
(A3)
P1
P2
KE ═
[u '2 + v '2 ]
dp
2g
(A4)
[T '2 ]
dp
2"
(A5)
!
P1
P2
AE ═
!
P1
Ck ═ Ck1 +Ck2 +Ck3 +Ck3
P2
═ $
!
"[u ] dp
[u ' v' ]
$
"y g
P1
P2
Cpk ═ "
R
!p
[# 'T ' ]
P1
!
P1
"[u ] dp
[u #% #]
$
"p g
!
P1
P1
!
[v' T ' ] "[T ]
dp #
$
"y
P2
!
P1
[% ' T ' ] "[T ] *
dp
$
"p
"[v] dp
g
! [v#% #] "p
(A6)
P1
(A8)
CA2
[T "Q "]
dp
#c P
P2
DE = # [u "Fx" + v "F y" ]
!
P2
(A7)
CA1
GE =
!
"[v] dp
[v # ]
$
"y g
2
g
p1
P2
P2
dp
P2
CA═ CA1 +CA2 = #
P2
(A9)
dp
g
(A10)
P1
1
BKE =
Lx
+
P2
'
2
2
% u (u . + v . )
%
2g
%
P1 &
!
& ) (u ' 2 + v ' 2 ) #
$
!
2g
$%
!"
x1
u (u . 2 + v . 2 )
/
2g
& ) (u ' 2 + v ' 2 ) #
($
!
2g
$%
!"
p1
$
1
"
"dp + L
"
y
x2 #
P2
' - v(u . 2 + v . 2 ) *
- v(u . 2 + v . 2 ) * $"
%
/
(
+
( "dp
%% +
2g
2g
+
(
+
,
)
,
)( y 2 "#
y1
P1 &
!
(A11)
p2
31
1
BΦE =
Lx
1
BAE=
Lx
P2
+
P1
*( u #" # $ u #" # '% dp + 1
x1
x2 & g
)
Ly
P2
'
2
% uT (
% 2*
%
P1 &
!
p2
HO(KE) =
!
p1
"[u #v #] dp
u#
+
"y g
p2
HO(AE) =
x1
uT ( 2
)
2*
!
p1
P2
dp 1
(
[v #" #] y1 $ [v #" #] y 2 ) + ([" #! #] P1 $ [" #! #] P 2 )
+
g g
P1
$
1
"
"dp + L
"
y
x2 #
p2
!
p1
' T ) ([T )v )] $
%
"dp +
& + (y #
P2
' [vT ( 2 ]
%
%% 2*
p1 &
!
"[u #$ #] dp
u#
+
"p
g
p2
!
p1
(A12)
p2
!
p1
)
y1
[vT ( 2 ]
2*
"[v # 2 ] dp
v#
+
"y g
&
$
2
"dp + $ [*T ' ]
$
""
$ 2)
y2 #
%
p2
! v#
p1
(
p1
[*T ' 2 ]
2)
"[v #$ #] dp
"p
g
' T ) ([T )* )] $
%
"dp
(p #
&+
#
!
! (A13)
!
p2 "
(A14)
(A15)
In the above formulation, [( )] represents a zonal average while [( )] represents a meridional
average of the zonal mean. Primes represent deviations from the zonal average, and asterisks
deviations from the area mean on an isobaric surface. They are related by ( ) = [( )]+ ( )! and
[( )]= [( )] + ( )! .
KZ is the area mean of the zonal kinetic energy, and AZ is zonal available potential energy,
which is generated by differential heating in the meridional direction.
Ck consists of four
conversion terms, all of which are associated with zonal and meridional wind shears. The first two
terms Ck1 and Ck2 are the fluxes of eddy momentum weighted by the horizontal and vertical shears
of the zonal wind. Similarly, Ck3 and Ck3 are the fluxes of eddy momentum weighted by the
meridional wind shears. Cpk, DE, CA and GE are defined in the text. HO(KE) and HO(AE) are higher
order terms involving triple products of perturbations in the eddy kinetic and eddy available
potential energy equations, respectively.
32
Zonal and meridional averages are calculated on pressure surfaces and then vertically
integrated.
The lower and upper integration limits are P1 =125 hPa and P2 = PS.
Vertical
integration uses the trapezoidal rule for the 21 pressure levels from the surface to 125 hPa.
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37
Figure Captions
Figure 1. 725 hPa composite winds and geopotential height deviation contours for 3-5-day waves.
The contour interval is 0.5 m, and the vector scale is indicated in m/s. Wind vectors with magnitude
greater than 0.2 m/s are displayed. Composites were computed for June 15th-September 14th by
selecting those 725 hPa fields in which the 3-5-day filtered meridional wind at 15˚N-0˚W is greater
than 1 m/s. Shading indicates regions with average vertical p-velocity magnitudes, ! , between 725
hPa and 325 hPa greater than 0.36 hPa/hr. Darker and lighter shadings indicate rising and sinking
motion, respectively.
Figure 2. The time series of the filtered meridional wind at 15°N-0°W. Values greater than 1 ms-1
are marked with open circles and represent the snapshots used to compute the wave composites.
Unit is in ms-1.
Figure 3. Longitude-pressure cross-sections of the 3-5-day composite waves along 12˚N for the (a)
meridional wind perturbations with a contour interval of 0.3 m/s and geopotential height
perturbations with a contour interval of 15m, (b) vertical pressure velocity perturbations with
contour intervals of 0.5 hPa/hr and 0.3 hPa/hr for regions with negative and positive values,
respectively, (c) geopotential height deviations with a contour interval of 0.5 m, (d) temperature
deviations with a contour interval of 0.1 Kelvin. Shading indicates regions with negative values.
Figure 4. Wavelet analysis for the time series of the daily mean meridional wind at 725 hPa at
12°N-0°W. (a) time series. (b) wavelet modulus (m s-1) of the time series in (a). Contour interval is
0.5 ms-1. Shading indicates regions with magnitude greater than 1.3 ms-1.
38
Figure 5. The time series of zonal mean precipitation from 10°W to 20°E averaged between 9°N
and 15°N. Unit is in mmday-1.
Figure 6. Daily mean precipitation for (a) 30 June, (b) 16 July, (c) 17 July, (d) 18 July, (e) 19 July,
(f) 20 July, (g) 21 July, (h) 6 September. Units are mmday-1.
Figure 7. The time series of energy conversion rates for (a) barotropic conversions Ck (dashed
lines), baroclinic conversion Cpk (thin solid lines), and sum of Ck and Cpk (thick solid lines), (b) Ck1
(solid line) and Ck2 (dashed line), (c) dissipation DE (thick solid line), boundary flux of eddy kinetic
energy BKE (thin solid line), boundary pressure work B! E (dashed lines), (d) left hand side of Eq.1
(solid lines) and right hand side of Eq. 1 (dashed lines). Units are W/m2.
Figure 8. July-August time series of energy conversion rates for (a) diabatic generation of eddy
available potential energy GE, (b) baroclinic conversions CA1 (solid line) and CA2 (dashed line), (c)
boundary flux of eddy available potential energy BAE, and (d) lhs of Eq. 2 (solid lines) and rhs of
Eq. 2 (dashed lines). Units are W/m2.
Figure 9. Seasonal mean energy cycle diagram for 3-5 day waves in the model. The energy
conversion rates are in W/m2. Numbers in the squares are energy amounts in units of J m-2.
Symbols are defined in the text.
Figure 10. The time series of energy conversion rates for the domain over East Africa. Cpk, Ck1,
and Ck2 are represented by thick, thin solid lines and dash lines, respectively.
39
Figure 11. Seasonal mean meridional distributions of energy conversions averaged between 10˚W
and 20˚E for (a) #
(c) #
[v "T "] ![T ]
$
!y
[u "v "] ![u ]
g !y
and (d) #
and (b) #
[& "T "] ![T ]$
%
!p
[u "$ "] ![u ]
.
g
!p
Contour intervals are 0.3 ! 10-6 ms-1 in (a) and (b).
. Contour intervals are 0.5 ! 10-6 ms-1 in (c) and (d).
Figure 12. Seasonal mean meridional distributions of energy conversions averaged between 10˚W
and 20˚E for (a) "
R
[# !T !] ,
pg
(b)
[T "Q "]
!C p
, (c)
"[u !Fx! + v !F y! ]
kinetic energy and net zonal eddy pressure work (
g
and (d) the net zonal fluxes of eddy
u (u ! 2 + v ! 2 )
2g
"
x1
u (u ! 2 + v ! 2 )
2g
) + (u "! " x1 # u "! " x 2 ) .
x2
Contour intervals are 0.5 ! 10-6 m/s.
Figure 13. Seasonal mean meridional distribution of eddy kinetic energy
[u ! 2 + v ! 2 ]
2g
averaged
between 10˚W and 20˚E. Contour intervals are 0.02 m. Shading denotes values greater than 0.08
m.
Figure 14. Latitudinal mean of the wavenumber versus frequency spectra of the meridional wind at
725 hPa between 5˚N and 20˚N. Contour intervals are 40 m2/s2. The dispersion relation curves of
mixed Rossby-gravity waves (MRG) and equatorial Rossby waves (ER) for five equivalent depths
of h= 25, 50, 100, 200, and 400 m are superimposed.
40
Figure 1. 725 hPa composite winds and geopotential height deviation contours for 3-5-day waves.
The contour interval is 0.5 m, and the vector scale is indicated in m/s. Wind vectors with magnitude
greater than 0.2 m/s are displayed. Composites were computed for June 15th-September 14th by
selecting those 725 hPa fields in which the 3-5-day filtered meridional wind at 15˚N-0˚W is greater
than 1 m/s. Shading indicates regions with average vertical p-velocity magnitudes, ! , between 725
hPa and 325 hPa greater than 0.36 hPa/hr. Darker and lighter shadings indicate rising and sinking
motion, respectively.
41
Figure 2. The time series of the filtered meridional wind at 15°N-0°W. Values greater than 1
ms-1 are marked with open circles and represent the snapshots used to compute the wave
composites. Unit is in ms-1.
42
Figure 3. Longitude-pressure cross-sections of the 3-5-day composite waves along 12˚N for the (a)
meridional wind perturbations with a contour interval of 0.3 m/s and geopotential height
perturbations with a contour interval of 15m, (b) vertical pressure velocity perturbations with
contour intervals of 0.5 hPa/hr and 0.3 hPa/hr for regions with negative and positive values,
respectively, (c) geopotential height deviations with a contour interval of 0.5 m, (d) temperature
deviations with a contour interval of 0.1 Kelvin. Shading indicates regions with negative values.
43
Figure 4. Wavelet analysis for the time series of the daily mean meridional wind at 725 hPa near
12°N-0°W. (a) time series. (b) wavelet modulus (m s-1) of the time series in (a). Contour interval is
0.5 ms-1. Shading indicates regions with magnitude greater than 1.3 ms-1.
44
Figure 5. The time series of zonal mean precipitation from 10°W to 20°E averaged between 9°N
and 15°N. Unit is in mm/day.
45
Figure 6. Daily mean precipitation for (a) 30 June, (b) 16 July, (c) 17 July, (d) 18 July, (e) 19 July,
(f) 20 July, (g) 21 July, (h) 6 September. Units are mm/day.
46
Figure 7. The time series of energy conversion rates for (a) barotropic conversions Ck (dashed
lines), baroclinic conversion Cpk (thin solid lines), and sum of Ck and Cpk (thick solid lines), (b) Ck1
(solid line) and Ck2 (dashed line), (c) dissipation DE (thick solid line), boundary flux of eddy kinetic
energy BKE (thin solid line), boundary pressure work B! E (dashed lines), (d) left hand side of Eq.1
(solid lines) and right hand side of Eq. 1 (dashed lines). Units are W/m2.
47
Figure 8. The time series of energy conversion rates for (a) diabatic generation of eddy available
potential energy GE, (b) baroclinic conversions CA1 (solid line) and CA2 (dashed line), (c) boundary
flux of eddy available potential energy BAE, and (d) lhs of Eq. 2 (solid lines) and rhs of Eq. 2
(dashed lines). Units are W/m2.
48
BAE (0.001)
CA (0.025)
AE (1487)
AZ (113886)
CZ (0.87)
KZ (340506)
GE (0.012)
Cpk (0.037)
DE (0.045)
Ck (0.023)
KE (5230)
BK E (0.008)
B! E (-0.0002)
Figure 9. Seasonal mean energy cycle diagram for 3-5 day waves in the model. The energy
conversion rates are in W/m2. Numbers in the squares are energy amounts in units of J m-2.
Symbols are defined in the text.
49
Figure 10. The time series of energy conversion rates for the domain over East Africa. Cpk,
Ck1, and Ck2 are represented by thick, thin solid lines and dash lines, respectively.
50
Figure 11. Seasonal mean meridional distributions of energy conversions averaged between 10˚W
and 20˚E for (a) #
(c) #
[v "T "] ![T ]
$
!y
[u "v "] ![u ]
g !y
and (d) #
and (b) #
[& "T "] ![T ]$
%
!p
[u "$ "] ![u ]
.
g
!p
Contour intervals are 0.3 ! 10-6 ms-1 in (a) and (b).
. Contour intervals are 0.5 ! 10-6 ms-1 in (c) and (d).
51
Figure 12. Seasonal mean meridional distributions of energy conversions averaged between 10˚W
and 20˚E for (a) "
R
[# !T !] ,
pg
(b)
[T "Q "]
!C p
, (c)
"[u !Fx! + v !F y! ]
kinetic energy and net zonal eddy pressure work (
Contour intervals are 0.5 ! 10-6 m/s.
52
g
and (d) the net zonal fluxes of eddy
u (u ! 2 + v ! 2 )
2g
"
x1
u (u ! 2 + v ! 2 )
2g
) + (u "! " x1 # u "! " x 2 ) .
x2
Figure 13. Seasonal mean meridional distribution of eddy kinetic energy
[u ! 2 + v ! 2 ]
2g
averaged
between 10˚W and 20˚E. Contour intervals are 0.02 m. Shading denotes values greater than 0.08
m.
53
Figure 14. Latitudinal mean of the wavenumber versus frequency spectra of the meridional wind at
725 hPa between 5˚N and 20˚N. Contour intervals are 40 m2/s2. The dispersion relation curves of
mixed Rossby-gravity waves (MRG) and equatorial Rossby waves (ER) for five equivalent depths
of h= 25, 50, 100, 200, and 400 m are superimposed.
54