• Berry, G. J., , and C. D. Thorncroft, 2005: Case study of an intense African easterly wave. Mon. Wea. Rev., 133, 752766, doi:10.1175/MWR2884.1.

    • Search Google Scholar
    • Export Citation
  • Berry, G. J., , and C. D. Thorncroft, 2012: African easterly wave dynamics in a mesoscale numerical model: The upscale role of convection. J. Atmos. Sci., 69, 12671283, doi:10.1175/JAS-D-11-099.1.

    • Search Google Scholar
    • Export Citation
  • Berry, G. J., , C. D. Thorncroft, , and T. Hewson, 2007: African easterly waves during 2004—Analysis using objective techniques. Mon. Wea. Rev., 135, 1251–1267, doi:10.1175/MWR3343.1.

    • Search Google Scholar
    • Export Citation
  • Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325334, doi:10.1002/qj.49709239302.

    • Search Google Scholar
    • Export Citation
  • Briggs, R. J., 1964: Electron-Stream Interaction with Plasmas. The MIT Press, 187 pp.

  • Burpee, R. W., 1972: The origin and structure of easterly waves in the lower troposphere of North Africa. J. Atmos. Sci., 29, 7790, doi:10.1175/1520-0469(1972)029<0077:TOASOE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Carlson, T. N., 1969: Some remarks on African disturbances and their progress over the tropical Atlantic. Mon. Wea. Rev., 97, 716726, doi:10.1175/1520-0493(1969)097<0716:SROADA>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Cornforth, R. J., , B. J. Hoskins, , and C. D. Thorncroft, 2009: The impact of moist processes on the African Easterly Jet–African Easterly Wave system. Quart. J. Roy. Meteor. Soc., 135, 894913, doi:10.1002/qj.414.

    • Search Google Scholar
    • Export Citation
  • Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553597, doi:10.1002/qj.828.

    • Search Google Scholar
    • Export Citation
  • Diaz, M., , and A. Aiyyer, 2013a: The genesis of African easterly waves by upstream development. J. Atmos. Sci., 70, 34923512, doi:10.1175/JAS-D-12-0342.1.

    • Search Google Scholar
    • Export Citation
  • Diaz, M., , and A. Aiyyer, 2013b: Energy dispersion in African easterly waves. J. Atmos. Sci., 70, 130145, doi:10.1175/JAS-D-12-019.1.

  • Dickinson, M., , and J. Molinari, 2000: Climatology of sign reversals of the meridional potential vorticity gradient over Africa and Australia. Mon. Wea. Rev., 128, 38903900, doi:10.1175/1520-0493(2001)129<3890:COSROT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1A, 3352, doi:10.1111/j.2153-3490.1949.tb01265.x.

  • Farrell, B. F., 1982: Pulse asymptotics of the Charney baroclinic instability problem. J. Atmos. Sci., 39, 507517, doi:10.1175/1520-0469(1982)039<0507:PAOTCB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., 1983: Pulse asymptotics of three-dimensional baroclinic waves. J. Atmos. Sci., 40, 22022210, doi:10.1175/1520-0469(1983)040<2202:PAOTDB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Grist, J. P., , S. E. Nicholson, , and A. I. Barcilon, 2002: Easterly waves over Africa. Part II: Observed and modeled contrasts between wet and dry years. Mon. Wea. Rev., 130, 212–225, doi:10.1175/1520-0493(2002)130<0212:EWOAPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hakim, G. J., 2003: Developing wave packets in the North Pacific storm track. Mon. Wea. Rev., 131, 2824–2837, doi:10.1175/1520-0493(2003)131<2824:DWPITN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hall, N. M. J., , G. N. Kiladis, , and C. D. Thorncroft, 2006: Three-dimensional structure and dynamics of African easterly waves. Part II: Dynamical modes. J. Atmos. Sci., 63, 22312245, doi:10.1175/JAS3742.1.

    • Search Google Scholar
    • Export Citation
  • Hill, C. M., , and Y.-L. Lin, 2003: Initiation of a mesoscale convective complex over the Ethiopian Highlands preceding the genesis of Hurricane Alberto (2000). Geophys. Res. Lett., 30, 1232, doi:10.1029/2002GL016655.

    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., , M. E. McIntyre, , and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877946, doi:10.1002/qj.49711147002.

    • Search Google Scholar
    • Export Citation
  • Hsieh, J.-S., , and K. H. Cook, 2007: A study of the energetics of African easterly waves using a regional climate model. J. Atmos. Sci., 64, 421440, doi:10.1175/JAS3851.1.

    • Search Google Scholar
    • Export Citation
  • Hsieh, J.-S., , and K. H. Cook, 2008: On the instability of the African easterly jet and the generation of African waves: Reversals of the potential vorticity gradient. J. Atmos. Sci., 65, 21302151, doi:10.1175/2007JAS2552.1.

    • Search Google Scholar
    • Export Citation
  • Huerre, P., , and P. A. Monkewitz, 1990: Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech., 22, 473537, doi:10.1146/annurev.fl.22.010190.002353.

    • Search Google Scholar
    • Export Citation
  • Janiga, M. A., , and C. D. Thorncroft, 2013: Regional differences in the kinematic and thermodynamic structure of African easterly waves. Quart. J. Roy. Meteor. Soc., 139, 1598–1614, doi:10.1002/qj.2047.

    • Search Google Scholar
    • Export Citation
  • Kiladis, G. N., , C. D. Thorncroft, , and N. M. J. Hall, 2006: Three-dimensional structure and dynamics of African easterly waves. Part I: Observations. J. Atmos. Sci., 63, 22122230, doi:10.1175/JAS3741.1.

    • Search Google Scholar
    • Export Citation
  • Leroux, S., , and N. M. J. Hall, 2009: On the relationship between African easterly waves and the African easterly jet. J. Atmos. Sci., 66, 23032316, doi:10.1175/2009JAS2988.1.

    • Search Google Scholar
    • Export Citation
  • Leroux, S., , N. M. J. Hall, , and G. N. Kiladis, 2010: A climatological study of transient–mean-flow interactions over West Africa. Quart. J. Roy. Meteor. Soc., 136, 397410, doi:10.1002/qj.474.

    • Search Google Scholar
    • Export Citation
  • Leroux, S., , N. M. J. Hall, , and G. N. Kiladis, 2011: Intermittent African easterly wave activity in a dry atmospheric model: Influence of the extratropics. J. Climate, 24, 53785396, doi:10.1175/JCLI-D-11-00049.1.

    • Search Google Scholar
    • Export Citation
  • Lin, Y.-L., , K. E. Robertson, , and C. M. Hill, 2005: Origin and propagation of a disturbance associated with an African easterly wave as a precursor of Hurricane Alberto (2000). Mon. Wea. Rev., 133, 3276–3298, doi:10.1175/MWR3035.1.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., , B. Farrell, , and A. J. Rosenthal, 1983: Absolute barotropic instability and monsoon depressions. J. Atmos. Sci., 40, 11781184, doi:10.1175/1520-0469(1983)040<1178:ABIAMD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Mass, C., 1979: A linear primitive equation model of African wave disturbances. J. Atmos. Sci., 36, 20752092, doi:10.1175/1520-0469(1979)036<2075:ALPEMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Mekonnen, A., , C. D. Thorncroft, , and A. R. Aiyyer, 2006: Analysis of convection and its association with African easterly waves. J. Climate, 19, 54055421, doi:10.1175/JCLI3920.1.

    • Search Google Scholar
    • Export Citation
  • Merkine, L.-O., 1977: Convective and absolute instability of baroclinic eddies. Geophys. Astrophys. Fluid Dyn., 9, 129157, doi:10.1080/03091927708242322.

    • Search Google Scholar
    • Export Citation
  • Mozer, J. B., , and J. A. Zehnder, 1996: Lee vorticity production by large-scale tropical mountain ranges. Part II: A mechanism for the production of African waves. J. Atmos. Sci., 53, 539549, doi:10.1175/1520-0469(1996)053<0539:LVPBLS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • O’Brien, J. J., 1970: A note on the vertical structure of the eddy exchange coefficient in the planetary boundary layer. J. Atmos. Sci., 27, 12131215, doi:10.1175/1520-0469(1970)027<1213:ANOTVS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Orlanski, I., , and J. Katzfey, 1991: The life cycle of a cyclone wave in the Southern Hemisphere. Part I: Eddy energy budget. J. Atmos. Sci., 48, 19721998, doi:10.1175/1520-0469(1991)048<1972:TLCOAC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Orlanski, I., , and E. K. M. Chang, 1993: Ageostrophic geopotential fluxes in downstream and upstream development of baroclinic waves. J. Atmos. Sci., 50, 212225, doi:10.1175/1520-0469(1993)050<0212:AGFIDA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 1976: Finite-amplitude baroclinic disturbances in downstream varying currents. J. Phys. Oceanogr., 6, 335344, doi:10.1175/1520-0485(1976)006<0335:FABDID>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pierrehumbert, R., 1984: Local and global baroclinic instability of zonally varying flow. J. Atmos. Sci., 41, 21412162, doi:10.1175/1520-0469(1984)041<2141:LAGBIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pierrehumbert, R., , and K. Swanson, 1995: Baroclinic instability. Annu. Rev. Fluid Mech., 27, 419467, doi:10.1146/annurev.fl.27.010195.002223.

    • Search Google Scholar
    • Export Citation
  • Pytharoulis, I., , and C. Thorncroft, 1999: The low-level structure of African easterly waves in 1995. Mon. Wea. Rev., 127, 22662280, doi:10.1175/1520-0493(1999)127<2266:TLLSOA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Reed, R. J., , D. C. Norquist, , and E. E. Recker, 1977: The structure and properties of African wave disturbances as observed during phase III of GATE. Mon. Wea. Rev.,105, 317–333, doi:10.1175/1520-0493(1977)105<0317:TSAPOA>2.0.CO;2.

  • Rennick, M. A., 1976: The generation of African waves. J. Atmos. Sci., 33, 19551969, doi:10.1175/1520-0469(1976)033<1955:TGOAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Shapiro, L. J., 1980: The effect of nonlinearities on the evolution of barotropic easterly waves in a nonuniform environment. J. Atmos. Sci., 37, 26312643, doi:10.1175/1520-0469(1980)037<2631:TEONOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., 1977: A note on the instability of the African easterly jet. J. Atmos. Sci., 34, 16701674, doi:10.1175/1520-0469(1977)034<1670:ANOTIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., , and B. J. Hoskins, 1979: The downstream and upstream development of unstable baroclinic waves. J. Atmos. Sci., 36, 12391254, doi:10.1175/1520-0469(1979)036<1239:TDAUDO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Springer, 670 pp.

  • Swanson, K., , and R. T. Pierrehumbert, 1994: Nonlinear wave packet evolution on a baroclinically unstable jet. J. Atmos. Sci., 51, 384396, doi:10.1175/1520-0469(1994)051<0384:DCCISF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., 1995: An idealized study of African easterly waves. III: More realistic basic states. Quart. J. Roy. Meteor. Soc., 121, 15891614, doi:10.1002/qj.49712152706.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , and B. J. Hoskins, 1994a: An idealized study of African easterly waves. I: A linear view. Quart. J. Roy. Meteor. Soc., 120, 953982, doi:10.1002/qj.49712051809.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , and B. J. Hoskins, 1994b: An idealized study of African easterly waves. II: A nonlinear view. Quart. J. Roy. Meteor. Soc., 120, 9831015, doi:10.1002/qj.49712051810.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , and M. Blackburn, 1999: Maintenance of the African easterly jet. Quart. J. Roy. Meteor. Soc., 125, 763786, doi:10.1002/qj.49712555502.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , N. M. J. Hall, , and G. N. Kiladis, 2008: Three-dimensional structure and dynamics of African easterly waves. Part III: Genesis. J. Atmos. Sci., 65, 35963607, doi:10.1175/2008JAS2575.1.

    • Search Google Scholar
    • Export Citation
  • Tupaz, J. B., , R. T. Williams, , and C.-P. Chang, 1978: A numerical study of barotropic instability in a zonally varying easterly jet. J. Atmos. Sci., 35, 12651280, doi:10.1175/1520-0469(1978)035<1265:ANSOBI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
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    (top) Absolute and (bottom) convective instability represented in the xt plane for an easterly flow. A hypothetical disturbance begins at the origin and spreads into a wave packet as time increases. Its spread in terms of group velocity , is bounded by and , and the region enclosed by these two rays marks its cone of influence in time and space. If cg = 0 falls within this cone, the flow is absolutely unstable. In this case, part of the traveling disturbance is left behind and undergoes exponential growth at its initial location. In the convectively unstable case, the traveling disturbance eventually leaves its source region.

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    Model domain showing boundaries and time-mean eddy kinetic energy (shaded; m2 s−2) at 650 hPa after the model has reached a steady state. The sponge boundary condition is applied only to the region outside of the thick rectangle.

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    (a) Meridional cross section of zonal wind (shaded; m s−1) and potential temperature (contours; K) at eastern edge of domain. (b) Meridional cross section of in Eq. (1) (shaded; K day−1) and potential temperature (contours; K) at eastern edge of domain.

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    Regressed composite AEWs from ERA-Interim. (a) Total 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (b) Perturbation 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5) (c) Total 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). (d) Perturbation 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). The solid contour in (b) shows the climatological d(PV)/dy = 0 line.

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    Sample AEWs from the simulation at day 99. (a) Total 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (b) Perturbation 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (c) Total 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). (d) Perturbation 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). The solid contour in (b) shows the climatological d(PV)/dy = 0 line. The background map is for scale only.

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    Average EKE and winds from ERA-Interim. (a) Average 650-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2) and (b) average 925-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2).

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    Average EKE and winds from simulation. (a) Average 650-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2) and (b) average 925-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2). The background map is for scale only.

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    Examples of AEW packets from ERA-Interim during 2008. Shaded values are the square of the 2-day low-pass-filtered meridional wind (m2 s−2) at (a) 925 hPa averaged from 13° to 23°N and (b) 650 hPa averaged from 2° to 10°N.

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    Hovmöller plot of the square of the meridional wind (m2 s−2) from the simulation averaged from 15° to 19°N at 925 hPa. The y axis denotes days into the simulation. (a) Once quasi-steady state is reached and (b) beginning of simulation.

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    Individual terms in the EKE budget: a snapshot of the simulation. The contours in each panel are 950–500-hPa-averaged EKE (contoured on log2 scale beginning at 2 J kg−1), and the shading is 950–500-hPa-averaged EKE conversion rate (J kg−1 day−1). The specific terms and vectors plotted are (a) geopotential flux convergence [first term on rhs of Eq. (9)] and ageostrophic geopotential fluxes (vectors; m3 s−3), (b) barotropic conversion [second and third terms on rhs of Eq. (8)], (c) pressure work [first term on rhs of Eq. (8)], (d) friction [fourth term on rhs of Eq. (8)], (e) baroclinic [second term on rhs of Eq. (9)], and (f) total nonadvective [sum of (a),(b),(d), and (e)]. The EKE maxima are labeled for discussion, and the background map is for scale only.

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    Example of procedure to define a wavelength. EKE (m2 s−2) is contoured, and geopotential flux convergence (J kg−1 day−1) is shaded.

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    EKE conversion [d(EKE)/dt; J day−1] and growth rates {[1/(EKE)][d(EKE)/dt]; day−1} following a wavelength. (a) Conversion rates [the specific terms from Eqs. (8) and (9) are defined in Fig. 10], (b) barotropic terms, (c) EKE growth rate, and (d) EKE growth rate for barotropic terms. In (a) and (c), the blue line is spatial EKE redistribution, the red line is temporal EKE growth, and the black line is total EKE tendency. For presentation, (a) is truncated (the baroclinic term reaches a maximum of 967 J day−1, and friction reaches a minimum of −899 J day−1).

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    The value of H0 given in Eq. (11) (K day−1).

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    Sample AEWs from the ITCZ heating experiment at day 97 simulation. (a) Total 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (b) Perturbation 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (c) Total 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). (d) Perturbation 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). The background map is for scale only.

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    Average EKE and winds from ITCZ heating simulation. (a) 650-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2). (b) 925-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2). Contours show regions of diabatic heating in fraction of maximum heating rate. Contours begin at 0.2 and end at 0.8 with a contour interval of 0.2. The background map is for scale only.

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    As in Fig. 10, but for the ITCZ heating experiment. The background map is for scale only.

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    EKE conversion [d(EKE)/dt; J day−1] and growth rates {[1/(EKE)][d(EKE)/dt]; day−1} following a wavelength for the ITCZ heating experiment. (a) Conversion rates, (b) barotropic terms, (c) EKE growth rate, and (d) EKE growth rate for barotropic terms. In (a) and (c), the blue line is EKE redistribution, the red line is temporal EKE growth, and the black line is total EKE tendency.

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    Total group velocity (m s−1) as a function of longitude. The solid line is for the original experiment, and the dashed line is for the ITCZ heating experiment.

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    Energy fluxes for different layers of the atmosphere for the ITCZ heating experiment. (a) Ageostrophic geopotential fluxes (vectors; m3 s−3) averaged over 950–800 hPa with 925-hPa time-mean potential temperature, (b) ageostrophic geopotential fluxes (vectors; m3 s−3) averaged over 800–450 hPa with 650-hPa time-mean potential vorticity (contours) and PV gradient reversal line (thick contours), (c) total EKE flux (mean advective and ageostrophic; m3 s−3) for 950–800-hPa layer (vectors) with regions of eastward flux shaded gray, and (d) total EKE flux (mean advective and ageostrophic; m3 s−3) for 800–450-hPa layer with regions of eastward flux shaded gray. The background map is for scale only.

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Absolute and Convective Instability of the African Easterly Jet

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  • 1 North Carolina State University, Raleigh, North Carolina
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Abstract

The stability of the African easterly jet (AEJ) is examined using idealized numerical simulations. It is found that a zonally homogeneous representation of the AEJ can support absolute instability in the form of African easterly waves (AEWs). This finding is verified through a local energy budget, which demonstrates the presence of both upstream and downstream energy fluxes. These energy fluxes allow unstable wave packets to spread upstream and downstream relative to their initial point of excitation. This finding is further verified by showing that the ground-relative group velocity of these wave packets has both eastward and westward components. In contrast with normal-mode instability theory, which emphasizes wave growth through energy extraction from the basic state, the life cycle of the simulated AEWs is strongly governed by energy fluxes. Convergent fluxes at the beginning of the AEW storm track generate new AEWs, whereas divergent fluxes at the end of the storm track lead to their decay. It is argued that, even with small normal-mode growth rates and a short region of instability, the presence of absolute instability allows AEWs to develop through the mixed baroclinic–barotropic instability mechanism, because upstream energy fluxes allow energy extracted through baroclinic and barotropic conversion to be recycled between successive AEWs.

Corresponding author address: Michael Diaz, Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, 2800 Faucette Drive, Raleigh, NC 27695. E-mail: mldiaz@ncsu.edu

Abstract

The stability of the African easterly jet (AEJ) is examined using idealized numerical simulations. It is found that a zonally homogeneous representation of the AEJ can support absolute instability in the form of African easterly waves (AEWs). This finding is verified through a local energy budget, which demonstrates the presence of both upstream and downstream energy fluxes. These energy fluxes allow unstable wave packets to spread upstream and downstream relative to their initial point of excitation. This finding is further verified by showing that the ground-relative group velocity of these wave packets has both eastward and westward components. In contrast with normal-mode instability theory, which emphasizes wave growth through energy extraction from the basic state, the life cycle of the simulated AEWs is strongly governed by energy fluxes. Convergent fluxes at the beginning of the AEW storm track generate new AEWs, whereas divergent fluxes at the end of the storm track lead to their decay. It is argued that, even with small normal-mode growth rates and a short region of instability, the presence of absolute instability allows AEWs to develop through the mixed baroclinic–barotropic instability mechanism, because upstream energy fluxes allow energy extracted through baroclinic and barotropic conversion to be recycled between successive AEWs.

Corresponding author address: Michael Diaz, Department of Marine, Earth, and Atmospheric Sciences, North Carolina State University, 2800 Faucette Drive, Raleigh, NC 27695. E-mail: mldiaz@ncsu.edu

1. Introduction

a. Background

African easterly waves (AEWs) are synoptic-scale weather systems that form over tropical Africa during the summer monsoon and have long been recognized as precursors to the majority of Atlantic hurricanes (Carlson 1969). One of the widely accepted hypotheses for their origin is the instability of the African easterly jet (AEJ). This midlevel jet is associated with a reversal in the meridional gradient of potential vorticity (PV) and therefore satisfies the Charney–Stern necessary criterion for a mixed baroclinic–barotropic instability (Burpee 1972). In support of the instability hypothesis, observations confirm that AEWs extract energy from the AEJ through both baroclinic and barotropic conversion (Burpee 1972; Reed et al. 1977). Subsequent idealized numerical simulations further strengthened this hypothesis by demonstrating that the structure of the most unstable normal mode of idealized representations of the AEJ closely resembles AEWs (e.g., Rennick 1976; Simmons 1977; Mass 1979; Thorncroft and Hoskins 1994a).

Although the structure of the most unstable normal mode resembles AEWs, are the implied growth rates high enough to account for observed AEW amplitudes? Thorncroft and Hoskins (1994a) reason that, given a growth rate of 0.4 day−1, a life span of 7 days, and a final amplitude of 5 m s−1, AEWs could originate from initial perturbations of only 0.3 m s−1. Based on these estimates, the instability hypothesis seems reasonable. However, a number of recent studies have questioned the role of instability of the AEJ in initiating AEWs (Hall et al. 2006; Thorncroft et al. 2008; Hsieh and Cook 2008). Hall et al. (2006) found that a modest amount of surface damping can bring the growth rate of the most unstable normal mode to near zero. They suggest that hydrodynamic instability alone cannot account for the existence of AEWs because small perturbations cannot undergo sufficient amplification before leaving the short region of instability.

The experiments of Hall et al. (2006) present a significant challenge to the normal-mode instability paradigm for AEW genesis. Using the reasoning of Thorncroft and Hoskins (1994a), either growth rates must be significantly higher than predicted by linear normal-mode instability, or the initial perturbations must be significantly larger.

One possible candidate for increasing growth rates is condensational heating from moist convection. Because the strength of the PV gradient reversal can be partly attributed to ITCZ convection (e.g., Thorncroft and Blackburn 1999; Hsieh and Cook 2008), increased ITCZ convection increases normal-mode growth rates (Grist et al. 2002; Cornforth et al. 2009). However, the mean state of Hall et al. (2006) was derived from a reanalysis dataset and implicitly includes the impact of ITCZ convection. It can, however, be argued that the relatively long time averaging smooths the shear of the AEJ and yields a more stable basic state. This possibility is supported by Leroux and Hall (2009), who demonstrated substantial sensitivity to the AEWs developing on basic states averaged over smaller time intervals. Nevertheless, an average of their basic states leading to strong AEW responses yields only 0.035 day−1, which still seems too small to account for AEW genesis. Moist convection can also increase AEW growth rates by enhancing cyclonic PV within AEW troughs (e.g., Berry and Thorncroft 2012; Janiga and Thorncroft 2013). However, convection is likely more important later in the life cycle of AEWs because they cannot organize convection before their genesis.

Other studies have argued that AEW genesis requires large-amplitude initial perturbations. Thorncroft et al. (2008) explored the possibility that mesoscale convective systems (MCSs) can serve this role. Using the same neutrally stable basic state as Hall et al. (2006), they prescribe circular regions of diabatic heating and cooling meant to mimic convection and found that AEWs can develop downstream from these perturbations. This finding supports the view that MCSs initiated over the higher terrain of East Africa generate perturbations that transform into AEWs (e.g., Berry and Thorncroft 2005; Mekonnen et al. 2006; Kiladis et al. 2006; Leroux et al. 2010). Other studies have suggested that AEWs can be generated by mountain-induced lee vortices (Mozer and Zehnder 1996; Hill and Lin 2003; Lin et al. 2005) or perturbations originating from the North Atlantic storm track (Leroux et al. 2011).

Based on the preceding discussion, there is strong evidence that mature AEWs amplify by extracting energy from the AEJ and that moist convection can enhance their growth rate. In this respect, normal-mode instability theory has been quite successful, especially given its generally correct account of the structure and energetics of mature AEWs. The discrepancy lies with their initiation. Can they develop spontaneously from small, random perturbations through a natural selection process of the most unstable normal mode, or do they require large-amplitude precursors? The slow growth rates implied by normal-mode instability theory combined with the short time they spend within the unstable region makes the former seem less plausible. However, the problem may be that normal-mode instability theory itself is unsuitable to explain instabilities on localized jets, such as the AEJ. Perhaps one of its more questionable aspects is that it assumes the simultaneous development of an infinite series of identical disturbances of a single wavelength and period on an infinitely long jet. In reality, AEWs evolve as localized disturbances, with only about two at a time fitting within the short unstable region of the AEJ. Given this discrepancy, it is questionable how applicable normal-mode growth rates are to the amplification of individual AEWs or to assessing the instability of the AEJ.

b. Absolute and convective instabilities

In recent decades, a number of studies have contributed to extending hydrodynamic instability theory to the development of localized instabilities of baroclinic flows (e.g., Merkine 1977; Farrell 1982; Pierrehumbert 1984; Huerre and Monkewitz 1990). Whereas normal-mode instability emphasizes the space-independent temporal growth of the most unstable normal mode, “local instability” considers both the temporal and spatial growth of an initially isolated disturbance, with particular focus at its leading and trailing edge as it disperses into a wave packet. From a normal-mode perspective, this spatial growth results from the constructive and destructive interference of the large spectrum of normal modes of different wavenumbers required to localize a disturbance. One important concept in local instability is the distinction between convective and absolute instability1 (Merkine 1977). Qualitatively, for absolute instability, the trailing edge of a dispersing wave packet spreads upstream faster than its downstream advection and thus undergoes exponential growth at and upstream of its initial location. For convective instability, the trailing edge of the wave packet is advected away from its source region before it can begin exponential growth at its initial location. Figure 1 shows a schematic of these two instability scenarios for an easterly flow.

Fig. 1.
Fig. 1.

(top) Absolute and (bottom) convective instability represented in the xt plane for an easterly flow. A hypothetical disturbance begins at the origin and spreads into a wave packet as time increases. Its spread in terms of group velocity , is bounded by and , and the region enclosed by these two rays marks its cone of influence in time and space. If cg = 0 falls within this cone, the flow is absolutely unstable. In this case, part of the traveling disturbance is left behind and undergoes exponential growth at its initial location. In the convectively unstable case, the traveling disturbance eventually leaves its source region.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

The distinction between absolute and convective instability becomes important when a region of instability is localized (Pierrehumbert 1984). If a flow is convectively unstable, a developing wave packet will eventually leave the region of instability, possibly before undergoing significant growth. If a flow is absolutely unstable, the wave packet will remain within the unstable zone, where it can continue to amplify. These ideas were originally introduced into the atmospheric sciences to help understand baroclinic instability of midlatitude westerly jets and storm tracks. Although early studies using simple models of baroclinic instability suggested that midlatitude jets can support weak absolute instability (e.g., Merkine 1977; Farrell 1982, 1983), the current consensus is that they are only convectively unstable because midlatitude baroclinic waves disperse most of their energy downstream and friction damps out any shallow upstream-dispersing waves (e.g., Simmons and Hoskins 1979; Swanson and Pierrehumbert 1994; Pierrehumbert and Swanson 1995; Hakim 2003).

Nevertheless, absolute instability may be a suitable model for AEWs developing on the AEJ. In contrast with midlatitude baroclinic waves, they disperse much of their energy slowly upstream back into the region of instability (Diaz and Aiyyer 2013a,b). Additionally, with monsoon southwesterlies underlying an easterly jet, the basic-state flow over Africa reverses direction, which satisfies the necessary criterion for an absolute instability in the Eady model (Eady 1949) of baroclinic instability (Merkine 1977). If indeed the AEJ can support absolute instabilities, then an initially isolated disturbance evolving into a wave packet would remain within the localized unstable region and continue to grow, even as the individual AEW troughs composing it move out of the unstable region. Hence, despite slow normal-mode growth rates and a short unstable region, the AEJ may still be unstable to small perturbations as disturbance energy gets “recycled” between successive AEWs. The goal of the present study is to test the hypothesis that the AEJ is absolutely unstable.

2. Experimental design

a. Overview

The primary tool for testing our hypothesis is a set of idealized numerical simulations using a modified version of the Weather Research and Forecasting (WRF) Model. With this model, we will construct a simplified representation of the AEJ and AEWs in which the essential dynamical processes involved in AEW development can be more easily isolated and diagnosed. Our philosophy is to make the model as simple as possible to test our hypothesis while still retaining enough complexity to realistically simulate AEWs. As such, we will exclude factors that would complicate the interpretation of our results, such as the diurnal cycle, convection, topography, and zonal inhomogeneities in the AEJ, while including those factors that are essential for assessing the instability of the AEJ, such as frictional dissipation of momentum. Out of necessity, we will also need to develop a special technique to maintain the basic state.

After comparing the idealized AEWs with observed AEWs to ensure that they are sufficiently realistic (section 3a), we will approach our hypothesis from two different angles. We will first use a local energetics budget to diagnose the process of upstream development, as explored in Diaz and Aiyyer (2013a,b) (section 3b). As suggested by Fig. 1, upstream development is an integral component of an absolute instability. We will then verify that the developing wave packet satisfies the criterion of absolute instability given by Orlanski and Chang (1993): namely, that it spreads both upstream and downstream relative to a fixed point (section 3d). Additionally, we will test a very simplified representation of ITCZ convective heating (section 3c).

b. Configuration

Our experiments use the dynamical core of WRF, version 3.0, with all of the prepackaged physics parameterizations switched off. The grid spacing used for all experiments is 120 km × 120 km in the x and y directions, respectively. Our conclusions are insensitive to decreasing the grid spacing to 40 km × 40 km. The model domain is shown in Fig. 2. The grid contains 240 points in the east–west direction and 49 points in the north–south direction and uses a Mercator projection. The model grid contains 28 vertical levels, and our results are insensitive to increasing the vertical resolution. The lateral boundaries are placed at 158.31°W, 138.32°E, 13°S, and 41.82°N. The model domain is illustrated in Fig. 2. We use a flat terrain set to a height of 400 m, which is the approximate average height of the terrain in the region of Africa that we are simulating.

Fig. 2.
Fig. 2.

Model domain showing boundaries and time-mean eddy kinetic energy (shaded; m2 s−2) at 650 hPa after the model has reached a steady state. The sponge boundary condition is applied only to the region outside of the thick rectangle.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

c. Basic state

As in many previous idealized studies using the normal-mode approach (e.g., Thorncroft and Hoskins 1994a), we begin with a zonally homogeneous basic state. This basic state is obtained using the National Centers for Environmental Prediction Final Analysis (NCEP-FNL) operational global data. We construct the basic state using the August 2004–08 time-mean meridional cross section averaged over 0°–10°E (Fig. 3a). Basic states derived over different regions of Africa all produce AEW activity, albeit with slightly different structures. The 0°–10°E basic state is a compromise between East and West Africa and produces AEWs most similar to observed ones. Basic states derived from the eastern Atlantic are stable in our modeling framework, consistent with their tendency to decay after crossing the West African coast.

Fig. 3.
Fig. 3.

(a) Meridional cross section of zonal wind (shaded; m s−1) and potential temperature (contours; K) at eastern edge of domain. (b) Meridional cross section of in Eq. (1) (shaded; K day−1) and potential temperature (contours; K) at eastern edge of domain.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

To adapt WRF for idealized modeling, we add two forcing terms to the thermodynamic energy equation:
e1
where θ is potential temperature, τ is a time scale, and F is a time-independent forcing term. The overbar represents the basic state.

The first term on the right-hand side (rhs) of Eq. (1) relaxes the atmosphere back to the basic state. Its main impact is to dampen the low-level temperature anomalies associated with the developing waves. Physically, this damping corresponds to the tendency for displaced air parcels to approach radiative–convective equilibrium with the ambient environment. In our simulations τ is set to 10−6 s−1. This value produces AEW temperature perturbations that are comparable with observations. Making small changes to this value or even excluding the term altogether does not affect the qualitative aspects of our conclusions. However, without low-level damping, the temperature perturbations become unrealistically large, and the basic state becomes much more unstable.

The second term on the rhs of Eq. (1) represents the diabatic forcing needed to maintain the basic state. Its inclusion is necessary because a time-mean cross section of a basic state derived from a reanalysis is, in general, not a steady-state solution to the primitive equations and will quickly self-destruct unless some forcing maintains it (e.g., diabatic heating in the Sahara is needed to counteract cold advection from the southwesterly monsoon). This term also partly accounts for the missing parameterizations of radiation, condensational heating, and changes in temperature caused by boundary layer processes. Since it is time independent, it does not damp the developing waves. This simple method of maintaining a basic state with a constant forcing term has a precedent in studies such as Pedlosky (1976), Tupaz et al. (1978), and Shapiro (1980) as an external vorticity source and in Hall et al. (2006) as a heat and momentum source.

We compute F by integrating the simulation forward for 15 days with τ in Eq. (1) set to 10−5 (1-day time scale) and F set to zero. This strong forcing prevents unstable waves from forming and holds the temperature field nearly constant. In essence, this procedure determines how much diabatic heating is needed to balance advection and keep the temperature field constant. Over the course of these 15 days, the temperature and wind fields reach an equilibrium, and the forcing term becomes nearly constant in time. The value of this forcing term at day 15 becomes F in Eq. (1) and is held constant through the remainder of the simulation. After day 15, τ is set to back to 10−6 s−1, and waves are free to develop. A cross section of F is given in Fig. 3b.

d. Boundary layer dissipation of momentum

To assess the instability properties of this basic state, it is essential that frictional damping be included, since it has been shown to drastically decrease growth rates (e.g., Hall et al. 2006). To simulate frictional drag on the wind, we implement a simple first-order turbulence closure scheme using chapter 6.4 from Stull (1988) as a resource. We resort to this approach instead of using the WRF prepackaged boundary layer parameterizations because our testing suggests that they do not function correctly unless used in conjunction with the other WRF parameterizations, which we have switched off.

The effect of subgrid-scale turbulence on the grid scale wind can be represented as
e2
where is the horizontal wind vector, w is the vertical velocity component, primes denote subgrid-scale values, and overbars indicate averages over a grid cell [Eq. (6.4.1a) in Stull (1988) without the pressure gradient and Coriolis terms, which are already included in the WRF dynamical core]. The turbulent momentum fluxes are parameterized by
e3
where K is the mixing coefficient [Eq. (6.4.1b) in Stull (1988)]. The mixing coefficient is determined using the O’Brien cubic polynomial within the boundary layer (O’Brien 1970):
e4
where zSL is the height of the surface layer, h is the height of the boundary layer, and z is the vertical height coordinate. The height of the boundary layer in our experiments is the time-average height from the FNL analysis used as input into the model. The surface-layer K is determined using the following relationship:
e5
where k is the von Kármán constant, z is height above ground level, and is the friction velocity. For our experiment, the surface layer is defined as the lowest two model levels. This definition places the top of the surface layer at 98 m. According to Stull (1988, p. 10), the surface layer is defined as the lowest 10% of the boundary layer. For a boundary layer approximately 1 km high, this definition would yield a 100-m-deep surface layer, which is roughly equivalent to the lowest two model layers in our experiments. The turbulent momentum fluxes at the lower boundary are determined using a surface drag law:
e6
where is the surface drag coefficient, which is set to 0.004 for our simulations.

e. Lateral boundary conditions

As in many previous idealized studies using the normal-mode approach (e.g., Thorncroft and Hoskins 1994a), our initial basic state is zonally homogeneous. However, rather than simulating a single normal mode, we want to simulate a localized wave packet. Thus, instead of the usual periodic boundary condition used in normal-mode studies, we impose a sponge boundary condition on the eastern and western boundaries. With this design, clean, unperturbed flow enters through the upstream boundary, and new AEWs must develop and reach sufficient amplitude within a zonally confined channel.

The sponge boundary condition is implemented at the eastern and western boundaries. It relaxes all fields back to the initial state. The relaxation coefficient is formulated as follows:
e7
where is the relaxation time scale at the outer edge of the sponge, is the time scale at the inner edge, is the x coordinate of the outer edge, and is the x coordinate of the inner edge. In the upstream sponge, and . In the downstream sponge, and . The downstream boundary is “spongier” because it must absorb the wave activity, and the upstream boundary is more rigid because it must maintain the basic state. The eastern and western edges of the sponge are placed at 108°W and 66°E (Fig. 2). This yields a region of instability with a zonal span of 174°. Though this is much wider than the real AEJ, it will be shown that nonlinear equilibration produces localized wave packets by removing the instability downstream. Repeating the experiment for a channel width of , which is more representative of the region where AEWs grow, yields similar results but with the undesirable outcome that large-amplitude AEWs hit the downstream boundary. The large channel width allows the waves to evolve freely with minimal interaction with the lateral boundaries, which can be seen by comparing the region of wave activity with the east–west boundaries in Fig. 2. The northern and southern boundaries use the WRF exponential lateral boundary condition and are 11 grid points wide.

Additional sensitivity tests were conducted to ensure that the east–west boundary condition does not adversely impact our conclusions. In one experiment, we increased the longitudinal width of the undamped channel to 270°, which yields 210° between the peak of the wave packet of interest and the start of the downstream damping region. This test resulted in a dominant wave packet almost identical to that in the original experiment, though with several weaker wave packets much farther downstream. To establish that the boundary condition itself is not the cause of the instability, an additional test was performed that excluded the boundaries altogether (i.e., a periodic east–west boundary condition). This experiment also yielded growing waves of similar structure. However, as one would expect, the waves develop nearly simultaneously across the entire channel.

f. Steady state

The simulation is run until the wave activity reaches a steady state. This process takes several weeks, before which AEW activity experiences intermittent periods of growth and decay. At steady state, AEW activity forms a localized, stationary wave packet trapped between the east–west boundaries, and all AEWs are structurally identical (i.e., analyzing the energetics of any particular AEW will yield essentially the same results). The long integration time removes the sensitivity to initial conditions. Running the experiment out to 1 year results in no change in this steady state.

3. Results

a. Validation

To establish that our modeling framework produces realistic AEWs, we will compare their structure and behavior to that of observed AEWs. All of the comparisons will use the ERA-Interim (Dee et al. 2011). Figures showing simulated AEWs will include a background map of Africa for ease of comparison. However, the map’s longitudinal position is arbitrary, though its latitudinal position does correctly correspond with the observed meridional cross section.

1) Disturbance structure

We begin with a comparison of the horizontal structure of observed and simulated AEWs. To describe the structure of a typical AEW in the ERA-Interim, we construct a composite AEW using single-point lag regressions. Following Kiladis et al. (2006), the meridional wind is filtered for periods of 2–6 days and wavenumbers 6–20 with westward phase velocities. For the regressions, the reference time series is the filtered 650-hPa meridional wind at 12°N, 0°. Other relevant fields (e.g., zonal wind and temperature) are obtained through a linear regression on the reference time series. The period of analysis is 1990–2010 for July–September. Recognizing that AEWs involve interactions between low-level and AEJ-level disturbances, we focus on 650-hPa relative vorticity, 925-hPa potential temperature, and winds at both levels. These fields are shown for the ERA-Interim data (Fig. 4) and for a single time (day 99) of the simulation (Fig. 5). Figures 4a,c and 5a,c show the full fields, and Figs. 4b,d and 5b,d show the anomalies from the time mean. For reference, the line is marked in Figs. 4b and 5b.

Fig. 4.
Fig. 4.

Regressed composite AEWs from ERA-Interim. (a) Total 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (b) Perturbation 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5) (c) Total 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). (d) Perturbation 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). The solid contour in (b) shows the climatological d(PV)/dy = 0 line.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Fig. 5.
Fig. 5.

Sample AEWs from the simulation at day 99. (a) Total 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (b) Perturbation 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (c) Total 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). (d) Perturbation 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). The solid contour in (b) shows the climatological d(PV)/dy = 0 line. The background map is for scale only.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Figure 5 shows the corresponding AEW structure from our simulation after the steady state is reached. The undisturbed flow entering the eastern edge of the domain features a zonal ribbon of enhanced 650-hPa cyclonic vorticity south of the AEJ (Fig. 5a). Beneath the AEJ lies a strong temperature gradient (Fig. 5c). This gradient marks the transition from the dry Sahara in the north to the wetter regions in the south. North of the AEJ lies the monsoon trough, where cool, moist southwesterly flow converges with hot, dry northeasterly flow in a region of enhanced cyclonic vorticity (Fig. 5c). Moving from east to west, the zonal flow of the AEJ evolves into a series of troughs and ridges. Beneath the troughs, low-level cyclones form within the vorticity-rich monsoon trough. These structures represent the model’s version of AEWs. Making concessions for the large zonal variations in the real atmosphere over Africa and all the physical processes neglected in the model, the simulated AEWs compare reasonably well with the ERA-Interim composite (Figs. 4a,c).

We now show that the simulated AEWs exhibit characteristics of baroclinic and barotropic instabilities by referring to the framework of counterpropagating Rossby waves on distinct PV gradients (Bretherton 1966; Hoskins et al. 1985). For baroclinic instability, midlevel PV anomalies interact with surface potential temperature anomalies. For barotropic instability, PV anomalies on either side of interact with each other in the horizontal plane. The fields of relevance are the temperature and vorticity anomalies from the time mean (Figs. 5b,d). Within and beneath the region of negative 650-hPa PV gradient, the 925-hPa potential temperature anomalies are phase shifted 6° west of the 650-hPa vorticity anomalies, a configuration suggesting baroclinic instability. At 650 hPa, the vorticity anomalies on either side of are phase shifted 12° with respect to each other. Their upshear tilt suggests the AEWs are extracting energy from the zonal shear of the AEJ.

Although the simulated AEW is broadly consistent with the ERA-Interim regression (Figs. 4b,d), there are important differences. The simulated 650-hPa vorticity anomalies on either side of are much farther separated from each other, indicating a weaker interaction. This separation, which increases toward the west, is the effect of the growing AEWs removing the horizontal shear instability by mixing out the midlevel PV gradients. Hence, the simulated AEWs are driven primarily by baroclinic instability, with barotropic instability playing a minor role compared with the ERA-Interim AEWs. Although these structures are within the realm of observed AEWs, we will modify our experiments to simulate AEWs more strongly driven by barotropic instability in section 3c.

2) Storm track

To assess the AEW storm track in the simulation, we examine the time-mean eddy kinetic energy [EKE; ] and compare it against the ERA-Interim. Although we should not expect to reproduce all of the details of the real storm track given the simplicity of our model, we should at least reproduce the latitudinal distribution of EKE and its zonal localization. For the reanalysis data, we use 2–10-day filtered zonal and meridional winds to calculate the EKE and average it over July–September 2000–09. For the simulation, we use unfiltered perturbation winds averaged for 1 month (once the steady state is reached, month-to-month variations in time-mean EKE vanish). Recognizing that AEW amplitude is maximized at low levels within the baroclinic zone and at midlevels south of the AEJ, we focus on EKE at 650 and 925 hPa. These fields are shown in Fig. 6 for the reanalysis and in Fig. 7 for the simulation.

Fig. 6.
Fig. 6.

Average EKE and winds from ERA-Interim. (a) Average 650-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2) and (b) average 925-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2).

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Fig. 7.
Fig. 7.

Average EKE and winds from simulation. (a) Average 650-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2) and (b) average 925-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2). The background map is for scale only.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

The ERA-Interim shows two zonally elongated regions of maximum time-mean EKE. One is located within the 925-hPa monsoon trough north of the AEJ (15°–25°N; Fig. 6b), and the other follows the 650-hPa PV gradient reversal within and south of the AEJ (5°–15°N; Fig. 6a). These regions represent the well-known northern and southern tracks of AEWs (e.g., Pytharoulis and Thorncroft 1999). Our simulation reproduces the northern-track EKE maximum and the EKE maximum within the AEJ core but does not reproduce the EKE maximum south of AEJ (Fig. 7). As suggested earlier, this deficiency is due to the lack of EKE generation through barotropic conversion south of the AEJ. Thus, the northern track is well simulated, while the southern track is much weaker.

A striking feature of the simulated storm track is that it abruptly terminates even though we started with a zonally homogeneous basic state. Its termination results from the interaction of the waves with the basic state. As the AEWs extract energy from the basic state, they remove the instability by mixing out the PV gradient reversal. As energy decay through dispersion and friction exceeds growth through baroclinic and barotropic conversion, the AEWs quickly weaken. These factors will be explored in detail in subsequent sections. This result raises the possibility that at least some observed AEWs have an intrinsic life cycle, which is limited by nonlinearity (e.g., Thorncroft and Hoskins 1994b), friction, or dispersion. In the real atmosphere, however, the fact that the diabatic circulations that generate the instability end near the West African coast is probably a more important factor in terminating the storm track. Nevertheless, in order to test our hypothesis that this basic state can support absolute instability, it is essential that the wave packet and resulting storm track remain localized.

3) Hovmöller

The wave packet nature of AEWs and the possible presence of absolute instability can be further illustrated through Hovmöller plots. We start with a sample of well-developed AEW packets from the ERA-Interim for two periods during 2008 (Fig. 8). Figure 8a shows the square of the 925-hPa meridional wind averaged from 15°–25°N, which represents low-level northern-track disturbances. Figure 8b shows the square of the 650-hPa meridional wind averaged from 5°–10°N, which represents midlevel southern-track disturbances. We apply a 2-day low-pass filter to remove the small-scale noise of the diurnal cycle but avoid using a high-pass filter so that the waviness is not imposed. Both the northern and southern tracks resemble nearly stationary to slowly eastward-moving wave packets, indicating a potential for absolute instability (cf. Fig. 1). Further spread of the wave packets is limited by the basic state becoming dynamically stable to the east and west. However, there is also some tendency for the northern track to propagate southward and the southern track to propagate northward, in both cases leaving the latitudinal averaging band of the Hovmöller plot.

Fig. 8.
Fig. 8.

Examples of AEW packets from ERA-Interim during 2008. Shaded values are the square of the 2-day low-pass-filtered meridional wind (m2 s−2) at (a) 925 hPa averaged from 13° to 23°N and (b) 650 hPa averaged from 2° to 10°N.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

We now examine similar diagrams for the simulation (Fig. 9). Our discussion will be limited to the northern track because it is better developed in this simulation. Figure 9b shows the initial development of the wave packet, and Fig. 9a shows the steady state reached after several weeks. The initial development takes the form of an eastward-dispersing wave packet which undergoes several periods of growth and decay. With time, AEWs form a stationary wave packet that somewhat resembles the northern track shown in Fig. 8a, albeit without temporal variability. Its further spread is limited by the flow stabilizing to the west and the sponge boundary to the east. This stationary wave packet is the steady state discussed in section 2f. Based on this Hovmöller plot, the average phase speed of the simulated AEWs is 7.8 m s−1. By comparison, observed AEWs typically have phase speeds between 8 and 12 m s−1 (Reed et al. 1977; Kiladis et al. 2006; Berry et al. 2007), and most previous idealized numerical simulations of AEWs have yielded phase speeds between 7.5 and 9.6 m s−1 [see Table 1 in Hall et al. (2006)].

Fig. 9.
Fig. 9.

Hovmöller plot of the square of the meridional wind (m2 s−2) from the simulation averaged from 15° to 19°N at 925 hPa. The y axis denotes days into the simulation. (a) Once quasi-steady state is reached and (b) beginning of simulation.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Comparing Figs. 9 and 8 and relating them with Fig. 1, both the simulated and observed wave packets qualitatively suggest absolute instability; new AEWs continuously form on the upstream side of the wave packet and maintain a localized storm track. We will verify this behavior in the following section using a local energetics budget.

b. Energetics

The most rigorous test to distinguish between absolute and convective instability involves examining the dispersion relation in complex phase space. As this approach is not readily possible for numerical simulations, we will deduce the instability properties using local energetics (e.g., Orlanski and Katzfey 1991; Orlanski and Chang 1993). The equation for the time tendency of EKE is as follows:
e8
where is the EKE, Vm is the time-averaged velocity, is the 2-day perturbation velocity, is the 3-day perturbation velocity, ϕ is the perturbation geopotential, disse refers to dissipative forcing, and Fo denotes the forcing that maintains the time-averaged circulation. The terms on the left-hand side of Eq. (8) are respectively the local EKE tendency, advection of EKE by the mean wind, and advection of EKE by the eddies. The first term on the rhs is work done by the pressure field. The second and third terms are Reynolds stress terms. They denote the exchange of kinetic energy between the mean flow and the eddies and are often referred to as barotropic conversion. The fourth term is dissipation of kinetic energy by the eddies. The last term is the impact of the steady-state forcing on the eddies and is generally found to be small.
Following Orlanski and Katzfey (1991), the pressure work term can be decomposed into the geopotential flux convergence term and the baroclinic term:
e9
where ω is vertical velocity in pressure coordinates and α is specific density. The nondivergent part of Eq. (9) can be isolated by removing the geostrophically balanced part of the flux:
e10
where is the Coriolis parameter at a reference latitude. This term is referred to as the ageostrophic geopotential flux. For small-amplitude quasigeostrophic waves, it represents an energy flux, which, when averaged over a full wavelength, points in the direction of the group velocity.

As discussed in the introduction, normal-mode instability considers only the space-independent temporal growth of EKE. To determine the instability of a localized mode, the spatial redistribution of the EKE must also be considered. Equations (8)(10) are useful for this purpose because they partition local EKE growth into both a temporal component, which amplifies the wave packet, and a spatial component, which spreads it out. The temporal component consists of both frictional dissipation and energy exchanges between the perturbations and the time-mean state (i.e., baroclinic and barotropic conversion). The spatial component, which consists of ageostrophic geopotential fluxes and their divergence, shows how EKE is redistributed. It has no counterpart in the energetics of individual normal modes, which rely on temporal growth. Over a sufficiently large domain, it averages to zero.

1) Wave packet energetics

Figure 10 shows a snapshot of the individual terms in Eqs. (8) and (9). As very little EKE production occurs above 500 hPa, each term is averaged from 950 to 500 hPa. For ease of discussion, EKE maxima are labeled from A (northerlies) to F (southerlies). Globally, the primary energy source is baroclinic conversion (Fig. 10e), and the primary energy sink is friction (Fig. 10d). Throughout the wave packet, the ageostrophic geopotential fluxes redistribute this baroclinically generated EKE eastward (Fig. 10a). These fluxes diverge from regions of concentrated baroclinic conversion in downstream EKE maxima and converge on upstream EKE maxima. At the upstream edge of the wave packet, center A receives EKE from center B, where the EKE is being generated. Center B then receives EKE from center C, which in turn receives EKE from center D, and so on. At the downstream edge, center F, despite continued baroclinic conversion, is actually decaying, as energy flux divergence and friction consume its EKE. As discussed in section 3a(2), this process helps to terminate the storm track (Fig. 7). The total nonadvective EKE tendency resembles most closely the geopotential flux divergence (Fig. 10a,f). Thus, although baroclinic conversion dominates the global energetics, geopotential fluxes dominate EKE growth and decay locally.

Fig. 10.
Fig. 10.

Individual terms in the EKE budget: a snapshot of the simulation. The contours in each panel are 950–500-hPa-averaged EKE (contoured on log2 scale beginning at 2 J kg−1), and the shading is 950–500-hPa-averaged EKE conversion rate (J kg−1 day−1). The specific terms and vectors plotted are (a) geopotential flux convergence [first term on rhs of Eq. (9)] and ageostrophic geopotential fluxes (vectors; m3 s−3), (b) barotropic conversion [second and third terms on rhs of Eq. (8)], (c) pressure work [first term on rhs of Eq. (8)], (d) friction [fourth term on rhs of Eq. (8)], (e) baroclinic [second term on rhs of Eq. (9)], and (f) total nonadvective [sum of (a),(b),(d), and (e)]. The EKE maxima are labeled for discussion, and the background map is for scale only.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Based on the local EKE budget, the simulated AEW life cycle is similar to that of observed AEWs (Diaz and Aiyyer 2013a,b) and is consistent with what one would expect from absolute instability: As individual EKE maxima move westward and grow by extracting energy from the basic state, they disperse some of their energy upstream. This dispersed energy initiates new troughs and ridges at the upstream edge of the wave packet.

2) Energetics following an AEW

To better understand the energetics from the perspective of an individual AEW, we calculate a volume average of each energetics term over a box enclosing a single wavelength of an AEW and follow it through time as it progresses through the wave packet. Defining the boundaries of a wavelength is somewhat arbitrary. Our approach, however, will attempt to minimize the splitting of regions with strong EKE generation. Based on Fig. 10, the cleanest way to define a wavelength is to cut along a line between successive EKE maxima where the geopotential flux convergence is equal to zero. This procedure is illustrated in Fig. 11. These boundaries coincide with minima in EKE between EKE maxima and do not cut through regions of strong baroclinic conversion. These boundaries enclose two EKE maxima, one for northerlies and one for southerlies, and tend to split through the middle of successive anticyclones. Growth rates are averaged over a volume bounded to the north and south by 26° and 5° N, to the east and west by the zero geopotential flux convergence line, and on the top and bottom by 200 and 950 hPa. Results are summarized in Fig. 12 in terms of both EKE conversion rates and EKE growth rates. Day 0 on the plot is chosen based on when the AEW could first be easily followed with the tracking algorithm.

Fig. 11.
Fig. 11.

Example of procedure to define a wavelength. EKE (m2 s−2) is contoured, and geopotential flux convergence (J kg−1 day−1) is shaded.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Fig. 12.
Fig. 12.

EKE conversion [d(EKE)/dt; J day−1] and growth rates {[1/(EKE)][d(EKE)/dt]; day−1} following a wavelength. (a) Conversion rates [the specific terms from Eqs. (8) and (9) are defined in Fig. 10], (b) barotropic terms, (c) EKE growth rate, and (d) EKE growth rate for barotropic terms. In (a) and (c), the blue line is spatial EKE redistribution, the red line is temporal EKE growth, and the black line is total EKE tendency. For presentation, (a) is truncated (the baroclinic term reaches a maximum of 967 J day−1, and friction reaches a minimum of −899 J day−1).

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Given their importance in understanding spatiotemporal instability, we will first compare the contributions of temporal EKE growth (red line) and spatial EKE growth (blue line) to total EKE growth (thick black line), as discussed in section 3b. Specifically, the temporal components include the second, third, and fourth terms on the rhs of Eq. (8) and the second term on the rhs of Eq. (9). The spatial component includes geopotential flux convergence, which is the first term on the rhs of Eq. (9). During the first few days of development, convergent geopotential fluxes (i.e., spatial growth) dominate the total EKE tendency, with their contribution to the growth rate exceeding 1 day−1 (Fig. 12c). Before day 3, temporal growth over this wavelength is actually negative, because frictional EKE dissipation exceeds EKE generation through baroclinic and barotropic conversion. Thus, without the flux of energy from upstream, this disturbance would be decaying. With time, the AEW begins to extract EKE from the basic state. From about day 4 to day 10, temporal growth rates remain nearly steady between 0.08 and 0.14 day−1. Although our model is nonlinear, we speculate that this is a reasonable estimate for the growth rate of the most unstable normal mode of this basic state, because individual normal modes can contribute only to temporal growth.

Comparing the temporal and spatial growth rates with the total growth rate leads to an interesting result: the AEW does not grow at the rate it extracts EKE from the basic state; it grows substantially faster in its early phase and later begins to quickly weaken despite baroclinic and barotropic EKE generation remaining strongly positive. Its growth rate is more strongly linked to its position within the wave packet, which impacts whether energy dispersion is a source or sink of energy. This analysis suggests that individual AEWs can grow much faster than a normal-mode instability analysis would imply.

Though previously combined, the individual EKE temporal growth terms vary substantially with time (Fig. 12). The earliest significant contributions come from . Though often overlooked for the energetics of AEWs, Hsieh and Cook (2007) found this term to be important in central and eastern Africa in a regional climate simulation, and Thorncroft (1995) found it to be important for his desert jet. Additionally, convergence in the monsoon trough contributes to EKE through . This was the largest barotropic term in the simulations of Hall et al. (2006) and contributed substantially to the early low-level growth of the AEW, which lead to hurricane Alberto (2000) (Diaz and Aiyyer 2013a). Some energy is also extracted from the horizontal shear of the AEJ (). The remaining barotropic terms are small because the mean state is approximately zonally symmetric. The largest source of EKE is through baroclinic conversion. Growth rates from baroclinic conversion increase throughout much of the simulation and peak at 0.85 day−1. However, with friction consuming large amounts of EKE (−0.8 day−1), the net EKE conversion is much smaller. This large cancellation occurs because baroclinic instability requires a surface disturbance (Fig. 5d), which is strongly damped by friction.

c. Southern-track AEWs

A notable deficiency of the previous simulation is the lack of an EKE maximum south of the AEJ (6°–10°N) associated with barotropic instability (Fig. 7). One problem is that F in Eq. (1) was designed to maintain the basic state against its own self-destruction, but not against its destruction by AEWs. Another is that the long time averaging used to construct the basic state smooths the horizontal shear of the AEJ and likely yields a more barotropically stable AEJ than would be seen at shorter time intervals. Noting that the AEJ is maintained in part by a meridional overturning circulation driven by ITCZ convection (e.g., Thorncroft and Blackburn 1999; Dickinson and Molinari 2000), simulating this circulation should destabilize the AEJ and produce more realistic AEWs (e.g., Grist et al. 2002; Hsieh and Cook 2008). To simulate an ITCZ, we place a zonally elongated heating anomaly centered at 7.5°N to mimic the impact of condensational heating. The heating distribution is prescribed by
e11
where represents a heating profile, y represents the meridional distance from the central latitude of the heating, and is set to 480 km. Beyond a distance of from the central latitude of heating, the value of H is set to zero. The vertical structure of the heating profile follows that of Janiga and Thorncroft (2013) for 20°E for the ERA-Interim (their Fig. 14) and is scaled to a maximum value of 12 mm day−1 (note that their heating rate is a function of precipitation rate). The vertical structure of is shown in Fig. 13. The amplitude of the simulated AEWs increases with larger heating rates, though the results are qualitatively similar for realistic values. In the zonal direction, decreases linearly to zero from 3.7°E to 21.2°W.
Fig. 13.
Fig. 13.

The value of H0 given in Eq. (11) (K day−1).

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

It is important to note that the AEWs generated in this experiment are not convectively coupled (e.g., Berry and Thorncroft 2012); the heating is temporally invariant and does not favor any particular wave phase. Its role is only to enhance the meridional PV gradient reversal, which is related to the horizontal shear of AEJ (Thorncroft and Blackburn 1999; Grist et al. 2002; Hsieh and Cook 2008).

We reproduce the vorticity and potential temperature snapshots and time-mean EKE for the new experiment in the same manner as the original (Figs. 14, 15). As anticipated, this simulation’s AEWs are stronger and more similar to observed AEWs (Fig. 4); the EKE maximum south of the AEJ is recovered (Fig. 15a), and the 650-hPa vorticity perturbations on either side of are closer together (cf. Figs. 14b and 5b). The diabatic heating has enhanced the cyclonic vorticity strip south of the AEJ, making the basic state more unstable for the growing AEWs (Fig. 14a).

Fig. 14.
Fig. 14.

Sample AEWs from the ITCZ heating experiment at day 97 simulation. (a) Total 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (b) Perturbation 650-hPa wind (vectors; m s−1) and relative vorticity (shaded; s−1 10−5). (c) Total 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). (d) Perturbation 925-hPa wind (vectors; m s−1) and potential temperature (shaded; K). The background map is for scale only.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Fig. 15.
Fig. 15.

Average EKE and winds from ITCZ heating simulation. (a) 650-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2). (b) 925-hPa winds (vectors; m s−1) and average EKE (shaded; m2 s−2). Contours show regions of diabatic heating in fraction of maximum heating rate. Contours begin at 0.2 and end at 0.8 with a contour interval of 0.2. The background map is for scale only.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Following Fig. 10, Fig. 16 shows the individual terms in Eqs. (8) and (9) averaged from 950 to 400 hPa for a single time of the simulation. In contrast with the original simulation, barotropic conversion south of the AEJ is a much larger source of EKE (Fig. 16b). It will later be shown that this barotropic conversion is primarily associated with . Similar to the regions of baroclinic conversion to the north, this barotropically generated EKE is being fluxed upstream south of the line, where it contributes to the growth of new disturbances upstream (Fig. 16a). North of this line, however, the pattern is more complicated, with fluxes directed upstream and downstream. This pattern is linked to the local PV gradients and will be discussed in section 3d. Interestingly, EKE maxima associated with barotropic conversion south of the AEJ are less subject to frictional dissipation than those to the north associated with baroclinic conversion (Fig. 16d). This difference occurs because they are above the boundary layer and suggests that barotropic conversion may be a more efficient generator of EKE, even though it is smaller than baroclinic conversion. These results agree with earlier experiments, in which barotropic instability dominated (e.g., Thorncroft and Hoskins 1994a) and friction had little effect on growth rates. In experiments such as ours and Hall et al. (2006), where baroclinic instability dominates, AEWs are more prone to friction because they have stronger low-level amplitude.

Fig. 16.
Fig. 16.

As in Fig. 10, but for the ITCZ heating experiment. The background map is for scale only.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

The same procedure as in section 2b is performed to obtain AEW-following growth rates (Fig. 17). Compared with the original simulation, EKE conversion through is much higher, with growth rates of 0.2 day−1 compared with 0.08 day−1 in the original simulation. The interplay between temporal and spatial EKE growth is similar to that in the original. Early growth is dominated by geopotential flux convergence (i.e., spatial growth). Energy extraction from the basic state (i.e., baroclinic and barotropic conversion) does not exceed its frictional dissipation until after day 3. Temporal growth rates during the mature phase are slightly higher than in the original, about 0.14–0.18 day−1. However, frictional dissipation is much less in this case, beginning from 0.3 to 0.5 day−1 as the AEWs become more baroclinic and surface intensified.

Fig. 17.
Fig. 17.

EKE conversion [d(EKE)/dt; J day−1] and growth rates {[1/(EKE)][d(EKE)/dt]; day−1} following a wavelength for the ITCZ heating experiment. (a) Conversion rates, (b) barotropic terms, (c) EKE growth rate, and (d) EKE growth rate for barotropic terms. In (a) and (c), the blue line is EKE redistribution, the red line is temporal EKE growth, and the black line is total EKE tendency.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

d. Absolute and convective instability

We will now verify more rigorously that the basic state is indeed absolutely unstable. As shown in Fig. 1, absolute instability requires that a wave packet spreads both upstream and downstream relative to a fixed point. Following Orlanski and Chang (1993), the distinction between absolute and convective instability can be formally expressed as
e12
where is the total group velocity, which can be interpreted as the velocity at which the energy of the wave packet is being transported by both the flow and the ageostrophic fluxes (Orlanski and Chang 1993). Starting from Eq. (6.3) in Orlanski and Chang (1993) and rewriting it in terms of pressure coordinates, the total group velocity can be expressed as
e13
where TEe, the total eddy energy including kinetic and potential, is defined as
e14
which is Eq. (4.3) in Orlanski and Katzfey (1991).

For our simulation, we evaluate the volume integral in Eq. (13) over half wavelengths in the zonal direction, from 0° to 28°N in the meridional direction, and throughout the depth of the model in the vertical direction. Results are then averaged over 1 month during the steady state and displayed as a function of longitude for both the original simulation (solid line) and ITCZ heating experiment (dashed line) (Fig. 18).

Fig. 18.
Fig. 18.

Total group velocity (m s−1) as a function of longitude. The solid line is for the original experiment, and the dashed line is for the ITCZ heating experiment.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

Figure 18 confirms that the wave packet in our simulations is absolutely unstable; the western edge of the wave packet is spreading westward (downstream), and the eastern edge of the wave packet is spreading eastward (upstream). The group velocity curve for the ITCZ experiment is shifted eastward compared with the original simulation simply because the wave packet extends farther eastward (cf. Figs. 7 and 15). It should also be noted that the westward phase velocity for the mature AEWs (−7.8 m s−1) is faster than the total westward group velocity at the downstream edge of the wave packet. This difference implies that AEWs should decay through dispersion in the storm-track exit region because individual troughs and ridges move westward faster than the speed at which their energy is transported.

The observed pattern of wave packet spreading can be explained using the energy flux terms themselves. We will first focus on the ageostrophic fluxes (Figs. 19a,b). The direction of Rossby wave energy dispersion, which is related to the ageostrophic geopotential flux, depends on the local PV gradient. Because the meridional PV gradient beneath the AEJ reverses sign near 800 hPa, we average the fluxes over two different layers, one bounded between 950 and 800 hPa and one between 800 and 450 hPa. The fluxes are averaged over half wavelengths and averaged for 1 month in time. For reference, time-mean 925-hPa potential temperature contours are shown in Fig. 19a and time-mean 650-hPa PV is shown in Fig. 19b. The meridional PV gradient reversal is marked with thick lines in Fig. 19b. For brevity, we will limit the analysis to the case with ITCZ heating.

Fig. 19.
Fig. 19.

Energy fluxes for different layers of the atmosphere for the ITCZ heating experiment. (a) Ageostrophic geopotential fluxes (vectors; m3 s−3) averaged over 950–800 hPa with 925-hPa time-mean potential temperature, (b) ageostrophic geopotential fluxes (vectors; m3 s−3) averaged over 800–450 hPa with 650-hPa time-mean potential vorticity (contours) and PV gradient reversal line (thick contours), (c) total EKE flux (mean advective and ageostrophic; m3 s−3) for 950–800-hPa layer (vectors) with regions of eastward flux shaded gray, and (d) total EKE flux (mean advective and ageostrophic; m3 s−3) for 800–450-hPa layer with regions of eastward flux shaded gray. The background map is for scale only.

Citation: Journal of the Atmospheric Sciences 72, 5; 10.1175/JAS-D-14-0128.1

As expected from theory, the ageostrophic fluxes point upstream in regions where the meridional PV gradient is positive [i.e., south of 10°N (Fig. 19b) and within the low-level baroclinic zone (Fig. 19a)]. At midlevels within the negative PV gradient, they point downstream. This pattern is consistent with Rossby wave dispersion and roughly matches that seen in real AEWs [see Fig. 11 in Diaz and Aiyyer (2013a)]. The vertical reversal in the direction of the ageostrophic fluxes is similar to that of unstable baroclinic waves (Orlanski and Chang 1993). The meridional reversal on either side of is associated with barotropic instability. It should also be noted that the strength of the PV gradient reversal, which is eroded by the growing AEWs, substantially weakens in the simulation toward the west (Fig. 19b). As previously discussed, this weakening helps to limit the storm track.

We now add the mean advective fluxes of EKE to the ageostrophic fluxes (Figs. 19c,d). Regions of eastward energy flux are shaded. Even after adding the advective fluxes, a large region of upstream energy flux remains in the lower layer (Fig. 19c) and a small region south of in the middle layer (Fig. 19d). By contrast, energy is fluxed quickly downstream within the core of the AEJ. The net result is that the lower portion of the wave packet spreads upstream, while the upper portion of the wave packet spreads downstream. South of , the wave packet spreads upstream throughout its entire depth. This pattern leads to the zonal distribution of group velocity seen in Fig. 18.

The dispersion pattern may provide insight into explaining the distribution of average EKE in the storm track that develops in Fig. 15. South of the line, where energy is fluxed upstream, the time-mean EKE extends much farther upstream. North of this line, where both the AEJ and the ageostrophic fluxes point westward, it extends much farther downstream. This pattern bears some resemblance to the real storm track (Fig. 6) and can be seen in the regressed composite (Fig. 4).

4. Summary and discussion

Our results suggest that the African easterly jet can support absolute instabilities. As an unstable wave packet begins to grow by extracting energy from the basic state, it disperses some of its energy upstream. Under realistic conditions, this upstream energy flux can overcome downstream advection by the AEJ and generate new troughs and ridges upstream, which subsequently undergo unstable growth. These new waves initially grow much faster than they extract energy through baroclinic and barotropic conversion because their main energy source is convergent energy fluxes from a larger-amplitude wave downstream. This scenario is a departure from normal-mode instability theory, which emphasizes wave growth solely through energy extraction from the basic state. For a localized mode, it is important to also consider how the extracted wave energy is advected and redistributed.

This view of instability is especially relevant to AEWs because the region where they can undergo unstable growth is relatively short and growth rates based on normal-mode instability theory are relatively slow. If the AEJ were convectively unstable to the point that no wave energy is fluxed upstream relative to the flow [corresponding to the example given in the introduction from Thorncroft and Hoskins (1994a) and an implicit assumption behind the “triggering hypothesis” (Thorncroft et al. 2008)], it would be difficult for disturbances to grow from small-amplitude perturbations because they would be quickly advected out of the unstable region before undergoing significant amplification. On the other hand, if the flow is absolutely unstable, any small perturbation may eventually lead to large-amplitude AEWs because part of the amplifying wave packet would remain within the unstable region until the absolute instability is removed. Between these two extremes, convectively unstable flow that can still support flow-relative upstream development could lead to a prolonged period of unstable growth before the wave packet is fully advected out of the unstable zone. Under the latter two scenarios, hydrodynamic instability can account for the observed amplitudes of AEWs.

Using these distinctions, we can speculate on the results of several previous studies. Basic states, such as those of Thorncroft et al. (2008) or Leroux et al. (2011), or the zonally varying basic states of Hall et al. (2006), which have negative normal-mode growth rates, are convectively unstable and thus support intermittent growth given a sufficiently large initial perturbation. If the basic state is just below the threshold of absolute instability, wave activity could be maintained for many wave periods and continue to grow for a long time before eventually decaying. As all of these studies use dry models, it is possible that a small amount of condensational heating could increase growth rates enough to make these basic states absolutely unstable. By contrast, the basic states of Hall et al. (2006) and Leroux and Hall (2009), which support small positive normal-mode growth could be classified as absolutely unstable. Despite very small normal-mode growth rates, AEW activity can develop from small perturbations and persist until the instability is removed.

Our results can also be cast in the terms of Pierrehumbert (1984). If the basic state can support absolute instabilities, then the unstable modes can be classified as local. As such, the region of absolute instability serves as the “seed” for instabilities growing downstream, and a localized storm track could exist without seeding from external perturbations. One distinguishing characteristic of a local mode is that its stability depends not only on the basic-state vertical shear, but also on the vertically averaged zonal wind (Pierrehumbert 1984). As its magnitude increases, the local growth rate decreases because the developing disturbances are advected more quickly downstream. Therefore, for a given amount of vertical and horizontal shear, a slower AEJ would be more unstable to small perturbations.

The ideas presented in this study may also have implications for other regions of the tropics. In contrast with the midlatitudes, the zonal flow in the tropics is much weaker and in some places reverses direction (e.g., near the ITCZ). Such conditions may favor the development of absolute instabilities. Indeed, Lindzen et al. (1983) suggest that, in contrast with normal-mode barotropic instability, absolute instability of localized perturbations is a more convincing model to explain the development of monsoon depressions over the Bay of Bengal.

Nevertheless, there are several caveats to consider before extrapolating our results to the real AEJ. One artificial aspect of our experiment is that the developing AEWs have a constant supply of unstable flow from the upstream boundary. For AEWs developing on the real AEJ, this is unrealistic. If the diabatic circulations maintaining the basic state cannot generate the instability as fast as it is removed by AEWs, it would become convectively unstable or even completely stable. It is probable that intraseasonal AEW intermittency can be partly explained by this transition. Additionally, although we argue that AEWs do not require large initial perturbations, our analysis does not exclude the possibility that some AEWs can grow from external perturbations, such as topographically forced waves, disturbances from the midlatitudes, or convectively induced circulations. Finally, as our model does not exclude nonmodal processes, it is possible that some of the growth of the simulated disturbances is nonmodal. In future work, it would be interesting to determine to what extent nonmodal processes are involved in their growth and decay.

Acknowledgments

This work was supported by NSF Grants ATM-0847323 and ATM-1433763. We are thankful to three anonymous reviewers for numerous substantial comments and suggestions.

REFERENCES

  • Berry, G. J., , and C. D. Thorncroft, 2005: Case study of an intense African easterly wave. Mon. Wea. Rev., 133, 752766, doi:10.1175/MWR2884.1.

    • Search Google Scholar
    • Export Citation
  • Berry, G. J., , and C. D. Thorncroft, 2012: African easterly wave dynamics in a mesoscale numerical model: The upscale role of convection. J. Atmos. Sci., 69, 12671283, doi:10.1175/JAS-D-11-099.1.

    • Search Google Scholar
    • Export Citation
  • Berry, G. J., , C. D. Thorncroft, , and T. Hewson, 2007: African easterly waves during 2004—Analysis using objective techniques. Mon. Wea. Rev., 135, 1251–1267, doi:10.1175/MWR3343.1.

    • Search Google Scholar
    • Export Citation
  • Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325334, doi:10.1002/qj.49709239302.

    • Search Google Scholar
    • Export Citation
  • Briggs, R. J., 1964: Electron-Stream Interaction with Plasmas. The MIT Press, 187 pp.

  • Burpee, R. W., 1972: The origin and structure of easterly waves in the lower troposphere of North Africa. J. Atmos. Sci., 29, 7790, doi:10.1175/1520-0469(1972)029<0077:TOASOE>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Carlson, T. N., 1969: Some remarks on African disturbances and their progress over the tropical Atlantic. Mon. Wea. Rev., 97, 716726, doi:10.1175/1520-0493(1969)097<0716:SROADA>2.3.CO;2.

    • Search Google Scholar
    • Export Citation
  • Cornforth, R. J., , B. J. Hoskins, , and C. D. Thorncroft, 2009: The impact of moist processes on the African Easterly Jet–African Easterly Wave system. Quart. J. Roy. Meteor. Soc., 135, 894913, doi:10.1002/qj.414.

    • Search Google Scholar
    • Export Citation
  • Dee, D. P., and Coauthors, 2011: The ERA-Interim reanalysis: Configuration and performance of the data assimilation system. Quart. J. Roy. Meteor. Soc., 137, 553597, doi:10.1002/qj.828.

    • Search Google Scholar
    • Export Citation
  • Diaz, M., , and A. Aiyyer, 2013a: The genesis of African easterly waves by upstream development. J. Atmos. Sci., 70, 34923512, doi:10.1175/JAS-D-12-0342.1.

    • Search Google Scholar
    • Export Citation
  • Diaz, M., , and A. Aiyyer, 2013b: Energy dispersion in African easterly waves. J. Atmos. Sci., 70, 130145, doi:10.1175/JAS-D-12-019.1.

  • Dickinson, M., , and J. Molinari, 2000: Climatology of sign reversals of the meridional potential vorticity gradient over Africa and Australia. Mon. Wea. Rev., 128, 38903900, doi:10.1175/1520-0493(2001)129<3890:COSROT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Eady, E. T., 1949: Long waves and cyclone waves. Tellus, 1A, 3352, doi:10.1111/j.2153-3490.1949.tb01265.x.

  • Farrell, B. F., 1982: Pulse asymptotics of the Charney baroclinic instability problem. J. Atmos. Sci., 39, 507517, doi:10.1175/1520-0469(1982)039<0507:PAOTCB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Farrell, B. F., 1983: Pulse asymptotics of three-dimensional baroclinic waves. J. Atmos. Sci., 40, 22022210, doi:10.1175/1520-0469(1983)040<2202:PAOTDB>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Grist, J. P., , S. E. Nicholson, , and A. I. Barcilon, 2002: Easterly waves over Africa. Part II: Observed and modeled contrasts between wet and dry years. Mon. Wea. Rev., 130, 212–225, doi:10.1175/1520-0493(2002)130<0212:EWOAPI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hakim, G. J., 2003: Developing wave packets in the North Pacific storm track. Mon. Wea. Rev., 131, 2824–2837, doi:10.1175/1520-0493(2003)131<2824:DWPITN>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Hall, N. M. J., , G. N. Kiladis, , and C. D. Thorncroft, 2006: Three-dimensional structure and dynamics of African easterly waves. Part II: Dynamical modes. J. Atmos. Sci., 63, 22312245, doi:10.1175/JAS3742.1.

    • Search Google Scholar
    • Export Citation
  • Hill, C. M., , and Y.-L. Lin, 2003: Initiation of a mesoscale convective complex over the Ethiopian Highlands preceding the genesis of Hurricane Alberto (2000). Geophys. Res. Lett., 30, 1232, doi:10.1029/2002GL016655.

    • Search Google Scholar
    • Export Citation
  • Hoskins, B. J., , M. E. McIntyre, , and A. W. Robertson, 1985: On the use and significance of isentropic potential vorticity maps. Quart. J. Roy. Meteor. Soc., 111, 877946, doi:10.1002/qj.49711147002.

    • Search Google Scholar
    • Export Citation
  • Hsieh, J.-S., , and K. H. Cook, 2007: A study of the energetics of African easterly waves using a regional climate model. J. Atmos. Sci., 64, 421440, doi:10.1175/JAS3851.1.

    • Search Google Scholar
    • Export Citation
  • Hsieh, J.-S., , and K. H. Cook, 2008: On the instability of the African easterly jet and the generation of African waves: Reversals of the potential vorticity gradient. J. Atmos. Sci., 65, 21302151, doi:10.1175/2007JAS2552.1.

    • Search Google Scholar
    • Export Citation
  • Huerre, P., , and P. A. Monkewitz, 1990: Local and global instabilities in spatially developing flows. Annu. Rev. Fluid Mech., 22, 473537, doi:10.1146/annurev.fl.22.010190.002353.

    • Search Google Scholar
    • Export Citation
  • Janiga, M. A., , and C. D. Thorncroft, 2013: Regional differences in the kinematic and thermodynamic structure of African easterly waves. Quart. J. Roy. Meteor. Soc., 139, 1598–1614, doi:10.1002/qj.2047.

    • Search Google Scholar
    • Export Citation
  • Kiladis, G. N., , C. D. Thorncroft, , and N. M. J. Hall, 2006: Three-dimensional structure and dynamics of African easterly waves. Part I: Observations. J. Atmos. Sci., 63, 22122230, doi:10.1175/JAS3741.1.

    • Search Google Scholar
    • Export Citation
  • Leroux, S., , and N. M. J. Hall, 2009: On the relationship between African easterly waves and the African easterly jet. J. Atmos. Sci., 66, 23032316, doi:10.1175/2009JAS2988.1.

    • Search Google Scholar
    • Export Citation
  • Leroux, S., , N. M. J. Hall, , and G. N. Kiladis, 2010: A climatological study of transient–mean-flow interactions over West Africa. Quart. J. Roy. Meteor. Soc., 136, 397410, doi:10.1002/qj.474.

    • Search Google Scholar
    • Export Citation
  • Leroux, S., , N. M. J. Hall, , and G. N. Kiladis, 2011: Intermittent African easterly wave activity in a dry atmospheric model: Influence of the extratropics. J. Climate, 24, 53785396, doi:10.1175/JCLI-D-11-00049.1.

    • Search Google Scholar
    • Export Citation
  • Lin, Y.-L., , K. E. Robertson, , and C. M. Hill, 2005: Origin and propagation of a disturbance associated with an African easterly wave as a precursor of Hurricane Alberto (2000). Mon. Wea. Rev., 133, 3276–3298, doi:10.1175/MWR3035.1.

    • Search Google Scholar
    • Export Citation
  • Lindzen, R. S., , B. Farrell, , and A. J. Rosenthal, 1983: Absolute barotropic instability and monsoon depressions. J. Atmos. Sci., 40, 11781184, doi:10.1175/1520-0469(1983)040<1178:ABIAMD>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Mass, C., 1979: A linear primitive equation model of African wave disturbances. J. Atmos. Sci., 36, 20752092, doi:10.1175/1520-0469(1979)036<2075:ALPEMO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Mekonnen, A., , C. D. Thorncroft, , and A. R. Aiyyer, 2006: Analysis of convection and its association with African easterly waves. J. Climate, 19, 54055421, doi:10.1175/JCLI3920.1.

    • Search Google Scholar
    • Export Citation
  • Merkine, L.-O., 1977: Convective and absolute instability of baroclinic eddies. Geophys. Astrophys. Fluid Dyn., 9, 129157, doi:10.1080/03091927708242322.

    • Search Google Scholar
    • Export Citation
  • Mozer, J. B., , and J. A. Zehnder, 1996: Lee vorticity production by large-scale tropical mountain ranges. Part II: A mechanism for the production of African waves. J. Atmos. Sci., 53, 539549, doi:10.1175/1520-0469(1996)053<0539:LVPBLS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • O’Brien, J. J., 1970: A note on the vertical structure of the eddy exchange coefficient in the planetary boundary layer. J. Atmos. Sci., 27, 12131215, doi:10.1175/1520-0469(1970)027<1213:ANOTVS>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Orlanski, I., , and J. Katzfey, 1991: The life cycle of a cyclone wave in the Southern Hemisphere. Part I: Eddy energy budget. J. Atmos. Sci., 48, 19721998, doi:10.1175/1520-0469(1991)048<1972:TLCOAC>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Orlanski, I., , and E. K. M. Chang, 1993: Ageostrophic geopotential fluxes in downstream and upstream development of baroclinic waves. J. Atmos. Sci., 50, 212225, doi:10.1175/1520-0469(1993)050<0212:AGFIDA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 1976: Finite-amplitude baroclinic disturbances in downstream varying currents. J. Phys. Oceanogr., 6, 335344, doi:10.1175/1520-0485(1976)006<0335:FABDID>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pierrehumbert, R., 1984: Local and global baroclinic instability of zonally varying flow. J. Atmos. Sci., 41, 21412162, doi:10.1175/1520-0469(1984)041<2141:LAGBIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Pierrehumbert, R., , and K. Swanson, 1995: Baroclinic instability. Annu. Rev. Fluid Mech., 27, 419467, doi:10.1146/annurev.fl.27.010195.002223.

    • Search Google Scholar
    • Export Citation
  • Pytharoulis, I., , and C. Thorncroft, 1999: The low-level structure of African easterly waves in 1995. Mon. Wea. Rev., 127, 22662280, doi:10.1175/1520-0493(1999)127<2266:TLLSOA>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Reed, R. J., , D. C. Norquist, , and E. E. Recker, 1977: The structure and properties of African wave disturbances as observed during phase III of GATE. Mon. Wea. Rev.,105, 317–333, doi:10.1175/1520-0493(1977)105<0317:TSAPOA>2.0.CO;2.

  • Rennick, M. A., 1976: The generation of African waves. J. Atmos. Sci., 33, 19551969, doi:10.1175/1520-0469(1976)033<1955:TGOAW>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Shapiro, L. J., 1980: The effect of nonlinearities on the evolution of barotropic easterly waves in a nonuniform environment. J. Atmos. Sci., 37, 26312643, doi:10.1175/1520-0469(1980)037<2631:TEONOT>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., 1977: A note on the instability of the African easterly jet. J. Atmos. Sci., 34, 16701674, doi:10.1175/1520-0469(1977)034<1670:ANOTIO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Simmons, A. J., , and B. J. Hoskins, 1979: The downstream and upstream development of unstable baroclinic waves. J. Atmos. Sci., 36, 12391254, doi:10.1175/1520-0469(1979)036<1239:TDAUDO>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Stull, R. B., 1988: An Introduction to Boundary Layer Meteorology. Springer, 670 pp.

  • Swanson, K., , and R. T. Pierrehumbert, 1994: Nonlinear wave packet evolution on a baroclinically unstable jet. J. Atmos. Sci., 51, 384396, doi:10.1175/1520-0469(1994)051<0384:DCCISF>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., 1995: An idealized study of African easterly waves. III: More realistic basic states. Quart. J. Roy. Meteor. Soc., 121, 15891614, doi:10.1002/qj.49712152706.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , and B. J. Hoskins, 1994a: An idealized study of African easterly waves. I: A linear view. Quart. J. Roy. Meteor. Soc., 120, 953982, doi:10.1002/qj.49712051809.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , and B. J. Hoskins, 1994b: An idealized study of African easterly waves. II: A nonlinear view. Quart. J. Roy. Meteor. Soc., 120, 9831015, doi:10.1002/qj.49712051810.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , and M. Blackburn, 1999: Maintenance of the African easterly jet. Quart. J. Roy. Meteor. Soc., 125, 763786, doi:10.1002/qj.49712555502.

    • Search Google Scholar
    • Export Citation
  • Thorncroft, C. D., , N. M. J. Hall, , and G. N. Kiladis, 2008: Three-dimensional structure and dynamics of African easterly waves. Part III: Genesis. J. Atmos. Sci., 65, 35963607, doi:10.1175/2008JAS2575.1.

    • Search Google Scholar
    • Export Citation
  • Tupaz, J. B., , R. T. Williams, , and C.-P. Chang, 1978: A numerical study of barotropic instability in a zonally varying easterly jet. J. Atmos. Sci., 35, 12651280, doi:10.1175/1520-0469(1978)035<1265:ANSOBI>2.0.CO;2.

    • Search Google Scholar
    • Export Citation
1

These terms originated from plasma physics, where the theory was first developed (Briggs 1964), and have no relationship with convective and absolute instabilities of vertical temperature gradients.

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