Structural Changes in the African Easterly Jet and Its Role in Mediating the Effects of Saharan Dust on the Linear Dynamics of African Easterly Waves

Dustin F. P. Grogan; Terrence R. Nathan; Shu-Hua Chen

doi:10.1175/JAS-D-19-0104.1

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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Model, basic states, and experimental design
3. Analytical analysis
4. Numerical results
- a. Growth rates and phase speeds
- b. Energetics
5. Conclusions

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Fig. 1.

Reference AEJ (solid) and corresponding potential temperature field (dashed). Contour intervals are 2 m s⁻¹ for the wind and 5 K for the temperature.

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Fig. 2.

(a) Basic-state dust mixing ratio and (b) daily averaged dust-induced heating rate. The dust mixing ratio is the sum of the 12 particle sizes. Contour intervals are 100 μg kg⁻¹ for the dust and 0.2 K day⁻¹ for the heating rate. The AEJ core is denoted by ⊗.

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Fig. 3.

Growth rates as a function of zonal wavenumber k for (a),(b) NODUST and (c),(d) DUST and for (top) peak meridional shear and (bottom) peak vertical shear. The solid line is the reference jet, and the dashed and dotted lines are the reduced and enhanced shear jets, respectively.

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Fig. 4.

Latitude–height plots of ${\hat{C}}_{K}$ for (a),(c) $U_{y}^{-}$ and (b),(d) $U_{y}^{+}$ for the k = 12 (left) NODUST and (right) DUST experiments. Contour interval is 1 × 10⁻⁵ m² s⁻³.

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Fig. 5.

As in Fig. 4, but for ${\hat{C}}_{E}$ .

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Fig. 6.

As in Fig. 4, but for ${\hat{C}}_{A}$ .

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Fig. 7.

Latitude–height plots of ${\hat{G}}_{E}$ for (a) $U_{y}^{-}$ and (b) $U_{y}^{+}$ for the k = 12 DUST experiments. Superimposed in each panel is the corresponding critical surface ( $\bar{u} k - ω_{r} \to 0$ ; bold circle) and the latitude of the peak meridional gradient of the zonal-mean dust ( $\partial \bar{γ} / \partial y$ ; bold dashed line). Contour interval is 1 × 10⁻⁵ m² s⁻³.

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Article Type:: Research Article

Structural Changes in the African Easterly Jet and Its Role in Mediating the Effects of Saharan Dust on the Linear Dynamics of African Easterly Waves

Dustin F. P. Grogan

Dustin F. P. Grogan Department of Atmospheric and Environmental Sciences, University at Albany, State University of New York, Albany, New York

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Terrence R. Nathan

Terrence R. Nathan Atmospheric Science Program, Department of Land, Air, and Water Resources, University of California, Davis, Davis, California

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Shu-Hua Chen

Shu-Hua Chen Atmospheric Science Program, Department of Land, Air, and Water Resources, University of California, Davis, Davis, California

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Print Publication:: 01 Nov 2019

DOI:: https://doi.org/10.1175/JAS-D-19-0104.1

Page(s):: 3351–3365

Received:: 22 Apr 2019

Accepted:: 08 Aug 2019

Published Online:: Nov 2019

Displayed acceptance dates for articles published prior to 2023 are approximate to within a week. If needed, exact acceptance dates can be obtained by emailing amsjol@ametsoc.org.

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Abstract

Analytical and numerical analyses are used to examine how structural changes to the African easterly jet (AEJ) mediate the effects of Saharan mineral dust aerosols on the linear dynamics of African easterly waves (AEWs). An analytical expression for the generation of eddy available potential energy (APE) is derived that exposes how the AEJ and dust combine to affect the energetics of the AEWs. The expression is also used to interpret the numerical results, which are obtained by radiatively coupling a simplified version of the Weather Research and Forecasting Model to a conservation equation for dust. The WRF-Dust model is used to conduct linear simulations based on five observationally consistent zonal-mean AEJs: a reference AEJ and four other AEJs that are obtained by perturbing the maximum meridional and vertical shear. For a dust distribution consistent with summertime observations over North Africa, the numerical simulations show the following: (i) Irrespective of the AEJ structure or the zonal scale of the AEWs, the dust increases the growth rates of the AEWs. (ii) The growth rates of the AEWs are optimized when the ratio of baroclinic to barotropic energy conversions is largest. (iii) When the energy conversions are sufficiently large, the zonal scale of the fastest-growing AEW shortens. The numerical results confirm the analytical analysis, which shows that the dust effects, which are modulated by the Doppler-shifted frequency, are strongest north of the AEJ axis, a region where the dust augments the preexisting meridional temperature gradient.

Corresponding author: Dustin Grogan, dgrogan@albany.edu

Abstract

Corresponding author: Dustin Grogan, dgrogan@albany.edu

Keywords: Atmosphere; Africa; Dynamics; Large-scale motions; Dust or dust storms; Aerosol radiative effect

1. Introduction

African easterly waves (AEWs) dominate the synoptic-scale circulation over North Africa during summer. AEWs occur every 2–6 days and travel westward along two tracks that straddle the African easterly jet (AEJ) (Pytharoulis and Thorncroft 1999), which is located at midtroposphere and aligned along ~15°N. The AEJ is the major source of energy for the AEWs and therefore affects their spatial–temporal evolution (Burpee 1972, 1974; Reed et al. 1977, 1988).

The AEWs that migrate along the north track extend their influence into the Sahara Desert, where they contribute to the emission, mixing, and transport of Saharan mineral dust aerosols (Jones et al. 2003; Knippertz and Todd 2010; Grogan and Thorncroft 2019; Nathan et al. 2019). The dust episodically coalesces into synoptic-scale plumes that absorb, scatter, and emit radiation, processes that affect the strength and structure of both the AEJ (Tompkins et al. 2005; Chen et al. 2010; Reale et al. 2011) and the AEWs (see Table 1).

Table 1.

Select papers on dust-coupled AEWs.

As discussed in Bercos-Hickey et al. (2017), dust-induced changes to the AEWs can be envisaged as operating along two pathways. One pathway is associated with the zonal-mean portion of the dust field (pathway I) and the other pathway with the eddy portion (pathway II).^¹ Along pathway I the zonal-mean dust alters the zonal-mean temperature field, which, due to thermal wind balance, alters the location, strength, and structure of the AEJ. As shown by Grogan et al. (2017), the AEWs then experience the effects of the zonal-mean dust through their interaction with the dust-modified AEJ.

Along pathway II, as shown by Nathan et al. (2017), the dust-induced eddy diabatic heating rate is a function of the zonal-mean aerosol optical depth (AOD), and the advection of zonal-mean dust by the eddy wind field. Together these radiative–dynamical processes change the growth, propagation, and structure of the AEWs (Grogan et al. 2016; Bercos-Hickey et al. 2017; Nathan et al. 2017).

Pathways I and II together constitute the dust-modified wave–mean flow interaction, a fundamentally nonlinear process (Grogan et al. 2017). If the background distributions of wind, temperature, and dust are specified, as in a linear model, then only pathway II operates on the dust-modified AEWs.

Grogan et al. (2016), for example, examined pathway II. To do so, they coupled an idealized version of the Weather Research and Forecasting (WRF) Model with a simplified dust model to understand the radiative–dynamical feedbacks that control the dust-modified linear dynamics of AEWs. The dust model, which was coupled radiatively to the WRF Model, included the effects of advection and sedimentation. The WRF-Dust model was used to conduct linear simulations around a zonal-mean background state that was chosen consistent with the observed zonal-mean distributions of wind, temperature, and dust over North Africa during summer. The model AEJ was chosen supercritical with respect to the threshold for combined barotropic–baroclinic instability. Grogan et al. (2016) showed that for the fastest-growing AEW, the growth rate increased monotonically with AOD, ranging from 15% to 90% for AODs ranging from 1.0 to 2.5. To explain the model results, Grogan et al. (2016) derived an analytical expression for the generation of eddy available potential energy (APE) that showed how the dust most effectively feeds back on the AEWs to affect their growth. The expression shows that the maximum generation of eddy APE occurs where the Doppler-shifted frequency^² is minimized and the meridional gradient of zonal-mean dust is maximized. This unique region occurs between the AEJ axis (~15°N) and the maximum in the dust distribution (~20°N).

Nathan et al. (2017) combined a theoretical framework with WRF-Dust model experiments to examine the subcritical destabilization of AEWs by the dust. The theoretical framework yielded analytical expressions for the dust-modified growth rate and frequency of the modeled AEWs. The expressions not only expose how the dust couples to the circulation to affect the AEWs, but they also serve as a tool for predicting what background wind and dust distributions can destabilize the AEWs. The theoretical predictions were confirmed through WRF-Dust model simulations, which showed that otherwise neutral AEWs can be destabilized by the dust, with growth rates similar to those obtained in supercritical AEJs (cf. Rennick 1976; Thorncroft and Hoskins 1994; Grogan et al. 2016).

The idealized studies of Grogan et al. (2016) and Nathan et al. (2017) examined different background jets—one supercritical and the other subcritical—yet they shared two key results: the dust enhances the growth of AEW-like waves; and the maximum generation of eddy APE is located between the AEJ axis and the dust maximum. Despite the simplicity of their models, the results are robust. This was confirmed by Bercos-Hickey et al. (2017), who used the full nonlinear, three-dimensional version of the WRF-Dust model, which includes processes excluded in the linear studies cited above, that is, wind-generated emission of dust, cloud microphysical processes, subgrid cumulus and boundary layer mixing of dust, and wet and dry deposition. Bercos-Hickey et al. (2017) showed that for an AEJ representative of summer 2006, the dust strengthens the AEWs and maximizes both the generation of eddy APE and the baroclinic energy conversion between the AEJ and dust maximum. The fact that the full WRF-Dust model reproduces the key results from the idealized version lends confidence in the ability of the idealized WRF-Dust model to not only expose the dust-modified dynamics of the AEWs, but to also replicate the results obtained from more complicated models.

The above dust–AEW studies each conducted their analyses around a single, though observationally representative, AEJ. Observations show, however, that the AEJ has pronounced intraseasonal variability (Newell and Kidson 1984; Afiesimama 2007) as well as interannual variability (Grist and Nicholson 2001; Grist et al. 2002; Dezfuli and Nicholson 2011). Such variability manifests in structural changes to the AEJ, which has been shown in previous idealized, dust-free modeling studies to significantly affect the linear dynamics of AEWs (Rennick 1981; Kwon 1989; Thorncroft 1995; Paradis et al. 1995).

It is unclear, however, to what extent structural changes in the AEJ mediate the effects of dust on the growth of AEWs. For example, are the dust-modified growth rates of the AEWs more sensitive to changes in the vertical or meridional shear of the background jet? How does the jet structure affect the dust modified energetics of the AEWs? And is there an optimal jet structure that maximizes the effects of the dust on the growth of AEWs?

To answer these questions, we combine an analytical analysis of the dust-modified energetics with idealized WRF-Dust model experiments. Such an approach aids in the exposure of important physical–dynamical processes, processes that may be obscured in observational data or difficult to identify in complex models. Moreover, the idealized WRF-Dust model is easily modified and thus is particularly useful for sensitivity analyses of different background jets, such as those examined in this study.

2. Model, basic states, and experimental design

a. Model

We use the same model as Grogan et al. (2016). The model has a WRF dynamical core (v3.7) that is coupled to the dust model developed by Chen et al. (2010, 2015). The dust model simulates the advection and sedimentation of 12-bin dust mixing ratios; the particles’ radii range from 0.15 to 5.0 μm. The radiative properties of the dust are calculated using the NASA Goddard Space Flight Center radiation model (Chou and Suarez 1999; Chou et al. 2001), which uses the Optical Properties of Aerosols and Clouds (OPAC) software package (Hess et al. 1998) to obtain the dust optical properties—extinction, single scattering albedo, and asymmetric parameter. The radiation model computes the daily averaged dust-induced heating rates, which includes the absorption and scattering due to dust, and the reabsorption of dust emission by other constituents.

The domain is a global channel projected on an equidistant cylindrical grid. In the horizontal, the resolution is 0.5° and the boundary conditions are periodic in the zonal direction and symmetric at the channel sidewalls (40°N and 10°S). In the vertical, there are 50 levels that extend from a flat surface up to 100 hPa, with no-slip conditions at the top and bottom. To reduce noise and boundary reflections, we impose a 30-min hyperdiffusion on the wind fields throughout the model domain, and a Rayleigh damping on the wind and temperature fields at the model top (300–100 hPa).

b. Basic states

Our linear stability analysis pivots on a zonal-mean background jet that is identical to that used in Grogan et al. (2016) (Fig. 1). The jet is consistent with the observed AEJ: it is symmetric in latitude, asymmetric in height, and has a maximum easterly wind of 15 m s⁻¹ centered at 15°N and ~650 hPa. The linear stability of this model AEJ will serve as our reference, about which we construct four other background states: two correspond to either a 25% reduction or a 25% enhancement in the maximum meridional shear zones, which are located north and south of the AEJ axis. The other two AEJs correspond to either a 25% reduction or a 25% enhancement of the maximum vertical shear below 650 hPa. These are the percentages used by Rennick (1981) and Paradis et al. (1995) in their idealized modeling studies of the sensitivity of the stability of the AEJ to changes in its structure.

Fig. 1.

Reference AEJ (solid) and corresponding potential temperature field (dashed). Contour intervals are 2 m s⁻¹ for the wind and 5 K for the temperature.

Citation: Journal of the Atmospheric Sciences 76, 11; 10.1175/JAS-D-19-0104.1

We denote the five background jets by U_REF (reference jet), $U_{y}^{-}$ (reduced meridional shear), $U_{y}^{+}$ (enhanced meridional shear), $U_{p}^{-}$ (reduced vertical shear), and $U_{p}^{+}$ (enhanced vertical shear). Table 2 lists, for each AEJ, the maximum values for the wind speed, and the meridional and vertical shear. Table 2 shows, for example, that the meridional shear jets $U_{y}^{-}$ and $U_{y}^{+}$ retain the same maximum vertical shear as U_REF (i.e., −5.9 × 10⁻³ s⁻¹). As in Rennick (1981) and Paradis et al. (1995), the meridional and vertical structures that characterize the AEJs in this study are coupled. This means the jet’s adjustment to the maximum shear in one direction (i.e., meridional or vertical) can modify the local shear in the other direction. For example, the meridional shear jets modify the vertical shear on the flanks of the AEJ axis, above and below the AEJ core. These changes in the local vertical shear are inversely proportional to the changes in the maximum meridional shear. For example, a decrease in $U_{y}^{-}$ increases the local vertical shear, and an increase in $U_{y}^{+}$ decreases the local vertical shear.

Table 2.

Maximum values of zonal-mean wind speed, meridional shear, and vertical shear for each of the five basic states. The maximum meridional shear on the north and south flanks are the same in magnitude but opposite in sign.

Figure 2 shows the background distribution of the dust mixing ratio and daily averaged dust-induced heating rate. As in Grogan et al. (2016), each of the 12 particle sizes that constitute the dust field have the same horizontal distribution, which is symmetric in the horizontal and centered at 20°N. In contrast to Grogan et al. (2016), we use a more realistic vertical profile and particle size distribution for the dust field. The vertical profile is guided by dust extinction profiles from CALIPSO lidar measurements over North Africa from Yu et al. (2010) (see their Fig. 10), while the particles follow the more realistic logarithmic size distribution described in Kok (2011, see their Fig. 2). The size distributions are scaled to produce a maximum AOD of ~1.0, which yields a dust maximum mixing ratio of ~800 μg kg⁻¹ at 20°N near the surface (Fig. 2a).

Fig. 2.

Citation: Journal of the Atmospheric Sciences 76, 11; 10.1175/JAS-D-19-0104.1

Figure 2b shows the basic-state daily averaged dust-induced heating rate. The weak dust-induced warming at the surface obtained by Grogan et al. (2016) is absent in Fig. 2b, a consequence of our choosing a more realistic logarithmic particle size distribution. More importantly, the absence of surface heating is more consistent with other dust-induced heating rate profiles over North Africa (e.g., Quijano et al. 2000). Above the surface, the heating rate increases with height, reaching a maximum at ~700 hPa, which is ~50 hPa lower and ~15% stronger than the maximum heating rate given in Grogan et al. (2016). Above 700 hPa, the heating rates decrease with height.

c. Experimental design

For each AEJ presented in Table 2, numerical experiments are run without dust (NODUST) and with dust (DUST). For the NODUST experiments, the radiative effects of the dust are decoupled from the thermodynamics so that the dust is simply a passive tracer. For the DUST experiments, the dust is interactive; that is, it absorbs, scatters, and emits radiation to affect the thermal field, which, in turn, affects the circulation and the transport of dust. Because all of the other parameterizations, such as microphysics, convection, and boundary layer physics, are deactivated in the model, the differences between NODUST and DUST isolate the direct dust-radiative feedbacks on the linear dynamics of the AEWs.

The linear stability characteristics of the AEWs are obtained using the same methodology as in previous studies (Thorncroft and Hoskins 1994; Simmons and Hoskins 1976; Grogan et al. 2016). For each experiment, a wave packet is superimposed onto the zonal-mean wind field. The packet contains zonal wavelengths spanning 2500–5000 km (nondimensional wavenumbers 8–16), which are typical for observed AEWs (Burpee 1975). The initial amplitude of the wave packet is small (~10⁻⁴ m s⁻¹), which ensures linearity throughout the simulation [see Grogan et al. (2016) for details]. The model is integrated in time until the domain-averaged perturbation energy for each wavelength grows exponentially to an accuracy of 10⁻³ for at least 12 h. A fast Fourier transform algorithm is used to obtain the desired disturbance structures from each field. The fields are then scaled to produce a peak meridional wind of 5 m s⁻¹.

3. Analytical analysis

To aid in the interpretation of the numerical results, we first examine the global and local energetics of the AEWs. The energetics equations can be written as (Hsieh and Cook 2007)

\frac{\partial K_{E}}{\partial t} = C_{K} + C_{E},

(3.1)

\frac{\partial A_{E}}{\partial t} = C_{A} - C_{E} + G_{E},

(3.2)

where the global eddy kinetic energy K_E, eddy available potential energy A_E, barotropic energy conversion C_K, baroclinic energy conversion C_E, conversion between zonal and eddy available potential energy C_A, and generation of eddy available potential energy due to diabatic heating G_E are defined as

K_{E} = \int_{p_{1}}^{p_{2}} 〈 \frac{u^{' 2} + υ^{' 2}}{2 g} 〉 d p,

(3.3a)

A_{E} = \int_{p_{1}}^{p_{2}} 〈 \frac{\bar{T^{' 2}}}{2 \bar{σ}} 〉 d p,

(3.3b)

C_{K} = - \int_{p_{1}}^{p_{2}} 〈 \underset{{\hat{C}}_{K}}{\underset{︸}{\bar{u^{'} υ^{'}} \frac{\partial \bar{u}}{d y} + \bar{u^{'} ω^{'}} \frac{\partial \bar{u}}{d p}}} 〉 \frac{d p}{g},

(3.3c)

C_{E} = - \int_{p_{1}}^{p_{2}} 〈 \underset{{\hat{C}}_{E}}{\underset{︸}{\frac{R}{p} \bar{ω^{'} T^{'}}}} 〉 \frac{d p}{g},

(3.3d)

C_{A} = - \int_{p_{1}}^{p_{2}} 〈 \underset{{\hat{C}}_{A}}{\underset{︸}{g (\frac{\bar{υ^{'} T^{'}}}{\bar{σ}} \frac{\partial \bar{T}}{d y} + \bar{\frac{ω^{'} T^{'}}{\bar{σ}}} \frac{\partial {\bar{T}}^{*}}{d p})}} 〉 \frac{d p}{g},

(3.3e)

G_{E} = 〈 \underset{{\hat{G}}_{E}}{\underset{︸}{g \frac{\bar{T^{'} {\dot{q}}^{'}}}{\bar{σ}}}} 〉 \frac{d p}{g} .

(3.3f)

The local energy conversions are denoted by

{\hat{C}}_{K} (y, p)

{\hat{C}}_{E} (y, p)

{\hat{C}}_{A} (y, p)

, and

{\hat{G}}_{E} (y, p)

. In (3.3) the overbars denote a zonal average, primes denote perturbations from the zonal average, angle brackets denote a meridional average, and the asterisk denotes deviations from the area average; that is,

(\overset{}{¯}) = 〈 \overset{}{¯} 〉 + {()}^{*}

. The remaining symbols are defined in Table 3.

Table 3.

List of variables.

Except for slight differences in notation, (3.3a)–(3.3f) are the same as in Hsieh and Cook (2007). But in contrast to their study, our global energy conversions are devoid of boundary terms, a consequence of our boundary conditions: zonal periodicity, and the application of the kinematic boundary condition at the channel sidewalls and at the lower and upper boundaries.

To show how G_E affects the energetics, we begin with the sum

C_{A} + G_{E} = - \int_{p_{1}}^{p_{2}} 〈 \frac{\bar{υ^{'} T^{'}}}{\bar{σ}} \frac{\partial \bar{T}}{d y} + \bar{\frac{ω^{'} T^{'}}{\bar{σ}}} \frac{\partial {\bar{T}}^{*}}{d p} 〉 d p + \int_{p_{1}}^{p_{2}} 〈 \frac{\bar{T^{'} {\dot{q}}^{'}}}{\bar{σ}} 〉 d p,

(3.4)

where we have used the expressions for C_A and G_E in (3.3e) and (3.3f), respectively. Following Nathan et al. (2017, their appendix A), for a trace shortwave radiative absorber such as dust, the linearized eddy diabatic heating rate

{\dot{q}}^{'}

can be written as the product between a transmissivity parameter

\bar{Γ} (y, p; \bar{γ})

and the eddy dust field γ′(x, y, p, t):

{\dot{q}}^{'} = \bar{Γ} γ^{'},

(3.5)

where

\bar{Γ} (y, p; \bar{γ}) = S_{0} σ_{a} ρ {\bar{T}}_{R} .

(3.6)

In (3.6) S₀ is the solar constant; σ_a is the specific absorption coefficient; ρ(z) is density;

{\bar{T}}_{R} = \exp (- \bar{τ})

is the zonal-mean transmissivity; and

\bar{τ} (y, p)

is the zonal-mean aerosol optical depth defined by

\bar{τ} = \frac{1}{μ} \int_{p}^{0} ρ σ_{a} \bar{γ} d p^{'},

(3.7)

where μ is the cosine of the solar zenith angle and

\bar{γ}

is the zonal-mean dust field. In (3.5) we have neglected the effects of scattering and longwave emission, which, as discussed in Nathan et al. (2017), are not needed to qualitatively assess the effects of the dust on the heating rate.

To further expose the effects of the dust on the heating rate, we follow Grogan et al. (2016) and Nathan et al. (2017) and write the linearized conservation equation for the eddy dust field in the following form:

(\frac{\partial}{\partial t} + \bar{u} \frac{\partial}{\partial x}) γ^{'} + υ^{'} \frac{\partial \bar{γ}}{\partial y} + ω^{'} \frac{\partial \bar{γ}}{\partial p} = - D γ^{'},

(3.8)

where D > 0 is a (constant) dust damping coefficient. Equation (3.8) states that local changes in the eddy dust field are due to the advection of eddy dust by the zonal-mean background wind; the advection of zonal-mean dust by the eddy meridional and vertical motions fields; and an eddy dust sink, which can be envisaged, for example, as due to dry deposition or gravitational settling.

To obtain an expression for the dust field, we assume normal-mode solutions of the form

χ^{'} = \tilde{χ} \exp i (k x - ω t) + c . c .,

(3.9)

where

\tilde{χ} (y, p)

represents the complex amplitude for a given eddy field; ω = ω_r + iω_i, where ω_r is the frequency and ω_i is the growth rate; k is the (real) zonal wavenumber; and c.c. denotes the complex conjugate of the preceding term. If the form of (3.9) is used for each eddy field and inserted into (3.8), we obtain

γ^{'} = - \frac{1}{| P |} e^{i ξ} (υ^{'} \frac{\partial \bar{γ}}{\partial y} + ω^{'} \frac{\partial \bar{γ}}{\partial p}),

(3.10)

where

| P | = {[{(\bar{u} k - ω_{r})}^{2} + {(ω_{i} + D)}^{2}]}^{1 / 2},

(3.11a)

ξ = \tan^{- 1} [\frac{(\bar{u} k - ω_{r})}{ω_{i} + D}] .

(3.11b)

Equation (3.10) shows that when the eddy wind is directed down the zonal-mean dust gradient the total amount of eddy dust is larger and smaller otherwise. For example, in a region where the zonal-mean dust gradient increases with latitude

(\partial \bar{γ} / \partial y > 0)

, a northerly wind (υ′ < 0) will increase the amount of the eddy dust.

If (3.10) and (3.11a) are combined with (3.5) and inserted into (3.3f), we obtain the following expression for the local generation of eddy APE:

{\hat{G}}_{E} \propto e^{i ξ} \frac{[- {\bar{α}}_{y} \bar{υ^{'} T^{'}} - {\bar{α}}_{p} \bar{ω^{'} T^{'}}]}{{[{(\bar{u} k - ω_{r})}^{2} + {(ω_{i} + D)}^{2}]}^{1 / 2}},

(3.12)

where the parameters

({\bar{α}}_{y}, {\bar{α}}_{p})

, which originate from the advection of the mean dust by the eddy field, are proportional to the products between the mean dust transmissivity and the mean dust gradients:

({\bar{α}}_{y}, {\bar{α}}_{p}) = (\bar{Γ} \frac{\partial \bar{γ}}{\partial y}, \bar{Γ} \frac{\partial \bar{γ}}{\partial p}) .

(3.13)

Equation (3.12) states that for fixed growth rate ω_i, the local generation of eddy APE occurs when the eddy heat flux is directed down the background dust gradient, and is largest near the critical surface, that is, where

(\bar{u} k - ω_{r}) \to 0

Insertion of (3.5) and (3.10) into (3.4) yields for the global energy conversion

C_{A} + G_{E} = - \int_{p_{1}}^{p_{2}} 〈 \underset{A}{\underset{︸}{\frac{\bar{υ^{'} T^{'}}}{\bar{σ}} (\frac{\partial \bar{T}}{d y} + \frac{1}{c_{p}} \frac{e^{i ξ}}{| P |} {\bar{α}}_{y})}} + \underset{B}{\underset{︸}{\bar{\frac{ω^{'} T^{'}}{\bar{σ}}} (\frac{\partial {\bar{T}}^{*}}{d p} + \frac{1}{c_{p}} \frac{e^{i ξ}}{| P |} {\bar{α}}_{p})}} 〉 d p .

(3.14)

The dust enters the global energy conversion implicitly through the flux terms,

\bar{υ^{'} T^{'}}

and

\bar{ω^{'} T^{'}}

, and explicitly through the product between the transmissivity parameter and the dust gradients, given by

({\bar{α}}_{y}, {\bar{α}}_{p})

. The explicit terms can be viewed as “corrections” to the background temperature gradients; the corrections are modulated by the Doppler-shifted frequency, growth rate, and dust damping coefficient, which are contained in P.

If we note that A ≫ B in (3.14), which is confirmed by our numerical calculations shown in section 4, we can then write $C_{A} + G_{E} \approx - \int 〈 A 〉 d p$ . Thus, the meridional gradient of the zonal-mean dust field is the primary control on the global eddy APE conversion by the dust field, a fact that will be used to interpret our numerical results in the following section. Because the zonal-mean dust field is symmetric about its maximum at ~20°N (see Fig. 2a), the zonal-mean dust gradient augments the zonal-mean temperature gradient south of 20°N and opposes it to the north of 20°N. Therefore, the regions where the dust dominates the local and global APE conversion for a given AEJ will depend on the projection of the zonally averaged heat flux onto the dust gradient.

4. Numerical results

a. Growth rates and phase speeds

Figure 3 shows the growth rates as a function of zonal wavenumber without dust (left column) and with dust (right column) for each of the background AEJs. Figure 3a shows that for U_REF, the NODUST growth rates are, as expected, the same as those in Grogan et al. (2016); for the fastest-growing AEW (k = 12) the growth rate is 0.34 day⁻¹. Figure 3a also shows that adjustments to the peak meridional shear affects the growth rates across the wave spectrum, and shifts the zonal scale of the fastest-growing wave. For example, compared to U_REF, the growth rates obtained from $U_{y}^{-}$ (dashed) are reduced by 15%–45% and the scale of the fastest-growing wave increases (k = 11). In contrast, for $U_{y}^{+}$ (dotted) the growth rates are enhanced by 6%–77% and the scale of the fastest-growing wave decreases (k = 13).

Fig. 3.

Citation: Journal of the Atmospheric Sciences 76, 11; 10.1175/JAS-D-19-0104.1

Figure 3b compares the NODUST growth rates obtained from $U_{p}^{-}$ (dotted) and $U_{p}^{+}$ (dashed) with U_REF (solid). For the larger-scale waves (k = 8–11), adjustment to the peak vertical shear has a slight effect on the growth rates (<5%). For the smaller wave scales (k = 12–16), the growth rates for $U_{p}^{-}$ are similar to U_REF, while the growth rates for $U_{p}^{+}$ increase by 9%–50%. In contrast to adjustments to the meridional shear, which affect the zonal scale of the fastest-growing wave, an adjustment to the vertical shear has no such effect; the fastest-growing wave remains at k = 12.

Comparison of Figs. 3a and 3b shows that adjustments to the meridional shear produce greater responses to the growth rates than adjustments to the vertical shear. This result is consistent with Rennick (1981) but not with Paradis et al. (1995), who attributed the difference between their study and Rennick’s (1981) to structural differences between their background jets. Here, however, we find that the growth rates are less sensitive to changes in the vertical shear, as in Rennick (1981), yet our reference jet is closer to that of Paradis et al. (1995), who found that the growth rates were more sensitive to adjustments in the vertical shear. To resolve this apparent conundrum, we compared our model with that of Paradis et al. (1995) and found that although our jets are similar, our mean vertical temperature profiles are not; our vertical temperature profile produces a more statically stable lower troposphere. To examine how the differences in the static stability between Paradis et al. (1995) and our study affect the response of the growth rates to adjustments in the vertical shear, we carried out sensitivity tests that involved reducing the static stability for the NODUST simulations. The reduced static stability increased the baroclinic energy conversions, which resulted in growth rates that were more sensitive to adjustments in the vertical shear, consistent with Paradis et al. (1995).

Figures 3c and 3d compare the DUST growth rates for U_REF with the perturbed meridional shear jets $U_{y}^{-}$ and $U_{y}^{+}$ and the perturbed vertical shear jets $U_{p}^{-}$ and $U_{p}^{+}$ , respectively. For U_REF, the dust increases the growth rates across the wave spectrum (cf. Figs. 3a and 3c, bold lines). This is consistent with Grogan et al. (2016), but in this study, the dust-enhanced growth rates are 20%–80% larger, a consequence of our stronger and more realistic dust distribution (see section 2). Moreover, irrespective of the AEJ, the dust increases the growth rates across the wave spectrum (cf. Figs. 3a and 3c, and cf. Figs. 3b and 3d). The increases range from 10% to 100%; the largest increases (>50%) occur for the slow-growing waves (<0.2 days⁻¹). For the fast-growing AEWs, however, Table 4 shows a more modest increase of 12%–35% for the growth rates, with the largest increases (27%–35%) occurring for $U_{y}^{-}$ and $U_{p}^{+}$ . Table 4 also shows that for $U_{y}^{-}$ and $U_{p}^{+}$ , dust reduces the scale of the fastest-growing wave.

Table 4.

Growth rates, easterly phase speeds, and zonal wavelengths of the fastest-growing AEW for each of the AEJs listed in Table 2.

The phase speeds for the NODUST and DUST were also calculated for each AEJ profile. As in previous dust-free studies (e.g., Thorncroft 1995), the phase speeds are fastest for flows with reduced meridional and vertical shear (i.e., reduced $U_{y}^{-}$ and $U_{p}^{-}$ ). For each AEJ, the dust-induced changes to the phase speed of the fastest-growing AEWs are <5% (Table 4). Similar phase speed changes occur for the other wave scales (values not shown).

b. Energetics

The global and local energetics are calculated for zonal wave k = 12 (wavelength = 3300 km). The energetics of this wave is representative of the other zonal waves examined in the previous subsection.

Table 5 shows the global energy conversions for the NODUST and DUST experiments. There are three general results. First, for each AEJ, with or without dust, the barotropic conversions are larger than the baroclinic conversions, that is, C_E/C_K < 1, which agrees with Grogan et al. (2016) and the studies cited in their Table 2. For this study, C_E/C_K is largest for $U_{y}^{-}$ and $U_{p}^{+}$ , which are the AEJs that also yield the largest dust-induced increases in the growth rates (Table 4). Second, the dust can either increase or decrease C_K depending on the jet structure. For example, for $U_{y}^{-}$ and $U_{p}^{+}$ , the dust increases C_K by ~60% and ~2%, respectively; whereas for U_REF, $U_{y}^{+}$ , and $U_{p}^{-}$ , the dust decreases C_K by ~13%–23%. Third, for each AEJ, C_E, C_A, and G_E are all larger with dust. The largest dust-induced increases in C_E and C_A occur for $U_{y}^{-}$ and $U_{p}^{+}$ , the same AEJs in which the dust increases C_K, and that have the largest C_E/C_K. The largest increases in G_E occur for $U_{y}^{-}$ ; the increases in G_E for the $U_{p}^{+}$ AEJ are similar to U_REF.

Table 5.

Global energy conversions (1 × 10⁻³ W m⁻²) for the k = 12 NODUST and DUST experiments. For NODUST, G_E = 0.0 for each AEJ.

Figures 4–7 show the local energetics, ${\hat{C}}_{K}$ , ${\hat{C}}_{E}$ , ${\hat{C}}_{A}$ , and ${\hat{G}}_{E}$ , for the reduced and enhanced meridional shear AEJs, $U_{y}^{-}$ and $U_{y}^{+}$ . We focus on $U_{y}^{-}$ and $U_{y}^{+}$ because the effects of dust on the growth rates are the strongest and weakest, respectively (Table 4). We do not present the local energetics for $U_{p}^{+}$ and $U_{p}^{-}$ because the results are similar to those obtained with $U_{y}^{-}$ and $U_{y}^{+}$ . The similarities between ( $U_{p}^{+}$ , $U_{p}^{-}$ ) and ( $U_{y}^{-}$ , $U_{y}^{+}$ ) are not surprising since the functional form of our observationally representative AEJ is such that the meridional and vertical shears are inversely proportional (see section 2a). Moreover, we do not present the results for U_REF since they are similar to those presented in Grogan et al. (2016).

Fig. 4.

Latitude–height plots of ${\hat{C}}_{K}$ for (a),(c) $U_{y}^{-}$ and (b),(d) $U_{y}^{+}$ for the k = 12 (left) NODUST and (right) DUST experiments. Contour interval is 1 × 10⁻⁵ m² s⁻³.

Citation: Journal of the Atmospheric Sciences 76, 11; 10.1175/JAS-D-19-0104.1

Fig. 5.

As in Fig. 4, but for ${\hat{C}}_{E}$ .

Citation: Journal of the Atmospheric Sciences 76, 11; 10.1175/JAS-D-19-0104.1

Fig. 6.

As in Fig. 4, but for ${\hat{C}}_{A}$ .

Citation: Journal of the Atmospheric Sciences 76, 11; 10.1175/JAS-D-19-0104.1

Fig. 7.

Citation: Journal of the Atmospheric Sciences 76, 11; 10.1175/JAS-D-19-0104.1

Figure 4 shows ${\hat{C}}_{K}$ for $U_{y}^{-}$ (top) and $U_{y}^{+}$ (bottom) for the NODUST and DUST experiments. For each experiment, Fig. 4 shows two positive regions on either side of the AEJ axis (15°N), with the largest values on the north side. For $U_{y}^{-}$ , comparison of Figs. 4a and 4c shows that the dust increases ${\hat{C}}_{K}$ north of the AEJ axis in two regions: midlevels south of the plume axis (850–700 hPa, 16°–20°N) and near the plume top (700–550 hPa, 18°–22°N). For $U_{y}^{+}$ , however, Figs. 4b and 4d show that dust decreases ${\hat{C}}_{K}$ on both sides of the AEJ axis, but the decreases are larger on the north side. The changes to ${\hat{C}}_{K}$ are due to the zonal-mean momentum flux, whereby the dust adjusts the amplitudes of the eddy meridional and zonal wind fields on both sides of the AEJ axis (not shown).

Figures 5 and 6 show, respectively, ${\hat{C}}_{E}$ and ${\hat{C}}_{A}$ for the same experiments shown in Fig. 4. For each experiment, ${\hat{C}}_{E}$ and ${\hat{C}}_{A}$ each have a positive region below the AEJ core (~650 hPa). Without dust, Figs. 5a and 5b and 6a and 6b show that there are large differences between the positive regions, which are accompanied by a negative region (see Figs. 5b and 6b). But despite the differences, Figs. 5c and 5d and 6c and 6d show that the dust produces a well-defined positive region for each AEJ. The dust also shifts the peak values farther north (1°–2°) and to higher levels (25–75 hPa). These enhancements to ${\hat{C}}_{E}$ and ${\hat{C}}_{A}$ are due to stronger heat fluxes that are driven by increases in the amplitude of the eddy temperature field in the dust plume (not shown).

Figure 7 shows ${\hat{G}}_{E}$ corresponding to the DUST experiments in Figs. 4–6. For each AEJ, Fig. 7 shows a single generation region that is collocated with the positive values of ${\hat{C}}_{E}$ and ${\hat{C}}_{A}$ , that is, at midlevels north of the AEJ axis (650–850 hPa, 16°–20°N). This result is consistent with the local energetics presented in Grogan et al. (2016). Moreover, the location of ${\hat{G}}_{E}$ can be explained by (3.12) and (3.13), which show that ${\hat{G}}_{E}$ is positive where the eddy heat flux is directed down the background dust gradient. For each AEJ, the jet axes are fixed at 15°N so that $\bar{υ^{'} T^{'}} < 0$ below the AEJ core. Thus ${\hat{G}}_{E}$ is positive where $\partial \bar{γ} / \partial y > 0$ , which occurs south of the dust plume axis (16°–20°N). This agrees with the reanalysis results of dust-coupled AEWs examined by Grogan and Thorncroft (2019).

Figure 7 also shows that the analytical analysis predicts the location of peak ${\hat{G}}_{E}$ in the numerical analysis. Equations (3.12) and (3.13) show that for a fixed growth rate and dust distribution, ${\hat{G}}_{E}$ is largest near the critical surface. For each AEJ in Fig. 7, the peak ${\hat{G}}_{E}$ occurs near the intersection of the latitudinal axis of the peak $\partial \bar{γ} / \partial y$ (bold dashed line) and the critical surface (bold circular contour). As expected, Fig. 7 show that the generation is stronger for $U_{y}^{-}$ , which is the AEJ that imparts the strongest dust effects on the growth rates. This is predicted by (3.12), which shows that for a fixed dust distribution, ${\hat{G}}_{E}$ increases with larger dust-modified heat fluxes and smaller fixed growth rates, conditions associated with $U_{y}^{-}$ .

5. Conclusions

Analytical and numerical approaches are used to examine how structural changes in the African easterly jet (AEJ) mediate the direct radiative effects of Saharan mineral dust aerosols on the growth and propagation of African easterly waves (AEWs). The analytical approach hinged on deriving an expression for the generation of eddy available potential energy (APE), an expression that exposes how the structure of the AEJ, dust distribution and Doppler-shifted frequency combine to affect the energetics of the AEWs. The analytical expression was used to interpret the numerical results, which were obtained by coupling a simplified version of the Weather Research and Forecasting (WRF) Model with a simplified conservation equation for the eddy dust field. The WRF-Dust model was used to conduct linear simulations around a zonal-mean background state that was chosen consistent with observations over North Africa during summer. The peak meridional and vertical shears of the background AEJ were varied and the linear dynamics of the AEWs subsequently examined.

The key findings are as follows: (i) Irrespective of the AEJ structure or the zonal scale of the AEWs, dust increases the growth rates of the AEWs. (ii) The growth rates of the AEWs are optimized when the ratio of baroclinic to barotropic energy conversions is largest. (iii) When the energy conversions are sufficiently large, the zonal scale of the fastest-growing AEW shortens.

For each of the AEJs that were examined, the dust generates eddy APE (G_E) at midlevels north of the AEJ axis (850–650 hPa, 15°–20°N) and increases the conversion of zonal to eddy APE (C_A) and baroclinic energy conversion (C_E) in the same region. Grogan and Thorncroft (2019) used MERRA-2 reanalysis to infer the distribution of G_E and C_E associated with dust-coupled AEWs. Our model calculations qualitatively agree with their results.

For AEJs with enhanced vertical shear and reduced meridional shear, the fastest-growing AEWs have the largest increase in growth rate (28%–35%), a consequence of the larger increase in C_A, C_E, and barotropic energy conversions (C_K) For AEJs with reduced vertical shear and enhanced meridional shear, however, the fastest-growing waves have smaller increases (12%–18%), a consequence of smaller increases in C_A and C_E, and decreases in C_K.

As discussed in section 3 and noted above, the interpretation of the energetics results obtained from the WRF-Dust model was aided by our analytical analysis of G_E. The analysis shows that the strength of G_E depends on the zonal-mean transmissivity, and the zonal-mean meridional and vertical gradients of the background dust field, which are modulated by the Doppler-shifted frequency. The expression for G_E can be interpreted as a correction to C_A, where the gradients of zonal-mean dust modify the preexisting zonal-mean temperature gradients. For the background states examined in this study, C_A + G_E shows that the dust always increases the production of eddy APE (and thus growth rates). The increase is due to the positive meridional dust gradient north of the AEJ axis, which aligns with the low-level positive meridional temperature gradient, which is also in the vicinity of a critical surface. Consequently, the differential heating associated with the dust gradient reinforces the temperature gradient, which is also where the equatorward heat fluxes of the unstable AEWs are largest (Norquist et al. 1977). Therefore, knowledge of the background distributions of wind, temperature and dust over Africa can be used as a predictive tool to assess the influence of dust radiative effects on the growth of AEWs.

Here we have examined the linear response of AEWs to changes in the structure of AEJs that are supercritical with respect to combined barotropic–baroclinic instability. A logical extension of this work would be to examine the linear dynamics of AEWs in subcritical background flows, which, as shown by Nathan et al. (2017), can support the dust-induced growth of otherwise neutral AEWs. Nathan et al. (2017), however, only considered a single, though observationally representative, AEJ. It would therefore be interesting to determine which subcritical AEJ structures optimize the dust-induced instabilities of the AEWs. Equally important is the consideration of the dust-modified nonlinear evolution of AEWs in different initial AEJ configurations. In such a problem the background distributions of zonal-mean and eddy dust would each feed back on the circulation to affect the life cycles of the AEWs. In Grogan et al. (2017), for example, it was shown for an initial supercritical jet, the total dust field (mean plus eddy) enhanced the linear growth, weakened the nonlinear stabilization, and caused larger peak amplitude. Throughout most of the AEW life cycle, the Eliassen–Palm flux divergence and residual mean circulation were both enhanced, but operated in opposition to each other. To what extent these results are sensitive to changes in the initial structure of the AEJ is unclear, but will be examined in a future study.

Acknowledgments

The authors thank Emily Bercos-Hickey and Scott Nelson for their comments on the manuscript. We also acknowledge high-performance computing support from Cheyenne (doi:10.5065/D6RX99HX) provided by NCAR’s Computational and Information Systems Laboratory. This work was supported by NSF Grant 1624414-0.

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The two pathways are not unique to dust; they operate in any system that involves the interaction between the circulation and a trace shortwave radiative absorber, such as stratospheric ozone (cf. Albers and Nathan 2012).

For a wave propagating in a zonal-mean current, the Doppler-shifted frequency is defined as $ω_{D} = \bar{u} k - ω_{I}$ , where $\bar{u} k$ is the Doppler shift and ω_I is the intrinsic frequency. The locus of points along which the Doppler-shifted frequency vanishes defines a critical surface.

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Journal of the Atmospheric Sciences

Abstract
1. Introduction
2. Model, basic states, and experimental design
3. Analytical analysis
4. Numerical results
- a. Growth rates and phase speeds
- b. Energetics
5. Conclusions

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Fig. 1.

Reference AEJ (solid) and corresponding potential temperature field (dashed). Contour intervals are 2 m s⁻¹ for the wind and 5 K for the temperature.

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Fig. 2.

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Fig. 3.

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Fig. 4.

Latitude–height plots of ${\hat{C}}_{K}$ for (a),(c) $U_{y}^{-}$ and (b),(d) $U_{y}^{+}$ for the k = 12 (left) NODUST and (right) DUST experiments. Contour interval is 1 × 10⁻⁵ m² s⁻³.