Necessary conditions for the instability of quasigeostrophic waves induced by trace shortwave radiative absorbers

The derivation of necessary conditions for hydrodynamic instability of parallel jet flows occupies a lofty position in the history of fluid mechanics. This history begins in the late nineteenth century when Lord Rayleigh (1879) established a necessary condition for the instability of a parallel jet flow in a non-rotating, barotropic fluid. Rayleigh showed that instability can only occur if the jet flow possesses an inflection point.

Nearly 70 years would elapse before Lord Rayleigh's inflection point theorem would be extended by Kuo (1949) to include the rotational effects, an extension that would find application to the development of waves in Earth's and other planetary atmospheres. The stability condition derived by Kuo (1949) states that the gradient of absolute vorticity, which is comprised of the jet and planetary vorticity gradients, must change sign within the domain. Yet, like Lord Rayleigh's (1879) study, the study of Kuo (1949) was for a barotropic fluid and, thus, was limited to jet flows possessing only lateral shear.

By the 1960s, Charney and Stern (1962) and Pedlosky (1964) extended the Rayleigh–Kuo stability criteria to atmospheric and oceanic currents possessing both lateral and vertical shear. They showed that for quasigeostrophic (QG) flow, a necessary condition for instability is that the “potential vorticity gradient must either change sign or be balanced by surface terms incorporating the surface potential temperature gradient and topographical variations” (Pedlosky, 1964).

The beauty of the necessary conditions for instability cited above is that the instability can be established simply from the knowledge of the basic state; details regarding the wave are not required. Although the above-mentioned studies and their descendants have provided key insight into the development of waves in geophysical fluid systems (Pedlosky, 1987; Vallis, 2017), they are all based on conservative flow and, thus, are limited in their application.

This study is novel in that it is the first to develop necessary conditions for instability for non-conservative flow, where the non-conservative effects are due to heating associated with the absorption of solar (shortwave) radiation by trace constituents. This study not only represents a significant extension of prior studies that have developed necessary conditions for the instability of parallel jet flow, it exposes the physical mechanisms associated with the radiative–dynamical instability of quasigeostrophic waves induced by trace shortwave absorbers.

Earth's atmosphere is replete with trace shortwave radiative absorbers. Among the most important are smoke aerosols, stratospheric ozone, and mineral dust aerosols. African biomass burning over southern West Africa, for example, has resulted in dry aerosol concentrations of ∼6 μg m⁻³ over the Atlantic Ocean (Haslett et al., 2019). In the Northern Hemisphere stratosphere at ∼6 hPa, 45° N, the January zonally averaged ozone mixing ratio is ∼7 ppmv (Brasseur and Solomon, 2005). Over the Sahara desert at ∼850 hPa, 18° N, the dust concentration is ∼18.0 μg kg⁻¹ (Grogan et al., 2016). Despite the fact that these absorbers constitute only a tiny fraction of the total atmospheric mass, their effects on the thermal structure and atmospheric circulation are far-reaching with consequences for weather and climate on both regional and global scales. Among the consequences are the emergence of absorber-induced instabilities, which arise from the coupling between the absorbers and the wave fields.

The coupling occurs when a wave generated in the far field enters a region occupied by the absorber or when a wave that already exists within the region, either as a free mode of oscillation or as a result of hydrodynamic instability, coexists with the absorber. Irrespective of the origin of the wave, however, it produces a wave response in the absorber. The phasing between the waves, i.e., among the absorber, temperature, and wind waves, directly affects the structure, propagation, and instability of the waves. Such absorber-induced wave instabilities have yet to be shown for smoke aerosols, but they have been shown for stratospheric ozone and Saharan dust (Lindzen, 1965; Gruzdev, 1985; Ghan, 1989a; 1989b; Nathan, 1989; Nathan and Li, 1991; Nathan et al., 1994; Jones et al., 2004; Ma et al., 2012; Grogan et al., 2016; 2017; Nathan et al., 2017).

Ghan (1989a; 1989b), for example, showed that in a gray atmosphere in which the basic state is horizontally uniform, an arbitrary trace shortwave absorber destabilizes both Rossby waves and inertia-gravity waves. Some of the growth rates of the absorber-induced instabilities rivaled those that arise from the hydrodynamic instabilities that grow on zonal-mean background currents. Internal Rossby waves, which are neutral in the absence of radiation-absorber feedbacks, were destabilized when the basic state absorber concentration increased with altitude.

Ghan's (1989a; 1989b) assumptions of a gray atmosphere and horizontally uniform background state, however, do not hold for stratospheric ozone, a molecule whose shortwave absorption is highly wavelength dependent, and whose zonal-mean background concentration is horizontally and vertically non-uniform. The radiative–dynamical interactions associated with stratospheric ozone have been shown to destabilize internal Rossby waves (Gruzdev, 1985; Nathan, 1989; Nathan and Li, 1991; Nathan et al., 1994; Li and Nathan, 1997). Nathan and Li (1991), for example, analytically and numerically showed that the ozone-induced Rossby wave instabilities are rooted in the advection of zonal-mean ozone by the wave field. For a basic state consistent with boreal winter, the destabilizing effects of the ozone reduced the damping rate due to Newtonian cooling by as much as 50% for internal Rossby waves of large vertical extent with large amplitude in the lower stratosphere.

Like stratospheric ozone, mineral dust aerosols have been shown to have a destabilizing effect on waves. This is most apparent over North Africa, the largest dust source region on Earth (Tanaka and Chiba, 2006). Over this region, the barotropic–baroclinic instability of the African easterly jet (AEJ) spawns synoptic-scale disturbances known as African easterly waves (AEWs), which contribute to the formation of synoptic-scale dust plumes that feedback on the waves to affect their stability and intensification (Jones et al., 2004; Ma et al., 2012; Hosseinpour and Wilcox, 2014; Grogan et al., 2017; 2019; Nathan et al., 2017; Grogan and Thorncroft, 2019; Bercos-Hickey et al., 2022). Grogan et al. (2016), for example, analytically showed that the zonal-mean generation of available potential energy (APE) by dust-radiative effects was maximized in regions where basic state gradients of dust were largest near a critical surface (where the Doppler-shifted frequency vanished). The analytical results were confirmed by numerical simulations, which Grogan et al. (2016) conducted with an idealized version of the Weather Research and Forecasting (WRF) model coupled to an online dust model. The simulations showed that for a supercritical basic state jet, that is, one that exceeds the threshold for combined barotropic–baroclinic instability, the dust increased the growth rates of the most unstable AEWs from 15% to 90%.

Nathan et al. (2017) presented a theoretical framework that exposes the radiative–dynamical relationships that govern the destabilization of AEWs in subcritical (hydrodynamically neutral) zonal-mean basic states. The framework, which was derived analytically and confirmed numerically, showed that for otherwise neutral AEWs, the dust destabilizes the waves and slows their westward propagation. Experiments carried out with the WRF-dust model showed that the dust-induced growth rates that emerged were commensurate with, and sometimes exceeded, those obtained in previous dust-free studies in which the AEWs grow on supercritical AEJs.

In this study, we do not seek to establish the existence of shortwave absorber-induced instabilities and their associated growth rates; this has already been done in the studies cited above. Rather, we seek to derive necessary conditions for the radiative–dynamical instability of quasigeostrophic waves, conditions that will obviate the need for detailed instability calculations in order to establish the existence of trace absorber-induced instabilities. We will show that the conditions have two virtues. First, the instabilities induced by trace shortwave radiative absorbers can be established based solely on knowledge of the zonal-mean distributions of the wind and absorber fields. Second, the conditions expose the physical processes and basic state flow structures that control the instabilities, processes not exposed in previous studies.

The analysis presented here is built on a continuously stratified, quasigeostrophic (QG) model atmosphere. QG model atmospheres have been used for more than 70 years to address the dynamics of fluid systems that are characterized by hydrostatic balance and small Rossby number, which measures the ratio of the inertial force to the Coriolis force (Pedlosky, 1987; Vallis, 2017; Reinaud, 2020). In Earth's atmosphere, for example, QG waves, which generally span about 1000 km or more in the horizontal, are associated with synoptic-scale weather systems that characterize mid-latitude weather and climate. Our focus on the QG circulation precludes consideration of the absorber-induced heating associated with localized patches of an absorber, since such patches would be too small to effectively project onto the waves that operate on the QG scale.

The QG model atmosphere used in this study is confined to a zonally periodic, β-channel of width L. The atmosphere extends from a flat, rigid horizontal lower boundary at z = 0 to infinity. As is standard in linear instability studies, we prescribe the background state (Pedlosky, 1987; Harris et al., 2022), which is a steady, zonally averaged state that is in radiative–dynamical equilibrium. We superimpose a small amplitude perturbation on this background state whose stability characteristics we subsequently examine.

The linear dynamics of this model atmosphere are governed by coupled conservation equations for QG potential vorticity, thermodynamic energy, and absorber concentration. The governing QG perturbation equations take the following form in log-pressure coordinates (Nathan and Li, 1991; Nathan et al., 2017):

where

is the perturbation potential vorticity, and

is the northward gradient of the basic state potential vorticity, where the overbar denotes a zonal average. The basic state wind is $u¯(y,z)$⁠; the perturbation geostrophic streamfunction is $ϕ(x,y,z,t),$ which in the QG framework is related to the geostrophic wind and temperature fields by $(u,v)=(−∂ϕ/∂y,∂ϕ/∂x)$ and $T=(Hf0Rd−1)∂ϕ/∂z$⁠, respectively, where H is the density scale height, f₀ is the Coriolis parameter evaluated at a central latitude, and R_d is the gas constant for dry air. In Eqs. (1) and (2), $ḣ$ is the diabatic heating rate per unit mass; $δ=κ/f0SH>0;$$κ=Rd/cp;$c_p is the specific heat capacity; and $S=N2/f02$⁠, where N² is the Brunt–Väisälä frequency squared (assumed constant without loss of generality). These symbols and the others appearing in the remainder of this section are listed in the Nomenclature.

In the absorber conservation Eq. (3), $γ¯(y,z)$ and $γ(x,y,z,t)$ are the background and perturbation trace absorber mass mixing ratios, respectively, and $ḋ$ represents the local depletion rate of the absorber. The depletion rate depends on the characteristics of the absorber. For stratospheric ozone, $ḋ$ is a function of the photodissociation rate per ozone molecule, ozone concentration, and temperature (Nathan and Li, 1991). For Saharan dust, $ḋ$ depends, for example, on sedimentation, scavenging, and dry and wet deposition (Chen et al., 2010). The specific form of $ḋ$ is important if growth rates and wave structures are sought. In this study, however, we seek general stability conditions, not growth rates. Consequently, in the analysis that follows, we neglect $ḋ$⁠. This will have no effect on the stability conditions but will simplify the analysis and ease the interpretation of the conditions.

The solutions that are obtained by combining Eqs. (1)–(5) satisfy the following boundary conditions. In the zonal direction, we require periodicity. At the channel sidewalls, we impose the kinematic boundary condition such that the meridional velocity, $∂ϕ/∂x,$ vanishes at $y=±L/2$⁠. At the lower (flat) horizontal boundary, we evaluate Eq. (2) at z = 0 and apply the kinematic boundary condition, which requires that the vertical velocity, w, vanishes at z = 0, so that

At the upper boundary, located at $z=∞$⁠, we demand the upward energy flux, $ρwϕ,¯$ remain bounded (Pedlosky, 1987).

Equations (1)–(3) and corresponding boundary conditions constitute a closed set that governs the linear dynamics of QG disturbances subject to absorber-induced diabatic heating. A physical interpretation of the equations is eased if we note that the wind and temperature fields are related to the streamfunction, as defined above, and the vertical wind shear is proportional to the meridional temperature gradient in accordance with thermal wind balance (Pedlosky, 1987; Nathan and Li, 1991). With these relationships in mind, Eq. (1) states that the local time changes in QG potential vorticity are due to three effects: advection of the perturbation potential vorticity by the background zonal wind; meridional advection of background potential vorticity by the perturbation; and the vertical gradient of the density-weighted diabatic heating rate. Equation (2), the thermodynamic equation, states that local time changes in temperature are due to the advection of temperature by the background zonal wind, the advection of background temperature by the perturbation, adiabatic cooling and heating associated with rising and sinking motion, and absorber-induced diabatic heating. Equation (3), the linear conservation equation for the absorber, states that local time changes in the absorber concentration are due to advection of the perturbation absorber by the background zonal wind, advection in the latitude-height plane of the background absorber by the perturbation, and sinks of the absorber.

To facilitate our analysis, we supplement Eqs. (1)–(3) with a conservation equation for pseudomomentum, which is a measure of wave-activity that is quadratic in the wave amplitude and is conserved in the absence of external forcing and non-conservative effects (Andrews and McIntyre, 1978). Pseudomomentum is a property of a wave, analogous to momentum, that determines the force exerted by a wave when it interacts with a material continuum (McIntyre, 2019; Singh and Hanna, 2021). Pseudomomentum has an advantage over energy, as made clear by Held et al. (1986) in their study of damping-induced wave instability; they state: “The key point is that the pseudomomentum equation is a conservation law, whereas the eddy energy equation includes an eddy-mean flow conversion term whose modification by the damping [absorber-induced heating in our case] must be included to compute the effect of the dissipation on the wave energetics.” So, with our pseudomomentum approach, there is no need to calculate the eddy conversion terms, which enables us to isolate the effects of the absorber-induced heating on the wave growth rate.

To obtain the pseudomomentum equation, multiply Eq. (1) by $ρq/βe$ and integrate over the domain, which, after combining with Eq. (6) multiplied by $(∂u¯/∂z)−1∂ϕ/∂z$⁠, yields, as in Held et al. (1986), the following order amplitude squared equation:

The integrand in the first term on the left-hand side is the pseudomomentum. The second term is the generation of pseudomomentum by lower boundary effects. In this study, the two terms on the right-hand side arise from the diabatic heating rate due to the direct radiative effects of the absorber.

Trace shortwave radiative absorbers also emit and scatter radiation (Liou, 2002). The emission is often modeled as Newtonian cooling, which is proportional to the temperature difference between an object (molecule) and a reference value (Liou, 2002). The effects of Newtonian cooling on the stability of neutral quasigeostrophic perturbations has been examined by Held et al. (1986), who showed that Newtonian cooling can destabilize external Rossby waves. Because we are focusing on absorption rather than emission of radiation, we will not consider Newtonian cooling in our analysis.

Scattering of solar radiation by ozone is weak and can be ignored, but the scattering of solar radiation by mineral dust aerosols may be large (Ghan, 1989b). If accurate quantitative measures of the effects of scattering by dust are sought, then it should be accounted for the calculation of diabatic heating rates. In the dust-coupled stability calculations carried out by Grogan et al. (2016) and Nathan et al. (2017), the scattering was important to accurately calculate the heating rates. The scattering effects, however, were subordinate to the absorption effects. Thus, for simplicity, we ignore scattering, an assumption that will simplify the analysis of the stability criteria to be derived in Sec. III.

To obtain the form of the absorber-induced perturbation heating rate, we follow Nathan et al. (2017) and consider a gray, plane-parallel atmosphere without scattering. In this case, the net diabatic heating rate, $ḣT$⁠, which is the sum of the basic state and perturbation heating rates, can be written as (Liou, 2002; Coakley and Yang, 2014)

where F_d is the radiant flux density, $μ$ is the cosine of the solar zenith angle, S₀ is the solar constant, $σa$ is the specific absorption coefficient, $γT$ is the total mass mixing ratio, and $τT$ is the total aerosol optical depth (AOD), given by

Equation (8) states that the absorber-induced heating rate is due to the change in the radiant flux caused by the absorption of solar radiation by a trace constituent within a column of the atmosphere.

The total mass mixing ratio and total AOD are each portioned into a basic state and perturbation

Insertion of Eqs. (10) and (11) into Eq. (8) and linearizing yield the perturbation heating rate

The perturbation heating rate consists of two terms: a local heating rate and a shielding effect, where the latter represents the contribution to the heating rate due to perturbations in the absorber above a given level. Previous studies have shown that the shielding effect can be neglected if the perturbations are relatively shallow (Ghan, 1989b; Nathan, 1989; Nathan and Li, 1991; Echols and Nathan, 1996; Grogan et al., 2012). This is the case for North Africa, where the dust is usually confined below ∼500 hPa (Grogan et al., 2016). In the case of stratospheric ozone, Nathan and Li (1991) have shown that the local heating rate dominates over the heating rate due to the shielding effect. Because our analysis only seeks to provide qualitative relationships rather than quantitative measures of the absorber instability of the waves, the shielding effect is neglected. Thus, the perturbation heating rate can be written as

where the heating rate coefficient is defined as

The stability criteria to be derived in Sec. III do not depend on the detailed characteristics of $Γ,$ only that $Γ$ and $∂Γ/∂z$ are both positive.

To evaluate the heating rate function given in Eq. (14), we choose the prescribed background dust distribution shown in Fig. 1. This distribution compares very well with the dust distribution obtained from multiple simulations carried out with a regional climate model by Konare et al. (2008). We also choose a solar zenith angle corresponding to summer solstice and an absorption coefficient $σa$ = 369.0 m² kg⁻¹, which yields an AOD of 1.0, a typical summertime value over North Africa (Bercos-Hickey et al., 2020). Figure 2 shows, based on Eq. (14), the vertical variation of $Γ.$ Because $Γ>0$ and increases monotonically with height at each latitude, $∂Γ¯/∂z>0.$

For stratospheric ozone, $Γ$ is a complicated function that depends on the absorption of solar radiation per ozone molecule; the catalytic destruction of odd oxygen by nitrogen, hydrogen, and chlorine compounds; the flux of solar energy, which depends on the spectral wavelength; the absorption cross sections, column and number densities of oxygen and ozone; and the solar zenith angle. The functional form of $Γ$ is shown in Nathan and Li (1991; their Appendix A). As seen in their Fig. 3, which is based on climatological distributions of background ozone, $Γ$ and $∂Γ/∂z$ are both positive.

The derivation of the conditions for the radiative–dynamical instability of quasigeostrophic waves pivots on two assumptions. First, the zonal mean background flow is hydrodynamically neutral; this isolates the radiative–dynamical instability, so, should this instability arise, it is due solely to the absorber-induced heating. Second, the absorber-induced diabatic heating rate is small but significant, such that

where $ε≪1$ is a nondimensional scaling parameter that makes explicit the smallness of the heating rate. This scaling is consistent with the heating-induced growth rates obtained in modeling studies of the effects of ozone on internal Rossby waves (Nathan, 1989; Nathan and Li, 1991; Nathan et al., 1994) and on the effects of Saharan dust aerosols on AEWs (Ma et al., 2012; Grogan et al., 2016; Nathan et al., 2017; Grogan et al., 2019). In the case of dust, for example, Ma et al. (2012) and Grogan et al. (2016a) showed, despite sharply different modeling approaches, that the dust heating effects can increase the eddy kinetic energy of AEWs by ∼10%–15%.

The necessary conditions for the radiative–dynamical instability of quasigeostrophic waves embedded in a hydrodynamically neutral background jet are derived by first writing the dependent variables in a perturbation series (Bender and Orszag, 1978)

where $ϕn(y,z), γn(y,z),$ and $wn(y,z)$ are the amplitudes of the respective wave fields, k is the (real) zonal wavenumber, $cn=cnr+icni$ is the (complex) phase speed, and the asterisk denotes the complex conjugate of the preceding term.

Insertion of Eq. (16) into Eqs. (1)–(3) and (6) yields the O(1) balance

Solving Eqs. (17) and (20) for $q0$ and $∂ϕ0/∂z,$ respectively, and inserting the results into the O(1) pseudomomentum equation, Eq. (7), yield

where

is the domain averaged pseudomomentum of the unforced O(1) wave field. If the horizontal temperature gradient vanishes at the lower boundary, then, in accordance with thermal wind balance, $∂u¯/∂z$ must also vanish at z = 0. In this special case, the second term in Eq. (22) vanishes.

Equation (21) is a statement of the Charney and Stern (1962) necessary condition for the linear instability of a parallel jet flow possessing both meridional and vertical shear. The condition states that for an exponentially amplifying normal mode, corresponding to $c0i≠0$⁠, the domain averaged pseudomomentum, $Aw$⁠, must vanish. This statement demands that one of the following conditions be satisfied: (i) the background potential vorticity gradient $βe$ must change sign somewhere in the domain; (ii) the meridional temperature gradient, which is proportional to $∂u¯/∂z$ through thermal wind balance, must change sign on the lower boundary; and (iii) $βe$ and $∂u¯/∂z$ must have the same sign.

In this context, where the background flow is assumed hydrodynamically neutral to quasigeostrophic waves, $Aw≠0.$ Thus Eqs. (17)–(20) yield, at O(1), neutral waves (⁠$c0i=0$⁠) with amplitude structures $ϕ0(y,z), γ0(y,z),$ and $w0(y,z),$ and real phase speed, $c0=c0r$⁠.

At $O(ε)$⁠, the absorber-induced diabatic heating rate, $ḣ$⁠, enters the pseudomomentum equation, so that Eq. (7) can be written as

where

Equation (18) yields

where the geostrophic relationship, $v0=∂ϕ/∂x$⁠, has been used. The second bracket in Eq. (25) is the advection of the zonal-mean background dust by the perturbation, which is responsible for the absorber-induced heating rate, Eq. (24). The background absorber gradients, $∂γ¯/∂y$ and $∂γ¯/∂z$⁠, will be central to the stability criteria to be shown later.

To establish the stability criteria, it requires that Eq. (25) be written in terms of the perturbation streamfunction, which, after solving for $w0$ in Eq. (19) and inserting into Eq. (25), yields

If Eqs. (24) and Eq. (26) are combined and then inserted into Eq. (23), we obtain, after integration by parts

where $Aw$⁠, which is defined by in Eq. (22), is positive, as shown by Held et al. (1986) for neutral modes in westerly flow and by Nathan et al. (2017) for neutral modes in easterly flow. Equation (27) states that in a hydrodynamically neutral basic state, where $Aw>0$⁠, exponentially amplifying $(c1i>0)$ waves induced by trace shortwave radiative absorbers are only possible if $Af>0$⁠, where

and

where

The coefficients defined in Eqs. (30a)–(30e) are all positive, provided $(u¯−c0r)>0.$ The proof that $(u¯−c0r)>0$ begins by writing (Pedlosky, 1964)

where $χ0(y,z)$ is an arbitrary function. Insertion of (31) into (17) yields, after some algebraic manipulation,

with boundary conditions: $χ0=0$ at y = $±L/2$⁠; $∂χ0/∂z=0$ at z = 0; and $ρw0χ0<∞$ as $z→∞.$

Next, multiply (32) by $ρχ0$ and integrate over the y–z plane to obtain

The left-hand side is positive definite, so $(u¯−c0r)$ must be positive.

The conditions for instability, which are embodied in Eqs. (29a) and (29b), depend only on the structure of the basic state: the zonal-mean vertical shear and the meridional and vertical gradients of the absorber. Examination of Eqs. (28) and (29) reveals the following stability conditions: A sufficient condition for the stability of quasigeostrophic waves induced by trace shortwave radiative absorbers is that for all points in the latitude-height plane, the following must be satisfied:

In this section, we confirm that the predictions for absorber-induced instabilities given in Eqs. (34a)–(34e) agree with the absorbed-induced instabilities that have been obtained in previous analytical and modeling studies (Ghan, 1989a; Nathan and Li, 1991; Nathan et al., 2017). To do so, we examine four cases. The four cases correspond to different combinations of background wind and absorber, which, as shown by Eq. (26), control the O(1) absorber concentration, $γ0.$ Consequently, because the absorber-induced heating rate depends on $γ0$⁠, as seen in Eq. (24), each case will correspond to a different heating rate.

Case I: $∂γ¯/∂z≠0, ∂γ¯/∂y=0, ∂u¯/∂z=0.$ For this case, Eq. (27) becomes

This case corresponds to an absorber heating rate that is produced solely by the vertical advection of the background absorber by the wave. Ghan (1989a) examined this case analytically for an arbitrary shortwave absorber whose heating rate was O(1). Because the background state in this case is horizontally uniform, Ghan (1989a) was able to choose a plane wave solution for the disturbance field, which yielded an analytical solution for the growth rate. Ghan (1989a) showed that within the QG framework, a westward-propagating internal Rossby wave, which is stable in the absence of radiative–dynamical feedbacks, is unstable if $∂γ¯/∂z>0.$ Nathan et al. (2017) derived the same result, though with a slightly different approach. They assumed that the background state was slowly varying, and the absorber heating rate was small. The analytical results of Ghan (1989a) and Nathan et al. (2017) both agree with the prediction given in Eq. (34e): in a horizontally uniform basic state, a sufficient condition for the instability of a QG wave induced by a trace shortwave absorber is $∂γ¯/∂z>0$⁠.

Case II: $∂γ¯/∂z≠0, ∂γ¯/∂y=0, ∂u¯/∂z≠0.$ For this case, Eq. (27) becomes

In this case, the square of the background vertical shear, $(∂u¯/∂z)2,$ enters $Λ4$⁠. Thus, irrespective of the zonal-mean wind's vertical structure, $∂γ¯/∂z>0$ $(∂γ¯/∂z<0)$ is destabilizing (stabilizing). This agrees with the analytical analysis of Nathan and Li (1991), who extended the analytical analysis of Ghan (1989a) to include vertical wind shear. The analysis of Nathan and Li (1991), however, was incomplete; it missed the important fact that the vertical shear, which appears in the first term on the right-hand side of Eq. (36), augments the growth rate.

Case III: $∂γ¯/∂z=0, ∂γ¯/∂y≠0, ∂u¯/∂z≠0$⁠. For this case, Eq. (27) becomes

This case corresponds to an absorber heating rate that is produced solely by the meridional advection of the background absorber by the wave. Inspection of Eqs. (29a) and (29b) shows that the effects of $∂γ¯/∂z$ and $∂γ¯/∂y$ on the stability of the wave are sharply different. Whereas the effects of $∂γ¯/∂z$ on the stability depend only on its sign, the effects of $∂γ¯/∂y$ on the stability depend on its sign as well as the sign of the vertical shear.

If $∂u¯/∂z=0,$ Eq. (37) shows that instability (stability) requires $∂γ¯/∂y<0$ $(∂γ¯/∂y>0).$ This agrees with Nathan et al. (2017), who obtained an analytical expression for the absorber-induced growth rate under the assumption that the absorber gradients were constant. Equation (37) requires no such assumption to establish instability.

If $∂u¯/∂z≠0$⁠, Eq. (37) shows that the vertical shear is vital to the instability. For example, if $∂γ¯/∂y<0$ and $∂u¯/∂z<0$ is sufficiently large, the instability is squelched. If, however, $∂γ¯/∂y>0$ and $∂u¯/∂z<0$ is sufficiently large, then instability is assured. These twin conditions, $∂γ¯/∂y>0$ and $∂u¯/∂z<0,$ are satisfied over North Africa, in the region between the Saharan dust maximum (∼$20°$ N) and the African easterly jet axis (∼$15°$ N). It is within this region that the generation of eddy available potential energy (APE) by Saharan dust aerosols is concentrated, as shown in previous analytical analyses (Grogan et al., 2016; Nathan et al., 2017) and WRF model simulations (Grogan et al., 2016; Bercos-Hickey et al., 2017). As shown in these studies, the dust-induced generation of APE is central to the growth in eddy kinetic energy, a measure of the growth rate. None of the studies, however, were able to expose the importance of the coupling between $∂γ¯/∂y$ and $∂u¯/∂z$ in the dust-modified energetics or growth rate, which is explicit in Eq. (37).

Case IV: $∂γ¯/∂z≠0, ∂γ¯/∂y≠0, ∂u¯/∂z≠0$⁠. This is the most general case. The first thing to notice is $∂2γ¯/∂y∂z$ now enters Eq. (29a) and, thus, will affect the stability conditions. The potential importance of this term to the absorber-induced instability of QG waves has not been identified before. For example, Ghan (1989a; 1989b) only considered horizontally uniform background states. Nathan and Li (1991) and Nathan et al. (2017) included meridional variations in the background absorber, but their highly simplified analytical analyses and numerical model experiments were unable to expose $∂2γ¯/∂y∂z$ as a contributor to the absorber-induced instability by either ozone or dust.

The utility of the necessary conditions for instability derived above is that instability can be established based solely on knowledge of the background state; knowledge of the wave structure is not required. However, without knowledge of the wave structure, it is not possible to compare the relative importance of the two integrals that appear in Eq. (28), since each integral contains a different function of the wave structure (⁠$|ϕ0|2, |∂ϕ0/∂z|2$⁠). We can, however, assess separately the contribution of each integral to the stability conditions. For the first integral in Eq. (28), we can compare the four contributions to A₁.

To place the interpretation of A₁ on a firm physical footing, we will use the background dust and wind distributions given in Nathan et al. (2017). The background dust distribution has already been shown Fig. 1; the hydrodynamically neutral background wind is shown in Fig. 3. The details regarding the background state are given in Nathan et al. (2017).

The background dust and wind distributions shown in Figs. 1 and 3, respectively, are representative of summertime conditions over North Africa. As shown in Fig. 1, the dust maximum is centered at $20°$ N, is well-mixed in the lower layer, and decreases monotonically with height and with distance from the maximum. The hydrodynamically neutral background wind, which is representative of the African easterly jet, is centered at $10°$ N and 4 km in height. For the calculations, we choose a scale height of 9 km, which is representative of the region over North Africa, and $c0r=−8.45$ ms⁻¹, which is the value used in Nathan et al. (2017) for their numerical simulations.

Figure 5 shows the spatial distribution of A₁, Eq. (29a), which is the sum of the four panels shown in Fig. 4. There are two large positive regions, one located below ∼3 km south of the dust maximum (at 20°N), and the other above ∼2.5 km north of the dust maximum. The fact that there are both negative and positive regions associated with A₁ means the first integral in Eq. (28a) alone satisfies the necessary conditions for instability. Determining if the first integral in Eq. (28) produces a positive growth rate requires knowledge of the product between A₁ and the wave structure, $|ϕ0|2$⁠. Because African easterly waves have their largest amplitude in the lower troposphere, at ∼1.5 km (∼850 hPa) (Reed et al., 1988), we can anticipate that in the region, where A₁ is relatively large and positive (Fig. 5), the integrated effect of A₁$|ϕ0|2$ would be positive, so that the first integral in Eq. (28) would be destabilizing. This destabilizing effect is offset by the second integral, however, since $∂γ¯/∂z<0$ for the dust distribution in Fig. 1. We note that Nathan et al. (2017) have shown that for the same background dust and wind distributions used here, the dust is destabilizing and yields growth rates that can sometimes exceed those obtained in previous dust-free studies in which the AEWs grow on AEJs that are supercritical with respect to the threshold for barotropic–baroclinic instability.

The radiative–dynamical destabilization of quasigeostrophic waves in Earth's atmosphere by trace shortwave radiative absorbers has been established in several studies over the past half century. As discussed in the Introduction, those studies have employed both analytical and numerical methods to solve the absorber-induced stability problem using standard linear approaches. The approaches require the imposition of a specific background state for the wind, temperature, and absorber, and the subsequent determination of the growth rate and wave structure.

In this study, we have derived necessary conditions for the radiative–dynamical instability of quasigeostrophic waves induced by trace shortwave radiative absorbers. The conditions have two virtues. First, they only require knowledge of the zonally averaged background distributions of the wind and absorber to establish the possibility of absorber-induced instability; knowledge of the wave structure is not required. Second, the conditions expose the physical processes involved in the destabilization, processes not exposed in previous analytical and modeling studies of trace absorber-induced instabilities.

For example, the conditions show that for a horizontally uniform basic state, irrespective of the zonal-mean wind's vertical structure, a background absorber that increases with height $(∂γ¯/∂z>0)$ is destabilizing, where the destabilization is enhanced by vertical shear, an important result that was missed in previous studies. In sharp contrast, the instability induced by a background absorber that varies with latitude, $(∂γ¯/∂y)≠0$⁠, depends on its coupling with the background vertical shear, another important result that was missed in previous studies.

The instability conditions were discussed in light stratospheric ozone and Saharan mineral dust aerosols, among the most prominent trace shortwave radiative absorbers in Earth's atmosphere. The predictions made by the necessary conditions for instability agree with prior studies of ozone and dust-induced instability of quasigeostrophic waves.

The simplicity of the instability conditions underscores their utility as a tool that is both interpretive of the physics that produces the instabilities and predictive of the types of background wind and absorber structures that may produce the instabilities in Earth's or other planetary atmospheres, such as Mars, where dust storms are commonplace (Fernandez, 1997).

The authors acknowledge high-performance computing support from Cheyenne (doi:10.5065/D6RX99HX), which is provided by NCAR's Computational and Information Systems Laboratory. This work was supported by NSF Grant No. 1624414–0 (T.R.N.) and NSF Grant No. 2108233 (D.F.P.G.).

The authors have no conflicts to disclose.

Terrence R. Nathan: Conceptualization; methodology; mathematical analysis; original draft preparation and writing. Dustin F. P. Grogan: Data preparation; visualization; reviewing and editing.

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

List of Symbols

t $x,y,z=−H ln (p/p0)$

Time; eastward, northward, and vertical directions

$ρ(z)=ρ0 exp (−z/H)$

Air density

H, $p0$⁠, $ρ0$

Constant density scale height, sea level reference pressure and density

$f0=2|Ω| sin θ0$

Coriolis parameter; $|Ω|=$ angular frequency of Earth; $θ0=$ central latitude

$S=N2/f02$

$N2≡$ Brunt–Väisälä frequency squared (assumed constant)

$u¯(y,z), T¯(y,z), γ¯(y,z)$

Basic state zonal mean wind, temperature, and absorber mass mixing ratio

$ϕ(x,y,z,t)$ $γ(x,y,z,t)$

Perturbation streamfunction and absorber mass mixing ratio

$w(x,y,z,t)$

Perturbation vertical wind

$βe$

Northward gradient of the background potential vorticity

$β=re−12|Ω| cos θ0$

Northward gradient of the Coriolis parameter; $re=$ Earth's radius

$ḣ$

Diabatic heating rate per unit mass

$Γ(y,z)$

Absorber heating rate coefficient

$κ=Rd/cp$

$Rd≡$ gas constant for dry air; $cp≡$ specific heat capacity at constant pressure

$δ=κ/f0SH$

Constant parameter

$τ(y,z)$

Aerosol optical depth

$S0,μ,σa$

Solar constant, cosine of the solar zenith angle, specific absorption coefficient