## 1. Introduction

Spontaneous adjustment, the mechanism whereby a well-balanced flow radiates gravity waves (GWs) in the course of its near-balanced evolution (Ford et al. 2000; Vanneste and Yavneh 2004; Vanneste 2008), is a possible source of atmospheric GWs. It can be distinguished from other mechanisms, including topographic forcing and the classical adjustment originally described in Rossby (1937), by the fact that it involves no process external to the flow itself. In realistic configurations, however, spontaneous adjustment is mixed with other mechanisms. For instance, large mountain GWs produce potential vorticity (PV) anomalies when they break (Schär and Smith 1993), as well as secondary GWs (Scavuzzo et al. 1998). To measure the relative importance of these two signals, Lott (2003) studied the large-scale response to mountain-wave breaking near critical levels and showed that substantial GWs are reemitted during the breaking itself, while the long-term evolution is dominated by the balanced response. Martin (2008) subsequently found that the PV field associated with the balanced response radiates GWs well after the end of the initial breaking. This emission, although weaker than the initial one, is potentially more persistent since it is tied to the slowly evolving PV. It is plausible, therefore, that it contributes to the low-frequency GWs observed in the wakes of breaking topographic waves (Plougonven et al. 2010).

To quantify this emission by PV more precisely, Lott et al. (2010, hereafter LPV10) examined the GWs emitted by small-amplitude PV anomalies in a shear. In this scenario, the separation between balanced motion and GWs does not hold: because of the Doppler shift, motions that are balanced in the vicinity of the anomalies become, in the far field, gravity waves [see also Plougonven et al. (2005); Mamatsashvili et al. (2010)]. In the linear approximation, and assuming constant wind shear Λ and constant Brunt–Väisälä frequency *N*, LPV10 found that substantial GWs reach the far field when the Richardson number *J* = *N*^{2}/Λ^{2} is not too large (say, between 1 and 10). By substantial, we mean that, for PV anomalies representative of those likely to be found when thin layers of stratospheric air enter the troposphere, the Eliassen–Palm (EP; or pseudomomentum) flux associated with the GWs is comparable to that measured in the stratosphere far from mountains (Hertzog et al. 2008).

*g*is the gravity constant,

*f*the Coriolis parameter,

*ρ*and

_{r}*θ*background reference values for the density and potential temperature,

_{r}*q*the amplitude of the PV anomaly, and

_{r}*σ*its depth. In a GCM, these last two quantities could be related to the grid-scale PV value and to the vertical grid spacing.

_{z}A limitation of LPV10 is the restriction to two-dimensional PV anomalies, with no structure in the direction transverse to the basic shear. This is a significant limitation since the absorption of GWs at inertial levels strongly depends on the orientation of the horizontal wave vector. This is known from the investigations on GW propagating upward toward inertial levels: Grimshaw (1975) and Yamanaka and Tanaka (1984) showed that the absorption at the lowest inertial level is large for *ν*Λ < 0, where *ν* = *l*/*k* is the ratio between the transverse and parallel horizontal wavenumbers, and much smaller for *ν*Λ > 0. This results in a “valve effect,” which Yamanaka (1985) interpreted by analyzing the tilt of the phase lines of the GWs (i.e., of particle displacements) relative to the isentropes. The configuration that we analyze is quite different since the GW associated with a PV disturbance is generated between the critical levels and propagates outwards of them. Nevertheless, the argument of Yamanaka (1985) applies and we find strong absorption at the inertial level if *ν*Λ > 0 and much weaker absorption if *ν*Λ < 0.

The motivation of the present paper is to extend the results in LPV10 to three-dimensional PV anomalies. Accordingly, its first aim is to obtain the vertical structure of the 3D singular modes associated with PV anomalies that have the form of a Dirac function in the vertical. The analytic results derived for monochromatic anomalies can then be integrated to obtain the vertical structure associated with anomalies of arbitrary horizontal structure and show, in particular, that a horizontally isotropic PV anomaly produces a very specific anisotropic GW signature beyond the inertial levels. A second aim is to deduce further, by integration over the continuous spectrum, the GW response to a vertically smooth, localized PV distribution. A third aim is to extend the EP flux predictions in (1.1) and (1.2) to the 3D case. In this case, the (vertical component of the) EP flux, which can also be interpreted as a wave stress, is a horizontal vector. For a horizontally isotropic shear in the Northern Hemisphere, this vector is shown to make an angle with the shear that decreases with altitude, from zero at the anomaly level to some negative value in the far field. This implies that a PV anomaly in a westerly shear exerts a drag that is oriented to the southwest in the upper inertial region and to the northwest where the associated GW dissipates aloft.

The plan of the paper is as follows. The general formulation of the problem and its transformation to a dimensionless form are given in section 2. There we discuss both the exact response to a *δ*-PV distribution in the vertical, and its Wentzel–Kramers–Brillouin (WKB) approximation valid for *J* ≫ 1. The WKB analysis extends that of LPV10 by resolving the Stokes phenomenon associated with the existence of an exponentially small (in *J*) contribution to the solution that grows exponentially between the PV anomaly and the inertial levels. Taking this contribution into account, we obtain a 3D generalization of the EP flux estimate (1.1). Section 3 presents the vertical structure of the response in some detail. It emphasizes the directional aspects and relates them to the tilt of the solution about isentropes in the meridional plane. Section 4 recasts the results in dimensional form and considers PV distributions that are localized horizontally and that have a finite depth, in which case the GW response is transient. Section 5 summarizes the results. Appendixes A and B provide technical details on the exact and WKB solutions, respectively.

## 2. Formulation

### a. Disturbance equations and potential vorticity

*u*′,

*υ*′, and

*w*′ are the three components of the velocity disturbance,

*p*′ is the pressure disturbance,

*θ*′ is the potential temperature disturbance, and

### b. Normal-mode decomposition

*δ*(

*ξ*) is the Dirac function of the variable

*z*′. The scaling used in (2.8) places the inertial levels

*z*=

*z*′ ±

*f*/(

*k*Λ) of these modes at

*ξ*= ±1 (Inverarity and Shutts 2000).

*w*′ corresponding to the expansion (2.7) of the PV can be written as

*W*(

*ξ*) is its vertical structure. Note that this expansion describes the part of

*w*′ slaved to the PV: an additional continuum of singular modes, representing free sheared GWs, would need to be added to the expansion to solve an arbitrary initial-value problem.

*u*′ and

*υ*′ and the potential temperature

*θ*′ have expansions analogous to (2.9), with

*W*replaced by

*U*,

*V*, and Θ, respectively. Introducing these expansions into (2.1a)–(2.1e) and choosing

*ν*=

*l*/

*k*. We now introduce (2.10)–(2.11), into the expressions (2.3) and (2.7) for the PV. Choosing the vertical-velocity amplitude

*W*(

*ξ*). Note that

*W*depends on

*J*and

*ν*in addition to

*ξ*, and that we use the notation

*W*(

*ξ*) as a shorthand for the more complete but more cumbersome

*W*(

*J*,

*ν*;

*ξ*).

*ξ*→ +∞ (

*ξ*→ −∞). An explicit expression for the amplitude

*E*of this GW is given in (A.13).

### c. WKB approximation

In the limit *J* ≫ 1, it is possible to derive an approximation to *W*(*ξ*) using a WKB approach. This approximation, which we now derive, is more transparent than the exact solution in terms of hypergeometric functions and proves remarkably accurate for moderately large *J*.

We focus on the region *ξ* > 0 since the solution for *ξ* < 0 follows immediately from (2.15). The WKB approximation does not provide a single solution that is valid uniformly in *ξ* > 0; instead, four regions, which we label (i)–(iv), need to be distinguished. The form of the solution in each of these regions, given below, is derived in appendix B.

*ξ*=

*O*(

*J*

^{−1/2}) ≪ 1, the quasigeostrophic (QG) approximation applies, leading to

*ξ*=

*O*(1) and

*ξ*< 1,

*ξ*− 1| =

*O*(

*J*

^{−1}), the solution is expressed in terms of the scaled variable

*ζ*=

*J*(1 +

*ν*

^{2})(

*ξ*− 1) as

*ξ*=

*O*(1) and

*ξ*> 1, the solution is

*A*

^{(i)}⋅⋅⋅

*B*

^{(iv)}are fixed by imposing a jump conditions at

*ξ*= 0 given in (A.1), a radiation condition as

*ξ*→ ∞, and continuity of the solution across the four regions. Starting with the radiation condition, we obtain from (2.14) and (2.19) that

*B*

^{(ii)}= 0, in agreement with the expectation from quasigeostrophic theory of a solution that decays exponentially with altitude above

*ξ*= 0. This solution is not entirely satisfactory, however, in that it fails to capture the feedback that radiation (as

*ξ*→ ∞) has on the solution in regions (ii) and (i). In particular, a single exponentially decaying solution [in regions (i) and (ii)] has a zero EP flux, inconsistent with the nonzero flux from the exact solution. To resolve this apparent difficulty, we need to recognize that

*B*

^{(ii)}= 0 is only an approximation. In fact,

*B*

^{(ii)}takes an exponentially small, nonzero value, which can be captured using the more sophisticated matching procedure applied in appendix B. This procedure yields

*ξ*→ 1. This combination of exponentially growing and decaying solutions is enforced by the radiation condition and is consistent with the consequent nonzero EP flux. By retaining the exponentially small

*B*

^{(ii)}[in spite of the neglect of much larger

*O*(

*J*

^{−1/2}) terms in the dominant solution], we capture this important part of the physics of the problem. A comparable situation arises for the Schrödinger equation in quantum mechanics, in the semiclassical study of wave propagation through a potential barrier (e.g., Bender and Orszag 1999). In this context, a wavelike solution radiating outside the barrier is associated with a combination of exponentially decaying and exponentially growing solutions inside the barrier. While the solution that decays toward the interior of the barrier (and corresponds to

*B*

^{(ii)}in our problem) is usually neglected (Bender and Orszag 1999), it can be retained (e.g., to obtain a direct estimate of the so-called decay width; Shepard 1983).

*ξ*= 0 yields

*O*(1) terms. It then follows that

### d. EP flux

*J*

^{3/2}(1 +

*ν*

^{2})

*W** and integrating by parts results in a conservation for the nondimensional EP flux,

*ξ*= 0, ±1. The scaling factor on the left of the real part symbol is introduced so that

*J*-independent dimensional factor (1.2).

*ξ*→ ∞ and the asymptotics in (A.14a) and (A.14b) for

*ξ*≪ 1 gives

*A*,

*B*, and

*E*are given explicitly in terms of Γ functions in appendix A. A more convenient expression is obtained by using the WKB form of

*W*. To compute

*J*approximation

*ξ*| < 1 relies on our estimate (2.26) of the exponentially small constant

*B*

^{(i)}.

## 3. Results for *W*(*ξ*)

In this section we examine the structure of *W*(*ξ*) and compare the exact solution with the WKB approximation.

### a. Vertical structure

The four panels in Fig. 1 show *W*(*ξ*) for *J* = 4 and for four different orientations of the horizontal wave vector **k** = (*k*, *l*) = *K*(cos*φ*, sin*φ*), with *φ =* −45°, −15°, −15°, 45°, that is for *ν* = −1, −0.267 … , 0.267 … , 1 and *J* = 4. In all cases, the real part of *W* is approximated by its geostrophic estimate *W*^{(i)} in (2.16) some distance away from the neighborhood of *ξ* = 0 where it is strictly valid. The imaginary part of *W* is substantially smaller than the real part, in particular near and around *ξ* ≈ 0 where the quasigeostrophic approximation predicts a purely real *W*^{(i)}. The real and imaginary parts of *W* only become comparable near the inertial levels, where balanced approximations do not apply.

Between (and away from) the inertial levels *ξ* = ±1, Im(*W*) follows in quadrature Re(*W*) when *ν* < 0, but precedes Re(*W*) in quadrature when *ν* > 0. As discussed in the next subsection, this behavior implies that the solutions always tilt along the isentropes in the (*y*, *z*) plane. Note that this behavior is well captured by the WKB solution in (2.17) but that can also be captured by correcting the QG solution to higher order as in Plougonven et al. (2005). Beyond *ξ* = ±1 the solution almost behaves as a pure gravity wave solution, in agreement with the asymptotic approximation in (2.14). The real part of the latter is shown by the gray dots in Fig. 1.

The most striking feature in Fig. 1 is the strong sensitivity of *W* to *ν*. According to the WKB estimates, *ν* affects the amplitude of *W* in three ways. First near *ξ* = 0, *W*, decreases with increasing |*ν*|, according to the approximation *W* in region (ii), is given by *ν*|. Third, the amplitude of *W* in the GW region (iv) depends strongly on *ν* through the factor *e*^{−νπ/2} in (2.27). The first two effects explain the decrease in *W* between *ξ* = ±1 from Figs. 1a,d to Figs. 1b,c. The third effect depends on the sign of *ν*; this introduces a meridional asymmetry and explains the changes from Fig. 1a to Fig. 1d and from Fig. 1b to Fig. 1c. We discuss this effect further in the next section.

### b. Meridional asymmetry and valve effect

A strong meridional asymmetry in absorption was highlighted by Grimshaw (1975) and Yamanaka and Tanaka (1984) in their studies of GWs propagating upward toward a critical level surrounded by two inertial levels. The latter authors showed that there is a very strong absorption at the lowest inertial level if *ν*Λ < 0. If *ν*Λ > 0, the wave crosses the first inertial level with little attenuation, but it is almost entirely reflected downward at a turning point located between the critical level in *ξ* = 0 and the lowest inertial level in *ξ* = −1. The reflected wave is then strongly absorbed as it returns to the lowest inertial level. Even though in both scenarios the initial upward GW is ultimately absorbed at the lowest inertial level, this potential intrusion of the GW signal between the inertial levels is quite remarkable and was referred to as a “valve” effect by these authors. This effect was interpreted heuristically by Yamanaka (1985), who analyzed with detail the behavior of two independent solutions near the lowest inertial level. He pointed out that the phase lines of one of the solutions change direction rapidly around the inertial level, and lie between the horizontal plane and the isentropes in a narrow region. Applying a static-stability method to analyze the stability of the air parcels displaced along phase lines leads to the conclusion that, for this solution, the region is baroclinically unstable (Pedlosky 1987).

*ν*> 0 than Λ

*ν*< 0 (recall that we assume Λ > 0). If we follow Yamanaka and Tanaka (1984) and translate their description of the valve effect in our context, this sensitivity is related to the mathematical fact that around the inertial level in

*ξ*= 1, the two independent solutions of (2.13),

*ξ*= 1 whereas the second varies sharply and jumps by a multiplicative factor equal to

*e*at

^{νπ}*ξ*= 1 [see the analytical continuation in (A.8)].

Following Plougonven et al. (2005), a good way to assess the significance of these two solutions is to visualize them in the (*y*, *z*) plane (Figs. 2b,c,f,g). Figure 2b indicates that the smooth solution always tilts in the direction of the isentropes. In contrast, the other solution also tilts in the direction of the isentropes for *ξ* < 1 but tilts in the other direction for *ξ* > 1. The structure of the upward waves above *ξ =* 1, namely *ξ*^{1/2+iμ}*e ^{ily}*, also tilts in the direction of the isentropes when

*ν*< 0 but in the opposite direction when

*ν*> 0 (Figs. 2a,e). It is therefore not a surprise that the smooth solution plays the greater role to match the PV anomaly and the upward wave when

*ν*< 0, and that the other, rapidly changing solution plays the greater role when

*ν*> 0. Of course this can be checked analytically since the parameters

*α*′ and

*β*′ in (A.6b) exactly control the role of the rapidly changing and of the smooth solution in the connection through the upper critical level, respectively. It turns out that |

*α*′/

*β*′| =

*e*, consistent with our argument.

^{νπ}Interestingly, the structures of the rapidly changing and smooth solutions are not much different well below *ξ* = 1 (e.g., cf. Figs. 2b and 2g between *ξ* = 0.4 and *ξ* = 0.6). In fact, the two solutions have the same Taylor expansion near *ξ* = 0 up to *O*(*ξ*^{3}). According to (A.14a), this means that as *ξ* → 0 both can equally be used to produce the *δ*-PV anomaly, which is consistent with the fact that the exact solutions around and above *ξ* = 0 are not much sensitive to the sign of *ν* (see Figs. 2d,h).

### c. GW amplitude and EP flux

The combined effect of the two parameters *J* and *ν* on the GW emission is shown in Fig. 3, which compares the exact values of the GW amplitude |*E*| with the WKB approximation (2.27). For a fixed value of *ν*, |*E*| decreases with *J* as in LPV10. For fixed values of *J*, the cases with *ν* > 0 and *ν* < 0 need to be distinguished. For *ν* > 0 and increasing, |*E*| decreases monotonically as a result of increasing exponential decay in region (ii) and increasing inertial-level absorption. When *ν* < 0 those two effects oppose: increasing |*ν*| increases the exponential decay but decreases the inertial-level absorption. Accordingly, |*E*| is maximized for some *ν*(*J*) < 0. Importantly, the WKB approximation (2.27) provides a good approximation for |*E*| for *J* ≳ 1, well beyond the theoretical range of validity *J* ≫ 1 of the asymptotics.

The EP flux within and outside the inertial levels are shown in Figs. 4a and 4b, respectively. The exact and WKB solutions (2.31) are compared. Figure 4a indicates that the EP flux between the inertial levels is only weakly sensitive to the angle *φ* = tan^{−1}*ν* of the wave vector. It remains almost constant, for instance, for −45° ≲ *φ* ≲ 45° when *J* ≈ 3, or for −30° ≲ *φ* ≲ +30° and *J* ≈ 10. For larger values of *φ*, however, the EP flux decreases rapidly and vanishes for *φ* = ±*π*/2. An important aspect of Fig. 4a is that the EP flux between the inertial levels is symmetric about the axis *φ* = 0.

*ξ*| > 1) as a result of the asymmetric absorption at the inertial levels. This is clear from Fig. 4b: when

*φ*≲ −30° the EP fluxes for |

*ξ*| > 1 are almost equal to the fluxes for |

*ξ*| < 1, but they are much smaller for

*φ*≳ 30°. This is well captured by the WKB approximation (2.31), which again provides a good estimate for

*J*≳ 1 both for |

*ξ*|< 1 and |

*ξ*| > 1. In particular, it leads to the prediction

*ν*for which

*ξ*| > 1. The corresponding angle

*φ*is shown as a dotted line in Fig. 4b.

_{M}## 4. Application to localized PV distributions

### a. Horizontally localized δ-PV

*σ*gives the characteristic horizontal width of the PV anomaly,

_{H}*q*is its characteristic amplitude, and

_{r}*σ*is its characteristic depth. The introduction of the scale

_{z}*σ*naturally follows from the fact that

_{z}*δ*(

*z*) scales as an inverse length. For such a distribution, the vertical velocity field in (2.9) reads

*W*on

*φ*, and where

To evaluate the double integral in (4.2) we next proceed numerically by tabulating in the vertical direction the structure function *W*(*φ*; *ξ*) for 180 discrete values of *φ*. This yields an angular resolution Δ*φ* = 2°. We also consider 50 discrete values for the horizontal wavenumber *K*, with a resolution Δ*K* = *π*/(10*σ _{H}*). For the physical grid we take for both horizontal directions the resolution Δ

*x*= Δ

*y*= 0.2

*σ*.

_{H}In the following, we express our results in dimensional form. We consider a *σ _{z}* = 1-km-thick layer of stratospheric air entering in the troposphere. We therefore take a PV amplitude of

*ρ*= 1 potential vorticity units (PVU; 1 PVU ≡ 1 × 10

_{r}q_{r}^{−6}K kg

^{−1}m

^{2}s

^{−1}) and assume

*σ*= 55 km. Assuming that this air enters the troposphere at midlatitudes, we take

_{H}*ρ*= 1 kg m

_{r}^{−3},

*N*= 0.01 s

^{−1},

*θ*= 300 K,

_{r}*f*= 10

^{−4}s

^{−1}, and

*J*= 4.

### b. Vertical velocity field

The vertical velocity calculated from (4.2) is shown in Fig. 5 for six different altitudes. Near the PV anomaly [i.e., for *z* = 0 km (Fig. 5a)], the vertical velocity is positive to the east of the positive PV anomaly and negative to the west. This is of course consistent with the balanced picture that the meridional geostrophic winds are toward the north on the eastern flank of a positive PV anomaly, and to the south on the western flank (not shown). As the advective terms are very small in the thermodynamic equation (2.1d) at this altitude, the vertical velocity balances the meridional advection of the background potential temperature (*f*Λ*υ*′ ≈ *N*^{2}*w*′). At the higher altitude *z* = 1 km (Fig. 5b), the signal in vertical velocity decays in magnitude and spreads in horizontal scale, consistent with the QG predictions that all wavelengths decay exponentially with altitude, with the long wavelengths decaying less rapidly than the short ones. Note, however, the two large-scale lobes of opposite sign of the vertical velocity that have moved slightly to the north, which is a first sign that the QG prediction starts to break down (the QG prediction is insensitive to the sign of *ν*; see appendix B). At *z* = 2 km (Fig. 5c), the signal in vertical velocity has decayed further in magnitude and spread farther horizontally (note the contour interval decrease between Figs. 5b and 5c), again somehow in agreement with the QG prediction. Nevertheless, the two large-scale lobes of vertical velocity start to be modulated by a smaller-scale oscillatory signal, clearly apparent aloft the PV disturbance. Higher up in altitude this oscillatory signal entirely dominates the response; its lines of constant phase make a positive angle with the longitude axis because the waves with *ν* < 0 are less absorbed at the inertial levels than those with *ν* > 0. Note also that the amplitude between *z* = 3 and 10 km increases in agreement with the *z*^{1/2} dependence predicted in (2.27). Because of the superposition of wavenumbers, the transition between decaying and wavelike perturbation does not occur sharply at a single inertial level but rather smoothly across an inertial-layer region. The altitude of the center of this region is given by the estimate *σ _{H}f*/Λ ≈ 1.1 km, consistent with Fig. 5.

### c. EP-flux vector

*x*component of the force exerted by the GWs on the (transformed Eulerian) mean flow (Andrews et al. 1987). Because our model (2.1) is both

*x*and

*y*independent and the GWs are plane waves in both directions, the two horizontal components of the force can in fact be obtained from the EP-flux vector (or, up to a sign, vertical pseudomomentum-flux vector)

*k*and

*l*, since these are the proportion of the

*x*and

*y*components of the corresponding pseudomomentum density (Bühler 2009). The nondimensional EP-flux vector can therefore be written as (1,

*l*/

*k*)

*η*′ denotes the vertical displacement satisfying

*D*′ =

_{t}η*w*′, and the factor

**F**has the dimension of a pressure. Using

*F*

_{0}is given in (1.2) and is exactly the same as in LPV10. For the parameters chosen, it is about

The exact results for the EP-flux vector in (4.8) are shown in Fig. 6 for two different values of *J*. When *J* = 4, **F** at *z* = 0 is purely zonal, with a magnitude near 5 mPa. The zonal orientation follows from the symmetry of the PV distribution about the *x* axis. At higher altitudes, **F** decreases in amplitude and changes direction. These two effects result from the absorption of an increasingly large portion of the wave spectrum at inertial levels, and from the fact that waves with *ν* < 0 are much less absorbed than those with *ν* > 0. When *J* = 4, **F** as *z* → ∞ makes an angle close to *φ* ≈ −30° with the *x* axis, almost the angle for which the normalized EP flux has a maximum according to (3.2) (see also Fig. 4b). For *J* = 10 (Fig. 6b), **F** in the far field has an amplitude that is about half that at *z* = 0, and it makes an angle with the *x* axis that is close to *φ* ≈ −15°, a value again consistent with (3.2).

*J*≫ 1 is large, these expressions can be further simplified using Laplace’s method to obtain

*z*≪ 1 and

*z*≫ 1 when

*J*≫ 1 as in Fig. 6b, but they underestimate it by a factor of almost 2 when

*J*≈ 1.

Comparing (4.11) to the 2D results in LPV10 [see also (1.1) and (1.2)] shows that the orders of magnitude of **F** are comparable in 2D and 3D (the *F*_{0} term), and that in both cases about half of the EP flux in the direction of the shear is deposited in the inertial layer. The most remarkable difference is that the EP-flux vector rotates with altitude. Consider, for example, a westerly shear in the Northern Hemisphere: the EP-flux vector tends, for large *z*, to an angle close to

### d. Horizontally localized, finite depth PV

*t*= 0 that is separable in the horizontal and the vertical directions and that has the same vertical integral as (4.1):

*z*=

_{m}*m*Δ

*z*and Δ

*z*=

*σ*/

_{z}*M*. In this case, the vertically discretized equivalent of the vertical velocity in (2.9) reduces to

*x*and

*t*enter (4.14) only in the combination

*x*− Λ

*z*, the computation of the sum over the indices

_{m}t*m*involves straightforward vertical and horizontal translations of

Figure 7 shows the evolution of the integral of the disturbance PV, *z* = 10 km, for *σ _{z}* = 1 km and

*J*= 4. All of the other parameters are as in the previous sections. The solution is only shown for negative values of

*t*: for positive

*t*it is almost symmetric to that at negative

*t*. The background velocity shears the PV whose horizontal extent therefore decreases with time until

*t*= 0 before increasing again. When it is more spread out horizontally (that is at large negative or positive time), its vertical integral is also relatively small compared to its value at

*t*= 0. As a result, the vertical velocity increases as

*t*increases toward 0.

Comparing the four panels in Fig. 7 to the time-independent disturbance produced by the *δ*-PV of Fig. 5 indicates that the amplitude of the GW patterns are comparable at *t* = 0 and ±6 h but substantially smaller at *t* = ±12 and ±18 h. Accordingly, it is only in time intervals of half a day or so that the values for GWs emission and for the associated EP flux given in the previous sections apply.

## 5. Conclusions

The linear motion associated with 3D localized potential vorticity (PV) anomalies in the presence of an unbounded vertical shear Λ has been analyzed in the linear approximation. Exact and approximate solutions were obtained analytically for PV anomalies that are monochromatic in *x* and *y*, and vary as a Dirac delta function *δ*(*z*) in the vertical. Combinations of these yield solutions for more general PV anomalies.

*σ*at

_{H}*z*= 0 induces two inertial critical layers at

*z*= ±

*σ*/Λ. Through these levels, the intrinsic frequency of the disturbance increases from subinertial to superinertial. Correspondingly, there is a transition from balanced near

_{H}f*z*= 0 (where the solutions can be described as quasigeostrophic) to sheared GW for |

*z*| >

*σ*/

_{H}f*λ*. The amplitude of the GW is approximately

*J*=

*N*

^{2}/Λ

^{2}is the Richardson number and

*ν*=

*l*/

*k*is the ratio of the

*y*and

*x*components of the wave vector. As previously noted (LPV10), these waves can be substantial for moderate Richardson numbers, say

*J*between 1 and 10. The present analysis reveals a new, remarkable result: the emitted waves have a strong meridional asymmetry, with larger amplitudes for waves with

*ν*< 0. For example, in a westerly shear in the Northern Hemisphere, waves aloft having their wave vector pointing to the southeast will be larger than waves with their wave vector pointing to the northeast (see Figs. 5 and 7). Using the exact analytical solutions we show how this asymmetry, in a symmetric background flow, is related to the meridional slope of the isentropes (see Fig. 2). This asymmetry has been identified previously in studies of gravity waves propagating toward critical levels in a constant shear (the valve effect) (Yamanaka and Tanaka 1984).

One implication is a strong sensitivity to orientation (i.e., to *ν*) of the absorption of the Eliassen–Palm flux through the inertial levels: there is almost no jump when *ν* is large and negative, in contrast to nearly complete absorption when *ν* is large and positive [see (2.31) and (2.32) and Fig. 4]. Hence the drag due to the waves absorbed within the upper inertial layer has a substantial component oriented to the right of the shear in the Northern Hemisphere (southeast in the above example). The WKB solutions provide simple expressions for the fluxes, the angle maximizing them, and the drag, in very good agreement with the exact analytical solutions (see Fig. 4).

The relevance of this emission in real flows remains to be assessed. Nonetheless, two points are worth noting: first, it has been noted from satellite observations (Wu and Eckermann 2008) and from high-resolution numerical weather prediction (NWP) models (Shutts and Vosper 2011) that gravity waves in the midlatitudes have a favored orientation: phase lines with a northeast-to-southwest tilt in the Northern Hemisphere, and with a northwest-to-southeast tilt in the Southern Hemisphere. Waves with these orientations are conspicuous in the stratospheric polar night jets of both hemispheres (i.e., in regions with strong positive vertical shear). The reasons for this favored orientation are not clear.^{1} It is noteworthy that this orientation is consistent with that expected in the case of emission from sheared PV anomalies. Whether this emission is occurring or this is only a coincidence due to a more fundamental property of GW in shear remains to be investigated. Second, at smaller scales, we can expect this mechanism to play a role where the breaking of intense orographic gravity waves produces small-scale PV anomalies (Plougonven et al. 2010).

As discussed in LPV10, our results could be used for parameterizations in GCMs of GW emission by fronts at the tropopause (Charron and Manzini 2002; Richter et al. 2010), where substantial intrusion of stratospheric air occurs and where strong shears are common. In this context the predictor given in LPV10 seems adapted, providing we add the transverse component of the EP flux as in (4.11). The factor ¼ for the flux emitted by 2D PV disturbance in (1.1) should more accurately be

The present paper has shown that the formula given in LPV10 and recalled here in (1.1) and (1.2) applies quite well in the 3D case. To take directional effects into account one should use (4.11) rather than (1.1), keeping unchanged the dimensional factor (1.2).

## Acknowledgments

FL was supported by the EU-FP7 project EMBRACE (Grant Agreement 282672), and RP and JV by the Alliance Programme of the French Foreign Affairs Ministry and British Council. JV also acknowledges the support of a NERC grant. We also thank Oliver Bühler for pointing out a useful reference.

## APPENDIX A

### Exact Solution for *W*(*ξ*)

*ξ*> 0 and impose a radiation condition for

*ξ*≫ 1 to obtain a solution that represents an upward-propagating GW. We deduce from this a solution valid for

*ξ*< 1 that represents a downward-propagating GW for

*ξ*≪ −1. The amplitudes of these two solutions are then chosen to satisfy the jump conditions

#### a. Homogeneous solution for ξ > 0

*W*= (1 +

*ξ*)

^{−iν}and

*η*=

*ξ*

^{2}transform (2.13) into the canonical form of the hypergeometric equation [(15.5.1) in Abramowitz and Stegun (1964), hereafter AS]:

*ξ*> 1 the two solutions of the hypergeometric equation (A.2a) are given by (15.5.7) and (15.5.8) in AS. We retain the second solution

*F*is the hypergeometric function and

*a*=

*a*′,

*b*′ =

*b′*−

*c*+ 1,

*c*′ =

*b*−

*a*+ 1, because its asymptotic form,

*ξ*< 1 the solution to (A.2a) is best written as a linear combination of the two independent solutions (15.5.3) and (15.5.4) in AS:

*a*″ =

*a*−

*c*+ 1,

*b*″ =

*b*−

*c*+1,

*c*″ = 2 −

*c*, and

*A*and

*B*are two complex constants.

*F*[(15.3.6) in AS] and obtain the asymptotic approximations

*α*′,

*β*′) and (

*α*″,

*β*″) are defined by the same formulas with (

*a*,

*b*) replaced by (

*a*′,

*b*′) and (

*a*″,

*b*″), respectively.

*ξ*= 1, we follow Booker and Bretherton (1967) and introduce an infinitely small linear damping that shifts the real

*ξ*axis into the lower half of the complex plane so that

*A*and

*B*and completes the evaluation of

*W*

^{(u)}(

*ξ*).

#### b. Solution over the entire domain

*ξ*< 0 can be deduced from

*W*

^{(u)}(

*ξ*) by noting that (2.13) applies to

*W** when

*ξ*is changed in −

*ξ*. A possible solution is simply

*ξ*→ −∞ since

*W*

^{(u)}and

*W*

^{(d)}can be combined to obtain a solution valid over the entire domain that satisfies the jump condition (A.1). This is given by

*E*is found by imposing the jump condition (A.1) and given by

*ξ*| ≪ 1, the upper and lower solutions in (A.12) have the asymptotic expansions

*ξ*> 0 and

*ξ*< 0, respectively. For the value of

*E*in (A.13),

*EA*is real, which implies that the first terms on the left-hand sides of (A.14a) and (A.14b) are identical, ensuring that

*W*/

_{ξ}*ξ*

^{2}jumps by 1 at

*ξ*= 0 as required. Note also that near |

*ξ*| = 0,

*W*(

*ξ*) approaches the value

## APPENDIX B

### WKB Approximation

In this appendix, we derive the WKB approximations (2.16)–(2.19) for *W* in regions (i)–(iv) and provide details of the matching procedure.

*ξ*> 0 by (2.16).

*O*(

*J*) and

*O*(

*J*

^{1/2})

*ζ*=

*J*(1 +

*ν*

^{2})(

*ξ*− 1) =

*O*(1). To leading order (2.13) then reduces to

*ξ*→ ∞. Matching with the limiting behavior of (2.16) as

*ξ*→ 1 gives (2.21).

To match the solutions between regions (ii) and (iii), we need to consider the limit of the Hankel functions for *ζ* = |*ζ*|*e*^{−iπ} with |*ζ*| → ∞, in accordance with the analytic continuation (A.5). Proceeding in similar fashion as above using (B.5) yields relation (2.22) between *A*^{(ii)} and *A*^{(iii)}, but *B*^{(iii)} = 0. As mentioned, *B*^{(ii)} = 0 is inconsistent with the nonzero EP flux expected because of the wave radiation as |*ζ*| → ∞. To resolve this difficulty, we need to employ a more sophisticated matching that recognizes that *B*^{(ii)} takes in fact a nonzero exponentially small value and provides an estimate for this value.

*B*

^{(ii)}arises as a result of a Stokes phenomenon (e.g., Ablowitz and Fokas 1997): the line

*ζ*< 0 (arg

*ζ*= −

*π*) is a Stokes line, where one solution (here multiplied by

*A*

^{(ii)}) is maximally dominant over the other, recessive solution (multiplied by

*B*

^{(ii)}). Across this Stokes line, the dominant solution switches on the recessive solution with an amplitude given by an exponentially small Stokes multiplier. Thus, below the Stokes line the amplitude

*B*

^{(ii)}= 0, and above it

*B*

^{(ii)}≠ 0 is given by the Stokes multiplier; on the Stokes line itself,

*B*

^{(ii)}is half the Stokes multiplier (Berry 1989). To obtain the Stokes multiplier, we need a large |

*ζ*| formula for (2.18) that is valid for

**−**3

*π*< arg

*ξ*< −

*π*so that it holds immediately above the Stokes line and also on the anti-Stokes line arg

*ζ*= −2

*π*where the two solutions have the same order and hence can be identified unambiguously. Such a formula is obtained using the connection equation [(9.1.37) in AS] to obtain

*π*< arg

*ζ*< −

*π*,

*ζ*| asymptotics

*B*

^{(iii)}= 0 and using that

*ξ*=

*J*(1 +

*ν*

^{2}) (1 −

*ξ*)

*e*

^{−iπ}leads to

*ξ*→ 1 gives (2.22), and (2.23) on taking into account that Stokes multiplier on the Stokes line is half its value away from it.

The matching between regions (i) and (ii) yielding (2.25) and (2.26) is straightforward.

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^{1}

Shutts and Vosper (2011) suggested that this tilt could be tied to the orientation of surface cold fronts, but gravity waves generated in idealized baroclinic life cycles (O’Sullivan and Dunkerton 1995; Plougonven and Snyder 2007) show the opposite tilt.