3.1 Preliminaries

We examine small-amplitude waves governed by the linearisation of the primitive equations (2.2)–(2.3) about the background flow. This is in geostrophic balance and given by

We assume a small Rossby number based on the envelope scale $\unicode[STIX]{x1D6E5}$ of the topography:

where $U_{b}=\unicode[STIX]{x1D6EC}H$ . We also assume that $NH/(f\unicode[STIX]{x1D6E5})=O(1)$ , corresponding to an order-one Burger number based on the horizontal and vertical scales $\unicode[STIX]{x1D6E5}$ and $H$ . Together with (3.2), this implies a large Richardson number, specifically

where we have introduced the parameter $J=Ri^{1/2}$ as a convenient substitute for the Richardson number $Ri$ .

We are interested in mountain waves, in the distinguished regime where the Doppler-shifted frequency $U_{b}k_{\ast }$ is of the same order as the Coriolis frequency. This corresponds to the scaling

with $r$ a Rossby number based on the mountain wavelength rather than the envelope scale that appears in $Ro$ . Since $r=k_{\ast }\unicode[STIX]{x1D6E5}Ro$ , this implies that

To ensure that the waves are propagating rather than evanescent at the bottom boundary we further require that

We emphasise that the scaling specified by (3.2), (3.3) and (3.5) corresponds to a distinguished limit, that is, it leads to results valid for a broad range of relative values of $Ro$ , $J$ and $k_{\ast }\unicode[STIX]{x1D6E5}$ . This becomes apparent in the asymptotic derivation of § 3.2 and is discussed further in the Conclusion, § 5.

Linearising (2.2) around the background flow (3.1) leads to the equations

with $\boldsymbol{e}_{x}$ the unit vector in the $x$ -direction, which govern the leading-order wave fields, denoted here by the subscript ‘1’. The amplitude of the waves is determined by the linearisation of the boundary condition (2.3) around the background flow (3.1),

This indicates that $w_{1}=O(U_{b}k_{\ast }h)$ . The polarisation relation for standard internal waves (see also (3.14a ) below) can then be used to estimate $u_{1}=O(Nw_{1}/(k_{\ast }U_{b}))=O(Nh)$ . The linearisation based on the small-amplitude condition $u_{1}\ll U_{b}$ thus requires that the inverse Froude number be small:

Based on this small parameter, we introduce the convention of using a subscript ‘ $n$ ’ $(n=1,2,3\ldots \,)$ to denote the $n$ th-order flow variables such that

After applying a Fourier transform, (3.7) can be reduced to the single equation

where $\unicode[STIX]{x1D708}=l/k$ . (This is equation (25) in Jones (Reference Jones1967) and equation (4) in Yamanaka & Tanaka (Reference Yamanaka and Tanaka1984).) The independent variable is the scaled vertical coordinate

We emphasise that for $k>0$ , as will be assumed when interpreting the results, $\unicode[STIX]{x1D701}$ has a sign opposite to that of $z$ and is positive below the inertial level and increasing downwards. The dependent variable ${\hat{w}}$ is the horizontal Fourier transform of $w_{1}$ , defined by

The other dependent variables are related to ${\hat{w}}$ through the polarisation relations

One of the key characteristics of inertia–gravity waves (IGWs) in shear flow is the presence of singularities: two inertial levels, where the Doppler-shifted frequency matches the Coriolis frequency, and a critical level, where the Doppler-shifted frequency and hence the background velocity vanish (Jones Reference Jones1967). These singularities are readily identified from (3.11): the two inertial levels and one critical level correspond to $\unicode[STIX]{x1D701}=\pm 1$ and $\unicode[STIX]{x1D701}=0$ , respectively. The critical level is an apparent singularity that can be removed by a variable transformation. The inertial levels, by contrast, have a marked physical impact since the wave solution switches abruptly from an oscillatory to an evanescent behaviour and back across them (Yamanaka & Tanaka Reference Yamanaka and Tanaka1984; Lott, Millet & Vanneste Reference Lott, Millet and Vanneste2015). As we discuss in § 4, this abrupt change underpins the forcing of a mean flow by the wavepacket.

Rotation plays a crucial role. The position $z=\pm f/(k\unicode[STIX]{x1D6EC})$ of the inertial levels depends on the wavenumber $k$ ; as a result, the singularities associated with each wavenumber making up the wavepacket are smeared out over a range of altitude, and the wavepacket solution is smooth even in the absence of dissipation (or more precisely in the limit of vanishing dissipation since dissipation is important to determine physically relevant branches of solution; Shutts Reference Shutts2001). This is in contrast with the non-rotating scenario, best thought of as the limit $f\rightarrow 0$ of the general situation. In this limit, the inertial levels coalesce with the critical level, leading to a stronger, $k$ -independent singularity and to a singular behaviour of the wavepacket unless dissipation is introduced. (A similar smearing out of singularities across different altitudes also occurs without rotation when more complicated flows, such as the directional shear flow, are considered; Shutts Reference Shutts1995, Reference Shutts2003, Martin & Lott Reference Martin and Lott2007.)

The term ‘inertial layer’ is used to describe the region where the effect of the inertial-level singularities is distributed. It is centred around the dominant inertial level,

determined by the central wavenumber $k_{\ast }$ of the topography. Note that, for the problem under consideration, only the lower inertial levels matter since the waves are exponentially small in $J$ at the upper inertial levels. Condition (3.6) ensures that $z_{\ast }>-H$ , that is, the dominant inertial level lies in the fluid domain. The characteristic thickness of the inertial layer is found as

on using that, according to expression (2.1) for the mountain height, the spectral width of the wavepacket is $\unicode[STIX]{x1D6FF}k=\unicode[STIX]{x1D6E5}^{-1}$ .

Equation (3.11) for ${\hat{w}}$ can be solved explicitly in terms of hypergeometric functions (Yamanaka & Tanaka Reference Yamanaka and Tanaka1984; Shutts Reference Shutts2001). We rely instead on the approximate WKB solution derived in Lott et al. (Reference Lott, Plougonven and Vanneste2012) (see also Lott et al. Reference Lott, Millet and Vanneste2015) and valid in the large Richardson number limit $J\gg 1$ . In this solution, ${\hat{w}}$ is approximated as

where

Note that we have normalised ${\hat{w}}$ so that its bottom boundary value is ${\hat{w}}(\unicode[STIX]{x1D701}_{b})=1$ .

Expression (3.17) holds for all real values of $\unicode[STIX]{x1D701}$ except in small regions of $O(J^{-2})$ thickness around the inertial levels $\unicode[STIX]{x1D701}=\pm 1$ . The sign of the argument of the exponential is taken to be negative because this ensures that the wave is propagating upwards above the upper inertial level, that is, for $\unicode[STIX]{x1D701}<-1$ (see Booker & Bretherton (Reference Booker and Bretherton1967); note that the opposite, positive sign is found when the background velocity is increasing with altitude). The fractional powers and logarithms involved in (3.17) are multivalued functions for which suitable branches need to be selected. This selection is dictated by causality and is most easily settled by adding small damping terms in (3.7). The upshot is that the multivalued functions should be continued from $\unicode[STIX]{x1D701}<-1$ to $\unicode[STIX]{x1D701}>1$ along a contour in the complex plane that passes below the singularities at $\unicode[STIX]{x1D701}=\pm 1$ (Booker & Bretherton Reference Booker and Bretherton1967; Jones Reference Jones1967; Lott et al. Reference Lott, Millet and Vanneste2015). In this way, $\sqrt{\unicode[STIX]{x1D701}^{2}-1}=-\text{i}\sqrt{1-\unicode[STIX]{x1D701}^{2}}$ between the inertial levels so that the solution is decreasing exponentially with $\unicode[STIX]{x1D701}$ there, like $\exp (-J\sqrt{1+\unicode[STIX]{x1D708}^{2}}\cos ^{-1}\unicode[STIX]{x1D701})$ . Overall, the solution experiences an absorption by the factor $\exp (-J\sqrt{1+\unicode[STIX]{x1D708}^{2}}\unicode[STIX]{x03C0})$ (see also Lott et al. Reference Lott, Plougonven and Vanneste2012) known to apply to both the rotating and non-rotating cases. Note that the reliance on causality means that the problem is not treated as strictly inviscid but rather as a vanishing viscosity limit.

The WKB approximation (3.17) breaks down near the inertial levels, specifically for $||\unicode[STIX]{x1D701}|-1|=O(J^{-2})$ where it should be replaced by an expression in terms of Hankel functions (Lott et al. Reference Lott, Plougonven and Vanneste2012). These regions are narrow enough and the singularities of (3.17) at $\unicode[STIX]{x1D701}=\pm 1$ are mild enough that they can be ignored when computing the vertical velocity of the complete wavepacket.

Substituting the form (2.1) of the topography into the boundary condition (3.8), we obtain the vertical velocity at the boundary as the Fourier expansion

Here and henceforth, a real part is implied. Combining this with (3.13) and  (3.17) leads to the vertical velocity of the wavepacket in the form

Note that, because ${\hat{w}}$ is exponentially small for $\unicode[STIX]{x1D701}<1$ as a result of wave absorption, the lower limit of the integral in $k$ could be taken as $-f\unicode[STIX]{x1D6EC}^{-1}z^{-1}$ . This absorption is crucial for the impact of the wavepacket on the mean flow.

3.2 Saddle-point approximations

The solution (3.20) can be further simplified by taking advantage of the assumptions $J\gg 1$ and $k_{\ast }\unicode[STIX]{x1D6E5}\gg 1$ to apply a saddle-point approximation. The key is to identify the dominant terms in the argument of the exponential, including a contribution from ${\hat{w}}$ . To avoid defining several new dimensionless numbers measuring the relative size of $J$ , $k_{\ast }\unicode[STIX]{x1D6E5}$ and $Ro$ , it is expedient to introduce a bookkeeping parameter $\unicode[STIX]{x1D716}$ which keeps track of the orders of various terms. This parameter is treated as formally small and used as a basis for a saddle-point approximation, but it is set to $1$ at the end of the computation to obtain asymptotic formulas in a convenient dimensional form. The bookkeeping parameter $\unicode[STIX]{x1D716}$ is introduced through the replacements

in accordance with the scalings (3.2), (3.3) and (3.5). Thus the formal smallness of $\unicode[STIX]{x1D716}$ captures at once the mountain scale separation, the large Richardson number, and the small Rossby number. We emphasise that the bookkeeping device is completely equivalent to using, say, $J$ as a large parameter and treating $\boldsymbol{k}_{\ast }\unicode[STIX]{x1D6E5}/J$ and $JRo$ as order one; it is employed here for the economy and transparency of notation it brings.

Introducing (3.21) into (3.20) leads to

where

and makes the dependence on $\unicode[STIX]{x1D716}$ explicit. In writing this expression, we have made a first approximation by replacing the wavenumber $k$ outside the exponential functions by its leading-order approximation $k_{\ast }$ . The error introduced is negligible as can be verified below once the size of the neighbourhood of $k_{\ast }$ controlling the integral is estimated.

A second approximation is made in carrying out the integration with respect to $l$ in (3.22). Because the integral is dominated by values of $l$ near $\unicode[STIX]{x1D716}^{-1}l_{\ast }$ , we write

where $\unicode[STIX]{x1D716}L/l_{\ast }\ll 1$ , leading to the following expressions

where $\unicode[STIX]{x1D716}k$ is treated as $O(1)$ since $k$ is close to $\unicode[STIX]{x1D716}^{-1}k_{\ast }$ . Substituting these into (3.22) and neglecting $O((\unicode[STIX]{x1D716}L/l_{\ast })^{2})$ , we obtain

In the second line, we have ignored a phase factor and introduced the symbol $\doteq$ to denote an equality in modulus only, ignoring phase factors. In what follows, we pay only attention to the modulus of $w_{1}$ since this controls the wave–mean-flow interaction properties: the spatially averaged EP flux, which is quadratic in wave quantities, only depends on the wave amplitude and on the relative phase of various fields which is easily worked out.

The appearance of the bookkeeping parameter $\unicode[STIX]{x1D716}^{-1}$ in the exponential in (3.27) motivates the saddle-point approximation. To apply this, we need to compare the leading terms in the exponential, namely

The first term stems from the finite spectral width of the mountain height; the second term, which depends on $\unicode[STIX]{x1D701}$ and hence $z$ , captures the vertical structure of the wave. We seek distinguished regimes, where the leading-order terms in (3.28) balance. This requires approximating $D(\unicode[STIX]{x1D701})$ to determine its order, a non-trivial task since the order of $D(\unicode[STIX]{x1D701})$ depends on the value of $\unicode[STIX]{x1D701}$ , that is, on the particular range of altitude considered. Mathematically, different altitude ranges are captured by different values of $\unicode[STIX]{x1D6FC}$ in the scaling

where $Z=O(1)$ .

Note that, since $z_{\ast }<0$ , $Z$ and $z$ have different signs, with $Z>0$ below the dominant inertial level and increasing downwards. Because $\unicode[STIX]{x1D701}$ depends on both $z$ and $k$ , we also need to scale $k$ to find range of wavenumbers controlling the integral in (3.27). Since the wavepacket is concentrated around $\unicode[STIX]{x1D716}^{-1}k_{\ast }$ , we write

Combining (3.29) and (3.30), we obtain

We now need to distinguish two situations: (i) for $\unicode[STIX]{x1D6FC}=0$ , that is, away from the dominant inertial level,

(ii) for $\unicode[STIX]{x1D6FC}>0$ , that is, asymptotically close to the dominant inertial level,

Because the leading-order terms in these two expansions are independent of $k$ , hence do not contribute to the integration over $k$ in (3.27), the order of the second terms is crucial. Using the scalings of the second terms of (3.32)–(3.33) in the second expression in (3.28), and balancing with the first expression (scaling like $\unicode[STIX]{x1D716}^{2\unicode[STIX]{x1D6FD}-2}$ ) leads to two distinguished regimes: Regime I, with $\unicode[STIX]{x1D6FC}=0$ and $\unicode[STIX]{x1D6FD}=1$ , and Regime II, with $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}=2/3$ (since $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$ gives a distinguished regime). In each regime, the coordinate $x$ appearing in $\exp (\text{i}kx)$ should be scaled so that the $K$ -dependent contribution to $kx$ , proportional to $\unicode[STIX]{x1D716}^{\unicode[STIX]{x1D6FD}-1}$ , be of the same order as the $K$ -dependent terms in (3.28). This leads to $x=O(\unicode[STIX]{x1D6E5})$ in Regime I and $x=O(\unicode[STIX]{x1D716}^{-1/3}\unicode[STIX]{x1D6E5})$ in Regime II.

We carry out the saddle-point expansion of (3.27) in these two regimes in appendix A and only quote the final results here. In Regime I, after setting $\unicode[STIX]{x1D716}=1$ , we find that

where

$D_{I}=\ln (1+Z_{I}+\sqrt{2Z_{I}+Z_{I}^{2}})-\ln (r+\sqrt{r^{2}-1})$ , $Z_{I}=z/z_{\ast }-1$ and we have used that $r=k_{\ast }\unicode[STIX]{x1D6EC}H/f$ . This makes clear that the wavepacket retains the bell shape of topography, with scale $\unicode[STIX]{x1D6E5}$ , throughout its propagation across Regime I. The path of the wavepacket in the $(x,z)$ -plane is determined by setting $X_{I}=0$ in (3.34), which corresponds to standard ray tracing. It shows in particular that the wavepacket diverges to infinity as it approaches the dominant inertial level, with $x\sim Jk_{\ast }^{-1}(1+\unicode[STIX]{x1D708}_{\ast }^{2})^{1/2}(2Z_{I})^{-1/2}$ as $Z_{I}\rightarrow 0$ . This is a limitation of the approximation made in Regime I (also a limitation of ray tracing) rather than a physical effect as the analysis of Regime II shows.

In Regime II, and again with $\unicode[STIX]{x1D716}=1$ , we find that

where the $y$ -dependence is controlled by the Gaussian

with $D_{II}=-\ln (r+\sqrt{r^{2}-1})$ , and is decoupled from the dependence in $x$ and $z$ . In (3.36), the function $p$ is defined by

and $K_{s}$ is one of its saddle points, satisfying $p^{\prime }(K_{s})=0$ , where the prime denotes derivative. The other symbols introduced are

and $Z_{II}=(k_{\ast }\unicode[STIX]{x1D6E5})^{2/3}(z/z_{\ast }-1)$ . The saddle point $K_{s}$ satisfies a cubic equation whose analytic solution is not particularly illuminating; we will solve it numerically. It is selected among the three roots of the cubic by the condition that it be accessible by a steepest-descent path connecting $-\infty$ to $\infty$ (e.g. Bender & Orszag Reference Bender and Orszag1999). For large $Z_{II}$ , the expansion $\sqrt{2(K_{s}+Z_{II})}\sim \sqrt{2Z_{II}}+K_{s}/\sqrt{2Z_{II}}$ can be used to confirm the matching between Regime I and Regime II. The behaviour for small $Z_{II}$ is key for the mean-flow forcing. As discussed in § 3.1, the change in EP flux is concentrated in the inertial layer, such that $|z-z_{\ast }|=O(\unicode[STIX]{x1D6FF}_{\ast })$ with $\unicode[STIX]{x1D6FF}_{\ast }$ given in (3.16). This corresponds to the scaling

hence to $\unicode[STIX]{x1D6FC}=1$ in (3.29) and thus $Z_{II}=O(\unicode[STIX]{x1D716}^{1/3})\ll 1$ . For this range of $Z$ , the integral in (3.27) is dominated by wavenumbers $k$ in an $O(1)$ neighbourhood of the central wavenumber $\unicode[STIX]{x1D716}^{-1}k_{\ast }$ , corresponding to $\unicode[STIX]{x1D6FD}=1$ in (3.30). The associated regime, which we term Regime II $_{B}$ , is a limit of Regime II, obtained when some terms are negligible (notably the first term in the phase function (3.38)). This makes it possible to derive an expression for $w_{1}$ simpler than (3.36) which we will subsequently use to compute the mean-flow forcing. This expression is derived in § A.3 and given by

where

and

According to this, the wavepacket is centred on the curve

in the $(X,Z)$ -plane (obtained by maximising (3.42)) and localised in a region of order-one size in both the $X$ and $Z$ direction. In view of (3.41) and (3.44), this corresponds to a region of streamwise extent $O((k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5})$ , thus much larger than the size $\unicode[STIX]{x1D6E5}$ of the mountain, located an $O((k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5})$ distance downstream of the topography, and of $O(\unicode[STIX]{x1D6FF}_{\ast })$ vertical extent. Thus, the prediction of ray tracing of a wavepacket that diverges to infinity as the dominant inertial level is approached is replaced in Regime II $_{B}$ (and hence in Regime II) by a large-but-finite horizontal shift. Note that in the limit of $f\rightarrow 0$ , the wavepacket propagates vertically with a vertical velocity that tends to zero as it approaches the critical level $z=0$ towards which the inertial levels coalesce, with $w_{1}\propto z^{1/2}$ as $z\rightarrow 0$ . This can be deduced from (3.34) by letting $r\rightarrow \infty$ and is consistent with a direct computation assuming $f=0$ . For a small-but-finite $f$ , however, there is an extremely thin inertial layer of size $O((k_{\ast }\unicode[STIX]{x1D6E5})^{-2}Ro^{-1}H)$ , in which the wavepacket experiences the horizontal displacement described by (3.44).

We remark that the three regimes identified by the saddle-point analysis can be interpreted physically. Regime I is the ray-tracing regime, which is unaffected by the singularities of the wave solution. Regime II is controlled by the singularity at the lower inertial level, and Regime II $_{B}$ is its part dominated by inertial-level absorption.

3.3 Numerical results

In this section, we compare the asymptotic predictions for $w_{1}$ with direct numerical computations of the integral in (3.27). We first take the parameters

so $r=k_{\ast }\unicode[STIX]{x1D6E5}Ro=2$ . A choice of physical parameters leading to these values is $N=1.4\times 10^{-2}~\text{s}^{-1}$ , $f=10^{-4}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6EC}=1.4\times 10^{-4}~\text{s}^{-1}$ , $H=5~\text{km}$ , $\unicode[STIX]{x1D6E5}=3.5\times 10^{2}~\text{km}$ and $k_{\ast }=2.8\times 10^{-4}~\text{m}^{-1}$ . The weak shear and short wavelengths make this choice somewhat contrived, but it has the advantage of enabling a comparison in conditions where the asymptotic assumptions hold unambiguously. Parameters corresponding to a stronger shear and longer wavelengths are considered at the end of the section. We concentrate on the amplitude $|w_{1}|$ in the $(x,z)$ cross-section where it is maximum, since the structure in $y$ is simply the Gaussian structure of the mountain envelope, albeit with a shift.

Figure 2.

Vertical velocity amplitude $|w_{1}|$ in the $(x,z)$ plane obtained by numerical integration of (3.22) for the parameters in (3.46) (7 equispaced contours with velocity in the range $[0.04,0.28]\times Uhk_{\ast }/\sqrt{2\unicode[STIX]{x03C0}}$ are shown). The inset is a zoom on the rectangle region indicated in the main panel.

Figure 2 shows a contour plot of $|w_{1}|$ obtained numerically. Away from the dominant inertial level, $k_{\ast }\unicode[STIX]{x1D6EC}z/f=-1$ , the wavepacket is only slightly deflected from the vertical. Closer to the dominant inertial level, the wavepacket shows a significant bend; we can read off from the inset that the peak of the wavepacket at the dominant inertial level is approximately $x/\unicode[STIX]{x1D6E5}\approx 12$ , in agreement with the peak value predicted in (3.44) which gives $X_{c}=2^{1/4}$ . This figure is more a qualitative illustration than a quantitative comparison which we carry out next.

Figure 3.

Vertical velocity amplitude $|w_{1}|$ normalised by $Uhk_{\ast }/\sqrt{2\unicode[STIX]{x03C0}}$ as a function of $x/\unicode[STIX]{x1D6E5}$ for the altitudes given by $-k_{\ast }\unicode[STIX]{x1D6EC}z/f=1.5$ (a), 1.05 (b), 1.01 (c) and 1 (d), and the parameters in (3.46). Numerical results (circles) are compared with the asymptotic predictions of Regime I (solid line, shown in panels a–c), Regime II (dashed line) and Regime II $_{B}$ (dash-dotted line, shown in panels b–d).

Figure 3 provides a detailed comparison of the numerical and asymptotic predictions for $|w_{1}|$ . Its four panels show $|w_{1}|$ as a function of $x/\unicode[STIX]{x1D6E5}$ at four different altitudes. Panel (a) corresponds to $-k_{\ast }\unicode[STIX]{x1D6EC}z/f=1.5$ , sufficiently far below the dominant inertial level for the Regime I asymptotics to apply. As expected, the asymptotic predictions of Regime I (solid line) matches the numerical results (circles), with a wavepacket that takes the bell shape of the mountain envelope, while the predictions of Regime II (dashed line) do not. Panel (b) shows the wavepacket closer to the dominant inertial level, for $-k_{\ast }\unicode[STIX]{x1D6EC}z/f=1.05$ , in a region where Regime I and Regime II overlap: the predictions of both regimes match the numerical results. The Regime II $_{B}$ approximation is also shown (dash-dotted line) and, unsurprisingly, is found to be invalid. Closer still to the dominant inertial level, as shown in panel (c) for $-k_{\ast }\unicode[STIX]{x1D6EC}z/f=1.01$ , the Regime I approximation breaks down. The Regime II predictions match the numerical results closely, while those of Regime II $_{B}$ are accurate for $x$ large enough. As expected, the wavepacket is no longer bell shaped, and its peak is shifted by an $O((k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5})$ amount to the right (since $x/\unicode[STIX]{x1D6E5}\approx 10=(k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}$ ) in agreement with (3.44) and (3.45). Finally, at the dominant inertial level as shown in panel (d), the predictions of both Regime II and Regime II $_{B}$ coincide and match the numerical results in most of range of $x$ , except to the very left of the peak where Regime II $_{B}$ underestimates the amplitude. As discussed above, the peak of the wavepacket remains at a finite $O((k_{\ast }\unicode[STIX]{x1D6E5})^{1/2}\unicode[STIX]{x1D6E5})$ position. Crucially for mean-flow forcing, the maximum amplitude is also strongly reduced as a result of absorption.

We now consider a more realistic parameter choice relevant to the atmosphere. Taking $N=1.4\times 10^{-2}~\text{s}^{-1}$ , $f=10^{-4}~\text{s}^{-1}$ , $\unicode[STIX]{x1D6EC}=5.6\times 10^{-3}~\text{s}^{-1}$ , $H=5~\text{km}$ , $\unicode[STIX]{x1D6E5}=3.5\times 10^{2}~\text{km}$ and $k_{\ast }=1.43\times 10^{-5}~\text{m}^{-1}$ gives

so $r=k_{\ast }\unicode[STIX]{x1D6E5}Ro=2$ . Since the parameters $Ro^{-1}$ , $k_{\ast }\unicode[STIX]{x1D6E5}$ and $J$ are only moderately large, this choice provides a strict test on the applicability of the asymptotic results. The results for the form of the wavepacket are shown in figure 4. The accuracy of the asymptotic approximations has degraded considerable compared with that in figure 4, unsurprisingly, perhaps, given that the error scales like $(k_{\ast }\unicode[STIX]{x1D6E5})^{-1/2}\approx 0.45$ . Nonetheless, there remains a reasonable qualitative match between asymptotic and numerical results which suggests our approximations remain useful.

Figure 4.

Same as figure 3 but for the parameters in (3.47) and for $-k_{\ast }\unicode[STIX]{x1D6EC}z/f=1.5$ (a), 1.35 (b), 1.2 (c), 1 (d).

4.1 Governing equation

Taking advantage of the small Rossby number, the mean flow is calculated using quasi-geostrophic theory. By taking the horizontal curl of the horizontal momentum equation (2.2a ), applying $\unicode[STIX]{x2202}_{z}f/N^{2}$ to (2.2c ) and using the incompressibility (2.2d ), we obtain

The waves and mean flow are separated by the small-scale average defined as

where $k_{\ast }^{-1}\ll D\ll \unicode[STIX]{x1D6E5}$ . Because of their small spatial scale, the waves have zero average. Applying this average to (4.1), and using the smallness of the wave amplitude and Rossby number to retain the leading-order terms for both wave and mean flow, we obtain

where $\unicode[STIX]{x1D6FB}^{\bot }=(-\unicode[STIX]{x2202}_{y},\unicode[STIX]{x2202}_{x})$ denotes the horizontal curl, and

is the vertical part of the EP flux (Eliassen & Palm Reference Eliassen and Palm1961). In (4.3), the mean flow is in geostrophic balance, with

where the streamfunction $\unicode[STIX]{x1D6F9}$ remains to be determined.

If there is no wave effect, $\boldsymbol{{\mathcal{F}}}=0$ , (4.3) reduces to the quasi-geostrophic potential-vorticity (QGPV) equation. Note that term (c) in (4.1) is the only quadratic wave term to contribute to (4.3): term (a) is $O(\unicode[STIX]{x1D6FF}_{\ast }/H)$ smaller than term (c) because of the vertically thinness of inertial layer; term (b) has zero average because $w_{1}$ and $b_{1}$ are out of phase in the limit of large $J$ (see (3.14c )).

The boundary condition associated with (4.3) is obtained by taking the average of (2.3), retaining terms up to second order in the wave amplitude to find

where $\boldsymbol{u}_{1}=(u_{1},v_{1})$ , and the incompressibility condition (3.7d ) is used.

One significant feature of the EP flux is its conservation when the background flow possesses certain symmetries, leading to the non-acceleration theorem (Charney & Drazin Reference Charney and Drazin1961): waves do not force the mean flow unless there exists a singularity or some dissipation. In our set-up, the background shear flow has $x$ - and $y$ -symmetry, and EP flux conservation is easily demonstrated for a plane wave with wavevector $\boldsymbol{k}=(k,l)$ : applying the polarisation relation (3.14) to (4.4), we obtain the plane-wave expression of the EP flux,

Its conservation is deduced from (3.11) by multiplication of ${\hat{w}}$ by its complex conjugate ${\hat{w}}^{\ast }$ and subtraction of the conjugate of the resulting equation to find

This conservation does not hold across the inertial-level singularities $\unicode[STIX]{x1D701}=\pm 1$ . Across the lower one, $\unicode[STIX]{x1D701}=1$ , the plane-wave EP flux attenuates to an exponentially small value, leading to the wave forcing of the mean flow in the inertial layer (Regime II $_{B}$ ). We now compute the EP flux in this layer.

4.2 Eliassen–Palm flux

The derivation of the EP flux is greatly simplified by observing that, in the saddle-point approximations valid in Regimes I and II, the relations of the various wave fields $u_{1}$ , $v_{1}$ , etc. associated with the wavepacket to $w_{1}$ mirror the polarisation relations (3.14). This is because the rapid dependence in $x$ of the wave solution corresponds to a plane wave with (possibly complex) wavenumber $\boldsymbol{k}_{s}$ given by a saddle-point value of $\boldsymbol{k}$ (see appendix A). Using the notation $\unicode[STIX]{x1D701}_{s}=-k_{s}\unicode[STIX]{x1D6EC}z/f$ , we obtain

where we also have used that $\unicode[STIX]{x1D708}_{s}=l_{s}/k_{s}\sim l_{\ast }/k_{\ast }=\unicode[STIX]{x1D708}_{\ast }$ to leading order. Correspondingly, a derivation that parallels that of (4.7) gives the $x$ -component of the wavepacket EP flux as

This can be simplified further. Focussing on Regime II $_{B}$ , we observe that the $O(\unicode[STIX]{x1D716})$ vertical scale implies that the first term in the brackets in (4.10) dominates the second, that $k_{s}$ can be approximated by $k_{\ast }$ , and that $\unicode[STIX]{x1D701}_{s}$ can be approximated by $1$ except in the factor $1-\unicode[STIX]{x1D701}_{s}$ of $1-\unicode[STIX]{x1D701}_{s}^{2}$ . This leads to the simple expression

Now, using that $\unicode[STIX]{x2202}_{z}=-k^{\ast }\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6EC}\unicode[STIX]{x2202}_{Z}/f$ , we obtain from the form of $w_{1}$ in (A 15) that, to leading order,

Introducing this result into (4.11) and using that

together with the explicit form (A 14) of $K_{s}$ , we find that

The explicit form of $w_{1}$ in (3.42) can finally be used to obtain the explicit expression

where

is a constant controlling the amplitude of $F$ . This expression only holds for $X>0$ ; for $X<0$ , the EP flux is exponentially small. We emphasise the remarkably simple form of (4.15): notwithstanding the many parameters involved, the formula is well suited for practical use in parameterisations.

We illustrate the form of the EP flux in figure 5 by showing its contours in the $(x,z)$ cross-section where it is maximised, using the parameters in (3.46). The EP flux is computed from the WKB linear solution (3.20) and polarisation relation (3.14). The validity of the asymptotic approximation is confirmed by figure 6 which compares the EP flux obtained numerically with the asymptotic approximation at the dominant inertial level.

Figure 5.

Contours of the EP flux in (4.15) normalised by the largest value at the bottom $(3/4)^{3/4}\text{e}^{-3/4}E$ for the parameters in (3.46); 7 equispaced contours in the range $[0.125,0.875]$ are shown in the main figure, and in the inset, 9 equispaced contours are in the range $[0.03,0.27]$ . The contours can be compared with those of the vertical velocity $|w_{1}|$ shown in figure 2: although the two fields obey the same scaling, the maximum of the EP flux is closer to the mountain. The inset zooms on the rectangular box indicated in the main panel.

Figure 6.

EP flux at the dominant inertial level for the parameters in (3.46): numerical results (circles) are compared with the prediction of the Regime II $_{B}$ approximation (solid line).

By integrating (4.15) over $x$ and $y$ , we obtain the horizontally integrated EP flux

where the decay of the error function to $0$ as $Z\rightarrow -\infty$ (above the inertial layer) clearly captures wave absorption. As $Z\rightarrow \infty$ , i.e. well below the dominant inertial level, $\overline{F}$ tends to the constant value

This matches the horizontally integrated EP flux at the ground, since the flux is conserved below the inertial layer. We check this in appendix B. We note that $\overline{F}_{tot}$ is finite in the limit $f\rightarrow 0$ , where it is given by $\overline{F}_{tot}=\unicode[STIX]{x03C0}N\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D6E5}^{2}h^{2}k_{\ast }H$ . The integrated EP flux $\overline{F}$ becomes discontinuous in this limit, jumping from $\overline{F}_{tot}$ to $0$ across the critical level.

From (4.17), we can also estimate the wave drag per unit area, defined as the vertical derivative of horizontally integrated EP flux divided by $\unicode[STIX]{x1D6E5}^{2}$ :

where $|\boldsymbol{k}_{\ast }|=\sqrt{k_{\ast }^{2}+l_{\ast }^{2}}$ is the amplitude of the dominant wavenumber.

4.3 Mean-flow response

We now consider the mean-flow response to the wave drag associated with the $z$ -dependent EP flux (4.15). We compute the steady flow response by solving the QGPV equation (4.3) asymptotically, taking advantage of the thinness of the inertial layer where the wave drag acts to apply matched asymptotics. Thus the domain $z>-H$ is separated into an inner region around the inertial level $z\approx z_{\ast }$ , specifically $Z=-k_{\ast }^{2}\unicode[STIX]{x1D6E5}\unicode[STIX]{x1D6EC}(z-z_{\ast })/f=O(1)$ , and an outer region where the wave drag is absent. Simplifications arise because the spatial scale in the $x$ -direction is longer than in the $y$ -direction; this is made explicit using the variable $X$ defined in (A 16), with $X=O(1)$ implying that $x=\unicode[STIX]{x1D6E5}\times O((k_{\ast }\unicode[STIX]{x1D6E5})^{1/2})$

We denote the streamfunction associated with the wave-induced mean flow in the inner and outer regions by $\unicode[STIX]{x1D6F9}(X,y,Z)$ and $\unicode[STIX]{x1D713}(X,y,z)$ , respectively. In the inner region, considering the scalings of Regime II $_{B}$ , the steady ( $\unicode[STIX]{x2202}_{t}=0$ ) QGPV equation (4.3) becomes

where $\unicode[STIX]{x1D735}$ and $\unicode[STIX]{x1D6FB}^{\bot }$ are gradients with respect to the scaled variables $(X,y)$ , $U_{\ast }=-\unicode[STIX]{x1D6EC}z_{\ast }=f/k_{\ast }$ is the background velocity at the dominant inertial level, and we have included the bookkeeping parameter $\unicode[STIX]{x1D716}$ . On the right-hand side we have neglected the $x$ derivative of $\unicode[STIX]{x2202}_{z}F$ against the $y$ -derivative, owing to the asymptotically larger scales in $x$ .

For sufficiently small mountain height, equation (4.20) can be linearised; we make explicit below the condition for this approximation to hold. Retaining only the leading-order terms in (4.20) reduces this to

Since $F$ is exponentially small for $X<0$ , we can integrate (4.21) for $X>0$ to obtain

where

can be interpreted as a scaled wave-induced QGPV. This can be computed explicitly using (4.15) to find

with $g(y)$ defined in (3.37).

Equation (4.22) is readily integrated, leading to

where $C_{1}$ and $C_{2}$ are integration ‘constants’ that are determined by matching the outer solution (cf. Plougonven & Vanneste Reference Plougonven and Vanneste2010). We have anticipated that the $Z$ -independent term is an order $\unicode[STIX]{x1D716}^{-1}$ larger than the other terms. Matching requires the asymptotic behaviour of $\unicode[STIX]{x1D6F9}$ as $Z\rightarrow \pm \infty$ , found to be

as $Z\rightarrow \pm \infty$ .

In the outer region, the QGPV equation (4.3) is

which, to the leading order, reduces to

Integrating in $X$ , we find that

This is best solved using a Fourier transform in the $y$ direction. Denoting this transform by a hat, we have

applying a vanishing boundary condition as $z\rightarrow \infty$ . Combining the condition (4.6) of zero mean vertical velocity with the buoyancy equation $U\unicode[STIX]{x2202}_{X}\unicode[STIX]{x1D713}_{z}=0$ stemming from (2.2c ), we find the condition $\unicode[STIX]{x1D713}_{z}=0$ at the lower boundary $z=-H$ . This implies

We now match (4.30) to (4.26) to determine ${\hat{C}}_{3}$ , ${\hat{C}}_{4}$ and ${\hat{C}}_{5}$ , and hence the outer solution, completely. Substituting $z-z_{\ast }=-\unicode[STIX]{x1D716}fZ/(k_{\ast }^{2}\unicode[STIX]{x0394}\unicode[STIX]{x1D6EC})$ into (4.30), expanding in powers of $\unicode[STIX]{x1D716}$ and matching with (4.26), we find that

and

Thus, (4.31), (4.32) and (4.33) provide three equations for ${\hat{C}}_{3}$ , ${\hat{C}}_{4}$ and ${\hat{C}}_{5}$ and hence determine the mean flow. The solution to these equations is straightforward but leads to lengthy expressions which we relegate to appendix C.

Remarkably, the right-hand side of (4.33) and hence ${\hat{C}}_{3}$ , ${\hat{C}}_{4}$ and ${\hat{C}}_{5}$ do not depend on $X$ . As a result, $\unicode[STIX]{x1D713}$ , $\unicode[STIX]{x1D6F9}$ and thus the entire mean-flow response does not change downstream of the mountain. Our solution suggests that there is a jump in this response, from a zero value for $X<0$ to the $X$ -independent value for $X>0$ . This is an artefact of the asymptotic approximation: the transition to a non-zero mean flow is in fact smooth. Its detailed form could be obtained using the approximation of the wave fields in Regime II. Here we only note that the scaling of Regime II indicates that the transition region has a characteristic length $x/\unicode[STIX]{x1D6E5}=O((k_{\ast }\unicode[STIX]{x1D6E5})^{1/3})$ , asymptotically smaller than the $O((k_{\ast }\unicode[STIX]{x1D6E5})^{1/2})$ scale that is resolved by the Regime IIB approximation used in our computation of the mean-flow response, hence the apparent discontinuity.

Equation (4.33) provides an estimate for the order of magnitude of the mean-flow response. Recalling that the change in mean velocity is $-\unicode[STIX]{x2202}_{y}\unicode[STIX]{x1D713}$ with the $y$ -scale $\unicode[STIX]{x1D6E5}$ , and noting that the maximum value of $\unicode[STIX]{x1D713}$ is $O({\hat{C}}_{3})$ , we estimate the wave-induced mean velocity as

assuming that $(r^{2}-1)^{1/2}=O(1)$ . Since $(k_{\ast }\unicode[STIX]{x1D6E5})Ro=O(1)$ , this indicates that $U_{w}\ll U_{b}$ , as required for the linearisation of the QGPV equation, provided that $J(h/H)\ll 1$ .

Figure 7.

Mean-flow response. (a) Contours of the wave-induced mean flow $\langle u_{2}\rangle$ , with the sign indicated by $\pm$ ; 20 equispaced contours in the range $[-0.52,0.13]$ are shown. (b)  $\langle u_{2}\rangle$ at $z/H=-0.9,-0.5,0$ , with $z=-0.5$ corresponding to the dominant inertial level. The mean flow is normalised by $\sqrt{3\unicode[STIX]{x03C0}}k_{\ast }J^{2}\unicode[STIX]{x1D6EC}^{2}h^{2}/f$ . (c) Mean pressure $\langle \unicode[STIX]{x1D719}_{2}\rangle$ at ground level ( $z=-H$ ), normalised by $\sqrt{3\unicode[STIX]{x03C0}}k_{\ast }\unicode[STIX]{x0394}J^{2}\unicode[STIX]{x1D6EC}^{2}h^{2}$ .

To illustrate our results, we have calculated the mean-flow response for $J=k_{\ast }\unicode[STIX]{x1D6E5}$ , $l_{\ast }=0$ and $r=2$ . The linear system (4.31), (4.32) and (4.33) is readily solved for ${\hat{C}}_{3}$ , ${\hat{C}}_{4}$ and ${\hat{C}}_{5}$ , leading to $\hat{\unicode[STIX]{x1D713}}$ and, after Fourier inversion, to the mean flow $\langle u_{2}\rangle =-\unicode[STIX]{x2202}_{y}\unicode[STIX]{x1D713}$ and mean pressure $\langle \unicode[STIX]{x1D719}_{2}\rangle$ . This is displayed in figure 7. Observe that the mean-flow response to the wave drag localised in the thin inertial layer is distributed through the entire depth of the fluid, and that the total mean-flow change $\int \langle u_{2}\rangle \,\text{d}y$ vanishes at each altitude since the streamfunction $\unicode[STIX]{x1D713}$ vanishes as $y\rightarrow \pm \infty$ .

In the limit of $f\rightarrow 0$ , i.e.  $r\rightarrow \infty$ , (4.33) indicates that $U_{w}\rightarrow \infty$ . This is because the quasi-geostrophic approximation breaks down: the wave forcing associated with the jump in the EP flux cannot be balanced by the Coriolis force but instead leads to an acceleration of the mean flow. A more meaningful limit treats $f$ as finite for the mean-flow response but sets it to $0$ for the evaluation of the EP flux. The fact, noted in § 3.2, that the EP flux undergoes a discontinuous jump in this case has little impact on the mean-flow response since, with the QGPV equation (4.3) remaining valid, the mean flow away from the inertial layer depends only on the magnitude of the jump and not on the details of its structure. Note however that the right-hand side of (4.34) needs to multiplied by an extra factor $r$ to account for the fact that $(r^{2}-1)^{1/2}\sim r\gg 1$ .