2.1. Incident beam structure

The incident beam is assumed to be described by the similarity solution of Moore & Saffman (Reference Moore and Saffman1969) and Thomas & Stevenson (Reference Thomas and Stevenson1972).

Using the notation introduced in Le Dizès & Le Bars (Reference Le Dizès and Le Bars2017), this harmonic solution can be written as ${{\boldsymbol {X}}} ^{(i)} = {\rm Re} (({{\boldsymbol {v}}}^{(i)},b^{(i)},p^{(i)}) \textrm {e}^{-{\textrm {i}} \omega t})$, where the velocity $v_{\parallel }^{(i)}$ along the direction of propagation, the azimuthal velocity $v_{\phi }^{(i)}$ and the buoyancy $b^{(i)}$ are all proportional to

(2.3)\begin{equation} H_{\mu}(x_{\perp}^{(i)},x_\parallel^{(i)},r) = \frac{1}{\sqrt{r} \left(x_\parallel^{(i)}\right)^{\mu /3}} {\boldsymbol{h_{\mu }}}\left(\frac{x_\perp^{(i)}}{\left(L_d^{2} x_\parallel^{(i)}\right)^{1/3}}\right) \end{equation}

with

(2.4a–c)\begin{equation} v_{\parallel}^{(i)}= \varepsilon^{(i)} H_{\mu},\quad v_{\phi}^{(i)} = -\frac{2\Omega{\textrm{i}} \cos\theta}{\omega} \varepsilon^{(i)} H_{\mu},\quad b^{(i)} = \frac{ N^{2} {\textrm{i}} \sin \theta}{\omega} \varepsilon^{(i)} H_{\mu}. \end{equation}

In (2.3), $r$ is the radial coordinate and $x_\parallel ^{(i)}$ and $x_\perp ^{(i)}$ are the coordinates parallel and normal to the direction of propagation of the incident wave beam in the $(r,z)$ plane (see figure 2$a$).

Figure 2.

Reflection process. $(a)$ Definition of the coordinate systems for $0 < \theta < \alpha {\rm \pi} -\theta$. Here the reflected beam is contracted because ${\rm \pi} /2<\alpha < {\rm \pi} -\theta$. The relationships between the various coordinate systems are provided in appendix A. $(b)$ Characteristic of the reflection process according to $\alpha$ (assuming $0<\theta <{\rm \pi} /2$).

The function ${\boldsymbol {h_{\mu }}}$ is given by

(2.5)\begin{equation} {\boldsymbol{h_{\mu }}}(\zeta) = \frac{\textrm{e}^{-{\textrm{i}} \mu {\rm \pi}/2}}{(\mu -1)!} \int_0^{+\infty} \textrm{e}^{{\textrm{i}}p \zeta - p^{3}} p^{\mu -1}\, \textrm{d}p. \end{equation}

The index $\mu$ characterizes the nature of the singularity which has given birth to the beam (Moore & Saffman Reference Moore and Saffman1969; Thomas & Stevenson Reference Thomas and Stevenson1972). For instance, an axisymmetric Dirac source is known to be associated with an index $\mu =1$ (Tilgner Reference Tilgner2000; Le Dizès Reference Le Dizès2015). Stronger singularities associated with larger values of $\mu$ can also be found. Le Dizès & Le Bars (Reference Le Dizès and Le Bars2017) demonstrated that the singularity associated with the critical latitude on a sphere gives rise to an index $\mu = 5/4$ (this situation is illustrated in figure 1$a$). Expression (2.4a--c) is also encountered in the far field of any localized source (Voisin Reference Voisin2003). In that case, the index $\mu$ characterizes the multipole order of the source (Machicoane etal. Reference Machicoane, Cortet, Voisin and Moisy2015); it is an integer in two dimensions, with $\mu =1$ for a dipole, but a half-integer in three dimensions, with $\mu =3/2$ for a dipole (Voisin Reference Voisin2003).

The constant $L_d$ is given by

(2.6)\begin{equation} L_d^{2}= \frac{ N^{2}\sin^{2}\theta (\nu +\kappa) + 8 \Omega^{2} \nu \cos^{2}\theta}{2 \sin\theta\cos\theta (4 \Omega^{2}-N^{2}) \omega}. \end{equation}

Its norm defines a characteristic diffusion scale. In the non-stratified case ($N=0$), $L_d^{2}= \nu /(2\Omega \sin \theta )$. In the non-rotating case ($\Omega =0$), $L_d^{2}= -(\nu +\kappa )/(2 N \cos \theta )$. When $L_d^{2}<0$, the quantity $(L_d^{2} x_\parallel ^{(i)})^{1/3}$ in (2.3) is defined as $-|L_d^{2} x_\parallel ^{(i)} |^{1/3}$.

The beam gets wider and wider as it propagates. We consider its characteristics at a typical distance $l_c= x_{\parallel c}^{(i)}$ from its source. If we use a characteristic frequency $f_c$ (say $\max (2 \Omega ,N)$) and define the Ekman number by

(2.7)\begin{equation} E=\frac{\nu}{f_c l_c^{2}}, \end{equation}

the characteristic width of the beam is given by

(2.8)\begin{equation} L^{(i)} = E^{1/3} \Lambda l_c, \end{equation}

where $\Lambda$ is a positive dimensionless number given by

(2.9)\begin{equation} \Lambda= \left| \frac{ N^{2}f_c\sin^{2}\theta (1 +{Pr}^{-1}) + 8 \Omega^{2} f_c\cos^{2}\theta}{2 \sin\theta\cos\theta (4 \Omega^{2}-N^{2}) \omega} \right|^{1/3}, \end{equation}

in which ${Pr}= \nu /\kappa$ is the Prandtl number.

In the following, we assume that $E\ll 1$ and that $\Lambda$ is neither zero nor infinite. The Prandtl number ${Pr}$ is assumed to be a fixed non-zero constant. The beam is therefore thin with a width scaling as $E^{1/3}$. This similarity solution is a leading-order approximation of a viscous solution in the limit $E\rightarrow 0$. The pressure $p^{(i)}$ and the normal velocity $v_\perp ^{(i)}$ are $O(E^{1/3})$ as the first-order correction to the similarity solution.

Note that the present similarity solution is a special case of a general expression of the form

(2.10)\begin{equation} v_\parallel = \frac{1}{\sqrt{r}} \int_{0}^{+\infty} G(k) \exp\left({{\textrm{i}} \epsilon_o k x_{\perp}/L^{(i)}}\right) \exp\left({- k^{3} x_{\parallel} /l_c}\right)\,\textrm{d}k, \end{equation}

which describes a viscous beam propagating in the direction of positive $x_{\parallel }$. For our similarity solution, $G(k)= \varepsilon ^{(i)} \textrm {e}^{-{\textrm {i}} \mu {\rm \pi} /2} k^{\mu -1}/(\mu -1)!$. With the frame convention shown in figure 2$(a)$, the parameter $\epsilon _o$ is equal to $1$ if $N<2 \Omega$ and to $-1$ if $N>2 \Omega$. The singular case $N=2 \Omega$ is not considered in the present work.

In the following, all the spatial and temporal variables are assumed to have been non-dimensionalized using $l_c$ and $f_c$.

2.2. Reflection on an inclined boundary

After having covered a distance $l_c$, the incident beam reaches an inclined boundary and is reflected as illustrated in figure 2$(a)$. This reflection process is analogous to that described by Phillips (Reference Phillips1966) for a plane wave in a non-rotating fluid, and further analysed by Kistovich & Chashechkin (Reference Kistovich and Chashechkin1994) and Tabaei etal. (Reference Tabaei, Akylas and Lamb2005). According to the boundary inclination angle $\alpha$, the beam is expected to expand or contract (see figure 2$b$). As in Kistovich & Chashechkin (Reference Kistovich and Chashechkin1994), we assume that the beam keeps its self-similar structure (2.3) with new variables $x_{\parallel }^{(r)}$ and $x_\perp ^{(r)}$ associated with the direction of the reflected beam. We consider the situation illustrated in figure 2$(a)$ where $\theta ^{(i)}= \theta$ and $\theta ^{(r)}= {\rm \pi} -\theta$ and look for a solution ${{\boldsymbol {X}}} ^{(r)} = {\rm Re} (({{\boldsymbol {v}}}^{(r)},b^{(r)},p^{(r)}) \textrm {e}^{-{\textrm {i}} \omega t})$ with

(2.11a–c)\begin{equation} v_{\parallel}^{(r)}= \varepsilon^{(r)} H_{\mu},\quad v_{\phi}^{(r)} = \frac{2\Omega{\textrm{i}} \cos\theta}{\omega} \varepsilon^{(r)} H_{\mu},\quad b^{(r)} = \frac{ N^{2} {\textrm{i}} \sin \theta}{\omega} \varepsilon^{(r)} H_{\mu}. \end{equation}

The reflected beam amplitude $\varepsilon ^{(r)}$ and the value $x_{\parallel c}^{(r)}$ of $x_{\parallel }^{(r)}$ at the impact point $I_c(r_c,z_c)$ should be chosen such that the sum of incident and reflected beams satisfies the condition of non-penetrability on the surface. To apply this condition, it is convenient to introduce local variables tangential and normal to the boundary (see figure 2a):

(2.12a,b)\begin{equation} x_t= \cos\alpha (r-r_c) + \sin\alpha (z-z_c),\quad x_n= -\sin\alpha (r-r_c) + \cos\alpha (z-z_c), \end{equation}

such that the condition $v_n(x_n=0)=0$ gives

(2.13)\begin{align} & -\sin(\alpha -\theta) \varepsilon^{(i)} H_{\mu}(\sin(\alpha-\theta) x_t, x_{\parallel _c}^{(i)},r_c)\nonumber\\ &\qquad + \sin(\alpha +\theta) \varepsilon^{(r)} H_{\mu} (\sin(\alpha + \theta) x_t,x_{\parallel _c}^{(r)},r_c ) =0. \end{align}

This equation is satisfied if and only if

(2.14a,b)\begin{equation} \frac{x_{\parallel c}^{(i)}}{x_{\parallel c}^{(r)}} = K^{3},\quad \frac{\varepsilon^{(i)}}{\varepsilon^{(r)}} =K^{\mu -1}, \end{equation}

with

(2.15)\begin{equation} K= \frac{\sin(\alpha -\theta)}{\sin(\alpha +\theta)}. \end{equation}

Equations (2.14) were already obtained by Kistovich & Chashechkin (Reference Kistovich and Chashechkin1994). They noted that the reflected beam can be interpreted as a beam generated by a virtual source of the same nature as the incident beam but located at a distance $x_{\parallel c}^{(r)}$ from the reflection point given by (2.14). Close to the impact point, the width of the reflected beam is given by $L^{(r)} = L^{(i)}/K$. The coefficient $K$ is therefore the contraction factor of the reflected beam. We can indeed check that the reflected beam expands ($0<K<1$) for $\theta <\alpha <{\rm \pi} /2$ and contracts ($K>1$) for ${\rm \pi} /2<\alpha < {\rm \pi} - \theta$ in agreement with figure 2(b). This formula for the contraction factor is not new and is related to the wavelength contraction factor that a plane wave experiences at reflection (Phillips Reference Phillips1966).

In a region of order $E^{1/3}$ around the impact point that we call the interaction region, a compact expression of the total harmonic field composed of the incident and reflected beams can be obtained. If we redefine $\varepsilon = \varepsilon ^{(i)}/r_c^{1/2}/\left (x_{\parallel c}^{(i)}\right )^{\mu /3}$, and introduce the local variables $\tilde {r} = (r-r_c)/E^{1/3}$ and $\tilde {z} =(z -z_c)/E^{1/3}$, we can write the leading-order expression of the velocity and buoyancy fields in the interaction region as

(2.16)\begin{equation} ({{\boldsymbol{v}}}, b) \sim \varepsilon \left(\boldsymbol{\tilde{v}}_0, \tilde{b}_0\right) = \varepsilon \left(\boldsymbol{\tilde{v}}_0^{(i)}+ \boldsymbol{\tilde{v}}_0^{(r)}, \tilde{b}_0^{(r)} + \tilde{b}_0^{(i)}\right), \end{equation}

(2.17a)\begin{gather} {\tilde{v}}_{0r} = \cos \theta \left( \tilde{h}_{\mu}^{(i)} - \tilde{h}_{\mu}^{(r)} \right), \end{gather}

(2.17b)\begin{gather} {\tilde{v}}_{0\phi } = - \frac{2 \Omega \cos\theta }{\omega} \left( \tilde{h}_{\mu}^{(i)} - \tilde{h}_{\mu}^{(r)} \right), \end{gather}

(2.17c)\begin{gather} {\tilde{v}}_{0z} = \sin \theta \left( \tilde{h}_{\mu}^{(i)} + \tilde{h}_{\mu}^{(r)} \right), \end{gather}

(2.17d)\begin{gather} \tilde{b}_0= {\textrm{i}} \frac{N^{2} \sin \theta}{\omega} \left( \tilde{h}_{\mu}^{(i)} + \tilde{h}_{\mu}^{(r)} \right) \end{gather}

and

(2.18a,b)\begin{equation} \tilde{h}_{\mu }^{(i)} = {\boldsymbol{h_{\mu }}}\left(\frac{-\sin\theta \tilde{r} + \cos \theta \tilde{z} }{\Lambda} \right) ,\quad \tilde{h}_{\mu }^{(r)} = K {\boldsymbol{h_{\mu }}}\left(K\frac{\sin\theta \tilde{r} + \cos \theta \tilde{z} }{\Lambda} \right). \end{equation}

Note that in terms of local boundary layer variables $\tilde {x}_t=x_t/E^{1/3}$ and $\tilde {x}_n=x_n/E^{1/3}$, $\tilde {h}_{\mu }^{(i)}$ and $\tilde {h}_{\mu }^{(r)}$ can also be written as

(2.19a)\begin{gather} \tilde{h}_{\mu}^{(i)}= {\boldsymbol{h_{\mu }}} \left(\frac{\sin(\alpha-\theta)}{\Lambda}(\tilde{x}_t + \cot (\alpha -\theta) \tilde{x}_n ) \right) , \end{gather}

(2.19b)\begin{gather} \tilde{h}_{\mu} ^{(r)} = K {\boldsymbol{h_{\mu }}}\left(\frac{\sin(\alpha-\theta)}{\Lambda}\left(\tilde{x}_t + \cot (\alpha +\theta) \tilde{x}_n \right) \right) , \end{gather}

and the tangential and normal velocity components read

(2.20a)\begin{gather} \tilde{v}_{0t} = \cos ( \alpha-\theta) \tilde{h}_{\mu}^{(i)} - \cos(\alpha+\theta) \tilde{h}_{\mu} ^{(r)} , \end{gather}

(2.20b)\begin{gather} \tilde{v}_{0n} = -\sin ( \alpha-\theta) \tilde{h}_{\mu}^{(i)} +\sin (\alpha+\theta) \tilde{h}_{\mu} ^{(r)}. \end{gather}

It is worth mentioning that the above solution is a leading-order approximation obtained by applying slip boundary conditions. This solution does not satisfy the vanishing buoyancy and no-slip boundary conditions. This means that a viscous/diffusive boundary layer is expected close to boundary. In the next subsection, we show that this viscous boundary layer is responsible for a viscous correction to the reflected beam. More precisely, we shall see that the reflected beam expands in the interaction region as

(2.21)\begin{equation} ({{\boldsymbol{v}}}^{(r)}, b^{(r)}) = \varepsilon \left[\left(\boldsymbol{\tilde{v}}_0^{(r)}, \tilde{b}_0^{(r)}\right) + E^{1/6} \left(\boldsymbol{\tilde{v}}_1^{(r)}, \tilde{b}_1^{(r)}\right) + O\left(E^{1/3}\right) \right], \end{equation}

whereas the incident beam is just

(2.22)\begin{equation} ({{\boldsymbol{v}}}^{(i)}, b^{(i)}) = \varepsilon \left[\left(\boldsymbol{\tilde{v}}_0^{(i)}, \tilde{b}_0^{(i)}\right) + O\left(E^{1/3}\right) \right]. \end{equation}

2.3. Viscous correction to the reflected beam

As long as the slope inclination is not critical, that is, $\alpha$ is different from $\theta$ and ${\rm \pi} - \theta$, the width of the viscous boundary layer is $O(E^{1/2})$. In the viscous boundary layer, the relevant variables are the local tangent variable $\tilde {x}_t$ and the viscous boundary variable $\bar {x} _n = E^{-1/6}\tilde {x}_n = E^{-1/2} x_n$. The form of the solution can be obtained by expanding the leading-order expression (2.17) as $\tilde {x} _n \rightarrow 0$ using the viscous boundary variable. Using (2.20), we observe that $\tilde {v}_{0t}$, $\tilde {v}_{0\phi }$, $\tilde {b}_0$ remain $O(1)$ whereas $\tilde {v}_{0n}\sim \tilde {x}_n \partial _{\tilde {x}_n}\tilde {v}_{0n}(\tilde {x}_t,0) = E^{1/6} \bar {x}_n \partial _{\tilde {x}_n}\tilde {v}_{0n}(\tilde {x}_t,0)$ as $\tilde {x} _n \rightarrow 0$. It follows that one can look for a viscous boundary solution of the form

(2.23a)\begin{gather} v_t = \varepsilon \big(\tilde{v}_{0t0}(\tilde{x}_t) + \bar{v}_{0t}(\tilde{x}_t,\bar{x}_n) \big) , \end{gather}

(2.23b)\begin{gather} v_n= \varepsilon E^{1/6} \bar{v}_{1n}\left(\tilde{x}_t,\bar{x}_n\right), \end{gather}

(2.23c)\begin{gather} v_\phi = \varepsilon \left(\tilde{v}_{0\phi 0}\left(\tilde{x}_t\right) + \bar{v}_{0\phi}\left(\tilde{x}_t,\bar{x}_n\right) \right) , \end{gather}

(2.23d)\begin{gather} b = \varepsilon \left(\tilde{b}_{00}\left(\tilde{x}_t\right) + \bar{b}_{0}\left(\tilde{x}_t,\bar{x}_n\right)\right) , \end{gather}

(2.23e)\begin{gather} p = \varepsilon E^{1/3} \left(\tilde{p}_{00}\left(\tilde{x}_t\right) + E^{1/6} \bar{p}_1\left(\tilde{x}_t,\bar{x}_n\right) \right), \end{gather}

where $\tilde {v}_{0t0}$, $\tilde {v}_{0\phi 0}$, $\tilde {b}_{00}$ and $\tilde {p}_{00}$ are the expressions as $\tilde {x}_n$ goes to 0 of the leading-order solution in the interaction region. Only $\tilde {p}_0$ has not been provided above. Its exact expression will turn out not to be necessary for the analysis.

The governing equations reduce to

(2.24a)\begin{gather} -{\textrm{i}} \omega \bar{v}_{0t} -2 \Omega \bar{v}_{0\phi} \cos\alpha - \bar{b}_0 \sin\alpha - \displaystyle\frac{\partial ^{2}{\bar{v}_{0t}}}{\partial {\bar{x}_n}^{2}} =0 , \end{gather}

(2.24b)\begin{gather} -{\textrm{i}} \omega \bar{v}_{0\phi} +2 \Omega \bar{v}_{t 0} \cos\alpha - \displaystyle\frac{\partial ^{2}{\bar{v}_{0\phi}}}{\partial {\bar{x}_n}^{2}} = 0 , \end{gather}

(2.24c)\begin{gather} 2 \Omega \bar{v}_{0\phi} \sin\alpha - \bar{b}_0 \cos\alpha + \displaystyle\frac{\partial {\bar{p}_1}}{\partial {\bar{x}_n}} =0, \end{gather}

(2.24d)\begin{gather} -{\textrm{i}} \omega \bar{b}_{0} -2 \Omega \bar{v}_{0\phi} \cos\alpha - N^{2} \bar{v}_{0t} \sin\alpha - \frac{1}{{Pr}} \displaystyle\frac{\partial ^{2}{\bar{b}_{0}}}{\partial {\bar{x}_n}^{2}} = 0 , \end{gather}

(2.24e)\begin{gather} \displaystyle\frac{\partial {\bar{v}_{1n}}}{\partial {\bar{x}_n}} + \displaystyle\frac{\partial {\left(\tilde{v}_{0t0} +\bar{v}_{0t}\right)}}{\partial {\tilde{x}_t}} = 0 , \end{gather}

which give

(2.25)\begin{align} & \left[\left( {\textrm{i}} \omega + \displaystyle\frac{\partial ^{2}{}}{\partial {\bar{x}_n}^{2}} \right)^{2}\left( {\textrm{i}} \omega + \displaystyle\frac{1}{{Pr}}\displaystyle\frac{\partial ^{2}{}}{\partial {\bar{x}_n}^{2}} \right) + 4\Omega^{2} \cos^{2}\alpha \left( {\textrm{i}} \omega + \displaystyle\frac{1}{{Pr}}\displaystyle\frac{\partial ^{2}{}}{\partial {\bar{x}_n}^{2}} \right) \right.\nonumber\\ &\qquad \left. + N^{2} \sin^{2}\alpha \left( {\textrm{i}} \omega + \displaystyle\frac{\partial ^{2}{}}{\partial {\bar{x}_n}^{2}} \right) \right] \bar{v} _{0t} =0. \end{align}

Among the six independent solutions, three correspond to exponentially decreasing solutions of the form $\textrm {e}^{-\lambda _j \bar {x}_n}$, $j=1,2,3$ with $\mathrm {Re}(\lambda _j) >0$. The solution for $\bar {v}_{0t}$ can then be written as $\bar {v}_{0t} =\sum _{j=1}^{3} \bar {a}_j(\tilde {x}_t) \textrm {e}^{-\lambda _j\bar {x}_n}$, which gives similar expressions for $\bar {v}_{0\phi }$ and $\bar {b}_{0}$ using (2.24a), (2.24b) and (2.24d). The conditions that $v_{t}$, $v_\phi$ and $b$ vanish at $\bar {x}_n=0$ then provide the functions $\bar {a}_j$, $j=1,2,3$ as linear combinations of $\tilde {v}_{0t0}$, $\tilde {v}_{0\phi 0}$ and $\tilde {b}_{00}$. Note that if ${Pr} =\infty$, (2.25) is of fourth order. There are then two unknown functions, which are obtained by applying the boundary conditions at $\bar {x} _n=0$ on $v_{t}$ and $v_\phi$ only.

To get $\bar {v}_{1n}$, we use (2.24e) which gives

(2.26)\begin{equation} \bar{v}_{1n} = -\tilde{v}_{0t0}'(\tilde{x}_t) \bar{x}_n - \sum_{j=1}^{3} \bar{a}'_j(\tilde{x}_t) \frac{1 - \textrm{e}^{-\lambda_j \bar{x}_n}}{\lambda_j}. \end{equation}

The first term automatically matches the expression of the normal velocity in the interaction region as $\tilde {x}_n\rightarrow 0$. It is the second term that gives the viscous correction for which we are looking. As $\bar {x}_n \rightarrow \infty$, this term tends to a function $\bar {u}_{1n}^{\infty }(\tilde {x}_t)= - \sum _{j=1}^{3} \bar {a}'_j(\tilde {x}_t) /\lambda _j$ that should match a contribution in the interaction region. Using the fact that $\bar {a}_j(\tilde {x}_t)$, $j=1,2,3$ are a linear combination of $\tilde {v}_{0t0}$, $\tilde {v}_{0\phi 0}$ and $\tilde {b}_{00}$, they can be written as $C_j {\boldsymbol {h_{\mu }}}(\sin (\alpha -\theta ) \tilde {x}_t/\Lambda )$ (see (2.17) and (2.19)). It then immediately follows that

(2.27)\begin{equation} \bar{u}_{1n}^{\infty}(\tilde{x}_t)= C_{1n}^{\infty} {\boldsymbol{h_{\mu +1}}}( \sin(\alpha-\theta) \tilde{x}_t/\Lambda), \end{equation}

where $C_{1n}^{\infty }$ is a complex constant. Such a normal flow can be matched in the interaction region to a solution of the same form as the reflected beam (2.11) by changing $\mu$ into $\mu +1$. More precisely, if we have (2.27), the viscous correction to the reflected beam would read in the interaction region as

(2.28)\begin{equation} \left( \begin{array}{@{}c@{}} v_{1r}^{(r)} \\ v_{1\phi}^{(r)} \\ v_{1z}^{(r)} \\ b_1^{(r)} \end{array} \right) = \varepsilon E^{1/6} C_{1n}^{\infty} \left( \begin{array}{@{}c@{}} \cos\theta \\ (2{\textrm{i}} \Omega / \omega) \cos\theta \\ \sin\theta \\ ({\textrm{i}}N^{2} /\omega) \sin\theta \end{array} \right) \tilde{h}_{\mu +1}^{(r)} , \end{equation}

where $\tilde {h}_{\mu }^{(r)}$ has been defined in (2.19b). It is worth mentioning that this $O(E^{1/6})$ viscous correction is larger than the first viscous correction to the similarity solution describing the beam, which is $O(E^{1/3})$.

A viscous correction to the reflected beam was calculated by Beckebanze etal. (Reference Beckebanze, Brouzet, Sibgatullin and Maas2018) in the purely stratified case. By comparing their expression (6.9) with ours, one can see that both expressions are similar. They obtained that the spectral density of the viscous correction is $k\hat {U}(k)$ if the spectral density of the incident beam is $\hat {U}(k)$ (our function $G(k)$ in (2.10)): this corresponds to the passage from ${\boldsymbol {h_{\mu }}}$ to ${\boldsymbol {h_{\mu +1}}}$ in our analysis.

2.4. Reynolds stress

In this section, our objective is to estimate the nonlinear terms $\boldsymbol {v} \boldsymbol{\cdot} \boldsymbol {\nabla } \boldsymbol {v}$ and $\boldsymbol {v} \boldsymbol{\cdot} \boldsymbol {\nabla } b$, or Reynolds stress, that appear in the governing equations (2.1a)–(2.1c). Owing to the particular form of the similarity solution, it is straightforward to show that these terms are at most $O(\varepsilon ^{2})$ everywhere except close to the source, the rotation axis and the reflection point. As already mentioned, our interest is in a reflection point distinct from the beam source and the rotation axis.

In the interaction region, the two beams interact with each other, making the Reynolds stress of order $\varepsilon ^{2} E^{-1/3}$. Simple expressions for the Reynolds stress can be obtained using (2.17a)–(2.17d) and (2.28). As expected, the Reynolds stress is composed of a steady and a second-harmonic part:

(2.29)\begin{equation} \boldsymbol{\mathcal{N}} = \boldsymbol{\mathcal{N}}^{(0)}+ \left(\boldsymbol{\mathcal{N}}^{(2)}\textrm{e}^{-2{\textrm{i}} \omega t} +\hbox{c.c.}\right), \end{equation}

where

(2.30a,b)\begin{equation} \boldsymbol{\mathcal{N}}^{(0)} = \boldsymbol{v} \boldsymbol{\cdot} \boldsymbol{\nabla} \left( \begin{array}{@{}c@{}} v_r ^{*}\\ v_\phi^{*} \\ v_z ^{*}\\ b^{*} \end{array} \right) +\hbox{c.c.};\quad \boldsymbol{\mathcal{N}}^{(2)} = \boldsymbol{v}\boldsymbol{\cdot} \boldsymbol{\nabla} \left( \begin{array}{c} v_r \\ v_\phi \\ v_z \\ b \end{array} \right), \end{equation}

the superscript $^{*}$ denoting the complex conjugate. The expressions of $\boldsymbol {\mathcal {N}}^{(0)}= \varepsilon ^{2}E^{-1/3}(\boldsymbol {\mathcal {N}}_0^{(0)} + E^{1/6}\boldsymbol {\mathcal {N}}_1^{(0)} + \cdots )$ and $\boldsymbol {\mathcal {N}}^{(2)} = \varepsilon ^{2}E^{-1/3}\boldsymbol {\mathcal {N}}_0^{(2)} + \cdots$ are given in appendix B.

2.5. Notation

We have introduced a number of notations that we shall use in the rest of the paper.

Whereas the harmonic solution is denoted without a superscript, the second-harmonic correction as well as the amplitude of the terms oscillating at a frequency $2\omega$, such as the second-harmonic part of the Reynolds stress $\boldsymbol {\mathcal {N}}^{(2)}$, will be denoted with the superscript ${(2)}$. The meanflow correction and the steady terms such as $\boldsymbol {\mathcal {N}}^{(0)}$ will always be indicated by the superscript ${(0)}$.

The subscripts will be of three types. The letters and special characters $\perp$ and $\parallel$ indicate the system of coordinates used for the velocity components. Three different systems of coordinates are used: $(r,\phi ,z)$ for the global cylindrical coordinate system, $(x_t,\phi ,x_n)$ for the boundary coordinate system and $(x_\parallel , \phi , x_\perp )$ for the beam-associated coordinate system (see figure 2a and appendix A for the relations between the coordinate systems). In this last case, the superscripts ${(i)}$, ${(r)}$, ${(2)}$ or ${(0)}$ indicate whether it corresponds to the incident, reflected, second-harmonic or steady beam. For the Reynolds stress, the subscript $b$ indicates the fourth component of the vector $\boldsymbol {\mathcal {N}}$.

The first subscript number is associated with the order of the expansion. The subscript $0$ always indicates the leading-order expression; larger numbers correspond to higher-order terms ordered in decreasing amplitude order. A second subscript number $0$ or $\infty$ will exceptionally be used to indicate the value taken as one of the variables goes to zero or infinity.

The symbols placed above the characters correspond to the regions where the solution is considered:

1. (i) tilde ($\tilde {\ \ }$) for the interaction region;

2. (ii) overbar ($\bar {\ \ }$) for the viscous boundary layer;

3. (iii) check ($\check {\ \ }$) for the $E^{4/9}$ viscous layer; and

4. (iv) without symbol for the outer region, that is, in a beam region far from the interaction region.

For the spatial variables, the symbol will imply a particular rescaling with respect to the global outer variable, by $E^{1/3}$ in the interaction region (i.e. $\tilde {x}_n= x_n/E^{1/3}$, $\tilde {r}=(r-r_c)/E^{1/3}$, etc.), by $E^{1/2}$ in the viscous boundary layer for the normal variable ($\bar {x} = x_n/E^{1/2}$), by $E^{4/9}$ in the $E^{4/9}$ viscous layer for the normal variable ($\check {x}_n=x_n/E^{1/4}=\check {r} = (r-r_c)/E^{4/9}$ for a vertical boundary).

4.1. Generic configuration: rotating and stratified fluid ($N\neq 0$, $\Omega \neq 0$)

In this section, we consider the generic configuration where neither $N$ nor $\Omega$ vanishes. The main contribution is obtained in the interaction region. In this region, the meanflow correction is found to be given by the following ansatz:

(4.1a)\begin{gather} v_r^{(0)} = \varepsilon^{2} E^{-1/3} \left( \tilde{v}_{0r}^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/6} \tilde{v}_{1r}^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/3} \tilde{v}_{2r}^{(0)} \left(\tilde{r},\tilde{z}\right)\right) , \end{gather}

(4.1b)\begin{gather} v_\phi^{(0)} = \varepsilon^{2} E^{-1/3} \left(\tilde{v}_{0\phi}^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/6} \tilde{v}_{1\phi}^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/3} \tilde{v}_{2\phi}^{(0)} \left(\tilde{r},\tilde{z}\right)\right) , \end{gather}

(4.1c)\begin{gather} v_z^{(0)} = \varepsilon^{2} E^{-1/3} \left(\tilde{v}_{0z}^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/6} \tilde{v}_{1z}^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/3} \tilde{v}_{2z}^{(0)} \left(\tilde{r},\tilde{z}\right)\right) , \end{gather}

(4.1d)\begin{gather} b^{(0)} = \varepsilon^{2} E^{-1/3} \left(\tilde{b}_0^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/6} \tilde{b}_{1}^{(0)} \left(\tilde{r},\tilde{z}\right) + E^{1/3} \tilde{b}_{2}^{(0)} \left(\tilde{r},\tilde{z}\right)\right), \end{gather}

(4.1e)\begin{gather} p^{(0)} = \varepsilon^{2} \tilde{p}_0^{(0)} (\tilde{r},\tilde{z}) , \end{gather}

where $\tilde {r}$ and $\tilde {z}$ are the local variables. Inserting (4.1) in (2.1) using expression (B 2) for the Reynolds stress, the meanflow amplitudes are found to satisfy the equations

(4.2a)\begin{gather} -2 \Omega \tilde{v}_{0\phi}^{(0)} + \displaystyle\frac{\partial {\tilde{p}_0^{(0)}}}{\partial {\tilde{r}}} = -\tilde{{\mathcal{N}}}_{0r}^{(0)} , \end{gather}

(4.2b)\begin{gather} 2 \Omega \tilde{v}_{0r}^{(0)} = -\tilde{{\mathcal{N}}}_{0\phi}^{(0)} , \end{gather}

(4.2c)\begin{gather} \tilde{b}_0^{(0)} + \displaystyle\frac{\partial {\tilde{p}_0^{(0)}}}{\partial {\tilde{z}}} = -\tilde{{\mathcal{N}}}_{0z}^{(0)}, \end{gather}

(4.2d)\begin{gather} -N^{2} \tilde{v}_{0z}^{(0)} = -\tilde{{\mathcal{N}}}_{0b}^{(0)} , \end{gather}

(4.2e)\begin{gather} \displaystyle\frac{\partial {\tilde{v}_{0r}^{(0)}}}{\partial {\tilde{r}}} + \displaystyle\frac{\partial {\tilde{v}_{0z}^{(0)}}}{\partial {\tilde{z}}} =0. \end{gather}

Equations (4.2b) and (4.2d) readily give $\tilde {v}_{0r}^{(0)}$ and $\tilde {v}_{0z}^{(0)}$. These expressions are compatible with (4.2e) which is automatically satisfied because of (B 4a). They give normal and tangential velocities as

(4.3a)\begin{gather} \tilde{v}_{0n}^{(0)} = \sin \alpha \frac{\tilde{{\mathcal{N}}}_{0\phi}^{(0)}}{2\Omega} + \cos\alpha \frac{\tilde{{\mathcal{N}}}_{0b}^{(0)}}{N^{2}} = -2\sin 2\theta \displaystyle\frac{\partial {}}{\partial {\tilde{x}_t}}{\rm Im} \left(\tilde{h}_{\mu}^{(r)} \tilde{h}_{\mu}^{(i)*}\right), \end{gather}

(4.3b)\begin{gather} \tilde{v}_{0t}^{(0)} = - \cos \alpha \frac{\tilde{{\mathcal{N}}}_{0\phi}^{(0)}}{2\Omega} + \sin\alpha \frac{\tilde{{\mathcal{N}}}_{0b}^{(0)}}{N^{2}} = -2\sin 2\theta \displaystyle\frac{\partial {}}{\partial {\tilde{x}_n}}{\rm Im} \left(\tilde{h}_{\mu}^{(r)} \tilde{h}_{\mu}^{(i)*}\right). \end{gather}

Using (2.19), it is immediately seen that $\tilde {v}_{0n}^{(0)} (\tilde {x}_n =0) = 0$. The condition of non-penetrability is therefore automatically satisfied at leading order. The first-order corrections $\tilde {v}_{1r}^{(0)}$ and $\tilde {v}_{1z}^{(0)}$ are also directly obtained from the governing equations at the next order:

(4.4a,b)\begin{equation} 2\Omega \tilde{v}_{1r}^{(0)} = -\tilde{{\mathcal{N}}}_{1\phi}^{(0)} ,\quad -N^{2} \tilde{v}_{1z}^{(0)}= -\tilde{{\mathcal{N}}}_{1b}^{(0)} , \end{equation}

which gives

(4.5a)\begin{gather} \tilde{v}_{1n} ^{(0)} = -2 \sin 2\theta \displaystyle\frac{\partial {}}{\partial {\tilde{x}_t}}{\rm Im} \left(C_{1n}^{\infty}\tilde{h}_{\mu +1}^{(r)} \tilde{h}_{\mu}^{(i)*}\right) , \end{gather}

(4.5b)\begin{gather} \tilde{v}_{1t} ^{(0)} = -2 \sin 2\theta \displaystyle\frac{\partial {}}{\partial {\tilde{x}_n}}{\rm Im} \left(C_{1n}^{\infty}\tilde{h}_{\mu +1}^{(r)} \tilde{h}_{\mu}^{(i)*}\right). \end{gather}

In contrast to $\tilde {v}_{0n}^{(0)}$, $\tilde {v}_{1n}^{(0)}$ does not cancel as $\tilde {x}_n\rightarrow 0$.

Equations (4.2a) and (4.2c) are not sufficient to determine the three other amplitudes $\tilde {v}_{0\phi }^{(0)}$, $\tilde {b}_0^{(0)}$ and $\tilde {p}_0^{(0)}$. For this purpose, one should consider the governing equations at the third order:

(4.6a)\begin{gather} 2 \Omega \tilde{v}_{2r}^{(0)} - \bar{{\nabla}}^{2} \tilde{v}_{0\phi}^{(0)} = -\tilde{{\mathcal{N}}}_{2\phi}^{(0)} , \end{gather}

(4.6b)\begin{gather} -N^{2} \tilde{v}_{2z}^{(0)} - \frac{1}{{Pr}}\bar{{\nabla}}^{2} \tilde{b}_0^{(0)} = -\tilde{{\mathcal{N}}}_{2b}^{(0)} , \end{gather}

(4.6c)\begin{gather} \displaystyle\frac{\partial {\tilde{v}_{2r}^{(0)}}}{\partial {\tilde{r}}} + \displaystyle\frac{\partial {\tilde{v}_{2z}^{(0)}}}{\partial {\tilde{z}}} + \frac{\tilde{v}_{0r}^{(0)}}{r_c} =0, \end{gather}

where $\tilde {{\mathcal {N}}}_{2\phi }^{(0)}$ and $\tilde {{\mathcal {N}}}_{2b}^{(0)}$ are second-order corrections to the steady Reynolds stress. This system gives the missing equation by plugging the expressions of $\tilde {v}_{2r}^{(0)}$ and $\tilde {v}_{2z}^{(0)}$ obtained from the first two equations into the third one. If we then replace $\tilde {v}_{0\phi }^{(0)}$ and $\tilde {b}_0^{(0)}$ by their expression in terms of $\tilde {p}_0^{(0)}$ using (4.2a) and (4.2c), we obtain a non-homogeneous fourth-order equation for $\tilde {p}_0^{(0)}$ of the form

(4.7)\begin{equation} \left(\frac{1}{4\Omega^{2}} \displaystyle\frac{\partial ^{2}{}}{\partial {\tilde{r}}^{2}} + \frac{1}{N^{2} {Pr}} \displaystyle\frac{\partial ^{2}{}}{\partial {\tilde{z}}^{2}} \right) \left( \displaystyle\frac{\partial ^{2}{}}{\partial {\tilde{r}}^{2}} + \displaystyle\frac{\partial ^{2}{}}{\partial {\tilde{z}}^{2}} \right) \tilde{p}_0 ^{(0)} = \tilde{{\mathcal{M}}}(\tilde{r},\tilde{z}). \end{equation}

The function $\tilde {{\mathcal {M}}}$ can be expressed in terms of the components of $\boldsymbol {\tilde {{\mathcal {N}}}_0^{(0)}}$, $\boldsymbol {\tilde {{\mathcal {N}}}_1^{(0)}}$ and $\boldsymbol {\tilde {{\mathcal {N}}}_2^{(0)}}$. We shall not provide its expression. For the present analysis, it is only important to know that ${\tilde {{\mathcal {M}}}}$ is a function localized in the interaction region (i.e. $|\tilde {{\mathcal {M}}}| \rightarrow 0$ exponentially fast as $\tilde {r}^{2}+\tilde {z}^{2}\rightarrow \infty$). Solutions to (4.7) localized in the interaction region therefore exist. The solution is fully determined upon prescribing $\partial _{\tilde {r}} \tilde {p}_0^{(0)}$ and $\partial _{\tilde {z}} \tilde {p}_0^{(0)}$ at the boundary $\tilde {x}_n=0$. As shown in appendix C, this last step can be made by applying a condition of matching with the solution in the viscous boundary layer.

To summarize this section, we have seen that the radial and axial components are directly forced by the azimuthal and buoyancy component of the Reynolds stress which is of order $\varepsilon ^{2} E^{-1/3}$ in the interaction region. The azimuthal velocity and the buoyancy turn out to be also $O(\varepsilon ^{2} E^{-1/3})$ but they depend on higher corrections of the Reynolds stress and also on the solution in the viscous boundary layer. Velocity components and buoyancy are all found to be localized in the interaction region. This property is also verified by the first-order meanflow correction. This guarantees that no meanflow correction is generated outside the interaction region (at least up to $O(\varepsilon ^{2})$ terms).

As we shall see in the next sections, a different conclusion is reached when either $N$ or $\Omega$ cancels.

4.2. Non-stratified case ($N=0$)

In the non-stratified case, the flow is characterized by its velocity and pressure field only and we have $\omega =2\Omega \cos \theta$ and $L_d^{2}= \nu / (2 \Omega \sin \theta )$. The meanflow correction depends on whether the boundary is vertical or not.

4.2.1. Non-vertical boundary ($\alpha \neq {\rm \pi} /2$)

When $\alpha \neq {\rm \pi} /2$, in the interaction region, the meanflow correction is still given by an ansatz of the form (4.1) with $N=0$. It follows that (4.2) applies. The previous expressions for $\tilde {v}_{0r}^{(0)}$ and $\tilde {v}_{0z}^{(0)}$ are still solutions of the problem, and are such that $\tilde {v}_{0n}^{(0)}$ vanishes on the boundary. However, the pressure $\tilde {p}_0^{(0)}$ and the azimuthal velocity $\tilde {v}_{0\phi }^{(0)}$ are now directly obtained from (4.2a) and (4.2c) because $\tilde {b}^{(0)}=0$. We then obtain from these equations

(4.8a)\begin{gather} \tilde{p}_0^{(0)} = -4 \sin^{2}\theta{\rm Re} \left(\tilde{h}_{\mu}^{(r)}\tilde{h}_{\mu}^{(i)*}\right) + F(\tilde{r}) , \end{gather}

(4.8b)\begin{gather} \tilde{v}_{0\phi}^{(0)} = -\frac{2}{\Omega} \displaystyle\frac{\partial { }}{\partial {\tilde{r}}}{\rm Re} \left(\tilde{h}_{\mu}^{(r)}\tilde{h}_{\mu}^{(i)*}\right) +\frac{1}{2\Omega} \displaystyle\frac{\partial {F}}{\partial {\tilde{r}}} , \end{gather}

where $F(\tilde {r})$ is an unknown function. The function $F$ is obtained by considering the outer region $|z-z_c|=O(1), |r-r_c|=O(E^{1/3})$ where a meanflow could a priori be created. In this region, the Reynolds stress is small and we expect the solution to be a homogeneous solution of the governing equations. Such a solution is given by an ansatz of the form $(E^{1/3}u^{(0)},v^{(0)},w^{(0)},E^{1/3}p^{(0)})$ and satisfies the homogeneous equations

(4.9a)\begin{gather} -2\Omega v^{(0)} +\displaystyle\frac{\partial {p^{(0)}}}{\partial {\tilde{r}}} =0 , \end{gather}

(4.9b)\begin{gather} 2\Omega u^{(0)} - \displaystyle\frac{\partial ^{2}{v^{(0)}}}{\partial {\tilde{r}}^{2}} = 0 , \end{gather}

(4.9c)\begin{gather} \displaystyle\frac{\partial {p^{(0)}}}{\partial {z}} - \displaystyle\frac{\partial ^{2}{w^{(0)}}}{\partial {\tilde{r}}^{2}} = 0 , \end{gather}

(4.9d)\begin{gather} \displaystyle\frac{\partial {u^{(0)}}}{\partial {\tilde{r}}} + \displaystyle\frac{\partial {w^{(0)}}}{\partial {z}} =0. \end{gather}

This system leads to the following equation for $p^{(0)}$:

(4.10)\begin{equation} 4\Omega^{2} \displaystyle\frac{\partial ^{2}{p^{(0)}}}{\partial {z}^{2}} + \frac{\partial^{6} p^{(0)}}{\partial \tilde{r}^{6}} =0. \end{equation}

Using Fourier transform in $\tilde {r}$, and requiring the solution to vanish far from the boundary, that is as $z \rightarrow - \infty$ assuming $\alpha >{\rm \pi} /2$, we get for $p^{(0)}$

(4.11)\begin{equation} p^{(0)} = \int_{-\infty}^{\infty} {\mathcal{P}}(k) \exp\left({|k|^{3} (z-z_c)/(2\Omega) }\right) \textrm{e}^{{\textrm{i}}k\tilde{r}} \, \textrm{d}k. \end{equation}

The other components can all be expressed in terms of ${\mathcal {P}}(k)$:

(4.12a)\begin{gather} u^{(0)}= \int_{-\infty}^{\infty} {\textrm{i}}k^{3}{\mathcal{P}}(k)/(2\Omega) \exp\left({|k|^{3} (z-z_c)/(2\Omega) }\right) \textrm{e}^{{\textrm{i}}k\tilde{r}} \, \textrm{d}k , \end{gather}

(4.12b)\begin{gather} v^{(0)} = -\int_{-\infty}^{\infty} {\textrm{i}}k{\mathcal{P}}(k)/(2\Omega) \exp\left({|k|^{3} (z-z_c)/(2\Omega) }\right) \textrm{e}^{{\textrm{i}}k\tilde{r}} \, \textrm{d}k , \end{gather}

(4.12c)\begin{gather} w^{(0)} = \int_{-\infty}^{\infty} |k| {\mathcal{P}}(k)/(2\Omega) \exp\left({|k|^{3} (z-z_c)/(2\Omega) }\right)\textrm{e}^{{\textrm{i}}k\tilde{r}} \, \textrm{d}k. \end{gather}

As these expressions should match the forced solution in the interaction region, it is immediate that $F(\tilde {r}) =0$ and that the forced solution does not excite any outer meanflow at the order $\varepsilon ^{2} E^{-1/3}$. The meanflow in this region is created at the next order and is such that ${\mathcal {P}}(k) = \varepsilon ^{2} E^{-1/6} {\mathcal {P}}_1(k)$. To obtain ${\mathcal {P}}_1(k)$, we must consider the solution in the viscous boundary layer and the forced solution in the interaction region at the next order. The viscous boundary layer solution is given in appendix C Equations (C 1a), (C 1b), (C 1c), (C 5), (C 7) and (C 10) together with (C 6) still apply when $N=0$. However, these expressions do not fully define the solution. A second equation for the functions $a_t$ and $b_t$ is missing. Contrarily to the case $N\neq 0$, this equation is obtained by the condition of matching of the azimuthal velocity component. In the viscous boundary layer, the azimuthal velocity can be written as

(4.13)\begin{align} & \bar{v}_{0\phi}^{(0)} = {\displaystyle \int_0^{\bar{x}_n} \ \int_{+\infty}^{s} \left(2\Omega \cos\alpha \, \bar{u}_t^{(0)}(\tilde{x}_t,x) + \bar{{\mathcal{N}}}_{0\phi}^{(0)}(\tilde{x}_t, x )\right) \,\textrm{d}x\, \textrm{d}s}\nonumber\\ &\qquad + \displaystyle\frac{a_t(1 -\exp\left({ (-1)^{3/4} \lambda \bar{x}_n})\right) }{ {\textrm{i}} \lambda^{2}} -\displaystyle\frac{b_t(1 -\exp\left({ (-1)^{5/4} \lambda \bar{x}_n}) \right) }{ {\textrm{i}} \lambda^{2}}. \end{align}

In the interaction region, this component is given by (4.1b) and (4.8b) with $F=0$. The condition of matching then reads

(4.14)\begin{align} -\frac{2}{\Omega} \displaystyle\frac{\partial {}}{\partial {\tilde{r}}} \mathrm{Re}\left(\tilde{h}_{\mu}^{(r)}\tilde{h}_{\mu}^{(i)*}\right)\bigg|_{_{\tilde{x}_n=0}} = {\displaystyle \int_0^{+\infty} \ \int_{+\infty}^{s} \left(2\Omega \cos\alpha \, \bar{u}_t^{(0)} + \bar{{\mathcal{N}}}_{0\phi}^{(0)}\right)\,\textrm{d}x \,\textrm{d}s } -\frac{{\textrm{i}}}{\lambda^{2}}(a_t- b_t). \end{align}

This is the second equation that fully determines the solution in the viscous boundary layer. The weak meanflow that is created in the outer region is obtained by matching the normal velocity. The two first orders of the solution in the interaction region are needed to perform such a matching. By collecting the forced solution at the first and second order given by (4.3a) and (4.5a), respectively, and an $O(\varepsilon ^{2} E^{-1/6})$ homogeneous solution of the form (4.12) that gives a normal velocity component in the outer region

(4.15)\begin{equation} v_n^{(0)} = \varepsilon^{2} E^{-1/6} \frac{\cos\alpha}{2\Omega} \int_{-\infty}^{+\infty} |k| {\mathcal{P}}_1(k)\exp\left({|k|^{3} (z-z_c)/(2\Omega) }\right)\textrm{e}^{{\textrm{i}}k\tilde{r}} \, \textrm{d}k , \end{equation}

we obtain, by matching normal velocities, the condition

(4.16)\begin{equation} - \int_{0}^{\infty} \partial_{\tilde{x}_t}\bar{u}_t^{(0)}\,\textrm{d}\bar{x}_n + \frac{a'_t -{\textrm{i}}b'_t}{ (-1)^{3/4} \lambda} = \tilde{v}_{1n}^{(0)}|_{_{\tilde{x}_n=0}}+ \frac{\cos\alpha}{2\Omega} \int_{-\infty}^{+\infty} |k| {\mathcal{P}}_1(k)\textrm{e}^{{\textrm{i}}k\tilde{r}} \, \textrm{d}k. \end{equation}

This condition gives the function ${\mathcal {P}}_1(k)$ that defines the homogeneous solution in the outer region. In the outer region, we therefore expect an axial velocity and an azimuthal velocity of order $\varepsilon ^{2}E^{-1/6}$. The different regions for this case are summarized in figure 4$(b)$.

4.2.2. Vertical boundary ($\alpha ={\rm \pi} /2$)

This case requires a special treatment because vertical boundaries correspond to critical boundaries for meanflow corrections in a non-stratified fluid. In the interaction region, the forced solution is unchanged and is still given by (4.1a)–(4.1d) with $N=0$ and

(4.17a)\begin{gather} \tilde{v}_{0r}^{(0)} = 2\sin 2\theta \displaystyle\frac{\partial {}}{\partial {\tilde{z}}}{\rm Im} \left(\tilde{h}_{\mu}^{(r)} \tilde{h}_{\mu}^{(i)*}\right) , \end{gather}

(4.17b)\begin{gather} \tilde{v}_{0\phi}^{(0)} = -\frac{2}{\Omega} \displaystyle\frac{\partial { }}{\partial {\tilde{r}}}{\rm Re} \left(\tilde{h}_{\mu}^{(r)}\tilde{h}_{\mu}^{(i)*}\right) , \end{gather}

(4.17c)\begin{gather} \tilde{v}_{0z}^{(0)} = 2\sin 2\theta \displaystyle\frac{\partial {}}{\partial {\tilde{r}}}{\rm Im} \left(\tilde{h}_{\mu}^{(r)} \tilde{h}_{\mu}^{(i)*}\right) , \end{gather}

(4.17d)\begin{gather} \tilde{p}_0^{(0)} = -4 \sin^{2}\theta{\rm Re} \left(\tilde{h}_{\mu}^{(r)}\tilde{h}_{\mu}^{(i)*}\right). \end{gather}

The difference appears at higher order and close to the boundary owing to a change of nature of the viscous boundary layer solution (e.g. $\lambda =0$ in (C 5) when $\alpha ={\rm \pi} /2$). A larger viscous layer of width $O(E^{4/9})$ appears close to the boundary. To find the solution in this layer, we introduce the new boundary layer variable $\check {r}= (r-r_c)/E^{4/9}$ and the ansatz

(4.18a)\begin{gather} v_r^{(0)} = \varepsilon^{2} E^{-2/9} \left(\partial_{\tilde{r}}\tilde{v}_{0r}^{(0)}|_{_{\tilde{r}=0}} \,\check{r} + \check{v}_{0r}^{(0)} (\check{r},\tilde{z}) \right), \end{gather}

(4.18b)\begin{gather} v_\phi^{(0)} = \varepsilon^{2} E^{-1/3} \left( \tilde{v}_{0\phi}^{(0)}|_{_{\tilde{r}=0}} \,, + \check{v}_{0\phi}^{(0)} (\check{r},\tilde{z}) \right) , \end{gather}

(4.18c)\begin{gather} v_z^{(0)} = \varepsilon^{2} E^{-1/3} \left(\tilde{v}_{0z}^{(0)}|_{_{\tilde{r}=0}} \,,+ \check{v}_{0z}^{(0)} (\check{r},\tilde{z}) \right) , \end{gather}

(4.18d)\begin{gather} p^{(0)} = \varepsilon^{2} \left(\tilde{p}_0^{(0)}|_{_{\tilde{r}=0}} \, + E^{1/9}( \partial_{\tilde{r}}\tilde{p}_0^{(0)}|_{_{\tilde{r}=0}} \, \check{r} + \check{p}_1^{(0)}(\check{r},\tilde{z}) \right), \end{gather}

where the functions $\check {v}_{0r}^{(0)}$, $\check {v}_{0\phi }^{(0)}$, $\check {v}_{0z}^{(0)}$ and $\check {p}_{1}^{(0)}$ vanish as $\check {r} \rightarrow -\infty$ such that the matching with the interaction region is guaranteed. We obtain the system

(4.19a)\begin{gather} -2 \Omega \check{v}_{0\phi}^{(0)} + \displaystyle\frac{\partial {\check{p}_{1}^{(0)}}}{\partial {\check{r}}} =0 , \end{gather}

(4.19b)\begin{gather} 2 \Omega \check{v}_{0r}^{(0)} - \displaystyle\frac{\partial ^{2}{\check{v}_{0\phi}^{(0)}}}{\partial {\check{r}}^{2}} = 0 , \end{gather}

(4.19c)\begin{gather} \displaystyle\frac{\partial {\check{p}_{1}^{(0)}}}{\partial {\tilde{z}}} - \displaystyle\frac{\partial ^{2}{\check{v}_{0z}^{(0)}}}{\partial {\check{r}}^{2}} = 0 , \end{gather}

(4.19d)\begin{gather} \displaystyle\frac{\partial {\check{v}_{0r}^{(0)}}}{\partial {\check{r}}} + \displaystyle\frac{\partial {\check{v}_{0z}^{(0)}}}{\partial {\tilde{z}}} =0 , \end{gather}

which gives

(4.20)\begin{equation} \frac{\partial^{6} \check{p}_{1}^{(0)} }{\partial \check{r}^{6}} + 4\Omega^{2} \displaystyle\frac{\partial ^{2}{\check{p}_{1}^{(0)}}}{\partial {\tilde{z}}^{2}} = 0. \end{equation}

Note that this equation is similar to (4.10) in the outer region. When $\alpha ={\rm \pi} /2$, the $E^{4/9}$ viscous layer thus plays the same role as the outer region. Here, because the boundary corresponds to $\check {r}=0$, it is convenient to integrate this equation using Fourier transform in $\tilde {z}$. It leads to

(4.21)\begin{equation} \check{p}_{1} ^{(0)} = \int_{-\infty}^{+\infty}\left( a_1(k)\textrm{e}^{-\alpha_1\check{r}} + a_2(k)\textrm{e}^{-\alpha_2\check{r} } + a_3(k)\textrm{e}^{-\alpha_3\check{r}}\right)\textrm{e}^{{\textrm{i}}k \tilde{z} } \,\textrm{d}k , \end{equation}

where $\alpha _j$, $j=1,2,3$, are the three roots of $\alpha _j^{6} = 4 \Omega ^{2} k^{2}$ with a negative real part (such that $\check {p}_{1}^{(0)}$ does vanish as $\check {r} \rightarrow -\infty$). The other components are then given by

(4.22a)\begin{gather} \check{v}_{0r}^{(0)}= -\int_{-\infty}^{+\infty}\left(\sum_{j=1}^{3} \frac{\alpha_j^{3}a_j(k)}{4\Omega^{2}} \textrm{e}^{-\alpha_j\check{r}}\right) \textrm{e}^{{\textrm{i}}k \tilde{z} } \,\textrm{d}k , \end{gather}

(4.22b)\begin{gather} \check{v}_{0\phi}^{(0)}= -\int_{-\infty}^{+\infty}\left(\sum_{j=1}^{3} \frac{\alpha_ja_j(k)}{2\Omega} \textrm{e}^{-\alpha_j\check{r}}\right)\textrm{e}^{{\textrm{i}}k \tilde{z} } \,\textrm{d}k , \end{gather}

(4.22c)\begin{gather} \check{v}_{0z}^{(0)}= \int_{-\infty}^{+\infty}\left(\sum_{j=1}^{3} \frac{{\textrm{i}}k a_j(k)}{ \alpha_j^{2}} \textrm{e}^{-\alpha_j\check{r}}\right)\textrm{e}^{{\textrm{i}}k \tilde{z} } \,\textrm{d}k. \end{gather}

The functions $a_j$, $j=1,2,3$ are obtained by matching this solution with the solution in the viscous boundary layer. In the viscous boundary layer, the solution expands, using $\bar {r}= \tilde {r}/E^{1/6} = \check {r}/E^{1/18}$, as

(4.23a)\begin{gather} v_r^{(0)}= \varepsilon^{2} E^{-1/6} \bar{v}_{0r} ^{(0)}(\bar{r},\tilde{z}) , \end{gather}

(4.23b)\begin{gather} v_\phi^{(0)}= \varepsilon^{2} E^{-1/3} \bar{v}_{0\phi} ^{(0)}(\bar{r},\tilde{z}) , \end{gather}

(4.23c)\begin{gather} v_z^{(0)}= \varepsilon^{2} E^{-1/3} \bar{v}_{0z} ^{(0)}(\bar{r},\tilde{z}) , \end{gather}

(4.23d)\begin{gather} p^{(0)}= \varepsilon^{2} \left(\bar{p}_{0} ^{(0)}(\tilde{z}) + E^{1/6} \bar{p}_{1}^{(0)}(\bar{r},\tilde{z})\right) , \end{gather}

and satisfies

(4.24a)\begin{gather} -2\Omega \bar{v}_{0\phi}^{(0)} + \displaystyle\frac{\partial {\bar{p}_{1}^{(0)}}}{\partial {\bar{r}}} =0, \end{gather}

(4.24b)\begin{gather} - \displaystyle\frac{\partial ^{2}{\bar{v}_{0z}^{(0)}}}{\partial {\bar{r}}^{2}} + \displaystyle\frac{\partial {\bar{p}_{0}^{(0)}}}{\partial {\tilde{z}}} = - \tilde{{\mathcal{N}}}_{0z}^{(0)}|_{_{\tilde{r}=0}} - \bar{{\mathcal{N}}}_{0z}^{(0)} , \end{gather}

(4.24c)\begin{gather} -\displaystyle\frac{\partial ^{2}{\bar{v}_{0\phi}^{(0)}}}{\partial {\bar{r}}^{2}} = -\bar{{\mathcal{N}}}_{0\phi}^{(0)}, \end{gather}

(4.24d)\begin{gather} \displaystyle\frac{\partial {\bar{v}_{0r}^{(0)}}}{\partial {\bar{r}}} + \displaystyle\frac{\partial {\bar{v}_{0z}^{(0)}}}{\partial {\tilde{z}}} = 0. \end{gather}

It follows, using $\bar {p}_0^{(0)} = \tilde {p}_0^{(0)}|_{_{\tilde {r}=0}}$ and the no-slip condition on the boundary, that

(4.25a)\begin{gather} \bar{v} _{0\phi}^{(0)} = \int_0^{\bar{r}}\!\!\int_{+\infty}^{s} \bar{{\mathcal{N}}}_{0\phi}^{(0)}(x,\tilde{z})\,\textrm{d}x\,\textrm{d}s , \end{gather}

(4.25b)\begin{gather} \bar{v} _{0z}^{(0)} =\int_0^{\bar{r}}\!\!\int_{+\infty}^{s} \bar{{\mathcal{N}}}_{0z}^{(0)}(x,\tilde{z})\,\textrm{d}x\,\textrm{d}s , \end{gather}

which then gives

(4.26)\begin{equation} \bar{v} _{0r}^{(0)} = - \int_0^{\bar{r}} \partial_{\tilde{z}} \bar{v} _{0z}^{(0)}(x,\tilde{z})\,\textrm{d}x. \end{equation}

The matching of the three velocity components between the viscous boundary layer and the $E^{4/9}$ viscous layer gives

(4.27a)\begin{gather} -\int_{-\infty}^{+\infty}\left(\sum_{j=1}^{3} \frac{\alpha_j^{3}a_j(k)}{4\Omega^{2}} \right) \textrm{e}^{{\textrm{i}}k \tilde{z} } \,\textrm{d}k =0 , \end{gather}

(4.27b)\begin{gather} -\int_{-\infty}^{+\infty}\left(\sum_{j=1}^{3} \frac{\alpha_ja_j(k)}{2\Omega} \right) \textrm{e}^{{\textrm{i}}k \tilde{z} } \,\textrm{d}k + \tilde{v}_{0\phi}^{(0)}|_{_{\tilde{r}=0}} = \int_0^{\infty}\ \int_{+\infty}^{s} \bar{{\mathcal{N}}}_{0\phi}^{(0)}(x,\tilde{z})\,\textrm{d}x\,\textrm{d}s , \end{gather}

(4.27c)\begin{gather} \int_{-\infty}^{+\infty}\left(\sum_{j=1}^{3} \frac{{\textrm{i}}k a_j(k)}{ \alpha_j^{2}} \textrm{e}^{-\alpha_j\check{r}}\right)\textrm{e}^{{\textrm{i}}k \tilde{z} } \,\textrm{d}k + \tilde{v}_{0z}^{(0)}|_{_{\tilde{r}=0}} = \int_0^{\infty}\ \int_{+\infty}^{s} \bar{{\mathcal{N}}}_{0z}^{(0)}(x,\tilde{z})\,\textrm{d}x\,\textrm{d}s. \end{gather}

These three equations permit one to obtain the functions $a_j$, $j=1,2,3$ and therefore the solution in the $E^{4/9}$ viscous layer. As the solution in the interaction region, this solution vanishes as $|\tilde {z}|\rightarrow \infty$. No solution is therefore created in the outer region for $N=0$ when the boundary is vertical. The different regions where the meanflow correction is important are summarized in figure 4$(c)$.

4.3. Non-rotating case $(\Omega =0)$

This case is similar to the case $N=0$, except that critical surfaces for the meanflow correction are now horizontal surfaces. We should then consider the case $\alpha =0$ separately. Both cases can be analysed in exactly the same way as for $N=0$. The characteristic regions defining the solution for $\Omega =0$ are shown in figure 4(d,e).