3.1 The amplitude-modulated singular mode

The linearized, inviscid and stationary system (2.5) can be reduced to a single equation, the Howard–Gupta modal equation generalized to the $f$ -plane and related to the modal radial velocity $U_{r}^{(0)}$ (Howard & Gupta Reference Howard and Gupta1962). This is expressed as ${\mathcal{L}}_{0}(U_{r}^{(0)})=0$ , namely

(3.3) $$\begin{eqnarray}\displaystyle \text{D}[S(r)\text{D}^{\ast }U_{r}^{(0)}]+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{2}U_{r}^{(0)}+\left((r\text{D}-2)[S(r)Q_{0}(r)]+\frac{k^{2}r^{2}\unicode[STIX]{x1D71A}_{0}(r)}{m\unicode[STIX]{x1D714}_{0}^{D}(r)}S(r)Q_{0}(r)\right)\frac{m\,U_{r}^{(0)}}{\unicode[STIX]{x1D714}_{0}^{D}(r)r^{2}}=0. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The superscript $D$ defines the Doppler-shifted frequency

(3.4a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D714}^{D}(r)=\tilde{\unicode[STIX]{x1D714}}-m\unicode[STIX]{x1D6FA}(r)-\tilde{k}\,\overline{W}(r),\quad S(r)=\frac{r^{2}}{m^{2}+k^{2}r^{2}}. & & \displaystyle\end{eqnarray}$$

Equation (3.3) is singular at the critical radius $r_{c}$ where the Doppler-shifted frequency $\unicode[STIX]{x1D714}_{0}^{D}(r)$ vanishes (that is, $\unicode[STIX]{x1D714}_{0}=m\unicode[STIX]{x1D6FA}_{0,c}$ ). But the solution is weakly singular around $r_{c}$ , at the end of the strong vortex/wave packet interaction, since the local equivalent Richardson number $J(r)=(kr/m)^{2}\,\unicode[STIX]{x1D71A}_{0}(r)Q_{0}(r)/[Q_{0}(r)-\unicode[STIX]{x1D71A}_{0}(r)]^{2}$ becomes small near the CL symmetry axis $r_{a}$ , as the zeroth-order mean axial vorticity vanishes at $r_{a}$ . The connection of this number with the linear stability of a rotating sheared flow is evoked in Caillol (Reference Caillol2014, Reference Caillol2015). The subscript $c$ characterizes the critical radius.

The neutral-mode radial velocity evaluated on the streamline $\unicode[STIX]{x1D702}$ is expressed by two Frobenius series of the variable $r-r_{c}=\unicode[STIX]{x1D702}+h(\unicode[STIX]{x1D702})$ :

(3.5) $$\begin{eqnarray}\displaystyle U_{r}^{(0)}=\left(\frac{a}{r_{c}}\unicode[STIX]{x1D719}_{a}[\unicode[STIX]{x1D702}+h(\unicode[STIX]{x1D702})]+\unicode[STIX]{x1D719}_{b}[\unicode[STIX]{x1D702}+h(\unicode[STIX]{x1D702})]\right)\,U_{0}(\unicode[STIX]{x1D6F7})\,\sin \unicode[STIX]{x1D709}+\frac{C}{r_{c}}\unicode[STIX]{x1D719}_{a}[\unicode[STIX]{x1D702}+h(\unicode[STIX]{x1D702})]U_{0}(\unicode[STIX]{x1D6F7})\,\cos \unicode[STIX]{x1D709}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where both Frobenius series $\unicode[STIX]{x1D719}_{a}$ and $\unicode[STIX]{x1D719}_{b}$ are expanded around $r_{c}$ in the following way:

(3.6a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{a}(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D702}+\mathop{\sum }_{n=2}^{\infty }a_{0,n}\unicode[STIX]{x1D702}^{n},\quad \unicode[STIX]{x1D719}_{b}(\unicode[STIX]{x1D702})=1+\mathop{\sum }_{n=2}^{\infty }b_{0,n}\unicode[STIX]{x1D702}^{n}+b_{0}\unicode[STIX]{x1D719}_{a}(\unicode[STIX]{x1D702})\ln \unicode[STIX]{x1D702}^{\ast }. & & \displaystyle\end{eqnarray}$$

The radius $\unicode[STIX]{x1D702}$ inside the logarithm is normalized by $\unicode[STIX]{x1D702}^{\ast }=\unicode[STIX]{x1D702}/\unicode[STIX]{x1D702}_{0}$ , where $\unicode[STIX]{x1D702}_{0}=r_{c}/\sqrt{2}$ . The complex logarithmic function in (3.6b ) induces an additional, in-phase quadrature term in (3.5) as $r<r_{c}$ , caused by the logarithmic phase shift $\unicode[STIX]{x1D719}$ defining $C$ as follows:

(3.7a,b ) $$\begin{eqnarray}\displaystyle r>r_{c},\quad C^{+}=0,\quad \text{and}\quad r<r_{c},\quad C^{-}=b_{0}r_{c}\unicode[STIX]{x1D719}(\unicode[STIX]{x1D706}). & & \displaystyle\end{eqnarray}$$

The superscript $s=-$ will be used to characterize the vortex core, where $r<r_{c}$ , and the superscript $s=+$ the outer core, where $r>r_{c}$ .

3.2 Order $\unicode[STIX]{x1D716}^{3/2}$ outer flow

The first-order inviscid disturbance is decomposed into five contributions. The radial velocity $U_{r,i}^{(1)}$ can be then written as

(3.8) $$\begin{eqnarray}\displaystyle U_{r,i}^{(1)}=U_{r,h}^{(1)}+U_{r,l}^{(1)}+U_{r,d}^{(1)}+U_{r,ad}^{(1)}+U_{r,\unicode[STIX]{x1D714}d}^{(1)}. & & \displaystyle\end{eqnarray}$$

The first term constitutes a CL feedback into the outer flow. It is required while matching the inner flow on the separatrices. The second term $U_{r,l}^{(1)}$ is generated by the advection of the mode by the first-order mean flow. The three next terms $U_{r,d}^{(1)}$ , $U_{r,ad}^{(1)}$ and $U_{r,\unicode[STIX]{x1D714}d}^{(1)}$ are respectively created by the vertical gradients of the modal amplitude $U_{0}$ , the phase jump $a$ and the frequency $\unicode[STIX]{x1D714}_{1}$ . The velocities $U_{r,h}^{(1)}$ and $U_{r,\unicode[STIX]{x1D714}d}^{(1)}$ are not $\unicode[STIX]{x1D6E9}$ -secular at the lowest expansion order near $\unicode[STIX]{x1D702}=0$ . So their secularity can be neglected in the study. Differentiation with respect to $\unicode[STIX]{x1D6E9}$ in the system (2.5) leads to the presence of coupled sine and cosine eigenfunctions. The dependence of $U_{r,i}^{(1)}$ on the phase $\unicode[STIX]{x1D709}$ is hence as follows:

(3.9) $$\begin{eqnarray}\displaystyle U_{r,i}^{(1)}(r,\unicode[STIX]{x1D709}) & = & \displaystyle \{[\unicode[STIX]{x1D719}_{l}^{(1,1)}(r)+\unicode[STIX]{x1D714}_{1,\unicode[STIX]{x1D6F7}}T_{1}\,\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D714}d}^{(1,1)}(r)+a_{\unicode[STIX]{x1D6F7}}\unicode[STIX]{x1D719}_{ad}^{(1,1)}(r)]\,U_{0}+\unicode[STIX]{x1D719}_{d}^{(1,1)}U_{0,\unicode[STIX]{x1D6F7}}\}\sin \unicode[STIX]{x1D709}\nonumber\\ \displaystyle & & \displaystyle +\,\{[\unicode[STIX]{x1D719}_{l}^{(1,2)}(r)+a_{\unicode[STIX]{x1D6F7}}\unicode[STIX]{x1D719}_{ad}^{(1,2)}(r)]\,U_{0}+\unicode[STIX]{x1D719}_{d}^{(1,2)}(r)\,U_{0,\unicode[STIX]{x1D6F7}}\}\cos \unicode[STIX]{x1D709}+\unicode[STIX]{x1D719}_{h}^{(1)}(r)\,U_{0}\cos (2\unicode[STIX]{x1D709}).\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The feedback flow satisfies the homogeneous equation

(3.10) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{0}(U_{r,h}^{(1)})=0,\quad \text{with}~\unicode[STIX]{x1D719}_{h}^{(1)}(\unicode[STIX]{x1D702})=\frac{\unicode[STIX]{x1D6FE}_{h,1}}{r_{c}}\unicode[STIX]{x1D719}_{a}(\unicode[STIX]{x1D702})+\unicode[STIX]{x1D705}_{h,1}\unicode[STIX]{x1D719}_{b}(\unicode[STIX]{x1D702}). & & \displaystyle\end{eqnarray}$$

The other terms satisfy non-homogeneous linear equations. The second contribution $U_{r,l}^{(1)}$ satisfies the equation

(3.11) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}{\mathcal{L}}_{0}(U_{r,l}^{(1,1)})=-{\mathcal{L}}_{1,1}(U_{r}^{(0)})-{\mathcal{L}}_{1,\unicode[STIX]{x1D703}}(\boldsymbol{U}_{l}^{(1,2)},P_{l}^{(1,2)}),\\ {\mathcal{L}}_{0}(U_{r,l}^{(1,2)})=-{\mathcal{L}}_{1,2}(U_{\unicode[STIX]{x1D703}}^{(0)})-{\mathcal{L}}_{1,\unicode[STIX]{x1D703}}(\boldsymbol{U}_{l}^{(1,1)},P_{l}^{(1,1)}),\end{array}\right\} & & \displaystyle\end{eqnarray}$$

with the operators ${\mathcal{L}}_{1,1}$ , ${\mathcal{L}}_{1,2}$ and ${\mathcal{L}}_{1,\unicode[STIX]{x1D703}}$ defined by

(3.12) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{1,1}(U) & = & \displaystyle \left\{(r\text{D}-2)\left[S(r)\left(Q_{z,1}(r)-\frac{kr}{m}Q_{\unicode[STIX]{x1D703},1}(r)\right)\right]\right.\nonumber\\ \displaystyle & & \displaystyle +\left.kr\frac{S(r)\unicode[STIX]{x1D71A}_{0}(r)}{\unicode[STIX]{x1D714}_{0}^{D}(r)}\left(\frac{kr}{m}Q_{z,1}(r)+Q_{\unicode[STIX]{x1D703},1}(r)\right)-\frac{\unicode[STIX]{x1D714}_{1}^{D}(r)}{\unicode[STIX]{x1D714}_{0}^{D}(r)}(r\text{D}-2)[S(r)Q_{0}(r)]\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{k^{2}r^{2}}{m}\frac{S(r)\unicode[STIX]{x1D71A}_{0}(r)}{\unicode[STIX]{x1D714}_{0}^{D}(r)}\left[\frac{\unicode[STIX]{x1D71A}_{1}(r)}{\unicode[STIX]{x1D71A}_{0}(r)}-2\frac{\unicode[STIX]{x1D714}_{1}^{D}(r)}{\unicode[STIX]{x1D714}_{0}^{D}(r)}\right]Q_{0}(r)\right\}\frac{mU}{\unicode[STIX]{x1D714}_{0}^{D}(r)r^{2}},\end{eqnarray}$$

and

(3.13) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{1,2}(U) & = & \displaystyle \frac{km\unicode[STIX]{x1D71A}_{0}(r)}{r\unicode[STIX]{x1D714}_{0}^{D2}(r)}S(r)U\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left[k\overline{V}_{1}(r)-\frac{m}{r}\overline{W}_{1}(r)\right]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{m}{\unicode[STIX]{x1D714}_{0}^{D}(r)}\text{D}\left[\frac{S(r)}{r}U\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left[k\overline{W}_{1}(r)+\frac{m}{r}\overline{V}_{1}(r)\right]\right]\end{eqnarray}$$

(3.14) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{1,\unicode[STIX]{x1D703}}(U,P) & = & \displaystyle \frac{m\unicode[STIX]{x1D71A}_{0}(r)}{\unicode[STIX]{x1D714}_{0}^{D2}(r)}\left[\unicode[STIX]{x1D6FA}_{0}(r)-\unicode[STIX]{x1D714}_{0}\frac{m}{r^{2}}S(r)\right]\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}U_{\unicode[STIX]{x1D703}}-m\frac{\unicode[STIX]{x1D6FA}_{0}(r)}{\unicode[STIX]{x1D714}_{0}^{D}(r)}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}U_{r}\nonumber\\ \displaystyle & & \displaystyle +\,k\frac{m}{r}\frac{\unicode[STIX]{x1D71A}_{0}(r)}{\unicode[STIX]{x1D714}_{0}^{D2}(r)}S(r)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}[kP-m\unicode[STIX]{x1D6FA}_{0}(r)U_{z}]\nonumber\\ \displaystyle & & \displaystyle +\,\frac{m\text{D}}{\unicode[STIX]{x1D714}_{0}^{D}(r)}\left[S(r)\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left(\frac{m}{r^{2}}P+\frac{\unicode[STIX]{x1D714}_{0}}{r}U_{\unicode[STIX]{x1D703}}+k\unicode[STIX]{x1D6FA}_{0}(r)U_{z}\right)\right].\end{eqnarray}$$

The first-order Doppler-shifted frequency is $\unicode[STIX]{x1D714}_{1}^{D}(r)=\unicode[STIX]{x1D714}_{1}-m\unicode[STIX]{x1D6FA}_{1}(r)-k\overline{W}_{1}(r)$ . The first-order equivalent Richardson number, derived from (3.12), is given at $r=r_{c}$ by

(3.15) $$\begin{eqnarray}\displaystyle J_{1,c}=\frac{\unicode[STIX]{x1D71B}}{\unicode[STIX]{x1D71A}_{0,c}}(\unicode[STIX]{x1D71B}Q_{z,1,c}+Q_{\unicode[STIX]{x1D703},1,c})-(1+2\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D714}_{1,c}^{D}\frac{r_{c}Q_{z,0,c}^{\prime }}{m\unicode[STIX]{x1D71A}_{0,c}^{2}}. & & \displaystyle\end{eqnarray}$$

The radial velocities $U_{r,d}^{(1)}$ , $U_{r,ad}^{(1)}$ and $U_{r,\unicode[STIX]{x1D714}d}^{(1)}$ satisfy the equations

(3.16) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}{\mathcal{L}}_{0}(U_{r,d}^{(1,1)})=-{\mathcal{L}}_{1,\unicode[STIX]{x1D703}}(\boldsymbol{U}_{d}^{(1,2)},P_{d}^{(1,2)}),\\ {\mathcal{L}}_{0}(U_{r,d}^{(1,2)})=-{\mathcal{L}}_{1,d}(\unicode[STIX]{x1D719}^{(0)})U_{0,\unicode[STIX]{x1D6F7}}\cos \unicode[STIX]{x1D709}-{\mathcal{L}}_{1,\unicode[STIX]{x1D703}}(\boldsymbol{U}_{d}^{(1,1)},P_{d}^{(1,1)}),\end{array}\right\} & \displaystyle\end{eqnarray}$$

(3.17) $$\begin{eqnarray}\displaystyle & \displaystyle \left.\begin{array}{@{}c@{}}{\mathcal{L}}_{0}(U_{r,ad}^{(1,1)})=-{\mathcal{L}}_{1,\unicode[STIX]{x1D703}}(\boldsymbol{U}_{ad}^{(1,2)},P_{ad}^{(1,2)}),\\ {\mathcal{L}}_{0}(U_{r,ad}^{(1,2)})=-{\mathcal{L}}_{1,d}(\unicode[STIX]{x1D719}_{a})a_{\unicode[STIX]{x1D6F7}}U_{0}\cos \unicode[STIX]{x1D709}-{\mathcal{L}}_{1,\unicode[STIX]{x1D703}}(\boldsymbol{U}_{ad}^{(1,1)},P_{ad}^{(1,1)}),\end{array}\right\} & \displaystyle\end{eqnarray}$$

(3.18) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{L}}_{0}(U_{r,\unicode[STIX]{x1D714}d}^{(1)})=-\unicode[STIX]{x1D714}_{1,\unicode[STIX]{x1D6F7}}T_{1}{\mathcal{L}}_{1,d}(\unicode[STIX]{x1D719}^{(0)})U_{0}\sin \unicode[STIX]{x1D709}, & \displaystyle\end{eqnarray}$$

with

(3.19) $$\begin{eqnarray}\displaystyle {\mathcal{L}}_{1,d}(\unicode[STIX]{x1D719})=\frac{2km}{r_{c}}\left[\frac{m}{r}\frac{Q_{0}(r)}{\unicode[STIX]{x1D714}_{0}^{D}(r)}-\text{D}\right]\left[S^{2}(r)\left(\frac{m}{r}\frac{Q_{0}(r)}{\unicode[STIX]{x1D714}_{0}^{D}(r)}\unicode[STIX]{x1D719}(r)+\text{D}^{\ast }[\unicode[STIX]{x1D719}(r)]\right)\right]. & & \displaystyle\end{eqnarray}$$

The Frobenius series related to each radial velocity have the same structure:

(3.20) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{j}^{(1,1)}(\unicode[STIX]{x1D702}) & = & \displaystyle \mathop{\sum }_{n=0}^{\infty }(b_{j,1,n}\ln ^{2}|\unicode[STIX]{x1D702}^{\ast }|+c_{j,1,n}\ln |\unicode[STIX]{x1D702}^{\ast }|+d_{j,1,n})\,\unicode[STIX]{x1D702}^{n}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6FC}_{j,1}}{r_{c}}\unicode[STIX]{x1D719}_{a}+\unicode[STIX]{x1D6FD}_{j,1}\unicode[STIX]{x1D719}_{b},\quad j=l,d,ad~\text{and}~\unicode[STIX]{x1D714}d,\end{eqnarray}$$

and

(3.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{j}^{(1,2)}(\unicode[STIX]{x1D702}) & = & \displaystyle \mathop{\sum }_{n=0}^{\infty }(e_{j,1,n}\ln ^{2}|\unicode[STIX]{x1D702}^{\ast }|+f_{j,1,n}\ln |\unicode[STIX]{x1D702}^{\ast }|+g_{j,1,n})\,\unicode[STIX]{x1D702}^{n}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6FE}_{j,1}}{r_{c}}\unicode[STIX]{x1D719}_{a}+\unicode[STIX]{x1D705}_{j,1}\unicode[STIX]{x1D719}_{b},\quad j=l,d~\text{and}~ad.\end{eqnarray}$$

The coefficients of the first terms in the series (3.6), (3.20), (3.21) are given in appendix A. The integration of equations (3.11), (3.16) and (3.17) is simplified owing to the assumption of $\unicode[STIX]{x1D6E9}$ -secularity; any physical quantity in both outer and inner flows is a polynomial of the variable $\unicode[STIX]{x1D6E9}$ . As a result, the quantity $q(\unicode[STIX]{x1D6E9})$ can be written exactly in a finite Taylor expansion:

(3.22) $$\begin{eqnarray}\displaystyle q(\unicode[STIX]{x1D6E9})=q(0)+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}q(0)\,\unicode[STIX]{x1D6E9}+{\textstyle \frac{1}{2}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}q(0)\,\unicode[STIX]{x1D6E9}^{2}+\cdots \,. & & \displaystyle\end{eqnarray}$$

From now on, any $n$ th-order $\unicode[STIX]{x1D6E9}$ -differentiation at $\unicode[STIX]{x1D6E9}=0$ of $q$ will be denoted by $\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{n}q(0)$ .

4.1 CL variables and equations

When the amplitude-modulated neutral mode is weakly singular, the local equivalent Richardson number at $r_{c}$ , $J_{c}$ is small and is assumed to be $O(\unicode[STIX]{x1D716}^{1/2})$ : $J_{c}=J_{1,c}\,\unicode[STIX]{x1D716}^{1/2}+O(\unicode[STIX]{x1D716})$ . The critical-layer width is then maximum and of order $O(\unicode[STIX]{x1D716}^{1/2})$ ; we will thus take the inner radial scaling $r-r_{c}=\unicode[STIX]{x1D716}^{1/2}R$ . The outer-flow expansion following $R$ gives the way in which the inner expansion must proceed:

(4.1) $$\begin{eqnarray}\displaystyle & \displaystyle U=\unicode[STIX]{x1D716}\,\overline{U}_{2}(r_{a})+\unicode[STIX]{x1D716}\{U^{(0)}+\unicode[STIX]{x1D716}^{1/2}\ln \unicode[STIX]{x1D716}\,U^{(1)}+\unicode[STIX]{x1D716}^{1/2}\,U^{(2)}+\cdots \}, & \displaystyle\end{eqnarray}$$

(4.2) $$\begin{eqnarray}\displaystyle & \displaystyle V=\unicode[STIX]{x1D6FA}_{0}(r_{a})\,r+\unicode[STIX]{x1D716}^{1/2}\unicode[STIX]{x1D6FA}_{1}(r_{a})\,r+\unicode[STIX]{x1D716}^{1/2}\{V^{(0)}+\unicode[STIX]{x1D716}^{1/2}\ln \unicode[STIX]{x1D716}\,V^{(1)}+\unicode[STIX]{x1D716}^{1/2}\,V^{(2)}+\cdots \}, & \displaystyle\end{eqnarray}$$

(4.3) $$\begin{eqnarray}\displaystyle & \displaystyle W=\unicode[STIX]{x1D716}^{1/2}\overline{W}_{1}(r_{a})+\unicode[STIX]{x1D716}^{1/2}\{\unicode[STIX]{x1D716}^{1/2}\ln \unicode[STIX]{x1D716}\,W^{(1)}+\unicode[STIX]{x1D716}^{1/2}\,W^{(2)}+\cdots \}, & \displaystyle\end{eqnarray}$$

(4.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{P}}={\textstyle \frac{1}{8}}(\unicode[STIX]{x1D71A}_{0}^{2}(r_{c})-Co_{0}^{2})r^{2}+\unicode[STIX]{x1D716}\{{\mathcal{P}}^{(0)}+\unicode[STIX]{x1D716}^{1/2}\ln \unicode[STIX]{x1D716}\,{\mathcal{P}}^{(1)}+\unicode[STIX]{x1D716}^{1/2}\,{\mathcal{P}}^{(2)}+\cdots \}. & \displaystyle\end{eqnarray}$$

In order to simplify the CL equations, let us make the following rescaling using the zeroth-order mean flow evaluated at $r_{c}$ :

(4.5) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle X=\unicode[STIX]{x1D709}+\frac{\unicode[STIX]{x03C0}}{2}(1-s_{i}),\quad R=r_{c}{\mathcal{R}},\quad \unicode[STIX]{x1D70F}_{n}=m\unicode[STIX]{x1D71A}_{0,c}\unicode[STIX]{x1D716}^{n/2}t,\quad s_{i}=-\text{sgn}[m\unicode[STIX]{x1D71A}_{0,c}U_{0}],\\ \displaystyle V^{(n)}=\unicode[STIX]{x1D71A}_{0,c}r_{c}\,\hat{V}^{(n)},\quad W^{(n)}=\unicode[STIX]{x1D71B}\unicode[STIX]{x1D71A}_{0,c}r_{c}{\hat{W}}^{(n)},\quad U^{(n)}=s_{i}m\unicode[STIX]{x1D71A}_{0,c}r_{c}\hat{U} ^{(n)},\quad {\mathcal{P}}^{(n)}=s_{i}\unicode[STIX]{x1D71A}_{0,c}^{2}r_{c}^{2}\hat{{\mathcal{P}}}^{(n)},\\ \displaystyle \overline{V}_{n,c}={\mathcal{V}}_{n}\unicode[STIX]{x1D71A}_{0,c}r_{c},\quad \overline{W}_{n,c}=\unicode[STIX]{x1D71B}{\mathcal{W}}_{n}\unicode[STIX]{x1D71A}_{0,c}r_{c},\quad \overline{U}_{n,c}=m{\mathcal{U}}_{n}\unicode[STIX]{x1D71A}_{0,c}r_{c},\quad \overline{\unicode[STIX]{x1D6E4}}_{n,c}=\mathfrak{P}_{n}\unicode[STIX]{x1D71A}_{0,c}^{2}r_{c}^{2},\\ \displaystyle \overline{\unicode[STIX]{x1D6E4}}_{n,c}^{\prime }=\mathfrak{P}_{n}^{\prime }\unicode[STIX]{x1D71A}_{0,c}^{2}r_{c},\quad \unicode[STIX]{x1D714}_{n}=m{\mho}_{n}\unicode[STIX]{x1D71A}_{0,c},\quad Co_{n}={\mathcal{C}}o_{n}\unicode[STIX]{x1D71A}_{0,c},\quad \unicode[STIX]{x1D71A}_{n,c}={\mathcal{S}}_{n}\unicode[STIX]{x1D71A}_{0,c},\\ \displaystyle Q_{z,n,c}={\mathcal{Q}}_{z,n}\unicode[STIX]{x1D71A}_{0,c},\quad \overline{W}_{n,c}^{\prime }=-{\mathcal{Q}}_{\unicode[STIX]{x1D703},n}\unicode[STIX]{x1D71A}_{0,c},\quad r_{c}Q_{0,c}^{\prime }=\unicode[STIX]{x1D701}\unicode[STIX]{x1D71A}_{0,c},\quad r_{c}Q_{z,n,c}^{\prime }=\unicode[STIX]{x1D701}_{z,n}\unicode[STIX]{x1D71A}_{0,c},\\ \displaystyle r_{c}Q_{\unicode[STIX]{x1D703},n,c}^{\prime }=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},n}\unicode[STIX]{x1D71A}_{0,c},\quad r_{c}^{2}Q_{0,c}^{\prime \prime }=\unicode[STIX]{x1D701}^{\prime }\unicode[STIX]{x1D71A}_{0,c},\quad r_{c}^{2}Q_{z,n,c}^{\prime \prime }=\unicode[STIX]{x1D701}_{z,n}^{\prime }\unicode[STIX]{x1D71A}_{0,c},\quad r_{c}^{2}Q_{\unicode[STIX]{x1D703},n,c}^{\prime \prime }=\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},n}^{\prime }\unicode[STIX]{x1D71A}_{0,c},\\ \displaystyle \unicode[STIX]{x1D706}^{\prime }=\frac{\unicode[STIX]{x1D706}}{U_{0}r_{c}},\quad A(\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D70F}_{2})=\left|\frac{U_{0}(\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D70F}_{2})}{m\unicode[STIX]{x1D71A}_{0,c}r_{c}}\right|.\end{array}\right\} & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

In the following analysis, the hat is dropped with the understanding that we are dealing with the new variables. The rescaled wave amplitude $A(\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D70F}_{2})$ is the solution of a nonlinear space–time evolution equation in the asymptotic quasi-steady regime, a long time after the CL formation, when the vorticity wave packet is slowly exchanging momentum and kinetic energy with the basic-state vortex. In this regime, where the zeroth-order mean axial vorticity is taken to be zero on the CL axis, the strength of the CL singularity is measured by the axial-vorticity radial gradient at $r_{c}$ (that is, the dimensionless number $\unicode[STIX]{x1D701}$ ). The radius $r_{c}$ and $\unicode[STIX]{x1D701}$ are finite, but $r_{c}>1$ while $\unicode[STIX]{x1D701}$ is small, according to observations (Caillol Reference Caillol2015). If we consider the limit $r_{c}\rightarrow 0$ , we suppose a critical layer in the eye of the tropical cyclone or the tornado. The basic-vortex axial vorticity is nearly constant in the eye, so vorticity waves are strongly damped there. Taking this limit would imply $\unicode[STIX]{x1D701}\rightarrow 0$ ; the CL interaction would be weaker since the leading-order CL equations would be postponed to a higher order. If we now consider the limit $r_{c}\rightarrow \infty$ , we suppose a critical layer far away from the eyewall, where the basic vorticity radial gradient is quasi-zero and vorticity waves are also strongly damped. Taking this limit would lead to $\unicode[STIX]{x1D701}\rightarrow 0$ and a weak WMFI as well, as the vorticity radial gradient decreases rapidly with $r$ and $r_{c}Q_{z,0}^{\prime }\rightarrow 0$ .

A general expression for the streamline radius ${\mathcal{R}}$ inside the critical layer is

(4.6) $$\begin{eqnarray}\displaystyle {\mathcal{R}}=s_{i}\unicode[STIX]{x1D70C}+H(\unicode[STIX]{x1D70C},X,\unicode[STIX]{x1D6E9},\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D70F}_{2}), & & \displaystyle\end{eqnarray}$$

with the function $H$ being expanded in terms of $\unicode[STIX]{x1D716}^{1/2}$ : $H=\unicode[STIX]{x1D716}^{1/2}H_{2}+\unicode[STIX]{x1D716}H_{3}+O(\unicode[STIX]{x1D716}^{3/2})$ .

We undertake a change of radial variables $R\rightarrow \unicode[STIX]{x1D70C}.$ The radius $\unicode[STIX]{x1D70C}$ is a function of ${\mathcal{R}}$ , $X$ , $\unicode[STIX]{x1D6E9}$ , $\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D70F}_{2}$ and its partial derivatives are

(4.7a-c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}_{{\mathcal{R}}}=\frac{1}{s_{i}+H_{\unicode[STIX]{x1D70C}}},\quad \unicode[STIX]{x1D70C}_{X}=-\frac{H_{X}}{s_{i}+H_{\unicode[STIX]{x1D70C}}},\quad \unicode[STIX]{x1D70C}_{\unicode[STIX]{x1D6E9}}=-\frac{H_{\unicode[STIX]{x1D6E9}}}{s_{i}+H_{\unicode[STIX]{x1D70C}}},\ldots & & \displaystyle\end{eqnarray}$$

The resulting system of inner equations is given in appendix B.

4.2 Zeroth- and first-order CL solutions

The zeroth-order inner equations lead to a simple solution as follows:

(4.8a-c ) $$\begin{eqnarray}\displaystyle U^{(0)}=-s_{i}A\sin X,\quad V^{(0)}=-s_{i}\unicode[STIX]{x1D70C},\quad {\mathcal{P}}^{(0)}={\textstyle \frac{1}{2}}[{\mathcal{S}}_{1}-{\mathcal{C}}_{0}{\mathcal{C}}_{1}-s_{i}\unicode[STIX]{x1D70C}]\unicode[STIX]{x1D70C}+s_{i}\mathfrak{P}_{2}. & & \displaystyle\end{eqnarray}$$

Two compatibility conditions arise from the inner angular and axial momentum equations

(4.9a,b ) $$\begin{eqnarray}\displaystyle V_{\unicode[STIX]{x1D6E9}}^{(2)}=0,\quad W_{\unicode[STIX]{x1D6E9}}^{(2)}=0. & & \displaystyle\end{eqnarray}$$

The first-order flow is straightforwardly deduced from the Frobenius series (3.6), (3.20) and (3.21):

(4.10) $$\begin{eqnarray}\displaystyle U^{(1)} & = & \displaystyle \frac{1}{2}[\unicode[STIX]{x1D701}(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D70C}-s_{i}(c_{l,1,0}+c_{ad,1,0}a_{\unicode[STIX]{x1D6F7}})A-s_{i}c_{d,1,0}A_{\unicode[STIX]{x1D6F7}}]\,\sin X\nonumber\\ \displaystyle & & \displaystyle -\,\frac{s_{i}}{2}[(f_{l,1,0}+f_{ad,1,0}a_{\unicode[STIX]{x1D6F7}})A+f_{d,1,0}A_{\unicode[STIX]{x1D6F7}}]\cos X,\end{eqnarray}$$

(4.11) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{{\mathcal{P}}^{(1)}}{\unicode[STIX]{x1D70C}}+\frac{s_{i}}{2}\unicode[STIX]{x1D701}\frac{{\mho}_{0}}{\unicode[STIX]{x1D70C}}\{[(\unicode[STIX]{x1D6FD}_{l,1,\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D6FD}_{ad,1,\unicode[STIX]{x1D6E9}}a_{\unicode[STIX]{x1D6F7}})A+\unicode[STIX]{x1D6FD}_{d,1,\unicode[STIX]{x1D6E9}}A_{\unicode[STIX]{x1D6F7}}]\sin X\nonumber\\ \displaystyle & & \displaystyle \quad +\,[(\unicode[STIX]{x1D705}_{l,1,\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D705}_{ad,1,\unicode[STIX]{x1D6E9}}a_{\unicode[STIX]{x1D6F7}})A+\unicode[STIX]{x1D705}_{d,1,\unicode[STIX]{x1D6E9}}A_{\unicode[STIX]{x1D6F7}}]\cos X\}=W^{(1)}=V^{(1)}=\frac{\unicode[STIX]{x1D701}}{2}A\,\cos X.\end{eqnarray}$$

The resonance condition ${\mho}_{1}^{D}=0$ results from the solvability conditions of the inner momentum equations as follows:

(4.12) $$\begin{eqnarray}\displaystyle {\mho}_{1}={\mathcal{V}}_{1}+\unicode[STIX]{x1D71B}^{2}{\mathcal{W}}_{1}, & & \displaystyle\end{eqnarray}$$

in addition to two other conditions:

(4.13) $$\begin{eqnarray}\displaystyle V_{\unicode[STIX]{x1D6E9}}^{(4)} & = & \displaystyle W_{\unicode[STIX]{x1D6E9}}^{(4)}=\frac{\unicode[STIX]{x1D701}}{2}\{[(\unicode[STIX]{x1D6FD}_{l,1,\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D6FD}_{ad,1,\unicode[STIX]{x1D6E9}}a_{\unicode[STIX]{x1D6F7}})A+\unicode[STIX]{x1D6FD}_{d,1,\unicode[STIX]{x1D6E9}}A_{\unicode[STIX]{x1D6F7}}]\cos X\nonumber\\ \displaystyle & & \displaystyle -\,[(\unicode[STIX]{x1D705}_{l,1,\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D705}_{ad,1,\unicode[STIX]{x1D6E9}}a_{\unicode[STIX]{x1D6F7}})A+\unicode[STIX]{x1D705}_{d,1,\unicode[STIX]{x1D6E9}}A_{\unicode[STIX]{x1D6F7}}]\sin X\}.\end{eqnarray}$$

The inner flow inside the separatrices is denoted by the superscript $\odot$ , while the jump and the mean of a quantity $q$ on either side of $r_{c}$ are respectively denoted by $[q]_{-}^{+}=q^{+}-q^{-}$ and $\{q\}_{-}^{+}=(q^{+}+q^{-})/2$ . The leading-order flow matchings on the separatrices show that the flows are identical inside and outside the separatrices, so $U^{(n,\odot )}=U^{(n)},V^{(n,\odot )}=V^{(n)},\ldots$ , with $n=0,1$ . A first consequence is the absence of a jump for the mean flow at $r_{c}$ : $[{\mathcal{V}}_{1}]_{-}^{+}=[{\mathcal{W}}_{1}]_{-}^{+}=[{\mathcal{U}}_{2}]_{-}^{+}=[\mathfrak{P}_{2}]_{-}^{+}=0$ . Continuity of the pressure leads to

(4.14a,b ) $$\begin{eqnarray}\displaystyle [(\unicode[STIX]{x1D6FD}_{l,1,\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D6FD}_{ad,1,\unicode[STIX]{x1D6E9}}a_{\unicode[STIX]{x1D6F7}})A+\unicode[STIX]{x1D6FD}_{d,1,\unicode[STIX]{x1D6E9}}A_{\unicode[STIX]{x1D6F7}}]_{-}^{+}=0,\quad [(\unicode[STIX]{x1D705}_{l,1,\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D705}_{ad,1,\unicode[STIX]{x1D6E9}}a_{\unicode[STIX]{x1D6F7}})A+\unicode[STIX]{x1D705}_{d,1,\unicode[STIX]{x1D6E9}}A_{\unicode[STIX]{x1D6F7}}]_{-}^{+}=0. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Continuity of the radial velocity implies

(4.15) $$\begin{eqnarray}\displaystyle [(c_{l,1,0}+c_{ad,1,0}a_{\unicode[STIX]{x1D6F7}})A+c_{d,1,0}A_{\unicode[STIX]{x1D6F7}}]_{-}^{+}=0, & & \displaystyle\end{eqnarray}$$

and

(4.16) $$\begin{eqnarray}\displaystyle [(f_{l,1,0}+f_{ad,1,0}a_{\unicode[STIX]{x1D6F7}})A+f_{d,1,0}A_{\unicode[STIX]{x1D6F7}}]_{-}^{+}=0. & & \displaystyle\end{eqnarray}$$

5.1 First-order mean flow

From (2.6d ), (2.6c ) and (2.6b ), respectively, we get independence of $\unicode[STIX]{x1D6E9}$ for the $O(\unicode[STIX]{x1D716}^{1/2})$ induced mean velocities and geopotential at $r\simeq r_{a}\simeq r_{c}$ :

(5.1a-c ) $$\begin{eqnarray}\displaystyle \overline{V}_{1,\unicode[STIX]{x1D6E9}}(r)=0,\quad \overline{W}_{1,\unicode[STIX]{x1D6E9}}(r)=0,\quad \overline{\unicode[STIX]{x1D6E4}}_{1,\unicode[STIX]{x1D6E9}}(r)=0. & & \displaystyle\end{eqnarray}$$

While differentiating (2.6b ) and (2.6c ) with respect to $r$ , we obtain the same result for the mean vorticities:

(5.2a,b ) $$\begin{eqnarray}\displaystyle Q_{z,1,\unicode[STIX]{x1D6E9}}(r)=0,\quad Q_{\unicode[STIX]{x1D703},1,\unicode[STIX]{x1D6E9}}(r)=0. & & \displaystyle\end{eqnarray}$$

However, the mean vorticity radial gradient is dependent on $\unicode[STIX]{x1D6E9}$ at the singularity $r=r_{c}$ . After differentiating (2.6b ) and (2.6c ) a second time with respect to $r$ , and simplifying using the above independence from $\unicode[STIX]{x1D6E9}$ for the first-order flow, we get

(5.3) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\displaystyle m\unicode[STIX]{x1D6FA}_{0}\overline{V}_{1,\unicode[STIX]{x1D6E9}}^{\prime \prime }(r)=-\frac{\unicode[STIX]{x1D716}^{3/2}}{r^{2}}\text{D}^{3}[r^{2}\overline{U_{r}U_{\unicode[STIX]{x1D703}}}]-2\unicode[STIX]{x1D716}^{3/2}\text{D}^{2}\left[\frac{1}{r}\overline{U_{r}U_{\unicode[STIX]{x1D703}}}\right],\\ \displaystyle m\unicode[STIX]{x1D6FA}_{0}\overline{W}_{1,\unicode[STIX]{x1D6E9}}^{\prime \prime }(r)=-\unicode[STIX]{x1D716}^{3/2}\text{D}^{2}[\text{D}^{\ast }[\overline{U_{r}U_{z}}]].\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The right-hand sides of equation (5.3) are made up of the divergence of the vorticity radial gradient mean radial wave flux, which is singular at $r=r_{c}$ . The CL flow smooths this singularity out, so we have to compute the flux expressed with the inner variables. The triple radial gradient is $O(\unicode[STIX]{x1D716}^{-3/2})$ inside the CL; as a result, some nonlinear inner terms on the right-hand side are $O(1)$ . Next, rescaling and taking at the left-hand side of (5.3) the upper CL limit $r\rightarrow r_{a}^{+}$ of the outer flow, corresponding on the right-hand side to the inner boundary $\unicode[STIX]{x1D70C}_{B}^{+}$ , we then have

(5.4a,b ) $$\begin{eqnarray}\displaystyle {\mho}_{0}\unicode[STIX]{x1D701}_{z,1,\unicode[STIX]{x1D6E9}}^{+}=-\overline{[U^{(0)}V_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{+},X)},\quad {\mho}_{0}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,\unicode[STIX]{x1D6E9}}^{+}=\unicode[STIX]{x1D71B}\overline{[U^{(0)}W_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{+},X)}. & & \displaystyle\end{eqnarray}$$

Taking the lower limit yields similar relations, so the spiralling part of the first-order axial-vorticity radial gradient admits a jump across the CL since

(5.5) $$\begin{eqnarray}\displaystyle {\mho}_{0}[\unicode[STIX]{x1D701}_{z,1,\unicode[STIX]{x1D6E9}}]_{-}^{+}=-\{\overline{[U^{(0)}V_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{+},X)}-\overline{[U^{(0)}V_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{-},X)}\}, & & \displaystyle\end{eqnarray}$$

with a similar relation for the azimuthal vorticity radial gradient $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,\unicode[STIX]{x1D6E9}}$ .

Section 7.1 will show that the $\unicode[STIX]{x1D6E9}$ -secular equations can be written on either side of the CL in the general way

(5.6a,b ) $$\begin{eqnarray}\displaystyle {\mho}_{0}\unicode[STIX]{x1D701}_{z,1,\unicode[STIX]{x1D6E9}}^{s}=\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s},\quad {\mho}_{0}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,\unicode[STIX]{x1D6E9}}^{s}=-\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}. & & \displaystyle\end{eqnarray}$$

As the zeroth-order flow must remain steady in the quasi-steady regime, in addition to the laminar force $F$ , we must assume the existence of an eddy force with azimuthal and axial components that prevents the basic-state vortex from being dependent on $\unicode[STIX]{x1D6E9}$ :

(5.7) $$\begin{eqnarray}\displaystyle & \displaystyle {\mho}_{0}{\mathcal{Q}}_{z,0,\unicode[STIX]{x1D6E9}}^{\prime \prime }=-s_{i}\,\overline{[U^{(0)}V_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{+},X)}+F_{ed}^{+}=-s_{i}\,\overline{[U^{(0)}V_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{-},X)}+F_{ed}^{-}=0, & \displaystyle\end{eqnarray}$$

(5.8) $$\begin{eqnarray}\displaystyle & \displaystyle {\mho}_{0}{\mathcal{Q}}_{\unicode[STIX]{x1D703},0,\unicode[STIX]{x1D6E9}}^{\prime \prime }=s_{i}\unicode[STIX]{x1D71B}\,\overline{[U^{(0)}W_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{+},X)}-\unicode[STIX]{x1D71B}F_{ed}^{+}=s_{i}\unicode[STIX]{x1D71B}\,\overline{[U^{(0)}W_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{-},X)}-\unicode[STIX]{x1D71B}F_{ed}^{-}=0. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The ratio of between the eddy force azimuthal and axial components is $-\unicode[STIX]{x1D71B}$ and the forcing $F_{ed}$ is undistorted.

5.2 Second-order mean flow

Next, from (2.6b–d ), we get the rescaled equations of the $O(\unicode[STIX]{x1D716})$ induced mean flow at  $r_{a}$ :

(5.9a-c ) $$\begin{eqnarray}\displaystyle \mathfrak{P}_{2,\unicode[STIX]{x1D6E9}}+{\mho}_{0}{\mathcal{V}}_{2,\unicode[STIX]{x1D6E9}}=0,\quad \mathfrak{P}_{1,\unicode[STIX]{x1D6F7}}+\unicode[STIX]{x1D71B}{\mho}_{0}{\mathcal{W}}_{2,\unicode[STIX]{x1D6E9}}=0,\quad {\mathcal{U}}_{2}^{\prime }+{\mathcal{U}}_{2}+\unicode[STIX]{x1D71B}{\mathcal{W}}_{1,\unicode[STIX]{x1D6F7}}+{\mathcal{V}}_{2,\unicode[STIX]{x1D6E9}}=0. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Note that the geopotential axial gradient in (2.6c ) generates a vertical wind at low orders. However, this motion is not relevant to the wave packet dynamics. The mean geopotential is therefore assumed uniform to the first order at $r_{a}$ : as a result, $\mathfrak{P}_{1,\unicode[STIX]{x1D6F7}}=\mathfrak{P}_{2,\unicode[STIX]{x1D6E9}}=\mathfrak{P}_{2,\unicode[STIX]{x1D6F7}}=\mathfrak{P}_{3,\unicode[STIX]{x1D6E9}}=0$ . We thus obtain the $\unicode[STIX]{x1D6E9}$ -independence of the second-order mean velocities:

(5.10a,b ) $$\begin{eqnarray}\displaystyle {\mathcal{V}}_{2,\unicode[STIX]{x1D6E9}}=0,\quad {\mathcal{W}}_{2,\unicode[STIX]{x1D6E9}}=0, & & \displaystyle\end{eqnarray}$$

while the radial-velocity radial gradient at $r_{a}$ is given by

(5.11) $$\begin{eqnarray}\displaystyle {\mathcal{U}}_{2}^{\prime }=-{\mathcal{U}}_{2}-\unicode[STIX]{x1D71B}{\mathcal{W}}_{1,\unicode[STIX]{x1D6F7}}. & & \displaystyle\end{eqnarray}$$

After $r$ -differentiating equation (2.6b ), and simplifying with (2.6d ) and (2.11c ), we obtain the $\unicode[STIX]{x1D6E9}$ -secular equation for the axial vorticity ${\mathcal{Q}}_{z,2}$ evaluated at $r\simeq r_{a}$ :

(5.12) $$\begin{eqnarray}\displaystyle m\unicode[STIX]{x1D6FA}_{0}(r)Q_{z,2,\unicode[STIX]{x1D6E9}}(r) & = & \displaystyle \frac{m}{r_{c}}Q_{0}(r)\overline{W}_{1,\unicode[STIX]{x1D6F7}}(r)-m[\unicode[STIX]{x1D6FA}_{1}(r)\,Q_{z,1}(r)]_{\unicode[STIX]{x1D6E9}}-Q_{0}^{\prime }(r)\overline{U}_{2}(r)\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x1D716}}{r^{2}}\{\text{D}[r^{2}\overline{U_{r}D^{\ast }[U_{\unicode[STIX]{x1D703}}]}]-\text{D}[\overline{(r^{2}\unicode[STIX]{x2202}_{z}U_{z}+mr\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}U_{\unicode[STIX]{x1D703}})U_{\unicode[STIX]{x1D703}}}]\}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D716}}{r^{3}}\text{D}[r^{2}\overline{U_{r}U_{\unicode[STIX]{x1D703}}}]-\frac{m}{r_{c}}\unicode[STIX]{x1D716}^{3/2}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6F7}}\text{D}^{\ast }[\overline{U_{z}U_{\unicode[STIX]{x1D703}}}].\end{eqnarray}$$

The two first terms on the right-hand side are negligible since $Q_{0}(r)\simeq 0$ , $\overline{V}_{1,\unicode[STIX]{x1D6E9}}(r)=0$ and $Q_{z,1,\unicode[STIX]{x1D6E9}}(r)=0$ . The fourth term on the right-hand side is $O(1)$ , since the double radial gradient is $O(\unicode[STIX]{x1D716}^{-1})$ inside the CL, whereas the three following terms are negligible. Finally, two terms remain on the right-hand side: the third term is the divergence of the radial flux of the axial vorticity $Q_{0}$ multiplied by the velocity $\overline{U}_{2}$ and the fourth term is the divergence of the leading-order mean radial wave flux of the axial vorticity. Next, introducing the inner variables in the flux divergence and taking the limit $\unicode[STIX]{x1D70C}\rightarrow \unicode[STIX]{x1D70C}_{B}^{+}$ , then taking the upper CL limit $r\rightarrow r_{a}^{+}$ for the other terms, and rescaling, we have

(5.13) $$\begin{eqnarray}\displaystyle {\mho}_{0}{\mathcal{Q}}_{z,2,\unicode[STIX]{x1D6E9}}^{+}=-\unicode[STIX]{x1D701}{\mathcal{U}}_{2}-s_{i}\,\overline{[U^{(0)}V_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{+},X)}. & & \displaystyle\end{eqnarray}$$

As for the mean azimuthal vorticity, the $\unicode[STIX]{x1D6E9}$ -gradient of the vorticity $Q_{\unicode[STIX]{x1D703},2}$ is generated by the divergence of the leading-order mean radial wave flux of the azimuthal vorticity and the vertical gradient of the leading-order mean geopotential radial gradient, so we have, after $r$ -differentiating (2.6c ), simplifying with (2.11b ), and rescaling as $r\rightarrow r_{a}^{+}$ ,

(5.14) $$\begin{eqnarray}\displaystyle {\mho}_{0}{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,\unicode[STIX]{x1D6E9}}^{+}={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}+s_{i}\unicode[STIX]{x1D71B}\overline{[U^{(0)}W_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{+},X)}. & & \displaystyle\end{eqnarray}$$

Section 7.1 will show that the second-order mean vorticity $\unicode[STIX]{x1D6E9}$ -secular equation can be written on either side of the CL in the general way

(5.15a,b ) $$\begin{eqnarray}\displaystyle {\mho}_{0}{\mathcal{Q}}_{z,2,\unicode[STIX]{x1D6E9}}^{s}=\unicode[STIX]{x1D6F4}_{q}^{s}-\unicode[STIX]{x1D701}{\mathcal{U}}_{2},\quad {\mho}_{0}{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,\unicode[STIX]{x1D6E9}}^{s}={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}-\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F4}_{q}. & & \displaystyle\end{eqnarray}$$

Section 7.1 will also prove that, for a symmetric CL extent such as $\unicode[STIX]{x1D70C}_{B}^{-}=-\unicode[STIX]{x1D70C}_{B}^{+}$ , we have $\unicode[STIX]{x1D6F4}_{q}^{-}=\unicode[STIX]{x1D6F4}_{q}^{+}$ . The second-order mean vorticity $\unicode[STIX]{x1D6E9}$ -gradient thus admits no jump across the CL. Taking the mean radial eddy vorticity flux-driven forcings into account in the inner equations then comes down to considering altered zeroth-order mean vorticity radial gradients at $r_{c}$ such as

(5.16a,b ) $$\begin{eqnarray}\displaystyle {\mho}_{0}{\mathcal{Q}}_{z,2,\unicode[STIX]{x1D6E9}}=-\hat{\unicode[STIX]{x1D701}}\,{\mathcal{U}}_{2},\quad {\mho}_{0}{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,\unicode[STIX]{x1D6E9}}={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}-\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}\,{\mathcal{U}}_{2}, & & \displaystyle\end{eqnarray}$$

with the modifications of the vorticity radial gradients given by

(5.17a,b ) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D701}}=\unicode[STIX]{x1D701}-\frac{\unicode[STIX]{x1D6F4}_{q}}{{\mathcal{U}}_{2}},\quad \hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}=\unicode[STIX]{x1D71B}\frac{\unicode[STIX]{x1D6F4}_{q}}{{\mathcal{U}}_{2}}. & & \displaystyle\end{eqnarray}$$

As the first-order vorticity radial gradient and the second-order vorticity vary following the azimuthal coordinate under the effect of the wave packet, they therefore evolve with the wave modulation. To prevent the viscous diffusion from damping these corrections, the definition of the rescaled force $\boldsymbol{F}$ near $r_{c}$ has to be slightly modified and an axial component has to be added:

(5.18a,b ) $$\begin{eqnarray}\displaystyle F_{\unicode[STIX]{x1D703}}=s_{i}\unicode[STIX]{x1D706}^{\prime }\left({\mathcal{Q}}_{z,0}^{\prime }(r)-\frac{\unicode[STIX]{x1D6F4}_{q}}{{\mathcal{U}}_{2}}\right)A\unicode[STIX]{x1D716}^{3/2},\quad F_{z}=s_{i}\unicode[STIX]{x1D706}^{\prime }\frac{\unicode[STIX]{x1D6F4}_{q}}{{\mathcal{U}}_{2}}A\unicode[STIX]{x1D716}^{3/2}. & & \displaystyle\end{eqnarray}$$

5.3 Evolution equations of the first-order mean velocity at $r_{a}$

We now determine the $\unicode[STIX]{x1D70F}_{2}$ -evolution equations of ${\mathcal{V}}_{1}$ and ${\mathcal{W}}_{1}$ , knowing that these velocities are inviscidly undistorted and that they do not depend on $\unicode[STIX]{x1D6E9}$ . The $O(\unicode[STIX]{x1D716}^{3/2})$ momentum equations (2.6b ) and (2.6c ) evaluated at $r$ near $r_{a}$ are

(5.19) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{V}_{1,T_{2}}+m\unicode[STIX]{x1D6FA}_{0}\overline{V}_{3,\unicode[STIX]{x1D6E9}}+\frac{m}{r_{c}}\overline{W}_{1}\overline{V}_{1,\unicode[STIX]{x1D6F7}}=-\frac{\unicode[STIX]{x1D716}^{1/2}}{r^{2}}\text{D}[r^{2}\overline{U_{r}U_{\unicode[STIX]{x1D703}}}]-Q_{z,1}\overline{U}_{2}-m\unicode[STIX]{x1D6FA}_{1}\overline{V}_{2,\unicode[STIX]{x1D6E9}}-\frac{m}{r}\overline{\unicode[STIX]{x1D6E4}}_{3,\unicode[STIX]{x1D6E9}}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.20) $$\begin{eqnarray}\displaystyle & \displaystyle \overline{W}_{1,T_{2}}+m\unicode[STIX]{x1D6FA}_{0}\overline{W}_{3,\unicode[STIX]{x1D6E9}}+\frac{m}{r_{c}}\overline{W}_{1}\overline{W}_{1,\unicode[STIX]{x1D6F7}}=-\frac{\unicode[STIX]{x1D716}^{1/2}}{r}\text{D}[r\overline{U_{r}U_{z}}]+Q_{\unicode[STIX]{x1D703},1}\overline{U}_{2}-m\unicode[STIX]{x1D6FA}_{1}\overline{W}_{2,\unicode[STIX]{x1D6E9}}-\frac{m}{r_{c}}\overline{\unicode[STIX]{x1D6E4}}_{2,\unicode[STIX]{x1D6F7}}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The divergences of the mean radial wave momentum fluxes on the right-hand side of (5.19) and (5.20) are $O(1)$ inside the CL. Introducing the inner variables in these terms, taking the limit $r\rightarrow r_{a}^{s}$ otherwise, assuming a uniform pressure at $r_{a}$ , rescaling, and eliminating ${\mathcal{V}}_{2,\unicode[STIX]{x1D6E9}}$ and ${\mathcal{W}}_{2,\unicode[STIX]{x1D6E9}}$ , we finally obtain

(5.21) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{V}}_{1,\unicode[STIX]{x1D70F}_{2}}+\unicode[STIX]{x1D71B}{\mathcal{W}}_{1}{\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}=\unicode[STIX]{x1D6F4}_{v}^{s}-{\mathcal{Q}}_{z,1}^{s}{\mathcal{U}}_{2}-{\mho}_{0}{\mathcal{V}}_{3,\unicode[STIX]{x1D6E9}}^{s}, & \displaystyle\end{eqnarray}$$

(5.22) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{W}}_{1,\unicode[STIX]{x1D70F}_{2}}+\unicode[STIX]{x1D71B}{\mathcal{W}}_{1}{\mathcal{W}}_{1,\unicode[STIX]{x1D6F7}}=\unicode[STIX]{x1D6F4}_{v}^{s}+\frac{{\mathcal{Q}}_{\unicode[STIX]{x1D703},1}^{s}}{\unicode[STIX]{x1D71B}}{\mathcal{U}}_{2}-{\mho}_{0}{\mathcal{W}}_{3,\unicode[STIX]{x1D6E9}}^{s}, & \displaystyle\end{eqnarray}$$

where

(5.23) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}_{v}^{s}=-\overline{[U^{(0)}V_{\unicode[STIX]{x1D70C}}^{(2)}](\unicode[STIX]{x1D70C}_{B}^{s},X)}. & & \displaystyle\end{eqnarray}$$

Let us decompose ${\mathcal{Q}}_{z,1}$ into an undistorted term and a distorted term so that we can separate the first- and third-order velocity evolutions in (5.21) and (5.22):

(5.24) $$\begin{eqnarray}\displaystyle {\mathcal{Q}}_{z,1}^{s}=\{{\mathcal{Q}}_{z,1}\}_{-}^{+}+{\mathcal{Q}}_{z,1,d}^{s},\quad \text{with}~{\mathcal{Q}}_{z,1,d}^{s}=\frac{s}{2}[{\mathcal{Q}}_{z,1}]_{-}^{+}. & & \displaystyle\end{eqnarray}$$

By considering a symmetric CL thickness, $\unicode[STIX]{x1D70C}_{B}^{-}=-\unicode[STIX]{x1D70C}_{B}^{+}$ , § 7.1 will show that $\{\unicode[STIX]{x1D6F4}_{v}\}_{-}^{+}=0$ , so $\unicode[STIX]{x1D6F4}_{v}=\unicode[STIX]{x1D6F4}_{v,d}$ . The third-order terms ${\mho}_{0}{\mathcal{V}}_{3,\unicode[STIX]{x1D6E9}}^{s}$ and ${\mho}_{0}{\mathcal{W}}_{3,\unicode[STIX]{x1D6E9}}^{s}$ are defined so that they can remove the distorted terms in (5.21) and (5.22), and thus we get

(5.25a,b ) $$\begin{eqnarray}\displaystyle {\mho}_{0}{\mathcal{V}}_{3,\unicode[STIX]{x1D6E9}}^{s}=\unicode[STIX]{x1D6F4}_{v}^{s}-{\mathcal{Q}}_{z,1,d}^{s}\,{\mathcal{U}}_{2},\quad {\mho}_{0}{\mathcal{W}}_{3,\unicode[STIX]{x1D6E9}}^{s}=\unicode[STIX]{x1D6F4}_{v}^{s}+\frac{{\mathcal{Q}}_{\unicode[STIX]{x1D703},1,d}^{s}}{\unicode[STIX]{x1D71B}}\,{\mathcal{U}}_{2}. & & \displaystyle\end{eqnarray}$$

As was done in the preceding subsection, the contribution of the mean wave momentum flux is included inside the first-order mean vorticity. The modified vorticities at $r_{c}$ are then denoted in the following way:

(5.26a,b ) $$\begin{eqnarray}\displaystyle \hat{{\mathcal{Q}}}_{z,1}={\mathcal{Q}}_{z,1}-\frac{\unicode[STIX]{x1D6F4}_{v}}{{\mathcal{U}}_{2}},\quad \hat{{\mathcal{Q}}}_{\unicode[STIX]{x1D703},1}={\mathcal{Q}}_{\unicode[STIX]{x1D703},1}+\unicode[STIX]{x1D71B}\frac{\unicode[STIX]{x1D6F4}_{v}}{{\mathcal{U}}_{2}}. & & \displaystyle\end{eqnarray}$$

The evolution equations of the CL-induced velocity $({\mathcal{V}}_{1},{\mathcal{W}}_{1})$ are therefore

(5.27a,b ) $$\begin{eqnarray}\displaystyle {\mathcal{V}}_{1,\unicode[STIX]{x1D70F}_{2}}+\unicode[STIX]{x1D71B}{\mathcal{W}}_{1}{\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}=-\{{\mathcal{Q}}_{z,1}\}_{-}^{+}{\mathcal{U}}_{2},\quad {\mathcal{W}}_{1,\unicode[STIX]{x1D70F}_{2}}+\unicode[STIX]{x1D71B}{\mathcal{W}}_{1}{\mathcal{W}}_{1,\unicode[STIX]{x1D6F7}}=\frac{\{{\mathcal{Q}}_{\unicode[STIX]{x1D703},1}\}_{-}^{+}}{\unicode[STIX]{x1D71B}}{\mathcal{U}}_{2}. & & \displaystyle\end{eqnarray}$$

The first-order mean velocity at $r_{a}$ (that is, ${\mathcal{V}}_{1},{\mathcal{W}}_{1}$ ) is vertically advected by ${\mathcal{W}}_{1}$ and is driven by the undistorted divergence of its radial flux generated by ${\mathcal{U}}_{2}$ . The forcing $\unicode[STIX]{x1D6F4}_{v}$ is absent since it is antisymmetric and it appears in the definitions of ${\mathcal{V}}_{3}$ and ${\mathcal{W}}_{3}$ . Jointly with the distorted mean vorticities, it generates a $\unicode[STIX]{x1D6E9}$ -secular growth of the third-order mean velocities. In C17, the latter have a $\unicode[STIX]{x1D70F}_{2}$ -linear growth.

6.1 Outside the separatrices

The CL flow is decomposed between a $\unicode[STIX]{x1D6E9}$ -dependent spiralling contribution and a purely oscillating one. For instance, for the fifth-order velocity $V^{(5)}$ , we have

(6.6) $$\begin{eqnarray}\displaystyle V^{(5)}=V_{sp}^{(5)}(\unicode[STIX]{x1D70C},X,\unicode[STIX]{x1D6E9})+V_{os}^{(5)}(\unicode[STIX]{x1D70C},X). & & \displaystyle\end{eqnarray}$$

The matchings of $V_{sp}^{(5)}$ and $W_{sp}^{(5)}$ on the CL edges as $\unicode[STIX]{x1D70C}\rightarrow \unicode[STIX]{x1D70C}_{B}^{\pm }$ give the asymptotic behaviours

(6.7) $$\begin{eqnarray}\displaystyle V_{sp}^{(5)} & \rightarrow & \displaystyle {\textstyle \frac{1}{2}}\unicode[STIX]{x1D701}_{z,1,sp}\unicode[STIX]{x1D70C}^{2}+s_{i}[{\mathcal{Q}}_{z,2,sp}+(\hat{\unicode[STIX]{x1D701}}+2)H_{2}]\unicode[STIX]{x1D70C}+\hat{\unicode[STIX]{x1D701}}[(\unicode[STIX]{x1D6FD}_{l,1}A+\unicode[STIX]{x1D6FD}_{d,1}A_{\unicode[STIX]{x1D6F7}}+\unicode[STIX]{x1D6FD}_{ad,1}a_{\unicode[STIX]{x1D6F7}}A)\cos X\nonumber\\ \displaystyle & & \displaystyle -\,(\unicode[STIX]{x1D705}_{l,1}A+\unicode[STIX]{x1D705}_{d,1}A_{\unicode[STIX]{x1D6F7}}+\unicode[STIX]{x1D705}_{ad,1}a_{\unicode[STIX]{x1D6F7}}A)\sin X]\ln |\unicode[STIX]{x1D70C}^{\ast }|\nonumber\\ \displaystyle & & \displaystyle +\,{\mathcal{V}}_{3,sp}+V_{5,sp}(X)+s_{i}\hat{\unicode[STIX]{x1D701}}A\frac{\cos X}{\unicode[STIX]{x1D70C}}\,H_{2}+O(\unicode[STIX]{x1D70C}^{-2}),\end{eqnarray}$$

(6.8) $$\begin{eqnarray}\displaystyle W_{sp}^{(5)} & \rightarrow & \displaystyle -\frac{1}{2}\frac{\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,sp}}{\unicode[STIX]{x1D71B}}\unicode[STIX]{x1D70C}^{2}-s_{i}({\mathcal{Q}}_{\unicode[STIX]{x1D703},2,sp}+\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}H_{2})\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D71B}}+\hat{\unicode[STIX]{x1D701}}[(\unicode[STIX]{x1D6FD}_{l,1}A+\unicode[STIX]{x1D6FD}_{d,1}A_{\unicode[STIX]{x1D6F7}}+\unicode[STIX]{x1D6FD}_{ad,1}a_{\unicode[STIX]{x1D6F7}}A)\cos X\nonumber\\ \displaystyle & & \displaystyle -\,(\unicode[STIX]{x1D705}_{l,1}A+\unicode[STIX]{x1D705}_{d,1}A_{\unicode[STIX]{x1D6F7}}+\unicode[STIX]{x1D705}_{ad,1}a_{\unicode[STIX]{x1D6F7}}A)\sin X]\ln |\unicode[STIX]{x1D70C}^{\ast }|\nonumber\\ \displaystyle & & \displaystyle +\,{\mathcal{W}}_{3,sp}+W_{5,sp}(X)+s_{i}\hat{\unicode[STIX]{x1D701}}A\frac{\cos X}{\unicode[STIX]{x1D70C}}\,H_{2}+O(\unicode[STIX]{x1D70C}^{-2}),\end{eqnarray}$$

with the expressions of $V_{5,sp}$ and $W_{5,sp}$ being given in appendix C.

The radial advections ${\mathcal{U}}_{2}\,\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ and ${\mathcal{U}}_{2}\,W_{\unicode[STIX]{x1D70C}}^{(2)}$ , the spiralling motions ${\mathcal{P}}_{sp}^{(2)}$ , $V_{sp}^{(5)}$ and $W_{sp}^{(5)}$ , and the forcing $\unicode[STIX]{x1D6F4}_{v}$ bring about a secularity with respect to the phase $X$ in the angular and axial momentum equations (6.2) and (6.3), which makes their integration difficult. The second-order outer flow gives a spiralling shift $h_{2}(\unicode[STIX]{x1D702})=\overline{U}_{2}(\unicode[STIX]{x1D702})/[m\unicode[STIX]{x1D6FA}_{0}(\unicode[STIX]{x1D702})]\unicode[STIX]{x1D6E9}$ . This deformation can be found heuristically. Indeed, an air particle located on a streamline $\unicode[STIX]{x1D702}$ undergoes a radius variation $\unicode[STIX]{x1D6FF}r$ after a time interval $\unicode[STIX]{x1D6FF}t$ , such that $\unicode[STIX]{x1D6FF}r=\unicode[STIX]{x1D716}\unicode[STIX]{x1D6FF}h_{2}(\unicode[STIX]{x1D702})=\unicode[STIX]{x1D716}\overline{U}_{2}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6FF}t$ , while it rotates by an angle $\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}=\unicode[STIX]{x1D6FA}_{0}(\unicode[STIX]{x1D702})\unicode[STIX]{x1D6FF}t$ . An expression for $H_{2}$ can be determined by removing the advection terms in (6.2) and (6.3) as

(6.9) $$\begin{eqnarray}\displaystyle H_{2}(\unicode[STIX]{x1D6E9})=\frac{{\mathcal{U}}_{2}}{{\mho}_{0}}\unicode[STIX]{x1D6E9}, & & \displaystyle\end{eqnarray}$$

which can be easily matched with $h_{2}$ as $\unicode[STIX]{x1D702}\rightarrow 0$ : $H_{2}=h_{2}(0)$ in rescaled variables. The deformation $H_{2}$ has been found a second time in appendix F by integrating the streamline equations. Note that inside the separatrices we have $H_{2}^{\odot }=H_{2}$ .

The remaining secularity must be eliminated by determining $V_{sp}^{(5)}$ and $W_{sp}^{(5)}$ properly. Let us now define $V_{sp}^{(5)}$ and $W_{sp}^{(5)}$ as

(6.10) $$\begin{eqnarray}\displaystyle & \displaystyle V_{sp}^{(5)}=ss_{i}\unicode[STIX]{x1D701}_{z,1,sp}\,{\mathcal{K}}_{0,\unicode[STIX]{x1D70C}^{-1}}A^{1/2}+s_{i}({\mathcal{Q}}_{z,2,sp}+2H_{2})\unicode[STIX]{x1D70C}+{\mathcal{V}}_{3,sp}+V_{5,sp}(X)+\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}, & \displaystyle\end{eqnarray}$$

(6.11) $$\begin{eqnarray}\displaystyle & \displaystyle W_{sp}^{(5)}=-ss_{i}\frac{\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,sp}}{\unicode[STIX]{x1D71B}}\,{\mathcal{K}}_{0,\unicode[STIX]{x1D70C}^{-1}}A^{1/2}-s_{i}[{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,sp}+(\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}+\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0})H_{2}]\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D71B}}+{\mathcal{W}}_{3,sp}+W_{5,sp}(X)+\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}. & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The unknown function ${\mathcal{K}}_{0}\,A^{1/2}$ must tend to $|\unicode[STIX]{x1D70C}|\gg 1$ as $\unicode[STIX]{x1D70C}\rightarrow \unicode[STIX]{x1D70C}_{B}^{\pm }.$ Substituting the velocities (6.10) and (6.11) into (6.2) and (6.3) simplifies the momentum equations to

(6.12) $$\begin{eqnarray}\displaystyle & & \displaystyle A\sin X\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}+\unicode[STIX]{x1D70C}\,\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}X}^{(2)}-\unicode[STIX]{x1D713}_{X}^{(2)}=s_{i}{\mho}_{0}[\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}+V_{5,sp}]_{\unicode[STIX]{x1D6E9}}-\hat{\unicode[STIX]{x1D701}}\,{\mathcal{U}}_{2}\unicode[STIX]{x1D70C}\nonumber\\ \displaystyle & & \displaystyle \quad +\,s\unicode[STIX]{x1D6F4}_{q^{\prime }}{\mathcal{K}}_{0,\unicode[STIX]{x1D70C}^{-1}}A^{1/2}+s_{i}\unicode[STIX]{x1D6F4}_{v}+\unicode[STIX]{x1D706}^{\prime }(\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}-\hat{\unicode[STIX]{x1D701}})A+\unicode[STIX]{x1D6F1}_{2,\unicode[STIX]{x1D6E9}}+\unicode[STIX]{x1D6F1}_{2,X},\end{eqnarray}$$

(6.13) $$\begin{eqnarray}\displaystyle & & \displaystyle A\sin XW_{\unicode[STIX]{x1D70C}}^{(2)}+\unicode[STIX]{x1D70C}\,W_{X}^{(2)}-\unicode[STIX]{x1D713}_{X}^{(2)}=s_{i}{\mho}_{0}[\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}+W_{5,sp}]_{\unicode[STIX]{x1D6E9}}-\hat{\unicode[STIX]{x1D701}}\,{\mathcal{U}}_{2}\unicode[STIX]{x1D70C}\nonumber\\ \displaystyle & & \displaystyle \quad +\,s\unicode[STIX]{x1D6F4}_{q^{\prime }}{\mathcal{K}}_{0,\unicode[STIX]{x1D70C}^{-1}}A^{1/2}+s_{i}\unicode[STIX]{x1D6F4}_{v}+\unicode[STIX]{x1D706}^{\prime }\left(W_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}+\frac{\unicode[STIX]{x1D6F4}_{q}}{{\mathcal{U}}_{2}}\right)A+\unicode[STIX]{x1D6F1}_{2,X}.\end{eqnarray}$$

A small equivalent Richardson number at $r_{a}$ enables one to uncouple the axial motion. The vertical motion is thus absent from (6.12), which models the leading-order, non-trivial dynamics inside a two-dimensional, cylindrical, steady and unstratified critical layer (Haberman (Reference Haberman1972), Caillol & Maslowe (Reference Caillol and Maslowe2007), Caillol (Reference Caillol2014)). However, this equation possesses extra terms, owing to the spiralling motion generated by the mean radial velocity ${\mathcal{U}}_{2}$ , which makes this CL flow more complex. Solving (6.12) gives the second-order azimuthal velocity, then the resolution of (6.13) yields the axial velocity. To solve (6.12), a second change of radial variables $(\unicode[STIX]{x1D70C},X)\rightarrow (Z,x)$ is performed where

(6.14) $$\begin{eqnarray}\displaystyle Z={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}^{2}+A\cos X. & & \displaystyle\end{eqnarray}$$

Each streamline is described by a unique value of the variable $Z$ , but one value of $Z$ yields two streamlines located on either side of the CL axis, each one characterized by $s=\pm$ (cf. figure 2). The variable $Z$ increases from $Z=-A$ in the cat’s eye centre to $Z=A$ on the separatrices, to reach large values on the edges of the CL. See in figure 3 the two helical and spiralling sheets corresponding to the outer and inner separatrices for a $m=2$ CL and for an arbitrary profile $A(\unicode[STIX]{x1D6F7})$ . Inside the frame $(\unicode[STIX]{x1D70C},X)$ , the relationship (6.14) characterizes the classic cat’s eye CL pattern. Inside the standard frame $(r,\unicode[STIX]{x1D703})$ , the CL pattern is even more deformed with respect to the tangential basic flow; the symmetry with respect to the axis $r=r_{a}$ is lost. The CL thickness $\unicode[STIX]{x1D6FF}_{cl}$ scales as $\unicode[STIX]{x1D6FF}_{cl}=4r_{c}\,\unicode[STIX]{x1D70E}(A\unicode[STIX]{x1D716})^{1/2}$ , with $1\ll \unicode[STIX]{x1D70E}\ll \unicode[STIX]{x1D716}^{-1/2}$ ; $\unicode[STIX]{x1D70E}$ is the ratio of the CL thickness over the cat’s eye thickness, whose value remains undetermined by the nonlinear CL theory. The radius $\unicode[STIX]{x1D70C}$ expressed with the variables $(Z,x)$ is denoted $Z_{\unicode[STIX]{x1D70C}}(Z,x)=ss_{i}[2(Z-A\cos x)]^{1/2}$ . The absolute value of $\unicode[STIX]{x1D70C}$ on the separatrix is denoted $\unicode[STIX]{x1D70C}^{X}(x)=\sqrt{2A}[1-\cos x]^{1/2}$ . We use the superscript $^{\ast }$ to indicate normalization with respect to $A$ of $\unicode[STIX]{x1D70C}^{X}$ and $Z$ : $\unicode[STIX]{x1D70C}^{X\ast }=\unicode[STIX]{x1D70C}^{X}/A^{1/2}$ and $Z^{\ast }=Z/A$ .

Differentiating (6.12) with respect to $\unicode[STIX]{x1D70C}$ and introducing the variables $Z$ and $x$ , we get

(6.15) $$\begin{eqnarray}\displaystyle Z_{\unicode[STIX]{x1D70C}}\,\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}x}^{(2)}=s_{i}{\mho}_{0}\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6E9}}^{(5)}-\hat{\unicode[STIX]{x1D701}}\,{\mathcal{U}}_{2}+s\unicode[STIX]{x1D6F4}_{q^{\prime }}{\mathcal{K}}_{0}A^{1/2}+\unicode[STIX]{x1D706}^{\prime }Z_{\unicode[STIX]{x1D70C}}\,\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}Z}^{(2)}\,A. & & \displaystyle\end{eqnarray}$$

The average over $x$ at constant $Z$ of the quantity $q$ is defined by $\langle q(Z)\rangle =1/(2\unicode[STIX]{x03C0})\int _{0}^{2\unicode[STIX]{x03C0}}q(Z,x)\,\text{d}x$ . Let us decompose $\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}$ between its $x$ -averaged and oscillating parts:

(6.16) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}=\langle \unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}\rangle +\unicode[STIX]{x1D6FF}\!\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}H_{2}. & & \displaystyle\end{eqnarray}$$

Using the antisymmetry property of the forcing generated by the mean radial wave flux of the axial-vorticity radial gradient $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{-}=-\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}$ (cf. § 7), the inviscid secularity condition applied to (6.15) yields $\langle \unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}\rangle$ as follows:

(6.17) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{Z}\langle \unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}\rangle =s_{i}\left(\hat{\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}{\mathcal{K}}_{0}[Z^{\ast }]\right)\left\langle \frac{H_{2}}{Z_{\unicode[STIX]{x1D70C}}}\right\rangle . & & \displaystyle\end{eqnarray}$$

Equation (6.15), after the $\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}$ substitution, admits the general inviscid solution

(6.18) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}(Z,x)={\mathcal{F}}_{2}(Z^{\ast })+s_{i}\int _{\unicode[STIX]{x03C0}}^{x}\unicode[STIX]{x2202}_{Z}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}(Z,x_{1})+s\,\unicode[STIX]{x1D6EF}(Z^{\ast })\,G_{5}(Z^{\ast },x_{1})A^{-1/2}\,\text{d}x_{1}\,{\mathcal{U}}_{2}, & & \displaystyle\end{eqnarray}$$

where

(6.19a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}(Z^{\ast })=\hat{\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}{\mathcal{K}}_{0}[Z^{\ast }]\quad \text{and}\quad G_{5}(Z^{\ast },x)=\text{S}\text{M}[Z^{\ast }]-[2(Z^{\ast }-\cos x)]^{-1/2}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

All the mathematical functions needed in the paper are defined in appendix H.

The viscous secularity condition applied to (6.15), namely $\unicode[STIX]{x2202}_{Z}\langle \unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}\rangle =0$ , as well as the inner axial-vorticity matching with the outer flow permit one to define ${\mathcal{F}}_{2}$ :

(6.20) $$\begin{eqnarray}\displaystyle {\mathcal{F}}_{2}(Z^{\ast })=ss_{i}\hat{\unicode[STIX]{x1D701}}\,{\mathcal{K}}_{0}[Z^{\ast }]\,A^{1/2}+s_{i}\hat{{\mathcal{Q}}}_{z,1},\quad \text{where}~{\mathcal{K}}_{0}[Z]=\sqrt{2Z}+K_{0}[Z]. & & \displaystyle\end{eqnarray}$$

From (6.7)–(6.8) and (6.10)–(6.11), the matching on the CL edges requires the following asymptotic behaviour for $\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}$ :

(6.21) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}(\unicode[STIX]{x1D70C},X)\rightarrow s_{i}\hat{\unicode[STIX]{x1D701}}\left(\unicode[STIX]{x1D70C}+A\frac{\cos X}{\unicode[STIX]{x1D70C}}\right)H_{2}-\frac{1}{2}\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}}{{\mathcal{U}}_{2}}\,\cos X\,AH_{2}\quad \text{as}~\unicode[STIX]{x1D70C}\rightarrow \unicode[STIX]{x1D70C}_{B}^{\pm }. & & \displaystyle\end{eqnarray}$$

According to (6.17), the $x$ -average of $\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}$ asymptotes as

(6.22) $$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}(Z)\rangle & \rightarrow & \displaystyle s_{i}\hat{\unicode[STIX]{x1D701}}\,\left(\unicode[STIX]{x1D70C}+A\frac{\cos X}{\unicode[STIX]{x1D70C}}+O(\unicode[STIX]{x1D70C}^{-3})\right)H_{2}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}}{{\mathcal{U}}_{2}}\left(\frac{1}{2}\unicode[STIX]{x1D70C}^{2}+A\cos X+O(\unicode[STIX]{x1D70C}^{-2})\right)H_{2}\quad \text{as}~\unicode[STIX]{x1D70C}\rightarrow \unicode[STIX]{x1D70C}_{B}^{\pm }.\end{eqnarray}$$

As a result, the expression for $\unicode[STIX]{x1D6FF}\!\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}$ is first determined as

(6.23) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}\!\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5)}=\frac{1}{2}\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}}{{\mathcal{U}}_{2}}\,A\cos x. & & \displaystyle\end{eqnarray}$$

Second, we deduce that the critical layer generates a small distorted and spiralling mean outer flow near $r_{c}$ that opposes the $\unicode[STIX]{x1D6E9}$ -gradient of the mean vorticity radial gradient $(\unicode[STIX]{x1D701}_{z,1},\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1})$ :

(6.24) $$\begin{eqnarray}\displaystyle \frac{\overline{V}(r,\unicode[STIX]{x1D6E9})}{\unicode[STIX]{x1D71A}_{c}r_{c}}=\frac{\overline{W}(r,\unicode[STIX]{x1D6E9})}{\unicode[STIX]{x1D71B}\unicode[STIX]{x1D71A}_{c}r_{c}}=-\frac{1}{2}\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}}{{\mathcal{U}}_{2}}\left(\frac{r}{r_{c}}-1\right)^{2}H_{2}(\unicode[STIX]{x1D6E9})\,\unicode[STIX]{x1D716}^{1/2}; & & \displaystyle\end{eqnarray}$$

the resulting vorticity radial gradient $(\hat{\unicode[STIX]{x1D701}}_{z,1},\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},1})$ is thus $\unicode[STIX]{x1D6E9}$ -independent.

The vorticity $\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ is invariant under the transformations ${\mathcal{U}}_{2}\rightarrow -{\mathcal{U}}_{2}$ and $x\rightarrow 2\unicode[STIX]{x03C0}-x$ . As a result, the CL study can be restricted to positive values of ${\mathcal{U}}_{2}$ . The function ${\mathcal{F}}_{2}$ in (6.20) is similar to that used previously (Caillol & Maslowe Reference Caillol and Maslowe2007; Caillol Reference Caillol2012, Reference Caillol2014) because it was again derived from the secularity condition applied to the viscous axial-vorticity equation. However, the axial-vorticity radial gradient $\hat{\unicode[STIX]{x1D701}}$ and the induced mean vorticity $\hat{{\mathcal{Q}}}_{z,1}$ are now corrected by the mean radial wave fluxes. In quasi-steady, three-dimensional helical or two-dimensional critical layers, each isoaxial-vorticity contour has been defined so far by a unique value of the variable $Z$ (Maslowe Reference Maslowe1986; Caillol & Grimshaw Reference Caillol and Grimshaw2007, Reference Caillol and Grimshaw2012; Caillol Reference Caillol2014, C17). Here, streamlines do not coincide with isovorticity lines because the CL flow has stronger three-dimensional features. The mean velocity ${\mathcal{U}}_{2}$ considerably alters the inner axial vorticity. The expression obtained in C17 is slightly different because it was derived from the inviscid secularity condition appearing at the next-order axial-vorticity equation. The chosen inner variables did not define weakly spiralling streamlines. As the leading-order mean radial velocity was $O(\unicode[STIX]{x1D716}^{3/2})$ , the present formulation applied to that study would imply taking $H_{2}=0$ and defining a first spiralling deviation: $H_{3}(\unicode[STIX]{x1D6E9})={\mathcal{U}}_{3}/{\mho}_{0}\unicode[STIX]{x1D6E9}$ . The same function ${\mathcal{F}}_{2}$ in (6.20) would be then deduced, but the integral in (6.18) would appear in the expression for the $O(\unicode[STIX]{x1D716})$ axial vorticity. The mean flow at $r_{c}$ would, however, be modified in the same way: $\hat{\unicode[STIX]{x1D701}}=\unicode[STIX]{x1D701}-\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{3}$ , $\hat{{\mathcal{Q}}}_{z,1}={\mathcal{Q}}_{z,1}-\unicode[STIX]{x1D6F4}_{v}/{\mathcal{U}}_{3}$ .

The function $G_{5}$ is singular on the separatrices and indeed the average $\langle 1/Z_{\unicode[STIX]{x1D70C}}\rangle$ is infinite there. The streamlines therefore need to be regularized near the meeting points of the separatrices. To do so, we consider that the corner angle made by both separatrices is equal to $180^{\circ }$ at this meeting point. For phases close to $0\,[2\unicode[STIX]{x03C0}]$ (that is, for $|x|\leqslant \unicode[STIX]{x1D707}$ or $|2\unicode[STIX]{x03C0}-x|\leqslant \unicode[STIX]{x1D707}$ , with $\unicode[STIX]{x1D707}\ll 1$ ), and for $|Z-A|\ll 1$ , we assume that the inner radius $\unicode[STIX]{x1D70C}$ has the following behaviour:

(6.25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D70C}=ss_{i}\sqrt{2}[Z-A(1-b\,x^{1+\unicode[STIX]{x1D6FE}})]^{1/2},\quad \text{with}~b=(1-\cos \unicode[STIX]{x1D707})\unicode[STIX]{x1D707}^{-(1+\unicode[STIX]{x1D6FE})}\simeq \frac{\unicode[STIX]{x1D707}^{1-\unicode[STIX]{x1D6FE}}}{2}. & & \displaystyle\end{eqnarray}$$

Regularization requires an exponent $\unicode[STIX]{x1D6FE}\leqslant 0$ . Integrating $\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ over $\unicode[STIX]{x1D70C}$ yields the second-order azimuthal velocity

(6.26) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}}^{(2)}(Z,x) & = & \displaystyle \hat{\unicode[STIX]{x1D701}}\left(\int _{\infty }^{Z^{\ast }}\frac{K_{0}[z]\,\text{d}z\,A}{[2(z-\cos x)]^{1/2}}+[Z(Z-A\cos x)]^{1/2}+A\cos x\ln [\unicode[STIX]{x1D6EC}(Z,x)]\right)\nonumber\\ \displaystyle & & \displaystyle +\,ss_{i}\int _{\infty }^{Z^{\ast }}\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\frac{\unicode[STIX]{x1D6EF}(z)\,\text{d}z}{Z_{\unicode[STIX]{x1D70C}}(Az,x)}A^{1/2}{\mathcal{U}}_{2}+s_{i}\hat{{\mathcal{Q}}}_{z,1}Z_{\unicode[STIX]{x1D70C}}+V_{2}(x),\end{eqnarray}$$

with

(6.27) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}\unicode[STIX]{x1D6EC}(Z,x)=\sqrt{Z}+\sqrt{Z-A\cos x}\quad \text{and}\\ \displaystyle V_{2}(x)=-\left[a+1-\frac{\hat{\unicode[STIX]{x1D701}}}{2}(1+3\unicode[STIX]{x1D71B}^{2})-\unicode[STIX]{x1D71B}\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}\right]\frac{A\cos x}{1+\unicode[STIX]{x1D71B}^{2}}.\end{array}\right\} & & \displaystyle\end{eqnarray}$$

The solvability condition (4.9) implies $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6E9}}^{(2)}=0$ , which leads to $A_{\unicode[STIX]{x1D6E9}}=a_{\unicode[STIX]{x1D6E9}}={\mathcal{U}}_{2,\unicode[STIX]{x1D6E9}}=0$ . The leading-order induced mean flow at $r_{a}$ : ${\mathcal{V}}_{1},{\mathcal{W}}_{1},{\mathcal{Q}}_{z,1},{\mathcal{Q}}_{\unicode[STIX]{x1D703},1},\mathfrak{P}_{1}$ , ${\mathcal{V}}_{2}$ , ${\mathcal{W}}_{2}$ , $\mathfrak{P}_{2}$ and ${\mathcal{U}}_{2}$ , the phase jump $a$ and the wave amplitude $A$ are therefore independent of $\unicode[STIX]{x1D6E9}$ . The mean velocity is thus very weakly spiralling since it starts to depend on $\unicode[STIX]{x1D6E9}$ at the third order, whereas the mean vorticity becomes dependent on $\unicode[STIX]{x1D6E9}$ at the second order and the mean vorticity radial gradient at the first order.

Integrating once again over $\unicode[STIX]{x1D70C}$ gives $\unicode[STIX]{x1D713}^{(2)}$ as follows

(6.28) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{i}^{(2)}(Z,x) & = & \displaystyle \hat{\unicode[STIX]{x1D701}}\left(\frac{ss_{i}}{6}(2Z)^{3/2}+\{Z_{\unicode[STIX]{x1D70C}}\ln [\unicode[STIX]{x1D6EC}(Z,x)]-ss_{i}\sqrt{2Z}\}A\cos x\right)\nonumber\\ \displaystyle & & \displaystyle +\,\hat{\unicode[STIX]{x1D701}}\left(Z_{\unicode[STIX]{x1D70C}}\!\int _{\infty }^{Z^{\ast }}\frac{K_{0}[z]\,\text{d}z}{\sqrt{2(z-\cos x)}}-ss_{i}\int _{\infty }^{Z^{\ast }}K_{0}[z]\,\text{d}z\,A^{1/2}\right)A\nonumber\\ \displaystyle & & \displaystyle -\,\frac{s_{i}}{2}\unicode[STIX]{x1D6F4}_{q^{\prime }}\ln Z^{\ast }\,A\sin x+ss_{i}\left(Z_{\unicode[STIX]{x1D70C}}\int _{\infty }^{Z^{\ast }}\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\frac{\unicode[STIX]{x1D6EF}(z)\,\text{d}z}{Z_{\unicode[STIX]{x1D70C}}(Az,x)}\right.\nonumber\\ \displaystyle & & \displaystyle -\,\int _{\infty }^{Z^{\ast }}\left(\hat{\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}K_{0}[z]\right)\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\,\text{d}z\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}\int _{\infty }^{Z^{\ast }}\sqrt{2z}\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}+\frac{\sin x}{2z}\,\text{d}z\right)A^{1/2}{\mathcal{U}}_{2}\nonumber\\ \displaystyle & & \displaystyle +\,s_{i}\hat{{\mathcal{Q}}}_{z,1}(Z-A\cos x)+V_{2}(x)Z_{\unicode[STIX]{x1D70C}}.\end{eqnarray}$$

The matchings of (6.12) and (6.13) on the edges of the CL give two distinct expressions for the integration function $\unicode[STIX]{x1D6F1}_{2}$ :

(6.29) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{2,X} & = & \displaystyle s_{i}\{\hat{{\mathcal{Q}}}_{z,1}\sin X-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6F4}_{q^{\prime }}\ln (4A)\cos X\}A-s_{i}{\mho}_{0}V_{5,sp,\unicode[STIX]{x1D6E9}}(X)-\unicode[STIX]{x1D6F1}_{2,\unicode[STIX]{x1D6E9}}\nonumber\\ \displaystyle & = & \displaystyle s_{i}\{\hat{{\mathcal{Q}}}_{z,1}\sin X-{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6F4}_{q^{\prime }}\ln (4A)\cos X\}A-s_{i}{\mho}_{0}W_{5,sp,\unicode[STIX]{x1D6E9}}(X).\end{eqnarray}$$

Pressure matching on the edges of the CL gives a third expression. Equating the three expressions comes down to determining the $\unicode[STIX]{x1D6E9}$ -gradients of the first-order coefficients $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D705}$ . From the definitions of the Frobenius series terms in appendix A, the third expression for $\unicode[STIX]{x1D6F1}_{2}$ is

(6.30) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{2}(X) & = & \displaystyle -s_{i}\left\{\hat{{\mathcal{Q}}}_{z,1}\cos X+\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}}{2}[\ln (4A)-2]\sin X\right\}A\nonumber\\ \displaystyle & & \displaystyle +\,s_{i}{\mho}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left[\left(\unicode[STIX]{x1D6FC}_{l,1}-\unicode[STIX]{x1D717}\unicode[STIX]{x1D6FD}_{l,1}+\unicode[STIX]{x1D71B}\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},1}+(1-\unicode[STIX]{x1D71B}\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0})\frac{\hat{\unicode[STIX]{x1D701}}_{z,1}}{\hat{\unicode[STIX]{x1D701}}}\right.\right.\nonumber\\ \displaystyle & & \displaystyle +\,\frac{{\mho}_{0}^{2}}{6\hat{\unicode[STIX]{x1D701}}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}\{\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }+[3(1-\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}\unicode[STIX]{x1D71B})-2\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}^{2}]\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }\}(0)\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{{\mho}_{0}^{4}}{120\hat{\unicode[STIX]{x1D701}}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}\{\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{IV}+[5(1-\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}\unicode[STIX]{x1D71B})-4\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}^{2}]\unicode[STIX]{x1D701}_{z,1}^{IV}\}(0)\right)A\nonumber\\ \displaystyle & & \displaystyle +\left.(\unicode[STIX]{x1D6FC}_{d,1}-\unicode[STIX]{x1D717}\unicode[STIX]{x1D6FD}_{d,1})A_{\unicode[STIX]{x1D6F7}}+(\unicode[STIX]{x1D6FC}_{ad,1}-\unicode[STIX]{x1D717}\unicode[STIX]{x1D6FD}_{ad,1})a_{\unicode[STIX]{x1D6F7}}A\right]\frac{\sin X}{1+\unicode[STIX]{x1D71B}^{2}}\nonumber\\ \displaystyle & & \displaystyle +\,s_{i}{\mho}_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left[\left(\unicode[STIX]{x1D6FE}_{l,1}-\unicode[STIX]{x1D717}\unicode[STIX]{x1D705}_{l,1}-\frac{{\mho}_{0}}{\hat{\unicode[STIX]{x1D701}}}\left(\frac{\hat{\unicode[STIX]{x1D701}}}{2}\unicode[STIX]{x1D71B}^{2}+\unicode[STIX]{x1D71B}\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}-1\right)\unicode[STIX]{x1D701}_{z,1,\unicode[STIX]{x1D6E9}}^{\prime }+\frac{{\mho}_{0}}{2}\unicode[STIX]{x1D71B}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,\unicode[STIX]{x1D6E9}}^{\prime }\right.\right.\nonumber\\ \displaystyle & & \displaystyle -\left.\frac{{\mho}_{0}^{3}}{24\hat{\unicode[STIX]{x1D701}}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{3}\{\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime \prime }+[4(1-\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}\unicode[STIX]{x1D71B})-3\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}^{2}]\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }\}(0)\right)A\nonumber\\ \displaystyle & & \displaystyle +\left.(\unicode[STIX]{x1D6FE}_{d,1}-\unicode[STIX]{x1D717}\unicode[STIX]{x1D705}_{d,1})A_{\unicode[STIX]{x1D6F7}}+(\unicode[STIX]{x1D6FE}_{ad,1}-\unicode[STIX]{x1D717}\unicode[STIX]{x1D705}_{ad,1})a_{\unicode[STIX]{x1D6F7}}A\right]\frac{\cos X}{1+\unicode[STIX]{x1D71B}^{2}}+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{6}),\end{eqnarray}$$

with $\unicode[STIX]{x1D717}=\hat{\unicode[STIX]{x1D701}}(2+3\unicode[STIX]{x1D71B}^{2})+\unicode[STIX]{x1D71B}\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}-1$ .

The double radial gradients of the first-order mean vorticity ( $\unicode[STIX]{x1D701}_{z,1}^{\prime }$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime }$ ) cannot be calculated here because they are deduced from the next-order radial wave fluxes. The higher gradients $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }$ , $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime \prime },\ldots$ are calculated through higher-order wave fluxes. As the $\unicode[STIX]{x1D6E9}$ -dependence increases algebraically with the order, $\unicode[STIX]{x1D701}_{z,1}^{\prime }$ and $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime }$ are then assumed to have a quadratic dependence on $\unicode[STIX]{x1D6E9}$ . For higher-order mean-flow radial derivatives, the mean flow $\unicode[STIX]{x1D6E9}$ -expansion is carried out up to the fifth power and is assumed to converge rapidly. Supposing the first-order coefficients $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ , $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D705}$ linear in $\unicode[STIX]{x1D6E9}$ , we deduce that $\unicode[STIX]{x1D6F1}_{2}$ is $\unicode[STIX]{x1D6E9}$ -independent and the velocities $V_{5,sp}$ and $W_{5,sp}$ are linear in $\unicode[STIX]{x1D6E9}$ too. The outer flow asymptotic solution of § 3.2 then imposes

(6.31) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }(0)=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }(0)=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }(0)=\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime \prime }(0)=0. & & \displaystyle\end{eqnarray}$$

Introducing the expressions (C 1) and (C 2) of $V_{5,sp}$ and $W_{5,sp}$ from appendix C in the equality (6.29), the sums of the $\unicode[STIX]{x1D6E9}$ -gradients of the coefficients $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D705}$ appear as functions of the successive $\unicode[STIX]{x1D6E9}$ -derivatives of the mean flow

(6.32) $$\begin{eqnarray}\displaystyle & & \displaystyle \hat{\unicode[STIX]{x1D701}}(\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}+\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left[\unicode[STIX]{x1D6FD}_{l,1}+\unicode[STIX]{x1D6FD}_{d,1}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}+\unicode[STIX]{x1D6FD}_{ad,1}a_{\unicode[STIX]{x1D6F7}}\right]=\frac{{\mho}_{0}^{2}}{6}\,\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{3}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }-(3\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+2\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }](0)\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}^{4}}{120}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{IV}-(5\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+4\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{IV}](0)+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{7}),\end{eqnarray}$$

(6.33) $$\begin{eqnarray}\displaystyle & & \displaystyle \hat{\unicode[STIX]{x1D701}}(\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}+\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0})\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left[\unicode[STIX]{x1D705}_{l,1}+\unicode[STIX]{x1D705}_{d,1}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}+\unicode[STIX]{x1D705}_{ad,1}a_{\unicode[STIX]{x1D6F7}}\right]=\frac{{\mho}_{0}}{2}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime }-(2\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{\prime }]\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}^{3}}{24}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime \prime }-(4\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+3\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }](0)+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{6}).\end{eqnarray}$$

Then, comparing (6.29) and (6.30), we find that the right-hand side of (6.33) vanishes and we get

(6.34) $$\begin{eqnarray}\displaystyle {\mho}_{0}\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\left[\unicode[STIX]{x1D6FD}_{l,1}+\unicode[STIX]{x1D6FD}_{d,1}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}+\unicode[STIX]{x1D6FD}_{ad,1}a_{\unicode[STIX]{x1D6F7}}\right]=\unicode[STIX]{x1D6F4}_{q^{\prime }}. & & \displaystyle\end{eqnarray}$$

Next, the combination of (6.32) and (6.34), then the right-hand side of (6.33) yield two relationships, a first one between the odd-order $\unicode[STIX]{x1D6E9}$ -derivatives of the high-order radial derivatives of the mean flow: $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }$ , $\unicode[STIX]{x1D701}_{z,1}^{IV}$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{IV},\ldots ,$ a second one between the even-order $\unicode[STIX]{x1D6E9}$ -derivatives of $\unicode[STIX]{x1D701}_{z,1}^{\prime }$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime }$ , $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime \prime },\ldots$

(6.35) $$\begin{eqnarray}\displaystyle & & \displaystyle \frac{{\mho}_{0}^{3}}{6}\,\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{3}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }-(3\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+2\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }](0)\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}^{5}}{120}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{IV}-(5\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+4\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{IV}](0)+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{7})=\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F4}_{q^{\prime }},\end{eqnarray}$$

(6.36) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime }-(2\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{\prime }]=\frac{{\mho}_{0}^{2}}{12}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}[\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime \prime }-(4\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}+3\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B})\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }](0)+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{6}). & & \displaystyle\end{eqnarray}$$

According to the expressions of the coefficients $c_{j,1,0}$ , $j=l,d$ and $ad$ in appendix A, the no-jump condition of (4.15) imposes

(6.37) $$\begin{eqnarray}\displaystyle [J_{1,c}]_{-}^{+}=0. & & \displaystyle\end{eqnarray}$$

The no-jump condition (4.14b ) is trivially satisfied, but insuring the no-jump condition (4.14a ) and the condition (4.16) impose the existence of two small logarithmic-order outer flows to respective orders: $O(\unicode[STIX]{x1D716}\ln \unicode[STIX]{x1D716}\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+})$ and $O(\unicode[STIX]{x1D716}^{3/2}\ln \unicode[STIX]{x1D716}\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+})$ and whose radial velocity expressions are

(6.38) $$\begin{eqnarray}\displaystyle U_{r}=\ln \unicode[STIX]{x1D716}\,\left\{\left(\frac{\unicode[STIX]{x1D6FC}_{h,0a}}{r_{c}}\unicode[STIX]{x1D719}_{a}+\unicode[STIX]{x1D719}_{b}\right)\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{[\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}}+\unicode[STIX]{x1D716}^{1/2}\left(\frac{\unicode[STIX]{x1D6FE}_{h,1a}}{r_{c}}\unicode[STIX]{x1D719}_{a}+\unicode[STIX]{x1D705}_{h,1a}\unicode[STIX]{x1D719}_{b}\right)\right\}U_{0}\cos \unicode[STIX]{x1D709}. & & \displaystyle\end{eqnarray}$$

For simplicity’s sake, we will not mention the other modifications brought about by these small corrections in the first-order inner flow. After the coefficients $f_{j,1,0}$ , $j=l,d$ and $ad$ in appendix A, the condition (4.16) implies the following distortion of the coefficient $\unicode[STIX]{x1D705}_{h,1a}$

(6.39) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x1D705}_{h,1a}]_{-}^{+}=(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}. & & \displaystyle\end{eqnarray}$$

In order to derive the most general evolution equations, any dependence relationships must be avoided between $A$ , $A_{\unicode[STIX]{x1D6F7}}$ and $a_{\unicode[STIX]{x1D719}}A$ , which implies

(6.40) $$\begin{eqnarray}\displaystyle {\mho}_{0}\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D6FD}_{l,1,\unicode[STIX]{x1D6E9}}^{s}=\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s},\quad \unicode[STIX]{x1D705}_{l,1,\unicode[STIX]{x1D6E9}}=0,\quad \unicode[STIX]{x1D6FD}_{j,1\unicode[STIX]{x1D6E9}}=0,\quad \text{and}\quad \unicode[STIX]{x1D705}_{j,1,\unicode[STIX]{x1D6E9}}=0,\quad j=d~\text{and}~ad. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The general solution of $W_{i}^{(2)}$ is determined by combining (6.12) and (6.13). The particular solution is obtained through the viscous secularity condition $\unicode[STIX]{x2202}_{Z}\langle W_{i,\unicode[STIX]{x1D70C}}^{(2)}-\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}+\unicode[STIX]{x1D701}Z_{\unicode[STIX]{x1D70C}}\rangle =0$ and the CL edge matching; the axial velocity found is then roughly proportional to the azimuthal velocity, once again highlighting the helical motion inside the nonlinear critical layer

(6.41) $$\begin{eqnarray}\displaystyle W_{i}^{(2)}(Z,x)=\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}}^{(2)}(Z,x)-\unicode[STIX]{x1D701}Z-ss_{i}\frac{J_{1,c}}{\unicode[STIX]{x1D71B}^{2}}{\mathcal{K}}_{0}[Z^{\ast }]A^{1/2}+{\mathcal{W}}_{2}. & & \displaystyle\end{eqnarray}$$

Note that the matchings of $V^{(2)}$ and $W^{(2)}$ on the CL edges with the velocities $U_{\unicode[STIX]{x1D703}}^{(1)}$ and $U_{z}^{(1)}$ , respectively, give the relationships (6.34) and (6.35).

The expression (6.5) for the radial velocity $U^{(2)}$ can then be calculated as

(6.42) $$\begin{eqnarray}\displaystyle U^{(2)}(Z,x) & = & \displaystyle \hat{\unicode[STIX]{x1D701}}(1+\unicode[STIX]{x1D71B}^{2})\left(Z_{\unicode[STIX]{x1D70C}}\,\int _{\infty }^{Z^{\ast }}\frac{K_{0}[z]\,\text{d}z\,A^{3/2}}{|Z_{\unicode[STIX]{x1D70C}}(Az,x)|^{3}}\right.\nonumber\\ \displaystyle & & \displaystyle +\left.\left\{\ln [\unicode[STIX]{x1D6EC}(Z,x)]+\frac{1}{2}\right\}Z_{\unicode[STIX]{x1D70C}}-ss_{i}\sqrt{2Z}\right)A\sin x\nonumber\\ \displaystyle & & \displaystyle -\,ss_{i}(1+\unicode[STIX]{x1D71B}^{2})\left(Z_{\unicode[STIX]{x1D70C}}\int _{\infty }^{Z^{\ast }}\unicode[STIX]{x1D6EF}(z)\frac{G_{5}(z,x)}{Z_{\unicode[STIX]{x1D70C}}(Az,x)}\,\text{d}z\right.\nonumber\\ \displaystyle & & \displaystyle -\,Z_{\unicode[STIX]{x1D70C}}\int _{\infty }^{Z^{\ast }}\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\frac{\unicode[STIX]{x1D6EF}(z)\,\text{d}z}{Z_{\unicode[STIX]{x1D70C}}(Az,x)^{3}}A\sin x\nonumber\\ \displaystyle & & \displaystyle -\,\int _{\infty }^{Z^{\ast }}\left(\hat{\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}K_{0}[z]\right)G_{5}(z,x)\,\text{d}z\nonumber\\ \displaystyle & & \displaystyle +\left.\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}\int _{\infty }^{Z^{\ast }}\sqrt{2z}\,G_{5}(z,x)+\frac{\cos x}{2z}\,\text{d}z\right)A^{1/2}{\mathcal{U}}_{2}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{s_{i}}{2}(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D6F4}_{q^{\prime }}\left((\ln Z^{\ast }-2)\,\cos x-\frac{\sin ^{2}x}{Z^{\ast }}\right)A\nonumber\\ \displaystyle & & \displaystyle +\,[(1-\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}^{2})A\sin x-(1+\unicode[STIX]{x1D71B}^{2})V_{2}^{\prime }(x)+{\mathcal{U}}_{2}^{\prime }]Z_{\unicode[STIX]{x1D70C}}\nonumber\\ \displaystyle & & \displaystyle -\,ss_{i}J_{1,c}(K_{1}[Z^{\ast },x]+\ln [\unicode[STIX]{x1D6EC}(Z,x)])A\sin x+\unicode[STIX]{x1D6F6}_{2}(x)+s_{i}{\mathcal{U}}_{3}.\end{eqnarray}$$

The matching of $U^{(2)}$ with the outer flow gives the integration function

(6.43) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}_{2}(x) & = & \displaystyle -s_{i}\{[\unicode[STIX]{x1D6FD}_{l,1}+d_{l,1,0}+(\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D714}d,1}+d_{\unicode[STIX]{x1D714}d,1,0}){\mho}_{1,\unicode[STIX]{x1D6F7}}\unicode[STIX]{x1D70F}_{1}]A+(\unicode[STIX]{x1D6FD}_{d,1}+d_{d,1,0})A_{\unicode[STIX]{x1D6F7}}\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x1D6FD}_{ad,1}+d_{ad,1,0})a_{\unicode[STIX]{x1D6F7}}A]\sin x-s_{i}[(\unicode[STIX]{x1D705}_{l,1}+g_{l,1,0})A+(\unicode[STIX]{x1D705}_{d,1}+g_{d,1,0})A_{\unicode[STIX]{x1D6F7}}\nonumber\\ \displaystyle & & \displaystyle +\,(\unicode[STIX]{x1D705}_{ad,1}+g_{ad,1,0})a_{\unicode[STIX]{x1D6F7}}A]\cos x\}-\unicode[STIX]{x1D705}_{h,1}A\cos (2x),\end{eqnarray}$$

with the different coefficients $d_{j,1,0}$ and $g_{j,1,0}$ , $j=l,d,ad$ and $\unicode[STIX]{x1D714}d$ being given in appendix A.

Owing to the extra terms in (6.18) generated by the velocity ${\mathcal{U}}_{2}$ and the mean radial wave fluxes, the symmetry with respect to the CL axis is broken due to the appearance of a Fourier sine series in $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}(\unicode[STIX]{x1D70C},x)$ , $V^{(2)}$ and $W^{(2)}$ , which allows for non-zero leading-order inviscid mean radial wave fluxes, since the zeroth-order radial velocity is simply taken as $U^{(0)}\equiv A\sin x$ . The wave/vortex interaction is thus enhanced in the quasi-steady regime when the wave envelope is taken with a short height scale $O(\unicode[STIX]{x1D716}^{-1/2})$ , whereas the scale $O(\unicode[STIX]{x1D716}^{-1})$ postpones the appearance of the inviscid mean wave fluxes to the next order.

6.2 Within the separatrices

Here we solve the second-order inner-flow equations (6.2), (6.3) inside the cat’s eye and match the solution on the separatrices with the inner flow from the preceding subsection. First, we match the leading-order vorticity, followed by pressure and velocity. We will finally obtain an unique inviscid CL solution at the first non-trivial order.

The second-order velocity inside the cat’s eye is defined through the function $\unicode[STIX]{x1D713}^{(2,\odot )}$ as

(6.44) $$\begin{eqnarray}\displaystyle V^{(2,\odot )}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70C}^{2}-s_{i}{\mathcal{S}}_{1}\unicode[STIX]{x1D70C}+\{{\mathcal{V}}_{2}\}_{-}^{+}+\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}}^{(2,\odot )}. & & \displaystyle\end{eqnarray}$$

An explicit and exact solution for the axial vorticity $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2,\odot )}$ cannot be found. We then seek an approximate solution through a Galerkin method. According to the symmetry properties of $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ , the vorticity $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2,\odot )}$ is expanded as a Fourier sine series as follows:

(6.45) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2,\odot )}(\unicode[STIX]{x1D70C},X)={\mathcal{F}}_{2}^{\odot }(Z,\unicode[STIX]{x1D6F7})+\unicode[STIX]{x1D6EF}(1)\mathop{\sum }_{n=1}^{\infty }b_{n}(\overline{\unicode[STIX]{x1D70C}})\,\sin (nX)\,\frac{{\mathcal{U}}_{2}}{A^{1/2}}. & & \displaystyle\end{eqnarray}$$

The $n$ th-order term $b_{n}$ is itself a weighted sum of odd Chebyshev polynomials of the variable $\overline{\unicode[STIX]{x1D70C}}=\unicode[STIX]{x1D70C}/\unicode[STIX]{x1D70C}^{X}$ . Applying the Prandtl–Batchelor theorem inside the cat’s eye (cf. appendix E) leads to the following restrictive condition:

(6.46) $$\begin{eqnarray}\displaystyle {\mathcal{F}}_{2}^{\odot }(Z,\unicode[STIX]{x1D6F7})=Q^{(2,\odot )}(\unicode[STIX]{x1D6F7}). & & \displaystyle\end{eqnarray}$$

The fifth-order azimuthal and axial spiralling velocities are defined in the following way:

(6.47) $$\begin{eqnarray}\displaystyle V_{sp}^{(5,\odot )}=s_{i}(2H_{2}+\{{\mathcal{Q}}_{z,2,sp}\}_{-}^{+})\unicode[STIX]{x1D70C}+\{{\mathcal{V}}_{3,sp}\}_{-}^{+}+\{V_{5,sp}(x)\}_{-}^{+}+\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5,\odot )}, & & \displaystyle\end{eqnarray}$$

(6.48) $$\begin{eqnarray}\displaystyle W_{sp}^{(5,\odot )}=-s_{i}((\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}+\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0})H_{2}+\{{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,sp}\}_{-}^{+})\frac{\unicode[STIX]{x1D70C}}{\unicode[STIX]{x1D71B}}+\{{\mathcal{W}}_{3,sp}\}_{-}^{+}+\{W_{5,sp}(x)\}_{-}^{+}+\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5,\odot )}. & & \displaystyle\end{eqnarray}$$

Differentiating the equation (6.2) with respect to $\unicode[STIX]{x1D70C}$ and substituting the expression (6.47) for $V_{sp}^{(5,\odot )}$ , we obtain the axial-vorticity equation inside the separatrices in the frame $(Z,x)$ as follows:

(6.49) $$\begin{eqnarray}\displaystyle Z_{\unicode[STIX]{x1D70C}}\,\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}x}^{(2,\odot )}=s_{i}{\mho}_{0}\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D6E9}}^{(5,\odot )}-\hat{\unicode[STIX]{x1D701}}\,{\mathcal{U}}_{2}+\unicode[STIX]{x1D706}^{\prime }Z_{\unicode[STIX]{x1D70C}}\,\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}Z}^{(2,\odot )}\,A. & & \displaystyle\end{eqnarray}$$

The velocity $\unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5,\odot )}$ is defined by introducing the function $G_{5}^{\odot }$ such that

(6.50) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{sp,\unicode[STIX]{x1D70C}}^{(5,\odot )}(Z,x,\unicode[STIX]{x1D6E9})=s_{i}\hat{\unicode[STIX]{x1D701}}Z_{\unicode[STIX]{x1D70C}}H_{2}+s\,\unicode[STIX]{x1D6EF}(1)\int _{1}^{Z^{\ast }}G_{5}^{\odot }(z,x)\,\text{d}z\,A^{1/2}H_{2}+V_{5,sp}^{\odot }(x,\unicode[STIX]{x1D6E9}), & & \displaystyle\end{eqnarray}$$

which results in the solution $\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2,\odot )}$ of (6.49):

(6.51) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2,\odot )}(Z,x)=Q^{(2,\odot )}+ss_{i}\,\unicode[STIX]{x1D6EF}(1)\int _{\unicode[STIX]{x03C0}}^{x}G_{5}^{\odot }(Z^{\ast },x_{1})\,\text{d}x_{1}\,\frac{{\mathcal{U}}_{2}}{A^{1/2}}. & & \displaystyle\end{eqnarray}$$

We identify $G_{5}^{\odot }$ by differentiating the expression (6.45) for $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2,\odot )}$ with respect to $x$ as

(6.52) $$\begin{eqnarray}\displaystyle G_{5}^{\odot }(Z^{\ast },x)=ss_{i}\mathop{\sum }_{n=1}^{\infty }[b_{n}[\overline{\unicode[STIX]{x1D70C}}(Z^{\ast },x)]\,\sin (nx)]_{x},\quad \text{with}~\overline{\unicode[STIX]{x1D70C}}(Z^{\ast },x)=ss_{i}\left(\frac{Z^{\ast }-\cos x}{1-\cos x}\right)^{1/2}. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

The inviscid secularity condition applied to (6.49) is then straightforwardly satisfied on a streamline $Z\leqslant A$ . The axial-vorticity matching involving the expressions (6.18) and (6.45) leads to the determination of the terms $b_{n}(1)$ as follows:

(6.53) $$\begin{eqnarray}\displaystyle b_{n}(1)=\frac{2}{n}\langle G_{5}(1,x)\cos (n\,x)\rangle ,\quad \forall n\geqslant 1. & & \displaystyle\end{eqnarray}$$

The same matching also yields the first-order mean vorticity jump and the uniform cat’s eye vorticity $Q^{(2,\odot )}$ as

(6.54a-c ) $$\begin{eqnarray}\displaystyle [\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}=-2{\mathcal{C}}\,\hat{\unicode[STIX]{x1D701}}\,A^{1/2},\quad [{\mathcal{Q}}_{z,1}]_{-}^{+}=2\frac{\unicode[STIX]{x1D6F4}_{v}^{+}}{{\mathcal{U}}_{2}}-2{\mathcal{C}}\,\hat{\unicode[STIX]{x1D701}}\,A^{1/2},\quad Q^{(2,\odot )}=s_{i}\{{\mathcal{Q}}_{z,1}\}_{-}^{+}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where ${\mathcal{C}}={\mathcal{K}}_{0}[1]=\sqrt{2}+K_{0}[1]\simeq 1.3788$ . The jump of ${\mathcal{Q}}_{z,1}$ is altered by the mean-wave-flux-induced corrections. The C17 jump is thus erroneous since the mean radial fluxes were omitted. We shall see in § 7 that this jump is still proportional to $\unicode[STIX]{x1D701}\,A^{1/2}$ like in previous studies (Caillol & Maslowe Reference Caillol and Maslowe2007; Caillol Reference Caillol2014). The jump (6.54) again shows the dynamical coupling between the induced mean flow and the wave packet; this coupling has been omitted until now in wave packet studies in the presence of a critical layer (Benney & Maslowe Reference Benney and Maslowe1975). The uniform vorticity $Q^{(2,\odot )}$ is still the average of the mean vorticities on both sides of the CL. Moreover, the wave packet assumption leaves the steady nonlinear CL logarithmic phase shift unchanged (Caillol Reference Caillol2014); it is proportional to $\unicode[STIX]{x1D706}^{\prime }$ as follows:

(6.55) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D719}(\unicode[STIX]{x1D706}^{\prime })}{\unicode[STIX]{x1D706}^{\prime }}=4s_{i}\,{\mathcal{C}}A^{-1/2},\quad \unicode[STIX]{x1D706}^{\prime }\rightarrow 0. & & \displaystyle\end{eqnarray}$$

After (6.2) and (6.3), the inviscid axial velocity can be written in a general way within the separatrices:

(6.56) $$\begin{eqnarray}\displaystyle W_{i}^{(2,\odot )}(Z,x)=\unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}}^{(2,\odot )}(Z,x)-\unicode[STIX]{x1D701}Z+{\mathcal{J}}_{2}(Z^{\ast })+\{{\mathcal{W}}_{2}\}_{-}^{+}, & & \displaystyle\end{eqnarray}$$

where

(6.57) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{i,\unicode[STIX]{x1D70C}}^{(2,\odot )}(\unicode[STIX]{x1D70C},X)=\unicode[STIX]{x1D6EF}(1)\mathop{\sum }_{n=1}^{\infty }\left[\int _{1}^{\overline{\unicode[STIX]{x1D70C}}}b_{n}(r)\,\text{d}r\,\sin (nX)\right]\unicode[STIX]{x1D70C}^{X\ast }{\mathcal{U}}_{2}+Q^{(2,\odot )}\unicode[STIX]{x1D70C}+V_{2}^{\odot }(X). & & \displaystyle\end{eqnarray}$$

The function ${\mathcal{J}}_{2}$ must be independent of $s$ owing to the continuity at $\unicode[STIX]{x1D70C}=0$ (see more details in Caillol (Reference Caillol2014)). Using (6.37), the azimuthal- and radial-vorticity matchings on the separatrices yield the same results:

(6.58a,b ) $$\begin{eqnarray}\displaystyle {\mathcal{J}}_{2}^{\prime }(1)=0,\quad J_{1,c}=0. & & \displaystyle\end{eqnarray}$$

The leading-order induced mean vorticities are then proportional in such a way that the first-order local equivalent Richardson number is zero:

(6.59) $$\begin{eqnarray}\displaystyle {\mathcal{Q}}_{\unicode[STIX]{x1D703},1}=-\unicode[STIX]{x1D71B}{\mathcal{Q}}_{z,1}. & & \displaystyle\end{eqnarray}$$

This relationship is identical to the previous studies: the steady vortical mode without any mean radial velocity (Caillol Reference Caillol2014) and the $O(\unicode[STIX]{x1D716}^{3/2})$ mean radial-velocity wave packet (C17). The equivalent Richardson number is therefore of order $O(\unicode[STIX]{x1D716})$ at $r_{c}$ . This amplitude modulated vortical mode is however weakly singular unlike in these previous studies. Indeed, the $O(\unicode[STIX]{x1D716}^{3/2})$ outer radial velocity of § 3.2 admits a non-vanishing leading-order cosine-logarithmic term around $\unicode[STIX]{x1D702}=0$ :

(6.60) $$\begin{eqnarray}\displaystyle U_{r,i}^{(1)} & = & \displaystyle \{[(c_{l,1,0}+c_{ad,1,0}a_{\unicode[STIX]{x1D6F7}})U_{0}+c_{d,1,0}U_{0,\unicode[STIX]{x1D6F7}}]\sin \unicode[STIX]{x1D709}+[(f_{l,1,0}+f_{ad,1,0}a_{\unicode[STIX]{x1D6F7}})U_{0}\nonumber\\ \displaystyle & & \displaystyle +\,f_{d,1,0}U_{0,\unicode[STIX]{x1D6F7}}]\cos \unicode[STIX]{x1D709}\}\ln |\unicode[STIX]{x1D702}^{\ast }|\equiv s_{i}(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D6F4}_{q^{\prime }}A\cos X\ln |\unicode[STIX]{x1D702}^{\ast }|,\end{eqnarray}$$

according to the definitions of these coefficients in appendix A and the relationships (6.35), (6.36), (6.40) and (6.58b ).

The distorted parameter $(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D6F4}_{q^{\prime }}$ can be hence assimilated to a $O(\unicode[STIX]{x1D716}^{1/2})$ pseudo equivalent Richardson number. Moreover, the relationship (6.59) implies the same driven term on the right-hand side of equations (5.27) for the evolution of ${\mathcal{V}}_{1}$ and ${\mathcal{W}}_{1}$ . The trivial solution ${\mathcal{W}}_{1}={\mathcal{V}}_{1}+Cst$ arises, which orientates the CL-induced mean flow towards a helical motion perpendicular to the isophase lines, since on a line $\unicode[STIX]{x1D709}=Cst$ , axial and azimuthal velocities are linked by $v\equiv -\unicode[STIX]{x1D71B}w$ at $r\simeq r_{a}$ . The evolution equations are also identical to those in C17, the slow time $\unicode[STIX]{x1D70F}_{2}$ is faster here.

Next, we match the pressure ${\mathcal{P}}^{(2)}$ , which is given inside the cat’s eyes by

(6.61) $$\begin{eqnarray}\displaystyle {\mathcal{P}}_{i}^{(2,\odot )} & = & \displaystyle \frac{1}{2}\unicode[STIX]{x1D70C}^{3}-\frac{s_{i}}{4}(3{\mathcal{S}}_{1}+{\mathcal{C}}_{0}{\mathcal{C}}_{1})\unicode[STIX]{x1D70C}^{2}+\left[\frac{1}{4}({\mathcal{S}}_{1}^{2}-{\mathcal{C}}_{1}^{2})+\{{\mathcal{V}}_{2}\}_{-}^{+}-H_{2}\right]\unicode[STIX]{x1D70C}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{s_{i}}{2}({\mathcal{S}}_{1}-{\mathcal{C}}_{0}{\mathcal{C}}_{1})H_{2}+\unicode[STIX]{x1D713}^{(2,\odot )}+\unicode[STIX]{x1D6F1}_{2}^{(\odot )}(X)+s_{i}\{\mathfrak{P}_{3}\}_{-}^{+}.\end{eqnarray}$$

Continuity of pressure through the separatrices enables one to determine $V_{2}^{\odot }(x)$ as

(6.62) $$\begin{eqnarray}\displaystyle V_{2}^{\odot }(x) & = & \displaystyle \hat{\unicode[STIX]{x1D701}}\left\{[1-\cos x]^{1/2}+\cos x\ln [\unicode[STIX]{x1D6EC}(A,x)]-\int _{1}^{\infty }\frac{K_{0}[z]\,\text{d}z}{|Z_{\unicode[STIX]{x1D70C}}(Az,x)|}A^{1/2}\right\}A\nonumber\\ \displaystyle & & \displaystyle -\,\int _{1}^{\infty }\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\,\frac{\unicode[STIX]{x1D6EF}(z)\,\text{d}z\,{\mathcal{U}}_{2}}{[2(z-\cos x)]^{1/2}}+[\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}\frac{\unicode[STIX]{x1D70C}^{X}}{2}+\{V_{2}(x)\}_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle +\,\int _{1}^{\infty }\left(\hat{\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}K_{0}[z]\right)\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\,\frac{\text{d}z\,{\mathcal{U}}_{2}}{\unicode[STIX]{x1D70C}^{X\ast }(x)}\nonumber\\ \displaystyle & & \displaystyle -\,\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}\int _{1}^{\infty }\sqrt{2z}\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}+\frac{\sin x}{2z}\,\frac{\text{d}z\,A^{1/2}}{\unicode[STIX]{x1D70C}^{X\ast }(x)}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{\unicode[STIX]{x1D6EF}(1)}{2}\mathop{\sum }_{n=1}^{\infty }\left[\int _{-1}^{1}b_{n}(r)\,r\text{d}r\sin (nx)\right]\,\unicode[STIX]{x1D70C}^{X\ast }{\mathcal{U}}_{2};\end{eqnarray}$$

it also yields the following jumps through the expression for $\unicode[STIX]{x1D6F1}_{2}$  (6.30) jointly with the relationships (6.35), (6.36) and (6.40):

(6.63) $$\begin{eqnarray}\displaystyle [\mathfrak{P}_{3}]_{-}^{+}=2\,\hat{\unicode[STIX]{x1D701}}\left(\frac{2^{3/2}}{3}-\int _{1}^{\infty }K_{0}[z]\,\text{d}z-{\mathcal{C}}\right)A^{3/2}, & & \displaystyle\end{eqnarray}$$

(6.64) $$\begin{eqnarray}\displaystyle {\mho}_{0}\hat{\unicode[STIX]{x1D701}}\left[\left[\unicode[STIX]{x1D6FC}_{l,1}+\unicode[STIX]{x1D6FC}_{d,1}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}+\unicode[STIX]{x1D6FC}_{ad,1}a_{\unicode[STIX]{x1D6F7}}\right]_{\unicode[STIX]{x1D6E9}}\right]_{-}^{+} & = & \displaystyle \{\hat{\unicode[STIX]{x1D701}}(1+\unicode[STIX]{x1D71B}^{2})[\ln (4A)+2]-2\}\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\mho}_{0}^{3}}{2}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{3}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }(0)]_{-}^{+}+\frac{{\mho}_{0}^{5}}{24}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}\unicode[STIX]{x1D701}_{z,1}^{IV}(0)]_{-}^{+}+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{7}),\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

and

(6.65) $$\begin{eqnarray}\displaystyle \hat{\unicode[STIX]{x1D701}}\left[\left[\unicode[STIX]{x1D6FE}_{l,1}+\unicode[STIX]{x1D6FE}_{d,1}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}+\unicode[STIX]{x1D6FE}_{ad,1}a_{\unicode[STIX]{x1D6F7}}\right]_{\unicode[STIX]{x1D6E9}}\right]_{-}^{+}=-{\mho}_{0}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}\unicode[STIX]{x1D701}_{z,1}^{\prime }]_{-}^{+}+\frac{{\mho}_{0}^{3}}{6}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }(0)]_{-}^{+}+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{6}). & & \displaystyle\end{eqnarray}$$

Note that the ratios of the second and third double integrals in (6.62) over $\unicode[STIX]{x1D70C}^{X}$ are not singular at $x=0\,[2\unicode[STIX]{x03C0}]$ . Continuity of pressure also permits one to determine the function

(6.66) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F1}_{2}^{(\odot )}(x)=\{\unicode[STIX]{x1D6F1}_{2}(x)\}_{-}^{+}+\frac{s_{i}}{2}[V_{2}(x)+{\mathcal{V}}_{2}]_{-}^{+}\unicode[STIX]{x1D70C}^{X}. & & \displaystyle\end{eqnarray}$$

Next, the rotational- and axial-velocity matchings on the separatrices yield the absence of jumps for the phase and the second-order mean velocity, and then the value of ${\mathcal{J}}_{2}(1)$ ,

(6.67a-d ) $$\begin{eqnarray}\displaystyle [a]_{-}^{+}=0,\quad [{\mathcal{V}}_{2}]_{-}^{+}=0,\quad [{\mathcal{W}}_{2}]_{-}^{+}=0,\quad {\mathcal{J}}_{2}(1)=0, & & \displaystyle\end{eqnarray}$$

as well as a second expression for $V_{2}^{\odot }(x)$ :

(6.68) $$\begin{eqnarray}\displaystyle V_{2}^{\odot }(x) & = & \displaystyle \hat{\unicode[STIX]{x1D701}}\left\{[1-\cos x]^{1/2}+\cos x\ln [\unicode[STIX]{x1D6EC}(A,x)]-\int _{1}^{\infty }\frac{K_{0}[z]\,\text{d}z}{|Z_{\unicode[STIX]{x1D70C}}(Az,x)|}A^{1/2}\right\}A\nonumber\\ \displaystyle & & \displaystyle -\,\int _{1}^{\infty }\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\frac{\unicode[STIX]{x1D6EF}(z)\,\text{d}z\,{\mathcal{U}}_{2}}{[2(z-\cos x)]^{1/2}}+[\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}\frac{\unicode[STIX]{x1D70C}^{X}}{2}+\{V_{2}(x)\}_{-}^{+}.\end{eqnarray}$$

Equating (6.62) with (6.68), a second relationship constraining the series terms $b_{n}$ arises, whose projection on the $n$ th harmonics $\sin (nx)$ is

(6.69) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6EF}(1)\int _{-1}^{1}[b_{n+1}(r)+b_{n-1}(r)-2b_{n}(r)]\,r\,\text{d}r\nonumber\\ \displaystyle & & \displaystyle \quad =4\left\langle \int _{1}^{\infty }\left(\hat{\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}K_{0}[z]\right)\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}\,\text{d}z\,\sin (nx)\right\rangle \nonumber\\ \displaystyle & & \displaystyle \qquad -\,4\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}\left\langle \int _{1}^{\infty }\sqrt{2z}\int _{\unicode[STIX]{x03C0}}^{x}G_{5}(z,x_{1})\,\text{d}x_{1}+\frac{\sin x}{2z}\,\text{d}z\,\sin (nx)\right\rangle \quad n\geqslant 1.\end{eqnarray}$$

Equation (6.5) evaluated within the cat’s eye gives the radial velocity

(6.70) $$\begin{eqnarray}\displaystyle U^{(2,\odot )}(Z,x) & = & \displaystyle \unicode[STIX]{x1D6EF}(1)(1+\unicode[STIX]{x1D71B}^{2})\mathop{\sum }_{n=1}^{\infty }\left[\frac{\overline{\unicode[STIX]{x1D70C}}}{2}\int _{1}^{\overline{\unicode[STIX]{x1D70C}}}b_{n}(r)\,\text{d}r\,[\cos (n-1)x-\cos (n+1)x]\right.\nonumber\\ \displaystyle & & \displaystyle -\left(\overline{\unicode[STIX]{x1D70C}}\int _{1}^{\overline{\unicode[STIX]{x1D70C}}}b_{n}(r)\,\text{d}r-\int _{0}^{\overline{\unicode[STIX]{x1D70C}}}b_{n}(r)\,r\,\text{d}r\right)\nonumber\\ \displaystyle & & \displaystyle \times \left.\vphantom{\frac{\overline{\unicode[STIX]{x1D70C}}}{2}}[2n\cos (nx)-(n+1)\cos (n+1)x-(n-1)\cos (n-1)x]\right]A^{1/2}{\mathcal{U}}_{2}\nonumber\\ \displaystyle & & \displaystyle +\,\unicode[STIX]{x1D71B}^{2}\int _{\cos x}^{Z^{\ast }}\frac{{\mathcal{J}}_{2}^{\prime }(z)\,\text{d}z}{Z_{\unicode[STIX]{x1D70C}}(Az,x)}\,A\sin x\nonumber\\ \displaystyle & & \displaystyle +\,[(1-\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}^{2})A\sin x-(1+\unicode[STIX]{x1D71B}^{2})V_{2}^{\odot ^{\prime }}(x)+{\mathcal{U}}_{2}^{\prime }]Z_{\unicode[STIX]{x1D70C}}+\unicode[STIX]{x1D6F6}_{2}^{\odot }(x)+s_{i}\{{\mathcal{U}}_{3}\}_{-}^{+}.\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

After a tedious calculation involving equations (6.42), (6.68) and (6.70), continuity of the radial velocity on the separatrices yields an equation determining ${\mathcal{J}}_{2}^{\prime }$ as

(6.71) $$\begin{eqnarray}\displaystyle 2\unicode[STIX]{x1D71B}^{2}\int _{1}^{cos\,x}\frac{{\mathcal{J}}_{2}^{\prime }(z)\,\text{d}z}{|Z_{\unicode[STIX]{x1D70C}}(Az,x)|}A\sin x & = & \displaystyle \frac{1}{2}(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}A\,[1+4\cos x-\cos (2x)]-[{\mathcal{U}}_{3}]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle -\,s_{i}[\unicode[STIX]{x1D6F6}_{2}(x)]_{-}^{+}-(1+\unicode[STIX]{x1D71B}^{2})[\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}A\,\sin x,\end{eqnarray}$$

and the integration function $\unicode[STIX]{x1D6F6}_{2}^{\odot }$ as

(6.72) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F6}_{2}^{\odot }(x)=\{\unicode[STIX]{x1D6F6}_{2}(x)\}_{-}^{+}-\frac{s_{i}}{2}(1+\unicode[STIX]{x1D71B}^{2})[V_{2}^{\prime }(x)]_{-}^{+}\unicode[STIX]{x1D70C}^{X}. & & \displaystyle\end{eqnarray}$$

Averaging the equation (6.71) over $x$ leads to the third-order mean radial velocity distortion. According to the expressions of $\unicode[STIX]{x1D6F6}_{2}$ in (6.43) and the terms $g_{j,1,0}$ , $j=l,d,ad$ , in appendix A, the projection of the same equation on $\cos (nx),n=1,2$ gives the jumps of the $O(\unicode[STIX]{x1D716}^{3/2})$ outer flow coefficients: $\unicode[STIX]{x1D705}_{h,1}$ and the sum of the $\unicode[STIX]{x1D705}_{j,1}$ , $j=l,d$ and $ad$ . The latter is a function of the mean radial wave flux $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}$ and the odd-order $\unicode[STIX]{x1D6E9}$ -derivatives of the mean flow distortion $\unicode[STIX]{x1D701}_{z,1}^{\prime }$ , $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }$ , $\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }$ , $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime },\ldots$

(6.73a,b ) $$\begin{eqnarray}\displaystyle [{\mathcal{U}}_{3}]_{-}^{+}=\frac{1}{2}(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}A,\quad [\unicode[STIX]{x1D705}_{h,1}]_{-}^{+}=\frac{s_{i}}{2}(1+\unicode[STIX]{x1D71B}^{2})\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}. & & \displaystyle\end{eqnarray}$$

By using the relationships (6.35), (6.40) and (6.64), the jump of the sum of the coefficients $\unicode[STIX]{x1D705}$ gets simplified as

(6.74) $$\begin{eqnarray}\displaystyle & & \displaystyle \hat{\unicode[STIX]{x1D701}}\left([\unicode[STIX]{x1D705}_{l,1}]_{-}^{+}+[\unicode[STIX]{x1D705}_{d,1}]_{-}^{+}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}+[\unicode[STIX]{x1D705}_{ad,\unicode[STIX]{x1D6F7}}]_{-}^{+}a_{\unicode[STIX]{x1D6F7}}\right)=-2\left[(\unicode[STIX]{x1D707}+1)\left(\frac{1}{2}\ln (4A)+1-\frac{1}{\hat{\unicode[STIX]{x1D701}}(1+\unicode[STIX]{x1D71B}^{2})}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \quad +\left.\hat{\unicode[STIX]{x1D701}}(1+\unicode[STIX]{x1D71B}^{2})+\frac{{\mathcal{M}}}{1+\unicode[STIX]{x1D71B}^{2}}\right]\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}-{\mho}_{0}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}\unicode[STIX]{x1D701}_{z,1}^{\prime }(0)]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}^{2}}{72(1+\unicode[STIX]{x1D71B}^{2})}\left[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{3}\left(\left[{\mathcal{M}}_{z4}-36\frac{{\mho}_{0}}{\hat{\unicode[STIX]{x1D701}}}(\unicode[STIX]{x1D707}+1)\right]\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }-{\mho}_{0}\unicode[STIX]{x1D71B}{\mathcal{M}}_{\unicode[STIX]{x1D703}4}\,\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime }\right)(0)\right]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{{\mho}_{0}^{3}}{6}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{3}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }(0)]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\mho}_{0}^{4}}{7200(1+\unicode[STIX]{x1D71B}^{2})}\left[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}\left(\left[{\mathcal{M}}_{z5}+300\frac{{\mho}_{0}}{\hat{\unicode[STIX]{x1D701}}}(\unicode[STIX]{x1D707}+1)\right]\unicode[STIX]{x1D701}_{z,1}^{IV}+{\mho}_{0}\unicode[STIX]{x1D71B}{\mathcal{M}}_{\unicode[STIX]{x1D703}5}\,\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{IV}\right)(0)\right]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}^{5}}{120}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}\unicode[STIX]{x1D701}_{z,1}^{V}(0)]_{-}^{+}+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{7}).\end{eqnarray}$$

The coefficients $\unicode[STIX]{x1D707}$ , ${\mathcal{M}}_{z4}$ , ${\mathcal{M}}_{\unicode[STIX]{x1D703}4}$ , ${\mathcal{M}}_{z5}$ and ${\mathcal{M}}_{\unicode[STIX]{x1D703}5}$ are given in appendix A. $\unicode[STIX]{x1D6E9}$ -derivating (6.71) projected on $\cos x$ then gives a relationship between various even $\unicode[STIX]{x1D6E9}$ -derivatives of the mean flow distortion

(6.75) $$\begin{eqnarray}\displaystyle [\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}\unicode[STIX]{x1D701}_{z,1}^{\prime }(0)]_{-}^{+}=\frac{{\mho}_{0}^{2}}{6}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }(0)]_{-}^{+}+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{6}). & & \displaystyle\end{eqnarray}$$

Finally, the projection of (6.71) on $\sin x$ gives the following integro-differential equation

(6.76) $$\begin{eqnarray}\displaystyle & & \displaystyle 2\unicode[STIX]{x1D71B}^{2}\int _{1}^{cos\,x}\frac{{\mathcal{J}}_{2}^{\prime }(z)\,\text{d}z}{[2(z-\cos x)]^{1/2}}\nonumber\\ \displaystyle & & \displaystyle \quad =\{[\unicode[STIX]{x1D6FD}_{l,1}+d_{l,1,0}+(\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D714}d,1}+d_{\unicode[STIX]{x1D714}d,1,0}){\mho}_{1,\unicode[STIX]{x1D6F7}}\unicode[STIX]{x1D70F}_{1}]_{-}^{+}+[\unicode[STIX]{x1D6FD}_{d,1}+d_{d,1,0}]_{-}^{+}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}\nonumber\\ \displaystyle & & \displaystyle \qquad +\,[(\unicode[STIX]{x1D6FD}_{ad,1}+d_{ad,1,0}]_{-}^{+}a_{\unicode[STIX]{x1D6F7}}-(1+\unicode[STIX]{x1D71B}^{2})[\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}\}A^{1/2}.\end{eqnarray}$$

The previous matching conditions (6.58a ) and (6.67d ) imply the trivial solution ${\mathcal{J}}_{2}(Z)=0$ ; as a result, the right-hand side of (6.76) vanishes. The expressions of $d_{j,1,0}$ , $j=l,d,ad$ and $\unicode[STIX]{x1D714}d$ , in appendix A, jointly with the relationships (6.40), (6.54), (6.59) and (6.65) leads to a first continuous coefficient $[\unicode[STIX]{x1D6FD}_{\unicode[STIX]{x1D714}d,1}]_{-}^{+}=0$ , whereas the sum of the other coefficients $\unicode[STIX]{x1D6FD}$ admits a jump linked to the distortions of $\hat{{\mathcal{Q}}}_{z,1}$ , $\hat{\unicode[STIX]{x1D701}}_{z,1}$ , the even-order $\unicode[STIX]{x1D6E9}$ -derivatives of the mean flow $\unicode[STIX]{x1D701}_{z,1}^{\prime }$ , $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }$ , $\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime },\ldots$ The jump of the sum of the coefficients $\unicode[STIX]{x1D6FD}$ is namely

(6.77) $$\begin{eqnarray}\displaystyle & & \displaystyle \hat{\unicode[STIX]{x1D701}}\left([\unicode[STIX]{x1D6FD}_{l,1}]_{-}^{+}+[\unicode[STIX]{x1D6FD}_{d,1}]_{-}^{+}\frac{A_{\unicode[STIX]{x1D6F7}}}{A}+[\unicode[STIX]{x1D6FD}_{ad,1}]_{-}^{+}a_{\unicode[STIX]{x1D6F7}}\right)(0)=[\hat{\unicode[STIX]{x1D701}}_{z,1}]_{-}^{+}-2{\mathcal{C}}\hat{\unicode[STIX]{x1D701}}[\hat{\unicode[STIX]{x1D701}}(2+3\unicode[STIX]{x1D71B}^{2})+\unicode[STIX]{x1D71B}\hat{\unicode[STIX]{x1D701}}_{\unicode[STIX]{x1D703},0}]\,A^{1/2}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}}{12(1+\unicode[STIX]{x1D71B}^{2})}\left[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}\left(\left[{\mathcal{M}}_{z1}+12\frac{{\mho}_{0}}{\hat{\unicode[STIX]{x1D701}}}(\unicode[STIX]{x1D707}+1)\right]\unicode[STIX]{x1D701}_{z,1}^{\prime }+{\mho}_{0}\unicode[STIX]{x1D71B}{\mathcal{M}}_{\unicode[STIX]{x1D703}1}\,\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime }\right)\right]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}^{2}}{2}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{2}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }(0)]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{{\mho}_{0}^{3}}{1440(1+\unicode[STIX]{x1D71B}^{2})}\left[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}\left(\left[{\mathcal{M}}_{z2}-240\frac{{\mho}_{0}}{\hat{\unicode[STIX]{x1D701}}}(\unicode[STIX]{x1D707}+1)\right]\unicode[STIX]{x1D701}_{z,1}^{\prime \prime \prime }+{\mho}_{0}\unicode[STIX]{x1D71B}{\mathcal{M}}_{\unicode[STIX]{x1D703}2}\,\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1}^{\prime \prime \prime }\right)(0)\right]_{-}^{+}\nonumber\\ \displaystyle & & \displaystyle \quad +\frac{{\mho}_{0}^{4}}{24}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{4}\unicode[STIX]{x1D701}_{z,1}^{IV}(0)]_{-}^{+}+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{6}).\end{eqnarray}$$

The coefficients ${\mathcal{M}}_{z1}$ , ${\mathcal{M}}_{\unicode[STIX]{x1D703}1}$ , ${\mathcal{M}}_{z2}$ and ${\mathcal{M}}_{\unicode[STIX]{x1D703}2}$ are given in appendix A. Next, $\unicode[STIX]{x1D6E9}$ -differentiating (6.71) projected on $\sin x$ gives the jump relationship

(6.78) $$\begin{eqnarray}\displaystyle -\frac{1}{2}{\mho}_{0}^{3}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{3}\unicode[STIX]{x1D701}_{z,1}^{\prime \prime }(0)]_{-}^{+}+\frac{{\mho}_{0}^{5}}{24}[\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{5}\unicode[STIX]{x1D701}_{z,1}^{IV}(0)]_{-}^{+}+O(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D6E9}}^{7})=2\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}. & & \displaystyle\end{eqnarray}$$

The velocity $V_{5,sp}^{\odot }$ in (6.50) is determined in such a way that the velocities $V_{sp}^{(5,\odot )}$ and $W_{sp}^{(5,\odot )}$ may not admit a jump on the CL axis at $\unicode[STIX]{x1D70C}=0$ (that is, $V_{5,sp}^{\odot }(x)=-s\unicode[STIX]{x1D6EF}(1)\int _{1}^{cos\,x}G_{5}^{\odot }(z,x)\,\text{d}z\,A^{1/2}H_{2}$ ). The expression for $V_{5,sp}^{\odot }$ can be then deduced from the relationship (6.69) as

(6.79) $$\begin{eqnarray}\displaystyle \frac{V_{5,sp}^{\odot }(x)}{A^{1/2}H_{2}} & = & \displaystyle \frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}}{{\mathcal{U}}_{2}}A^{1/2}\int _{1}^{\infty }\sqrt{2z}\,G_{5}(z,x)+\frac{\cos x}{2z}\,\text{d}z\nonumber\\ \displaystyle & & \displaystyle -\,s\int _{1}^{\infty }\left(\hat{\unicode[STIX]{x1D701}}-\frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}}{{\mathcal{U}}_{2}}A^{1/2}K_{0}[z]\right)G_{5}(z,x)\,\text{d}z.\end{eqnarray}$$

The matchings of the fifth-order azimuthal and axial spiralling velocities on the separatrices – that is, the conditions $V_{\unicode[STIX]{x1D6E9}}^{(5)}=V_{\unicode[STIX]{x1D6E9}}^{(5,\odot )}$ and $W_{\unicode[STIX]{x1D6E9}}^{(5)}=W_{\unicode[STIX]{x1D6E9}}^{(5,\odot )}$ as $Z=A$ , according to (6.10)–(6.11), (6.16), (6.47)–(6.48) and (6.50) – are only possible if $[V_{5,sp}(x)]_{-}^{+}=-\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A\ln (4A)\,H_{2}\cos x$ , which is compatible with the relationships (6.64) and (6.65), and if the mean velocities ${\mathcal{V}}_{3,sp}$ , ${\mathcal{W}}_{3,sp}$ are undistorted. However, the third-order mean velocity distortion (5.25) implies

(6.80) $$\begin{eqnarray}\displaystyle [{\mathcal{V}}_{3,sp}]_{-}^{+}=[{\mathcal{W}}_{3,sp}]_{-}^{+}=2{\mathcal{C}}\hat{\unicode[STIX]{x1D701}}\,A^{1/2}H_{2}, & & \displaystyle\end{eqnarray}$$

which means that $V_{sp}^{(5)}$ and $W_{sp}^{(5)}$ are not matchable on the separatrices owing to the presence of the third-order distorted induced mean flow. The matching can be carried out by adding a boundary layer along the separatrices.

The choice of the radial variable $\unicode[STIX]{x1D70C}$ in (4.6) jointly with a secular azimuthal angle is here very convenient because it permits one to describe the spiralling streamlines and remove the time secularity used in C17, where the induced mean flow secularly evolved over $\unicode[STIX]{x1D70F}_{2}$ as soon as the first order for the mean vorticity $({\mathcal{Q}}_{z,1},{\mathcal{Q}}_{\unicode[STIX]{x1D703},1})$ and ${\mathcal{Q}}^{(2,\odot )}$ . The subsequent CL blow-up highlighted a wave-breaking modelling weakness. These variables structure all the other fields not modelled here, such as reflectivity, convection and so on, in spiral bands as well. The diffusion boundary layers located at either side of the nonlinear critical layer are strongly coupled to the latter and similarly exhibit a helical motion, as was proved in C17. The same variables describe the spiralling motion in these layers.

Matching conditions have linked the Frobenius series term jumps with the mean flow distortions. Those relationships will be relevant while deriving the evolution equations, because most of the coefficients of those equations are proportional to the mean flow distortions.

7.1 Computation of the mean radial wave-flux-induced forcings

7.1.1 Forcing $\unicode[STIX]{x1D6F4}_{q}$ driven by the mean wave vorticity flux

According to (5.13)–(5.14) and § 6, the $\unicode[STIX]{x1D6E9}$ -gradient of the second-order mean vorticity is given for a CL symmetric with respect to the axis $r_{a}$ ( $\unicode[STIX]{x1D70C}_{B}^{+}=-\unicode[STIX]{x1D70C}_{B}^{-}$ ), by

(7.1) $$\begin{eqnarray}\displaystyle & \displaystyle {\mho}_{0}{\mathcal{Q}}_{z,2,\unicode[STIX]{x1D6E9}}^{s}=\overline{\sin X\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}(\unicode[STIX]{x1D70C}_{B}^{s},X)}\,A-\unicode[STIX]{x1D701}{\mathcal{U}}_{2}=\unicode[STIX]{x1D6F4}_{q}^{s}-\unicode[STIX]{x1D701}{\mathcal{U}}_{2}, & \displaystyle\end{eqnarray}$$

(7.2) $$\begin{eqnarray}\displaystyle & \displaystyle {\mho}_{0}{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,\unicode[STIX]{x1D6E9}}^{s}={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}-\unicode[STIX]{x1D71B}\,\overline{\sin X\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}(\unicode[STIX]{x1D70C}_{B}^{s},X)}\,A={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}-\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F4}_{q}^{s}. & \displaystyle\end{eqnarray}$$

We define $\unicode[STIX]{x1D70C}_{B}=|\unicode[STIX]{x1D70C}_{B}^{+}|$ and the corresponding variable $Z$ : $Z_{B}=1/2\unicode[STIX]{x1D70C}_{B}^{2}+A\cos X$ . The forcing is obtained after $\unicode[STIX]{x1D70C}$ -differentiating the axial vorticity $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ in (6.18); its expression (G 1) is given in appendix G, showing a first relationship between the forcings $\unicode[STIX]{x1D6F4}_{q}^{s}$ and $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}$ as follows:

(7.3) $$\begin{eqnarray}\displaystyle (1+{\mathcal{A}}_{3})\unicode[STIX]{x1D6F4}_{q}^{s}+s{\mathcal{A}}_{4}\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}A^{1/2}=\unicode[STIX]{x1D701}{\mathcal{A}}_{3}{\mathcal{U}}_{2}, & & \displaystyle\end{eqnarray}$$

where the various coefficients ${\mathcal{A}}_{i}$ are $X$ -averages at constant $\unicode[STIX]{x1D70C}_{B}$ and are functions of $\unicode[STIX]{x1D70C}_{B}^{\ast }$ only. Their expressions are given in appendix G.

7.1.2 Forcing $\unicode[STIX]{x1D6F4}_{q^{\prime }}$ driven by the mean wave flux of the vorticity radial gradient

After (5.4) and § 6, the $\unicode[STIX]{x1D6E9}$ -gradient of the first-order mean vorticity radial gradient is given at either side of $r_{a}$ , for a symmetric CL, by

(7.4) $$\begin{eqnarray}\displaystyle & \displaystyle {\mho}_{0}\unicode[STIX]{x1D701}_{z,1,\unicode[STIX]{x1D6E9}}^{s}=s_{i}\,\overline{\sin X\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}(\unicode[STIX]{x1D70C}_{B}^{s},X)}\,A=\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}, & \displaystyle\end{eqnarray}$$

(7.5) $$\begin{eqnarray}\displaystyle & \displaystyle {\mho}_{0}\unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,\unicode[STIX]{x1D6E9}}^{s}=-s_{i}\unicode[STIX]{x1D71B}\,\overline{\sin X\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}(\unicode[STIX]{x1D70C}_{B}^{s},X)}\,A=-\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}. & \displaystyle\end{eqnarray}$$

The calculation yields, after twice $\unicode[STIX]{x1D70C}$ -differentiating $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ in (6.18), the expression (G 2) in appendix G. A second relationship between the forcings $\unicode[STIX]{x1D6F4}_{q}^{s}$ and $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}$ is thus deduced as

(7.6) $$\begin{eqnarray}\displaystyle s[\unicode[STIX]{x1D70C}_{B}^{\ast }+{\mathcal{A}}_{4}+\unicode[STIX]{x1D70C}_{B}^{\ast 3}{\mathcal{A}}_{6}]\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}A^{1/2}+({\mathcal{A}}_{3}+\unicode[STIX]{x1D70C}_{B}^{\ast 3}{\mathcal{A}}_{5})\unicode[STIX]{x1D6F4}_{q}^{s}=\unicode[STIX]{x1D701}({\mathcal{A}}_{3}+\unicode[STIX]{x1D70C}_{B}^{\ast 3}{\mathcal{A}}_{5}){\mathcal{U}}_{2}, & & \displaystyle\end{eqnarray}$$

involving the similar coefficients ${\mathcal{A}}_{i}$ expressed in appendix G.

The forcings $\unicode[STIX]{x1D6F4}_{q}$ and $\unicode[STIX]{x1D6F4}_{q^{\prime }}$ are solutions of the linear system (7.3) and (7.6), whose variables are only $A$ , ${\mathcal{U}}_{2}$ and $\unicode[STIX]{x1D70C}_{B}^{\ast }$ . For a symmetric CL extent, the rescaled radius $\unicode[STIX]{x1D70C}_{B}^{\ast }$ is linked to the CL thickness: $\unicode[STIX]{x1D6FF}_{cl}\simeq 2r_{c}\unicode[STIX]{x1D70C}_{B}^{\ast }\,(\unicode[STIX]{x1D716}A)^{1/2}$ . This radius is roughly twice the ratio $\unicode[STIX]{x1D70E}$ of the CL thickness over the cat’s eye thickness, and is thus bounded such that $\unicode[STIX]{x1D70C}_{B}^{\ast }>2$ . The ratio can be taken constant (that is, we assume a CL extent that varies according to the wave modulation, which is more realistic than a constant-extent CL). Both forcings are hence only functions of the slow modulation scales $\unicode[STIX]{x1D6F7}$ and $\unicode[STIX]{x1D70F}_{2}$ , namely

(7.7a,b ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}}{{\mathcal{U}}_{2}}=s\unicode[STIX]{x1D701}\,\mathbb{B}\,A^{-1/2},\quad \frac{\unicode[STIX]{x1D6F4}_{q}}{{\mathcal{U}}_{2}}=\unicode[STIX]{x1D701}\frac{{\mathcal{A}}_{3}}{1+{\mathcal{A}}_{3}}-\unicode[STIX]{x1D701}\frac{{\mathcal{A}}_{4}}{1+{\mathcal{A}}_{3}}\mathbb{B}, & & \displaystyle\end{eqnarray}$$

where $\mathbb{B}$ is a ratio of various coefficients ${\mathcal{A}}_{i}$ and whose long expression is given in appendix G. For a symmetric CL, the forcing $\unicode[STIX]{x1D6F4}_{q^{\prime }}$ is antisymmetric vis-à-vis the CL axis $r_{a}$ and proportional to $\unicode[STIX]{x1D701}{\mathcal{U}}_{2}\,A^{-1/2}$ , whereas $\unicode[STIX]{x1D6F4}_{q}$ is symmetric and proportional to $\unicode[STIX]{x1D701}{\mathcal{U}}_{2}$ .

The expressions of the first-order spiralling mean vorticity radial gradient and second-order spiralling mean vorticity are now derived by

(7.8a,b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}_{z,1,sp}^{s}=s\unicode[STIX]{x1D701}\,\mathbb{B}\,\frac{H_{2}}{A^{1/2}},\quad \unicode[STIX]{x1D701}_{\unicode[STIX]{x1D703},1,sp}^{s}=-s\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}\,\mathbb{B}\,\frac{H_{2}}{A^{1/2}}, & & \displaystyle\end{eqnarray}$$

(7.9a ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{Q}}_{z,2,sp}=-\unicode[STIX]{x1D701}[1+{\mathcal{A}}_{4}\mathbb{B}]\frac{H_{2}}{1+{\mathcal{A}}_{3}}, & \displaystyle\end{eqnarray}$$

(7.9b ) $$\begin{eqnarray}\displaystyle & \displaystyle {\mathcal{Q}}_{\unicode[STIX]{x1D703},2,sp}=\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}\left((1+{\mathcal{A}}_{3})\frac{{\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}}{\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}{\mathcal{U}}_{2}}-{\mathcal{A}}_{3}+{\mathcal{A}}_{4}\mathbb{B}\right)\frac{H_{2}}{1+{\mathcal{A}}_{3}}. & \displaystyle\end{eqnarray}$$

The leading-order mean radial velocity was determined in the second-order inner analysis in C17 ( ${\mathcal{U}}_{3}={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}/(\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B})$ ) but, in the present approach, it will be necessary to go to the next order to find ${\mathcal{U}}_{2}$ as a function of the first-order induced mean flow at $r_{a}$ . We can heuristically determine a close value to ${\mathcal{U}}_{2}$ by introducing $\hat{\unicode[STIX]{x1D701}}$ in the mean azimuthal vorticity $\unicode[STIX]{x1D6E9}$ -secular equation as follows:

(7.10) $$\begin{eqnarray}\displaystyle {\mho}_{0}{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,\unicode[STIX]{x1D6E9}}={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}-\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F4}_{q}={\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}-\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}{\mathcal{U}}_{2}+\hat{\unicode[STIX]{x1D701}}\unicode[STIX]{x1D71B}{\mathcal{U}}_{2}. & & \displaystyle\end{eqnarray}$$

Equating ${\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}$ with $\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B}\,{\mathcal{U}}_{2}$ then permits one to write the $\unicode[STIX]{x1D6E9}$ gradient of the azimuthal vorticity as a slight correction of the vertical rotational-velocity shear ${\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}$ : ${\mho}_{0}{\mathcal{Q}}_{\unicode[STIX]{x1D703},2,\unicode[STIX]{x1D6E9}}=\hat{\unicode[STIX]{x1D701}}/\unicode[STIX]{x1D701}\,{\mathcal{V}}_{1,\unicode[STIX]{x1D6F7}}$ . Finally, we obtain the same result as in C17, that the radial velocity is generated by the small vertical wind shear, then we get the relationship ${\mathcal{Q}}_{\unicode[STIX]{x1D703},2,sp}=-\unicode[STIX]{x1D71B}{\mathcal{Q}}_{z,2,sp}$ .

The advantage of taking the azimuthal variable as the secular variable is evident as soon as the second order, since ${\mathcal{V}}_{2}$ and ${\mathcal{Q}}_{z,2}$ vary quadratically over $\unicode[STIX]{x1D70F}_{2}$ while ${\mathcal{Q}}_{\unicode[STIX]{x1D703},2}$ evolves cubically over $\unicode[STIX]{x1D70F}_{2}$ in C17. The dependence on $\unicode[STIX]{x1D70F}_{2}$ increases fast algebraically, with the expansion order making the expansions divergent if the vertical shear is not assumed very small.

7.1.3 Forcing $\unicode[STIX]{x1D6F4}_{v}$ driven by the mean wave momentum flux

According to §§ 5.3 and 6, the forcing on the first-order velocity ( ${\mathcal{V}}_{1},{\mathcal{W}}_{1}$ ) is given on either side of $r_{a}$ , for a symmetric CL, by

(7.11) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}_{v}^{s}=s_{i}\,\overline{\sin X\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}(\unicode[STIX]{x1D70C}_{B}^{s},X)}\,A. & & \displaystyle\end{eqnarray}$$

Substituting the expression (6.18) for the axial vorticity $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ , we straightforwardly get the forcing (G 3) in appendix G. A relationship linking the forcing $\unicode[STIX]{x1D6F4}_{v}^{s}$ to the two previous ones, $\unicode[STIX]{x1D6F4}_{q}^{s}$ and $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}$ , is then found as follows:

(7.12) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F4}_{v}^{s}+s\unicode[STIX]{x1D6F4}_{q}^{s}{\mathcal{A}}_{1}\,A^{1/2}+\unicode[STIX]{x1D6F4}_{q^{\prime }}^{s}{\mathcal{A}}_{2}\,A=s\unicode[STIX]{x1D701}{\mathcal{A}}_{1}\,A^{1/2}{\mathcal{U}}_{2}. & & \displaystyle\end{eqnarray}$$

We deduce the expression for $\unicode[STIX]{x1D6F4}_{v}^{s}$ as a function of only $\unicode[STIX]{x1D701}$ , $A$ and ${\mathcal{U}}_{2}$ , as well as of the coefficients ${\mathcal{A}}_{i}$ and ${\mathcal{B}}_{i}$ (cf. appendix G).

(7.13) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6F4}_{v}^{s}}{{\mathcal{U}}_{2}}=s\frac{\unicode[STIX]{x1D701}{\mathcal{A}}_{1}}{1+{\mathcal{A}}_{3}}[1+{\mathcal{A}}_{4}\mathbb{B}]A^{1/2}-s\unicode[STIX]{x1D701}{\mathcal{A}}_{2}\mathbb{B}\,A^{1/2}. & & \displaystyle\end{eqnarray}$$

The forcing is antisymmetric with respect to $r_{a}$ and is proportional to $\unicode[STIX]{x1D701}A^{1/2}{\mathcal{U}}_{2}$ . As for the first-order mean axial vorticity, its jumps follows from (6.54b ) as

(7.14) $$\begin{eqnarray}\displaystyle \frac{[{\mathcal{Q}}_{z,1}]_{-}^{+}}{2\unicode[STIX]{x1D701}A^{1/2}}=-\frac{{\mathcal{C}}-{\mathcal{A}}_{1}}{1+{\mathcal{A}}_{3}}[1+{\mathcal{A}}_{4}\mathbb{B}]-{\mathcal{A}}_{2}\mathbb{B}; & & \displaystyle\end{eqnarray}$$

this is a correction of the steady-state jump $-2{\mathcal{C}}\unicode[STIX]{x1D701}A^{1/2}$ (Caillol Reference Caillol2014).

The eddy forcing $F_{ed}$ applied at $r_{a}$ , in order to keep the double radial gradient of the basic-state axial-vorticity constant there, scales like $\unicode[STIX]{x1D701}{\mathcal{U}}_{2}/A$ . Let us write it as

(7.15) $$\begin{eqnarray}\displaystyle F_{ed}=\unicode[STIX]{x1D701}\,f_{ed}(\unicode[STIX]{x1D70C}_{B}^{\ast })\,{\mathcal{U}}_{2}A^{-1}. & & \displaystyle\end{eqnarray}$$

Exerting a constant forcing $F_{ed}$ at $r_{a}$ therefore links ${\mathcal{U}}_{2}$ and $A$ proportionally:

(7.16) $$\begin{eqnarray}\displaystyle {\mathcal{U}}_{2}=\frac{F_{ed}}{\unicode[STIX]{x1D701}\,f_{ed}(\unicode[STIX]{x1D70C}_{B}^{\ast })}\,A. & & \displaystyle\end{eqnarray}$$

The coefficients ${\mathcal{A}}_{i}$ being decreasing functions of the CL thickness, computation shows that these are small for a value of $\unicode[STIX]{x1D70C}_{B}^{\ast }>2$ . As a result, in practice, the rescaled forcings $\unicode[STIX]{x1D6F4}_{q^{\prime }}/(\unicode[STIX]{x1D701}{\mathcal{U}}_{2})A^{1/2}$ , $\unicode[STIX]{x1D6F4}_{q}/(\unicode[STIX]{x1D701}{\mathcal{U}}_{2})$ and $\unicode[STIX]{x1D6F4}_{v}/(\unicode[STIX]{x1D701}{\mathcal{U}}_{2})A^{-1/2}$ are small, and they are actually very small for a value of $\unicode[STIX]{x1D70C}_{B}^{\ast }\geqslant 4$ . Figure 4(a) shows these forcings on a logarithmic scale as a function of $\unicode[STIX]{x1D70C}_{B}^{\ast }$ . Their magnitude decreases steeply with $\unicode[STIX]{x1D70C}_{B}^{\ast }$ . The wave-flux-induced vorticity radial gradient $-\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}$ is thus a slight correction term with respect to the basic-state vorticity radial gradient at $r_{c}$ , $\unicode[STIX]{x1D701}$ in the expression of ${\mathcal{Q}}_{z,2,sp}$ . The wave-flux-induced vorticity $-\unicode[STIX]{x1D6F4}_{v}/{\mathcal{U}}_{2}$ is also a slight correction with respect to the mean vorticity ${\mathcal{Q}}_{z,1}$ . For example, as $\unicode[STIX]{x1D70C}_{B}^{\ast }=3$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}$ induces a 2 % correction to $\unicode[STIX]{x1D701}$ while $\unicode[STIX]{x1D6F4}_{v}/{\mathcal{U}}_{2}$ and $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}$ induce a 0.6 % correction to the axial-vorticity distortion $[{\mathcal{Q}}_{z,1}]_{-}^{+}$ . As $\unicode[STIX]{x1D70C}_{B}^{\ast }=4$ , the first correction only reaches 0.6 % and the second shrinks to 0.03 %. The forcing $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}$ is negative, so $\hat{\unicode[STIX]{x1D701}}\gtrsim \unicode[STIX]{x1D701}$ ; the basic-state vortex stability at $r_{c}$ has then slightly diminished, but the $\unicode[STIX]{x1D6E9}$ -gradient of the second-order mean vorticity has decreased. The forcing $\unicode[STIX]{x1D6F4}_{v}^{+}/{\mathcal{U}}_{2}$ is positive, so $[\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}\lesssim [{\mathcal{Q}}_{z,1}]_{-}^{+}$ ; the first-order mean vorticity distortion is hence reduced. In the literature, the eddy contribution to vortex intensification is generally weaker than the axisymmetric contribution. Numerical simulations of a full-physics model of a tropical cyclone-like vortex examine the intensification generated by moist-convection-driven vorticity waves (Wang Reference Wang2002a ; Qiu et al. Reference Qiu, Tan and Xiao2010). The mean rotational-wind acceleration is observed near $r_{c}$ , while the mean wave fluxes bring about contributions one order of magnitude smaller. The critical layer was unfortunately not resolved in these numerical schemes; we wonder what the eddy contributions would be if the CL was properly modelled. An absolute angular momentum budget, averaged over time and azimuthally, was computed from mesoscale-resolving Doppler observations of Hurricane Guillermo (1997); the tendency also results in stronger axisymmetric contributions (Reasor, Eastin & Gamache Reference Reasor, Eastin and Gamache2009).

Figure 4.

(a) Forcings: $\unicode[STIX]{x1D6F4}_{q}/(\unicode[STIX]{x1D701}{\mathcal{U}}_{2})$ (solid line), $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/(|\unicode[STIX]{x1D701}|{\mathcal{U}}_{2})A^{1/2}$ (dashed line) and $\unicode[STIX]{x1D6F4}_{v}^{+}/(|\unicode[STIX]{x1D701}|{\mathcal{U}}_{2})A^{-1/2}$ (dotted line) as a function of $\unicode[STIX]{x1D70C}_{B}^{\ast }$ . (b)  $b_{n}(\overline{\unicode[STIX]{x1D70C}})$ as a function of $\overline{\unicode[STIX]{x1D70C}}$ as $\unicode[STIX]{x1D70C}_{B}^{\ast }=5$ , $\unicode[STIX]{x1D6FE}=0$ , $\unicode[STIX]{x1D707}=0.2$ and $N=20$ .

7.2 Resolution of the spectral method

The Fourier sine series in $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2,\odot )}$ of (6.45) is truncated at $n=N$ . Every $b_{n}$ , $n=1,\ldots ,N$ is a weighted sum of the odd Chebyshev polynomials $T_{1}$ and $T_{3}$ , whose weights are computed by using both matching conditions (6.53) and (6.69). Note that these weights are independent of $\unicode[STIX]{x1D701}$ , $A$ and ${\mathcal{U}}_{2}$ . They depend mainly on the mathematical properties of the function $G_{5}$ . They also depend slightly on the CL boundary $\unicode[STIX]{x1D70C}_{B}^{\ast }$ and on the corner parameters $\unicode[STIX]{x1D6FE}$ and $\unicode[STIX]{x1D707}$ . As new matching conditions involving the series terms $b_{n}$ are encountered at higher orders, new polynomials $T_{5}$ , $T_{7}$ etc. must be added. The computation shows that convergence of the series is slow; the fundamental mode $n=1$ is weakly dominating (cf. figure 4 b). For $N=20$ , adding the next harmonic $n=21$ changes the integrated series $L^{2}$ norm $[\int _{0}^{1}\sum _{n=1}^{N}b_{n}^{2}(r)\,\text{d}r]^{1/2}$ by 5 % for any $\unicode[STIX]{x1D70C}_{B}^{\ast }\geqslant 3$ ; for $N=30$ , adding another harmonic changes this integrated norm by 3 %. From now on, we will take truncation at $N=20$ . If $\unicode[STIX]{x1D70C}_{B}^{\ast }\geqslant 4$ , the effect of the thickness of the CL is negligible. For example, with $\unicode[STIX]{x1D6FE}=0$ , $\unicode[STIX]{x1D707}=0.2$ , $N=20$ , the departure between the series computed with distinct thicknesses, $\unicode[STIX]{x1D70C}_{B}^{\ast }=3$ and $\unicode[STIX]{x1D70C}_{B}^{\ast }=\infty$ , in the sense of the $L^{2}$ norm is only 2 %.

Figure 5.

Isovorticity lines: $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)+}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=Cst$ , $A=1$ , $\unicode[STIX]{x1D701}=-0.1$ , $s_{i}=1$ , $\unicode[STIX]{x1D6FE}=0$ , $\unicode[STIX]{x1D707}=0.2$ , $\unicode[STIX]{x1D70C}_{B}^{\ast }=5$ . (a)  $U_{2}=0$ , (b)  $U_{2}=1$ , $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A^{1/2}=1.957\times 10^{-4}$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}=-2.420\times 10^{-4}$ . The vorticity level is indicated on each curve.

Figure 5 displays the contour lines of the axial-vorticity $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}$ . In figure 5(a), the radial velocity ${\mathcal{U}}_{2}$ is zero and in figure 5(b), ${\mathcal{U}}_{2}=A$ . The isovorticity contours coincide with the streamlines in (a) and are symmetric with respect to the axis $x=\unicode[STIX]{x03C0}$ . Inside the separatrices, the axial vorticity is constant. The symmetry is broken in figure 5(b) for the contours outside and near the separatrix. Indeed, the new terms brought about by the mean radial velocity only alter the flow near the separatrix. Far from the separatrix, the vorticity is given by the radially monotonic function ${\mathcal{F}}_{2}(Z)$ and the inner vorticity profile then follows the basic-vorticity profile. We study a linearly stable basic-state vortex; as a result, $\unicode[STIX]{x1D701}<0$ and the inner axial vorticity decreases with the radius. However, near the separatrix and inside the cat’s eye, the axial vorticity is no longer monotonic. Inside the separatrices, the axial vorticity is quadripolar, with one negative pole at $x<\unicode[STIX]{x03C0}$ for $s=+$ and another one at $x>\unicode[STIX]{x03C0}$ for $s=-$ as ${\mathcal{U}}_{2}>0$ ; as a result, the fundamental mode $n=1$ prevails ( $\sin x$ ). Both upper poles are not of opposite strengths, the negative pole is stronger. The emergence of the CL in the presence of a $O(\unicode[STIX]{x1D716})$ mean radial velocity has thus generated secondary vorticity extrema outside the eyewall. In spiral rainbands, these extrema are usually associated in pairs of local positive and negative PV anomalies (Chen & Yau Reference Chen and Yau2001). The contours intersect the separatrix at an angle $x\gtrsim \unicode[STIX]{x03C0}$ for ${\mathcal{U}}_{2}>0$ . The contours $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)+}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=-0.177,-0.152$ penetrate the cat’s eye at $x>\unicode[STIX]{x03C0}$ , then go out at $x\lesssim 2\unicode[STIX]{x03C0}$ on forming a bend to fulfil the $2\unicode[STIX]{x03C0}$ periodicity at $x=2\unicode[STIX]{x03C0}$ . The particular contour $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=-s_{i}/2[\hat{{\mathcal{Q}}}_{z,1}]_{-}^{+}\simeq -0.138$ starts at $x=0$ , on the meeting point of the separatrices, and crosses the separatrix at $x=\unicode[STIX]{x03C0}$ ; it coincides with the vertical axis $x=\unicode[STIX]{x03C0}$ inside the cat’s eye. This contour coincides with the separatrix as ${\mathcal{U}}_{2}=0$ in figure 5(a). Two new branches depart at either side of this axis in a nearly symmetric way and join both of the saddle points, forming a closed lobe at $x<\unicode[STIX]{x03C0}$ . The contour $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=-0.198$ has a first branch that does not cut the separatrix. A second and closed branch is located on either side of the separatrix for $x>\unicode[STIX]{x03C0}$ . As the mean radial velocity ${\mathcal{U}}_{2}$ is stronger, isovorticity lines are even more deformed, as we can see in figure 6(a), where ${\mathcal{U}}_{2}=10$ ; more contours penetrate the cat’s eye. The contour $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)+}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=0$ starts on the separatrix near the corner and forms a positive-vorticity lobe that cuts the separatrix a second time at $x<\unicode[STIX]{x03C0}$ . The contours $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)+}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=-0.244,-0.198,-0.177$ intersect the separatrix at $x>\unicode[STIX]{x03C0}$ , then go out and bend at $x\lesssim 2\unicode[STIX]{x03C0}$ . The contour $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=-0.339$ has a branch that does not cut the separatrix and another one that is located on either side of the separatrix at $x>\unicode[STIX]{x03C0}$ . The contour $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=-0.315$ has a single branch that penetrates the cat’s eye at $x\gtrsim \unicode[STIX]{x03C0}$ after making a large bend, then goes out at $x\lesssim 2\unicode[STIX]{x03C0}$ , making such a large second bend that it quasi-connects with the first. For greater clarity, the part of the contour inside the cat’s eye is not drawn. The level $-0.315$ is thus nearly the strongest negative-vorticity level for which a $2\unicode[STIX]{x03C0}$ -range contour can penetrate the cat’s eye. Since $A$ , $\unicode[STIX]{x1D70C}_{B}^{\ast }$ and $\unicode[STIX]{x1D701}$ have not changed in figures 5 and 6, the same contour $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=-0.138$ divides the cat’s eye at $x=\unicode[STIX]{x03C0}$ into two parts in the same way. The vorticity extrema are higher in each pole, of magnitude greater than one, so they may be much stronger than the neighbouring basic-state vorticity.

Figure 6.

$A=1$ , $s_{i}=1$ , $\unicode[STIX]{x1D6FE}=0$ , $\unicode[STIX]{x1D707}=0.2$ , $\unicode[STIX]{x1D70C}_{B}^{\ast }=5$ . (a) Isovorticity lines: $\unicode[STIX]{x1D713}_{\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70C}}^{(2)+}-s_{i}\hat{{\mathcal{Q}}}_{z,1}^{+}=Cst$ , $\unicode[STIX]{x1D701}=-0.1$ , $U_{2}=10$ , $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A^{1/2}=1.957\times 10^{-4}$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}=-2.420\times 10^{-4}$ . The vorticity level is indicated on each curve. (b) Mode: $m=1$ , $\unicode[STIX]{x1D71B}=20$ , $\unicode[STIX]{x1D716}=0.1$ , $a=-1$ , ${\mho}_{0}=0.49$ , $\unicode[STIX]{x1D6FD}_{l,1}^{+}+d_{l,1,0}^{+}+(\unicode[STIX]{x1D6FD}_{d,1}^{+}+d_{d,1,0}^{+})A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D6FD}_{ad,1}^{+}+d_{ad,1,0}^{+})a_{\unicode[STIX]{x1D6F7}}=1.5A^{1/2}$ , $\unicode[STIX]{x1D705}_{l,1}^{+}+g_{l,1,0}^{+}+(\unicode[STIX]{x1D705}_{d,1}^{+}+g_{d,1,0}^{+})A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D705}_{ad,1}^{+}+g_{ad,1,0}^{+})a_{\unicode[STIX]{x1D6F7}}=10^{-4}\unicode[STIX]{x1D6E9}$ , mean flow: $\unicode[STIX]{x1D701}=-0.0005$ , ${\mathcal{Q}}_{z,1}^{+}=0.001$ , ${\mathcal{S}}_{1}=1$ , ${\mathcal{U}}_{2}=1$ , ${\mathcal{U}}_{2}^{\prime }=-2.04$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}=-1.0299\times 10^{-6}$ , $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A^{1/2}=9.783\times 10^{-7}$ and $\unicode[STIX]{x1D6F4}_{v}^{+}/{\mathcal{U}}_{2}A^{-1/2}=2.0\times 10^{-6}$ . The critical radius $r_{c}=2$ is shown as a dotted line.

Owing to the inner-flow distortions, the cat’s eye is not strictly symmetric with respect to the level $\unicode[STIX]{x1D70C}=0$ . As a result, though we have used a coordinate $Z$ describing the streamlines in a refined way, this variable $Z$ only gives a first approximation of the dividing-streamline location. A better variable, the stressed coordinate $\tilde{Z}$ , is needed in order to correctly describe the deformed trajectory of the streamlines inside the critical layer. Thus, it is necessary to ensure that the tangential and axial velocities are linked in the frame moving with the vertical linear-wave speed, at the core of the cat’s eye and at the crossing point of the separatrices, by the relationship (D 4), a topological property which has previously been overlooked (see, for instance, Benney & Bergeron (Reference Benney and Bergeron1969), Maslowe (Reference Maslowe1986)). The variable $Z$ does not allow one to satisfy (D 4) as soon as the first inner-flow order. As a result, $Z$ is rescaled in an asymptotic expansion following $\tilde{Z}$ ; the method is explained in appendix D. The new fields, denoted by a tilde, are thus expanded in the following way, for instance, for the radial velocity

(7.17) $$\begin{eqnarray}\displaystyle \tilde{U} (\tilde{Z},x)=\unicode[STIX]{x1D716}\,[\,\tilde{U} ^{(0)}+\unicode[STIX]{x1D716}^{1/2}\ln \unicode[STIX]{x1D716}\,\tilde{U} ^{(1)}+\unicode[STIX]{x1D716}^{1/2}\,\tilde{U} ^{(2)}+\cdots \,]. & & \displaystyle\end{eqnarray}$$

Using these analytical expressions of the three-dimensional velocity field, we were able to analytically compute the CL streamlines (cf. appendix F). For instance, the characteristic pattern associated with a nonlinear vorticity wave packet is shown in figures 6–9, in a horizontal plane at a constant height $\unicode[STIX]{x1D6F7}$ and in a vertical plane at a constant angle $\unicode[STIX]{x1D703}$ . For the three lowest azimuthal wavenumbers, the charts exhibit dipolar, tripolar and quadrupolar vortices whose satellites wind and spiral along the vortex axis and are asymmetric with respect to both the critical radius and the axis $\unicode[STIX]{x1D703}=0\,[2\unicode[STIX]{x03C0}/m],\unicode[STIX]{x03C0}/m\,[2\unicode[STIX]{x03C0}/m]$ (the azimuthal angle being calculated from a reference axis passing by a meeting point of the separatrices). These low-azimuthal-wavenumber patterns dominate asymmetric disturbances in a rapidly rotating vortex (Reasor et al. Reference Reasor, Montgomery, Marks and Gamache2000). The contours in these figures are not the trajectories of air particles but their envelopes, the surfaces on which particles slide but cannot cross, given by the radial coordinate ${\mathcal{R}}_{st}(x,\unicode[STIX]{x1D6F7})$ in the parameterized representation $({\mathcal{R}}_{st}(x,\unicode[STIX]{x1D6F7}),z_{st}(x,\unicode[STIX]{x1D6F7}))$ of a three-dimensional $Z_{st}$ -streamline. These contours are drawn in a local reference system moving with the vertical linear-wave speed $c_{z}=\hat{\unicode[STIX]{x1D714}}/k=\tilde{\unicode[STIX]{x1D714}}/k+\overline{W}_{c}/\unicode[STIX]{x1D71B}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D6F7}}T_{1}\equiv \tilde{{\mho}}/\unicode[STIX]{x1D71B}+{\mathcal{W}}{\mho}_{\unicode[STIX]{x1D6F7}}\unicode[STIX]{x1D70F}_{1}$ .

Figure 7.

Mode: $m=2$ , $\unicode[STIX]{x1D71B}=0.5$ , $\unicode[STIX]{x1D716}=0.05$ , ${\mho}_{0}=0.49$ , $a=-1$ , $A=(1+\cos \unicode[STIX]{x1D6F7}/2)^{2}$ , $s_{i}=1$ , $\unicode[STIX]{x1D6FD}_{l,1}^{+}+d_{l,1,0}^{+}+(\unicode[STIX]{x1D6FD}_{d,1}^{+}+d_{d,1,0}^{+})~A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D6FD}_{ad,1}^{+}+d_{ad,1,0}^{+})~a_{\unicode[STIX]{x1D6F7}}=-2A^{1/2}$ , $\unicode[STIX]{x1D705}_{l,1}^{+}+g_{l,1,0}^{+}+(\unicode[STIX]{x1D705}_{d,1}^{+}+g_{d,1,0}^{+})~A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D705}_{ad,1}^{+}+g_{ad,1,0}^{+})a_{\unicode[STIX]{x1D6F7}}=10^{-4}\unicode[STIX]{x1D6E9}$ , mean flow: $\unicode[STIX]{x1D701}=-0.1$ , ${\mathcal{Q}}_{z,1}^{+}=A^{1/2}$ , ${\mathcal{S}}_{1}=0.4A^{1/2}+0.2$ , ${\mathcal{U}}_{2}=-0.1\sin \unicode[STIX]{x1D6F7}/(\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B})$ , ${\mathcal{U}}_{2}^{\prime }=-{\mathcal{U}}_{2}/{\mho}_{0}$ , $\unicode[STIX]{x1D70C}_{B}^{\ast }=5$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}=-2.4198\times 10^{-4}$ , $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A^{1/2}=1.9566\times 10^{-4}$ and $\unicode[STIX]{x1D6F4}_{v}^{+}/{\mathcal{U}}_{2}A^{-1/2}=4.0085\times 10^{-4}$ . (a)  $\unicode[STIX]{x1D6F7}=0$ , (b)  $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/2$ . The critical radius $r_{c}=1$ is shown as a dotted line.

Figure 8.

Mode: $m=2$ , $k=1$ , $a=-1$ , ${\mho}_{0}=0.49$ , $\unicode[STIX]{x1D716}=0.05$ , $A=(1+\cos \unicode[STIX]{x1D6F7}/2)^{2}$ , $s_{i}=1$ , $\unicode[STIX]{x1D6FD}_{l,1}^{+}+d_{l,1,0}^{+}+(\unicode[STIX]{x1D6FD}_{d,1}^{+}+d_{d,1,0}^{+})~A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D6FD}_{ad,1}^{+}+d_{ad,1,0}^{+})~a_{\unicode[STIX]{x1D6F7}}=-2A^{1/2}$ , $\unicode[STIX]{x1D705}_{l,1}^{+}+g_{l,1,0}^{+}+(\unicode[STIX]{x1D705}_{d,1}^{+}+g_{d,1,0}^{+})~A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D705}_{ad,1}^{+}+g_{ad,1,0}^{+})~a_{\unicode[STIX]{x1D6F7}}=10^{-4}\unicode[STIX]{x1D6E9}$ , mean flow: $\unicode[STIX]{x1D701}=-0.1$ , ${\mathcal{Q}}_{z,1}^{+}=A^{1/2}$ , ${\mathcal{S}}_{1}=0.4A^{1/2}+0.2$ , ${\mathcal{U}}_{2}=-0.1\sin \unicode[STIX]{x1D6F7}/(\unicode[STIX]{x1D701}\unicode[STIX]{x1D71B})$ , ${\mathcal{U}}_{2}^{\prime }=-{\mathcal{U}}_{2}/{\mho}_{0}$ , $\unicode[STIX]{x1D70C}_{B}^{\ast }=5$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}=-2.4198\times 10^{-4}$ , $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A^{1/2}=1.9566\times 10^{-4}$ and $\unicode[STIX]{x1D6F4}_{v}^{+}/{\mathcal{U}}_{2}A^{-1/2}=4.0085\times 10^{-4}$ . (a)  $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}$ , (b)  $\unicode[STIX]{x1D703}=0$ between the heights $\unicode[STIX]{x1D6F7}=0$ and $\unicode[STIX]{x1D6F7}=6\unicode[STIX]{x03C0}/\unicode[STIX]{x1D71B}\unicode[STIX]{x1D716}^{1/2}\simeq 2.68\,\unicode[STIX]{x03C0}$ . The critical radius $r_{c}=1$ is shown as a dotted line.

Figure 9.

Mode: $m=3$ , $A=1$ , $s_{i}=1$ , $a=-1.5$ , ${\mho}_{0}=0.49$ , $\unicode[STIX]{x1D6FD}_{l,1}^{+}+d_{l,1,0}^{+}+(\unicode[STIX]{x1D6FD}_{d,1}^{+}+d_{d,1,0}^{+})~A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D6FD}_{ad,1}^{+}+d_{ad,1,0}^{+})~a_{\unicode[STIX]{x1D6F7}}=-1.5A^{1/2}$ , $\unicode[STIX]{x1D705}_{l,1}^{+}+g_{l,1,0}^{+}+(\unicode[STIX]{x1D705}_{d,1}^{+}+g_{d,1,0}^{+})~A_{\unicode[STIX]{x1D6F7}}/A+(\unicode[STIX]{x1D705}_{ad,1}^{+}+g_{ad,1,0}^{+})~a_{\unicode[STIX]{x1D6F7}}=10^{-4}\unicode[STIX]{x1D6E9}$ , mean flow: ${\mathcal{Q}}_{z,1}^{+}=0.2$ , ${\mathcal{U}}_{2}^{\prime }=-2.04$ , $\unicode[STIX]{x1D70C}_{B}^{\ast }=5$ , $\unicode[STIX]{x1D707}=0.2$ , $\unicode[STIX]{x1D6FE}=0$ . (a)  $k=1$ , $r_{c}=1$ , $\unicode[STIX]{x1D701}=-0.5$ , $\unicode[STIX]{x1D716}=0.07$ , ${\mathcal{U}}_{2}=0.5$ , ${\mathcal{S}}_{1}=1$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}=-1.2099\times 10^{-3}$ , $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A^{1/2}=9.7830\times 10^{-4}$ and $\unicode[STIX]{x1D6F4}_{v}^{+}/{\mathcal{U}}_{2}A^{-1/2}=2.0043\times 10^{-3}$ . (b)  $k=10$ , $r_{c}=2$ , $\unicode[STIX]{x1D701}=-0.0125$ , $\unicode[STIX]{x1D716}=0.04$ , ${\mathcal{U}}_{2}=1$ , ${\mathcal{S}}_{1}=-1$ , $\unicode[STIX]{x1D6F4}_{q}/{\mathcal{U}}_{2}=-3.025\times 10^{-5}$ , $\unicode[STIX]{x1D6F4}_{q^{\prime }}^{+}/{\mathcal{U}}_{2}A^{1/2}=2.446\times 10^{-5}$ and $\unicode[STIX]{x1D6F4}_{v}^{+}/{\mathcal{U}}_{2}A^{-1/2}=5.0011\times 10^{-5}$ . The critical radius $r_{c}$ is shown as a dotted line.

The parameters $\unicode[STIX]{x1D716}A$ , $\hat{\unicode[STIX]{x1D701}}$ , $\unicode[STIX]{x1D71B}^{2}$ and ${\mathcal{U}}_{2}/A$ are crucial for the convergence of the asymptotic expansion. If these are large in absolute value then the expansions may diverge. Increasing the phase tilt $|\unicode[STIX]{x1D71B}|$ strongly reduces the values of $\unicode[STIX]{x1D701}$ and/or the wave amplitude $\unicode[STIX]{x1D716}A$ ; for example, in figure 9(a) $\unicode[STIX]{x1D71B}=1/3$ , $\unicode[STIX]{x1D716}A=0.07$ and $\unicode[STIX]{x1D701}=0.5$ , in figure 9(b) $\unicode[STIX]{x1D71B}=20/3$ , $\unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D701}$ are then reduced to $\unicode[STIX]{x1D716}A=0.04$ and $\unicode[STIX]{x1D701}=-0.0125$ , and in figure 6(b), $\unicode[STIX]{x1D71B}=20$ , so $\unicode[STIX]{x1D701}$ is shrunk to $\unicode[STIX]{x1D701}=-0.0005$ for $\unicode[STIX]{x1D716}A=0.1$ . We observe that the satellites become larger as $|\unicode[STIX]{x1D71B}|$ increases. This weakly nonlinear approach does not permit one to model singular vorticity waves with large phase tilts (see Caillol (Reference Caillol2014, Reference Caillol2015) for more details about the limitations on the parameter $\unicode[STIX]{x1D71B}$ ). Small phase tilts are usually related to small-size intense vortices with small vertical wavenumbers $k$ and moderate $r_{c}\geqslant 1$ . For instance, the wavenumber $k=1$ in a tornado characterized by $R_{mw}=100$ m is linked to a vertical wavelength $L_{z}\simeq 628$ m. The wavenumber $k=10$ in a hurricane with $R_{mw}=20$ km is related to a wavelength $L_{z}\simeq 12.6$ km. The critical radius is chosen as $r_{c}\geqslant 1$ , since it is associated with the outer eyewall edge where the rotational-wind profile is decreasing monotonically. The plots of the rotational and axial velocity (not shown) reveal that the cat’s eye is not the location of a velocity maximum. The azimuthal velocity decreases gradually with radius inside the critical layer.

Figures 7 and 8 exhibit a vortex perturbed by a $m=2$ vortical wave whose chosen envelope is $A=(1+0.5\cos \unicode[STIX]{x1D6F7})^{2}$ . Figure 7 shows the related tripole at various heights: (a) $\unicode[STIX]{x1D6F7}=0$ , where $A=9/4$ is maximum and ${\mathcal{U}}_{2}=0$ ; (b) $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}/2$ , where $A=1$ and ${\mathcal{U}}_{2}=2$ is maximum. Figure 8(a) shows the horizontal plane $\unicode[STIX]{x1D6F7}=\unicode[STIX]{x03C0}$ where $A=1/4$ is minimum and ${\mathcal{U}}_{2}=0$ ; the vortex is quasi-axisymmetric and both satellites are thin. For the heights where ${\mathcal{U}}_{2}=0$ , the CL pattern is symmetric vis-à-vis the axis $\unicode[STIX]{x1D703}=\unicode[STIX]{x03C0}/2$ and $\unicode[STIX]{x03C0}$ , but not with respect to the critical radius. These symmetries are broken in figure 7(b). Though $A$ in figure 7(a) is more than twice as large as in (b), the cat’s eye thickness is roughly equal; the radial velocity ${\mathcal{U}}_{2}$ then affects the CL thickness too. Figure 8(b) shows a vertical cross-section at $\unicode[STIX]{x1D703}=0$ that spans three vertical wave periods, from $\unicode[STIX]{x1D6F7}=0$ up to $\unicode[STIX]{x1D6F7}=6\unicode[STIX]{x03C0}/\unicode[STIX]{x1D71B}\unicode[STIX]{x1D716}^{1/2}>2\unicode[STIX]{x03C0}$ . Note that as the ratio of the envelope period to the vertical wave period $=\unicode[STIX]{x1D71B}\unicode[STIX]{x1D716}^{-1/2}=\sqrt{5}$ is here irrational, the vertical pattern is then not periodic and all cat’s eyes have a distinct shape. For $0<\unicode[STIX]{x1D6F7}<\unicode[STIX]{x03C0}\,[2\unicode[STIX]{x03C0}]$ , the spiral extends cyclonically and, for $\unicode[STIX]{x03C0}<\unicode[STIX]{x1D6F7}<2\unicode[STIX]{x03C0}\,[2\unicode[STIX]{x03C0}]$ anticyclonically.

We can notice from figures 6–9 that, though the critical layer is vertically unbounded and spiralling, the particles are trapped within the separatrices. As a result, a generalization of the Batchelor integral condition applied to a steady and closed two-dimensional flow (Batchelor Reference Batchelor1956) can be carried out here within the quasi-steady three-dimensional cat’s eyes in order to find a unique inviscid inner flow there.

The vorticity wave packet/vortex CL interaction could partly explain the formation of spiral rainbands. Indeed, the latter are strongly associated with vorticity wave dynamics. Propagating PV waves characterized by spiral structures in PV fields in the mid-lower troposphere are either neutral or nearly neutral discrete modes, or the continuous modes described by the CL theory. The radial wind shear gradually axisymmetrizes outward propagating PV waves that turn into circular filaments (Guinn & Schubert Reference Guinn and Schubert1993; Montgomery & Enagonio Reference Montgomery and Enagonio1998; Moller & Montgomery Reference Moller and Montgomery2000; Wang Reference Wang2002a ,Reference Wang b ; Wang, Zhong & Wang Reference Wang, Zhong and Wang2018). Montgomery & Kallenbach (Reference Montgomery and Kallenbach1997) suggested that PV waves are responsible for the initiation of inner spiral rainbands. In the inner core, continuous modes decay to the benefit of the vortex through the WMFI. Discrete unstable modes may be excited by the subsequent basic-state vorticity profile modification, favour the formation of unstable inner spiral rainbands, mix vorticity between the eyewall and the eye, and then lead to the vortex weakening (Moller & Montgomery Reference Moller and Montgomery1999; Ruan, Zha & Huang Reference Ruan, Zha and Huang2014). Reasor et al. (Reference Reasor, Montgomery, Marks and Gamache2000) found evidence in the inner core of weakening Hurricane Olivia (1994) of a $m=2$ spiral PV wave packet. Spiral rainbands detected by an enhanced reflectivity were observed in the wave vicinity. The majority of inner vorticity bands observed in intensifying Hurricane Guillermo (1997) were associated with enhanced reflectivity, and all coincided with an enhanced upward motion (Reasor et al. Reference Reasor, Eastin and Gamache2009). Inner spiralling rainbands in Hurricane Katrina (2005) were also manifestations of vorticity waves (Judt & Chen Reference Judt and Chen2010). Chen & Yau (Reference Chen and Yau2001) detected inner spiral rainbands near spiral PV bands in their numerical intensifying TC-like vortex model. A modal analysis inside the spiral rainbands showed that the leading wave activity $m=1,2$ modes were continuous PV modes (Chen et al. Reference Chen, Brunet and Yau2003). No discrete unstable PV mode was found in spite of the unstable vorticity profile. Inertia-gravity waves had a smaller wave activity contribution, and their main components were unstable. Their propagation velocity is much faster than the observed moving speed of spiral cloud bands (Wang et al. Reference Wang, Koblinsky, Howden and Beckley1998). An earlier modal analysis of a $f$ -plane shallow water vortex model showed that banded features were composed almost entirely of vorticity modes (Guinn & Schubert Reference Guinn and Schubert1993). Montgomery & Kallenbach (Reference Montgomery and Kallenbach1997) suggested as well that outer spiral rainbands could form around the stagnation radius, while the basic flow is intensifying. In the numerical TC-like vortex simulations, Wang (Reference Wang2002b ) observed the presence of outer spiral rainbands at this radius. Radar observation of intensifying Hurricane Elena (1985) showed four inner convective bands whose azimuthal and radial propagation speeds were consistent with a vorticity–wave dispersion relationship. They spiralled outwards to a stagnation radius where they began to thin and lose their convective signature (Corbosiero et al. Reference Corbosiero, Molinari and Aiyyer2006).

Spiral vorticity anomaly bands are here produced by continuous vortex erosion inside the critical layer. During the slowly evolving WMFI, the CL gradually expels vorticity into the diffusion boundary layers. The erosion is generated by the presence of hyperbolic points in the related Lagrangian flow located near the saddle points, where the separatrices meet, in the quasi-steady regime. CL vorticity then flows away along the unstable manifolds on forming filaments (Velasco Fuentes Reference Velasco Fuentes2005). Owing to the nonlinear CL coupling, the diffusion layers are also helical and elongated by the mean radial velocity. They generate, far from the vortex centre, through a slow viscous diffusion process, long, thin and spiralling high-vorticity filaments. In the earlier, unsteady and stronger vortex/wave interaction, the expelled vorticity rate is clearly higher, but the mean radial velocity is lower, so filaments are not likely to be initially perfect spiralling helices.