2.1. The $z$-Fourier series

We seek $z$-Fourier series solutions of the form

(2.6)\begin{equation} \left[\begin{array}{@{}c@{}} \chi \\ v \end{array}\right]={-}\sum_{m=1}^\infty\displaystyle\frac{({-}1)^m}{m {\rm \pi}} \left[\begin{array}{@{}c@{}} {\tilde \chi}_m(r,t)\sin\left(m{\rm \pi} (z-1)\right) \\ {\tilde{v}}_m(r,t)\cos\left(m{\rm \pi} (z-1)\right) \end{array}\right] \end{equation}

for which (2.5b), noting

(2.7)\begin{equation} \frac12 (z-1) ={-}\sum_{m=1}^\infty\displaystyle\frac{({-}1)^m}{m{\rm \pi}}\sin(m{\rm \pi} (z-1)) \quad (0 < z\leqslant 1), \end{equation}

leads to the boundary condition

(2.8)\begin{equation} r{\tilde \chi}_m=\ell {\mathfrak{W}}(t) \quad \mbox{at} \ r=\ell. \end{equation}

The LT-solution obtained from (${O}$: 2.16$a$), following the change ${\hat {\mathfrak {E}}}(p)\mapsto {\hat {\mathfrak {W}}}(p)$, is

(2.9a)\begin{equation} \left[\begin{array}{@{}c@{}} {\hat {\tilde \chi}}_m \\ {\hat {\tilde{v}}}_m \end{array}\right]= \left[\begin{array}{@{}c@{}} 1 \\ 2m{\rm \pi}/{\mathfrak{p}} \end{array}\right]{\hat {\mathfrak{W}}}(p) \displaystyle\frac{{\mathrm J}_1\left(m{\rm \pi} qr\right)}{{\mathrm J}_1\left(m{\rm \pi} q\ell\right)}. \end{equation}

Here, ${\mathrm J}_1$ is the Bessel function, while the functions ${\mathfrak {p}}={\mathfrak {p}}(p)$ and $q=q({\mathfrak {p}})= q({\mathfrak {p}}(p))$ are determined by the dispersion relation (${O}$: 2.17$a$–$d$), which gives them as the solutions of

(2.9b,c)\begin{equation} \left.\begin{array}{r@{}} {\mathfrak{p}}^2={-}4\left/\left(q^2+1\right)\!,\right.\\ {\mathfrak{p}}=p+(q^2+1){\mathsf{d}}_m \end{array} \right\} \ \Longleftrightarrow\ \left\{\begin{array}{@{}l} q^2+1={-}4/{\mathfrak{p}}^2,\\ p={\mathfrak{p}}+4{\mathsf{d}}_m/{\mathfrak{p}}^2, \end{array}\right. \end{equation}

where

(2.9d)\begin{equation} {\mathsf{d}}_m=E(m{\rm \pi})^2. \end{equation}

In terms of the Bessel function of imaginary argument ${\mathrm I}_1$, the initial behaviour

(2.10)\begin{equation} {\tilde \chi}_m\approx\displaystyle\frac{8t^{3/2}}{3\sqrt{\rm \pi}}\,\displaystyle\frac{{\mathrm I}_1\left(m{\rm \pi} r\right)}{{\mathrm I}_1\left(m{\rm \pi} \ell\right)}\quad (t\ll 1) \end{equation}

is recovered on expanding the integrand of the inverse-LT ${\mathcal {L}}^{-1}\{{\hat {\tilde \chi }}_m\}$ of ${\hat {\tilde \chi }}_m$ defined by (2.9a) under the limit $p\to \infty$, for which $q\to {\mathrm i}$ (see (2.9c)) and noting that ${\mathfrak {W}}(t)\approx 8t^{3/2}/(3\sqrt {\rm \pi})$ for $t\ll 1$ (see (1.13a,c)). The initial response (2.10), $\propto t^{3/2}$, is ‘softer’ than the impulsive response (${O}$: 2.9$c$) to the ${\mathfrak {E}}$-trigger.

For $t>0$ the LT-inversion of (2.9a) involves consideration of the contributions from various poles, which include the zeros of ${\mathrm J}_1(m{\rm \pi} q\ell )$, as well as the cuts at $p=\pm 2{\mathrm i}$ exhibited by ${\hat {\mathfrak {W}}}(p)$. Since we have omitted the stress boundary conditions at $r=\ell$, following Oruba et al. (Reference Oruba, Soward and Dormy2020), we disregard the shear layer responses, both QG and ageostrophic, abutting the outer boundary $r=\ell$ linked to the poles $p=0$ (in ${\hat {\mathfrak {W}}}(p)$) and ${\mathfrak {p}}=0$ (in the factor ${\mathfrak {p}}^{-1}$ of the expression for ${\hat {\tilde {v}}}_m$). Their proper treatment using the full boundary conditions (5.2) was undertaken by Oruba et al. (Reference Oruba, Soward and Dormy2017).

To understand the nature of the triggered waves, we begin by focusing attention on the remaining set $\daleth$ of poles $p=p_{mn}$, $p^*_{mn}$ (the superscript ‘$*$’ denotes the complex conjugate) identified by

(2.11)\begin{equation} q=q_{mn}={j_n}/({m{\rm \pi}\ell})\ (>0), \end{equation}

determined by the real zeros $j_n$ of ${\mathrm J}_1(m{\rm \pi} q \ell )$ (i.e. ${\mathrm J}_1(\kern1.5pt j_n)=0$). In turn, they define

(2.12a,b)\begin{gather} {\mathfrak{p}}_{mn}={\mathrm i}\omega_{mn},\quad \omega_{mn}={2}/\sqrt{q_{mn}^2+1}, \end{gather}

(2.12c,d)\begin{gather} p_{mn}={\mathrm i}\omega_{mn}-d_{mn},\quad d_{mn}=4{\mathsf{d}}_m/\omega_{mn}^2, \end{gather}

and, following the differentiation of $p={\mathfrak {p}}+4{\mathsf{d}}_m/{\mathfrak {p}}^2$ (2.9c), they determine the value

(2.12e)\begin{equation} \left[\displaystyle\frac{{\mathfrak{p}}}{p}\displaystyle\frac{{\mathrm d}{p}}{{\mathrm d}{{\mathfrak{p}}}}\right]_{{\mathfrak{p}}={\mathfrak{p}}_{mn}}= \displaystyle\frac{{\mathrm i}\omega_{mn}+2d_{mn}}{{\mathrm i}\omega_{mn}-d_{mn}} \quad \mbox{at} \ p=p_{mn}, \end{equation}

needed to evaluate pole residues in § 4.1 below. From (2.12b), we may generate

(2.13a)\begin{gather} \varphi_{mn}=\sqrt{1-(\omega_{mn}/2)^2}=\cos (2\alpha_{mn}), \end{gather}

(2.13b)\begin{gather}\tfrac12\omega_{mn}=\sqrt{1-\varphi_{mn}^2}=\sin(2\alpha_{mn}), \end{gather}

(2.13c)\begin{gather}q_{mn}=2\varphi_{mn}/\omega_{mn}=\cot(2\alpha_{mn}), \end{gather}

cf. $\varphi _{mn}$ with $\varphi _\pm$ in (C8b) in Appendix C.1 below. The role of the aforementioned cuts re-emerges in (2.19) below.

Our disregard of the QG-response linked to the poles $p=0$ and ${\mathfrak {p}}=0$ has consequences on the boundary values (2.8) of individual triggered waves, which actually satisfy

(2.14)\begin{equation} r{\tilde \chi}_m=\ell {\mathfrak{W}}_{{{\textit{MF}}}}(t) \quad \mbox{at} \ r=\ell, \end{equation}

i.e. the oscillatory waves cannot exhibit the steady features of the E-trigger $\ell \,{\mathfrak {W}}_{{{{E}}}}$ constituent (1.30b) (a curiosity noted and explained in the last paragraph of § 2.3 of Oruba et al. (Reference Oruba, Soward and Dormy2020), in connection with the ${\mathfrak {E}}$-trigger).

2.2. The $r$-Fourier–Bessel series

As in Oruba et al. (Reference Oruba, Soward and Dormy2020), we take advantage of the Fourier–Bessel series expansion

(2.15)\begin{equation} \displaystyle\frac{{\mathrm J}_1(m{\rm \pi} q r)}{{\mathrm J}_1(m{\rm \pi} q l)}={-} \dfrac12\sum_{n=1}^\infty\dfrac{q_{mn}^2}{\omega_{mn}^{{-}2}+{\mathfrak{p}}^{{-}2}} \, \displaystyle\frac{{\mathrm J}_1(\kern1.5pt j_nr/\ell)}{j_n{\mathrm J}_0(\kern1.5pt j_n)} \quad \mbox{on} \ 0\leqslant r < \ell, \end{equation}

which follows from (${O}$: A3) on noting that (2.9c), (2.12b) together imply $q_{mn}^2-q^2=4(\omega _{mn}^{-2}+{\mathfrak {p}}^{-2})$. So we write

(2.16a)\begin{equation} \left[\begin{array}{@{}c@{}} {\tilde \chi}_m \\ {\tilde{v}}_m \end{array}\right] =\sum_{n=1}^\infty \left[\begin{array}{@{}c@{}} {\mathring\chi}^{\mathsf{d}}_{mn} \\ {\mathring v}^{\mathsf{d}}_{mn} \end{array}\right]\displaystyle\frac{{\mathrm J}_1(\kern1.5pt j_nr/\ell)}{j_n{\mathrm J}_0(\kern1.5pt j_n)} \quad \mbox{on} \ 0\leqslant r < \ell. \end{equation}

Here, the superscript ‘${\mathsf{d}}$’ distinguishes our general ${\mathsf{d}}_m(=E(m{\rm \pi} )^2)\not =0$ (2.9d) usage from the ${\mathsf{d}}_m\downarrow 0$ limit considered in the following § 3, where the superscript ‘${\mathsf{d}}$’ will be omitted. It follows that the LT of (2.16a) is determined via

(2.16b)\begin{equation} \left[\begin{array}{@{}c@{}} \widehat{{\mathring\chi}_{mn}^{\mathsf{d}}} \\ \widehat{{\mathring v}_{mn}^{\mathsf{d}}} \end{array}\right]={-}2\varphi^2_{mn} \left[\begin{array}{@{}c@{}} {\mathfrak{p}} \\ 2m{\rm \pi} \end{array}\right] \displaystyle\frac{{\mathfrak{p}}\,{\hat {\mathfrak{W}}}(p)}{{\mathfrak{p}}^2+\omega_{mn}^2} \end{equation}

on substitution of $q_{mn}=2\varphi _{mn}/\omega _{mn}$ (2.13c) into (2.15). An unfortunate feature of the Fourier–Bessel series expansion (2.15) is that it necessarily fails at $r=\ell$, where each eigenfunction ${\mathrm J}_1(\kern1.5pt j_nr/\ell )$ vanishes. So it is not possible for (2.16a) to satisfy the reduced boundary condition ${\tilde \chi }_m(\ell ,t)={\mathfrak {W}}_{{\textit {MF}}}(t)$ (2.14) except in the limiting sense $r\uparrow \ell$. That such a limit is achievable is exemplified by the early time inverse-LT (2.10) of (2.9a).

We express the inverse-LT of (2.16b) as

(2.17)\begin{equation} \left[\begin{array}{@{}c@{}} {\mathring\chi}^{{\mathsf{d}}}_{mn} \\ {\mathring v}^{{\mathsf{d}}}_{mn} \end{array}\right]={-}{\mathcal{L}}^{{-}1}\left\{\varphi^2_{mn} \left[\begin{array}{@{}c@{}} {\mathfrak{p}} \\ 2m{\rm \pi} \end{array}\right]\displaystyle\frac{{\hat {\mathfrak{W}}}(p)}{{\mathfrak{p}}-{\mathrm i}\omega_{mn}} \right\}+\mbox{c.c.}, \end{equation}

where ‘c.c.’ denotes complex conjugate. A further reduction of the LT $\widehat {\,{\mathring v}_{mn}^{\mathsf{d}}}$, exhibited by the right-hand side of (2.17), is possible upon using the partial fraction decomposition

(2.18)\begin{equation} \displaystyle\frac{1}{{\mathfrak{p}}-{\mathrm i}\omega_{mn}}=\displaystyle\frac{1}{{\mathrm i}\omega_{mn}}\left[\displaystyle\frac{{\mathfrak{p}}}{{\mathfrak{p}}-{\mathrm i}\omega_{mn}}-1\right]. \end{equation}

The contribution to the inverse-LT ${\mathcal {L}}^{-1}\{\widehat {\,{\mathring v}_{mn}^{\mathsf{d}}}\}$ from the second term $-1/({\mathrm i}\omega _{mn})$ is pure imaginary and so when added to its complex conjugate vanishes leaving

(2.19a)\begin{equation} \left[\begin{array}{@{}c@{}} {\mathring\chi}^{\mathsf{d}}_{mn} \\ {\mathring v}^{\mathsf{d}}_{mn} \end{array}\right] =\varphi^2_{mn}\left[\begin{array}{@{}c@{}} -1\\ 2 {\mathrm i} m{\rm \pi}/\omega_{mn} \end{array}\right]{\mathfrak{W}}^{{\mathsf{d}}}_{mn}(t)+\mbox{c.c.}, \end{equation}

where ${\mathfrak {W}}^{{\mathsf{d}}}_{mn}(t)$ has LT

(2.19b)\begin{equation} {\hat {\mathfrak{W}}}^{{\mathsf{d}}}_{mn}(p)=\displaystyle\frac{{\mathfrak{p}}{\hat {\mathfrak{W}}}(p)}{{\mathfrak{p}}-{\mathrm i}\omega_{mn}}. \end{equation}

The ${\mathfrak {W}}^{{\mathsf{d}}}_{mn}$-notation is motivated by the property ${\mathfrak {W}}^{{\mathsf{d}}}_{mn}(t)={\mathfrak {W}}(t)$, when $\omega _{mn}=0$. So employing (1.13c), we write

(2.19c)\begin{equation} {\mathfrak{W}}^{{\mathsf{d}}}_{mn}(t)=\sum_{{\pm}}\left\{{\mathfrak{W}}_{mn}^{{{\mathsf{d}}}\pm}(t)\right\}, \end{equation}

where ${\mathfrak {W}}^{{{\mathsf{d}}}\pm }(t)$ has LT

(2.19d)\begin{equation} {\hat {\mathfrak{W}}}^{{{\mathsf{d}}}\pm}_{mn}(p)=\displaystyle\frac{{\mathfrak{p}}{\hat {\mathfrak{W}}}^\pm(p)}{{\mathfrak{p}}-{\mathrm i}\omega_{mn}}=\displaystyle\frac{\pm{\mathrm i}\,{\mathfrak{p}}}{p(p\pm 2{\mathrm i})^{1/2}({\mathfrak{p}}-{\mathrm i}\omega_{mn})} \end{equation}

on use of (1.12b).

We must stress that our use of the ${\mathring \chi }_{mn}$, ${\mathring v}_{mn}$ notation, although similar in spirit to Oruba et al. (Reference Oruba, Soward and Dormy2020), differs in detail. For, whereas the factor ${\mathfrak {F}}_{mn}=q_{mn}^2\omega _{mn}^2/2=2\varphi ^2_{mn}$ appears multiplying $[{\mathring \chi }_{mn}, {\mathring v}_{mn}]$ in (${O}$: 2.23$a$), we have removed it from (2.16a), including it instead in (2.16b) (cf. (${O}$: 2.23$b$)).

3.1. The pole part, $\daleth$

On consideration of the explicit form for ${\hat {\mathfrak {W}}}^{\pm }_{mn}(p)$, given (3.1a) the residue at the pole $p={\mathrm i}\omega _{mn}$ is determined from (3.1b), noting that ${\mathfrak {W}}^\pm _{{{E}}}=(\pm {\mathrm i}/2)^{1/2}$ (1.15b), as

(3.3a)\begin{align} {\mathfrak{W}}^\daleth_{mn}(t)&=\left[({\mathrm i}/{\boldsymbol{\varpi}}_{mn}^+)^{1/2}+(-{\mathrm i}/{\boldsymbol{\varpi}}_{mn}^-)^{1/2}\right] \exp\left({\mathrm i}\omega_{mn}t\right) \end{align}

(3.3b)\begin{align} &=\varphi_{mn}^{{-}1}\exp\left({\mathrm i}\omega_{mn}t-\alpha_{mn}\right), \end{align}

in which we have introduced $\alpha _{mn}$ via the relation

(3.3c)\begin{equation} 2\varphi_{mn}=\sqrt{{\boldsymbol{\varpi}}^+_{mn}{\boldsymbol{\varpi}}^-_{mn}}=2\cos (2\alpha_{mn}) \end{equation}

(recall (2.13a) and (3.1d)). The result (3.3b) may be checked by squaring both right-hand sides of (3.3a,b) and noting (2.13b), (3.3c). Substitution of (3.3b) into (2.19a) determines

(3.4a,b)\begin{equation} \left[\begin{array}{@{}c@{}} {\mathring\chi}^\daleth_{mn} \\ {\mathring v}^\daleth_{mn} \end{array}\right]={-}2\varphi_{mn} \left[\begin{array}{@{}c@{}} \cos (\omega_{mn}t-\alpha_{mn})\\ (2m{\rm \pi}/\omega_{mn}) \sin(\omega_{mn}t-\alpha_{mn}) \end{array}\right], \quad \left\{\begin{array}{@{}l} 1>\varphi_{mn}>0,\\ 0<\alpha_{mn}<{\rm \pi}/4. \end{array}\right. \end{equation}

Cederlöf (Reference Cederlöf1988) solved this normal mode problem but from a different perspective. At first sight his method appears different to ours, because driving stems directly from the transient Ekman pumping ${\mathfrak {W}}(t)$ (1.10) (see his (5.21) in which the LT-form ${\tilde {F}}(s)$ is essentially our ${\hat {\mathfrak {W}}}(p)$) and not from the boundary condition at $r=\ell$. Nevertheless, after appropriate variable changes, his solution (5.25) follows directly from (3.3a) via (2.19a), (2.16a). He does not, however, introduce the instructive phase angles $\alpha _{mn}$.

It is interesting to compare the above pole response ${\mathring \chi }_{mn}^{\daleth }=-2\varphi _{mn}\cos (\omega _{mn}t-\alpha _{mn})$ to our ${\mathfrak {W}}$-trigger (1.30a), with the corresponding ${\mathfrak {E}}$-trigger (1.29) response, which from (${O}$: 4.2$a$) is ${\mathring \chi }_{mn}^{\daleth {\mathfrak {E}}}=-\tfrac 12 q_{mn}^2\omega _{mn}^2\cos (\omega _{mn}t)=-2\varphi ^2_{mn}\cos (\omega _{mn}t)$. The responses differ by a factor $\varphi _{mn}$ but coincide when $\varphi _{mn}=1 \ (\alpha _{mn}=0)$. That is the QG-limit $\omega _{mn}\downarrow 0$, which occurs as $q_{mn}=j_n/(m{\rm \pi} \ell )\to \infty$, namely the short radial-$r$ length scale limit ($\kern1.5pt j_n\gg 1$). As the frequency $\omega _{mn}$ increases, $\varphi _{mn}$ decreases ($\alpha _{mn}$ increases) in concert. Accordingly, relative to the ${\mathfrak {E}}$-trigger response ${\mathring \chi }_{mn}^{\daleth {\mathfrak {E}}}$, the phase shifted ($\alpha _{mn}\not =0$) ${\mathfrak {W}}$-trigger response ${\mathring \chi }_{mn}^{\daleth }$ increases in amplitude by a factor $1/\varphi _{mn}$. The increasing trend of the phase shift and mode amplitude terminates as $\varphi _{mn}\downarrow 0$ ($\alpha _{mn}\uparrow {\rm \pi}/4$). That is the MF-limit $\omega _{mn}\uparrow 2$, which occurs as $q_{mn}\downarrow 0$, namely, the short axial-$z$ length scale limit, which is of ever decreasing magnitude

(3.5)\begin{equation} \left[\begin{array}{@{}c@{}} {\mathring\chi}^\daleth_{mn} \\ {\mathring v}^\daleth_{mn} \end{array}\right]\approx{-} 2\sqrt{{\boldsymbol{\varpi}}^-_{mn}}\left[\begin{array}{r} \cos (2t-{\rm \pi}/4)\\ m{\rm \pi} \sin(2t-{\rm \pi}/4) \end{array}\right], \quad \mbox{as}\ {\boldsymbol{\varpi}}^-_{mn} \downarrow 0, \end{equation}

with ${\mathring \chi }^{\daleth }_{mn}$ yet large compared to ${\mathring \chi }_{mn}^{\daleth {\mathfrak {E}}}$ by the factor $\varphi _{mn}^{-1}\approx 1/\sqrt {{\boldsymbol {\varpi }}^-_{mn}}$. Both the phase shift and increased amplitude of the $m=1$ mode are evident in the well-developed solution sufficiently far from the outer boundary $r=\ell$ in ${\mathfrak {W}}$-panels ($c$) of figures 1–4, when compared to the ${\mathfrak {E}}$-panels ($d$), albeit at $E=10^{-4}$.

3.2. The cut part, $\leftrightarrows$

The remaining cut contribution ${\mathfrak {W}}_{mn}(t)-{\mathfrak {W}}^\daleth _{mn}(t)$ is

(3.6a)\begin{equation} {\mathfrak{W}}^\leftrightarrows_{mn}(t)=\sum_{{\pm}}\left\{{\mathfrak{W}}^{\leftrightarrows\pm}_{mn}(t)\right\}, \end{equation}

where the results (3.1b) give

(3.6b)

(3.6c)\begin{align} &={-}{\mathfrak{P}}^\pm_{mn}[f(T^{{\pm}}_{mn})\pm{\mathrm i} g(T^{{\pm}}_{mn})]\exp(\mp2{\mathrm i} t) \end{align}

on application of (1.17b,c), noting $T^{\pm }_{mn}=({\boldsymbol {\varpi }}^{\pm }_{mn}T/2)^{1/2}$ (3.1f) (but cf. (3.1c)). The large $T^{\pm }_{mn}$ asymptotic behaviour $f(T^{\pm }_{mn})\approx ({\rm \pi} T^{\pm }_{mn})^{-1/2}$, $g(T^{\pm }_{mn})\approx 0$ (see (A4a)) determines

(3.7)\begin{equation} {\mathfrak{W}}^{\leftrightarrows}_{mn}(t)\approx{-}\varphi^{{-}2}_{mn}({\rm \pi} t)^{{-}1/2}\left[\cos (2t)-\tfrac12 {\mathrm i}\omega_{mn}\sin(2t)\right] \quad \mbox{for} \ T^{{\pm}}_{mn}\gg 1. \end{equation}

Substitution into (2.19a) yields

(3.8)\begin{equation} \left[\begin{array}{@{}c@{}} {\mathring\chi}^\leftrightarrows_{mn} \\ {\mathring v}^\leftrightarrows_{mn} \end{array}\right] \approx\displaystyle\frac{2}{\sqrt{{\rm \pi} t}} \left[\begin{array}{r} \cos (2t)\\ m{\rm \pi} \sin(2t) \end{array}\right], \end{equation}

similar to (3.5) except for the phase shift ${\rm \pi} /4$ but notably smaller by a factor $({\rm \pi} {\boldsymbol {\varpi }}^-_{mn}t)^{-1/2}\approx \sqrt {2}/({\rm \pi} T^-_{mn})$ whenever $T^-_{mn}\gg 1$.

The simplicity of (3.8) suggests the possibility of constructing from it the large $t$ asymptotic form of the cut solution. On use of the identity

(3.9a)\begin{equation} \displaystyle\frac{r}{2\ell}={-}\sum_{n=1}^\infty\,\displaystyle\frac{{\mathrm J}_1(\kern1.5pt j_nr/\ell)}{j_n{\mathrm J}_0(\kern1.5pt j_n)}, \end{equation}

determined from the $q\to 0$ limit of (${O}$: A3), substitution of (3.8) into the $r$-Fourier–Bessel series (2.16a) yields

(3.9b)\begin{equation} \left[\begin{array}{@{}c@{}} {\tilde \chi}_m^{\leftrightarrows} \\ {\tilde{v}}_m^{\leftrightarrows} \end{array}\right] \approx{-}\displaystyle\frac{r}{\ell}\displaystyle\frac{1}{\sqrt{{\rm \pi} t}} \left[\begin{array}{@{}c@{}} \cos(2t) \\ m{\rm \pi}\sin (2t) \end{array}\right]\quad \mbox{as}\ t\to\infty. \end{equation}

Substitution into the $z$-Fourier series (2.6), noting (1.19b), determines

(3.10)\begin{equation} \left[\begin{array}{@{}c@{}} \chi^{\leftrightarrows} \\ v^{\leftrightarrows} \end{array}\right]\approx{-} \displaystyle\frac{1}{\sqrt{4{\rm \pi} t}}\,\displaystyle\frac{r}{\ell}\left[\begin{array}{@{}c@{}} (z-1) \cos(2t) \\ \sin (2t) \end{array}\right]\approx{-}E^{{-}1/2} \left[\begin{array}{@{}c@{}} {\bar\chi}_{{{\textit{MF}}}} \\ {\bar v}_{{{\textit{MF}}}} \end{array}\right]\quad (z>0) \end{equation}

(see § 1.2.1 and (A5)).

The result (3.10) raises disturbing issues. It suggests that the dominant cut contribution determines a flow contribution of the same size (but opposite in sign) as the original Greenspan and Howard MF-flow responsible for the MF-trigger. However, because of the discontinuous nature of the Fourier series for $(z-1)$ and $1$ at $z=0$ and the Fourier–Bessel series for $r$ at $r=\ell$, the series have to reach extremely large values of $m$ and $n$ to achieve convergence. This is an issue because of the large $T^{-}_{mn}$ requirement $t\gg 1/{\boldsymbol {\varpi }}^-_{mn}=1/(2-\omega _{mn})$ (see (3.1d) and (3.7)). So at any finite $t$ (however large), it is unclear how good the approximation is. Certainly at the values of $t$, for which results are reported here, some evidence of an MF-contribution from the cut was evident in tests (not illustrated). However, surprisingly a similar MF-contribution from the poles was also found (again not illustrated), which when combined with the cut-contribution led to their cancellation, i.e. there is no evidence of an MF-contribution in panels ($c$) of figures 1–4, or for that matter in panels ($a$), ($b$) of those figures at small but finite $E$. The large time MF-issues raised provide a focal point for our discussion of the $\ell \gg 1$ limit in Appendix C, which explains the curious cancellation evident in the aforementioned numerical results (not illustrated).

Remarkably, the ‘method of images’ approach of Appendix C.1, identifies immediately the main cut contribution (3.10) that prompted our concerns, via the MF-part $-{\mathfrak {U}}_{{\textit {MF}}}(t)$ of the very first term $-{\mathfrak {U}}(t)$ on the right-hand side of (C1a). The role of the poles is subtler, because the estimate $|{\mathring \chi }^\daleth _{mn}| \sim T_{mn} |{\mathring \chi }^\leftrightarrows _{mn}|$ below (3.8) suggests that the pole contribution will be even bigger. However, the second term on the right-hand side of final result (C6b) demonstrates that wave interference must occur reducing the accumulated pole effect to exactly that from the cuts. The fact, that $-{\mathfrak {U}}_{{\textit {MF}}}(t)=O(t^{-1/2})$ is indeed the dominant cut contribution, is established by (C6c), which shows the remainder to be even smaller $O(t^{-1})$ for $t\gg 1$.

An alternative perspective is gleaned from the steepest descent results of Appendix C.2, which for $t\gg 1$ identify two wave families ${\tilde \chi }_\pm$ (C7). The triggered MF-wave ${\tilde \chi }_-$, characterised by (C10c,d,f), is only pertinent close to the outer boundary $r=\ell$. There, it remains small compared to the low frequency mode ${\tilde \chi }_+$, characterised by (C10a,b,e). Of even greater importance is the identification, in the last paragraph of Appendix C.2, of a wave front at which the ${\tilde \chi }_\pm$-waves merge and beyond which the waves are evanescent, a feature clearly visible in figures 1–4.

4.1. Pole solution

The pole contribution is determined from (2.19d) by the residue

(4.2)\begin{equation} {\mathfrak{W}}^{{\mathsf{d}}\daleth\pm}_{mn}(t)=\left[\displaystyle\frac{\pm{\mathrm i}}{(p\pm 2{\mathrm i})^{1/2}}\,\displaystyle\frac{{\mathfrak{p}}}{p}\displaystyle\frac{{\mathrm d}{p}}{{\mathrm d}{{\mathfrak{p}}}}\exp(pt)\right]_{{\mathfrak{p}}={\mathrm i}\omega_{mn}} \end{equation}

of the inverse-LT at $p=p_{mn}={\mathrm i}\omega _{mn}-d_{mn}$ (2.12c,d), corresponding to ${\mathfrak {p}}={\mathrm i}\omega _{mn}$ (2.12a). On use of the expression (2.12e) for $[({\mathfrak {p}}/p)({\mathrm d} p/{\mathrm d}{\mathfrak {p}})]_{{\mathfrak {p}}={\mathrm i}\omega _{mn}}$ and noting from (3.1d) that $p_{mn}\pm 2{\mathrm i}=\pm {\mathrm i}{\boldsymbol {\varpi }}^{\pm }_{mn}-d_{mn}$, the residue (4.2) becomes

(4.3a)\begin{equation} {\mathfrak{W}}^{{\mathsf{d}}\daleth\pm}_{mn}(t)=\tfrac12 (1 \pm{\mathrm i})\, {\mathfrak{P}}^{{\mathsf{d}}\pm}_{mn}\exp\left({\mathrm i}\omega_{mn}t-d_{mn}t\right)\!, \end{equation}

where

(4.3b)\begin{equation} {\mathfrak{P}}^{{\mathsf{d}}\pm}_{mn}=\left(\displaystyle\frac{2}{{\boldsymbol{\varpi}}^{{\pm}}_{mn}\pm {\mathrm i} d_{mn}}\right)^{1/2}\displaystyle\frac{\omega_{mn}-2{\mathrm i} d_{mn}}{\omega_{mn}+{\mathrm i} d_{mn}}, \end{equation}

in which $\mbox {Re}\{({\boldsymbol {\varpi }}^{\pm }_{mn}\pm {\mathrm i} d_{mn})^{1/2}\}>0$ as required by analytic continuation on increasing $d_{mn}$ from zero. The decay rate $d_{mn}=4E(m{\rm \pi} )^2/\omega _{mn}^2$ (see (2.9d), (2.12d)) is a consequence of internal friction (identified by the first term in (4.5) of Zhang & Liao Reference Zhang and Liao2008).

4.2. Composite solution

To obtain an approximate solution of the full problem, we make the ansatz

(4.4a)\begin{equation} {\mathfrak{W}}^{{\mathsf{d}}\pm}_{mn}(t)={\mathfrak{P}}^{{\mathsf{d}}\pm}_{mn}\,{\mathfrak{W}}^{{\pm}}\left({\boldsymbol{\varpi}}^{{\pm}}_{mn}t\big/2\right) \exp\left({\mathrm i}\omega_{mn}t-d_{mn}t\right)\!, \end{equation}

which has the properties

(4.4b,c)\begin{equation} {\mathfrak{W}}^{{\mathsf{d}}\pm}_{mn}(t)\to \left\{\begin{array}{@{}ll} {\mathfrak{W}}^{{\pm}}_{mn}(t) \quad & \mbox{as} \ d_{mn}\to 0,\\ {\mathfrak{W}}^{{\mathsf{d}}\daleth\pm}_{mn}(t) & \mbox{as}\ T^{{\pm}}_{mn}\to \infty. \end{array}\right. \end{equation}

The former limit (4.4b) is a simple consequence of the fact that , as $d_{mn}\downarrow 0$, so recovering the $E\downarrow 0$ result (3.1b). The latter limit (4.4c) applies because ${\mathfrak {W}}^{\pm }({\boldsymbol {\varpi }}^{\pm }_{mn}t\big /2)\to {\mathfrak {W}}_{{{E}}}^\pm =\tfrac 12 (1 \pm {\mathrm i})$ (1.15b), as $T^{\pm }_{mn}\to \infty$, leaving the pole contribution (4.3), i.e. the remaining cut contribution ${\mathfrak {W}}^{{\mathsf{d}}\pm \leftrightarrows }_{mn}(t)$ is assumed negligible in this limit.

Our construction of the composite (4.4a) is in much the same spirit as Greenspan and Howards’ unbounded flow composite ${\bar {\mathfrak {u}}}_{{\textit {GH}}}(t)$ (1.31b), i.e. neither is a true asymptotic formula, but instead both capture the dominant flow characteristics.

4.3. Ekman layer damping

Consistent with our construction of the composite (4.4a), we implement Ekman damping by replacing the exponential $\exp ({\mathrm i}\omega _{mn}t-d_{mn}t)$ with

(4.5a)\begin{equation} \exp\left[{\mathrm i}\left(\omega_{mn}+\omega^E_{mn}\right)t-\left(d_{mn}+d^E_{mn}\right)t\right], \end{equation}

where the frequency and damping increments $\omega ^E_{mn}$ and $d^E_{mn}$ adopted are those for pure inertial waves, given by (${O}$: 2.25) (also Kerswell & Barenghi Reference Kerswell and Barenghi1995; Zhang & Liao Reference Zhang and Liao2008). The increments have the form

(4.5b)\begin{equation} \left[\begin{array}{@{}c@{}} d^E_{mn}\\ \omega^E_{mn} \end{array}\right]=\sqrt{\displaystyle\frac{E}{2}}\,\varphi_{mn} \left[\begin{array}{@{}c@{}} (2-\varphi_{mn})\sqrt{1+\varphi_{mn}} \\ (2+\varphi_{mn})\sqrt{1-\varphi_{mn}} \end{array}\right], \end{equation}

where $\varphi _{mn}=\tfrac 12\sqrt {{\boldsymbol {\varpi }}^+_{mn}{\boldsymbol {\varpi }}^-_{mn}}=\sqrt {1-(\omega _{mn}/2)^2}$ (see (3.3c)), as can be verified by squaring both sets of expressions for $d^E_{mn}$ and $\omega ^E_{mn}$, given by (4.5b) and (${O}$: 2.25$b$).

The oscillatory Ekman layer adjacent to $z=0$, frequency $\omega _{mn}$, has a double layer structure of respective widths

(4.6)\begin{equation} \varDelta_{mn}^\pm={\mathfrak{P}}_{mn}^\pm\sqrt{E}, \end{equation}

where ${\mathfrak {P}}_{mn}^\pm =\sqrt {2/{\boldsymbol {\varpi }}_{mn}^\pm }$ (3.1e) (cf., e.g. Kerswell & Barenghi Reference Kerswell and Barenghi1995, (2.8)). The two widths are readily identifiable on setting $p=2{\mathrm i} \omega _{mn}$ in the exponential of the LT-solution (1.12a) for ${\mathfrak {z}}^\pm (z,t)$. In the geostrophic limit $\omega _{mn}=0$ (${\mathfrak {P}}_{mn}^\pm =1$), the steady Ekman layer width $\varDelta _{{{E}}}=\sqrt {E}$ is recovered. The width

(4.7a,b)\begin{equation} \varDelta^-_{mn}=\sqrt{Et_{mn}} \quad \mbox{with} \ t_{mn}=({\mathfrak{P}}^-_{mn})^2=2/{\boldsymbol{\varpi}}_{mn}^->0, \end{equation}

of the broader layer increases indefinitely as ${\boldsymbol {\varpi }}_{mn}^-\downarrow 0$ (${\mathfrak {P}}^-_{mn}\to \infty$), i.e. $\omega _{mn}\uparrow 2$. So the boundary layers of inertial modes, possessing frequencies close to $2$, fill the entire gap $0\leqslant z\leqslant 1$ of the layer.

In addition to the MF matters associated particularly with the final steady state just discussed, there are also delicate issues concerning how those states are reached. The key feature of the transient evolution of inertial modes is their expanding viscous boundary layer width $\varDelta (t)=\sqrt {Et}$ (1.14). It eventually splits into two parts, each of which ceases to grow at time $t=t^\pm _{mn}$ that solves $\varDelta (t)=\varDelta ^\pm _{mn}$. Since $\varDelta ^-_{mn}>\varDelta ^+_{mn}$ with $t^-_{mn}>t^+_{mn}$, the splitting occurs at $t=t_{mn}^+$, at which the thinner reaches its final width $\varDelta ^+_{mn}$. For $t>t^+_{mn}$, the thicker broadens until terminating with width $\varDelta ^-_{mn}$ at $t=t^-_{mn}$. Furthermore, whenever $t^-_{mn}\gg 1$, this time may be longer than the times reached in our numerical investigations. In any event, when either $\varDelta ^-_{mn}=O(1)$ or $1\ll t \leqslant O(t^-_{mn})$, formula (4.5b) for $d^E_{mn}$ and $\omega ^E_{mn}$ ceases to apply. Nevertheless, since (4.5b) predicts $d^E_{mn}\to 0$ and $\omega ^E_{mn}\to 0$ as $|\omega _{mn}|\uparrow 2$ ($\varphi _{mn}\downarrow 0$), their use in that limit, although inappropriate, ought to be harmless.

The defects, just described are of exactly the same nature as the failure of the Greenspan & Howard (Reference Greenspan and Howard1963) LT-solution (1.26) near the cut points $p=\pm 2{\mathrm i}$, elucidated in the antepenultimate paragraph of § 1.2.2 which ends with the definition (1.30) of the ${\mathfrak {W}}$-trigger. In short, the failure of both the trigger and the waves pertain to boundary layers that expand to fill the entire domain, and as such are two sides of the same coin.

Our appraisal of the situation indicates that it is impossible to produce asymptotic ($0 < E \ll 1$) results that are justifiable in all space or for all time. As explained, our proposed solution (4.4a) modified by (4.5a), namely

(4.8)\begin{equation} {\mathfrak{W}}^{E{\mathsf{d}}\pm}_{mn}(t)={\mathfrak{P}}^{{\mathsf{d}}\pm}_{mn}\,{\mathfrak{W}}^{{\pm}}\left(\left.{\boldsymbol{\varpi}}^{{\pm}}_{mn}t\right/2\right) \exp\left[{\mathrm i}\left(\omega_{mn}+\omega^E_{mn}\right)t-\left(d_{mn}+d^E_{mn}\right)t\right], \end{equation}

is weakest for disturbances with $\omega _{mn}\approx 2$. Furthermore, in view of the approximate nature of the ${\mathfrak {W}}$-trigger (1.30), nothing is gained by using the primitive form ${\mathfrak {W}}^{{\mathsf{d}}\pm }_{mn}(t)$ of the damped (internal friction only) wave solution defined by its LT (2.19). We, therefore, simply adopt (4.8). Indeed, despite many approximations, our guiding consideration is to maintain accuracy compatible with our ${\mathfrak {W}}$-trigger assumption. The very tight comparisons with DNS reported in the following § 5 fully endorse this strategy.

5.1. Our new FNS

We consider the decompositions

(5.3a)\begin{gather} {\boldsymbol{v}}={\boldsymbol{v}}_{{\textit{{QG}}}}+{\boldsymbol{v}}_{{{\textit{MF}}}}+E^{1/2}{\boldsymbol{v}}^\sim, \end{gather}

(5.3b)\begin{gather}{\boldsymbol{v}}_{{\perp}}={\bar {\boldsymbol{v}}}_{{{\textit{{QG}}}}\perp}+{\boldsymbol{v}}^{{{\varDelta}}}_{{{\textit{{QG}}}}\perp}+{\bar {\boldsymbol{v}}}_{{{\textit{MF}}}\perp}+ {\boldsymbol{v}}^{{{\varDelta}}}_{{{\textit{MF}}}\perp}+E^{1/2}{\boldsymbol{v}}^\sim_{{\perp}} \end{gather}

of the entire velocity ${\boldsymbol{v}}$ and the horizontal velocity ${\boldsymbol{v}} _{\perp }$. Note that there are also ageostrophic sidewall boundary layer contributions adjacent to $r=\ell$, ignored in (5.3a,b). The $z$-average of (5.3b) yields

(5.3c)\begin{equation} \langle{\boldsymbol{v}}_{{\perp}}\rangle=\langle{\boldsymbol{v}}_{{{\textit{{QG}}}}\perp}\rangle+\langle{\boldsymbol{v}}_{{{\textit{MF}}}\perp}\rangle+O(E), \end{equation}

where the wave Ekman layer contribution $\langle {\boldsymbol{v}} ^\sim _{\perp } \rangle =O(E^{1/2})$ is included in the $O(E)$ error. As the filter only pertains to the flow outside all boundary layers, we are only interested in the contribution ${\bar {\boldsymbol{v}} }_{{{\textit {{QG}}}}\perp }+{\bar {\boldsymbol{v}} }_{{{\textit {MF}}}\perp }+E^{1/2}{\boldsymbol{v}} ^\sim _{\perp }$ to (5.3b). Oruba et al. (Reference Oruba, Soward and Dormy2020) filtered the DNS by simply removing the $z$-independent QG-contribution ${\bar {\boldsymbol{v}} }_{{{\textit {{QG}}}}\perp }$. However, a more useful filter is obtained by removing, in addition, the MF-contribution ${\bar {\boldsymbol{v}} }_{{{\textit {MF}}}\perp }$, which like ${\bar {\boldsymbol{v}} }_{{{\textit {{QG}}}}\perp }$ is $z$-independent. By this devise we are left with our new filtered horizontal velocity

(5.4)\begin{equation} {\boldsymbol{v}}_{{{\textit{FNS}}}\perp}=E^{{-}1/2}\left({\boldsymbol{v}}_{{{\textit{DNS}}}\perp}-{\bar {\boldsymbol{v}}}_{{{\textit{{QG}}}}\perp}-{\bar {\boldsymbol{v}}}_{{{\textit{MF}}}\perp}\right). \end{equation}

To evaluate our new filter (5.4), we assume that the needed features of the MF-part of the DNS are

(5.5a)\begin{gather} E^{{-}1/2}{\bar {\boldsymbol{v}}}_{{{\textit{MF}}}\perp}\approx(r/\ell)[{\mathfrak{U}}_{{{\textit{MF}}}}, {\mathfrak{V}}_{{{\textit{MF}}}}+{\mathfrak{R}}_{{{\textit{MF}}}}], \end{gather}

(5.5b)\begin{gather}E^{{-}1/2}\langle{\boldsymbol{v}}_{{{\textit{MF}}}\perp}\rangle\approx(r/\ell)[0,{\mathfrak{R}}_{{{\textit{MF}}}}], \end{gather}

as given by the analytic results (1.21) and (1.24). Thus, with ${\boldsymbol{v}} _{{{\textit {MF}}}}$ assumed known, we remove it from both sides of (5.3a) and so obtain ${\boldsymbol{v}} _{{{\textit {DNS}}}\perp }-{\boldsymbol{v}} _{{{\textit {MF}}}\perp }={\boldsymbol{v}} _{{{\textit {{QG}}}}\perp }+ E^{1/2}{\boldsymbol{v}} ^\sim _{\perp }$ instead of (5.3b). Then, as in Oruba et al. (Reference Oruba, Soward and Dormy2020), we use the recipe implicit in (${O}$: 3.4) and (${O}$: 3.7$a$) to relate ${\bar v}_{{\textit {{QG}}}}$ and ${\bar u}_{{\textit {{QG}}}}$ to $\langle v_{{\textit {{QG}}}} \rangle$

(5.6a)\begin{equation} E^{{-}1/2}{\bar {\boldsymbol{v}}}_{{{\textit{{QG}}}}\perp}\approx\mu^{{-}1}\left[\tfrac12\sigma,E^{{-}1/2}\right]\langle v_{{\textit{{QG}}}} \rangle, \end{equation}

where $\sigma \approx 1+\tfrac 34 E^{1/2}$ (1.28e), $\mu \approx 1-\tfrac 12 E^{1/2}$. Next, (5.3c) determines

(5.6b)\begin{equation} \langle v_{{{\textit{{QG}}}}}\rangle\,\approx\,\langle v_{{{\textit{DNS}}}}\rangle-\langle v_{{{\textit{MF}}}}\rangle\approx\langle v_{{{\textit{DNS}}}}\rangle-E^{1/2}(r/\ell){\mathfrak{R}}_{{{\textit{MF}}}}, \end{equation}

on use of (5.5b). Substitution of $E^{-1/2}{\bar {\boldsymbol{v}} }_{{{\textit {MF}}}\perp }$ (5.5a) and $E^{-1/2}{\bar {\boldsymbol{v}} }_{{{\textit {{QG}}}}\perp }$ (5.6a,b) into (5.4) then provides an explicit representation of ${\boldsymbol{v}} _{{{\textit {FNS}}}\perp }$ in terms of ${\boldsymbol{v}} _{{{\textit {DNS}}}\perp }$ and known analytic results. Finally we note that

(5.7)\begin{equation} \chi_{{{\textit{FNS}}}}= E^{{-}1/2} \chi_{{{\textit{DNS}}}}-E^{{-}1/2}({\bar u}_{{\textit{{QG}}}}+{\bar u}_{{\textit{MF}}})(1-z), \end{equation}

in which $E^{-1/2}{\bar u}_{{\textit {{QG}}}}\approx \tfrac 12(\sigma /\mu )\langle v_{{\textit {{QG}}}} \rangle$ (see (5.6a)) and $E^{-1/2}{\bar u}_{{\textit {MF}}}\approx (r/\ell ){\mathfrak {U}}_{{{\textit {MF}}}}$ (see (5.5a)).

Following the neglect of $\langle v^\sim \rangle =O(E^{1/2})$ and other similar approximations such as the reliability of the trigger itself (see e.g. (1.32)), we expect the filter values $\chi _{{{\textit {FNS}}}}$ (5.7) and $v_{{{\textit {FNS}}}}$ (the azimuthal component of (5.4)) to agree with the analytic predictions for $\chi _{{{{\mathfrak {W}}}}}^\sim$ and $v_{{{{\mathfrak {W}}}}}^\sim$, with $O(E^{1/2})$ error.

5.2. FNS results

For $E=10^{-4}$, FNS solutions (using the new filter described above in § 5.1) are compared to those obtained using the analytic results of §§ 3, 4 in figures 1–4. All figures are arranged in three blocks (each consisting of four panels (a–d)) corresponding to different time instants. We adopt the times employed in figures ${O}$:$1$–${O}$:$4$, close to which either $\chi _{{\textit {MF}}}$ or $v_{{\textit {MF}}}$ take their stationary values (i.e. the zeros of the time derivative of ${\mathfrak {U}}_{{\textit {MF}}}$ or ${\mathfrak {V}}_{{\textit {MF}}}$ given by (1.18a)). As $t\to \infty$, they coincide with the vanishing of the corresponding asymptotic forms (A5). Although, having filtered out the MF-contribution, those instants are no longer special, we limit our attention to them in order to facilitate comparison of our current ${\mathfrak {W}}$-trigger results with the earlier ${\mathfrak {E}}$-trigger results displayed in figures ${O}$:$1$–${O}$:$4$, albeit for the larger $E=10^{-3}$. On the one hand, the effects of dissipation are less evident in our $E=10^{-4}$ figures 1–4. On the other hand and related, our new displays pertain to a very early stage of the spin-down process, i.e. short compared to the QG spin-down time $O(E^{-1/2})=O(100)$ for $E=10^{-4}$. As our filter removes the QG flow, the time stage is largely irrelevant to the filter output.

The FNS values for the ${\mathfrak {W}}$-trigger are portrayed in panel ($a$) of every block. The dark region at the bottom of those panels reflects the Ekman layer (thin and very dark) and the expanding MF shear layer, when present in figures 1, 4 (thicker with some contours visible), that the filter does not remove. There is also an ageostrophic $E^{1/3}$-sidewall Stewartson layer adjacent to the outer $r=\ell$ boundary that the filter does not remove either. Further, it is important to realise that, when the MF-contributions are negligible as in figures 2, 3, the new filter (5.4) is essentially the same as the previous filter. Our ${\mathfrak {W}}$-triggered wave solution portrayed in panel ($b$) of every block is based on the Ekman layer damped composite solution (4.8). Outside the aforesaid boundary layers, the agreement with panels ($a$) is remarkable. For comparison, we show ${\mathfrak {E}}$-triggered wave solution, based on the results of Oruba et al. (Reference Oruba, Soward and Dormy2020) (previously portrayed for $E=10^{-3}$, but see the following paragraph) in panel ($d$). Although the ${\mathfrak {E}}$-results identify all the major wave processes involved, the ${\mathfrak {E}}$-results are clearly found wanting, and, unlike the robust ${\mathfrak {W}}$-results, do not capture the FNS solution in detail.

The comparisons just described are all for $E=10^{-4}$. We can see the effect of changing $E$ from $10^{-3}$ to $10^{-4}$ by comparing the FNS, IW ($\textrm {MF}+$‘wave’) panels (b), (c) (etc.) of figures ${O}$:$2$, ${O}$:$3$ with our FNS, waves($^\sim$) panels (a), (b) (all blocks) of figures 2, 3, because at the instants (with no MF-contribution) chosen the two filters coincide, as do the IW and waves.

In panels ($c$) of figures 1–4, we portray $E=0$ results obtained by use of (3.1b) (equivalently (3.1c)), namely the $E=0$ version of (4.8). Their comparison with panels ($b$) shows how the small dissipation damps the waves. The large scale features are weakly damped, whereas the small scale structures near $r=\ell$ are strongly damped. Their further comparison of panels ($b$) with ($a$) shows how well our damping ansatz (4.5) works outside the ageostrophic $E^{1/3}$-sidewall layer, and particularly close to it, where the $E=0$ panels ($c$) show considerable fan structure, the smoothing of which by dissipation is captured accurately. We mention also that figures ${O}$:$6$, ${O}$:$7$ show the $E=0$ ${\mathfrak {E}}$-response that correspond to the ${\mathfrak {W}}$-results portrayed in panels ($c$) of figures 2, 3. Direct correspondence is not easy. Still, as the $E=10^{-4}$ results are very similar to the $E=0$ results except near $r=\ell$, the major differences are simply illustrated in figures 1–4 by comparing the ${\mathfrak {W}}$-wave results panels ($b$) with the ${\mathfrak {E}}$-wave results panels ($d$).

Other than the effects of damping, all other features are explained by the $E=0$ analysis of § 3. There we identify the main modification of the ${\mathfrak {E}}$-wave results of Oruba et al. (Reference Oruba, Soward and Dormy2020) that lead to our present ${\mathfrak {W}}$-wave results and so do not repeat them here.

5.3. Quantitative tests

In this subsection, our objective is to assess quantitatively how well the FNS is approximated by the analytic ${\mathfrak {W}}$-triggered wave results on decreasing $E$. To achieve that goal, we analyse the results for $v_{{\textit {FNS}}}(E)$ and $v_{{{{\mathfrak {W}}}}}^\sim (E)$ portrayed in the final $t=18.07$ block of panels ($a$)–($c$) in figure 4, extending the values of $E$ considered to include $E=10^{-n}$, $n=3$, 4, 5, 6. In figures 5 and 6, various quantitative tests are made over the range $5 \leqslant r\leqslant 9$ of significant wave activity, yet outside the $E^{1/3}$-sidewall layer abutting the outer boundary $r=10$. The height $z=0.8$ chosen for the horizontal cross-sections is

1. (i) sufficiently far above the lower boundary $z=0$ to avoid possible corruption of the results by the tail of the lower $\sqrt {Et}$-MF boundary layer; and

2. (ii) well inside the upper cells, illustrated in figure 4, in order to capture substantial azimuthal $v$-wave motion.

Figure 5. Horizontal cross-sections of $v$-field measures over the range $5\leqslant r\leqslant 9$ at $z=0.8$, at $t=18.07$. ($a$) Shows the linearly interpolated values $v^{{lerp}}_{{{\textit {FNS}}}}(E_1,E_2)$ (5.10) that estimate $v_{{{\textit {FNS}}}}(0)$; $(E_1,E_2)=(10^{-3},10^{-4})$ (red), $(10^{-4},10^{-5})$ (green), $(10^{-5},10^{-6})$ (blue). ($b$) Shows the discrepancy $[\![ v ]\!](E)$ (5.11) between the FNS and ${\mathfrak {W}}$-trigger solution; $E= 10^{-3}$ (red), $10^{-4}$ (green), $10^{-5}$ (blue), $10^{-6}$ (light blue).

Figure 6. Horizontal cross-sections at $t=18.07$, as in figure 5, showing the $E^{-1/2}$-scaled ($a$) increment of $v_{{{{\mathfrak {W}}}}}^\sim (E)$, namely $\delta v_{{{{\mathfrak {W}}}}}^\sim (E)/E^{1/2}$ (5.12a); ($b$) increment of $v_{{{\textit {FNS}}}}(E)$, namely $\delta v_{{{\textit {FNS}}}}(E)/E^{1/2}$ (5.12b); ($c$) difference $[\![ v ]\!](E)$, namely $[\delta v_{{{\textit {FNS}}}}(E)-\delta v_{{{{\mathfrak {W}}}}}^\sim (E)]/E^{1/2}$ (5.12c) (illustrated previously in figure 5($b$), albeit without the $E^{-1/2}$-scaling).

The reason for avoiding both the $E^{1/3}$-sidewall and $\sqrt {Et}$-MF boundary layers is that our filter cannot resolve them.

Of particular interest is the $E\downarrow 0$ value of the horizontal FNS-velocity ${\boldsymbol{v}} _{{{\textit {FNS}}}\perp }(E)$, which being a measure of $E^{1/2}{\boldsymbol{v}} _{{{\textit {FNS}}}\perp }(E)$, is only defined for $E>0$. All our estimates suggest that its value is determined analytically by the $E=0$ value of ${\boldsymbol{v}} ^\sim _{{\mathfrak {W}}\perp }(E)$ determined by the results (3.2a,b) of § 3. So our expectation is that

(5.8)\begin{equation} {\boldsymbol{v}}_{{{\textit{FNS}}}\perp}(0)\equiv\lim_{E\downarrow 0}{\boldsymbol{v}}_{{{\textit{FNS}}}\perp}={\boldsymbol{v}}^\sim_{{\mathfrak{W}}\perp}(0). \end{equation}

For $0 < E \ll 1$, various analytic estimates suggest that exterior to all boundary layers ${\boldsymbol{v}} _{{{\textit {FNS}}}\perp }(E)$ and ${\boldsymbol{v}} ^\sim _{{\mathfrak {W}}\perp }(E)$ have, at fixed $t$, expansions of the form

(5.9a)\begin{gather} {\boldsymbol{v}}_{{{\textit{FNS}}}\perp}(E)={\boldsymbol{v}}_{{{\textit{FNS}}}\perp}(0)+E^{1/2} {\boldsymbol{v}}_{{{\textit{FNS}}}\perp}^{\prime}+O(E), \end{gather}

(5.9b)\begin{gather}{\boldsymbol{v}}^\sim_{{\mathfrak{W}}\perp}(E)={\boldsymbol{v}}^\sim_{{\mathfrak{W}}\perp}(0)+E^{1/2} {\boldsymbol{v}}^{{\sim}\prime}_{{\mathfrak{W}}\perp}+O(E), \end{gather}

where the notation $\bullet ^{\prime }$ is intended to suggest ${\mathrm d} \bullet /{\mathrm d} E^{1/2}|_{E\downarrow 0}$. As our filter has $O(E^{1/2})$ errors, there is no asymptotic reason for ${\boldsymbol{v}} _{{{\textit {FNS}}}\perp }^{\prime }$ and ${\boldsymbol{v}} ^{\sim \prime }_{{\mathfrak {W}}\perp }$ to agree. The main purpose of our tests in figures 5 and 6 is to assess the worth of our ansatz (5.9).

As we cannot determine $v_{{\textit {DNS}}}(0)$ (and hence $v_{{\textit {FNS}}}(0)$) other than via the limit $E\downarrow 0$, we consider instead its linearly interpolated value

(5.10)\begin{equation} v^{{lerp}}_{{{\textit{FNS}}}}(E_1,E_2)\equiv\displaystyle\frac{E_2^{1/2}v_{{\textit{FNS}}}(E_1)-E_1^{1/2}v_{{\textit{FNS}}}(E_2)}{E_2^{1/2}-E_1^{1/2}}, \end{equation}

which we plot in figure 5($a$) for various pairs $(E_1,\,E_2)$. In (5.9), we have been cautious to offer only a two term Maclaurin series. In the unlikely event of the series remaining valid up to the next term $O(E)$, it would follow that $v^{{lerp}}_{{{\textit {FNS}}}}(E_1,E_2)= v_{{\textit {FNS}}}(0)+ O((E_1E_2)^{1/2})$, and then only for sufficiently small $E_1$ and $E_2$. Be that as it may, the $v^{{lerp}}_{{{\textit {FNS}}}}(E_1,E_2)$ cross-section in figure 5($a$) converges well. The limiting curve ${\boldsymbol{v}} ^\sim _{{\mathfrak {W}}\perp }(0)$ is not plotted being indistinguishable to graph plotting accuracy, consistent with our expectation (5.8).

To obtain a measure of the relative sizes of $v_{{\textit {FNS}}}^{\prime }$ and $v^{\sim \prime }_{{{{\mathfrak {W}}}}}$, we plot in figure 5($b$) the difference

(5.11)\begin{equation} [\![ v ]\!](E)\equiv v_{{\textit{FNS}}}(E)-v^\sim_{{{{\mathfrak{W}}}}}(E)= E^{1/2}[v_{{\textit{FNS}}}^{\prime}-v^{{\sim}\prime}_{{{{\mathfrak{W}}}}}]+O(E) \end{equation}

with $O(E)$ error, as estimated by (5.9). Reassuringly, $[\![ v ]\!](E)$ decreases with decreasing $E$ consistent with the implications of convergence in figure 5($a$), but not very fast. So in figure 6($c$), we plot the scaled value $[\![ v ]\!]/E^{1/2}$, which by (5.11) ought to tend to an $O(1)$ limit. Instead it shows sign of increasing with decreasing $E$. It should be emphasised, however, that we are attempting to identify very small effects. For with $v_{{\textit {FNS}}}=O(E^{1/2}v_{{\textit {DNS}}})$, $v_{{\textit {FNS}}}^{\prime }=O(E^{1/2}v_{{\textit {DNS}}})$, $v^{\sim \prime }_{{{{\mathfrak {W}}}}}=O(E^{1/2}v_{{\textit {DNS}}})$, the relation (5.11) implies that $[\![ v ]\!]=O(E v_{{\textit {DNS}}})$, an accuracy which is perhaps hard to achieve from the numerics.

As a further test of the ansatz (5.9) we plot respectively

(5.12a)\begin{gather} \delta v^\sim_{{\mathfrak{W}}}(E)\equiv v^\sim_{{\mathfrak{W}}}(E)-v^\sim_{{\mathfrak{W}}}(0)=E^{1/2} v_{{\mathfrak{W}}}^{\prime}+O(E), \end{gather}

(5.12b)\begin{gather}\delta v_{{{\textit{FNS}}}}(E)\equiv v_{{\textit{FNS}}}(E)-v^\sim_{{\mathfrak{W}}}(0)=E^{1/2} v^{\prime}_{{{\textit{FNS}}}} +O(E), \end{gather}

scaled by $E^{-1/2}$ in figure 6(a,b). There is some evidence that both $\delta v^\sim _{{\mathfrak {W}}}(E)/E^{1/2}$ and $\delta v_{{{\textit {FNS}}}}(E)/E^{1/2}$ are approaching the proposed respective limiting values $v_{{\mathfrak {W}}}^{\prime }$ and $v^{\prime }_{{{\textit {FNS}}}}$. To assess this tendency from a slightly different perspective, we plot $[\![ v ]\!](E)/E^{1/2}$ (5.11) in figure 6($c$). Being related to (5.12a,b) via

(5.12c)\begin{equation} [\![ v ]\!](E)=\delta v_{{{\textit{FNS}}}}(E)-\delta v^\sim_{{\mathfrak{W}}}(E), \end{equation}

its small size relative to both $\delta v^\sim _{{\mathfrak {W}}}(E)$ and $\delta v_{{{\textit {FNS}}}}(E)$ is very reassuring, despite lacking evidence of convergence. The lack of convergence may be attributable to the aforementioned numerical error. More seriously, however, the implied presumption that the $O(E^{1/2})$ value of $[\![ v ]\!](E)=O(E^{1/2})$ defined by (5.11) is meaningful at that order is flawed, because our ${\mathfrak {W}}$-trigger (1.30), used to obtain $v^\sim _{{{{\mathfrak {W}}}}}(E)$, itself possesses $O(E^{1/2})$ errors, as explained in § 1.3. Essentially, the accuracy of our ${\mathfrak {W}}$-trigger is strictly limited and what figure 6($c$) attempts to illustrate is beyond the accuracy claimed by our asymptotics. Despite the caveat about figure 6($c$), the quantitative measures in figures 5 and 6, strongly support the convergence of the FNS results as $E\downarrow 0$ to the $E=0$ analytic results.