2.1 The $z$-Fourier series

We seek a $z$-Fourier series solution

(2.7a)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D712}\\ \unicode[STIX]{x1D6FE}\end{array}\right]=-\unicode[STIX]{x1D705}\unicode[STIX]{x1D70E}\mathop{\sum }_{m=1}^{\infty }{\displaystyle \frac{(-1)^{m}}{m\unicode[STIX]{x03C0}}}\left[\begin{array}{@{}c@{}}\widetilde{\unicode[STIX]{x1D712}}_{m}\\ \unicode[STIX]{x1D6FE}_{m}\end{array}\right]\sin (m\unicode[STIX]{x03C0}(z-1)), & & \displaystyle\end{eqnarray}$$

chosen so that $\unicode[STIX]{x1D712}(r,z,t)$ satisfies the top and bottom boundary conditions (2.6c), and use

(2.7b)$$\begin{eqnarray}\displaystyle {\textstyle \frac{1}{2}}(z-1)=-\mathop{\sum }_{m=1}^{\infty }{\displaystyle \frac{(-1)^{m}}{m\unicode[STIX]{x03C0}}}\sin (m\unicode[STIX]{x03C0}(z-1))\quad (0<z\leqslant 1) & & \displaystyle\end{eqnarray}$$

in our application of the boundary condition (2.6b). The horizontal velocity is

(2.7c)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}u\\ v\end{array}\right]=-\unicode[STIX]{x1D705}\unicode[STIX]{x1D70E}\mathop{\sum }_{m=1}^{\infty }{\displaystyle \frac{(-1)^{m}}{m\unicode[STIX]{x03C0}}}\left[\begin{array}{@{}c@{}}\widetilde{u}_{m}\\ \widetilde{v}_{m}\end{array}\right]\cos (m\unicode[STIX]{x03C0}(z-1)). & & \displaystyle\end{eqnarray}$$

A further property of $v$, due to its assumed form (2.7c), is

(2.7d)$$\begin{eqnarray}\displaystyle \langle v\rangle =0,\quad \text{strictly }O(E^{1/2}) & & \displaystyle\end{eqnarray}$$

when the consequences of the Ekman layer are taken into account. The series (2.7a,c) satisfy (2.4) when $\widetilde{\unicode[STIX]{x1D712}}_{m}(r,t)$ and $\widetilde{v}_{m}(r,t)$ are governed by

(2.8a,b)$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{v}_{m}}{\unicode[STIX]{x2202}t}}+2\widetilde{u}_{m}=E\mathsf{D}_{m}\widetilde{v}_{m},\quad \widetilde{u}_{m}=-m\unicode[STIX]{x03C0}\widetilde{\unicode[STIX]{x1D712}}_{m}, & & \displaystyle\end{eqnarray}$$

(2.8c,d)$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\unicode[STIX]{x2202}\widetilde{\unicode[STIX]{x1D6FE}}_{m}}{\unicode[STIX]{x2202}t}}+2m\unicode[STIX]{x03C0}\widetilde{v}_{m}=E\mathsf{D}_{m}\widetilde{\unicode[STIX]{x1D6FE}}_{m},\quad \widetilde{\unicode[STIX]{x1D6FE}}_{m}=-\mathsf{D}_{m}\widetilde{\unicode[STIX]{x1D712}}_{m}, & & \displaystyle\end{eqnarray}$$

in which

(2.8e)$$\begin{eqnarray}\displaystyle \mathsf{D}_{m}\bullet ={\displaystyle \frac{1}{r}}{\displaystyle \frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}r}}\left(r{\displaystyle \frac{\unicode[STIX]{x2202}\bullet }{\unicode[STIX]{x2202}r}}\right)-\left({\displaystyle \frac{1}{r^{2}}}+(m\unicode[STIX]{x03C0})^{2}\right)\bullet . & & \displaystyle\end{eqnarray}$$

They are to be solved subject to the initial ($t=0$) conditions

(2.9a-c)$$\begin{eqnarray}\displaystyle \widetilde{v}_{m}=0,\quad \widetilde{\unicode[STIX]{x1D6FE}}_{m}=0\quad \Longrightarrow \quad \widetilde{\unicode[STIX]{x1D712}}_{m}(r,0)=\text{I}_{1}(m\unicode[STIX]{x03C0}r)/\text{I}_{1}(m\unicode[STIX]{x03C0}\ell ) & & \displaystyle\end{eqnarray}$$

(see (2.5) together with (2.8d) and (2.10) below), and for $t\geqslant 0$ the boundary conditions

(2.10a)$$\begin{eqnarray}\displaystyle & \displaystyle r\widetilde{\unicode[STIX]{x1D712}}_{m}=0,\quad \text{at }r=0~(0<z<1), & \displaystyle\end{eqnarray}$$

(2.10b)$$\begin{eqnarray}\displaystyle & \displaystyle r\widetilde{\unicode[STIX]{x1D712}}_{m}=\ell \mathfrak{E}(t),\quad \text{at }r=\ell ~(0<z<1) & \displaystyle\end{eqnarray}$$

In the following §§ 2.2 and 2.3, we solve the problem posed by (2.8)–(2.10) by the Laplace transform (henceforth LT) method. To that end it is helpful to note that the initial value, $\widetilde{\unicode[STIX]{x1D712}}_{m}(r,0)=\text{I}_{1}(m\unicode[STIX]{x03C0}r)/\text{I}_{1}(m\unicode[STIX]{x03C0}\ell )$ (2.9c) can be represented, via the use of the Fourier–Bessel series (A 3) with $q=\text{i}$ (giving $\text{J}_{1}(\text{i}m\unicode[STIX]{x03C0}r)=\text{i}\text{I}_{1}(m\unicode[STIX]{x03C0}r)$), in the form

(2.11a)$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{I}_{1}(m\unicode[STIX]{x03C0}r)}{\text{I}_{1}(m\unicode[STIX]{x03C0}\ell )}}=-\mathop{\sum }_{n=1}^{\infty }\mathfrak{F}_{mn}{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{j_{n}\text{J}_{0}(j_{n})}}\quad \text{on }0\leqslant r<\ell , & & \displaystyle\end{eqnarray}$$

where $j_{n}$ denotes the $n$th zero (${>}0$) of $\text{J}_{1}(x)$, and

(2.11b-d)$$\begin{eqnarray}\displaystyle \mathfrak{F}_{mn}={\displaystyle \frac{q_{mn}^{2}\unicode[STIX]{x1D714}_{mn}^{2}}{2}},\quad \unicode[STIX]{x1D714}_{mn}={\displaystyle \frac{2}{\sqrt{q_{mn}^{2}+1}}},\quad q_{mn}={\displaystyle \frac{j_{n}}{m\unicode[STIX]{x03C0}\ell }}. & & \displaystyle\end{eqnarray}$$

2.2 The Laplace transform solution

We employ the LT

(2.12)$$\begin{eqnarray}\displaystyle \widehat{\bullet }(z,p)={\mathcal{L}}_{p}\{\bullet (z,t)\}\equiv \int _{0}^{\infty }\bullet (z,t)\exp (-pt)\,\text{d}t, & & \displaystyle\end{eqnarray}$$

in which the subscript ‘$p$’ to the LT-operator ${\mathcal{L}}$ identifies the independent transform variable. The LT of the governing equations (2.8a–d) and initial conditions (2.9) determine

(2.13a)$$\begin{eqnarray}\displaystyle & \displaystyle p\widehat{\widetilde{v}}_{m}-2m\unicode[STIX]{x03C0}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}=E\mathsf{D}_{m}\widehat{\widetilde{v}}_{m}, & \displaystyle\end{eqnarray}$$

(2.13b)$$\begin{eqnarray}\displaystyle & \displaystyle p\mathsf{D}_{m}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}-2m\unicode[STIX]{x03C0}\widehat{\widetilde{v}}_{m}=E\mathsf{D}_{m}^{2}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}, & \displaystyle\end{eqnarray}$$

where the differential operator $\mathsf{D}_{m}$ is defined by (2.8e),

(2.13c)$$\begin{eqnarray}\displaystyle [\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m},\widehat{\widetilde{v}}_{m}](z,p)={\mathcal{L}}_{p}\{[\widetilde{\unicode[STIX]{x1D712}}_{m},\widetilde{v}_{m}]\} & & \displaystyle\end{eqnarray}$$

and ${\mathcal{L}}_{p}$ is defined by (2.12). Elimination of $\widehat{\widetilde{v}}_{m}$ leads to a single equation for $\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}$,

(2.14)$$\begin{eqnarray}\displaystyle (p-E\mathsf{D}_{m})^{2}\mathsf{D}_{m}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}-4(m\unicode[STIX]{x03C0})^{2}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}=0. & & \displaystyle\end{eqnarray}$$

As already stressed, we ignore viscous boundary layers and solve (2.14) on the basis that, when $E=0$, it is second order in $r$, for which the endpoint boundary conditions

(2.15a,b)$$\begin{eqnarray}\displaystyle r\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}=0\quad \text{at }r=0\text{ and}\quad \left\{\begin{array}{@{}l@{}}r\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}=\ell \widehat{\mathfrak{E}}(p)\quad \text{at }r=\ell ,\\ \widehat{\mathfrak{E}}(p)=(p+Q)^{-1},\end{array}\right. & & \displaystyle\end{eqnarray}$$

namely the LTs of (2.10) and (1.8d), suffice. Upon seeking modal solutions of the form

(2.16a)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}\\ \widehat{\widetilde{v}}_{m}\end{array}\right]=\left[\begin{array}{@{}c@{}}1\\ 2m\unicode[STIX]{x03C0}/\mathfrak{p}\end{array}\right]\widehat{\mathfrak{E}}(p){\displaystyle \frac{\text{J}_{1}(m\unicode[STIX]{x03C0}qr)}{\text{J}_{1}(m\unicode[STIX]{x03C0}q\ell )}}, & & \displaystyle\end{eqnarray}$$

that meet the boundary condition (2.15b) at $r=\ell$ and have the property

(2.16b)$$\begin{eqnarray}\displaystyle \mathsf{D}_{m}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}=-(q^{2}+1)(m\unicode[STIX]{x03C0})^{2}\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}, & & \displaystyle\end{eqnarray}$$

it follows from (2.14) that $p$ and $q$ are related by the ‘dispersion relation’

(2.17a,b)$$\begin{eqnarray}\displaystyle \mathfrak{p}^{2}=-4/(q^{2}+1),\quad \text{where }\mathfrak{p}=p+E(q^{2}+1)(m\unicode[STIX]{x03C0})^{2}, & & \displaystyle\end{eqnarray}$$

equivalently

(2.17c,d)$$\begin{eqnarray}\displaystyle q^{2}+1=-4/\mathfrak{p}^{2}\quad \text{and}\quad p=\mathfrak{p}+E(2m\unicode[STIX]{x03C0})^{2}/\mathfrak{p}^{2}, & & \displaystyle\end{eqnarray}$$

from which we obtain the useful result

(2.17e)$$\begin{eqnarray}\displaystyle \mathfrak{p}{\displaystyle \frac{\text{d}p}{\text{d}\mathfrak{p}}}=\mathfrak{p}-2E{\displaystyle \frac{(2m\unicode[STIX]{x03C0})^{2}}{\mathfrak{p}^{2}}}. & & \displaystyle\end{eqnarray}$$

We utilise the inverse-LT of (2.12), namely

(2.18)$$\begin{eqnarray}\displaystyle \bullet (z,t)={\mathcal{L}}_{p}^{-1}\{\widehat{\bullet }(z,p)\}\equiv {\displaystyle \frac{1}{2\unicode[STIX]{x03C0}\text{i}}}\int _{-\text{i}\infty +c}^{\text{i}\infty +c}\widehat{\bullet }(z,p)\exp (pt)\,\text{d}p, & & \displaystyle\end{eqnarray}$$

with c sufficiently large that the integration contour is to the right of all poles. When applied to $\widehat{\widetilde{\unicode[STIX]{x1D712}}}_{m}$, defined by (2.16a), we may simply write

(2.19)$$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D712}}_{m}={\mathcal{L}}_{p}^{-1}\left\{\widehat{\mathfrak{E}}(p){\displaystyle \frac{\text{J}_{1}(m\unicode[STIX]{x03C0}qr)}{\text{J}_{1}(m\unicode[STIX]{x03C0}q\ell )}}\right\}. & & \displaystyle\end{eqnarray}$$

The initial condition (2.9c) is recovered on expanding the corresponding integrand of the inverse-LT integral (2.18) about the limit $p\rightarrow \infty$ (appropriate to $t\downarrow 0$), for which $q\rightarrow \text{i}$ (see (2.17a,b)) and $p\widehat{\mathfrak{E}}(p)\rightarrow 1$. For $t>0$ the inverse of (2.16a) has two parts:

(2.20)$$\begin{eqnarray}\displaystyle [\widetilde{\unicode[STIX]{x1D712}}_{m},\widetilde{v}_{m}]=[\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth },\widetilde{v}_{m}^{\daleth }]+[\widetilde{\unicode[STIX]{x1D712}}_{m}^{AG},\widetilde{v}_{m}^{AG}]. & & \displaystyle\end{eqnarray}$$

The former inertial wave part $[\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth },\widetilde{v}_{m}^{\daleth }]$ stems from the residues (denoted by Res$\{\}$) at the set $\daleth$ of poles $p=p_{mn}(\not =0)$ linked to the zeros $j_{n}$ of the denominator $\text{J}_{1}(m\unicode[STIX]{x03C0}q\ell )$ in (2.19), i.e. $q=q_{mn}=j_{n}/(m\unicode[STIX]{x03C0}\ell )$ (see (2.11d)); for $t>0$, explicit evaluation in (2.21) below shows that each inertial wave part has the property $\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth }(\ell ,t)=0$. The poles $p_{mn}(\not =0)$ are pure imaginary, when $E=0$ (see (2.17a,b)), and are thus associated with wave motion. The latter ageostrophic part $[\widetilde{\unicode[STIX]{x1D712}}_{m}^{AG},\widetilde{v}_{m}^{AG}]$ stems from the residues at the poles of $\widehat{\mathfrak{E}}(p)=(p+Q)^{-1}$ and $\mathfrak{p}^{-1}$. When internal friction is included ($d_{mn}\not =0$), this ageostrophic part determines a Stewartson $E^{1/3}$-layer and alone meets the boundary condition $\widetilde{\unicode[STIX]{x1D712}}_{m}^{AG}(\ell ,t)=\mathfrak{E}(t)$ (see (2.15b)). However, since we have not applied any stress related boundary conditions, the flow so determined is unphysical and we consider it no further. Hence, the wave part of the velocity $\boldsymbol{v}^{wave}$ alluded to in (2.1) is simply $\boldsymbol{v}^{\daleth }$, valid on the entire range $0\leqslant r\leqslant \ell$.

2.3 The $r$-Fourier–Bessel series

The residue calculation outlined above is messy to implement, because evaluation of the residues involves first determining the $p$-derivatives of the denominator $\text{J}_{1}(m\unicode[STIX]{x03C0}q\ell )$ at the poles $p_{m,n}\in \daleth$. It can be done but that complication is best bypassed by use of the Fourier–Bessel series (A 3) with $q^{2}=-1-4/\mathfrak{p}^{2}$ (2.17a), $q_{mn}^{2}=-1+4/\unicode[STIX]{x1D714}_{mn}^{2}$ (see (2.11c)) giving $2q_{mn}^{2}/(q_{mn}^{2}-q^{2})=\mathfrak{F}_{mn}\mathfrak{p}^{2}/(\mathfrak{p}^{2}+\unicode[STIX]{x1D714}_{mn}^{2})$ on use of (2.11b). It enables us to express the residue sum for $[\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth },\widetilde{v}_{m}^{\daleth }]$ derived from (2.16a) directly in the form

(2.21)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth }\\ \widetilde{v}_{m}^{\daleth }\end{array}\right] & = & \displaystyle -\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{j_{n}\text{J}_{0}(j_{n})}}\mathfrak{F}_{mn}\underset{p\in \daleth }{\text{Res}}\left\{\left[\begin{array}{@{}c@{}}\mathfrak{p}\\ 2m\unicode[STIX]{x03C0}\end{array}\right]{\displaystyle \frac{\mathfrak{p}\exp (pt)}{\mathfrak{p}^{2}+\unicode[STIX]{x1D714}_{mn}^{2}}}{\displaystyle \frac{1}{p+Q}}\right\}\nonumber\\ \displaystyle & = & \displaystyle -\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{j_{n}\text{J}_{0}(j_{n})}}\mathfrak{F}_{mn}\left(\underset{p\in \daleth +}{\text{Res}}\left\{\left[\begin{array}{@{}c@{}}\mathfrak{p}/2\\ m\unicode[STIX]{x03C0}\end{array}\right]{\displaystyle \frac{\exp (pt)}{\mathfrak{p}-\text{i}\unicode[STIX]{x1D714}_{mn}}}{\displaystyle \frac{1}{p+Q}}\right\}+\text{c.c.}\right).\qquad\end{eqnarray}$$

The pole structure is immediately apparent and identified by the pole half-set, $\daleth +$,

(2.22a,b)$$\begin{eqnarray}\displaystyle \mathfrak{p}=\mathfrak{p}_{mn}=\text{i}\unicode[STIX]{x1D714}_{mn}\quad \Longleftrightarrow \quad p=p_{mn}=\text{i}\unicode[STIX]{x1D714}_{mn}-d_{mn} & & \displaystyle\end{eqnarray}$$

having $\unicode[STIX]{x1D714}_{mn}=\text{Im}\{\mathfrak{p}_{mn}\}>0$, which when combined with their complex conjugates (denoted by c.c.) form the complete set $\daleth$. On use of (2.17d,e), we determine

(2.22c,d)$$\begin{eqnarray}\displaystyle d_{mn}={\displaystyle \frac{E(2m\unicode[STIX]{x03C0})^{2}}{\unicode[STIX]{x1D714}_{mn}^{2}}}\quad \Longrightarrow \quad \left[\mathfrak{p}{\displaystyle \frac{\text{d}p}{\text{d}\mathfrak{p}}}\right]_{\mathfrak{p}=\text{i}\unicode[STIX]{x1D714}_{mn}}=\text{i}\unicode[STIX]{x1D714}_{mn}+2d_{mn}. & & \displaystyle\end{eqnarray}$$

Evaluation of the residues in (2.21), noting that the factor $\text{d}\mathfrak{p}/\text{d}p|_{\mathfrak{p}=\text{i}\unicode[STIX]{x1D714}_{mn}}$ is needed in the denominator, yields

(2.23a)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth }\\ \widetilde{v}_{m}^{\daleth }\end{array}\right]=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{j_{n}\text{J}_{0}(j_{n})}}\mathfrak{F}_{mn}\left[\begin{array}{@{}c@{}}{\unicode[STIX]{x1D712}\unicode[STIX]{x0030A}}_{mn}^{\daleth }\\ {v\unicode[STIX]{x0030A}}_{mn}^{\daleth }\end{array}\right]\exp (-\unicode[STIX]{x1D706}_{mn}t), & & \displaystyle\end{eqnarray}$$

in which $\mathfrak{F}_{mn}$ is defined by (2.11b) and

(2.23b)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}{\unicode[STIX]{x1D712}\unicode[STIX]{x0030A}}_{mn}^{\daleth }\\ {v\unicode[STIX]{x0030A}}_{mn}^{\daleth }\end{array}\right]=\left[\begin{array}{@{}c@{}}-{\textstyle \frac{1}{2}}\\ \text{i}m\unicode[STIX]{x03C0}/\unicode[STIX]{x1D714}_{mn}\end{array}\right]{\displaystyle \frac{\text{i}\unicode[STIX]{x1D714}_{mn}+2d_{mn}}{\text{i}\unicode[STIX]{x1D714}_{mn}-d_{mn}+Q}}\exp (\text{i}\unicode[STIX]{x1D714}_{mn}t)+\text{c.c.} & & \displaystyle\end{eqnarray}$$

Written explicitly, equation (2.23a) is

(2.24a)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth }=-\mathop{\sum }_{n=1}^{\infty }\mathfrak{F}_{mn}{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{j_{n}\text{J}_{0}(j_{n})}}(\mathsf{C}_{mn}^{\mathfrak{E}}\cos \unicode[STIX]{x1D719}_{mn}+\mathsf{S}_{mn}^{\mathfrak{E}}\sin \unicode[STIX]{x1D719}_{mn})\exp (-\unicode[STIX]{x1D706}_{mn}t), & \displaystyle\end{eqnarray}$$

(2.24b)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{v}_{m}^{\daleth }=-\mathop{\sum }_{n=1}^{\infty }\mathfrak{F}_{mn}{\displaystyle \frac{2m\unicode[STIX]{x03C0}}{\unicode[STIX]{x1D714}_{mn}}}{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{j_{n}\text{J}_{0}(j_{n})}}(\mathsf{C}_{mn}^{\mathfrak{E}}\sin \unicode[STIX]{x1D719}_{mn}-\mathsf{S}_{mn}^{\mathfrak{E}}\cos \unicode[STIX]{x1D719}_{mn})\exp (-\unicode[STIX]{x1D706}_{mn}t), & \displaystyle\end{eqnarray}$$

(2.24c,d)$$\begin{eqnarray}\displaystyle \mathsf{C}_{mn}^{\mathfrak{E}}=1-{\displaystyle \frac{(3d_{mn}-Q)(d_{mn}-Q)}{\unicode[STIX]{x1D714}_{mn}^{2}+(d_{mn}-Q)^{2}}},\quad \mathsf{S}_{mn}^{\mathfrak{E}}={\displaystyle \frac{(3d_{mn}-Q)\unicode[STIX]{x1D714}_{mn}}{\unicode[STIX]{x1D714}_{mn}^{2}+(d_{mn}-Q)^{2}}}, & & \displaystyle\end{eqnarray}$$

and

(2.24e,f)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{mn}(t)=\unicode[STIX]{x1D714}_{mn}t,\quad \unicode[STIX]{x1D706}_{mn}=d_{mn}. & & \displaystyle\end{eqnarray}$$

Since the boundary condition (2.10b) on $\widetilde{\unicode[STIX]{x1D712}}_{m}$ at $r=\ell$ is non-zero, it is counter-intuitive that each $\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth }(\ell ,t)$ vanishes. The apparent paradox is resolved by noting the non-zero value of $\widetilde{\unicode[STIX]{x1D712}}_{m}(\ell ,t)$ is accommodated by the quasi-steady ageostrophic part $\widetilde{\unicode[STIX]{x1D712}}_{m}^{AG}(\ell ,t)$ that we disregard.

2.4 Ekman layer damping

The various frictional damping effects that we need to consider are encapsulated by equation (4.5) of Zhang & Liao (Reference Zhang and Liao2008), which consists of three sets of terms. The first corresponds to our internal friction decay rate $d_{mn}$. The second, proportional to their $\unicode[STIX]{x1D6E4}^{-1}$ (our $\ell ^{-1}$), corresponds to decay caused by the end wall boundaries, which is negligible in our large aspect $\ell \gg 1$ limit. Indeed, that friction is absent for our stress-free outer boundary. The third, namely the remaining pair of terms, identifies the Ekman layer decay rate $d_{E}$. For that, we halve the Zhang & Liao result because we only have an Ekman layer on $z=0$ and no layer on $z=1$. Accordingly, to accommodate Ekman layer dissipation, we add the complex growth rate

(2.25a)$$\begin{eqnarray}\displaystyle p_{mn}^{E\pm }=-d_{mn}^{E}\pm \text{i}\unicode[STIX]{x1D714}_{mn}^{E}, & & \displaystyle\end{eqnarray}$$

where

(2.25b,c)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}d_{mn}^{E}\\ \unicode[STIX]{x1D714}_{mn}^{E}\end{array}\right]={\textstyle \frac{1}{2}}E^{1/2}\unicode[STIX]{x1D70E}_{mn+}\unicode[STIX]{x1D70E}_{mn-}\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D70E}_{mn+}^{3}+\unicode[STIX]{x1D70E}_{mn-}^{3}\\ \unicode[STIX]{x1D70E}_{mn+}^{3}-\unicode[STIX]{x1D70E}_{mn-}^{3}\end{array}\right],\quad \unicode[STIX]{x1D70E}_{mn\pm }=\sqrt{1\pm \unicode[STIX]{x1D714}_{mn}/2}\quad & & \displaystyle\end{eqnarray}$$

(c.f. also Scott Reference Scott2014, equation (2.22), but in Cartesian geometry and with time unit $(2\unicode[STIX]{x1D6FA})^{-1}$ rather than our $\unicode[STIX]{x1D6FA}^{-1}$). Here, the correction $\unicode[STIX]{x1D714}^{E}$ to the frequency is not given by Zhang & Liao, but can be determined from the formula (2.12) of Kerswell & Barenghi (Reference Kerswell and Barenghi1995). To conclude, the formula (2.24a,b) continues to hold, but with (2.24e,f) replaced by

(2.26a,b)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{mn}(t)=(\unicode[STIX]{x1D714}_{mn}+\unicode[STIX]{x1D714}_{mn}^{E})t+\unicode[STIX]{x1D716}_{mn}^{E},\quad \unicode[STIX]{x1D706}_{mn}=d_{mn}+d_{mn}^{E}, & & \displaystyle\end{eqnarray}$$

in which the small phase corrections $\unicode[STIX]{x1D716}_{mn}^{E}$ are not determined by the aforementioned results. Although $\unicode[STIX]{x1D716}_{mn}^{E}$ is the same size as $\unicode[STIX]{x1D714}_{mn}^{E}$, as time proceeds it becomes small compared to the secular phase $\unicode[STIX]{x1D714}_{mn}^{E}t$. Accordingly, we believe $\unicode[STIX]{x1D716}_{mn}^{E}$ to be unimportant and set $\unicode[STIX]{x1D716}_{mn}^{E}=0$ in all our numerical evaluations.

3.1 The filtered DNS-velocity $\boldsymbol{v}_{FNS}$

The most dominant feature of the spin-down, exterior to the bottom Ekman layer, is the $z$-independent azimuthal QG-flow $\overline{v}_{QG}$, which is larger by a factor of at least $O(E^{-1/2})$ than almost all other contributions to the complete flow description. So, to make comparison with results based on our § 2 theory for $E^{1/2}v^{wave}$, we need to remove $\overline{v}_{QG}$ from $v$. As $\overline{v}_{QG}$ is not easily identifiable from the numerics, we determine it indirectly from the $z$-average $\langle v\rangle$ of $v$. To this end, we note that, on ignoring all wave motion, equation (1.11) indicates that $\langle v\rangle =\unicode[STIX]{x1D707}\overline{v}_{QG}+O(E)$, a result that even holds in the expanding QG-shear layer adjacent to the outer boundary $r=\ell$. Interestingly, for $t\gg 1$, although the MF-wave has $\overline{v}_{MF}=O((E/t)^{1/2})$ (see (1.17c), its $z$–average $\langle \overline{v}_{MF}\rangle$ is smaller by a factor $O(t^{-1})$: $\langle v_{MF}\rangle =O(E^{1/2}t^{-3/2})$ (see (1.18)). Furthermore the inertial waves $E^{1/2}v^{wave}$ in their assumed form (2.7c) have zero $z$-average. That assumption is based on neglect of their associated Ekman layer. In practice, these Ekman layers carry an azimuthal flux smaller by a factor $O(E^{1/2})$ so that $E^{1/2}\langle v^{wave}\rangle =O(E)$. This fortuitous estimate indicates that the (IW-)contribution $E^{1/2}v_{IW}$ (3.3b), outside the Ekman layer, is related to the full solution by

(3.4)$$\begin{eqnarray}\displaystyle v_{IW}=E^{-1/2}v_{MF}+v^{wave}=E^{-1/2}(v-\unicode[STIX]{x1D707}^{-1}\langle v\rangle )+O(E^{1/2}) & & \displaystyle\end{eqnarray}$$

on a spin-down time $t=O(E^{-1/2})$ large compared to unity.

We also assume that the quasi-steady $z$-dependent correction to $\overline{v}_{QG}$ is relatively small $O(E|\overline{v}_{QG}|)$ (see Oruba et al. Reference Oruba, Soward and Dormy2017, equation (2.11a)) so that its presence on the right-hand side of (3.4) does not corrupt the recipe for the IW-part $v_{IW}$, at any rate to the order of accuracy needed. Importantly, we may evaluate $v-\unicode[STIX]{x1D707}^{-1}\langle v\rangle$ directly from the DNS results and refer to

(3.5)$$\begin{eqnarray}\displaystyle v_{FNS}=E^{-1/2}(v_{DNS}-\unicode[STIX]{x1D707}^{-1}\langle v_{DNS}\rangle ) & & \displaystyle\end{eqnarray}$$

as the ‘filtered DNS’ or simply FNS. It should be emphasised that this filter is delicate as it needs to determine the difference of the $O(1)$ quantities $v_{DNS}$ and $\unicode[STIX]{x1D707}^{-1}\langle v_{DNS}\rangle$ accurately to $O(E^{1/2})$. In figures 3 and 4 (below), we portray $v_{FNS}$ in the FNS panels (b,e,h), derived from $E^{-1/2}v_{DNS}$ illustrated in the DNS panels (a,d,g), while $v_{IW}$ is shown in the IW panels (c,f,i).

All contributions to the radial flow $u$ are $O(E^{1/2})$. Nevertheless, just as for $v$, we need to first identify the QG-part $\overline{u}_{QG}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70E}E^{1/2}\overline{v}_{QG}={\textstyle \frac{1}{2}}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D707})E^{1/2}\langle v_{QG}\rangle$ (the same recipe as in (1.8a,b), also (1.11)), and note that the IW-contribution, outside the Ekman layer, is

(3.6a)$$\begin{eqnarray}\displaystyle u_{IW}=E^{-1/2}u_{MF}+u^{wave} & = & \displaystyle E^{-1/2}(u-\overline{u}_{QG})+O(E^{1/2})\nonumber\\ \displaystyle & = & \displaystyle E^{-1/2}u-{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D707})\langle v\rangle +O(E^{1/2})\end{eqnarray}$$

on the spin-down time $t=O(E^{-1/2})$. Exactly as before in our consideration of $\overline{v}_{QG}$, we neglect the small quasi-steady $z$-dependent correction $O(E|\overline{u}_{QG}|)$ to $\overline{u}_{QG}$ (see Oruba et al. Reference Oruba, Soward and Dormy2017, equation (2.11b)) on the right-hand side of (3.6a). On defining the mainstream streamfunction as $r\unicode[STIX]{x1D712}=r\int _{z}^{1}u\,\text{d}z$, we may extract the IW-part via the recipe

(3.6b)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{IW}=E^{-1/2}\unicode[STIX]{x1D712}_{MF}+\unicode[STIX]{x1D712}^{wave}=E^{-1/2}\unicode[STIX]{x1D712}-{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D707})(1-z)\langle v\rangle +O(E^{1/2}). & & \displaystyle\end{eqnarray}$$

The results (3.6a,b) suggest that we define the radial FNS-velocity and streamfunction by

(3.7a)$$\begin{eqnarray}\displaystyle & \displaystyle u_{FNS}=E^{-1/2}u_{DNS}-{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D707})\langle v_{DNS}\rangle , & \displaystyle\end{eqnarray}$$

(3.7b)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}_{FNS}=E^{-1/2}\unicode[STIX]{x1D712}_{DNS}-{\textstyle \frac{1}{2}}(\unicode[STIX]{x1D70E}/\unicode[STIX]{x1D707})(1-z)\langle v_{DNS}\rangle . & \displaystyle\end{eqnarray}$$

$E^{-1/2}\unicode[STIX]{x1D712}_{DNS}$

$\unicode[STIX]{x1D712}_{FNS}$

$\unicode[STIX]{x1D712}_{IW}$

3.2 The inertial wave $\boldsymbol{v}_{IW}$ comparison with $\boldsymbol{v}_{FNS}$

In the FNS and IW panels of figures 1–4, contours are scaled consistently as in (3.4)–(3.7) so that amplitude comparisons are readily discernible. The full DNS results, however, exhibit a wider amplitude range, because they contain, in addition to the IW-contribution, the generally large QG-part. It is therefore impractical to employ the same scaling in the DNS panels as used in the FNS and IW panels. The DNS panels are, however, important as they illustrate the entire spin-down process and provide a visual measure of the relevance of the IW-contribution. This is particularly pertinent to $E^{-1/2}v_{DNS}$ which is $O(E^{-1/2})$ larger than both $v_{FNS}$ and $v_{IW}$.

The results portrayed in figures 1–4 all concern $E=10^{-3}$. The lower Ekman layer has width $E^{1/2}\doteqdot 0.03$, which is perhaps most readily identifiable in the azimuthal velocity $E^{-1/2}v$ contour plots of figures 3 and 4. The well-known Ekman spiral is evident in the DNS panels, whereas on the blown up scale of the FNS panels it blurs and appears as thin black shaded layer. There is also a persistent ageostrophic $E^{1/3}$-sidewall layer at $r=\ell$ of width $0.1$. Our FNS- and IW-results are only meaningful in the regions exterior to those quasi-static boundary layers. Note that neither the Ekman layer nor the sidewall layer appears on the IW panels as they are not part of either of the constituents ($\unicode[STIX]{x1D712}_{MF}$, $v_{MF}$) or ($\unicode[STIX]{x1D712}^{wave}$, $v^{wave}$) that together compose the IW-solution.

The time range of our plots starts at $t=4.72>1$ (i.e. large compared to the spin-down time) in figures 1 and 3, panels (a–c) and ends at $t=18.07<10^{3}$ (i.e. short compared to the MF boundary-layer (width $\unicode[STIX]{x1D6E5}(t)=\sqrt{Et}$ (1.4b)) diffusion time $E^{-1}$ needed to fill $0<z<1$) in figures 2 and 4, panels (g–i). Essentially, the results apply on the spin-down time $t_{sd}=E^{-1/2}\doteqdot 30$. The actual times chosen in figure 1 (3) are $t=N\unicode[STIX]{x03C0}/2$ ($N=3,7,11$) at which the MF-wave contribution $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}\propto \cos (2t)$ given by (1.17a) is maximised (for $E^{-1/2}\overline{v}_{MF}\propto \sin (2t)=0$, see (1.17c)). The times $t=(N+{\textstyle \frac{1}{2}})\unicode[STIX]{x03C0}/2$ ($N=3,7,11$) used in figure 2 (4) are when $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}\propto \cos (2t)=0$ ($E^{-1/2}\overline{v}_{MF}\propto \sin (2t)$ is maximised). The idea is that at times when $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}=0$ ($E^{-1/2}\overline{v}_{MF}=0$), the FNS and IW panels for $E^{-1/2}\unicode[STIX]{x1D712}$ ($E^{-1/2}v$) in figure 2 (3) simply describe $\unicode[STIX]{x1D712}^{wave}$ ($v^{wave}$). However, at times, when $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}$ ($E^{-1/2}\overline{v}_{MF}$) are maximised, the FNS and IW panels for $E^{-1/2}\unicode[STIX]{x1D712}$ ($E^{-1/2}v$) in figure 1 (4), through comparison with figure 2 (3), identify the role of the $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}$ ($E^{-1/2}\overline{v}_{MF}$) contribution. Perhaps the most striking characteristic of this comparison is that $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}$ ($E^{-1/2}\overline{v}_{MF}$) identified in figure 1 (4) is non-zero throughout the entire domain, just as predicted by (1.17a (c)). By contrast $\unicode[STIX]{x1D712}^{wave}$ ($v^{wave}$) identified in figure 2 (3) is only non-zero for a limited radial extent from the outer boundary $r=\ell$. In the following § 4 we ignore all damping and in § 6 explain this phenomenon. There is also much detailed structure in a subdomain close to $r=\ell$, which we explain in § 5.

As explained in § 1.1.2(b) and noted earlier in this subsection in the context of time scales, the MF-wave possesses a spreading boundary-layer width $\unicode[STIX]{x1D6E5}(t)=\sqrt{Et}$ adjacent to the lower boundary $z=0$ quantified by (1.17b,d). This layer is most clearly evident in the IW panels of figure 4 (sufficiently far to the left for $v^{wave}$ to be negligible), for which the Ekman layer is absent. It is also evident in the FNS panels, where it extends beyond the prominent Ekman layer. These features can also be identified, but less obviously, in the corresponding panels of figure 1 (sufficiently far to the left for $u^{wave}$ to be negligible).

The values of $\unicode[STIX]{x1D712}^{wave}$, $v^{wave}$ used for our IW-plots are given by the $z$-Fourier series (2.7a,c) using $\widetilde{\unicode[STIX]{x1D712}}_{m}$, $\widetilde{v}_{m}$ determined by (2.24a,b). Since we found that the slow decay of the QG-flow has virtually no effect on the result, we set

(3.8)$$\begin{eqnarray}\displaystyle Q=E^{1/2}\unicode[STIX]{x1D70E}=0 & & \displaystyle\end{eqnarray}$$

in (2.24c,d), which define the parameters $\mathsf{C}_{mn}^{\mathfrak{E}}$, $\mathsf{S}_{mn}^{\mathfrak{E}}$ (2.24a,b). As the formulae (2.24c,d) for the phase $\unicode[STIX]{x1D719}_{mn}(t)$ and decay rate $\unicode[STIX]{x1D706}_{mn}$ account only for the damping by internal friction, we used instead (2.26a,b), which also takes into account Ekman damping, but with the slight change of phase $\unicode[STIX]{x1D716}_{mn}^{E}$ set to zero, as explained there. To assess whether or not our damping predictions are reasonable, we need to compare the FNS IW panels for $\unicode[STIX]{x1D712}_{FNS}$ and $\unicode[STIX]{x1D712}_{IW}$ in figure 2 (noting that $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}\approx 0$) and $v_{FNS}$ and $v_{IW}$ in figure 3 (noting that $E^{-1/2}\overline{v}_{MF}\approx 0$). Our theoretical model, although generally good, appears to slightly overestimate damping on the shorter length scales. This appears to be a shortcoming of our choice of the QG-trigger $\overline{u}_{QG}(\ell ,t)={\textstyle \frac{1}{2}}\unicode[STIX]{x1D70E}\unicode[STIX]{x1D705}E^{1/2}\mathfrak{E}(t)$ (1.20a). As we will report elsewhere, the inclusion of the MF-trigger $\overline{u}_{MF}(\ell ,t)$ improves the comparison.

4.1 Formulation and results

On setting $E=0$ in (2.4a–d), these governing equations together with the initial conditions (2.5a,b) and boundary condition $\unicode[STIX]{x1D712}=0$ at $z=1$ (2.6c) determine

(4.1a-c)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}=\int _{z}^{1}u\,\text{d}z,\quad v=-2\int _{0}^{t}u\,\text{d}t=2\int _{0}^{t}{\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D712}}{\unicode[STIX]{x2202}z}}\,\text{d}t,\quad \unicode[STIX]{x1D6FE}=2\int _{0}^{t}{\displaystyle \frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}z}}\,\text{d}t. & & \displaystyle\end{eqnarray}$$

We continue to consider the representations (2.7a,c) and (2.24a,b), in which $\mathfrak{F}_{mn}=q_{mn}^{2}\unicode[STIX]{x1D714}_{mn}^{2}/2$ (2.11b) and $\unicode[STIX]{x1D719}_{mn}=\unicode[STIX]{x1D714}_{mn}t$ (2.24e), but set $E^{1/2}\unicode[STIX]{x1D70E}=Q=0$, $\unicode[STIX]{x1D705}=\unicode[STIX]{x1D70E}=1$, $d_{mn}=0$ so that the coefficients become $\mathsf{C}_{mn}^{\mathfrak{E}}=1$, $\mathsf{S}_{mn}^{\mathfrak{E}}=0$ (see (2.24c,d)) and $\unicode[STIX]{x1D706}_{mn}=0$ (see (2.24f)). In this way, equation (2.24a,b) yield

(4.2a)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{\unicode[STIX]{x1D712}}_{m}=\widetilde{\unicode[STIX]{x1D712}}_{m}^{\daleth }=-\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{q_{mn}^{2}\unicode[STIX]{x1D714}_{mn}^{2}}{2}}{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{j_{n}\text{J}_{0}(j_{n})}}\cos (\unicode[STIX]{x1D714}_{mn}t), & \displaystyle\end{eqnarray}$$

(4.2b)$$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}c@{}}\widetilde{u}_{m}\\ \widetilde{v}_{m}\end{array}\right]=\left[\begin{array}{@{}c@{}}\widetilde{u}_{m}^{\daleth }\\ \widetilde{v}_{m}^{\daleth }\end{array}\right]=\mathop{\sum }_{n=1}^{\infty }{\displaystyle \frac{q_{mn}\unicode[STIX]{x1D714}_{mn}^{2}}{2}}{\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{\ell \text{J}_{0}(j_{n})}}\left[\begin{array}{@{}c@{}}\cos (\unicode[STIX]{x1D714}_{mn}t)\\ -(2/\unicode[STIX]{x1D714}_{mn})\sin (\unicode[STIX]{x1D714}_{mn}t)\end{array}\right]. & \displaystyle\end{eqnarray}$$

4.1.1 The $E\downarrow 0$ results

Some solutions $\unicode[STIX]{x1D712}^{wave}=\unicode[STIX]{x1D712}$ and $v^{wave}=v$, realised by substitution of (4.2) into (2.7a,c), are illustrated in figures 6 and 7, respectively. The times $t=5.50$, 11.79, 18.07 adopted in the first three panels (a–c) of figure 6 correspond to the prescription $t=(N+{\textstyle \frac{1}{2}})\unicode[STIX]{x03C0}/2$ ($N=3,7,11$) adopted in figure 2, for which $E^{-1/2}\overline{\unicode[STIX]{x1D712}}_{MF}=0$. By this choice, we see how the small scale structure of $\unicode[STIX]{x1D712}^{wave}$ visible in figure 6(a–c) particularly near the outer boundary $r=\ell$ is largely eliminated by dissipation in the contour plots of $\unicode[STIX]{x1D712}_{IW}$ in figure 2(c,f,i). Likewise, at times $t=4.72$, 11.00, 17.28, i.e. $t=N\unicode[STIX]{x03C0}/2$ ($N=3,7,11$) when $E^{-1/2}\overline{v}_{MF}=0$, a similar comparison of figure 7(a–c) with figure 3(c,f,i) can be made. To understand in detail this small scale structure, we formulate an asymptotic approach in § 4.2 based on $\ell \gg 1$ (see (4.5)), which we apply in §§ 5 and 6.

Figure 6.

The analytic $\unicode[STIX]{x1D712}^{wave}$-contours in the $E\downarrow 0$ limit at sequential times $t=(N+{\textstyle \frac{1}{2}})\unicode[STIX]{x03C0}/2$. (a) $N=3$, (b) $N=7$, (c) $N=11$, cf. figures 2(c), (f), (i) respectively. (d) $N=15$ (colour scale from $-0.1$ to 0.1).

Figure 7.

The analytic $v^{wave}$-contours in the $E\downarrow 0$ limit at sequential times $t=N\unicode[STIX]{x03C0}/2$. (a) $N=3$, (b) $N=7$, (c) $N=11$, cf. figures 3(c), (f), (i), respectively. (d) $N=15$ (colour scale from $-0.5$ to 0.5).

A striking feature of $\unicode[STIX]{x1D712}^{wave}$ plotted in figure 6, but also evinced by the $v^{wave}$ plots in figure 7, is the limited distance, $x(t)=\ell -r(t)$ (say), reached by the wave disturbance triggered at the outer boundary $r=\ell$. For that reason, it is instructive to consider results for the individual Fourier modes

(4.3)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{m}^{wave}(r,z,t)=-(m\unicode[STIX]{x03C0})^{-1}\widetilde{\unicode[STIX]{x1D712}}_{m}\sin (m\unicode[STIX]{x03C0}z) & & \displaystyle\end{eqnarray}$$

(see (2.7a)) for which contour plots with $m=1,2$ are illustrated in figure 8 at times $t=15$ and $25$. Indeed, on comparing the $m=1$ panels (a,b) with the $m=2$ panels (c,d), suggests that each of these two $m$-modes reaches a distance $x_{m}(t)=\ell -r_{m}(t)$ related by $x_{1}(t)\approx 2x_{2}(t)$, with the possible implication that the larger $m$-modes reach a distance decreasing with $m$. This idea is explored further in the space–time ($0<t\leqslant 30$) contour plots of $\unicode[STIX]{x1D712}_{m}^{wave}(r,z_{m},t)$, for the $m=1,2$ modes, in figure 9(a,b), at their respective maxima $z_{1}=0.5$, $z_{2}=0.25$. From these the spatial extent $r_{m}(t)\lesssim r<\ell$ is clearly visible.

In view of the above remarks, we reassess the wave activity in figures 6 and 7. Sufficiently far to the left, the waves, when existent, are clearly dominated by the $m=1$ mode. On halving the distance to the right-hand outer boundary, $r=\ell$, some interference from the $m=2$ mode is visible. Yet further reduction of that distance leads to interference from successive higher harmonics, that complicates the picture more. Note too that, though the waves penetrate further to the left with time, a feature of their negative group velocity, the waves themselves propagate to the right with positive phase velocity.

Figure 8.

The analytic $\unicode[STIX]{x1D712}_{m}^{wave}$-contours in the $E\downarrow 0$ limit (see (4.3)) for the cases $m=1$ at (a) $t=15$, (b) $t=25$, and $m=2$ also at (c) $t=15$, (d) $t=25$ (colour scale from $-0.1$ to 0.1).

Figure 9.

The analytic $\unicode[STIX]{x1D712}_{m}^{wave}$-contours in the $E\downarrow 0$ limit (see (4.3)) at fixed $z=z_{m}$, in the $r$–$t$ plane for $\ell =10$. Pertaining to the asymptotic $\ell \gg 1$ results of § 6.3: the solid black line identifies the critical group velocity, i.e. $c_{gc}t+(\ell -r)=0$ with $c_{gc}\doteqdot -0.245/m$ (see (6.21c)) valid for $\ell -r=O(1)$. The dashed black line corresponds to the critical phase velocity $c_{pc}t=r$ with $c_{pc}=-3c_{gc}\doteqdot 0.735/m$ (cf. (6.21d)) also only valid for $\ell -r=O(1)$. (a) $m=1$ with $z_{1}=0.5$; (b) $m=2$ with $z_{2}=0.25$ (colour scale from $-0.1$ to 0.1).

4.1.2 The $E\ll 1$ results

Now the complete inertial wave response is the sum of the MF-wave and the triggered waves,

(4.4)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}_{IW}=E^{-1/2}\unicode[STIX]{x1D712}_{MF}+\mathop{\sum }_{m=1}^{\infty }\unicode[STIX]{x1D712}_{m}^{wave} & & \displaystyle\end{eqnarray}$$

(see (3.6b) and (4.3)). In order to make comparison with the DNS, we plot $\unicode[STIX]{x1D712}_{FNS}$-contours at $z=z_{1}=0.5$ in figure 10 for the cases $E=10^{-3}$, $10^{-5}$ in panels (a,b) respectively, in the same style as $\unicode[STIX]{x1D712}_{1}^{wave}$ in figure 9(a). So whereas figure 9(a) concerns $\unicode[STIX]{x1D712}_{1}^{wave}$, figure 10 for $\unicode[STIX]{x1D712}_{IW}$ comprises $E^{-1/2}\unicode[STIX]{x1D712}_{MF}$ and the other odd harmonics $\unicode[STIX]{x1D712}_{m}^{wave}$ ($m=3,5,7,\ldots$) non-zero at $z=z_{1}=0.5$. To the left of the line $r\approx r_{1}(t)$, namely $0<r\lesssim r_{1}(t)$, where the triggered waves have not reached, only the MF-wave is visible. To the right of the line, namely $r_{1}(t)\lesssim r<\ell$, much of the pattern in figure 9(a) is reproduced in figure 10(b) for $E=10^{-5}$, broken up to some extent by the MF-wave. However, there is little evidence of the higher $m\geqslant 3$ harmonics, which only penetrate a short distance from $r=\ell$. The suggestion from figure 10(a,b) is that, although the MF-waves dominate initially, the triggered inertial waves with frequency less than 2 are more persistent, a suggestion that must be tempered by the following considerations. At the end of the introduction (§ 1), we noted the importance of the ratio of the time the triggered waves take to reach the axis $r=0$ (figure 10 suggests $t_{axis}\approx 40=4\ell$) and the spin-down time, namely $t_{axis}/t_{sd}=O(E^{1/2}4\ell )$. With $\ell =10$, the ratio is $O(1)$ for $E=10^{-3}$ but a factor $1/10$ smaller for $E=10^{-5}$, a consideration which suggests that figure 10(a) concerns events on the spin-down time, whereas figure 10(b) concerns events at a relatively early stage of spin-down. For that our estimate (1.19b) is also relevant, in the sense that the MF-wave decays algebraically while the inertial wave decays exponentially like the QG-flow with virtually no decay when $E^{1/2}t\ll 1$.

In the above discussion, we introduced the distance $x_{m}(t)=\ell -r_{m}(t)$, that each $m$-mode reaches from the boundary $r=\ell$, in a qualitative way. In § 6 we use our asymptotic theory, based on $\ell \gg 1$ (see (4.5)), to derive robust results based on $x=O(1)$, i.e. valid far from the symmetry axis $r=0$.

Figure 10.

The filtered DNS (FNS) $\unicode[STIX]{x1D712}_{FNS}$-contours at fixed $z=0.5$, as in 9(a). (a) $E=10^{-3}$, (b) $E=10^{-5}$ (colour scale from $-0.1$ to 0.1).

4.2 The Cartesian limit, $\ell =L/H\gg 1$, $\ell -r=O(1)$

The wave solutions are best understood by their behaviour at large $r$. So throughout this section and the following §§ 5 and 6, we restrict our attention to

(4.5a,b)$$\begin{eqnarray}\displaystyle x=\ell -r=O(1)\quad \text{for }\ell \gg 1, & & \displaystyle\end{eqnarray}$$

for which two key approximations follow:

(4.6a)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{J}_{1}(j_{n}r/\ell )}{\text{J}_{0}(j_{n})}}\approx -\sin (n\unicode[STIX]{x03C0}x/\ell ),\quad j_{n}\approx n\unicode[STIX]{x03C0}\quad \text{for }n\gg 1, & \displaystyle\end{eqnarray}$$

(4.6b)$$\begin{eqnarray}\displaystyle & \displaystyle {\displaystyle \frac{\text{I}_{1}(m\unicode[STIX]{x03C0}r)}{\text{I}_{1}(m\unicode[STIX]{x03C0}\ell )}}\approx \exp (-m\unicode[STIX]{x03C0}x). & \displaystyle\end{eqnarray}$$

$x$

$r$

$x$

$r$

The essential idea is that for $x=O(1)$, the $r=0$ axis is unimportant. So, with $\ell \gg 1$, we may regard $n/\ell$ as a continuous rather than discrete variable and approximate the sum $\sum _{n=1}^{\infty }\bullet _{n}$ in (4.2) by the integral $\int _{0}^{\infty }\bullet _{n}\,\text{d}n$ instead. In this way, from (2.11c,d) and (4.6a), we obtain

(4.7a,b)$$\begin{eqnarray}\displaystyle n\approx j_{n}/\unicode[STIX]{x03C0}=(\ell /\unicode[STIX]{x03C0})k,\quad \text{d}n\approx (\ell /\unicode[STIX]{x03C0})\text{d}k, & & \displaystyle\end{eqnarray}$$

(4.7c,d)$$\begin{eqnarray}\displaystyle q=q_{mn}=k/(m\unicode[STIX]{x03C0}),\quad \unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{mn}=2m\unicode[STIX]{x03C0}/\sqrt{k^{2}+(m\unicode[STIX]{x03C0})^{2}}. & & \displaystyle\end{eqnarray}$$

Accordingly, equation (4.2b) is approximated by

(4.8a,b)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\widetilde{u}_{m}\\ \widetilde{v}_{m}\end{array}\right]\approx -{\displaystyle \frac{2}{\unicode[STIX]{x03C0}}}\int _{0}^{\infty }{\displaystyle \frac{\unicode[STIX]{x1D714}^{2}k\sin (kx)}{4m\unicode[STIX]{x03C0}}}\left[\begin{array}{@{}c@{}}\cos (\unicode[STIX]{x1D714}t)\\ -(2/\unicode[STIX]{x1D714})\sin (\unicode[STIX]{x1D714}t)\end{array}\right]\text{d}k,\quad \widetilde{\unicode[STIX]{x1D712}}_{m}=-{\displaystyle \frac{\widetilde{u}_{m}}{m\unicode[STIX]{x03C0}}}. & & \displaystyle\end{eqnarray}$$

On noting that ${\mathcal{L}}_{p}\{\exp (\text{i}\unicode[STIX]{x1D714}t)\}=(p+\text{i}\unicode[STIX]{x1D714})/(p^{2}+\unicode[STIX]{x1D714}^{2})$, the Fourier sums (2.7c) for $u$ and $v$, based on (4.8), have Laplace transforms

(4.9a)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\widehat{u}\\ \widehat{v}\end{array}\right]\approx {\displaystyle \frac{2}{\unicode[STIX]{x03C0}p}}\left[\begin{array}{@{}c@{}}1\\ -2/p\end{array}\right]\mathop{\sum }_{m=1}^{\infty }\left[\int _{0}^{\infty }{\displaystyle \frac{k\sin (kx)}{k^{2}+(m\unicode[STIX]{x03C0}s)^{2}}}\text{d}k\right]\cos (m\unicode[STIX]{x03C0}z), & & \displaystyle\end{eqnarray}$$

where, on setting $E=0$ in (2.17) to obtain

(4.9b,c)$$\begin{eqnarray}\displaystyle s=-\text{i}q=(p^{2}+4)^{1/2}/p\quad \Longleftrightarrow \quad 4/p^{2}=s^{2}-1, & & \displaystyle\end{eqnarray}$$

we have noted from (4.7d) and (4.9c) that $4/\unicode[STIX]{x1D714}^{2}+4/p^{2}=k^{2}/(m\unicode[STIX]{x03C0})^{2}+s^{2}$. Evaluation of the integral in (4.9a) (use § 2.2 equation (15) of Erdélyi et al. Reference Erdélyi, Magnus, Oberhettinger and Tricomi1954) yields

(4.10a)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\widehat{u}\\ \widehat{v}\end{array}\right] & = & \displaystyle {\displaystyle \frac{1}{p}}\left[\begin{array}{@{}c@{}}1\\ -2/p\end{array}\right]\mathop{\sum }_{m=1}^{\infty }\exp (-m\unicode[STIX]{x03C0}sx)\cos (m\unicode[STIX]{x03C0}z)\end{eqnarray}$$

(4.10b)$$\begin{eqnarray}\displaystyle & = & \displaystyle {\displaystyle \frac{1}{2p}}\left[\begin{array}{@{}c@{}}1\\ -2/p\end{array}\right]\left[-1+{\displaystyle \frac{\sinh (s\unicode[STIX]{x03C0}x)}{\cosh (s\unicode[STIX]{x03C0}x)-\cos (\unicode[STIX]{x03C0}z)}}\right].\end{eqnarray}$$

$x$

$s$

$p$

$t=0$

$t>0$

$x$

$\widehat{\unicode[STIX]{x1D712}}=\int _{z}^{1}\widehat{u}\,\text{d}z$

(4.10c,d)$$\begin{eqnarray}\displaystyle \widehat{\unicode[STIX]{x1D712}}\approx -\frac{1}{\unicode[STIX]{x03C0}p}\left[-{\displaystyle \frac{\unicode[STIX]{x03C0}z}{2}}+\tan ^{-1}\left({\displaystyle \frac{\tan (\unicode[STIX]{x03C0}z/2)}{\tanh (s\unicode[STIX]{x03C0}x/2)}}\right)\right],\quad \widehat{\unicode[STIX]{x1D6FE}}\approx {\displaystyle \frac{2}{p}}{\displaystyle \frac{\unicode[STIX]{x2202}\widehat{v}}{\unicode[STIX]{x2202}z}}. & & \displaystyle\end{eqnarray}$$

In order to invert the Laplace transforms, we need to note that $s=(p^{2}+4)^{1/2}/p\rightarrow 1$ as $|p|\rightarrow \infty$, i.e. $s$ is defined by a cut connecting $p=-2\text{i}$ to $p=2\text{i}$ along the $\text{Im}\{\,p\}$-axis and by analytic continuation elsewhere. This consideration is essential to guarantee that we take the correct sign of the square root of $(p^{2}+4)^{1/2}$. Indeed, this property may be used to extract the initial values of $\unicode[STIX]{x1D712}$ and $u$, which are determined by the form of $\widehat{u}$ and $\widehat{\unicode[STIX]{x1D712}}$ in the limit $|p|\rightarrow \infty$. In that limit, evaluation of the inverse-LTs is achieved by simply setting $s=1$ and then evaluating the residues of (4.10b,c) at the only remaining singularity, the pole at $p=0$ ($s=1$), so determining

(4.11a)$$\begin{eqnarray}\displaystyle & \displaystyle u(x,z,0)\approx {\displaystyle \frac{1}{2}}\left[-1+{\displaystyle \frac{\sinh (\unicode[STIX]{x03C0}x)}{\cosh (\unicode[STIX]{x03C0}x)-\cos (\unicode[STIX]{x03C0}z)}}\right], & \displaystyle\end{eqnarray}$$

(4.11b)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D712}(x,z,0)\approx -{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\left[-{\displaystyle \frac{\unicode[STIX]{x03C0}z}{2}}+\tan ^{-1}\left({\displaystyle \frac{\tan (\unicode[STIX]{x03C0}z/2)}{\tanh (\unicode[STIX]{x03C0}x/2)}}\right)\right], & \displaystyle\end{eqnarray}$$

with the property

(4.11c)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D712}(x,z,0)\approx -{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\exp (-\unicode[STIX]{x03C0}x)\sin (\unicode[STIX]{x03C0}z),\quad \text{as }x\rightarrow \infty & & \displaystyle\end{eqnarray}$$

(essentially the $m=1$ mode $\unicode[STIX]{x1D712}_{1}^{wave}$ defined by (4.3), but see also (4.10a) and (C 4)). Initially $v$ and $\unicode[STIX]{x1D6FE}$ are zero, but for $t\ll 1$ (4.1b,c) determine

(4.12a,b)$$\begin{eqnarray}\displaystyle v\approx -2tu(x,z,0),\quad \unicode[STIX]{x1D6FE}\approx -2t^{2}{\displaystyle \frac{\unicode[STIX]{x2202}u}{\unicode[STIX]{x2202}z}}(x,z,0). & & \displaystyle\end{eqnarray}$$

Despite the apparent simplicity of the Laplace transforms $\widehat{u}$, $\widehat{v}$ and $\widehat{\unicode[STIX]{x1D712}}$ given by (4.10b,c), their direct inversion is not straightforward. That is partly due to the essential singularity of $\tanh (s\unicode[STIX]{x03C0}x)$ at $p=0$ which leads to some apparently suspicious results following LT-inversion. For example, the forms (4.10a,b) hint at a pole at $p=0$, where none exists in the primitive form (4.9a) (recall that $p^{2}s^{2}\rightarrow 4$ as $p\rightarrow 0$). An alternative approach is suggested by the formula

(4.13)$$\begin{eqnarray}\displaystyle -1+{\displaystyle \frac{\sinh (s\unicode[STIX]{x03C0}x)}{\cosh (s\unicode[STIX]{x03C0}x)-\cos (\unicode[STIX]{x03C0}z)}}=-1+{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\mathop{\sum }_{l=-\infty }^{\infty }{\displaystyle \frac{2sx}{(z-2l)^{2}+s^{2}x^{2}}} & & \displaystyle\end{eqnarray}$$

(Gradshteyn & Ryzhik Reference Gradshteyn, Ryzhik, Jeffrey and Zwillinger2007, § 1.445, equation (9)), which we substitute into (4.10b). Due to the invariance of the sum (4.13) under the shift $z\mapsto z+2$, there is only one independent solution linked to $l=0$. We refer to the others, for $l\not =0$, as the ‘image system’.

The various LT-representations suggest two distinct strategies for their inversion. In § 5, we adopt the ‘method of images’, based on (4.10b) and (4.13), to explain detailed features of the solution particularly evident at small $x$. In § 6, we study the evolution of the individual $z$-Fourier $m$-modes (4.8). The smallest, $m=1$, identifies the dominant structure at large $x$. Indeed, the initial evanescent behaviour identified by (4.11c) continues to be a feature for sufficiently large $x$ (see appendix C and § 6.2.2).

5.1 The primary $l=0$ mode in $x>0$, $-\infty <z<\infty$

On the introduction of the unit vector

(5.2a-c)$$\begin{eqnarray}\displaystyle [\mathsf{x},\mathsf{z}]=[x,z]/\unicode[STIX]{x1D71B},\quad \unicode[STIX]{x1D71B}=\sqrt{x^{2}+z^{2}},\quad \mathsf{x}=\sqrt{1-\mathsf{ z }^{2}}, & & \displaystyle\end{eqnarray}$$

the primary mode, defined via (4.10b) and (4.13) and expressed in the form

(5.2d)$$\begin{eqnarray}\displaystyle [\breve{u},\breve{v}](x,z,t)=(\unicode[STIX]{x03C0}x)^{-1}[\mathsf{u},\mathsf{v}](\mathsf{z},t)\quad (-1<\mathsf{z}<1) & & \displaystyle\end{eqnarray}$$

has LT:

(5.2e)$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x03C0}x)\left[\begin{array}{@{}c@{}}\widehat{\breve{u}}\\ \widehat{\breve{v}}\end{array}\right](x,z,p)=\left[\begin{array}{@{}c@{}}\widehat{\mathsf{u}}\\ \widehat{\mathsf{v }}\end{array}\right](\mathsf{z},p)={\displaystyle \frac{(p^{2}+4)^{1/2}\mathsf{x}^{2}}{p^{2}+4\mathsf{x}^{2}}}\left[\begin{array}{@{}c@{}}1\\ -2/p\end{array}\right]. & & \displaystyle\end{eqnarray}$$

In view of our remarks in the penultimate paragraph of § 4.2, the pole at $p=0$ determines an unexpected steady geostrophic flow $[\breve{u}_{G},\breve{v}_{G}]$ given by

(5.3)$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x03C0}x)[\breve{u}_{G},\breve{v}_{G}]=[\mathsf{u}_{G},\mathsf{v}_{G}]=[0,-1]. & & \displaystyle\end{eqnarray}$$

When, however, we consider the full solution in the following § 5.2, we see that this unwelcome contribution is eliminated under accumulation with the image flows. Indeed, the entire flow evolves indefinitely with no identifiable non-oscillatory part.

The inverse-LT of (5.2e) at $z=0$ for $x>0$ ($\Longrightarrow$$[\mathsf{x},\mathsf{z}]=[1,0]$) is

(5.4a,b)$$\begin{eqnarray}\displaystyle \mathsf{u}(0,t)=\text{J}_{0}(2t),\quad \mathsf{v}(0,t)-\mathsf{v}_{G}=2\int _{t}^{\infty }\text{J}_{0}(2\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}, & & \displaystyle\end{eqnarray}$$

alternatively $\mathsf{v}(0,t)=-2\int _{0}^{t}\text{J}_{0}(2\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}$. Elsewhere (indeed, $\forall z$) it is

(5.5a)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\mathsf{u}\\ \mathsf{ v }\end{array}\right]=\left[\begin{array}{@{}c@{}}\mathsf{x}^{2}\cos (2\mathsf{x}t)\\ -\mathsf{x}\sin (2\mathsf{x}t)\end{array}\right]+\int _{0}^{t}{\displaystyle \frac{\text{J}_{1}(2\unicode[STIX]{x1D70F})}{\unicode[STIX]{x1D70F}}}\left[\begin{array}{@{}c@{}}\mathsf{x}\mathsf{E}_{i}(2\mathsf{x}(t-\unicode[STIX]{x1D70F}))\\ \mathsf{E}_{r}(2\mathsf{x}(t-\unicode[STIX]{x1D70F}))\end{array}\right]\text{d}\unicode[STIX]{x1D70F} & & \displaystyle\end{eqnarray}$$

in which

(5.5b,c)$$\begin{eqnarray}\displaystyle \mathsf{E}(\unicode[STIX]{x1D711})=-1+\exp (\text{i}\unicode[STIX]{x1D711}),\quad \left[\begin{array}{@{}c@{}}\mathsf{E}_{i}(\unicode[STIX]{x1D711})\\ \mathsf{E}_{r}(\unicode[STIX]{x1D711})\end{array}\right]=\left[\begin{array}{@{}c@{}}\sin \unicode[STIX]{x1D711}\\ -1+\cos \unicode[STIX]{x1D711}\end{array}\right]. & & \displaystyle\end{eqnarray}$$

On use of ${\mathcal{L}}_{p}\{t^{-1}\text{J}_{1}(2t)\}=2/[(p^{2}+4)^{1/2}+p]$ (see § 4.14 equation (5) of Erdélyi et al. Reference Erdélyi, Magnus, Oberhettinger and Tricomi1954), it is readily verified that the Laplace transform of (5.5a) is (5.2e). In view of the unlikely relevance of $2/[(p^{2}+4)^{1/2}+p]$ to (5.2e), the direct derivation (without hindsight) of (5.5a) was not obvious to us. The primitive form (5.5a) is useful for $t=O(1)$. However, the identity $\int _{0}^{\infty }\unicode[STIX]{x1D70F}^{-1}\text{J}_{1}(2\unicode[STIX]{x1D70F})\mathsf{E}(2\mathsf{x}\unicode[STIX]{x1D70F})\,\text{d}\unicode[STIX]{x1D70F}=-1+|\mathsf{z}|+\text{i}\mathsf{x}$ (use § 1.12 equation (4) and § 2.12 equation (5) of Erdélyi et al. Reference Erdélyi, Magnus, Oberhettinger and Tricomi1954) permits the alternative representation

(5.6a)$$\begin{eqnarray}\displaystyle [\mathsf{u},\mathsf{v}]=(\unicode[STIX]{x03C0}x)([\breve{u},\breve{v}]_{ms}+[\breve{u},\breve{v}]_{bl})=[\mathsf{u},\mathsf{v}]_{ms}+[\mathsf{u},\mathsf{v}]_{bl}, & & \displaystyle\end{eqnarray}$$

useful for $t\gg 1$, where

(5.6b)$$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}c@{}}\mathsf{u}\\ \mathsf{ v }\end{array}\right]_{ms}-\left[\begin{array}{@{}c@{}}0\\ \mathsf{v }_{G}\end{array}\right]=|\mathsf{z}|\left[\begin{array}{@{}c@{}}\mathsf{x}\sin (2\mathsf{x}t)\\ \cos (2\mathsf{x }t)\end{array}\right]=|\mathsf{z}|\left[\begin{array}{@{}c@{}}\mathsf{x}\mathsf{E}_{i}(2\mathsf{x}t)\\ 1+\mathsf{E}_{r}(2\mathsf{x}t)\end{array}\right], & \displaystyle\end{eqnarray}$$

(5.6c)$$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}c@{}}\mathsf{u}\\ \mathsf{ v }\end{array}\right]_{bl}=-\int _{t}^{\infty }{\displaystyle \frac{\text{J}_{1}(2\unicode[STIX]{x1D70F})}{\unicode[STIX]{x1D70F}}}\left[\begin{array}{@{}c@{}}\mathsf{x}\mathsf{E}_{i}(2\mathsf{x}(t-\unicode[STIX]{x1D70F}))\\ \mathsf{E}_{r}(2\mathsf{x}(t-\unicode[STIX]{x1D70F}))\end{array}\right]\text{d}\unicode[STIX]{x1D70F}. & \displaystyle\end{eqnarray}$$

At given position $[x,z]$, the mainstream motion $[\breve{u},\breve{v}]_{ms}$ is composed of the steady flow $[0,\breve{v}_{G}]$ and the oscillatory flow amplitude (5.6b) frequency $2\mathsf{x}$ stemming from the poles $p=0$ and $\pm 2\text{i}\mathsf{x}$. For $t\gg 1$, the remaining $[\breve{u},\breve{v}]_{bl}$ (5.6c), stemming from the cut points $p=\pm 2\text{i}$, defines an ever thinning boundary layer width $\unicode[STIX]{x1D6E5}_{bl}=xt^{-1/2}$, whose detailed character is described in the supplementary material B. For $t\gg 1$, its width $\unicode[STIX]{x1D6E5}_{bl}=xt^{-1/2}$ is small, so that this thin transient boundary layer concerns $\mathsf{x}-1\approx -{\textstyle \frac{1}{2}}\mathsf{z}^{2}=O(t^{-1})$, i.e. of quasi-MF type, frequency $2\mathsf{x}\approx 2$. In the $z>0$ half-space, it carries the volume flux

(5.7a)$$\begin{eqnarray}\displaystyle [\langle \breve{u}_{bl}\rangle ,\langle \breve{v}_{bl}\rangle ]\equiv \int _{0}^{\infty }[\breve{u},\breve{v}]_{bl}\,\text{d}z={\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\int _{0}^{1}{\displaystyle \frac{1}{\mathsf{x}^{2}\mathsf{z}}}[\mathsf{u},\mathsf{v}]_{bl}\,\text{d}\mathsf{x}. & & \displaystyle\end{eqnarray}$$

Using (5.6c), the time derivative of $\langle \breve{u}_{bl}\rangle$ is

(5.7b)$$\begin{eqnarray}\displaystyle {\displaystyle \frac{\text{d}\langle \breve{u}_{bl}\rangle }{\text{d}t}}=-\int _{t}^{\infty }{\displaystyle \frac{\text{J}_{1}(2\unicode[STIX]{x1D70F})}{\unicode[STIX]{x1D70F}}}\text{J}_{0}(2(\unicode[STIX]{x1D70F}-t))\,\text{d}\unicode[STIX]{x1D70F}\approx -{\displaystyle \frac{\sin (2t)}{\unicode[STIX]{x03C0}t}}\quad \text{for }t\gg 1 & & \displaystyle\end{eqnarray}$$

(use Gradshteyn & Ryzhik Reference Gradshteyn, Ryzhik, Jeffrey and Zwillinger2007, § 3.753, equation (2)). Since $\text{d}\langle \breve{v}_{bl}\rangle /\text{d}t=-2\langle \breve{u}_{bl}\rangle$, two successive integrations determine

(5.7c)$$\begin{eqnarray}\displaystyle [\langle \breve{u}_{bl}\rangle ,\langle \breve{v}_{bl}\rangle ]\approx (2\unicode[STIX]{x03C0}t)^{-1}[\cos (2t),-\sin (2t)]\quad \text{for }t\gg 1, & & \displaystyle\end{eqnarray}$$

giving the estimate $[\breve{u},\breve{v}]_{bl}=O(\unicode[STIX]{x1D6E5}_{bl}^{-1}t^{-1})$. Moreover, when $z=O(\unicode[STIX]{x1D6E5}_{bl})$, we also estimate from (5.6b) that $[\breve{u}_{ms},\breve{v}_{ms}-\breve{v}_{G}]=O(\unicode[STIX]{x1D6E5}_{bl}x^{-2})$. Hence, both flow velocities are the same size $O(x^{-1}t^{-1/2})$ within the evaporating boundary layer.

An interesting variant of the problem just solved was explored by Davidson et al. (Reference Davidson, Staplehurst and Dalziel2006). They considered the evolution of a Gaussian eddy, their equation (2.11), close to the origin relative to cylindrical polar coordinates in the half-space $z>0$. They identify contours of swirl velocity (their figure 1) that resemble our fan structure (5.6b) in the vicinity of the outer corner $[x,z]=[0,0]$ for our limiting Cartesian geometry. We suspect that they have no boundary-layer structure like (5.6c) because their eddy source is of finite size. The resemblance to our mainstream solution (5.6b) is unsurprising as wave propagation in both cases is similar though the geometry and source differ.

5.2 The full solution on $0\leqslant z\leqslant 1$

With $[0,-\breve{v}_{G}]$ removed, the mainstream solution, $[\breve{u}_{ms},\breve{v}_{ms}-\breve{v}_{G}]$ (5.6b), has the remarkable feature that the frequency $2\mathsf{x}$ of the oscillation at any given point $[x,z]$ is constant, independent of time. Nevertheless, consideration of the temporally evolving lines $2\mathsf{x}t$ of constant phase, emanating from the corner $[x,z]=\mathbf{0}$, reveals that the nodes $\mathsf{x}_{n}=n\unicode[STIX]{x03C0}/(2t)$$(n=1,2,\ldots )$ for $\breve{u}$ lie on a fan, which contracts with time as illustrated in figures 6 and 7. When internal friction is included, the associated shortening of the length scale leads to considerable dissipation. This is a well-known characteristic of phase mixing (Heyvaerts & Priest Reference Heyvaerts and Priest1983), which occurs whenever there is a frequency gradient (here $\unicode[STIX]{x1D735}(2\mathsf{x})\not =\mathbf{0}$).

Also visible in figures 6 and 7 are the waves reflected at $z=1$. They correspond to the $l=1$ image fan emanating from the image sink $[x,z]=[0,2]$ and are particularly evident in panels (b,c). Owing to the intensity of the reflections including further interference from other images, the last panel (d) (longest time) is ‘busy’ and a little confused. The fan and its images (reflections) are also visible in the small-$E$ FNS-plots of figure 5(b,c) and unsurprisingly in the $\unicode[STIX]{x1D712}^{wave}$-plot ($E=0$), figure 5(d). By contrast unexpected strong reflections are visible in the DNS-plot, figure 5(e), which must follow from some reinforcement by the QG-meridional flow in certain locations; in others it must be destructive.

Though much of what is visible in figures 6 and 7 may be understood in terms of the primary mode $[\breve{u},\breve{v}]_{ms}$ and its reflections, the complete mathematical description, at least within the asymptotic approximations (4.7), (4.8) for $x=O(1)$ and $\ell \gg 1$, is given by the sum (5.1a). As already remarked $[\breve{u},\breve{v}]_{bl}$ is small for $t\gg 1$. So we omit its contribution to the sum (5.1a) and define what remains,

(5.8)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}u\\ v\end{array}\right]_{ms}=\left[\begin{array}{@{}c@{}}-{\textstyle \frac{1}{2}}\\ t\end{array}\right]+\mathop{\sum }_{l=-\infty }^{\infty }\left[\begin{array}{@{}c@{}}\breve{u}_{l}\\ \breve{v}_{l}\end{array}\right]_{ms}, & & \displaystyle\end{eqnarray}$$

as the mainstream solution.

A disconcerting feature of (5.8) is the presence of the divergent contribution $t$ to $v_{ms}$. To test the worth of the approximation (5.8), which ignores the $[\breve{u}_{l},\breve{v}_{l}]_{bl}$ contributions, we consider the $z$-average of that mainstream solution. Since $[\breve{u},\breve{v}](x,z,t)$ is symmetric in $z$, we note that $[\langle \breve{u}\rangle ,\langle \breve{v}\rangle ](x,t)={\textstyle \frac{1}{2}}\int _{-1}^{1}[\breve{u},\breve{v}]\,\text{d}z$ with the implication $[\langle \breve{u}_{l}\rangle ,\langle \breve{v}_{l}\rangle ](x,t)={\textstyle \frac{1}{2}}\int _{-1-2l}^{1+2l}[\breve{u},\breve{v}]\,\text{d}z$. This property permits us to express the integral of the infinite sum in (5.8) as a single infinite integral:

(5.9a)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}\langle u_{ms}\rangle \\ \langle v_{ms}\rangle \end{array}\right] & = & \displaystyle \left[\begin{array}{@{}c@{}}-{\textstyle \frac{1}{2}}\\ t\end{array}\right]+{\displaystyle \frac{1}{2\unicode[STIX]{x03C0}x}}\int _{-\infty }^{\infty }\left[\begin{array}{@{}c@{}}\mathsf{x}|\mathsf{z}|\mathsf{E}_{i}(2\mathsf{x}t)\\ -1+|\mathsf{z}|+|\mathsf{z}|\mathsf{E}_{r}(2\mathsf{x}t)\end{array}\right]\text{d}z\nonumber\\ \displaystyle & = & \displaystyle \left[\begin{array}{@{}c@{}}-{\textstyle \frac{1}{2}}\\ t\end{array}\right]+{\displaystyle \frac{1}{2\unicode[STIX]{x03C0}}}\int _{-1}^{1}\left[\begin{array}{@{}c@{}}\mathsf{x}\mathsf{E}_{i}(2\mathsf{x}t)\\ -|\mathsf{z}|^{-1}+1+\mathsf{E}_{r}(2\mathsf{x}t)\end{array}\right]{\displaystyle \frac{\text{d}\mathsf{x}}{\mathsf{ x }^{2}}}\end{eqnarray}$$

(5.9b)$$\begin{eqnarray}\displaystyle & = & \displaystyle -{\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\int _{2t}^{\infty }\left[\begin{array}{@{}c@{}}\unicode[STIX]{x1D719}^{-1}\sin \unicode[STIX]{x1D719}\\ 2t\unicode[STIX]{x1D719}^{-2}\cos \unicode[STIX]{x1D719}\end{array}\right]\text{d}\unicode[STIX]{x1D719}.\end{eqnarray}$$

$\breve{v}_{G}=-(\unicode[STIX]{x03C0}x)^{-1}$

$x^{-1}\text{d}z=-\mathsf{x}^{-2}|\mathsf{z}|^{-1}\text{d}\mathsf{x}$

$\int _{0}^{\infty }[\unicode[STIX]{x1D719}^{-1}\sin \unicode[STIX]{x1D719},\unicode[STIX]{x1D719}^{-2}(1-\cos \unicode[STIX]{x1D719})]\,\text{d}\unicode[STIX]{x1D719}=[\unicode[STIX]{x03C0}/2,\unicode[STIX]{x03C0}/2]$

For $t\gg 1$, equation (5.9b) behaves like

(5.10)$$\begin{eqnarray}\displaystyle [\langle u_{ms}\rangle ,\langle v_{ms}\rangle ]\approx (2\unicode[STIX]{x03C0}t)^{-1}[-\cos (2t),\sin (2t)]+O(t^{-2}). & & \displaystyle\end{eqnarray}$$

Reassuringly, the diverging contribution $t$ to $v$ in (5.9a) is eliminated by the summation and its mean value (5.10), being $O(t^{-1})$, decays. By implication, since $[\langle u\rangle ,\langle v\rangle ]=\mathbf{0}$, the $z$-average of the remaining $[u,v]_{bl}=\sum _{l=-\infty }^{\infty }[\breve{u}_{l},\breve{v}_{l}]_{bl}$, namely

(5.11)$$\begin{eqnarray}\displaystyle [\langle u_{bl}\rangle ,\langle v_{bl}\rangle ]=-[\langle u_{ms}\rangle ,\langle v_{ms}\rangle ], & & \displaystyle\end{eqnarray}$$

decays at the same rate. Moreover the value of $[\langle u_{bl}\rangle ,\langle v_{bl}\rangle ]$ predicted by (5.10) and (5.11) coincides, at least to leading order, with the value of $[\langle \breve{u}_{bl}\rangle ,\langle \breve{v}_{bl}\rangle ]$ for the primary $l=0$ mode alone given by (5.7c). By implication, we may ignore entirely the $l\not =0$ contributions $[\breve{u}_{l},\breve{v}_{l}]_{bl}$ and make the approximation $[u,v]_{bl}\approx [\breve{u},\breve{v}]_{bl}$, when $t\gg 1$.

Appendix B explores the boundary layer matter further by considering values on the boundary $z=0$. Our main conclusion is that, though the primary $l=0$ mode $[\breve{u},\breve{v}]_{ms}$ vanishes there, the mainstream flow $[u,v]_{ms}$, stemming from the accumulation of the images $l\not =0$, is an order of magnitude larger than the boundary-layer flow $[u,v]_{bl}$. Specifically, equation (B 3) confirms the domination of the mainstream solution by showing that

(5.12)$$\begin{eqnarray}\displaystyle [u_{ms},v_{ms}]_{z=0}=O(1)\gg [\breve{u}_{bl},\breve{v}_{bl}]_{z=0}=O(x^{-1}t^{-1/2}) & & \displaystyle\end{eqnarray}$$

(see also (5.7c) and the discussion that follows it).

The most striking feature of our solutions is the contracting (due to the secular phase angle $2\mathsf{x}t$, i.e. phase mixing) fan structure (5.6b) of the primary $l=0$ mode, discussed in § 5.1. The interaction of the infinite sum of image fans $l\not =0$ together with the $l=0$ mode causes secularity that is compensated (i.e. eliminated) by the secular contribution $[-{\textstyle \frac{1}{2}},t]$ to the complete solution $[u,v]$ in (5.1a). Although complicated, the representation (B 2) on $z=0$ clearly indicates how the compensation is achieved but it is unfortunate that the sum–integral difference in (B 2d) is difficult to evaluate.

Finally, the apparatus of this section is not suited to explain the cell structure visible at moderate-$x$ nor for that matter the absence of wave motion at large-$x$. These are matters resolved in § 6 by consideration of individual $z$-Fourier $m$-modes.

6.1 Case $X\gg 1$: large $t$ asymptotics

We evaluate the inverse-LT (6.1d) asymptotically, in the limit $t\gg 1$ or more precisely

(6.3a,b)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D717}=\sqrt{m\unicode[STIX]{x03C0}x/(2t)}=O(1),\quad X=\sqrt{2m\unicode[STIX]{x03C0}xt}\gg 1, & & \displaystyle\end{eqnarray}$$

by the method of steepest descent, as in appendix C. To encompass the notation, we express the exponent $\unicode[STIX]{x1D709}$ (6.1e) as the product

(6.4a,b)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D709}=\unicode[STIX]{x1D6EF}X,\quad \text{with}\quad \unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D717},p)=-\unicode[STIX]{x1D717}s(p)+{\textstyle \frac{1}{2}}\unicode[STIX]{x1D717}^{-1}p. & & \displaystyle\end{eqnarray}$$

It may be shown (although we omit details) that the dominant contributions to the integral (6.1d) stem from the saddle points located, where the $p$-derivative

(6.5)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{,p}\equiv {\displaystyle \frac{\text{d}\unicode[STIX]{x1D6EF}}{\text{d}p}}={\displaystyle \frac{4\unicode[STIX]{x1D717}}{p^{3}s}}+{\displaystyle \frac{1}{2\unicode[STIX]{x1D717}}} & & \displaystyle\end{eqnarray}$$

vanishes. The relevant saddle points occur at purely imaginary locations defined parametrically by

(6.6a,b)$$\begin{eqnarray}\displaystyle p=p_{sa}=2\text{i}(1-\unicode[STIX]{x1D711}^{2})^{1/2},\quad s=s_{sa}=2\unicode[STIX]{x1D711}/p_{sa} & & \displaystyle\end{eqnarray}$$

together with their complex conjugates, all chosen to satisfy (6.1c). The condition $\unicode[STIX]{x1D6EF}_{,p}=0$ implies that $p_{sa}^{3}s_{sa}=-8\unicode[STIX]{x1D717}^{2}$, from which (6.4b) determines

(6.6c)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}=\unicode[STIX]{x1D6EF}_{sa}=\text{i}\unicode[STIX]{x1D711}^{-1/2}(1+\unicode[STIX]{x1D711}^{2}), & & \displaystyle\end{eqnarray}$$

as well as establishing that $\unicode[STIX]{x1D711}$ is one of the two real positive roots $\unicode[STIX]{x1D711}_{+}$ and $\unicode[STIX]{x1D711}_{-}$ of the cubic

(6.6d)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D711}^{3}-\unicode[STIX]{x1D711}+\unicode[STIX]{x1D717}^{2}=0. & & \displaystyle\end{eqnarray}$$

We order them, $0<\unicode[STIX]{x1D711}_{-}<\unicode[STIX]{x1D711}_{+}<1$, such that they define

(6.6e,f)$$\begin{eqnarray}\displaystyle p_{sa}=p_{\pm }=\text{i}\unicode[STIX]{x1D714}_{\pm },\quad \unicode[STIX]{x1D714}_{\pm }=2(1-\unicode[STIX]{x1D711}_{\pm }^{2})^{1/2},\quad 2>\unicode[STIX]{x1D714}_{-}>\unicode[STIX]{x1D714}_{+}>0. & & \displaystyle\end{eqnarray}$$

At $\unicode[STIX]{x1D717}=0$, we have $\unicode[STIX]{x1D714}_{+}=0$ and $\unicode[STIX]{x1D714}_{-}=2$. On increasing $\unicode[STIX]{x1D717}$, $\unicode[STIX]{x1D714}_{+}$ increases and $\unicode[STIX]{x1D714}_{-}$ decreases until they coalesce when $\unicode[STIX]{x1D717}$ reaches

(6.7a)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D717}=\unicode[STIX]{x1D717}_{c}=2^{1/2}3^{-3/4}, & & \displaystyle\end{eqnarray}$$

at which

(6.7b-d)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6EF}_{,pp}=0,\quad \unicode[STIX]{x1D711}=\unicode[STIX]{x1D711}_{c}=3^{-1/2},\quad \unicode[STIX]{x1D714}=\unicode[STIX]{x1D714}_{c}=2^{3/2}3^{-1/2}. & & \displaystyle\end{eqnarray}$$

The roots $\unicode[STIX]{x1D719}_{\pm }$ only form a real pair for $0<\unicode[STIX]{x1D717}<\unicode[STIX]{x1D717}_{c}$. They coalesce at $\unicode[STIX]{x1D717}=\unicode[STIX]{x1D717}_{c}$ and emerge for $\unicode[STIX]{x1D717}>\unicode[STIX]{x1D717}_{c}$ as a single complex root $p_{sa}$ with $\text{Re}\{\,p_{sa}\}>0$. The dominant contribution to the integral from the emergent $p_{sa}$ saddle is small in proportion to $\exp (X\unicode[STIX]{x1D6EF}_{sa})$ with $\text{Re}\{\unicode[STIX]{x1D6EF}_{sa}\}<0$ (recall that $X\gg 1$ (6.3b)). Of course, this is supplemented by its complex conjugate.

Our formulation in terms of $\unicode[STIX]{x1D717}$ and $X$ permits us to identify asymptotic ranges unambiguously. It is well suited to the large $\unicode[STIX]{x1D717}$ case investigated in appendix C, where the result (C 5a) is derived valid for $X\gg \unicode[STIX]{x1D717}^{1/3}(\gg 1)$ (see (C 2b) and (C 5b)). However, when $\unicode[STIX]{x1D717}\leqslant \unicode[STIX]{x1D717}_{c}$, the stationary phase approach adopted in the following section leads to a clearer physical interpretation. The values of $\unicode[STIX]{x1D711}_{\pm }$, $\unicode[STIX]{x1D711}_{c}$ and related results identified in (6.6), (6.7) provide the cornerstones on which the analysis builds.

6.2 A stationary phase formulation

Our alternative approach to evaluating the inverse-LT (6.1d) begins by shrinking ${\mathcal{C}}$ as much as possible, specifically to two lines either side of the imaginary $p$-axis connecting the cut points $p=\pm 2\text{i}$. On them, we set $p=\text{i}\unicode[STIX]{x1D714}$, $m\unicode[STIX]{x03C0}s=-\text{i}k$ and $\unicode[STIX]{x1D709}=\text{i}(kx+\unicode[STIX]{x1D714}t)$, and change the integration variable from $p$ to $s$, noting also that $-(m\unicode[STIX]{x03C0})^{-1}\text{d}p/\text{d}s=\text{d}\unicode[STIX]{x1D714}/\text{d}k=1/(\unicode[STIX]{x1D714}^{3}k)$. Then, on taking considerable care with the signs of $k$ and $\unicode[STIX]{x1D714}$ (real) on each of the four sections of ${\mathcal{C}}$ (recall that the cut point $p=2\text{i}$, and the essential singularity at $p=0$ are now at $k=0$ and $k=\infty$, respectively), we may express (6.1d) in the form

(6.8)$$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D712}}_{m}={\displaystyle \frac{1}{\unicode[STIX]{x03C0}}}\int _{0}^{\infty }{\displaystyle \frac{k\sin (kx+\unicode[STIX]{x1D714}t)+k\sin (kx-\unicode[STIX]{x1D714}t)}{k^{2}+(m\unicode[STIX]{x03C0})^{2}}}\text{d}k & & \displaystyle\end{eqnarray}$$

equivalent to (4.8), where $\unicode[STIX]{x1D714}=2m\unicode[STIX]{x03C0}/\sqrt{k^{2}+(m\unicode[STIX]{x03C0})^{2}}$ as defined in (4.7d).

On the basis that $k>0$ and $\unicode[STIX]{x1D714}>0$, the waves with phase $kx+\unicode[STIX]{x1D714}t$ ($kx-\unicode[STIX]{x1D714}t$) travel outwards (inwards) in the direction of $x$ decreasing (increasing). The integrals in the complex $p$-plane from which the $kx+\unicode[STIX]{x1D714}t$ ($kx-\unicode[STIX]{x1D714}t$) contribution originates stem from the sections of ${\mathcal{C}}$ with $\text{Re}\{\,p\}>({<})0$. Only the first set of waves with phase $kx+\unicode[STIX]{x1D714}t$ have points of stationary phase, which correspond to the saddle points $p=\text{i}\unicode[STIX]{x1D714}_{\pm }$ (see (6.6e,f)), and so we limit our attention to them. That saddle point analysis identifies the two dominant waves, linked to $\unicode[STIX]{x1D719}_{\pm }$, at given $x$ and time $t$. In order to take advantage of the $\unicode[STIX]{x1D717}$, $X$ formalism (6.4)–(6.7) of the § 6.1 steepest descent problem, we introduce the new variables $K$ and $\unicode[STIX]{x1D6F6}$ defined by

(6.9a-c)$$\begin{eqnarray}\displaystyle k/(m\unicode[STIX]{x03C0})=K/\unicode[STIX]{x1D717},\quad \unicode[STIX]{x1D714}=2\unicode[STIX]{x1D717}\unicode[STIX]{x1D6F6}\quad \Longrightarrow \quad kx+\unicode[STIX]{x1D714}t=(K+\unicode[STIX]{x1D6F6})X. & & \displaystyle\end{eqnarray}$$

Following the parallel formalisms, the phase velocity $c_{p}=\unicode[STIX]{x1D714}/k({>}0)$ is given by

(6.10)$$\begin{eqnarray}\displaystyle m\unicode[STIX]{x03C0}c_{p}=\left\{\begin{array}{@{}l@{}}m\unicode[STIX]{x03C0}\unicode[STIX]{x1D714}/k,\quad \text{where }\unicode[STIX]{x1D714}=2m\unicode[STIX]{x03C0}/\sqrt{k^{2}+(m\unicode[STIX]{x03C0})^{2}},\\ 2\unicode[STIX]{x1D717}^{2}\unicode[STIX]{x1D6F6}/K,\quad \text{where }\unicode[STIX]{x1D6F6}=1/\sqrt{K^{2}+\unicode[STIX]{x1D717}^{2}}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

The group velocity $c_{g}=\unicode[STIX]{x2202}\unicode[STIX]{x1D714}/\unicode[STIX]{x2202}k$ (${<}0$, see below) is given by

(6.11)$$\begin{eqnarray}\displaystyle m\unicode[STIX]{x03C0}c_{g}=\left\{\begin{array}{@{}l@{}}m\unicode[STIX]{x03C0}\unicode[STIX]{x1D714}^{\prime },\quad \text{where }\unicode[STIX]{x1D714}^{\prime }=-\unicode[STIX]{x1D714}^{3}k/(2m\unicode[STIX]{x03C0})^{2},\\ 2\unicode[STIX]{x1D717}^{2}\dot{\unicode[STIX]{x1D6F6}},\quad \text{where }\dot{\unicode[STIX]{x1D6F6}}=-\unicode[STIX]{x1D6F6}^{3}K,\end{array}\right. & & \displaystyle\end{eqnarray}$$

where the prime and dot denote the partial derivatives with respect to $k$ and $K$ respectively. Being negative the group velocity is directed inwards ($x$ increasing, $r$ decreasing), i.e. opposite to the phase velocity. Incidentally, the prefactor to $\sin (kx+\unicode[STIX]{x1D714}t)$ in the integral (6.8) may now be expressed as

(6.12)$$\begin{eqnarray}\displaystyle k/[k^{2}+(m\unicode[STIX]{x03C0})^{2}]=-\unicode[STIX]{x1D714}^{\prime }/\unicode[STIX]{x1D714}=-c_{g}/\unicode[STIX]{x1D714}. & & \displaystyle\end{eqnarray}$$

On further differentiation of (6.11), we obtain

(6.13)$$\begin{eqnarray}\displaystyle (m\unicode[STIX]{x03C0})^{2}c_{g}^{\prime }=\left\{\begin{array}{@{}l@{}}(m\unicode[STIX]{x03C0})^{2}\unicode[STIX]{x1D714}^{\prime \prime },\quad \text{where }k\unicode[STIX]{x1D714}^{\prime \prime }/\unicode[STIX]{x1D714}^{\prime }=1+3k\unicode[STIX]{x1D714}^{\prime }/\unicode[STIX]{x1D714},\\ 2\unicode[STIX]{x1D717}^{3}\ddot{\unicode[STIX]{x1D6F6}},\quad \text{where }K\ddot{\unicode[STIX]{x1D6F6}}/\dot{\unicode[STIX]{x1D6F6}}=1-3(K\unicode[STIX]{x1D6F6})^{2}.\end{array}\right. & & \displaystyle\end{eqnarray}$$

The points $k=k_{\pm }$ ($k_{-}<k_{+}$ with $\unicode[STIX]{x1D714}^{\prime }<0\Longrightarrow \unicode[STIX]{x1D714}_{-}>\unicode[STIX]{x1D714}_{+}$) of stationary phase occur where the $k$-derivative of the phase $kx+\unicode[STIX]{x1D714}t$ vanishes,

(6.14)$$\begin{eqnarray}\displaystyle m\unicode[STIX]{x03C0}(kx+\unicode[STIX]{x1D714}t)^{\prime }|_{\pm }=\left\{\begin{array}{@{}c@{}}m\unicode[STIX]{x03C0}(x+c_{g\pm }t)\\ (1+\dot{\unicode[STIX]{x1D6F6}}_{\pm })\unicode[STIX]{x1D717}X\end{array}\right\}=0\quad \Longrightarrow \quad \left\{\begin{array}{@{}c@{}}c_{g\pm }=-x/t,\\ \dot{\unicode[STIX]{x1D6F6}}_{\pm }=-1.\end{array}\right. & & \displaystyle\end{eqnarray}$$

Substitution of $\dot{\unicode[STIX]{x1D6F6}}_{\pm }=-1$ into (6.11) determines $\unicode[STIX]{x1D6F6}_{\pm }^{3}K_{\pm }=1$ which together with (6.10) yields $K_{\pm }^{2/3}=K_{\pm }^{2}+\unicode[STIX]{x1D703}^{2}$. A comparison of this identity with (6.6d) shows that

(6.15a-c)$$\begin{eqnarray}\displaystyle K_{\pm }=\unicode[STIX]{x1D711}_{\pm }^{3/2}\quad \Longrightarrow \quad \unicode[STIX]{x1D6F6}_{\pm }=\unicode[STIX]{x1D711}_{\pm }^{-1/2},\quad \ddot{\unicode[STIX]{x1D6F6}}_{\pm }=(3\unicode[STIX]{x1D711}_{\pm }^{2}-1)/\unicode[STIX]{x1D711}_{\pm }^{3/2}\gtrless 0. & & \displaystyle\end{eqnarray}$$

Here, $\unicode[STIX]{x1D711}_{\pm }=K_{\pm }\unicode[STIX]{x1D6F6}_{\pm }=k_{\pm }\unicode[STIX]{x1D714}_{\pm }/(2m\unicode[STIX]{x03C0})$ (cf. (6.6b)) are the positive roots ($0<\unicode[STIX]{x1D711}_{-}<\unicode[STIX]{x1D711}_{+}<2$) of (6.6d). Together with (6.13) and (6.14), the inequalities (6.15c) imply that

(6.15d,e)$$\begin{eqnarray}\displaystyle k_{\pm }c_{g\pm }^{\prime }/c_{g\pm }=1+3c_{g\pm }/c_{p\pm }\gtrless 0\quad \text{with }m\unicode[STIX]{x03C0}c_{g\pm }=-2\unicode[STIX]{x1D717}^{2}. & & \displaystyle\end{eqnarray}$$

A routine stationary phase evaluation of the integral in (6.8) involving $\sin (kx+\unicode[STIX]{x1D714}t)$, noting (6.12), determines

(6.16)$$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D712}}_{m}(x,t)\approx \left\{\begin{array}{@{}l@{}}\displaystyle \sum _{k=k_{\pm }}{\displaystyle \frac{-c_{g\pm }}{\unicode[STIX]{x1D714}_{\pm }}}\left({\displaystyle \frac{2}{\pm \unicode[STIX]{x03C0}c_{g\pm }^{\prime }t}}\right)^{1/2}\sin (k_{\pm }x+\unicode[STIX]{x1D714}_{\pm }t\pm \unicode[STIX]{x03C0}/4),\\ \displaystyle \sum _{K=K_{\pm }}{\displaystyle \frac{-1}{\unicode[STIX]{x1D6F6}_{\pm }}}\left({\displaystyle \frac{2}{\pm \unicode[STIX]{x03C0}\ddot{\unicode[STIX]{x1D6F6}}_{\pm }X}}\right)^{1/2}\sin [(K_{\pm }+\unicode[STIX]{x1D6F6}_{\pm })X\pm \unicode[STIX]{x03C0}/4].\end{array}\right. & & \displaystyle\end{eqnarray}$$

In the following two subsections we describe the nature of the solution (6.16) as $\unicode[STIX]{x1D717}$ is increased from zero.

6.2.1 Case $\unicode[STIX]{x1D717}\ll 1$ : wave-like solutions for $-m\unicode[STIX]{x03C0}c_{g\pm }\ll 1$

The simultaneous limits $X\gg 1$ and $\unicode[STIX]{x1D717}\ll 1$ restrict $m\unicode[STIX]{x03C0}x$ to the range

(6.17)$$\begin{eqnarray}\displaystyle (2t)^{-1}\ll m\unicode[STIX]{x03C0}x\ll 2t, & & \displaystyle\end{eqnarray}$$

which only exists for $t\gg 1$. With $\unicode[STIX]{x1D717}$ small, the roots of (6.6d) are $\unicode[STIX]{x1D711}_{+}\approx 1$ and $\unicode[STIX]{x1D711}_{-}\approx \unicode[STIX]{x1D717}^{2}$. Accordingly (6.15) determines

(6.18a-e)$$\begin{eqnarray}\displaystyle K_{+}\approx 1,\quad \unicode[STIX]{x1D6F6}_{+}\approx 1,\quad \ddot{\unicode[STIX]{x1D6F6}_{+}}\approx 2,\quad k_{+}\approx m\unicode[STIX]{x03C0}/\unicode[STIX]{x1D717},\quad \unicode[STIX]{x1D714}_{+}\approx 2\unicode[STIX]{x1D717}, & & \displaystyle\end{eqnarray}$$

(6.18f-j)$$\begin{eqnarray}\displaystyle K_{-}\approx \unicode[STIX]{x1D717}^{3},\quad \unicode[STIX]{x1D6F6}_{-}\approx \unicode[STIX]{x1D717}^{-1},\quad \ddot{\unicode[STIX]{x1D6F6}_{-}}\approx -\unicode[STIX]{x1D717}^{-3},\quad k_{-}\approx m\unicode[STIX]{x03C0}\unicode[STIX]{x1D717}^{2},\quad \unicode[STIX]{x1D714}_{-}\approx 2, & & \displaystyle\end{eqnarray}$$

with which (6.16) becomes

(6.19)$$\begin{eqnarray}\displaystyle \widetilde{\unicode[STIX]{x1D712}}_{m}\approx {\displaystyle \frac{\sin (2X+\unicode[STIX]{x03C0}/4)}{(\unicode[STIX]{x03C0}X)^{1/2}}}+{\displaystyle \frac{\unicode[STIX]{x1D717}^{5/2}\sin (2t-\unicode[STIX]{x03C0}/4)}{(\unicode[STIX]{x03C0}X/2)^{1/2}}}. & & \displaystyle\end{eqnarray}$$

The former $\widetilde{\unicode[STIX]{x1D712}}_{m}^{+}$-mode is linked to the essential singularity at $p=0$ ($\unicode[STIX]{x1D711}_{+}$). In the restricted limit $t\gg z^{2}/x\gg x$, the $z$-Fourier series (2.7a) is dominated by terms with $m=O(xt/z^{2})$ large and may by summed asymptotically, so recovering (5.6b) in the limited domain $\mathsf{x}\ll 1$ ($\mathsf{x}$ defined by (5.2a–c)). The complete summation over all $m$ can only be understood via the analysis of § 5. By implication the $\widetilde{\unicode[STIX]{x1D712}}_{m}^{+}$-modes are linked to the fine structure visible in figures 6 and 7 in the vicinity of the corner $[r,z]=[0,\ell ]$.

The latter $\widetilde{\unicode[STIX]{x1D712}}_{m}^{-}$-mode, smaller by a factor $O(\unicode[STIX]{x1D717}^{5/2})$, is linked to the cuts at $p=\pm 2\text{i}$ ($\unicode[STIX]{x1D711}_{-}$). There is no asymptotic regime on which the ensuing $z$-Fourier series (of small terms) may by summed over $m$, but the very small relative size $O(\unicode[STIX]{x1D717}^{5/2})$ of the $\widetilde{\unicode[STIX]{x1D712}}_{m}^{-}$-modes would seem to render them irrelevant anyway.