2.1 $E^{2/7}$ - and $E^{1/4}$ -quasigeostrophic (QG) shear layers

Away from the Ekman layers, QG-flow exists in the region outside the AG shear layer, which contains the inner sphere tangent cylinder $\mathscr{C}$ . Here the flow velocity components $\breve{\boldsymbol{v}}_{\bot }\equiv (\breve{u},\breve{v},0)$ , in the plane perpendicular to $\breve{z}$ , are dependent on $\breve{s}$ alone. In view of (1.2) this implies that $\breve{\unicode[STIX]{x1D713}}$ is linear in $\breve{z}$ and given by

(2.1) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D713}}=\left\{\begin{array}{@{}ll@{}}[\breve{\unicode[STIX]{x1D713}}_{+}(\breve{z}-\breve{z}_{-})+\breve{\unicode[STIX]{x1D713}}_{-}(\breve{z}_{+}-\breve{z})]/(\breve{z}_{+}-\breve{z}_{-})\quad & \text{in }\mathscr{D}_{in},\\ \breve{\unicode[STIX]{x1D713}}_{+}\breve{z}/\breve{z}_{+}\quad & \text{in }\mathscr{D}_{out},\end{array}\right.\end{eqnarray}$$

where $\breve{z}_{\pm }$ are defined by (1.9c,d ).

Now the azimuthal component of the momentum equation

(2.2) $$\begin{eqnarray}2\breve{u}(=E\unicode[STIX]{x1D60B}\breve{v})=\displaystyle \frac{E}{\breve{s}^{2}}\displaystyle \frac{\text{d}}{\text{d}\breve{s}}\left(\breve{s}^{3}\displaystyle \frac{\text{d}}{\text{d}\breve{s}}\left(\displaystyle \frac{\breve{v}}{\breve{s}}\right)\right)\end{eqnarray}$$

(see (1.3b )) integrated with respect to $\breve{z}$ determines

(2.3a ) $$\begin{eqnarray}\displaystyle \frac{E}{\breve{s}}\displaystyle \frac{\text{d}}{\text{d}\breve{s}}\left(\breve{s}^{3}\displaystyle \frac{\text{d}}{\text{d}\breve{s}}\left(\displaystyle \frac{\breve{v}}{\breve{s}}\right)\right)=\left\{\begin{array}{@{}ll@{}}2(\breve{\unicode[STIX]{x1D713}}_{-}-\breve{\unicode[STIX]{x1D713}}_{+})/(\breve{z}_{+}-\breve{z}_{-})\quad & \text{in }\mathscr{D}_{in},\\ -2\breve{\unicode[STIX]{x1D713}}_{+}/\breve{z}_{+}\quad & \text{in }\mathscr{D}_{out}.\end{array}\right.\end{eqnarray}$$

Substitution of $\breve{\unicode[STIX]{x1D713}}_{\pm }$ defined by (1.10) into the right-hand side of (2.3a ) and noting that $\breve{v}_{\pm }=\breve{v}$ gives

(2.3b ) $$\begin{eqnarray}\displaystyle \frac{E}{\breve{s}^{2}}\displaystyle \frac{\text{d}}{\text{d}\breve{s}}\left(\breve{s}^{3}\displaystyle \frac{\text{d}}{\text{d}\breve{s}}\left(\displaystyle \frac{\breve{v}}{\breve{s}}\right)\right)=\left\{\begin{array}{@{}ll@{}}[(\unicode[STIX]{x1D6FF}_{-}+\unicode[STIX]{x1D6FF}_{+})\breve{v}-\unicode[STIX]{x1D6FF}_{+}\breve{s}]/(\breve{z}_{+}-\breve{z}_{-})\quad & \text{in }\mathscr{D}_{in},\\ \unicode[STIX]{x1D6FF}_{+}(\breve{v}-\breve{s})/\breve{z}_{+}\quad & \text{in }\mathscr{D}_{out},\end{array}\right.\end{eqnarray}$$

to be solved subject to $\breve{v}$ and $\text{d}\breve{v}/\text{d}\breve{s}$ continuous at $\breve{s}=1$ .

In the mainstream, the left-hand side of (2.3b ) is smaller than its right-hand side by a factor $E^{1/2}$ (recall that $\unicode[STIX]{x1D6FF}_{\pm }=O(E^{1/2})$ ; see (1.9a,b )) so recovering the Proudman solution (1.11). The discontinuities of $\breve{v}$ and $\text{d}\breve{v}/\text{d}\breve{s}$ at $\breve{s}=1$ described by (1.11) are smoothed out across a thin QG shear layer, as indicated in § 1.2. A key feature of these QG-layers is the remaining weak singular behaviour of $\breve{v}=\breve{s}\breve{\unicode[STIX]{x1D714}}$ :

(2.4) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}-\breve{\unicode[STIX]{x1D714}}_{G}\approx \left\{\begin{array}{@{}ll@{}}\displaystyle \frac{16}{21}\displaystyle \frac{\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}(-\breve{x})^{7/4}}{E^{1/2}2^{1/4}\breve{z}_{+}}\quad & \text{in }\mathscr{D}_{in},\\ 0\quad & \text{in }\mathscr{D}_{out}\end{array}\right\}\quad \text{as }\breve{s}\rightarrow 1,\end{eqnarray}$$

where $\breve{\unicode[STIX]{x1D714}}_{G}=\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}+\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\breve{x}$ (see (1.15a )), that we identify in (2.12). The discontinuity of the second derivative $\text{d}^{2}\breve{\unicode[STIX]{x1D714}}/\text{d}\breve{s}^{2}$ at $\breve{s}=1$ , implied by (2.4), is smoothed out in a thinner AG $E^{1/3}$ -sublayer considered in § 2.2. There, matching of $\breve{\unicode[STIX]{x1D714}}_{A}$ to $\breve{\unicode[STIX]{x1D714}}-\breve{\unicode[STIX]{x1D714}}_{G}$ defined by (2.4), when $\breve{x}=\breve{s}-1=O(E^{1/3})$ , provides the estimate $\breve{\unicode[STIX]{x1D714}}_{A}=O(E^{1/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}})$ invoked in (1.16c ) and predicted by (2.31a ).

2.1.1 The $\mathscr{D}_{out}$ solution

Outside the tangent cylinder, $\mathscr{D}_{out}:\breve{x}=\breve{s}-1>0$ , we make the approximations

(2.5a,b ) $$\begin{eqnarray}\breve{s}\approx 1,\quad \breve{z}_{+}\approx \sqrt{\unicode[STIX]{x1D6FC}^{2}-1}\end{eqnarray}$$

to obtain the leading order solution

(2.6a ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}\approx \breve{v}=1-(1-\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}})\exp (-\breve{x}/\unicode[STIX]{x1D6FF}_{out}),\end{eqnarray}$$

where

(2.6b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{out}^{2}=E\breve{z}_{+}/\unicode[STIX]{x1D6FF}_{+}=(E\breve{z}_{-}^{3}/\unicode[STIX]{x1D6FC})^{1/2}\end{eqnarray}$$

(see (1.9b )), of (2.3b ) subject to the boundary conditions $\breve{\unicode[STIX]{x1D714}}\rightarrow 0$ as $\breve{x}\rightarrow \infty$ and $\breve{\unicode[STIX]{x1D714}}=\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ at $\breve{x}=0$ . There, the value of $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }=\text{d}\breve{\unicode[STIX]{x1D714}}/\text{d}\breve{s}$ determined by (2.6a ) ( $\breve{x}=\breve{s}-1$ ) is

(2.7) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }=(1-\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}})/\unicode[STIX]{x1D6FF}_{out}=O(E^{-1/4}).\end{eqnarray}$$

With the help of (1.10b ) and (2.1), the formula (2.6a ) for $\breve{\unicode[STIX]{x1D714}}$ determines

(2.8) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D713}}={\textstyle \frac{1}{2}}(E/\unicode[STIX]{x1D6FF}_{out}^{2})(1-\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}})\breve{z}\exp (-\breve{x}/\unicode[STIX]{x1D6FF}_{out})=O(E^{1/2})\end{eqnarray}$$

(see Stewartson Reference Stewartson1966, equation (5.3a,b)).

2.1.2 The $\mathscr{D}_{in}$ solution and $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$

The solution inside the tangent cylinder, $\mathscr{D}_{in}$ : $\breve{x}=\breve{s}-1<0$ , is rather more complicated because of the singular behaviour of $\breve{z}_{-}$ and $\unicode[STIX]{x1D6FF}_{-}$ described by (1.12a,b ). To appreciate its complexity we write out the leading order form of (2.3b ) explicitly:

(2.9a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{out}^{2}\displaystyle \frac{\text{d}^{2}\breve{\unicode[STIX]{x1D714}}}{\text{d}\breve{x}^{2}}-\displaystyle \frac{E^{1/2}}{\unicode[STIX]{x1D6FF}_{+}}(-2\breve{x})^{-1/4}\breve{\unicode[STIX]{x1D714}}=-(1-\breve{\unicode[STIX]{x1D714}}),\quad \displaystyle \frac{E^{1/2}}{\unicode[STIX]{x1D6FF}_{+}}=\left(\frac{\breve{z}_{+}}{\unicode[STIX]{x1D6FC}}\right)^{1/2}\end{eqnarray}$$

(cf. Stewartson Reference Stewartson1966, equation (5.5)). On balancing the two terms on the left-hand side of (2.9a ) and noting the $(-\breve{x})^{-1/4}$ singularity in the coefficient of $\breve{\unicode[STIX]{x1D714}}$ , we may identify the $\mathscr{D}_{in}$ length scale $E^{2/7}$ (short compared to the $\mathscr{D}_{out}$ length scale $E^{1/4}$ ) and so introduce the stretched coordinate

(2.10a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D639}=\breve{x}/\unicode[STIX]{x1D6FF}_{in},\quad \unicode[STIX]{x1D6FF}_{in}=(2E^{2}\breve{z}_{+}^{4})^{1/7}.\end{eqnarray}$$

We note that the ratio of the inner ( $\unicode[STIX]{x1D6FF}_{in}$ ) to outer ( $\unicode[STIX]{x1D6FF}_{out}$ ) shear layer length scales

(2.11a ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{in}/\unicode[STIX]{x1D6FF}_{out}=E^{1/28}\aleph\end{eqnarray}$$

is small, where

(2.11b,c ) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x1D6FF}_{+}}{E^{1/2}}(2\unicode[STIX]{x1D6FF}_{in})^{1/4}=E^{1/14}\aleph ^{2}=\left(\displaystyle \frac{\unicode[STIX]{x1D6FF}_{in}}{\unicode[STIX]{x1D6FF}_{out}}\right)^{2},\quad \aleph =\displaystyle \frac{2^{1/7}\unicode[STIX]{x1D6FC}^{1/4}}{(\unicode[STIX]{x1D6FC}^{2}-1)^{5/56}}.\end{eqnarray}$$

The power series solution of (2.9a ), that meets the $\unicode[STIX]{x1D639}=0$ boundary conditions, begins

(2.12) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}=\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}+\unicode[STIX]{x1D6FF}_{in}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\unicode[STIX]{x1D639}+(16/21)\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}(-\unicode[STIX]{x1D639})^{7/4}+\cdots \,.\end{eqnarray}$$

Matching with the mainstream Proudman solution (1.11a ), identified by neglecting the term $\unicode[STIX]{x1D6FF}_{out}^{2}\text{d}^{2}\breve{\unicode[STIX]{x1D714}}/\text{d}\breve{x}^{2}$ in (2.9a ), requires

(2.13a ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}\approx E^{1/14}\aleph ^{2}\mathscr{G}_{\infty }(\unicode[STIX]{x1D639})\quad \text{as }\unicode[STIX]{x1D639}\rightarrow -\infty ,\end{eqnarray}$$

where

(2.13b ) $$\begin{eqnarray}\mathscr{G}_{\infty }(\unicode[STIX]{x1D639})=(-\unicode[STIX]{x1D639})^{1/4}/[1+E^{1/14}\aleph ^{2}(-\unicode[STIX]{x1D639})^{1/4}].\end{eqnarray}$$

Guided by (2.12) and (2.13), the solution of (2.9a ) may be expressed as the sum

(2.14) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}=\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\mathscr{F}(\unicode[STIX]{x1D639})+E^{1/14}\aleph ^{2}\mathscr{G}(\unicode[STIX]{x1D639})\end{eqnarray}$$

(cf. Stewartson Reference Stewartson1966, equation (5.7)). Under this partition, $\mathscr{F}(\unicode[STIX]{x1D639})$ is chosen to solve

(2.15a ) $$\begin{eqnarray}\mathscr{F}^{\prime \prime }-[(-\unicode[STIX]{x1D639})^{-1/4}+E^{1/14}\aleph ^{2}]\mathscr{F}=0\end{eqnarray}$$

(here, the $\text{}^{\prime }$ denotes the $\unicode[STIX]{x1D639}$ -derivative) subject to

(2.15b,c ) $$\begin{eqnarray}\mathscr{F}(0)=1,\quad \mathscr{F}\approx 0\quad \text{for }1\ll -\unicode[STIX]{x1D639}<O(E^{-2/7}),\end{eqnarray}$$

while $\mathscr{G}(\unicode[STIX]{x1D639})$ solves

(2.16a ) $$\begin{eqnarray}\mathscr{G}^{\prime \prime }-[(-\unicode[STIX]{x1D639})^{-1/4}+E^{1/14}\aleph ^{2}]\mathscr{G}=-1\end{eqnarray}$$

subject to

(2.16b,c ) $$\begin{eqnarray}\mathscr{G}(0)=0,\quad \mathscr{G}\approx \mathscr{G}_{\infty }(\unicode[STIX]{x1D639})\quad \text{for }1\ll -\unicode[STIX]{x1D639}<O(E^{-2/7})\end{eqnarray}$$

(cf. Stewartson Reference Stewartson1966, equations (5.8), (5.9)). Finally, using (2.7) and (2.11a ), continuity of $\text{d}\breve{\unicode[STIX]{x1D714}}/\text{d}\breve{s}$ requires

(2.17a ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\mathscr{F}^{\prime }(0)+E^{1/14}\aleph ^{2}\mathscr{G}^{\prime }(0)=\unicode[STIX]{x1D6FF}_{in}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }=E^{1/28}\aleph (1-\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}})\end{eqnarray}$$

(cf. Stewartson Reference Stewartson1966, equations (6.13), (6.14)) implying that

(2.17b ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}=E^{1/28}\aleph \displaystyle \frac{1-E^{1/28}\aleph \mathscr{G}^{\prime }(0)}{\mathscr{F}^{\prime }(0)+E^{1/28}\aleph }.\end{eqnarray}$$

The complete asymptotic solution of the combined $E^{1/4}$ and $E^{2/7}$ QG shear layers involves expansions in powers of the parameter $E^{1/28}$ identified by (2.11a ). Accordingly in (2.14), we introduce the expansions

(2.18a ) $$\begin{eqnarray}\displaystyle \mathscr{F}(\unicode[STIX]{x1D639}) & = & \displaystyle \mathscr{F}_{0}(\unicode[STIX]{x1D639})+E^{1/14}\aleph ^{2}\mathscr{F}_{1}(\unicode[STIX]{x1D639})+\cdots \,,\end{eqnarray}$$

(2.18b ) $$\begin{eqnarray}\displaystyle \mathscr{G}(\unicode[STIX]{x1D639}) & = & \displaystyle \mathscr{G}_{0}(\unicode[STIX]{x1D639})+E^{1/14}\aleph ^{2}\mathscr{G}_{1}(\unicode[STIX]{x1D639})+\cdots \,.\end{eqnarray}$$

The dominant contribution $\mathscr{F}_{0}(\unicode[STIX]{x1D639})$ to (2.14), which solves $\mathscr{F}_{0}^{\prime \prime }-(-\unicode[STIX]{x1D639})^{-1/4}\mathscr{F}_{0}=0$ and decays to zero as $\unicode[STIX]{x1D639}\rightarrow -\infty$ , may be expressed in terms of the modified Bessel function:

(2.19a,b ) $$\begin{eqnarray}\mathscr{F}_{0}(\unicode[STIX]{x1D639})=\unicode[STIX]{x1D6F6}(-\unicode[STIX]{x1D639})^{1/2}\text{K}_{4/7}(\unicode[STIX]{x1D70E})\quad \text{with }\mathscr{F}_{0}^{\prime }(x)=\unicode[STIX]{x1D6F6}(-\unicode[STIX]{x1D639})^{3/8}\text{K}_{3/7}(\unicode[STIX]{x1D70E}),\end{eqnarray}$$

where

(2.19c,d ) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=(8/7)(-\unicode[STIX]{x1D639})^{7/8}\quad \text{and}\quad \unicode[STIX]{x1D6F6}=2(4/7)^{4/7}/\unicode[STIX]{x1D6E4}(4/7)\end{eqnarray}$$

ensures that $\mathscr{F}_{0}(0)=1$ (use (A 3b )) as well as fixing

(2.20) $$\begin{eqnarray}\mathscr{F}_{0}^{\prime }(0)=(4/7)^{1/7}\unicode[STIX]{x1D6E4}(3/7)/\unicode[STIX]{x1D6E4}(4/7)\end{eqnarray}$$

(see Stewartson Reference Stewartson1966, equation (5.11)) sufficient to determine the zeroth-order approximation

(2.21) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}0}=E^{1/28}\aleph /\mathscr{F}_{0}^{\prime }(0)\end{eqnarray}$$

to (2.17b ) (cf. Stewartson Reference Stewartson1966, equation (6.15)).

Figure 2.

The $E^{2/7}$ -layer DN-solutions (solid lines) for $\breve{\unicode[STIX]{x1D714}}/(E^{1/28}\aleph )$ (see (2.14); $\unicode[STIX]{x1D6FC}^{-1}=0.35$ ) of the problem (2.15)–(2.17) plotted versus $\unicode[STIX]{x1D639}$ for $E=10^{-7}$ (upper), $10^{-14}$ (middle), $10^{-28}$ (lower); the $E$ -values listed in table 1. The Stewartson solution $\mathscr{F}_{0}(\unicode[STIX]{x1D639})/{\mathscr{F}_{0}}^{\prime }(0)$ (see also (2.20), (2.21)) and the Proudman solutions $E^{1/28}\aleph \mathscr{G}_{\infty }(\unicode[STIX]{x1D639})$ (see (2.13)) are identified by the dash-dotted and dashed curves, respectively.

Table 1.

The values of $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ for $\unicode[STIX]{x1D6FC}^{-1}=0.35$ at various values of $E$ (see (2.24b ) for $\aleph E^{1/28}$ in column 2). Asymptotic results truncated at the three levels indicated in (2.25) are listed in middle columns 3–5, respectively, the first of which, column 3, gives the values of $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}0}$ defined by (2.21). The DN-results are listed in the last column 6.

In appendix A we extend Stewartson’s zeroth-order results and construct the next-order contributions $\mathscr{G}_{0}(\unicode[STIX]{x1D639})$ and $\mathscr{F}_{1}(\unicode[STIX]{x1D639})$ , from which we obtain the $\unicode[STIX]{x1D639}=0$ values

(2.22a,b ) $$\begin{eqnarray}-\!\mathscr{G}_{0}^{\prime }(0)=(7/4)^{1/7}\unicode[STIX]{x1D6E4}(8/7),\quad \mathscr{F}_{1}^{\prime }(0)=(7/4)^{1/7}\displaystyle \frac{\unicode[STIX]{x1D6E4}(12/7)[\unicode[STIX]{x1D6E4}(8/7)]^{2}}{\unicode[STIX]{x1D6E4}(16/7)\unicode[STIX]{x1D6E4}(4/7)}\end{eqnarray}$$

needed to determine $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ from (2.17b ) correct to $O(E^{3/28})$ (see also (2.25)). Note too that even the smallest contribution $E^{1/14}\aleph ^{2}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\mathscr{F}_{1}(\unicode[STIX]{x1D639})$ to $\breve{\unicode[STIX]{x1D714}}$ in (2.14) is just larger than the AG-contribution $\breve{\unicode[STIX]{x1D714}}_{A}=O(E^{1/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}})$ (see (1.16c )) that we will consider in § 2.2.

Interpretation of the leading-order solution $\breve{\unicode[STIX]{x1D714}}=\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}0}\mathscr{F}_{0}(\unicode[STIX]{x1D639})$ requires care. For though it dominates for $\unicode[STIX]{x1D639}=O(1)$ , it decays exponentially for large $-\unicode[STIX]{x1D639}$ and so does not match up with the Proudman solution (2.13a ). The needed composite solution valid in the overlap domain $1\ll -\unicode[STIX]{x1D639}\ll E^{-2/7}$ , determined by the large $\unicode[STIX]{x1D70E}$ asymptotic form of $\text{K}_{4/7}(\unicode[STIX]{x1D70E})$ and leading order approximation $\mathscr{G}_{\infty }=(-\unicode[STIX]{x1D639})^{1/4}$ of (2.13b ), is

(2.23) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}=E^{1/28}\aleph \left[\displaystyle \frac{\sqrt{\unicode[STIX]{x03C0}}(-\unicode[STIX]{x1D639})^{1/16}}{(4/7)^{1/14}\unicode[STIX]{x1D6E4}(3/7)}\exp \left(-\displaystyle \frac{8}{7}(-\unicode[STIX]{x1D639})^{7/8}\right)+E^{1/28}\aleph (-\unicode[STIX]{x1D639})^{1/4}\right].\end{eqnarray}$$

It shows that, though $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}0}=O(E^{1/28})$ (see (2.21)), the value of $\breve{\unicode[STIX]{x1D714}}$ continues to decrease in size with increasing $-\unicode[STIX]{x1D639}$ by an order of magnitude, attaining its minimum of $O(E^{1/14})$ at large $-\unicode[STIX]{x1D639}=-\unicode[STIX]{x1D639}_{M}$ (say, and see figure 2, also Dormy et al. Reference Dormy, Cardin and Jault1998, figure 4, left panel). As the value of $\breve{\unicode[STIX]{x1D713}}$ on the outer boundary is given correct to leading order by $\breve{\unicode[STIX]{x1D713}}_{+}\approx \breve{\unicode[STIX]{x1D713}}_{P}-(1/2)\unicode[STIX]{x1D6FF}_{+}\breve{\unicode[STIX]{x1D714}}$ (see (1.10b ) and (1.13c )), it follows that the maximum of $\breve{\unicode[STIX]{x1D713}}_{+}$ is also located at $-\unicode[STIX]{x1D639}\approx -\unicode[STIX]{x1D639}_{M}\gg 1$ .

2.1.3 The direct numerical (DN-)solution: $\unicode[STIX]{x1D6FC}^{-1}=0.35$

The complete resolution of the QG shear layers relies on the solution of the problem (2.14)–(2.17). The expansions (2.18a,b ) of $\mathscr{F}(\unicode[STIX]{x1D639})$ and $\mathscr{G}(\unicode[STIX]{x1D639})$ in powers of $E^{1/28}$ limit their applicability to extremely small $E$ . Nevertheless, the direct numerical (DN-)solution at fixed $E$ is not impeded by this consideration. Such DN-solutions for the oft-used value

(2.24a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}^{-1}=0.35\quad \Longrightarrow \quad \aleph \doteqdot 1.2040\end{eqnarray}$$

(see (2.11c )) are plotted in figure 2 for $E=10^{-7}$ , $10^{-14}$ , $10^{-28}$ , while the corresponding DN-values of $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ (see (2.17b )) are listed in table 1. They are compared with the asymptotic values of (2.17b ) with $[\mathscr{F}^{\prime }(0),\mathscr{G}^{\prime }(0)]$ truncated at various levels:

(2.25) $$\begin{eqnarray}[\mathscr{F}^{\prime }(0),\mathscr{G}^{\prime }(0)]=\left\{\begin{array}{@{}ll@{}}[{\mathscr{F}_{0}}^{\prime }(0),0],\quad & O(\aleph E^{1/28}),\\ \text{}[{\mathscr{F}_{0}}^{\prime }(0),{\mathscr{G}_{0}}^{\prime }(0)],\quad & O(\aleph E^{1/14}),\\ \text{}[{\mathscr{F}_{0}}^{\prime }(0)+E^{1/14}\aleph ^{2}{\mathscr{F}_{0}}^{\prime }(0),{\mathscr{G}_{0}}^{\prime }(0)],\quad & O(\aleph E^{3/28})\end{array}\right.\end{eqnarray}$$

giving $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ accurate to the orders of magnitude indicated. Though the comparison at $E=10^{-7}$ is fair, at $E=10^{-14}$ it is excellent. Certainly the values of $\breve{\unicode[STIX]{x1D714}}/(E^{1/28}\aleph )$ on the $\unicode[STIX]{x1D639}=0$ axis of figure 2 (namely, the tangent cylinder $\mathscr{C}$ ) for $E=10^{-14}$ and $10^{-28}$ are indistinguishable from the lowest order value $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}0}/(E^{1/28}\aleph )=1/{\mathscr{F}_{0}}^{\prime }(0)$ (see (2.21)). We also show the limiting forms of the Proudman solutions $E^{1/28}\aleph \mathscr{G}_{\infty }(\unicode[STIX]{x1D639})$ (see (2.13)), because they provide the $-\unicode[STIX]{x1D639}\gg 1$ asymptotes to the numerical solutions. Evidently, on each finite $E$ curve, a minimum at negative $\unicode[STIX]{x1D639}_{M}$ (dependent on $E$ ) can be identified, as predicted by the discussion in § 2.1.2 following (2.23).

2.2 $E^{1/3}$ -ageostrophic (AG) shear layer

The relatively thick QG shear layers have ensured continuity of the geostrophic flow close to the tangent cylinder $\mathscr{C}$ , where $\breve{\unicode[STIX]{x1D714}}\approx \breve{\unicode[STIX]{x1D714}}_{G}(\breve{s})$ (see (1.15a )). Nevertheless, since both $\breve{\unicode[STIX]{x1D713}}$ , $\unicode[STIX]{x2202}^{2}\breve{\unicode[STIX]{x1D714}}/\unicode[STIX]{x2202}\breve{s}^{2}$ and their derivatives remain discontinuous at $\breve{s}=1$ , it is the role of the AG shear layer to smooth them out over the $\breve{s}$ -directed length scale $E^{1/3}$ . As anticipated in (1.16a,b ), we set $\breve{\unicode[STIX]{x1D714}}=\breve{\unicode[STIX]{x1D714}}_{G}+\breve{\unicode[STIX]{x1D714}}_{A}$ , $\breve{\unicode[STIX]{x1D713}}=\breve{\unicode[STIX]{x1D713}}_{A}$ and solve for $\breve{\unicode[STIX]{x1D714}}_{A}$ , $\breve{\unicode[STIX]{x1D713}}_{A}$ . In view of the estimates (1.16c ), it is evident that the leading-order approximation to (1.15a ) is

(2.26) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}_{G}\approx \breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}.\end{eqnarray}$$

2.2.1 The Fourier transform solution: $\breve{x}=O(E^{1/3})$ , $\breve{z}=O(1)$

On the short radial length scale $\breve{x}=O(E^{1/3})$ , the governing equations (1.3a,b ) reduce to

(2.27a,b ) $$\begin{eqnarray}2\displaystyle \frac{\unicode[STIX]{x2202}\breve{v}_{A}}{\unicode[STIX]{x2202}\breve{z}}=E\displaystyle \frac{\unicode[STIX]{x2202}^{4}\breve{\unicode[STIX]{x1D713}}_{A}}{\unicode[STIX]{x2202}\breve{x}^{4}},\quad -2\displaystyle \frac{\unicode[STIX]{x2202}\breve{\unicode[STIX]{x1D713}}_{A}}{\unicode[STIX]{x2202}\breve{z}}=E\displaystyle \frac{\unicode[STIX]{x2202}^{2}\breve{v}_{A}}{\unicode[STIX]{x2202}\breve{x}^{2}}.\end{eqnarray}$$

Correct to leading order the boundary conditions (1.10) determine

(2.28a,b ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D713}}_{A+}={\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}_{+}\approx \breve{\unicode[STIX]{x1D713}}_{P},\quad \breve{\unicode[STIX]{x1D713}}_{A-}=\left\{\begin{array}{@{}ll@{}}{\textstyle \frac{1}{2}}\unicode[STIX]{x1D6FF}_{-}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\quad & \text{in }\mathscr{D}_{in},\\ 0\quad & \text{in }\mathscr{ D}_{out}\end{array}\right.\end{eqnarray}$$

(see also (1.13c ), (1.17)). On use of the estimates (1.12b ) and (1.14c ), we obtain

(2.29) $$\begin{eqnarray}(\unicode[STIX]{x1D6FF}_{-}/E^{1/2})\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}=O(E^{-1/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}})=O(E^{-1/21})\gg (\unicode[STIX]{x1D6FF}_{+}/E^{1/2})=O(1).\end{eqnarray}$$

So despite the small size of $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ , the value of $\breve{\unicode[STIX]{x1D713}}_{A}$ on the bottom boundary in $\mathscr{D}_{in}$ exceeds $\breve{\unicode[STIX]{x1D713}}_{P}$ on the top by a moderate factor $O(E^{-1/21})$ . With $\breve{\unicode[STIX]{x1D713}}_{P}$ neglected, we are left with

(2.30a ) $$\begin{eqnarray}\displaystyle \breve{\unicode[STIX]{x1D713}}_{A} & = & \displaystyle 0\hspace{155.00023pt}\text{at }\breve{z}=\sqrt{\unicode[STIX]{x1D6FC}^{2}-1},\end{eqnarray}$$

(2.30b ) $$\begin{eqnarray}\displaystyle \breve{\unicode[STIX]{x1D713}}_{A} & = & \displaystyle \left\{\begin{array}{@{}ll@{}}\displaystyle \frac{1}{2}\left[\displaystyle \frac{E^{1/3}}{(-2\breve{x})}\right]^{1/4}E^{5/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}\quad & \text{in }\mathscr{D}_{in},\\ 0\quad & \text{in }\mathscr{D}_{out}\end{array}\right\}\quad \text{at }\breve{z}=0.\end{eqnarray}$$

Here, since the bottom boundary is located at $\breve{z}_{-}\approx \sqrt{-2\breve{x}}=O(E^{1/6})$ , the boundary condition (2.30b ) in $\mathscr{D}_{in}$ may be applied, correct to leading order, at $\breve{z}=0$ as stated. Together with natural boundedness requirements as $E^{-1/3}\breve{x}\rightarrow \pm \infty$ , the boundary conditions (2.30) are sufficient to determine the Fourier transform solution $(\breve{v}_{A}\approx \breve{\unicode[STIX]{x1D714}}_{A})$

(2.31a ) $$\begin{eqnarray}\displaystyle \breve{\unicode[STIX]{x1D714}}_{A} & = & \displaystyle -\displaystyle \frac{E^{1/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}}{2^{7/4}\unicode[STIX]{x1D6E4}(1/4)}\int _{-\infty }^{\infty }\displaystyle \frac{\widetilde{k}^{1/4}\cosh \left[{\textstyle \frac{1}{2}}\widetilde{k}^{3}(\widetilde{z}_{+}-\widetilde{z})\right]}{\sinh \left[{\textstyle \frac{1}{2}}\widetilde{k}^{3}\widetilde{z}_{+}\right]}\exp \left[\text{i}\left(\widetilde{k}\widetilde{x}+\displaystyle \frac{3\unicode[STIX]{x03C0}}{8}\right)\right]\,\text{d}\widetilde{k},\quad\end{eqnarray}$$

(2.31b ) $$\begin{eqnarray}\displaystyle \breve{\unicode[STIX]{x1D713}}_{A} & = & \displaystyle \displaystyle \frac{E^{5/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}}{2^{7/4}\unicode[STIX]{x1D6E4}(1/4)}\int _{-\infty }^{\infty }\displaystyle \frac{\sinh \left[{\textstyle \frac{1}{2}}\widetilde{k}^{3}(\widetilde{z}_{+}-\widetilde{z})\right]}{\widetilde{k}^{3/4}\sinh \left[{\textstyle \frac{1}{2}}\widetilde{k}^{3}\widetilde{z}_{+}\right]}\exp \left[\text{i}\left(\widetilde{k}\widetilde{x}+\displaystyle \frac{3\unicode[STIX]{x03C0}}{8}\right)\right]\,\text{d}\widetilde{k}\end{eqnarray}$$

(see Stewartson Reference Stewartson1966, equation (6.22a,b)), where

(2.32a-c ) $$\begin{eqnarray}\widetilde{x}=E^{-1/3}\breve{x},\quad \widetilde{z}=\breve{z},\quad \widetilde{z}_{+}=\sqrt{\unicode[STIX]{x1D6FC}^{2}-1}\end{eqnarray}$$

and $\widetilde{k}^{1/4}=|\widetilde{k}|^{1/4}\text{e}^{\text{i}\unicode[STIX]{x03C0}/4}$ and $\widetilde{k}^{-3/4}=|\widetilde{k}|^{-3/4}\text{e}^{-\text{i}3\unicode[STIX]{x03C0}/4}$ for $\widetilde{k}<0$ .

Significantly, the result (2.31a ) shows that $\breve{\unicode[STIX]{x1D714}}_{A}=O(E^{1/12}\breve{\unicode[STIX]{x1D714}}_{G})$ , where $\breve{\unicode[STIX]{x1D714}}_{G}=\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ (see (2.26)). The small relative size of $\breve{\unicode[STIX]{x1D714}}_{A}$ to $\breve{\unicode[STIX]{x1D714}}_{G}$ , albeit by a factor $O(E^{1/12})$ , ensures that any Ekman suction produced by $\breve{\unicode[STIX]{x1D714}}_{A}$ is small compared to that assumed in our applied bottom boundary condition (2.30b ) by an $O(E^{1/12})$ factor. This estimate is sufficient to confirm the consistency of the approximations made.

Finally, we consider the $\breve{z}$ -average

(2.33) $$\begin{eqnarray}\langle \breve{\unicode[STIX]{x1D714}}_{A}\rangle =\displaystyle \frac{1}{\widetilde{z}_{+}}\int _{0}^{\widetilde{z}_{+}}\breve{\unicode[STIX]{x1D714}}_{A}\,\text{d}\widetilde{z}=\left\{\begin{array}{@{}ll@{}}\displaystyle \frac{E^{1/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}}{2^{1/4}\widetilde{z}_{+}}\displaystyle \frac{16}{21}(-\widetilde{x})^{7/4}\quad & \text{in }\mathscr{D}_{in},\\ 0\quad & \text{in }\mathscr{D}_{out},\end{array}\right.\end{eqnarray}$$

which may be determined from (2.31a ) or more easily by direct integration of the $\breve{z}$ -average of (2.27b ).

The combination $\breve{\unicode[STIX]{x1D714}}_{G}+\langle \breve{\unicode[STIX]{x1D714}}_{A}\rangle =\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}+\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\breve{x}+\langle \breve{\unicode[STIX]{x1D714}}_{A}\rangle$ (see (1.15a )) for $\mathscr{D}_{in}$ , determined by (2.33), is simply the QG asymptotic behaviour in the $E^{2/7}$ -layer identified in (2.4) and established by (2.12). So if we denote the entire QG-solution of (2.3b ), obtained by combining the Proudman mainstream solution (1.11a ) and the $E^{2/7}$ - and $E^{1/4}$ -QG shear layer solutions (2.6a ) and (2.14), as $\breve{\unicode[STIX]{x1D714}}_{PQG}(\breve{s})$ , it means that the $\breve{z}$ -average of $\breve{\unicode[STIX]{x1D714}}$ is given correct to leading order by

(2.34a ) $$\begin{eqnarray}\langle \breve{\unicode[STIX]{x1D714}}\rangle \approx \breve{\unicode[STIX]{x1D714}}_{PQG}(\breve{s})\end{eqnarray}$$

everywhere outside the Ekman layers (including the equatorial Ekman layers); a very useful result. It has the important non-trivial consequence that

(2.34b ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}\approx \breve{\unicode[STIX]{x1D714}}_{PQG}+(\breve{\unicode[STIX]{x1D714}}_{A}-\langle \breve{\unicode[STIX]{x1D714}}_{A}\rangle )\end{eqnarray}$$

everywhere outside Ekman layers too. Here the singularities of $\langle \breve{\unicode[STIX]{x1D714}}_{A}\rangle$ and $\breve{\unicode[STIX]{x1D714}}_{PQG}$ on $\mathscr{C}$ cancel each other out, while $\breve{\unicode[STIX]{x1D714}}_{A}$ defined by (2.31a ) is continuous. We conclude that, whereas $\langle \breve{\unicode[STIX]{x1D714}}\rangle$ is singular on $\mathscr{C},\breve{\unicode[STIX]{x1D714}}$ itself is smooth across $\mathscr{C}$ except for singular behaviour in the immediate neighbourhood of the equator, $(\widetilde{x},\widetilde{z})=(0,0)$ (see §§ 2.2.2, 2.2.3).

2.2.2 The similarity sublayer: $\breve{x}=O(E^{1/3}\breve{z}^{1/3})$ , $E^{1/5}\ll \breve{z}\ll 1$

Close to the equator, where

(2.35a,b ) $$\begin{eqnarray}\widetilde{x}=O(\widetilde{z}^{1/3})\quad \text{when }\widetilde{z}\ll 1,\end{eqnarray}$$

the solutions (2.31a,b ) may be approximated as

(2.36a ) $$\begin{eqnarray}\displaystyle \breve{\unicode[STIX]{x1D714}}_{A} & = & \displaystyle -E^{1/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\widetilde{z}^{\,-5/12}V_{0}(\unicode[STIX]{x1D6F7}),\end{eqnarray}$$

(2.36b ) $$\begin{eqnarray}\displaystyle \breve{\unicode[STIX]{x1D713}}_{A} & = & \displaystyle -E^{5/12}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\widetilde{z}^{\,-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7}),\end{eqnarray}$$

in which

(2.36c ) $$\begin{eqnarray}\displaystyle V_{0}(\unicode[STIX]{x1D6F7}) & = & \displaystyle \displaystyle \frac{2^{-3/4}}{\unicode[STIX]{x1D6E4}(1/4)}\int _{0}^{\infty }\unicode[STIX]{x1D71B}^{1/4}\cos \left(\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F7}+\displaystyle \frac{3\unicode[STIX]{x03C0}}{8}\right)\exp \left(-\frac{1}{2}\unicode[STIX]{x1D71B}^{3}\right)\,\text{d}\unicode[STIX]{x1D71B},\end{eqnarray}$$

(2.36d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7}) & = & \displaystyle -\displaystyle \frac{2^{-3/4}}{\unicode[STIX]{x1D6E4}(1/4)}\int _{0}^{\infty }\unicode[STIX]{x1D71B}^{-3/4}\cos \left(\unicode[STIX]{x1D71B}\unicode[STIX]{x1D6F7}+\displaystyle \frac{3\unicode[STIX]{x03C0}}{8}\right)\exp \left(-\frac{1}{2}\unicode[STIX]{x1D71B}^{3}\right)\,\text{d}\unicode[STIX]{x1D71B}\end{eqnarray}$$

(cf. Stewartson Reference Stewartson1966, equations (6.24), (6.25a,b)), where

(2.37a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D71B}=\widetilde{k}\widetilde{z}^{1/3},\quad \unicode[STIX]{x1D6F7}=\widetilde{x}/\widetilde{z}^{1/3}.\end{eqnarray}$$

2.2.4 Full shell DNS-results: $\unicode[STIX]{x1D6FC}^{-1}=0.35$ , $E=10^{-5}$ and $10^{-7}$

DNS-results for the full shell equations (1.3), again at $\unicode[STIX]{x1D6FC}^{-1}=0.35$ , are illustrated in figure 3(a,b) for the particular case $E=10^{-5}$ .

Figure 3.

Contours of (a) the azimuthal angular velocity $\breve{\unicode[STIX]{x1D714}}$ and (b) the streamfunction $\breve{\unicode[STIX]{x1D713}}$ obtained from the full shell DNS for the case $\unicode[STIX]{x1D6FC}^{-1}=0.35$ , $E=10^{-5}$ . A weak reversed flow eddy is identified by the single black $\breve{\unicode[STIX]{x1D713}}$ -contour just outside the tangent cylinder.

Well inside the tangent cylinder $\mathscr{C}$ , the Proudman solution is illustrated by the contours parallel to the rotation axis both for $\breve{\unicode[STIX]{x1D714}}=\breve{\unicode[STIX]{x1D714}}(\breve{s})$ and $\breve{\unicode[STIX]{x1D713}}=\breve{\unicode[STIX]{x1D713}}(\breve{s})$ . For $\breve{\unicode[STIX]{x1D714}}$ , this feature continues into the QG-layers either side of $\mathscr{C}$ , where $\breve{\unicode[STIX]{x1D714}}\approx \breve{\unicode[STIX]{x1D714}}_{PQG}(\breve{s})$ (defined above (2.34a )). There, $\breve{\unicode[STIX]{x1D713}}$ is linear in $\breve{z}$ (see (2.1)), with the streamline structure in figure 3(b) consistent with the QG-solution described in § 2.1.2. Just outside the tangent cylinder, $\mathscr{D}_{out}$ , the small tipping of the $\breve{\unicode[STIX]{x1D713}}$ -contours near the outer sphere is implied by the $E^{1/4}$ -layer solution (2.8). Inside, $\mathscr{D}_{in}$ , closed $\breve{\unicode[STIX]{x1D713}}$ -contours identify a clockwise eddy (cf., figure 5, middle panel of Wei & Hollerbach Reference Wei and Hollerbach2008, where a very similar eddy structure is visible albeit in an alternative parameter range). The analysis at the end of § 2.1.2 reveals that the outermost closed contour touches the outer sphere at $\breve{s}=\breve{s}_{M}=1+\unicode[STIX]{x1D6FF}_{in}\unicode[STIX]{x1D639}_{M}$ (see explanation below (2.23)), which since $-\unicode[STIX]{x1D639}_{M}\gg 1$ is at the edge of the $E^{2/7}$ -layer. It means that almost all of $E^{2/7}$ -layer embraces return flow from the outer Ekman layer (a striking feature anticipated remarkably by Stewartson (Reference Stewartson1966) at the end of his § 5).

Interestingly, virtually none of the structure associated with the $E^{1/3}$ -AG shear layer is distinguishable on either figure 3(a) or 3(b). Nevertheless, we should note that the more intense part of the $E^{2/7}$ -layer clockwise eddy (close to the equator of the inner sphere) is located within the smaller $(E\breve{z})^{1/3}$ -Stewartson similarity sublayer, which also contains a weaker counterclockwise eddy outside, in $\mathscr{D}_{out}$ . As we will see (figure 6 b), the entire double-eddy structure continues into the equatorial Ekman layer.

Asymptotically, the $\breve{z}$ -dependence of $\breve{\unicode[STIX]{x1D714}}(\breve{s},\breve{z})\approx \breve{\unicode[STIX]{x1D714}}_{PQG}(\breve{s})+(\breve{\unicode[STIX]{x1D714}}_{A}(\breve{s},\breve{z})-\langle \breve{\unicode[STIX]{x1D714}}_{A}(\breve{s})\rangle )$ predicted by (2.34b ) in the $E^{1/3}$ -layer is small except in and near the equatorial Ekman layer. However, the value $E=10^{-5}$ used to obtain the DNS-results illustrated in figure 3(b) is not small enough to reach that asymptotic regime. So, though not visible in the figure, there is significant $\breve{z}$ -dependence of $\breve{\unicode[STIX]{x1D714}}$ on the tangent cylinder $\mathscr{C}$ . Nevertheless, as explained at the end of § 2.2.1 with particular reference to (2.34a ), the $\breve{z}$ -average $\langle \breve{\unicode[STIX]{x1D714}}\rangle$ of $\breve{\unicode[STIX]{x1D714}}$ is approximated to a high order of accuracy by $\breve{\unicode[STIX]{x1D714}}_{PQG}(\breve{s})$ throughout the shear layers. The comparison on $\mathscr{C}$ of

(2.38) $$\begin{eqnarray}\begin{array}{@{}c@{}}\langle \breve{\unicode[STIX]{x1D714}}\rangle _{\mathscr{C}}\\ \text{DNS}\end{array}=\left\{\begin{array}{@{}l@{}}0.5609\\ 0.4965\end{array}\right.\qquad \quad \text{with }\qquad \begin{array}{@{}c@{}}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\\ \text{DN}\end{array}=\left\{\begin{array}{@{}l@{}}0.5628\\ 0.5012\end{array}\right.\qquad \text{ at }\qquad E=\left\{\begin{array}{@{}l@{}}10^{-5}\\ 10^{-7}\end{array}\right.\end{eqnarray}$$

(DN-values are given in the last column of table 1) is excellent.

3.1 The governing equations and boundary conditions

The far-field boundary condition on $\breve{\unicode[STIX]{x1D714}}$ , as either $x\rightarrow \infty$ or $z\rightarrow \infty$ , is

(3.5) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D714}}/\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}\rightarrow \breve{\unicode[STIX]{x1D714}}_{G}/\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}=1+E^{2/5}(\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}^{\,\prime }/\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}})x=1+O(E^{4/35}x)\end{eqnarray}$$

(see (1.15a ) and (1.14b ): $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }/\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}=O(E^{-2/7})$ , but cf. (1.20)). As in § 2.2, we will neglect the linear shear flow correction and simply apply $\breve{\unicode[STIX]{x1D714}}\rightarrow \breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ (see (2.26)). The error estimate is included in (3.5) only to make clear the level of accuracy possible inside the equatorial Ekman layer at finite $E$ . We dimensionalise velocity in the form

(3.6a ) $$\begin{eqnarray}(u^{\star },v^{\star },w^{\star })=\unicode[STIX]{x1D700}L\unicode[STIX]{x1D6FA}(\breve{u},\breve{v},\breve{w})=-\unicode[STIX]{x1D700}L\unicode[STIX]{x0394}\unicode[STIX]{x1D6FA}(E^{1/5}u,v-1,w)\end{eqnarray}$$

based on the angular velocity $\unicode[STIX]{x1D700}\unicode[STIX]{x0394}\unicode[STIX]{x1D734}=\unicode[STIX]{x1D700}\unicode[STIX]{x1D734}\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ rather than $\unicode[STIX]{x1D700}\unicode[STIX]{x1D734}$ (see (1.14a )); by introducing $-\unicode[STIX]{x1D700}$ we have essentially reversed the sign of the velocity. However, the ploy has the merit that $v=1$ on $\mathscr{S}$ and $v\rightarrow 0$ far from $\mathscr{S}$ . By implication our new rest frame $\boldsymbol{v}=\mathbf{0}$ rotates with the angular velocity $\unicode[STIX]{x1D734}+\unicode[STIX]{x1D700}\unicode[STIX]{x0394}\unicode[STIX]{x1D734}$ of the dominant flow in the far field. The corresponding streamfunction $\breve{\unicode[STIX]{x1D713}}$ becomes

(3.6b ) $$\begin{eqnarray}\breve{\unicode[STIX]{x1D713}}=-E^{2/5}\breve{\unicode[STIX]{x1D714}}_{\mathscr{ C}}\unicode[STIX]{x1D713},\end{eqnarray}$$

so that from (1.2) we have

(3.7a,b ) $$\begin{eqnarray}u=-\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}z},\quad w=\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x}.\end{eqnarray}$$

Hence the $E^{1/3}$ -layer azimuthal vorticity and momentum equations (2.27a,b ) become

(3.8a,b ) $$\begin{eqnarray}2\displaystyle \frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}z}=\displaystyle \frac{\unicode[STIX]{x2202}^{4}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x^{4}},\quad 2\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}z}=-\displaystyle \frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}x^{2}}.\end{eqnarray}$$

On regarding $v$ , $w$ and $\unicode[STIX]{x1D713}$ as functions of $(y,z)$ (see (3.3b )) rather than $(x,z)$ , we may recast (3.8) as

(3.9a,b ) $$\begin{eqnarray}2\left(\displaystyle \frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}z}+z\displaystyle \frac{\unicode[STIX]{x2202}v}{\unicode[STIX]{x2202}y}\right)=\displaystyle \frac{\unicode[STIX]{x2202}^{4}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}y^{4}},\quad 2\left(\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}z}+z\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}y}\right)=-\displaystyle \frac{\unicode[STIX]{x2202}^{2}v}{\unicode[STIX]{x2202}y^{2}}\end{eqnarray}$$

(cf. Philander Reference Philander1971, equation (4.3), who was concerned with the outer sphere Ekman layer). This has the advantage that the fluid domain becomes simply $y>0$ .

The system (3.8) (or equivalently (3.9)) is to be solved subject to

(3.10a ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}v/\unicode[STIX]{x2202}z=0, & \unicode[STIX]{x1D713}=0\qquad \text{on} & \displaystyle z=0,\;\qquad \,x>0,\end{eqnarray}$$

(3.10b ) $$\begin{eqnarray}\displaystyle v=1,\qquad w=0,\qquad & \unicode[STIX]{x1D713}=0\qquad \text{on} & \displaystyle z=\sqrt{-2x},\,\,x<0\nonumber\\ \displaystyle & \text{or simply} & \displaystyle \;y=0,\end{eqnarray}$$

(3.10c ) $$\begin{eqnarray}\displaystyle v\rightarrow 0, & \unicode[STIX]{x1D713}\rightarrow 0\qquad \text{as} & \displaystyle x\rightarrow \infty \text{or }\,\,y\rightarrow \infty ,\end{eqnarray}$$

(3.10d ) $$\begin{eqnarray}\displaystyle v\sim v_{0}^{bl}(\unicode[STIX]{x1D701}),\quad w\sim w_{0}^{bl}(\unicode[STIX]{x1D701}),\quad & \unicode[STIX]{x1D713}\rightarrow 0\qquad \text{as} & \displaystyle z\rightarrow \infty ,\;\quad \,\unicode[STIX]{x1D701}>0,\end{eqnarray}$$

(3.11a,b ) $$\begin{eqnarray}v_{0}^{bl}=\text{e}^{-\unicode[STIX]{x1D701}}\cos \unicode[STIX]{x1D701},\quad w_{0}^{bl}=-\text{e}^{-\unicode[STIX]{x1D701}}\sin \unicode[STIX]{x1D701},\end{eqnarray}$$

in which $\unicode[STIX]{x1D701}$ is the Ekman layer coordinate (3.3a ). The boundary conditions (3.10) are homogeneous except for the requirement $v=1$ (3.10b ) $_{1}$ on the sphere boundary $\mathscr{S}$ and the persistence of the associated Ekman boundary layer as $z\rightarrow \infty$ (3.10d ) $_{1,2}$ . Also (3.10a ) $_{1,2}$ follow from the symmetry conditions (1.6) across the equatorial plane in $\mathscr{D}_{out}$ . These are essentially the boundary conditions proposed in Stewartson (Reference Stewartson1966, § 7) for the terminal $E^{2/5}$ Stewartson layer. There, he also raises the possibility that (3.10d ) is not enough and that matching, as $z\rightarrow \infty$ , with the similarity forms (2.36a,b ), namely

(3.12a,b ) $$\begin{eqnarray}v=z^{-5/12}V_{0}(\unicode[STIX]{x1D6F7}),\quad \unicode[STIX]{x1D713}=z^{-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7})\end{eqnarray}$$

in our dimensionless notation, may be needed (but see also Stewartson Reference Stewartson1966, equation (6.29)). Our view is that (3.12) follows consistently from the solution of the problem posed by (3.8a,b ) and (3.10a – d ).

On introduction of the complex variable

(3.13) $$\begin{eqnarray}\unicode[STIX]{x1D638}=v(x,z)+\text{i}w(x,z),\end{eqnarray}$$

and differentiation of (3.8b ) with respect to $x$ , the system of equations (3.8a,b ) may be expressed compactly, noting (3.7b ), as

(3.14) $$\begin{eqnarray}\displaystyle \frac{\unicode[STIX]{x2202}^{3}\unicode[STIX]{x1D638}}{\unicode[STIX]{x2202}x^{3}}=2\text{i}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D638}}{\unicode[STIX]{x2202}z}\end{eqnarray}$$

(see, e.g., Moore & Saffman Reference Moore and Saffman1969), while the boundary conditions (3.10) become

(3.15a ) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}Im\{\unicode[STIX]{x1D638}\}\\ Re\{\unicode[STIX]{x2202}\unicode[STIX]{x1D638}/\unicode[STIX]{x2202}z\}\end{array}\right\} & = & \displaystyle 0,\qquad \quad \text{on }z=0,\;x>0,\end{eqnarray}$$

(3.15b ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D638} & = & \displaystyle 1,\qquad \quad \text{on }y=0,\end{eqnarray}$$

(3.15c ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D638} & \rightarrow & \displaystyle 0,\qquad \quad \text{as }x\rightarrow \infty ,\end{eqnarray}$$

(3.15d ) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D638} & {\sim} & \displaystyle \unicode[STIX]{x1D638}_{0}^{bl}(\unicode[STIX]{x1D701}),\quad \text{as }z\rightarrow \infty ,\;\unicode[STIX]{x1D701}>0,\end{eqnarray}$$

where

(3.15e ) $$\begin{eqnarray}\unicode[STIX]{x1D638}_{0}^{bl}(\unicode[STIX]{x1D701})=\exp [-(1+\text{i})\unicode[STIX]{x1D701}]\end{eqnarray}$$

$(\unicode[STIX]{x1D701}=z^{1/2}y,y=x+(z^{2}/2),z>0)$ , together with

(3.15f ) $$\begin{eqnarray}Re\left\{\int _{-z^{2}/2}^{\infty }\unicode[STIX]{x1D638}\,\text{d}x\right\}=0,\end{eqnarray}$$

which implements the zero axial flux condition: $\unicode[STIX]{x1D713}=0$ on $\mathscr{S}$ with $\unicode[STIX]{x1D713}\rightarrow 0$ far from  $\mathscr{S}$ .

3.2 A far-field (including the similarity sublayer) boundary condition at large $z$

Anticipating the numerical solution on a finite domain, our objective here is to replace the far-field boundary conditions $v\rightarrow 0$ , $\unicode[STIX]{x1D713}\rightarrow 0$ on $-\infty <x<\infty$ as $z\rightarrow \infty$ by a boundary condition at finite $z=H(\gg 1)$ . To that end, we restrict attention to the far field $z\geqslant H(=\text{const.})$ on which we consider the Fourier transforms $\widehat{\unicode[STIX]{x1D713}}(k,z)$ , $\widehat{v}(k,z)$ :

(3.16) $$\begin{eqnarray}[\unicode[STIX]{x1D713},v](x,z)=\int _{-\infty }^{\infty }[\widehat{\unicode[STIX]{x1D713}},\widehat{v}](k,z)\exp (\text{i}kx)\,\text{d}k.\end{eqnarray}$$

The Fourier transform of (3.8a,b ) determines

(3.17a,b ) $$\begin{eqnarray}2\displaystyle \frac{\unicode[STIX]{x2202}\widehat{v}}{\unicode[STIX]{x2202}z}=k^{4}\widehat{\unicode[STIX]{x1D713}},\quad 2\displaystyle \frac{\unicode[STIX]{x2202}\widehat{\unicode[STIX]{x1D713}}}{\unicode[STIX]{x2202}z}=k^{2}\widehat{v}.\end{eqnarray}$$

The solution non-divergent as $z\rightarrow \infty$ is

(3.18) $$\begin{eqnarray}\widehat{v}(k,z)=a(k)\exp \left(-{\textstyle \frac{1}{2}}|k|^{3}z\right),\end{eqnarray}$$

which together with (3.17a ) leads to the relation

(3.19) $$\begin{eqnarray}\widehat{\displaystyle \frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x^{2}}}=-k^{2}\widehat{\unicode[STIX]{x1D713}}=-\displaystyle \frac{2}{k^{2}}\displaystyle \frac{\unicode[STIX]{x2202}\widehat{v}}{\unicode[STIX]{x2202}z}=|k|\widehat{v}.\end{eqnarray}$$

At $z=H$ , it gives

(3.20) $$\begin{eqnarray}\widehat{v}(k,H)=\displaystyle \frac{1}{|k|}\widehat{\displaystyle \frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x^{2}}}(k,H),\end{eqnarray}$$

whose inverse determines the convolution integral

(3.21) $$\begin{eqnarray}v(x,H)=-\displaystyle \frac{1}{\unicode[STIX]{x03C0}}\int _{-\infty }^{\infty }\ln |x-x^{\prime }|\displaystyle \frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x^{2}}(x^{\prime },H)\,\text{d}x^{\prime }=-\displaystyle \frac{1}{\unicode[STIX]{x03C0}}\unicode[STIX]{x2A0D}_{-\infty }^{\infty }\displaystyle \frac{1}{x-x^{\prime }}\displaystyle \frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D713}}{\unicode[STIX]{x2202}x}(x^{\prime },H)\,\text{d}x^{\prime }\end{eqnarray}$$

( $\unicode[STIX]{x2A0D}$ denotes the principal part). It will provide the basis of our boundary condition (B 1e ), which is implemented (see appendix B) in the solution of our numerical model.

Of course, the Stewartson similarity solution (3.12), which determines

(3.22a,b ) $$\begin{eqnarray}v(x,H)=H^{-5/12}V_{0}(x/H^{1/3}),\quad \unicode[STIX]{x1D713}(x,H)=H^{-1/12}\unicode[STIX]{x1D6F9}_{0}(x/H^{1/3}),\end{eqnarray}$$

satisfies (3.21), but significantly is only an approximation to the true far-field solution at finite $z=H$ . (‘Mainstream’ is a natural description, which we adopt in the equatorial Ekman layer context, but in truth is a misnomer as the region lies entirely within $E^{1/3}$ -layer.).

3.3 The numerical model

In order to capture the equatorial structure of the Ekman layer and the emergence for $z\gg 1$ of Stewartson’s self-similar solution (3.12), we consider a 2D numerical domain of finite extent, and focus attention on a region bounded below by the equatorial plane $z=0$ and on its left-hand side by the parabolic contour $\mathscr{S}$ : $x=-z^{2}/2$ (see (3.2)). By adopting the alternative $(y,z)$ -coordinates, recall that $x(y,z)=y-z^{2}/2$ (see (3.3b )), $\mathscr{S}$ becomes the left-hand ‘vertical’ boundary of our rectangular numerical box, $0\leqslant y\leqslant L$ , $0\leqslant z\leqslant H$ (see figure 4), on which we solve (3.9). The true problem associated with the boundary conditions (3.10) is on an otherwise unbounded domain $L\rightarrow \infty$ , $H\rightarrow \infty$ . Evidently $L$ and $H$ must be sufficiently large that any approximate boundary conditions applied on both the right-hand boundary $y=L$ and the top boundary $z=H$ do not seriously influence the nature of the solution for $x=O(1)$ , $z=O(1)$ .

Figure 4.

The map from the $x$ – $z$ plane to the numerical box $[0,L]\times [0,H]$ in the $y$ – $z$ plane. The tangent cylinder $\mathscr{C}$ (dashed line) intersects the top boundary $z=H$ at $y=H^{2}/2$ , as indicated. The other locations $y=a$ , $y_{\pm }$ , $b$ (labelled red in the online version) are used in our implementation, described in appendix B, of the top boundary condition (B 1e ) based on (3.21).

Our choice of domain dimensions is guided by the analytic results for $z\gg 1$ . For the true unbounded domain, an Ekman layer forms against the $y=0$ boundary, outside which the amplitude of the motion is rather small and tends to zero as $z\rightarrow \infty$ . In reality, the decay to zero is slow in the similarity sublayer on the tangent cylinder $\mathscr{C}$ : $x=0$ , where $\unicode[STIX]{x1D713}=O(z^{-1/12})$ (see (3.12b )). Recall too that the thinning Ekman boundary layer of width $O(z^{-1/2})$ , the thickening similarity sublayer of width $O(z^{1/3})$ and the remaining geostrophic (or mainstream) flow domain are only distinguishable for $z\gg 1$ . Accordingly, a minimal requirement is that the top right-hand corner $(L,H)$ of our $(y,z)$ -rectangle should lie well outside and to the right of the similarity sublayer, i.e., $x(L,H)=L-H^{2}/2\gg O(H^{1/3})$ , so as not to interfere with it.

We discretise the differential system (3.9) on a regular grid by means of finite differences, which for all the $y$ -derivatives utilise a symmetric, second-order scheme, while to approximate the $z$ -derivatives of the streamfunction, $\unicode[STIX]{x1D713}$ (azimuthal velocity, $v$ ) we use a third-order backward (forward) scheme.

We solve the discretised system subject to no-slip, impermeable boundary conditions $\unicode[STIX]{x1D713}=0$ , $\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}y=0$ , $v=1$ on $\mathscr{S}$ : $y=0$ (implemented in appendix B: (B 1a )), matching with the outer flow velocity, which essentially vanishes with $\unicode[STIX]{x1D713}=0$ (equivalently $\int _{0}^{L}w\,\text{d}y=0$ ), $\unicode[STIX]{x2202}\unicode[STIX]{x1D713}/\unicode[STIX]{x2202}y=0$ , $v=0$ on $y=L$ (implementation (B 1b )) and equatorial symmetry: $\unicode[STIX]{x1D713}=0$ (implying $\unicode[STIX]{x2202}v/\unicode[STIX]{x2202}z=0$ ) on $z=0$ (implementation (B 1c )). The correct (or rather practical) choice of the remaining top boundary condition for $v$ is, however, more challenging, because the natural prescription of the homogeneous Dirichlet boundary condition $v=0$ on $z=H$ (correct as $H\rightarrow \infty$ ) needs a very large value of $H$ to give a solution for $z=O(1)$ independent of $H$ , i.e., a domain size far too large for numerical computation. Indeed, the most obvious difficulty is manifest by the similarity sublayer. For, if the value of $v$ on the top boundary, albeit small, is not the value realised by the true unbounded solution, then it will act as the source of a reflected disturbance. This spurious reflection is due to the propagative nature of the advection-like terms on the left-hand side of (3.9a,b ) along geostrophic cylinders, $x\;$ constant, in both the positive and negative $z$ -directions. Unfortunately, even improving the accuracy by replacing $v(x,H)=0$ with the Stewartson similarity solution $v(x,z)=H^{-5/12}V_{0}(x/H^{1/3})$ (see (3.22a )) helps little to eliminate the spurious reflection. A better option is to apply the non-local boundary condition (3.21) that essentially provides the needed non-reflective (i.e., ‘soft’) boundary condition. However, a number of difficulties are immediately apparent, the most obvious being that (3.21) is a mainstream boundary condition which ignores the Ekman boundary layer on $\mathscr{S}$ . It also involves an integral over $-\infty <x<\infty$ , whereas the numerical domain size is finite. Neither of these shortcomings lead to unresolvable difficulties, and the implementation of (3.21) using the iterative procedure described in appendix B appears to be surprisingly effective.

Finally, we emphasise that the top boundary condition for $v_{0}(y,H)$ neither includes nor anticipates the Stewartson similarity form $v\approx z^{-5/12}V_{0}(\unicode[STIX]{x1D6F7})$ , $\unicode[STIX]{x1D6F7}=x(y,z)/z^{1/3}$ (see (3.12a ) with (2.36c )), valid for $z\gg 1$ . However, in the next § 4 we generate the series (4.9a ) involving higher order terms each of similarity form (see (4.12a )) with an arbitrary constant. Consideration of the Ekman boundary layer in the later § 5 determines the constants, which are contained in each of the formulae (4.13a ) and (4.14a ) for $z^{-5/6}V_{1}(\unicode[STIX]{x1D6F7})$ and $z^{-15/12}V_{2}(\unicode[STIX]{x1D6F7})$ that comprise the sum (4.12a ). The numerical solution for $v(y,H)$ and $\unicode[STIX]{x1D713}(y,H)$ is compared with the asymptotic expansions (4.12a,b ) in § 6.

5.1 Zeroth-order problem for $\mathscr{W}_{0}$ , $\mathscr{P}_{0}$

The $n=0$ solution of (5.6a ) subject to $\mathscr{W}_{0}(0)=1$ is

(5.7a,b ) $$\begin{eqnarray}\mathscr{W}_{0}(\unicode[STIX]{x1D701})=E(\unicode[STIX]{x1D701}),\quad \text{where }E(\unicode[STIX]{x1D701})=\exp [-(1+\text{i})\unicode[STIX]{x1D701}],\end{eqnarray}$$

which from (5.6b ) determines

(5.7c,d ) $$\begin{eqnarray}\mathscr{P}_{0}(\unicode[STIX]{x1D701})=Im\left\{-{\textstyle \frac{1}{2}}(1-\text{i})E(\unicode[STIX]{x1D701})\right\}.\end{eqnarray}$$

At $\unicode[STIX]{x1D701}=0$ it gives $\mathscr{P}_{0}(0)=1/2$ , consistent with the boundary condition (5.2b ) and the leading-order term in (4.11b ). It should be emphasised that this result is not automatic but the reason why Stewartson (Reference Stewartson1966) chose his normalisation for his zeroth-order similarity sublayer solution (3.12a,b ).

5.2 First-order problem for $\mathscr{W}_{1}$ , $\mathscr{P}_{1}$

With $\mathscr{W}_{0}(\unicode[STIX]{x1D701})=E(\unicode[STIX]{x1D701})$ the first order ( $n=1$ ) (5.6a ) becomes

(5.8) $$\begin{eqnarray}\mathscr{W}_{1}^{\prime \prime \prime }-2\text{i}\mathscr{W}_{1}^{\prime }=\text{i}\mathscr{W}_{0}^{\prime }=(1-\text{i})\unicode[STIX]{x1D701}E(\unicode[STIX]{x1D701}).\end{eqnarray}$$

The $\unicode[STIX]{x1D701}=0$ boundary condition determined by the $O(z^{-5/2})$ term in (4.10b ) and (5.2a ) is $\mathscr{W}_{1}(0)=-((1-\text{i})/4)$ . Subject to that and $\mathscr{W}_{1}(\unicode[STIX]{x1D701})\rightarrow 0$ as $\unicode[STIX]{x1D701}\rightarrow \infty$ , the solution of (5.8) is

(5.9a ) $$\begin{eqnarray}\mathscr{W}_{1}(\unicode[STIX]{x1D701})=-{\textstyle \frac{1}{4}}\left[(1-\text{i})+{\textstyle \frac{3}{2}}\unicode[STIX]{x1D701}+{\textstyle \frac{1}{2}}(1+\text{i})\unicode[STIX]{x1D701}^{2}\right]E(\unicode[STIX]{x1D701}).\end{eqnarray}$$

Then the $n=1$ integral (5.6b ) determines

(5.9b ) $$\begin{eqnarray}\mathscr{P}_{1}(\unicode[STIX]{x1D701})=Im\left\{{\textstyle \frac{1}{16}}[-9\text{i}+5(1-\text{i})\unicode[STIX]{x1D701}+2\unicode[STIX]{x1D701}^{2}]E(\unicode[STIX]{x1D701})\right\}.\end{eqnarray}$$

At $\unicode[STIX]{x1D701}=0$ , it gives $\mathscr{P}_{1}(0)=-9/16$ , which by (5.2b ) and the $O(z^{-3})$ term in (4.11b ) yields

(5.10) $$\begin{eqnarray}a_{1}=\displaystyle \frac{9}{8\sqrt{2}}.\end{eqnarray}$$

5.3 Second-order problem for $\mathscr{W}_{2}$ , $\mathscr{P}_{2}$

With $\mathscr{W}_{1}(\unicode[STIX]{x1D701})$ given by (5.9a ) the second-order ( $n=2$ ) equation (5.6a ) becomes

(5.11) $$\begin{eqnarray}\displaystyle \mathscr{W}_{2}^{\prime \prime \prime }-2\text{i}\mathscr{W}_{2}^{\prime } & = & \displaystyle \text{i}(\unicode[STIX]{x1D701}\mathscr{W}_{1}^{\prime }-5\mathscr{W}_{1})\nonumber\\ \displaystyle & = & \displaystyle {\textstyle \frac{1}{4}}[5(1+\text{i})+8\text{i}\unicode[STIX]{x1D701}-3(1-\text{i})\unicode[STIX]{x1D701}^{2}-\unicode[STIX]{x1D701}^{3}]E(\unicode[STIX]{x1D701}).\end{eqnarray}$$

The $\unicode[STIX]{x1D701}=0$ boundary condition determined by the $O(z^{-5})$ term in (4.10b ) and (5.2a ) is $\mathscr{W}_{2}(0)=-(3\text{i}/\sqrt{2})a_{1}=-(27/16)\text{i}$ (see (5.10)). The solution of (5.11) that decays to zero as $\unicode[STIX]{x1D701}\rightarrow \infty$ is

(5.12a ) $$\begin{eqnarray}\mathscr{W}_{2}(\unicode[STIX]{x1D701})=-\displaystyle \frac{1}{16}\left[27\text{i}-\displaystyle \frac{145}{16}(1-\text{i})\unicode[STIX]{x1D701}-\displaystyle \frac{89}{8}\unicode[STIX]{x1D701}^{2}-\displaystyle \frac{7}{4}(1+\text{i})\unicode[STIX]{x1D701}^{3}-\displaystyle \frac{\text{i}}{4}\unicode[STIX]{x1D701}^{4}\right]E(\unicode[STIX]{x1D701}).\end{eqnarray}$$

Then the $n=2$ integral (5.6b ) determines

(5.12b ) $$\begin{eqnarray}\mathscr{P}_{1}(\unicode[STIX]{x1D701})=Im\left\{\displaystyle \frac{1}{64}\left[\displaystyle \frac{863}{8}(1+\text{i})+\displaystyle \frac{431}{4}\text{i}\unicode[STIX]{x1D701}-\displaystyle \frac{143}{4}(1-\text{i})\unicode[STIX]{x1D701}^{2}-9\unicode[STIX]{x1D701}^{3}-\displaystyle \frac{1+\text{i}}{2}\unicode[STIX]{x1D701}^{4}\right]E(\unicode[STIX]{x1D701})\right\}.\end{eqnarray}$$

At $\unicode[STIX]{x1D701}=0$ it gives $\mathscr{P}_{2}(0)=-863/512$ , which by (5.2b ) and the $O(z^{-11/2})$ term in (4.11b ) yields

(5.13) $$\begin{eqnarray}a_{2}=\displaystyle \frac{863}{256}-\displaystyle \frac{45}{16}=\displaystyle \frac{143}{256}.\end{eqnarray}$$

6.1 Comparison with full spherical shell DNS

Much of the solution obtained resembles the DNS-results already portrayed on figure 3(a,b). To emphasise the behaviour associated with the equatorial Ekman layer, we provide contour plots of our solution for the azimuthal velocity $v$ and streamfunction $\unicode[STIX]{x1D713}$ on a small $[0,10]\times [0,4]$ box in figure 5(a,b). We also show results for a DNS-solution of the full shell problem in figure 6(a,b) for $\unicode[STIX]{x1D6FC}^{-1}=0.35$ , as in figure 3 but now for $E=10^{-7}$ , with contour values expressed in the units of $v$ and $\unicode[STIX]{x1D713}$ introduced in (3.6).

In view of the extremely small powers of $E$ in our expansions and the approximations based on them leading to our equatorial Ekman layer problem, detailed quantitative agreement between figures 5 and 6 is not to be expected. Nevertheless, the $\unicode[STIX]{x1D713}$ -comparison (figures 5 b, 6 b) is rather good. In contrast, the $v$ -comparison (figures 5 a, 6 a) shows good qualitative agreement well inside the equatorial Ekman layer (say, $0\leqslant z\lessapprox 2$ ), in the sense that the closed contour on the equator is visible in both figures, but detailed comparison for large $z(\gtrapprox 2)$ is poor. This does not mean the asymptotics is incorrect, but rather that higher-order terms ignored in the asymptotics are significant at the finite $E$ used to produce figure 6(a).

The culprit is readily identified from the complete form $\breve{\unicode[STIX]{x1D714}}=\breve{\unicode[STIX]{x1D714}}_{G}+\breve{\unicode[STIX]{x1D714}}_{A}$ (see (1.16a )), in which $\breve{\unicode[STIX]{x1D714}}_{G}=\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}+\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\breve{x}$ (see (1.15a )). In formulating our equatorial Ekman layer problem, we have approximated $\breve{\unicode[STIX]{x1D714}}_{G}$ by $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}$ and ignored the linear shear $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\breve{x}$ . This is justified by (3.5), which provides the error estimate $O(E^{4/35}x)$ ; not particularly small at finite $E$ . Indeed, the ever-increasing size of the linear shear for large $|x|$ is overwhelming. The consequent limitation on the validity of the ensuing asymptotics in the far field is clearly highlighted by (1.16c ), which indicates that $|\breve{\unicode[STIX]{x1D714}}_{A}|\ll |\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\breve{x}|$ in the bulk of the $E^{1/3}$ -layer. Only within the very thin similarity sublayer region $\breve{x}=O((E\breve{z})^{1/3})$ for $\breve{z}\ll O(E^{1/21})$ , whose width shrinks with decreasing $\breve{z}$ , is the ratio $|\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\breve{x}|/|\breve{\unicode[STIX]{x1D714}}_{A}|$ small (see (1.19)), reaching its smallest size $O(E^{4/35})$ at the equatorial Ekman layer $\breve{z}=O(E^{1/5})$ (see (1.20)). This is borne out by the DNS-results illustrated by the figure 7 plots of $v$ versus $x$ on the equatorial plane $z=0$ . The curves at each value of $E$ all exhibit the linear shear $\breve{\unicode[STIX]{x1D714}}_{\mathscr{C}}^{\,\prime }\breve{x}$ at large $x$ . Not surprisingly, therefore, this shear significantly modifies the contours of constant- $v$ on figure 6(a) and, for that matter, figure 3(a).

Figure 7.

Profiles, for various values of $E$ , of the azimuthal velocity $v$ versus $x$ at $z=0$ . The dash-dotted line, corresponds to the equatorial Ekman layer solution portrayed in figure 5(a). The DNS-results are identified by the continuous lines of increasing width: $E=10^{-5}$ (lower), $10^{-6}$ (middle), $10^{-7}$ (upper). The thickest line ( $E=10^{-7}$ ) is determined by the equatorial cross-section through figure 6(a).

6.2 The far-field solution $z\gg 1$

The nature of the far-field asymptotic behaviour as both $x(y,z)=y-z^{2}/2\rightarrow \infty$ and $z\rightarrow \infty$ has already been discussed at length in the earlier sections. Still one feature of the leading order Stewartson (Reference Stewartson1966) similarity solution $v^{sl}=V(x,z;5/4)=z^{-5/12}V_{0}(\unicode[STIX]{x1D6F7})$ , $\unicode[STIX]{x1D713}^{sl}=\unicode[STIX]{x1D6F9}(x,z;5/4)=z^{-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7})$ (see (4.8)) worth stressing here is that, for $|x|\gg z^{1/3}$ (equivalently $|\unicode[STIX]{x1D6F7}|\gg 1$ ), (C 3a,b ) determines

(6.1a,b ) $$\begin{eqnarray}v^{sl}\approx \left\{\begin{array}{@{}l@{}}2^{-11/4}x^{-5/4},\quad \\ 2^{-15/4}(-x)^{-5/4},\quad \end{array}\right.\unicode[STIX]{x1D713}^{sl}\approx \left\{\begin{array}{@{}ll@{}}0\quad & (x>0),\\ -2^{-3/4}(-x)^{-1/4}\quad & (x<0)\end{array}\right.\end{eqnarray}$$

for the mainstream flow outside both the Ekman layer and similarity sublayer. Indeed, even by $z=4$ , there is clear evidence of the tendency towards this $z$ -independent QG-behaviour in the figure 8 plots, which include the $v^{sl}$ and $\unicode[STIX]{x1D713}^{sl}$ asymptotes, lower entries of (6.1), valid for $-x\gg z^{1/3}$ . A reminder of the comment in § 1.3, that the equatorial Ekman layer and its immediate surrounds lie within the $E^{1/3}$ -layer, is pertinent, having two implications. First, the term ‘mainstream’ is used here in the context of the equatorial Ekman layer (§ 3) rather than the Proudman (§ 1.1) problem. Second, it simply happens that the leading-order approximation (6.1) to the $E^{1/3}$ -layer flow, outside the similarity sublayer ( $|\unicode[STIX]{x1D6F7}|\gg 1$ ), is almost geostrophic.

Figure 8.

Profiles of (a) azimuthal velocity $v$ and (b) streamfunction $\unicode[STIX]{x1D713}$ versus $x$ for various values of $z=\text{const.}$  identified by the inset key. Their $v^{sl}$ and $\unicode[STIX]{x1D713}^{sl}$ asymptotes (6.1) for $-x\gg z^{1/3}$ are illustrated by the red dashed line on the left-hand sides.

We display $v$ and $\unicode[STIX]{x1D713}$ versus the self-similar variable $\unicode[STIX]{x1D6F7}=x/z^{1/3}$ at $z=6$ (beneath the top boundary $z=7$ of our numerical box but above the top $z=4$ of figures 5 and 6) by the solid lines in figures 9(a) and 10(a), respectively. They appear to feature a clearly defined Ekman layer at the sphere boundary: $\unicode[STIX]{x1D6F7}=-z^{5/3}/2$ ( ${\approx}-9.9$ for $z=6$ ). For though, following our non-dimensionalisation (3.1)–(3.6), the Ekman number is removed from our problem, it is evident from (3.9) that, when $z\gg 1$ and $z$ -derivatives are neglected, the coefficient $z$ plays the role of the inverse Ekman number $E^{-1}$ . Away from the Ekman layer, motion is largely slow except in the similarity sublayer $\unicode[STIX]{x1D6F7}=O(1)$ , where according to the Stewartson similarity solution (3.12a,b ) we expect $v=O(z^{-5/12})$ , $\unicode[STIX]{x1D713}=O(z^{-1/12})$ , $w=O(z^{-5/12})$ . Indeed the comparison of the numerical solution and (3.12a,b ) in figures 9(a) and 10(a), though not perfect, is encouraging.

Figure 9.

Profiles of azimuthal velocity $v$ at $z=6.0$ versus the self-similar variable $\unicode[STIX]{x1D6F7}$ . The numerical solutions for $v$ in the finite box (solid line) are compared to the asymptotic solutions $v^{sl}$ (see (4.12a )) (dashed lines): (a) first order, $z^{-5/12}V_{0}(\unicode[STIX]{x1D6F7})$ ; (b) plus the second order, $z^{-5/6}V_{1}(\unicode[STIX]{x1D6F7})$ ; (c) plus the third order, $z^{-15/12}V_{2}(\unicode[STIX]{x1D6F7})$ .

Figure 10.

As in figure 9 but for the streamfunction $\unicode[STIX]{x1D713}$ . The asymptotic solutions $\unicode[STIX]{x1D713}^{sl}$ (see (4.12b )) used are (a) the first order, $z^{-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7})$ ; (b) plus the second order, $z^{-1/2}\unicode[STIX]{x1D6F9}_{1}(\unicode[STIX]{x1D6F7})$ ; (c) plus the third order, $z^{-11/12}\unicode[STIX]{x1D6F9}_{2}(\unicode[STIX]{x1D6F7})$ .

Numerical results for the full box height $H=7$ show that the intense Ekman layer visible on figures 9 and 10 bordering the inner sphere $\mathscr{S}$ does indeed thicken $(\propto z^{-1/2})$ with decreasing latitude. From the constant- $v$ contours at the top, $z=4$ , of figure 5(a) we see the normal Ekman layer feature that, on moving away from the boundary, the strong eastward azimuthal velocity associated with the moving boundary is rapidly damped and reverses sign towards the fluid interior. At approximately $z=3$ the Ekman layer starts to merge with the thinning similarity sublayer, width $O(z^{1/3})$ , the small reverse (westward) flow eventually vanishing at $z=2.4$ . For $z\leqslant 1.5$ approximately, the shear and Ekman layers are certainly no longer distinguishable. Figure 5(b) displays the double-eddy structure of the equatorial meridional flow, with a strong, counterclockwise-rotating eddy located roughly inside the tangent cylinder $\mathscr{C}$ and a much weaker, clockwise-rotating eddy outside; features that are already evident from the DNS-results portrayed on figures 3(b) and 6(b).

6.3 The relevance of the Moore and Saffman similarity solutions

With respect to the comparisons of the far-field similarity and numerical solutions, two matters need to be emphasised. Firstly, these similarity solutions were not used in the specifications of the boundary condition at $z=H$ . Rather the non-local Green’s function integral (3.21) was implemented numerically by use of the form (B 1e ). Thus, our finding that the numerical solution compares so well with the asymptotics is most reassuring. Secondly, we must be cautious about how reliable the far-field asymptotics actually are. In deriving the asymptotic series (4.12a,b ) we have only applied a limited set of boundary conditions. Within that class, we show in appendix C by reference to (C 4) that any similarity solution

(6.2) $$\begin{eqnarray}V+\text{i}W=\unicode[STIX]{x1D61E}(x,z;p)\equiv -2^{-1/2}(p-1)2^{-(p-1)}z^{-p/3}\unicode[STIX]{x1D621}(x/z^{1/3};p)\end{eqnarray}$$

(see (4.1a,b )) will satisfy the lowest order homogeneous sphere boundary condition $W$ or $\unicode[STIX]{x1D6F9}=0$ on $x<0$ , $z\approx 0$ (or more precisely as $x/z^{1/3}\downarrow -\infty$ ) whenever $p$ is an integer greater or equal to unity. These are essentially ‘complementary functions’ that can be added in arbitrary proportions to the ‘particular’ Stewartson solution (4.8). To assess relative sizes, we note that the complementary $\unicode[STIX]{x1D6F9}$ -functions are functions of $\unicode[STIX]{x1D6F7}$ multiplied by $z^{-(p-1)/3}$ (see (4.7b )), whereas the Stewartson form $\unicode[STIX]{x1D6F9}(x,z;5/4)=z^{-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7})$ (see (4.8b )) is proportional to $z^{-1/12}$ . Of course, the application of the true sphere boundary condition on $\mathscr{S}$ generates an asymptotic series similar to (4.12) for each integer $p$ .

The important exceptional case $p=1$ requires some care, as the integral on the right-hand side of (4.7b ) is not properly defined. Nevertheless, Moore & Saffman (Reference Moore and Saffman1969) explained that it corresponds to an infinite boundary $z=0$ on which $W=0$ , but with a line source of fluid at $x=0$ . Such a configuration is relevant to many situations, such as the split discs considered by Stewartson (Reference Stewartson1957). Even so, this $p=1$ solution with $\unicode[STIX]{x1D713}=0$ both on $\mathscr{S}$ and as $x\rightarrow \infty$ is excluded by our boundary conditions (3.10b,c ). Since the $p=2$ solution is generated by differentiating the $p=1$ 2D-point source solution with respect to $x$ (see (4.5) with $p=2$ ) the $p=2$ solution corresponds to a 2D-dipole source. Larger integer $p$ likewise produce 2D-multipoles. All these multipole solutions $p\geqslant 2$ must be present to a greater or lesser extent and are generated by the flow in the equatorial Ekman layer region $|(x,z)|=O(1)$ . As that flow is unknown to us from an analytic point of view, so a priori are the magnitudes of the 2D-multipoles. Instead their magnitudes are determined, in principle, by the full numerical solution, a task that we have not attempted, because the discrepancies between the numerical and high-order asymptotic solutions illustrated in figures 9(b,c) and 10(b,c) appear to show apparent convergence without the addition of any multipole contribution. This finding is most intriguing and deserves further comment. The explicit powers of $z$ for $\unicode[STIX]{x1D713}$ -multipoles are $-1/3$ ( $p=2$ ), $-2/3$ ( $p=3$ ), $-1$ ( $p=4$ ), $\ldots$  . Fortunately the modulus of dipole index $1/3$ for $p=2$ is larger than the corresponding $1/12$ for the Stewartson similarity solution $z^{-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7})$ , which must reassuringly dominate as $z\rightarrow \infty$ . This is not the case for the higher-order corrections $z^{-1/2}\unicode[STIX]{x1D6F9}_{1}(\unicode[STIX]{x1D6F7})$ and $z^{-11/12}\unicode[STIX]{x1D6F9}_{2}(\unicode[STIX]{x1D6F7})$ to it with index moduli $1/2$ and $11/12$ , respectively, which interlock the free mode sequence $1/3$ , $2/3$ , $1$ . So, for sufficiently large $z$ , the $p=2$ free mode must dominate over the $\unicode[STIX]{x1D6F9}_{1}$ correction to the Stewartson solution. We therefore conclude that, for our plots at $z=6$ in figures 9 and 10, the magnitudes of the multipoles are relatively small and on increasing $z$ their contributions must continue to decrease in magnitude relative to the dominant Stewartson solution.

The above remarks prompt us to review the status of figures 9(a–c) and 10(a–c). Evidently the lowest-order asymptotic similarity profiles portrayed in (a) are reliable. The marked improvement in the comparison of the asymptotics and the numerics in (b) is a bonus which from a formal mathematical point of view is surprising as we would reasonably expect the $p=2$ dipole contribution to influence the numerical result. Indeed, the apparent continuing small improvement in (c) is remarkable because both the $p=2$ dipole and the $p=3$ quadrupole ought (by any reasonable expectation) to be influencing the numerical results by amounts at least comparable to the small improvement mentioned. As a further technical point, the power $z^{-4/3}$ of the next $p=5$ multipole coincides with the next-order term $q=3$ in our expansion of the Stewartson solution with power $z^{-(p-1)/3}\;(p=5(q+1)/4)$ (see (4.9d ) and (4.12b )). It leads to a resonance generating a similarity mode proportional to $z^{-4/3}\ln z$ , which, in view of our comments and being of such high order, is largely an irrelevance and not considered here.

6.4 Related studies

All the features mentioned about figure 5(a,b) are very similar to those also found numerically by van de Vooren (Reference van de Vooren1993) and illustrated in his figures 4 and 3, respectively, for the simpler rotating disc problem. His geometry differs from ours through the simple expedient of replacing the sphere by a flat disc lying in the equatorial plane $z=0$ (of our sphere) with its outer edge on $x<0$ , thus having its surface at $z=0$ rather than $z=-x^{2}/2$ . Indeed close inspection of the two sets of figures reveals that we replicate his illustrated topological features, including the noticeable kink of the $v=0$ contour, which occurs at approximately $z=1.5$ on figure 5(a). This is not particularly surprising when one recalls that, in his far field, $-x\gg 1$ , the Ekman jump condition determines the mainstream boundary condition $\unicode[STIX]{x1D713}$ const.  on the disc, in contrast to our $\unicode[STIX]{x1D713}$ proportional to $(-x)^{1/4}$ . So whereas our Stewartson similarity sublayer solution in complex form is $\unicode[STIX]{x1D61E}(\unicode[STIX]{x1D6F7},z;p)$ (see (6.2)) with $p=5/4$ , the disc self-similar solution is the 2D-point source version $p=1$ . As the integrands in the two integrals defining each $\unicode[STIX]{x1D621}(\unicode[STIX]{x1D6F7};p)$ (see (4.3)) only differ by a factor $\unicode[STIX]{x1D71B}^{-1/4}$ , we do not expect much qualitative difference in the resulting functions in the far field. With similar far-field features and topologically similar flow structures visible for $x=O(1)$ , $z=O(1)$ , we may speculate that flow topology in the small equatorial Ekman layer responds to the flow in the much larger region outside it, rather than the local geometry of the boundary.

At this point it is helpful to briefly digress to a related detached MHD shear layer in spherical Couette flow studied by Dormy, Jaultc & Soward (Reference Dormy, Jaultc and Soward2002). This layer also exhibits a flow region known as the equatorial Hartmann layer, which in many respects resembles our equatorial Ekman layer. As the equations governing the equatorial Hartmann layer are by comparison simpler, Dormy et al. (Reference Dormy, Jaultc and Soward2002) were able to find an analytic solution based on a method previously developed by Roberts (Reference Roberts1967). Knowledge of this analytic solution enables its asymptotic form at a large distance to be recovered as a similarity solution, cf. our Stewartson solution $v\approx z^{-5/12}V_{0}(\unicode[STIX]{x1D6F7})$ and $\unicode[STIX]{x1D713}\approx z^{-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7})$ (see (3.12a,b )).

The common features do not end there. Inspection of figures 9(a) and 10(a) reveals that the Stewartson similarity solution is offset by a small, but readily perceived, amount from the full numerical boundary layer solution. A similar shift appears in the corresponding MHD problem, where a higher-order approximation of the analytic solution reveals similarity function corrections involving larger inverse powers of the distance, cf. our $z^{-5/6}V_{1}(\unicode[STIX]{x1D6F7})$ and $z^{-1/2}\unicode[STIX]{x1D6F9}_{1}(\unicode[STIX]{x1D6F7})$ in (4.12a,b ). Those extra terms significantly improved comparison with the numerical solution for the full MHD free shear layer. Unfortunately, as we have no analytic solution for the equatorial Ekman layer, we have needed to resort to asymptotic solutions of the governing equations valid for $z\gg 1$ . Interestingly, when the $V_{1}$ and $\unicode[STIX]{x1D6F9}_{1}$ corrections, just mentioned, are included, we too find considerable (indeed remarkable) improvement in the comparison of the asymptotic solution (see figures 9(b) and 10(b)) with the numerical one. Dormy et al. (Reference Dormy, Jaultc and Soward2002) noted that the sum of the zeroth- and first-order terms could be expressed to the same order of accuracy by the zeroth-order solution alone, provided the coordinate origin is shifted by a small amount $\overline{\unicode[STIX]{x1D6FF}}$ (say) from $(x,z)=(0,0)$ to $(\overline{\unicode[STIX]{x1D6FF}},0)$ . Here the origin shift has the classical boundary layer interpretation of a ‘displacement’ thickness. However, even though our figure 5(a,b) clearly display such a similarity sublayer displacement and figures 9(b) and 10(b) hint at a similar interpretation, this is not formally possible. Indeed, no origin shift $\overline{\unicode[STIX]{x1D6FF}}$ effected by redefining $\unicode[STIX]{x1D6F7}$ as $(x-\overline{\unicode[STIX]{x1D6FF}})/z^{1/3}$ can absorb the first-order corrections $z^{-5/6}V_{1}(\unicode[STIX]{x1D6F7})$ and $z^{-1/2}\unicode[STIX]{x1D6F9}_{1}(\unicode[STIX]{x1D6F7})$ into the zeroth-order terms $z^{-5/12}V_{0}(\unicode[STIX]{x1D6F7})$ and $z^{-1/12}\unicode[STIX]{x1D6F9}_{0}(\unicode[STIX]{x1D6F7})$ . The reason for the difference can be traced to the far-field Ekman layer, across which the Ekman suction produces a $\unicode[STIX]{x1D713}\propto (-x)^{1/4}$ (significantly dependent of $x$ ), so that the flow never forgets the $x=0$ location of the origin. In figures 9(c) and 10(c) we include the second-order corrections $z^{-15/12}V_{2}(\unicode[STIX]{x1D6F7})$ and $z^{-11/12}\unicode[STIX]{x1D6F9}_{2}(\unicode[STIX]{x1D6F7})$ in (4.12a,b ).