3.1. The spectrum of $\boldsymbol{{\mathcal{C}}}$

The spectrum of $\boldsymbol{{\mathcal{C}}}$ may be defined in terms of its resolvent operator $(\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}$ on $\boldsymbol{{\mathcal{V}}}^{1}(V)$ . The resolvent set ${\it\Gamma}$ of $\boldsymbol{{\mathcal{C}}}$ is the set of ${\it\lambda}$ in the complex plane $\mathbb{C}$ for which $(\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}$ is a bounded operator. The forced solution of (2.6) is given in terms of the resolvent by $\boldsymbol{v}=(\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}\boldsymbol{f}$ . The solution is well behaved, and in particular no resonance occurs, if ${\it\lambda}$ lies in the resolvent set. The resolvent has been useful in the analysis of stability problems with non-normal operators (see the review by Trefethen Reference Trefethen1997).

Proof. The inner product of (2.6) with $\boldsymbol{v}$ gives

(3.11) $$\begin{eqnarray}(\widehat{{\it\bf\Omega}}\times \boldsymbol{v},\boldsymbol{v})-{\it\lambda}(\boldsymbol{v},\boldsymbol{v})=(\boldsymbol{f},\boldsymbol{v}),\end{eqnarray}$$

since $(\boldsymbol{{\rm\nabla}}p,\boldsymbol{v})=0$ by the divergence theorem, $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0$ in $V$ and $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{v}=0$ on $\partial V$ . The identity

(3.12) $$\begin{eqnarray}(\boldsymbol{v}\times \boldsymbol{v}^{\ast })^{\ast }=-\boldsymbol{v}\times \boldsymbol{v}^{\ast },\end{eqnarray}$$

implies $\text{Re}(\widehat{{\it\bf\Omega}}\times \boldsymbol{v},\boldsymbol{v})=0$ and thus, if ${\it\lambda}_{r}$ , ${\it\lambda}_{i}$ are the real and imaginary parts of ${\it\lambda}$ ,

(3.13a,b ) $$\begin{eqnarray}{\it\lambda}_{r}\Vert \boldsymbol{v}\Vert ^{2}=-\text{Re}(\boldsymbol{f},\boldsymbol{v}),\quad {\it\lambda}_{i}\Vert \boldsymbol{v}\Vert ^{2}=-\text{i}(\widehat{{\it\bf\Omega}}\times \boldsymbol{v},\boldsymbol{v})-\text{Im}(\boldsymbol{f},\boldsymbol{v}).\end{eqnarray}$$

By the Cauchy–Schwartz inequality $|(\boldsymbol{f},\boldsymbol{v})|\leqslant \Vert \boldsymbol{f}\Vert \,\Vert \boldsymbol{v}\Vert$ and $|(\widehat{{\it\bf\Omega}}\times \boldsymbol{v},\boldsymbol{v})|\leqslant \Vert \boldsymbol{v}\Vert ^{2}$ . Hence

(3.14a,b ) $$\begin{eqnarray}\Vert \boldsymbol{v}\Vert \leqslant |{\it\lambda}_{r}|^{-1}\,\Vert \boldsymbol{f}\Vert ;\quad \Vert \boldsymbol{v}\Vert \leqslant (|{\it\lambda}_{i}|-1)^{-1}\,\Vert \boldsymbol{f}\Vert ,\quad \text{if }|{\it\lambda}_{i}|>1.\end{eqnarray}$$

Since $\boldsymbol{v}=(\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}\boldsymbol{f}$ , $\Vert (\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}\Vert \leqslant |{\it\lambda}_{r}|^{-1}$ and $\Vert (\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}\Vert \leqslant (|{\it\lambda}_{i}|-1)^{-1}$ if $|{\it\lambda}_{i}|>1$ . Hence the resolvent set contains all ${\it\lambda}$ such that ${\it\lambda}_{r}\neq 0$ or $|{\it\lambda}_{i}|>1$ .

The spectrum of $\boldsymbol{{\mathcal{C}}}$ is the complement $\mathbb{C}\setminus {\it\Gamma}$ of the resolvent set ${\it\Gamma}$ in the complex plane $\mathbb{C}$ . Theorem 3.1 shows that the spectrum lies in $[-\text{i},\text{i}]$ on the imaginary axis. In general, the spectrum is the disjoint union of the point (or discrete) spectrum ${\it\sigma}_{p}(\boldsymbol{{\mathcal{C}}})$ , the continuous spectrum ${\it\sigma}_{c}(\boldsymbol{{\mathcal{C}}})$ and the residual spectrum ${\it\sigma}_{r}(\boldsymbol{{\mathcal{C}}})$ . The discrete spectrum ${\it\sigma}_{p}(\boldsymbol{{\mathcal{C}}})$ consists of the eigenvalues of $\boldsymbol{{\mathcal{C}}}$ , i.e. the values of ${\it\lambda}\in \mathbb{C}$ for which $\boldsymbol{{\mathcal{C}}}\boldsymbol{v}={\it\lambda}\boldsymbol{v}$ for some non-zero $\boldsymbol{v}\in \boldsymbol{{\mathcal{V}}}^{1}(V)$ . These discrete modes, which we will refer to simply as Coriolis modes, are the free modes ( $\boldsymbol{f}=\mathbf{0}$ ) of $\boldsymbol{{\mathcal{C}}}$ and can be physically interpreted as inertial modes or geostrophic modes, or Rossby modes, of frequency ${\it\omega}=-\text{i}{\it\lambda}$ . As indicated in the introduction we regard the Coriolis modes of zero and non-zero frequency as the geostrophic modes and inertial spatial modes respectively. There is a further discussion of ${\it\sigma}_{c}(\boldsymbol{{\mathcal{C}}})$ and ${\it\sigma}_{r}(\boldsymbol{{\mathcal{C}}})$ in § 9. The following theorem is important for the spectrum of $\boldsymbol{{\mathcal{C}}}$ , but particularly for ${\it\sigma}_{p}(\boldsymbol{{\mathcal{C}}})$ .

The values ${\it\lambda}=\pm \text{i}$ are not eigenvalues of $\boldsymbol{{\mathcal{C}}}$ , i.e.  $\pm \text{i}\notin {\it\sigma}_{p}(\boldsymbol{{\mathcal{C}}})$ . Physically, there are no Coriolis modes of frequency ${\it\omega}=\pm 1$ , but if the fluid occupies all space $E^{3}$ , modes with frequency ${\it\omega}=\pm 1$ are possible. Equation (2.6) with ${\it\lambda}=\text{i}$ and $\boldsymbol{f}=\mathbf{0}$ implies $\widehat{{\it\bf\Omega}}\times \boldsymbol{v}=\text{i}\boldsymbol{v}-\boldsymbol{{\rm\nabla}}p$ and hence $(\widehat{{\it\bf\Omega}}\times \boldsymbol{v},\widehat{{\it\bf\Omega}}\times \boldsymbol{v})=(\text{i}\boldsymbol{v}-\boldsymbol{{\rm\nabla}}p,\text{i}\boldsymbol{v}-\boldsymbol{{\rm\nabla}}p)$ . Simplifying using $(\boldsymbol{v},\boldsymbol{{\rm\nabla}}p)=(\boldsymbol{{\rm\nabla}}p,\boldsymbol{v})^{\ast }=0$ yields $\Vert \boldsymbol{{\rm\nabla}}p\Vert ^{2}+\Vert \widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v}\Vert ^{2}=0$ . Thus $p$ is constant and $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v}=0$ . Equation (2.6) reduces to $\pm \text{i}\boldsymbol{v}+\widehat{{\it\bf\Omega}}\times \boldsymbol{v}=\mathbf{0}$ . From $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0$ , it follows that $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\times \boldsymbol{v}=0$ . Hence, there exists a potential ${\it\varphi}(x,y,z)$ such that $\boldsymbol{v}=\boldsymbol{{\rm\nabla}}_{\bot }{\it\varphi}$ , where $\boldsymbol{{\rm\nabla}}_{\bot }$ is the gradient perpendicular to $\widehat{{\it\bf\Omega}}$ and ${\rm\nabla}_{\bot }^{2}{\it\varphi}=0$ . Multiplying the last equation by ${\it\varphi}^{\ast }$ and integrating by parts over a plane section of $V$ with $z$ constant yields $\int _{z=\text{const}}|\boldsymbol{v}|^{2}\text{d}A=0$ , since $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{v}=0$ on the boundary $\partial V$ . It follows that $\boldsymbol{v}=\mathbf{0}$ everywhere in $V$ . The case ${\it\lambda}=-\text{i}$ is similar.

3.2. Eigenvalues and orthogonality of Coriolis modes

Using the symmetry of $-\text{i}\boldsymbol{{\mathcal{C}}}$ from Theorem 3.2, we now show the eigenvalues of $\boldsymbol{{\mathcal{C}}}$ are imaginary, i.e. the frequencies ${\it\omega}=-\text{i}{\it\lambda}$ are real, and that Coriolis modes with different eigenvalues are orthogonal. We also use the boundedness of $\boldsymbol{{\mathcal{C}}}$ to show $|{\it\omega}|=|{\it\lambda}|\leqslant 1$ . These results are consistent with the restriction on the spectrum found in Theorem 3.1. We denote the pressure of the Coriolis modes by $P$ .

The orthogonality property of Theorem 3.3 does not apply to modes of the same eigenvalue. Thus, for any eigenvalue with an eigenspace of dimension greater than one, as is true for the geostrophic modes in a sphere, an orthogonal basis must be constructed, for example using the Gram–Schmidt method. For this purpose it is useful to express the inner product of two inertial modes $(\boldsymbol{v}_{1},P_{1},{\it\lambda}_{1}=\text{i}{\it\omega}_{1})$ and $(\boldsymbol{v}_{2},P_{2},{\it\lambda}_{2}=\text{i}{\it\omega}_{2})$ in terms of the pressures $P_{1}$ and $P_{2}$ . Equation (2.2) with $\boldsymbol{f}=\mathbf{0}$ or the dot product of (2.6) with $\widehat{{\it\bf\Omega}}$ if ${\it\omega}^{2}\neq 1$ gives the useful result

(3.15) $$\begin{eqnarray}\text{i}{\it\omega}\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v}=\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P.\end{eqnarray}$$

Using (2.8) with $\boldsymbol{f}=\mathbf{0}$ , (3.15), integrating over $V$ and using the boundary condition (1.6) yields, if ${\it\omega}_{1}{\it\omega}_{2}\neq 0$ ,

(3.16) $$\begin{eqnarray}(\boldsymbol{v}_{1},\boldsymbol{v}_{2})=(1-{\it\omega}_{1}{\it\omega}_{2})^{-1}\int _{V}\left\{\boldsymbol{{\rm\nabla}}P_{1}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P_{2}^{\ast }+\frac{1}{{\it\omega}_{1}{\it\omega}_{2}}(\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P_{1})(\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P_{2}^{\ast })\right\}\!\text{d}V.\end{eqnarray}$$

Formula (3.16) fails if one or both modes are geostrophic, since then ${\it\omega}_{1}{\it\omega}_{2}=0$ . The formula for the inner product of two geostrophic modes is derived in § 3.3 below.

3.3. Inertial modes

In some simple geometries inertial modes can be constructed analytically using the pressure. With $\boldsymbol{f}=\mathbf{0}$ , (2.8) reduces to

(3.17) $$\begin{eqnarray}\text{i}{\it\omega}(1-{\it\omega}^{2})\boldsymbol{v}=-{\it\omega}^{2}\boldsymbol{{\rm\nabla}}P+(\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P)\widehat{{\it\bf\Omega}}+\text{i}{\it\omega}\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P\quad \text{in }V\end{eqnarray}$$

and

(3.18) $$\begin{eqnarray}-{\it\omega}^{2}\partial _{n}P+(\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P)\boldsymbol{n}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}+\text{i}{\it\omega}\boldsymbol{n}\times \widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P=0\quad \text{in }\partial V.\end{eqnarray}$$

Poincaré’s equation (1.7) must be solved for $P$ subject to the boundary condition (3.18) and the velocity then determined from (3.17). The construction for a sphere (Bryan Reference Bryan1889) is outlined in § 4 during the enumeration of the spherical inertial modes.

3.4. Geostrophic modes

The Coriolis modes of zero frequency, i.e. the geostrophic modes, require special treatment. Their properties can occasionally be found from those of inertial modes by taking the limit ${\it\omega}\rightarrow 0$ . Thus the limit as ${\it\omega}\rightarrow 0$ of (3.15) yields $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P=0$ , i.e. $P=P(\boldsymbol{r}-\boldsymbol{r}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}\widehat{{\it\bf\Omega}})$ , so these modes have no vertical structure.

The velocity $\boldsymbol{v}$ can be found by substituting (3.15) into (3.17), dividing by $\text{i}{\it\omega}$ and then taking the limit ${\it\omega}\rightarrow 0$ to obtain $\boldsymbol{v}=\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P+(\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v})\widehat{{\it\bf\Omega}}$ , or by taking the cross product of (2.6) with $\widehat{{\it\bf\Omega}}$ . The axial component $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v}$ of the flow is still undetermined. Letting ${\it\omega}_{1},{\it\omega}_{2}\rightarrow 0$ in the inner product formula (3.16) using (3.15) leads to the undetermined axial components of $\boldsymbol{v}_{1}$ , $\boldsymbol{v}_{2}$ . We derive an expression for $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v}$ in terms of the pressure and the geometry following Greenspan (Reference Greenspan1968). Let $(x,y,z)$ be Cartesian coordinates with the $z$ -axis aligned along $\widehat{{\it\bf\Omega}}$ . We assume that any line parallel to $\widehat{{\it\bf\Omega}}$ intersects $\partial V$ in the top $z=z_{t}(x,y)$ and bottom $z=z_{b}(x,y)$ boundaries, with outward normals $\boldsymbol{n}_{t}:=\boldsymbol{{\rm\nabla}}(z-z_{t})$ and $\boldsymbol{n}_{b}:=-\boldsymbol{{\rm\nabla}}(z-z_{b})$ . Clearly $\boldsymbol{n}_{t}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}=1$ and $\boldsymbol{n}_{b}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}=-1$ . Imposing $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{v}=0$ on $\partial V$ yields $-\boldsymbol{n}_{t}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P=\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v}=\boldsymbol{n}_{b}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P$ or $\boldsymbol{{\rm\nabla}}z_{t}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P=\boldsymbol{{\rm\nabla}}z_{b}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P$ , i.e. $\boldsymbol{{\rm\nabla}}h\times \boldsymbol{{\rm\nabla}}z\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P=0$ , where $h(x,y):=z_{t}-z_{b}$ is the height of the boundary. Hence, $P=P(h,z)$ , if $\boldsymbol{{\rm\nabla}}h\neq \mathbf{0}$ . But $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P=0$ implies $P=P(h)$ and so $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{v}=\boldsymbol{{\rm\nabla}}z_{t}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P=\boldsymbol{{\rm\nabla}}z_{t}\times \boldsymbol{{\rm\nabla}}z_{b}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}\,P^{\prime }(h)$ . So the horizontal structure of geostrophic flows is restricted to one degree of freedom. Finally, using $\boldsymbol{{\rm\nabla}}z_{t}\times \boldsymbol{{\rm\nabla}}z_{b}=-\boldsymbol{{\rm\nabla}}((z_{t}+z_{b})/2)\times \boldsymbol{{\rm\nabla}}h$ ,

(3.19) $$\begin{eqnarray}\boldsymbol{v}=[\widehat{{\it\bf\Omega}}-\boldsymbol{{\rm\nabla}}{\textstyle \frac{1}{2}}(z_{t}+z_{b})]\times \boldsymbol{{\rm\nabla}}P=P^{\prime }(h)\,\boldsymbol{t},\end{eqnarray}$$

where $\boldsymbol{t}:=\boldsymbol{{\rm\nabla}}[z-(z_{t}+z_{b})/2]\times \boldsymbol{{\rm\nabla}}h$ is tangent to the streamlines of $\boldsymbol{v}$ . The inner product of two geostrophic modes with pressures $P_{1}$ and $P_{2}$ is

(3.20) $$\begin{eqnarray}(\boldsymbol{v}_{1},\boldsymbol{v}_{2})=\int _{V}\left[\widehat{{\it\bf\Omega}}-\boldsymbol{{\rm\nabla}}{\textstyle \frac{1}{2}}(z_{t}+z_{b})\right]\times \boldsymbol{{\rm\nabla}}P_{1}\boldsymbol{\cdot }\left[\widehat{{\it\bf\Omega}}-\boldsymbol{{\rm\nabla}}{\textstyle \frac{1}{2}}(z_{t}+z_{b})\right]\times \boldsymbol{{\rm\nabla}}P_{2}^{\ast }\text{d}V.\end{eqnarray}$$

If $V$ is symmetrical about some $z$ -plane, then $z_{t}+z_{b}$ is constant and the inner product reduces to

(3.21) $$\begin{eqnarray}(\boldsymbol{v}_{1},\boldsymbol{v}_{2})=\int _{V}\boldsymbol{{\rm\nabla}}P_{1}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P_{2}^{\ast }\text{d}V.\end{eqnarray}$$

An orthogonal set of geostrophic modes can be constructed by applying Gram–Schmidt orthogonalisation to a linearly independent set of geostrophic pressure modes using the inner product (3.21).

The spectrum of $\boldsymbol{{\mathcal{C}}}$ depends only on the geometry of $V$ and $\widehat{{\it\bf\Omega}}$ , not on the rotation rate nor on any flow parameters. The simplest dependence is between closed curves and geostrophic modes. The intersection of the level surfaces of $z-(z_{t}+z_{b})/2$ and $h$ , which depend only on $V$ and $\widehat{{\it\bf\Omega}}$ , are called geostrophic curves, or contours if they lie on the boundary $z=z_{t}$ or $z=z_{b}$ . The streamlines of the geostrophic flow (3.19) are geostrophic curves. If $G$ is a geostrophic curve,

(3.22) $$\begin{eqnarray}\oint _{G}\boldsymbol{v}\boldsymbol{\cdot }\text{d}\boldsymbol{l}=P^{\prime }(h)\oint _{G}|\boldsymbol{t}|\text{d}l\neq 0.\end{eqnarray}$$

There is no non-zero geostrophic flow along a curve which terminates on the boundary with $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{t}\neq 0$ . If all curves are of this form there is no geostrophic flow, e.g. as in a sliced cylinder rotating about its axis, and therefore no Coriolis mode of frequency zero. Of course, curves need not terminate on the boundary, as in the case of a sliced cone rotating about its axis.

Geostrophic and inertial modes can be separated by frequency or orthogonality, but also geometrically by geostrophic cylinder average. The geostrophic contours that have the same boundary projections along $\widehat{{\it\bf\Omega}}$ generate a geostrophic cylinder of constant height, which need not be of circular cross-section. The geostrophic cylinder average of a vector field $\boldsymbol{F}$ over a geostrophic cylinder $C$ is

(3.23) $$\begin{eqnarray}\langle \boldsymbol{F}\rangle _{C}:=\frac{1}{|C|}\int _{C}\boldsymbol{F}\boldsymbol{\cdot }\widehat{\boldsymbol{t}}\text{d}S,\end{eqnarray}$$

where $|C|$ is the surface area of $C$ and $\widehat{\boldsymbol{t}}$ is the unit tangent field along the geostrophic curves generating $C$ . We show that, if $C$ divides the fluid into two disjoint components $V=V_{1}\cup V_{2}$ , then

(3.24) $$\begin{eqnarray}{\it\omega}\langle \boldsymbol{v}\rangle _{C}=0,\end{eqnarray}$$

and hence for the inertial modes ( ${\it\omega}\neq 0$ ) the geostrophic cylinder average of $\boldsymbol{v}$ vanishes: $\langle \boldsymbol{v}\rangle _{C}=0$ . The proof follows from the geostrophic cylinder average of (2.6), $-\text{i}{\it\omega}\langle \boldsymbol{v}\rangle _{C}+\langle \widehat{{\it\bf\Omega}}\times \boldsymbol{v}\rangle _{C}=-\langle \boldsymbol{{\rm\nabla}}p\rangle _{C}$ and

(3.25) $$\begin{eqnarray}0=\int _{V_{i}}\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}\text{d}V=\int _{C}\boldsymbol{v}\boldsymbol{\cdot }\text{d}\boldsymbol{S}=\iint _{C}\boldsymbol{v}\boldsymbol{\cdot }\text{d}\boldsymbol{l}\times \widehat{{\it\bf\Omega}}\text{d}z=|C|\langle \widehat{{\it\bf\Omega}}\times \boldsymbol{v}\rangle _{C},\end{eqnarray}$$

using $\boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0$ in $V_{i}$ and $\boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{v}=0$ on the boundary. Also $\langle \boldsymbol{{\rm\nabla}}p\rangle _{C}=0$ , since $\oint _{G}\boldsymbol{{\rm\nabla}}p\boldsymbol{\cdot }\text{d}\boldsymbol{l}=0$ for any geostrophic contour $G$ on $C$ . So, unlike geostrophic modes, inertial modes have vertical structure $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P\neq 0$ and hence an axial flow along the rotation axis by (3.15). The result (3.24) is also known as the mean circulation theorem (Greenspan Reference Greenspan1968), that the circulation of a depth-averaged inertial (geostrophic) mode around a closed geostrophic contour is zero (non-zero).

3.5. Angular momentum of Coriolis modes

Inertial and geostrophic modes also differ in angular momentum effects. Geostrophic modes transmit the component of the pressure torque along the rotation axis between the ends of a geostrophic cylinder or annulus. Inertial modes lose torque to the angular momentum of the fluid. This mechanism of torque transfer is significant if the region is periphractic, i.e. the outer fluid boundary encloses one or more inner fluid boundaries so not every closed surface in $V$ can be continuously shrunk to a point without leaving $V$ (Maxwell Reference Maxwell1891, pp. 18, 24, 25). For then the geostrophic modes transmit the axial component of the pressure torque from the outer boundary to the inner boundary (Hide Reference Hide1995). The results follow from the identity

(3.26) $$\begin{eqnarray}\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\int _{\partial C}\boldsymbol{r}\times P\text{d}\boldsymbol{S}=\text{i}{\it\omega}\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\unicode[STIX]{x1D647}_{C}-\frac{\text{d}(\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\unicode[STIX]{x1D644}_{C}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}})}{\text{d}t},\end{eqnarray}$$

where $\unicode[STIX]{x1D644}_{C}$ and $\unicode[STIX]{x1D647}_{C}$ are the inertia tensor of $C$ and the angular momentum of $C$ in the rotating frame, noting that $\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\unicode[STIX]{x1D644}_{C}\boldsymbol{\cdot }\widehat{{\it\bf\Omega}}$ is constant if $C$ is a geostrophic cylinder or annulus. In dimensional form, $\unicode[STIX]{x1D644}_{C}:=\int _{C}{\it\rho}(r^{2}\unicode[STIX]{x1D644}-\boldsymbol{r}\boldsymbol{r})\text{d}V$ and $\unicode[STIX]{x1D647}_{C}:=\int _{C}\boldsymbol{r}\times {\it\rho}\boldsymbol{v}\text{d}V$ . Since $\widehat{{\it\bf\Omega}}$ is constant the last term in (3.26) vanishes, giving the result. For a sphere or spherical shell the result is trivial. Identity (3.26) is established by combining the dimensional identities

(3.27) $$\begin{eqnarray}\frac{\text{d}\unicode[STIX]{x1D644}_{C}}{\text{d}t}=\int _{C}{\it\rho}D_{t}(r^{2}\unicode[STIX]{x1D644}-\boldsymbol{r}\boldsymbol{r})\text{d}V+\int _{\partial C}{\it\rho}(r^{2}\unicode[STIX]{x1D644}-\boldsymbol{r}\boldsymbol{r})(\boldsymbol{v}_{\partial C}-\boldsymbol{v})\boldsymbol{\cdot }\text{d}\boldsymbol{S},\end{eqnarray}$$

$D_{t}(r^{2}\unicode[STIX]{x1D644}-\boldsymbol{r}\boldsymbol{r})=2\boldsymbol{r}\boldsymbol{\cdot }\boldsymbol{v}\unicode[STIX]{x1D644}-\boldsymbol{r}\boldsymbol{v}-\boldsymbol{v}\boldsymbol{r}$ and $\boldsymbol{r}\times (\dot{{\it\bf\Omega}}\times \boldsymbol{r})=(r^{2}\unicode[STIX]{x1D644}-\boldsymbol{r}\boldsymbol{r})\boldsymbol{\cdot }\dot{{\it\bf\Omega}}$ to yield the following identity for the moments of the Coriolis and Poincaré forces,

(3.28) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{C}\boldsymbol{r}\times ({\it\rho}2{\it\bf\Omega}\times \boldsymbol{v})\text{d}V+\int _{C}\boldsymbol{r}\times ({\it\rho}\dot{{\it\bf\Omega}}\times \boldsymbol{r})\text{d}V\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\int _{\partial C}{\it\bf\Omega}\boldsymbol{\cdot }{\it\rho}(r^{2}\unicode[STIX]{x1D644}-\boldsymbol{r}\boldsymbol{r})(\boldsymbol{v}_{\partial C}-\boldsymbol{v})\boldsymbol{\cdot }\text{d}\boldsymbol{S}=\frac{\text{d}({\it\bf\Omega}\boldsymbol{\cdot }\unicode[STIX]{x1D644}_{C})}{\text{d}t}+{\it\bf\Omega}\times \unicode[STIX]{x1D647}_{C}.\end{eqnarray}$$

If $\dot{{\it\bf\Omega}}=\mathbf{0}$ and $C$ moves so that $(\boldsymbol{v}_{\partial C}-\boldsymbol{v})\boldsymbol{\cdot }\boldsymbol{n}=0$ , which implies the angular momentum flux across the boundary $\partial C$ due to ${\it\bf\Omega}$ vanishes, the (dimensionless) identity reduces to

(3.29) $$\begin{eqnarray}\frac{\text{d}(\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\unicode[STIX]{x1D644}_{C})}{\text{d}t}+\widehat{{\it\bf\Omega}}\times \unicode[STIX]{x1D647}_{C}=\int _{C}\boldsymbol{r}\times (\widehat{{\it\bf\Omega}}\times \boldsymbol{v})\text{d}V=\text{i}{\it\omega}\int _{C}\boldsymbol{r}\times \boldsymbol{v}\text{d}V-\int _{\partial C}\boldsymbol{r}\times P\text{d}\boldsymbol{S}.\end{eqnarray}$$

The component along the rotation axis gives (3.26).

3.6. Inversion of $\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644}$ and resonance

Applications usually require, in principle, calculation of the resolvent $(\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}$ and forced solutions $\boldsymbol{v}=(\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}\boldsymbol{f}$ , where $\boldsymbol{f}$ satisfies (1.16). For this the velocity can be eliminated as in (2.7) and (2.9), which can be solved for $p$ . Equation (2.8) then gives $\boldsymbol{v}$ . Alternatively, elimination of $p$ from (1.19) yields the vorticity equation,

(3.30) $$\begin{eqnarray}\partial _{t}\boldsymbol{{\rm\nabla}}\times \boldsymbol{v}-\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{v}=\boldsymbol{{\rm\nabla}}\times \boldsymbol{f},\end{eqnarray}$$

or from (2.6) the modal form,

(3.31) $$\begin{eqnarray}\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{v}+{\it\lambda}\boldsymbol{{\rm\nabla}}\times \boldsymbol{v}=-\boldsymbol{{\rm\nabla}}\times \boldsymbol{f}.\end{eqnarray}$$

Uncurling recovers (1.19) or (2.6). In principle, if ${\it\lambda}$ is in the resolvent set ${\it\Gamma}$ of $\boldsymbol{{\mathcal{C}}}$ , this can be done for all $\boldsymbol{f}\in \boldsymbol{{\mathcal{V}}}^{1}(V)$ . However, if ${\it\lambda}$ is in the spectrum $\mathbb{C}\setminus {\it\Gamma}$ , $\boldsymbol{f}$ may have to satisfy non-resonance conditions.

We consider only the point spectrum ${\it\lambda}\in {\it\sigma}_{p}(\boldsymbol{{\mathcal{C}}})$ . If ${\it\lambda}=0$ , (1.19) reduces to the steady form $\boldsymbol{{\mathcal{C}}}\boldsymbol{v}=\boldsymbol{f}$ and the resolvent is then the inverse of $\boldsymbol{{\mathcal{C}}}$ . If a geostrophic mode exists, then $0\in {\it\sigma}_{p}(\boldsymbol{{\mathcal{C}}})$ and $\boldsymbol{{\mathcal{C}}}$ is not invertible. Taylor (Reference Taylor1963) considered this important case for the dynamo problem in the rapid-rotation limit $\mathit{E},\mathit{Ro}\rightarrow 0$ . In this case the force $\boldsymbol{f}$ is the (projected) Lorentz force $\boldsymbol{f}_{L}$ and the force balance is magnetostrophic. We state a general existence theorem, Theorem 3.4, for solutions of $\boldsymbol{{\mathcal{C}}}\boldsymbol{v}=\boldsymbol{f}$ ; a constructive proof is given in appendix A. We assume the region $V$ is axisymmetric with $\widehat{{\it\bf\Omega}}$ oriented along the symmetry $z$ -axis. We assume the top and bottom boundaries, $z=z_{t}(s)$ and $z=z_{b}(s)$ , where the cylindrical radius $s$ satisfies $s_{0}\leqslant s\leqslant s_{1}$ , are smooth. For a given $\widehat{{\it\bf\Omega}}$ , $V$ is geostrophically guided if there is exactly one geostrophic contour through each point on the boundary. We assume $V$ is geostrophically guided: $\boldsymbol{{\rm\nabla}}(z_{t}-z_{b})\neq \mathbf{0}$ except at isolated values of $s$ .

Theorem 3.4. A necessary and sufficient condition for the existence of a single-valued solution $\boldsymbol{v}\in \boldsymbol{{\mathcal{V}}}^{1}(V)$ of

(3.32a−c ) $$\begin{eqnarray}\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}\boldsymbol{v}=-\boldsymbol{{\rm\nabla}}\times \boldsymbol{f},\quad \boldsymbol{{\rm\nabla}}\boldsymbol{\cdot }\boldsymbol{v}=0\quad \text{in }V,\quad \boldsymbol{n}\boldsymbol{\cdot }\boldsymbol{v}=g\quad \text{on }\partial V,\end{eqnarray}$$

where $\boldsymbol{f}$ and $g$ are prescribed subject to the consistency condition $\int _{\partial V}g\,\text{d}S=0$ , is

(3.33) $$\begin{eqnarray}|C(s)|\langle \boldsymbol{f}\rangle _{C(s)}+\int _{S(s)}g\text{d}S=0,\end{eqnarray}$$

for all geostrophic cylinders $C(s)$ , where $S(s)$ are the caps of $C(s)$ . The solution is only determined up to an arbitrary additive function of $s$ to $v_{{\it\phi}}$ . If the further condition,

(3.34) $$\begin{eqnarray}\langle \boldsymbol{v}\rangle _{C(s)}=0\quad \text{for all }s_{0}\leqslant s\leqslant s_{1},\end{eqnarray}$$

is imposed, then the solution is unique.

The function $g$ allows for Ekman suction or a moving boundary. Imposing $g=0$ gives the result for the Coriolis operator.

Corollary 3.1. A necessary and sufficient condition for the existence of a single-valued solution $\boldsymbol{v}\in \boldsymbol{{\mathcal{V}}}^{1}(V)$ of $\boldsymbol{{\mathcal{C}}}\boldsymbol{v}=\boldsymbol{f}$ is that all geostrophic cylinder averages of $\boldsymbol{f}$ vanish,

(3.35) $$\begin{eqnarray}\langle \boldsymbol{f}\rangle _{C(s)}=0.\end{eqnarray}$$

The non-resonance or solvability condition (3.35) is well known in dynamo theory as Taylor’s (Reference Taylor1963) condition for the magnetostrophic momentum equation. The theorem fails if $z_{t}(s)$ or $z_{b}(s)$ are not smooth, e.g. in a spherical shell. The proof produces a smooth solution $\boldsymbol{v}$ over each interval of $s$ for which $z_{t}(s)$ and $z_{b}(s)$ are smooth but the smooth pieces of $\boldsymbol{v}$ do not fit together smoothly. This remains an outstanding problem (Hollerbach & Proctor Reference Hollerbach, Proctor, Matthews, Proctor and Rucklidge1993; Hollerbach Reference Hollerbach1994a ,Reference Hollerbach b ; Kleeorin et al. Reference Kleeorin, Rogachevskii, Ruzmaikin, Soward and Starchenko1997; Dormy, Cardin & Jault Reference Dormy, Cardin and Jault1998; Soward & Hollerbach Reference Soward and Hollerbach2000; Livermore & Hollerbach Reference Livermore and Hollerbach2012).

The invertibility of $\boldsymbol{{\mathcal{C}}}$ can also be treated spectrally:

Theorem 3.5. If $\boldsymbol{f}$ has an expansion in discrete Coriolis modes, $\boldsymbol{f}=\sum _{{\it\alpha}}f_{{\it\alpha}}\boldsymbol{v}_{{\it\alpha}}$ with $f_{{\it\alpha}}=(\boldsymbol{f},\boldsymbol{v}_{{\it\alpha}})$ , then $\boldsymbol{{\mathcal{C}}}\boldsymbol{v}=\boldsymbol{f}$ has the solution $\boldsymbol{v}=\sum _{{\it\alpha}}\text{i}f_{{\it\alpha}}\boldsymbol{v}_{{\it\alpha}}/{\it\omega}_{{\it\alpha}}$ , if the non-resonance or solvability conditions,

(3.36) $$\begin{eqnarray}(\boldsymbol{f},\boldsymbol{v}_{{\it\beta}})=0,\end{eqnarray}$$

hold for all ${\it\omega}_{{\it\beta}}=0$ , i.e. for all geostrophic modes, and $\sum _{{\it\alpha}}|f_{{\it\alpha}}/{\it\omega}_{{\it\alpha}}|^{2}\Vert \boldsymbol{v}_{{\it\alpha}}\Vert ^{2}<\infty$ .

The proof is elementary. The solvability conditions in Theorem 3.5 are equivalent to condition (3.34) in Theorem 3.4, but may be more tractable. If the Coriolis modes are complete, so that any $\boldsymbol{f}$ can be expanded in discrete Coriolis modes, then the spectral inversion in Theorem 3.5 works for any $\boldsymbol{f}$ which satisfies the solvability conditions. This highlights the usefulness and importance of completeness for the Coriolis modes, which we prove in § 8 if $V$ is a sphere. For the general forced problem, $(\boldsymbol{{\mathcal{C}}}-{\it\lambda}\unicode[STIX]{x1D644})^{-1}\boldsymbol{f}=\sum _{{\it\alpha}}\text{i}({\it\omega}-{\it\omega}_{{\it\alpha}})^{-1}f_{{\it\alpha}}\,\boldsymbol{v}_{{\it\alpha}}$ , where ${\it\omega}=-\text{i}{\it\lambda}$ , if $(\boldsymbol{f},\boldsymbol{v}_{{\it\beta}})=0$ for all ${\it\beta}$ for which ${\it\omega}={\it\omega}_{{\it\beta}}$ .

4.1. Inertial pressure modes

Poincaré’s equation (2.7) in cylindrical polar coordinates $(s,{\it\phi},z)$ with $\boldsymbol{f}=\mathbf{0}$ is

(4.1a,b ) $$\begin{eqnarray}s^{-1}\partial _{s}(s\partial _{s}P)+s^{-2}\partial _{{\it\phi}}^{2}P-({\it\omega}^{-2}-1)\partial _{z}^{2}P=0,\quad s^{2}+z^{2}<1\end{eqnarray}$$

and the boundary condition (2.8) is

(4.2a,b ) $$\begin{eqnarray}{\it\omega}^{2}s\,\partial _{s}P+\text{i}{\it\omega}\partial _{{\it\phi}}P-(1-{\it\omega}^{2})z\,\partial _{z}P=0,\quad s^{2}+z^{2}=1,\end{eqnarray}$$

since $\boldsymbol{n}=s\mathbf{1}_{s}+z\mathbf{1}_{z}$ . We transform the problem to the non-orthogonal frequency-dependent spheroidal coordinates $({\it\xi},{\it\zeta},{\it\phi})$ (Bryan Reference Bryan1889) given by

(4.3a,b ) $$\begin{eqnarray}{\it\sigma}s=\sqrt{1-{\it\zeta}^{2}}\sqrt{1-{\it\xi}^{2}},\quad {\it\omega}z={\it\xi}{\it\zeta},\quad 0\leqslant {\it\zeta}\leqslant {\it\omega},~{\it\omega}\leqslant {\it\xi}\leqslant 1,\end{eqnarray}$$

where ${\it\sigma}^{2}+{\it\omega}^{2}=1$ . The relation between ${\it\sigma}$ and ${\it\omega}$ removes the mixed second derivative from Poincaré’s equation, allowing the variables to separate into Legendre’s equation in ${\it\zeta}$ and in ${\it\xi}$ , and also ensures the spherical boundary $s^{2}+z^{2}=1$ is a ${\it\zeta}$ -surface, ${\it\zeta}={\it\omega}$ . The transformation is degenerate if ${\it\omega}=0$ . The ${\it\zeta}$ -surfaces ( ${\it\xi}$ -surfaces) are non-confocal oblate (prolate) spheroids with semi-axes $\sqrt{1-{\it\zeta}^{2}}/\sqrt{1-{\it\omega}^{2}}>1$ and ${\it\zeta}/{\it\omega}<1$ (respectively, $\sqrt{1-{\it\xi}^{2}}/\sqrt{1-{\it\omega}^{2}}<1$ and ${\it\xi}/{\it\omega}>1$ ), co-axial with the $z$ -axis. Thus the spherical inertial modes are

(4.4) $$\begin{eqnarray}P=P_{n,{\it\omega}}^{m}:=P_{n}^{m}({\it\zeta}_{{\it\omega}})\,P_{n}^{m}({\it\xi}_{{\it\omega}})\text{e}^{\text{i}m{\it\phi}},\end{eqnarray}$$

where $P_{n}^{m}$ is the associated Legendre function of degree $n$ and order $m$ , $m\in \mathbb{Z}$ , $|m|\leqslant n$ , ${\it\omega}\neq 0$ and the ${\it\omega}$ -dependence of ${\it\zeta}$ and ${\it\xi}$ is indicated.

The product $P_{n}^{m}({\it\xi})P_{n}^{m}({\it\zeta})$ is a polynomial in $s$ and $z$ of degree $n$ , and $P_{n}^{m}({\it\xi})P_{n}^{m}({\it\zeta})\text{e}^{\text{i}m{\it\phi}}$ is a polynomial in $x,y,z$ of degree $n$ (Kudlick Reference Kudlick1966; Zhang et al. Reference Zhang, Earnshaw, Liao and Busse2001), even if ${\it\sigma}^{2}+{\it\omega}^{2}\neq 1$ . In fact,

(4.5) $$\begin{eqnarray}P_{n,{\it\omega}}^{m}=\left\{\begin{array}{@{}ll@{}}({\it\sigma}s\text{e}^{\text{i}{\it\phi}})^{m}({\it\omega}z)^{q}U_{n}^{m}({\it\sigma}^{2}s^{2},{\it\omega}^{2}z^{2}),\quad & \text{if }m\geqslant 0;\\ ({\it\sigma}s\text{e}^{-\text{i}{\it\phi}})^{|m|}({\it\omega}z)^{q}U_{n}^{m}({\it\sigma}^{2}s^{2},{\it\omega}^{2}z^{2}),\quad & \text{if }m<0;\end{array}\right.\end{eqnarray}$$

where $q:=n-|m|(\text{mod}~2)$ is either 0 or 1, and $U_{n}^{m}({\it\sigma}^{2}s^{2},{\it\omega}^{2}z^{2})$ is a polynomial in ${\it\sigma}^{2}s^{2}$ and ${\it\omega}^{2}z^{2}$ of degree $(n-|m|-q)/2$ . To establish this, note the associated Legendre function $P_{n}^{m}(w)=(1-w^{2})^{|m|/2}w^{q}u_{n}^{m}(w^{2})$ , where $u_{n}^{m}(w^{2})$ is a polynomial in $w^{2}$ of degree $(n-|m|-q)/2$ . Thus $P_{n,{\it\omega}}^{m}=({\it\sigma}s\text{e}^{\text{i}{\it\phi}})^{m}({\it\omega}z)^{q}u_{n}^{m}({\it\xi}^{2})u_{n}^{m}({\it\zeta}^{2})$ if $m\geqslant 0$ ; and $P_{n,{\it\omega}}^{m}=({\it\sigma}s\text{e}^{-\text{i}{\it\phi}})^{|m|}({\it\omega}z)^{q}u_{n}^{m}({\it\xi}^{2})u_{n}^{m}({\it\zeta}^{2})$ if $m<0$ . The result (4.5) follows, since the product $u_{n}^{m}({\it\xi}^{2})u_{n}^{m}({\it\zeta}^{2})$ is a polynomial $U_{n}^{m}({\it\sigma}^{2}s^{2},{\it\omega}^{2}z^{2})$ in $({\it\sigma}s)^{2}$ and $({\it\omega}z)^{2}$ of degree $(n-|m|-q)/2$ with coefficients independent of ${\it\omega}$ and ${\it\sigma}$ by the following lemma. Expressions (4.5) are polynomials in $x,y,z$ since $s\text{e}^{\pm \text{i}{\it\phi}}=x\pm \text{i}y$ . The $z$ -parity of $P_{n,{\it\omega}}^{m}$ , which is not immediately apparent from ${\it\zeta}$ and ${\it\xi}$ , is determined by $q$ .

Proof. Let $p(w)=\sum _{k=0}^{K}p_{k}w^{k}$ , where the $p_{k}$ are constants. Then

(4.6) $$\begin{eqnarray}p({\it\xi}^{2})p({\it\zeta}^{2})=\mathop{\sum }_{k=0}^{K}\left\{p_{k}^{2}z_{1}^{2k}+\mathop{\sum }_{l=1}^{K-k}p_{k}p_{k+l}z_{1}^{2k}({\it\xi}^{2l}+{\it\zeta}^{2l})\right\}.\end{eqnarray}$$

By rearranging the binomial expansion of $({\it\xi}^{2}+{\it\zeta}^{2})^{l}$ in the form,

(4.7) $$\begin{eqnarray}{\it\xi}^{2l}+{\it\zeta}^{2l}=({\it\xi}^{2}+{\it\zeta}^{2})^{l}-\mathop{\sum }_{j=1}^{\left\lfloor \frac{l-1}{2}\right\rfloor }~^{l}C_{j}({\it\xi}{\it\zeta})^{2j}({\it\xi}^{2(l-2j)}+{\it\zeta}^{2(l-2j)})\end{eqnarray}$$

and using the identity ${\it\xi}^{2}+{\it\zeta}^{2}=1-s_{1}^{2}+z_{1}^{2}$ , it follows by induction that ${\it\xi}^{2l}+{\it\zeta}^{2l}$ is a polynomial in $s_{1}^{2}$ and $z_{1}^{2}$ of degree $l$ . Hence $p({\it\xi}^{2})p({\it\zeta}^{2})$ is a polynomial in $s_{1}^{2}$ and $z_{1}^{2}$ of degree $K$ .

The boundary $s^{2}+z^{2}=1$ is given by $s=\sin {\it\theta}$ and $z=\cos {\it\theta}$ , or in oblate spheroidal coordinates by ${\it\xi}=\cos {\it\theta}$ and ${\it\zeta}={\it\omega}$ . The boundary condition implies ${\it\omega}$ is a root of

(4.8) $$\begin{eqnarray}(1-{\it\omega}^{2})\frac{\text{d}P_{n}^{m}({\it\omega})}{\text{d}{\it\omega}}+mP_{n}^{m}({\it\omega})=0.\end{eqnarray}$$

Now $P_{n}^{m}({\it\omega})=(1-{\it\omega}^{2})^{|m|/2}p_{n}^{m}({\it\omega})$ , where $p_{n}^{m}({\it\omega})$ is a polynomial in ${\it\omega}$ of degree $n-|m|$ . Moreover, $\{p_{n}^{m}({\it\omega})\}_{n=|m|}^{\infty }$ form an orthogonal set of polynomials on $[-1,1]$ with weight function $(1-{\it\omega}^{2})^{|m|}$ even in ${\it\omega}$ , since

(4.9) $$\begin{eqnarray}\frac{1}{2}\int _{-1}^{1}P_{n}^{m}({\it\omega})P_{N}^{m}({\it\omega})\text{d}{\it\omega}={\it\delta}_{nN}.\end{eqnarray}$$

Thus $p_{n}^{m}({\it\omega})$ has $n-|m|$ simple zeros ${\it\omega}_{k},k=1:n-|m|$ , on $(-1,1)$ symmetric about ${\it\omega}=0$ and hence $p_{n}^{m}({\it\omega})=c_{n}^{m}\prod _{k=1}^{n-|m|}({\it\omega}-{\it\omega}_{k})$ , where $c_{n}^{m}$ is a constant. The frequency equation (3.4) can be written as $\text{d}\ln P_{n}^{m}({\it\omega})/\text{d}{\it\omega}=-m/(1-{\it\omega}^{2})$ or

(4.10) $$\begin{eqnarray}\mathop{\sum }_{k=1}^{n-|m|}\frac{1}{{\it\omega}-{\it\omega}_{k}}=-\frac{{\textstyle \frac{1}{2}}(|m|+m)}{{\it\omega}+1}-\frac{{\textstyle \frac{1}{2}}(|m|-m)}{{\it\omega}-1}.\end{eqnarray}$$

For $m\neq 0$ this equation has $n-|m|$ zeros; for $m=0$ there are $n-1$ zeros. Figure 1 shows the generic case for $n-|m|=4$ . The zeros are not symmetrically distributed about ${\it\omega}=0$ . Geostrophic modes ${\it\omega}=0$ occur if and only if $m=0$ and $n$ is even. There are no $m=\pm n$ modes: in fact, the dispersion relation gives ${\it\omega}=1~(-1)$ for $m>0~({<}0)$ in these cases. Thus we label the non-zero frequencies for $0<m\leqslant n-1$ by ${\it\omega}_{n,j}^{m}$ , where $j\in \mathbb{Z}$ , $j\neq 0$ and $j=-(n-m+1)/2:(n-m-1)/2$ , if $n-m$ is odd, or $j=-(n-m)/2:(n-m)/2$ , if $n-m$ is even. The zero frequencies are ${\it\omega}_{n,0}^{0}$ , where $n$ is even, $m=0$ and $j=0$ . For $-(n-1)\leqslant m<0$ , ${\it\omega}_{n,j}^{m}={\it\omega}_{n,-j}^{-m}$ . We define the associated inertial pressure mode for the non-zero frequency ${\it\omega}_{n,j}^{m}$ by

(4.11) $$\begin{eqnarray}P_{n,j}^{m}:=P_{n,{\it\omega}_{n,j}^{m}}^{m}.\end{eqnarray}$$

Note that $P_{n,j}^{m}=P_{n,-j}^{-m\ast }$ . We construct the geostrophic modes in § 4.2.

Figure 1.

Schematic plot of the two sides of (5.6) for inertial frequencies ${\it\omega}$ with $n-|m|=4$ . The right-hand side curves are labelled by $m>0$ and $m<0$ ; the $m=0$ curve is the ${\it\omega}$ axis.

The number of Coriolis pressure modes of degree $N+1$ or less, which equals the number of Coriolis velocity modes of degree $N$ or less, is

(4.12) $$\begin{eqnarray}C_{N}=\mathop{\sum }_{n=1}^{N+1}\mathop{\sum }_{m=-n}^{n}(n-|m|-{\it\delta}_{0}^{m})=\frac{1}{6}N(N+1)(2N+7).\end{eqnarray}$$

For $N=1,2,3$ there are $3,11,26$ modes and for $N\gg 1~C_{N}\approx N^{3}/3$ . Greenspan (Reference Greenspan1968) counts $n-|m|~(n-|m|-1)$ inertial modes if $n-m$ is even (odd), in agreement with (4.12).

4.2. Geostrophic pressure modes

We construct the geostrophic modes by taking the limit ${\it\omega}\rightarrow 0$ in the solutions (4.4). They are the natural choice, since they are hierarchical in polynomial degree. These geostrophic modes can also be constructed by applying Gram–Schmidt orthogonalisation with the inner product (1.11) to the polynomials $\{1,s^{2},s^{4},\dots \}$ . See figure 2. Other geostrophic bases are possible.

Lemma 4.2. For $m=0$ and ${\it\omega}\rightarrow 0$ the solutions (4.4) reduce to the geostrophic modes,

(4.13) $$\begin{eqnarray}P_{n,0}^{0}(s)=P_{n}(0)\,P_{n}(\sqrt{1-s^{2}})-[P_{n}(0)]^{2},\end{eqnarray}$$

where $n$ is even. Here, $P_{n,0}^{0}(s)$ is a polynomial in $s^{2}$ of even degree $n$ in $s$ and $P_{n}$ is the Legendre polynomial of degree $n$ .

Proof. From (4.3) ${\it\xi}^{2}+{\it\zeta}^{2}=1-{\it\sigma}^{2}s^{2}+({\it\xi}{\it\zeta})^{2}=1-{\it\sigma}^{2}s^{2}+{\it\omega}^{2}z^{2}$ . Thus as ${\it\omega}\rightarrow 0$ , ${\it\xi}^{2}+{\it\zeta}^{2}\rightarrow 1-s^{2}$ and, by (4.7), ${\it\xi}^{2l}+{\it\zeta}^{2l}\rightarrow ({\it\xi}^{2}+{\it\zeta}^{2})^{l}$ , since ${\it\sigma}\rightarrow 1$ and $({\it\xi}{\it\zeta})^{2}={\it\omega}^{2}z^{2}\rightarrow 0$ . If $n$ is even, $P_{n}(z)=\sum _{k=0}^{n/2}p_{n,2k}z^{2k}$ , where the $p_{n,2k}$ are constants, and

(4.14) $$\begin{eqnarray}P_{n}({\it\xi})\,P_{n}({\it\zeta})=p_{n,0}\left[\mathop{\sum }_{k=0}^{n/2}p_{n,2k}({\it\xi}^{2k}+{\it\zeta}^{2k})-p_{n,0}\right]+\mathop{\sum }_{k=1}^{n/2}\mathop{\sum }_{l=1}^{n/2}p_{n,2k}p_{l,2k}{\it\xi}^{2k}{\it\zeta}^{2l}.\end{eqnarray}$$

The double sum is $O({\it\xi}{\it\zeta})^{2}$ . Thus, for $n$ even, $P_{n}({\it\xi})\,P_{n}({\it\zeta})\rightarrow P_{n,0}^{0}$ in (4.13) as ${\it\omega}\rightarrow 0$ , since $p_{n,0}=P_{n}(0)$ . If $n$ is odd, $P_{n}({\it\xi})\,P_{n}({\it\zeta})\rightarrow 0$ as ${\it\omega}\rightarrow 0$ .

Linearly independent Coriolis modes of the same frequency may be non-orthogonal. The modes (4.13) are orthogonal: evaluating the inner product $(\boldsymbol{v}_{1},\boldsymbol{v}_{2})$ of two modes using (3.21), noting $n_{1}$ and $n_{2}$ are even and substituting $v^{2}=1-s^{2}$ yields, apart from the factor $P_{n_{1}}(0)P_{n_{2}}(0)$ ,

(4.15) $$\begin{eqnarray}\int _{V}\boldsymbol{{\rm\nabla}}P_{n_{1}}(\sqrt{1-s^{2}})\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P_{n_{2}}(\sqrt{1-s^{2}})\text{d}V=2{\rm\pi}\int _{-1}^{1}P_{n_{1}}^{\prime }(v)(1-v^{2})P_{n_{2}}^{\prime }(v)\text{d}v.\end{eqnarray}$$

An integration by parts and Legendre’s differential equation shows the last integral vanishes if $n_{1}\neq n_{2}$ .

Figure 2.

The geostrophic modes $P_{n,0}^{0}$ as a function of cylindrical radius $s$ for $n=2,4,6,8,10$ .

4.3. Velocity modes

The velocity modes are constructed from the pressure modes using (3.17). Thus associated with the pressure mode $P_{n+1,j}^{m}$ is the velocity mode $\boldsymbol{v}_{n,j}^{m}$ . Define

(4.16) $$\begin{eqnarray}\boldsymbol{v}_{n}^{m}({\it\omega}):=\left\{\begin{array}{@{}l@{}}\displaystyle \frac{\text{i}{\it\omega}}{1-{\it\omega}^{2}}(\boldsymbol{{\rm\nabla}}P_{n+1,{\it\omega}}^{m}-{\it\omega}^{-2}\widehat{{\it\bf\Omega}}\widehat{{\it\bf\Omega}}\boldsymbol{\cdot }\boldsymbol{{\rm\nabla}}P_{n+1,{\it\omega}}^{m})+\frac{1}{1-{\it\omega}^{2}}\widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P_{n+1,{\it\omega}}^{m},\quad {\it\omega}\neq 0;\quad \\ \widehat{{\it\bf\Omega}}\times \boldsymbol{{\rm\nabla}}P_{n+1,0}^{0},\quad {\it\omega}=0,m=0,n\text{ odd}.\quad \end{array}\right.\end{eqnarray}$$

The Coriolis velocity modes are thus

(4.17) $$\begin{eqnarray}\boldsymbol{v}_{n,j}^{m}:=N_{n,j}^{m}\boldsymbol{v}_{n}^{m}({\it\omega}_{n+1,j}^{m}),\quad |m|\leqslant n,\end{eqnarray}$$

where $j=-(n-m+1)/2:(n-m-1)/2$ if $n-m$ is odd, $j=-(n-m)/2:(n-m)/2$ if $n-m$ is even, and $j\neq 0$ unless $m=0$ and $n$ is odd. The normalisation constant $N_{n,j}^{m}>0$ is chosen so that the mode is normalised to $(\boldsymbol{v}_{n,j}^{m},\boldsymbol{v}_{n,j}^{m})=1$ . The complex conjugate $\boldsymbol{v}_{n,j}^{m\ast }=\boldsymbol{v}_{n,-j}^{-m}$ . No modes $\boldsymbol{v}_{n,{\it\omega}}^{m}$ arise from $m=n+1$ , i.e.  $P_{n+1}^{\pm (n+1)}$ . For $m\neq 0$ the associated inertial waves propagate eastward (westward) if $j>0~(j<0)$ . Equatorial symmetries are determined by $n-m$ : the velocity modes are quadrupole symmetric if $n-m=0(\text{mod}~2)$ and dipole symmetric if $n-m=1~(\text{mod}~2)$ ; and rotational symmetries about $\widehat{{\it\bf\Omega}}$ are determined by $m$ . The velocities of the geostrophic modes (4.13) agree with Liao & Zhang (Reference Liao and Zhang2010a ) up to normalisation and phase: in terms of Jacobi polynomials their ${\it\phi}$ -components can be written as

(4.18) $$\begin{eqnarray}v_{2n-1,0}^{0}=N_{2n-1,0}^{0}\frac{\text{d}P_{2n,0}^{0}(s)}{\text{d}s}=N_{2n-1,0}^{0}(-)^{n}(2n+1)P_{2n}(0)sP_{n}^{(1/2,1)}(2s^{2}-1).\end{eqnarray}$$

Below we label the Coriolis modes using a 3-index Greek letter, e.g. $\boldsymbol{v}_{{\it\alpha}}\equiv \boldsymbol{v}_{n_{{\it\alpha}},j_{{\it\alpha}}}^{m_{{\it\alpha}}}$ , and denote the degree of the polynomial flow $\boldsymbol{v}_{{\it\alpha}}$ by $\deg {\it\alpha}$ .