^{a}

^{*}

^{b}

The eigenfrequencies of freely propagating divergent barotropic planetary and gravity waves in a spherical polar cap are discussed. The key amplitude equation is derived with the full spherical geometry maintained and leads to a second-order differential equation with coefficients functions of the co-latitude. Previous study of this problem has derived approximations to the requisite frequencies by evaluating these coefficients at some chosen fixed value of the co-latitude thereby reducing the problem to that of a constant coefficient differential equation solved easily using routine methods. Here, we demonstrate that such a simplification can be avoided since the full equation can be solved by standard asymptotic methods based on the latitudinal limit of the polar basin as the natural small parameter. Three-term asymptotic series are developed which are in remarkably good accord with numerical solutions of the full equation.

An accurate approximation for the eigenfrequencies of freely propagating barotropic divergent planetary and gravity waves in a circular polar cap - an archetype representation of the Arctic Ocean basin - is the subject of this paper. The analytical treatment of atmospheric or ocean dynamics near the pole is frustrated by the variation in the meridional gradient of the Coriolis parameter

LeBlond (

Consider an ocean of uniform depth centred at the pole; in terms of a spherical co-ordinate system in which

Schematic of the spherical polar co-ordinate system showing the unit vectors

We point out that by taking the fluid to be of uniform density we have focused attention on gravity and Rossby free waves. However, the adoption of the thin-shell approximation eliminates a class of sub-inertial waves that exist even if the surface is rigid, see Stewartson and Rickard (

Azimuthally propagating wave solutions of (1) are sought in the form

If we eliminate

It is

Rather than solving _{B}_{0}; typically WPA17 chose _{0} and it is not apparent how to optimise its value

The purpose of our work here is to show how the conclusions of WPA17 can be improved markedly by constructing formal asymptotic solutions of _{0} and we demonstrate the accuracy of our findings by comparison with numerical simulations which show that our results are likely to be useful over a wide range of parameters. To this end we arrange the remainder of the work as follows. Presently, in § 2 we develop the form of the low-frequency planetary modes while the analogous workings for the higher-frequency gravity modes is deferred to § 3. The paper closes with a few final remarks in § 4.

WPA17 noted that low-frequency planetary waves are only possible when

We remark that these forms follow from a simple balance of terms within _{B}_{B}_{M}_{M}^{th}

Moving to next order shows that

This solution automatically satisfies the requirement at

We proceed one stage further. The boundary condition at

The right-hand side of this equation can be written as a linear combination of _{1}–_{4}. The satisfaction of boundary condition (14) leads to

We now have the first three terms in the asymptotic expression for the planetary wave frequency _{0}–_{2} given by (9), (12) and

The potential usefulness of these results is best assessed by comparison with a few numerical simulations of the full

Parameter values used by LeBlond (

_{B}

Some sample results are given in _{0} given by (9) is often markedly superior to the IT result; inclusion of the values of (12) and (16) gives predictions that are very impressive indeed. For a given azimuthal wavenumber the accuracy of our prediction seems to improve with increasing _{B}

Summary of calculated results for the frequencies of the planetary waves corresponding to azimuthal wavenumbers _{IT}_{1}–_{3}).

_{IT}

_{1}(%)

_{2}(%)

_{3}(%)

^{–3}

^{–3}

^{–2}

^{–3}

^{–3}

^{–3}

^{–4}

^{–4}

^{–4}

^{–4}

^{–4}

^{–4}

^{–3}

^{–3}

^{–3}

^{–3}

^{–3}

^{–4}

^{–4}

^{–4}

^{–4}

^{–4}

^{–4}

^{–4}

^{–3}

^{–3}

^{–3}

^{–3}

^{–3}

^{–4}

^{–4}

^{–3}

^{–4}

^{–4}

^{–4}

^{–5}

In

The form of the leading order eigenfunction ^{th}

Gravity wave solutions are characterised by

The equation for

Again, this is a scaled Bessel equation with solution
^{th}

At next order it follows that

which admits the solution

Routine, though lengthy, manipulation shows that

Some sample numerical simulations were conducted for the gravity wave modes. Full solutions of ^{2}, the sign of

Summary of calculated results for gravity wave frequencies for azimuthal wavenumbers

For completeness some representative leading order eigenfunctions

The form of the leading order eigenfunction ^{th}

The purpose of this article has been to develop simple asymptotic expressions for the frequencies of both planetary and gravity waves that may be present in a polar basin. While previous estimates have relied on various largely _{B}

It is worth pointing out that the structure the governing

Mention should be made of the work by Harlander (

The motivation for this study is to advance our understanding of the free wave dynamics in the Arctic Ocean basin. Of course, we recognise that this basin has complex topography characterised by wide continental shelves and a trans-polar ridge separating two deep interior basins. The results we have presented here form the basis of extension to more realistic representations of the Arctic Ocean basin. Indeed, LeBlond (

The referees are thanked for their encouraging comments that led to improvements in the presentation of this work.

No potential conflict of interest was reported by the authors.