4.4 Comparison with the Antarctic Circumpolar Wave

The Antarctic Ocean circles the globe without continents and is periodic in the zonal direction. Here the progress of oceanic Rossby waves are less impeded by meridional boundaries than in the gyre regimes of ocean basins, so perhaps the unbounded model described above is more directly applicable here than elsewhere. Let us see whether the present model can support coupled oscillations in the Antarctic Ocean.

Our previous discussions show that conditions for growth depend crucially on the sign of r and a. However, these quantities remain positive definite in the southern hemisphere despite changes in the sign of f and $\frac{ \partial }{\partial y}\overline{\mbox{SST}}$. All the results of section 3 still apply. According to our model, growth of decadal-scale coupled waves could occur in the Southern Ocean if the atmospheric response to SST forcing is equivalent barotropic and if highs are located above warm water.

Recently, White & Peterson (1996) and Jacobs & Mitchell (1996) described an ``Antarctic Circumpolar Wave'' (ACW) which takes the form of a wavenumber-2 perturbation of SST, surface air pressure, sea-surface height, windstress, and sea ice extent, circling eastward around Antarctica with a period of around 4 years. Jacobs & Mitchell report that sea-surface height (a proxy for oceanic streamfunction $\psi _{o}$) is coincident with SST. Both White & Peterson and Jacobs & Mitchell report that wind-stress curl (and hence, to the extent that the geostrophic approximation is appropriate at the surface, surface air pressure anomaly) appears to lead SST by 90$^{\circ}~$phase in the observations. This configuration is summarized in figure 12.

  

Figure: Schematic summary of the Antarctic Circumpolar Wave based on observed correlations between SST (W=warm, C=cold), atmospheric sea-level pressure (H=high, L=low),meridional wind stress ($\tau$), and sea-surface height observed by White & Peterson and Jacobs & Mitchell. The wave encircles Antarctica with wavenumber 2, and travels eastward at 10 cm/s.

By using parameters appropriate to the Antarctic Ocean (U₁ = 15 m/s, U₂ = 5 m/s, $l = \pi/3100$ km, $\beta = 1.6\cdot10^{-11} \mbox{(m s)}^{-1}$, other parameters as in Table 1), we obtain a growing mode of wavenumber 2 around the globe, a growth rate of 0.35 yrs^-1 and a westward phase speed of 4 cm/s. Our model assumes an ocean at rest: to adapt it to the Antarctic Ocean, we simply suppose our model dynamics occur in a frame moving eastward with the Antarctic Circumpolar Current at 10-15 cm/s: the resultant phase speed ``over ground'' for our waves is 5-10 cm/s eastward. SST, $\psi _{o}$, and $\widehat{\psi}$ are all approximately in phase.

This wave has some similarity to the ACW, but also some important differences. Phase speed and wavelength are in good agreement, as is the phase match between SST and $\psi _{o}$. However, our model predicts that the surface air pressure (and therefore wind-stress curl) should be in phase with SST. Observations of the ACW show a 90$^{\circ}~$phase shift.

Our model can produce phase-shifted growing modes in two ways. An off-resonant wave would have a significant phase shift (since $\nu/\Gamma \not= 0$) between atmosphere and ocean; such an off-resonant wave might be demanded by periodicity constraints. Furthermore, the tendency term in the SST equation (20) can allow the SST response to lag behind the forcing produced by the dynamic ocean. Moreover the requirement that the amplitude of SST grow over time means some phase-shifting must occur to allow the dynamic ocean to supply additional warmth to regions where SST is already large.

The model can support growing modes with phase shifts, but it is difficult to generate phase shifts much larger than 45$^{\circ}~$. In addition, we note that if the atmosphere-ocean phase shift is truly 90$^{\circ}~$, we must have $\nu/\Gamma \rightarrow \infty$ (see (45)), which means that the atmospheric response to SST anomalies (see (34)) is zero, and growth does not occur (see 44). While this could be an artifact of the atmospheric model chosen, we note that a 90$^{\circ}~$lag between wind-stress curl and $\psi _{o}$ implies that the windstress cannot increase the amplitude of the oceanic streamfunction. The windstress is zero when the currents are maximum and vice versa, so no work is done on the current, again making growth impossible. We conclude that either the phase relationships in nature are not as the presently-available observations suggest, or the Antarctic Circumpolar Wave does not grow through windstress feedback coupling.

While preparing this paper for submission, we became aware of a study by Qiu and Jin (1997) which applies a model similar to ours to the Antarctic Circumpolar Wave. Their SST equation resembles that of section 3.2.3, but allows cooling of SST anomalies by air-sea flux. They employ a greatly simplified atmosphere which ignores $\beta$-effects and Rossby waves (essentially a thermodynamic equation plus thermal wind), in which the response is assumed a priori to be equivalent barotropic. Their ocean dynamics and coupling assumptions are similar to ours, but with two oceanic levels and a mean zonal current. A coupled growing mode and a damped uncoupled mode are found, just as in this study. However, our use of a more dynamically-based, albeit still highly simplified, description of the atmosphere leads to differences that cannot be ignored. The meridional wavelength and zonal windspeeds chosen by Qiu and Jin are so small that any reasonable choice of the baroclinic component of the mean winds (a factor not part of their model) generates a baroclinic response in our model, with $\mu > 0$ (see (31)). This leads to a decaying mode in our equations. Their assumption that the atmosphere responds barotropically agrees with observations of the ACW, but it is not trivial to explain or generate such a response through atmospheric dynamics. Most importantly, however, our model and that of Qiu & Jin adopt the same mechanical forcing of the ocean by wind stress, and so theirs, like ours, must prohibit growth when wind stress curl leads oceanic streamfunction by 90$^{\circ}~$.

The model described here, that of Qiu & Jin, and the observations have their limitations. We note that Christoph and Barnett (1996) have observed an ACW in their ECHAM4 + OPYC3 coupled numerical model. Because the model may provide a continuous record of all relevant fields over many decades (particularly wind and surface air pressure fields, which are difficult to measure remotely) it may be fruitful to test our analytical model against this numerically simulated ACW.