Thanks for sending me Schneider (1996) "A note on the annual cycle of SST at the equator" (hereafter S96). Shukla told me of the conclusions of that paper in Goa last year, and this text and associated figures are in response to that discussion.

You can read this text and click the embedded links to see the figures, but I recommend either opening two browser windows, one to this text and the other to the figure list, or printing the text and browsing from the figure list.

1. I talked in Goa about a paper I have submitted to J.Climate on the annual cycle of east Pacific cold tongue SST, based on observations and the Gent/Cane/Chen OGCM. One of the main findings is that for the annual cycle the 3-d ocean advection terms tend to cancel, and as a result SST tends to follow the net input from the surface, which is mostly the solar radiation as modified by cloudiness. The ocean cancellation is primarily between equatorial upwelling and warming due to tropical instability waves (TIW). In the second half of each year, the winds are strongest, and the entire upper circulation quickens. Increased upwelling cools the surface near the equator, and sharpens the SST front near 3-4N. Also at this time the TIW grow both because of increasing meridional shear and meridional SST gradient (both barotropic and baroclinic instabilities). This eddy mixing tends to erode the SST front and thereby warms the equator. Upwelling and the eddy mixing processes have roughly equal and opposite annual amplitude at the equator (about 60 W/m²). A second finding is that variations of mixed layer depth (MLD) are a key component of the annual variation of SST. (Kessler et al. 1998).

2. Shukla heard my talk and immediately realized that the ocean cancellation I found is consistent with the argument in the S96 energy balance model, namely that one does not need to invoke complicated ocean-atmosphere processes to explain the dominance of the 1 cpy cycle of SST in the Pacific cold tongue. That is partly true, but I believe it is too simple.

3. The most important point is that variations of cloudiness are a key aspect of the annual cycle of net solar radiation received at the sea surface in the Pacific cold tongue region. There is a large annual cycle of cloudiness over the cold tongue, with greater cloud cover (mostly stratus decks) in the cool part of the year (Aug-Jan) (Fig.1). The result is to greatly reduce the second solar maximum in September (Fig.2), and the net solar radiation received at the sea surface is primarily at 1 cpy in the eastern equatorial Pacific. At the top of the equatorial atmosphere the 1 cpy solar amplitude is about 0.63 times as large as the 2 cpy amplitude (Fig.3), but at the sea surface in this region cloudiness changes the relative importance so 1 cpy variability becomes 1.15 times larger than 2 cpy (Fig.4). Therefore one might argue that the reason for the dominant 1 cpy variation of SST in the cold tongue is due to clouds, and there is no reason to invoke any oceanic processes. But of course this is false since the stratus decks that modify the incoming solar radiation are probably strongly coupled to the SST (though no one can say precisely how), and there is no alternative to looking at the coupled ocean- atmosphere problem.

4. Nevertheless, the S96 energy balance (EB) model is intriguing, and it is certainly true (and possibly overlooked by some) that simply time- integrating a mixed 1 and 2 cpy forcing doubles the relative amplitude of the lower frequency. The result of this integration should be to double the 1 cpy/2 cpy ratio from 0.63 (for solar forcing at the top of the atmosphere) to 1.26 for SST. This change is just about the same as the effect of clouds.

5. I repeated the EB model calculations as detailed in S96, for the equatorial Pacific strip only. My forcing parameters (SST and solar flux) were slightly different than in S96. I used solar radiation based on harmonic fitting (values of harmonic coefficients from Reed (1977, JPO p482-485)), and SST from the Reynolds weekly satellite product (average annual cycle over 1984-93). The Reed formula gives radiation values fairly similar to the NCEP model solar radiation shown in Fig 1 of S96, although the peaks are about 15% smaller (Fig.5). The 10-year average Reynolds SST I used appears similar to the field shown in Fig 2 of S96, although this SST field has somewhat larger amplitude (range from -2C to 3.5C at 95W, whereas S96 gives -1.5C to 2.5C) (Fig.6). Otherwise I used the same parameter values as in S96. In the following, equation numbers are those of S96, viz:

           rho*c*H * dT/dt = F - est * T                (1)

where F is observed solar forcing, H is the MLD and T is the SST solution. est = 4*epsilon*sigma*T_bar³ is a constant with the value 1.90.

           H = H_hat * V_mod/V_obs                      (2)

is the procedure given in S96 to estimate H from observed SST variance and an initial estimate of SST made using uniform H_hat. When an estimate of H is found from (2), then used to find a new solution to (1), this constrains model SST variance to equal that observed.

6. I followed this S96 procedure for estimating H, making an initial guess of 50m and re-estimating based on the SST variance ratio. The results are similar to those shown in Fig 3 of S96 (Fig.7). Values of H are less than 10m in the east, deepening to a sharp trough (about 120m depth) at about 160E. In Fig 3 of S96, the corresponding trough is deeper but appears not to be as sharp. In the far western Pacific my H-solution rises back up to almost 20m, whereas S96 shows a similar shallowing but only to 50m. For reference, my figure also shows MLD determined from the TAO buoy network, where MLD is defined as the depth where the temperature is 0.5C less than the SST. These observed MLDs also deepen towards the central Pacific before shoaling in the far west.

7. Continuing along the same path as S96, I determined SST from the EB model using the H-field found as above (Fig. 8). The results are again quite similar to those shown in Fig.4 of S96, with slightly higher amplitude in the east. An exception is that my solution has weak semi- annual variance resulting in a second annual peak of SST, but the S96 solution apparently does not (Fig 4 of S96). It seems to me that, in the heat storage limit appropriate here, the integration in eqn (1) must transmit some amplitude of each frequency in the solar forcing F, and therefore it is hard for me to see how the S96 solution does not appear to have any second peak, which should be visible in the zero contour of Fig.4 of S96, even if small. Am I missing something here? In any case, slices at various longitudes through my solution (Fig.9) show the semi-annual variance clearly. Fig.10 compares the modeled and observed time series at 95W. The model solution clearly has much more semi-annual energy than do the observations. Fig.11 compares the observed and modeled 1 to 2 cpy variance ratios across the Pacific. East of about 170W, observed SST has about four times as much 1 cpy amplitude as semi-annual, bearing out the visual impression from Fig.10. The model ratio is nearly constant across the basin with a value of 1.27, just what is expected from the integration of a function (F) whose 1 to 2 cpy ratio is 0.63. In the far east, the model SST diverges slightly from this value, indicating a small departure from the heat storage limit.

8. Since I am only working on the equatorial strip, I did not repeat the analogue of Fig 6 from S96 (and the copy I got was missing that figure anyway).

9. Going further, I have an independent observation of MLD (from the TAO moorings, as mentioned above in regard to Fig.7. It is not really any less justifiable to use observed MLD than to estimate H from the observed SST (as in eqn (2) of S96), so I repeated the integration of eqn (1) using the H-field from TAO.

10. The first calculation with TAO MLD uses mean values shown in Fig.7 for H in eqn (1). MLD was linearly interpolated between the buoys and the east and west ends were extrapolated as constant values. The SST solution in this case is shown in Fig.12. Since the TAO H-field is similar in overall character to the model-determined field, the SST solution in Fig.12 is also similar to the original solution in Fig.8 (but note the difference in contour levels, sorry), although of smaller amplitude in the east where the TAO mean MLD is deeper by a factor of about two than the S96 model estimate (Fig.7).

11. The second calculation is more interesting, using the time-varying average annual cycle of observed TAO MLD (Fig.13) for H in (1). The annual variation of TAO MLD is large (Fig.14) with a peak-to-peak range greater than the mean value east of 125W, and it has been suggested by several authors that this variability plays a significant role in the east Pacific heat balance. The time series of TAO MLD have both annual and semi-annual components, as shown by the light and dark blue curves in Figs.15 and 16.

15. Introducing time variation of H in eqn (1) changes the character of the solution, since a wider variety of behavior is possible, including phase changes and secular growth when H is correlated with F. The form of (1) becomes:

		dT   F(t)    T
		-- = ---- - ----				(1')
		dt   H(t)   H(t)

16. Fig.17 shows time series of SST from the modified energy balance model (1'). The heavy lines (1-year running means) show that low- frequency warming occurs in the east, with weaker cooling in the west. The magnitude of warming/cooling is about 1/3rd the annual amplitude. These low-frequency signals are due to the correlation (r) of F and H, as shown by Fig.20. In this figure, mean SST during year 10 (the value of which demonstrates the low-frequency change in SST) is closely tied to the F:H correlation, and r < 0 (positive solar:shallow MLD) is associated with SST warming. Negative correlation occurs at 110W and 95W (note that r in Fig.20 is plotted inverted), along with warming model SST, while west of 140W the correlations are positive and model SST cools. SST growth has relatively larger magnitude at 110W and 95W because mean H is shallower there. Without the thermal damping term (2nd term on the RHS of (1')), SST would grow without bound, but damping causes the growth rate to decay exponentially (note that H is always positive) and produce the low-frequency solutions in Fig.17.

17. Clearly the observed variation of MLD alters the solutions to a heat balance model like (1) in fundamental ways, suggesting that the it remains necessary to consider other aspects than just the direct response to solar forcing. One might speculate that the F:H correlation in the east is not fortuitous but represents a process in which more solar surface warming increases the stratification of the upper ocean, tending to create a shallow mixed layer and consequent greater sensitivity to heat fluxes. However, it is not obvious why the F:H correlations should be positive in the central and western Pacific, resulting in low-frequency cooling in the solution to (1') (Fig.20).

18. Although MLD variations have a large rectifying effect onto low frequencies, their effect on the annual cycles is not nearly as great. Fig.21 compares the solution to (1') against SST from a run with constant H (in which there is no mean warming or cooling) (shown in Fig.12). Mean H is the same in both cases. For this comparison the SST from (1') is demeaned by the year 10 mean to remove the low-frequency signals. The phasing and overall character of the two solutions are similar, but the variation of H results in an increase of total standard deviation by a factor of about 1.25, with the 1 cpy amplitude enlarged more than the 2 cpy signals.

19. In summary, the EB model of S96 is able to capture many of the aspects of the annual variability of equatorial SST, in particular by reducing the semi-annual influence of solar variability on SST. This follows just because integration is a low-pass filter. However, the modeled reduction is not nearly as much as actually occurs in the ocean. The observed semi-annual component of SST in the east-central Pacific is about 4 times smaller than the 1 cpy component, whereas the EB model implies a ratio of about 1.27, not nearly enough to justify the conclusion of S96 that processes more complex than the direct response to solar radiation are not needed to explain the variability of SST. Therefore, I conclude that more complex processes are necessary, and that prominent among these are the feedback between cool SST and stratus clouds, which seems to account for a reduction in semi-annual variability at least as strong as the low-pass due to integration.

20. I am a little afraid that the conclusion that the 3-d ocean advection terms tend to cancel in the annual cycle will be all people notice about my paper, and really that is an overly-simple summary (suitable for a 15 minute talk as I gave in Goa). In fact the processes involved have sharp meridional gradients and there is a lot of regional complexity. In particular, equatorial upwelling has a strong annual cycle with 1 cpy amplitude greater than that of solar radiation. This is to a large degree balanced by TIW mixing away the resulting SST gradients, but there is still a substantial cooling influence, particularly just south of the equator, during the second half of the year.

21. A final issue concerns the difficulties in positing a model in which the solar flux is entirely taken up within a shallow mixed layer. When the layer gets very thin (the EB model suggests that the layer shoals to 5-6m in the cold tongue region), much of the solar flux will penetrate through this layer and the assumptions of the model are thereby violated.

Thanks for stimulating me to think about this! I look forward to hearing from you. I would be happy to provide the TAO MLD fields that I used to make these calculations.

Regards, Billy Kessler

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William S. Kessler
NOAA / Pacific Marine Environmental Laboratory
7600 Sand Point Way NE
Seattle Wa 98115 USA
 
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