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.

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**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/m^{2}). 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_bar^{3} 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)where F (the solar forcing) and H (the observed MLD) are both composed of mixed 1 and 2 cpy frequencies, and H is always positive. Qualitatively the effect of the term F/H is that solar anomalies during the shallow-MLD periods are enhanced compared to those in the deep-MLD times of year. Figs.15 and 16 show that both positive and negative correlation of F and H occur at annual and semi-annual frequencies. For the observed configuration of solar and MLD variations in the eastern Pacific, at semi-annual frequencies positive anomalies of solar radiation coincide with shallow MLD and are augmented by this signal, but for the annual component, negative solar anomalies are emphasized (Figs.15 and 16). Since observed MLD varies by up to a factor of three in the east, even in these highly averaged annual cycles, it is a major influence on the integration of eqn (1).

**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

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

William S. Kessler NOAA / Pacific Marine Environmental Laboratory 7600 Sand Point Way NE Seattle Wa 98115 USA Tel: 206-526-6221 o__ ____ Fax: 206-526-6744 _,>/'_ ----- E-mail: kessler@pmel.noaa.gov (_) \(_) ------~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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