Sunspots and Sea Surface Temperature

Guest Post by Willis Eschenbach

I thought I was done with sunspots … but as the well-known climate scientist Michael Corleone once remarked, “Just when I thought I was out … they pull me back in”.  In this case Marcel Crok, the well-known Dutch climate writer, asked me if I’d seen the paper from Nir Shaviv called “Using the Oceans as a Calorimeter to Quantify the Solar Radiative Forcing”, available here. Dr. Shaviv’s paper claims that both the ocean heat content and the ocean sea surface temperature (SST) vary in step with the ~11 year solar cycle. Although it’s not clear what “we” means when he uses it, he says: thumb its the sun“We find that the total radiative forcing associated with solar cycles variations is about 5 to 7 times larger than just those associated with the TSI variations, thus implying the necessary existence of an amplification mechanism, though without pointing to which one.” Since the ocean heat content data is both spotty and incomplete, I looked to see if the much more extensive SST data actually showed signs of the claimed solar-related variation.

To start with, here’s what Shaviv2008 says about the treatment of the data:

Before deriving the global heat flux from the observed ocean heat content, it is worth while to study in more detail the different data sets we used, and in particular, to better understand their limitations. Since we wish to compare them to each other, we begin by creating comparable data sets, with the same resolution and time range. Thus, we down sample higher resolution data into one year bins and truncate all data sets to the range of 1955 to 2003.

I assume the 1955 start of their data is because the ocean heat content data starts in 1955. Their study uses the HadISST dataset, the “Ice and Sea Surface Temperature” data, so I went to the marvelous KNMI site and got that data to compare to the sunspot data. Here are the untruncated versions of the SIDC sunspot and the HadISST sea surface temperature data.

sidt sunspots and HadISST sea surface temperature 1870 2013Figure 1. Sunspot numbers (upper panel) and sea surface temperatures (lower panel).

So … is there a solar component to the SST data? Well, looking at Figure 1, for starters we can say that if there is a solar component to SST, it’s pretty small. How small? Well, for that we need the math. I often start with a cross-correlation. A cross-correlation looks not only at how well correlated two datasets might be. It also shows how well correlated the two datasets are with a lag between the two. Figure 2 shows the cross-correlation between the sunspots and the SST:

cross correlation sidc sunspots hadISST 1870 2013Figure 2. Cross-correlation, sunspots and sea surface temperatures. Note that they are not significant at any lag, and that’s without accounting for autocorrelation.

So … I’m not seeing anything significant in the cross-correlation over full overlap of the two datasets, which is the period 1870-2013. However, they haven’t used the full dataset, only the part from 1955 to 2003. That’s only 49 years … and right then I start getting nervous. Remember, we’re looking for an 11-year cycle. So results from that particular half-century of data only represent three complete solar cycles, and that’s skinny … but in any case, here’s cross-correlation on the truncated datasets 1955-2003:

cross correlation sidc sunspots hadISST 1955 2003Figure 3. Cross-correlation, truncated sunspots and sea surface temperatures 1955-2003. Note that while they are larger than for the full dataset, they are still not significant at any lag, and that’s without accounting for autocorrelation.

Well, I can see how if all you looked at was the shortened datasets you might believe that there is a correlation between SST and sunspots. Figure 3 at least shows a positive correlation with no lag, one which is almost statistically significant if you ignore autocorrelation.

But remember, in the cross-correlation of the complete dataset shown back in Figure 2, the no-lag correlation is … well … zero. The apparent correlation shown in the half-century dataset disappears entirely when we look at the full 140-year dataset.

This highlights a huge recurring problem with analyzing natural datasets and looking for regular cycles. Regular cycles which are apparently real appear, last for a half century or even a century, and then disappear for a century …

Now, in Shaviv2008, the author suggests a way around this conundrum, viz:

Another way of visualizing the results, is to fold the data over the 11-year solar cycle and average. This reduces the relative contribution of sources uncorrelated with the solar activity as they will tend to average out (whether they are real or noise).

In support of this claim, he shows the following figure:

Shaviv Figure 5Figure 4. This shows Figure 5 from the Shaviv2008 paper. Of interest to this post is the top panel, showing the ostensible variation in the averaged cycles.

Now, I’ve used this technique myself. However, if I were to do it, I wouldn’t do it the way he has. He has aligned the solar minimum at time t=0, and then averaged the data for the 11 years after that. If I were doing it, I think I’d align them at the peak, and then take the averages for say six years on either side of the peak.

But in any case, rather than do it my way, I figured I’d see if I could emulate his results. Unfortunately, I ran into some issues right away when I started to do the actual calculations. Here’s the first issue:

sidc sunspots hadISST 1955 2003Figure 5. The data used in Shaviv2008 to show the putative sunspot-SST relationship.

I’m sure you can see the problem. Because the dataset is so short (n = 49 years), there are only four solar minima—1964, 1976, 1986, and 1996. And since the truncated data ends in 2003, that means that we only have three complete solar cycles during the period.

This leads directly to a second problem, which is the size of the uncertainty of the results of the “folded” data. With only three full cycles to analyze, the uncertainty gets quite large. Here are the three folded datasets, along with the mean and the 95% confidence interval on the mean.

sst anomaly folded over solar cycle 1955-2003Figure 6. Sea surface temperatures from three full solar cycles, “folded” over the 11-year solar cycle as described in Shaviv2008

Now, when I’m looking for a repetitive cycle, I look at the 95% confidence interval of the mean. If the 95%CI includes the zero line, it means the variation is not significant. The problem in Figure 6, of course, is the fact that there are only three cycles in the dataset. As a result, the 95%CI goes “from the floor to the ceiling”, as the saying goes, and the results are not significant in the slightest.

So why does the Shaviv2008 result shown in Figure 4 look so convincing? Well … it’s because he’s only showing one standard error as the uncertainty in his results, when what is relevant is the 95%CI. If he showed the 95%CI, it would be obvious that the results are not significant.

However, none of that matters. Why not? Well, because the claimed effect disappears when we use the full SST and sunspot datasets. Their common period goes from 1870 through 2013, so there are many more cycles to average. Figure 7 shows the same type of “folded” analysis, except this time for the full period 1870-2013:

full sst anomaly folded over solar cycle 1955-2003Figure 7. Sea surface temperatures from all solar cycles from 1870-2013, “folded” over the 11-year solar cycle as described in Shaviv2008

Here, we see the same thing that was revealed by the cross-correlation. The apparent cycle that seemed to be present in the most recent half-century of the data, the apparent cycle that is shown in Shaviv2008, that cycle disappears entirely when we look at the full dataset. And despite having a much narrower 95%CI because we have more data, once again there is no statistically significant departure from zero. At no time do we see anything unexplainable or unusual at all

And so once again, I find that the claims of a connection between the sun and climate evaporate when they are examined closely.

Let me be clear about what I am saying and not saying here. I am NOT saying that the sun doesn’t affect the climate.

What I am saying is that I still haven’t found any convincing sign of the ~11-year sunspot cycle in any climate dataset, nor has anyone pointed out such a dataset. And without that, it’s very hard to believe that even smaller secular variations in solar strength can have a significant effect on the climate.

So, for what I hope will be the final time, let me put out the challenge once again. Where is the climate dataset that shows the ~11-year sunspot/magnetism/cosmic rays/solar wind cycle? Shaviv echoes many others when he claims that there is some unknown amplification mechanism that makes the effects “about 5 to 7 times larger than just those associated with the TSI variations” … however, I’m not seeing it. So where can we find this mystery ~11-year cycle?

Please use whatever kind of analysis you prefer to demonstrate the putative 11-year cycle—”folded” analysis as above, cross-correlation, wavelet analysis, whatever.

Regards,

w.

My Usual Request: If you disagree with someone, myself included, please QUOTE THE EXACT WORDS YOU DISAGREE WITH. This prevents many flavors of misunderstanding, and lets us all see just what it is that you think is incorrect.

Subject: This post is about the quest for the 11-year solar cycle. It is not about your pet theory about 19.8 year Jupiter/Saturn synoptic cycles. If you wish to write about them, this is not the place. Take it to Tallbloke’s Talkshop, they enjoy discussing those kinds of cycles. Here, I’m looking for the 11-year sunspot cycles in weather data, so let me ask you kindly to restrict your comments to subjects involving those cycles.

Data and Code: I’ve put the sunspot and HadISST annual data online, along with the R computer code, in a single zipped folder called “Shaviv Folder.zip

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459 Comments
Bernie Hutchins
June 12, 2014 2:58 pm

Willis said in part –
“….until a man can demonstrate that he can do it right, I must confess that I pay little attention to his claims that I’m doing it wrong.”
Way too tough Willis! What if our climate modeling friends says this to us when we criticize their predictions?
Yes – I know what you mean.

milodonharlani
June 12, 2014 2:58 pm

Willis Eschenbach says:
June 12, 2014 at 2:42 pm
Tree rings, ice core data & radionuclides. Dendro isn’t much use for reconstructing T, but better at precip. Papers containing the relevant data have been commented or blogged upon here over the years.
Here’s a recent one:
A 3,500-year tree-ring record of annual precipitation on the northeastern Tibetan Plateau
http://www.pnas.org/content/111/8/2903.long
While not specifically dealing with the average 11-year cycle, it does provide annual data from which the signal is recoverable. The paper is more oriented toward trying to associate precip with T than with short-term solar activity.

milodonharlani
June 12, 2014 3:02 pm

Some other hydrological cycle correlations:
http://www.co2science.org/subject/m/summaries/dsccsi.php

milodonharlani
June 12, 2014 3:31 pm

I didn’t know that observations of this correlation were even controversial, so often have I seen it made in paleoclimatic literature:
http://www.clim-past-discuss.net/1/121/2005/cpd-1-121-2005-print.pdf
“Abstract
Spectral and wavelet analysis were performed on a tree ring width time series obtained
from a 2500 yr old cypress tree (Fitzroya cupressoides) from Costa del Osorno, Chile.
The periods for analysis were selected at 95% confidence level. Both periodicities
characteristic 5 of solar activity and climatic variations were found in this tree ring width
series. The 11 and 22 years solar cycle periods were present in tree ring data with a
confidence level above 98%. This indicates the solar modulation of climatic variations
is being recorded by the tree ring grown. However wavelet analysis shows that these
are present only sparsely. Short-term variations, between 2–5 years, are also present
10 in tree ring data, and are shown by wavelet maps to be a more permanent characteristic.
This time scale is a signature of ENSO events. Long-term variations, above 200
years, are also present in tree ring data. The spectral analysis performed in this work
shows that this species has the ability to record solar-ENSO variations that seems to be
affecting the local environment of tree growth, and also that this region was influenced
15 by ENSO events at least in the past 2500 yr interval covered by this study.”
We’re lucky that any “alerce” giants survived the mass burning of bosque nativo in southern Chile during the 19th century & subsequent logging.

milodonharlani
June 12, 2014 3:40 pm

Enigmatic cycles detected in subfossil and modern bog-pine chronologies from southern Sweden
http://treering.de/sites/default/files/TRACE_pdf/Volume_9/Edvardsson_et_al_TraceVol_9.pdf
Golden oldie, in which, perhaps anomalously, no correlation could be found between precipitation & tree growth, but an 11-year cycle was observed:
http://www.treeringsociety.org/TRBTRR/TRBvol21_16-27.pdf
And another Chilean study:
Solar and climate signal records in tree ring width from Chile (AD 1587–1994)
http://faculty.fgcu.edu/twimberley/EnviroPol/EnviroPhilo/AD.pdf
“Abstract
Tree growth rings represent an important natural record of past climate variations and solar activity effects registered on them. We
performed in this study a wavelet analysis of tree ring samples of Pilgerodendron cupressoides species, from Glaciar Pio XI (Lat: 491120S;
741550W; Alt: 25 m), Chile. We obtained an average chronology of about 400 years from these trees. The 11-yr solar cycle was present
during the whole period in tree ring data, being more intense during Maunder minimum (1645–1715). The short-term periods, around
2–7 yr, that were found are more likely associated with ENSO effects. Further, we found significant periods around 52 and 80–100 yr.
These periodicities are coincident with the fourth harmonic (52 yr) of the Suess cycle (208 yr) and Gleissberg (80–100 yr) solar cycles.
Therefore, the present analysis shows evidence of solar activity effect/modulation on climatic conditions that affect tree ring growth.
Although we cannot say with the present analysis if this effect is on local, regional or global climate, these results add evidence to an
important role of solar activity over terrestrial climate over the past 400 yr.
r 2006 Elsevier Ltd. All rights reserved.”
Maybe enough for now. I have to go increase the growth rings of the vegetation in my yard.

milodonharlani
June 12, 2014 3:44 pm

As a quondam historian of science, had to add this oldie but goodie, from 1933, finding that cycle:
http://ltrr.arizona.edu/sites/ltrr.arizona.edu/files/bibliodocs/Douglass,%20AE_Evidences%20of%20Cycles%20in%20Tree%20RIng%20Records_1933.pdf

milodonharlani
June 12, 2014 4:28 pm

The plants will survive another day.
Here’s what NASA has to say about precipitation & the sun. Note the usual ring of treed suspects. Hope all readers get a big kick out of the first quoted sentence:
http://science.nasa.gov/science-news/science-at-nasa/2013/08jan_sunclimate/
“In recent years, researchers have considered the possibility that the sun plays a role in global warming. After all, the sun is the main source of heat for our planet. The NRC report suggests, however, that the influence of solar variability is more regional than global. The Pacific region is only one example.
“Caspar Amman of NCAR noted in the report that “When Earth’s radiative balance is altered, as in the case of a change in solar cycle forcing, not all locations are affected equally. The equatorial central Pacific is generally cooler, the runoff from rivers in Peru is reduced, and drier conditions affect the western USA.”
“Raymond Bradley of UMass, who has studied historical records of solar activity imprinted by radioisotopes in tree rings and ice cores, says that regional rainfall seems to be more affected than temperature. “If there is indeed a solar effect on climate, it is manifested by changes in general circulation rather than in a direct temperature signal.” This fits in with the conclusion of the IPCC and previous NRC reports that solar variability is NOT the cause of global warming over the last 50 years.
“Much has been made of the probable connection between the Maunder Minimum, a 70-year deficit of sunspots in the late 17th-early 18th century, and the coldest part of the Little Ice Age, during which Europe and North America were subjected to bitterly cold winters. The mechanism for that regional cooling could have been a drop in the sun’s EUV output; this is, however, speculative.”

1sky1
June 12, 2014 5:13 pm

MiloDonHarlani:
Before taking up your suggestion that I elucidate the challenges of establishing or disproving any relationship between surface temperatures and SSNs, there are two corrections to my last comment that need to be made:
1) The over-estimation of sample VARIANCES of x and y series due to
extraneous components outside the narrow SSN frequency bands is by a factor
of ~4 when the ratio of total spectral power there is ~4, but the effect
upon the normalization of sample cross-correlation is the product of square
roots of ~.8 and ~.2, which is ~.4. Thus the cross-correlation between
coherent signal components in that narrow band may be UNDERESTIMATED by the
reciprocal of that factor, i.e., ~2.5, by the sample values shown here.
2) It’s Figure 2, not Figure 3, that I had in mind when talking about
LAGGING cross-correlation. I suspect that if the computations were
extended to, say, 22yrs, even higher values than those presently shown would
be obtained.
An elucidation to an audience that has scant grasp of spectral methods of
random signal analysis is daunting, but I’ll try to prepare one by Saturday.

milodonharlani
Reply to  1sky1
June 12, 2014 6:35 pm

Good for you, although IMO you’re underestimating the statistical erudition of your audience.
Thanks.

Konrad.
June 12, 2014 5:39 pm

Pamela Gray says:
June 12, 2014 at 11:20 am
———————————
Pamela,
The point about UV-A below the thermocline is that although shorter wavelengths of visible light
also penetrate this deep, they do not vary as much as UV between solar cycles.
In terms of detecting the mechanism of solar variation effecting ocean temps over the long term, below the diurnal thermocline is the place to look. This is where the ocean surface thermostat effect has low influence. This is where variation in UV-A can become a cumulative effect. If you just measure SST, you will only see diurnal and seasonal/annual solar variation. (unless you look at a full 150 years)

Pamela Gray
June 12, 2014 6:30 pm

Konrad, we have measures below SST and below the thermocline. We have physics related to solar insolation at the surface and at depth, meaning below the thermocline. Visible light variations represented as heat will bury UVa variations represented by heat. You have at best a mathematical calculation resulting in a tiny several decimal places fraction of other components far more capable of driving ocean heat trends below the thermocline. Your piddly small UVa driver will never be observed in situ and will never be a left over component after everything else is removed. The figure would be too small for serious scientists to write about.

Konrad.
June 12, 2014 8:12 pm

Pamela Gray says:
June 12, 2014 at 6:30 pm
——————————-
“Konrad, we have measures below SST and below the thermocline.”
The only thing vaguely accurate below diurnal thermocline we have is ARGO with only around 10 years of data (and that was corrupted to remove cooling).
“We have physics related to solar insolation at the surface and at depth, meaning below the thermocline.”
This is questionable. Most climastrologists still consider the oceans to be a “near blackbody” 😉
“Visible light variations represented as heat will bury UVa variations represented by heat.”
True, but the variation above the thermocline can be subtracted from the record below the thermocline, removing SW variation from cloud and season.
“The figure would be too small for serious scientists to write about.”
We are talking about a 25% variation in over 10 w/m2. Make believe “radiative forcing” from a CO2 doubling from make believe pre industrial levels is only supposed to be around 3 w/m2. I would suggest that both figures should be too small for serious scientists to worry about. But that doesn’t seem to stop anyone.

milodonharlani
June 12, 2014 8:19 pm

Willis Eschenbach says:
June 12, 2014 at 7:43 pm
As you must know, the 11 year cycle is an average.
I wonder why you didn’t pick some of the citations in which the authors themselves found at a high confidence level a signal near this average &/or its 2x.
It’s your reputation as a statistical analyst seriously interested in science rather than special pleading that’s at issue here, not my reputation as a provider of scientific studies.
Which is it? Are you really interested, as you state, in finding out if there is a signal, or in supporting your belief that there isn’t? Ignoring studies that counter your faith opens you to the charge that you are indeed intentionally sailing south from Europe when the New World lies to the west.
I can proffer just as many based upon ice core papers as tree rings. But how about just addressing those studies I linked which found statistically significant signals? You seem heavily invested in no signal. That’s not just an unscientific attitude, but anti-scientific on a Mannian scale. Which I’m sorry to say, since I expected a more thorough analysis from you.
IMO the average 11 year cycle is well supported in precipitation studies since early in the last century. As I noted, this conclusion isn’t even controversial. To vitiate this conclusion, well supported by at least 80 years of observations & even admitted by the Team to some extent, will require more than two dismissive hand waving drive-bys. Well, one drive by & one analysis of a paper whose point wasn’t the 11 year cycle, although I found it in there on that one, too.
I’d be happy to offer five of the many tree ring studies I’ve cited, if you’re willing to undertake them. How about these, in chronological order:
1) Various, 1933:
http://ltrr.arizona.edu/sites/ltrr.arizona.edu/files/bibliodocs/Douglass,%20AE_Evidences%20of%20Cycles%20in%20Tree%20RIng%20Records_1933.pdf
2) Anyone of the following, which include not just tree rings but other vegetative proxies, ie peat bogs:
Rigozo et al. (2002) detected an 11-year cycle in tree-ring width data from Brazil over the period 1837-1996; and Black et al. (1999) reported finding a 12.5- to 13-year signal of climatic variability in the North Atlantic Ocean over the past 825 years. Additionally, Dean et al. (2002) found an approximate 10-year cycle in a lake sediment core obtained from Elk Lake, Minnesota, USA, covering the past 1500 years. Both Rigozo et al. and Dean et al. implicate the sun as the likely source of the approximately 11-year periodicity noted in their records. Black et al. are less enthusiastic about this possibility, but they feel the sun is responsible for driving centennial-scale climate oscillations in their record.
In an analysis of tree-ring chronologies from northeastern Mongolia, Pederson et al. (2001) report “possible evidence for solar influences” on the regional hydrologic cycle. For the period 1651-1995, they reconstructed annual precipitation and streamflow histories for this region, which upon subjection to spectral analysis revealed significant periodicities of 12 and 20-24 years that are believed to be solar-induced.
Nearby in China, Xu et al. (2002) examined plant cellulose ð18O variations in cores retrieved from peat deposits at the northeastern edge of the Qinghai-Tibetan Plateau (32° 46’N, 102° 30’E). Power spectrum analyses of these data revealed multi-decadal periodicities of 79 and 88 years, “suggesting,” in the words of the authors, “that the main driving force of Hongyuan climate change is from solar activities.”
Neff et al. (2001) also provide evidence for a solar-induced influence on the hydrologic cycle. For the period 9,600-6,100 years before present, they investigated the relationship between a 14C tree-ring record and a proxy record of monsoon rainfall intensity inferred from calcite ð18O data obtained from a stalagmite in northern Oman. Their investigation revealed an “extremely strong” correlation between the two data sets; and spectral analyses revealed statistically significant decadal and multi-decadal periodicities of 10.4, 26 and 89 years for the 14C tree-ring record, and 87 years for the ð18O record.
3) Chile, 2005:
http://www.clim-past-discuss.net/1/121/2005/cpd-1-121-2005-print.pdf
4) Chile, 2007:
http://faculty.fgcu.edu/twimberley/EnviroPol/EnviroPhilo/AD.pdf
5) Some modeling to test observations, but pretty suggestive, 2013:
http://journals.ametsoc.org/doi/abs/10.1175/JCLI-D-12-00843.1
To which I’d add Joe d’Aleo’s solar study, but that’s about T rather than precip.
But if you’re satisfied that you’ve made your case, you’re welcome to that cocoon. IMO my citations aren’t a garbage heap & should be of interest to anyone actually trying to find the signal you claim to be after.

Reply to  milodonharlani
June 13, 2014 6:40 am

@milodonharlani
interesting discussion
from my own investigations, I reported some of my own results here:
http://wattsupwiththat.com/2014/06/06/sunspots-and-sea-surface-temperature/#comment-1660775
as far as rainfall goes I analysed some good data from a South African station going back to 1927. (-25 degrees latitude)
I report the average annual rainfall in the years:
1927-1950 611.7
1951-1971 587
1972-1995 596.1
1996-2013 641.2
Note the difference in sign of the binomials (100% correlation) for the periods indicated for (minimum) temperature and rainfall?
Clearly there are cycles, for everyone to find for him or herself

milodonharlani
June 12, 2014 8:21 pm

Willis Eschenbach says:
June 12, 2014 at 8:00 pm
Because they found a statistically significant association, even if weak, which is what a reasonable analyst would expect. IMO.

Catherine Ronconi
June 12, 2014 8:39 pm

Willis Eschenbach says:
June 12, 2014 at 8:30 pm
What about all those other citations?

Pamela Gray
June 13, 2014 11:05 am

Konrad, here is my statement: “Visible light variations represented as heat will bury UVa variations represented by heat.”
You responded, “True, but the variation above the thermocline can be subtracted from the record below the thermocline, removing SW variation from cloud and season.”
Huh???? Go chew on this paper. Something as simple as changes in “chlorophyll a” produced such a variance in shortwave visible light penetration through the mixing layer that the overturning circulation flow slowed. The limit to the paper in terms of your speculation is that they used visible light (300-750nm) as their source which is longer than the narrow UV (10-310nm) you are focused on, yet it is still considered shortwave and able to penetrate far more deeply than infrared. Infrared is considered to be longwave (700nm to 1 mm) and solar infrared is a piece of that (750 nm to 2500nm). That’s why we call solar infrared “shortwave” infrared, because it is in the shorter wave length band of infrared, but it is the shortwave visible light band, not the infrared band, that penetrates deeply into the ocean water column with considerable energy to heat it compared to UVa.
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&cad=rja&uact=8&ved=0CB0QFjAA&url=http%3A%2F%2Fwww.ldeo.columbia.edu%2F~csweeney%2Fpapers%2FSWpen_Sweeney.pdf&ei=SC2bU42DNouyyASt8YDABg&usg=AFQjCNFkYbr5CXptTpORygbB0K6ax21mCg&sig2=Wx2ZpMNVhvdtyUMEjwG1DA
So why did these clearly well-informed scientists endeavor such a task using the visible lightband and ignore UVa, which you seem to think is such a big deal? Here are a few reasons. Below the thermocline layer, there is an infinitesimally small amount of mixing (with the exception of the overturning locations) due to the strong thermocline boundary found between the deep layer and the always expanding and contracting mixing layer. The sub-thermocline is by far the deepest most voluminous layer, strongly replacing warm water which rises through the thermocline barrier hastily into the mixing layer and beyond at the overturning circulation locations in the oceans. Your piddly amount of UVa heating riding that overturning would be undetectable there, if it exists at all, and certainly nowhere else. Seriously.

Pamela Gray
June 13, 2014 11:27 am

HenryP, your investigation is extremely poor from what I have read so far. Do you have this written up with introduction, literature review, problem statement (which usually is stated in terms of either new information needed for a new or potentially new observation, or an accepted hypothesis is not sufficient to explain a current observation), your hypothesis against the null hypothesis, methods, results, discussion, and conclusion? You said you have investigated but I can’t find it anywhere.

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