## Foundation SST: how to define?

In support of ToR, foundation SST and near surface gradients topics

### Foundation SST: how to define?

Alexey Kaplan sent this email to the GHRSST Science Team on August 1, 2011:

Dear Colleagues,

First of all I, personally, think that the current SSTfnd definition on the GHRSST website is fine. I thought so as well about it's previous version, which was on the website until a year ago or so, and about the definition given in Donlon et al BAMS paper, etc. I am happy with all these definitions b/c i find foundation SST a beautiful liberating concept, helping to relate the richness of the temperature distribution in the ocean surface layer with the simplified conceptual view of ocean heat balance on longer than a day time scales, which traditionally uses mixed layer concept and its temperatures, etc. And b/c i think i know what Craig, Peter, and others mean by SSTfnd.

I however, have a substantial history of frustration trying to get people from outside of GHRSST community (especially ocean modelers but not only) to accept SSTfnd as a legitimate physical variable. The impression I've received from them about their problem with it (although not in as many words) is that we do not provide a rigorous definition of SSTfnd.

1. What do I mean by a rigorous definition?

Let me give you an idea of what could be a general rigorous definition of such a variable. Suppose, hypothetically, that in a certain location x of the ocean we know, continuosly in time and depth, all temperatures from z=z_skin to some significant depth z=H: T(z,t). We identify the following important time instants t=t_r of the local sunrise, t=t_s of the following sunset, and t=t_sp of the previous day sunset. Now, could anybody write for me a formula that computes SSTfnd in terms of these data: SSTfnd=F[T(z,t), t_sp, t_r, t_s]?? That would be a rigorous definition for SSTfnd.

2. Why is it important?

As soon as we have such a formula, modelers will have no way of not accepting SSTfnd as a legit variable. They will only have to find a good way to relate our definition to their model variable(s) -- so the ball will be in their court (even if it would require incorporating diurnal variability models into ocean models). Ditto for the SST intercomparison studies accross platforms and data sets.

The problem with how things are presented on the website at the moment is that the SSTfnd definition is somehow mixed with the SSTfnd properties. I guess what turns some people off is that they suspect that the quantity we define might not sometimes exist. Then what a modeler to do? Such things should be addressed heads-on in the definition: if we want to define a variable that sometimes or someplaces does not exist (though I don't think this is the case), formally we can do that (like a depth of the 22 degree isotherm, for example), but we have to explain in the definition or in the follow-up to it, under what conditions the variable we have defined exists.

3. What are the possible problems and what do we do next after we rigorously defined SSTfnd

Now, it might be that the functional F[*] for SSTfnd that I have made up above to depend only on the time-dependent temperature profile and on sunset/sunrize instances, just cannot be written, but has to depend on surface heat fluxes, wind, etc. I don't think so, but if it is the case, this will be fine too. Let's have that complicated definition introduced first. Because after there is a rigorous definition of SSTfnd that depends on whatever it must depend on, we always can discuss simpler, more convenient, definitions for its approximations.

But what we need to start with is a rigorous definition of SSTfnd; after that we can discuss various properties of SSTfnd under this definition, for example that at the sunrise t=t_r all T(t,z) for z=subskin--10m collapse to roughly SSTfnd, or that if wind>2m, then instantaneous T(t,z) for the entire bunch of depth (say up to 10m) are very close to SSTfnd (ideally, we would have accuracy for these statements too).

Finally, and I am afraid that this issue might not be quite settled for the entire comunity yet, what is the time grid of SSTfnd? Does it depend on time t instantaneously or it is a variable that is defined as constant for a day -- in a manner of a daily average, or max, or min daily temperatures in meteorology? If so, the day for the SSTfnd is defined how: from a sunrise to a sunrise? or as a period between midpoints of the two sequential sunrise pairs (i.e. ~24hr surrounding a sunrise instant).

4. A specific proposal for the SSTfnd definition

Enough of requests and problems, here is my specific proposal. Let us define SSTfnd as a subskin temperature at (right before) sunrise: SSTfnd=T(z_subskin,t_r). From reading various definition versions and from the G12 discussion it seems to me that this simple definition is consistent with however else SSTfnd was ever presented. In other words, we can define SSTfnd as basically a sunrise subskin ocean temperature and then disuss/claim that it has all the good simplifying (collapsing to) properties compared to temperatures at other depths and times. To clinch the time-grid definition as well, I'd attribute suchly-defined SSTfnd to the period from the previous to the current sunrise: [t_rp, t_r].

If this definition for SSTfnd is not acceptable for, say, esthetic reasons, i.e., if SSTfnd should not be defined in a such narrow (in terms of time & space) way, I'd avocate to simply introduce subskin-sunrise SSTsssr as a proxy for SSTfnd; the latter, if necessary for the sake of a concept, could be defined more loosely, but for practical purposes it could always be approximated via SSTsssr... And that would be a big step for the practical value of the SSTfnd definition.

Thank you for reading this long email,
Alexey

In the period August 2-3, 2011, there were email comments on this topic from Mike Chin, Jorge Vazquez, Doug May, Peter Cornillon, and Peter Minnett. Jorge Vazquez suggested to have further discussion of the definitiond for SSTfnd on this Forum rather than over the email. Therefore I've posted below all email contributions made to this discussion up to now. (I've posted them in their chronological order, but, slightly confusingly, they appear on this page from the bottom up, the "freshest" one placed right below this message.) So please post all further comments here (by "replying" to this topic or to replies to it that are already posted)! --Alexey (August 4, 2011)
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### Re: Foundation SST: how to define?

Lurking on this topic

Hi All... as a sort of test post to the forum, I wanted to say I have been following the discussion but have nothing to add yet.

Thanks, Ken
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### Re: Foundation SST: how to define?

Comments from Peter Minnett

On Thu, Aug 4, 2011 at 1:20 PM Peter Minnett wrote:

Dear Alexey et al.

I've been following the email exchanges with interest and agree with much of what has been written. (And I am very gratified to know that in some small way I have contributed to Alexey's "liberation"!).

I have also exchanged emails with Jorge about migrating the discussion to the GHRSST Forum, and am awaiting my password. I encourage others to register and continue the dialogue there.

It is important, I think, that Tfnd be defines as a concept and not through a measurement process. Temperature itself is defined through the Zeroth Law of Thermodynamics, and "thermometer" or "degrees Celsius", for example, are not part of the definition. It is also important that it not be defined by a time of day or a depth....

More on the Forum.... Keep your thoughts and comments coming.

Best regards,

Peter
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### Re: Foundation SST: how to define?

Qualitative Concepts versus Quantitative Formal Definition

Dear Peter,

I, of course, agree that SSTfnd should be defined "as a concept and not through a measurement process." But I am not sure that I understand what you mean when you say that it "should not be defined by a time of day or a depth," b/c after all SSTfnd is not as fundamental a concept as the temperature itself, hence it's bound to be related to the surface ocean structure, which we do conceptualize in terms of time (day vs night) and depth (skin, subskin, diurnal thermocline, etc.) If you are saying that SSTfnd := T(1mm, 3am LST) is unacceptable as the principal definition for SSTfnd, I understand that. But if you'd fail SSTfnd := T(z_subskin, T_sunrise) for the same general reason, I understand this less. Could you explain your thinking a bit more?

I went over many published references to SSTfnd. The concept is explained in similar but sometimes subtly different ways. In one of the earliest published (in a monograph!) references, Ian Robinson says in his 2004 book, with the reference to the personal communication with yourself, that "the temperature measured or estimated at the base of the diurnal thermocline should be referred to as the foundation temperature, SSTf, or Tfnd." Is this still applicable? (I know that the wording in his 2010 book is different, but don't quite see that it contradicts this earlier definition.)

Let me for now proceed assuming that this definition is still OK. It is a reasonably clear conceptual definition: SSTfnd is the temperature at the base of the diurnal thermocline. It assumes that an identifiable diurnal thermocline structure is present (or else a surface temperature profile is flat below the skin layer), but it's all right. However, if we are to produce data sets of the SSTfnd, we still need a formal definition of the SSTfnd, i.e. a description allowing us to say unambiguously, for example, what "true" SSTfnd values would be if the full true T(z,t) profiles were available.

This is not a trivial task: let us look at a similar case, for the seasonal thermocline depths and temperatures. I don't think that very many people are producing such data sets now, despite their tremendous importance and reasonable data availability. I think this happens b/c of the inherent ambiguity of such values for realistic ocean temperature profiles. People would rather report the entire profile, or settle for an isothermal substitute (e.g. reporting the depth of the 20C isotherm in the tropics rather than thermocline depth). I think sometimes people would define the thermocline depth as the depth at which the derivative T'(z) is largest (by abs value) or, conceptually equivalently, where the second derivative T''(z) is zero, but I don't know if there are popular data sets based on such definitions now. I suspect that these definitions are not actively used in data set production b/c they are somewhat contrived but also b/c they can give strange results when the real temperature profile is not well-approximated by a two-layer ocean...

There is a somewhat similar situation with the ocean mixed layer depth (MLD). But there are popular data sets of MLD now. The producers are very explicit how their various versions of MLD are defined from ocean profiles: these are sometimes obtained using temperature profiles, or salinity profiles, or density, and the threshholds for these properties used in the definition are made clear (and are bound to be somewhat arbitrary).

I think that with the SSTfnd we are in similar shoes with the MLD people and with would-be seasonal thermocline temperature producers, in the sense that we have a good conceptual definition: SSTfnd := T(z_bdt,t), where bdt stands for the "base of the diurnal thermocline," but it is still a qualitative description, b/c even with the full T(z,t) in hand, we cannot unambiguously decide what z_bdt is (especially in some non-standard cases). Therefore we still need a formal quantitative definition for the SSTfnd. But we do get a bit of a break, b/c if it can be assumed that the diurnal thermocline for sure collapses by dawn, z_bdt cab be any shallow enough depth below the skin layer (e.g., z_bdt = z_subskin), as long as we take t=t_sunrise... That is, unless, there is a need to use a more sophisticated definition in order to accommodate less standard cases.

Cheers,
Alexey
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### Re: Foundation SST: how to define?

Agree + emphasize + another strawman

Dear Peter,

I think we are pretty much in agreement on everything so far. But I want to highlight one of your arguments, b/c I think it is very important for getting to the bottom of this (or at least to the bottom of the diurnal thermocline -- forgive the bad pun). You wrote:

My point is that cooling can continue past sunrise. And, although I haven't thought it through, it seems conceivable to me that warming can also begin before sunrise if a very warm air mass moves over colder water although the physics are quite complicated here I think. This would be very rare however and I don't want to get stuck on it; it's the continued cooling that I was thinking about originally.

Well, if we want to have a universally acceptable definition, we cannot ignore such cases either.

The issue really is that in introducing the concept of SSTfnd people seem to make some assumptions of various levels of implicitness, namely that the surface heatflux into the ocean is always negative at night and that outside of the skin layer temperature is always non-increasing with depth, etc. We could agree on such a set of assumptions and then to define in the easiest possible way the SSTfnd that goes with it. In this case, if assumptions do not hold, we either say that SSTfnd is undetermined or give a separate definition for SSTfnd in such cases. We have to agree on that too (if SSTfnd has to be defined in all cases).

Naturally we could have a few different sets of assumptions and thus a few different definitions of SSTfnd that correspond to them. It's not unlike defining ocean mixed layer depth (MLD) according to the exceeded threshhold in temperature, or salinity, or density. And BTW, in case of MLD the threshhold value is a free parameter too...

Anyway, why did i want to emphasize your argument above? B/c it describes a possibility that the diurnal thermocline could start building up before sunrise, which contradicts a standard set of assumptions, I believe. In case of this happening, SSTfnd either should be undefined or re-defined by some special provision.

I'm starting to think that
SSTfnd := min{SSTsubskin}, where the minimum is taken over the period from a sunrise to sunrise,
could be a more universal definition for the SSTfnd, as it only uses an assumption of the non-increasing T with depth (from the subskin level and below).

Best,
Alexey
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### Re: Foundation SST: how to define?

Comments from Peter Cornillon + a reply from Alexey

On Wed, Aug 3, 2011 at 9:32 AM Peter Cornillon wrote:

Hi Alexey,

I like (probably should have a little hand with a thumbs up here, but I'm not good at the imotocons or whatever they are called) your insistence on a formal definition for the foundation temperature. My only concern with the definition that you propose is that no satellite-borne sensors make a measurement that fits this definition at present: infrared radiometers on geosynchronous satellites measure the skin temperature just before sunrise and microwave instruments on polar orbiting satellites measure the subskin at around 2AM LST. At the end of your message below you say that how the SSTfnd is observed is for another e-mail. It might be worth folding this into the discussion at this point since a quantity that is not directly observable (or at least as directly observable as temperatures in the ocean are from satellite-borne sensors) may lead to more confusion in the future - might as well come up with a definition that is the least dependent on ocean model assumptions as possible and that includes observations from the broadest range of satellite-borne microwave and infrared radiometers. I realize that the problem with such a definition is that it might applicable to different instruments flown on satellites in different orbits in the future, but the new observations could be corrected to the definition that we agree on now. Although maybe not quite as satisfying, the above would tend to push us toward a definition of the foundation temperature based on observations near 2AM LST. Infrared observations could be corrected to the subskin with some error (probably similar to the error that would be made by such a correction just before sunrise) and microwave observations would require no correction in space or time. Just before sunrise is a better time to estimate the foundation temperature, but it is still just a point on a continuously varying curve. On very cloudy, cold days, the subskin temperature could well continue to decrease beyond sunrise. Parts of the globe with longer nights than other parts of the globe will have likely have somewhat colder foundation temperatures both because there is less solar insolation during the day and because the cooling period has not been as long... And, as was noted in the discussion at Edinburgh, the foundation temperature often increases over several days before quickly falling back to a lower levels when winds pick up.

Peter

On Wed, Aug 3, 2011 at 12:02 PM Alexey replied

Hi Peter,

My understanding is that SSTfnd is a concept (introduced for the purpose of simplifying the description of ocean surface temperature variations). It should be what its authors say it is, and should be defined accordingly. (I'm not the author; I'm just proposing strawmen for the rigorous definition, based on what I think the authors mean)

It would be great if we could define SSTfnd so that it was easily directly observable, but on the other hand, there are plenty of useful conceptual variables that are never directly observable: global mean temperature, available potential energy (APE), convective APE, ocean volume transports, z500 streamfunctions, meridional overturning streamfunctions, NINO3 at the last glacial maximum,... -- none of these are observable, but all of them are precisely defined as soon as the full (or partial) state of the ocean/atmosphere is described. Because all of them have these precise definitions, people can estimate them from available observations. The formulas for various estimates are legion, they are all different and depend on the observational data set(s) used; these estimation formulas cannot substitute and should not be confused with the fundamental definition of a target conceptual variable.

As to how to estimate this never observable SSTfnd if it's defined as for example SSTsubskin(sunrise), you wrote about possible approaches better than I could, in particular:

Infrared observations could be corrected to the subskin with some error (probably similar to the error that would be made by such a correction just before sunrise) and microwave observations would require no correction in space or time.

Further,

Just before sunrise is a better time to estimate the foundation temperature, but it is still just a point on a continuously varying curve.

t_sunrise and "just before sunrise" (t_sunrise-0 ?) are the same thing for me. But sunrise is not "just a point on a continuously varying curve": it is a point at which diurnal heating potentially can commence (before sunrise, it cannot).

On very cloudy, cold days, the subskin temperature could well continue to decrease beyond sunrise.

Therefore the SSTfnd for the day that starts with this sunrise will probably end up lower than the previous day's SSTfnd.

Parts of the globe with longer nights than other parts of the globe will have likely have somewhat colder foundation temperatures both because there is less solar insolation during the day and because the cooling period has not been as long... And, as was noted in the discussion at Edinburgh, the foundation temperature often increases over several days before quickly falling back to a lower levels when winds pick up.

To me these all sound as legit properties of SSTfnd: they could/should be discussed and documented.
BTW, the definition SSTfnd := SSTsubskin(t_sunrise) will have all these properties, I believe.

Cheers,
Alexey
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### Re: Foundation SST: how to define?

Comments from Doug May + a reply from Alexey

On Tue, Aug 2, 2011, at 1:58 PM Doug May wrote:

Thoughtful discussion. Rather than limit the thought process to ocean modelers, you need to seek consistency with the atmospheric modelers. Both are trying to improve forecasts of heat flux exchange to better couple together. I am not aware of atmospheric modelers attempting to define air temperature, wind speed or specific humidity as some value at sunrise at a specific height above the ocean. They appear to be going with a 10m value that varies every hour despite the fact that few in situ and no satellite sensors measure those parameters at that height.

Doug

On Tue, Aug 2, 2011, at 5:07 PM Alexey replied:

Dear Doug,

I'm discussing the issue not from a particular modeler's prospective (ocean or atm) but from a data provider's prospective for whom modelers (incl. ocean modelers) are clients. Observational / data providers community needs a concept of SSTfnd - no question about that. As far as modelers are concerned, the issue here is helping provide a practical interface to their variables. An ocean modeler who receives a data set of SSTfnd wants to know how it is connected to model variables. We are not going to answer this question for any of their models completely, but we've got to go halfway by defining SSTfnd explicitly in terms of relevant physical variables. After this is done, it's their (modeler's) job to cover the second half of the way, from model variables to our SSTfnd definition, since ocean models (theoretically) do describe a physical reality of the ocean.

Of course, the need of rigorous (formal) definition for SSTfnd is bigger, in my view, than the interface to the models.

Regarding your analogy of variable usage/definition in atmospheric models, indeed, 10m height is taken purely for standardization purposes, to fix a point on the surface boundary profile; 5m and 15m, used consistently throughout, would work as well, I suspect. But a right analogy here is not what modelers tend to use within the models, as physical variables, but what they have to use in order to compare model results to the reality. There, whenever a quasi-cyclic physical process is involved (in our case - diurnal cycle), for example, seasonal cycle, for the interpretation of model results they would often use specific cycle phases for some index definitions, e.g., winter values, as opposed to summer, or all-season values.

As an example of possible communication between data producers and atm modelers that is more analogous to our case, take an atmospheric modeler who wants to verify surface air temperature simulation by the daily USHCN station data. There are three daily temperature variables in this observational data set: Tmin, Tmax, and Tmean. The documentation for this data set explains, in effect, that these are meant to be observations (after various adjustments) of the following functions of surface air temperatures at station locations T(t): Tmin=min{T(t)}, Tmax=max{T(t)}, where min and max are taken over all times from local midnight to local midnight; further,
Tmean = (Tmin+Tmax)/2. The latter might be a surprise for some modelers; nevertheless, all three variables have been rigorously defined. What a modeler will do with this info, i.e. would they actually implement the calculation of Tmin, Tmax, Tmean from the model time-step data, or will do a cheaper approximation from already saved 6- or 3-hourly means, is their business. It is not of the most crucial concern for data producers, since they've done their part: provided for a public use a data set of rigorously defined atmospheric variables.

I'd like to have SSTfnd formally defined at the same level of clarity as weather stations' daily air temperature variables Tmin, Tmax, Tmean...

Thanks,
Alexey
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### Re: Foundation SST: how to define?

Comments from M.Chin and J.Vazquez + a reply from Alexey

On 8/1/11 7:13 PM, Mike Chin wrote:

Hello Alexey,
I also think this seems to be an important issue for the ocean modelers.

Question: would SSTsubskin be more rigorous than our present
SSTfnd because the subskin depth is better defined? (e.g. 1mm depth?)

With regards,
Mike

On Tue, Aug 2, 2011 at 1:00 AM, Jorge Vazquez wrote:

Alexey,
I really appreciate your thoughtfulness in writing this email. Let me say
right off the bat I like your definition, basically because at least it is
honest. From a user perspective (my limited emails and exchanges with the
modeling community) when we say we are providing a foundation temperature,
are we really being true to the definition. From all I have heard it seems
like we are not. Can we really, we degree of confidence say, we are
providing the temperature at the depth where the SST is not affected by
the diurnal cycle.

What I like about your definition is that at least the user community
knows exactly what they are getting and we know we can provide it
accurately. The sub-skin definition is a bit worrisome to me because it
might get confused with the definition for what the microwave instrument
measures.

My two cents worth.

Thank you,
Jorge.

On Tue, Aug 2, 2011 at 3:41 PM, Alexey replied

Dear Mike and Jorge,

First, I was using z_subskin meaning the shallowest water depth at which the temperature is free of the skin-effect. I think this usage is consistent with the definition for SSTsubskin as given on the GHRSST website: the subskin temperature (SSTsubskin) represents the temperature at the base of the conductive laminar sub-layer of the ocean surface.

Second, as a useful analogy, let's talk about SSTsubskin for second. In my view we (almost) have a rigorous definition there: I assume there are people here who can easily define depth of the skin-layer and thus z_subskin; and they can offer a formula for it: either it is a mean or conservative constant (1mm?) or a function of some parameters. In either case, if I'm given a profile T(z) and these parameters, I can compute z_subskin and return SSTsubskin :=T(z_subskin). Therefore I have a rigorous definition for SSTsubskin here, for what it's worth (conditional on the existence of the rigorous definition for z_subskin).

Note that the 2nd sentence in the SSTsubskin paragraph on the website says "For practical purposes, SSTsubskin can be well approximated to the measurement of surface temperature by a microwave radiometer operating in the 6-11 GHz frequency range, but the relationship is neither direct nor invariant to changing physical conditions or to the specific geometry of the microwave measurements." This is VG too: after SSTsubskin is defined we are saying how it is usually measured, and the caveats are given. However, a small step is missing here: we could also define SSTmw, i.e. a function of surface temperature that would be measured by a microwave instrument, in the ideal settings (with no error in the instrument or in the atmospheric corrections). Such SSTmw would be dependent on the instrument characteristics through its depth-weighting function W(z):
SSTmw := Integral_0^infty {W(z)T(z)} dz (W is assumed to integrate to 1 from 0 to infinity). So this SSTmw is what microwave instruments actually measure, albeit with various errors.

Therefore we have here three rigorously defined variables: SSTsubskin, SSTmw, and SST1mm:=T(1mm). The first and third are useful descriptors b/c they naturally appear on the temperature profile pictures; the second is useful b/c that's a true value that a given MW instrument attempts to measure. Having these definitions, one can discuss how and why and with what discrepancies these three variables are close to each other.

I'd like to have the similar level of rigor (or if you will, explicitness) for the SSTfnd. In other words I'd like to have a formula (or a computer algorithms that uses fairly fundamental variables and relationships) that takes in T(z,t) and whatever else parameters it needs (times of sunrise/sunset, winds, fluxes, whatever) and returns SSTfnd. In other words, a formal definition SSTfnd := F[ T(z,t),...].

Finally, please note that I did not have equated SSTfnd=SSTsubskin.
Instead, I proposed (only as a strawman, mind you)
SSTfnd := T(z_subskin,t_sunrise),
b/c I thought it fits the physical description we now have on the website (and particularly illustrated by the bottom figure there: https://www.ghrsst.org/images/Tfnd.jpg ).

So in effect I did propose
SSTfnd := SSTsubskin(t_sunrise),
but note that SSTsubskin here is to be taken at the sunrise (this is an attempt at a definition, not a recommendation for an observational strategy!) and to be attributed to the entire previous day (in the picture on the website it seems to be attributed to the following day). Therefore in this definition it is not a continuous function of time, but has one value per day (which day: the previous or the following -- has to be decided).

In case that it is not easy to give a fundamental definition of z_subskin, I can propose a second strawman
is SSTfnd := T(1mm,t_sunrise) = SST1mm(t_sunrise)

How to estimate suchly (or differently) defined SSTfnd from available observations is a question for another email.
A formal definition for SSTfnd, in a sense of SSTfnd := F[ T(z,t),...], is all I care about for now.

Best regards,
Alexey
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