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Lucia has a bad week on partial derivatives

Posted By __David Evans__ On October 20, 2015 @ 3:12 pm In __Global Warming__ | __Comments Disabled__

We welcome collaboration, but empty, uninformed ill-will doesn’t help the unresourced skeptics to beat the billion dollar green machine. It’s time for Lucia to admit she got it wrong. Lucia’s second post failed to clarify anything. She didn’t acknowledge that she had not found a single real mistake David’s work, nor did she apologize for getting so much wrong. Having decided everything David was doing was “crud” after reading two paragraphs, she now has the onerous and pointless task of trying to defend a hasty uninformed position.

Lucia didn’t *have* to dig the hole deeper but she tried. To turn her mistaken accusations into something useful she transparently shifts the goals and won’t join the dots. Evans was critiquing Held, Soden, and Pierrehumbert. He described how they relied on partial derivatives of dependent variables, impossibly holding everything else constant in climate and thereby incurring unknown errors. Lucia now says “but they could’ve done it a different way without them” and perhaps hopes no one notices the unspoken admission that David Evans was right.

The bizarre thing is that you don’t need a maths degree to know her method is silly on its face. In the real world there is no way to hold temperature, clouds, humidity or anything constant while changing the surface temperature – and mathematical trickery won’t make it so. (Lucia packs the changing variables under a term she calls *R*_{p} — which is a bit like hiding income in an offshore tax avoidance scheme.) The bane of basic climate analysis that inevitably it has to use partial derivatives while unrealistically holding all else constant — the issue is ignoring the risk and uncertainty that brings.

Notably, Lucia didn’t link to our reply to her first post (despite the post being a request for a reply from David). Nor did she email us either time she published (despite having our emails). Does she want the truth, or just to indulge in point scoring snark? If she’s hoping for an easy target, she’s picked the wrong people. – Jo

by Dr David Evans, 20 October 2015**, David Evans’ Basic Climate Models Home, Nomenclature**

David’s post 3: New Science 3: The Conventional Basic Climate Model — In Full

David’s post 4: New Science 4: Error 1: Partial Derivatives

Lucia’s first post on our posts, at the Blackboard: Questions to David Evans: What do you mean about partial derivatives?

Our response to Lucia’s first post: Lucia has a bad day with partial derivatives

Lucia’s second post, at the Blackboard: Held & Soden without “hypothetical partials”

Our response to Lucia’s second post: Lucia has a bad week on partial derivatives <—- This post

As with Lucia’s first post, having read carefully through her second post and its comments, I’m still waiting for Lucia to find any mistakes in my posts or even made any informed criticism of them.

Lucia’s first post alleged I made had mistakes with differentials in post 3 of the series of posts about basic climate models. We showed her in more detail how to do them in our reply, applying an online class note from MIT that I had referenced in the first post. Lucia’s second post has no mention at all about any mistakes with differentials, which I’ll take as implicit acknowledgement that I was right — as were Held and Soden ^{[1]}, and Pierrehumbert ^{[2]}, whose model development I was copying. No retraction or apology from Lucia though. No one reading Lucia’s two posts would know that I was correct about differentials all along and Lucia was wrong; they’d get quite the opposite impression.

In her second post, Lucia moves on to attacking my claim in post 4 that the partial derivatives in the conventional basic climate model, such as the Planck feedback (the reciprocal of the Planck sensitivity, λ_{0}), are “strictly hypothetical”, and she claims that the basic model “can be developed without resorting to these “strictly hypothetical” partials”. As with her first post, Lucia was quite disparaging of me but did not bother to email us — we weren’t so discourteous (we email Lucia immediately we post about her).

As anticipated (see Comment 37.1.1), Lucia makes her own derivation of the basic model, aiming to avoid partial derivatives where “everything is held constant”. She thinks she succeeds, but she fails. An alternative approach that got around the obvious problem with the Planck feedback (that it impossibly relies on holding all climate variables constant except OLR and surface temperature*) would clearly be an improvement, but it would almost certainly have been discovered decades ago in a problem space as small and as intensively researched as this.

We’ll skip over most of the detail of Lucia’s more complicated approach, and just focus on the crucial piece of her development. Consider the OLR function, *R*. Lucia does not want to use the arguments used in post 4 for *G* (or downward flux, ASR – OLR):

where the surface temperature is *T*_{S}, there are *n* driver variables *V*_{1},…,*V _{n}*, and there are

and let *R* depend on feedbacks via *T*_{S} . Fair enough.

But Lucia wants to avoid using the partial derivative ∂*R*/∂*T*_{S} where all the drivers and feedbacks are held constant — this is the “Planck feedback” (which isn’t really a “feedback” as the term is used elsewhere in this series — see post 2 and post 3 where it is discussed and defined). This is important so I’ll stress it: in the standard development in post 3, ∂*R*/∂*T*_{S} means the derivative of *R* with respect to *T*_{S} when all the drivers and feedbacks are held constant — and since (nearly?) every climate variable depends on feedback, basically this means holding “everything else constant” except the OLR *R* and temperature *T*_{S}.

Lucia splits *R* into two parts, one where drivers and feedbacks are held constant and one where they can vary, by setting

where she defines *R*_{p} (which she also writes as “*R*_{pe}“) as the OLR that “would arise on an earth whose temperature is *T*_{S}” and the values of the feedbacks are the same as the feedbacks “of the current earth”. Also, *R*_{p} “does not vary with” the drivers. That is, *R*_{p} is OLR as a function of temperature only, and which is independent of the drivers and feedbacks — so *R*_{p} is the part of the OLR where the drivers and feedbacks are held constant. The other bit, R-tilda (the *R* with the squiggly line on top in Eq. (3), which cannot be typed in this text), is just the remainder of the OLR, namely *R* less *R*_{p} — it is the part of OLR that depends on the drivers and feedbacks. Fair enough.

Lucia then develops the basic model, successfully — which is no great achievement because she is basically copying the standard approach in Held and Soden but with her different notation.

Lucia then goes on to claim, in several variations, that in her formulation of the basic model “partial differential are not taken holding “everything about the climate” constant”. Not so. Lucia’s equations (7) and (9) both contain *dR*_{p}/*dT*_{S}, and her result for the ECS in her Eq. (8) thus also contains it. Because she defined *R*_{p} as *the part of the OLR that holds drivers and feedbacks constant*,

Yep, she has the conventional Planck feedback in her formulation too — the derivative of OLR with respect to temperature, holding all drivers and feedbacks constant.

Merely due to her definition of *R*_{p} , she can *write* the Planck feedback with straight derivative symbols instead of partial derivative symbols. * This is mere notational trickery and legerdemain;* Lucia is fooling herself and her readers with her multiple claims to the effect that her formulation does not contain partial derivatives that hold everything constant. “Her Planck feedback” is the just same as the conventional Planck feedback –

For those interested in more details, here is my analysis of her post.

**UPDATE, 20 Oct 2015:** Lucia has added an update to her post, a few hours after this post went live. She says

My formulation does not hold “

everythingabout the climate” constant while taking a differential. It either holds “T” constant or “CO2” constant. Nothing else.

Early in her post Lucia defined *R*_{p(e)} with

where

R_{pe}(T) is defined as the outgoing radiation that would arise on an earth whose temperature isTand has the ice, cloud ,water vapor and CO_{2}of the current earth; this does not vary with log_{2}(CO_{2}).

Is not that holding ice, cloud, and water vapor constant? Does that not imply that *R*_{p} holds constant both the feedbacks (ice, cloud, and water vapor are the feedbacks considered in the Held and Soden treatment she is following) and the drivers (CO2 is the only driver in her context)?And does not holding feedbacks and drivers constant hold (pretty much) everything about the climate constant — except for what is involved in the differential ratio, of course?

Holding *T* constant in Lucia’s formulation holds the feedbacks and thus “everything in climate” constant. And yes, Lucia’s Planck feedback is the same number as the conventional one that explicitly holds “everything about climate constant”.

In a new development, Lucia did at least email us when she added the update: her entire email reads

You are so confused.

Well, at least she emailed us.

*The Planck conditions for evaluating the Planck feedback or sensitivity conventionally are that all else besides tropospheric temperatures and OLR are held constant — so there are no feedbacks, all tropospheric temperatures (including the surface temperature) change in unison, and stratospheric temperatures are unchanged (Soden & Held, 2006, pp. 3355-56). There are some arbitrary choices to be made, such as whether it is the specific or the relative humidities that remain unchanged as the troposphere warms, or what happens at the tropopause.

[1^] Held, I. M., & Soden, B. J. (2000). Water Vapor Feedback and Global Warming. *Annu. Rev. Energy. Environ.*, 25:441–75. This seems to be legit: http://cips.berkeley.edu/events/rocky-planets-class09/ClimateVol1.pdf. See section 3.4.2, pages 135 – 138 of this pdf.

[2^] Pierrehumbert, R. T. (2010). *Principles of Planetary Climate.* Cambridge: Cambridge University Press. http://www.dgf.uchile.cl/~ronda/GF3004/helandsod00.pdf

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