More learned about water — science still not settled

Scientists have long debated the impact on global climate of water evaporated from vegetation. New research from Carnegie’s Global Ecology department concludes that evaporated water helps cool the earth as a whole, not just the local area of evaporation, demonstrating that evaporation of water from trees and lakes could have a cooling effect on the entire atmosphere. These findings, published on 14 September in Environmental Research Letters, have major implications for land-use decision making.

The researchers even thought it was possible that evaporation could have a warming effect on global climate, because water vapour acts as a greenhouse gas in the atmosphere.

Using a climate model, they found that increased evaporation actually had an overall cooling effect on the global climate.

Increased evaporation tends to cause clouds to form low in the atmosphere, which act to reflect the sun’s warming rays back out into space. This has a cooling influence.

The only thing I understand about climate models is that they reflect our (human) understanding of the climate. The IPCC expresses our understanding of many climate processes as “poor”. That’s a good guide when considering the output of climate models, in my opinion.

But how different this is from the usual tale we hear from defenders of CAGW, that increased water vapour leads inevitably to increased warming. But there’s a sting in the tail of the lead author’s comments.

“This shows us that the evaporation of water from trees and lakes in urban parks, like New York’s Central Park, not only help keep our cities cool, but also helps keep the whole planet cool,” Caldeira said. “Our research also shows that we need to improve our understanding of how our daily activities can drive changes in both local and global climate. That steam coming out of your tea-kettle may be helping to cool the Earth, but that cooling influence will be overwhelmed if that water was boiled by burning gas or coal.”

Boil your water by approved means and there’ll be no effect on global temperature!

There must be some scientific reason for that, but we’ll just have to trust them.

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13 Thoughts on “More learned about water — science still not settled

  1. Richard C (NZ) on 16/09/2011 at 1:43 pm said:

    Richard T, great that you’ve opened up this avenue of investigation.

    You say:-

    But how different this is from the usual tale we hear from defenders of CAGW, that increased water vapour leads inevitably to increased warming

    This is exactly the problem I see with climate science, they just assume certain actions take place without finding out what other disciplines already know. Climate science assumes that DLR (down-welling long-wave radiation from clouds and GHG’s) “warms the earth” (geologic material including ocean) according to AGW and to NASA here:-

    High, thin clouds primarily transmit incoming solar radiation; at the same time, they trap some of the outgoing infrared radiation emitted by the Earth and radiate it back downward, thereby warming the surface of the Earth

    That action they say causes increased evaporation, increased water vapour and increased warming.

    I’ve yet to find a peer-reviewed paper that actually measures ocean evaporation by DLR (could be done by experiment in a lab) by I have come across 2 papers that investigate evaporation from forest and willows, one paper being from our own NZ Forest Research Institute (FRI). I’ve documented this here at CCG but it bears repeating here:-

    Downwelling long-wave infrared (LWIR) radiation (DLR) is supposed to increase evaporation and evapotranspiration according to AGW due to increasing GHG effect from CO2 levels. DLR penetrates 10 microns into the ocean so to find out if it causes evaporation (and it’s increasing) a study of night-time ocean evaporation would be required, has anyone come across such a study?

    The significance of NH land (greater landmass vs SH) evapotranspiration can be seen in the following link where column water vapour levels are highest when NH rates are highest:- (click “Greenhouse Gasses)

    But does DLR play any part in evapotranspiration (or ocean evaporation)?


    “Nighttime, Wet Canopy Evaporation Rates and the Water Balance of an Evergreen Mixed Forest”

    Pearce FRI, Christchurch, NZ, Rowe, FRI,

    Christchurch, NZ, Stewart, Inst of Hydrology,

    Oxon, UK.

    “The similarity of daytime and nighttime evaporation rates indicates that evaporation from the wet canopy is driven by advected energy not by radiation”


    “Night-time evaporation from a short-rotation willow stand”

    Iritz and Landroth,

    Swedish University of Agricultural Sciences.

    “Night-time evaporation was controlled mainly by vapour pressure deficit and ventilation whereas net radiation had only a minor influence.”

    I.e. DLR from GHGs (or any other radiation) was not the evapotranspiration driver.

    AGW takes a big hit here.

    I suspect that a similar situation exists in regard to ocean.

    I’m pursuing the related possibility that climate science is making gigantic errors in respect to the ineffectiveness of DLR to “warm the earth” and that it is solar SW that does that almost exclusively. Also that Trenberth, Fasullo and Kiehl’s (TF&K) “Earth’s Global Energy Budget” 2009 Figure 1 is misnamed (should be a Radiation Budget) and a separate Energy Budget prepared in units of Joules (not W.m2). I’ve put this to Lubos Motl at The Reference Frame (TRF) and started a thread header on the topic at CCG here.

    • Richard C (NZ) on 16/09/2011 at 2:04 pm said:

      From the EurekAlert article (linked in the post):-

      Evaporative cooling is the process by which a local area is cooled by the energy used in the evaporation process

      What Pierce, Rowe and Stewart along with Iritz and Landroth indicate is that “the energy used” is NOT radiative energy (it’s a “minor influence” only).

  2. Richard C (NZ) on 16/09/2011 at 2:14 pm said:

    From the EurekAlert article (linked in the post):-

    Increased evaporation tends to cause clouds to form low in the atmosphere, which act to reflect the sun’s warming rays back out into space. This has a cooling influence.

    From what I can gather so far, this supports Spencer-Braswell 2011 but is contrary to Dessler 2011.

    Also at odds with Trenberth because according to him “clouds don’t cause climate change”

    Might be a bit wrong here – have to think about it some more.

    • Richard C (NZ) on 17/09/2011 at 2:52 pm said:

      The Carnegie Institution for Science, Department of Global Ecology study focuses on LOW cloud but the Spencer – Braswell vs Dessler – Trenberth argument is about the NETT effect of all clouds.

      The argument however is being couched in warmist terms by Team cheerleaders i.e in a “warming world” blah-de-blah, see this Live Science article: Clouds: The Wild Card of Climate Change”:-

      Currently, all of the Earth’s clouds exert a net cooling effect on our planet. But the substantial and opposing influences of clouds begs the question: What will be the net effect of all of the Earth’s clouds on climate as the Earth continues to warm in the future? Will clouds accelerate warming or help offset, or dull, warming?”


      Most scientists doubt that the net cooling effect of clouds will ever be large enough to completely offset ongoing warming.

      [What on-going warming? And “most scientists” – who cares?]

      But many scientists say that if warming were to increase cooling clouds or decrease warming clouds, the current net cooling effect of clouds on the Earth’s climate would probably increase.

      [The author (Lily Whiteman, National Science Foundation) omits the negative feedback loop here and non-temperature drivers, see below]

      This would moderate, or offset, ongoing warming. The result: The Earth’s end-of-the-century temperature may be pulled down toward the lower end of its predicted range.

      But if, on the other hand, warming were to increase warming clouds or decrease cooling clouds, scientists say the current net cooling effect of clouds on the Earth’s climate would probably decrease, and an important moderating force on ongoing warming would thereby diminish. The result: The Earth’s end-of-the-century temperature may be pushed up toward the upper end of its predicted range.

      The resulting rise in temperature would, in a positive feedback loop, tend to promote the formation of even more warming clouds or further reduce the presence of cooling clouds.

      [Ah yes, the temperature-only driven “positive feedback” loop]

      Either way, temperatures would rise even higher. ”

      [The validity of the last sentence being predicated on the notion that CO2 is the major climate driver]

      Therefore, the controversy is in five parts:

      1) The cause of cloud variation

      2) The sign of the feedback from the nett effect (CLD) of the variation.

      3) Relationship between CLD forcing and temperature

      4) Difference between models and observations

      5) Climate change

      Opposing positions:-

      Team says 1) is caused by temperature only,

      Others say 1) is caused by a solar-cosmic dynamic.

      Team says 2) is positive only (Dessler 0.54 w/m2/K)

      Others say 2) is weakly negative (McIntyre -0.96 w/m2/K)

      Team says 3) is temperature >>> clouds only

      Spencer-Braswell says 3) is temperature >>> <<< clouds

      Team says 4) is agreement using instantaneous relationship

      Others say 4) is no agreement using lagged relationship that is more significant than instantaneous

      Team say climate warms only

      Others say climate warms and cools

      [I’ll need to check 2) and 4) at some stage]

      An increase in LOW cloud (see Carnegie study) caused by 1) is this action in 1), 2), 3), 4) and 5) depending on other (e.g. high cloud) variations:-

      Team: temperature >>> + CLD >>> temperature increase >>> disagreement >>> climate warms

      Others: solar-cosmic >>> – or + CLD >>> temperature decrease or increase >>> disagreement >>> climate cools or warms

      Obviously the Team say there is agreement in models vs observations but Dessler neglects the bigger picture.

      Obvious also is that the cause of evaporation in the Carnegie study (let alone an increase) and low cloud formation as a result of it is very complex in 1) for both Team and Others and we need to know as much about the process as possible.

    • Richard C (NZ) on 17/09/2011 at 3:12 pm said:

      More from Live Science, this time by Stephanie Pappas, LiveScience Senior Writer:-

      Within days of the editor’s resignation, Dessler published a study refuting Spencer’s claims in the journal Geophysical Research Letters.

      “I said, ‘Let’s quantitatively measure how much energy the clouds are trapping and how much energy it takes to change the climate, and see if the clouds are trapping enough energy to change the climate,’” Dessler said. “The answer is, they’re not.”

      [McIntyre (using Dessler’s data) says implicitly, they are]

      The changes Spencer saw in his model are explained by El Niño/La Niña cycles, Dessler said, not caused by clouds.

      The El Niño/La Niña cycles seem to be a point-of-difference for the Team but I can’t yet get my head around how this fits into the controversy given the deficiency in Dessler’s paper (lag vs instantaneous CLD forcing – temperature relationship).

    • Richard C (NZ) on 17/09/2011 at 11:12 pm said:

      The El Niño/La Niña point-of-difference.

      Spencer responds to this:-

      The changes Spencer saw in his model are explained by El Niño/La Niña cycles, Dessler said, not caused by clouds.

      With this:-

      In his paper, Dessler dismissed all of the evidence we presented with a single claim: that since (1) the global temperature variations which occurred during the satellite record (2000-2010) were mostly caused by El Nino and La Nina, and (2) no one has ever demonstrated that “clouds cause El Nino”, then there could not be a clouds-causing-temperature-change contamination of his cloud feedback estimate.

      But we now have clear evidence that El Nino and La Nina temperature variations are indeed caused in large measure by changes in clouds, with the cloud changes coming months in advance of the temperature changes.

      And this:-

      At the heart of this debate is whether cloud changes, through their ability to alter how much sunlight is allowed in to warm the Earth, can cause temperature change.

      We claim they can, and have demonstrated so with both phase space plots of observed temperature versus Earth radiative budget variations here [SB10], and with lag-regression plots of the same data here [SB11], and with a forcing-feedback model of the average climate system in both of those publications. (The model we used was suggested to us by Isaac Held, Princeton-GFDL, who is hardly a global warming “skeptic”.)


      Dessler says: El Niño/La Niña “not caused by clouds”

      Spencer says: El Niño/La Niña “caused in large measure by changes in clouds, with the cloud changes coming months in advance of the temperature changes

      And Dessler refuses SB10 and SB11 as demonstrations of El Niño/La Niña being caused by changes in clouds.

      Now Trenberth has a comment on SB11 in Remote Sensing (linked at CA) that is the subject of this article by Steve McIntyre at Climate Audit:-

      “More Hypocrisy from the Team”

      Trenberth wildly overstates Dessler 2011 as well by saying that it “quantifies the magnitude and role of clouds and shows that cloud effects are small”. “Quantifying the magnitude and role of clouds” is an enormous undertaking and would take hundreds of pages of analysis. Dessler 2011 is a short little article addressing a narrow issue. It did not pretend to “quantify the magnitude and role of clouds” nor did it do so.

      See also more background here:-

      “More Thoughts on the War Being Waged Against Us” by Roy W. Spencer, Ph. D.

      It is obvious to many people what is going on behind the scenes. The next IPCC report (AR5) is now in preparation, and there is a bust-gut effort going on to make sure that either (1) no scientific papers get published which could get in the way of the IPCC’s politically-motivated goals, or (2) any critical papers that DO get published are discredited with any and all means available.

    • Richard C (NZ) on 20/09/2011 at 10:30 pm said:

      Phase Shift in Spencer’s Data

      Dr David Stockwell

      The LW and SW_clr components lead the global surface temperature. There are three possible explanations:

      1. Changes in cloud cover actually do drive changes in global temperature due to gamma-ray flux (GRF) or other effects, or

      2. The changes in cloud cover are caused by changes in global temperature, with the derivative mechanism described above.

      3. Both 1 and 2.

      Spencer argues that it is impossible to distinguish between 1 and 2. Both Spencer and Lindzen both consider the lags important because correlation is greatly improved (and determines whether feedback is positive to negative). Neither seem to have mentioned the 3 month phase relationships emerging from integral/derivative system dynamics.

      I can’t see how it is possible perform a valid analysis without this insight.

    • Richard C (NZ) on 09/10/2011 at 2:23 pm said:

      I’ve Looked at Clouds from Both Sides Now -and Before

      October 8th, 2011 by Roy W. Spencer, Ph. D.

      …sometimes, the most powerful evidence is right in front of your face…..

      I never dreamed that anyone would dispute the claim that cloud changes can cause “cloud radiative forcing” of the climate system, in addition to their role as responding to surface temperature changes (“cloud radiative feedback”). (NOTE: “Cloud radiative forcing” traditionally has multiple meanings. Caveat emptor.)

      But that’s exactly what has happened. Andy Dessler’s 2010 and 2011 papers have claimed, both implicitly and explicitly, that in the context of climate, with very few exceptions, cloud changes must be the result of temperature change only.

      Shortly after we became aware of Andy’s latest paper, which finally appeared in GRL on October 1, I realized the most obvious and most powerful evidence of the existence of cloud radiative forcing was staring us in the face. We had actually alluded to this in our previous papers, but there are so many ways to approach the issue that it’s easy to get sidetracked by details, and forget about the Big Picture.

      Well, the following graph is the Big Picture……..

      [See plot and details]

      What this graph shows is very simple, but also very powerful: The radiative variations CERES measures look nothing like what the radiative feedback should look like. You can put in any feedback parameter you want (the IPCC models range from 0.91 to 1.87…I think it could be more like 3 to 6 in the real climate system), and you will come to the same conclusion.

      And if CERES is measuring something very different from radiative feedback, it must — by definition — be radiative forcing (for the detail-oriented folks, forcing = Net + feedback…where Net is very close to the negative of [LW+SW]).

      The above chart makes it clear that radiative feedback is only a small portion of what CERES measures. There is no way around this conclusion.


      I just wanted to put this evidence out there for people to see and understand in advance. It will be indeed part of our response to Dessler 2011, but Danny Braswell and I have so many things to say about that paper, it’s going to take time to address all of the ways in which (we think) Dessler is wrong, misused our model, and misrepresented our position.

  3. anthony on 18/09/2011 at 6:49 am said:

    It seems that any effort to save the environment will collide with other development projects and above all urbanization. I was surprised when I found the other day that the drainage systems we all have in our gardens contribute a great deal to the way water circulates in the environment and that their design may have a damaging impact upon the eco-friendliness of our homes’ exteriors.

  4. Richard C (NZ) on 20/09/2011 at 9:55 pm said:

    New paper shows clouds have a large negative-feedback cooling effect

    A paper published last week in the journal Meteorological Applications undermines a key assumption of the theory of man-made global warming, finding that the cooling effect of clouds far outweighs a supposed ‘greenhouse’ warming effect. Alarmists claim clouds have an overall ‘positive-feedback’ warming effect upon climate due to ‘back-radiation’ of the ‘greenhouse’ gas water vapor. This new paper based on satellite measurements finds instead that clouds have a large net cooling effect by blocking solar radiation and increasing radiative cooling outside the tropics. The cooling effect is found to be -21 Watts per meter squared, more than 17 times the supposed warming effect from a doubling of CO2 concentrations [1.2 W/m2]. Another key assumption of the AGW theory topples in the face of real-world data showing the net feedback from clouds is strongly negative.

    Combining satellite data and models to estimate cloud radiative effect at the surface and in the atmosphere

    Richard P. Allan, 2011

    Abstract: Satellite measurements and numerical forecast model reanalysis data are used to compute an updated estimate of the cloud radiative effect on the global multi-annual mean radiative energy budget of the atmosphere and surface. The cloud radiative cooling effect through reflection of short wave radiation dominates over the long wave heating effect, resulting in a net cooling of the climate system of − 21 Wm−2. The short wave radiative effect of cloud is primarily manifest as a reduction in the solar radiation absorbed at the surface of − 53 Wm−2. Clouds impact long wave radiation by heating the moist tropical atmosphere (up to around 40 Wm−2 for global annual means) while enhancing the radiative cooling of the atmosphere over other regions, in particular higher latitudes and sub-tropical marine stratocumulus regimes. While clouds act to cool the climate system during the daytime, the cloud greenhouse effect heats the climate system at night. The influence of cloud radiative effect on determining cloud feedbacks and changes in the water cycle are discussed.

    Poke in the eye to Dessler and the Team.

    • Richard C (NZ) on 21/09/2011 at 9:31 pm said:

      Multi-institutional study group finds strong negative-feedback cooling effect from clouds

      The Newsletter of the multi-institution Climate Process Team on Low-Latitude Cloud Feedbacks on Climate Sensitivity outlines their significant findings to date:

      1. Clouds have a strong negative-feedback cooling effect on climate in both the tropics and extra-tropics
      2. A warmer climate enhances [increases] boundary layer clouds resulting in increasing negative-feedback
      3. Due to this strong negative-feedback, global climate sensitivity is only 0.41 K/(W m-2) – HALF the 0.8 K/(W/m2) + assumed by the IPCC


      The Climate Process Team on Low-Latitude Cloud Feedbacks on Climate Sensitivity (cloud CPT) includes three climate modeling centers, NCAR, GFDL, and NASA’s Global Modeling and Assimilation Office (GMAO), together with 8 funded external core PIs led by Chris Bretherton of the University of Washington (UW). Its goal has been to reduce uncertainties about the feedback of low-latitude clouds on climate change as simulated in atmospheric general circulation models (GCMs). To coordinate this multi-institution effort, we have hired liaison scientists at NCAR and GFDL, had regular teleconferences and annual meetings, and developed special model output datasets for group analysis. The cloud CPT web site provides links to all its publications and activities. The cloud CPT has had many interesting subplots; here we focus on two of interesting recent results and its future plans. The results showcase a key CPT strategy – gaining insight from the
      use of several complementary modeling perspectives on the cloud feedbacks problem.

      Two recent findings of the cloud CPT:

      (1) The world’s first superparameterization climate sensitivity results show strong negative cloud feedbacks driven by enhancement of boundary layer clouds in a warmer climate.

      Superparameterization is a recently developed form of global modeling in which the parameterized moist physics in each grid column of an AGCM is replaced by a small cloud-resolving model (CRM). It holds the promise of much more realistic simulations of cloud fields associated with moist convection and turbulence. Superparameterization is computationally expensive, but multiyear simulations are now feasible. The Colorado State University and UW cloud CPT groups collaborated on the first climate sensitivity analysis of a superparameterized AGCM (Wyant et al. 2006b). The Khairoutdinov-Randall (2001, 2005) superparameterized CAM3, hereafter CAM-SP, was used. Each CRM in CAM-SP has the same vertical levels as CAM3, 4 km horizontal resolution, and one horizontal dimension with 32 horizontal gridpoints.

      Following Cess et al. (1989), climate sensitivity was assessed by examining the TOA radiative response to a uniform SST increase of 2K, based on the difference between control and +2K 3.5 year CAMSP simulations. Fig. 2 compares the results to standard versions of the NCAR CAM3, GFDL AM2 and GMAO AGCMs. All these models have similar clear-sky responses, so we just plot the +2K changes in longwave (greenhouse) and shortwave (albedo) cloud radiative forcings (ΔLWCF and ΔSWCF). Since ΔSWCF tends to be larger than ΔLWCF. boundary-layer cloud changes (which have little greenhouse effect compared to their albedo enhancement) appear to be particularly important. The CAM-SP shows strongly negative net cloud feedback in both the tropics and in the extratropics, resulting in a global climate sensitivity of only 0.41 K/(W m-2), at the low end of traditional AGCMs (e.g. Cess et al. 1996), but in accord with an analysis of 30- day SST/SST+2K climatologies from a global aquaplanet CRM run on the Earth Simulator (Miura et al. 2005). The conventional AGCMs differ greatly from each other but all have less negative net cloud forcings and correspondingly larger climate sensitivities than the superparameterization. The coarse horizontal and vertical resolution of CAM3-SP means that it highly under-resolves the turbulent circulations that produce boundary layer clouds. Thus, one should interpret its predictions with caution. With this caveat, cloud feedbacks are arguably more naturally simulated by superparameterization than in conventional AGCMs [conventional climate models], suggesting a compelling need to better understand the differences between the results from these two approaches.

      The cloud CPT findings of Wyant and Bretherton (2006) were known prior to AR4 but the IPCC studiously ignored them. Will they do the same with this report for AR5?

    • Richard C (NZ) on 22/09/2011 at 4:19 pm said:


      “Oh dear, now we have three peer reviewed papers (Lindzen and Choi, Spencer and Braswell, and now Richard P. Allan) based on observations that show a net negative feedback for clouds, and a strong one at that. What will Trenberth and Dessler do next? Maybe the editor of Meteorological Applications can be persuaded to commit professional suicide and resign? The key paragraph from the new paper:

      …the cloud radiative cooling effect through reflection of short wave radiation is found to dominate over the long wave heating effect, resulting in a net cooling of the climate system of −21 Wm−2.

      After all the wailing and gnashing of teeth over the Spencer and Braswell paper in Remote Sensing, and the stunt pulled by its former editor who resigned saying the peer review process failed, another paper was published last week in the journal Meteorological Applications that agrees well with Spencer and Braswell.

      This new paper by Richard P. Allan of the University of Reading discovers via a combination of satellite observations and models that the cooling effect of clouds far outweighs the long-wave or “greenhouse” warming effect. While Dessler and Trenberth (among others) claim clouds have an overall positive feedback warming effect upon climate due to the long-wave back-radiation, this new paper shows that clouds have a large net cooling effect by blocking incoming solar radiation and increasing radiative cooling outside the tropics. This is key, because since clouds offer a negative feedback as shown by this paper and Spencer and Braswell plus Lindzen and Choi, it throws a huge monkey wrench in climate model machinery that predict catastrophic levels of positive feedback enhanced global warming due to increased CO2.”


      Make that three based on observations and the Bretherton CPT report based on models Anthony.

    • Richard C (NZ) on 23/09/2011 at 1:55 pm said:

      Alan Siddons on the negative-feedback cooling effect of clouds
      email from Alan Siddons:

      Someone asked me today to comment on the recent Richard P Allan paper, which is discussed on WUWT [and originally posted on The Hockey Schtick]. Allan’s finding is that clouds have a negative influence on the earth’s temperature (duh). But here is what wrote back. Hope you like it.

      Anthony Watts offers an important statement about the Allan paper:

      While Dessler and Trenberth (among others) claim clouds have an overall positive feedback warming effect upon climate due to the long-wave back-radiation, this new paper shows that clouds have a large net cooling effect by blocking incoming solar radiation and increasing radiative cooling outside the tropics.

      Now, I always focus on the basic claim that back-radiating greenhouse gases make the earth’s surface warmer. The earth’s SURFACE. Liquid clouds are often included as greenhouse agents because at nighttime they’re thought to reflect heat rays back to the earth and at least retard surface cooling if not actually raise the temperature.

      In other words, the same heating mechanism, that of back-radiation, is attributed to clouds because ‘radiative forcing’ theory assumes that opposing flows of infrared actively warm the surface —and always the surface alone, please note. Thus, for instance, Lindzen argues that 240 watts from the surface matched by 240 from the sky will make the surface radiate 480 watts per each square meter. But won’t make the sky radiate 480 in turn.

      Similarly, Trenberth adds 324 sky watts to the surface’s 66, resulting in 390 surface watts.

      If you aim your attention on temperature, however, the impossibility of such a thing becomes apparent. For Lindzen’s scenario has a 255 Kelvin sky facing a 255 Kelvin surface — yet it is known that two bodies at the same temperature aren’t able to transfer heat to each other. Nor, of course, can one of those bodies raise the other’s temperature. 303 Kelvin resulting on the surface simply cannot happen, then. Likewise, Trenberth’s sky is at 275 K while the surface (radiating 66 watts per square meter) is at 185 K. Thus the surface can’t possibly get warmer than 275 K, or 1.8° Celsius. And at 1.8° the surface could only radiate 324 W/m² at most, not 390.

      From this it’s obvious that opposing flows of radiation do NOT add, which exposes ‘radiative forcing’ as a fiction.

      [See diagrams]


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