climate models
jshe...@vnet.ibm.com
>This example is getting tedious. Let me see if I can dispose of it.
and also:
>This is interesting but I don't think your equations could possibly be
>a model for the effect of CO2 on global temperature. Since Mr. Shearer
>seems to like two variable linear systems of differential equations let
>me show what happens when one tries to do climate modeling with such a
>system. Note, I'm not a climatologist so I claim no proof from
>authority. I could even be wrong.
and also:
>Which leads to the conclusion that with a positive CO2 forcing (e > 0)
>both temperature (x1) and CO2 concentration (x2) will increase relative
>to the present state. Which leads to the further conclusion that the
>equations Mr. Shearer presented above make no sense as a climate model.
Your post is not exactly wrong, but it is irrelevant.
Perhaps an analogy will make my point clearer. Suppose that
Tobis said "all cars in California are white because all cars every-
where are white" and then Shearer said "Tobis is wrong, all cars
everywhere are not white, for example my car is black" and finally
Swanson points out "Shearer's car is not in California". Swanson is
correct but Tobis is still wrong. Note all cars in California could
be white and Tobis would still be wrong.
In the case at hand Tobis claimed in effect that we could be
certain that the response of temperature to CO2 forcing would be
positive because feedback can not change the sign of the response of a
system to forcing. I posted an example to show that Tobis's argument
was incorrect. I do not claim that this example in fact models the
climate system or that the response of temperature to CO2 forcing will
be negative just that Tobis's argument that it must be positive is
wrong.
Also note that showing my example does not model climate is
not enough to show that the response of temperature to CO2 forcing is
positive. One must show that any system that responds opposite to
forcing does not model climate (ie that all nonwhite cars are not in
California). This cannot possibly be done without a detailed analysis
of the climate system which is what Tobis is claiming is unnecessary.
James B. Shearer (email j...@watson.ibm.com)
jshe...@vnet.ibm.com
>Remind me what your mechanism was supposed to illustrate. If it was
>supposed to illustrate a case where feedback reverses the sign of the
>response (i.e. yields a cooling) then your example fails. In 1D RC
>models with adjustment to the moist adiabat, the lapse rate feedback
>somewhat reduces the temperature increase you get from doubling CO2, but
>it does not turn it into a cooling. This feedback is a good example of
>the typical sort of feedback, where the feedback only depends on CO2
>through its effect on temperature, and not directly (as in my unphysical
>example of an imaginary water-destroying chemical reaction involving
>CO2).
The context of the Halliwell challenge was:
Dave Halliwell posted:
> There is no obvious reason for the lapse rate to _decrease_,
>eiether. The radiative forcing from increasing CO2 is to _increase_
>the lapse rate. By assuming the 6.5C constraint is unchanged, the
>model is implicitly assuming that there is a negative feedback
>sufficient to _completely_ counteract that radiative forcing. A more
>realistic assumption would be that such a strong negative feedback is
>unlikely, and the model should allow the lapse rate to increase,
>leading to greater warming. To decrease the lapse rate would require
>a feedback strong enough to *reverse* the sign of the initial forcing.
>Shearer is back to claiming that feedbacks in the atmosphere can
>reverse forcing!
>
> I challenge Shearer to present a physically-plausible mechanism
>by which such a feedback can exist in the atmosphere. ...
Halliwell was asking for a physically-plausible way that
forcing could reduce the lapse rate. I hope you will agree that
I have presented such a mechanism. Incidently in this example
feedback has reversed the sign of the response of the lapse rate
to forcing (since as Halliwell points out by heating the surface
radiative forcing is trying to increase the lapse rate).
R. T. Pierrehumbert also posted:
>If you want an example where increasing CO2 can decrease temperature
>without invoking anything truly impossible, see my J. Atmos. Sci.
>paper from last Spring, "Thermostats, radiator fins, and the local
>runaway greenhouse." If the subsiding regions of the tropics were
>completely dry so that all their emissivity were due to CO2, and if the
>convective region of the tropics remained moist, then increasing CO2 in
>the subsiding regions increases their ability to radiate to space,
>and hence cools the atmosphere. I don't think this is a very
>plausible scenario, given what is known about the moisture dynamics
>of the subsiding regions, but there it is. Note that this example
>refers to the temperature of the tropical atmosphere. The mean surface
>temperature itself need not decrease, since the part of the planet under
>the subsidence warms (due to the increased CO2 greenhouse) at the same
>time the atmosphere cools. I leave it as an exercise to Mr. Shearer to
>do the algebra to say whether there are circumstances when the mean
>surface temperature decreases.
You agree then that Tobis is wrong in saying that we can
be certain without looking at the details of the climate system
that CO2 forcing will increase temperature?
Dave Halliwell
> The context of the Halliwell challenge was:
> Dave Halliwell posted:
>> There is no obvious reason for the lapse rate to _decrease_,
>>eiether. The radiative forcing from increasing CO2 is to _increase_
>>the lapse rate. By assuming the 6.5C constraint is unchanged, the
>>model is implicitly assuming that there is a negative feedback
>>sufficient to _completely_ counteract that radiative forcing. A more
>>realistic assumption would be that such a strong negative feedback is
>>unlikely, and the model should allow the lapse rate to increase,
>>leading to greater warming. To decrease the lapse rate would require
>>a feedback strong enough to *reverse* the sign of the initial forcing.
>>Shearer is back to claiming that feedbacks in the atmosphere can
>>reverse forcing!
>>
>> I challenge Shearer to present a physically-plausible mechanism
>>by which such a feedback can exist in the atmosphere. ...
I will again make the point that this challenge is with regard to the
mean global lapse rate, as used in radiative-convective models. Shearer
seems to want to ignore this aspect.
> Halliwell was asking for a physically-plausible way that
>forcing could reduce the lapse rate. I hope you will agree that
>I have presented such a mechanism.
No, you have not, and the following paragraph illustrates where you
are making your mistake (or, _one_ of the many places where you are
making your mistakes):
> Incidently in this example
>feedback has reversed the sign of the response of the lapse rate
>to forcing (since as Halliwell points out by heating the surface
>radiative forcing is trying to increase the lapse rate).
Anything that induces radiative forcing AT THE SURFACE that leads to a
change in lapse rate ABOVE THE SURFACE is not causing lapse rate changes
via a feedback mechanism: there is no closed loop in the physical process.
A feedback loop requires that radiative forcing that *directly* alters the
lapse rate above the surface _also_ have *indirect* effects that lead to
_further_ changes in the lapse rate.
What I point out in the challenge above is that adding CO2 to the
atmosphere causes changes in the _local_ radiative transfer regime such
that it tends to force a greater lapse rate _locally_. The reason for
this is that the increased absorption by CO2 leads to a reduced energy
transfer given the same temperature gradient. In order to transfer the
same amount of energy (with increased CO2) via IR radiation, a greater
temperature gradient will be required. This is the forcing.
Now, if this *local* radiative forcing leads to changes in the system
that cause further changes that then modify the *local* temperature
gradient to something other than what the local radiative forcing did
_directly_, we have a feedback loop. If that feedback causes an even
_steeper_ temperature gradient, it is positive. If it causes a less steep
gradient, it is negative.
Of course, in the radiative-convective model, there *is* only one lapse
rate: the global mean. In such a model "local" is _global_, so we need an
argument that the _global_ forcing leads to a _global_ lapse rate change
that is reversed _globally_. Arguing that a _portion_ of what constitutes
the global mean will change in the direction desired is insufficient.
Shearer seems to think that he has provided an argument that the
feedback can actually _reverse_ the sign of the original forcing: that
the feedback not only completely _negates_ the change caused by adding
CO2 (which was to increase the gradient), but to actually overwhelm the
CO2 and cause a decrease in gradient.
His latest incarnation of this argument (above) is one that doesn't
even _have_ a feedback loop. His earlier incarnation is also wrong, but
for a more subtle reason. Let's review it again.
Shearer feels that the possibility of a decreased lapse rate can be
attributed to the fact that warmer temperatures correspond to a decrease
in the saturated adiabatic lapse rate (SALR). Where the SALR controls the
observed (environmental) lapsed rate, a warmer climate should have a lower
lapse rate. I have pointed out that the SALR exerts such a control *only*
where we have rising air that has reached saturation, and that such a
situation is the norm in the tropics.
Now, to get a CO2-induced forcing with a feedback loop, we need to
consider what happens if CO2 is added to this rising air. If CO2 tends to
force an increase in lapse rate in the rising air, and the warming leads
to a lapse rate that ends up being less steep than the original, we would
have Shearer's desired reversal due to feedback. However, in every
single situation where rising saturated air limits the lapse rate, we form
clouds. Why? Well, the reason the lapse rate is limited to the SALR is
_because_ the energy released by condensation reduces the cooling that
would normally happen in unsaturated rising air. At warmer temperatures,
more condensation occurs, so the cooling rate (lapse rate) is further
reduced. Where we have condensation, we have clouds.
So what? Well, with clouds present, we have liquid water. Liquid water
is an almost-perfect absorber (and emitter) of IR radiation. Within
clouds, IR radiative energy transfer is completely controlled by the
presence of water. As a result, adding CO2 to a cloud is going to have
*no* effect on radiative transfer within the cloud. If it has no effect
on radiative transfer, it has no tendency to force an increase in the
lapse rate. If it has no tendency to increase the lapse rate, then the
eventual reduction in the lapse rate had nothing to do with the local
increase in CO2! There is *no* feedback loop present, so we obviously
don't have a feedback that is reversing the sign of the forcing.
However, we do seem to be getting a decrease in lapse rate as the
result of a warmer climate, which is supposedly being caused by radiative
forcing by CO2, so where is the decreased lapse rate coming from? Well,
in _other_ parts of the globe, where the _local_ lapse rate is not
controlled by the SALR (because we don't have rising saturated air that
is forming clouds), CO2 *does* have a radiative forcing. That alters the
overall flow of energy, causing warming at the surface, and the effects
of this extend to the tropics and the rising air. As the result of the
SALR limit, the tropical lapse rate should be decreased. However, this
decrease is not the result of _tropical_ radiative forcing, but
non-tropical radiative forcing. We radiatively induce a lapse rate
increase _outside_ the tropics, and this coincidentally leads to a lapse
rate _decrease_ *in* the tropics. Once again, this is _not_ a feedback
loop, since the decreased lapse rate in the tropics is a _direct_ result
of things happening outside the tropics.
Lindzen's argument for a low global sensitivity to CO2-induced warming
has to do with the balance between radiative effects in rising and
descending air: the two cases discussed above where lapse rates are
limited by the SALR, and where they are not. Lindzen only argues that the
strong negative feedback will lead to minor warming; he makes no argument
that the sign of the forcing will be reversed.
I had originally suggested that Shearer look to spatial considerations
in the search for his alleged feedback that reversed the global mean
lapse rate forcing. Raymond Pierrehumbert has also made suggestions,
which Shearer has included at the end of his post. I'll leave it in, so
that others can see how it relates to the mechanisms I have discussed above.
Unfortunately, Shearer seems to insist upon continuing to demonstrate his
ignorance regarding climate processes.
> R. T. Pierrehumbert also posted:
>>If you want an example where increasing CO2 can decrease temperature
>>without invoking anything truly impossible, see my J. Atmos. Sci.
>>paper from last Spring, "Thermostats, radiator fins, and the local
>>runaway greenhouse." If the subsiding regions of the tropics were
>>completely dry so that all their emissivity were due to CO2, and if the
>>convective region of the tropics remained moist, then increasing CO2 in
>>the subsiding regions increases their ability to radiate to space,
>>and hence cools the atmosphere. I don't think this is a very
>>plausible scenario, given what is known about the moisture dynamics
>>of the subsiding regions, but there it is. Note that this example
>>refers to the temperature of the tropical atmosphere. The mean surface
>>temperature itself need not decrease, since the part of the planet under
>>the subsidence warms (due to the increased CO2 greenhouse) at the same
>>time the atmosphere cools. I leave it as an exercise to Mr. Shearer to
>>do the algebra to say whether there are circumstances when the mean
>>surface temperature decreases.
I would also suggest that Shearer begin to ask questions regarding the
stuff that he doesn't understand, instead of blindly continuing to
demonstrate his ignorance.
--
Dave Halliwell I don't speak for my employers, and you
Edmonton, Alberta shouldn't expect them to speak for me.
Michael Tobis
: I posted an example to show that Tobis's argument
: was incorrect. I do not claim that this example in fact models the
: climate system or that the response of temperature to CO2 forcing will
: be negative just that Tobis's argument that it must be positive is
: wrong.
I do not make quite so strong a claim. I claim that *feedback* cannot
reverse the sign (at least in a sensibly linearizable situation) of
the system without feedback. That is, if the output is temperature, there
is no temperature-dependent phenomenon which can reverse the sign of
the response. This is a very solid result in elementary systems theory.
At first glance, Shearer's counter-example seems to contradict this.
I intend to figure out why.
The argument does NOT guarantee positive temperature response absent
physical reasoning. It only guraantees that no negative response is
possible through phenomena which are temerature dependent.
Steinn Sigurdsson pointed out that temperature feedback at one place
is not excluded by this argument from causing cooling at another
place. (The NADW formation instability being the prominent example.)
Despite Peixoto & Oort, I conceded that this at least seemed to me
possible. I haven't spent a lot of time thinking about how to
extend the result to a multidimensional variable.
I'm still confident that Shearer's counterexample is invalid, though.
As far as I know, that is the point of disagreement that we are discussing.
I am pleased that Shearer's expectations have converged roughly onto
the same ballpark as the IPCC's. I am not sure what exactly we are
arguing about in the large. He claims that others have made extravagant
claims about climate models. As far as I know, none of the folks in
relevant professions have done so here. I'd like to see a counterexample.
mt
R. T. Pierrehumbert
> You agree then that Tobis is wrong in saying that we can
>be certain without looking at the details of the climate system
>that CO2 forcing will increase temperature?
No, since Tobis' remark is limited to temperature-dependent feedbacks,
and refers to sensitivity to the amount of increase of heating
of the planet.
My radiator-fin example is rather different. In that case, in
the unrealistic and extreme parameter limit I mentioned, increasing
CO2 increases the outgoing infrared radiation, rather than decreasing
it. Hence the "DF" is negative, rather than a positive 4W/m**2 as
usually quoted. It isn't a matter of feedback reversing the
sign of the response. Rather, it is an example of physics
reversing the sign of the forcing from what it usually is
expected to be.
jshe...@vnet.ibm.com
> My argument is, if t = x + y we can not say increasing y
>by 2 will increase t by 2 when x is a function of y.
> Consider the following example. We have a bucket with a hole
>in the bottom. The bucket is sitting on a damped spring. Water is
>flowing into the bucket from a pipe. Let t be the height (above the
>ground) of the water surface in the bucket. Let x be the height (above
>the ground) of the bottom of the bucket. Let y be the height (above
>the bottom of the bucket) of the water level in the bucket. Now
>t = x + y. In equilibrium y will be determined so that the amount
>water flowing out through the hole equals the amount flowing in
>through the pipe. Now increase the flow in. This will cause y to
>increase until the greater pressure causes to flow out to increase
>until it balances the flow in. Does this mean t will increase the
>same amount as y. Clearly not, the increased weight of water in the
>bucket will cause the spring to contract and x to decrease. Hence t
>will increase by less than y. In fact given a sufficiently elastic
>spring t will move in the opposite direction (at equilibrium), yet
>another illustration that feedback can reverse the sign of a
>response.
Dave Halliwell responded
> I think I've finally figured out what your problem is: you seem to be
>utterly incapable of telling the difference between a mathematical
>expression and the description of physical processes that exist in the
>real world.
>
> Let's see if I can create a bit of ASCII art:
>
> \
> / spring
> \
> /
> | | | |
> | | bucket | Y |
> |_| | |
> | | T
> | |
> | X |
> | |
> | |
> _____________________________________ ground
Your problem seems to be: you can't read. I said the bucket
was sitting on a spring not suspended from a spring. Therefore the
increased weight of water in the bucket will cause the spring to
contract as I said. Of course the bucket moves down in either case.
If the spring is flexible enough this will cause T to decrease
reversing the sign of the response to the forcing.
Dave Halliwell added in part:
> Now, do you want to try and restrict the rest of your comments
>regarding climate to mathematics that actually has some relationship to
>the physics involved?
How about restricting your responses to things that actually
have some relationship to what I post.
jshe...@vnet.ibm.com
> You agree then that Tobis is wrong in saying that we can
>be certain without looking at the details of the climate system
>that CO2 forcing will increase temperature?
>No, since Tobis' remark is limited to temperature-dependent feedbacks,
>and refers to sensitivity to the amount of increase of heating
>of the planet.
Ok, how about the following model (which I am not claiming is
physically realistic). Suppose the ability of the atmosphere to absorb
visible light (ie shortwave radiation) were temperature dependent
increasing as the temperature of the atmosphere increased. Then
suppose we increase the solar constant. The initial response will be
to heat the surface. The surface will then heat the atmosphere. The
hotter atmosphere will absorb more incoming solar radiation causing
further heating. Meanwhile the atmosphere will shade the surface
causing the surface to begin to cool. If we choose the proper
parameters I believe we can end up with a hotter atmosphere and a
colder surface. Then feedback has reversed the sign of the response
of surface temperature to forcing which is what Tobis says is
impossible.
Dave Halliwell
> I posted:
>> My argument is, if t = x + y we can not say increasing y
>>by 2 will increase t by 2 when x is a function of y.
>> Consider the following example. We have a bucket with a hole
>>in the bottom. The bucket is sitting on a damped spring. Water is
>>flowing into the bucket from a pipe. Let t be the height (above the
>>ground) of the water surface in the bucket. Let x be the height (above
>>the ground) of the bottom of the bucket. Let y be the height (above
>>the bottom of the bucket) of the water level in the bucket. Now
>>t = x + y.
[remainder of non-existent feedback and my response deleted]
> Your problem seems to be: you can't read. I said the bucket
>was sitting on a spring not suspended from a spring. Therefore the
>increased weight of water in the bucket will cause the spring to
>contract as I said. Of course the bucket moves down in either case.
>If the spring is flexible enough this will cause T to decrease
>reversing the sign of the response to the forcing.
...and you _still_ haven't provided a mathematical description that
matches the _physics_ of the problem.
The height of water surface above the ground is completely irrelevant
as far as the behaviour of the system is concerned.
The height of the water in the bucket is _only_ significant with
regard to the pressure forcing the water out the hole, and to the extent
that it relates to the weight of water in the bucket.
The height of the bottom of the bucket above the ground is only
significant to the extent that it relates to the compression of the
spring. (It may equal the compression, but only if the spring has no
non-compressible attachments at either end. Otherwise you must subtract
those non-compressible dimensions to get a proper description of the
behaviour of the spring.)
If you actually write a system of equations that describes the
_physics_ of the problem you describe, your mythical "feedback reversing
the original forcing" disappears completely.
> How about restricting your responses to things that actually
>have some relationship to what I post.
How about posting some mathematics that actually have some
relationship to the physical systems you are describing?
Nobody questions your ability to provide mathematical expressions that
have the behavour you want them to have. It's just that none of those
mathematical expressions seem to have anything to do with the way the
real world works.
Dave Halliwell
> I posted:
>> You agree then that Tobis is wrong in saying that we can
>>be certain without looking at the details of the climate system
>>that CO2 forcing will increase temperature?
> R.T. Pierrehumbert responded:
>>No, since Tobis' remark is limited to temperature-dependent feedbacks,
>>and refers to sensitivity to the amount of increase of heating
>>of the planet.
> Ok, how about the following model (which I am not claiming is
>physically realistic).
Would you even be able to _tell_ if it was physically realistic?
> Suppose the ability of the atmosphere to absorb
>visible light (ie shortwave radiation) were temperature dependent
>increasing as the temperature of the atmosphere increased. Then
>suppose we increase the solar constant. The initial response will be
>to heat the surface. The surface will then heat the atmosphere. The
>hotter atmosphere will absorb more incoming solar radiation causing
>further heating. Meanwhile the atmosphere will shade the surface
>causing the surface to begin to cool.
...and since the surface was able to warm the overlying atmosphere
before, it will now _cool_ the overlying atmosphere, and the cooler
atmosphere will absorb less incoming solar radiation. ...and that will
then let the surface heat up, and....
...ad infinitum. Of course, in the real world you have to build
significant time delays into the system to get it to overshoot and swing
back. For example, if you could get the atmosphere to heat up for several
months _without_ increasing atmospheric absorption, then suddenly allow
the temperature-dependent absorption to kick in with a bang, then you
might see the response you describe. Physics doesn't work that way,
though.
Without time delays, the warming of the surface will immediately lead
to atmospheric warming and increased atmospheric absorption, and this will
just simply _limit_ the surface warming to less than you would expect
without the atmospheric feedback. The system will never have a chance to
warm up to the point where you will cause enough absorption to cool the
surface to such an extent.
What Shearer has described is a simple negative feedback that will
reduce the amount of warming to a level lower than expected without
feedback. It can't reverse the initial surface warming. If such a system
existed, it wouldn't be stable to begin with. It would likely be chaotic.
If it's chaotic, then short-term statistics can't be implied to describe
the bounds of the system.
The only reason he thinks that a _reversal_ of surface warming would
occur is because he keeps the feedback shut off until a lot of warming
has occurred, and then instantaneously turns on a negative feedback at high
strength. He then shuts off all further change in the system (read
that as saying that he ignores it) and pretends that an equilibrium has
been reached and that the feedback and forcing will no longer play any
role in the dynamics of the system.
In the real world, the feedback exerts its influence as soon as the
system starts to change, and a negative feedback simply fights against the
forcing _all_the_time_. As a result, the system doesn't reach a point
where it can do the flip-flop.
>If we choose the proper
>parameters I believe we can end up with a hotter atmosphere and a
>colder surface.
I challenge you to actually quantify such a system, for the solar
radiation/surface/atmosphere system that you have described above. If you
can actually come up with such a system, using parameterizations that
actually relate to the physical processes involved, with parameters
chosen to fall within the bounds of physically-plausible values (e.g. no
absorption coefficients greater than 1 or less than 0), then you have a
case.
Please, just for once, try to come up with a mathematical description
that deals with the _physics_, instead of something that is just a
figment of your imagination.
Your "belief" means nothing if it doesn't correspond to reality.
You've claimed that you think you can do it. Either put up or shut up.
> Then feedback has reversed the sign of the response
>of surface temperature to forcing which is what Tobis says is
>impossible.
...and Tobis says so because he (unlike you) actually has some
understanding of physics.
R. T. Pierrehumbert
>visible light (ie shortwave radiation) were temperature dependent
>increasing as the temperature of the atmosphere increased. Then
>suppose we increase the solar constant. The initial response will be
>to heat the surface. The surface will then heat the atmosphere. The
>hotter atmosphere will absorb more incoming solar radiation causing
>further heating. Meanwhile the atmosphere will shade the surface
>parameters I believe we can end up with a hotter atmosphere and a
>of surface temperature to forcing which is what Tobis says is
>impossible.
I honestly hope I don't sound patronizing, but I think Shearer is
learning something from these discussions. This exchange is beginning
to assume some semblance of physical interest.
Next time Shearer will have to do his own arithmetic, but this
time, by special dispensation, I'll draw it out for him.
Model A: Isothermal slab atmosphere with temperature
Ta; absorption gam(Ta) monotonically increasing in Ta;
surface temperature Te; surface albedo = 0 (i.e. perfectly
absorbing); solar flux S.
The zero albedo simplifies the algebra, but keeps the main behavior.
I leave it to somebody else to discuss the case of imperfect
surface absorption.
Solution:
gam*S + sigma*Te^4 = 2sigma*Ta^4
(1-gam)*S + sigma*Ta^4 = sigma*Te^4
(Note I assumed also the atmosphere has unit emissivity).
Hence, the atmospheric temperature is determined by
sigma*Ta^4 = S
and the surface temperature is
sigma*Te^4 = S*(2-gam(Ta))
By taking the derivative w.r.t. S, you find that increasing
S decreases Te near an equilibrium Ta where
2-gam(Ta)-.25Ta* (dgam/dTa) <0
which can clearly be satisfied with a sufficiently steep
choice of the function gam(Ta). Hence a case where increasing
atmospheric absorption in a temperature dependent way cools
the surface.
If you do a case with nonzero surface albedo, the absorption
feedback enhances the increase of Ta with S, while retarding or
reversing the increase of Te with S.
I have lost track by now of what Tobis, Halliwell and Shearer
were arguing about, so I have no idea whether this kind of
feedback is of the sort to satisfy anybody, according to
the terms of the original argument. Over to you, gents, on
that one. But it is clearly of the class of feedbacks which
enhances (or is neutral towards) the particular temperature
it depends on (Ta in this case), while decreasing SOME OTHER
temperature in the system. It's not unlike the situation
where global warming increases atmospheric temperature,
but shuts of North Atlantic Deep Water formation, leading
to a local cooling of the N. Atlantic.
========================
Model B: (quite appropriate for the tropics)
Instead of a slab atmosphere, let the atmosphere sit
on a set of profiles T0 - az, e.g., with outgoing radiation
determined by solving the rad. transfer eqn for the
outgoing IR ("OLR") as a function of the low level temperature
T0. Let the surface albedo be alpha. Assume the atmosphere
is optically thick in the IR so that the OLR is independent
of the surface temperature Te. As before, assume atmospheric
absorption with a coefficient gam(To), monotonically increasing.
Finally, assume that surface heat exchanges are efficient enough
to keep Te=T0. This is roughly true in the tropics, owing
to the efficiency of evaporative heat exchange.
With this model, temperature is determined by solving the
equation:
net absorbed solar = OLR(T0)
the rhs being monotonically increasing in T0. If you increase
S, you increase T0, but this leads to an increase in net
absorbed solar, owing to gam(T0). Hence, T0 increases more
than it would without the absorption feedback. However, in
this case, the surface temperature is dragged along with To,
and so it also increases more than it would without feedback.
Thus, whether the temperature-dependent absorption feedback
warms or cools the surface depends on the surface heat exchange
processes. In Model A, these were purely radiative. Model B
assumes strong heat fluxes due to, say, evaporative transfer.
Dave Halliwell
text, to keep things clear. I've already responded once to Shearer's
original post. Pierrehumbert's response adds a few twists.
"R. T. Pierrehumbert" <rt...@midway.uchicago.edu> writes:
[In response to a suggestion from Shearer]
First, I'll point out a few aspects of Shearer's original proposal.
>> Ok, how about the following model (which I am not claiming is
>>physically realistic). Suppose the ability of the atmosphere to absorb
>>visible light (ie shortwave radiation) were temperature dependent
>>increasing as the temperature of the atmosphere increased. Then
>>suppose we increase the solar constant. The initial response will be
>>to heat the surface.
Note here that increased absorption of solar radiation by the
*atmosphere* has been ignored. If we start out with a completely
transparent atmosphere, this would be appropriate. However, this would
presume that the "current" atmosphere is at a delicate balance point,
where there is _currently_ no absorption, but the slightest warming would
lead to absorption. Such a situation is highly unlikely, so the
description _should_ include direct atmospheric heating as the result of
increased solar output.
>> The surface will then heat the atmosphere. The
>>hotter atmosphere will absorb more incoming solar radiation causing
>>further heating. Meanwhile the atmosphere will shade the surface
>>causing the surface to begin to cool. If we choose the proper
>>parameters I believe we can end up with a hotter atmosphere and a
>>colder surface. Then feedback has reversed the sign of the response
>>of surface temperature to forcing which is what Tobis says is
>>impossible.
>Next time Shearer will have to do his own arithmetic, but this
>time, by special dispensation, I'll draw it out for him.
>Model A: Isothermal slab atmosphere with temperature
>Ta; absorption gam(Ta) monotonically increasing in Ta;
>surface temperature Te; surface albedo = 0 (i.e. perfectly
>absorbing); solar flux S.
>Solution:
>gam*S + sigma*Te^4 = 2sigma*Ta^4
>(1-gam)*S + sigma*Ta^4 = sigma*Te^4
>(Note I assumed also the atmosphere has unit emissivity).
Note that this also means unit absorptivity: IR emitted upwards from
the surface is entirely absorbed in the overlying atmosphere. In
descriptive terms, the first equation is the energy balance for the
atmosphere: the left side is absorbed solar plus absorbed IR (emitted from
the surface), and the right side is IR emitted from the atmosphere (where
the factor of 2 is because the atmosphere emits both upwards and
downwards). The second equation is the energy balance for the surface:
again the left side is absorption (the solar that passed thorugh the
atmosphere, plus IR emitted downwards by the atmosphere), while the right
side is emission (this time only a factor of one because the surface only
emits upwards).
>Hence, the atmospheric temperature is determined by
> sigma*Ta^4 = S
Again, descriptively this means that the IR emitted to space (which
has to be from the atmosphere), balances the total solar (absorbed by
both the atmosphere and the surface). It represents the boundary
condition at the top of the atmosphere.
>and the surface temperature is
> sigma*Te^4 = S*(2-gam(Ta))
>By taking the derivative w.r.t. S, you find that increasing
>S decreases Te near an equilibrium Ta where
> 2-gam(Ta)-.25Ta* (dgam/dTa) <0
>which can clearly be satisfied with a sufficiently steep
>choice of the function gam(Ta). Hence a case where increasing
>atmospheric absorption in a temperature dependent way cools
>the surface.
Let's look at some reasonable limits for the "current" climate. S is
342 W/sq.m (1368/4). This gives Ta=279K. Obviously, gam(Ta) has to be
limited between 0 and 1. For gam(Ta) starting at 0, dgam/dTa>0.029 is
required; i.e. gam(Ta) has to increase by 0.029 for each 1K rise in Ta.
For such a selection of gam(Ta), a change in Ta of only 35K is sufficient
to take gam(Ta) from 0 to 1. I would consider such a sensitivity to be
quite high. If we take the other limit (gam(Ta)=1), dgam/dTa need only be
greater than 0.014 to have Ts decreasing as S increases, but it isn't
actually possible to have gam(Ta) continue to increase at that point,
since we are already at the limiting value of gam.
>If you do a case with nonzero surface albedo, the absorption
>feedback enhances the increase of Ta with S, while retarding or
>reversing the increase of Te with S.
In such a case, the overall temperatures would be cooler. One needs to
be careful in setting up the energy balance equations for such a system,
since the atmosphere will be absorbing a fraction of the incoming solar
radiation _plus_ a fraction of the solar radiation reflected by the
surface.
For anyone interested in pursuing this sort of simple radiative energy
balance model a bit further, I would suggest reading John Harte's
"Consider a Spherical Cow", where a similar model is presented. The
assumption Pierrehumbert makes of a single atmospheric layer is a bit
simplistic (to say the least). Harte considers a model with no
atmospheric absorption, but "n" layers, and presents closed form
solutions for the temperature of each layer. Oddly, the appropriate
number of layers for the current atmosphere is about 2. Adding gases that
absorb in the IR has the effect of increasing the apparent number of
layers, and Harte considers the effects of doubling CO2 in his book.
(Mathematically, there is no problem in handling fractional layers!) Even
more oddly, such a simplistic model isn't far off the results you'd get
for a 1d-RCM with the convective adjustment shut off.
>I have lost track by now of what Tobis, Halliwell and Shearer
>were arguing about, so I have no idea whether this kind of
>feedback is of the sort to satisfy anybody, according to
>the terms of the original argument. Over to you, gents, on
>that one.
...and of course this post is an attempt to address that question. :-)
Unfortunately, the argument is not over whether a forcing (in this
case, solar radiation) can lead to different signs for T response in
different parts of the system. The argument is over whether or not a
forcing that _initially_ causes direct warming (in the case, allegedly the
surface) can later be affected by a feedback that _reverses_ the original
forcing, leading to a final result of cooling. [Also note that the
original argument is with regard to fairly small perturbations.]
Remeber that in Shearer's original description of the problem, the
_initial_ surface response is supposed to be *heating*. He then adds a
feedback that _requires_ the heating to manifest itself, followed by a
cooling that more than offsets the original heating. The finla result is
supposed to be a cooler surface.
In the model that you discuss, the only way to create such a
time-dependent temperature change would be to allow the initial solar
increase to cause a warming by ignoring the rest of the energy balance
terms for a period of time, followed by the addition of the atmospheric
radiative feedback later on.
I consider such a system to be physically unrealistic. Mathematically,
it is easy to establish, but I don't see it matching any realistic
Paul Swanson
>> I posted:
[cute results about a 2-d linear ode deleted]
jshe...@VNET.IBM.COM writes:
> Your post is not exactly wrong, but it is irrelevant.
[remainder deleted]
Actually what is irrelevant is your whole debate with (Dr.?)Tobis. Essentially
you two are arguing over the definition of linear feedback. Your example
had nothing to do with climate and that was the point of my post. For
what its worth, I'm skeptical that Tobis' version of feedback in a linear
system is relevant to climate modeling but that remains to be seen.
Let me summarize your debate with Tobis (with side comments by Len Evans)
Shearer: Here is an example of a feedback system where the new steady state
is in the opposite direction of the forcing.
Tobis: Its not a linear feedback system.
Len Evans: (to Shearer) But can you come up with a physically sensible model
that does this?
while(1) {
Shearer: Is too.
Tobis: Is not.
Len Evans: (to Shearer) But can you come up with a physically sensible model
that does this?
}
My post was prompted by a desire to see an answer to Prof. Evans' question.
Paul.
Steve Emmerson
>... Suppose the ability of the atmosphere to absorb
>visible light (ie shortwave radiation) were temperature dependent
>increasing as the temperature of the atmosphere increased. Then
>suppose we increase the solar constant.
To me, this system doesn't have feedback, which, as I understand it, is
the "feeding back" of an output response to the input side of a system
via some mechanism. The system you describe has solar forcing as the
input forcing and temperature (either surface or atmospheric or both) as
the output response. To obtain feedback, you would have to couple the
temperature back to the solar constant -- which is unlikely.
What is your definition of "feedback"?
--
Steve Emmerson st...@unidata.ucar.edu ...!ncar!unidata!steve
Dave Halliwell
>jshe...@VNET.IBM.COM writes:
>>... Suppose the ability of the atmosphere to absorb
>>visible light (ie shortwave radiation) were temperature dependent
>>increasing as the temperature of the atmosphere increased. Then
>>suppose we increase the solar constant.
>To me, this system doesn't have feedback, which, as I understand it, is
>the "feeding back" of an output response to the input side of a system
>via some mechanism. The system you describe has solar forcing as the
>input forcing and temperature (either surface or atmospheric or both) as
>the output response. To obtain feedback, you would have to couple the
>temperature back to the solar constant -- which is unlikely.
However, each _component_ of the system has its own inputs and
outputs. In an electrical sense, where is the "real" input to a circuit?
At the equipment's power supply? Where it is plugged into the wall? At
the main panel? Back at the regional distribution centre? Each can be
treated as an input (in a local sense), depending on the scale of the
problem being discussed.
In the climate system, there is a similar cascade of energy flow, with
the output of one portion acting as input to another portion. All that is
required for feedback to exist is that the output from one component
eventually lead to changes in input to that same component.
In Shearer's example, solar radiation changes affect surface
temperature, surface temperatures affects other factors, and those other
factors can eventually influence surface temperature again, closing the
feedback loop. That feedback loop exists _regardless_ of what causes the
intital surface temperature change. The example suggested solar output,
but geothermal heat flux would work as well (as would a number of other
things).
The problem with Shearer's example is that he then analyzes the system
as if solar forcing only affects surface temperature (it doesn't - there
is atmospheric absorption in his hypothesized world). He also describes
the time-dependent response as if the surface warming exerts itself
without any feedback (solely in response to solar forcing) for a period
of time, and then the feedback acts alone for a period of time. What
needs to be done is that all processes act concurrently, and that the
surface temperature be in balance with all processes that affect it
(instead of treating one process as if it acts independently).
jshe...@vnet.ibm.com
proposed in an earlier post.
I had posted:
> Ok, how about the following model (which I am not claiming is
>visible light (ie shortwave radiation) were temperature dependent
>increasing as the temperature of the atmosphere increased. Then
>hotter atmosphere will absorb more incoming solar radiation causing
>further heating. Meanwhile the atmosphere will shade the surface
>causing the surface to begin to cool. If we choose the proper
>parameters I believe we can end up with a hotter atmosphere and a
>colder surface. Then feedback has reversed the sign of the response
>of surface temperature to forcing which is what Tobis says is
>impossible.
Halliwell observed:
> Note here that increased absorption of solar radiation by the
>*atmosphere* has been ignored. If we start out with a completely
>transparent atmosphere, this would be appropriate. However, this would
>presume that the "current" atmosphere is at a delicate balance point,
where there is _currently_ no absorption, but the slightest warming would
>lead to absorption. Such a situation is highly unlikely, so the
>description _should_ include direct atmospheric heating as the result of
>increased solar output.
I realized after posting the original scenario that it was
open to this objection. I should have chosen a means of heating the
surface that only has indirect effects on the atmosphere. Assume
say a network of nuclear power stations or as Halliwell suggested in
a later post increased geothermal heat flow. In what follows I will
discuss such a revised model (ie keep the solar constant constant and
force the surface directly in some way).
First we will look at the equilibrium behavior. Following
Pierrrehumbert let:
Ta - temperature of isothermal slab atmosphere
Te - temperature of surface
a(T) - shortwave absorption of atmosphere at temperature T
S - solar flux
c*T**4 - blackbody radiation at temperature T
f - surface forcing
Assume surface albedo 0 and atmospheric emissivity 1.
Assume radiative energy transfer only. Then the energy leaving
the atmosphere will be 2*c*Ta**4, the energy leaving the surface
will be c*Te**4, the energy absorbed in the atmosphere will be
a(Ta)*S+c*Te**4 and the energy absorbed in the surface will be
(1-a(Ta))*S+c*Ta**4+f. At equilibrium the energy emitted and
absorbed will balance in the atmosphere and at the surface. So
we have the following equations.
a(Ta)*S + c*Te**4 = 2*c*Ta**4 <1>
(1-a(Ta))*S + c*Ta**4 + f = c*Te**4 <2>
Adding <1> and <2> we may solve for Ta**4 ie:
Ta**4 = (S+f)/c <3>
Note Ta is independent of the form of a(Ta). We now choose a(Ta)
as follows:
a(Ta) = b*((c/S)*Ta**4 - 1) + d <4>
Note when f=0, Ta**4=(S/c) and a(Ta)=d. a(Ta) must lie between
0 and 1 which will be true for values of Ta near (S/c)**.25 if
0<d<1.
Now plug <4> and <3> into <1> and solve for Te. We
obtain:
Te**4 = ((2-d)*S + (2-b)*f)/c <5>
Hence when b is greater than 2 the surface temperature at
equilibrium moves in a direction opposite to the forcing f.
Note however we have ignored questions of stability. The
equilibrium solution may not be stable in which case the above
simple analysis is invalid. Also Halliwell objects:
> The problem with Shearer's example is that he then analyzes the system
>as if solar forcing only affects surface temperature (it doesn't - there
>is atmospheric absorption in his hypothesized world). He also describes
>the time-dependent response as if the surface warming exerts itself
>without any feedback (solely in response to solar forcing) for a period
>of time, and then the feedback acts alone for a period of time. What
>needs to be done is that all processes act concurrently, and that the
>surface temperature be in balance with all processes that affect it
>(instead of treating one process as if it acts independently).
To investigate stability and deal with Halliwell's second
objection (we already dealt with the first when we changed the way we
introduced forcing) we need to model the nonequilibrium dynamics of
the system. We do this by assuming when the energy fluxes for the
surface or atmosphere do not balance the temperature of the surface
or atmosphere changes at a rate proportional to the amount of the
imbalance (with constants of proportionality dependent on the heat
capacities). Let Ta' and Te' be the time derivatives of Ta and Te.
Let Ha and He be the constants of proportionality. Then <1> and
<2> yield the following differential equations:
Ta' = ( a(Ta)*S + c*Te**4 - 2*c*Ta**4 ) * Ha <6>
Te' = ( (1 - a(Ta))*S + c*Ta**4 + f - c*Te**4 ) * He <7>
These equations are nonlinear. However we are interested in small
perturbations around the f=0 equilibrium solution <3> and <5>.
Hence we may linearize the system as follows. Let
Ta = (S/c)**.25 + Ra <8>
Te = ((2-d)*S/c)**.25 + Re <9>
where Ra and Re are small perturbation terms. Next plug <8> and <9>
into <6> and <7> using <4> and dropping all higher order terms in
Ra and Re. If I did the algebra correctly this gives the following
linear system of differential equations in Ra and Rb.
Ra' = (b-2)*g*Ha*Ra + (2-d)**.75*g*Ha*Re <10>
Re' = (1-b)*g*He*Ra - (2-d)**.75*g*He*Re + f*He <11>
where g = 4. * (c**.25) * (S**.75) <12>
Now the stability of this linear system depends on eigenvalues
of the matrix M:
(b-2)*g*Ha +(2-d)**.75*g*Ha
(1-b)*g*He -(2-d)**.75*g*He
The determinant is (g**2)*Ha*He*(2-d)**.75 which is always positive.
Hence the system will be stable iff the trace is negative. This is
clearly true when b<2. However we are interested in the case b > 2.
This produces the following stability condition:
(b-2) * Ha < (2-d)**.75 * Hb <13>
This means the rate at which the temperature of the atmosphere
responds (Ha) must not be too large compared to the rate at which
the temperature of the surface responds (He). Intuitively what
going on is this. When b > 2 the atmosphere considered by itself
is an unstable system. If it begins to heat, the feedback caused
by increased solar absorption will cause a runaway temperature rise.
However the interaction with the temperature of the surface is a
potentially stabilizing negative feedback loop (since the increased
solar absorption in the atmosphere may cool the surface which in
turn may cool the atmosphere). However for this loop to succeed
in stabilizing the system the atmosphere must not respond too
quickly compared to the surface. This is the stability condition
<13>. Note the larger b is, the more unstable the atmosphere is by
itself and the more stringent the stability condition becomes.
Now consider the effect of the forcing term f. For small f
we may continue to approximate the nonlinear system <6>,<7> by the
linear system <10>,<11>. It is easy to verify that the equilibrium
response of Re in the linear system is opposite to the direction of the
forcing when b>2 (just as in the nonlinear system considered above).
It is possible to write down the exact solution of the linear system
with forcing turned on at time 0 and initial conditions Ra=Re=0 in
terms of the eigenvalues and eigenvectors of M which will show how the
system moves to the new equilibrium. I will not bother as I have
already done this for the original order 2 system I posted as a
counterexample to Tobis and the behavior will be qualitatively
similar. Intuitively what happens is this. Suppose the system is
in a stable equilibrium with Ra=Re=0. When forcing is turned on Re'
and Ra'' immediately become positive. However Ra' remains 0 (to first
order). This means the initial response to the forcing will be much
greater for Re than for Ra. However Ra will eventually respond as
well. Now once Ra starts to increase it will try to runaway (since
the atmosphere is unstable by itself). However the interaction with
Re prevents this. First the increase in Ra will cause the increase in
Re to halt. However this is not enough to stop the increase in Ra. As
Ra continues to increase Re will start to decrease. Eventually the
decrease in Re will halt the rise in Ra. Note however this cannot
occur until Re becomes negative since when Re is 0, Ra' remains
positive.
The case where the stability condition is not satisfied may
also be of interest. In this case, I believe Ta will move so that
a(Ta) flips back and forth between 0 and 1 (it can't get stuck in
either position because then the long term energy balance of the earth
will be wrong). When a(Ta) is 1 the surface will slowly cool reducing
Ta as well until a(Ta) moves below 1 at which point feedback rapidly
cools the atmosphere forcing a(Ta) to 0. The surface will now begin
to slowly warm eventually warming the atmosphere enough so that a(Ta)
moves above 0. Then feedback will rapidly warm the atmosphere until
a(Ta) becomes 1. The cycle can then repeat. One could even speculate
that ice ages and interglacials might be caused by some similar
mechanism that does not require any external forcing.
R. T. Pierrehumbert
Note that to get a forcing of this form, one doesn't need to
appeal to anything as unrealistic as geothermal heat flux. If
the surface is the ocean, this could represent changes in the
oceanic heat transport into the region (e.g. reductions in
cold upwelling in the tropics).
Thanks for doing the algebra on this; can't say I've checked
it in detail.
While I approve of slab atmospheres as a model of a somewhat
atmospheric physical system, which is OK for this discussion,
I wish to emphasize that they are very misleading with regard
to the behavior of the real atmosphere. The vertical profile
of temperature, and the surface heat flux, are both crucial
to the way the real greenhouse effect works. This is especially
true in the tropics.
I urge the participants in this discussion to work out the
same example for the more realistic case (I think it's Halliwell's
turn this time).
Dave Halliwell
>> (1-a(Ta))*S + c*Ta**4 + f = c*Te**4 <2>
> ^
>Note that to get a forcing of this form, one doesn't need to
>appeal to anything as unrealistic as geothermal heat flux. If
>the surface is the ocean, this could represent changes in the
>oceanic heat transport into the region (e.g. reductions in
>cold upwelling in the tropics).
In the short term (10-20 years?), the ocean mixed-layer can be taken
largely as a heat source/sink. For longer periods, deeper ocean currents
come into play. This in an area where different time constants (and
non-linearities) become very important.
>While I approve of slab atmospheres as a model of a somewhat
>atmospheric physical system, which is OK for this discussion,
>I wish to emphasize that they are very misleading with regard
>to the behavior of the real atmosphere. The vertical profile
>of temperature, and the surface heat flux, are both crucial
>to the way the real greenhouse effect works. This is especially
>true in the tropics.
...as is the fact that we _have_ a "tropics", and poles, etc. The
equator-pole heat transfer is a major determinant of the climate system.
>I urge the participants in this discussion to work out the
>same example for the more realistic case (I think it's Halliwell's
>turn this time).
Done. I've posted an analysis that illustrates an instability in
Shearer's model. From my perspective, it looks as if his linear
approximation was misleading. His model isn't stable in _any_
configuration, let alone in the configuration where initial surface
warming leads to long-term surface cooling.
Dave Halliwell
deal of the original text. Given that it has been a few days since this
was posted, I'll try to leave as much of the model intact as possible.
jshe...@VNET.IBM.COM writes:
This definitely avoids the question of ignoring secondary effects
associated with the original forcing. Essentially, you are modifying the
heat flux at the lower boundary.
> First we will look at the equilibrium behavior. Following
>Pierrrehumbert let:
> Ta - temperature of isothermal slab atmosphere
> Te - temperature of surface
> a(T) - shortwave absorption of atmosphere at temperature T
> S - solar flux
> c*T**4 - blackbody radiation at temperature T
Note that in this case "c" is not an arbitrary constant: it is the
Stefan-Boltzman constant (5.67E-8 W/m^2/K^-4).
> f - surface forcing
> Assume surface albedo 0 and atmospheric emissivity 1.
>Assume radiative energy transfer only. Then the energy leaving
>the atmosphere will be 2*c*Ta**4, the energy leaving the surface
>will be c*Te**4, the energy absorbed in the atmosphere will be
>a(Ta)*S+c*Te**4 and the energy absorbed in the surface will be
>(1-a(Ta))*S+c*Ta**4+f. At equilibrium the energy emitted and
>absorbed will balance in the atmosphere and at the surface. So
>we have the following equations.
> a(Ta)*S + c*Te**4 = 2*c*Ta**4 <1>
> (1-a(Ta))*S + c*Ta**4 + f = c*Te**4 <2>
>Adding <1> and <2> we may solve for Ta**4 ie:
> Ta**4 = (S+f)/c <3>
>Note Ta is independent of the form of a(Ta). We now choose a(Ta)
>as follows:
> a(Ta) = b*((c/S)*Ta**4 - 1) + d <4>
>Note when f=0, Ta**4=(S/c) and a(Ta)=d. a(Ta) must lie between
>0 and 1 which will be true for values of Ta near (S/c)**.25 if
>0<d<1.
Note that this particular function makes for an absorptivity that
changes from 0 to 1 over a rather narrow change in temperature (for the
suggested values of b and d), but this is moot. I am still quite willing
to examine the chracteristics of this system.
> Now plug <4> and <3> into <1> and solve for Te. We
>obtain:
> Te**4 = ((2-d)*S + (2-b)*f)/c <5>
>Hence when b is greater than 2 the surface temperature at
>equilibrium moves in a direction opposite to the forcing f.
All of the above appears to be in order.
>Note however we have ignored questions of stability. The
>equilibrium solution may not be stable in which case the above
>simple analysis is invalid. Also Halliwell objects:
[objection deleted for brevity]
> To investigate stability and deal with Halliwell's second
>objection (we already dealt with the first when we changed the way we
>introduced forcing) we need to model the nonequilibrium dynamics of
>the system. We do this by assuming when the energy fluxes for the
>surface or atmosphere do not balance the temperature of the surface
>or atmosphere changes at a rate proportional to the amount of the
>imbalance (with constants of proportionality dependent on the heat
>capacities). Let Ta' and Te' be the time derivatives of Ta and Te.
>Let Ha and He be the constants of proportionality. Then <1> and
><2> yield the following differential equations:
> Ta' = ( a(Ta)*S + c*Te**4 - 2*c*Ta**4 ) * Ha <6>
> Te' = ( (1 - a(Ta))*S + c*Ta**4 + f - c*Te**4 ) * He <7>
>These equations are nonlinear. However we are interested in small
>perturbations around the f=0 equilibrium solution <3> and <5>.
>Hence we may linearize the system as follows. Let
...and this is where we start to run into problems. Through analysis
of a linear approximation (deleted) of the non-linear system, Shearer
arrives at the following conclusions:
>The determinant is (g**2)*Ha*He*(2-d)**.75 which is always positive.
>Hence the system will be stable iff the trace is negative. This is
>clearly true when b<2. However we are interested in the case b > 2.
>This produces the following stability condition:
> (b-2) * Ha < (2-d)**.75 * Hb <13>
(I am assuming that Hb is a typo, and should be He as in the rest of
the analysis.)
Note two claims here: the system is _always_ stable if b<2, and if b>2
it will still be stable if certain conditions are met, namely the
inequality expressed in equation <13>. Since the inequality only applies
for b>2, and d has been constrained to be between 0 and 1, all terms in
<13> are positive and it can be restated as:
(Ha/He) < (2-d)^0.75/(b-2)
As an example for b=4 and d=0.5, (Ha/He) < 0.68 is the resulting
requirement for stability.
Now, rather than doing a linearization, I set up the differential
equations <6> and <7> as difference equations, with delta(T)/delta(t)
expressed as functions of the previous Ta and Te. I can then do an
iterative calculation of the time-dependent temperature evolution of the
system of equations.
I actually did the calculations for a series of combinations of b
(0,1,2,4), d (0,0.5,1), and Ha/He (0.5,1,2). I used S=342 (current
climate) in all simulations. I present only two results here, both with
d=0.5 and Ha/He=0.5.
First, b=4. This simulation should be stable according to Shearer's
linear analysis. I started the simulation at time 0 with the _equilibrium
value_, calculated directly from the input data. (The results were done in
a spreadsheet. I don't know machine precision for the real numbers used by
this particular spreadsheet.) Here are the results for the first 10 times
steps.
t Te Ta
0 308.4 278.7
1 308.4 278.7
2 308.4 278.7
3 308.4 278.7
4 308.4 278.7
5 308.4 278.7
6 308.4 278.7
7 308.4 278.7
8 306.6 280.1
9 -2659.8 965.7
10 ************ 817942852.0
Initially, the system seems stable at the equilibrium value, but then
value of Ta seems to start to increase, which immediately throws off the
value of Te, and we rapidly go into an unstable state.
The second example is for b=1 and d=0.5, which should be
unconditionally stable (regardless of Ha/He) according to Shearer's
analysis.
t Te Ta
0 308.4 278.7
1 308.4 278.7
2 308.4 278.7
3 308.4 278.7
4 308.4 278.7
5 308.4 278.7
6 308.4 278.7
7 308.4 278.7
8 308.4 278.5
9 285.3 516.9
10 42662.6 -1070083.9
Several points to note: the equilibrium solution hasn't changed. The
first seven iterations display the same results (to one decimal place) as
the first example. In time step 8, we see the instability developing.
Note that it is opposite in sign to the first example: the second shows a
drop in Ta at time 8, whereas the first showed an increase. Note also
that the first example showed Ta cointinuing to increase, while the
second shows it osscilating above and below the equilibrium value. This
behaviour is the result of the warming/cooling relationship associated
with the value of b.
All of the combinations of b, d, and Ha/He that I considered showed
the instability demonstrated above. I did not bother to set up the
spreadsheet to limt a(Ta) between 0 and 1: they'll end up that way at
equilibrium because of the formulation of the equations, and all I want
to do is demonstrate the instability.
I had originally set up the spreadsheet so that I could start at time
0 with a value that was x degrees away from equilibrium, to see how the
system approached equilibrium. I never bothered to actually introduce an
offset, since the system was not stable when starting with initial values
calculated to the precision of the real variable in the spreadsheet.
It is clear that _any_ deviation away from the equilibrium value leads
to a situation where the system is dynamically unstable. Now, it may be
that I have introduced a _numerical_ instability by casting the
differential equation as a difference equation. However, Shearer's
analysis using a linear approximation should be equally valid in
determining numerical instability. Any theoretical anlysis of numerical
stability is easily refuted by a *demonstration* of instability. I
haven't examined Shearer's linear approximation in detail, but I presume
that the problem is one where ignoring the higher-order terms leads to
errors.
The system Shearer has described is "stable" in the same sense that a
ball can be balanced on top of another ball: there is a theoretical
equilibrium that can be demonstrated mathematically, but in practice the
ball is doomed to fall off eventually.
It may be possible to select a combination of values for which the
equilibrium temperatures are exact powers of two (and therefor can be
stored exactly in the computer). Such a combination would not show
instability, but that is essentially irrelevant.
[remainder of stability discussion of the linear approximation deleted]
> Now consider the effect of the forcing term f. For small f
>we may continue to approximate the nonlinear system <6>,<7> by the
>linear system <10>,<11>.
Even without introducing a a forcing term (which would shift
equilibrium and require an adjustment), the non-linear system is shown to
be unstable.
> The case where the stability condition is not satisfied may
>also be of interest. In this case, I believe Ta will move so that
>a(Ta) flips back and forth between 0 and 1 (it can't get stuck in
>either position because then the long term energy balance of the earth
>will be wrong). When a(Ta) is 1 the surface will slowly cool reducing
>Ta as well until a(Ta) moves below 1 at which point feedback rapidly
>cools the atmosphere forcing a(Ta) to 0. The surface will now begin
>to slowly warm eventually warming the atmosphere enough so that a(Ta)
>moves above 0. Then feedback will rapidly warm the atmosphere until
>a(Ta) becomes 1. The cycle can then repeat. One could even speculate
>that ice ages and interglacials might be caused by some similar
>mechanism that does not require any external forcing.
Such systems _have_ been discussed in the climatological literature.
Lorenz is one of the most notable authors in that respect (after all - he
started much of the ideas of chaos in atmospheric processes). The primary
source of _possible_ mechanisms with these characteristics is ocean
circulation, not atmospheric ciruclation, as far as climate is concerned.
The Morgan and Keith paper surveying 16 climatologists, discussed
recently, looks at questions of this nature: is there more than one
possible climate state, and what is the probability of a change in state
as the result of doubling CO2?
If such a change of state _does_ occur, then we probably have very
little idea regarding what climate to expect. It most likely would be
drastically different from the current one. One of the experts in the
Morgan and Keith paper suggested a warming as much as 16C, which is
nearly three times the change between the last glacial climate and the
current climate.
I do not consider such possibilities as "reassuring".
In summary, once again Shearer has shown that he can derive a
mathematical expression that demonstrates the behaviour he desires, but
that it does not represent the system he is trying to model. Oddly, in
this case it was another mathematical system that he was trying to model!
jshe...@vnet.ibm.com
atmosphere to absorb shortwave radiation is temperature dependent.
I had posted an analysis of (among other things) when this
situation would be stable (using a two component atmosphere, earth
model suggested by Pierrehumbert). Halliwell responded by disputing
my stability conclusions based on his numerical calculations showing
instability.
Briefly Halliwell's calculations prove nothing because they
are using too big a timestep (or equivalently too large values for Ha
and He). This is apparent from the negative temperature values he
obtains.
In more detail Halliwell claimed in part:
> It is clear that _any_ deviation away from the equilibrium value leads
>to a situation where the system is dynamically unstable. Now, it may be
>that I have introduced a _numerical_ instability by casting the
>differential equation as a difference equation. However, Shearer's
>analysis using a linear approximation should be equally valid in
>determining numerical instability. Any theoretical anlysis of numerical
>stability is easily refuted by a *demonstration* of instability. I
>haven't examined Shearer's linear approximation in detail, but I presume
>that the problem is one where ignoring the higher-order terms leads to
>errors.
This is incorrect, numerical instability can occur when the
original system is stable. Often a numerical method will only be
stable for sufficiently small timesteps. (Climate modeling would be
simpler if you could use arbitrarily large timesteps.) In this case
we are approximating a set of differential equations by a set of
difference equations. Suppose we start with a set of linear
differential equations:
x'(t) = M * x(t) <1>
where x is a vector and M is a matrix. I have already observed that
0 is a stable solution of this system if all the eigenvalues of M
have real part < 0. Suppose we approximate x'(t) in <1> by (x(t+h)-
x(t))/h. Then we obtain the following set of difference equations:
y(n+1) = y(n) + h * M * y(n) <2>
where y(0)=x(0) and y(k) approximates x(k*h). <2> may be
rewritten as:
y(n+1) = (I+ h * M) * y(n) <3>
Now the stability condition for <3> is that the eigenvalues of
(I + h * M) must lie within the unit circle. Now if the eigenvalues
of M lie in the left half plane, a little thought will show that
the eigenvalues of (I + h * M) will lie in the unit circle iff h is
sufficiently small.
Hence I believe if Halliwell repeats his calculations
with a much smaller timestep the instability he is seeing will go
away.
Halliwell also posted:
> The system Shearer has described is "stable" in the same sense that a
>ball can be balanced on top of another ball: there is a theoretical
>equilibrium that can be demonstrated mathematically, but in practice the
>ball is doomed to fall off eventually.
This would be an unstable equilibrium. I believe I have
established mathematically that the system has a stable equilibrium.
As noted above you have not yet shown otherwise.
Dave Halliwell
> I continue to discuss a model in which the ability of the
>atmosphere to absorb shortwave radiation is temperature dependent.
> I had posted an analysis of (among other things) when this
>situation would be stable (using a two component atmosphere, earth
>model suggested by Pierrehumbert). Halliwell responded by disputing
>my stability conclusions based on his numerical calculations showing
>instability.
> Briefly Halliwell's calculations prove nothing because they
>are using too big a timestep (or equivalently too large values for Ha
>and He). This is apparent from the negative temperature values he
>obtains.
If you examine the output for b=4 in my tabulated results, you will
notice that the "correction" value sends the temperatures further _away_
from the equilibrium value. No change in Ha, He, or time step will alter
that fact.
> This is incorrect, numerical instability can occur when the
>original system is stable. Often a numerical method will only be
>stable for sufficiently small timesteps.
This is particularly true for simple, explicit formulations such as
the one I used. For the case of b=1, where the times series oscillates on
either side of equilibrium - the system is "overcorrecting" - a smaller
time step (or an underrelaxation adaptation) could solve that problem. It
is because of this behaviour that my original response acknowledged that
I "may have introduced a numerical instability".
> Hence I believe if Halliwell repeats his calculations
>with a much smaller timestep the instability he is seeing will go
>away.
It can't possibly go away in the case where a positive departure from
equyilibrium leads to a further positive departure. All that a smaller
time step will do is make the process take longer for the instability to
show up.
> Halliwell also posted:
>> The system Shearer has described is "stable" in the same sense that a
>>ball can be balanced on top of another ball: there is a theoretical
>>equilibrium that can be demonstrated mathematically, but in practice the
>>ball is doomed to fall off eventually.
> This would be an unstable equilibrium. I believe I have
>established mathematically that the system has a stable equilibrium.
>As noted above you have not yet shown otherwise.
As noted, your argument does not apply to the results I posted for
b=4.
jshe...@vnet.ibm.com
I had posted.
> I continue to discuss a model in which the ability of the
>atmosphere to absorb shortwave radiation is temperature dependent.
> I had posted an analysis of (among other things) when this
>situation would be stable (using a two component atmosphere, earth
>model suggested by Pierrehumbert). Halliwell responded by disputing
>my stability conclusions based on his numerical calculations showing
>instability.
> Briefly Halliwell's calculations prove nothing because they
>are using too big a timestep (or equivalently too large values for Ha
>and He). This is apparent from the negative temperature values he
>obtains.
> If you examine the output for b=4 in my tabulated results, you will
>notice that the "correction" value sends the temperatures further _away_
>from the equilibrium value. No change in Ha, He, or time step will alter
>that fact.
So what? A trajectory which converges to 0 may be moving away
from 0 at some times (even in linear systems). Think of spiraling
in along near ellipses.
I had also posted:
> Hence I believe if Halliwell repeats his calculations
>with a much smaller timestep the instability he is seeing will go
>away.
Halliwell responded:
> It can't possibly go away in the case where a positive departure from
>equyilibrium leads to a further positive departure. All that a smaller
>time step will do is make the process take longer for the instability to
>show up.
As noted above this is wrong. I tried your b=4, d=.5 case with
Ha=.5, He=1.0. A timestep of 1.0 is quite unstable. .1 appears to be
stable (although not a good approximation to the differential equations).
.01 seems to be stable and approximate the differential equations. (I
calculated that timesteps less than .1068 will be stable).
Btw what values of Ha and He were you using? I was unable to
reproduce your numbers. Please try smaller timesteps (or equivalently
smaller Ha and He values). If you still see instability please post
enough details so your results can be reproduced.
R. T. Pierrehumbert
eigenvalues of the linearized problem yield decay, then the
corresponding nonlinear system is also stable to sufficiently small
amplitude perturbations. (See Holmes and Guckenheimer).
Nonlinear instability can only arise when one or more eigenvalues
are purely oscillatory (neutral stability). In that case, resonance
can give rise to nonlinear instability, and one must construct
the center manifold.
I didn't save enough of Shearer's stability analysis to see
which case applies.
Note that a reasonable approximation in the time-dependent
equations is to assume the atmosphere has no thermal inertia
(i.e. its temperature adjusts instantaneously to equilibrium)
while the surface has a finite thermal inertia, corresponding
to a mixed layer of water of depth (say) 50 meters. The
atmosphere responds much more quickly than the ocean.
Over land, the opposite approximation would be appropriate.