### Of Warm Poles and Cold Tropics

Yes, you read it correctly. And no, it’s not about the good citizens of Poland. Why are the poles warm and the tropics cold, and compared to what? An equivalent question is: what determines the difference in temperature between tropics and poles, and can we understand why that difference has the magnitude it has on the real planet Earth? Just for the record, there is not a single climate simulation model that tells us what that difference should be, let alone why its magnitude is what it is; it is one of the boundary conditions at the start of a climate ‘run’.

One of the delights of discussions with climate scientists is that it gives you an insight in their way of thinking. For instance, take the often posited: ‘one should look at the big picture’. What many climate scientists mean by that is considering the latest super-duper ocean-atmosphere coupled models with line-by-line radiative transfer codes and what not more. Without that, the claim is, you cannot understand how the climate system works. Curiously, to a physicist such as yours truly ‘looking at the big picture’ means precisely the opposite: stripping down the system under consideration to its knuckle-bare essentials and glean insights about its inner workings from the underlaying physics without being distracted by detail.

So what’s here the big picture? In its simplest form the Earth is a heat-transfer system in non-equilibrium (but in a steady state) where heat (predominantly) enters in the equatorial regions and heat is (predominantly) lost at the poles. Inside, heat is transported from the tropics to the poles by atmospheric and oceanic currents, cooling the tropics and warming the polar regions. Without such transport the local temperature would have been determined by the balance between insolation and radiative losses only, which would make the tropics warmer than they are and the poles colder. The tropics are cooler than they would have been without that transport, while the polar regions are warmer. Hence, cold tropics and warm poles.

Now here is the conundrum: the heat transfer between tropics and polar regions is determined by their temperature difference. The larger it is, the bigger the heat flow. But the heat transfer reduces that temperature difference and thus acts against its own causation. For instance consider the limiting case of a very large heat flow in which the heat transfer is practically instantaneous. As a result the poles would have the same temperature as the tropics, but then there can be no heat flow at all, because that is driven by a temperature difference. Thus we have a contradiction. Starting from the converse situation of no heat flow is also untenable because the large temperature difference will induce a heat flow. Yet, there is a heat flow, which therefore must be consistent with the temperature difference it causes. A smaller temperature difference will not be possible, because that would decrease the heat flow and thus increase that difference again; and a larger temperature difference would increase the heat flow, thus decreasing it. This suggests strongly that the atmosphere as a whole settles in a state where the temperature difference between tropics and poles is minimal while concurrently the heat flow it causes between them is maximal. We can use this insight to calculate its magnitude by applying the principle of Maximum Entropy Production MEP, a working hypothesis in non-equilibrium thermodynamics which has increasing support in observations (see Kleidon for an excellent introduction, refs 1, 2).

The MEP principle states that a system far from (thermodynamic) equilibrium will adapt to a steady state in which energy is dissipated and entropy produced at the maximum possible rate. Notice that ‘maximum entropy production’ is something completely different from ‘maximum entropy’. The latter relates to a closed system in thermodynamic equilibrium. Here we are dealing with an open system not in thermodynamic equilibrium but in a steady state of non-equilibrium.

I’m afraid I’ll have to introduce here some formulae. What I’ll do is
give the basic equation and it’s solution here, while putting how you
can find that solution in an appendix for the aficionados. Those
familiar with thermodynamics will know this expression for the
production, that is the change, dS of entropy S by a change of heat
content *dQ *at temperature* T*:

From this follows directly the expression for the entropy production by a heat flow *Q *between the equatorial regions at temperature *T _{e} *and the poles at

*T*:

_{p}The heat leaves the equatorial regions at temperature *T _{e}*, hence the minus sign, and arrives at the poles at temperature

*T*. Maximising

_{p}*S*is akin to finding the optimum combination of

*Q*and

*T*. The solution of this equation is rather straightforward (see the appendix). The actual difference

_{e}-T_{p}*T*comes out at very nearly half the difference there would have been without energy flow. Half, hence not 0.3 or 0.7 times.

_{e}-T_{p}Let’s test this result against models and observations. It appears to
be well established – from calculations with standard atmospheric
models – that without the heat transfer the equatorial regions would be
about 15 degrees warmer and the poles about 25 degrees colder
(Centigrade, ref 3). I take these estimates at face value; they imply an
asymmetry *β* = 0.25 for which 1 – 2*α* ~ 0.51 (re: appendix). Then the prediction is that the temperature difference between tropics and the poles *with *the
heat transfer is about 41 degrees (being (15 + 25) × 0.51∕0.49) while
the difference without would be about 80 degrees. The table lists some
real observations.

Region | Temperature range | T mean | Notes |

Arctic | -32 +6 | -13 | Danish Arctic Survey (> 80^{o}L) |

Antarctic | -32 -17 | -28 | Byrd Station |

Tropics | 21 29 | 24 |

Observed Polar and Equatorial Temperatures (Centigrade)

For the Antarctic I used data from the Byrd station because it was located on a plateau and therefore presumably less affected by possible shielding by mountains. The averaged polar temperature appears to be about -21C. For the tropics the lower end of the range corresponds predominantly to oceanic data and should therefore have a larger weight in the mean, here taken as 24C. Thus the observed temperature difference between polar and equatorial regions comes out at about 45 degrees, against a prediction of 41 degrees. Not bad for such a basic model.

It may be a simple model but the MEP principle has an important
implication. Maximisation of entropy production for steady-state
conditions implies a strong negative feedback to perturbations (see
Kleidon). Any perturbation will lead to a decrease in *S*, by
definition, and the system will adjust trying to optimise it again. One
such perturbation in climate models is well-known and long established
and widely accepted as a valid concept: the ‘polar amplification’. It
says that in a warming atmosphere the polar regions warm faster than the
equatorial regions; hence that *T _{e} – T_{p}*
decreases. However, it is also one of the oddest concepts to be found in
climate modelling because it has no limiting condition: nowhere in the
literature does it say something like ‘under this and that conditions,
at this level of warming, the polar amplification will stop’. This could
lead to an absurdity: should the atmosphere get hot enough the
temperature difference between equator and pole would disappear
completely or even change sign. Therefore there must be conditions when
the amplification does not happen anymore. Why is this important? If
there is such a mechanism which kicks in under certain circumstances
then one has to explain why it does not work at this very moment and
that could be embarrassing. Whatever the case may be, from the foregoing
it should be obvious that the concept of polar amplification is in fact
rather questionable: the decreased difference of polar and equatorial
temperatures reduces the heat flow to the polar regions which would
result in a cooling. A realistic climate model should behave as the
underlaying physics dictates. It can be deduced, therefore, that such a
feedback is absent in current climate models and that consequently the
whole concept could be an artefact of an incomplete model. [

*You read it here first.*]

But, as luck would have it, the Arctic has warmed over the past 3 or 4 decades. That would prove the polar amplification concept, wouldn’t it? Well, that warming may be the case in the Arctic, but it should also apply to the Antarctic at the same time, but there the concept appears to fail the test against observations (ref 4).

So what may be going on then in the Arctic? If I’m allowed, I put here my two cents worth on the subject. To me, the key is in the fact that there are two poles. It would be extraordinary if at all times precisely the same fraction of equatorial heat would be partitioned to each pole. The transfer mechanisms, atmospheric and in particular ocean currents, are inherently chaotic systems, prone to switch from one particular quasi-stable flow pattern to another. For ocean currents we know this behaviour as the Pacific Decadal Oscillation, the Atlantic Multi-decadal Oscillation, the Indian Ocean Dipole and the like. The timescale for such changes is many decades, from 3 for the PDO, to 6-8 for the AMO, because there is an enormous mass and momentum of moving water involved and water is an incompressible fluid which means that the change occurs over the whole flow pattern simultaneously. Therefore it is to be expected that the partitioning of heat to the north and south is subject to changes on a timescale of decades in an erratic seesaw fashion. We happen to live in a time when the northern hemisphere gets a bit more than its fair share of the heat; in a few decades it may be completely different. To disentangle such changes from a possible underlaying long-term warming trend will require data over a much longer timespan then we have now.

References:

(1) Axel Kleidon (2009): ”Non-equilibrium thermodynamics and maximum entropy production in the Earth system”, https://link.springer.com/article/10.1007/s00114-009-0509-x

(2) A Kleidon, R D Lorenz (editors) ”Non-equilibrium thermodynamics and the production of entropy” (2005) Springer, ISBN 978-3-540-22495

(3) e.g. http://earthguide.ucsd.edu/virtualmuseum/virtualmuseum/EarthsClimateMachine.shtml

(4) e.g. https://wattsupwiththat.com/2019/04/10/the-curious-case-of-the-southern-ocean-and-the-peer-reviewed-journal/

Solving the EP equation

The entropy production *S* for a heat flow *Q *between the equatorial regions of temperature *T _{e} *and the polar regions of temperature

*T*is given by:

_{p}The MEP hypothesis is that the atmospheric system tends to maximise
S. This optimum can be found analytically as follows. Introduce *Θ _{e} *and

*Θ*, the temperatures of the equatorial and polar regions respectively if there were no heat transport. That is,

_{p}*Θ*and

_{e}*Θ*stand for the radiative equilibrium temperatures of those regions. The actual temperatures (

_{p}*T*and

_{e}*T*) are different because of the heat transport. Hence define

_{p}*ΔT*=

_{e}*Θ*and

_{e}– T_{e}*ΔT*=

_{p}*T*. Let’s for the moment assume that they are both equal and equal to a fraction α of the range

_{p}– Θ_{p}*Θ*:

_{e}– Θ_{p}Notice that the difference *T _{e} – T_{p}* equals:

Obviously for *α* = 0 there is no heat transfer, hence *S* = 0, while for *α* = 0.5 we have *T _{e} *equal to

*T*and thus no entropy production either. In between these extremes the heat flux Q is somehow proportional to

_{p}*ΔT*, the larger the drop in temperature of the equatorial region, the larger the heat loss from it and one can assume the relation to be (close to) linear. This gives us an expression for

_{e}*S*in terms of

*ΔT*, hence as function of

*α :*

The maximum of *S* in the range [0.0 – 0.5] is
straightforwardly found with basic calculus techniques – you need to
solve a cubic equation in α – and turns out to be almost exactly at α =
0.25. Alternatively, one could use the R analysis package, encode the
function for S, find the maximum with Newton’s method and get a figure
as a bonus.

The upshot of all this is that the MEP temperature difference *T _{e} -T_{p}* is very nearly half the difference

*Θ*which we would have had in the absence of any heat flow.

_{e}– Θ_{p}Let’s now revisit *ΔT _{e}* and

*ΔT*. Adopting equality of

_{p}*ΔT*and

_{e}*ΔT*means assuming that for every degree drop in temperature of the equatorial regions the polar temperature rises by one degree. This is not very realistic; quite likely the polar temperature rises more because of area and heat capacity differences. We can include such an imbalance by slightly modifying the definitions of

_{p}*ΔT*and

_{e}*ΔT*and introduce an asymmetry parameter

_{p}*β:*

With a value of *β *= 0.25, for instance, this would mean that
for every 3 degrees drop in equatorial temperature the poles would gain
5 degrees. Notice that the difference *T _{e} – T_{p}*
remains as given earlier. Repeating the foregoing exercise for a range
of β values presents us with an interesting result: the corresponding α
values are practically unchanged: for

*β*= 0 the maximum is at

*α*= 0.2487, for

*β*= 0.25 at 0.2441 and

*β*= 0.33 at 0.2427. This means that irrespective of a possible asymmetry between equator and polar regions the temperature difference between the two remains firmly at about half the difference

*Θ*, but that the centre of the range shifts according to the value of

_{e}– Θ_{p}*β*.

This article appeared on the Watts Up With That? website at https://wattsupwiththat.com/2020/12/21/of-warm-poles-and-cold-tropics/

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