<p>If we consider that by &quot;Assume daily cycle and convection and so on operate as usual&quot; you meant that all heat transport from/to the pole remain as it is today. Then, we can do a back of the envelope calculation. This calculation will at least give you an order-of-magnitude answer, and we can then consider everything that would affect the result.</p>
<p>Currently both poles radiate more energy than they receive trough direct solar irradiation. This is because they are colder than the rest of the planet. Therefore, there is a neat heat transport from the equatorial latitudes to the poles. That heat travels on atmospheric and oceanic currents. This figure nicely summarize this radiation balance:</p>
<p><a href="https://i.sstatic.net/AjGDk.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/AjGDk.png" alt="enter image description here" /></a></p>
<p>Figure from <a href="https://www.meted.ucar.edu/search/details.php?id=24464" rel="nofollow noreferrer">here</a> ©The COMET Program</p>
<p>If we consider that the heat transport depicted at the bottom of the figure remains the same. The only change in the energy budget would be from solar irradiation, and once they start warming the only way to release the additional heat and reach equilibrium would be trough infrared radiation (black body radiation).</p>
<p>The poles receive zero energy on the equinox and the winter solstice and <a href="http://www.applet-magic.com/insolation.htm" rel="nofollow noreferrer">12.64 kWh/m<span class="math-container">$^2$</span> per day on the summer solstice</a>. Therefore a rough yearly average would be 3.16 kWh/m<span class="math-container">$^2$</span> per day. So, if we were locked on a summer solstice state you would receive four time more energy, and none if locked in a winter solstice state.</p>
<p><strong>Summer solstice locked pole</strong></p>
<p>Let's consider the summer solstice case first. If we call <span class="math-container">$E_1$</span> the current amount of outgoing radiation from the pole, and <span class="math-container">$T_1$</span> the current temperature. The <a href="https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law" rel="nofollow noreferrer">Stefan–Boltzmann law</a> states that</p>
<p><span class="math-container">$E_1 = \sigma T_1^4$</span></p>
<p>Analogously, if <span class="math-container">$E_2$</span> and <span class="math-container">$T_2$</span> represent the summer solstice locked state, you have</p>
<p><span class="math-container">$E_2 = \sigma T_2^4$</span></p>
<p>Considering that <span class="math-container">$E_2 = 4 E_1$</span></p>
<p>You can write that</p>
<p><span class="math-container">$4 = \frac{T_2^4}{T_1^4}$</span></p>
<p>Or</p>
<p><span class="math-container">$T_2 = (4)^{1/4}\ T_1 = \sqrt{2}\ T_1 \approx 1.41\, T_1$</span></p>
<p>Therefore the temperature would increase 41%. It doesn't sounds much, but considering that the temperatures above are in Kelvin, it es a lot! In fact, <strong>with the current <a href="https://en.wikipedia.org/wiki/North_Pole#Climate" rel="nofollow noreferrer">mean temperature in the north pole of -16 °C</a> (257K), it would go up to +89°C!! And the South pole would go from the <a href="https://en.wikipedia.org/wiki/South_Pole#Climate_and_day_and_night" rel="nofollow noreferrer">current -49°C</a> (224K) to +42°C.</strong></p>
<p>This diverges from any posible reality because with 89°C at the north pole there would not be a net influx of energy from the equatorial latitudes as we are assuming. Instead, atmospheric and oceanic currents would take a lot of that heat down south and the temperature would equilibrarte at a much lower value.</p>
<p>Note that in the above calculation I've ignored the energy flux due to advection in atmospheric/oceanic currents. I did so because that flux is about five time smaller than the flux from solar radiation, so it won't change much the results and ignoring it simplifies the math. You can do the calculation considering those flows if you want.</p>
<p><strong>Winter solstice locked pole</strong></p>
<p>In the case of a pole locked to a winter solstice. There would be no incoming solar radiation at all. Therefore, all incoming energy would be transported by currents from equatorial latitudes. In that way, we can see in the above figure that the north pole receives about 150 W/m<span class="math-container">$^2$</span>, and the south pole about 100 W/m<span class="math-container">$^2$</span>. The equilibrium temperature would be reached when they radiate the same amount. So, using <a href="https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law" rel="nofollow noreferrer">Stefan–Boltzmann law</a> again we can do some rough calculations.
Let's call the south pole temperature <span class="math-container">$T_{SP}$</span> and <span class="math-container">$T_{NP}$</span> for the north pole. Then:</p>
<p><span class="math-container">$T_{SP} = \left(\frac{E_{SP}}{\sigma}\right)^{1/4} = \left(\frac{100 \,W/m^2}{5.67\times 10^{-8}}\right)^{1/4} = 205 K= -68°C$</span></p>
<p>And for the north pole</p>
<p><span class="math-container">$T_{NP} = \left(\frac{E_{NP}}{\sigma}\right)^{1/4} = \left(\frac{150 \,W/m^2}{5.67\times 10^{-8}}\right)^{1/4} = 227 K= -46°C$</span></p>
<p>But again, with such cold temperatures the actual inflow of energy would be grater and the equilibrium temperature would not be that extreme.
In any case, that condition would be favourable to the growth of a massive ice sheet in the corresponding polar area, in the same way I describe in <a href="https://earthscience.stackexchange.com/questions/15158/what-would-happen-to-the-earth-if-there-were-no-seasons/15159#15159">this question and answer</a>.</p>
<p>As a final thought. For this condition to happen in real life you would need something analogous to <a href="https://en.wikipedia.org/wiki/Tidal_locking" rel="nofollow noreferrer">Tidal locking</a> to lock the period of <a href="https://en.wikipedia.org/wiki/Axial_precession" rel="nofollow noreferrer">Earth's precession (currently about 26,000 years)</a> to exactly one year. And don't know of any mechanism that could achieve that.</p>

CC BY-SA 4.0
2018-09-25T21:40:01.650Z
2024-07-28T05:33:07.870Z
2024-07-28T05:33:07.870Z
Camilo Rada
<p>Inspired by: <a href="https://earthscience.stackexchange.com/questions/4589/how-does-antarctica-stay-frozen">How does Antarctica stay frozen?</a></p>

<p>If the Earth was in a fixed solstice state - northern winter and southern summer (e.g. the axis obliquity rotated with the Earth's orbit), what  would the equilibrium temperature be at each pole? Assume daily cycle and convection and so on operate as usual, just that it's permanent midnight in the Arctic, and permanent midday in the Antarctic.</p>

<p>Bonus questions: Would the equilibrium temperatures be significantly different if it was northern summer and southern winter (e.g. would land masses and orography have much effect, relative to the solar imbalance)?</p>

CC BY-SA 4.0
2018-09-19T11:22:35.450Z
You would have to run a climate model while forcing the sunlight from the solstice state.
Communisty
<hypothetical><poles><axial-obliquity><seasons>
What would the equilibrium temperature be at the poles in a world without seasonality?
naught101
# What would the equilibrium temperature be at the poles in a world without seasonality?

(Asked by: username_0 on 2015-04-01)

Inspired by: [How does Antarctica stay frozen?](https://earthscience.stackexchange.com/questions/4589/how-does-antarctica-stay-frozen)

If the Earth was in a fixed solstice state - northern winter and southern summer (e.g. the axis obliquity rotated with the Earth's orbit), what would the equilibrium temperature be at each pole? Assume daily cycle and convection and so on operate as usual, just that it's permanent midnight in the Arctic, and permanent midday in the Antarctic.

Bonus questions: Would the equilibrium temperatures be significantly different if it was northern summer and southern winter (e.g. would land masses and orography have much effect, relative to the solar imbalance)?

**Tags:** hypothetical, poles, axial-obliquity, seasons

> **Comments:**
> **username_1**: You would have to run a climate model while forcing the sunlight from the solstice state.  

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## 1 answers

### Answer by username_2 on 2018-09-25. Score: 4 | Accepted answer

If we consider that by "Assume daily cycle and convection and so on operate as usual" you meant that all heat transport from/to the pole remain as it is today. Then, we can do a back of the envelope calculation. This calculation will at least give you an order-of-magnitude answer, and we can then consider everything that would affect the result.

Currently both poles radiate more energy than they receive trough direct solar irradiation. This is because they are colder than the rest of the planet. Therefore, there is a neat heat transport from the equatorial latitudes to the poles. That heat travels on atmospheric and oceanic currents. This figure nicely summarize this radiation balance:

[![enter image description here](https://i.sstatic.net/AjGDk.png)](https://i.sstatic.net/AjGDk.png)

Figure from [here](https://www.meted.ucar.edu/search/details.php?id=24464) ©The COMET Program

If we consider that the heat transport depicted at the bottom of the figure remains the same. The only change in the energy budget would be from solar irradiation, and once they start warming the only way to release the additional heat and reach equilibrium would be trough infrared radiation (black body radiation).

The poles receive zero energy on the equinox and the winter solstice and [12.64 kWh/m$^2$ per day on the summer solstice](http://www.applet-magic.com/insolation.htm). Therefore a rough yearly average would be 3.16 kWh/m$^2$ per day. So, if we were locked on a summer solstice state you would receive four time more energy, and none if locked in a winter solstice state.

**Summer solstice locked pole**

Let's consider the summer solstice case first. If we call $E\_1$ the current amount of outgoing radiation from the pole, and $T\_1$ the current temperature. The [Stefan–Boltzmann law](https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) states that

$E\_1 = \sigma T\_1^4$

Analogously, if $E\_2$ and $T\_2$ represent the summer solstice locked state, you have

$E\_2 = \sigma T\_2^4$

Considering that $E\_2 = 4 E\_1$

You can write that

$4 = \frac{T\_2^4}{T\_1^4}$

Or

$T\_2 = (4)^{1/4}\ T\_1 = \sqrt{2}\ T\_1 \approx 1.41\, T\_1$

Therefore the temperature would increase 41%. It doesn't sounds much, but considering that the temperatures above are in Kelvin, it es a lot! In fact, **with the current [mean temperature in the north pole of -16 °C](https://en.wikipedia.org/wiki/North_Pole#Climate) (257K), it would go up to +89°C!! And the South pole would go from the [current -49°C](https://en.wikipedia.org/wiki/South_Pole#Climate_and_day_and_night) (224K) to +42°C.**

This diverges from any posible reality because with 89°C at the north pole there would not be a net influx of energy from the equatorial latitudes as we are assuming. Instead, atmospheric and oceanic currents would take a lot of that heat down south and the temperature would equilibrarte at a much lower value.

Note that in the above calculation I've ignored the energy flux due to advection in atmospheric/oceanic currents. I did so because that flux is about five time smaller than the flux from solar radiation, so it won't change much the results and ignoring it simplifies the math. You can do the calculation considering those flows if you want.

**Winter solstice locked pole**

In the case of a pole locked to a winter solstice. There would be no incoming solar radiation at all. Therefore, all incoming energy would be transported by currents from equatorial latitudes. In that way, we can see in the above figure that the north pole receives about 150 W/m$^2$, and the south pole about 100 W/m$^2$. The equilibrium temperature would be reached when they radiate the same amount. So, using [Stefan–Boltzmann law](https://en.wikipedia.org/wiki/Stefan%E2%80%93Boltzmann_law) again we can do some rough calculations.
Let's call the south pole temperature $T\_{SP}$ and $T\_{NP}$ for the north pole. Then:

$T\_{SP} = \left(\frac{E\_{SP}}{\sigma}\right)^{1/4} = \left(\frac{100 \,W/m^2}{5.67\times 10^{-8}}\right)^{1/4} = 205 K= -68°C$

And for the north pole

$T\_{NP} = \left(\frac{E\_{NP}}{\sigma}\right)^{1/4} = \left(\frac{150 \,W/m^2}{5.67\times 10^{-8}}\right)^{1/4} = 227 K= -46°C$

But again, with such cold temperatures the actual inflow of energy would be grater and the equilibrium temperature would not be that extreme.
In any case, that condition would be favourable to the growth of a massive ice sheet in the corresponding polar area, in the same way I describe in [this question and answer](https://earthscience.stackexchange.com/questions/15158/what-would-happen-to-the-earth-if-there-were-no-seasons/15159#15159).

As a final thought. For this condition to happen in real life you would need something analogous to [Tidal locking](https://en.wikipedia.org/wiki/Tidal_locking) to lock the period of [Earth's precession (currently about 26,000 years)](https://en.wikipedia.org/wiki/Axial_precession) to exactly one year. And don't know of any mechanism that could achieve that.


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