A column model of the Arctic atmosphere is developed including the nonlinear positive feedback responses of surface albedo and water vapour to temperature. The atmosphere is treated as a grey gas and the flux of longwave radiation is governed by the two-stream Schwarzschild equations. Water vapour concentration is determined by the Clausius–Clapeyron equation. Representative concentration pathways (RCPs) are used to model carbon dioxide concentrations into the future. The resulting 9D two-point boundary value problem is solved under various RCPs and the solutions analysed. The model predicts that under the highest carbon pathway, the Arctic climate will undergo an irreversible bifurcation to a warm steady state, which would correspond to annually ice-free conditions. Under the lowest carbon pathway, corresponding to very aggressive carbon emission reductions, the model exhibits only a mild increase in Arctic temperatures. Under the two intermediate carbon pathways, temperatures increase more substantially, and the system enters a region of bistability where external perturbations could possibly cause an irreversible switch to a warm, ice-free state.

Climate change is causing rapid temperature increases in the polar regions. A fundamental question is whether these temperature increases are reversible. If humanity fails to prevent a substantial warming of the planet in the next few decades, which is appearing to be more and more likely, will it be possible in the future to reverse our effects on climate enough to restore lower temperatures, or will we have passed a tipping point beyond which return to the present state is impossible? We address this question in particular for the Arctic, where the observed climate change is the most dramatic.

The Earth's climate is an extremely complex system. Modelling efforts range from simple models attempting to isolate the most pertinent features to very complicated numerical models trying to capture as many details as possible. The model presented here is close to the simple end of this spectrum, although not as simple as some, in that it is a 9D nonlinear two-point boundary value problem. The advantage of relatively simple models is that they allow more direct analysis of cause and effect, which is often obscured in highly complicated models.

The term “tipping point” is used by different researchers in various ways;
see

For a tipping point to be present, the underlying mathematical model will be characterized by
nonlinearity, generally in the form of a positive feedback that accelerates change once change has begun. For the Arctic, one of the primary positive feedbacks is the surface albedo.
When the Arctic Ocean is frozen, the surface reflects a significant portion
of the insolation back into space, but open water absorbs much more heat from the Sun. Timing of the melt in the spring has significant impact

Past studies on general circulation models (GCMs) have given mixed results regarding the presence of multiple stable states for ice conditions in the Arctic. Some indicate that there appears to be a continuous transition from perennial ice cover
to annually ice-free that is reversible

The Arctic climate model presented here is motivated by three observations.
First is the observation that the climate changes taking place on the Earth
today are most dramatic in the high Arctic. Therefore, it is prudent to put
a special focus on understanding Arctic climate change. Second,
irreversible change is inevitably the result of nonlinear geophysical processes.
So, while this model is kept very simple, it does include significant nonlinear
phenomena that can lead to tipping points.
Third, the 3D spherical shell of the atmosphere of the Earth is rotationally symmetric about the polar axis if annually and zonally averaged. Due to the rotation of the Earth,
Hadley, Ferrel and polar cells form in the global circulation. If perfect rotational symmetry is assumed, the polar axis becomes
flow-invariant, and this remains approximately true for the real Earth.
Thus, a 1D model restricted to the polar axis can be expected to give useful information about climate in a neighbourhood of the pole.
The study of a rotationally symmetric spherical shell model by

The present model builds on the simple energy balance slab model of Dortmans et al. (2019), which was applied to paleoclimate transitions, and the model of Kypke et al. (2020), which was applied to anthropogenic climate change.
The primary improvement of the present model is a more physically accurate
description of the atmosphere.
Instead of using a slab to represent a uniform atmosphere with absorption properties similar to the
real atmosphere, here we use the Schwarzschild two-stream equations to model absorption
in the atmosphere explicitly as a function of altitude

A bifurcation analysis is performed on the model, tracking the steady-state solutions as carbon
dioxide levels increase.
The question of reversibility is a question of whether the current cold state simply warms
but persists.
The disappearance of this cold state through a saddle-node bifurcation would result in an abrupt
change in climate that may be practically irreversible.
The simpler model of

Section

The model is developed from first principles and has the following features.

The atmosphere is a 1D column at the North Pole with physical properties that vary with altitude, from the surface to the tropopause.

The incoming solar radiation is annually averaged and undergoes reflection and absorption in the atmosphere as well as at the Earth's surface.

The surface albedo is a nonlinear function of the surface temperature.

A well-mixed surface boundary layer is included.

The Earth emits longwave radiation as a black body.

The atmosphere is considered to be a grey gas.

The Schwarzschild two-stream equations govern the absorption and emission of both upward- and downward-directed longwave radiation in the atmosphere.

The atmospheric absorption of longwave radiation is due to three factors: water vapour,

Water vapour concentration is governed by the nonlinear Clausius–Clapeyron equation.

Transfer of latent and sensible heat from the surface to the atmosphere is modelled.

Both ocean and atmospheric meridional heat transports to the Arctic are dictated by empirical values.

In the Arctic, there is a slow downward movement of air in the column
corresponding to the polar circulation cell near the pole

The radiation absorption coefficients are calibrated by fitting the model to global average data.

The functional forms of the mass transport and atmospheric heat transport are used to calibrate the model to an empirical Arctic temperature profile.

The annually and zonally averaged Earth atmosphere is rotationally symmetric around the polar axis, which is invariant under the flow.
Therefore, if one considers a column of the atmosphere near the North Pole, it is
reasonably approximated by a 1D model with altitude-varying quantities. This approximation becomes exact in the limit as the diameter of the column shrinks to zero.
Alternatively, one can view the model as a meridional and zonal average over a
cylinder centred at the North Pole.
Further, although the Arctic Ocean is not zonally symmetric, in the above view, the contribution
of ocean heat transport can be reasonably captured as a scalar quantity.
Thus our model is more precisely a model of the North Pole rather than the Arctic.
Nonetheless, we do use some empirical data for the region north of 70

The model domain is a vertical cylinder of cross-sectional area

Schematic illustration of the model. Symbols as described in the text.

The model equations in the troposphere are developed from the
fundamental transport theorem in one spatial dimension:

If the property

Now take the property

Finally, consider the case where the property in Eq. (

In order to complete the system, a constitutive relation between the density

The mass, momentum, and energy balance equations, Eqs. (

The model includes a boundary layer extending from

The primary reason for including a boundary layer is a numerical one.
As shown in Appendix

The total mass crossing from the atmosphere into the boundary layer per unit time is

Consider now the energy balance at the Earth's surface.
There is energy transport from the surface to the boundary layer in the form of sensible and latent
heat, which is modelled, as per

Now consider the energy balance for the combined surface and boundary layer (one could alternatively
consider just the boundary layer without the surface, but the chosen formulation results in
a slightly smaller equation).
Input energy to this combined surface and boundary layer includes ocean heat transport and shortwave and longwave radiation entering at

Since temperature, pressure, and relative humidity are constant in the boundary layer, the radiation
equations may be solved analytically inside the layer in order to relate the radiation
terms at

There are eight unknown dependent variables:

The model is nondimensionalized and put in standard form as detailed in
Appendix

For the Arctic parameter values given in Appendix

Results of the fully calibrated Arctic model at

For

The International Panel on Climate Change (IPCC) has published various

The curve of equilibria in Fig.

Total atmospheric heat transport,

Although the model presented here is clearly a simplification of the climate,
made possible by the near invariance of the vertical flow on the polar axis,
we believe it captures some of the most important aspects relevant for Arctic climate change.
The model predicts that if humanity keeps carbon emission levels close to RCP6.0 or lower,
then the Arctic will not likely undergo a sudden dramatic rise in annual average temperature.
However, if carbon emissions are much worse than RCP6.0, such a change is likely, and the cause is
a saddle-node bifurcation of the stable cold equilibrium.
Such a change would clearly have catastrophic effects on the Arctic environment, leading to massive global effects.
These results are in agreement with those of

Our model addresses the equilibrium state only and represents the Arctic temperature as an annual average. The real Arctic climate undergoes massive seasonal changes, which effectively means that the system is actually oscillating around the equilibrium temperatures of our model. Such temperature oscillations may be sufficient to effectively push a system located on a cold solution in the bistable regime to “above” the unstable solution and so into the basin of attraction of the warm solution.

Seasonal variations in Arctic temperatures and sea ice are studied in many works, including

This Appendix provides details regarding the model. The model is written in a nondimensional form in Sect.

Define the nondimensional variables

Applying the change in variables to the troposphere BVP, Eqs. (

The BVP given by Eqs. (

Now expanding the derivatives on the right-hand side of Eq. (

In summary, the BVP for the troposphere in standard form is given by

As mentioned above, of the nondimensional constants, all are close to order 1 except

In this section we describe the various functional forms that we used for relative humidity,
atmospheric heat transport, and mass flux. For the latter two, several different forms were tried, and these are detailed below.
Calibration to empirical data, described in Appendix

The relative humidity is modelled as a linear function decreasing with altitude from a higher surface value,

Atmospheric heat transport is primarily
due to large-scale turbulent mixing of the column with its environment. This mixing is not modelled explicitly, but, instead, it is incorporated into the model via
the function

Choose

Assume that

The remaining portion of

As a numerical issue, since the boundary condition value

The choice of the functional form of

The function

piecewise linear,

piecewise sine,

piecewise cosine,

This Appendix lists the parameter values used in the model and discusses how some of them were calibrated to empirical data.
Section

Values of the model parameters are given in
Tables

Physical constants used in the model.

Other model parameters. Some parameters are geographically dependent and have different values for the global and Arctic situations. An empty value in the “Arctic” column indicates the global value is used in both cases. Values that were fitted by the calibration steps described in the text are indicated in bold.

Here we provide justification and explanation of our choice of parameter values in
Table

Calibration of the other model parameters was done in two steps.
First, the absorption coefficients,

Global average energy fluxes (

Using these parameter settings, we minimized the sum of squares of the differences between the data from Table

Model calibrated to global average values.
The vertical axis in all the plots is the pressure.

Using the calibrated values for

Annual Arctic temperature data from Fig. 1 of

Temperature profiles for best fits for each of the forms of

Calibrated functions:

For the above calibrations, the albedo values

This section presents the calculation of the insolation used in the Arctic model, where,
in particular, the insolation is taken as an annual average over the region north of 70

Select a Cartesian coordinate system

Let

Analysis code is available from the authors on request.

All of the data used in this paper are publicly available from the references listed.

The research project was conceptualized by WFL. Funding acquisition was by WFL, GML, and ARW. The methodology was developed by KLK, GML, and ARW. KLK did the formal analysis, investigation and validation. Software was primarily written by KLK with support from GML and ARW. Figure visualization was done by KLK and ARW. ARW wrote the original draft; all the authors were involved in reviewing and editing.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We were supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). Kolja L. Kypke thanks the Ontario Ministry of Colleges and Universities and the University of Guelph for a Queen Elizabeth II Graduate Scholarship in Science and Technology.

This research has been supported by the Natural Sciences and Engineering Research Council of Canada (grant nos. 400450 and 006257).

This paper was edited by Vicente Perez-Munuzuri and reviewed by Marek Stastna and one anonymous referee.