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## Ice-Albedo Feedback, and Temperature Bifurcation Structure

In the previous post, I discussed the simplest of energy balance models that can yield insight into the temperature of a planet. I will elaborate on the arguments presented there to include a temperature-dependent albedo, $\alpha = \alpha(T)$, which allows the rate at which a planet absorbs starlight to depend itself on the climate state. We will again ignore the existence of a greenhouse effect in this discussion, and write the energy balance as before:

$\displaystyle \frac {Q (1-\alpha (T_{s}))}{4} = \sigma T_{s}^4$

where Ts is the surface temperature, and the other terms are defined as before. I will focus this discussion on the ice-albedo feedback, since the extent to which a planet is covered in ice will be intimately connected to temperature.  One can intuit that changing the ratio of ice surface to land/ocean surface, in response to climate change, will modify a planets reflectivity to sunlight and amplify the initial cause of the change.  One can also speak of albedo changes due to desertification or re-forestation, for example.  However, the ice-albedo feedback is a common example of thinking about surface albedo changes, and one that also enters prominently into the “snowball Earth” issue that I want to shed light on (and has broader connections to planetary habitability as one moves farther away from a star).  To move forward with the discussion, I work under two assumptions:

1) We are interested in a water version of the ice-albedo feedback on an ocean-covered world. This is because water is unique amongst expected planetary substances in that its ice form is less dense than its liquid form, so the ice can float on an ocean. Ice composed of carbon dioxide or methane floating on an ocean of CO2/methane would sink, preventing surface albedo fluctuations. It is also possible that one can develop glaciers on a land surface, although then the problem becomes more complicated than simply knowing the freezing point. Rather the details of where precipitation falls become vital. Glaciers will not form without precipitation, even if the local temperature resides below the freezing point of the substance in consideration. Mars, for example, does not have extensive glacial deposits on the surface even though it is always below the freezing point of water.

2) I assume there is a sufficient difference in reflectivity between liquid water and ice/snow. This difference in reflectivity generates the feedback. This assumption is true for Earth, where reflectivity of ice is high in the solar spectrum (liquid water is dark at those same wavelengths). One thus expects a colder planet with more ice in the ocean to raise the planetary albedo, a destabilizing tendency. However, if Earth were placed around another type of star (for example M-type stars, which are colder and have a spectrum shifted to infrared wavelengths relative to our sun) then the feedback would be suppressed. This is because ice becomes “dark” at wavelengths longer than solar, typically past ~1 micron wavelength, and colder stars emit a greater fraction of energy at those wavelengths. It is a consequence of something known as Wien’s displacement law that the emitted spectrum of colder bodes are typically shifted to longer wavelengths. See e.g., Joshi and Haberle (2012), Astrobiology for a description of this effect around redder, dwarf stars that compromise most of the stars that we know of (and have thus become an interesting target for planet hunters looking for prospective habitable worlds where liquid water might exist). Shown in the below figure is a typical reflectivity spectrum for various surfaces, including lake water and snow (fresh snow is a bit more reflective than ice at solar wavelengths, but in both cases much more so than standard ocean water conditions).

From Grant Petty’s “First Course in Atmospheric Radiation”

Ice-Albedo Feedback

We can think about the ice-albedo feedback from a global or local perspective. We can assume temperature increases equatorward and thus the ice line migrates in that direction as the planet becomes colder. In the local perspective, we aim our focus on the latitude at which the ice-line resides, which occurs when that latitude reaches a critical temperature (for example, -10 C). The ice-line is thought of as being zonally (east-west) uniform. Let $y_{i}$ be the latitude of the ice-line, and let the planetary albedo vary smoothly between a low-albedo state where ice is located only near the poles, and a high-albedo state where ice advances closer to the equator. One expects the albedo dependence (i.e., the slope of the albedo vs. latitude of ice-line) to vary more strongly as ice moves equatorward.

see Roe and Baker (2010), J.Climate and references therein for a mathematical outline of the parameterization used here (click for larger figure)

The local energy balance at the ice line is not just radiative, but depends also on heat convergence/divergence from the ocean and atmosphere. We can assume the heat convergence/divergence is proportional to the temperature difference at the ice line, $T_{i}$ and the global mean temperature, $\overline{T}$, so $F_{div}= c(T_{i} - \overline{T})$ where c is a positive constant. When the local temperature is higher than global-mean, the divergence term is positive indicating heat export from the source, and vice versa. Heat transport efficiency increases with higher values of c. Thus, the energy balance at the location of the ice-line is:

$\displaystyle Q(y_{i}) (1-\alpha_{i}) - c(T_{i} - \overline{T})= OLR_{i}$

where $Q(y_{i})$, $\alpha_{i}$, and $OLR_{i}$ denote the local insolation, local albedo, and local outgoing (longwave) radiation at the ice-line, respectively. The outgoing radiation here is a constant due to the imposed condition of the temperature being at a certain value at the ice-line.

It follows that as the ice-line advances, and because the local solar insolation increases equatorward, the heat flux divergence must also increase locally. Ice will advance if this cooling term exceeds the warming caused by increased sunlight at lower latitudes. From these arguments, it becomes possible to construct a plot of how the ice-line latitude varies as a function of the solar constant, Q (or some other parameter such as carbon dioxide concentration in the atmosphere). The argument here has been developed in a large number of paper (see e.g., Budyko, 1969 , Held and Suarez, 1974 , Ghil, 1976 , Lindzen and Farrell, 1977 , North, 1990 , a more recent discussion connecting to ancient Neoproterozoic climates is in Pierrehumbert et al, 2011 , see also more advanced modeling treatments, e.g., Yang et al., 2012). An example of this is in the following figure:

The y-axis is the sine of latitude T at which the ice-line resides, and x-axis is the ratio of the solar constant to the modern solar constant. Values greater than one indicate higher input of sunlight, and less than one, lower input of sunlight. Image from Shen, S., and G.R. North, 1999: A simple proof of the slope stability theorem for energy balance climate models. Canadian Apl. Math. Quaterly, 7, 203-215.

Note that the y-axis is the sine of the latitude, moving poleward as you go up the axis. It is a consequence of something known as the “slope-stability theorem” that regions with positive slope on this diagram are stable equilibria, and negative slopes are unstable equilibria. The albedo feedback thus gives rise to the possibility of a bifurcation in the system (loosely, a tipping point) in which a small change in a parameter such as the solar constant can produce a radical transition in climate state. Suppose, for instance, one was in a climate at 1.2 times the modern solar constant, residing in the ice-free branch at the top. Suppose further that the solar constant was gradually lowered (moving left along this horizontal branch); eventually, it will get cool enough that some ice will exist in the Polar Regions.

However, at some point, a further gradual reduction in sunlight will result in enough albedo feedback (as the ice-line advances equatorward) that one abruptly moves downward to the ice-covered branch. Because of the high-albedo, one can then increase the solar constant again to its original value (moving right along the bottom horizontal line now) yet still remain in the ice-covered state. This introduces dependence on the history that the climate took to get to a point.

From a global perspective

Let us focus on the case where we gradually lower the solar constant. From a global perspective, recall from the previous post that the outgoing radiation of a planet depends on temperature (to the fourth power). It follows that the cooling induced by decreasing the solar constant will be stabilized by less planetary emission to space. This “Stefan-Boltzmann response” is not often described as a feedback in the climate science community, but it is the classical way planets reach an equilibrium point when they are perturbed. All other feedbacks are thought of as modifying the strength of that restoring response. When ignoring any of those other feedbacks, such as albedo variation, the sensitivity is $dF=4 \sigma \overline{T}^3 d\overline{T}$. $dF$ is the forcing resulting from an incremental change in sunlight, greenhouse gases, or whatever. In our case it is equivalent to dQ.

Sensitivity expressed in this way is equivalent to the radiative restoring tendency by a planet that radiates according to Stefan-Boltzmann. It is the inverse of the no-feedback “climate sensitivity” metric, $\lambda _{o}= dT/dF$ that is commonly used to help diagnose the magnitude of future global warming. A more sensitive system will respond to greater temperature fluctuations for a given solar forcing; equivalently, the radiative restoring tendency will be less for a given temperature change than for a less sensitive system.

When we include the albedo change, the temperature sensitivity to a change in solar constant becomes:

$\displaystyle d \overline{T} = dQ \frac {\lambda_{o}}{1-f_{\alpha}}$

where,

$\displaystyle f_{\alpha}= \lambda_{o} \frac{\partial R}{\partial \alpha} \frac{d \alpha}{d \overline{T}}$

R is the net radiation flux change of the planet due to the change in the albedo. The feedback strength is thus a function of how sensitive albedo changes are to temperature, and how strongly that albedo change impacts the planets radiation balance. In this linear analysis, it is seen that when f = 1 the destabilizing albedo feedback becomes sufficient to outweigh the restoring response caused by decreased planetary emission to space, and a runaway snowball ensues. (I will have more to say about feedbacks, the origin of the above equation, and this interpretation of f=1 in a later post). It can be shown (e.g., in the Roe and Baker, 2010 paper linked in the first figure caption) that the global and local interpretations are equivalent- that is, f=1 occurs when the ice line is at some latitude, the same latitude as when the slope of yi vs. Q becomes negative. Equivalently, if one envisions a curve of radiative flux vs. surface temperature, the equilibrium is stable if the slope of the solar curve is less than the OLR curve at the intersection point.

Colored curves: Green = Solar constant = 1370 W/m2; Red = 2740 W/m2; blue = 685 W/m2 (values on y-axis are divided by 4 to account for spherical geometry of the Earth and an albedo parametrization that allows albedo to vary with temperature near a global-mean freezing point). Dashed black lines are two OLR curves for no greenhouse effect (top line) and one with a parametrized greenhouse effect. The intersection point of the curves correspond to the equilibrium climate. Note that only snowball solutions are stable for the blue curve, and no snowball exists for the red curve.

In thinking about contemporary global warming, the albedo feedback enters into the picture in a similar way. Less ice cover means more solar energy absorbed at the surface and amplified warming. However, the role that the surface albedo has on the total planetary albedo becomes somewhat muddied when you bring clouds and atmospheric scattering into the picture. Moreover, ice does not exist in substantial amounts on the present Earth except very near the Poles where there is already weak sunlight to begin with. So the modern albedo feedback should be thought of as very important locally but small globally, and is a decidedly secondary effect relative to some other feedbacks we will encounter in later posts. It is used as a illustration of feedbacks quite often however, since it behaves very intuitively. The bifurcation arguments play out prominently in studies of the “Snowball Earth” hypothesis, proposed to explain the Neoproterozoic glacial episodes that occurred twice in the period 750–550 million years ago, and suggesting that the Earth was nearly or globally covered by ice/snow during these events. I will return to this time period in later posts.