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## Climate Sensitivity and the Linearized Response

In discussions of climate change, it is often useful to think about the transition of the climate from one state to another, and ask how the magnitude of the response is related to changes in a control parameter (such as the solar constant, or CO2 concentration). This is the classical problem of climate sensitivity, which is intimately connected with assessing the degree to which Earth has the capacity to change. In such an analysis, we typically begin by reducing the “climate” to a single variable, commonly global mean temperature ($T$), and gauge its evolution as a function of the control parameter. Of general interest to society, for example, is how the global mean temperature responds to changes in CO2 concentration. I will build up some theoretical background into climate sensitivity in this article, and move on to observations and modeling results in subsequent posts.

## Building a Planet Part 2: Greenhouse Effects

Introduction

In Part 1 a first, simple model of planetary temperature was discussed, all based on knowledge of how much starlight a planet receives ($Q$) and how “reflective” that planet is (i.e, its albedo, $\alpha$, an effect elaborated on in this post).

In the first post, an “effective temperature” was solved for of the form:

$\displaystyle T_{e}= [\frac{Q (1- \alpha)} {\sigma \mu}]^{0.25}$

where $\mu$ is a geometrical redistribution term that accounts for how well the input of stellar energy is evened out across the planet by rotation/thermal inertia and planetary motions. For a sufficiently rotating planet with an advecting atmosphere, it takes the value of ~4, but it would be more appropriate to take on a different local value on an airless body incapable of transporting much heat around (concerning the question of habitability, this could set up a regime in which water could exist in liquid form over parts of a planet but not others).

We now introduce the effect of an atmosphere that can interact with the radiation entering or exiting the planet. On Earth, that interaction is predominately in the infrared (the outgoing energy) via absorption/emission processes and is accounted for by trace gases in the atmosphere, which we refer to as greenhouse gases (water vapor, carbon dioxide, methane, nitrous oxide, etc); the interaction can also occur with aerosol particles in the atmosphere and with clouds, though these latter two also tend to scatter shortwave solar energy and cause a net cooling effect on Earth (there are exceptions to this, such as black carbon). Ozone interacts in the infrared as a greenhouse gas, but also absorbs incoming UV radiation high in the stratosphere to set up a persistent thermal inversion where temperatures increase with height.

## The Venus Transit: an analog to exo-climate detection

There has been a large interest during the last couple of days in the Venus transit, where the second planet in our solar system passed directly between Earth and the sun, which was seen by many people and shown in the video above. For us, this phenomenon will not happen again until 2117. However, a viewer on a planet orbiting a distant star might see a Venus transit every Venusian year (~225 Earth days) if they resided in the same plane of orbit. Likewise, it would be possible for that observer (or for example, an observer on Mars) to see an Earth transit.

This leads to the subject of the post, which may be a bit more descriptive than others. Unknown to many yesterday, the world had a front row seat to a common method that is used to detect exoplanets (distant planets that orbit around stars other than our own, often tens of light-years away). The Venus transit will be used a test of the quality of the technique. The detection of such exoplanets is the goal of the Kepler satellite, one of the greatest scientific missions of our time. Kepler is NASA’s first mission capable of detecting Earth-size planets in orbit around other solar-like stars. So far, well over 1200 candidate planets have been discovered since 2009, with sizes ranging from less than Earth to twice as large as Jupiter (and with orbital periods shorter than a day to more than a year). It is a photometric space-based mission with the intent of finding bodies that orbit in the so-called habitable zone of their host star. This is the region where liquid water water is capable of being sustained on the surface of a planet (the factors that govern these limits will be discussed in later posts). From an observational and climate perspective, it is also possible to retrieve information about the atmospheres of such planets based on spectroscopy techniques of selected transiting planets.

## Thermal Wind

One of the fundamental characteristics of a planetary atmosphere is its wind distribution with height. Shown in the opening figure is the east-west component (or the u component) of wind speed on a latitude-height grid. The vertical axis is pressure, which decreases upward with height, since there is less air above you as one progresses higher in the air column.

One of the most noticeable features of the figure is that winds become more westerly (i.e., stronger towards the east) with height. The question motivating this post is what drives this observed phenomenon, and can we come up with a relationship between vertical wind shear (or the change in wind speed/direction with height) and horizontal temperature gradients? It is certainly not intuitive that such a relationship would exist. By the end of this post, we will conclude that the presence of westerly vertical shear is a direct consequence of the uneven solar heating of the Earth (more heating at the equator) coupled with the dominance of something known as the thermal wind balance, particularly in mid-latitudes. Broader application of this thermal wind balance will arise in future posts.

## 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:

## Building a Planet: A First Simple Model of Temperature

I expect that a recurring theme in my writing here will be to outline and describe simple models of climate. In doing so, I wish to highlight the assumptions/flaws in those models, as well as the properties of those models which appear robust to more complex calculations (i.e., those properties which emerge in a full hierarchy of model complexity, and are thus more likely to yield meaningful insight in the real world).

Keeping this goal in mind, one could argue that a reasonable starting point for understanding a planet is to simply know how much energy it receives and how that might translate into a temperature if you stood somewhere on the planet. By “energy it receives,” we are accustomed to thinking about the energy from the host star like our sun. Earth receives virtually 100% of its energy from the sun, although it is quite possible to receive significant amounts of energy from sources other than a star (this is the case for several of the gaseous planets in our own solar system). I will ignore those terms in this post, and will return to them in a later article.