Guest post by Kevin Kilty

### Introduction

Within the past week or two we have read posts from Dr. Spencer

(6/7/2019), Nick Stokes (6/6/2019), Lord Monckton (6/8/2019), and Willis Eschenbach (6/8/2019) covering a variety of topics involving simple block models; and each one involving, in one way or another, climate feedback. I have had a few thoughts banging around in my mind for a long time which relate to these topics and build on each of these recent works in a series of postings. This one presents a simple block model illustrating the Earth as a thermal solar panel or solar collector. It will have a direct, independent and supportive bearing on Dr. Spencer’s post.

### 1. Basics of a Collector

Earth bound solar panels are constructed to collect solar irradiance, reduce parasitic heat losses from conduction and convection, and transfer the solar energy they have collected as heat into a working fluid. For the Earth as a whole the only heat loss is outgoing longwave radiation. This makes analyzing the Earth as a solar collector particularly simple. Let’s begin with two important balance equations from thermodynamics.

Energy balance is our primary analytical tool. Stated in words energy balance is:

(1)

Solar energy in = Longwave (IR) energy out + energy being stored

Or symbolically for the Earth this becomes:

(2)

Where Is represents solar irradiance, αs is solar absorptivity, r is Earth radius, σ is the Stefan-Boltzmann constant (5.67 × 10−8 in S.I. units), e is the effective infra-red (IR) emissivity of the Earth, T is absolute surface temperature, and C represents a capacity for thermal storage. Since the final term representing rate of new storage in Equation 1 is quite small compared to the others, we can ignore it for our purposes, and write energy balance in terms of surface temperature as

(3)

I have separated out the ratio

because it is a common engineering figure of merit for solar collectors used to guide choice of materials.

The second balance equation, one I have mentioned in threads here before, is entropy balance.

(4) Entropy outgoing = Entropy incoming + Entropy generated

We have no particular use for this balance equation at present, but in terms of the operation of a real solar collector we could use it to calculate energy that might have been put to useful work but was wasted by parasitic losses instead. Be assured that global climate models and solar collectors alike have to adhere to both of these balance equations in order to be realistic and provide credible results.

### 2. Block Model of Our Collector

Let’s stipulate the following. The sun is a black body radiator with a surface temperature of 5900K, and solar irradiance of 1370W/m2 at the orbit of the Earth. Emissivity (e) is a parameter which we will determine from energy balance. Solar absorptivity (αs) equals (1 − A) where A is the Bond Albedo of Earth. We will use a value of 0.3 for A, recognizing that this is uncertain to a degree and varies with time.

Finally, we will use a temperature of 288K for the mean surface, recognizing that this temperature is of the atmosphere at about two meters above the surface. The surface must be, on average, different than 288K in order to allow heat transfer between the surface and the air, but using 288K for the mean surface temperature serves our purposes just fine.

A block model one might derive here is shown in Figure 1. It is much simpler than the block models with feedback loops. In engineering science such a model we refer to simply as a system or possibly as a black box. If this system is linear we can state its operation using an impulse response function, or a transfer function. If the system is non-linear we usually have to get down and dirty and specify the input/output relationship in detail. In control engineering the block is known as a plant, and represents the workings of a facility or machine. It makes good sense to think of Earth as a facility.

Figure 1: A system, plant, or facility in abstract form

In this case of our solar collector the input to the system is solar irradiance, which is, in fact, the only driver of the climate system (CO2 and water vapor feedbacks are internal to the system). The diagram shows that mean temperature is our only observable at present although we could choose to measure others. We use mean surface temperature as a proxy for what is happening in the climate system most of which is hidden from view inside the block. The block appears simple, but may contain great complexity including feedback loops, time constants, delays, and even additional blocks. I plan to address such hidden detail in a subsequent post building on Nick Stokes’ contribution on feedback.

### 3. Calculating Apparent Emissivity

Let’s put what numbers we can into the model of Equation 3.

In order to make this an equality and produce energy balance emissivity (e) must have a value of about 0.61.

We now arrive at what seems like a paradox. All the materials making up the Earth‘s surface are very black at infrared wavelengths. Pavement, water, soil, plants, skin, snow and ice all have emissivity in the range of 0.9 to 0.96. Yet, energy balance reveals that the effective emissivity is one-third lower. This is a robust result. The resolution of the paradox is that all the dark stuff on the surface is covered by an atmosphere containing infrared active gases. Just as we apply thin coatings to materials to change their radiative properties, the thin coating of atmosphere does the same for the Earth. One cannot use measured temperatures, solar irradiance, and absorptivity, and at the same time balance energy without including the effect of our greenhouse gases.

### 4. A Note about the Figure of Merit

The ratio

is known as the figure of merit for solar collectors. To make a solar collector that becomes very hot in sunlight, we choose to make it from materials in which is as large as possible. Think of a chrome alloy tool. Its figure of merit is approximately 6 – It lays in sunlight and…Ouch! On the other hand to fabricate a surface which stays cool in sunlight we seek materials with as small as possible. Some aerospace materials, like aluminum with a thin titanium dioxide coating have a ratio around 0.2. For our Earth solar collector the figure of merit is approximately one.

### 5. Including Disturbances in the Model

In this model of Earth as a facility we can modify the block diagram to include a separate input that allows for disturbances to the system (Figure 2). It alters the system parameters and changes how the system behaves. If we know enough about the function of the system we can

Figure 2: A system with a separate input for noise and disturbance.

write the disturbance to the output as a partial derivative like,

. This partial derivative is called a sensitivity of the facility to the disturbance. If we know so much as how e depends on other parameters like the partial pressure of CO2 we can use the chain rule of differentiation to find which is the climate sensitivity–an important, but currently unknown, parameter. For our simple block model we can calculate per 0.01 change in e; and as +1.02◦C per 0.01 change in αs. Unfortunately our simple model doesn‘t specify how e depends on pCO2; so we cannot calculate climate sensitivity from it.

### 6. Conclusion

The simple model of Earth as a solar collector shows conclusively that greenhouse gases in the atmosphere lower the effective emissivity of the Earth, which in turn raises the mean temperature of the surface in order to achieve energy balance. We can‘t balance energy using measured values of irradiance, albedo and temperature without a substantial greenhouse effect–A conclusion backing up Dr. Spencer‘s simple diurnal temperature model.

The model of disturbances to the Earth facility presented here is an alternative to block models containing explicit feedback loops. What I find attractive about eliminating feedback loops and making use of disturbance inputs instead, is that we can dispense with the complications which arise from the distinction between top of atmosphere values, and surface values. It also allows us to avoid feedback as an external forcing, which suggests it as a separate source of driving energy, when it is no such thing.

7. Notes:

The discussion about solar collectors and the figure of merit is available from most modern engineering textbooks on heat transport, Heat Transfer by Alan Chapman, MacMillan, 3rd edition, 1974 is an example.

For some discussion about entropy balance, and the engineering calculation of entropy transport and entropy generation, consult any engineering thermodynamics text, even one as old as Obert’s famous text from 1948, Engineering Thermodynamics; or better yet, Zemansky’s Heat and Thermodynamics in any of its eight editions.

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