Global warming: Sun and water

Abstract

This paper demonstrates that global warming can be explained without recourse to the greenhouse theory. This explanation is based on a simple model of the Earth's climate system consisting of three layers: the surface, a lower and an upper atmospheric layer. The distinction between the atmospheric layers rests on the assumption that the latent heat from the surface is set free in the lower atmospheric layer only. The varying solar irradiation constitutes the sole input driving the changes in the system's energy transfers. All variations in the energy exchanges can be expressed in terms of the temperature variations of the layers by means of an energy transfer matrix. It turns out that the latent heat transfer as a function of the temperatures of the surface and the lower layer makes this matrix next to singular. The near singularity reveals a considerable negative feedback in the model which can be identified as the ‘Klimaverstärker’ presumed by Vahrenholt and Lüning. By a suitable, yet realistic choice of the parameters appearing in the energy transfer matrix and of the effective heat capacities of the layers, the model reproduces the global warming: the calculated trend in the surface temperature agrees well with the observational data from AD 1750 up to AD 2000.

Keywords

Global warming climate model climate energetics hydrological cycle Klimaverstärker

Introduction

The mean temperature of the Earth's surface has gone up over the past 250 years. According to the Intergovernmental Panel on Climate Change (IPCC)¹ the temperature rise during 20th century was 0.78 ± 0.07 K. Most people accept the Panel's view that the warming owes to the so-called greenhouse effect, viz. the increase of the internal energy of the Earth's climate system (to be defined in ‘Preliminaries’ section) caused, through a complex of processes, by the growing CO₂ concentration in the atmosphere. However, a genuine proof of the effect has not been given in spite of many indications brought forward. Maybe the strongest point in favour of the greenhouse idea is the absence of a credible alternative, possibly additional explanation for the global warming.

Basically, as there are no other significant energy inputs to the Earth, we can conceive just one alternative: the varying solar irradiation. However, the IPCC has reported that the climate forcing by the solar radiation only accounts for about 10% of the full forcing. The minor role of the Sun is often illustrated by a calculation based on a very simple model of the Earth (e.g. in Beer et al.²). The model treats the Earth as a black body at a temperature of 255 K emitting long-wave radiation the power of which is balanced by the net solar energy input, about 239 W m⁻². Over the past century the solar irradiation, 1361 W m⁻², has changed by no more than 1 W m⁻², hence the net input has changed by about 0.17 W m⁻². Assuming the balance and using Stefan–Boltzmann's law we then find a surface temperature rise of roughly 0.05 K.

The simple model is just a very crude representation of the Earth. Only the infrared radiation emission determines the response of the system to its energy input. The model does not take into account the heat transfers between the atmosphere and the surface. Inclusion of these transfers may be important as Vahrenholt and Lüning argue in their book Die kalte Sonne: Warum die Klimakatastrophe nicht stattfindet.³ Their point, in brief, is that the greenhouse effect explains only part of the temperature increase. To make up for the deficiency they presume the existence of a ‘Klimaverstärker’, viz. a phenomenon that amplifies the temperature effect of the variable solar energy input (compared to the simple model). The authors suggest that this phenomenon might relate to the latent heat transfer from the surface into the atmosphere. The idea of a Klimaverstärker is taken up here.

Preliminaries

Global warming is believed to be brought about primarily by the increase of the amount of CO₂ in the atmosphere, viz. the greenhouse effect. Here, that proposition is left as it is. Rather the focus is on the varying solar irradiation affecting the temperatures in the Earth's climate system. It can be studied independently, viz. a possible greenhouse effect and an effect owing to the irradiation can be disentangled. First, it is because the one does not directly influence the other. Second, the effects are quite small (less than 1 K surface temperature change compared to a mean temperature of 288 K) permitting first-order approximations for both effects. Then they can be treated separately. Hence, as the focus is on the irradiation, we might as well take the atmospheric CO₂ content fixed. If one so desires he or she can add a greenhouse effect, if any, to the effect of the varying irradiation.

As is not uncommon the climate system is represented by a single column of 1 m² cross section. The properties of this column are averages over the globe and over a suitable running time interval, e.g. one year. We choose the boundaries of the column so that all processes pertaining to the energy transfers within the system and between the system and its surroundings are included. It means the variables determining the system's state only depend on the altitude in the atmosphere and on the depth in the surface. Lateral energy transfers can be left out.

By assumption, the climate system is stable if the amount of atmospheric CO₂ does not grow, as in pre-industrial times, and the irradiation is fixed. Stability means the system, initially in a stationary state, responds to a disturbance by re-attaining a stationary state (not necessarily the same as the initial state). The assumption is credible since the Earth's temperatures do not show a runaway trend, at least not prior to the outset of the anthropogenic CO₂ growth. (In general, physics tell us that a system deprived of interactions with its surroundings develops towards a stationary state. Chemists refer to Le Chatelier's principle.) Obviously, the conditions are hypothetical as the atmospheric CO₂ content changes with seasons and the irradiation exhibits small fluctuations. Yet, it makes sense to define a reference state under the indicated steady conditions and describe the real state by making small adjustments to this reference state.

Definition of the 3L model

As said in the ‘Introduction’ section, inclusion of the heat transfers in modelling the Earth's system may be important. That is why a simple and transparent model, a three-layer model (3L model), is developed. The model divides the column representing the system into three layers: a surface layer, a lower and an upper layer of the atmosphere. The boundaries of the layers are obvious (cf. the second paragraph in the previous section) except for the boundary between the two atmospheric layers. We define this boundary by the condition that the latent heat transferred from the surface to the atmosphere is fully absorbed by the lower layer, i.e. the latent heat does not reach the upper layer. The boundary can be chosen at an altitude of about 2.3 km where on average the atmospheric temperature is 273 K (0℃). Each layer is attributed a single (mean) temperature and a single heat capacity. Figure 1 schematically displays the Earth's system and its layers. The caption contains the specification of the energy transfer rates and the relevant properties of each layer.

Figure 1.

Schematic representation of the three-layer model including the energy transfers. The system consists of a surface layer, index 1; a lower atmospheric layer, index 2; and an upper atmospheric layer, index 3. J: the absorption rate of solar energy; J_k: the absorption rate of solar energy for layer k; Q_k: the sensible heat transfers between the layers; L: the latent heat transfer; E_k: the infrared emission by layer k, except for the surface which emits E₁ + E₁^d, E₁^d directly to outer space; R: the infrared emission into outer space. The arrows indicate the direction in which the transfers are taken positive. Each layer k is characterized by an average temperature T_k and a heat capacity C_k.

Energy conservation underlies the description of the system's behaviour. This principle applies to each of the layers at all times. It yields three coupled equations depending on time t. However, the equations are practically unsolvable. The specification of the terms is hard to give and non-linearity presents grave problems. A way out of this situation is to consider variations of the quantities involved. We express the variations as δU_k for the internal energies, as δE_k for the emission rate of long-wave radiation energy and so on. We define them with respect to the stable reference state of the Earth as introduced in the previous section. Then we can write the energy conservation equations in terms of the variations of the quantities (time t excepted). Of course, it implies we will end up with relative temperature variations. The equations for the variations read

\frac{d}{dt} δ U_{1} = δ J_{1} + δ E_{2} - δ Q_{1} - δ L - (δ E_{1} + δ {E_{1}}^{d})

(1)

\frac{d}{dt} δ U_{2} = δ J_{2} + δ E_{1} + δ E_{3} + δ Q_{1} + δ L - δ Q_{2} - 2 \cdot δ E_{2}

(2)

\frac{d}{dt} δ U_{3} = δ J_{3} + δ E_{2} + δ Q_{2} - 2 \cdot δ E_{3}

(3)

The strategy now is to convert these energy conservation relations into equations relating the temperature variations of the three layers. We can achieve this by making several assumptions. For one, it is assumed first-order variations will do. This is reasonable if the real state is close to the reference state. The variations of the internal energies U_k are expressed as $δ U_{k} = C_{k} \cdot δ T_{k}, T_{k}$ being the temperature of layer k. We assume the heat capacities $C_{k}$ constant because the composition of the layers does not change significantly (the share of CO₂ and H₂O in the content of the layers is very small). The layers are treated as grey bodies. This is permitted since the mean free path of photons, viz. the optical depth, is small compared to the dimensions of the bodies. Stefan–Boltzmann's law, $E_{k} = ɛ \cdot σ \cdot T_{k^{4}}$ provides us with the variations of the long-wave radiation emissions

δ E_{k} / E_{k} = 4 \cdot δ T_{k} / T_{k}

(4)

where E_k and T_k are the emission rates and the temperatures in the stable reference state of the Earth. It also applies to

E_{1}^{d}

since the ratio of

E_{1}^{d}

and E₁ can be taken fixed. Note that in this way the emissivity ε drops out. Yet, we cannot dispense with it because it relates E_k with T_k. In fact, we take ε = 0.9 as follows from the Earth's energy budget as presented by Kiehl and Trenberth⁴: out of the 390 W m⁻² the long-wave radiation emitted by the surface carries away per second only 40 W m⁻² makes it directly into outer space.

As to the sensible heat transfers Q_k, $k = 1, 2$ , Newton's law of cooling provides us with

δ Q_{1} = - κ_{1} \cdot (δ T_{2} - δ T_{1})

(5)

δ Q_{2} = - κ_{2} \cdot (δ T_{3} - δ T_{2})

(6)

the convection coefficients κ₁ and κ₂ assumed constant (since the composition of the atmosphere is fixed). These equations are particular forms of Fourier's law which are reasonable because in good an approximation the atmospheric temperature falls linearly with altitude (at least in the troposphere).

In general, the latent heat transfer L depends on the temperatures of the surface and the lower atmospheric layer. (More details will be given in ‘Quantification of the heat transfer parameters’ section.) Its variation then reads

δ L = ξ_{1} \cdot δ T_{1} + ξ_{2} \cdot δ T_{2}

(7)

ξ₁ and ξ₂ being constants.

With these specifications the energy conservation equations turn into a vector equation that is basic for the 3L model

C \frac{d}{dt} ϑ = δ J - (M - N) ϑ

(8)

The vector η has the components δT₁, δT₂ and δT₃, whereas the vector δJ consists of δJ₁, δJ₂ and δJ₃. The matrices C , M and N are given by

M = (4 \cdot (E_{1} + E_{1}^{d}) / T_{1} - 4 \cdot E_{2} / T_{2} 0 - 4 \cdot E_{1} / T_{1} 8 \cdot E_{2} / T_{2} - 4 \cdot E_{3} / T_{3} 0 - 4 \cdot E_{2} / T_{2} 8 \cdot E_{3} / T_{3})

(9)

C = (C_{1} 000 C_{2} 000 C_{3})

(11)

The essential elements of the basic equation are the matrix M − N , here called the energy transfer matrix, and the variation of the solar energy absorption δJ. Since the absorption rate can be taken proportional to the solar irradiation (because the amount of molecules absorbing solar radiation does not change), δJ is a function of time following the irradiation variations. Then the properties of M − N determine the general nature of the solution. As the temperatures are required not to run away, the matrix must have eigenvalues with positive real parts.⁵ As the condition on the eigenvalues is not self-evident, the energy transfer matrix needs special attention when we quantify the 3L model.

Quantification of the reference energy transfers

We intend to use the data on the Earth's energy budget as presented by Kiehl and Trenberth⁴ for quantifying the matrices. To that end we have to decide whether these data adequately describe the energy budget for the Earth in its reference state. Let the differential of the ratio E/T, as a typical element of M (indices omitted), with respect to the stable state be $δ (E / T)$ . It follows from $E / T = ɛ \cdot σ \cdot T$ ³ that $(T / E) \cdot δ (E / T) = 3 \cdot δ T / T$ . A temperature increase of 0.7 K over the last century and a surface temperature of 288 K yield 0.7% for the right-hand side and hence for the relative error in $δ (E / T)$ as far as the surface is concerned. Similar estimates can be made for the atmospheric layers. The percentage compares favourably with the uncertainties, a few W m⁻² at least, in the data on the radiation energy transfers. Since we only consider a first-order approximation the Kiehl and Trenberth data suffice for quantifying the elements of M .

The Kiehl and Trenberth data apply to the Earth's system consisting of the surface and the entire atmosphere. For the construction of a 3L model, some adjustments and additions are in order. In particular, the solar energy absorption rates, J_k, and the sensible heat transfer rate, Q₂, from the lower to the upper atmospheric layer have to be determined. They can be estimated by applying Lambert–Beer's law to the solar energy absorption under clear sky conditions followed by a correction for the mean cloudy conditions (the reflection of sunlight by the atmosphere is attributed to the clouds only). We arrive at $J_{2} = 13 W m^{- 2}$ and $J_{3} = 58 W m^{- 2}$ . These values say that in the atmosphere the major portion of the solar radiation is absorbed in the upper layer because of the layer's depth. This is not unreasonable since the solar radiation comes in from above.

The discussion in Kiehl and Trenberth⁴ on the energy transfers and their uncertainties shows the estimate on the sensible heat transfer, Q₁ in the 3L model, and the estimate of the infrared radiation emission rate of the atmosphere back to the surface, E₂, are indirect only and, moreover, interrelated. In fact, the authors estimate the emission rate to be 324 W m⁻² from which the sensible heat transfer rate follows: 24 W m⁻². They also refer to papers in which the latter rate amounts to 17–18 W m⁻².^6,7 The uncertainty is considerable and that is why we adjust the emission rate as follows. The temperature of the lower layer runs from 288 K at the surface to 273 K at its upper boundary. We take a mean temperature of 280 K which yields an infrared emission rate (with 0.9 for the emissivity) of, rounded, 313 W m⁻². Then the latent heat transfer rate becomes 14 W m⁻² and it follows that

Q_{2} = 27 W m^{- 2}

. This completes the evaluation of the reference energy transfers which we summarize in Table 1. It presents the base case as in the sequel we shall also consider other cases with slightly different data

Table 1.

The data representing the base case for the energy transfer rates in the reference system for the Earth. They are taken from Kiehl and Trenberth⁴ with some adjustments and additions.

Solar energy absorption:	J₁ = 168 W m⁻²	J₂ = 13 W m⁻²	J₃ = 58 W m⁻²
IR energy transfers:	E₁ = 350 W m⁻²	$E_{1}^{d} = 40 W m^{- 2}$	E₂ = 313 W m⁻²	E₃ = 199 W m⁻²
Heat transfers:	Q₁ = 14 W m⁻²	Q₂ = 27 W m⁻²	L = 77 W m⁻²

IR: infrared radiation.

The 3L model attributes a single temperature to each layer. By Stefan–Boltzmann's law (with emissivity 1 for the surface and 0.9 for the atmospheric layers) the surface temperature reads 288 K, the lower layer temperature is 279.9 K and that of the upper layer is 249.9 K. With these data we can evaluate the radiation energy transfer matrix. In ‘The energy transfer matrix’ section it is included in M − N .

Quantification of the heat transfer parameters

As to the sensible heat transfer the coefficients κ₁ and κ₂ follow since the temperatures of the layers and the sensible heat transfers are known: κ₁ = 1.722 W m⁻² K⁻¹ and κ₂ = 0.9013 W m⁻² K⁻¹. (Note: the number of digits does not reflect the level of inaccuracy. They merely serve consistency for the values of the parameters turn out to be quite critical. This note holds for all parameter values in the sequel.)

The identification and the quantification of the coefficients ξ₁ and ξ₂ appearing in the expression for the latent heat transfer require some details of the hydrological cycle, in particular of its ascending branch. Elements of the cycle pertaining to the latent heat transfer can be found in Peixoto and Oort⁸. But as they are insufficient for dealing with the problem at hand, some words are devoted to a simple presentation of the ascending branch, i.e. evaporation followed by condensation.

Evaporation of water occurs by virtue of a gradient of the vapour pressure (which is directly related to the chemical potential since the vapour behaves as a perfect gas). If the vapour would not condense, the vertical vapour pressure distribution would be the barometric distribution with a scale length of about 13 km (slightly depending on the atmospheric temperature). This distribution expresses the balance between buoyancy and gravitation. But water vapour does condense if its pressure is equal or greater than the saturated vapour pressure. We can construct the vertical profile of the saturated vapour pressure from the data in Schmidt⁹ and the approximately linear fall of the atmospheric temperature with increasing altitude (the lapse rate being about −6.49 K km⁻¹; see International Civil Aviation Organization¹⁰). This profile and the barometric distribution intersect in the dew point, at and above which the vapour condenses and clouds appear. Above the dew point the vapour pressure follows the saturated vapour profile and the balance represented by the barometric distribution is lost. The natural tendency to restore the balance implies that the difference between the two pressure distributions, at and above the dew point, constitutes the driving force for the water vapour going up to disappear by condensation. The freezing point is at an average altitude of about 2.3 km. It is assumed that no condensate, viz. no liquid water, is present at higher altitudes: all latent heat is set free between the dew point and an altitude of 2.3 km. This explains the choice of the imaginary boundary between the lower and the upper atmospheric layer.

In line with the phenomenological equations of thermodynamics¹¹, we take it that the latent heat transfer L, viz. the product of vapour mass transport with the evaporation heat, is proportional to the driving force. This driving force, in turn, is proportional to a (weighted) integral of the difference between the saturated vapour pressure p_v,sat and the barometric vapour pressure, p_v,bar, i.e. to an averaged difference $p_{v, bar} - p_{v, sat} = (ϕ - 1) \cdot p_{v, sat}, ϕ$ being the averaged relative humidity. In reasonable approximation the relative humidity is insensitive to changes of the temperature viz. T₂, provided these changes are small, of the order of 1 K or less (cf. Buck's equation¹²). It means the temperature dependence of the driving force, and hence of the latent heat transfer, is determined by $d p_{v, sat} / d T_{2}$ . From Schmidt⁹ we deduce $(1 / p_{v, sat}) \cdot (d p_{v, sat} / d T_{2}) = 0.0686 K^{- 1}$ for small variations around T₂ = 280 K. As the latent heat transfer L, the reference value of which is 77 W m⁻², varies by the same relative amount it follows $d L / d T_{2} = 5.284 W m^{- 2} K^{- 1}$ . Since the surface temperature did not enter the picture, we find $ξ_{2} = (\partial L / \partial T_{2})_{T} 1 = 5.284 W m^{- 2} K^{- 1}$ .

The other parameter to be determined is $ξ_{1} = (\partial L / \partial T_{1})_{T} 2$ , viz. the variation of L with the surface temperature T₁ whereby T₂ is kept fixed. In reality, a rise of the surface temperature makes the air temperature close to the surface increase. It would affect the vapour pressure. However, the condition is that the atmospheric temperature, viz. T₂, does not change. To fulfil this condition, we have to consider a process counteracting the air temperature rise or, for that matter, the increase of the local thermal energy right above the surface. As such a process cannot supply or demand energy to or from the surroundings, it must be a change of the vapour mass, viz. condensation. By condensation the energy in a volume right above the surface goes down and that is exactly what we need.

A very simple experiment shows it is not just theory. If we pour hot water, say at 70 or 80℃, into a relatively cold, preferably metal sink, we observe that mist, viz. condensed water, immediately appears to disappear again after a few seconds. It appears because the bottom film of the hot water heats up the sink while the air initially remains at the same temperature (because heat transfer is relatively slow). Apparently, the heating of the sink's surface while the air temperature is constant brings on condensation in accordance with the above argument. As the air temperature rises after a while by heat transfer, the mist disappears.

To quantify the effect we consider a fixed volume of the atmosphere right above the surface. The focus is on the water vapour. If the surface temperature T₁ increases by δT₁, the volume's temperature would follow, through transfer of sensible heat, by, say, δT₂. As the vapour behaves as a perfect gas, we know that the vapour pressure p_v changes by $δ p_{v} / p_{v} = δ T_{2} / T_{2}$ . But as the volume temperature is required to be fixed, because of the condition on ξ₁, some of the vapour has to condense whereby the volume's temperature increase is annulled. The heat that would be supplied from the surface reads (n is the mass in moles and c_V is the specific heat capacity): $δ H = n \cdot c_{V} \cdot δ T_{2}$ . For restoring T₂ the same amount of energy has to be taken away by condensation: $δ H = - δ n \cdot c_{V} \cdot T_{2}, δ n$ being the amount of vapour that disappears by condensation. It follows that $δ n / n = - δ T / T_{2}$ . As no exchange of energy with the surroundings is in play, the condensation process runs adiabatically. Consequently, we have $δ p_{v} / p_{v} = γ \cdot δ n / n$ , γ being $c_{p} / c_{V}$ , viz. the ratio of the heat capacities at constant pressure and at constant volume, respectively. (This relation is easily deduced from the well-known condition for an adiabatic change at constant volume: $p^{1 - γ} \cdot T^{γ} = α_{1} \cdot p \cdot n^{- γ} = α_{2}, α_{1}$ and α₂ being constants). Combining the variations we arrive at $δ p_{v} / p_{v} = - γ \cdot δ T_{2} / T_{2}$ . So, we have the same circumstances as those underlying the evaluation of ξ₂ and we can conclude that $ξ_{1} = - γ \cdot ξ_{2}$ . The above value for ξ₂ and $γ = 4 / 433$ (a value commonly used in engineering) yields ξ₁ = −7.045 W m⁻² K⁻¹.

The energy transfer matrix

This concludes the quantification of the model parameters (the heat capacities will be dealt with in the next section). The matrix M − N turns out to be

M - N = (0.0936 - 0.9116 0.0000 0.4620 6.2868 - 4.0867 0.0000 - 5.3751 7.2721) W m^{- 2} K^{- 1}

(12)

The determinant of this matrix is 5.29 (W m⁻² K⁻¹)³. Its eigenvalues are 0.24, 1.94 and 11.5 W m⁻² K⁻¹. As the determinant is non-zero and the eigenvalues are real and positive, the system is stable as anticipated. Note the first diagonal element is relatively small compared to the other diagonal elements. It means that if the surface temperature changes, the change of the infrared emission by the surface is almost cancelled by an opposite change in the heat transfer, primarily latent heat transfer, from the surface to the atmosphere.

Response to a one-step irradiation increase in the 3L model

As a first step in analysing global warming by the 3L model we take the Earth's reference system and consider a sudden one-step change of the solar irradiation. This change is set equal to the irradiation increase over the 20th century, i.e. 0.06%¹³, and a lower value of the absolute value was proposed in Kopp and Lean¹⁴; both papers agree quite well on the relative irradiance variation. The composition of the Earth's system remains unaltered. Then, the solar energy absorption rate jumps accordingly by 0.1434 W m⁻² which is distributed over the layers as

δ J_{step} = (0.1008 0.0078 0.0348) W m^{- 2}

(13)

Since stability is guaranteed, in the limit t → ∞ it holds that $C \frac{d}{dt} ϑ = 0$ and the temperature changes tend to the values $ϑ_{\infty} = (M - N)^{- 1} δ J_{step}$ . (Note that the heat capacity is irrelevant for $ϑ_{\infty}$ .) The calculation with the data from the previous section results in the asymptotic temperature changes

ϑ_{\infty} = (δ T_{1, \infty} δ T_{2, \infty} δ T_{3, \infty}) = (0.49 - 0.06 - 0.04) K

(14)

While the atmospheric temperatures hardly change, in fact they decrease minutely, the surface temperature has gone up considerably. The increase is about 10 times the increase as derived from the simple model mentioned in ‘Introduction’ section.

The results strongly depend on the parameter values used. To get an idea of the sensitivity we vary the sensible heat transfer, Q₁. It is taken because its value is not well determined by observations (cf. ‘Quantification of the reference energy transfers’ section). We assume the energy balances remain intact which means E₂ and Q₂ have to be varied along with Q₁ (other energy transfers are kept fixed). The outcomes for the surface temperature are displayed in Figure 2. We see δT_1,∞ agrees with the temperature increase reported by the IPCC for Q₁ between 13.2 and 13.5 W m⁻². If Q₁ is about 12.49 W m⁻² the energy transfer matrix M − N becomes singular. For smaller values of Q₁ one eigenvalue of this matrix is negative which corresponds to an unstable state of the Earth's system. The energy transfer matrix being next to singular constitutes the Klimaverstärker presumed by Vahrenholt and Lüning³ accounting for the relatively large value of δT_1,∞. This property of the matrix is connected to the nearly equal but opposite responses of the latent heat transfer and the rate by which the surface sends long-wave radiation into the atmosphere, to a change of, primarily, the surface temperature.

Figure 2.

Surface temperature rise, δT_1,∞, as a function of the sensible heat transfer from the surface to the atmosphere, Q₁. Included is the temperature rise as reported by the IPCC.¹ The grey area, typed n.s., indicates the sensible heat transfer domain in which the Earth's system is unstable.

Global warming driven by solar irradiation

It isn't really fair to compare the asymptotic values for the surface temperature with the temperature increase reported by the IPCC. It is because, owing to its heat capacities, the Earth's system responds to irradiation changes with some delay. To get the picture we have to solve the basic equation. For that we need to specify the heat capacity matrix C and the change of the solar energy absorption over time δJ(t).

The heat capacities of the atmospheric layers can be estimated by taking 1.0 kJ kg⁻¹ K⁻¹ for the specific heat capacity of air, and assuming an exponential density profile of the atmosphere. It follows that C₂ = 0.080 Wyr ⁻² K⁻¹ and C₃ = 0.22 Wyr m⁻² K⁻¹. The quantification of the surface heat capacity C₁ is less straightforward. Its value depends on the average depth of the oceans taken into account since the heat capacity of the oceans greatly dominates the surface heat capacity. Different values circulate in literature. For instance, Schwartz¹⁵ reports C₁ = 17 ± 7 Wyr m⁻² K⁻¹ in connection with a typical response time of 5 ± 1 year for the Earth's system. It has invited comments from other researchers.^16–18 The subsequent discussion makes clear that different values for C₁ can be deduced, and that one single heat capacity for the surface might be insufficient. For instance, although Boer et al.¹⁹ in studying the Earth's response to volcanic events mention heat capacities ranging between 6.4 and 8.4 Wyr m⁻² K⁻¹, they find indications for a smaller heat capacity (in the order of 1 Wyr m⁻² K⁻¹). Given the considerable uncertainty and the lack of unanimity we shall take a single C₁ as an adjustable parameter.

The absorption of solar energy is derived from the irradiation data provided by Krivova et al.¹³ These data are considered representative for the total solar irradiation although at present they are subject to discussion.²⁰ We assume the composition of the layers as to solar energy absorbers is constant. It means the absorption is distributed over the layers by 24.268 and 5.439% for the atmosphere and 70.293% for the surface. The absorption rates vary with the solar input rate at a fixed ratio: 0.1756. To keep the calculations simple and transparent, the irradiation variations are approximated by piecewise linear fits to the 11 years running mean of the data as shown in Figure 3.

Figure 3.

The total solar irradiation from AD 1600 up to AD 2010. It is assumed proportional to the absorption of solar energy which serves as the input to the basic equation. The TSI is approximated by a piecewise linear fit to the 11-year running mean (fat black line).

The basic differential equation, $\frac{d}{dt} ϑ = C^{- 1} δ J - C^{- 1} (M - N) ϑ$ , is solved by Euler's method with sufficiently small step size. As a first case Q₁ = 14 W m⁻² is chosen. For C₁ we take 1, 2, 3 and 5 Wyr m⁻² K⁻¹. Figure 4 shows the results in comparison with the temperature data from McIntyre and McKitrick^21,22 covering AD 1700 to about AD 1980 and from NASA GISS²³ for AD 1880 to AD 2010. These relative observational data have been joined by equating their mean over the first half of the past century. The offset of the solutions has been chosen so that they equal the observations around AD 1880. Several remarks can be made. The calculated temperature anomalies roughly correspond with the structure in the observational data over the time span from AD 1750 to present. We note that the heat capacity C₁ has considerable impact on the temperature variations: the smaller C₁, the larger the fluctuations. Contrarily, a large heat capacity adversely affects the agreement for times before AD 1750. Another point is that the 3L model does not disclose the finer details of the surface temperature variations.

Figure 4.

Solutions of the basic equation for the case Q₁ = 14 W m⁻². The solutions correspond to the heat capacities C₃ as indicated. For comparison the temperature data from McIntyre and McKitrick^21,22 (black) and from NASA GISS²³ (grey) are included. The relative offsets of the curves are discussed in the main text.

The graphical representations of the solution in Figure 4 show the calculated temperature variations lag behind to the observational data over the past decades. It has to do with the choice for Q₁ as indicated by Figure 2. That is why the calculations are repeated for Q₁ equal to 13.5 and 13.2 W m⁻². The results are displayed in Figure 5. Clearly they are significant improvements over the case considered above. In particular, the solution for Q₁ = 13.5 W m⁻² and C₁ = 1 Wyr m⁻² K⁻¹ comes close to the actual temperature anomalies. The solutions for larger values of C₁ are omitted because their correspondence with the data rapidly deteriorates with increasing C₁.

Figure 5.

Same as Figure 4 but for Q₁ = 13.2 and 13.5 W m⁻². The heat capacity C₁ is restricted to the given values because higher values yield solutions that do not closely follow the observational data.

We can exhibit the finer details the model does not cover, by taking the difference between the observed data and the model's results on δT₁. An example pertaining to Q₁ = 13.5 W m⁻² and C₁ = 1 Wyr m⁻² K⁻¹ is shown in Figure 6 where the difference is denoted by Δ(δT₁). Apparently, strong oscillations occur. Other realistic values for Q₁ and C₁ only affect the amplitude to some extent. Here, no attempt has been made to incorporate these phenomena in the modelling, because much research still is devoted to them. Several explanations have been proposed. The fluctuations might relate to more or less periodic phenomena like decadal oscillations, El Niño and La Niña (ENSO); see https://www.esrl.noaa.gov/psd/enso/.²⁴ Other studies^25,26 say that the 60-year cycle might correlate with astronomical phenomena.

Figure 6.

The difference between the observational data (black: McIntyre, McKitrick; grey: NASA GISS) and the model calculations for Q₁ = 13.5 W m⁻² and C₁ = 1 Wyr m⁻² K⁻¹. It is indicative for oscillatory phenomena in the Earth's climate system.

The input for solving the basic differential equation consisted of the 11-year running mean of the irradiation. However, the irradiation considerably fluctuates with the solar cycles.^13,14,20 We can take the fluctuations into account by superimposing them on the input. It turns out that the effect on the surface temperature is moderate: the temperature oscillations are about ± 0.15 K maximum, the largest value for the smallest heat capacity and for the last century (since the solar cycle fluctuations tend to get larger over this era). Such oscillations submerge in the overall oscillations shown in Figure 6.

Precipitation rate in the 3L model

A point of interest is the globally averaged precipitation rate. In the construction of the 3L model the variation of this rate is taken directly proportional to that of the latent heat transfer rate. From the results of the model calculations, we infer that over the 20th century the latent heat transfer falls by 4.9% in case Q₁ = 14 W m⁻², by 7.3% for Q₁ = 13.5 W m⁻² and by 10.5% for Q₁ = 13.2 W m⁻². (These percentages owe to the large value of ξ₁, i.e. to the negative feedback.) The precipitation rate then must decrease by the same relative amount. This seems to disagree with the position quite commonly held that the precipitation rate has not significantly changed over the past two centuries.²⁷ Indeed the EPA data²⁹ support this view although the data are valid for land only. However, if we consider the post-war era the IPCC data and the model results do agree, cf. Figure 7. As both data sets are of a relative nature, their averages have been set equal over the time interval. The correspondence for times prior to AD 1950 is not really satisfactory but that might relate to the reliability of the data. In fact, the measurements of the precipitation rate may suffer from systematic errors, the more so in the old days. (Some words on the uncertainties in measuring precipitation rates can be found at: http://mynasadata.larc.nasa.gov/global-precipitation/; see also IPCC fourth assessment report.²⁷)

Figure 7.

The trend of the globally averaged annual precipitation rate from AD 1950 up to AD 2008. The black line presents the data as given by the IPCC, whereas the blue line shows the results of the 3L model (Q₁ = 13.5 W m⁻², C₁ = 1 Wyr m⁻² K⁻¹).

Another indication for the trend of the precipitation rate can be derived from the change in the Earth's cloud coverage. The relation between the precipitation and the clouds is simple though not trivial. When the evaporation diminishes, so does the amount of water in the clouds because the precipitation responds with some delay (the residence time of atmospheric water, which may be affected by the evaporation rate, is a little over a week). Hence, the cloud cover goes down. The 3L model says the precipitation has decreased slowly over the past century (−7.3% over 100 years). It is slow compared to the response time and, hence, we can assume quasi-equilibrium conditions. Then the changes in precipitation and in cloud cover relate positively, i.e. less precipitation means less cloud coverage. Since AD 1983 the International Satellite Cloud Climatology Project (ISCCP) monitors the average cloud coverage. (Data obtained by the ISCCP can be found at: http://isccp.giss.nasa.gov/products, and graphs of the cloud coverage are found in http://climate4you.com). The observations show that from that day to the present the cloud coverage has gone down by about 3.5%. In particular, the low-level cloud cover accounts for the decrease. Accordingly, the evaporation and the precipitation must have dropped which is in line with the 3L model reconstruction.

The 3L model and the Milankovič cycles

From the model we can estimate the variability of the surface temperature along with the irradiation, i.e. dT₁/dI. The result is roughly 0.1 K (W m⁻²)⁻¹. The Milankovič cycles indicate that the full irradiation oscillations, viz. twice the amplitude, are of the order of 100 W m⁻².³⁰ It follows that the temperature variations are about 10 K which is close to the data derived from the Vostok ice cores³¹ commonly taken as a proxy for the average global temperatures. Hence, the model constitutes a direct explanation for the glacial and interglacial eras without recourse to other phenomena.

Conclusion

It has been demonstrated that by using a simple model of the Earth's climate system with realistic values for the energy transfers involved, the varying solar irradiation and the system's response fully explain the global warming over the past 250 years even without any greenhouse effect. The model surely is open for expansion, e.g. for introducing more layers. Still, the revealed main trends as to the Earth's surface temperature and the precipitation will not seriously be affected by such adjustments. The pivotal element in the model is a strong negative feedback brought on by the response of the latent heat transfer to the surface temperature variations. This feedback constitutes the ‘Klimaverstärker’, viz. the mechanism amplifying the effect of the irradiation, as presumed by Vahrenholt and Lüning.³

The results may be helpful for further research into climate forcing, the precipitation rate and the climatic oscillations. As to the first issue the forcing by the solar irradiation turns out to be grossly underestimated nowadays. Second, the variation of the globally averaged precipitation rate is shown to be inversely related to the irradiation variation due to the negative feedback. And finally, the residuals from the comparison of the temperature data with the model results may constitute additional input for studies on the decadal oscillations.

Footnotes

Acknowledgements

The author wishes to express his gratitude to Dr Sebastian Lüning for useful comments, for pointing out valuable references and for encouragement. Also, he is indebted to one of the reviewers for bringing up additional research information.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Harold J Blaauw holds a PhD in physics from the University of Amsterdam/AMOLF. He has worked in the energy policy field first within the Ministry of Economic Affairs and then as Secretary of the former Advisory Council for Energy Research, prior to joining the Science Policy Department of the Ministry of Education and Science. After being acting director of the former Foundation for Biophysics (NWO) till his retirement he has been an independent consultant on science management and energy.

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