The Science of Everything Podcast - Episode 143: Climate Modelling

Episode Date: April 28, 2024

An overview of climate models, beginning with a summary of the major types of models, and then a more detailed disussion of hte primitive equations and parameterised feedbacks that characterise the widely used Atmospheric General Circulation Models. We also discuss techniques for model validation and some evidence regarding the accuracy of various climate models. Recommended pre-listening is Episode 142: The Greenhouse Effect. If you enjoyed the podcast please consider supporting the show by making a PayPal donation or becoming a Patreon supporter. https://www.patreon.com/jamesfodor https://www.paypal.me/ScienceofEverything Check out out youtube channel! The Science of Everything Podcast - YouTube

you're listening to The Science of Everything podcast episode 143 climate models. I'm your host, James Fodor. So in this episode, we are going to talk about modeling the climate, and particularly focusing on the types of climate models that are used to inform our estimates about the magnitude of warming that will occur in response to a particular amount of greenhouse gas emissions. So this episode will build upon the discussion in the previous couple of episodes, In particular, recommended pre-listening is the previous episode, 142, the Greenhouse Effect, and the previous two episodes, a natural climate change and a history of Earth's climate.

Those will provide some of the background relevant to what we're discussing here. And in particular, in those episodes, I introduced the notion of climate sensitivity, which is important to understanding the relevance of what we're discussing today. So in this episode, we're going to talk about the different types of climate models. and introduce the primitive equations. I'll explain what those are in a little bit, which form the basis for most climate models. We'll then talk about some of the various climate feedbacks that need to be parameterized for the models, including things like water vapor, the lapse rate, convection, and ice albedo, some of which we've mentioned in previous episodes,

but we'll go over those again. We'll talk about the role of the oceans, which are very important for understanding and modeling the climate. And finally, we'll discuss model validation, which is how we establish the validity and accuracy of climate models. And as part of that, address some of the criticisms that have been levied against climate models. So, without further ado, let's get started and talk about climate modeling. So what is a climate model? Well, a climate model is a numerical simulation of the Earth's climate system, and it usually attempts to provide quantitative estimates of important aspects of the climate,

including things like temperature, wind patterns, rainfall, and so forth. In doing so, it usually attempts to incorporate the major drivers of climate, particularly the atmosphere, but also the oceans and the land surface and ice. Now, there are many different types of climate models, and it sort of depends exactly how you define what counts as a climate model, but I'm just going to go through from least sophisticated or most simplified to most sophisticated and least simplified, briefly some of the different classes or classifications of climate models

to help you understand how they relate to each other. So we'll start with what's sometimes called an idealised greenhouse model or a zero-dimensional model. This is a type of climate model that we've already discussed in the previous episode when we talked about the greenhouse effect. It's not really a model of the Earth's climate as a whole, it's really just the model of the greenhouse effect specifically, but it makes a good starting point to understand what we're talking about.

here. The idealized greenhouse model only considers radiative transfers between discrete layers of atmosphere, and so it's extremely simplified, but even such a simplified model, as we discussed in the previous episode, is sufficient to illustrate the key mechanisms of the greenhouse effect and help us to understand how and why it happens. Now, moving up from the idealized greenhouse model, we get to radiative transfer models. We discussed those in the previous episode as well. So these are kind of a generalization of the idealized greenhouse model, but in this case we use continuous rather than discrete layers of the atmosphere, and we allow for the modeling of multiple different wavelengths of light. So in the idealized model, you just sort of lump all of the light

together, all of the radiation together, and forget about different frequencies and so forth. In radiative transfer models, you now allow for separate wavelengths, which have different effects, and that allows you to better model the impact of greenhouse gases, because obviously different greenhouse gases absorb different wavelengths differently, and as we discussed in the previous episode as well. And radiative transfer models allow for a continuous treatment of the atmosphere, which is much more realistic than discrete layers. The next step up from radiative transfer models are radiative convective models, which we also touched on in the previous episode. So these add the effects of convection to radiative transfer models. So idealized

and radiative transfer models both only consider radiation. which is what's sort of most directly involved in the greenhouse effect, because it's radiation that is absorbed by greenhouse gases. But obviously, convection is extremely important in the atmosphere as well, and you need to incorporate that in order to have anything like an accurate representation of what's going on. So radiative convective models couple convection in with the radiation. And this allows temperatures to adjust much more realistically over the vertical dimension in the atmosphere. And they also allow for incorporating the feedback effect of increased water vapor concentrations in the atmosphere, which you mentioned last.

time and I'll discuss a bit more later because convection is one of the major mechanisms of heat transference in the atmosphere and you need to consider that in order to make sense of the movement of water vapor in the atmosphere because it's largely by convection that water vapor moves around in the atmosphere. So radiative convective models are useful for that. And radio of convective models already can do quite a good job in modeling the key components and key feedbacks in the climate system with regard to the greenhouse effect, as we discussed in the previous episode, radiative convective models can give a reasonably accurate estimate of the sensitivity parameter. As we said, it was estimated about 0.5 using radiative convective models

when best estimates are it's maybe 0.8. So even very simplified models, which aren't even full climate models, they're just models of really the greenhouse effect, can do fairly well there. But let's take the next step up then and move to atmospheric general circulation models, or GCMs for short. These are quite a large step up in complexity. Basically, what they involve is taking a radiated convective model, but then adding to it a grid-based, discrete modeling of the actual three-dimensional structure of the atmosphere over the surface of the Earth. So previous to this, we hadn't really made any effort to realistically model the actual shape or size of the Earth.

We modeled it using fairly crude methods. like, well, the zero-dimensional model doesn't incorporate distance or size at all, but radiative convective models might just model things based on a single square meter of the Earth's surface, which is considered as an average, and then you just multiply by the entire area of the Earth surface or something like that to get the total amount of energy transfer or something along those lines, but it doesn't realistically consider the actual size and shape of the Earth surface. In general circulation models, we will now do that. So what we do is we set up a grid over the entire surface of the Earth,

and over the entire atmosphere. So a typical model these days will have grid resolutions of between two, three or four degrees of latitude or longitude, and so we'll cover the earth in a grid of about 100 by 100. And they'll typically have maybe 20 vertical layers as well covering the atmosphere, and the vertical height varies in accordance with the altitude. So it's not constant with height. It depends on how far above ground you are. So the total number of variables that are modeled there, there are four key variables in each grid point.

We'll talk about what those are a bit later. Might be about 500,000. And what the model does is it computes all of the key primitive equations. We'll get to those in a moment, but it's like temperature and pressure and stuff like that. It computes the relationship between these key variables in each of these cells for each point in time. And then they integrate these over time so that you can. and create a trajectory of the value of these variables over time. And then you can tweak the model by introducing perturbations,

like you can inject extra greenhouse gases, for example, and see what effect this has on the system. So this is the point of these models, is that they allow you to see what the effect of changes are on the Earth's climate system, holding everything else constant. You can kind of produce these artificial experiments, which you obviously can't do with the real atmosphere. And they help us to understand different things such as the magnitude of the effects of different

perturbations, as well as the time span of how long they take to have their impacts. So as I said, I'll explain it a little bit about exactly what these equations are that are modeled at each of these grid points. But the main innovation of these general circulation models is the grid-like structure over the Earth's atmosphere, and then modeling the key set of equations within each grid and integrating that over time, allowing us to have actual trajectories with a spatial relationship over the surface of the earth. I should also mention that these models typically incorporate the topological layout of Earth's surface as well. So it will incorporate the effects of mountains and the difference between land and ocean and things like that. And that's something that the radiative convective models don't have that.

They don't have a particular notion of the actual shape of the Earth or the spatial layout and things like that. But there are major limitations of these models. So because they only model certain primitive equations directly, many aspects of the Earth's climate system, including frictions of the motions of air with the surface of the Earth, the formation of clouds, ice formation and changes in the ice coverage, and other such factors are not explicitly incorporated. And so they have to be separately parameterized. And I'll discuss that. That's a major aspect of constructing these models. I'll discuss that in a separate section in a moment. Now typically these days, the major models that are used for these sorts of simulation are not just atmospheric general circulation models, but they actually couple atmospheric and oceanic general circulation models. Basically, we have this grid-like system of simulating basic equations of fluid dynamics for the atmosphere. We can do the same thing with the ocean, right? In fact, many of the equations are much the same.

It's just that the fluid is water instead of air. And the grid light structure is applied, obviously, to the ocean basins instead of to the atmosphere. but it's much the same thing otherwise. What we can then do is get our ocean, our oceanic general circulation model and our atmospheric general circulation model and couple them together. So obviously we need equations as to how energy and water and things are interchanged at the interface between the atmosphere and the ocean, but that will obviously be part of the coupled atmospheric and oceanic general circulation model.

This allows for much more accurate modeling of the storage of energy in the oceans, which obviously is extremely important in the climate system. And so you really need these coupled atmospheric and oceanic general circulation models to get a decent picture and accurate simulations of the Earth's climate. Now there is a step up from coupled atmospheric and oceanic general circulation models, as if that weren't enough of a mouthful. These are sometimes called Earth system models. So these coupled atmospheric and oceanic general circulation models still leave out quite a lot of stuff. in particular they don't directly simulate the biosphere, so the impact of animal and plant life in particular, or the geosphere. So typically they'll still leave out geochemical interactions, so like weathering and the effects of weathering on take-up or release of carbon into the atmosphere, vegetation responses.

So obviously vegetation coverage will change with temperature and rainfall and other things like that. That's not incorporated in the typical coupled GCMs. changes in soil and soil distribution, that won't be included. Changes in ice cover as well may not be included in the general circulation models because the cryosphere is kind of a bit separate from the oceans. So there are a lot of things, especially relating to the impact of the geosphere and the biosphere on the oceans and the atmosphere that aren't incorporated into the couple GCMs. And Earth's system models attempt to incorporate those as well.

Of course, that makes them even more complicated and have even more things that you have to parameterize. So I'm not going to talk about them too much more here. I just wanted to mention them for completeness. So to summarize then, in terms of what we're talking about when we talk about climate modeling, most people are referring to these big coupled atmospheric and oceanic general circulation models, which divide the atmosphere and the oceans up into a series of fairly large grids with cells connected to each other. In each of these cells, we simulate a bunch of equations relating to fluid dynamics and energy transfers, and then integrate the values of the variables over time to get a trajectory of energy flows

as well as a transfer of water, vapor and air currents and other things in the atmosphere and the oceans over time. Let's now turn to talking about exactly what are these equations that are simulated in all of the separate grid cells. What are they, where do they come from and why are they important? And then after we discuss that, we'll talk about some of the things that aren't incorporated into these primitive equations, which are the so-called parameterized feedbacks, which have to be added kind of separately.

Okay, so these primitive equations are sets of nonlinear partial differential equations that are used to approximate the flow of energy within the atmosphere as well as the ocean. So basically, they describe the motion of fluids. These equations are also much the same as the equations that are used to model the weather as well. In fact, general circulation models are used for weather prediction as well as for climate modeling with just some tweaks for their particular purpose. But we'll discuss that in a bit more later. So what are these equations specifically? Essentially, it boils down to six core equations. So there are four conservation equations that describe the conservation of four different

key components of the climate system. So this is conservation of momentum, conservation of energy, conservation of mass and conservation of water. You might wonder how the last two are different. By conservation of mass, what I'm referring to is conservation of the mass of air molecules, essentially, whereas water is treated separately because it's quite a different substance, obviously, compared to oxygen and nitrogen, which makes up most of the atmosphere. So we've got four conservation equations. We then have an equation of state. So an equation of state essentially describes the relationship between key variables in the climate system, particularly pressure, volume and temperature. So for those who know PV equals NRT as the ideal gas equation,

well, that will be making an appearance here. So that's effectively the equation of state. The final equation is called the hydrostatic equation. And this represents an assumption about how pressure behaves in the atmosphere. And basically it relates to the fact that Earth has a gravitational pull, which pulls the atmosphere towards it, and therefore results in a pressure gradient with higher pressure closer to the surface and lower pressure the higher up you go. So the hydrostatic equation relates to that phenomenon and expresses a particular assumption about the way that the atmosphere is structured. So these are the six key equations we need for simulating the atmosphere. We'll also have a similar set of equations in simulating the oceans,

but here I'm going to focus on the atmosphere. So let's talk a bit about these six equations in more detail. So let's start with the four conservation equations. the first of which is the Navia Stokes equations, and these embody the conservation of momentum. Now, obviously, I'm not going to talk through all of the symbols and the precise mathematical formalisms of the equations, interesting though as that is, but I will give you a general sense of what the equations are actually saying and what we can do by sort of simulating them. So what the Navier Stokes equations state, the Naviastokes equations are quite general in fluid dynamics, and here we're concerned with a specific form of them that's relevant to the atmosphere.

the Navier-Stokes equations basically state that the acceleration of a parcel of air in the atmosphere is equal to the sum of three different forces that act on that parcel of air, namely the Coriolis force, so that's the force due to the rotation of the earth, the pressure gradient force, and the gravitational force. Now, it might seem like I just double-counter gravity there, but remember there's two different effects here. There's the direct effect of the gravity force pulling a parcel of air down, and then there's the effect of the pressure gradient on that parcel of air. The pressure gradient is not only caused by the gravitational force. The vertical pressure gradient is caused by the gravitational force, but there can be other

pressure gradients as well. I mean, we know that, right? Because air moves about the atmosphere in the form of wind, and wind is generated by differences in pressure. So just bear in mind that air pressure can vary because of different reasons. And the gravitational force is one of those reasons, which generates the vertical pressure gradient, but there can be horizontal pressure gradients as well. And so in this case, we're predominantly talking about horizontal pressure gradients, but there can be vertical pressure gradients as well that will have an effect on the vertical motion of air. So anyway, to repeat, the Navi-Stokes equations essentially says that the acceleration on a parcel of air is equal to some of the three key forces that act on that parcel of air, and those are the

Coriolis force, the pressure gradient force, and the gravitational force. And that should make good sense, right, because those are the three key forces that act on air as it's moving about in the atmosphere. Now, one thing you might be wondering is why we see the Coriolis force here, because you may know from previous episodes that I've done or from otherwise, that the Coriola's force is not a real force. It's an apparent force caused by the rotation of the earth, which leads to deflection of large bodies of air as they move across the surface of the earth due to its rotation. and that gives rise to the swirling, to the circular motion of hurricanes and so forth. Now, the reason why we see that here as an explicit force is because of the coordinate system

that's used for modeling the primitive equations. And what's used is a rotating coordinate system. So this allows us to effectively ignore the rotation of the earth and treat what's happening as if it's happening on a sort of a fixed coordinate system. So we don't have to explicitly model the rotation of the earth and worry about that. The price of that is that we then need to introduce the koreal Coriolis force. So there's sort of two ways of looking at it. You can either have a stationary non-rotating reference frame, and in that case, the Coriolis force is not real, it's just a pseudo

force, right? But having a stationary reference frame is difficult when the Earth itself is rotating. It's not convenient to model it that way, essentially. So the alternative is to adopt a rotating reference frame, which kind of moves with the rotation of the Earth. You can think of it that way. The price of doing so is that we need to then introduce the Coriolis force as an actual real force. So that's why we see it here, because of the coordinate system that we've chosen, essentially, which is convenient for the type of modeling that we're doing here. So that's the Nalvi-Stokes equations representing the conservation of momentum as different forces act on parcels of air as they're moving about. Now let's turn to the next equation, the thermodynamic equation. This represents

or embodies the conservation of energy. So essentially what this equation states is that the change in the internal energy of a parcel of air is equal to the work done on that parcel of air by the pressure force, plus the work done by gravity, plus heat transfer from various sources. Now, these sources of heat transfer will include radiation, conduction, and evaporation. Note that we don't see the Coriola's force here because it's a rotational force, and therefore it cannot do work. Work can only be done by forces that act in the direction of motion. The Coriola's force is a rotational force only. It doesn't act in the direction of motion, so we don't see it in the thermodynamic equation.

So you'll see that there's sort of similarities between the Naviostokes and thermodynamic equations, because momentum and energy are closely related to each other. Just as in the Naviostokes equations, we saw the roles of the Coriolis, pressure gradient, and gravitational forces. Here, likewise, we see the effects of the pressure force and gravity. We don't see the Coriolis force because it cannot do work, and the extra thing that we, the extra term that we see in the thermodynamic equation is heat transfer. Heat transfer does not appear in the Navi-Stokes equation because it doesn't transfer momentum. It just transfers energy. So this equation essentially represents all the ways that energy is transferred to an energy parcel in the atmosphere by pressure, by gravity, and by heat

transfer. So we've discussed Navier-Stokes for momentum and thermodynamic for energy. Now we discuss the continuity equation which embodies conservation of mass. This one's fairly simple. The continuity equation states that the rate at which air velocity exits a region is equal to the rate of change of density in that region. So basically the way to think about this is if on balance air is moving out of a region, that means that there is a change in the density of that region. Those are kind of two ways of saying the same thing, assuming there's conservation of mass. Now, if there is not conservation of mass, then you could have air moving out, but the density not changing. Well, that would imply that air molecules are either disappearing or appearing from thin

air, which obviously would violate the conservation of mass. And so it kind of makes sense that we would write down an equation saying that, well, if there's air coming out of a region, that means the density of that region is going down, like on net, or vice versa. If there's air going into a region, then the density goes up. So that's sort of a very sensible way of enforcing the conservation of mass. And specifically we're talking about conservation of air mass here. In these simulations, by the way, we're kind of ignoring the actual composition of the atmosphere, we're just treating it as a uniform fluid, which is appropriate for the level of granularity we require here.

Now, I mentioned that there's a similar conservation equation for water, so this will look much like the continuity equation, but we will have to add an extra term to deal with evaporation and condensation, because the continuity equation that we just mentioned for air does not hold for water, precisely because water can fall out of the atmosphere due to condensation. will be added because of evaporation. So those special terms will need to be included in the version of the water continuity equation. So we've now discussed our four conservation equations for momentum, energy, mass, and water. Now let's turn to the equation of state. So the equation of state links different key thermodynamic variables. It may not have been obvious because of the way

that I introduced these four conservation equations, particularly the first three that relate to the the air directly. But each of these equations sort of has a corresponding thermodynamic variable that it's most directly related to. I mean, the thermodynamic variables relate to all of them, but you can think of it as sort of one each relates to one of the equations. So the conservation of momentum relates to the thermodynamic variable of pressure, because pressure and momentum are very closely related, essentially. Pressure is imparted by molecules hitting a surface, and that also imparts momentum. So they're directly related to four. forces, and so you can sort of see how they're connected there.

So that's the relevant variable there, momentum. We then have energy, which is related to the thermodynamic variable of temperature, the temperature within a volume of air being directly related to its internal energy, and the final thermodynamic variable is density, which is directly related to mass, so the amount of air in a given volume. So these three thermodynamic variables, momentum, temperature, and density are related to each other through the equation of state. So they appear in the conservation equations, but they also appear in the equation of state,

which says how they relate to each other. And the equation of state is the ideal gas law, pv equals n RT. Or in this case, we're going to replace v and n with the density. If you think about it, the density is just the volume divided by the number of particles in it. N and V can kind of be replaced with the density. And the form of the ideal gas law that's used here basically just says that the pressure of the air is proportional to the density of the air times the temperature. And hopefully that makes intuitive sense, right? Basically, what it's saying is that if we put a bunch of gas in a box,

the pressure exerted by the gas on the sides of the box is going to be proportional to the amount of gas that we put in the box multiplied by the temperature of the gas, and the temperature is directly related to the kinetic energy of the gas molecules. So basically, in order to increase the pressure, you can either add more gas in the box, or you can make the gas move faster so that each molecule strikes the sides of the box more often, and with a higher force, and that's basically what temperature measures. So hopefully the ideal gas law, the equation of state is sort of intuitive as to why it's there. it describes the relationship of these variables to each other. The final equation, which I already mentioned, is the hydrostatic equation.

So this embodies an assumption of vertical stasis, which means essentially that there is limited vertical motion of air in the atmosphere. Obviously, there is vertical motion of air. That's largely due to convection. And convection largely is not incorporated in the primitive equations. It needs to be modeled, it needs to be parameterized separately. So I'll discuss that at a moment. But the hydrostatic equation effectively embodies the assumption that there is limited vertical motion of air, at least in this set of primitive equations.

And what it states is that specifically is that the rate of change of pressure with respect to altitude is dependent upon the density of air and the gravitational constant. I sort of explained this before that the hydrostatic equation represents the fact that, there is a vertical pressure gradient in the atmosphere caused by the gravity of Earth. So the Earth's gravitational field is pulling the atmosphere down. And that means that the closer you are at the surface of the Earth, the more dense the atmosphere will be. And so there's a pressure gradient as you move further away from the surface of the Earth, you move higher up. The pressure will decrease in direct proportion to how far you've moved up. So there's a proportional decrease in pressure with altitude relating directly to the lower density of the air.

as you move upwards, away from the surface of the earth, and that's directly determined by the gravitational pull of the earth. Now, as I said, that hydrostatic equation represents an assumption of limited vertical motion of air. That is obviously not true because we know that there is convection and cloud formation, which is largely what drives vertical motion of air on the Earth's surface, and so that should be a kind of a warning that these primitive equations do not incorporate convection. That will need to be a separately parameterized element. So at the point of these primitive equations isn't to incorporate all of the aspects that are important in the atmosphere. It's really just to describe the basic movement of air and water in the atmosphere.

In accordance with the equations of fluid dynamics, particularly the conservation equations of momentum, energy, mass, and then water, and incorporating the equation of state and a hydrostatic equation. There is much more to the atmosphere, but those need to be modeled separately because basically we don't have simple equations. that are derivable from basic physics that can describe these phenomena. The primitive equations can be described using fairly simple physics equations derivable from the very basic behavior of fluids and transfer of energy and such. And so we can model them using the primitive equations in each of the grid cells and integrate them over time. There are other aspects, but those will need to be modeled in different ways. So let's turn to that, right? We've talked about the primitive equations,

which form the sort of mathematical core, if you like, of the general circuit. models, but let's talk now about the parameterized feedbacks or the additional elements to the model which are not incorporated in the primitive equations. Now, I'm sort of putting parameterization and feedbacks in the same category. Technically, this isn't quite true. So, for example, radiation needs to be modeled as a parameterized component. It's not part of the primitive equations, but radiation isn't really a feedback, likewise surface friction. I'll make that distinction clear in a little bit, but most of the parameterizations are feedback mechanism, so it makes sense to talk about them together.

So I've talked about feedbacks in the past couple of episodes, and so I'll just briefly revisit the idea here. A climate change feedback is an effect of global warming or any change to the climate system that either amplifies or diminishes the initial force that initially caused the warming or the cooling for that matter. So positive feedbacks enhance warming, while negative feedback's weaken warming.

So feedbacks can't initiate a change to the climate, but they amplify or diminish that initials. change, right? Now, we've talked about some feedbacks already. We've talked about water vapor as one of the major positive feedbacks. As the earth warms up or as the atmosphere warms up, it's able to hold a higher water vapor content. Water vapor is a major greenhouse gas, and so that leads to the absorption of more long wave radiation that's emitted from the surface, thereby amplifying the greenhouse effect and causing further warming. As I've discussed previously, the water vapor feedback effect is strongly positive. It has a magnitude of about 2 watts per meter squared per Kelvin. I'll sort of

summarise these effect sizes a bit later, so don't worry too much about those numbers. But the point is it's a very large, really the largest single positive feedback mechanism. So it is very important. However, water vapor feedback does not directly need to be parameterized in global circulation models, because if you have a coupled ocean atmosphere global circulation model, then the movement of water and the holding of water in the atmosphere will be modeled by those basic equations. Much of that will occur at that level of the coupling between the atmosphere and the ocean, which we haven't directly talked about, but it will already be incorporated, and so it won't need to be sort of explicitly parameterized separately.

But it is certainly important to ensure that you have that coupling between the ocean and the land surface as well, but a lot of evaporation occurs over the ocean, so that you can have this water vapor effect properly incorporated into the model. Now, the next feedback mechanism is the lapse rate feedback. So the lapse rate is the rate at which temperature falls with altitude, at least in the troposphere. As you increase in altitude, the temperature declines. It gets colder. The emission of radiation falls with temperature.

This is just the Stefan Boltzmann law, right? So as the atmosphere gets cooler, it doesn't emit as much radiation. Now, this means that long wave radiation emissions from the upper atmosphere, which is colder, are less than those from the lower atmosphere. which are warmer. So there's less radiation emitted higher up where it's colder. This means that the strength of the greenhouse effect depends upon the rate at which temperature decreases with height. So the lapse rate affects the strength of the greenhouse effect. Now, theory and models both indicate that global warming tends to reduce the lapse rate.

This actually weakens the greenhouse effect and so constitutes a negative feedback effect. Now, truth be told, I'm not entirely sure how much of this effect is already incorporated into global circulation models or whether this needs to be separately parameterized, I would suspect that much of the effect of the weakened lapse rate would already be incorporated as part of the global circulation models, but I'm not 100% sure about that. Regardless, the lapse rate feedback is expected to be not enormous, but still sizable, and a negative effect, so about minus 0.5 watts per meter squared per Kelvin. So about a third to a quarter of the size of the water vapor feedback effect.

So it's important to incorporate as well. The next feedback effect is convection. I've sort of discussed this briefly before, and also cloud formation, convection being very directly related to cloud formation. So the flows of air, particularly horizontal air flows, are modeled by the primitive equations. A lot of the important convection behavior, so particularly vertical motion, occurs at a very fine resolution and at a much smaller spatial scale than can be modeled by the grid cells

in the models. So the types of airflow that are modeled in the, the grid cells occur at tens to hundreds of kilometers, whereas convection is much more localized than that, typically. So convection and cloud formation need to be explicitly parameterized. That behavior doesn't arise in the models by itself. Now, this is important. Obviously, we need to model convection to model accurately the vertical transfer of energy in the atmosphere. Cloud formation is also important because clouds effectively add to the greenhouse effect by trapping in infrared radiation, emitting it back to the surface. So they amplify the greenhouse effect. On the other hand,

clouds are also quite light. They reflect sunlight back to space, and so they exert a cooling effect. So it's unclear as to exactly what the net balance will be between the warming and the cooling effect of clouds. And also, it's difficult to model as to how much changed in cloud cover there will be as a result of warming of the atmosphere. So cloud formation and convection is a very complex and more uncertain component of these climate models. Best estimates are that clouds contribute an overall positive feedback effect of about 0.5 watts per meter squared per Kelvin. So about the same magnitude of the lapse rate, but in the opposite direction.

The next feedback effect that I want to discuss is the ice albedo effect. So ice is not incorporated in the primitive equations of the general atmospheric circulation models that we've just discussed. But obviously it's important because, first of all, it's a mechanism of heat transfer in the Earth system as ice melts and obviously as ice melts it absorbs energy, right? And so that's important. But also melting of ice reduces the ice cover of the Earth's surface, which reduces the albedo of the Earth, making it less reflective, meaning that the Earth absorbs more solar radiation. That's going to then contribute to a positive feedback effect. The effect is different in the

different hemispheres. So loss of Arctic sea ice results in a very rapid warming of the Arctic. while Antarctic's thick continental ice sheet doesn't readily melt, and so it's tended to maintain a much cooler temperature around the Antarctic, which hasn't warmed as much as the Arctic can. So there are some regional effects there. But overall, the loss of ice due to warming contributes a fairly large positive feedback effect. The magnitude is estimated to be about 0.4 watts per meter square per Kelvin, so slightly less than the effect of clouds and convection. So these are some of the major feedback effects that need to be parameterized in the global. circulation atmospheric models. In addition, there are other things that need to be parameterized as well that are not feedback effects as such, but they're still important for accurately characterizing the behavior of the atmosphere. So I mentioned a couple of these before. I'll just go through them briefly in a bit more detail. So surface friction is also important. Surface friction is not directly modeled in the primitive equations, but is important with the motion of air close to the earth's surface, there's a friction that affects those motions. Also, other types of atmospheric turbulence,

which is a bit more complicated than is typically incorporated into the primitive equations. Those also need to be added as parameterized effects. I mentioned that radiation is not incorporated into the primitive equations because those only really describe fluid motions around the atmosphere and the ocean. So radiation is neglected. So we need to use the radiative transfer equations, which are kind of a separate class of climate models, if you like, but we can incorporate them as part of alongside the primitive equations in the global circulation models. So we use those radiative transfer models to model the transfer of energy to the surface of the earth and then also from the surface of the earth back to the atmosphere.

I've already discussed those in more detail in modeling the greenhouse effect, so I won't go over those again here. But remember that we still need those radiative transfer equations. It's just that then we incorporate them alongside the grid-wise set of fluid dynamics equations to describe how energy and momentum is transferred across the atmosphere and also in the oceans. There are many other aspects of the Earth's climate system and also oceanic circulations and currents which need to be parameterized, and I won't discuss all of them here. The important point, though, is that this parameterization process is rather difficult because it's not based on fundamental physical laws in the way that the primitive equations are,

and so they're more subject to different modeling assumptions and disputes about the exact right way and the exact magnitudes and so forth of these different parameterizations. The biggest source of uncertainty that I'm aware of are those relating to the correct way to model convection and cloud formation and also modeling ice albedo. Those are subject to high levels of uncertainty. That being said, when we add up all of these effects of the different feedbacks and the parameterizations, we end up with a net climate feedback effect of approximately plus to watts per meter squared per Kelvin. So in other words, for each additional degree that the

atmosphere heats up, the feedback effects that are internal to the atmosphere and also interaction with oceans and so forth, these feedback effects will contribute a further roughly two watts per meter squared of additional equivalent warming effect. So the effect's equivalent to two watts per meter squared of radiative forcing on the atmosphere, which is quite a substantial effect. And overall, this means that if we consider when I discussed a couple of episodes ago the climate sensitivity, so the amount by which the Earth's temperature responds to a given amount of radiative forcing, the simplest way to compute the sensitivity, so the temperature response, is to use the Stefan Boltzmann law, which directly relates the energy emissions with the temperature of a radiating black body,

we get a sensitivity of about 2.7 Kelvin per watts per meter squared. So in other words, for each one watt per meter squared increase in solar forcing, the straightforward direct effect of that on the Earth's temperature is about 0.27 Kelvin. When we then add in the effect of the positive feedbacks, or the net effect of all of the feedbacks, that gives about an extra 2 watts per meter squared of forcing equivalent for every degree of warming. So working out the kind of final result of that is a bit tricky, obviously, because if you imagine, well, Earth increases in temperature because of an external forcing, that then leads to a feedback, which then leads to a further increase in temperature,

which then leads to extra feedback from that extra increase in temperature and so forth. And so there's a sort of an iterative increase over time, and this is why the climate system takes a while to reach the new equilibrium. But if you work through the maths, the total of these climate feedbacks, the net effect of these feedback mechanisms is to increase the estimated equilibrium climate sensitivity from about 0.27 to about 0.8. So that's an increase of roughly three times. So the net effect of all of these feedback effects on the Earth's climate is to mean that it's about three times as sensitive to solar, to radiative forcings as it otherwise would be. And so that is,

quite a large effect overall to increase the sensitivity by about a factor of three relative to what it would be without any of these feedbacks. So that's what the combination of these feedbacks is for the climate system. And that estimate is still fairly uncertain. But we have a reasonable idea about what it is approximately. And obviously that's why it's important to incorporate these feedback effects into the general circulation models, because otherwise we'll get quite inaccurate results. This concept then leads us on to the notion of impulse response modeling. And this is one of the very useful things that we can do with the climate model, once we've set up the primitive equations and parameterized everything and run the model and so forth, is that we can see what the trajectory

is of various parameters of interest, often temperature, in response to different types of forcings or perturbations in the model. This is what's called impulse response modeling. So you introduce an impulse, which basically is a sudden injection of something into the model or a sudden change in the model. It could be a sudden injection of CO2, or it could be a sudden increase in solar radiation, or a sudden increase in particulate matter in the atmosphere, like from a volcanic eruption. Anything like this could be an impulse, right? And then you see what the response of the model is to that as you run it for a certain period of time, because the response won't be instant, because of these different feedback effects and the interactions between the ocean and the atmosphere and everything

like that. It takes time for the effects to be manifest and for the equilibrium to be reached. One of the questions we would like to answer is, what will the effect of all of our greenhouse gas emissions be on, say, atmospheric temperature at different points in time? And how long will it take for temperatures to increase? How long will it take for the carbon dioxide to come out of the atmosphere and so forth? And impulse response modelling can be very useful for that. As I mentioned in our previous episodes, the climate sensitivity figure of about 0.8 is the equilibrium climate sensitivity. So that represents the sensitivity of Earth's temperature to a radiative forcing after equilibrium has been reached, which can often take decades or even centuries, right? So, for example, in this case, what it would mean is if we increase the carbon dioxide concentration of the atmosphere from 300 to 400 parts per million, what will the effect be on the Earth's temperature after equilibrium has been reached in, I don't know, several hundred years or longer.

That's what the equilibrium sensitivity will tell you. The actual warming that we see at present or in the near future is less than that equilibrium amount because the effect is delayed. The major reason for this delay is the high heat capacity of the oceans. They absorb a great amount of heat, which slows the warming of the atmosphere, and it takes time for the energy to be transferred from the oceans to the atmosphere. There's many complex aspects that are sort of relevant here. But the key point to understand is that the equilibrium response will be different to the response over a much shorter time. horizon. We have a mechanism for describing this, which is the transient climate sensitivity, and that is defined as the change in the global mean surface temperature averaged over a 20-year

period at the time of carbon dioxide doubling. So basically, how much does the temperature change in response to a doubling of carbon dioxide over a 20-year period within the time those emissions occurred? So it's estimated to be about half of the equilibrium effect, a sensitivity of about 0.4 compared to 0.8 for the whole equilibrium effect. The biggest reason for the differences here is that the transient climate sensitivity is related to the thermal timescale of the shallow ocean. So over a shorter time horizon, the atmosphere exchanges energy with only the most upper part of the ocean, like predominantly, because it takes much longer for vertical transfer of energy

to deeper parts of the ocean. So those deeper parts of the oceans have a much longer time scale of interacting with the atmosphere. And it's this longer time scale of interaction with the deep ocean that determines the equilibrium sensitivity. So basically transient climate sensitivity relates to the thermal scale of shallow ocean, but equilibrium to the time scale of the deep ocean, which is much longer. So that's why there's this difference of about a point eight for the sensitivity of the equilibrium sensitivity of the atmosphere, but only about a point four when we talk about the transient sensitivity. Now this discussion of

the shorter term and longer term effects can get a bit confusing because there's a couple of things going on at the same time. So the climate models that we've been discussing can help us to conduct these impulse response simulations of the effect of introducing, you know, say, a doubling of carbon dioxide concentration in the atmosphere. That will help us to show how the temperature will change over time, for example. But there's another process, another set of processes that are happening, which are largely outside the purview of the general circulation models, or at least the simpler versions of them. And these relate to the movement of carbon dioxide between the atmosphere and the oceans. So these can be parameterized as well. It's just not something that we've discussed, but it is a

separate process to modeling the change of temperature in the atmosphere. Separately, you can also model what happens to the CO2 once it's introduced into the atmosphere. And this is where the oceans are also important. The oceans both absorb energy. They're absorbing currently a lot of the extra energy that is retained because of the increased greenhouse gas concentrations in the atmosphere. So much of that is being absorbed by the oceans and not the atmosphere. In addition, the ocean is also absorbing CO2 directly. So the ocean absorbs energy and it absorbs carbon dioxide. They're separate processes. In fact, only about 40% of annual human greenhouse gas emissions are absorbed by the atmosphere or are released into the atmosphere and stay in the atmosphere.

about 30% are absorbed by plants and soils, and about another 30% are absorbed by the oceans. So much of the greenhouse gases were emitting into the atmosphere don't actually stay there, even in the short term, they're absorbed by either plants, soils, or the oceans. Now, in the short term, as I mentioned, there's only significant interaction with the upper region of the ocean, the shallow ocean, and it only has a certain amount of capacity for carbon oxide. There's only a certain amount that it can absorb in a certain unit of time. Over a period of centuries, however, the deep oceans will eventually absorb most of the added carbon dioxide that we've put into the atmosphere. So over a period of years to decades, only the top parts of the ocean will absorb much CO2.

That's what's been happening as we've continued to emit greenhouse gases into the atmosphere, CO2's been gradually absorbed, but mostly into only the upper parts of the ocean. Over many centuries, the deeper parts of the ocean will gradually absorb most of the added CO2, which will then be removed from the atmosphere. It's estimated that after CO2 is relation to the atmosphere, about 60% of it is removed after about a century, mostly by interaction with the oceans. However, even after about 1,000 years, only about 80% of it's been removed, meaning that 20% of it is still there in the atmosphere. So it's important to understand the scale of the ocean's importance with regard to both temperature directly as well as carbon dioxide concentrations.

The ocean is absorbing about a third of carbon dioxide emissions, plus it's absorbing the large majority of the excess thermal energy resulting from global heating. By some estimates I've seen about 90% of all of the extra energy is absorbed into the ocean. Only a small percentage is actually contributing directly to heating up the atmosphere. So you better be thankful for the ocean, because if also for the ocean, the atmosphere will be dramatically hotter than it was now. Of course, if the ocean didn't exist, then probably life would have never formed in the first place, but that's a separate question. And that reminds us of why it's so important to consider coupling of the general circulation models of the atmosphere and the oceans

because of the important role of the oceans in absorbing CO2 and absorbing energy and also interactions of obviously evaporation of water from the oceans into the atmosphere. So let's conclude this episode by talking a bit about the validation of climate models. So we've talked about how they're constructed and we've talked about one of the things that can be used for, which is impulse response modeling, and that's quite useful in looking at the trajectories of climate change over time.

But what else can we use them for? Well, one of the major uses is that we can use them to estimate the magnitude of warming would be as a result of anthropogenic greenhouse gas emissions over the course of the 19th and 20th centuries, and compare that to observed warming and see how well we understand the underlying mechanisms. We can also use these models to conduct experiments in terms of what are the relative magnitudes of the effect of human factors versus natural factors, particularly changes in solar radiation on Earth's temperatures over that period of time. And so those are all very useful in helping us to understand the magnitudes of the different

forces affecting the Earth's climate. And for example, those simulations show that over the course of the last two centuries, natural climate forcing, such as solar forcing or orbital changes, have had a negligible net effect on the Earth's temperature. And effectively, all of the warming that we've observed over the last 150 years has been due to increased greenhouse gas emissions. In fact, another finding is that there would have been more warming as a result of greenhouse gas emissions, but some of it has been masked by sulfate particulates that have also been released into the atmosphere

by humans as a result of industrial activity. But as a result of efforts to reduce those types of pollutants since the 1970s, air quality is generally improved, and so that has removed one factor that sort of masked temperature rise in the mid-20th century. I mentioned this a couple of episodes ago in the climate change series. So climate models have been very useful and have helped. us to understand the magnitudes of the different effects on the climate and really illustrate that all of the effects, or almost all of the effects on temperature that we've seen in the past two centuries are the result of human activity, and the magnitude of the temperature rise that

we've seen is roughly consistent with what we'd expect from modeling. But in order to be confident in these findings, we need to be sure that these models accurately model what they're supposed to, that they deliver qualitatively and quantitatively accurate outputs. And this is what model validation is for. Now, model validation can be done in a number of different ways. One argument in favor of the models is that the primitive equations they use are derived from basic physics and really aren't subject to any dispute. The main source of dispute or potential concern with the models would be the parameterizations, and of those relating to cloud formation and ice formation or ice melting are some of those that have greater uncertainties. There's also certain uncertainties relating to

certain ocean currents and other aspects that we haven't really discussed here. another general source of model validation is that the same sets of equations and general approach is used for weather prediction. Weather forecasting has become substantially more accurate every decade since World War II. It may not seem like the weatherman often knows what's going to happen very accurately. But in fact, we're actually quite good at predicting whether weather forecasts out to certainly one week are quite accurate these days and we're even able to get reasonable better than chance estimates up to, I think about two or three weeks out. So that provides some additional basis for trust in the climate models, which use fundamentally the same setup as the

weather forecasting models. We can also validate the models by comparing past predictions that they've made with the actual subsequent trajectory of temperature increase. One systematic review that I looked at, which is one of the best that I was able to find just from a couple of years ago, looked at 17 different predictions that have been actually published predictions from these global circulation models over the past 30 or 40 years. I think the earliest prediction was made in about 1970. So these are all looking at actual predictions, not retradictions after the fact of what the trajectory of temperatures will be. This is actually, they made a prediction about what would happen in the future, and we're then looking backwards and seeing how accurate was that prediction.

Now, one of the difficulties in predicting the future temperature trajectories is that you don't know at the time you make the prediction what the forcings will be, because you don't know what greenhouse gas emissions will look like 10, 20, 30, 40 years in the future. Of course, you can make an estimate, but you might be wrong on that front. But being wrong about the greenhouse gas emissions doesn't count against the climate model because that's not part of the climate model. That's a completely separate fact. That's just the magnitude of the forcings. So what this study did is actually factored out that component by using the estimated sensitivity of the models to provide a standardized measure of the response of the Earth's climate to a doubling of CO2. and then comparing that estimate to what we've actually measured over the past 40 or 50 years or so, depending on when the estimate was made. So basically, this study allows us to compare different estimates, even though different estimates were made at times when we weren't sure what greenhouse gas emissions would look like, and therefore that's an additional source of error that we don't really want to muddy the results here.

So the short-term response, short-term being over a period about 20 years or so, of doubling greenhouse gas emissions, is measured to be slightly less than two degrees, There's an uncertainty measure around that which is fairly broad. The uncertainty range is somewhere between 1 and 2.5, with the mean estimate being a bit less than 2. The predicted responses for the 17 different projections varied as well, the lowest being about a transient response of 1 degree, and the highest being about 3 degrees. But most of the estimates clustered in the range of between 1.5 to 2.5 degrees. And nearly all of them fell within the margin of error of the actual measured short-term sensitivity.

14 of the 17 models were consistent with the observation, meaning that for 14 of the 17 models, their predictions fell within the margin of error of the measured climate sensitivity. Of the three that fell outside of the margins of error of the measured sensitivity, two showed a higher implied sensitivity compared to observations, and one showed a lower. So there was no trend, systematic trend, I mean, not that you could really observe that with just three,

but the point is two were above and one below. So not really a trend either way. So on balance, paper concluded, and I agree with its conclusions, that the climate models were quite accurate, and it's especially impressive because some of these predictions were made in the early 1970s when the models were far cruder than they are today. And our knowledge of climate change as a whole was more primitive, and computing infrastructure was also much more primitive. So these models should be expected to be significantly worse than those that we have today.

But even then, they made fairly accurate predictions of the transient climate. climate response to a doubling of CO2. And in particular, there's also no indication that the models either systematically overestimate or underestimate the degree of warming. There is variation, of course, so there's a degree of uncertainty, but on average the models appear to have got it about right. Now, I should comment on the fact that there have been over the years certain misleading plots or graphs that have been shared online, which purport to show that climate models dramatically overestimate the warming in recent decades. These plots purport to show that, on average, models predict much more warming than we've actually observed.

So one of them that I have here purports to show that since the mid-1970s, so let's say in the past 40 years or so, the plot was made around 2015. So in the past 40 years to when the plot was produced, observed warming, and this was derived from satellite data, we'll talk about that in a moment, observed warming was about one-third of a degree in that period of time, whereas the predicts, predicted warming from these climate models was about three times that, so about 0.8 degrees. There's a few different versions of these graphs, obviously this is just the specific one that I'm looking at, but they're usually fairly similar. They show that the observed warming is much less than that predicted by these climate models or some of the major climate models. There are a few things to say about these plots, but I mean, basically they're deeply misleading. Most of these plots use satellite data, in particular satellite data that has not.

not been corrected for the warming effect of the stratosphere. We talked about satellite data way back in the first episode in this series on historical climatology. The important thing is that satellite data only dates back to the 1970s, and it's actually quite difficult to construct an accurate temperature reconstruction using satellite data, because satellites don't directly measure the temperature. They measure radiative transfers, which then you need to convert to a temperature measure by using a whole bunch of complicated statistical methods and a whole bunch of adjustments. And early series, even those not too long ago, was still not properly correcting for the fact that the stratosphere is cooling and it's known and expected that the stratosphere will cool down. I discussed that in the previous episode on the greenhouse effect.

But because of the way that the satellite measurements were taken, there was improper correction for this fact. So there were, you know, the attempt is to measure atmospheric temperature in the lower troposphere, or at least the mid-troposphere. but these measurements weren't properly correcting for the fact that some of their data was coming from the stratosphere, which was cooling, thereby leading to an underestimate of the warming in the lower or mid-troposphere. And that has been a long-standing issue with these satellite estimates. It's only in fairly recent last five or ten years or so, I think, that the series have really been properly corrected for this. There's a series of papers that demonstrated that this effect was an issue and that it's recently been corrected for in the most recent versions of this series. So some of these plots use these older uncorrected estimates of the warming trend. So effectively the plots show an artificially low amount of warming because of this improper correction.

Another paper that I looked at, which really illustrates this point very well, shows that without using this correction method, you have about a three-to-one ratio between the observed warming of the satellite data and the predicted warming of the models. But when you make this correction, the ratio goes from about three-to-one to about one and a half to one. So in other words, instead of the climate models allegedly overestimating warming by a factor of three, that factor goes down to about one and a half. So warming is still greater in these models compared to the satellite data after this correction is made, but the effect is much, much smaller. It's about half as much as initially stated.

The other thing to note is that often what these plots will do is they'll take a range of different climate models or runs of a climate model, because you know, you don't just run them once, you run them a bunch of times under slightly different conditions, slightly different assumptions. And what you should do is you take the average and show the margin of uncertainty. Because as I discussed previously with the paper that looked at the 17 different past projections, there's a fairly large range of uncertainty when it comes to both empirical measures of the climate sensitivity, but also predictions from the models, right? So you should ideally show that margin of error when showing predictions of models.

But often what these misleading graphs will do, they'll only show either the, the top most, like the model runs that show the greatest amount of heating, or they may show the average, but even the average is misleading because it doesn't show the uncertainty, and often will be, if it's the mean, maybe biased upwards by certain very high runs. So really you want to show the range of uncertainty and not just the average, or certainly not just the top. When you factor in the uncertainty of model predictions, as well as the corrections to the satellite data because of the stratospheric cooling that I mentioned, what you find, and as I said, this paper illustrates this very well, what you find is that the satellite data actually falls

comfortably within the margin of error of the climate models. And so really, the entire alleged effect of the climate models overstating warming disappears once you factor in the error in earlier satellite data sets because of the stratospheric cooling, plus the uncertainty of climate models. That being said, even when you do consider these two factors, it's still the case that the climate models predict higher warming than is measured by the satellite data. However, even after correcting for the stratosphere cooling issue, there are still issues about how reliable the satellite data are. I discussed these earlier when we talked about the last 200 years and the satellite data controversy. Essentially, it's very difficult to construct accurate satellite

data reconstructions and many climate scientists of the opinion that satellite data still somewhat underestimate the warming on the surface and that the data collected from the surface is potentially more reliable. There's issues of, for example, completeness of the coverage of the satellite data and it's important to have an accurate representation of the entire surface of the earth when making these average surface calculations. Another issue is that there are different series of satellite data. There's three main series that I'm aware of. I won't give names because they're pretty meaningless, but the point is that they've constructed by different groups using different statistical methods. As far as I understand, the raw data that they use is all the

same. It's all derived from the same set of satellites. It's just that they make different statistical assumptions. And there's one group in particular that produces abnormally low estimates of warming. So one has to be careful that the data one is seeing is not from this particular series, a dataset series that shows unusually low amounts of warming, because, that particular satellite series is, well, it's inconsistent with the other two sets of satellite data, but it's also inconsistent with surface temperature data. And to be honest, I don't find it very trustworthy myself. I'm a bit suspicious of all of the satellite data. The most up-to-date series are broadly consistent with what we see at the surface. I think there's still a bit of an

underestimate relative to the surface data. But I think on balance, the surface data is actually more reliable than the satellite data. At any rate, the point is that these plots typically use the satellite data and they don't use surface observations precisely because the satellite data are basically the lowest estimates of temperature increase that you can find. And so they best provide the contrast between what the models predict and what the measured warming actually is. On balance, it seems that the climate models accurately predict temperature increases that we've seen in the past 50 years or so. There is a hint that the current climate models may predict slightly higher warming, at least than we observe with the satellite data, though the effect is not very

large there. From the paper that I mentioned, the two satellite series that predicted the higher amount of warming, and I think that are more realistic, estimate trends of about 0.2 degrees Celsius per decade, compared to the model average, which was about 0.3 degrees Celsius per decade, they're bearing in mind that the very large margin of error encompass the 0.2, so there is a difference on average, which is still a question as to why, but there was no sort of statistically significant difference if you want to think of it that way. So it's unclear whether the climate models may be slightly overestimating warming on average, but again, if you look at the other paper that I mentioned, that which compared historical predictions using previous climate models, there was no evidence that

climate models there were overestimating warming on average. Now, those two studies use different methodology, so it's a bit hard to compare them directly. So I'm not entirely sure whether that issue has been resolved. Possibly current models are slightly overestimating on average, the rate of temperature increase. But if they are, it's not sort of statistically significant in the sense that the range of uncertainty encompasses measured values. And so there's no basis for saying that the models are wrong as such. It's possible that they are slightly biased high, but because the uncertainties are so large, it still encompasses the measured values. And overall, there is evidence that our predictions based on these models are on balance

quite accurate. So, that brings us to the end of what I wanted to talk about in this episode. We talked about different types of climate models and what they can be used for. We introduced the primitive equations for the global circulation models and also the parameterized feedbacks and how they relate to each other. We talked about some of the applications of climate models, including impulse response modeling, and we talked about the role of the oceans as well and how important they are for interaction with absorptions of heat and carbon dioxide and so forth over time. And we concluded with the discussion of validation of climate models, including a couple of studies which have compared past predictions to the observed trends since the predictions were made,

as well as some of the controversy relating to alleged overestimation of warming relative to satellite data. In the next episode, we will talk about some of the so-called fingerprints of human-produced warming, so further evidence that climate change is indeed caused by humans, and we'll talk about some of the impacts of climate change and what can be done to mitigate them. and that will bring our climate change series to a conclusion finally. So hopefully you've enjoyed this episode. If so, you can support the podcast in a number of ways. If you would like to make a financial contribution, you can donate via PayPal

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