 Great. Well, I was asked to wait a minute until the recordings. Can you hear me? Yes? Yes? Okay. Well, I'll just start by saying thank you to Ricardo and the other organizers for inviting me here. It's my first time at ICTP and it's a pleasure to be here and I'm going to try to follow Professor Kang's advice from this morning and give a good lecture so I get invited back for the next 30 years. So, also, the words multiple equilibria in the climate system appear in the title of this summer school so I thought I would give you a lecture all about multiple equilibria in the climate system and I will acknowledge several people who are here in the room today that have been involved in various parts of what I'm going to show you today including my former MIT collaborators David Ferrara and John Marshall and my student Cameron Renkrel is also here in the room and some other folks who I've worked with who have influenced my thinking on these topics over the years. So, I'm going to begin just with a few slides of, I don't know why the letter B appears there but anyhow I think that was supposed to say something like 60 million years of earth history and I'm showing you this in a couple other slides of paleo climate oriented material just as a way of setting the scene a little bit for why the problems we're going to look at in models are sort of interesting and I want us to, whoops, I want us to look at this curve on this side of the plot which is a compilation of oxygen isotope records from deep sea sediments and essentially what this tells us about is is temperature of the water at the bottom of the ocean which tells us about temperature of the polar regions where the deep water is formed and the reason I'm showing it to you is to emphasize that if we go back 60 70 million years ago we're in a time of great warmth this is a temperature scale here with zero Celsius and the history of the most recent few tens of millions of years of our planet has been one of gradual cooling so this has been drifting towards cold and the other thing that we get from a record like this is this idea that as the earth drifts into cooler regimes and you have some schematics here of when the various ice sheets first appeared in Antarctica and then the the permanent ice sheets in North America excuse me in the northern hemisphere as the earth drifts into cold climates the climate tends to become a lot more variable so we have all this noise in this time series that doesn't seem to be there during the times of warmth that's one interesting thing there's a few other records from much shorter time periods okay now we're looking in this case oh I gotta get used to this pointer here this is a record from over the looking at some details of the last ice age taken from Greenland again lots of noise in the glacial part of the record this is another record that's inferred to tell us something about temperature the most recent part of the story you have to watch how the time axis goes on plots like this because there's no good convention about whether time goes forward or backward so that's why I flipped one of these just so they're going on the same way on the same slide here icy climates noisy and variable warmer climates apparently less so that's a theme that shows up again and again in various paleo records and this goes back farther here you see this is another kind of sediment record showing the coming and going of the great ice sheets over these relic sort of hundred thousand year time periods here's today and here's a bunch of big ice ages that occurred and smaller more rapid ice ages that occurred before that one of the interesting themes that you can see on plots like this is kind of prevalence of sawtooth sort of signals warming happens more rapidly than cooling that's another we can see that in these curves here sort of difficult to point both sides but I'll try to move over there halfway through the talk how about that anyhow so last thing I want to point out in this whirlwind introduction to earth history if we go much farther back in time we're now talking about something like 700 million years ago the continents were first of all as indicated in these schematics largely arranged in tropical latitudes and a lot of work through field geology over a long period of time has revealed a lot of information about two periods in which there was extensive glaciation on these tropical land masses at these times and through the field geology and through a lot of geochemical work and a lot of other kinds of lines of evidence it's inferred that the reason these tropical land masses were covered by ice sheets was that in fact the whole planet was likely covered by ice and this we believe happened at least twice there's still some controversy over what the details looked like but this is called the snowball earth event or these these are two snowball earth events and here's an artist's depiction of what earth looks like if you paint it white okay and that's going to be a theme of what I'm talking about today is what happens when the earth gets partially painted white by ice and snow all right so here's a question a basic question we can ask along the theme of this summer school is the is the climate system unique in the sense of if we set up the continents in a certain way and set up a certain orbital arrangement and distance from the star and so on is the climate determined or not are there multiple equilibria in the system we've just seen a number of very quickly a number of big climate changes that earth has gone through in its history and one way I can summarize all of that is to say that the fraction of earth surface that's been covered by ice and snow has varied between zero and a hundred percent over over the time that earth has been around so fundamental questions what what determines the equilibrium surface temperature of the planet is it warm or cold is it a unique function of boundary conditions and a corollary to that is does a large climate change necessarily imply that there is a large forcing or can we instead think of some of these dramatic climate changes in earth history as a sort of mode switching between between different stable states and if that's true then climate modeling in some sense becomes an initial value problem the climate we get depends on what we start from and that's gonna be the theme that I'm gonna elaborate on today so here's a here's a sort of skeletal outline of what I'm gonna tell you about over the next hour and a half or so first of all some basics about this so-called ice albedo feedback mechanism and why it can give rise to multiple solutions multiple climates using very very simple models models that don't that aren't informed at all about our understanding of how the oceans work and how the oceans interact with sea ice I'm then going to talk about what's special about the oceans and why the answer we get from the very simple models is wrong in some sense and I'm going to then elaborate on that by showing various kinds of results from various kinds of climate models that point to the sort of robust result that we can expect to find multiple stable climates in systems that that let's say know about the oceans and how oceans interact with sea ice if we have time will then give a little bonus discussion about the climatic impact of oceans in worlds that don't have any ice at all which is another interesting topic okay so multiple equilibria a system with multiple stable states is a nonlinear system almost by definition a linear system if we put in the boundary conditions we get an answer to have multiple answers we're going to need some competition between positive and negative feedbacks and the earth has such competitions and the simplest and most classic example of a positive feedback in the earth system is the so-called ice albedo feedback and the ice albedo feedback in essence is very simple to understand it's that ice and snow tend to exist in places where it's cold and they tend to be bright and so ice and snow covered surfaces reflect a lot of sunlight and so they tend to remain cold because they're bright so here's a very simple geometrical argument that will help us think about why and how earth could slip into this snowball earth state completely covered by ice and snow so what I'm sketching here is half of a planet okay with a distribution of sunlight there's more sunlight at the equator than at the pole and it's coldest at the pole so if I have part of the planet covered by ice and snow it's a cap of ice that looks something like this we can in very simple cartoon sense talk about the difference in albedo between the ice covered and non ice covered region okay so I'm gonna ask this this stability question I'm gonna pause it that I have some some equilibrium state that looks like this and I'm gonna say something comes along that cools the earth down okay so let's say for the sake of argument we have a uniform cooling I'm gonna call it delta t we cool the earth down a bit because we've cooled the earth down a bit the ice and snow grows a little bit so the line here I'm calling phi sub i which is a latitude of the edge of this ice this ice edge will move toward the equator then I ask is that a stable or unstable perturbation and the simple way to think about the answer to that question is just to ask whether the the two competing tendencies which ones wins out so because the earth is trying to maintain energy balance and it's radiating away to space at something something like sigma t to the 4 if I if I linearize that and express the anomalous warming tendency from the fact that I've cooled the planet down this number b here is just it's a number let's say in watts per meter squared per degree that sort of measures the strength of that restoring force from emission to space so there's a warming tendency that's going to tend to return me back to my my original state but at the same time I introduce this cooling tendency from the fact that I've expanded the reach of the ice and snow so what I've written down here is just basically this extra slice of ice and snow that is now brighter than it was okay so I have my little delta albedo here factor and I have a latitude dependence which expresses two things one the fact that we live on a sphere so the the size of this slice gets larger as I move toward the equator I also have latitude dependence through the sunlight which is this s term right it's larger here and smaller here okay so simple argument the perturbation is stable if the cooling tendency is larger than the warming tendency right okay so I can go a little bit further and say I can relate the delta t the overall global cooling here to the local temperature gradient at the surface of the earth at the location where the ice edge is sitting so right here if I measure the temperature gradient as a function of latitude just sort of Taylor series expansion I can write this like this and so I can then write down a quantitative statement here for when my stability condition is met so the ice edge is going to be stable if the temperature gradient here is larger than some factor that depends on the albedo difference between the ice covered and not ice covered region and depends on how much sunlight I get and depends on this geometric factor so both of these terms here grow as I move toward the equator that's the key point we live on a sphere so we can't get away from the fact that this term gets larger as we move toward the equator this term the temperature gradient becomes close to zero at the equator for a lot of reasons but the most simple one is that it reflects the distribution of sunlight and the gradient of sunlight because again we live on a sphere is zero here and gets largest in the mid latitudes okay so this term is big in mid latitudes it goes to zero at the equator this is large at the equator so it's basically an unavoidable consequence of living on a sphere that's differentially heated by the sun that at some point if there's any albedo difference between the cold and the warm regions that this is going to become unstable and once it becomes unstable then the positive feedback from the albedo takes over and the ice runs away to the equator okay so that's the phenomenon that we call large ice cap instability and I'm trying to convince you with these couple slides that we find it in every climate model that has any representation of albedo feedback because it's unavoidable because we live on a sphere okay so I can quantify this a little bit more and it's worth doing well I think it's worth doing so I'm going to do it I'm going to elaborate that argument just a little bit and introduce what gets called the buddhiko cellars energy balance model so it's a one-equation model okay the it's an equation to solve for the temperature that's t here the surface temperature is a function of latitude so so we're going to try to think about t being the zonal average temperature we're going to solve for the temperature gradient from equator to pole okay so this is really just an expression of an energy budget for slices at each latitude I have a term that represents absorption of sunlight here's my albedo here here's you know s is still that distribution of sunlight here's a linear expression for the emission to space goes up with temperature okay you can think of it as sigma t to the four think of it as a linearization of sigma t to the four but with numbers that account for the presence of a greenhouse atmosphere and we have to introduce one more piece of physics here because we've introduced a spatial dimension we can't avoid thinking about the role of dynamics in communicating energy from one latitude band to another and one very simple way to do it which is often invoked is to introduce a parameterization like this okay this basically this is a heat equation right we we represent the flux of energy from warm regions to cold regions as a diffusion process okay so this this gradient operator is just saying where the temperature gradient is steep the flux is large and there's a number here I've called k that just measures the efficiency of all the motions in the system at stirring the system up okay and that's a number we're going to have to in some way tune empirically because this is a parameterization of all of the well all the dynamics okay we can write this in a time-dependent form right if we want to do a seasonal problem we have a seasonal storage here I'm going to talk about steady state solutions to this so we're going to set this term to zero and we're going to solve it but first let's talk about the albedo this is where we bring the idea of ice and snow into the model and we do it in let's say the dumbest and simplest way we can which is a great place to start we're going to say where the temperature is cold the albedo is high where the temperature is warm the albedo is low so it's a step function this is exactly the kind of thing one could simply or easily code into the ISCA model and get and I hope I inspire some people to do that today from some results I'm going to show today that it's it's worth thinking about so we're going to look at steady state solutions of this one equation model it's now a non-linear equation because we've introduced this temperature dependence into the albedo it turns out this is not important for this lecture but it turns out that although it's a non-linear partial differential equation it's still analytically solvable by basically solving on the warm side and solving on the cold side and matching in the middle because it's analytically solvable we can um get all kinds of glorious detail so the model then solves for these are somewhat complicated figures so let me step you through it here's a hemisphere here's the equator here's the pole what we're solving for is temperature as a function of latitude right this is degrees celsius we have an equator here at 30 degrees we have a pole sitting near minus 20 degrees um we've uh encoded this this ice threshold temperature into the model so this schematic indicates the size of the ice cap in this particular equilibrium solution of the model the other thing it tells us about is equator to pole heat transport that's the other thing that's sketched here um in and on in this axis in petawatts so if you remember john marshall's talk yesterday showed you many many curves of pole to equator heat transport curves that have essentially this kind of shape okay it doesn't take much to get this kind of shape from a model you basically just need the shape of the sphere and some differential heating but um what's actually let me skip ahead to the next slide i think it's easier to see this way the really interesting thing that we get from these models is multiple solutions and that's why we're talking about it here so what this graph shows it's in a non-dimensional uh it's expressed non-dimensionally here but this we can think of this as the solar constant or equivalently the amount of greenhouse gases in the atmosphere we're mapping out in a graph like this where is the ice edge okay this axis is latitude and so this is a branch of solutions of the model that are warm enough that they're completely free of ice and snow this is a branch of solutions that are so cold that they're completely frozen over okay so we can find multiple states by just drawing a straight line through this graph okay and we get these beautiful smooth curves because we have an analytical model but i'm going to refer to this as a as a as a bifurcation diagram or a hysteresis curve or something like that i'll use those terms kind of interchangeably what i mean by that is and i'm going to demonstrate if this little animation works let's imagine we cool our warm planet down gradually this this little funny little star here traces out the time history of the climate of our planet if we try to cool it down and warm it up we cool it down we reach a threshold where it gets glaciated and then the ice grows gradually until it hits a point where we have a bifurcation and it grows suddenly into the snowball state and if we try to warm it back up we have to warm it up a whole lot before we can return to the climate we started from so that's that's the classic notion of a hysteresis in a in a by stable system okay so this was the point this is the point here that we called large ice cap instability that i argued is a inevitable consequence of living on a sphere and here it is captured uh quantitatively in the this one-dimensional model okay great what else do we want to say i did that okay we have to see it go one more time okay so because we have just a few parameters in this model there's a couple things we can learn just by thinking about parameter variations in this same solution and what i'm showing you here i'm i'm uh playing with a parameter that controls that contrast in albedo between the bright icy part of the planet and the not icy part right so on this side of the graph the albedo feedback is strong because uh the bright parts are really bright relative to the dark parts and here we're on the other side so we can think for example of a planet that's very cloudy okay if a planet is very cloudy in both the ice covered and not ice covered regions then as far as the planetary energy budget whether there's ice and snow on the surface is not so important because the clouds are doing a lot of reflecting regardless so we might be more in this regime okay and what happens here is you see these curves sort of tilt over in different directions here on the strong feedback side of the of the graph this branch of solutions that represents stable ice caps right partial ice cover on the planet that is happy to remain the way it is basically goes away okay because this is tilted over almost entirely in this direction if i put my funny little star here it would go whoop whoop whoop so in other words the planet if it were subject to some changing boundary conditions would flip between warm and cold and it would not find any stable states in between okay does that make some sense all right we can play similar games with the other parameter that controls that mixing okay this is a diffusive model so we're representing all the dynamics through this mixing coefficient and i've color coded these here so that blue is very effective mixing and yellow is very weak mixing and similarly to that albedo feedback parameter if we get into large mixing or efficient mixing we get very unstable right this blue curve is always bent over in the unstable so there's no stable partial ice cover solutions in a very efficiently mixed system and think of that in terms of if we stir up the planet well enough it's isothermal so it either has to be above or below the temperature that permits ice okay so that's that's nice there's interesting things to think about there um and the i'm drawing these pictures from a recent paper where we actually compared this to what happens at high obliquity where the equator to pull temperature gradient reverses and instead of thinking about ice caps we're thinking about ice belts around the equator i'm not going to talk about that today but that's an application of this very simple model so to summarize here okay in this in this so-called buddhiko cellar's energy balance model what do we get well we get multiple states okay we get both stable and unstable um states um there's a there's a point at which we can't have stable ice edges any farther than a certain point um here's a critical conclusion from this model that we're gonna challenge as we go forward today is that in no case do we have any more than one stable solution with a finite amount of ice let me go back to these pictures for a moment to emphasize that right we can we can draw straight lines through these graphs and in some cases intersect the graph in several places right here we can integrate intersect the graph in five different places this region is unstable this region is unstable that's stable that's stable that's stable okay there's only ever one stable solution in any of these graphs that has a finite amount of ice okay so remember that okay so in other words we can't flip between a little bit of ice and a lot of ice but not fully ice covered okay all right that's what we want to remember okay so what's missing I mean John Marshall set this up very well the yesterday the thing that's missing from this very simple model is any description of the ocean and some might say well you know that parameter that k that represents the stirring of mixing of heat from where it's warm to where it's cold does include the ocean right the ocean contributes to the poleward transport of energy so it's wrapped up somehow in that k but I want to argue in various ways that that's that's a bad parameterization of how the ocean works and one fundamental reason for that is that the ocean is basically a mechanical system it's driven by winds and so it's not always sort of enforcing a down gradient kind of mixing process okay what I'm showing you here is you've seen curves like this in John's lecture yesterday but here are some observational estimates and some model results for poleward heat transport in a present daylight climate or now I'm showing you both hemispheres here's the total amount of energy carried poleward it has this beautiful smooth shape that peaks somewhere just poleward of 30 degrees latitude and the partition between the atmosphere and the ocean okay so as we saw yesterday the oceans are doing a lot of work to move heat out of the deep tropics there's disagreement both in observations and in models about the shape but of this curve the details of the partition are not 100 sorted out but we can all agree that the blue curves are small as we were in the mid to high latitudes relative to the red curves and so we could say off the bat we should assume that the ocean dynamics don't matter that much right for this problem that seems to involve transport into the high latitudes and what sets the temperature of the polar regions and whether we have ice or snow your intuition might be maybe ought to be that the blue curve is irrelevant because it's small okay you're happy with that should I stop oh okay so why do ocean dynamics matter okay I'm gonna just I've tried to put a lot of pretty pictures and movies in today's talk because I know it's the afternoon and if you're like me you're probably feeling sleepy but this is just an animation of a simulation with a fancy kitchen sink climate model relatively high resolution bells and whistles this is just a kind of control simulation of a present-day like climate and you're just seeing the seasonal cycle slosh back and forth and you're seeing the snow and ice advance and retreat seasonally I've got these polar views here that illustrate where the sea ice is let's look at that one more time the colors in these bottom panels actually indicate well they're what we would call the q flux okay it's the it's the monthly seasonally varying flux of heat into the top most mixed layer of the ocean the net effect of all the dynamics of the ocean okay both horizontal and vertical mixing processes the ocean dynamics as they're revealed in kind of the surface energy budget and the reason I'm showing these is to sort of draw our attention to the fact that we have we tend to have these big red features right at the edge of where the sea ice sits okay and it's no no no coincidence okay that the ocean is pumping a lot of heat out to the atmosphere right at the edge of where the sea ice is permitted to exist and I'm hope to convince you in various ways that it is no coincidence over the course of the next hour or so so what I want to do here is answer the question of do the ocean dynamics matter at all in my favorite way as a climate modeler which is to take them away right and and there's a there's a straightforward and sort of logical way to do that and that's to run the model in slab ocean configuration where I don't change anything about the rest of the model physics it's the same atmosphere it's the same sea ice model it's the same land surface model etc I just replace the ocean with 50 meters of water and I put no correction in it so the ocean is just a source of water it's a source of heat capacity but otherwise the model is free to find its own climate okay no q flux okay because I'm I want because if I put the q flux in that's actually what we're just seeing here as fact this this is a slab ocean simulation with a q flux diagnosed from a fully coupled simulation so it has the same climate as the fully coupled simulation right so what happens so I replace a full three-dimensional ocean with a pure slab so what's that permanent El Nino so remember that you know if I flip back a slide what are the oceans actually doing in fact I didn't point out this picture here this is kind of a spatially resolved map of the annual average flux of heat in and out of the mixed layer so if you like this is the this is the annual q flux that I would need to apply to an ocean model to get the right annual mean climate where it's blue here those are regions where the ocean is removing a lot of heat from the surface okay essentially that's that's we're seeing all this upwelling of cold water to the surface which then leads to this big net surface heat flux into the ocean in the deep tropics and where it's red in this picture those are regions where heat is coming out of the ocean it balances if you average over the planet it comes out to zero it's a it's a modeling equilibrium so if I take this away I'm taking away a big cooling signal in the deep tropics and I'm taking away a warming signal in the high latitudes so I don't know maybe we'll get a permanent El Nino let's find out so this is just now a free running simulation with a real swamp ocean turning into permanent El Nino so the gray and the white here is the sea ice okay so yeah in fact we have this tongue of sea ice extending right up to the equator in the in the eastern Pacific there's a whole lot of interesting stuff going on here but in broad brush what's happening is the planet got a whole lot colder it got a whole lot colder because the ice was allowed to expand well I told you yesterday I don't know no that's no it's a very good question because the simple theory that I presented at the beginning would suggest that once we allow this to happen it should be we should hit that unstable point and it should run away it hasn't it might be that if you run this out for longer you might trip it's hard to tell whether you're close to that instability or not and these are very expensive simulations to do so but to the best I've been able to gather it does sort of find an equilibrium that's summarized on the next page it's 24 degrees colder than present-day earth okay so the effect of ocean circulation today is a 24 degree warming according to this this is the latest and greatest end-carb model okay so the oceans are warming the planet by 24 degrees we call this the the ELSA world and this the oh did I get that backward wait that's Anna right my kids love the movie frozen so they gave me this terminology this is the ELSA world here it's very cold okay I'm sorry I didn't understand the question there's no ocean circulation so so it's a pure slab so the motionless 50 meter slab that's all we've got here one is q flux corrected to basically reproduce the coupled control simulation yes yes okay well the movie we just watched was about 60 years so something like that yeah okay so why doesn't your simple the first problem you presented of the large isotope instability so something is coming into play here that stabilizes this I I think it has something to do with something perverse going on with tropical clouds but I haven't dug deep enough in to figure it out you know the argument that I presented would say that you can't get away from the instability but I don't have a quantity of a fully robust quantitative prediction for where that instability lies it always depends on the details of what sets the albedo contrast and there's something going on here whether we believe it or not there's a yeah yes yeah I can't rule that out right so so the idea is when we're exploring these nonlinear phenomena and numerical models sometimes we just need a lot of patience and luck because um adjustments can take a long time and with an expensive model like this we we can only run it so long and then we do our best to say it's close to equilibrium and that's all I can do but it's impossible basically to rule out that it will suddenly jump into the snowball yes that's what it would tell you but it's the simple model is is predicated on this constant albedo contrast between the two regions right and that kind of goes out the window quantitatively in a model where the albedo is an interactive function of complicated cloud structures that are changing as the climate is changing so so we have to take that as a sort of qualitative prediction that there is an instability somewhere we just haven't reached it here okay so this is really interesting and fun I it's not really the main meat of my presentation today but it is my main argument for why it's important to think about the oceans okay it's through their interaction with sea ice that the heat transport by the ocean has a very strong effect on the climate that we live with okay and we find that out by taking it away okay so where does the simple model fail to account for that I said that oceans are not well represented by k times temperature gradients and the basic argument is that when I turned up k in the simple model and I showed you those curves tilting over transport is destabilizing the climate and the way to think about that is that the transport is trying to homogenize temperatures and so it's sharing energy from where it's warm to where it's cold so it's moving energy over the ice edge but that's not what the oceans do okay and they they can't do that essentially because the ice is like a layer of styrofoam sitting on the sea surface it's an insulator and so in something close to an equilibrium state I'm never going to find that the ocean is carrying a great deal of energy under the ice because it has nowhere to go so somewhere or other the circulation has to arrange itself so that the all the heat that's being carried forward by the ocean finds its way out of the ocean somewhere near the edge of the ice and we're going to see in various ways models kind of do that okay but we have to think about the structure of that transport the spatial structure of where and how the ocean is moving heat toward the ice edge so I'll take you quickly through work I did as part of my phd where we wanted to address this issue how how what's the simplest way to extend that buddhiko cellar's energy balance model to inform it about the spatial structure of ocean heat transport so we kind of build a model that looks like this right instead of a single slab with heat going this way we put two slabs in one that represents ocean and one atmosphere and we build in this insulating argument and we say that the model is constrained that the transport has to go to zero at the edge of the ice okay and so whatever equilibrium we find is going to be one that sort of respects the fact that heat can't pass readily through ice and then we come up with a simple parameterization we think about a planet where the ocean heat transport is dominated by wind driven gyres and that leads us through arguments I won't go into here to another sort of diffusion equation but one in which our k for the ocean is now a function of the wind stress curl the wind stress curl through Sverdrup theory tells us tells us how strong the mass transport in a gyre is okay so if we know the wind we can build a parameterization that looks something like this so how do we know about the wind and this is where we build this what you could think of as the the simplest coupled model in which the atmosphere and ocean are coupled energetically but also through angular momentum and so we're going to kind of marry these two points of view and the way we do that actually follows some old work by green and others um where we think about mixing of of potential vorticity in the atmosphere and again I'm not going to go through the details and I'm just going to give you the flavor of it but we mix potential vorticity as a diffusive process and in doing so we constrain the angular momentum budget and a product of that is that we get this surface wind stress that we can feed into our ocean model so again just a flavor of what we're talking about and this actually alludes to things that uh professor keng was talking about this morning that that uh if we do a column integrated budget of momentum for the atmosphere then the surface stress this thing the ocean is feeling is is an integral of the the the convergence of momentum transport in the atmosphere so if we have westerly stress at the surface blowing on the ocean that has to be supplied somehow through eddies in the atmosphere go into quasi-geostrophic theory that just turns into a flux of potential vorticity and here's where we follow green and we say well we can use a kind of mixing length argument to think about this as a diffusive process moving pv from where it's large to where it's small okay so we do that we get a model that in schematic form looks like this i'm not going to step you through any equations here but just to say the reason i'm presenting this is because what pops out of this is something qualitatively different than what we've seen previously so the model solves for pv in two layers and so that kind of accounts simultaneously for momentum and energy transport so we call it the energy momentum balance model which is kind of an awkward word but um that's what it is and the solution kind of gives us this wind stress that's westerly here in our mid latitudes and we get easterly trade winds that come up as a natural consequence of diffusing pv on the differentially heated sphere kind of neat so we plug this wind stress into our gyre model and we let the ocean and the sea ice talk to each other we let the sea ice talk to the energy budget that the atmosphere is feeling and so on so it's fully coupled what happens well so now i've kind of rotated things around here's the pole okay here's 30 degrees so the equator's over here the two colors are just different parameter sets not to worry about that the point is that we get multiple states that pop out of this model and that's why i've gone through this long song and dance what i'm showing you here is a solar constant so again we're thinking of if we start warm for example we start with a hot planet and for some reason it cooled down because the sun dimmed or the greenhouse gas amounts were reduced what would happen well we would get some ice and the ice would follow this path we ice caps would grow and grow and grow and keep growing keep growing and i've cut this graph off at the point where the large ice cap instability pops up if i continue drawing this it would go well it would go up down to sorry where would we go we would jump to zero okay jump to the equator okay so there's an unstable branch of this curve that would look something like that that i didn't plot okay but this gives us something different that we didn't have before right there are if i go back the other way if i warm up i actually follow this branch so what i've pointed out with the one and two are multiple states predicted by this model in addition to the snowball which is not on the graph two possibilities for how big the ice cap is okay remember that's what i said was the important thing to remember from the simple model was that it can only ever give you one state one site one state with partial ice cover okay there's only one size of ice cap that's ever possible given external conditions this model has two so what does that mean what do they look like i've just sketched out again these are very busy graphs but let me walk you through part of it here's the equator here's the pole okay here's the temperature of the surface in the red solid curve so warm equator here's an ice edge a cold ice top okay here's the edge of the ice sitting here somewhere around the polar region 70 some degrees okay so this is a sort of present day like climate this is the wind stress actually that we get in this solution and here's a partition of heat transport total atmosphere and here's the heat transport by the ocean okay so this model because it's representing gyres it knows about the curl of this stress and it knows that here at this point there's a zero wind stress curl line so we basically have transport in a subtropical gyre and we have transport in a subpolar gyre okay and it finds an equilibrium that looks something like present day but then there's this other equally valid solution for the same parameters that looks quite different because the ice comes down into the mid latitudes and the entire subpolar gyre here is frozen over okay so it finds an equilibrium where we have substantial heat transport in the subtropics in the ocean that comes into the mid latitudes dumps all its heat out to the atmosphere and the ice edge sits here and the whole planet is a lot colder see we've we've the tropics are now you know substantially colder as they were here and so on okay so multiple states we get two sizes of ice caps that's kind of the the big prize here is that by thinking about the spatial structure of how the ocean moves heat around something qualitatively different emerges from these toy models okay then we can flesh that idea out a little bit by doing things like say I imagine that over some long time scale there's an oscillation right I could have greenhouse gases coming up and down for example I impose a long sinusoidal forcing trying to warm up the planet and cool down the planet what this model predicts is actually not a smooth sort of linear response gradual warming and cooling actually gives us this sawtooth okay so warming is abrupt cooling is gradual it's kind of interesting because that's a feature that I said at the beginning shows up in various kinds of paleo records okay and the abrupt warming is that subpolar gyre abruptly melting away okay all right well that would be an interesting an interesting exercise in toy modeling and no more than that if I had no more evidence to present to you that this was a useful line of a useful way to think about the problem but as you've already seen from John Marshall yesterday we've actually found multiple states that have something to do with these predictions from this toy model in much more complex models and so we're going to look at those and sort of start to build this hierarchy okay it turns out that the insight from the simple model actually was very useful in sort of shaping our ideas of what to look for in the more complex model so I'm going to go to this numerical coupled atmosphere ocean sea ice GCM in these very idealized configurations that you heard about yesterday and I'm specifically here going to be looking at a pure aqua planet it's just 100 ocean no basins so zonal circulation as we heard about yesterday and the ridge world has this one stick continent that goes pole to pole the ridge world has giant gyres right it has a basin that the ocean can lean up against the side of the continent and so spheridic balance can exist and it can circulate in in the form of gyres okay so we saw some of these results yesterday and here's the sort of deterministic story which turns out to be incomplete in an important way that we can take the aqua planet and find oh it has ice caps right you saw I'm going to skip over this quickly because we saw yesterday we can look at the partition of heat transport in the aqua planet and say well the aqua planet doesn't have efficient ways of getting heat to the polar regions that's why the ice caps were allowed to grow and if we put the ridge in then suddenly we have these subpolar gyres which do provide this conduit for energy to get into the high latitudes and it's warm as a result okay but they're not unique and so the deterministic story is a nice story but it is a little bit misleading so we'll talk about the details of the model offline for those who are interested but let's look at this uh let's look at this animation so what i'm going to do now this is you're going to look at results from a simulation now with this ridge world model that's going to start in a warm climate and this graph down here is temperature surface temperature in this ridge world on this color scale in degrees celsius and this is sea ice here there is no sea ice at time zero okay and we're going to see some cross sections here of half of the ocean this is the equator in the pole and the experiment here is to do sort of like i was describing in the toy model is to turn down the sun gradually and then brighten it back up so i'm going to try to cool down the planet by turning down the sun and then warm it back up so when it gets back here at 4 000 years the parameter will be back to where it started so we're warm and we're going to try to cool it down takes a while to really get going but okay so we've undergone a very dramatic climate change and at 4 000 years the climate is cold and the ice cap is large right we warm it up again eventually we melt the ice i think this will loop for us if it doesn't i'll start it again okay so let's see it again and the the forcing here is deliberately slow and long enough that we can think of of it being well separated from most of the internal time scales of the system so um yeah so i have some plots later in the talk that will be better to look at to talk about that question but yeah it's it's a small four it's a it's a it's a it's a large when we compare it to well let's let's call this a couple of doublings of co2 something like that yeah so we see salinity and and and yes as the ice advances we get this salt stratification that shows up so we get these layers of of of relatively cold fresh water that's buoyant under the ice so there's all kinds of interesting dynamics going on the contours here are the overturning okay so this is well it's a residual overturning but it tells us something about the strength of the the mass flux in the ocean these are temperature contours of course in the in the gray yeah i think i have some plots that address your question pretty directly um this is just barely long enough that it's pretty well separated from most of the internal dynamics is i think the right answer um yeah that was the challenge here was it was deliberately chosen to be long but of course constrained because these are pretty lengthy calculations to do even with a relatively coarse resolution gcm like this doing 8 000 years of simulation takes a while okay all right so my reason for showing you this is well just because it's a neat animation and it has a lot of food for thought but it's a basic demonstration that the model is by stable right because when i get back to 4 000 years i've gone to a completely different climate so i'm gonna not going to say too more about that i think that there's david ferrer i will give a lecture tomorrow where he's going to say more about related work and he's going to go into more detail about the dynamics um this these results were written up in a paper involving all three of us a few years ago and um the basic cartoon idea here we have two stable states of the model and much like i was describing in the toy model that i went through earlier we can describe our cold state here as one in which the ice cap sort of advances right to the point where all the heat has been dumped out of the ocean and the reason that the ocean heat transport sort of looks the way it does is that it's strongly constrained by the winds and this cartoon is a cartoon of an overturning circulation that's essentially forced by trade winds blowing over the subtropical oceans that are pushing this warm water toward the poles and being subducted and returning and this shape is basically very hard to get away from when we have winds blowing over an ocean and so it shows up in similar ways actually in both the ridge world and the aqua planet we get essentially the qualitatively the same behavior so cold state has a big ice cap that extends to the point where this transport drops to zero okay all right i think what's next oh we're not going to do that section okay well here's my argument in in cartoon equation form and all the ingredients for multiple states are we need to have spatial structure in the ocean heat transport we need to acknowledge the fact that the ice is an insulator at the sea surface and so the ocean can't be carrying heat under this ice cap and we need wind to give us this spatial structure we put those ingredients together in both this toy model and a complex numerical model and we get these multiple states okay so let's take a five minute break and then i'm going to come back and tell you is that all right we take five minutes and then i'm going to i'm going to take apart the model a little bit in order to sort of understand a bit more this interaction between the ocean heat transport the ice and the global climate okay