 I'm being told that I can start again so blame the organizers Okay, so then some evidence that we are not totally off by thinking about monsoons through This convective cause equilibrium framework This is from Booz and Hurley 2013 it shows from reanalysis the July Climatology of the upper tropospheric temperatures here in the line contours and the more static energy in the boundary or near the surface at least for the South Asian monsoon this coupling between high more static energy content and where maximum in upper tropospheric precipitation is well verified, of course Let me hear it. Let me try to emphasize here. Of course in the boundary layer. There is much more structure In the more static energy than there is in the free upper troubles here again because in the tropics You cannot sustain strong temperature contrasts this coupling between regions of more static energy and High more static energy and higher or maximum upper tropospheric temperature in the North American monsoon is clearly not verified After I studied the North American monsoon as part of my PhD thesis I'm still doing a lot of work and my advisor Bjorn Stevens Used to make the joke that I spent half of my career tried to convince him that the North American Monsoon is a monsoon and then the rest of my career convincing him that in fact, it's not a real monsoon I leave it at that So how about other timescales? So for instance, let's look at internal timescales especially for the focusing on the South Asian Monsoon region. I don't have time to go into the details But again to reflect this new way of thinking about monsoons We have spent quite some time thinking about what are appropriate indices of monsoon strength And so this is work from Jennifer Walker student of mine who graduated last year where we looked at the internal variability of the South Asian monsoon and we used as a measure of the interannual strength of the monsoonal circulation the overall moisture flux converging in this big box estimated from the reanalysis and then the the motivation being that the moisture budget tells us that if we can neglect transient storage effect The net precipitation is equal to the let me do it this way the vertically the convergence of the vertically integrated Horizontal moisture flux convergence. So this quantity is strongly related to things that we care about P-C net precipitation in reanalysis Q is probably not very reliable But we showed that an interannual timescales fluctuations in the moisture flux are primarily due to fluctuations in winds And so we said well probably this is reliable large-scale winds are something that reanalysis are able to do So again interannual fluctuations in things that we care about in monsoon regions Which are not the winds but their precipitation are related to a quantity that is strongly constrained by the winds that we understand better Then we understand moisture Then based on this index we started looking at regressions of all possible fields. This is what the temperature near the surface look like looks like During the summer monsoon season on average. This is climatology again higher temperatures But then if you look at regressions and these you can think as being your typical Temperature anomaly near the surface associated with one standard deviation and monsoon that is stronger by one standard Deviation then you find that in fact strong monsoon years are characterized by colder temperature Over the land and warmer temperature over the ocean So if now you look at the overall near surface temperature gradient the blue is the one Strong monsoon years the red is the one in weak monsoon years strong monsoon years are characterized by your weaker Not a stronger meridional temperature gradient again If you take the meridional temperature gradient as an external forcing Driving the strength of the monsoon of flow Based on that temperature gradient you would say ha that here is a weaker monsoon year In fact, it's a stronger if you look at moist that you can't in fact you find the opposite Moistatic energy gradients are stronger in strong monsoon years, and they're all driven by positive anomalies In the near surface moisture. So moisture is at least as important if not more important in terms of its spatial gradients Then our temperatures so this is summer streaming Is because of course then once the monsoon gets absolutely and that is something Absolutely, but let me emphasize how this dashed line Represent the same meridional temperature gradient now average three months before monsoon onset again for a stronger Monsoon year and for a weaker monsoon year in fact even in the preceding winter a spring It's not clear that the temperature contrast is strongest stronger in a strong monsoon year But again that is to emphasize how and I think that this is accepted right the fact that of course you have lots Of feedbacks clouds evaporation from the surface one precipitation gets going but this is to emphasize there is still plenty of statements in the literature that one robust projection In climate models due to for instance W O CO2 is that land will warm up more than the ocean and that is not just a transient response That is an equilibrium response hence the monsoon of circulation will strengthen and in fact I hope that I'll get there But I'll show you results from models where precipitation still go up still goes up, right? We are in a moisture world, so precipitation will go up, but because moisture goes up at a very fast pace seven cal seven percent per degree of warming In fact because we have increased ability the circulation can weaken So it's not a baneer surface temperature gradient that drives the strength of the monsoon again we should go beyond this thinking as I Temperature gradients being an external forcing to the monsoon the extra the temperature gradients are influenced are affected by what the monsoon is doing Again a convictively couple view of the circulation also true one interest is on our time scales If you're interested in how here we define the timing of the onset of a monsoon now looking at the seasonal transitions Come to my port store next week. This is an off molar plot again We are averaging in longitude over the monsoon sector Lachitude versus time relative to the day of onset for each Individual season. This is what the temperature looks like again There's a large anomalies in temperatures at monsoon on set are clearly coupled to changes in here use the equivalent Potential temperature in the boundary layer. It's the same thing more or less as the Moisturic energy the two the two are strongly strongly coupled if we now look at the distribution of Near-surface temperature near-surface moistatic energy at day at the day of monsoon onset again This is an average or several many monsoon seasons again We see clearly see strong veridional temperature and gradients in both But if we now look at the change in temperature and change in moistatic energy In the 15 days following monsoon onset, we see that moistatic energy increases The temperature decreases at monsoon onset again this view that the Near-surface temperature drives the monsoon of flow is clearly flawed You have weaker Near-surface temperature gradients as the monsoon circulation is strongly intensifying Okay, so now the other part that I would like to discuss It's Mechanisms for rapid monsoon onset The rapidity or the abruptness of monsoon onset is something that I haven't discussed so far Let me show you another animation This is zooming over the South Asian monsoon region the vectors again. We started sorry I'll have to let you look through several times. We started January 1st The winds are the lower level winds the contours are precipitation these are long-term daily averages Again, let's start again notice or at the summer trade winds blowing from the northeast to the southwest precipitation over the Southern near equatorial ocean Okay, now we're going back to the cold season again noticing the cold season Precipitation likes to sit south of the equator notice how now around April things start to change and start to change very Rapid may start to change very rapidly notice that within two weeks You really see this strong reversal of the low level flow with the development is very strong cross a material jet called a Somali jet because it's costly trapped against Somalia and notice how this strong Development and reversal of the winds is accompanied by a really rapid shift in the precipitation patterns that basically jump really rapidly from the near equatorial ocean into the subcontinent of India nearby Solutions Again, there's there you can define non linearities in many in many ways But here it is clear the changes in the monsoon of circulation and Precipitation occur on time scales that are much more rapid that what can be interpreted just the linear response of very smoothly bearing installation forcing Another view of the rapid onset again here. I'm showing of molar of precipitation Starting in January 1st again only averaged during the Indian monsoon sector 6200 degrees east again notice how the beginning of the month in season really is accompanied by a rearrangement of this precipitation patterns primary convergence zone in the Cold season being south of the equator at the beginning of the warm season something is happening And this convergence on really shifts very rapidly into the tropical northern hemisphere Okay, so going back to this very strong statement that I really like to make that really the schematic from ruderman of Monson's a series circulations is not true Let me try to provide you more convincing evidence if what I showed you before is not enough And this is work that I've done already ten years ago. I'm still talking about it And it's the fact that monsoons can in fact be simulated either even over in aqua planet And I would like to emphasize that despite the fact that I really want to focus on theoretical framework So I really don't have time to develop my own interpretation of a hierarchical modeling approach I use it extensively as part of my research efforts So these results are based on a GCM that is idealized in many ways the physics that is represented It's very simple this model does not have cloud But really one very very important simplification is the fact that the atmospheric model is coupled to a completely uniform Slab ocean this means that my surface is covered by a completely uniform mix layer depth with a fixed Depth okay, it's just a slab of water the the ocean doesn't move We are only capturing the thermodynamic coupling we have at least the summer presentation of how the ocean transports energy by imposing a q-flux in the surface in the mix Layer as I'm the surface energy budget Okay, so can we simulate monsoons over such an aqua planet this is again the observations is a very similar plot to the one I showed you before again Rearrangement of the main convergence zones from near equatorial ocean into Subtropical land masses at the beginning of the monsoon Season so now what happens when we look at the simulated seasonal cycle of precipitation over at these aqua planet When we make that slope of ocean very very shallow and here what I mean by shallow I really mean shallow This is only a meter Actually in fact half a meter of water although a meter doesn't make much difference Of course, this is not a realistic depth for mixed layer depths in the oceans that are of course I mean the depth of the mixed layer in the ocean depends on where you are where When and through the seasonal cycle, but definitely it's around 20 25 and 50 at times So you can think of this being a planet that is covered by a swamp So it's completely saturated surface, but it's heat capacity is so small that this Boundary can adjust rapidly to the seasonally Imposed in solution forcing. Okay, so here I would like to argue that despite not having any lengthy contrast We can simulate monsoon like transitions in the precipitation patterns in the sense that in this simulation Tropical precipitation tends to be organized along subtropical conversion zones That shift quite rapidly from one summer hemisphere to the opposite summer if now you take the same model and Increase the mixed layer depth by an order of magnitude you make it 50 You see that tropical precipitation has a seasonal cycle this way more muted both in terms of intensity and in terms of location the Precipitation doesn't like to move much away from the equator when the mixed layer depth is large Whereas when the mixed layer depth is shallow you see much stronger seasonal migrations at latitudes that are not unlike to what you see in the observations in the Indian monsoon sector I know that thinking about an aqua planet is a little bit confusing. So now let me show you a snapshot From the model. This is really just a snapshot. It's a Five-day average the winds I mean that the vectors here represent the low-level winds This is what the precipitation distribution looks like notice that despite being an aqua planet at any instance There are zonal variations But these zonal variations average out when you take longer term averages because longitude doesn't have any meaning Okay, there is no difference in longitude because we have completely uniform aqua planet And this is again the observations from reanalysis for the month of July near surface winds precipitation patterns again Notice how even despite the fact that I don't have any subtropical land mass in this simulation I can get a complete reversal of the lower level flows that blows from the south Crosses the equator blows from the southwest And turns at subtropical latitudes very similar to what happens in the Indian monsoon region And again notice that even without accounting and let alone topography The convergence on can migrate at subtropical latitudes similar to what we see in Monson regions I have a lot but I don't have the line C contrast so again Let me maybe go back to this famous schematics that I take issues with and I I hope I'm don't sound too strong But again, and I also have to say that a lot of work previous work has tried to of course Exactly why is the monsoon onset rapid especially if you believe In this in this view, right? There's nothing non-linear, right? The lensy control builds up smoothly following the installation forcing So there must be some non-linearity that gives rise to this rapid Monson onset and what is this the relevant non-linearity has been discussed at length and let me say that in general though It is assumed that in any previous work that no matter what the non-linearity is the fact that you have a land mass that can warm up Faster than the surrounding ocean is implicit in all of these previous arguments. I don't have this Meridional gradient in that surface heat capacity the heat capacity is everywhere the same and I'll go back to the role That land because of course land I mean monsoons do develop over land So land must play a role and I'm hoping to get there in a minute Okay, so then how about the overturning circulation associated with this simulation again? Let me for the time being show you a snapshot in the middle of the summer for this aqua planet monsoon Forget the winds I'll get back to the winds in a minute This is the overturning circulation that develops ascending motion in the subtropics again cross equatorial flow You don't see any summer cell most of the precipitation is confined within the end Ascent of this cross equatorial Hadley cell here in the red I'm showing the more static energy in the boundary layer the blue is the precipitation again notice how well It's no surprise. We use we use a quasi convective a QE Convection scheme in this model, but notice how the poleward boundary of this circulation is indeed in agreement with the convective quasi equilibrium view of Monsons is collocated with Maximum in my static energy Located in the subtropics just the quaterward of which most of the precipitation is confined again. Let me emphasize this in this situation We do get a reversed Meridional my static energy or temperature gradient. We are an aqua planet. So temperature and moisture are Strongly coupled but again in general the existence of this reverse meridional temperature gradient with temperatures that are warmer As subtropical latitudes then at the equator is interpreted as really arising from again The fact that land in the subtropic can warm up relative to the ocean We don't have any of that imposed gradient in surface properties there And yet thanks in fact to also feedbacks with the circulation We do develop this reverse meridional temperature gradient There's also be a lot of discussion in the literature about the role that for instance elevated mountain ranges the dependent plateau plays In strengthening that gradient. We don't have any of that in this simulation and yet we do get a reverse meridional temperature gradient Okay, so then how do so what happens what makes this? What what what gives rise to is none? This very rapid monsoon transitions in this model again You flip from as Jeff was saying we really flip from one cross-sectorial circulation in one hemisphere to the opposite one So I apologize, but I'm gonna spend some time at the board to Guide you through what is the current thinking about the non-linearity is implicated in these rapid transitions in the shallow mixed layer depth But yeah, let me make a point because apart from all the math One thing that I want to emphasize is the fact that of course So in this view, what is it that we are trying to say? We're trying to say that you don't need the lensy contrast per se to drive monsoons But you need land in so far that it provides a boundary with a low enough Thermal inertia that it allows for rapid just adjustments through interactions with the circulation and of course surface fluxes of The lower level more aesthetic energy so that this maximum that controls where precipitation is located and where the polar boundary of The circulation is located can move to subtropical latitudes in a very rapid fashion. Okay Questions about that so then what drives the rapid development of a monsoon in these simulations, okay So this is where we go back to angular momentum budget And I'm not as organized as Jeff who has all his equations on the slides I also like to derive equations at the board because as Jeff said if you look them at slides You have no idea where they come from And I apologize because on top of having a strong Italian accent that becomes stronger when I get animated also have a terrible handwriting Okay, so then what drives those what is the nonlinearity in the simulation so Key to this is the angular momentum budget. I won't do it I'll just write it as normal momentum budget But as Jeff said, this is basically equivalent to the angular momentum budget in spherical coordinates. I'm lazy So I'm gonna do everything in local Cartesian coordinates. So don't have to worry about cosines and the likes Okay, so that's our momentum budget to make Jeff happy. I'm gonna use the coordinates I'm not gonna use pressure coordinates, but later on I'll use pressure coordinates But it's actually the same. Okay, so do you DT? plus affection you do the X Plus V do you do I? Plus W do you DZ? Minus FV Coriolis force on the meridional flow will have to be balanced by the pressure gradient the DP DX we are in the free troposphere. So we don't have any Discos effects. Okay, so now we're gonna do and I'm not gonna do all of the steps So we're gonna the following we're gonna do typical Reynolds the composition and averaging Okay, so every field like you for instance will be decomposed in Zonal and time means so the bar here represents both zonal and time mean plus Deviations from this my deviations now. I'm really thinking as the large-scale Extra tropical eddies that for instance we saw in the cloud field Okay, these are really the large-scale extra tropical eddies that are doing a lot of the momentum transport outside of the tropics As all Reynolds averaging of course then if for instance we take a u prime bar They will be this will be equal to zero because the bar is also zonal mean DDX of any Mean quantity will be equal to zero. Okay Then we're gonna use continuity Actually, we should use continuity before but it's fine continuity for both so non divergent field for both Mean and so then what I get If I then take another mean I get the following Equation for the evolution of the zonal means on a flow the u bar dt. Okay, so then I'll have the advection of The mean flow by the mean flow the u bar term is not there because any DDX goes away So I have the meridional Advection by the mean flow of mean zonal when I have the vertical advection w bar the u bar d z And then I have the fv bar term so far so good right nothing is too surprising But when of course, we are doing the averaging there are the nonlinear terms the terms that are associated with covariances between Basically the fluctuation terms in the nonlinear advective terms do not average it to zero I'm gonna take them on the left-hand side. There is no zonal pressure gradient and this will be equal to minus Ddy u prime v prime bar minus dd z U prime w prime bar. Okay So what do these terms represent these terms represent the convergence the minus sign here in the meridional and vertical direction of the zonal Momentum In the meridional and vertical direction. Okay, so this is again the meridional Westerly momentum convergence and this is the Vertical convergence of westerly momentum again these It really arises from covariances of fluctuations between eddies field any eddy field individually averages to zero But fluctuations do not average to zero and those again are associated with fluxes of westerly momentum by meridional fluctuations and Biotical fluctuations. Yes said again you where is the bar is over the fluctuations? Yes bar. Yes, let me write it Zonal and time average. Yes The what the No, the only thing that goes to zero is the ddx What is it that goes to zero? No, that is that the mean the mean can be slowly evolving is the u prime That one time average goes to zero. Okay, we're gonna do look at steady states But this is exactly telling you that apart from these terms that we understand advection by the mean and the Coriolis force in fact Convergence of momentum by large-scale eddies can give rise to a westerly tendency questions So again here the goal is really thinking about let me schematically represent again the Hadley cells in a very schematic way Put the equator here We're gonna try to understand The existence of upper level and the direction of upper level Meridional flow in the Hadley cells. Okay, and we also use some arguments to explain for instance the direction of The flow in the extra tropics at upper levels in the ferrel cells. Okay Okay, another way in which we can rewrite it is by noting that the relative vorticity the vertical component of the relative vorticity is dv dx minus the udy So when we take a zonal and time average this term goes to zero so I can notice how now we can rewrite the advection term in terms of a Meridional vorticity flux. Okay, and again, this is an equation that Jeff did show yesterday in his slides I think for statistically studied states So I think he put the du bar dt to zero, but then I have again du bar dt F plus z bar notice that this is the absolute vorticity the sum of the relative and the planetary vorticity Multiply by the mean Meridional flow I have the vertical advection that I will drop in a minute and then I have again the ddy u prime v prime bar minus d dz u prime w prime bar Okay Okay, let me say that we can interpret this as the Meridional absolute vorticity Can we say about the eddy momentum flux conversions and divergence patterns in the extra tropics and in the tropics? Turns out that this is really the most important term the Vertical term is really important only in the boundary layer So again, this represents how this large-scale eddies that are primarily generated in our through our clinical stability in mid latitudes are transporting angular momentum in this case the westerly momentum And these are things that In sick dick cover In his lecture, I'm gonna just draw a schematic at the end result. In fact, all of this is actually the Focus on a ten week class that I teach So I'm really trying to summarize in ten minutes what I Do in a much longer time But again, what actually so again, let me emphasize that where we have going back to this Equation that where we have a westerly convergence by the eddies where minus ddy You prime v prime s positive again. This is convergence of Westerly momentum by the extra tropical eddies Because again, they are depositing momentum in the region where there is not convergence They provide a westerly tendency on the means on a flow. Okay Corresponds to westerly tendency The opposite, of course, we do to where there is not divergence With the minus sign, sorry for the confusion Let me do this then So they don't confuse anybody Because the divergence is positive to find this way Because what appears there is the convergence the divergence Will be an easterly tendency on them in flow So then it turns out that this would that really behaves a large-scale Rosby waves a real alive Because of the existence of a meridional gradient in the absolute vorticity Are such that they actually tend and I don't know how to represent this there are very nice schematics in Jeff's books, but they are generated in the extra tropics as the start propagating both Forward and upward they actually tend to converge momentum In the region where they originate. Okay, so again, they propagate meridianally outside of the source region And it turns out that again in the source region. There is net convergence of westerly Where these waves break at either higher or lower latitudes there is net divergence of momentum Okay, so where this bar clinica is are generated you would expect a Westerly tendency on the mean flow and again This is really at the at the core of theories for the development of jets in extra Tropical atmospheres and one of the reason is really as in in sequels Was discussing how the lines of constant faces are Tilted as this wave propagated outside of the source region again here being where Our clinic instability will determine them their faces are tilted from the Northwest to the southeast north of the source region they are tilted in the opposite direction and so these results in negative Momentum so Wesley momentum transported southward north of the source region U prime v prime will be positive South of the source region again this results in net Convergence and that is I can't draw it But in sick did a very nice job But but basically you can look at the correlation between the u prime and v prime fluctuations and convince that this is the case Very important. Is there an explanation for this too? Yes, if you you that's in sick did provide It's a theoretical explanation. You can use it. You can actually there are different ways of trying to explain this Raspberry waves theory will will give this Yeah, yeah, right. I don't know if there is any intuitive would have to go through some derivations, but yes Yes, nothing is happening you need Will be zero is average to zero Yes Yeah That tilt here, okay, and I will show you plots of the fact that in fact we do see this How this okay, so where these Waves are generated the convergent momentum where they're dissipated they diverge Momentum these waves also tend to behave as barotropic rosby waves They like to propagate in regions of upper level westerlies and they again break and dissipate where they meet their critical Latitude for simplicity just where the u does not mean so long when it goes to zero, okay? Okay, great. So then now let's do some more simplifications to the zonal momentum budget We're gonna look at statistically study states We're gonna put as was suggested before the DU bar dt who I should not have if you can still see in my schematic of the hardly cell and The Pharrell cells what we're gonna assume is the fact that we're gonna be looking at Again, we're looking at the free troposphere. So we're gonna be looking. So this is the Pharrell. This is hardly This is hardly. This is the Pharrell. We're gonna be looking at regions of upper level flow in the upper branches of the Circulations where more or less the flow is primarily meridional So if you look at the stream function the stream function the lines of the stream function are parallel for instance pressure contours Which means that their vertical velocity is small and we can neglect this term, okay? so then again the Zonal momentum budget simply becomes I'm gonna change the sign a Balance between the meridional flux by possibly the mean meridional circulation of absolute vorticity And the adding momentum flux divergence ddy U prime V prime bar, okay? Okay, so let's go back and think about The held and how model that of the hardly circulation that Jeff discussed yesterday All those theories are based on Axi-symmetry what it means is that they completely neglect any fluctuations in longitudes in the held and how model There is no eddies whatsoever. Okay, so in axis in metric theories such as the The held and how I constructed the right-hand side is zero. Okay. We don't consider any It's really a two-dimensional theory of the hardly circulation so Angular momentum conservation is simply a statement that f plus e bar v bar is equal to zero So how can this be verified two ways? And the way we want is that to have So one is V is equal to zero right, but we want a model for the hardly cell So we want and a returning with V bar less different than zero, right? And so then the only way in which this angular momentum budget is verified is for the absolute vorticity Again in the upper branch of this hardly cell to be equal to zero So why this is this equivalent to a statement of conservation of angular momentum? It is because the absolute vorticity is proportional to the meridional gradient of the angular momentum So zero meridional gradients of angular momentum means that an air parcel that moves Along a streamline in the upper branch of the hardly cell is moving along contours of angular momentum Okay, is this clear so angular momentum as Jeff Described yesterday. This is the Popsicle at the angular momentum per unit mass around the earth spin axis Has a planetary component omega is squared cosine squared of theta plus a relative component U bar a cosine of theta without all these terms again if we go back to our sphere This is our planet. We're considered a ring of parcels again zone of symmetry at the latitude theta Their distance from the axis of rotation is a cosine of theta if we also make the thin shell approximation Any parcel in the atmosphere will be 10 kilometers from the surface is much smaller than the radius of our planet Okay, so this is the pin shell approximation So again in addition to the fact that of course this air parcel rotates together with the planet It might also have an additional component associated with the direction of the west women Okay, and again Jeff yesterday described the angular momentum conserving winds For instance for a hardly cell in which air rises right at the equator in a region of negligible Zonal winds those angular momentum conserving winds are equal to omega a sine square of theta divided by the cosine of theta and So they increase the angular momentum conserving winds increase with latitude and why is that the case? It's really a very simple concept of conservation of angular momentum and let's suppose we are in our parcel that starts here at the equator with zero So no mean wind as it moves Along latitudes its distance from the axis of rotation will decrease right To conserve angular momentum Because its distance from the axis of rotation is decreasing it will want to spin faster Exactly the same way in which a skater and Katrina knows that very well because she used to be a nice or maybe she's still a nice skater and a skater starts Spinning with her arms wide open as she brings her arms closer She spins faster because she's decreasing her moment of inertia And so she can spin she spins faster to conserve angular momentum That's exactly what is happening to this parcel that moving meridianally is getting closer to the axis of rotation and for its spinning faster Means acquire acquiring a westerly component relative to the solid body rotation Okay Okay, so in this And that is held and how it's really based on assuming that the angular momentum conserving winds that the winds in the upper Troubles here are angular momentum conserving and that is really from this basic angular momentum budget a requirement for angular momentum to be conserved And then other properties of the Hadley circulations follow through for instance assumption of conservation of energy within the cell and continuities of temperatures at the edges of the cell Okay, so The other thing that I want to maybe define is something that we it's a local rosby number a local rosby number row not defined as the Ratio change sign of the relative to the planetary vorticity Again that that means that the absolute vorticity can be rewritten as f that multiplies one minus this rosby number So notice that angular momentum conservation implies the limit Rosby number going to one Okay, this is a local rosby number because it has your traditional scaling of a rosby number right vorticity Relative vorticity divided by the planetary vorticity is local because it depends on the local value of f Okay, so large rosby numbers we are as probably would expect in the tropics This is not a limit that is going to be valid in the extra tropics. Okay. This is an all linear limit right in this the maintenance of this angular momentum budget the This term that basically represents the advection of relative momentum by the mean meridians circulation takes a dominant role Okay, this is a non-linear momentum Okay, so then Questions Okay, so then I don't know where to write anymore. Let's look at Another limit that we might think of again, let me rewrite the Zonal momentum budget as f that multiplies one minus rho not v bar Approximately balanced by the eddy momentum flux divergence ddy u prime v prime Okay, now let's suppose that we are in the mid-latitudes in the mid-latitudes. We know that there has been number oh Sorry, it's much less than one Right, it's really the fact that f Becomes important. So in the extra tropics This limit becomes f v bar Balanced by the eddy moment Okay, so notice how and I have a little bit of a hard time calling it literally a linear limit because of course you have To be able to say something about the eddy fluxes But to the extent that we know how to perma trice the as eddy fluxes maybe through downgradient diffusion although that might be Maybe debatable for momentum for a state momentum, but putting that aside basically this limit is a limit in which The strength of the circulation is really slaved by the eddy momentum flux divergence. Okay In the sense that any change in the eddy momentum flux Divergence will have to be balanced by changes in the mean meridiana circulation Possibly by shifts in the cell and the relevant value of f. So this is the limit that holds in the ferrule cells in the ferrule cells that are sitting in the Extra tropics. This is where we have net convergence of eddy momentum flux because these are our source regions For these eddies that are generated there So these eddies are converging westly momentum in these regions and so the westerly Acceleration due to the eddy momentum flux convergence there Needs to be balanced by a quitter or flow in both hemispheres. Okay, so easterly Acceleration by the Coriolis force on the equatoral flow At both levels in both hemispheres. Okay, so that is why in one of my first slides I said the ferrule cells are really eddy driven They exist because they are driven by eddy momentum flux convergence There Again for f positive you can balance the convergence you need be negative Equator workflow same thing for f negative southern hemisphere V needs to be positive also Equator workflow and this is what sustains this indirect cells. Okay, so then the Let's say that we understand these two limits fairly well, right? This limit is somewhat I don't want to say simple But again, we need to be able to say something about the eddies to say something about the strength of the circulation in this limit Well, if the circulation needs to exist All we need to know is that once it exists it will want to conserve angular momentum also notice that when the circulation is angular momentum conserving this The angular momentum budget does not provide any constraint on the strength of the cell, right? Once f plus z bar is equal to zero It's really a trivial balance zero equals zero. It doesn't tell you anything about the strength of the circulation We will have to use another balance to say something about the strength of the circulation That is for instance the energy balance. Okay, so an angular momentum conserving circulation is also strongly energetically constrained Okay, so then the question is where is the Hadley cell sitting in observations and the truth is that is Probably not in the regime of axisymmetric theories as We maybe Would have expected or hoped for oh I forgot to say something, but I returned to that in a minute and that is exactly from observations from reanalysis Eschematics showing the Seasonal evolution DJF JJA so these are the solstice seasons the equinox seasons The color again, this is the circulation color contours represent at the momentum flux divergence in warm colors Convergence in blue unfortunately. It's cut out here, but basically those are your regions of generation of of large-scale eddies The shading here represents regions where the rosby number is larger than 0.5 So notice that in the equinox season Rosby numbers tend to be small almost everywhere, right? Rosby numbers mall if anything is closer to the extra tropical limit than the axisymmetric limit, okay also notice how this Summer cells here. You don't see a summer cell these summer cells are really not angular momentum Conserving at all rosby numbers in the upper branches of the summer cells are really very very small maybe where These angular momentum conserving theories start being relevant He's in the strong cross equatorial Hadley cell especially the Sustitial Hadley cell especially for north and hemisphere summer and this is what Jeff yesterday tried schematically to say that Probably the angular momentum conserving limit is not Relevant that much for the annual mean for sure not in the summer cells More so in the cross equatorial winter cells and actually it's even more so in monsoonal circulation So I haven't figured out a good way to show this, but maybe one way that might be convincing is looking at the Streamlines of the circulation at upper levels In the monsoon region notice how this is a very strong broad Anticyclone F in the subtropics is not insignificant This is an anti-cyclonic relative vorticity that is part largely Capable to balance F So that you start approaching conservation of angular momentum Of course when you really go from axis symmetry to a three-dimensional circulation things are a little bit more complicated But there is evidence that in fact PD Tends to approach zero within the center of this anti-cyclone Okay, so can I just say one more thing and then we'll break So one thing that I forgot to mention and that is something that because it will be important for What I'll gonna be discussing later on again, let's go back to the cross equatorial Hadly-cell picture from Lyndsen and how this is also completely axis in metric But instead of considering annual forcing it considers a forcing that is off equatorial in this case The forcing maximizes as phi not Again, the picture that emerges is a stronger cross equatorial winter cell and a weaker summer cell In this solution notice how the poleward extent of this The dividing branch between the two cells is actually at a latitude larger than the forcing Despite the fact that the ascent tends to remain concentrated Closer to the forcing and it turns out that you need to this extra parameter to solve for to close your system Of equations so the important thing in this case again We also assume in this case that in the upper branches of the circulations angular momentum is conserved But now it's the angular momentum of air parcel originating from the surface Negligible winds goes up. So the angular momentum conserving winds for this cross equatorial Hadly-cell is equal to omega a Cosine squared of theta one. Sorry Lyndsen and how use phi I'm using theta to be consistent with Jeff's notation rest today minus the cosine squared of latitude divided by the cosine of Latitude so what does this distribution look like it is exactly this these are the angular momentum conserving winds for Cell whose phi one is right where the winds go to zero the important thing that I want to emphasize is the fact that for a cross equatorial Hadly-cell The zone of wind distribution is symmetric about the equator But the interesting things is that I start having upper level easterlies in a broad Latitude in Alban and in fact between phi one Theta one which is the latitude of the poleward boundary of the cross equatorial Hadly-cell and the minus theta one And through thermal wind balance you need to develop a reverse meridional temperature gradient To balance the easterly wind shear minimum at the equator Maximize the subtropics and exactly where the zone of wind goes to zero. I'll stop here I will gonna use these arguments to understand the aqua planet simulations and we're gonna be back at around 11 Certainly