Keynote talk: Wayne Schubert, Colorado State University

he had just finished his analysis of the eastern waves of the western Pacific, which he did with a triangle of stations. I remember we called it kep, yes and kwajalein triangle. I knew we

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ed and ponape or something like that. like this and made this triangle, he made a beautiful study of those composite eastern waves. Now I would like to go ahead and

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about this article that I found really interesting and this article is by Sato and Matsuno and in my time I didn't do it. t Sorry, I didn't put your name there, it was published in 2008 and the results from it come from his non-hydrostatic icosahedral atmospheric model.

I always say night camera, some people say Nicoma, which is true, your camera, does it matter? Yeah, so I was fascinated by this article and at the end they presented the results of this simulation of Aqua Planet at about three and a half kilometers and then they interpreted the results as Two Worlds, so they said there was a mid-latitude world here that was It is based on PV and is where the PV is materially. conserved, so they said the PV world is mid to high latitudes, it's a quasi-2d Vortex world and it's governed by PV conservation and then there's this other world and I think for a catchy name, they reversed the letters and called it the VP world which means velocity potential and this is the low latitude wave and convection world and this is a world where PV is not conserved due to convection and you can see schematically there they have some clouds and they have the top branch of the Hadley circulation there and I thought this was a really interesting world and the interesting way to look at the world and the way they explained it is that the mid-latitude world here is driven by radiation processes that generate a polar temperature gradient . and therefore generates baroclinic instability and this more equatorial world is driven by long-term radiative processes that change the vertical gradient of the temperature film, so some examples are found in the mid-latitudes of the PV world.

Cyclones and acyclones, cold fronts and warm fronts. the polar vortex and the mid-latitude jet stream and in the VP world there is the mjo, the itcc, the Hadley circulation, the Walker circulation and tropical cyclones, although in their article they put a question mark, at least that It was my interpretation, they put a question mark. about tropical cyclones because they were talking about vorticity in tropical cyclones, so here's a three-world view, so this is a slight expansion of what they proposed and I thought, do you know that quantum physicists have a three-world view? miniature and actually there are In your opinion, there are so many worlds that they are not even countably infinite, so I thought that meteorologists should be able to have at least three worlds, so here is a three-world view: you have the conservative PB world just as they described it and then you have a non-conservative PB world, so this is a typical example of what a PV fanatic would do when trying to expand the PV world, so you have a non-conservative PV world, so which photovoltaic energy is not materially conserved and, in fact, can grow to a very large size. values ​​like in a hurricane 250 photovoltaic units, we are in the conservative photovoltaic world, it simply advects around the photovoltaic on isentropic surfaces and if it starts between two and five photovoltaic units, it stays there, it simply reorganizes and that is why it is a photovoltaic.

In the world, if PV is not preserved well, the idea is that the principle of invertibility for PV still holds and so it used to, then you have the world without PV and examples of that would be Kelvin waves near of the equator that vertically propagate inertial gravity waves. and mountain waves, so let's see how you could justify something like this here is the definition of PV, it is an absolute vorticity above the dotted density on the gradient of theta, by the way, this is a traditional way of defining PV, but there are other ways and you should talk to Kerry about this about using something besides Theta.

There have been some interesting ideas put forward about using saturation equivalent potential temperature there, but this is just traditional PV and keep in mind that it is a scalar because it is a DOT product, so what does that mean? it only considers part of the absolute vorticity, considers the part of absolute vorticity that is perpendicular to the Theta surface, and discards the part of absolute vorticity that is parallel to the Theta surface. No, this is strange. I'm sure the students are really baffled by this. it's the unit of PV if you define it that way it's square meter Kelvin per kilogram per second what the heck it's so cool if you rearrange it and you can rearrange it like this in the bottom box here and you can think of the numerator here it's vorticity and the denominator is a Density, not a true density but a pseudodensity, is a kilogram per square meter, which is a horizontal area times Kelvin and Kelvin is the vertical coordinate if you're in Theta, so that's a more reasonable way to write it.

This is true for invisible adiabatic flows anyway. The PV is materially conserved and this is true for synoptic scale midlatitude flows above the boundary layer now, if you have inviscid flows but you have diabetic heating, you get a term on the right hand side, which looks like the PV , except that Theta at the end has a DOT on top of it, so that would be its source term, so it would be more valid for tropical cyclones in the itcz above the boundary layer because there is no friction now, what there is in This definition of PV is the wind fields, the U vector, and the mass field through rho and Theta, but remember that Theta is p zero over p multiplied by temperature and temperature can be written as p over rho r. so you can think of theta as a function of rho and P so what's in oh, please recognize that there's an uppercase and a lowercase p here? uppercase is PV, lowercase is pressure, so what happens if you have hydrostatic balance in the motion you're in?

Talking about a well, rho is given in terms of pressure and if the flow is balanced in some way, it could be a non-linear geostrophic gradient equilibrium, so somehow the flow is related to the pressure gradient, which means that everything on the right side of this definition of p can be written in terms of pressure, so you get this box down here, the PV is L sub 2 of the pressure L sub 2 means an elliptic operator of second order and what that operator is depends on what you're dealing with with geostrophic or gradient equilibrium or whatever, so what does that mean?

If you know the PV you can invert it to find the pressure and if you get the pressure you can get the wind and the density so you know all the mass. Field, this is a surprising quantity, the information that is carried there in that scalar quantity here is an example of PV inversion. This is a simple 2D example, so there's an and the iso lines there, the solid isolines are the Theta field every four degrees Kelvin starting about 2 300 at the bottom and 374 at the top and the PV is uniform everywhere here outside of this oval and inside the oval, but inside the oval the PV is higher so what happens is when you invert it inside the oval the PV splits into vorticity and stability so high PV means high stability and high vorticity and high vorticity is go in this color.

The contours and the yellow ones are on the page and the blue ones here are off the page, so you have cyclonic vorticity and high stability there, but the interesting thing is that if you go below the PV anomaly, the static stability is altered and the vorticity extends below the PV anomaly. This is typical when you invert an elliptical operator, the influence spreads correctly, so how does this apply to tropical cyclones? There are three main questions about tropical cyclones, when they will form, what their path will be and how intense they will be. and vikuyama once described it this way: a tropical cyclone is a mesoscale power plant with a synoptic scale supporting structure, so what he meant is that the inner core, where the eyewall is on the order of 100 kilometers, is the power plant and the larger scale circulation is a The thousand kilometer scale is the synoptic scale support structure and it turns out that TC track forecasting requires accurate modeling of the synoptic scale support structure and intensity forecasting requires accurate modeling of the power plant itself; that's basically the reason why trajectory forecasting is easier than intensity forecasting. but anyway, TCS are basically a balanced PV phenomenon and the balanced wind and mass fields are invertible from the PV field, PV is not materially conserved if we define it in the traditional way with just Theta.

Friction effects are limited to the boundary layer and the evolution of PV above the boundary layer is determined by advection and diabetic processes. Here is the result of a complete physical model, it is a cross section with radius and height and these are PV isolines and this is a photovoltaic tower that was formed in this model and has PV of approximately 250 photovoltaic units and, now, here is A famous Hurricane Patricia occurred a few years ago and my colleague Michael Bell tells me that this is the biggest forecast failure in history, that it was a tropical storm and it passed. category one, two, three, four, five in 24 hours and no model caught this and they were in a field program and they did almost no flights there, but it happened in October, so they were near the end of the program and they had these flights . hours for them to decide that they needed to use them and they flew this thing which was a great luck or genius and this was observed by this NASA plane it is a B-57 it looks like a U2 to me it has very long wings It flies subsonically just above the tropopause, which means it has to have big wings to do that and you can't put any scientists on it, it's too small for that, but it has this thing under the fuselage that you can put on. there and it contains these Drop Zones and these are the most amazing Drop Zones, they're small, they have GPS receivers for the winds and you can drop these things less than 30 seconds apart so you're flying over the hurricane and you're dropping these things every 30 seconds and you can actually reconstruct the wind field because if things go through the entire troposphere correctly, the result was, I think, the best observed hurricane ever, Hurricane Patricia and Michael Bell and his graduate student.

Jonathan Martínez made these cross sections, this is radius here and height here and the one above is the vorticity field, absolute vorticity field and the static stability field is drawn with isolines here, but anyway you can see that there is a tower with a very high absolute vorticity and when you put those two fields together and create the PV field, you get this bottom diagram here and you can see that it's yellow with some oranges and it has up to about 240 PV units. It's amazing and the photovoltaic structure is within 10. kilometers of maximum wind right, so could I ever produce such a structure with a toy model?

Consider the reversed flow symmetric in a plane F and here is the equation for PV governed by that, so that you have radial advection U DDR and vertical advection wddz and have the source term PV is the Theta point on the right side, so, what if we consider a transformation from this to this different coordinate system here, this uppercase r coordinate system that is defined in this bottom box here? Well, the right side of that is absolute angular momentum, so this is absolute angular momentum, so the capital r can be thought of as a coordinate of absolute angular momentum, but it has the correct units of length because the units of that term agree with battery units, so so does a coordinate. transformation from radius height time to potential radius potential temperature and time Tau where tau is equal to T you would need a new symbol Tau because partial with respect to Tau is different than partial with respect to T you can see that as I mean with respect to Tyler keeping the big r and Theta fixed with respect to T, you're keeping the Z level of small r fixed, so tau is T, but the partials differ well if you do that complicated equation there, the top one, that's awesome. how it is simplified comes to that which is just a coordinate transformation, now you can see well the detail of DP, there is a term P multiplied by the derivative of theta Dot, so there is an exponential growth built right there, dpd tau is proportional a p, and what happens if you just divide that equation by capital p and you get this equation here, well, this equation on the left is a partial differential equation in Theta and Tau, it's first order, if you look in any book onmathematics, it says well the way to solve an equation like that is by the method of characteristics, so you convert it into two equations.

That doesn't seem like progress at all, but they are ordinary differential equations, so you have an equation for the characteristics of theta. Point and then you have this equation over this ordinary differential equation that tells you how the PV varies along the characteristic so you can easily write analytical solutions for that and then you get something like this, the diagram on the left is capital. r on the abscissa and Theta on the ordinate and there are isolines of the theta point, that dimensionless version of the theta point anyway, just think that the Theta point is maximized in the mid-troposphere, so if you solve this pair of ordinary differential equations you have With the feature method, you can obtain not only the features but also the variation of PVD along the feature.

This is an isolation of the PV, it's actually P divided by some reference P, so that's its dimensions, so this ISO line means two times the reference value and this means four. times the difference so what you are doing is producing a large PV in the lower troposphere and a small PV in the upper troposphere and I have three times here. These are dimensionless times, but watch what happens as time goes by, remember that these PV islands are logarithmic 2 4. Now it's 16 here and it's uh, it's 1 16 up there at the top and then, a little later, it's like this, it's like a photovoltaic tower, there is a tall photovoltaic tower, but in real physical space.

It is tilted because this is the potential radius and in the hurricane the angular momentum surface tilts outward with the radius, so if you convert this back to small r, all this and it tilts outwards, this is how you can get a photovoltaic tower like in Hurricane Patricia. Now what happens when you do that? It becomes unstable because it comes here like this and the PV increases and then here the PV decreases, so the radial gradient of PV is opposite here. and here it means that the oxidized waves propagate in opposite directions relative to the flow and can become barotropically unstable thanks and this leads to polygonal and mesoborthic eye walls and PV rearrangement, so this has been studied with a hierarchy of models not divergent. to the full 3D non-hydrostatic physics model and, uh, Eric Hendricks, who now works at incar, found this example.

This is Hurricane Dolly and it just hit the Texas coast. The Texas coast is on the left and this is what it looked like. I was going through this barotropic instability of the polygonal eyewall. This is a national weather service radar, uh, and the radar is at its lowest elevation, half a degree above the horizon, sweeping around and the blue is very low reflectivity and the red is very reflective. This is what that kind of jitter sometimes looks like in satellite images, here's a close up of that and Jim Carson, who worked with Carrie on a lot of things recently, he's at the University of Wisconsin, he's actually fine right now.

He was an employee of Noah, but he called this the starfish pattern because he looks at this and sees these things coming out like this and he says there are five of them like a starfish and you can pattern that and like that now. There is an interesting question here: what process would produce actual confinement of the eyewall and one possibility is that the boundary layer, the frictional boundary layer, is very involved in this and so what if you think about this in the most simple? slab model, which means there is no vertical structure in it as only a layer and it has a tangential velocity and a radial velocity and the flow in the boundary layer is forced by the pressure field that the flow above the layer limit feels so This is the standard Ekman idea just when you do the Ekman Eckerman spiral solutions, you impose the pressure field well.

Here is an interesting example. This is the infamous case of Hurricane Hugo on September 15, 1989. So think that Hurricane Hugo is here and on the plane. is in Barbados and they fly there and they will fly into the hurricane from the southwest to the northeast with that red arrow there and they will fly at 1500 feet so as they fly as they fly They are measuring the tangential wind that is on that red curve so they are in the entry section, they are in the boundary layer just 500 meters above the ocean and the wind is picking up and reaching up to 85 meters per second. that's 170 knots and it fluctuates around there and then look at what happens in that radial interval there where the yellow is, which has a radius of about seven kilometers, the tangential wind drops from 80 meters per second to 20 meters per second in approximately a kilometer and, uh, you know Greg Holland, he said, let me tell me about this, he said vorticity was gained and I said, what do you mean vorticity was one? and he said, well, you normally think of vorticity as 10 to minus 4 seconds for the minus one was once seconds at least one this is 10 to a quarter higher than what you normally think of and then what happened was the plane It flew around the eye and gained altitude and then exited the eye in this blue curve. which is what's there and you can see the blue curve, it doesn't have a lot of interesting things, so this wild behavior is in the boundary layer, it's not in the fluid at the top and the vertical velocities if you look down here at the bottom. here the vertical speeds went up to over 20 meters per second and then decreased here and by the way this was the end of flying the P3 in the boundary layer because the plane, the wind, the wings flexed so much that the plane has a de-icing sleeve at the front of the wing and the de-icing sleeve broke towards the end of the wing and it was hanging there and they thought if it got tangled in the propeller, yeah, it would be a disaster. and they're only 500 meters above the ocean and the plane has a stress meter that measures the G forces in the wings and they showed it to the Lockheed engineers and I think they said the engineers were shocked. about how high the g forces of this flight were so that they no longer fly there at 500 meters it is too dangerous Gary told me if we have my talk one two fight with your word shock what you have is surface convergence and secondary circulation there is the peak of that W and extreme cyclonic vorticity is a front is a threat yes Carrie has interpreted this in the past in terms of a front is that so Carrie yes it is a strong feedback and collapse you are beautiful coordinates collapse Yes, well, let me present a counterargument to that in just a second and I don't know if it's right, but it's interesting to think about.

Let me do it from this equation here, so this is the slab boundary layer model and Carrie told me. once you knew there is a simpler model than this and I said no and he said yes it's called a founding layer model you remember yes but that model is for bums he said anyway it has a radial equation and a tangential equation B and you have horizontal diffusion, you have a surface drag here, the coriolister here and this is a flow in and out of the slab and then this is a radial infection here, so what you could argue, I would be interested to hear your reaction to this is that Beauty Plus u d u d r is what you have in the Burger equation and therefore you have an embedded Burger equation, you could say an embedded Burger equation here, well, you get burgers, you have to include this diffusion in she too so the radial equation has this Burger effect and uh this is the specified pressure gradient force coming down from above imposed on the model so let me show you what you get from this if you take that pressure gradient think on it in terms of a wind gradient and think.

In this regard it specifies three wind gradients corresponding to category one, three and five hurricanes, so this is a large wind based on the art, so impose the pressure field associated with these and run this model from a domain of zero to a thousand kilometers and doing this consideration of a hundred meters and a time step of one second, so if you take 100 meters in a thousand kilometers, that's ten thousand grid points and this simulation will run around 10,000 time steps, so , why would you do it? It seems excessive, but What happens is that the top diagram is the radial wind field and it goes to a costly steady

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, so this is the steady

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and for category one, three and five, as you can see in the category Five, a radial inflow is obtained. here about 12 kilometers and it's a radial information of about 30 meters per second and what happens is it stops abruptly there about 11 kilometers or something like that and that's what the tangential wind field looks like in the boundary layer, so and this dashed The red line is the green wind that prevails, so in the boundary layer is this, so there is a subgradient there, but there is a supergradient there, now here it is oh and by the way, this is the motion field vertical or the eclipse pumping you come out of. which you can see how it is concentrated and then this is the vorticity couple and it is concentrated where the pumping is high, so let me argue this why I don't call it a front because I think of the Hoskins Brotherton front model. and I think about how geostrophic equilibrium is used in that model, so what happens in that model is that the flow along the front is geostrophic and that means that this upper equation is replaced by an equilibrium between that Coriolis term there and that pressure gradient term there. so on the hospital Brotherton front model you don't have that UDR, well I know Hoskins Brotherton is a very idealized front model and of course the real fronts do have UDR, but that Hoskins Brotherton model doesn't have that , so the behavior we're seeing comes from that UDR which is like a Burgers shock and well, I know sometimes when I say shock people say it has nothing to do with compressibility as a shock. that a plane forms that's true, it's not, it's not like that, it's something more general about collisions as described in Widham's book on linear and nonlinear waves anyway and Hoskins Brotherton's geostrophic approximation actually falls apart quickly in front of Genesis, yeah, so both of them, huh. -Eh, yeah, well, let me continue with this last topic.

I only have a couple of minutes left. This is about itcz and do you know that Alex is amazing at finding the theoretical insight of itcz and looking for the most beautiful image? out of thousands of images he finds things like this, so this is the only symmetric itcc from a theorist here and I just want to say that if you solve that problem, you can see the itcz as essentially a balanced photovoltaic phenomenon and it's just symmetrical. as if the hurricane were really symmetrical, it is invisible above the boundary layer and you can write the PB equation for it and you can do a coordinate transformation analogous to what you did in the hurricane case, you can get a simplified PV equation that looks like identical to the one in the hurricane case, you can solve it by the method of characteristics and you get things like this here is the itz at 10 North, so the latitude is in the abscess here and Theta is in the vertical coordinate and then this is how the PV develops in time so this is after a short period of time then a little oops a little later is this a little later is this then a photovoltaic anomaly forms there and starts to form a tower, but it doesn't do anything like a hurricane can build a photovoltaic tower and uh and the ideas about barotropic instability are there, the photovoltaic gradient on a Theta surface reverses the signs on the two sides of the itcz so you can get easterly waves now here's the mjo this this is strange here this is a linearized primitive equation model based on Matsuno's equatorial beta plane theory and I just want to show that here is a complete solution that arises from this: the equator is right in the middle here and then this is x in thousands of kilometers, so this is 8,000 kilometers going in the opposite direction and here is the flow field and these ISO lines are the geopotential anomaly.

Now, in Matsuno theory, you can divide this into three parts, the first part is the part of the oxidized wave that looks like this second one. The part is the part of the inertial gravity wave that looks like this quite confined and the third part is the part of the Kelvin wave that looks like this. Do you know that if you calculate the PV from this diagram you need the wind field and the mass field? to calculate the PV that you have here on the right so you can calculate the PV from this and you can calculate the PV of this rough wave part, if you do that, you will get the same PV feeling from there and from there so that you don't get no PV from the Rossby wave part, why?

Well, there's just a conceptual argument here. What's that? There is no PV in the Kelvin part of the wave. Yes Yes. I think I said that wrong. There is no PV in the Kelvin part of the wave. Well, there's just a shallow water argument here. Here is the definition of. PV in shallow water and divide the depth of this fluid into a mean depth H bar plus h Prime and then the F in the beta plane approximate that by Beta Y and in this Zeta field remember that a Kelvin wave does not have any V, so what there is no DV dvdx,it only has a dudy part, so PV becomes equal to Beta and plus P Prime, where P Prime is defined by this, it has a vorticity part and a mask field part.

Now think about the Kelvin wave, it has a zone flux maximum right at the equator, so the uby south of that is one and orthodox, it is a different sign because if U is declared maximum, then this du d and change the sine between the northern hemisphere Y and positive and the Pacific and the part of the mass field of also changes sign, not because H Prime changes. H Prime is the same on both sides, but the beta switch and is signed, so it turns out that in the Northern Hemisphere these two cancel P Prime is zero and then you go to the Southern Hemisphere, the sign of this switch is on the side of those switches, so it still cancels out, so you get zero PV, that's a hand waving argument but it turns out to be true and thus the idea is Kelvin wave packets, yes I mean a package. of the weight of Kelvin waves is basically almost invisible on a photovoltaic map and that word I almost think you have to put there because the argument is based on beta plane theory and linearization and it's interesting if you make a Kelvin wave on the sphere. instead of the beta plane it has a bit of meridian velocity which is interesting so the PV map has virtually no information on Kelvin waves so I have a friend Brian McNulty and he told me that Kelvin waves are the heel of Achilles of PV, thinking there is a decline of PV nerds and I think these same arguments are valid for uh for enso because enso has Kelvin waves, so anyway PV is a useful quantity to study understanding many of atmospheric phenomena and that's why I was thinking that there is a kind of three worlds, there is the conservative PV World PV is materially conserved.

The principle of invertibility is sustained. there is the non-conservative PB World PV is not materially conserved, but invertibility is still true and that applies to hurricanes in the itcz and part of the mjl and then there is this other world. the VP world that, uh, that was pointed out, so anyway, here's a shout out to Akio and I've had a great privilege working with them on this kind of thing over the years if you want more information on this.there a website there, thank you very much, so we had a discussion about how to do the talk, let's answer a couple more questions from the audience and then we can take a break after that, yeah, so my question was did you actually witness the breakup of invertibility in the second? world that you mentioned where television and the particular vertical are not considered, so how do you go from that board to the third world?

But with the conditions you need. Yes, that's a good question. You know, Kelvin waves at least in the equatorial and linear beta plane that they have. zero PV but they are kind of a balanced flow, it's a bit strange isn't it? Because in a Kelvin wave the flow is zonal and the high and low pressure is to its left in the northern hemisphere and is to its right in the southern hemisphere, so the zonal flow obeys the geostrophic relationship, so, in In a sense, the Kelvin wave looks like it should be a balanced flow, but it has zero PV, so it's a bit strange, so you might think that invertibility is always valid if the flow is hydrostatic and is in geostrophic equilibrium, but that does not seem to be true for the Kelvin wave, what is that zero?

Not important, yes, and Kelvin waves are not balanced in their direction of propagation. That's true, yes the Coriolis force in the zonal direction is zero because V is zero, yes, so d u d t has an f v term but V is zero, so yeah, maybe that's a good way to look at it Jim, the Kelvin wave in one direction looks like a geostrophic equilibrium in the other direction. It looks like a pure gravity wave, there is no Coriolis effect, yes, it is not non-dispersive, right, yes, that's right, aha, the Kelvin wave is not dispersive, you know?