Anelastic equations as a tool to understand gravity waves and meanflow effects
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Anelastic equations as a tool to understand gravity waves and meanflow effects
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Anelastic equations as a tool to understand gravity waves and meanflow effects

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CC Attribution 3.0 Germany:
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2019

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English

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Abstract 
Internal gravity waves (GWs) in the atmosphere play a major role in the global circulation of the middle atmosphere (10100 km height). The meanflow effects of GWs consist of 1) the momentum deposition (corrected by the Stokes drift of the GWs), 2) the energy deposition (corresponding to the sum of mechanical and thermal dissipation), and 3) the convergence of the GW heat flux. This set of meanflow effect is obtained from the anelastic equations with sensible heat as prognostic thermodynamc variable. Furthermore, we have to assume the Boussinesq limit for the waves, which means that the vertical wavelength, λz , is assumed to be small against 4π H, where H is the scale height. We show that these anelastic equations yield the usual dispersion and polarization relations for GWs. According to current wisdom, however, these solutions include the case λz ∼ 4π H. New linear theory for by Vadas (2013, J. Geophys. Res.) yields different polarization relations, but confirms the usual dispersion relation. The likely reason for different linear solutions in the case of λz ∼ 4π H is that usual linear theory starts out from the fully compressible equations, and the anelastic limit is calculated separately for the dispersion and polarization relations. In this contribution we present a set of anelastic equations where the Boussinesp limit for perturbations from the reference state is relaxed. We recover the more recent polarization relations for GWs, but obtain a different dispersion relation in the case of λz ∼ 4π H. The mean flow effects from GWs are consistently simulated in a highresolution circulation model, provided the model is based on the compressible equations and includes a hydrodynamically diffusion schemes. However, a consistent theoretical approach for the meanflow effects of GWs having large λz is not yet available. According to theory and highresolution modeling, such GWs occur frequently in the thermosphere.

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that is related to gravity waves and the mean flow effects of gravity waves in the background is that gravity waves so often cannot be resolved in model that in models they have to be parametrized in those polymer tries Asians go back to linear fury for the waves and then we use results from linear fury in order to
00:22
die way to the right the 2nd moments of these waves nonlinear effects and so what about these concepts
00:31
that's that's what this is about and
00:34
so I am I benefited a lot of from discussions was shown about us from Boulder and I would 1 from institute
00:44
so I have 3 points gravity waves so 1st a brief introduction to gravity waves in the middle atmosphere and thanks again to Meyer to the previous speaker who gave a nice introduction into middle atmosphere than physics then we will do a little bit with a traditional linear gravity where theory
01:03
and then we will consider the different limits for gravity waves so anelastic approximation and what the limits actually of these approximate of these anelastic the the equations are
01:16
so this year is to because and you'd cycles simulated with a model where gravity waves are parametrized and you see the would be called the sold only averaged fields off the temperature and the solar wind solar wind is the east by
01:31
component of the wind and so no leverage means you average along the latitude cycles and usually so each Snapchat she is also a temporal average over a couple of days so this is how this looks like and when you look for the precisely you see that during summer you have 4 always hear very cold temperatures and and during winter you have also very warm temperatures here in a region that we call the strut applause so how can we understand these things and it's mainly because of the
02:04
gravity waves you know other waves also very important and the basic concept to understand the ClO with circulation in the middle atmosphere are the socalled transform Kynurenine equations so follow purpose here these are simply the reached Eulerian equations completed by the Coriolis force and then also and then only the completed by would be called a wave track and she is also the gravity wave Drexel so that you have in the solar wind equation you have basically a wave track and the Coriolis force from the north would wind component efforts the Coriolis parameter then you have the continuity equation for the for the mean Maridi on a wind and the René were to go wind and then you have a very simplified sensible heat equation where you have few the edge about Keating from the word go away and and here's some diabetic heating which includes radiation chemistry all those kinds of things and you simply write this as as relaxation to what's already 8 of the equilibrium temperature and this year is known as the search for ruined equation this is married or the wind equation properly averaged and simplified and so here these gravitywave track is written here and you see this looks something like this looks a little bit like a rainout stress tensor course like were so like it's it's a momentum flux convergence of rain on stress tens of Daiwa directions and you'll be included in the singer column oxide approximation only the vertical the momentum flux from these waves so these equations are very powerful because they imply causality when you apply the climate to logical mean you you see that there's only mean married on its circulation when there is a wave train which is when they are waves which transport momentum from the light at you it's too high attitudes and then the circulation is balanced of course by some of by some vertical flow and the vertical flow actually induces the diabetic heating so it's quite the other way around according to our intuition that we from daily life the diabetic heating is not the cause of the circulation the circulation is the cause of the diabetic heating and then the solar wind is simply a in balance with the temperature structure that is induced by the circulation due to the thermal balance so this is
04:38
small family but this is years again from a model and then you see the wave track the ostream function that code that corresponds to this mean Marie on circulation again the temperature in the solar wind and you have here in the east what wave track you have here in Westwood where if track and and according to the Coriolis force the circulation has to be then here from the summer opposed to what's the winter pollen and this is from the gravity waves and then planetary Rossby waves also contribute in the winter stratosphere and in the troposphere and then the circulation simply goes like this and since it goes like this you have upwelling here which causes the courts is a policy of Dong where the year which due to edge about a compression causes the warm winter struck the parts so this is the basic concept of middle atmosphere dynamics in a very short and so this is
05:35
actually a comparison of of a fluid simulation was resolved gravity waves where you also see here the code some is a pause the warm winter struck oppose this was the gesticulation so and this is the corresponding simulation where I switched off all the dynamics of all the waves or in a model you can do these things so what do I get simply based on radiative transfer and forced convection and all these things and this is then what you get so such pick similar picture is also available in the literature of course but I picked this year in order to demonstrate this how the actual thermal structure in the middle atmosphere deviates from something where we ignore dynamics so the difference He's are simply substantial some more than 100 CAD in missions so that our
06:26
gravity waves and in order to see what protein ways are we have to know what are some ways sound waves are compressible waves of course or they are no Matutinal waves so we have here the warm and cold fronts and then the velocity perturbations actually perpendicular to the way faces and for gravity waves things are just different you also have you warm and cold phases and maybe the faces propagate propagate here to the east and downward but then the wind perturbations are perpendicular to the forms which means that gravity waves are transfers always and watch transfers a waves and they're also strongly dispersed and this process the depend on them so they wrote the wavelength and frequency depends on the back promptly and and so forth and in this case the buoyancy forces the restoring force so with art gravity there credulous of course and so the dispersion relation in in in its most simplest form you'd this this the vertical wave number and this the the buoyancy frequency and this year is the mean wind minus the phase speed of the waves so you see when the mean wind and the speech at get closer to each other than than the word at the wavelength of the wave compresses and this has a certain
07:51
effect on the stability of these waves this you see here this this year's probably or this is assumed to be the wave evidence buffer source somewhere and then the wafer supposed to propagate to the east here you have a to the Gowin profile for the summer middle atmosphere with West what winds in the stratosphere and measles sphere and then East wins here in the upper measles here in in the thermosphere and when the wave now comes into this regime then the wavefronts are or the word a wavelength is compressed and which becomes dynamically and collectively unstable and and then you get to lands and the wave has to deposit the transport of momentum and energy into the mean flow so there's another curiosity you have to face propagation in this way then the propagation of momentum and energy so the group velocity points perpendicular points into this direction so this is a typical something typical for the atmosphere when the phases propagate on what the energy and momentum properties of what so gravity waves break because of the dependence of the vertical wavelength on the mean wind and of course you also have an exponential growth of the amplitude of the wind and temperature perturbations with increasing attitude because the mean density data decreases with an attitude according to the barometric formula
09:25
so how do we deal with gravity waves and models and so I use here not is precisely the equations how they are coded in models but the socalled on elastic form and I will acquire a and I will explain this just in a minute so but you can remember this year all thermodynamic variables are expanded into a reference profile and then a perturbation from the reference for 5 and here long land of fire and Z are longitude latitude to end the war article called coordinate and here you have the horizontal momentum equation in hydrostatic model then you have here the hydrostatic approximation in the world to go this is the continuity equation and this is again or the sensible heat equation but since we wanna run a climate model we have to extend these equations by all kinds of additional terms and so radiating Fudan that cheating and especially sup with skater parametrizations and I write them here also as simply as possible just radical momentum diffusion horizontal momentum diffusion this year's vertical diffusion of sensible heat due to turbulence and molecular the school or the molecular diffusivity this hallways turbulent horizontal diffusion of heat and then you have fewer the frictional heating rates from the momentum increases for the from the momentum diffusion so this would be for high resolution model marry was off the gravity waves so what happens is the so what is our concept for the gravity waves if they have to be parametrized as well then we take this equations and the average over the gravity wave scale and that that we get the threat additional terms we get here the momentum to position we get here a convergence of the vertical heat flux from the gravity waves and then from this so from this and medicating term here we get this BN see production term with a negative sign and get in addition frictional heating rates from the gravity wave of evasions so this is the concept and the anelastic and the strength of the anelastic equations is that you can just write down these equations just by averaging you don't have to deal with all the density put variations and have to in wall complicated to discussions about that so this is really a strength of the anelastic equations and furthermore you can you get the kinetic energy equation for the gravity waves and then when you consider that in steadystate approximation you see immediately but the energy position of the gravity waves actually is namely its these dissipated feeding rates plus this buoyancy production terms which by the way can be shown to correspond to the thermal dissipation of waves but this would be too far right now and this year's the typical form of the energy to position Howard as included in gravity raise parametrizations but you see we have to write it in this way all we have to include all these to all these other terms to compute the energy position from accredited way of resolving on OK so this is
12:39
so maybe a few results regarding this this use the energy deposition a simulated with the model with parametrize gravity waves you have few contributions from the gravity waves and here from the from the tights and in these TEM equations that I mentioned and I think on my 2nd 3rd slide this these equations IC can all use direct filament effects you know there are substantial as well and now you can also run the model with all without energy to position and you see the energy deposition it's very important in order to get realistic results from a model from the above about 80 kilometers so now if we wanna
13:23
parametrize gravity waves so what does the traditional gratuitously TUIfly new theory tell us and so this year
13:31
is attritional inverse could fury for gravity waves you start with the equations you assume wave propagation just in 1 horizontal direction you linearize about and hydrostatic the mean flow say and to what's speaks direction and then you have you mean profiles for the temperature and the pressure and the and the density you have C dependent scale high and so which means you you invoke the WKB approximation that the background can still very slowly on the scale of the waves and then you get these set then you get the set of linearized equations and you can eliminate a few further terms that you get this set and then you block in the way starts for the waves this is also very usual you anticipate the exponential growth of the wave amplitudes and then he gets this set of equations here and this is basically also your set of polarization relations for the waves but right now we have not distinguish here between gravity waves and sound waves so sound waves still part of the solution and and all very interestingly in order to get then the real solution for the dispersion relation you have to demand that the reference temperature is constant otherwise you get to the imaginary parts and this is then the solutions that you get for the dispersion relation and Alfredsson Alexander and they're very famous refuge paper they simply say all let's assume the sound wave goes and to infinity and then we get the dispersion relation for gravity waves but there are a few
15:18
questions on this namely that the assumption of an infinite sound speed really provides what we may call the and elastic limit for the dispersion relation which then would correspond to the gravity waves which which the longitudinal sound waves into a transfer of gravity waves mainly because the scale height can also expressed in terms of 1 always squared so should be then ignore this time as well and and in fact in there when you look into the
15:50
literature the different polarization relations suggested for the gravity waves and so we may ask what are the correct polarization relations for gravity waves that have long wavelength so whether would wave number is of the order of 1 over 2 H and in fact this case is important in the film was here so these
16:11
are here on shouldn't know I need not
16:14
mention that so we can skip this so this
16:17
year from a gravity wave resolving model that extends up to 200 kilometers and you see here are just an arbitrary we daytime by during July for the vertical wind which is a marker for the gravity reflectivity and you see here the southern Andes which are actually the most or the strongest gravitywave hot spot on earth because you have the strong Eastwood winds that blow over the mountains and this generates the strongest orographic remedy waves so stronger than anywhere else on the planet and to this is just a tiny example where you get here an orographic event when the winter intensifies over the mountains and then at 65 kilometers you can see this year at the end of the simulation and then when you look into higher attitudes and I think this will just a fast and most of the here the faces of the waves are aligned here you see concentric rings which means these are no longer the same crime waves than those those gravity waves break here make a momentum deposition the momentum to position listen and balanced meaningful behind and the unbalanced being flow responds with new general with the generation of new gravity waves this concept has been suggested by day for its 15 20 years ago we applied it now for the middle atmosphere and showed that it is substantial in the middle atmosphere and then of course these waves break again and then high up in the firm was fewer you get these huge concentric rings is actually tertiary graduates and so forth and so forth and so the atmosphere makes it possible that
18:00
in the thermosphere you actually get the credit you waves with very very long Werdegar wavelength so if you wanna understand and parameter or even parametrize gratuities and the firm was here you have to have at least linear if you read that use correctly with long would wavelength and maybe so much about that and so very long wavelength in the thermosphere and this is not just here in artificial model results this is also predicted by linear theory and it is confirmed also by light measurements when they are events where the light can measure up to 140 150 kilometers
18:40
and so what on so so the so what is now how do we have to deal with all the new gravitywave is a theory given the questions that I get that I
18:51
raced 10 minutes ago so we started so we now just inspect the anelastic equations and I think I I already mentioned what the anelastic approximation is about and how these are the UN elastic it go you during equations in the in basket case and so it's that they are simply linearized about these deviations from the referee about these deviations of the thermodynamic variables from the reference state and the only place where we go farther than this is the continuity equation in the continuity equation the country we consider just the reference density and the reason is we have to do this in order to kill the gravity waves otherwise the continuity equation coupled with the momentum equation and we will have some ways this is this is what we have to do and then we can go and and of course their previous versions of the anelastic equations in in in a number of papers and I should note all of them include the Boussinesq limit so what does the further posing estimate not means
20:00
the Boussinesq limit means that when we have changes of state here for the relative density changes pressure changes potential temperature changes we say the pressure perturbations our negligible in that equation so which means relative density perturbations are proportional to minus their relative potential temperature perturbations and similar and b get a similar result from the equation of state you write this year in terms of perturbations then we see pressure or relative density perturbations are proportional to negative temperature perturbations which means negative temperature and negative relative potential temperature perturbations are also the same and when the black this in here into all vertical into the right hand side of the vertical wind equation we get the result that actually the vertical derivative of our water pressure perturbations it's very large against pressure perturbation along 1 H and this means that the Boussinesq limit implies that wave solutions should apply that the vertical wavelength very small against 2 pi times the scale height so let's use all on elastic equations in the Boussinesq limit so this is these you deviate only from the from minds by using the sensible heat equation instead of using another the thermodynamic variable
21:28
and then we make the same so the same efforts again we have few the wave equations we have the way funds arts as before and then we get this dispersion relation here and now this dispersion relation is the same as before in the fully compressible case where we made the limit sound speed against infinity but but I do have irrational includes only cases where the word at wavelength is very small against the 4 pi and the scale right so we now have assumed she explicitly that the word ago wavelength is not too large but we get the same result for the dispersion relation that the Alexander that Fritz and others can Alexander got in the general case so in
22:18
order to see clearly what happens we write on again we go back to the compressible case right on our way equations and I'll be mark those terms that correspond to the UN elastic limit into the Boussinesq limit in red and blue and then we just go forward and try to maintain everything and an entity and in particular we do not make the assumption of an eye for from background of atmosphere than these here our our generalized to polarization relations and this year
22:51
is the general dispersion relation in the compressible case so we see the get additional terms compared to the fruits and Alexander result and actually these 2 terms together that we have here and scored over G minus couple were age this is precisely the vertical derivative of the background temperature divided by the temperature and if we set this to see than these terms cancel and be recovered the old result if we neglect all the rat include terms then we get the the idea so and we can also right away they collect all the red and blue terms and then we get the anelastic dispersion relation in the Boussinesq limit where slowly varying tr is allowed but that say somebody wants to say all I want to so far away the sound waves but I wanna have gravity waves with arbitrary wavelength than you might say all I want to neglect only the rat terms on the terms that correspond to the compressibility but then be a completely unrealistic dispersion relation mainly because we also we always get here imagine parts which here we can more or less they collect because these are tool Ratched terms which are which 1 so which concentrates on which almost cancel each other but the 1 latched term comes from the compressibility assumption and this term here will would correspond to within a state so that's wrong and I would say nothing is wrong
24:31
all interpretation is wrong so when when this very large against 1 4 H squared all these terms are negligible and then we are on the safe side anyway in the term in the case when the is comparable to 1 over 2 and then we have to claim all we have to demand that the reference temperature is constant provided you because otherwise we would violate or WKB approximation so the background has 2 very slowly at the scale of the waves but if the wave scale is very large 100 kilometers them the reference temperature cannot change on that scale so this cases safe and but further but would be half such a notch would occur wavelength then he it and then the anelastic equations are basically useless we have to we have to admit that both gravity solutions always compressible and that the compressible polarization relations have to be used along just with the gravity with branch of the dispersion relation so when we have this equation you fall of all the for the for the intrinsic frequency then of course we get 2 solutions 1 with a plus and 1 with a minus and 1 of the blastocyst sound waves and 1 is with a our gravity waves so this is how it is but it is completely compressible polarization relations are not the ones that are used in gravity theory underlying any parametrizations this is worshiped emphasize at this point and I think this comes
26:12
also here on the last slide so this is my summary on elastic equations usually include the pool also the Boussinesq approximation for the wave solutions and if this assumption is relaxed that the anelastic equations I would say perhaps useless or the analysts' guns answers useless useless gravitywave solutions with men would occur wavelength are subject to the compressible dynamics and such gravity waves can still be hydrostatic so they can have intrinsic periods of of ours Villani so the only requirement is that the word wavelength is large so you can simulate those gravity waves with hydrostatic model so the hydrostatic equations as used in in climate models and also in the model results that I showed but the previous form of the gravity wave mean flow injection we discussed the energy to position and also the gravity wave track which here includes usually the core true a correction from the stomach strip of the gravity waves this all involves the anelastic theory so which means the set of wave mean flow action this is my take away from dealing with this during the last couple of weeks so so this set of wave me Florida actions has to be revised for applications in the thermosphere where the wavelength of the gravity waves can be very long and to so so the modeling community this has not been addressed the problem yet I think thanks for your attention at
27:55
the