The roles of imbalances for divergent and rotational modes in atmospheric flows

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The roles of imbalances for divergent and rotational modes in atmospheric flows
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A three-dimensional general balanced solution to the governing equations of the atmosphere is given. The deviation from this balanced, or inactive, wind solution is called the active wind. For the horizontal components, this active wind is comparable to the ageostrophic wind. The vertical active wind component is similar to the isentropic displacement vertical wind. Transformed governing equations are derived as functions of the active wind components. This is also possible for the vorticity and divergence equations, respectively. It turns out that the classical balance equation is not part of the transformed divergence equation, but reflects the balanced part which had been removed. The terms on the right of the transformed equations can be scrutinized with respect to their effects on the evolution of the atmospheric state. Exemplarily, this is done for an idealized baroclinic wave in a dry atmosphere. The different terms on the right of the transformed equations and the active wind components are visualised and interpreted in their meaning. Most importantly, the newly introduced vertical active wind component allows for an analysis of the vertical motion of isentropes. The baroclinic wave development is discussed with focus on the fronts and the generation or depletion of kinetic energy. The new method allows for a unique separation of gravity waves and vortical modes. This facilitates the analysis of gravity wave generation and propagation from jets and fronts.
Presentation of a group Numerical analysis
Mass flow rate Logic Chemical equation Divergence Algebraic structure Mach's principle
Dynamical system Characteristic polynomial Gradient Chemical equation Vector potential Rotation Divergence Mathematics Invariant (mathematics) Gravitation Vertical direction Gravitation Aerodynamics Thermodynamics
Slide rule Chemical equation Multiplication sign Price index Food energy Term (mathematics) Vector potential Friction Function (mathematics) Vector space Nichtlineares Gleichungssystem Iteration Absolute value Equation Bernoulli number Pressure
Slide rule Functional (mathematics) Dynamical system Multiplication sign Maxima and minima Price index 8 (number) Vector potential Mach's principle Plane (geometry) Term (mathematics) Atomic number Vector space Aerodynamics Nichtlineares Gleichungssystem Pressure Mass flow rate Chemical equation Gradient Food energy Price index Term (mathematics) Measurement Permutation Vector potential Function (mathematics) Kinetische Gastheorie Absolute value Bernoulli number Pressure Daylight saving time
Mass flow rate Closed set Multiplication sign Chemical equation Price index Food energy Term (mathematics) Vector potential Pi Function (mathematics) Vector space Phase transition Kinetische Gastheorie Absolute value Aerodynamics Pressure
Price index Food energy Term (mathematics) Mereology Vector potential Equations of motion Different (Kate Ryan album) Term (mathematics) Function (mathematics) Vector space Kinetische Gastheorie Nichtlineares Gleichungssystem Equation Absolute value Aerodynamics Nichtlineares Gleichungssystem Bernoulli number Pressure Daylight saving time
Momentum Euclidean vector Many-sorted logic Term (mathematics) Mass flow rate Equation Vertical direction Nichtlineares Gleichungssystem Equation Nichtlineares Gleichungssystem Pressure
Standard deviation Mass flow rate Connectivity (graph theory) Multiplication sign Chemical equation Physical law Line (geometry) Mereology Wind wave Graph coloring Mach's principle Different (Kate Ryan album) Velocity Phase transition Vertex (graph theory) Nichtlineares Gleichungssystem Equation Boundary value problem Pressure Physical system
Velocity Vector space Kinetische Gastheorie Vertical direction Aerodynamics Food energy Musical ensemble Boundary value problem Wind wave Mach's principle
Multiplication sign Algebraic structure Mereology Shift operator Graph coloring Mach's principle Residual (numerical analysis) Velocity Average Vector space p-adische Zahl Distribution (mathematics) Standard deviation Mass flow rate Chemical equation Food energy Line (geometry) Wind wave Vector potential Wave Arithmetic mean Velocity Kinetische Gastheorie Vertex (graph theory) Vertical direction Energy level
Standard deviation Distribution (mathematics) Chemical equation Modulform Algebraic structure Algebraic structure Term (mathematics) Wind wave Shift operator Variable (mathematics) Rotation Mach's principle Divergence Residual (numerical analysis) Velocity Gravitation Vertical direction Nichtlineares Gleichungssystem Equation Aerodynamics Energy level Pressure
Momentum Direction (geometry) Modulform Rotation Variable (mathematics) Mach's principle Vortex Mathematics Many-sorted logic Term (mathematics) Velocity Vector space Divergence Gravitation Aerodynamics Equation Nichtlineares Gleichungssystem Pressure Mass flow rate Term (mathematics) Divergence Velocity Vector space IRIS-T Vertex (graph theory) Equation Vertical direction Nichtlineares Gleichungssystem Turbulence Flux
Modulform Zeitdilatation Term (mathematics) Mereology Cartesian coordinate system Variable (mathematics) Cartesian coordinate system Divergence Velocity Vector space Autoregressive conditional heteroskedasticity Vector space Divergence Gravitation Vertical direction Gravitation Nichtlineares Gleichungssystem Equation Aerodynamics Pressure
Vector graphics Mereology Wind wave Graph coloring Order of magnitude Rotation Divergence Complex number Vector space Gravitation Equation Vertical direction Differential equation Gravitation Equation Block (periodic table)
Modulform Likelihood function Variable (mathematics) Rotation Vortex Term (mathematics) Gravitation Physical law Equation Aerodynamics Pressure Mass flow rate Projective plane Vector graphics Price index Mortality rate Line (geometry) Term (mathematics) Wind wave Divergence Velocity Vertical direction Gravitation Nichtlineares Gleichungssystem Flux
Divergence Helmholtz decomposition Gravitation Nichtlineares Gleichungssystem Mereology Thermodynamics Term (mathematics) Force Rotation Flux
so the topic of my talk is not numerics but I think it is important to presenters here because Interscience to light on what we are doing and I a P and
it it also focuses on a on
the importance of what this balanced and what a smart arbitration molds and what are divergent mode in atmospheric flows and I think about what to see these structures familiar to true to and therefore I thought the I think is saying something about
balances and of logical modes and along this could be also interesting for people who are here so
we are on in atmospheric dynamics and which is a kind of threw it at room and guests which is on 1 of us and therefore we have to consider who we are as far as so we are on a specific and gravity and therefore we have to think about to good to we all enough we should not forget about gravity and then what we should not forget about is found with someone dynamics and these are the the principles to shape our atmospheric motions and and 2 disk I'd also a to flow which is balanced so what does balance in the atmosphere is the 1 of the questions we want to ask and this talk and the and then we want to ask the question what but people use if they are all at this balance the how does the what is it evolve holders' divergent evolve and how they'll they'll do vertical motions involved and and if we are in balance nothing that happened we will not have any changes and what is it no changes in the divergent and know changes in the heat of or in in temperature will no changes in temperature so and
what is the concept of to balance solution is that we are writing down or atmospheric iterations as we know them and then we say OK what is a balance we have put a very everything which is friction and heating and we say we want to be stationary and then all what this happens just does not change in
time all the slides here balanced and so we can have such a
solution and the solution is just this that the solution and this is given here it's a gradient of the potential temperature therefore day the 1st love summer dynamics or at least all off had to do was the temperature is important so it's a gradient of 2 potential temperature times in rated of Tibetan the function and a divided by the atoms potential what is achieved so that means and others potential what is it is the Coriolis force in depend on the function is the as the gravid he entered the 1st of summer than it is expressed by this term 10 town but so now we have to this kind of these kind of equations and this kind of steady solution for the wind and also for the pressure of I do not want to say something about this ideas I indexes is a general measure a whole to say how imbalanced as an atmospheric flow
but I wanted to make you aware of and attentive to apply and of which is not quite known because usually if you talk about balances and yet was serious say OK again we have the geostrophic balance and we have to hydrostatic balance but what we are usually usually doing is we define the geostrophic balance on the plane all of which is the horizontal plane all which are you which which is actually there was something said plane but to dispel solution which I have written down on the previous slide here
but it also contains the fact that if house if we have a balanced solution then this balance solution must be found on an ISO attacked us face so on an idea about take of the on an idea about and this is in in in atmospheric close this is the potential temperature and ordered to Fayyad so phase of the same potential temperature so it means all flow is not balanced on the on the horizontal plane if if you have on the same time also the temperature in the play other
no defined transformed the creations in which we just and try to move a former late hour the equations of motion and the 1st of summer than makes independent on on the active wind which is the the difference between the actual wind and this a native all balanced solution and and this is what we get you that he that means we have removed all of the other parts of the equations which I balance so here we
have the 3 terms of which might have some parts which I balance so if you add them up they may add and up to serial
and here and these equations we have made this balanced part invisible and therefore all equations so do look a bit different especially in the momentum equation we have only Sweden instead of much more terms and this term here is so small that we can forget about and followers sort of motion so what it means to me draw the main driving term of all atmospheric motions as this the term which looks like a generalized core you as far as much as infect yup just 1 of the significant of forcing term in atmospheric flows of yeah and but as I said
the main difference of it is a balanced solution is that and has a vertical
component and this is what I wanted to show you here is the the color but the color shading is known a high at OFTA's isentropic so faced so do this through so face years no here and and plotted in the defined experiment of for the different types so the way you can see that that do balanced flawed part which is balanced that flows exactly on this black lines CEO which are depend really lines antecedent he's been really eyes lines go up and down and they do to laws and you go to the cells and so on and the interesting thing is no auditory see that this balance solution has a vertical component so was goes up and down but it cannot cool but down all the time even though it is barely eyes lines they crossed the so face you know at the end to end downfall to ideas fuel can no longer follow these generally eyes alliance because it hits the ground and they are the flow cannot be any longer balanced and what I have plotted in what it's like to call us Mr. of vertically in difference that means this is the balanced part off to the the balance part this is the Vedic part of this balanced solution here that would and if is negative out dispensed mint wants to go follow to what's the problem but this is not possible because at the who abundantly we have 2 vertical velocity secret to see what this means of while the active in solid deviation from the balance must be must going up here and it must going down here this and that and that means that the call as just a flashing over the so face and rising die Central and this is the whole of volcanic instability and it's actually working so all or where systems are just about trying to make to make does ISO so phase less steep as what it means our all our who color someone to flush intruders also so it wants to become less steep and at the same time that they have to rush over the ground and therefore a be shifted up and this is what we can see
here and this is the actual of the spectral band when where you mainly see that the idea of into goes up and down of you on the
size and trope them but did active
part which makes that makes that our atmosphere is evolving this is only governed by it is kind of a the black contour CEO pay you can see that upward movement this year and don't but movement is C and of what movement and if you think that it will fill is mainly going from west to east and you can see that the the green lines which are the ice and I central they just shift it here along by pushing them down on this eastern side and pushing them up wonders western side and therefore
in the end if you're looking but what our atmosphere p-adic flow is doing this even in the initial time the half the size and troops which others shit to this the dashed lines you know there are quite steeped in the poem in the mid latitudes and after the passage of a Bauer Clinic wave d use I send troops have a different shape with those of the solid line and they are both shifted up what's you know and shifted downwards here and this is indicated by the mean the deviation from this and in actively and if you would just plot the actual average after vertical velocity we would get this those color shading here and you would see that the wind is going down here and going up here but it does all in balance and therefore it does not change and the distribution of to potential temperature or anything else and then I have also I
wanted to say is this is what I wanted to say about its temperature structure which is governed by the deviation of off to of what good to do what I have
discussed here was the the related to the deviation of the vertical wind from the balanced amend and and distribution of about a given from the balance and go governs the temperature distribution in
atmosphere and no we are interested also in what governs the vorticity and what governs the divergencies and atmosphere and we can rewrite that Our transformed equations in terms of the vorticity and a divergency creation and we see that we have only about 1 forcing terms for both the what is it the equation and a divergent exploration and D forcing terms the vorticity flux yeah the horizontal vorticity flux and a vertical advection term for momentum and this this this seat age as the voice the vector what his city and this is mainly EU or DVD over the set so it's a a change in the velocity so that means they have mainly true only to drive us for the was sort of velocity equation and if if this vector here is divergent then it would drive toward his city and if this vector here is quotation and ended with a drive the divergent and so you can discriminate so you can see that the same vector governs both the what is a and 2 directions and today you can imagine and strongly turbulent fluid flow have an imprint on both the vorticity and divergent now let's look on
a DEC of the divergent
and what is achieved that the divergent and dilatation of this G. back of this vector here here you
see it but I of quotation is part of the spectra and which is the application of part of this vector at the end of the wood Haitian of this and you see that as an imprint of the of the gravity vase which you can see here and here you see the divergency of this vector and the red and the divergent part of Office vector and you see that is it forces the what is city and yeah that and so you can see that the same vector governs bolus the what is achieved and divergent of and you you can
see this in the different tied you see gravity waves are moving up it onto the stratosphere and also order what is it passed the book sold of what is it did that part which was is the what is it equation has also an imprint on the same scale as and 2 divergent part has so that means the order of magnitude is a bit smaller for the what is a key part but it does that at the same the locations s in the real parts the what is it you forcing is still not drawn so therefore we can
also received now dear movement of a gravity base which are divergent part of 2 forcing into the Borgia part of the phones imposing isn't colors and you can see that the relative base and move up into the stratosphere or they can also move against the vendor along of fronts so you have different and no you can ask yourself because this of forcing to here this
vorticity flux term projects on both the vorticity and divergent so can I ask
perhaps if Duflo is strongly nonlinear den the likelihood that it forces of bowl was the what is it and a divergent is large and this is what you for instance here the the shaded black con here did the black dashed lines day indicator forcing for the vorticity and a call-out our indicator forcing for the divergent and to see whether what is city forcing is large at the front and you also that he checked that gravity rates and this is what we want to know and all of research we want to know where the gravity vase to come from and obviously the nonlinearity of to flow is the most important aspect they have yeah it's
you you know