QG-Omega Equation

[LocoAko]

I know this might be kind of a big order, but I figured it might be interesting to have a thread for those who didn't do meteorology in school about things like the quasi-geostrophic omega equation where seasoned meteorologists and those good at explaining things (I'm looking at you, BI ) could explain things in non-technical terms and provide a physical basis for why certain things work. For starters, we know the QG-Omega equation,

(σ∇_p²+f²(∂²/∂p²))ω= -(R/p)∇_p²(-V_g∙∇_pT)+ f ∂/∂p[V_g∙∇_p (ξ_g+f)]

where (for reference purposes)

-(R/p)∇_p²(-V_g∙∇_pT) = Laplacian of temperature advection

f ∂/∂p[V_g∙∇_p (ξ_g+f)] = Differential vorticity advection

provides a qg way of diagnosing vertical motions, and while I have a decent grasp at it, I think it would be useful for someone to explain physically why the laplacian term and the differential vorticity advection term actually cause upward vertical motion to occur since I see vorticity advection, etc. referenced fairly often on the board (and so much time was spent actually deriving this thing that I was left a little befuddled as to the actual physical understanding of it, which I consider more useful). I have referenced http://www.cimms.ou.edu/~doswell/PVAdisc/PVA.html before which provides a good explanation but I'd like to hear some other explanations, if possible.

[tornadotony]

I'm going to try my hand at explaining this for those who haven't had an atmospheric dynamics course (or two, depending on the program).

Keep in mind that there are plenty of caveats to this and I am leaving many details out that would likely make dynamics profs cringe at this post. I will attempt to cover some of them but invariably not all of them. This is more for a basic explanation/understanding.

Here is the QG-Omega equation, rewritten as it is in Jake's post:

(σ∇_p²+f²(∂²/∂p²))ω= -(R/p)∇_p²(-V_g∙∇_pT)+ f ∂/∂p[V_g∙∇_p (ξ_g+f)]

∇_p² -> LaPlace operator with respect to constant pressure surfaces. Forget the mathematical scariness that it brings. The easiest way to think about it is that it reverses the sign of whatever value it modifies. Therefore, if we solve the QG-Omega equation and get an answer that is the positive LaPlacian of omega, then omega itself will solve to be negative.

ω -> Omega -> Vertical motion as referenced to constant PRESSURE (not HEIGHT) surfaces. Most forum readers here are familiar with the fact that we study the atmosphere on surfaces of constant pressure and not height (300mb, 500mb, 700mb, 850mb, etc.). Omega is the vertical motion with respect to those surfaces. Thus, RISING motion is associated with a NEGATIVE omega, and SINKING motion is associated with a POSITIVE omega.

(σ∇_p²+f²(∂²/∂p²))ω -> LaPlacian of Omega: Think of this as Omega with its sign reversed. Thus, a POSITIVE LAPLACIAN of Omega represents RISING motion, and a NEGATIVE LAPLACIAN of Omega represents SINKING motion.

T -> Temperature

-(R/p)∇_p²(-V_g∙∇_pT) -> Negative LaPlacian of temperature advection on constant pressure surfaces. This is pretty much the equivalent of a double negative of temperature advection. Warm air advection (WAA) is positive and cold air advection (CAA) is negative. Thus, if we have WAA at a given location, the QG-Omega response would be for air to RISE, because WAA is positive, the double-negative of a positive is positive, so solving for the LaPlacian of Omega would yield a positive value, which would indicate a negative Omega, and thus RISING motion would result. Oppositely, if we have CAA at a given location, the QG-Omega response would be for air to SINK, because CAA is negative, the double-negative of a negative is negative, so solving for the LaPlacian of Omega would yield a negative value, which would be a positive Omega, and positive Omega indicates SINKING motion.

(ξ_g+f) -> Absolute vorticity: This is the total vorticity in the atmosphere, due both to planetary (Coriolis) effects and relative effects.

f ∂/∂p[V_g∙∇_p (ξ_g+f)] -> Differential absolute vorticity advection on constant pressure surfaces. This is difficult to explain, but I'll give it my best shot. The differential part of this is crucial. We care more about how vorticity advection CHANGES WITH HEIGHT. Cyclonic vorticity advection itself is the horizontal advection of parcels with greater cyclonic (positive in the N Hemisphere) vorticity, while anticyclonic vorticity advection occurs with the advection of air parcels containing more anticyclonic (negative in the N Hem) vorticity. The focus here is not so much on whether or not the advection at any certain level is cyclonic or anticyclonic, but rather on HOW THAT ADVECTION CHANGES WITH HEIGHT. Thus, if I'm at 850mb and experiencing no vorticity advection, if I were to rise through the atmosphere, would the value for vorticity advection become more cyclonic, more anticyclonic, or would it remain 0? If I were to experience MORE CYCLONIC vorticity advection as I rose, then I would say I'm experiencing DIFFERENTIAL CYCLONIC VORTICITY ADVECTION (dCVA). Likewise, if I were to experience MORE ANTICYCLONIC vorticity advection, I would say I'm experiencing DIFFERENTIAL ANTICYCLONIC VORTICITY ADVECTION (dAVA). Note that I could easily be experiencing strong cyclonic vorticity advection at 850mb that weakens as I go higher aloft. If that were the case, even if the vorticity advection itself remained CYCLONIC through the atmosphere, I would be experiencing dAVA, not dCVA. The vice-versa is also true; I could experience strong anticyclonic vorticity advection at 850mb that weakens as I go aloft and it would be dCVA. For QG-Omega, dCVA is POSITIVE and dAVA is NEGATIVE. Thus, dCVA would result is a positive LaPlacian of Omega, a negative Omega, and RISING motion. Likewise, dAVA would result in a negative LaPlacian of Omega, a positive Omega, and SINKING motion.

NOTE: We cheat with dCVA and dAVA in forecasting. Because of how cyclones and anticyclones form (another long-winded post for some other time), we can use 500mb vorticity to estimate dCVA and dAVA, with cyclonic vorticity advection at 500mb typically indicative of dCVA, and anticyclonic vorticity advection at 500mb typically indicative of AVA.

Now, there's a big issue with QG-Omega. It is very possible to have the negative LaPlacian of Temperature Advection and differential vorticity advection terms work against each other (WAA and dAVA, CAA and dCVA). How do we deal with that? Theoretically, we have formulations that eliminate that potential issue (Q-Vectors and Trenberth) that go beyond the scope of this post. Operationally, it takes common sense. If you have dCVA and WAA working together at a location, you will have rising air. Likewise, if you have dAVA and CAA working together, you will have sinking motion. If you have conflicting forcings, then some common sense is required. For example, if you have weak CAA and strong DCVA, you may have some rising motion. If you have strong CAA and weak dCVA, you will likely have sinking motion instead of rising motion. If you have strong WAA and weak dAVA, you will likely have some rising motion instead of sinking motion. If you have weak WAA and strong dAVA, you will likely have some sinking motion. If all comes down to common sense and being able to recognize which factors may hold greater weight in a forecasting situation. This is partly why forecasting is such a challenge!

[tornadotony]

As for physically why WAA and dCVA lead to rising motion and CAA and dAVA lead to sinking motion, this link from Millersville contains some darn good, straightforward explanations.

http://www.atmos.millersville.edu/~lead/OMEGA/QG_Omega_Home.html

[LocoAko]

As for physically why WAA and dCVA lead to rising motion and CAA and dAVA lead to sinking motion, this link from Millersville contains some darn good, straightforward explanations.

http://www.atmos.millersville.edu/~lead/OMEGA/QG_Omega_Home.html

Good stuff. This link also seems worthwhile - http://marrella.aos.wisc.edu/~swetzel/winter/ingredients.html - although it still doesn't exactly explain physically what these processes do (your Millersville link comes much closer).

One thing I'd add on that is worth noting is the term in the parenthesis with the omega is a stability parameter, showing that for a given forcing (from the RHS), higher stability will result in lower omega and vice versa. This is intuitive, I suppose, but worth pointing out.

Also, while I'm hesitant to admit my confusion - I thought that the thermal advection term, having the laplacian operator, meant that it was non-uniform temperature advection that contributed to UVM/DVM. So if you had a field where there was temperature advection of exactly the same magnitude everywhere, it wouldn't contribute to UVM. Am I wrong on that? Of course, in the physical world, you couldn't have identical temperature advection everywhere as that wouldn't make sense and it would have to lessen somewhere, but I digress..

[usedtobe]

Good stuff. This link also seems worthwhile - http://marrella.aos....ngredients.html - although it still doesn't exactly explain physically what these processes do (your Millersville link comes much closer).

One thing I'd add on that is worth noting is the term in the parenthesis with the omega is a stability parameter, showing that for a given forcing (from the RHS), higher stability will result in lower omega and vice versa. This is intuitive, I suppose, but worth pointing out.

Also, while I'm hesitant to admit my confusion - I thought that the thermal advection term, having the laplacian operator, meant that it was non-uniform temperature advection that contributed to UVM/DVM. So if you had a field where there was temperature advection of exactly the same magnitude everywhere, it wouldn't contribute to UVM. Am I wrong on that? Of course, in the physical world, you couldn't have identical temperature advection everywhere as that wouldn't make sense and it would have to lessen somewhere, but I digress..

That correct. If all the air is moving the same speed there really is no cold air for the warm air to lift over. That's why you can infer stronger lifting during synoptic type event with warm advection when the thickness lines do not move to the north. Instead of warming , your getting lifting. During snowstorms, I always get excited when the thcikness lines were crashing south or east in the face of what looked like warm advection.However, it also does have implications about the sign as discussed earlier in the thread.

[kylemacr]

You can also think about it mathematically. A Laplacian is essentially a gradient of a gradient. You are really looking at the gradient of the gradient of temperature advection. If you have a uniform temperature advection field, then the gradient of temperature advection doesn't vary and therefore has no gradient and you are stuck with zero.

[baroclinic_instability]

Good stuff. This link also seems worthwhile - http://marrella.aos....ngredients.html - although it still doesn't exactly explain physically what these processes do (your Millersville link comes much closer).

One thing I'd add on that is worth noting is the term in the parenthesis with the omega is a stability parameter, showing that for a given forcing (from the RHS), higher stability will result in lower omega and vice versa. This is intuitive, I suppose, but worth pointing out.

Also, while I'm hesitant to admit my confusion - I thought that the thermal advection term, having the laplacian operator, meant that it was non-uniform temperature advection that contributed to UVM/DVM. So if you had a field where there was temperature advection of exactly the same magnitude everywhere, it wouldn't contribute to UVM. Am I wrong on that? Of course, in the physical world, you couldn't have identical temperature advection everywhere as that wouldn't make sense and it would have to lessen somewhere, but I digress..

Super important...something not noted very often, and this is a reason why a moist baroclinic zone/WCB is a common "ingredient" in rapid cyclogenesis. I discuss it briefly here. http://jasonahsenmac...lf-development/

I will also add stability plays a huge role in the froude number calculation and mtn meteorology/terrain meteorology. It isn't just the synoptic omega/chi equations. As you said, it is intuitive that a highly stable airmass will tend to subside/sink and vice versa.

And also midway through in this longer post.

http://jasonahsenmac...icity-thinking/

Hence why I find upper level PV maps/low level theta-e/theta maps so useful.

[isohume]

Doswell and others have had some good talks/papers on the physical interpretation of QG forcing. Mainly that QG Omega is a hypothetical vertical motion that is a response to instantaneously restore geostrophic and hydrostatic balance, since QG theory assumes geostrophic and hydrostatic balance to begin with. It has no reality and cannot be measured the same way as a real flow can be measured.

In any case...here are 3 qualitative and useful interpretations of the QG Omega equation related to vertical motions:

Vorticity Advection term

1. Trof embedded in westerlies....implies (in a real sense) a blob of vorticity.

A parcel entering this trof from behind means the air must converge to increase it's vorticity and conservation of angular momentum requires the spin to increase. From this reasoning...parcels upstream are in a region of anticyclonic vorticity advection (AVA) and are having to converge...and on the other side of the trof axis parcels are in cyclonic vorticity advection (CVA) and they are diverging. So...AVA implies convergence and CVA implies divergence.

According to the the Law Of Mass Continuity...llvl conv changes to upper div with a level of non-divegergence (LND) somewhere in between. If CVA is to imply upward motion, then, the level where vorticity advection is measured must be above the LND...ie: in the area of increasing div.

Hence, "CVA equals upward motion" depends on a series of assumptions:

1. Parcels are moving faster than the troughs

2. Parcels seek to maintain vorticity values in equilibrium with their surroundings

3. The level where vorticity advection is being evaluated is above the LND

4. QG theory is a good approximation to the real flow

5. The only contribution to vertical motion is related to vorticity advection

2. Another interpretation of the QG Omega equation involves imagining a vertically aligned column of vorticity embedded in sheared flow.

After a while you'd expect this column to be tilted. If you remove the mean wind (which only serves to move the system from west to east and does not affect the tilt) you get a revised look.

This arrives at the same flow described in interpretation #1 above...mainly AVA upstream and CVA downstream of the vort axis.

It is this conv/div pattern that counteracts the tendency of advection to tilt the tube....and the associated vertical motions are a direct consequence of CVA/AVA.

3. The 3rd interpretation is related to the QG Height Tendency Equation.

Basically that equation states that vorticity advection is directly associated with height changes (assuming the system is not deepening or filling from differential thermal advection).

In it's simplified form...this interpretation means that CVA produces height falls proportional to the magnitude of the vorticity advection. If CVA increases with height (which is normally the case), then the heights are falling more with height and the depth of the atmos btw the sfc and some upper level is decreasing.

According to the hypsometric equation, the thickness of a layer between two pressure levels is related to the mean temperature in between the pressure levels. So...if the thickness decreases like above, the mean temperature is also decreasing. In an adiabatic atmosphere and in the absence of thermal advection, the way a column must cool is by ascent. Thus vertical motion is created by height falls, which in turn are created by magnitude of the dCVA.

The Thermal Advection term

In many synoptic cases the thermal advection term is actually the dominant physical process associated with large scale ascent. Since the atmos becomes more barotropic with height due to increasingly w/ly flow...thermal advection is more of a llvl feature, but not always. Anyway...in QG theory, the flow is assumed to be adiabatic, which means the parcels potential temperature does not change. It is conserved, which means the parcel must rise into regions of the same potential temp under waa flow (vice versa for caa). Basically, parcels follow potential temperature surfaces, which rise up in waa and desend during caa.

Many times...the thermal and vorticity terms work against each other as far a contributing to overall vertical motions. But when they combine, or couple, there is deep layered omega (pos or neg) and the potential for strong cyclogenesis/storms (neg omega) or stable conditions with potentially high wind momentum transfers allowed to the surface (pos omega).

[Chinook]

Also, while I'm hesitant to admit my confusion - I thought that the thermal advection term, having the laplacian operator, meant that it was non-uniform temperature advection that contributed to UVM/DVM. So if you had a field where there was temperature advection of exactly the same magnitude everywhere, it wouldn't contribute to UVM. Am I wrong on that? Of course, in the physical world, you couldn't have identical temperature advection everywhere as that wouldn't make sense and it would have to lessen somewhere, but I digress..

The advection term (of any field) is the gradient of that field dot product with wind vectors. Since there's variations in the wind field, variations in the gradient vector field, and variations in the relative angles of the wind vector and the gradient vector, the advection of the field may change rapidly in space.

When you take the laplacian, it changes even more rapidly in space, since you are taking a type of a second derivative. Derivatives always make things more noisy.

If you look at a map of vorticity advection (at one level) and a map of the laplacian of the temperature advection, you would see rapid changes in space for both fields. That makes it messy to conceptualize. Then try to visualize the differential vorticity advection with height. That throws me off.