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The omega equation is a culminating result in synoptic-scale meteorology. It is an elliptic partial differential equation, named because its left-hand side produces an estimate of vertical velocity, customarily^[1] expressed by symbol $\omega$ , in a pressure coordinate measuring height the atmosphere. Mathematically, $\omega ={\frac {dp}{dt}}$ , where ${d \over dt}$ represents a material derivative. The underlying concept is more general, however, and can also be applied^[2] to the Boussinesq fluid equation system where vertical velocity is $w={\frac {dz}{dt}}$ in altitude coordinate z.

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Topics in Advanced Storm Spotting Welcome to advanced spotter training, in this video we will dive into the complicated, yet fascinating world of severe storm ingredients. It is our hope, that after completing this training, you will have a greater understanding, and respect of the complicated combination of ingredients that must exist for severe storms to develop and thrive. But before we begin Recall a few basic equations Alright, before we begin, we need to go ahead and recall a few basic equations, first off, we’ll need our set of variables, our equations of motion, different forecast equations, my favorite, the quasigeostrophic omega equation, the basic equations of conservation of momentum, energy, mass, and water, not forgetting the equation of state, and finally we’ll need the Hypsometric equation to complete the derivation of the second law of thermodynamics. OK, I know, right now you’re thinking, they’re kidding right? Well I’m not, or am I? The Ingredients for Severe Storms Well, we’re all in luck, because we’re not going to go that in depth in this training. In general, we can break down the ingredients of severe storms into four separate, but easily understood components. Moisture, Instability, Lift, and wind shear. With wind shear playing a vital role in the development of supercell thunderstorms. The Ingredients for Severe Storms So lets start from the beginning. Nearly all moisture for thunderstorms across the plains is transported from the Gulf of Mexico. Commonly, southerly winds ahead of a strong storm system moving off the western high plains drives this transportation of moisture. To the west, behind a cold front and/or dry line (which we will describe later) very dry air is transported off the desert southwest. So how do we measure the available moisture? We;;. we use the dew point temperature. In short, the dew point temperature, is the temperature at which water condenses, the higher the dew point, the more water vapor in the air. The dew point can never be more than the observed temperature, and if the temperature falls, the dew point will follow. When the temperature and the dew point are equal, the air is completely saturated. The Ingredients for Severe Storms Now that we have discussed moisture, lets discuss how it plays a major role in the development of Instability. In short, instability is the term meteorologists use to describe the atmosphere's tendency to encourage or deter vertical motion. In the simplest sense, we need warm moist air to rise for thunderstorms to develop. For instability to exist, we usually have warm moist air in the lowest levels of the atmosphere, with generally cooler air aloft. When you think instability, one only needs to picture a hot air balloon, which uses instability to its advantage. The air within the balloon is much warmer than that of the surrounding air. Therefore, the warm air creates buoyancy. Now working against buoyancy is gravity, when the buoyancy is greater than that of the downward force of gravity, the net gain is your instability. The Ingredients for Severe Storms Now that we know what instability is, how do we measure it in the atmosphere, how do we quantify it? We use a measurement called, CAPE, which stands for Convective Available Potential Energy. We measure CAPE by looking at the atmosphere in the vertical, one of the best ways of doing this is by releasing a weather balloon and actually sampling the atmosphere. The data that is returned by the weather balloon is plotted on a Skew-t diagram, which we call a sounding, here are two basic examples of a skew-t diagram. On the sounding we theoretically lift a parcel of air, either from the surface or from a desired height in the atmosphere, which is represented by the yellow line on the two soundings above. We then measure the area between the temperature, the red line, and the yellow line, the trajectory of the parcel. That area is the computed CAPE, which has units of Joules per Kilogram. The distribution of CAPE in the atmosphere plays a critical role in thunderstorm development, both Sounding A and Sounding B have the same value of CAPE, but notice their distribution is very different, the “Fatter” CAPE of Sounding A may be more beneficial for development than the “skinny” CAPE of Sounding B. The Ingredients for Severe Storms So, now that we have discussed what instability is, lets go ahead and discuss the cap and it’s role in the development and sometimes hindrance of thunderstorms. Given this is advanced training, many, if not all of you, have likely heard one time or another about the cap. Either the cap is limiting thunderstorm development, or the cap is eroding and development has started. So what is the cap? The cap is basically a region in the atmosphere where air temperature increases with height instead of decreasing, the more technical term is, temperature inversion. So lets say we have a parcel of warm air, and it’s rising through cooler air within the environment, so it’s unstable. But then our parcel rises into an environment that is warmer than itself. Once the parcel reaches a level where its temperature equals that of the inversion, it stops rising, our parcel has become stable. Here is quick way to visualize the cap in three dimensions. We have the surface, with our dry line and region of warm moisture laden air advecting into the region, and above, at approximately 3000 feet, we have a layer of warm or hot air. This is our cap, or temperature inversion if you will. For forecasters and spotters, the easiest way to observe the Cap is to look at either an operational or model sounding. Here we have a model sounding from April of 2013. Notice the temperature begins to rise with height once you get to a height of approximately 3000 to 4000 feet, this is your cap. The Ingredients for Severe Storms Sometimes you can actually visualize the temperature inversion with your own eyes. I would imagine many of you watching right now, have seen smoke stacks, either from factories or power plants where the rising smoke, which is unstable, suddenly begins to flatten out and spread like it’s hit a ceiling. The region where the smoke begins to spread, is the location of the Cap, or temperature inversion. The Ingredients for Severe Storms Ok, so we’ve discussed moisture, we’ve discussed instability, and we’ve covered the role and impact of the cap. Now, thunderstorms can form with only the lift available from instability, when no capping inversion is present. We typically refer to these storms as pulse thunderstorms, but we’ll dive more into thunderstorm types in another video. For the moment, lets focus on what many consider, the most severe type of thunderstorm, the supercell. Now from the previous slides, you might be thinking, well the Cap is bad, but au contraire, The cap plus instability play a vital role in the development of strong, severe, thunderstorms, typically supercellular thunderstorms. With that said, with a cap in place early in the day into the early afternoon, you can “build up” CAPE at the surface and above the cap, this is especially true on days when there is a fairly stout inversion. Now, this is where lift comes into play, we need a source of lift, to give a “nudge” upward, to help our parcel of air “bust through” the cap and rise into the unstable air above it. Lift comes in many forms, the most common being that of warm and cold fronts, dry lines, and outflow from current or previous thunderstorms. The Ingredients for Severe Storms Let us now focus on the outflow scenario. Thunderstorm development due to outflow is a particularly fascinating way of generating lift. This is because, for outflow boundaries to be present, there must already be thunderstorms or the remnants of thunderstorms in the general area. Take the example shown, we have a line of thunderstorms shown on radar. Extending out from the storms is a visual boundary picked up on the base reflectivity scan. The storm farthest to the left, likely developed along the outflow boundary which was generated by the storm to its right. Another example would be that of morning boundaries left by mesoscale convective systems or MCSs. In short, MCSs are just a large cluster of thunderstorms that usually develop late in the evening and persist through the morning hours, but as mentioned before, we’ll go more into thunderstorm types in another video. However, sometimes the boundaries left by MCSs can linger through the afternoon, providing a source of lift for thunderstorms. Ok, now that I’ve bored you with all that talk about lift, instability, the cap… and… what was it? Oh yeah, moisture… lets get into some serious fluid dynamics, recall those equations? Right then, only kidding, but seriously lets talk about wind shear, one of the most vital components in the development of severe thunderstorms. In the simplest terms, wind shear is generated by increasing wind speed and/or direction with height. We need deep, strong wind shear for the development of supercells. Wind shear is measured in knots and we usually look for deep wind shear within a layer from the surface to 6 km above the ground. Now for the development of tornadoes, we also need strong low level wind shear, and for this we usually look at the surface to about 1 to 3 km. Here is a simple example of how wind shear results in the development of rotation within storms. First off, you have very strong winds aloft, notice the high clouds are tilted in the direction of the upper level flow. Meanwhile, the winds near the surface are slower. The difference between these two wind speeds, the shear, and since we’re only dealing with changes in speed with height, we’d call this speed shear. This results in a rotational component, this rotation, is vital for the development of severe supercellular thunderstorms. The Ingredients for Supercells Ok, so lets take this a step further, what if we add a directional component to our shear profile. So we not only have winds increasing in speed with height, we also have a change in direction. Here our surface winds start out from the southeast and as we move up in the atmosphere, our winds veer, or turn clockwise, with height. So, as with the previous example, we still have our rotational component in the horizontal. Now, if we add in updraft, we begin to tilt this rotation into the vertical. This is a critical component to the development of supercellular thunderstorms. The Ingredients for Supercells Alright, so now we have our established rotating updraft. Recall from basic spotter training, that a supercell is defined as having a persistent deep rotating updraft, which we call the mesocylone. In tornadic supercells, the mesocyclone is the source of rotation for the tornado. Remember, all supercells have a mesocylcone, but not all supercells result in a tornado. In fact, there are thousands of supercellular storms every year across the world, but only a fraction of these generate a tornado. Now I know there is a lot going on in this image, and we’ll dive into the nuts and bolts of supercells in another video, but for right now, the major take away from this video is to understand and respect all of the intricate ingredients that have to combine for supercells, and severe storms to develop. In closing, I just want to remind all of you that safety is your number one priority while spotting! Don’t risk your life for a video, photograph, or report. Always report what you see, don’t ever assume its already been reported. Make sure you stay in touch, follow us on Twitter, like us on Facebook, and definitely subscribe to our YouTube channel. We will continue to post more videos diving into advanced spotter topics as the months move on. We have a lot of topics to cover, such as storm types and structure, safe spotting practices, where to find and how to monitor data. The topics are endless! And finally, thank you for your interest in advanced spotter training. Let us know what you think of this video in the comments section below. Spotters are our eyes in the field and we appreciate all you do. And with that, this concludes the Severe Storms Ingredients Advanced Spotter Training video from the National Weather Service Norman, Oklahoma Forecast Office.

Concept and summary

Vertical wind is crucial to weather and storms of all types. Even slow, broad updrafts can create convective instability or bring air to its lifted condensation level creating stratiform cloud decks. Unfortunately, predicting vertical motion directly is difficult. For synoptic scales in Earth's broad and shallow troposphere, the vertical component of Newton's law of motion is sacrificed in meteorology's primitive equations, by accepting the hydrostatic approximation. Instead, vertical velocity must be solved through its link to horizontal laws of motion, via the mass continuity equation. But this presents further difficulties, because horizontal winds are mostly geostrophic, to a good approximation. Geostrophic winds merely circulate horizontally, and do not significantly converge or diverge in the horizontal to provide the needed link to mass continuity and thus vertical motion.

The key insight embodied by the quasi-geostrophic omega equation is that thermal wind balance (the combination of hydrostatic and geostrophic force balances above) holds throughout time, even though the horizontal transport of momentum and heat by geostrophic winds will often tend to destroy that balance. Logically, then, a small non-geostrophic component of the wind (one which is divergent, and thus connected to vertical motion) must be acting as a secondary circulation to maintain balance of the geostrophic primary circulation. The quasi-geostrophic omega $\omega _{QG}$ is the hypothetical vertical motion whose adiabatic cooling or warming effect (based on the atmosphere's static stability) would prevent thermal wind imbalance from growing with time, by countering the balance-destroying (or imbalance-creating) effects of advection. Strictly speaking, QG theory approximates both the advected momentum and the advecting velocity as given by the geostrophic wind.

In summary, one may consider the vertical velocity that results from solving the omega equation as that which would be needed to maintain geostrophy and hydrostasy in the face of advection by the geostrophic wind.^[1]

The equation reads:

\sigma \nabla _{\text{H}}^{2}\omega +f^{2}{\frac {\partial ^{2}\omega }{\partial p^{2}}}=f{\frac {\partial }{\partial p}}\left[\mathbf {V} _{\text{g}}\cdot \nabla _{\text{H}}\left(\zeta _{\text{g}}+f\right)\right]-\nabla _{\text{H}}^{2}\left(\mathbf {V} _{\text{g}}\cdot \nabla _{\text{H}}{\frac {\partial \phi }{\partial p}}\right)

(1)

where $f$ is the Coriolis parameter, $\sigma$ is related to the static stability, $\mathbf {V} _{\text{g}}$ is the geostrophic velocity vector, $\zeta _{\text{g}}$ is the geostrophic relative vorticity, $\phi$ is the geopotential, $\nabla _{\text{H}}^{2}$ is the horizontal Laplacian operator and $\nabla _{\text{H}}$ is the horizontal del operator.^[3] Its sign and sense in typical weather applications^[4] is: upward motion is produced by positive vorticity advection above the level in question (the first term), plus warm advection (the second term).

Derivation

The derivation of the $\omega$ equation is based on the vertical component of the vorticity equation, and the thermodynamic equation. The vertical vorticity equation for a frictionless atmosphere may be written using pressure as the vertical coordinate:

{\frac {\partial \xi }{\partial t}}+V\cdot \nabla \eta -f{\frac {\partial \omega }{\partial p}}=\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)+{\hat {k}}\cdot \nabla \omega \times {\frac {\partial V}{\partial p}}

(2)

Here $\xi$ is the relative vorticity, $V$ the horizontal wind velocity vector, whose components in the $x$ and $y$ directions are $u$ and $v$ respectively, $\eta$ the absolute vorticity $\xi +f$ , $f$ is the Coriolis parameter, $\omega ={\frac {dp}{dt}}$ the material derivative of pressure $p$ , ${\hat {k}}$ is the unit vertical vector, $\nabla$ is the isobaric Del (grad) operator, $\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)$ is the vertical advection of vorticity and ${\hat {k}}\cdot \nabla \omega \times {\frac {\partial V}{\partial p}}$ represents the "tilting" term or transformation of horizontal vorticity into vertical vorticity.^[5]

The thermodynamic equation may be written as:

{\frac {\partial }{\partial t}}\left(-{\frac {\partial Z}{\partial p}}\right)+V\cdot \nabla \left(-{\frac {\partial Z}{\partial p}}\right)-k\omega ={\frac {R}{C_{\text{p}}\cdot g}}\cdot {\frac {Q}{p}}

(3)

where $k\equiv \left({\frac {\partial Z}{\partial p}}\right){\frac {\partial }{\partial p}}\ln \theta$ , in which $Q$ is the heating rate (supply of energy per unit time and unit mass), $C_{\text{p}}$ is the specific heat of dry air, $R$ is the gas constant for dry air, $\theta$ is the potential temperature and $\phi$ is geopotential $(gZ)$ .

The $\omega$ equation (1) is obtained from equation (2) and (3) by casting both equations in terms of geopotential Z, and eliminating time derivatives based on the physical assumption that thermal wind imbalance remains small across time, or d/dt(imbalance) = 0. For the first step, the relative vorticity must be approximated as the geostrophic vorticity:

\xi ={\frac {g}{f}}\nabla ^{2}Z

Expanding the final "tilting" term in (2) into Cartesian coordinates (although we will soon neglect it), the vorticity equation reads:

{\frac {\partial }{\partial t}}\left({\frac {g}{f}}\nabla ^{2}Z\right)+V\cdot \nabla \eta -f{\frac {\partial \omega }{\partial p}}=\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)+\left({\frac {\partial \omega }{\partial y}}{\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}{\frac {\partial v}{\partial p}}\right)

(4)

Differentiating (4) with respect to $p$ gives:

{\frac {g}{f}}{\frac {\partial }{\partial t}}\nabla ^{2}\left({\frac {\partial Z}{\partial p}}\right)+{\frac {\partial }{\partial p}}(V\cdot \nabla \eta )-f{\frac {\partial ^{2}\omega }{\partial p^{2}}}={\frac {\partial }{\partial p}}\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)+{\frac {\partial }{\partial p}}\left({\frac {\partial \omega }{\partial y}}\cdot {\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}\cdot {\frac {\partial v}{\partial p}}\right)

(5)

Taking the Laplacian ( $\nabla ^{2}$ ) of (3) gives:

{\frac {\partial }{\partial t}}\nabla ^{2}\left(-{\frac {\partial Z}{\partial p}}\right)+\nabla ^{2}[V\cdot \nabla \left(-{\frac {\partial Z}{\partial p}}\right)]-\nabla ^{2}k\omega ={\frac {R}{C_{\text{p}}\cdot g}}\cdot {\frac {\nabla ^{2}Q}{p}}

(6)

Adding (5) to g/f times (6), substituting $gk=\sigma$ , and approximating horizontal advection with geostrophic advection (using the Jacobian formalism) gives:

\nabla ^{2}\omega +{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega }{\partial p^{2}}}={\frac {1}{\sigma }}\left[{\frac {\partial }{\partial p}}J(\phi ,\eta )+{\frac {1}{f}}\nabla ^{2}J\left(\phi ,-{\frac {\partial \phi }{\partial p}}\right)\right]-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left({\frac {\partial \omega }{\partial y}}\cdot {\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}\cdot {\frac {\partial v}{\partial p}}\right)-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right){\frac {R\cdot \nabla ^{2}Q}{C_{\text{p}}\cdot S\cdot p}}

(7)

Equation (7) is now a diagnostic, linear differential equation for $\omega$ , which can be split into two terms, namely $\omega _{1}$ and $\omega _{2}$ , such that:

\nabla ^{2}\omega _{1}+{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega _{1}}{\partial p^{2}}}={\frac {1}{\sigma }}\left[{\frac {\partial }{\partial p}}J(\phi ,\eta )+{\frac {1}{f}}\nabla ^{2}J\left(\phi ,-{\frac {\partial \phi }{\partial p}}\right)\right]-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left({\frac {\partial \omega }{\partial y}}\cdot {\frac {\partial u}{\partial p}}-{\frac {\partial \omega }{\partial x}}\cdot {\frac {\partial v}{\partial p}}\right)-{\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\left(\xi {\frac {\partial \omega }{\partial p}}-\omega {\frac {\partial \xi }{\partial p}}\right)

(8)

and

\nabla ^{2}\omega _{2}+{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega _{2}}{\partial p^{2}}}={\frac {R\cdot \nabla ^{2}Q}{C_{\text{p}}\cdot \sigma \cdot p}}

(9)

where $\omega _{1}$ is the vertical velocity attributable to all the flow-dependent advective tendencies in Equation (8), and $\omega _{2}$ is the vertical velocity due to the non-adiabatic heating, which includes the latent heat of condensation, sensible heat fluxes, radiative heating, etc. (Singh & Rathor, 1974). Since all advecting velocities in the horizontal have been replaced with geostrophic values, and geostrophic winds are nearly nondivergent, neglect of vertical advection terms is a consistent further assumption of the quasi-geostrophic set, leaving only the square bracketed term in Eqs. (7-8) to enter (1).

Interpretation

Equation (1) for adiabatic $\omega _{QG}$ is used by meteorologists and operational weather forecasters to anticipate where upward motion will occur on synoptic charts. For sinusoidal or wavelike motions, where Laplacian operators act simply as a negative sign,^[4] and the equation's meaning can be expressed with words indicating the sign of the effect: Upward motion is driven by positive vorticity advection increasing with height (or PVA for short), plus warm air advection (or WAA for short). The opposite signed case is logically opposite, for this linear equation.

In a location where the imbalancing effects of adiabatic advection are acting to drive upward motion (where $\omega _{QG}<0$ in Eq. 1), the inertia of the geostrophic wind field (that is, its propensity to carry on forward) is creating a demand for decreasing thickness ${\frac {-\partial Z}{\partial p}}$ in order for thermal wind balance to continue to hold. For instance, when there is an approaching upper-level cyclone or trough above the level in question, the part of $\omega _{QG}$ attributable to the first term in Eq. 1 is upward motion needed to create the increasingly cool air column that is required hypsometrically under the falling heights. That adiabatic reasoning must be supplemented by an appreciation of feedbacks from flow-dependent heating, such as latent heat release. If latent heat is released as air cools, then an additional upward motion will be required based on Eq. (9) to counteract its effect, in order to still create the necessary cool core. Another way to think about such a feedback is to consider an effective static stability that is smaller in saturated air than in unsaturated air, although a complication of that view is that latent heating mediated by convection need not be vertically local to the altitude where cooling by $\omega _{QG}$ triggers its formation. For this reason, retaining a separate Q term like Equation (9) is a useful approach.^[6]

References

^ ^a ^b Holton, James (2004). An Introduction to Dynamic Meteorology. Elsevier Academic Press. ISBN 0123540151.
^ Davies, Huw (2015). "The Quasigeostrophic Omega Equation: Reappraisal, Refinements, and Relevance". Monthly Weather Review. 143 (1): 3–25. Bibcode:2015MWRv..143....3D. doi:10.1175/MWR-D-14-00098.1.
^ Holton, J.R., 1992, An Introduction to Dynamic Meteorology Academic Press, 166-175
^ ^a ^b "Quasi-Geostrophic Omega Equation Lab". METEd, CoMET program. Retrieved 10 November 2019.
^ Singh & Rathor, 1974, Reduction of the Complete Omega Equation to the Simplest Form, Pure and Applied Geophysics, 112, 219-223
^ Nie, Ji; Fan, Bowen (2019-06-19). "Roles of Dynamic Forcings and Diabatic Heating in Summer Extreme Precipitation in East China and the Southeastern United States". Journal of Climate. 32 (18): 5815–5831. Bibcode:2019JCli...32.5815N. doi:10.1175/JCLI-D-19-0188.1. ISSN 0894-8755.

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This page was last edited on 14 January 2024, at 23:10

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