Trying to understand baroclinic instabilities

[Steven Scott]

Hi everyone,

I understand that a baroclinic atmosphere is one in which the height contours and isotherms intersect one another, leading to a redistribution of heat around the globe through cold and warm air advection.

What I'm having trouble with is understanding what a baroclinic instability actually is and how this relates to the above definition. My understanding is that it's an instability arising out of a horizontal temperature gradient that extracts potential energy from the background flow which leads to its intensification as a low pressure system.

Does that mean that a baroclinic instability is what we see as a low pressure system, with the archetypal comma cloud, etc?

Thanks!

 

[Jeff Duda]

Baroclinic instability summed up as concisely as possible (without details given as to why): if a horizontal temperature gradient exists, "stuff" is going to happen to neutralize that gradient.

Slightly more detail: a horizontal temperature gradient (especially on an isobaric surface) necessarily implies there will be a density gradient. Density -> buoyancy -> vertical motion -> temperature change and potential diabatic heating through latent heating -> "change"

 

[Ashwin D]

Is there a reference you can provide that will show me how to calculate the waves in the 500 or 200 hPa surfaces ? My programming language is Python and I want to identify baroclinic waves in the 3 to 5 day period.

 

[Ashwin D]

No response so I thought maybe I could throw an answer to my own question and have the experts correct and suggest improvements.

In the mid latitudes we take the geopotential heights at the 500 hPa level (both GFS analysis and forecast).

So the thing is to calculate the anomalies by taking the difference of the forecast and analysis data at the 500 hPa level.

Then take a Fourier transform of the geopotential height anomalies and then look for periodicity in the Fourier transform. Generally baroclinic waves are present at the 3 and 5 day period (maybe 7 as well).

Obviously there is a lot more detail on what exactly goes into the Fourier transform. Is it just the anomalies or is it the anomalies plus a constant ? But this is what I am hoping experts can jump in and correct my approach.

 

[Jeff Duda]

Is there a reference you can provide that will show me how to calculate the waves in the 500 or 200 hPa surfaces ? My programming language is Python and I want to identify baroclinic waves in the 3 to 5 day period.

This question is too broad to offer specific help. I can guess that you are looking to create a program to objectively and automatically identify Rossby waves, and the only way I can think to do that is to identify ridge and trough axes. To do that you're going to have to use some sort of graphical or calculus technique searching for semi-1D-connected regions of maximum and minimum heights.

 

[Ashwin D]

This question is too broad to offer specific help. I can guess that you are looking to create a program to objectively and automatically identify Rossby waves, and the only way I can think to do that is to identify ridge and trough axes. To do that you're going to have to use some sort of graphical or calculus technique searching for semi-1D-connected regions of maximum and minimum heights.

Can you explain why it is too broad as that will be useful the next time I ask? I presumed in this question that there will be a very old paper that details how the baroclinic waves are numerically computed. Unfortunately I am not able to dig up that reference.

 

[Ashwin D]

This question is too broad to offer specific help. I can guess that you are looking to create a program to objectively and automatically identify Rossby waves, and the only way I can think to do that is to identify ridge and trough axes. To do that you're going to have to use some sort of graphical or calculus technique searching for semi-1D-connected regions of maximum and minimum heights.

From this response I presume by calculus technique you are probably suggesting that I calculate the curvature(https://en.wikipedia.org/wiki/Curvature) numerically of the geopotential heights to identify the troughs and ridges. However I want to know the periodicity of the baroclinic waves - is it a shortwave (like an eddy) that dissipates within few hours or is it long persisting of the order of 3 day or 5 day or 7 day ?

 

[Ashwin D]

https://rmets.onlinelibrary.wiley.com/doi/full/10.1002/qj.2139

This paper is fairly useful to me to get an understanding of what baroclinic instability/Rossby wave train is at least from a numerical perspective and it is fairly recent as well.

Taking that paper as a rough guide I looked at the latest reanalysis data from NCAR NOAA of the 200 hPa meridional wind. So one can see at least over the continental USA the undulating waves in the meridional wind. There is a pattern of alternating southerly and northerly wind(positive values representing southerly and negative values northerly) . Whether that contributes to surface level instability is another question.

 

[Marshall Stoner]

Baroclinic instability is more theoretical than practical. Unlike a lot of other types of instability, there's no simple number to define it in the real atmosphere. That said, in the mid-latitudes the conditions for baroclinic instability are almost always present. That doesn't mean every single wave embedded in the flow will grow into a monster low pressure system. Most of the time mid-latitude cyclones develop when an anomaly at the top of the troposphere (dynamic tropopause lowering or tropopause front) interacts with a zone of tropospheric temperature gradient (called a baroclinic zone). The full explanation involves a lot of math that I'm not fully familiar with though. I haven't ever really found a text book or article that explains it intuitively and accurately without using mathematical jargon and equations.

 

[Jeff Duda]

Baroclinic instability is more theoretical than practical. Unlike a lot of other types of instability, there's no simple number to define it in the real atmosphere. That said, in the mid-latitudes the conditions for baroclinic instability are almost always present. That doesn't mean every single wave embedded in the flow will grow into a monster low pressure system. Most of the time mid-latitude cyclones develop when an anomaly at the top of the troposphere (dynamic tropopause lowering or tropopause front) interacts with a zone of tropospheric temperature gradient (called a baroclinic zone). The full explanation involves a lot of math that I'm not fully familiar with though. I haven't ever really found a text book or article that explains it intuitively and accurately without using mathematical jargon and equations.

Lots of math games for sure. I took a few advanced dynamics courses in grad school and even I don't fully understand it. Something about a "longwave cutoff wavelength of 3000 km comes to mind, though". Wave behavior is different for length scales above and below that threshold.

 

[Ashwin D]

Baroclinic instability is more theoretical than practical. Unlike a lot of other types of instability, there's no simple number to define it in the real atmosphere. That said, in the mid-latitudes the conditions for baroclinic instability are almost always present. That doesn't mean every single wave embedded in the flow will grow into a monster low pressure system. Most of the time mid-latitude cyclones develop when an anomaly at the top of the troposphere (dynamic tropopause lowering or tropopause front) interacts with a zone of tropospheric temperature gradient (called a baroclinic zone). The full explanation involves a lot of math that I'm not fully familiar with though. I haven't ever really found a text book or article that explains it intuitively and accurately without using mathematical jargon and equations

Lots of math games for sure. I took a few advanced dynamics courses in grad school and even I don't fully understand it. Something about a "longwave cutoff wavelength of 3000 km comes to mind, though". Wave behavior is different for length scales above and below that threshold.

That wavelength cutoff seems to me the concept of a wavenumber. In a latitude circle around the world how many troughs and ridges are you going to have ?

From this link - https://en.wikipedia.org/wiki/Zonal_wavenumber

So from that link if I take a geopotential anomaly at the 500 hPa and then subject that to a periodic mathematical transform(Fourier series for e.g.) and I get the wavelength to be around 3000 km and I get a wavenumber of say 3. That means in the world at that instant there are three troughs and three ridges. So given the earth's circumference at the equator is around 40,000 kms and shrinks rapidly as we go towards the mid latitudes there maybe other wave numbers in that anomaly but there are only three long waves in that anomaly.

 

[Marshall Stoner]

I still have a text book that covers that stuff, but I never got to reading it. I didn't get that far in mid-latitude dynamics. I ended up specializing in equatorial waves since my thesis was on the MJO.

My understanding is that the planetary vorticity advection is more dominant for waves longer than 3000 km. In the QG vorticity equations the planetary vorticity advection component involves a latitude dependent "beta" coefficient, so it's referred to as the "beta term". The beta term counteracts the terms responsible for baroclinic wave growth to some degree, so long waves don't manifest baroclinic instability. The beta term is also responsible for the degree to which long waves move west against the mean flow. Short waves move more-or-less with the mean westerly flow, long waves move west against the mean flow - to the point where they can be stationary or retrograding westward over time.

 

[Ashwin D]

I still have a text book that covers that stuff, but I never got to reading it. I didn't get that far in mid-latitude dynamics. I ended up specializing in equatorial waves since my thesis was on the MJO.

My understanding is that the planetary vorticity advection is more dominant for waves longer than 3000 km. In the QG vorticity equations the planetary vorticity advection component involves a latitude dependent "beta" coefficient, so it's referred to as the "beta term". The beta term counteracts the terms responsible for baroclinic wave growth to some degree, so long waves don't manifest baroclinic instability. The beta term is also responsible for the degree to which long waves move west against the mean flow. Short waves move more-or-less with the mean westerly flow, long waves move west against the mean flow - to the point where they can be stationary or retrograding westward over time.

Very likely. In my post I was not making a case for the direction of the propagation of the wave. My point was when you have a geopotential height anomaly at either the 200 hPa or 500 hPa you can have wavenumbers of many values. If there is indeed a planetary Rossby wave (like when the polar vortex splits during a sudden stratospheric warming) you can have either a displacement towards southern latitudes(wavenumber = 1 event or the unbroken polar vortex is displaced equatorward) or the polar vortex splits into two(wavenumber = 2 event) and both the split vortices move equatorward. The impact then is likely to be planetary scale through the wave-mean flow interactions as you rightly mention.

There is also the notion of group velocity and phase velocity of the waves. The group velocity can be eastward or westward whereas the phase velocity is always westward.

 

[Marshall Stoner]

Very likely. In my post I was not making a case for the direction of the propagation of the wave. My point was when you have a geopotential height anomaly at either the 200 hPa or 500 hPa you can have wavenumbers of many values. If there is indeed a planetary Rossby wave (like when the polar vortex splits during a sudden stratospheric warming) you can have either a displacement towards southern latitudes(wavenumber = 1 event or the unbroken polar vortex is displaced equatorward) or the polar vortex splits into two(wavenumber = 2 event) and both the split vortices move equatorward. The impact then is likely to be planetary scale through the wave-mean flow interactions as you rightly mention.

There is also the notion of group velocity and phase velocity of the waves. The group velocity can be eastward or westward whereas the phase velocity is always westward.

I guess we're just discussing slightly different topics. Not sure where we diverged. I was talking about the cutoff wavelength/wave-number for baroclinic instability. Polar vortex breakup is planetary scale, so it isn't really related to baroclinic instability, at least not in the basic theory.

 

[Ashwin D]

I guess we're just discussing slightly different topics. Not sure where we diverged. I was talking about the cutoff wavelength/wave-number for baroclinic instability. Polar vortex breakup is planetary scale, so it isn't really related to baroclinic instability, at least not in the basic theory.

My point was waves in general in the atmosphere and how they are described in terms of wavenumbers , wavelengths and group velocity and phase velocity.

By the way a polar vortex breakup can influence an existing mid latitude storm(baroclinic instability) to become more equatorward. There is extensive research on this topic. The exact mechanism is one of debate but there is agreement that there is a vertical influence downward.