Tropical Cyclone Kinetic Energy (TCKE) [ATL - 7.623 EJ | EPAC - 15.284 EJ]

[kala]

Tropical Cyclone Kinetic Energy (or TCKE) is a measure of the kinetic energy dissipated by a tropical cyclone throughout its lifetime. TCKE is a function of the winds throughout a tropical cyclone and is calculated by integrating the winds within each quadrant of the cyclone. Based on this integral provided by Kerry Emanuel in this paper, we can determine the total power dissipated by a storm:



This would give us an index representing the instantaneous intensity of a storm, which I call Tropical Cyclone Power (or TCP), measured in real-world SI units: watts. Tropical cyclones are significantly powerful that we can use the prefix tera- (10^12) to describe the values we calculate. Thus a typical storm's TCP value is somewhere in the range of 1-2 terawatts (TW). I use the wind radii values provided by NHC in their running best track and archived best track and a trapezoidal approximation of the integral above to determine TCP. Unfortunately, as a result the TCP value is only calculated for areas in the tropical cyclone where winds are >34 kts. Much like ACE is only calculated for systems of TS strength or greater.

We can further integrate the TCP values (I actually use a left approximation, like ACE) to calculate total TCKE produced throughout the storm's lifetime. This is possible because watts (a unit of power) is just joules (a unit of energy) per second. TCKE is then measured in exajoules (EJ or 10^18 joules). So TCP*6 hours = some value of kinetic energy in EJ produced over that time period.

Thus, we arrive at a system of intensity measurement which describes (roughly) the energy "taken out of" the atmosphere/ocean to produce the winds within a tropical cyclone, which is measured both instantaneously (as a wind speed measurement would be) and cumulatively (as ACE would be). This way, storms can, within the same system, have both instantaneous intensities as well as count toward the total kinetic energy "used up" during a cyclone season.

One of my reasonings for devising this system is that ACE is a very "one-dimensional" measurement. A storm has much more to it than just being 100 kts. For example, much more energy has to be put into a large cyclone for it to reach a 100 kt intensity, whereas a smaller storm (like Danny or Hilda) takes significantly less energy. In a similar manner, a much larger storm is pushing a lot more wind around, and will thus have a higher TCP, and has a larger potential for damage caused by wind, storm surge, or rough waves than a smaller storm does.

In this thread, I will keep track of each storm's TCKE and TCP statistics. I'll post graphics for each storm after it dissipates and add their values to the cumulative season counts in the first post (this one), below. The data for basins outside of NHC/CPHC warning areas should be taken with a grain of salt as JTWC does not keep very good track of wind radii and I have to edit them a little bit (e.g. when a 35 kt storm has no gale-force wind radii).

2016
North Atlantic - 7.623 EJ (through Ian)
East Pacific - 15.284 EJ (through Orlene)
experimental
West Pacific - ?
North Indian - ?
Southwest Indian - 12.903 EJ
Australian region - 0.674 EJ
South Pacific - 16.038 EJ
South Atlantic - 0.000 EJ

[kala]

Since Gaston and Madeline dissipated recently, I'll post their graphics now.



Maximum Power Output: 9.487 TW
Tropical Cyclone Kinetic Energy: 3.797 EJ




Maximum Power Output: 5.110 TW
Tropical Cyclone Kinetic Energy: 0.816 EJ

Track point size is proportional to the average gale radius of the storm, not TCP, though there is significant correlation between the two.
Larger storms have higher radii and thus higher TCPs, as well as stronger storms (by wind speed) have higher radii and thus higher TCPs.

Something else you'll note is that Gaston produced 3.797 EJ TCKE, which is 67% of the season total so far of 5.612 EJ.

[kala]

I guess Hermine hasn't been tropical for a while either so, here's that:



Maximum Power Output: 6.210 TW
Tropical Cyclone Kinetic Energy: 0.809 EJ

[kala]

Maximum Power Output: 6.064 TW
Tropical Cyclone Kinetic Energy: 3.036 EJ



Maximum Power Output: 5.868 TW
Tropical Cyclone Kinetic Energy: 0.875 EJ

[Ptarmigan]

Tropical Cyclone Kinetic Energy (TCKE) would be useful to gauge how active hurricane season is. I wonder about radius of outer closed isobars (ROCI). I know some storms you cannot get hurricane and tropical storm force wind extent.

[kala]

Yes, that is part of the goal. I'll add some previous seasons (I have calculations for 2004-2015 ATL and EPAC) to the original post later.

ROCI would be interesting because it should be roughly proportional to gale force wind radii and TCP. I could maybe make some sort of conversion factor.

[kala]

The total TCKE is not correct (some points are missing) but the current peak TCP is correct:



Which basically puts it on par with most systems of its intensity. Exceptions to this are the most extreme systems (like Haiyan (~18 TW) and Patricia (7.585 TW)) which have such tight cores that their wind radii are pretty small, putting Meranti ahead of them.

[kala]

Maximum Power Output: 4.727 TW
Tropical Cyclone Kinetic Energy: 1.202 EJ



Maximum Power Output: 3.439 TW
Tropical Cyclone Kinetic Energy: 0.807 EJ

[kala]

Maximum Power Output: 25.915 TW
Tropical Cyclone Kinetic Energy: 4.707 EJ

[1900hurricane]

Bumping up an old thread here, but I would be interested in replicating how you derive TCP and eventually TCKE, and then compare them to other metrics, particularly IKE and TIKE, respectively. Not sure if you'll see this kala, but when you set up the formula to calculate TCP, did it look something like this?

[kala]

I probably wouldn't have seen this post if someone who reads this forum more than me (Darvince) hadn't linked it to me. (Thanks for that, btw)

Anyway, I had to go look through my TCKE scripts and spreadsheets to remember how I actually calculated it. The formula I use for a bin in a quadrant is this:



where a bin is some portion of the quadrant defined between some inner radius r_0 and outer radius r_1. I apparently assumed that the integral could be approximated with something like trapezoid rule, where the trapezoid has bases of length rV^3 and a height r_1 - r_0, the distance between the given radii. Something like your V vs r graph in your image.

I did work out the exact solution to the integral and found the same thing as you, and have started to implement it in a script to see how it compares. I do expect the results to be similar though, since the exact solution should just define some smooth monotonic curves between the given radii that are similar to the trapezoid rule line. Although perhaps it will be off by some constant factor (with some error to account for the approximation). However, I'm not really sure yet, because the script is acting weird between bins inside and outside of the RMW.

As an example, at 12z on 09/10, my script calculates Rene's TCP to be 1.885 TW, where only 0.167 TW of this comes from inside the RMW (50 nmi), while 1.718 TW is produced outside this to the R34 (60 nmi). It seems weird that the smaller outer ring would dissipate 10x the power as the inner region, even if the inner region includes the center... it has twice the area! So, I tried graphing the formulae to figure out the problem...



... but it looks like what I would expect, where they actually contribute nearly equal parts to the TCP for this case. Clearly something is up with the script.

As for TCKE, it is simply calculated as a left Reimann sum of the 6 hourly TCP. I'm also curious to see how TCKE compares to other similar metrics, because from what (albeit, little) I've seen, I believe it differs from them by some orders of magnitude.

[1900hurricane]

Rene at that time is an interesting case because three of the four TS radii are actually smaller than the RMW (how does that make sense?). If left unmodified, the inner and outer radius are switched from what is expected, and that might be a possible reason values are not what is anticipated. I ended up making a rat's nest of if statements to check for things like that and modify things to make the radii make at least some semblance of sense. In this case, if a TS radius is less than the rmw, I calculate from 0 at the center up to 34 kt at the rmw with no region outside of that. Not sure if it's ideal, but it is the best solution I could come up with. At that same Rene point, I got 0.203 TW of power, though I still need to do some checking to make sure everything is calculating the way I want it.

Something I have also tested a little is including the radius of the outermost closed isobar as the outermost radius, and assuming that is the point at which the work done by the wind of the TC is zero, ergo v = 0. That would also mean that you could possibly get TCP and TCKE for TDs and stuff like that since rmw and roci are included in the b-decks. Using the same Rene point, TCP jumps up to 0.776 TW when including the somewhat large 180 nm roci.

[kala]

I did figure out what the problem was. I didn't negate all four inner radius terms, just the first one, so now my results match what was graphed. I'm now getting 0.231 TW.

Here's a comparison of the exact and approximate versions:



Both measures of TCKE are quite similar, actually, they do not differ by any particular factor. The one major difference occurs at low intensities where the 0 to RMW calculation is a significant portion of the TCP. In the exact solution form this results in a lower power compared to the linear approximation. This makes sense, just see the graph in my previous post. The linear approximation overestimates the power dissipation near the center, where the exact solution produces very low values. This only makes for significantly different results for weak storms like Rene; stronger storms like Laura dissipate most of their power outside of the RMW, however, resulting in only small differences.

As for gale radii being inside of the RMW, my script was already set up to deal with that, and I handle it basically exactly opposite. I move the RMW to the gale radius, neglecting anything outside of that, and I use whatever the maximum wind speed is for the outer bound instead of 34 kt. As for which way (or any other) is better, I'm not sure.

[1900hurricane]

I duplicated and modified my script that computes IKE and TIKE so I could compute TCP and TCKE, so now I am able to generate the latter numbers just as fast as I can the former. Naturally, the first think I decided to do was compare the two metrics and see how similar or different they are. After graphing both maximum TCP vs maximum IKE and TCKE vs TIKE for NAtl systems from 2010-2019, it is easy to see a very strong linear relation between the two. This makes sense considering the same data is used for both metrics, but I'm actually surprised the relation is as strong as it is, my first guess was that TCP/TCKE would lean a little more in favor of strong TCs when compared to IKE/TIKE. Since IKE/TIKE is only computed for TS force winds and greater, in order for the most 1:1 comparison, I omitted the r34 to rOCI outermost term with these numbers for TCP/TCKE.





Worth noting is that as anticipated, the magnitude of values is considerably different between TCP/TCKE vs IKE/TIKE. Something that is unclear to me, and I'm not sure if it is ever stated, is what time frame exactly IKE is measured over. So, out of curiosity and given the strong correlation between IKE and TCP, I took the max IKE values from above and divided them by the max TCP values. Given IKE is an energy in TJ and TCP is a power in TW, this operation should result in a value with a unit in seconds: theoretically the missing time frame for IKE. This is what I got (note logarithmic x-axis for ease in viewing data):



So theoretically, the power output of a tropical cyclone would achieve the energy calculated in IKE in roughly ten seconds, although there is a little variance there mostly between 8-12 seconds. Seems like an unusual time frame to me.

[1900hurricane]

Also, for whatever reason, my peak Laura number is running a little lower than yours. I'm only getting 9.326 TW of power at 00Z August 27th when computing out to the TS radii.

[1900hurricane]

Turns out TCP does lean a little more towards intense TCs than IKE does. After getting all TCPs and IKEs in the NAtl from 2004-2019 instead of just the maximums of each, some trends can be picked out. The most notable is that the ratio of IKE over TCP decreases linearly as MSW increases. The ratio goes from about ten seconds as you approach hurricane intensity, but decreases to under six seconds once you make it to 140 kt. That means stronger systems are going to be found on the right side of the TCP vs IKE graph, while weaker systems will generally be found on the left side of scatter.







Honestly, I think I prefer this behavior to that of IKE, which I think it weighted too heavily towards radii size and too little towards maximum sustained winds.

[1900hurricane]

Like with Integrated Kinetic Energy, I was experimenting some with applying a "Surge Destruction Potential" function to Tropical Cyclone Power. I've been testing a few iterations (what I'm calling "version 0.9" is below), and I'm generally happy with the results. I ended up taking the percentile rank of SSHWS occurrences in best track through NHC best track 2004-2019 in the NAtl, and then applying those same percentiles to TCP. Up to "category 4," the points are more or less linear before diverging above that. I'm experimenting with both a square root function and a logarithmic function above that, and both honestly behave rather well. I'm capping it at "category 6": which is the top known case in that time frame.



The only complaint I have is that I feel like small, intense storms may be slightly underrepresented. For example, the fairly compact Michael '18 only scored a 2.3 with its 7.257 TW TCP. In response, I'm also playing with using MSW as a modifier to perhaps more accurately represent surge impacts. With both maximum height and areal coverage, surge isn't a one dimensional impact, so quantifying it with a single number is fairly difficult, but that doesn't stop me from trying!

[1900hurricane]

I made a rough image (and I mean rough, it's a NHC map overlain a python generated .csv file opened in Excel) of Tropical Cyclone Kinetic Energy Dissipated in the North Atlantic Basin from 2004-2021.

[1900hurricane]

 https://twitter.com/1900hurricane/status/1571688112400613378



 https://twitter.com/1900hurricane/status/1572760128914620416