Thermodynamics and Tornado Prediction. Tierney Dillon, Alex Lopez, Paddy Halloran. CAM Summer Research Summary Report August 31, 2015

Transcription

1 Thermodynamics and Tornado Prediction Tierney Dillon, Alex Lopez, Paddy Halloran CAM Summer Research Summary Report August 31, 2015 Advisors: Misha Shvartsman and Pavel Bělík

2 1 Introduction and Problem Statement. Meteorologists use a number of of indicators (or indices, or parameters) to decide whether to issue a tornado warning. Among the most important ones are CAPE (Convective Available Potential Energy), SRH (Storm Relative Helicity), EHI (Energy Helicity Index), LI (Lifting Index), and STP (Significant Tornado Parameter). We will provide details on the first two parameters as we introduce them throughout our work. For modern view and details on all of them we refer to [1] and [3]. In our work we undertake a case study of the June 17, 2010 Midwest tornado outbreak (Minnesota, Iowa, and South Dakota) using the data from the Minneapolis weather station. We analyze specifically the role of 2 indices, CAPE and SRH, and their influence on the strength of the storm (in our study strength is associated with free energy density). Theoretical tools that we use for analysis are based on the non equilibrium thermodynamics methods, in particular, the Helmholtz free energy evolution approach developed in [2]. We apply these methods through the lens of an abstract air parcel subject to a rapid change in the violent thunderstorm environment where standard equilibrium thermodynamics tools are not applicable. Why this research is important. Tornados are a common occurrence in the Midwest region of the United States. The great tri state tornado of 1925 for example, was the most devastating in US history. This tornado, in particular, left a 235 mile path, killed 747 people and left more than 2,300 injured. One can see how important it is to accurately predict such a tornado in a timely fashion in order to give people the maximum amount of time to evacuate the area. Famous University of Chicago professor Ted Fujita developed a tornadic strength scale and classified the tri state tornado as an F5. His original strength scale for a tornado was based on the air speed via the formula: According to this formula, an F5 classification (F = 5), for instance, would correspond to the speeds up to 117 m/s. These speeds can cause major damage to people and structures. Thus again, importance of being able to predict tornadic behavior is indisputable. Later (in 2007) structural damage was modified into the

3 2 new Extended Fujita ( EF ) scale based on 28 damage indicators [6]. The June 17 of 2010 outbreak had a total of 93 tornados with the following EF scale distribution: The distribution map of this outbreak including the EF classification is given below: Since our study will monitor the energy of the thunderstorm as well, we should note that

4 3 Thus even for a modest CAPE (Convective Available Potential Energy) of 1.5 kj/kg the total energy of 1.5 TJ is comparable to a nuclear bomb explosion energy. Of course, if it is dissipated slowly over 100 square km, its effect is barely noticeable. However, if it is concentrated in a small area (tornado vortex) over short periods of time, it can have a devastating effect. Thermodynamics of an air parcel We regard an air parcel as a thermodynamic system that can be subject to energy exchange with its environment, for example, the picture below shows energy flowing in and out of the system: Equilibrium Thermodynamics. There are 4 basic laws of Equilibrium Thermodynamics:

5 4 Here If we substitute the expression for work and heat into the first law of thermodynamics, we obtain the total differential for the internal energy U as a function of entropy S and volume V: The partial derivatives here do not depend on the size of the system and are called intensive variables. Thus temperature T and pressure P are intensive variables. The independent differentials here correspond to the extensive variables S and V that do depend on the size of the system. This pairing between intensive variables ( forces ) and differentials of extensive variables ( fluxes ) allows to account for all the sources of contributions to energy of the system. While the internal energy of the system is important to know, not all of it can be converted into kinetic energy of the flow. As a matter of fact, only a small portion of it can. For that reason, instead of U, one often uses the

6 5 that provides us with the information on available (free) energy that may be converted into kinetic energy of the flow. Using the product rule of differentiation we have so, according to our discussion the Helmholtz free energy is a function of T and V and Non Equilibrium Thermodynamics. The air parcel in the tornadic state is far from equilibrium and undergoes fast changes in the quantities that describe it. However, for a very short time and for a small volume we can assume the parcel to be in an equilibrium. For that reason, we use the In this case we have to talk about densities (per unit mass) instead of global values, so

7 6 Local evolution of the Helmholtz free energy density in time. The theoretical part of our work is analysis of the free energy density of an air parcel in a tornado like environment and evolution of this density in time. We adopt a horizontal layer assumption of [2] where all the significant fluxes act in horizontal layer (see [2]). While in the equilibrium case in the non equilibrium (local layer equilibrium) case we have: In this case all the variables can depend on time t. In particular, the non equilibrium temperature theta depends on the vector xi of parameters associated with thermodynamic fluxes. Then, using the chain rule of differentiation we have where we have corrected some terms on the right hand side given in [2]. If we multiply both sides by dt, we have where, again, the terms contributing to the Helmholtz free energy density change on the right have the same structure: Using our thermodynamic relations for F, we have

8 7 where sigma accounts for the explicit dependence of the free energy density on time, and where However, since we introduce new parameters and, therefore, new unknowns, we need some constitutive theory connecting forces and fluxes (like the Fourier law that connects heat flux and temperature gradient). In this context these additional equations are called Neglecting the terms related to the equilibrium theory and explicit dependence on time for our parameters we have In our work the thermodynamic fluxes are associated with CAPE and SRH. However, before we can approach CAPE notion we shall introduce the notion of virtual temperature :

9 8 From the Conservation of Mass Law for the given volume we have Now, by the Dalton s Law we have So

10 9 and Proof: Indeed, we notice that it is easier to transform the right hand side into the desired result on the left: CAPE (Convective Available Potential Energy) We define CAPE via the formula where The basic original reason for buoyancy in the atmosphere is that heavier (colder particles) want to get down and lighter (hotter particles) want to go up as the system tries to minimize its potential energy

11 10 Standard CAPE is usually calculated between 2 heights: LFC (level of free convection) and EL (equilibrium level). However, in our case we can (and will) estimate CAPE between any two given layers in our data (soundings)

12 11 If we integrate the last expression in the height z, we obtain the A nice geometric way to represent SRH is given by a hodograph

13 12 Results and Methodology: Our methods for this research started with importing soundings data from the National Oceanic and Atmospheric Administration (NOAA). The data that came from the soundings included wind speed, wind direction, temperature, dewpoint and a few others. We could use this data to approximate values of the terms provided in the free density rate equation in [2]. When doing this, we had a couple of mishaps in not realizing the time difference between UTC and CDT of the soundings and forgetting to convert Celsius into Kelvin. We first analyzed CAPE and SRH taking soundings from 4pm, 7pm and 10pm on June 17th 2010 and graphed the data in relation to height (see the attached appendix). We found that during the storm the higher SRH was compensating for the low levels of CAPE. Soundings. We obtained soundings from the NOAA (National Oceanic and Atmospheric Administration) website [4], where it provides 6 columns of data: Pressure, Height, Temperature, Dew Point, Wind Direction, Wind Speed. The example below is a partial snapshot of the soundings from 06/17/2010, Minneapolis Station, 7:00 PM

14 CAPE. Blue: 4PM, Red: 7PM, Green: 10PM 13

15 14 To confirm our calculation we provide the estimate of these quantities by CAPS (Center for Analysis and Prediction of Storms) [5] As we see the numbers for CAPE and SRH are in good agreement with ours. Helmholtz free energy density results.

16 15

17 16 In all three graphs we observe sensitivity of the free energy density to the changes in values of CAPE and SRH. We also notice that for the heights when CAPE is small we have a stronger SRH taking over. Evolution of the Helmholtz free energy density in time results. We analyzed the information at 2 heights where CAPE was significant enough not to be sensitive to replacement of virtual temperature by actual temperature. h=1200m :

18 17 h=2500m : At h=1200m the slope is roughly.003 J/(Kg*s) which fits the lower end of the estimate for tornadic events by [2] quoted above. Future Plans: 1. To calculate the CAPE using precise virtual temperature to avoid distorted values at the low CAPE values 2. To refine the calculation for F(t) taking into consideration the updraft component of the flow 3. Evaluate additional parameters such as EHI, STP, LI and their ability to influence F(t) Bibliography: 1. C. A. Doswell, D. M. Schultz, On the Use of Indices and Parameters in Forecasting Severe Storms, E Journal of Severe Storms Meteorology, Vol. 3, No. 3, G. P. Bystrai, I. A. Lykov, S. A. Okhotnikov, Thermodynamics of nonequilibrium processes in a tornado: synergistic approach, 09/23/2011

19 Indices definitions from NOAA 4. Archived Meteorology from NOAA 5. Center for Analysis and Prediction of Storms 6. Extended Fujita Scale Appendix: Attached excel and mathematica files.