Tornadogenesis: The Birth of a Tornadic Supercell (Sasaki’s Theory of Tornado Genesis) University of St. Thomas­ Center for Applied Mathematics Summer of 2015 Austin Swenson & Connor Theisen Advisor:​
​
Dr. Douglas Dokken 1 Problem Statement We want to derive the Euler Equation that Sasaki uses in his paper ​
Entropic Balance Theory and Variational Field Lagrangian Formalism: Tornadogenesis. ​
To derive the Euler Equation Sasaki uses the Calculus of Variations, which leads to the Euler­Lagrange Equations for each variable. We also want to test the validity of the Sasaki’s Entropic Balance Theory (EBT) and his Entropic Right­Hand Rule. We then want to see if if both the EBT and the Entropic Right­Hand Rule can be applied to different parts of the supercell which were not tested by Sasaki in his papers. Research Goals 1. Use the numerical weather model, ARPS (​
Advanced Regional Prediction System), we will simulate supercell thunderstorms through different code changes. 2. The ARPS simulation of storms and the variables, being viewed through Vis5d, will hopefully help us view variables that Sasaki uses to understand the process of how a supercell and tornado form. View entropic anomalies in the Vis5d simulations, both entropic Source and Sink, at visible levels. a. How to reach this? i.
Include all the necessary variables in our simulations and at various levels. ii.
Include other variables, such as hail, in some supercell simulations to find a relationship between other variables and current ones. 3. Find the correct Euler­Lagrange Equations of each variable from the Euler­Lagrange Density Equation we use Calculus of Variations on. a. How to reach this? i.
Using Calculus of Variations, we would add a variational “h” to the Euler­Lagrange Density Equation and subtract that by the original Euler­Lagrange Density Equation. ii.
Minimize the Action Lagrangian functional for each variable. 4. Find the Euler Equation, which is basically the F = ma (Newton’s second law) equation for fluids. a. How to reach this? i.
Taking the Euler­Lagrange Equations that we found for each variable, then, find the Euler Equation. 5. Apply the Entropic Balance Theory and Entropic Right­Hand Rule in different regions and ways for tornadic supercells. a. How to reach this? i.
Create different isosurfaces for various variables for the supercell simulations we will create using supercell data from ARPS. 2 ii.
Create different visual images of a supercell system from different perspectives. Sasaki’s Hypothesis [5, 6] 1. The phase change time scale is significantly shorter than the time scales of convective storms and tornadoes. 2. Variations of the initial entropy levels are small enough that they can be approximated their ensemble means. Results and Methodology Calculus of Variations​
: To find the Euler Equation that Sasaki uses in his tornadogenesis paper we will use the Calculus of Variations of the Euler­Lagrange Density Equation. Before we can begin with the actually performing the Calculus Variations we have to give a couple definitions. Our first definition has to do with functionals. A functional is function of curves, surfaces, or functions of more general variables ​
[1]​
. Our next definition has to do with a variation. A variation is when we differentiate a functional to find the best linear approximation of that functional ​
[1]​
. Let’s consider the following example: Φ(γ + h) − Φ(γ) = F + R (Eq. 1) For this equation, Φ(γ + h) and Φ(γ) represent functionals with Φ(γ + h) being the variation of Φ(γ) . With the difference between the functionals, we obtain the derivative of the functional where ‘F’ represents the linear approximation and the ‘R’ is higher power terms in (Eq. 1). ‘F’ depends linearly on the the variation ‘h’; for example: F (h1 + h2) = F (h1) + F (h2) and F (ch) = cF (h) where F (h) = F • h (Eq. 2) However, even clearly seen in (Eq. 2), the linear approximation ‘F’ and variation ‘h’ are separable. They’re separable because we desire to find the best linear approximation for the equation, regardless of the value of ‘h’. Therefore, we are able to factor out the ‘h’ from the equation. As for the higher power terms ‘R’, it’ll represent any form of high­exponential values greater than one. For example: 3 R(h, γ) = a2h2 + a3h3 + a4h4 + ⋅ ⋅ ⋅= 0(h2) (Eq. 3) However, since we desire the best linear approximation of the equation, we would imagine that the higher power terms of each equation will be too small and approach zero (as reference to (Eq. 3)). Therefore, we can omit that portion out of the equation. higher power terms can also include 2​
when the variation ‘h’ has a higher power terms, such as ‘h​
’. Thus, we can omit this ‘h’ as well. One of the keys concepts to Sasaki’s journal article ​
[5, 6]​
is the Euler­Lagrangian Density Equation: ​
ℒ:= ⍴ 12 |v|2 − U (⍴, S) − Φ − α {∂t⍴ + ∇ ∙ (⍴v)} − β {∂t (⍴S) + ∇ ∙ (⍴vS)} {
}
(Eq. 4) Key ρ : The density of the air U( ⍴, S ) : Internal energy S : Entropy (randomness) Φ : Gravitational Potential Energy v :​
​
Flow velocity (represented as a vector) α, β : Lagrange Multipliers to satisfy the constraints of conservation Using Calculus of Variations on (Eq. 4), we are able to find the correct Euler Equation that Sasaki presents in his paper. However, finding the Euler Equation will involve performing variations on different variables. Our different variations will be performed through a functional referred to as the Action Lagrangian L , which we want to minimize. Since we want to minimize the Action Lagrangian L , we set the derivative of the Lagrangian to zero. The variables involved with the Action Lagrangian are α, β, ⍴, S ​
and v , where we will be finding each variable’s minimized value such that: δαL = 0 , δβL = 0 , δSL = 0 , δvL = 0 , δ⍴L = 0 (Eq. 5) Variations of the Euler­Lagrange Density Equation: Now that we have described the Calculus of Variations and its importance to our research we can begin our variations of the Euler­Lagrange Density Equation. So to start with the first variation we start with α . Here is how we would write out the variation with respect to α : 4 δαL := ∫∫∫∫ ℒ(α + h) − ℒ(α) d4x (Eq. 6) If we break this up into the different functionals of ℒ(α + h) and ℒ(α) we would get: [ { |v| − U (⍴, S) − Φ} − (α + h) {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴S) + ∇ ∙ (⍴vS)}] ℒ(α + h) = ⍴
1
2
2
t
t
(Eq. 7) [ {
ℒ(α) = ⍴
1 2
2 |v| − U (⍴, S) − Φ
} − α {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴S) + ∇ ∙ (⍴vS)}] t
t
(Eq. 8) If we take (Eq. 7) and subtract (Eq. 8) we would get the simplified answer of: ∫∫∫∫ ℒ(α + h) − ℒ(α) d4x = ∫∫∫∫[− h {∂t⍴ + ∇ ∙ (⍴v)}] d4x (Eq. 9) So that our variation with respect to α would be: δαL := ∫∫∫∫[− h {∂t⍴ + ∇ ∙ (⍴v)}] d4x (Eq. 10) We want (Eq. 10) to equal zero regardless of the value of h so would set the coefficient of h to be equal to zero. By ensuring that the coefficient is always zero, that means that h will always equal zero. So this means that: ∂t⍴ + ∇ ∙ (⍴v) = 0 (Eq. 11) would be the variation of Lagrangian Equation with respect to α . The next variation that will deal with the other lagrange multiplier of β . The variation with respect to β would look like the following: δβL := ∫∫∫∫ ℒ(β + h) − ℒ(β) d4x (Eq. 12) If we break this up into the different functionals of ℒ(β + h) and ℒ(β) we would get: [ { |v| − U (⍴, S) − Φ} − α {∂ ⍴ + ∇ ∙ (⍴v)} − (β + h) {∂ (⍴S) + ∇ ∙ (⍴vS)}] ℒ(β + h) = ⍴
1
2
2
t
t
5 (Eq. 13) [ {
ℒ(β) = ⍴
1 2
2 |v| − U (⍴, S) − Φ
} − α {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴S) + ∇ ∙ (⍴vS)}] t
t
(Eq. 14) If we take (Eq. 13) and subtract (Eq. 14) we would get the simplified answer of: ∫∫∫∫ ℒ(β + h) − ℒ(β) d4x = ∫∫∫∫[− h {∂t(⍴S) + ∇ ∙ (⍴vS)}] d4x (Eq. 15) So that our variation with respect to β would be: δβL := ∫∫∫∫[− h {∂t(⍴S) + ∇ ∙ (⍴vS)}] d4x (Eq. 16) Again we will set (Eq. 16) equal to zero and we will set the coefficient of h to be zero so that regardless of the value of h , the variation will be zero. So this means that: ∂t(⍴S) + ∇ ∙ (⍴vS) = 0 (Eq. 17) would be the variation of the Lagrangian Equation with respect to β. Our next variation has to do with velocity ( v ). The variation with the respect to v would look like this: δvL := ∫∫∫∫ ℒ(v + h) − ℒ(v) d4x (Eq. 18) If we break this up into the different functionals of ℒ(v + h) and ℒ(v) we would get: [ { |v + h| − U (⍴, S) − Φ} − α {∂ ⍴ + ∇ ∙ (⍴(v + h)} − β {∂ (⍴S) + ∇ ∙ (⍴(v + h)S)}] ℒ(v + h) = ⍴
1
2
2
t
t
(Eq. 19) [ {
ℒ(v) = ⍴
1 2
2 |v| − U (⍴, S) − Φ
} − α {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴S) + ∇ ∙ (⍴vS)}] t
t
(Eq. 20) If we take (Eq. 19) and subtract (Eq. 20) we would get the simplified answer of: 6 (
)
∫∫∫∫ ℒ(v + h) − ℒ(v) d4x = ∫∫∫∫ ⍴ |vh| + 12h2 − α{∇ ∙ (⍴h)} − β{∇ ∙ (⍴h)} d4x (Eq. 21) By looking at (Eq. 21) we can see that the 12 h2 term would be a higher power term, and since we are looking for the best linear approximation, we can disregard this term. Also once integration by parts is performed on the α and β terms, we would get the simplified answer of: ∫∫∫∫ ℒ(v + h) − ℒ(v) d4x = ∫∫∫∫ h {⍴ (v + ∇α + S∇β)} d4x (Eq. 22) Since we want this variation to be zero for all values of h we would take its coefficient and set it equal to zero. Since we can distribute a ⍴ to every term in the coefficient, we can divide both side by ⍴ and get the equation of: v + ∇α + S∇β = 0 (Eq. 23) But we want to focus on the velocity so we will subtract the α and β terms to the other side to get the variation of the Lagrangian Equation with respect to v to be: v = − ∇α − S∇β (Eq. 24) This equation will be useful in some of our later in our paper when it comes to other variations and also applying it to Sasaki’s Entropic Right­hand rule and expanding the right­hand rule.
The variations so far have been simple with a lot of simplification based on like variables, but these next couple are more challenging to simplify. The next variation that we will do has to do with ⍴ (density of air). Here is how we would write out the variation with respect to ⍴ : δ⍴L := ∫∫∫∫ ℒ(⍴ + h) − ℒ(⍴) d4x (Eq. 25) If we break this up into the different functionals of ℒ(⍴ + h) and ℒ(⍴) we would get: [
ℒ(⍴ + h) = (⍴ + h)
{ |v| − U (⍴ + h, S) − Φ} − α {∂ (⍴ + h) + ∇ ∙ ((⍴ + h)v)} − β {∂ ((⍴ + h)S) + ∇ ∙ ((⍴ + h)vS)}] 1
2
2
t
t
(Eq. 26) [ {
ℒ(⍴) = ⍴
1 2
2 |v| − U (⍴, S) − Φ
} − α {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴S) + ∇ ∙ (⍴vS)}] t
t
(Eq. 27) 7 If we take (Eq. 26) and subtract (Eq. 27) we would get the answer of: ∫∫∫∫ h [12|v|2 − {U(⍴ + h, S)} − Φ] − ⍴ [U(⍴ + h, S) − U (⍴, S)] − α {∂th + ∇ ∙ (hv)} − β {∂t (hS) + ∇ ∙ (hvS)} d4x (Eq. 28) This variation doesn’t have the same look as the past variations so we will have to do some work to get the variation simplified. The first step we will do is to add h {U(⍴, S)} − h {U(⍴, S)} to (Eq. 28) since it does not change the value of (Eq. 28). Below is what (Eq. 28) would look like: ∫ ∫ ∫ ∫ h [ 12 |v|2 − {U(⍴ + h, S) − U (⍴, S)} − U (⍴, S) − Φ] − ⍴ [U(⍴ + h, S) − U (⍴, S)] − α {∂th + ∇ ∙ (hv)} − β {∂t (hS) + ∇ ∙ (hvS)} d4x (Eq. 29) Through a thermodynamic substitution below (look at the appendix under “Thermodynamic Equations” for more details), we can simplify (Eq. 29) to be: [U(⍴ + h, S) − U (⍴, S)] = hU ⍴(⍴, S) + higher order terms in h (that we can ignore). (Eq. 30) If we apply this substitution to (Eq. 29) then we would get: ∫∫∫∫ h [12|v|2 − hU ⍴(⍴, S) − U (⍴, S) − Φ] − ⍴[hU ⍴(⍴, S)] − α {∂th + ∇ ∙ (hv)} − β {∂t (hS) + ∇ ∙ (hvS)} d4x (Eq. 31) Now we can see that the first hU ⍴(⍴, S) when multiplied by the h would result in a higher power term which we can disregard. If we perform integration by parts to remove the derivatives of terms involving h , the α and β terms would would change into what we have below: ∫∫∫∫ h [12|v|2 − ⍴U ⍴(⍴, S) − U (⍴, S) − Φ] + h {∂tα + S∂tβ − v [− ∇α − S∇β]} d4x (Eq. 32) Through more simplifying using the velocity equation (Eq. 24) and thermodynamic equation (A. 19) we get the equation: {
}
δ⍴L := ∫∫∫∫ h ∂tα + S∂tβ − 12 |v|2 − U (⍴, S) − P⍴ − Φ d4x (Eq. 33) Again we want the variation to equal zero for all h, that means the coefficient of h needs to always equal zero meaning: 8 ∂tα + S∂tβ − 12 |v|2 − U (⍴, S) − P⍴ − Φ = 0 (Eq. 34) is the variation of the Lagrangian equation with respect to ⍴. Another variation involves one of the key concepts and properties of supercell formation and tornadogenesis, which is entropy ( S ​
). The concept of entropy will be explained later on, but basically, its positive and negative anomalies will be tools into creating the rotation of the vorticity (vortex of tornadogenesis). The variation with respect to entropy S ​
would look like: δSL := ∫∫∫∫ ℒ(S + h) − ℒ(S) d4x (Eq. 35) If we break (Eq. ) up into the different functionals of ℒ(S + h) and ℒ(S) we would get: [ { |v| − U (⍴, S + h) − Φ} − α {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴(S + h)) + ∇ ∙ (⍴v(S + h))}] ℒ(S + h) = ⍴
1
2
2
t
t
(Eq. 36) [ {
ℒ(S) = ⍴
1 2
2 |v| − U (⍴, S) − Φ
} − α {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴S) + ∇ ∙ (⍴vS)}] t
t
(Eq. 37) Before we begin subtraction of (Eq. 36) and (Eq. 37), we need to distribution of the (S + h) in (Eq. 36) to get its simplified form. Through distribution and simplification, (Eq. 36) will become: [ { |v| − U (⍴, S + h) − Φ} − α {∂ ⍴ + ∇ ∙ (⍴v)} − β {∂ (⍴S) + ∇ ∙ (⍴vS)} − β {∂ (⍴h) + ∇ ∙ (⍴vh)}] ℒ(S + h) = ⍴
1
2
2
t
t
t
(Eq. 38) Now, when we take (Eq. 38) and subtract (Eq. 37), we would get a simplified answer of: ∫∫∫∫ ℒ(S + h) − ℒ(S) d4x = ∫∫∫∫[⍴ {− U (⍴, S + h) + U (⍴, S)} − β {∂t (⍴h) + ∇ ∙ (⍴vh)}] d4x (Eq. 39) In (Eq. 39), we can simplify the “ − U (⍴, S + h) + U (⍴, S) ” portion of rho by using the definition of derivatives, in which it will change to “ − hU S(⍴, S) + higher order terms in h (that we can ignore)”. Thus, we are given: ∫∫∫∫ ℒ(S + h) − ℒ(S) d4x = ∫∫∫∫[⍴ {− hU S(⍴, S)} − β {∂t (⍴h) + ∇ ∙ (⍴vh)}] d4x (Eq. 40) 9 Taking (Eq. 40), we can now use integration by parts, which will cause gradients to switch their constants and change their signs. Thus, we are given: ∫∫∫∫ ℒ(S + h) − ℒ(S) d4x = ∫∫∫∫[⍴h {− U S(⍴, S)} + ⍴h {∂t(β)} + ⍴vh{∇ ∙ (β)}] d4x (Eq. 41) Simplifying (Eq. 41) more can be done by factoring out the ⍴ and h out of the equation such that: ∫∫∫∫ ℒ(S + h) − ℒ(S) d4x = ∫∫∫∫[⍴h {− U S(⍴, S) + (β) + v ∙ ∇β} ] d4x (Eq. 42) Since we want to find the best linear approximation for each of the variations, this will found for any value of ⍴ , so we can omit the ⍴ out of the equation and have our variational h present such that: ∫∫∫∫ ℒ(S + h) − ℒ(S) d4x = ∫∫∫∫[h {− U S(⍴, S) + ∂t(β) + v ∙ ∇β} ] d4x (Eq. 43) So that our variation with respect to S would be: δSL := ∫∫∫∫[− h {− U S(⍴, S) + ∂tβ + (v ∙ ∇β)}] d4x (Eq. 44) As with previous variations, we want the best linear approximation of the variation. Thus, we will set (Eq. 44) equal to zero and we will set the coefficient of h to be zero so that regardless of the value of h , the variation will be zero. So this means that: − U S(⍴, S) + ∂tβ + (v ∙ ∇β) = 0 (Eq. 45) would be the variation of the Lagrangian Equation with respect to S. However, through a thermodynamics substitution below (look at appendix for more details), we can simplify (Eq. 45) as: − T + ∂tβ + (v ∙ ∇β) = 0 (Eq. 46) where T is temperature. Therefore, (Eq. 46) can be the variation of the Lagrangian Equation with respect to S. 10 Taking all these Euler­Lagrange equations {(Eq. 11), (Eq. 17), (Eq. 24), (Eq. 34), and (Eq. 46)}, we can put them together to find the Euler Equation, which is represented by: ∂tv + (v ∙ ∇)v + ∇P
⍴ + g = 0 (Eq. 47) Within (Eq. 47), we don’t take friction into account because our research and Sasaki’s approach was viewing tornadogenesis before it fully touches down onto the surface. The Euler Equation nd​
(Eq. 47) is basically F = ma (Newton’s 2​
law) equation for fluids. Also, the g of (Eq. 47) represents the gradient of Φ ( ∇Φ ). To fully understand how we came to (Eq. 47) and what (Eq. 47) means, let’s take a look into the proof behind it. Using the Euler Lagrange equation with respect to ⍴ (Eq. 34), we will take the gradient of (Eq. 34). Thus, we would have: ∇(∂tα + S∂tβ − 12 |v|2 − U (⍴, S) − P⍴ − Φ) = 0 (Eq. 48) Otherwise, (Eq. 48) can simplified through distribution to certain parts to appear as: ∇(∂tα + S∂tβ) − ∇( 12 |v|2) − ∇[U(⍴, S) + P⍴ ] − ∇Φ = 0 (Eq. 49) Let’s focus on the first gradient of (Eq. 49), which is ∇ (∂tα + S∂tβ) . Once we apply the gradient rules through distribution, our result will be: ∇ (∂tα + S∂tβ) = ∂t(∇α) + (∇S)∂tβ + S∂t(∇β) (Eq. 50) However, (Eq. 50) can be simplified more. By adding ∂t(S∇β) − ∂t(S∇β) , which does not change the value of the equation, will help us further in the paper. Thus, our result is: ∇(∂tα + S∂tβ) = ∂t(∇α) + ∂t(S∇β) − ∂t(S∇β) + S∂t∇β + (∇S)∂tβ
(Eq. 51) Observing (Eq. 51), we can group both the ∇α and S∇β since they have a common multiplier of ∂t . Then by using the product rule on − ∂t(S∇β) and changing the order of differential integration on S∂t∇β , our result is: ∇(∂tα + S∂tβ) = ∂t(∇α + S∇β) − (∂tS)∇β − S∇(∂tβ) + S∇(∂tβ) + (∇S)∂tβ (Eq. 52) 11 By substituting in the negative value of (Eq. 24), which is equal to ∇α + S∇β , we can simplify (Eq. 52). We would substitute (A.24) for ∂tS and (A.25) for ∂tβ (look at the appendix under “Various Substitutions” for more details). Also − S∇(∂tβ) + S∇(∂tβ) is equal to zero so these values cancel out. If we apply all these changes to (Eq. 52) we would get: ∇(∂tα + S∂tβ) = − ∂tv + (v ∙ ∇S)∇β + ∇S(− v ∙ ∇β) + T ∇S (Eq. 53) ∇(∂tα + S∂tβ) = − ∂tv + T ∇S + (v ∙ ∇S)∇β − (v ∙ ∇β)∇S (Eq. 54) We’ll do a further substitution and simplification of (Eq. 54) by using a vector identity that uses cross­product, vector fields, and distribution. Through the vector identity (see the appendix under “Vector Fields and Identities” for more details) and applying it to the (v ∙ ∇S)∇β − (v ∙ ∇β)∇S portion of (Eq. 54) , we get: ∇ (∂tα + S∂tβ) = − ∂tv + T ∇S + v × (∇β × ∇S) (Eq. 55) To simplify (Eq. 55), we need to use a substitution using vorticity (ω) . Vorticity can be represented by ω = ∇β × ∇S (look at the appendix under “Various Substitutions” for more details, A.26). Therefore, our substitution gives us: ∇(∂tα + S∂tβ) = − ∂tv + T ∇S + v × ω (Eq. 56) Thus, (Eq. 56) is our result for the ∇ (∂tα + S∂tβ) portion of (Eq. 49). As for other portions of (Eq. 49), if we take ∇( 12 |v|2) of (Eq. 49) and apply the gradient rules to it, we will get: ∇( 12 |v|2) = (v ∙ ∇)v + v × ω (Eq. 57) Another portion of (Eq. 49) that can be simplified is the ∇[U(⍴, S) + P⍴ ] part. First, we can use distribution to get the gradient of both parts. Thus, we have: ∇[U(⍴, S) + P⍴ ] = ∇[U(⍴, S)] + ∇[ P⍴ ] (Eq. 58) Using thermodynamic substitution and gradient rules, we can simplify (Eq. 58) to: P
∇[U(⍴, S) + P⍴ ] = [U ⍴∇⍴ + U S∇S] + [ ∇P
⍴ − ⍴2 ∇⍴ ] 12 (Eq. 59) ∇[U(⍴, S) +
P
⍴]
=
[ ⍴P2 ∇⍴ + T ∇S] + [ ∇P
⍴
−
P
∇⍴ ] ⍴2
(Eq. 60) ∇[U(⍴, S) +
P
⍴]
= T ∇S +
∇P
⍴
(Eq. 61) Thus, by taking (Eq. 56, 57, 61), we can substitute them into (Eq. 49), simplify them, and find our Euler Equation. Through substitution and simplification, our result is: ∂tv − T ∇S − v × ω + (v ∙ ∇)v + v × ω + T ∇S + ∇P
⍴ + ∇Φ = 0 (Eq. 62) ∂tv + (v ∙ ∇)v +
∇P
⍴
+ ∇Φ = 0 (Eq. 63) Therefore, (Eq. 63) is our Euler Equation as previously shown in (Eq. 47). Applications to a Supercell: To fully understand the Euler Equation, one must look more in depth into the vorticity in several important regions of the supercell and variables from the Euler­Lagrange Density Equation (Eq. 4) and the Euler Lagrange equations (Eq. 11, 17, 24, 34, 46), especially with the Euler­Lagrange equation for velocity (Eq. 24). The velocity equation (Eq. 24) will be used to understand the vorticity (ω) , which can be defined as the curl of velocity (or rotation in the flow) [4, 5, 6]​
. Using the velocity equation (Eq. 24) we can gain a better understanding of vorticity: ω = curl(v) = curl(− ∇α − S∇β) (Eq. 64) To simplify (Eq. 64), the curl can be distributed into the parenthesis area, which will result in: ω = curl(− ∇α) − curl(S∇β) (Eq. 65) Recalling from multivariable Calculus, the curl of a gradient is zero, therefore: curl(− ∇α) = 0 (Eq. 66) Looking at the second part of (Eq. 64) and recalling properties of a curl we obtained: curl(− S∇β) = − ∇S × ∇β (Eq. 67) 13 Then, we multiply both sides by SS and distribute the S to the second gradient of our cross­product and the 1S to the first gradient of the cross­product, we obtained: ω = ( 1S ∇S) × (− S∇β) (Eq. 68) From (Eq. 68), we obtain two components within vorticity that will be essential into understanding Sasaki’s Entropic Right­Hand Rule that will be introduced later on in our paper. The first part of the cross­product in (Eq. 68, 1S ∇S) represents the entropic gradient. The entropic gradient is the gradient that goes from low entropy to high entropy. The second part of the cross­product in (Eq. 68, − S∇β) represent the flow of velocity in the system. For Sasaki’s Entropic Right­Hand Rule we use the Right­Hand Rule from multivariable Calculus. To begin, you start with your thumb, which represents the entropy gradient, points from the low entropy (Entropic Sink) to the higher entropy (Entropic Source) ​
[5]​
. Then you would use your pointer finger as the flow in velocity, this means that the pointer finger would point in the direction that the velocity vector is going ​
[5]​
. Using the Right­Hand rule, your middle finger would represent the cross­product of the entropy gradient and flow velocity to get the vorticity ​
[5]​
. The figure below shows how Sasaki represents the Entropic Right­Hand rule: Figure 1: Following Sasaki’s Entropic Right­Hand Rule ​
[5] Since we are using the Entropic Right­Hand Rule in different way than Sasaki, we switched around the fingers so that when we applied the Right­Hand Rule to different situations 14 it would be easier to use with our hands. Below is the Entropic Right­Hand Rule that we came up with: Figure 2: Our Representation of the Entropic Right­Hand Rule Along with vorticity, another important variable describing the supercell that involved velocity and vorticity is referred to as helicity. Helicity is defined as the dot product of the velocity and vorticity, in which we use to understanding and visualizing the development of tornadogenesis [6]​
. Helicity can be represented by: ɦ= v ∙ ω (Eq. 69) Now, by the definition of dot­product rule, (Eq. 69) can as be represented by: ɦ= |v| |ω| cos(θ) (Eq. 70) θ in (Eq. 70) represents the angle between velocity and vorticity. In helicity, it’s the angle and direction of velocity and vorticity that helps us determine the development stage of tornadogenesis. If we reach maximum helicity, then velocity and vorticity are going in the same direction, there is no angle between velocity and vorticity, and tornadogenesis is at full development. Under these circumstances velocity and vorticity fields of the tornado are in a 15 helical, rather tightly stretched cork­screw shape in the supercell. When we reach maximum helicity with the angle between velocity and vorticity being zero, then the cos(θ) in (Eq. 70) will be equal to one. Therefore, the maximum helicity will equal one. Any value of helicity between zero and one, the tornadogenesis is still in development. An easier way to interpret the helicity is write out the relative helicity equation represented by: cos(θ) = |vv∙ω
||ω| (Eq. 71) (Eq. 71) is basically the helicity over the product of the magnitudes velocity and vorticity equalling the cosine of the angle between velocity and vorticity. Entropic Right­Hand Rule​
: As previously mentioned, Sasaki’s Entropic Right­Hand Rule (Figure 1) is a tool to visualizing and practicing the components that help form a supercell and develop tornadogenesis. When it comes to using the Entropic Right­Hand Rule, there are two different levels and types of vorticity that occur in the supercell that affects how we move and use our right hand. These vorticities are the barotropic and baroclinic vorticity. In baroclinic, one can view it as a function of both temperature (or density) and pressure, so temperature (density) and pressure both affect this vorticity stage ​
[4, 5]​
. The baroclinic effects in a supercell are more dominant in the mesocyclone and formation of vorticity around the rear flank downdraft of a supercell (Figure 3). Baroclinic effects can even help lead to the development of a mesocyclone and hook echo ​
[4, 5]​
. As for barotropic, it would be a function of only pressure, so temperature would not have any affect on the barotropic vorticity ​
[4, 5]​
. Barotropic effects would be more dominant during the full development of the tornadogenesis or when we reach maximum helicity ​
[4, 5, 6]​
. Since we are reaching a maximum helicity and only having pressure affect the vorticity here, then the effects of entropy on vorticity generation is being minimized in the barotropic stage due to (Eq. 46, 48, 68). The dominance of the barotropic vorticity would appear mostly within the main tornado region of the visualized supercell in (Figure 3). 16 Figure 3: Hand­drawn near surface view of a tornadic supercell system To use Sasaki’s Entropic Right­Hand Rule, you have your thumb (the entropy gradient) pointing in the direction towards the entropic source of a supercell from the entropic sink, your pointer finger (the flow velocity) at a 90 degree angle with your middle finger (vorticity). Potential Temperature and Entropy​
: In this section we will be relating two important concepts that help us to better understand tornadogenesis. To begin with, we will start with potential temperature. Potential temperature is the temperature of a parcel of air, if it was moved adiabatically (no heat exchanged) from its current location to mean sea level​
[4]​
. Below is a figure that represents potential temperature visually: Figure 4: Representation of how to find potential temperature Below is the mathematical equation to find potential temperature ​
[4]: 17 R
θ = T ∙ ( PPs ) Cp (Eq. 72) Key ●
●
●
●
●
T : The temperature of the parcel at it’s current height (K) R : The gas constant of air; 8.3144621 J ∕ (mol • K) C p : The specific heat capacity at a constant pressure P s : Atmospheric pressure at sea level P : Pressure at the parcel’s original level and altitude Now that we have given a basic understanding of what potential temperature is, we can move on to what entropy is. Entropy, for our intuitive interpretation, is the measure of randomness. The relationship that we found between potential temperature and entropy is a positive relationship. This means that if potential temperature is low, entropy would also be low and vice versa. One way one could imagine this relationship is if we think of regular temperature that we use. Regular temperature that we think of kinetic energy in the system. When the temperature is higher, than the particles are moving faster. As the particles are moving faster, it becomes harder to know where the particles are going to be, thus the randomness in the system is high which makes the entropy also high. We can also think of if there is a low temperature, the kinetic energy is lower and that means the particles are not moving as fast as they would be in higher temperatures. Since the particles in the system are not moving as fast as before, we would be able to have a better understanding of where the particles are heading, thus making the system less random than the higher temperature. Since it is less random that means that the entropy is lower. Supercell and its Properties​
: (Continue to next page) 18 Figure 5: Following Sasaki’s idea of a supercell and its parts/properties ​
[5] The previous sections were to give the premise and explanation of what tools and principles we will be using to test and examine a tornadic supercell. Now we, discuss the structure of a supercell and explain some of its properties, we will be able to incorporate the tools and principles we are using to make the problem statement more clearer. A key property to understand is the environmental winds when supercells form, which are: Moist Southerly Flow, Dry Upper Westerlies, and South­Westerly Jet Stream in (Figure 5). For the Moist Southerly Flow winds, these carry moist air coming primarily from the Gulf of Mexico ​
[5, 6]​
. At a low level near the surface, that moist air is going to be carried into the updraft of the supercell, where it will begin a process of condensation ​
[5, 6]​
. The condensation of these moisture in the rising air parcels will cause the release of latent heat in the intensifying the updraft of the supercell. The release of latent heat increases the potential temperature of the parcels and causes the entropic anomaly called Entropic Source ​
[5, 6]​
. During this process of condensation, the air parcels are carrying high overshoot the level of neutral buoyancy levels of kinetic energy, so when the air parcels reach the level of neutral buoyancy at the top of the supercell, they will overshoot the dry upper level westerlies. The parcels then fall back down, some to the back­side of the supercell, some in front of the supercell forming the anvil ​
[5, 6]​
. At those high levels of altitude, the air parcels that are overshooting the supercell will be interact with the Dry Upper Westerlies ​
[5, 6]​
. The air parcels will against these Dry Upper Westerlies at overshoot the level of neutral buoyancy, the air parcels lose momentum and drop back down into the supercell. Some parcels fall into the back of the storm where they will enter the downdraft stage ​
[5, 6]​
. The air parcels will begin a process of evaporation and cooling 19 creating a downdraft on the back side of the storm ​
[5, 6]​
. The evaporation and cooling of air parcels will cause the other entropic anomaly called Entropic Sink ​
[5, 6]​
. This process of evaporation and cooling of air parcels will continue when downdraft encounters the South­Westerly Jet Stream winds at mid­levels of the storm ​
[5, 6]​
. The South­Westerly Jet Stream causes further evaporation and cooling of the air parcels, in which the cool air parcels will interact with/create the Rear Flank Downdraft to essentially help create the low level surface rotation of the supercell and/or the hook echo ​
[5, 6]​
. Referencing (Figure 3), we see an indication of rotation in a feature of a supercell called a hook echo when viewed on radar. This is basically where we see the rotation formation of a tornado occur in a supercell. Recall that back in (Figure 3), we see the main tornado hook echo has a Barotropic dominance in which the vorticity and velocity are in the same direction to cause maximum helicity. As previously mentioned back at (Figure 3), there are smaller vortices that form away from the main updraft where baroclinic vorticity is dominant. What is contributing the hook echo formation and rotation is the Rear Flank Downdraft (RFD) and Forward Flank Downdraft (FFD) as seen in (Figure 3). The RFD, as seen in (Figure 3), consists of relatively cool air parcels of the downdraft that at low levels will be interacting with the warm moist environmental air. This interaction will cause a rotation or wraparound in the back of the supercell via the entropic right hand rule. The FFD interaction with the warm air will keep the rotation between the entropic anomalies occurring, so basically, the FFD forces or keeps the hook echo in place so that the tornadogenesis can fully rotate and develop. Baker’s Transformation: To represent the hook echo in the supercell, Sasaki uses what is called a baker’s transformation to help represent the shape of the hook echo. Below is a figure that shows Sasaki’s baker’s transformation and our representation of a baker’s transformation: 20 Figure 6: Left side Sasaki’s Baker’s transformation and Right side our Baker’s transformation [5] To begin with, both Sasaki’s and our Baker’s transformation start out the same. We start with the box with the smile inside of it in the top left corner. The next step is like a baker kneading bread, we decrease the height of the box in half and double the length. The area of the new box is equal to the area of the original box that we started with. Here is where Sasaki’s Baker’s transformation and our Baker’s transformation differ. Sasaki, for his Baker’s transformation on the left side of the Figure , takes the flattened box cuts it down the middle and takes the right piece and places it on top of the left piece. For our Baker’s transformation what we did was take the flattened box and fold over the right piece onto the left piece. By doing this, we believe that we can represent the curve of the hook echo more accurate. Still Images of Vis5d Simulations: (Continue Figure 7 on next page) 21 Figure 7​
:​
Entropic Anomaly (Hail­storm supercell) Key: ● Updraft​
= 20 m/sec ​
(Orange) ● Positive Perturbation Potential Temp​
= 3 K​
(Blue) ● Negative Perturbation Potential Temp​
= ­1 K ​
(Green) ● Vorticity ​
= 0.010 1/s​
(Aqua) ● Hail​
: 2.0 g/kg (White) Note​
: This supercell was tested and changed to a hail­storm and did include a hail variable; however, the hail variable doesn’t appear now due to our desire to see the entropic anomaly occur more clearly to us. It’s still a hail­storm supercell. In (Figure 7) above, these are snapshots of our animated simulations that we ran through Vis5d in ARPS. We ran a coarse grid simulation, using the basic settings for the ARPS, May 20, 1977 run: delta x and delta y =1km, etc. This grid is too coarse to view the tornado, however it does show the development of the entropic anomalies in the regions consistent with Sasaki’s theory. We have variables of the updraft, a positive and negative perturbation temperature, vorticity, and hail in this supercell. The perturbation potential temperatures will represent the entropic anomalies occurring in this hail storm supercell, in which the positive perturbation PT represents the Entropic Source and the negative perturbation potential temperature represents the Entropic Sink. We also did runs using Kessler warm rain physics, and found the entropic anomalies to be weaker and less depth and vertical extent in the rear flank region. This suggests that hail formation might be an important precursor to tornado formation. It could also be that the scales used and the micro­physics options were showing some bias towards the hail option. 22 Much higher resolution runs would be necessary to investigate the role of hail in tornado formation. The phenomenon we are mostly looking for is the presence of the downdraft through the entropic anomalies, especially when it comes close to the mesocyclone as seen in (Figure 5). The positive perturbation potential temperature, or Entropic Source, is going up with the updraft and rotating around the vorticity to show the presence of a hook echo due to the RFD and FFD. Near the high levels of the updraft in (Figure 7), perturbation potential temperature changes from positive to negative. Although at low levels, the negative perturbation potential temperature can be seen coming down alongside the vorticity, which shows the presence the downdraft in this supercell and clear usage of the Entropic Sink. Thus, we have found the presence of the downdraft through the entropic anomalies, just as Sasaki had hypothesized and explained in his works. One little detail and relationship we noticed while creating this hail storm simulation is that hail and tornadic supercell have a proportional relationship. In a previous Vis5d simulation with the same data used for (Figure 7), there was no clear presence of the downdraft. However, we didn’t include hail in that previous simulation. Once we added the hail variable to this simulation to change the system from a rain/precipitation supercell to a hail storm supercell, we were able to view the downdraft. We realized and concluded that the presence of hail in a supercell would a strong indication that tornadogenesis is developing or nearly completed. Future Directions 1. Perform further simulation runs and examinations through the MSI (Minnesota Supercomputing Institute) supercomputer using the Itasca system at the University of Minnesota. 2. Create more Vis5d simulations with larger, clearer Entropic anomalies appearing, especially the Entropic Sink in the downdraft. a. Incorporate different variables, such as rain and wind directions, to find more relationships between them and the Entropic anomalies. 3. Calculate values for the Entropy Gradient based on the data used for the simulations and the values we used for the variables in the simulations. Bibliography [1] Arnold, V.I. (1978). Mathematical Methods of Classical Mechanics, Springer­Verlag, New York. [2] ARPS 4.5.2 (User Guide) http://www.caps.ou.edu/ARPS/arpsdoc.html 23 ARPS was written by Center for Analysis and Prediction of Storms (CAPS) at the University of Oklahoma (The Norman Campus). [3] Chorin, Alexandre Joel, and Jerrold E. Marsden. ​
A Mathematical Introduction to Fluid Mechanics​
. 3rd ed. New York: Springer­Verlag, 1993. Print. [4] Dutton, J. A., 1976: ​
The Ceaseless Wind: An Introduction to the Theory of Atmospheric Motion​
. McGraw­Hill, 579 pp. [5] Sasaki, Yoshi K. "Entropic Balance Theory and Variational Field Lagrangian Formalism: Tornadogenesis." ​
J. Atmos. Sci. Journal of the Atmospheric Sciences​
71 (2014): 2104­113. Print. [6] Sasaki, Yoshi K., Veress B., and Szigethy J. "Entropic Balance Theory and Tornadogenesis." Horizons in Earth Science Research​
(2010): 243­60. Print. [7] Vis5d http://www.ssec.wisc.edu/~billh/vis5d.html Vis5D was written by the Visualization Project at the University of Wisconsin­Madison Space Science and Engineering Center (SSEC) developed by Bill Hibbard, Johan Kellum, and other collaborators. Appendix Action Equation: Recall the Euler­Lagrange Density equation (Eq. 4), there are different principle formulas/equations in (Eq. 4). In the first part of (Eq. 4) involving ⍴ , the equation is: 1 2
2 |v| − U (⍴, S) − Φ (A.1) (A.1) is the Action equation involving all the energies involved with the supercell system, which are: Kinetic energy, Internal energy, and Gravitational Potential energy. The kinetic energy is represented by 12 |v|2 , in which the ⍴ (the density of air) on the outside of the brackets will represent the mass involved with kinetic energy. Internal energy is represented by U (⍴, S) , while the gravitational potential energy is represented by Φ . Conservation of Mass: As for the second part of (Eq. 4), involving α , we are dealing with the conservation of mass represented by: 24 ∂t⍴ + ∇ ∙ (⍴v) (A.2) However, (A.2) of itself if not real conservation of mass unless you were set it equal to zero. α is a Lagrangian multiplier that’s satisfying the constraints of conservation, but the mass could change due to the Lagrangian multiplier and energies could be changing the mass. Energy can’t be created or destroyed, so when you set them equal to zero, the energies will be at equilibrium. When you set (A.2) equal to zero, the mass won’t change. Therefore, we have: ∂t⍴ + ∇•(v⍴) = 0 (A.3) We changed (A.3) by switching the v and ⍴ , due to multiplication properties, to make it more simple to go through the proof behind the conservation of mass (A.3). Make notice that ⍴ in (A.3) is a function of x and t ; e.g. ⍴(x, t) . When it comes to understanding conservation of mass in tornadogenesis, consider a parcel of air such as a beachball or a glob. Air parcels may not always be a perfect sphere. When it comes to examining the air parcel’s surface and boundary, we can view it as: d
dt
V p
∂V ⍴
∫∫ ∫ ⍴ dV = − ∫ ∫ v⍴ • ndS (A.4) In (A.4), the n (in vector style) represents the unit normal to the surface (outward normal), the dS represents the surface integral, and the V p represents the boundary of the sphere’s surface. Further expanding on (A.4), let’s recall the Divergence Theorem in which we have: ∫ ∫ F • ndS = ∫ ∫ ∫ ∇ • F dV ∂V R V R (A.5) In (A.5), the ∂V R is a closed surface bounding the surface V R . Incorporating the Divergence Theorem (A.5) to (A.4), we will have: d
dt
∫
∫∫ ⍴ dV = − ∫
V p ∫
∂V p
v⍴ • ndS = − ∫∫∫ ∇ • (v⍴)dV (A.6) However, after interchanging the order of integration and differentiation in (A.6), our result will be: 25 d
dt
V R
V R ∫ ∫∫ ⍴ dV = ∫∫ ∫ ∂⍴∂t dV = − ∫ ∫ ∫ ∇ • (⍴v)dV V p (A.7) To help complete our proof and simplify (A.7), we can group the integrals to the left side of the equation by adding the ∫ ∫ ∫ ∇ • (v⍴)dV to both sides of (A.7). Thus, we would get: V R V R
V R ∫∫ ∫ ∂⍴∂t dV + ∫ ∫ ∫ ∇ • (v⍴)dV = 0 (A.8) ∫ ∫ ∫ ∂⍴∂t + ∇ • (v⍴)dV = 0 V R (A.9) ∂⍴
∂t
+ ∇ • (v⍴)dV = 0 (A.10) Therefore, we have our conservation of mass at (A.10). Conservation of Entropy: As for the final and third part of the Euler­Lagrange Density Equation (Eq. 4), involving
β , the equation is the conservation of entropy in the form of: ∂(⍴S)
∂t
+ ∇ • (v⍴S) (A.11) (A.11) would be involving no net loss of entropy, but like the conservation of mass section in (Eq. 4), this is not the true representation of conservation of entropy due to the Lagrange multiplier and changing energies. Thus, similar to conservation of mass, we need to set (A.11) to zero to prevent any mass from changing and put energies at equilibrium. Thus, we have: ∂(⍴S)
∂t
+ ∇ • (v⍴S) = 0 (A.12) To further understand the conservation of entropy in tornadogenesis, we will basically be imitating the derivation of conservation of mass from the previous section. Within the gradient of 26 (A.12), we have S added, so to understand its effect on the gradient, let’s do a physical example. Imagine in this situation that S is the entropy/unit mass and [⍴S] are units, then: [⍴S] = (entropy/mass)(mass/unit volume) = (entropy/unit volume) (A.13) Referencing (A.13), the air parcel would have an entropy represented by: d
dt
V R ∂V R ∫ ∫ ∫ ⍴S dV = − ∫ ∫ ⍴vS • n dS (A.14) Using a very similar process that was done in the conservation of mass section, which involves using the Divergence Theorem, interchanging the order of integration and differentiation, and grouping integrals, we will have a similar result of: ∫ ∫ ∫ dtd (⍴S) + ∇ • (⍴Sv) dV = 0 V R (A.15) d
dt (⍴S) + ∇ • (⍴Sv)
= 0 (A.16) Therefore, we have (A.16) as our conservation of entropy just like (A.12). Thermodynamic Equations: Here are some thermodynamic equations that we used to help simplify some of our equations: dU = T dS − P dV (A.17) ∂U
∂S
= U S(⍴, S) = T (A.18) ∂U
∂⍴
= U ⍴(⍴, S) =
P
⍴2
(A.19) Vector Fields and Identities: 27 In order for us to find the Euler Equation, which involved the velocity vector v , we had to reference vector fields and their identities. The vector identity that was used for simplifying the gradients and vectors within (Eq. 58) was: F × (G × H ) = (F ∙ H )G − H (F ∙ G) (A.20) in which the F is in vector form. The F would represent v , the H represents ∇S , and the G
represents ∇β from (Eq. 54). Various Substitutions​
: Here are some more equations that we used to simplify when dealing with the Euler Equation. To begin with we are going to go through how we found ∂tS . We start by using the product rule on (Eq. 17) and get: ∂t(⍴S) + ∇ ∙ (⍴vS) = (∂t⍴)S + ⍴(∂tS) + (⍴v) ∙ ∇S + S[∇ ∙ (⍴v)] = 0 (A.21) If we group like terms then we would get the equation of: S[(∂t⍴) + ∇ ∙ (⍴v)] + ⍴[(∂tS) + (v ∙ ∇S)] = 0 (A.22) We can simplify further with (Eq. 11), (∂t⍴) + ∇ ∙ (⍴v) = 0 , and divide both sides by ⍴ to get the final equation of: (∂tS) + (v ∙ ∇S) = 0 (A.23) ∂tS = − (v ∙ ∇S) (A.24) Now for the ∂tβ we just add T and subtract (v ∙ ∇β) to both sides to (Eq. 46) to get the equation of: ∂tβ = − (v ∙ ∇β) + T (A.25) For our last equation just has to do with a cross product identity that we used for (Eq. 55) and (Eq. 56): ω = ∇β × ∇S = − ∇S × ∇β = ( 1S ∇S) × (− S∇β) (A.26) 28 ARPS simulations​
: Here are the following links to the ARPS simulations that we created through Vis5d. In order to upload our simulations to our presentations, we had to create these youtube links. Please reference the copyright of the video address if these will be used in later works ​
[1, 7]. ● Supercell formation (Vd. 1) ○ https://www.youtube.com/watch?v=xQDzkOlb1kQ Key: Updraft​
= 20 m/sec ​
(Orange) Positive Perturbation Potential Temp ​
= 3 K​
(Blue) Negative Perturbation Potential Temp​
= ­3 K ​
(Green) Vorticity ​
= 0.010 1/s​
(Aqua) ●
●
●
●
● Hail storm supercell system (Vd. 2) ○ https://www.youtube.com/watch?v=icYnRMf_9U0 ○ There’s more of an entropic anomaly and downdraft appearing when the hail variable is included in the formation of tornadogenesis, especially with Entropic Sink. Key: Updraft​
= 20 m/sec ​
(Orange) Positive Perturbation Potential Temp​
= 2 K​
(Blue) Negative Perturbation Potential Temp​
= ­2 K ​
(Green) Vorticity ​
= 0.010 1/s​
(Aqua) Hail​
: 2.0 g/kg (White) ●
●
●
●
●
● Arching Vortex Lines of a supercell (Vd. 3) ○ https://www.youtube.com/watch?v=N1zNkP8jSSc ○ The visual representation is only part of the arching vortex lines ■ Theoretically, the vortex lines should appear as going around the vorticity on the surface to show the interaction between the winds and flank downdrafts. ■ The simulation fails to capture the whole picture of arching vortex lines. ●
●
●
●
Key: Updraft​
= 20 m/sec ​
(Orange) Positive Perturbation Potential Temp​
= 4 K​
(Blue) Vorticity ​
= 0.010 1/s​
(Aqua) Arching Vortex Lines ​
(Purple) ● Hail Storm Supercell (entropic anomalies & downdraft present) (Vd. 4) ○ https://www.youtube.com/watch?v=l66f7vYEgP4 ○ Pause at frames 3 & 4 to see Entropic anomaly occur with Entropic Source and Sink. 29 ○ This is a hail storm supercell system; however, the hail variable doesn’t appear because we desire to see the entropic anomaly occur more visibly to us. ●
●
●
●
●
Key: Updraft​
= 20 m/sec ​
(Orange) Positive Perturbation Potential Temp​
= 3 K​
(Blue) Negative Perturbation Potential Temp​
= ­2 K ​
(Green) Vorticity ​
= 0.010 1/s​
(Aqua) Hail​
: 2.0 g/kg (White) ● Hail Storm Supercell (more clearer entropic anomalies and downdraft present) (Vd. 5) ○ https://www.youtube.com/watch?v=DI6ciRdDZ0E&feature=youtu.be ○ Frames 26­28 will have the most clear example of the entropic anomalies (both the Entropic Source and Sink). ■ Frames 3, 4, 15, and 16 are also good representations ○ This is a hail storm supercell system; however, the hail variable doesn’t appear because we desire to see the entropic anomaly occur more visibly to us. ○ This is the simulation that provided the Vis5d simulation screenshots we used for (Figure 7). ●
●
●
●
●
Key: Updraft​
= 20 m/sec ​
(Orange) Positive Perturbation Potential Temp​
= 3 K​
(Blue) Negative Perturbation Potential Temp​
= ­1 K ​
(Green) Vorticity ​
= 0.010 1/s​
(Aqua) Hail​
: 2.0 g/kg (White)