Entropic Balance Theory and Radar Observation for Prospective Tornado Data Assimilation

Abstract

This article reports further theoretical development on the entropic balance theory applied to tornadogenesis (Sasaki 2009, 2010), and the first preliminary application of the theory to radar observations. The entropic balance is a newly found balance, different from the other balance conditions, such as hydrostatic, (quasi-)geostrophic, cyclostrophic, Boussinesq, and anelastic. The entropic balance condition is described as the sole diagnostic Euler-Lagrange equation derived from the Lagrangian of the variational formalism. The entropic balance is most general and involves no additional assumptions other than for the flow with high Reynolds and Rossby numbers estimated as appropriate for supercell storms and tornadoes. The entropic balance theory and the deduced wrap-around mechanism explain well the observations and simulations of tornado, RFD, hook-echo, upward tilting of horizontal vorticity, the vertical in-phase superimposition between upper and lower mesocyclones, and sudden transition from supercell, mesocyclones totornado. In the application, new variables DZ (temporal difference of radar reflectivity) and DZ_DR (temporal difference of differential reflectivity) are introduced to compute the entropy anomaly based on the entropic balance theory. The conditions necessary for the transition from supercell to tornado are clarified from the theory and verified from the DZ and DZ_DR analyses for a non-tornadic supercell case compared with VORTEX2 tornadic case.Since the entropic balance theory is found to fit well with all analyzed results of tornado and visual observations, it is suggested to use the entropic balance equation as a constraint for variational data assimilation in future development as a challenge.

Appendices

Appendix 1 Some Key Questions on Tornadogenesis

Accurate forecasting of tornadogenesis is one of the unsolved problems, in spite of a great number of observations and research made over many decades. In recent years, significant progress has been made to understand the mechanism of tornadogenesis. However, there still remain key questions and difficult problems in fully understanding tornadogenesis and tornado.

Some of the key questions that need to be answered by any proposed theory of tornadogenesis are:

• 1. How does the mesocyclone develop? Note that mesocyclone and wall cloud are known as observed with tornadogenesis.

• 2. Why are hydrometeors able to be overshot against the upper air westerlies of much stronger wind speed than that of low level south-easterly inflow to tornadic storm?

• 3. Why are the locations of tornado and major precipitation regions spatially separate, not coincident?

• 4. Why is dry air aloft important for tornadogenesis?

• 5. Why do multiple vortices sometime develop before and during tornadogenesis?

• 6. How does the tornado develop in a hook echo, at the south-west corner, not at the center, of a supercell, and why is it associated with wall cloud?

• 7. Why does the tornado touch down in the perpendicular direction to the ground? Note that tornado should touch down to the ground in a parallel direction if the tornado is generated by the upward tilting theory of horizontal vortex.

• 8. Why is the tornado a phenomenon of low altitudes ( < 2–3 km) of the atmosphere?

• 9. What is the role of upward tilting of low level horizontal vorticity for tornadogenesis ?

• 10. What are necessary and sufficient conditions to separate tornadogenesis from non-tornadogenesis, in spite of several favorable conditions (such as favorable environmental soundings, supercell, mesocyclones, and RFD)?

These questions seemed to be well answered by the entropic balance theory (Sasaki 2009, 2010).

Appendix 2 Answer to the Questions 1, 2, 3 and 4

Firstly we consider a simple case where the linear slope between the entropic source and sink as shown in Fig. 18.5 and the flow velocity (rotational component) is horizontal and southerly. For this case, the direction of the vorticity $\boldsymbol{\omega }$ is vertical as given by (18.5) that is expressed by the entropic right-hand rule (Fig. 18.1), and is shown in Fig. 18.5 where the rotational flow direction of the vorticity $\boldsymbol{\omega }$ is shown by an arrow with double solid lines. The vorticity $\boldsymbol{\omega }$ represents mesocyclone existed indeed in the linear entropy slope between the entropic source and sink. It may answer the question 1.

The entropic anomaly in the air above the top of convective supercell is negative due to evaporation of the hydrometeors in the dry westerlies and radiative cooling, while the having the positive entropic source below. Accordingly, the maximum spatial gradient of the entropy anomaly between the points just above the supercell (S^′ < 0) and at the convective center of the supercell (S^′ > 0) is directing straight downwards. With the upper air westerlies wind vector and the entropic spatial gradient vector, the vorticity $\boldsymbol{\omega }$ is that of the horizontal vortex tube as shown at the top of the supercell in Fig. 18.5. The upper vortex seems the key to overshoot the hydrometeors generated in the supercell updraft, westwards against the headwind westerly jet. This may answer the question 2. From the above answer to the questions 1, and 2, it is apparent that the question 3 is answered from the above answer to the questions 1 and 2. To answer the both questions 1, 2 and 3, it is noted that the dry air plays the key roles, and it may answer the question 4.

Appendix 3 Answer to the Questions 5, 6, 7, 8, 9 and 10

The “wrap-around mechanism” is explained in a simple way by using the diagnostic E-L equation (18.5) and the entropic right-hand rule (Fig. 18.1) of (18.6). Figure 18.6 is prepared which has opposite sign of spatial entropy gradient and the flow velocity direction (rotational component) and produces the same vorticity of Fig. 18.1. It is called as conjugate vortex (or vorticity) in this article. Combination of the original vortex and its conjugate produces multi-vortexes as well as single vortex. It answers the question 5. The answer for the questions 6 and 8 may be easily found from Fig. 18.4.

The answer for the questions 7 and 8 is discussed in Sect. 18.7. The question 9 may be answered in Sect. 18.5. The answer to the historically long-standing question 10 is a tough one, but may be hinted by the entropic balance theory discussed in this article and in an example shown in the following Appendix 4. It is challenging for continuing research to find a full answer for the question 10.

Appendix 4 Entropy Variation and Tornadogenesis

The entropy variation due to cloud-physical phase change is computed at the altitudes of 1–3 km where condensation and evaporation to provide thermodynamical effects for development of mesocyclones and tornado, and the atmospheric pressure of approximately 750 mb and temperature of 0^∘ C (273^∘K) as an example. For simplicity for this preliminary investigation, we assume also that S₀ = 0 and only consider the diabatic effects of water molecules on S of the surrounding air on a moving coordinates with tornado.

The entropy change Δ _c S of the surrounding air due to water vapor condensation measured at 100 ^∘C and 1,013 mb is estimated as 109.0 ${\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$, and that of evaporating of water droplet $-109\,{\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}\,{\mathrm{mol}}^{-1}$. Since moisture measurement is not considered in this preliminary investigation and insufficient measurement and knowledge on the cloud-physical phase changes of actual cloud, the estimates were made simply based on the measurements of heat in published chemical experiments. Their values are adjusted to the value of 0 ^∘C and 750 mb for representing the altitude of 1–3 km, using the standard adjustment processes (Atkins and de Paula 2002; Watanabe 2003).

The adjustment amount due to the temperature change Δ _T S (100^∘C — $> {0}^{\,\circ }\mathrm{C})\,=\,-\,16.6\,{\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$ and that due to pressure change Δ _p S (1,013 mb — > 750 mb) = 2.1 ${\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$.

After the adjustments, the entropy change of the surrounding air due to condensation of water vapor is; $\Delta _{\mathrm{c}}\,\mathrm{S} = (109.0 - 16.6 + 2.1)\ {\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1} = 94.5\,{\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$, and for that due to evaporation of water droplets is $\Delta _{\mathrm{e}}\,\mathrm{S} = (-109.0 - 16.6 + 2.1)\ {\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1} = -123.5\,{\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$.

Thus we get the entropy difference between the entropic source and sink separated by the distance d; $\Delta _{\mathrm{d}}\,\mathrm{S} = (94.5 - (-123.5)) = 218.0\,{\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$.

Similarly, the absolute entropy S is calculated by adding the entropy changes due to melting of ice, $22.0\,{\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$ and the residual entropy, $0.8\,{\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$ from the Boltzmann’s third law of thermodynamics, resulting $\mathrm{S} = (94.5 + 22.0 + 0.8)\ {\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1} = 117.3\ {\mathrm{J}}^{\circ }{\mathrm{K}}^{-1}{\mathrm{mol}}^{-1}$.

From (18.6), the vorticity ω is written as

$$\omega = \mathbf{v}_{\beta }\, \times (1/\mathrm{S)}\,\mathrm{S}.$$

(18.51)

where Vrot represents the rotational component of flow velocity.

Usin the estimated values of S and Δ S, (18.51) becomes

$$\omega = \mathrm{V}_{\beta }\, \times \, 1.86(= 273.0/117.3)/\mathrm{d}\quad {\mathrm{s}}^{-1}$$

(18.52)

where d is distance between the entropic source and sink.

For an example of mesocyclone cases, v _β is taken 10 m/s and d as 5 km, then (18.52) leads $\omega = 0.0037\,{\mathrm{s}}^{-1}$. For tornado by wrap-around mechanism cases, (18.52) with 50 m/s of V _β and 100 m of d leads $\omega = 0.93\,{\mathrm{s}}^{-1}$. The former and latter seem appropriate order of magnitudes for mesocyclones and tornado respectively.