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## Solid body rotation (XY):

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**Solid body rotation (XY):**low pressure Divergent flow (XY): high pressure Shear flow (XY): *no pressure perturbation**(stagnation pressures near saddle pts. in streamline**pattern) Splat: Spin: (eddy rotation) For flow in solid body rotation:**2D Supercells??**2D Vorticity Eq. No tilting, stretching!! 2D Diagnostic Pressure: No rotationally-induced mid-level mesolow**A Simple “Model” of a Tornado:**Cyclostrophic balance: Rankine Combined Vortex: Solid-body rotation in core: Potential vortex outside: For full vortex: *For core region:**For Vortex at 3 km AGL:***Vmax=10 ms-1 .5 hPa .5 K 2 K 2 hPa Vmax=20 ms-1 8 hPa 8 K Vmax=40 ms-1 *simply assuming pressure change from inner-core region**…for Rankine vortex:**at rmax: beyond rmax: V20= 20 ms-1 Vrmax=40 ms-1**Physical processes controlling cell types:**• Buoyancy processes: basic updraft/downdraft, (ordinary cells) • Gust front processes: triggering of new cells, upscale growth, (multicells) • Dynamic processes: rotating updraft, dynamic vertical pressure gradient forcing, (supercells)**Basic Equations:**-- (Buoyancy) + ice….**…real bubble in 3D simulation**Buoyancy is Scale-Dependent!!!**Density Currents**Theoretical speed of propagation: **RKW Theory**Rotunno et al. (JAS, 1988) C/∆u > 1 “Optimal”condition for cold pool lifting C/∆u = 1 C/∆u < 1**Dynamic Pressure Effects:**(take divergence) diagnostic pressure eq. Dynamic pressure Buoyancy pressure Vertical momentum:**~**Updraft growing in sheared environment:**Vorticity Equation:**Vertical Vorticity: tilting stretching**Vortex Tube**Circulation:**Supercell Hodographs:**Supercell processes are Galilean invariant!!!**Potential Vorticity:**= 0 for isentropic motions Equivalent Potential Vorticity:**Davies-Jones, 1984**…from linear theory of circular, convective cells in a sheared environment, covariance of vertical velocity and vertical vorticity is proportional to the storm-relative environmental helicity *assumes steady-state, propagating storm**Storm-relative Environmental Helicity (SREH)**(actually, streamwise vorticity)**Thompson et al., WAF 2012**EBWD Convective Modes EBWD: Effective Bulk Wind Difference (half storm depth)**Thompson et al., WAF 2012**ESRH Convective Modes ESRH: Effective Storm-Relative Helicity (effective inflow layer)**Vortex Tube**Circulation:**Ward Tornado Chamber (1972)**Ingredients for a tornado:1) source of rotation 2) updraft