Chapter 2: basic equations and tools

All sections are considered core material, but the emphasis is on the “new” elements in the last three sections: Chapter 2: basic equations and tools 2.5 – pressure perturbations 2.6 – thermodynamic diagrams 2.7 - hodographs

flow vsvorticity: the right-hand rule

hydrostatic pressure perturbations 2004 – 2046 UTC 2049 – 2130 UTC 2134-2214 UTC H H L H H L L L L H H H Measured height (1-m increments) of the 595-hPa surface during a winter storm over the Med Bow Range Wyoming (11 February 2008) (Parish and Geerts 2013)

hydrostatic pressure perturbations:stratified flow over an isolated peak wind H L H L

hydrostatic pressure perturbations:2D stratified flow over a mountain Fig. 6.6: polarization relations between q’, p’, u’, and w’ in a westward tilting, vertically-propagating internal gravity wave blue = cold, red = warm wind H H L examine column T’ above terrain

non-hydrostatic pressure perturbations pressure field in a density current (M&R, Fig. 2.6) pressure units: (Pa) density current L H H L L H H interpretation: use Bernoulli eqn along a streamline L H H Fig. 2.6 Note that p’ = p’h+p’nh = p’B+p’D p’h is obtained from and p’nh = p’-p’h p’B is obtained by solving with at top and bottom. p’D = p’-p’B

The pressure perturbation field in a buoyant bubble (e.g. a Cu tower) contains both a hydrostatic and a non-hydrostatic component. The hydrostatic component looks like this. p’h B < 0 B > 0

the buoyancy-induced pressure perturbation gradient acceleration (BPPGA): z x Shaded area is buoyant B>0 Analyze: This is like the Poisson eqn in electrostatics, with FB the charge density, p’B the electric potential, and p’B show the electric field lines. The + and – signs indicate highs and lows: where L is the width of the buoyant parcel

Where pB>0 (high), 2pB <0, thus the divergence of [- pB] is positive, i.e. the BPPGA diverges the flow, like the electric field. • The lines are streamlines of BPPGA, the arrows indicate the direction of acceleration. • Within the buoyant parcel, the BPPGA always opposes the buoyancy, thus the parcel’s upward acceleration is reduced. • A given amount of B produces a larger net upward acceleration in a smaller parcel • for a very wide parcel, BPPGA=B (i.e. the parcel, though buoyant, is hydrostatically balanced) (in this case the buoyancy source equals d2p’/dz2) Proof

This pressure field contains both a hydrostatic and a non-hydrostatic component. B < 0 B > 0 explanation: what is shown is a buoyantly-induced (p’b) perturbation pressure field with a high above and a low below the warm core. The spreading of the isobars in the warm core suggest that these pressure perturbations are at least partly hydrostatic: greater thickness in the warm core. Yet a pure hydrostatic component (p’h) would just have a low below the warm core, down to the surface (Fig. 2.7).

pressure field in a buoyant plume, e.g. growing cumulus pressure units: (Pa) 2K bubble, radius = 5 km, depth 1.5 km, released near ground in an environment with CAPE=2200 J/kg. Fields are shown at t=10 min H L L Note that p’ = p’h+p’nh = p’B+p’D L L L p’h is obtained from and p’nh = p’-p’h H -250 +225 p’Bis obtained by solving with at top and bottom. p’D = p’-p’B H H L L L H Fig. 2.7

Cumulus bubble observation Example of a growing cu on Aug. 26th, 2003 over Laramie. Two-dimensional velocity field overlaid on filled contours of reflectivity (Z [dBZ]); solid lines are selected streamlines. (source: Rick Damiani)

dBZ 8m/s Cumulus bubble observation 20030826, 18:23UTC • Two counter-rotating vortices are visible in the ascending cloud-top. • They are a cross-section thru a vortex ring, aka a toroidal circulation (‘smoke ring’) (Damiani et al., 2006, JAS)

Shear interacting with an updraft ambient wind p’D> 0 (a high or “H”) on the upshear side of a convective updraft, and p’D< 0 (L) on the downshear side

Shear, buoyant updraft, and linear dynamic pressure gradient S L H Shear deforms the parcel in the opposite direction as that due to a convective updraft on the upshear side of that updraft. In other words, the respective horizontal vorticity vectors point in opposite directions on the upshear side, yielding an erect upshear flank. The downshear flank is tilted downwind because the respective vorticity vectors supplement each other. Fig. 2.8

skew T log p CAPE, and CIN always use virtual temperature ! radiosonde analysis model sounding analysis Fig. 2.9

downdraft CAPE Fig. 2.10

hodographs c : storm motion vr= v-c : storm-relative mean flow S: mean shear wh : mean horizontal vorticity wh = ws+wc: streamwise & crosswise vorticity Fig. 2.11 and 2.12

horizontal vorticity shear vector S .M storm-relative flow vr c horizontal vorticity mass-weighted deep-layer mean wind

directional shear vs. speed shear x storm motion c Fig. 2.13

real hodographs near observed severe storms Fig. 2.14

streamwise vorticity cross-wise • streamwise • cross-wise streamwise

storm-relative flow horizontal vorticity definition of helicity (Lilly 1979) • the top is usually 2 or 3 km (low level !) • H is maximized by high wind shear NORMAL to the storm-relative flow •  strong directional shear • H is large in winter storms too, but static instability is missing

streamwisevorticityproduces helical flow more on this in chapter 7 Fig. 2.15

Chapter 2: basic equations and tools