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

flow vs vorticity: the right-hand rule

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

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

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

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. B > 0 B < 0 p’ h

Analyze: This is like the Poisson eqn in electrostatics, with F B 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 the buoyancy-induced pressure perturbation gradient acceleration (BPPGA): Shaded area is buoyant B>0 x z

Where p’ B >0 (high),  2 p’ B <0, thus the divergence of [-  p’ B ] 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 d 2 p’/dz 2 ) 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).

interpreting pressure perturbations Note that p’ = p’ h +p’ nh = p’ B +p’ D p’ B is obtained by solving with at top and bottom. p’ D = p’-p’ B p’ h is obtained from and p’ nh = p’-p’ h

pressure units: (Pa) H L H L H L H L H H interpretation: use Bernoulli eqn along a streamline Fig. 2.6 Pressure perturbations in a density current

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 H L H L H L H L Fig L L L Pressure perturbations in a buoyant bubble, e.g. a growing cumulus

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

dBZ 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’) , 18:23UTC 8m/s (Damiani et al., 2006, JAS) Cumulus bubble observation

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

Fig. 2.8 S Shear, buoyant updraft, and linear dynamic pressure gradient HL 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. 2.9 skew T log p CAPE, and CIN always use virtual temperature ! radiosonde analysis model sounding analysis

limitations of parcel theory (section 3.1.2)

Fig downdraft CAPE

Fig and 2.12 c : storm motion v r = v-c : storm-relative mean flow S: mean shear  h : mean horizontal vorticity  h =  s +  c : streamwise & crosswise vorticity hodographs

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

Fig directional shear vs. speed shear x storm motion c

Fig real hodographs near observed severe storms

streamwise vorticity streamwise cross-wise streamwise cross-wise

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 storm-relative flow horizontal vorticity

Fig streamwise vorticity produces helical flow more on this in chapter 7