the dynamics of convection 1. Cumulus cloud dynamics

1. the dynamics of convection1. Cumulus cloud dynamics • The basic forces affecting a cumulus cloud • buoyancy (B) • buoyancy-induced pressure perturbation gradient acceleration (BPPGA) • dynamic sources of pressure perturbations • entrainment: Simple models of entraining cumulus convection (E) Suggested readings: M&R Section 2.5: fundamentals of convection and buoyancy M&R Section 2.6: pressure perturbations Bluestein (Synoptic-Dynamic Met Pt II, 1993) Part III Houze (Cloud Dynamics, 1993), Chapter 7 Holton (Dynamic Met, 2004) sections 9.5 and 9.6 Cotton and Anthes (Storm and Cloud Dynamics, 1982) Emanuel (Atmospheric Convection, 1994)

2. Essential Role of Convection • You have learned about the role of baroclinic systems in the atmosphere: they transport sensible heat and water vapor poleward, offsetting a meridional imbalance in net radiation. • Similarly, thermal convection plays an essential role in the vertical transport of heat in the troposphere: • the vertical temperature gradient that results from radiative equilibrium exceeds that for static instability, at least in some regions on earth.

3. Range in size from cumuli (less than 1000 m in 3D diameter) congesti (~1 km wide, topping near 4-5 km) cumulonimbus clouds (~10 km) to thunderstorm clusters (~100 km) to mesoscale convective complexes (~500 km). All are ab initio driven by buoyancy B vertical equation of motion: PPGF: pressure perturbation gradient force Cumulus Clouds topping near the tropopause buoyancy force PPGF

4. buoyancy: we derive buoyancy from vertical momentum conservation equation vertical equation of motion, non-hydrostatic basic state is hydrostatically balanced terms cancel small term define buoyancy B:

5. scaling the buoyancy force • Complete expression for buoyancy: where qh is the mixing ratio of condensed-phase water (‘hydrometeor loading’). In terms of its effect on buoyancy (Bq/g), 1 K of excess heat q‘ is equivalent to ... ... 5-6 g/kg of water vapor (positive buoyancy) ... 3-4 mb of pressure deficit (positive buoyancy) ... 3.3 g/kg of water loading (negative buoyancy) This shows that q’ dominates. All other effects can be significant under some conditions in cumulus clouds. see (Houze p. 36)

6. buoyancy:outside of cloud, buoyancy is proportional to the virtual pot temp perturbation

7. The buoyancy force • To a first order, the maximum updraft speed can be estimated from sounding-inferred CAPE: This updraft speed is a vast over-estimate, mainly b/o two opposing forces. • pressure perturbations: Buoyancy-induced (or ‘convective’) ascent of an air parcel disrupts the ambient air. On top of a rising parcel, you ‘d expect a high (i.e. a positive pressure perturbation), simply because that rising parcel pushes into its surroundings. The resulting ‘perturbation’ pressure gradient enables compensating lateral and downward displacement as the parcel rises thru the fluid. Solutions show the compensating motions decaying away from the cloud, concentrated within about one cloud diameter. • entrainment this ignores effect of water vapor (+B) and the weight of hydrometeors (-B). w= sqrt(2CAPE) is the thermodynamic updraft strength limit

8. this pressure field contains both a hydrostatic and a non-hydrostatic component B < 0 B > 0 Fig. 2.2 in M&R

9. discuss pressure changes in a hydrostatic atmosphere (M&R 2.6.1) • mass conservation: • pressure tendency = vertically integrated mass divergence • hypsometric eqn: • pressure tendency = vertically integrated temperature change

10. 2nd force: pressure perturbation gradient acceleration (PPGA) (M&R section 2.6.3) anelastic continuity eqn FB: buoyancy source FD: dynamic source ** These equations are fundamental to understand the dynamics of convection, ranging from shallow cumuli to isolated thunderstorms to supercells. For now we focus on FB. Later, we ‘ll show that FD is essential to understand storm splitting and storm motion aberrations. tensor notation

11. It can be shown that Also, p’D> 0 (a high H) on the upshear side of a convective updraft, and p’D< 0 (a low L) on the downshear side

12. 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

13. 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)

14. (10 km) (3 km) t=13 min t=8 min H H L L L Fig. 2.3 in M&R (no entrainment)

15. pressure field in a density current (M&R, Fig. 2.5) pressure units: (Pa) L H H L L H H interpretation: use Bernoulli eqn along a streamline L H H 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

16. pressure field in a cumulus cloud (M&R, Fig. 2.6) pressure units: (Pa) 2K bubble, radius = 5 km, depth 1.5 km, released near ground in environment with CAPE=2200 J/kg. Fields shown at t=10 min H L Note that p’ = p’h+p’nh = p’B+p’D p’h is obtained from and p’nh = p’-p’h L H p’B is obtained by solving with at top and bottom. p’D = p’-p’B H H L L L H

17. Third force (also holding back buoyancy):entrainment • entrainment does two things: (a) both the upward momentum and the buoyancy of a parcel are dissipated by mixing (b) cloudy air is mixed with ambient dry air, causing evaporation • No elegant mathematical formulation exists for entrainment E. The reason is that we are entering the realm of turbulence. We are reduced to some simple conceptual models of cumulus convection. • Thermals or Bubbles • Plumes or Jets • This is a general expression for continuous, homogenous entrainment (Houze p. 227-230) (1D, steady state): (assumed) simplify to 1D, steady state & solve

18. effect of entrainment on a skew T-log p diagram M&R Figure 2.4. A possible trajectory (dashed) that might be followed by an updraft parcel on a skew T-log p diagram as a result of the entrainment of environmental air.

19. Thermals or Bubbles • Laboratory studies • negatively buoyant, dyed parcels are released, with small density difference relative to the environmental fluid • basic circulations look like this: At first vorticity is distributed throughout the thermal. Later it becomes concentrated in a vortex ring. • The thermal grows as air is entrained into the thermal, via: • turbulent mixing at the leading edge; • laminar flow into the tail of the thermal Results: shape oblate, nearly spherical; volume=3R3

20. note shear instability along leading boundary • entrainment rate seems small at first • undilute core persists for some time, developing into a vortex ring Sanchez et al 1989

21. 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)

22. 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)

23. 2.7 sounding analysis • CAPE • CIN • DCAPE (D for downdraft)

24. 2.8 hodographs height AGL (km) • total wind vh • shear vector S • storm motion c • storm-relative wind vr = vh-c S vh storm-relative flow vr c

25. 2.8.5 true & storm-relative wind near a supercell storm c real example hypothetical profiles: different wind vh, but identical vr

26. 2.8.6 horizontal vorticity storm-relative flow horizontal vorticity

27. streamwise vorticity cross-wise • streamwise • cross-wise streamwise note error in book error

28. 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