the dynamics of convection 1. 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)

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.

Cumulus Clouds 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 topping near the tropopause PPGF buoyancy force

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:

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

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

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 The buoyancy force this ignores effect of water vapor (+B) and the weight of hydrometeors (-B). w= sqrt(2CAPE) is the thermodynamic updraft strength limit

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

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

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

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

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 )

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

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

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 H L H L H L H L 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

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 ) (1D, steady state): simplify to 1D, steady state & solve (assumed)

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. effect of entrainment on a skew T-log p diagram

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=3R 3

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

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

2.7 sounding analysis CAPE CIN DCAPE (D for downdraft)

2.8 hodographs total wind v h shear vector S storm motion c storm-relative wind v r = v h -c height AGL (km) storm-relative flow v r vhvh S c

2.8.5 true & storm-relative wind near a supercell storm real example hypothetical profiles: different wind v h, but identical v r c

2.8.6 horizontal vorticity storm-relative flow horizontal vorticity

streamwise vorticity streamwise cross-wise streamwise cross-wise error note error in book

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