Q. Deng a, NYU Abu Dhabi G. Hernandez-Duenas b, UNAM A. Majda, Courant S. Stechmann, UW-Madison L.M. Smith, UW-Madison

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1 Minimal Models for Precipitating Turbulent Convection Q. Deng a, NYU Abu Dhabi G. Hernandez-Duenas b, UNAM A. Majda, Courant S. Stechmann, UW-Madison L.M. Smith, UW-Madison Contours of rain water in a simulated squall line a NSF b CMG p. 1/37

2 Goal Find the simplest possible model that captures convective organization at many different length scales, for example vortical hot towers: tall cumulonimbus clouds roughly 10 km (horizontal) by km (vertical) squall lines: horizontal scales 200 km convectively coupled waves: horizontal scales 2000 km p. 2/37

3 Examples from Observations Satellite data of hot towers during Hurricane Bonnie 1998: colors correspond to surface precipitation; 18 km high; 10 km wide. p. 3/37

4 Examples from Observations Radar image of a squall line near Hong Kong, roughly 200 km long, propagates normal to the line p. 4/37

5 Outline Derivation of the minimal model Test case: squall lines Dynamic stability analysis; relation to thermodynamic stability p. 5/37

6 The Anelastic Equations with Warm-Rain Bulk Cloud Physics ( ) Du p Dt + f sin(φ)u = h ρ(z) ( ) θ +k g θ(z) + ε oq v q c q r ( ρ(z)u) =0, Dθ +wd θ(z) Dt dz = L θ(z) c p T (z) (C d E r ) ρ(x,t)= ρ(z)+ρ (x,t), with ρ(z) prescribed, etc; valid for H ρ(d ρ/dz) 1 (the density scale height) p. 6/37

7 Bulk Cloud Physics Dq v Dt = C d + E r, Dq c Dt =C d A r C r Dq r Dt 1 ρ(z) z ( ρ(z)v T q r )=A r + C r E r C d : Condensation q v q c, E r : Evaporation q r q v A r, C r : Auto-conversion and Collection q c q r V T : Rainfall velocity p. 7/37

8 Now make the following approximations Boussinesq H ρ(d ρ/dz) 1 (not true here!), eliminates prescribed backgrounds Fast Auto-Conversion of cloud to rain water (in reality 15 minutes), eliminates q c,a r,c r Fast Condensation (in reality a few seconds) and Fast Evaporation p. 8/37

9 The Boussinesq Approximation un-differentiated background replaced by constants Du ( θ Dt + f sin(φ)u = p h + k g + ɛ o q v q r θ o ) u =0, Dθ Dt + d θ(z) dz w = L c p (C d E r ) θ = T (p o /p) R/c p R, R v gas constant for air, vapor; R v /R = ɛ o +1 p. 9/37

10 Equations for water vapor and rain Dq v Dt = (C d E r ), Dq r Dt V T q r z = C d E r C d E r =0, q v q vs, q r =0 unsat C d E r = w dq vs(z), q v = q vs, q r > 0 sat dz q vs (T,p) q vs (z) is the saturation profile p. 10/37

11 Basic physics C d E r > 0: Dw Dθ Dt + Bw = L c p (C d E r ) Dt = p z + g ( θ θ o + ε o q vs q r ) Standard values: L d Jkg 1, c p 10 3 Jkg 1 K 1, θ o = T o =300K, d θ(z)/dz B =3Kkm 1, etc. p. 11/37

12 Contours of Water Vapor Anomaly ( km) Moisture anomaly contour ( kg/kg) in a Vortical Hot Tower; 30 min after moisture bubble injection at low altitude p. 12/37

13 Constraints Either Unsaturated: q v <q vs : q r =0, q tot = q v Or Saturated: q v = q vs : q r = q tot q vs q tot = q v + q r p. 13/37

14 FARE Model: Fast Auto-Conversion & Rain Evaporation Using θ e = θ + L c p q v, q tot : Dθ e Dt =0, Dq tot Dt V T q r z =0 no explicit source terms Similar models for non-precipitating, shallow convection; q r replaced by q c : Grabowski & Clark 1993; Spyksma, Bartello & Yao 2006; Pauluis & Schumacher 2010 p. 14/37

15 Environmental conditions Low-altitude moistening, e.g. over a warm ocean Scattered convection; 3D contours of rain water p. 15/37

16 A background wind will organize the convection: 15 Background velocity for squall lines 10 z in km u in m/s p. 16/37

17 A squall line Fred Roswald & Judy Jensen, SquallApproachesSumatra.jpg p. 17/37

18 More Characteristics of Squall Lines (Wallace & Hobbs, 2006) Long lasting multi-cell storm Tilted profile Propagate long distances (Houze 2004) Low-altitude cold pool p. 18/37

19 References Observations, e.g. during GATE: Barnes & Sieckman 1984; during TOGA COARE: Jorgensen, LeMone & Trier 1997; LeMone, Zipser & Trier 1998; Review: Houze CRMs: Fovell & Ogura 1988; Lafore & Moncrieff 1989; Grabowski, Wu & Moncrieff 1996, 1998; Xu & Randall 1996; Lucas, Zipser & Ferrier 2000; Lu & Moncrieff 2001 Conceptual & Minimal Models: Moncrieff & Green 1972; Moncrieff & Miller 1976; Moncrieff 1981; Emanuel 1986; Rotunno, Klemp, Weisman 1988; Majda & Xing 2010 p. 19/37

20 FARE squall line: 3D Contours of Rain Water p. 20/37

21 Vertically averaged rain, Low-level cold pool z avg q r, T = 33.6 hrs 120 at T=33.6 hrs, z =0.6 km y (km) 60 y (km) x (km) x (km) x p. 21/37

22 Time Evolution q r ( kg/kg) x u horizontal averages Background 7.92hrs 16.08hrs 24hrs 31.92hrs 40.08hrs 48hrs time (hrs) Height (km) m/s x (km) m/s Propagates at the speed of the jet max (black dash) p. 22/37

23 Multi-cloud structure 15 q r contours. Time integration on [24,25.92] hrs x z (km) x (km) 0.5 p. 23/37

24 Stability: what is the probability for clouds/precipitation to form? Thermodynamic stability: parcels rising adiabatically, equilibrium thermodynamics Dynamic stability: linear/nonlinear stability analysis of the dynamical equations p. 24/37

25 Rising Dry vs Moist Parcels Dry Parcel Expands and cools Moist Parcel Cools less due to latent heat release Altitude (decreasing pressure) p. 25/37

26 Thermodynamic Concept of Conditional Stability Height B=0 Dry ALR Unstable Moist ALR e =0 Conditionally unstable Environmental Temp. Stable Environmental Temp. Dry parcel: Moist parcel Temperature 25 o C p. 26/37

27 Dynamical Linear Stability FARE with a saturated background gives the same right stability boundary: Γ e =(g/θ o ) d θ e /dz > 0, θ e = θ +(L/c p )q vs sufficient for stability; as well as a left boundary sufficient for instability in the saturated case. p. 27/37

28 Stability Diagram; fixed q vs (z) profile VT (m s 1 ) Unstable 1 km Stable UnStable 40000km d T/dz (K km 1 ) Grey denotes unstable scales; left of red line guaranteed unstable: Γ=(g/θ o )d( θ e θ o q t )/dz < 0 right of blue line guartd. stable: Γ e =(g/θ o )d θ e /dz > 0, [E86] p. 28/37

29 Growth rate of the unstable eigenmode 5 Growth rate (hr 1 ) V =5m/s T V =4.5m/s T V =4m/s T V T =3.5m/s V T =3m/s V T =2.5m/s V T =2m/s V =0.5m/s V =1m/s V =1.5m/s 0 T T T k h /(2 /40000km) Fixed background temp/saturation gradients; k z =2π/15 km 1 p. 29/37

30 (In)stability boundaries Numerically find Γ < 0, Γ e > 0 as sufficient conditions Analytically Γ=0is the single stability boundary for V T =0(a singular limit) Analytically Γ e =0is the single stability boundary for V T (nonlinear asymptotics) p. 30/37

31 Singular limit V T = kh (km 1 2π/40000) dq vs /dz (g kg 1 km 1 ) dq vs /dz (g kg km ) Left: V T =0ms 1 ; Right: V T =0.01 m s 1 ; Grey = unstable; k z =2π/15 km 1, B =3Kkm 1. p. 31/37

32 Equation Structure: Saturated with V T =0or V T Du Dt = p + ˆkb, 1 w Db Dt = Γ ( Γ e) with positive definite energy for Γ > 0: ( u 2 + (b ) 2 ) E = 1 2 Γ and eigenvalues σ ± = ±(k h /k)γ 1/2 p. 32/37

33 Stability Parameter Dry: b = gθ /θ o, Γ dry = N 2 =(g/θ o )d θ/dz Saturated, V T =0: b = g(θ e/θ o q t) Saturated, V T : b = gθ e/θ o p. 33/37

34 Energy Conservation in Saturated Regimes, V T 0 t E + E = 1 2 ( u(e + p) positive definite for ( u 2 + (gθ e/θ o ) 2 ) = V T z ( (gq r ) 2 2(Γ Γ e ) Γ e + (gq r/θ o ) 2 Γ Γ e Γ e = g θ o d θ e dz > 0 Γ Γ e strictly greater than zero ) ) p. 34/37

35 Conclusions The FARE model ignores detailed cloud physics, but retains conservations laws for q tot, θ e = θ + L c p q v (why it "works") Provides a framework for linear and nonlinear analysis p. 35/37

36 Conclusions FARE adds V T to a concept of conditional instability in a saturated domain Next: Derivation of a Moist Balanced Dynamics (the vortical mode + the moisture mode)? Nonlinear energy analysis p. 36/37

37 Nonlinear Energy Consistency ( ) [ ( )] u u u u t 2 +Π + u 2 +Π+p ] [V T g(z a)q r z = V T gq r Π(θ e,q t,z)= z a g θ o θ v (θ e,q t,η) dη θ v = θ v (θ e,q t,z)=θ e θ o q t + θ o ( ɛ o L ) +1 min(q t,q vs (z)). c p θ o θ v linear virtual potential temperature; integration assumes θ e, q t fixed; a arbitrary reference height with q vs (a) =0. p. 37/37