Report

The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard University Physical importance of convection • vertical flux of heat and moisture – i.e., fluxes of sensible and latent heat » (momentum ignored here) • Really we care about flux convergences – or heating and moistening rates • Q1 and Q2 are common symbols for these » (with different sign and units conventions) Kinds of atm. convection – Dry • Subcloud turbulent flux – may include or imply surface molecular flux too – Moist (i.e. saturated, cloudy) • shallow (cumulus) – nonprecipitating: a single conserved scalar suffices • middle (congestus) & deep (cumulonimbus) – precipitating: water separates from associated latent heat – Organized • ‘coherent structures’ even in dry turbulence • ‘multicellular’ cloud systems, and mesoscale motions “Deep” convection • Inclusive term for all of the above – Fsurf, dry turbulence, cu, cg, cb – multicellular/mesoscale motions in general case • Participates in larger-scale dynamics – Statistical envelopes of convection = weather • one line of evidence for near-linearity – Global models need parameterizations • need to reproduce function, without explicitly keeping track of all that form or structure deep convection: cu, cg, cb, organized http://www.rsmas.miami.edu/users/pzuidema/CAROb/ http://metofis.rsmas.miami.edu/~bmapes/CAROb_movies_archive/?C=N;O=D interacts statistically with large scale waves straight line = nondispersive wave 1998 CLAUS Brightness Temperature 5ºS-5º N A convecting “column” (in ~100km sense) is like a doubly-periodic LES or CRM • Contains many clouds – an ensemble • Horizontal eddy flux can be neglected – vertical is w'T' Atm. column heat and moisture budgets, as a T & q column vector equation: water phase LS dynamics rad. changes eddy flux convergence T (p) Tadvtend(p) Qr L /C p (c e) z(w'T') t q qadvtend (p) 0 (c e) z(w'q') (p) t Surface fluxes and all convection (dry and moist). What a cloud resolving model does. Outline • Linearity (!) of deep convection (anomalies) • The system in matrix form – M mapped in spatial basis, using CRM & inversion (ZK) – math checks, eigenvector/eigenvalue basis, etc. • Can we evaluate/improve M estimate from obs? Linearity (!) of convection My first grant! (NSF) Work done in mid- 1990s Hypothesis: low-level (Inhibition) control, not deep (CAPE) control (Salvage writeup before moving to Miami. Not one of my better papers...) A much better job of it Low-level Temperature and Moisture Inject stimuli suddenly Perturbations Imposed Separately Q1 anomalies Q1 Courtesy Stefan Tulich (2006 AGU) Linearity test of responses: Q1(T,q) = Q1(T,0) + Q1(0,q) ? both q T sum, assuming linearity T sum q both Yes: solid black curve resembles dotted curve in both timing & profile Courtesy Stefan Tulich (2006 AGU) Linear expectation value, even when not purely deterministic Ensemble Spread Courtesy Stefan Tulich (2006 AGU) Outline • Linearity (!) of deep convection (anomalies) • The system in matrix form – in a spatial basis, using CRM & inversion (ZK) • could be done with SCM the same way... – math checks, eigenvector/eigenvalue basis, etc. • Can we estimate M from obs? – (and model-M from GCM output in similar ways)? Multidimensional linear systems can include nonlocal relationships Can have nonintuitive aspects (surprises) Zhiming Kuang, Harvard: 2010 JAS Splice together T and q into a column vector (with mixed units: K, g/kg) Q1(z) TÝconv (z) T(z) M Q2(z) qÝconv (z) q(z) • Linearized about a steadily convecting base state • 2 tried in Kuang 2010 – 1. RCE and 2. convection due to deep ascent-like forcings Zhiming’s leap: Estimating M-1 w/ long, steady CRM runs Zhiming’s inverse casting of problem Ý(p) T(p) T c 1 convective M q(p) qÝc (p)heating/dryin g profiles Make bumps on CRM’s • ^^ With these, build M-1 column by column • Invert to get M! (computer knows how) • Test: reconstruct transient stimulus problem Specify LSD + Radiation as steady forcing. Run CRM to ‘statistically steady state’ (i.e., consider the time average) T (p) Tadvtend(p) Qr L /C p (c e) z(w'T') t q (p) qadvtend(p) 0 (c e) z(w'q') t 0 0 for a long time average Treat all this as a timeindependent large-scale FORCING applied to doubly-periodic CRM (or SCM) All convection (dry and moist), including surface flux. Total of all tendencies the model produces. Forced steady state in CRM T Forcing (cooling) CRM response (heating) Surface fluxes and all convection q Forcing (dryCRM andresponse moist). Total of all tendencies the CRM/ SCM (moistening) (drying) produces. Now put a bump (perturbation) on the forcing profile, and study the time-mean response of the CRM T Forcing (cooling) CRM time-mean response: heating’ balances forcing’ (somehow...) Surface fluxes and all convection (dry and moist). Total of all tendencies the CRM/ SCM produces. How does convection make a heating bump? anomalous convective heating = 1. cond 1. Extra net Ccondensation, in anomalous cloudy upward mass flux 2. EDDY: Updrafts/ downdrafts are extra warm/cool, relative to environment, and/or extra strong, above and/or below the bump How does convection ‘know’ it needs to do this? How does convxn ‘know’? Env. tells it! by shaping buoyancy profile; and by redefining ‘eddy’ (Tp – Te) (another linearity test: 2 lines are from heating bump & cooling bump w/ sign flipped) via vertically local soundingcond differences.. ... and nonlocal (net surface flux required, so surface T &/or q must decrease) 0 0 Kuang 2010 JAS ... this is a column of M-1 . . T( p) . q( p) . . . . . . . . . . . . . . . . . . . . . . . . . . . -1 M . . .TÝ c ( p) .qÝc ( p) . . a column of M-1 T'( p) q'( p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0 .1 .0 .0 .0 .0 Image of M-1 0 . . . T'(z) . . . Tdot T’ z . . . . . . z q'(z) . Tdot. q’. . . . . . . 0 Ý .qdot . T’.Tc . . . . . . .qdot . q’.qÝc -4 . . . Unpublished matrices in model stretched z coordinate, courtesy of Zhiming Kuang Units: K and g/kg for T and q, K/d and g/kg/d for heating and moistening rates -1 M and M from 3D 4km CRM from 2D 4km CRM ( ^^^ these numbers are the sign coded square root of |Mij| for clarity of small offdiag elements) -1 M and M from 3D 4km CRM from 3D 2km CRM ( ^^^ these numbers are the sign coded square root of |Mij| for clarity of small offdiag elements) Finite-time heating and moistening TÝ T M q qÝ has solution: T(0) T(t) exp(Mt) q(t) q(0) So the 6h time rate of change is: T(6h) T(0) exp(M 6h) exp(M 0) exp(M 6h) I 6h q(6h) q(0) 6h 6h 6h M Spooky action at a distance: How can 10km Q1 be a “response” to 2km T? (A: system at equilibrium, calculus limit) [exp(6h×M)-I] /6h Comforting to have timescale explicit? Fast-decaying eigenmodes affect this less. Fast-decaying eigenmodes are noisy (hard to observe accurately in equilibrium runs), but they produce many of the large values in M. Low-level Temperature and Moisture Inject stimuli suddenly Perturbations Imposed Separately Q1 anomalies Q1 Courtesy Stefan Tulich (2006 AGU) Columns: convective heating/moistening profile responses to T or q perturbations at a given altitude z (km) 120 1 2 5 8 12 . . . . . . . . . . . . . . . . . . T . . . q . 0 z (km) . . z (km) . . 1 2 5 8 12 . . 120 . . 1 2 5 8 . . . . . . 1 2 5 8 0 Ý T (z) c Ýc (z) q 0 z (km) square root color scale Image of M Units: K and g/kg for T and q, K/d and g/kg/d for Tdot and qdot 0 1 2 5 8 1 2 5 8 12 z (km) z (km) . . . . 1 2 5 8 12 .T’ . qdot . . . . . PBL. FT 120 Tdot . . q’ . . T Tdot 0 . . . . .q’ . q qdot . . 1 2 5 8 . . 0 . . .T’ . z (km) 120 PBL FT . . . . . . PBL FT Ý T (z) c Ýc (z) q z (km) square root color scale Image of M (stretched z coords, 0-12km) Units: K and g/kg for T and q, K/d and g/kg/d for Tdot and qdot . + . . Ý T’- Tdot T (z) . . . z c + . . . . . . +/-/+ : cooling at level and Ýc (z) zthat q .warming . of . T’ levels qdot adjacent (diffusion) . . . . positive . T’ . within PBL yields... q’ Tdot . . .T . . . . . . . q’. .q qdot . . . 0 square root color scale Off-diagonal structure is nonlocal (penetrative convection) Ý T (z) c Ýc (z) q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T 0 . . . q . square root color scale Off-diagonal structure is nonlocal (penetrative convection) 1 2 5 8 . . . q’ . .Tdot .T + effect of q’ on heating . its .level . above . . . . q’. .q qdot . . . z (km) 0 120 z (km) 1 2 5 8 12 T’+ (cap) qdot+ below 1 2 5 8 12 1 2 5 8 . . . T’ Tdot . . . T’ cap inhibits . deepcon . . . . . . . . T’ qdot . . . 120 z (km) 0 Ý T (z) c Ýc (z) q 0 z (km) square root color scale Integrate columns in top half x dp/g Ý T (z) c Ýc (z) q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T . . . q . 0 square root color scale sensitivity: d{Q1}/dT, d{Q1}/dq Integrate columns in top half x dp/g of [exp(6h×M)-I] /6h . Ý T (z) . c . . Ýc (z) compare. q plain old M . . . . . . . . . . . . . . . . . . . . . . . . . . . T . . . q . 0 A check on the method prediction for the Tulich & Mapes ‘transient sensitivity’ problem T( p,t) T( p,0) exp(Mt) q( p,t) q( p,0) initial ‘stimuli’ in the domainmean T,q profiles Example: evolution of initial warm blip placed at 500mb in convecting CRM Doing the actual experiment in the CRM: Computed as [exp( (M-1)-1 t)] [Tinit(p),0]: Appendix of Kuang 2010 JAS Example: evolution of initial moist blip placed at 700mb in convecting CRM Doing the actual experiment in the CRM: Computed via exp( (M-1)-1 t): Appendix of Kuang 2010 JAS Example: evolution of initial moist blip placed at 1000mb in convecting CRM Doing the actual experiment in the CRM: Computed via exp( (M-1)-1 t): Appendix of Kuang 2010 JAS Outline • Linearity (!) of deep convection (anomalies) • The system in matrix form – mapped in a spatial basis, using CRM & inversion (ZK) – math checks, eigenvector/eigenvalue basis • Can we estimate M from obs? – (and model-M from GCM output in similar ways)? Eigenvalues Sort according to inverse of real part (decay timescale) 10 Days 1 Day 3D CRM results 2D CRM results 2.4 h 15 min There is one mode with slow (2 weeks!) decay, a few complex conj. pairs for o(1d) decaying oscillations, and the rest (e.g. diffusion damping of PBL T’, q’ wiggles) Slowest eigenvector: 14d decay time 3D CRM results 2D CRM results 0 Column MSE anomalies damped only by surface flux anomalies, with fixed wind speed in flux formula. A CRM setup artifact.* *(But T/q relative values & shapes have info about moist convection?) eigenvector pair #2 & #3 ~1d decay time, ~2d osc. period 3D CRM results 2D CRM results congestus-deep convection oscillations Singular Value Decomposition • Formally, the singular value decomposition of an m×n real or complex matrix M is a factorization of the form M = UΣV* where U is an m×m real or complex unitary matrix, Σ is an m×n diagonal matrix with nonnegative real numbers on the diagonal, and V* (the conjugate transpose of V) is an n×n real or complex unitary matrix. The diagonal entries Σi,i of Σ are known as the singular values of M. The m columns of U and the n columns of V are called the left singular vectors and right singular vectors of M, respectively. Outline • Linearity (!) of deep convection (anomalies) • The system in matrix form – mapped in a spatial basis, using CRM & inversion anything a CRM can do, an SCM can do – math checks, eigenvector/eigenvalue basis, etc. • Evaluation with observations Put in anomalous sounding [T’,q’]. Get a prediction of anomalous rain. COARE IFA 120d radiosonde array data M prediction can be decomposed since it is linear. Most of the response is to q anomalies. Wrapup • Convection is a linearizable process – can be summarized in a timeless object M – tangent linear around convecting base states • How many? as many as give dynamically distinct M’s. • This unleashes a lot of math capabilities • ZK builds M using steady state cloud model runs – can we find ways to use observations? • Many cross checks on the method are possible – could be a nice framework for all the field’s tools » GCM, SCM, CRM, obs.