The linear algebra of atmospheric convection

1. The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard University

2. 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)

3. 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

4. “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

5. 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

6. interacts statistically with large scale waves straight line = nondispersive wave 1998 CLAUS Brightness Temperature 5ºS-5º N

7. 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

8. Atm. column heat and moisture budgets, as a T & q column vector equation: water phase changes eddy flux convergence LS dynamics rad. Surface fluxes and all convection (dry and moist). What a cloud resolving model does.

9. 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?

10. 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...)

11. A much better job of it

12. Low-level Temperature and Moisture Perturbations Imposed Separately Inject stimuli suddenly Q1 anomalies Q1 Courtesy Stefan Tulich (2006 AGU)

13. Linearity test of responses: Q1(T,q) = Q1(T,0) + Q1(0,q) ? both sum sum, assuming linearity q T q T both Yes: solid black curve resembles dotted curve in both timing & profile Courtesy Stefan Tulich (2006 AGU)

14. Linear expectation value, even when not purely deterministic Ensemble Spread Courtesy Stefan Tulich (2006 AGU)

15. 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)?

16. Multidimensional linear systems can include nonlocal relationships Can have nonintuitive aspects (surprises)

17. Zhiming Kuang, Harvard: 2010 JAS

18. Splice together T and q into a column vector (with mixed units: K, g/kg) • Linearized about a steadily convecting base state • 2 tried in Kuang 2010 • 1. RCE and 2. convection due to deep ascent-like forcings

19. Zhiming’s leap: Estimating M-1 w/ long, steady CRM runs

20. Zhiming’s inverse casting of problem measure time-mean, domain mean sounding anoms. • ^^ With these, build M-1 column by column • Invert to get M! (computer knows how) • Test: reconstruct transient stimulus problem Make bumps on CRM’s time-mean convective heating/drying profiles

21. Specify LSD + Radiation as steady forcing. Run CRM to ‘statistically steady state’ (i.e., consider the time average) 0 Treat all this as a time-independent 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.  0 for a long time average

22. Forced steady state in CRM T Forcing (cooling) CRM response (heating) Surface fluxes and all convection (dry and moist). Total of all tendencies the CRM/ SCM produces. q Forcing (moistening) CRM response (drying)

23. Now put a bump (perturbation) on the forcing profile, and study the time-mean response of the CRM CRM time-mean response: heating’ balances forcing’ (somehow...) T Forcing (cooling) Surface fluxes and all convection (dry and moist). Total of all tendencies the CRM/ SCM produces.

24. 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?

25. 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 sounding differences.. cond ... and nonlocal (net surface flux required, so surface T &/or q must decrease) 0 0 0 0 Kuang 2010 JAS

26.  ... this is a column of M-1 M-1

27. a column of M-1

28. Image of M-1 0 0 Tdot  T’ qdot  T’ z Tdot  q’ qdot  q’ z -4 Units: K and g/kg for T and q, K/d and g/kg/d for heating and moistening rates Unpublished matrices in model stretched z coordinate, courtesy of Zhiming Kuang

29. M-1 and M from 3D 4km CRM from 2D 4km CRM ( ^^^ these numbers are the sign coded square root of |Mij| for clarity of small off-diag elements)

30. M-1 and M from 3D 4km CRM from 3D 2km CRM ( ^^^ these numbers are the sign coded square root of |Mij| for clarity of small off-diag elements)

31. Finite-time heating and moistening has solution: So the 6h time rate of change is:

32. M [exp(6h×M)-I] /6h Spooky action at a distance: How can 10km Q1 be a “response” to 2km T? (A: system at equilibrium, calculus limit) Fast-decaying eigenmodes are noisy (hard to observe accurately in equilibrium runs), but they produce many of the large values in M. Comforting to have timescale explicit? Fast-decaying eigenmodes affect this less.

33. Low-level Temperature and Moisture Perturbations Imposed Separately Inject stimuli suddenly Q1 anomalies Q1 Courtesy Stefan Tulich (2006 AGU)

34. Columns: convective heating/moistening profile responses to T or q perturbations at a given altitude z (km) z (km) z (km) z (km) 0 1 2 5 8 12 0 1 2 5 8 12 0 1 2 5 8 12 0 1 2 5 8 12 0 square root color scale

35. Image of M Units: K and g/kg for T and q, K/d and g/kg/d for Tdot and qdot z (km) z (km) z (km) z (km) 0 1 2 5 8 12 0 1 2 5 8 12 0 1 2 5 8 12 0 1 2 5 8 12 q’  Tdot T’  Tdot 0 PBL FT PBL FT FT  PBL q’  qdot T’  qdot square root color scale

36. 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 + - + positive T’ within PBL yields... z T’  Tdot q’  Tdot 0 +/-/+ : cooling at that level and warming of adjacent levels (diffusion) z q’  qdot T’  qdot square root color scale

37. Off-diagonal structure is nonlocal(penetrative convection) 0 square root color scale

38. Off-diagonal structure is nonlocal(penetrative convection) z (km) z (km) z (km) z (km) 0 1 2 5 8 12 0 1 2 5 8 12 0 1 2 5 8 12 0 1 2 5 8 12 T’  Tdot q’  Tdot 0 + effect of q’ on heating above its level T’ cap inhibits deepcon T’+ (cap)  qdot+ below q’  qdot T’  qdot square root color scale

39. Integrate columns in top half x dp/g 0 square root color scale

40. sensitivity: d{Q1}/dT, d{Q1}/dq

41. Integrate columns in top half x dp/gof [exp(6h×M)-I] /6h 0 compare plain old M

42. A check on the method prediction for the Tulich & Mapes ‘transient sensitivity’ problem domain mean profile evolution (since Q1 and Q2 are trapped within the periodic CRM) initial ‘stimuli’ in the domain-mean T,q profiles

43. 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

44. 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

45. 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

46. 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)?

47. EigenvaluesSort according to inverse of real part(decay timescale) 3D CRM results 2D CRM results 10 Days 1 Day 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)

48. 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?)

49. eigenvector pair #2 & #3~1d decay time, ~2d osc. period 3D CRM results 2D CRM results congestus-deep convection oscillations

50. 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×ndiagonal 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.