The linear algebra of atmospheric convection

The linear algebra of atmospheric convection

Brian MapesRosenstiel School of Marine and

Atmospheric Sciences, U. of Miamiwith

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

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

1998 CLAUS Brightness

Temperature 5ºS-5º N

straight line = nondispersive wave

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:

LS dynamics rad.

water phasechanges

eddy fluxconvergence

Surface fluxes and all convection(dry and moist). What a cloud

resolving model does.

€

∂T ∂t

( p)

∂q ∂t

(p)

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

Tadvtend (p)qadvtend(p) ⎡ ⎣ ⎢

⎤ ⎦ ⎥+

Qr

0

⎡

⎣ ⎢

⎤

⎦ ⎥+

L /Cp (c − e)−(c − e)

⎡

⎣ ⎢

⎤

⎦ ⎥+

−∂ ∂z(w'T ')−∂ ∂z(w'q')

⎡

⎣ ⎢

⎤

⎦ ⎥

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 Perturbations Imposed Separately

Courtesy Stefan Tulich (2006 AGU)

Inject stimuli suddenly

Q1 anomalies

Q1

Linearity test of responses: Q1(T,q) = Q1(T,0) + Q1(0,q) ?

T qTq

sum, assuming linearityboth


both

sum

Yes: solid black curve resembles dotted curve in both timing & profile

Linear expectation value, even when not purely deterministic

Ensemble Spread



– 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)

• Linearized about a steadily convecting base state • 2 tried in Kuang 2010

– 1. RCE and 2. convection due to deep ascent-like forcings

€

˙ T conv (z)˙ q conv (z)

⎡

⎣ ⎢

⎤

⎦ ⎥= M[ ]

T(z)q(z) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

Q1(z)Q2(z) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

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

Zhiming’s inverse casting of problem

• ^^ With these, build M-1 column by column• Invert to get M! (computer knows how)• Test: reconstruct transient stimulus problem€

T(p)q(p) ⎡ ⎣ ⎢

⎤ ⎦ ⎥= M−1[ ]

˙ T c (p)˙ q c (p)

⎡

⎣ ⎢

⎤

⎦ ⎥Make bumps on CRM’s time-mean convective heating/drying profiles

measure time-mean,

domain mean

soundinganoms.

€

∂T ∂t

(p)

∂q ∂t

(p)

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

Tadvtend (p)qadvtend(p) ⎡ ⎣ ⎢

⎤ ⎦ ⎥+

Qr

0

⎡

⎣ ⎢

⎤

⎦ ⎥+

L /Cp (c − e)−(c − e)

⎡

⎣ ⎢

⎤

⎦ ⎥+

−∂ ∂z(w'T ')−∂ ∂z(w'q')

⎡

⎣ ⎢

⎤

⎦ ⎥

Specify LSD + Radiation as steady forcing. Run CRM to ‘statistically steady state’

(i.e., consider the time average)

All convection (dry and moist), including surface flux. Total of all tendencies the model produces.

0 for a long time average

Treat all this as a time-independent large-scale FORCING applied to doubly-periodic CRM

(or SCM)

0

Forced steady state in CRM

Surface fluxes and all convection(dry and moist). Total of all tendencies the CRM/ SCM

produces.

T Forcing (cooling)

q Forcing (moistening)

CRM response (heating)

CRM response (drying)

Now put a bump (perturbation) on the forcing profile, and study the

time-mean response of the CRM

Surface fluxes and all convection(dry and moist). Total of all tendencies the

CRM/ SCM produces.

T Forcing (cooling)

CRM time-mean response: heating’ balances forcing’ (somehow...)

How does convection make a heating bump?

anomalous convective heating

1. cond

1. Extra net Ccondensation, in anomalous cloudy upward mass flux

How does convection ‘know’ it needs to do this?

2. EDDY: Updrafts/ downdrafts are extra warm/cool, relative to environment, and/or extra strong, above and/or below the

bump

=

How does convxn ‘know’? Env. tells it!by shaping buoyancy profile; and by redefining ‘eddy’ (Tp – Te)

Kuang 2010 JAS

cond

0

... and nonlocal(net surface flux required, so

surface T &/or q must decrease)

00 0

(another linearity test: 2 lines are

from heating bump & cooling bump w/

sign flipped)

via vertically local sounding differences..

... this is a column of M-1

€

T(p)q(p) ⎡ ⎣ ⎢

⎤ ⎦ ⎥=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

˙ T c (p)˙ q c (p)

⎡ ⎣ ⎢

⎤ ⎦ ⎥M-1

a column of M-1

€

T '(p)

q'(p)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

010000

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

0

-4

Image of M-1

€

T '(z)

q'(z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

˙ T c

˙ q c

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Units: K and g/kg for T and q, K/d and g/kg/d for heating and

moistening rates

Tdot T’ qdot T’

Tdot q’ qdot q’

z

z

0

Unpublished matrices in model stretched z coordinate, courtesy of Zhiming Kuang

M-1 and M

from 3D4km CRM

from 2D4km CRM

( ^^^ these numbers are the sign coded square root of |Mij| for clarity of small off-diag elements)

M-1 and M

from 3D4km CRM

from 3D 2km CRM

( ^^^ these numbers are the sign coded square root of |Mij| for clarity of small off-diag elements)

Finite-time heating and moistening

has solution:

So the 6h time rate of change is: €

˙ T ˙ q

⎡

⎣ ⎢

⎤

⎦ ⎥= M[ ]

Tq ⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

T(t)q(t) ⎡ ⎣ ⎢

⎤ ⎦ ⎥= exp(Mt)[ ]

T(0)q(0) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

€

T(6h) − T(0)6h

q(6h) − q(0)6h

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥=

exp(M⋅ 6h) − exp(M⋅ 0)6h

=exp(M⋅ 6h) − I

6h

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.

Low-level Temperature and Moisture Perturbations Imposed Separately


Inject stimuli suddenly

Q1 anomalies

Q1

Columns: convective heating/moistening profile responses to

T or q perturbations at a given altitude

€

˙ T c (z)

˙ q c (z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

−

T

q

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

0

square root color scale

0 1 2 5 8 12 0 1 2 5 8 12 z (km) z (km)

0 1 2 5 8 12 0 1 2 5 8 12 z (km

)z (km

)

Image of M

€

˙ T c (z)

˙ q c (z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

−

T

q

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Units: K and g/kg for T and q, K/d and g/kg/d for Tdot and qdot

0 T’ Tdot

q’ Tdot

T’ qdot

q’ qdot


FT PBL

PBL FT

PBL FT

0 1 2 5 8 12 0 1 2 5 8 12 z (km) z (km)

0 1 2 5 8 12 0 1 2 5 8 12 z (km

)z (km

)

Image of M (stretched z coords, 0-12km)

€

˙ T c (z)

˙ q c (z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

T

q

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Units: K and g/kg for T and q, K/d and g/kg/d for Tdot and qdot

0z

z

T’ Tdot q’ Tdot

T’ qdotq’ qdot


positive T’ within PBL

yields...

+/-/+ : cooling at that level and

warming of adjacent levels

(diffusion)

+-+

€

˙ T c (z)

˙ q c (z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

−

T

q

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Off-diagonal structure is nonlocal(penetrative convection)

0


€

˙ T c (z)

˙ q c (z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

T

q

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Off-diagonal structure is nonlocal(penetrative convection)

0 T’ Tdot q’ Tdot

T’ qdotq’ qdot


+ effect ofq’ on heating above its level

T’+ (cap) qdot+ below

T’ cap inhibits deepcon

0 1 2 5 8 12 0 1 2 5 8 12 z (km) z (km)

0 1 2 5 8 12 0 1 2 5 8 12 z (km

)z (km

)

€

˙ T c (z)

˙ q c (z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

−

T

q

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Integrate columns in top half x dp/g

0


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

€

˙ T c (z)

˙ q c (z)

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

=

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

. . . . . .

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

−

T

q

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥

Integrate columns in top half x dp/gof [exp(6h×M)-I] /6h

0

compare plain old M

A check on the methodprediction 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

€

T(p, t)q(p,t) ⎡ ⎣ ⎢

⎤ ⎦ ⎥= exp( M[ ]t)[ ]

T(p,0)q( p,0) ⎡ ⎣ ⎢

⎤ ⎦ ⎥

Example: evolution of initial warm blip placed at 500mb in convecting CRM

Appendix of Kuang 2010 JAS

Doing the actual experiment in the CRM:

Computed as [exp( (M-1)-1 t)] [Tinit(p),0]:

Example: evolution of initial moist blip placed at 700mb in convecting CRM



Computed via exp( (M-1)-1 t):

Example: evolution of initial moist blip placed at 1000mb in convecting CRM



Computed via exp( (M-1)-1 t):


– 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)?

10 Days

1 Day

2.4 h

3D CRM results 2D CRM results

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)

EigenvaluesSort according to inverse of real part

(decay timescale)

15 min

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?)


Slowest eigenvector: 14d decay time

0


eigenvector pair #2 & #3~1d decay time, ~2d osc. period

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.

http://en.wikipedia.org/wiki/Unitary_matrix

http://en.wikipedia.org/wiki/Diagonal_matrix

http://en.wikipedia.org/wiki/Conjugate_transpose

http://en.wikipedia.org/wiki/Singular_value


– 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.

• 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.

Wrapup

Documents

The linear algebra of atmospheric convection