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)  .
.

.
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-1
M
.

.

.TÝ
c ( p)


.qÝc ( p) 
.

.
a column of M-1




T'(
p)








q'( p) 




.

.

.

.

.

.
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.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
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.  
  
. 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)  zthat
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




.

.

.

.

.

.
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.  
  
. 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




.

.

.

.

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.  
  
. 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
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 .
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.  
  
. T 
.  
  
.  
. q 
  
.  
0
A check on the method
prediction for the Tulich & Mapes ‘transient
sensitivity’ problem
T( p,t) 
T( p,0)

 exp(Mt)

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.