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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
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
Courtesy Stefan Tulich (2006 AGU)
both
sum
Yes: solid black curve resembles dotted curve in both timing & profile
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)
• 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)
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
0
1
0
0
0
0
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
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[ ]
T
q
⎡
⎣ ⎢
⎤
⎦ ⎥
€
T(t)
q(t)
⎡
⎣ ⎢
⎤
⎦ ⎥= exp(Mt)[ ]
T(0)
q(0)
⎡
⎣ ⎢
⎤
⎦ ⎥
€
T(6h) − T(0)
6hq(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
Courtesy Stefan Tulich (2006 AGU)
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
square root color scale
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
square root color scale
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
square root color scale
€
˙ T c (z)
˙ q c (z)
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
T
q
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Off-diagonal structure is nonlocal(penetrative convection)
0 T’ Tdot q’ Tdot
T’ qdotq’ qdot
square root color scale
+ 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
square root color scale
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
Appendix of Kuang 2010 JAS
Doing the actual experiment in the CRM:
Computed via exp( (M-1)-1 t):
Example: evolution of initial moist blip placed at 1000mb in convecting CRM
Appendix of Kuang 2010 JAS
Doing the actual experiment in the CRM:
Computed via exp( (M-1)-1 t):
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)?
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?)
3D CRM results 2D CRM results
Slowest eigenvector: 14d decay time
0
3D CRM results 2D CRM results
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.
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.
• 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
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