The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric...
Transcript of The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric...
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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
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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)
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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
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“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
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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
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interacts statistically with large scale waves
1998 CLAUS Brightness
Temperature 5ºS-5º N
straight line = nondispersive wave
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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 '
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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')
⎡
⎣ ⎢
⎤
⎦ ⎥
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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?
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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...)
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A much better job of it
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Low-level Temperature and Moisture
Perturbations Imposed Separately
Courtesy Stefan Tulich (2006 AGU)
Inject stimuli suddenly
Q1 anomalies
Q1
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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
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Linear expectation value, even when not purely deterministic
Ensemble Spread
Courtesy Stefan Tulich (2006 AGU)
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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)?
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Multidimensional linear systems can include nonlocal relationships
Can have nonintuitive aspects (surprises)
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Zhiming Kuang, Harvard: 2010 JAS
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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)
⎡
⎣ ⎢
⎤
⎦ ⎥
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Zhiming’s leap: Estimating M-1 w/ long, steady CRM runs
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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.
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€
∂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
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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)
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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...)
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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
=
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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..
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... this is a column of M-1
€
T(p)
q(p)
⎡
⎣ ⎢
⎤
⎦ ⎥=
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
˙ T c (p)
˙ q c (p)
⎡
⎣ ⎢
⎤
⎦ ⎥M-1
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a column of M-1
€
T '(p)
q'(p)
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
0
1
0
0
0
0
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
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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
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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)
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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)
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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
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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.
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Low-level Temperature and Moisture
Perturbations Imposed Separately
Courtesy Stefan Tulich (2006 AGU)
Inject stimuli suddenly
Q1 anomalies
Q1
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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
)
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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
)
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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)
+-+
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€
˙ T c (z)
˙ q c (z)
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
−
T
q
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Off-diagonal structure is nonlocal(penetrative convection)
0
square root color scale
![Page 38: The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard.](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649c9d5503460f9495d38f/html5/thumbnails/38.jpg)
€
˙ 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
)
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€
˙ T c (z)
˙ q c (z)
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
=
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
. . . . . .
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
−
T
q
⎡
⎣
⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢
⎤
⎦
⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥
Integrate columns in top half x dp/g
0
square root color scale
![Page 40: The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard.](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649c9d5503460f9495d38f/html5/thumbnails/40.jpg)
sensitivity: d{Q1}/dT, d{Q1}/dq
![Page 41: The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard.](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649c9d5503460f9495d38f/html5/thumbnails/41.jpg)
€
˙ 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
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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)
⎡
⎣ ⎢
⎤
⎦ ⎥
![Page 43: The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard.](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649c9d5503460f9495d38f/html5/thumbnails/43.jpg)
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]:
![Page 44: The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard.](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649c9d5503460f9495d38f/html5/thumbnails/44.jpg)
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):
![Page 45: The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard.](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649c9d5503460f9495d38f/html5/thumbnails/45.jpg)
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):
![Page 46: The linear algebra of atmospheric convection Brian Mapes Rosenstiel School of Marine and Atmospheric Sciences, U. of Miami with Zhiming Kuang, Harvard.](https://reader035.fdocuments.net/reader035/viewer/2022062518/56649c9d5503460f9495d38f/html5/thumbnails/46.jpg)
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)?
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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
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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
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3D CRM results 2D CRM results
eigenvector pair #2 & #3~1d decay time, ~2d osc. period
congestus-deep convection oscillations
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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.
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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
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Put in anomalous sounding [T’,q’]. Get a prediction of anomalous rain.
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COARE IFA 120d radiosonde array data
M prediction can be decomposed since it is linear. Most of the response is to q anomalies.
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• 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