Singular Vectors and Moist Physics

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Singular Vectors and Moist Physics

• Martin Ehrendorfer
• Institut für Meteorologie und Geophysik
• Fakultät für Geo- und Atmosphärenwissenschaften
• Universität Innsbruck

Seminar Laboratoire de Météorologie Dynamique du
Centre national de la recherche scientifique 
Paris, France 13 June 2006
http//www2.uibk.ac.at/meteo
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Katrina 28/8/2005
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(No Transcript)
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Outline

• moist SVs from a study with Ron Errico and
  Kevin Raeder
• SV definition and norms considered
• MAMS2 (mesoscale adjoint modeling system)
• optimal growth
• horizontal SV structures
• vertical SV structures
• nonlinear relevance
• a simple QG model
• dynamics
• physics
• properties (mean and transient)
• error growth
• summary

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Motivation for moist SV study

• What are the relative effects of initial
  perturbations (or errors) in the dynamical fields
  versus the moisture fields?
• Are similar structures optimal for affecting both
  perturbation energy and precipitation (as final
  norms)?
• Is convective or non-convective precipitation
  generally more sensitive to initial perturbations?

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SVs briefly revisited
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Norms considered
E
E_m
P
conv. nonconv.
V_d
V_m
a larger value of E can be produced with V_d1
compared with E_m1
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initial and final time norms
optimize energy E or precipitation P
constrain moisture E_m or dry fields V_d
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Upscale energy transfer, conversion APE to KE
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inverse-variance weighted norms
penalize large q perturbations at the top 2 m/s
1 K 0.001
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Mesoscale Adjoint Modeling System MAMS2

• PE with water vapor, B grid
• Bulk PBL (Deardorff)
• Stability-dependent vertical diffusion (see CCM3)
• Dry convective adjustment
• RAS scheme (Moorthi and Suarez)
• Stable-layer precipitation
• ?x80km, 20 s levels with equal thickness
• Nonlinear vertical NMI (Bourke McGgregor 1983)
• Moist TLM (Errico Raeder 1999)

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leading SVs from principal experiments
t_d 2.5 h
t_d 2.0 h
V_m moisture perturbations more effective than
V_d dry perturbations to maximize E
doubling time t_d OTI ln 2 / ln ?
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similar structures may maximize both E and P norm
v SV-2 V_d ? E
v SV-1 V_d ? P
s 0.575 r 0.81
T SV-2 V_d ? E (not shown)
Strongly correlated
q SV-1 V_m ? E
q SV-2 V_m ? P
s 0.525 r 0.76
initial time, case S2
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finaltime SVs (diff. init. norm) highly
correlated
initial time, case S2
different initial structures (dry-only and
moist-only) lead to high similarity at final
time (also other fields)
v SV-2 V_d ? E
v SV-1 V_m ? E
final time, case S2
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Explanation
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vertical structures I (4 curves are 4 cases)
surprisingly small impact through stronger
vertical constraint on q
initial time
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vertical structures II (4 curves are 4 cases)
T contribution larger than KE contribution (APE ?
KE)
initial time
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initial temperature perturbations
pronounced effect of vertical weights on initial
temperature structure
initial time
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a peculiar gravitywave SV SV1 of V_d ? P, W1
significant DIV and CONV
u t0
v t0
u t12h
rapidly propagating gravity wave
exciting vertical motion below 12 hours later
(100 m/s)
R_t t12h
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nonlinear relevance, SV1, V_m ? P
TLD
NLD
80 mm/day
R_n
nonconvective precipitation rate R_n corr.
r0.93
TLD
NLD
convective precipitation rate R_c corr. r0.27
R_c
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MAMS linearization
TLD
NLD (scaled)
NLD
Errico Raeder 1999
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moist SVs summary

• moisture perturbation alone may achieve larger E
  than dry perturbations
• given the same initial constraint, similar
  structures can be optimal for maximizing E and P
• in most cases structures are different
• dry-only (V_d) and moist-only (V_m) SVs may lead
  to nearly identical final-time fields
• through an inferred dependence on moist enthalpy
  a q perturbation is converted into a T
  perturbation through diabatic heating during
  nonconvective precipitation
• vertical structures vary from case to case (even
  for same norm)
• nonlinear relevance TLD and NLD may match
  closely (2 g/kg)
• enhanced sensitivity of non-convective
  precipitation is not universally dominant

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QG framework
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RRKF in QG T21L3 framework
A. Beck, 2003
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QG PV equation (filtered baroclinic)
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physics in the QG model (TNLn)
relaxation with 10-day or 20-day time scale
damping in lower model domain
tendency of relative QGPV
forcing taken as mean observed tendency from
20002005 January ECMWF analyses
diffusion with 2-day time-scale (on smallest
spatial scale) proportional to ?2 and ?6
QGPV dynamics
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10-3/103 0.0001
e/e0 exp (t ln2)/td
td 13 h
2138192
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geostrophic streamfunction
Observed (from ECMWF ERA-40 atlas) time-mean
flow
T106L9 (b1ku) time-mean stream- function 200
hPa
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Kinetic Energy (KE) of the Mean Flow
360 J/kg
observed
Observed (upper right at 500 hPa based on
2000-2005) and modeled (lower row) KE (in units
of J/kg) of time-mean flow. Lower left T106L9
(b1ku) KE of time-mean field at 500 hPa. Lower
right T45L6 (b1la) KE of time-mean field at 550
hPa. Note that fields have been normalized
slightly differently.
375 J/kg
250 J/kg
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issues and problems

• It is easy to reproduce the observed mean state
  by relaxing very strongly i.e., making t smaller
• Strong relaxation, however, counteracts the
  strength of the transient flow component
• A correct time-mean is, however, is needed to
• provide energy to the transient flow components
• It is thus necessary to balance the model between
• strength of relaxation and of transient flow
• The damping term k? reflects the important (see
  SV and adjoint studies) PBL damping effect
• It is important to reproduce spectral slope of 3
• A sufficiently large number of vertical levels is
  
• needed to resolve the vertically tilted and
  rather shallow leading SVs
• It is hard to get transient flow in summer
  hemi-sphere right, since it is convectively, as

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Time series
T45L6
T106L9
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observed and modeled energy spectra
observed
T45L6 model
T106L9 model
Energy spectra (KE and TE) obtained from model
integrations over 500 days (T45L6 left b1la) and
250 days (T106L9 right b1ku), in terms of
time-mean zonal and eddy, as well as transient
components (see legend). Middle panel shows
observed KE spectra taken from Boer (1994). Note
good quantitative agreement (e.g., in terms of
the transient KE spectrum peaking around n10).
Also, the transient spectrum follows a
minus-three slope.
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kinetic energy of transient flow
110 J/kg
215 J/kg
observed
85 J/kg
50 J/kg
KE of the transient flow component at around 500
hPa, in units of J/kg. Upper left observed
(derived from ECMWF analyses over 2000-2005)
Upper right T106L9 (b1ku) Lower right T45L6
(b1la) Lower left T45L6 (b1jj). The good
quantitative agreement of the results in the
right column with observations is noted. Note
also that the use of a relaxation time scale of
20 days in these experiments leads to
improvements, especially in the Southern
Hemisphere compared to the experiment shown in
the lower left using a 10-day relaxation time.
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SVs at T106L9 in QG model

• initial-time leading SVs obtained with T106L9,
  TE, OTI 1 day
• SV at 500 hPa and 800 hPa TE ratio ?173.97
  (b1ks)
• lower right SV at 800 hPa
• for experiment with weak low-level damping TE
  ratio ?1217.57 (b1jv)
• fields normalized slightly differently

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QG TE SV spectrum of G
?133.47
1642 13
tau_d 4.74 h
12690 x 12690
T45L6
?0.0212
decay by factor of 2 after 4.32 h
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spectra with MAMS
Errico/Ehrendorfer/Raeder 2001
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summary

• realistic models
• show high sensitivity wrt initial condition
  (nature of atmospheric flow)
• doubling times 1.5 to 2 days
• nonmodal moist SV growth
• moist perturbations impact E or P as much as
  dynamical fields
• implications for data assimilation
• conceptual QG model
• resembles some of observed atmospheric properties
• relatively cheap for experimentation (data
  assimilation)

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References
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QG model in pressure coordinates
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Seminar at LMD, 13 June 2006