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Zonal mean vertical mean momentum a data analysis and experimental prediction exercise motivated by impacts (work in progress) Brian Mapes MPO seminar September 26, 2012

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Outline Motivation: impacts on our latitude (25N) Background: – an old planetary physics problem – meaningful vs. arbitrary averaging Time series at one latitude A full latitude-time analysis – climatology – anomalies and their budgets Statistical prediction of anomalies by LIM Conclusions

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iBTracs data Barotropic zonal mean flow and TCs High-Low tercile 20-30°N composite (ASO)

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H 500 H

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Zonal mean "pushes" the subtropical highs (by exerting a PV tendency) Heating (PV source) Eddy Z 1000 no [u] Eddy Z 1000 w/ July [u] Chen, Hoerling & Dole 2001

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High-Low tercile 20-30°N composite (ASO) Barotropic zonal mean flow and TCs

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Outline Motivation: impacts on our latitude (25N) Background: – an old planetary physics problem – meaningful vs. arbitrary averaging Time series at one latitude A full latitude-time analysis – climatology – anomalies and their budgets Statistical prediction of anomalies by LIM Conclusions

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Basic u equation use x,y instead of lon,lat for clarity p hyd coordinate in vertical (no clutter) "usual approximations" (no f*w) PGF Cor Convergence of 3D flux (by both fluid and molecular motions)

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Break flux into components and group d/dx terms: Zonal average on a p hyd surface is []: "mountain torque" p surface L H L H

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Mass-weighted vertical average <> These 4 (+1) terms must add up (physics)... But in data (reanalysis-estimated budgets) there is a 6 th (implied) term: "residual" mtn torque transport (meridionally across latitude lines) transport (meridionally across latitude lines) TENDENCY = + + Friction Net Coriolis Net Coriolis + it is small

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"Meaningful" averaging and The averaged quantity obeys an equation w/ fewer, smaller, & simpler terms than u It is thus "harder to change" than local u, so it has an existence (and, we will see, a persisence) that is arguably more substantial than u This is quite unlike many averages of many quantities that we sum up in our computers! » many are just fictions or conveniences » to blend (but dilute) information or signals, or reduce "noise" I'll speak of as a wind that "advects" scalars

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, GLobal AAM and length of day Velocity * lever arm *circumference of lat line – GLAAM = (a cos( )) (2 a cos( )) GLAAM exchanged w/ solid Earth (conserved) measurable length of day fluctuations » semiannual, ENSO, MJO (much lit. as AAM)

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Zonal mean vs. “Annular modes” NAM or “Arctic Oscillation” are EOFs of SLP – Related to NAO = p Azores -p Iceland decomposition debate: semantic? profound? – NAM/SAM are patterns w/ max SLP’ 2 (variance) averaged over polar cap surface area – Some link to via geostrophy, but loose Example of arbitrary vs. meaningful averaging – SLP’ 2 does not obey a physics equation that area- averaging interestingly refines – If we end up calling it ‘annular’, why not use annuli? []

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Outline Motivation: impacts on our latitude (25N) Background: – an old planetary physics problem – meaningful vs. arbitrary averaging Global picture Time series at 25N A full latitude-time analysis – climatology – anomalies and their budgets Statistical prediction of anomalies by LIM Conclusions

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Data used NCEP-NCAR Reanalysis ("NCEP1") 1980-2011 (32 years) 94 equally spaced latitudes (~2 o, "Gaussian") Courtesy Dr. Klaus Weickman (NOAA PSD) who computed the zonal means and budget terms – for his papers over ~20 years in spherical, sigma coordinates rather than p (MERRA used for some climatology figs)

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textbook: 4 jets, with meanders

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u 200 annual mean [u 200 ] ann cyc Jan -------------------------- Dec

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climatology of NCEP1 sigma 1-21MERRA 1000-100 mb NCEP1 all levels 45 All Scales: +/- 25 m/s Annual mean map MERRA 1000-100 mb 30

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u 50 mb annual mean and cycle

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Outline Motivation: impacts on our latitude (26N) Background: – an old planetary physics problem – meaningless vs. meaningful averaging Time series at our latitude (26N) A full latitude-time analysis – climatology – anomalies and their budgets Statistical prediction of anomalies by LIM Conclusions

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at 26N

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data => mean clim(doy) + anomaly(doy,y) color = year (blue red rainbow)

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data => mean clim(doy) + anomaly(doy,y)

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data => mean clim(doy) + anomaly(poy,y)

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Quantify seasonality: RMS (or stdev) Winter has bigger anomalies on average than summer anomalies are strongly persistent from day to day – (5-day data pre-averaging doesn’t reduce their square very much) » (the heavy line) ?

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Winter anomalies are bigger Midwinter (ground hog day) minimum?

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Winter anomalies are bigger Midwinter (ground hog day) minimum?

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Quantify longevity of anomalies by lagged multiplication

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Quantify longevity by lagged multiplication 1980 mean of all years 1983

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Quantify longevity by lagged multiplication Mean of all March 1s mean of all March days' means over all years Repeat for all 30 days in March Mean of all March 31s

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Quantify longevity by lagged multiplication Jan mean of all months Repeat for all months of the year Feb Mar Dec Summer Magnitude difference is distracting. We already quantified that winter anomalies are larger: the lag-0 variance. So divide by that to normalize. (this is "auto-covariance")

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Quantify longevity by lagged multiplication Repeat for all months of the year Ja n mean acov of all months Fe b Ma r De c Summe r Ja n Fe b Ma r De c Apr mean acov of all months normalized by its lag=0 value May

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Repeat for all months of the year Ja n Fe b Ma r De c Apr IDEA: is it exponential decay C=C 0 exp(- d ?...a postulated reinterpretation of: May JJA Nov Oct Sep... a solution of this... (univariate) ? both

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data => mean clim(doy) + anomaly(doy,y)

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Power = | FT(autocov) | 2 broad hump @ "wave period" ~2 characteristic times (50d) broad hump @ "wave period" ~2 characteristic times (50d) 1 2 Period (months)

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Histogram of 26N anomalies INCONSISTENT w/ this univariate model, at least w/Gaussian, white noise (Ornstein-Uhlenbeck process )

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Outline Background: an old planetary physics problem Motivation: impacts on our latitude Our time series (26N) A full latitude-time analysis – climatology – anomalies Statistical prediction of anomalies by LIM Outstanding questions

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climatology of NCEP1 sigma 1-21MERRA 1000-100 mb NCEP1 all levels 45 All Scales: +/- 25 m/s Annual mean map MERRA 1000-100 mb 30

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Real time monitoring (Klaus Weickmann) http://www.esrl.noaa.gov/psd/map/clim/aam.rean.shtml JAN MAY SEP 2011 2012 25N

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Characterize timescale w/autocorr.

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autocov(lag,lat) = autocorr(lag,lat)*stdev 2 (lat)

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anomalies are more persistent on edges of jets MERRA 1000-100 mb Jan Dec 45 Annual mean map

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Timescale: extend this to lag x lat 1980 mean of all years 1983

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* apply with negtive weight * apply with pos. weight etc. etc. for 32 years (optionally for seasons) and sum it all up Composite anomalies in lag x lat weighted by 26N base time series

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This weighted composite is an array of regression coefficients (slopes of linear fits on base(t) vs. field(t+lag,lat) scatterplots at each altitude) same, along base lat... All seasons: "Characteristic" time scales AND latitude evolution

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Can do the same for tendency field, and torques m/s composite of d/dt = d/dt(composite)

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Budget (composite over anomalies in all seasons) TEND = MTN + TRANSPORT + (FRICTION+GW) + CORIOLIS + RESIDUAL TEND MTN TRANSPORT (FRICTION) RESIDUAL

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Outline Background: an old planetary physics problem Motivation: impacts on our latitude Our time series (26N) A full latitude-time analysis – climatology – anomalies Statistical prediction of anomalies by LIM Outstanding questions

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Understand, schmunderstand Can we predict it? Statistical prediction by Linear Inverse Modeling (LIM) Predicting from itself only – ENSO signals etc. are already in precisely to the extent that they affect it! » but mutiple signals are separately predictable only if they have distinct spatial structure, not by time scale per se... IDEA: Using 32 years of observed time evolution, estimate the linear matrix B in this eqn:

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Understand, schmunderstand Can we predict it? The vector of at all latitudes is the x in: The expectation solution (...if ‘noise’ averages to nothing, in an ensemble mean, which is what we’d want as a forecast... ) is: » x( ) = exp(B ) x(initial) where is the forecast lead

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Understand, schmunderstand Can we predict it? In the univariate case, we fit such an exp( t) form based on a dx/dt = - x + noise model. There we estimated from: cov( ) = exp(- ) cov(0) so = ln( cov( )/cov(0) )/

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Understand, schmunderstand Can we predict it? The multivariate version is the same – except that “matrix division” is on the left » in a Matlab syntax I admire: univariate: = ln( cov( )/cov(0) )/ multivariate: B = ln( cov(0)\cov( ) )/ Really: B = ln( [cov(0)] -1 cov( ) )/

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Covariance of x with x Eq. anoms are broad Midlat anoms are flanked by opposite values

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Covariance of x with x Note the diagonal is something weve seen before..

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Covariance of x with (x, days later) 25 N&S lobes hardly weaken w/ lag 25 N&S lobes hardly weaken w/ lag asymmetric (poleward prop.) symmetric by construction: Not: 0d 5d 10d 15d 20d

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B and cov(0)\cov( ) are inscrutable But this is efficient, precise “information plumbing” We know B’s character by its action: – x( ) = exp(B ) x(initial) 1.Make this hindcast for all for every day in the record 2.Use it to define the “noise” – discrete version, daily 3.Make white noise by randomly re-sampling that in time 4.Use randomized noise in the forward eq syn. data:

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Use B and randomized noise to make syn. data: B knows about typical structure and timescale, as a function of latitude, including latitudinal drifts, width dependence of an anomaly's longevity, etc. Which is real data?

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LIM expected (or ensemble) hindcast using (no noise) from initial spatial structure alone

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LIM hindcasts: skill eval (Work in progress...) Make a summer B and a winter B? – each fitted from ½ as much data... but the seasons may be different enough to justify 2? Why not a different B for every day (using a season centered around that day)? It’s just matrix calculations done once – cheap.

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Conclusions correlated with TC recurvature longitude hyp: via "pushing" the subtropical high E/W is a "meaningful" average – obeys a budget with fewer terms It is "hard to change" – & thus arguably has a solid "existence" (or at least persistence) Earth has 5-7 belts with enhanced persistence Long predictability may be possible – 10s of days... Residual is large in budget – models may be challenged? – statistical prediction appealing?

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