An examination of zonal mean geopotential variability

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Leslie Mitchell Bruce

B.Sc., University of Waterloo, 1996 B.Ed., University of Western Ontario, 1997

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of

MASTER OF SCIENCE

in the School of Earth and Ocean Sciences

c

Leslie Mitchell Bruce, 2011 University of Victoria

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An Examination of Zonal Mean Geopotential Variability

by

Leslie Mitchell Bruce

B.Sc., University of Waterloo, 1996 B.Ed., University of Western Ontario, 1997

Supervisory Committee

Dr. Adam H. Monahan, Supervisor

(School of Earth and Ocean Sciences, University of Victoria)

Dr. John C. Fyfe, Departmental Member

(School of Earth and Ocean Sciences, University of Victoria; Canadian Centre for Climate Modelling and Analysis, Environment Canada, University of Victoria)

Dr. John F. Scinocca, Departmental Member

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Dr. Adam H. Monahan, Supervisor

(School of Earth and Ocean Sciences, University of Victoria)

Dr. John C. Fyfe, Departmental Member

Dr. John F. Scinocca, Departmental Member

ABSTRACT

A systematic sectoral empirical orthogonal function (EOF) analysis of Southern Hemisphere (SH) extratropical tropospheric zonal-mean geopotential height (GH) is conducted in order to determine how EOF shapes and shape ordering is affected by a decrease in the width of the sector. Previous work (Kushner and Lee 2007) using surface pressure found that the two lead EOFs exchange shape as the sector width decreases below seventy degrees. In the present work, the 500hPa GH field is found to exhibit a similar feature. By fitting a idealized kinematic model, in the form of a Gaussian error function, to daily 500 hPa GH for each sector, the kinematic features of the shape reordering observed in the lead EOFs is shown to arise from the covariance structure of the fluctuating model parameters. The correlations between model parameters which are shown to influence the EOF shapes are further shown to be strongly influenced by statistical properties of daily mass and angular momentum fluctuations.

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List of Tables

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List of Figures

Figure 1.1 1a) The Southern Annular Mode for 500hPa geopotential (1980-2009) 1b) The projection of the Southern Annular Mode princi-pal component onto 500hPa geopotential. (Contour Interval is 103 _{meters) a) and b) are calculated using an areal weighting} factor of √cos φ in order to effect an equal area transformation. 2 Figure 2.1 a) Fluctuations in geopotential height associated with changes

in the baseline parameter. The thick black line shows model geopotential using δθ = 5◦ _{sectoral-mean parameters. The blue} dotted line shows the same but increases B by one standard deviation. The red dotted line shows the same but decreases B by one standard deviation. b) As in a) but for amplitude. c) As in a) but for position. d) As in a) but for width. . . 8 Figure 2.2 a) Root mean squared error as a function of sector width for the

austral winter curve fits. Individual thin coloured lines represent the set of sectors centered on particular longitudes(θ) while the thick black line is the sectoral-mean RMS error. b) As in a) but for austral summer. . . 12 Figure 2.3 a) Ratio of RMSE (from Figure 2.2a) and austral winter

geopo-tential height variability (σZ). A value of 1 indicates that the curve fitting error is comparable to the geopotential height vari-ability. b) As in a) but for the austral summer. . . 13 Figure 2.4 a) A comparison of extended winter time-mean geopotential

be-tween the best fit narrow (δθ = 5◦_{) sectors (θ = 0}◦_{− 140}◦_,₃₀₀◦₋ 350◦_{) and worst fit sectors. b) The time-mean zonal-mean zonal} wind for the sectors shown in a). c) As in a) but for the extended winter. d) As in a) but for the extended summer. . . 14

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Figure 2.5 Left: Representative sectors centered on 0◦ _{and 180}◦_{. Right:(thin} blue lines) Estimated time-mean jet amplitude for the two fam-ilies of sectors depicted on the left. Each data point on the blue lines corresponds to the time-mean amplitude for a particular sector. (Thick black line) The sectoral-mean (taken over all sec-tors) of the time-mean values as a function of sector width. . . 17 Figure 2.6 a) A contour plot of sectoral-mean EOFs normalized to unit

value. b) A representative narrow sector EOF. c) A represen-tative wide sector EOF. d) Robustness of the EOFs in a) as a function of sector width. . . 18 Figure 3.1 Contour plot of sectoral-mean EOFs derived from extended

win-ter zonal-mean GH (1980-2009) as a function of sector width. . 20 Figure 3.2 Eigenvalue fraction and robustness of mean EOFs in Figure 3.1. 21 Figure 3.3 Contour plot of sectoral-mean EOFs derived from extended

sum-mer zonal-mean GH (1980-1999) as a function of sector width. . 21 Figure 3.4 Eigenvalue fraction and robustness of mean EOFs in Figure 3.3. 22 Figure 3.5 a) Thin lines represent the extended winter time-mean amplitude

model parameter A(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) As in a) but for position P (δθ). c) As in a) but for width W (δθ). d) As in a) but for baseline or B(δθ) . . . 24 Figure 3.6 a) Thin lines represent the extended summer time-mean

ampli-tude model parameter A(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) As in a) but for position P (δθ). c) As in a) but for width W (δθ). d) As in a) but for baseline or B(δθ) . . . 25 Figure 3.7 a) Thin lines represent standard deviation of the extended winter

amplitude model parameter σA(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) Same as a) but for position or σP(δθ). c) same as a) but for width or σW(δθ). d) Same as a) but for baseline or σB(δθ) . . . 26

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mer amplitude model parameter σA(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) Same as a) but for position or σP(δθ). c) same as a) but for width or σW(δθ). d) Same as a) but for baseline or σB(δθ) . . . 27 Figure 3.9 a-f) Thin lines represent correlation between model parameters

during the extended winter for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. . . 28 Figure 3.10a-f) Thin lines represent correlation between model parameters

during the extended summer for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. . . 29 Figure 4.1 a) Contour plot of sector-mean lead EOFs derived from

corre-lated white noise with similar covariance to the extended winter curve fits, as a function of sector width. b) As in a) but for the second EOFs. . . 32 Figure 4.2 a) Explained Variance fraction for the EOFs in Figure 4.1. b)

Similarity for the two lead EOFs in Figure 4.1. . . 33 Figure 4.3 a) Contour plot of the sectoral-mean lead EOFs derived from

in-dependently fluctuating model parameters as a function of sector width. b) As in a) but for the second EOFs. . . 34 Figure 4.4 a) Explained Variance fraction for the EOFs in Figure 4.3. b)

Similarity for the two lead EOFs in Figure 4.3. . . 35 Figure 4.5 a) The similarity(M) between the first two base case model EOFs

and the model EOFs derived from a model excluding amplitude fluctuations. b) as a) but for baseline. c) as in a) but for position. d) as in a) but for width . . . 37 Figure 4.6 a) The similarity(M) between the first two base case model EOFs

and the model EOFs derived from a model excluding amplitude correlations. b) as a) but for baseline. c) as in a) but for position. d) as in a) but for width . . . 38

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Figure 5.1 a) Thin lines represent the extended winter mass variability es-timate for each individual sector in the data set, as a function of sector width. The thick line represents the sectoral-mean of the thin lines. b) As in a) but for the mass variability estimate calculated from the extended winter curve fits. . . 43 Figure 5.2 Thin lines represent the standardized model mass fluctuation

estimates, as a function of sector width. The thick line represents the sectoral-mean of the thin lines. . . 44 Figure 5.3 Scatter plots of position (P) vs. baseline (B). a) P and B

gen-erated from uncorrelated white noise with variances taken from the curve fits. b) As in a) but (P,B) were selected based on mass values lying within m_{± 3σ}m. c) As in b) but for m ± σm. d) P and B taken directly from the hemispheric curve fits. . . 45 Figure 5.4 a-f) Thin lines represent correlation between model parameters

for the CSm model sectors for each θ in the data set. The thick line represents the mean of the thin lines. . . 48 Figure 5.5 a) Thin lines represent the extended winter angular momentum

variability estimate for each sector in the data set as a function of sector width. The thick line represents the sectoral mean of the thin lines. b) As in a) but for the angular momentum variability estimate calculated from the extended winter curve fits. . . 50 Figure 5.6 a-f) Thin lines represent correlation between model parameters

for the CSL model sectors for each sector in the data set as a function of sector width. The thick line represents the mean of the thin lines. . . 51 Figure 5.7 a-f) Thin lines represent correlation between model parameters

for the CSm,L model sectors for each sector in the data set as a function of sector width. The thick line represents the mean of the thin lines. . . 53 Figure 5.8 Contour plot of zonal-mean EOFs derived from the CSm,L model

curves as a function of sector width. . . 54 Figure 5.9 Eigenvalue fraction and similarity (M) with respect to the

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I would like to thank my supervisor, Dr. Adam Monahan, for giving me the opportunity to work towards an M.Sc., for providing funding, showing great patience, helpful comments, and expert editing skills. I would also like to thank my committee members, Dr. John Fyfe, Dr. John Scinocca, and Dr. Paul Kushner.

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Introduction

1.1 Background and Previous Work

Annular Modes (AM) of atmospheric low frequency variability are ubiquitous on a hemispheric spatial scale[1]. While they have been studied for some time, the reason for their presence has yet to be completely agreed upon. They are spatial statistical patterns of hemispheric extent found in daily or monthly mean atmospheric circula-tion data. Defined as the lead Empirical Orthogonal Funccircula-tion (EOF) of mid-latitude tropospheric hemispheric zonal-mean geopotential height, they represent more of the variability of anomalous flow than any other sub-seasonal structure. Annular Modes are thought to reflect internal atmospheric variability, given that they appear similar in both Northern and Southern hemispheres even though there is a large difference in the degree of land-sea asymmetry and orography[2]. Other lower boundary effects such as sea surface temperatures appear to have a measurable but secondary influence on the daily to yearly time scales as confirmed by both data driven[3] and numerical modelling[4] studies.

The Southern Annular Mode (shown here in Figure 1.1a) is a dipole shaped zonal-mean geopotential height anomaly characteristic of a ‘see-saw’ of mass between middle (30◦_S-50◦_{S) and high (60}◦_S-90◦_{S) latitudes with a node at around 55}◦_{S. While shown} for geopotential, a similar shape is also found in zonal-mean surface pressure and in zonal-mean zonal wind fields where (to a first approximation) it is indicative of the intensification and weakening of the polar vortex and the North-South fluctuations of the eddy-driven jet respectively. The projection of the lead EOF onto the full hemi-spheric geopotential field (Figure 1.1b) shows the one standard deviation anomalous

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notably zonally symmetric (though not exactly) or ‘Annular’. However, the see-saw −80 −60 −40 −20 −0.2 −0.1 0 0.1 0.2 0.3 0.4 0.5 1a) SAM Latitude (degrees) Amplitude (dimensionless) −4 −4 −4 −4 −3 −3 −3 −3 −3 −3 −3 −2 −2 −2 −2 −2 −2 −1 −1 −1 0 0 1 1 2 2 3 3 4 1b) SAM Projected on Z 500

Figure 1.1: 1a) The Southern Annular Mode for 500hPa geopotential (1980-2009) 1b) The projection of the Southern Annular Mode principal component onto 500hPa geopotential. (Contour Interval is 103 _{meters) a) and b) are calculated using an areal} weighting factor of √cos φ in order to effect an equal area transformation.

motions are not in unison, i.e. there is little synchronized action between points on the outer ring (the mid-latitudes) and the South Pole. In fact, points on the outer ring are not necessarily positively correlated and can even show negative correlation to points on the opposite side of the pole[5]. One possible reason for this is that sig-nificant zonally asymmetric disturbances are present in the actual flow, for example, variability on synoptic timescales is dominated by propagating wavelike structures as inferred from a variety of techniques[6, 7, 8]. Given that zonal asymmetries exist, we can apply a statistical method such as EOF decomposition in order to determine if the spatial variability of geopotential is fundamentally different on smaller spatial scales and if so, at what width of sector does any change emerge and why? These questions, even in the Southern Hemisphere, have only been the subject of a limited number of studies, mostly concentrating on sectors with a specific spatial domain such as 90◦_-120◦ _{in width or geographic feature such as the South Atlantic Ocean} basin([9] for example).

Such a change in spatial statistics as a function of sector width is not purely spec-ulative. In Kushner and Lee[10](KL07) a systematic analysis of surface level pressure was carried out by calculating the EOFs of the extratropical SH region for sectors ranging in width from single longitudes to the full hemisphere. The hemispheric sec-tors showed a lead EOF with a dipolar shape similar to Figure 1.1 and a monopolar

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second EOF. For narrow sectors, those less than roughly 70◦_{, the ordering of these} patterns was reversed.

Since EOFs, in an atmospheric context, are characterizations of the spatial statis-tics of the flow, it is reasonable to make the connection between the synoptic features of day-to-day changes in that flow and these statistically derived spatial patterns. Using a kinematic model for the daily zonal mean zonal jet first proposed by Fyfe[11]

u(t, φ) = U(t)e−12

(Φ(t)−φ)2

η(t)2 (1.1)

Whittman et al.(WCP)[1] suggested that it is possible to link daily fluctuations in individual jet parameters with specific EOFs1_{. Their approach was to model a time} dependent zonal jet using three parameters representing its strength, width, and posi-tion. Each parameter was generated separately as a stationary stochastic time series with variance and relaxation parameters estimated from 1000hPa zonal wind obser-vations. Synthetic zonal wind data were generated and EOFs calculated numerically. Using orthogonality arguments WCP were able to show that a dipole shaped lead EOF similar to that of the annular modes arose specifically from the North-South fluctuations in jet position alone and not due to the other modes of variability such as fluctuations in jet strength or width. This result is particularly interesting since it shows that the annular mode-like dipole arises across a wide variety of physical assumptions and offers a potential explanation of why AM’s appear in so many dif-ferent types of models. Simply put, if a time dependent jet with a variable position can be resolved, then an AM is likely to result.

In the work of Monahan and Fyfe (2006, hereafter referred to as MF06)[12] the fluctuating jet model was extended by allowing for the possibility of correlation be-tween jet parameters. As previous models involving the eddy-driven jet invoked momentum conservation[13] it is natural to expect that fluctuations between jet posi-tion and jet strength will be inversely correlated. In a further refinement, EOFs were calculated analytically from a covariance matrix derived by direct integration of the fluctuating jet function (Eqn 1.1) over the probability space of the fluctuations. For the case where fluctuations in one single degree of freedom were considered, the result-ing EOFs compared favourably with WCP. When multiple degrees of freedom were considered, the authors were also able to show that it is not possible beyond leading order (if at all) to associate a particular EOF with a specific aspect of variability

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between parameters also introduces new results. When considering correlated fluctu-ations between strength and width, for example, EOF shape was found to depend on the degree of correlation.

By extending their fluctuating jet model to geopotential height (GH) (by integrat-ing the the EOF expansions of the zonal-mean zonal wind found in MF06) Monahan and Fyfe (2008, hereafter referred to as MF08)[14] were able to show that the AM cannot result from a single unique degree of freedom and that an accurate model requires fluctuations in both jet position and jet amplitude.

1.2 Statement of the Problem and Summary of

The main focus of this work is to subject geopotential height at the 500hPa pressure level to a sectoral analysis as done in KL07 to verify that a similar shape reordering occurs. Once verified, we ask “Can the daily fluctuations of geopotential height as a function of sector width, as seen by an idealized but physically motivated model similar to that used in MF06, be used to characterize the shape and shape ordering of the first two EOFs? If so, is it possible to show kinematically that these EOFs arise from the imposition of simple physical constraints on the model by the GH field?”

We proceed to curve fit and collect jet statistics in a similar manner as MF06 but model GH. We emphasize that in doing so our model still relates to the zonal-mean zonal wind through geostrophic balance. While working in this fashion we lose the ability to obtain analytical expressions for our EOFs, but gain the ability to associate the daily fluctuations of mass and angular momentum within a sector to correlations between model parameters in a kinematic way that is both simple and general. This modelling framework, like MF06, can be applied to sectors of any width allowing us to examine why lead EOFs change shape as they do in KL07. Since this is a kinematic approach, we are unable to present a fundamental dynamical mechanism for the EOFs shapes and the shape reordering, but suggest that these results may be useful as a guide to future work.

The work that follows is divided up into six chapters. In Chapter 2 the Idealized Fluctuating Gaussian Error Function model is introduced and parameters within are interpreted. We also discuss the data source used, the method of curve fitting

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em-ployed, and show examples of the diagrams used to present the results. In Chapter 3 we give observational EOFs for the data set under consideration, present the outcome of the curve fitting procedure, and describe these collected statistics. In Chapter 4 model EOFs are calculated from synthetic data derived solely from the collected statistics. These model EOFs are found to compare favourably to the observed and reproduce both their shapes and the order in which these shapes appear. In Chap-ter 5 we show that the correlations between the three model parameChap-ters required to model the Annular Mode can be accounted for by imposing two simple constraints on curve profiles representing daily fluctuations of mass and angular momentum within a sector. We also show that these constraints are sufficient to model the shapes and shape reordering of the two lead EOFs. Lastly, these results are summarized and discussed in Chapter 6.

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Chapter 2 Data, Methods, and Model

Introduction

2.1 Data Source

Geopotential height data for this study come from the National Center for Envi-ronmental Protection/ National Center for Atmospheric Research (NCEP/NCAR) reanalysis project[15]. It should be noted here that reanalysis products are a com-posite of several observational data sources, from land, aircraft, satellite, and others, integrated into an operational forecast model. For the sake of simplicity these re-analysis products will be referred to hereafter as data. Some of these rere-analysis data sets are more reliable than others. Even within a particular data set reliability is not homogeneous. With respect to the geopotential height used in this study, all data come from the post 1979 period when satellite observations were included. While the results of this study do not depend in detail on the time period used, post 1979 GH reanalysis has much better observational coverage for the high latitudes in the South-ern Hemisphere. Even so, the zonal band centered on 85◦_{S still has only 10-20 daily} observations per 2.5◦ _{by 2.5}◦ _{latitude-longitude box compared to a similar band} cen-tered on 35◦_{S with 120-240[16]. Some authors[17] have also criticized this reanalysis} set for containing spurious seasonal trends which tend to exaggerate the SAM in the Southern Hemisphere winter by up to a factor of two. While this concern cannot be ignored, the data used in this present study is split into seasons and detrended which reduces this effect. Subsequent experiments using ERA-40 reanalysis data from the European Center for Medium-term Weather Forecasting[18], which is thought not to

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exaggerate the wintertime SAM, found similar results to those obtained from NCEP-NCAR, and so these concerns are not considered further.

2.2 Data Preparation

Daily 2.5◦_{by 2.5}◦ _{GH fields from 1980-2009 were used for the sectoral analysis. Using} smaller twenty year periods (1980-1999 for example) produced similar results. Data were divided into two separate seasons, an extended summer (NDJFMA) and an extended winter (MJJASO). Data outside of the meridional boundaries (20◦_S-90◦_S) were neglected as were leap days. From the remaining data set, a series of sectors for each season at the 500hPa pressure level were formed for the following zonal widths (δθ), and central longitudes (θ):

δθ= [5, 10, 15, ..., 355, 360] (2.1) θ= [0, 10, 20, ..., 350, 360] (2.2) Any GH data falling within a particular sector was zonally averaged between θ−δθ 2 and θ + δθ

2; the resulting ‘sectoral zonal mean’ is denoted z(t, φ, δθ, θ).

2.3 Idealised Jet Model

In order to relate fluctuations in GH to the resulting EOFs, a simple four parameter model based on MF06 (expressed in terms of geopotential height rather than zonal wind) is employed. This model consists of a single idealized GH profile in the shape of a Gaussian error function. At any time, this profile is specified by four kinematic degrees of freedom: the baseline (mean of a daily geopotential profile), amplitude, position, and width; these are denoted respectively by (B, A, P, W ).

Zm(t, φ, δθ, θ) = B(t, δθ, θ) + A(t, δθ, θ)erf φ − P (t, δθ, θ)√ 2W (t, δθ, θ)

(2.3) Schematically, fluctuations in individual degrees of freedom are shown in Fig-ure 2.1. The model is fit to daily GH to characterize the day-to-day fluctuations in these parameters.

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−80 −60 −40 −20 5000

5500 6000

a) BaseLine

Geopotential Height (meters) ₋₈₀ ₋₆₀ ₋₄₀ ₋₂₀

5000 5500 6000 b) Amplitude −80 −60 −40 −20 5000 5500 6000 c) Position

Geopotential Height (meters)

Latitude (degrees) −80 −60 −40 −20 5000 5500 6000 d) Width Latitude (degrees)

Figure 2.1: a) Fluctuations in geopotential height associated with changes in the baseline parameter. The thick black line shows model geopotential using δθ = 5◦ sectoral-mean parameters. The blue dotted line shows the same but increases B by one standard deviation. The red dotted line shows the same but decreases B by one standard deviation. b) As in a) but for amplitude. c) As in a) but for position. d) As in a) but for width.

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We also note that Eqn. 2.3 and the model from MF06 are related by: 1

f(φ)

∂Zm(t, φ, δθ, θ)

∂φ ∝u(t, φ, δθ, θ). (2.4)

As a result, jet position (P and Φ) and jet width (W and η) are the same. Amplitude (A) and jet strength (U) are related by:

U_{(t, δθ, θ) ∝} A(t, δθ, θ)

W(t, δθ, θ). (2.5)

Baseline (B) has no analogue in MF06.

Our model, when fit to GH, produces a geostrophic wind that contains a single ‘effective jet’. Since the extended winter zonal wind often contains two distinct jets, the model will often produce values for position and width, for example, that are different from those obtained from the MF06 model that fits the eddy driven jet directly. In practice these differences are not large.

2.4 Curve Fitting

The model was fit to each daily GH profile (Z(t, φ, δθ, θ)) using a standard gradient descent algorithm[19]. This iterative minimization algorithm requires initial guesses for (B,A,P,W). To this end we set:

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BE(t, δθ, θ) = 1 n N X i=1 z(t, i, δθ, θ) (2.6) AE(t, δθ, θ) = 1 2{ 1 4 4 X i=1 z_{(t, i, δθ, θ) − B}E(t, δθ, θ) + 1 4 N X i=N −3 z_{(t, i, δθ, θ) − B}E_{(t, δθ, θ)}} (2.7) PE_{(t, δθ, θ) = minloc{z(t, φ, δθ, θ) − B}E_{(t, δθ, θ)}} (2.8) WE(t, δθ, θ) = 1 2[minloc{z(t, φ, δθ, θ) − 1₂(min(z(t, φ, δθ, θ)) + BE_{(t, δθ, θ))} + minloc{z(φ, t, δθ, θ)} − 1 2(max(z(t, φ, δθ, θ)) + B E (t, δθ, θ))}] (2.9)

where min(), max(), and minloc() are functions that when supplied with a set return the minimum, maximum, and location of the minimum respectively (the function notation and action are taken from their intrinsic FORTRAN 90 counterparts). The baseline is estimated by finding the average GH for the profile across latitude for that day. The amplitude is estimated by taking the average GH for the intervals φ=(20◦_S,27.5◦_{S) and φ=(82.5}◦_S,90.0◦_{S), measuring their distance from the baseline} estimate, and then averaging the two amplitude estimates together. Position is esti-mated by the latitude of the data point closest to where the GH profile crosses the baseline estimate. Finally, width is estimated by the distance, in degrees, between the position estimate and the GH half way (in terms of height) between the position estimate and the maximum (or minimum) for the curve, with the estimates from either side being averaged together.

The day-to-day GH profiles display a degree of variability in their shape that cannot always be modelled. In roughly 5% of the cases Eqns (2.4)-(2.7) do not give particularly good initial guesses. As a result, the above scheme is also applied to a three point and a five point simple moving average of Z(t, φ, δθ, θ) (along the φ direction) with the estimate that displays the lowest RMSE of the three being used. While costly in computing time the fitting procedure was numerically stable (i.e. always converged on a finite answer) and required almost no hand tuning aside

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from the initial determination of equations (2.4)-(2.7). All fitted parameters BE_{(t, δθ, θ). . .W}E_{(t, δθ, θ) are retained.}

Figure 2.2 depicts the Root Mean Squared error (RMSE) of the curve fits at 500hPa for both seasons as well as the sectoral-mean (the average over all sectors of the same δθ). Not surprisingly, narrow sectors have, on average, the greatest RMSE as well as the greatest spread of values between highest and lowest. The majority of the fitting error was concentrated in the high latitudes φ > 60◦_{. However, if we} account for the fact that the narrower sectors are more variable by dividing RMSE by the corresponding zonal mean standard deviation of GH, as shown in Figure 2.3, we see that in relative terms the model profiles for different sector widths are equally good fits in the winter and only slightly worse at smaller sectors in the summer. Fitting error as a percentage of the zonal mean standard deviation is in the range of 7%-12% for the winter and 5%-7% for the summer.

The method outlined is best applied to pressure levels in the middle troposphere. Below this, starting roughly at the 850hPa pressure level, the model is unable to accurately represent the shape of the time-mean GH in the high latitudes likely due to the topographic extent of the Antarctic continent. For upper tropospheric pressure levels there is increased curve fitting error due to the increased prominence of the subtropical jet. Figure 2.4 compares time mean GH (at 500hPa) from 5◦ _{sectors with} the highest and lowest fitting error to the corresponding zonal-mean zonal wind for the winter (2.4a,b) and summer (2.4c,d) seasons. For the winter sectors the worst fit occurs when there is a clear separation between the eddy-driven and subtropical jets. The summer season does not exhibit as prominent a feature.

2.5 EOF Calculation

Since the idealized model of geopotential height variations is expressed in terms of a Gaussian error function, the integral of which does not have a simple analytic form, the calculation of EOFs must procede numerically. To calculate the EOFs for a given sector, anomaly fields were formed by subtracting a 31-day moving average in order to supress timescales longer than those that are associated with the AM. This removes most of the influence of the seasonal cycle. Anomalies are then multiplied by an areal weighting factor of √cos φ at each latitude. This weighting results in a covariance matrix that is ‘equal area’ (important since a unit of unweighted variance at the pole would otherwise be considered equal to a unit of variance towards the equator even

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50 100 150 200 250 300 350 100 150 200 250 300 a) RMSE (meters) 50 100 150 200 250 300 350 100 150 200 250 300 b)

Sector Width (δθ) (degrees)

RMSE (meters)

Figure 2.2: a) Root mean squared error as a function of sector width for the austral winter curve fits. Individual thin coloured lines represent the set of sectors centered on particular longitudes(θ) while the thick black line is the sectoral-mean RMS error. b) As in a) but for austral summer.

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50 100 150 200 250 300 350 0.06 0.07 0.08 0.09 0.1 0.11 0.12 a) RMSE/ σ Z (dimensionless) 50 100 150 200 250 300 350 0.06 0.07 0.08 0.09 0.1 0.11 0.12 b)

Sector Width (δθ)(degrees)

RMSE/

σ Z

(dimensionless)

Figure 2.3: a) Ratio of RMSE (from Figure 2.2a) and austral winter geopotential height variability (σZ). A value of 1 indicates that the curve fitting error is comparable to the geopotential height variability. b) As in a) but for the austral summer.

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14 −90 −80 −70 −60 −50 −40 −30 −20 4800 5000 5200 5400 Height (meters) −90 −80 −70 −60 −50 −40 −30 −20 −5 0 5 10 Velocity (m/s) −90 −80 −70 −60 −50 −40 −30 −20 4800 5000 5200 5400 5600 5800 6000

Sector Width (δθ) (degrees)

Height (meters) NDJFMA Geopotential Best Fit Worst Fit −90 −80 −70 −60 −50 −40 −30 −20 −5 0 5 10 15 20 25

Sector Width (δθ) (degrees) NDJFMA Zonal Wind

Velocity (m/s)

Best Fit Worst Fit

Figure 2.4: a) A comparison of extended winter time-mean geopotential between the best fit narrow (δθ = 5◦_{) sectors}

(θ = 0◦_{− 140}◦_,₃₀₀◦_{− 350}◦_{) and worst fit sectors. b) The time-mean zonal-mean zonal wind for the sectors shown in a). c)}

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though it occurs over an much smaller domain).The result is denoted as Z0_{(φ, t; δθ, θ)} Each of the sector-averaged observed anomaly fields was decomposed into EOFs (ψi), eigenvalues (λi), and principal component time series (ai) using singular value decomposition[20].

Z0_{(φ, t; δθ, θ) =} X i

ψi(t; δθ, θ)λi(t; δθ, θ)ai(t; δθ, θ) (2.10)

By construction the ψi are mutually orthogonal in space and the ai are mutually uncorrelated in time. The subsequent analysis focuses on the leading two EOFs.

2.6 Sectoral Analysis and Sectoral Mean

The sectoral structure of parameter estimates is illustrated in Figure 2.5, in which each blue line on the graph represents the time-mean amplitude (A) for a family of sectors centered on a particular longitude and varying in width (depicted for central latitudes θ=0◦ _{and 180}◦_{). Each data point on a given blue line corresponds to the} time-mean jet amplitude in a sector with a particular width (δθ). The black line corresponds to the sectoral mean amplitude taken over all sectors. We will refer to the range of a particular model parameter’s value between sectors of similar width as its disparity. Hemispheric sectors, by definition, display no disparity whereas δθ = 5◦ sectors have large disparity. Figure 2.6a depicts the sectoral-mean EOFs for the extended summer GH as a function of δθ. Representitive cross sections of narrow and wide sectors (with widths of 20◦ _{and 350}◦ _{respectively) are depicted in 2.6b and} 2.6c. Clearly narrow sectors no longer exhibit a dipole AM-like spatial variability. The reason for this difference between wide and narrow sector spatial variability is the focus of this present work. The mean EOFs are normalized to unit amplitude so that the sectoral-mean shapes at different widths are more clearly comparable1_. The normalization factor is referred to (as in KL07) as the robustness (R) and can be interpreted as a measure of the consistency of a group of EOF shapes with the same width. As seen in Figure 2.6d the lowest value for R (and therefore the least consistency in EOF shape among different sectors) occurs when the shape of the EOFs are undergoing their most rapid change. We will refer to this region of smaller R values as the ‘transition zone’. ‘Narrow’ or ‘Wide’ if not otherwise specifically stated

1_{While SVD produces EOFs of unit norm within each sector, their sectoral mean is not in general} and thus normalization is used

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17 50 100 150 200 250 300 350 435 440 445 450 455 460 (meters) δθ (degrees)

Figure 2.5: Left: Representative sectors centered on 0◦ _{and 180}◦_{. Right:(thin blue lines) Estimated time-mean jet amplitude}

for the two families of sectors depicted on the left. Each data point on the blue lines corresponds to the time-mean amplitude for a particular sector. (Thick black line) The sectoral-mean (taken over all sectors) of the time-mean values as a function of sector width.

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18

Sector Width in Degrees

Latitude (EOF 1) 50 100 150 200 250 300 350 −80 −60 −0.2 0 −80 −60 −40 −20 −0.2 0 0.2 0.4 b) Amplitude Latitude(degrees) −80 −60 −40 −20 −0.2 0 0.2 0.4 c) Latitude(degrees) 0 50 100 150 200 250 300 350 0.6 0.8 1 d)

R (no dimension)

EOF1

Figure 2.6: a) A contour plot of sectoral-mean EOFs normalized to unit value. b) A representative narrow sector EOF. c) A representative wide sector EOF. d) Robustness of the EOFs in a) as a function of sector width.

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Chapter 3 Observational EOFs and Jet

Parameters

In this chapter we begin by presenting a sectoral analysis of the observed GH for the extended summer and winter seasons. These results confirm that, similar to surface pressure in KL07, the EOFs undergo a shape reordering roughly around δθ = 70◦_.

We also give curve fitting results. The process of estimating idealized jet model parameters from data yields a useful picture of geopotential height variability from the hemispheric to the regional scale in terms of the statistical properties of the individual model parameters and their mutual covariances. Later chapters will use the information presented here to show that the shape and shape ordering of the EOFs are strongly influenced by variations in these statistics.

3.1 Extended Winter and Summer EOFs

For both seasons, narrow sector lead EOFs (Figures 3.1 and 3.3) are found to be monopolar in shape while those of widths greater than 100◦_{(summer) and 75}◦_(winter) are dipolar. EOFs for these δθ regions are robust across sectors (R > 0.95; Figures 3.2 and 3.4). As both seasons display the shape reordering we can rule out seasonal differences as its cause.

Spatial statistics with these characteristics appear equivalent barotropic, in that they display a similar shape as a function of sector width across most of the tropo-sphere (not shown). Figures 3.1 and 3.3 are representative of sectoral analyzes of GH between 1000hPa and 300hPa.

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δθ < 120◦ _{for the summer) over which the shape reordering occurs. Figures 3.2} and 3.4 illustrate the characteristics of the eigenvalue fraction and robustness of the EOFs as a function of sector width. In particular, the dipole EOF, beginning at δθ = 360◦ _{(the Annular Mode) is dominant but progressively explains less variance} as the sector width narrows. The transition region coincides with the zone of greatest proximity between the eigenvalues as well as a dip in the robustness. Despite the transition region for the extended summer occurring over a wider range of sector widths, subsequent analysis shows no qualitative difference between the two seasons. As a result, we concentrate on the extended winter.

Lastly, we note that the wide sector second EOFs have two small oppositely signed anomalies close to the poleward and equatorward boundaries in the range 250◦ _{< δθ <} ₃₆₀◦_{. This is likely due to the relative proximity of the eigenvalues for} the second and third EOFs. While the resolution of EOFs with similar eigenvalues can sometimes be improved by using a longer time period for the study[21], using 35 years of seasonal data in place of 30 as used here produced similar results.

a) EOF 1 (MJJASO) Latitude (degrees) 50 100 150 200 250 300 350 −80 −60 −40 −20 −0.2 0 0.2 b) EOF 2 (MJJASO)

Latitude (degrees) 50 100 150 200 250 300 350 −80 −60 −40 −20 −0.2 0 0.2 0.4

Figure 3.1: Contour plot of sectoral-mean EOFs derived from extended winter zonal-mean GH (1980-2009) as a function of sector width.

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0 50 100 150 200 250 300 350 0.2

0.3 0.4 0.5

a) Mean Fraction of Explained Variance (MJJASO)

EV Frac. (no dimension)

EOF1 EOF2 0 50 100 150 200 250 300 350 0.6 0.8 1

b) Robustness of EOFs (MJJASO)

R (no dimension)

EOF1 EOF2

Figure 3.2: Eigenvalue fraction and robustness of mean EOFs in Figure 3.1.

a) EOF 1 (NDJFMA) Latitude (degrees) 50 100 150 200 250 300 350 −80 −60 −40 −20 −0.2 0 0.2 b) EOF 2 (NDJFMA)

Latitude (degrees) 50 100 150 200 250 300 350 −80 −60 −40 −20 −0.4 −0.2 0 0.2 0.4

Figure 3.3: Contour plot of sectoral-mean EOFs derived from extended summer zonal-mean GH (1980-1999) as a function of sector width.

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0 50 100 150 200 250 300 350 0.2

0.3 0.4 0.5

a) Mean Fraction of Explained Variance (NDJFMA)

EOF1 EOF2 0 50 100 150 200 250 300 350 0.6 0.8 1

b) Robustness of EOFs (NDJFMA)

R (no dimension)

EOF1 EOF2

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3.2 Extended Winter and Summer Curve Fits

Sectoral-mean parameter estimates for each extended season (Figures 3.5 and 3.6 for winter and summer respectively) show little if any trend as sector width decreases. Differences between δθ = 5◦ _{and δθ = 360}◦ _{are less than 5% for all parameters.} The time-mean amplitude and width for individual sectors (the blue lines in Fig-ures 3.5 and 3.6) show more disparity amongst sectors of similar width in the winter than in summer while disparity in position and baseline are roughly comparable. We also find that the sectoral mean amplitude and width show a winter eddy-driven jet that is both stronger and wider than its summer counterpart, with similarities in seasonal jet position indicating no corresponding shift North or South.

The standard deviations of the model parameters (Figures 3.7 and 3.8 for winter and summer respectively) show an increase as sector width narrows. The extended winter model parameters show much greater disparity and on average are larger than the summer values.

Correlation coefficients between model parameters (shown in Figures 3.9 and 3.10) are generally larger in wider sectors than in narrow sectors. In 3.9(e) for example, the correlation between the position and baseline is roughly 0.9 for δθ = 360◦_{. This} near perfect correlation is of particular interest as it may reflect some degree of mass conservation below 500 hPa. In the context of the model, and assuming equivalent barotropic dynamics, a poleward shift of the daily jet position requires a reduction of the overall baseline so that the mass below the pressure level is left unchanged. The connection between ρP B and daily mass fluctuations within a given sector will be further investigated in Chapter 5.

As sector width narrows we see the correlations become smaller in magnitude and show a greater disparity between sectors. Differences between the sectoral mean correlations for summer and winter are small (0.1 at most).

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100 200 300 460 480 500 520 a) Mean A (meters) 100 200 300 −54 −52 −50 −48 b) Mean P (degrees) 100 200 300 8 10 12 14 16 18 c) Mean W (degrees) δθ (degrees) 100 200 300 5250 5300 5350 5400 d) Mean B (meters) δθ (degrees)

Figure 3.5: a) Thin lines represent the extended winter time-mean amplitude model parameter A(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) As in a) but for position P (δθ). c) As in a) but for width W(δθ). d) As in a) but for baseline or B(δθ)

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100 200 300 400 410 420 430 a) Mean A (meters) 100 200 300 −54 −52 −50 −48 b) Mean P (degrees) 100 200 300 8 10 12 14 16 18 c) Mean W (degrees) δθ (degrees) 100 200 300 5400 5420 5440 5460 d) Mean B (meters) δθ (degrees)

Figure 3.6: a) Thin lines represent the extended summer time-mean amplitude model parameter A(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) As in a) but for position P (δθ). c) As in a) but for width W(δθ). d) As in a) but for baseline or B(δθ)

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100 200 300 50 60 70 80 90 a) σ A (meters) 100 200 300 4 6 8 b) σ P (degrees) 100 200 300 2 4 6 8 c) σ W (degrees) δθ (degrees) 100 200 300 50 60 70 80 90 d) σ B (meters) δθ (degrees)

Figure 3.7: a) Thin lines represent standard deviation of the extended winter ampli-tude model parameter σA(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) Same as a) but for position or σP(δθ). c) same as a) but for width or σW(δθ). d) Same as a) but for baseline or σB(δθ)

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100 200 300 50 60 70 80 90 a) σ A (meters) 100 200 300 4 6 8 b) σ P (degrees) 100 200 300 2 4 6 8 c) σ W (degrees) δθ (degrees) 100 200 300 50 60 70 80 90 d) σ B (meters) δθ (degrees)

Figure 3.8: a) Thin lines represent standard deviation of the extended summer am-plitude model parameter σA(δθ) for each θ in the data set. The thick line represents the sectoral-mean of the thin lines. b) Same as a) but for position or σP(δθ). c) same as a) but for width or σW(δθ). d) Same as a) but for baseline or σB(δθ)

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100 200 300 −0.8 −0.6 −0.4 −0.2 0 a) ρ AP (no dimension) 100 200 300 0.4 0.5 0.6 0.7 0.8 b) ρ AW 100 200 300 −0.8 −0.6 −0.4 −0.2 c) ρ AB (no dimension) 100 200 300 −0.6 −0.4 −0.2 0 d) ρ PW 100 200 300 0.4 0.6 0.8 e) ρ PB (no dimension) δθ (degrees) 100 200 300 −0.6 −0.4 −0.2 0 f) ρ WB δθ (degrees)

Figure 3.9: a-f) Thin lines represent correlation between model parameters during the extended winter for each θ in the data set. The thick line represents the sectoral-mean of the thin lines.

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100 200 300 −0.8 −0.6 −0.4 −0.2 0 a) ρ AP (no dimension) 100 200 300 0.4 0.5 0.6 0.7 0.8 b) ρ AW 100 200 300 −0.8 −0.6 −0.4 −0.2 c) ρ AB (no dimension) 100 200 300 −0.6 −0.4 −0.2 0 d) ρ PW 100 200 300 0.4 0.6 0.8 e) ρ_PB (no dimension) δθ (degrees) 100 200 300 −0.6 −0.4 −0.2 0 f) ρ_WB δθ (degrees)

Figure 3.10: a-f) Thin lines represent correlation between model parameters during the extended summer for each θ in the data set. The thick line represents the sectoral-mean of the thin lines.

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Chapter 4 Model EOFs

Using the curve-fitting results from the previous chapter we can now test the main question addressed in this study: Can EOF structures and the order in which those structures appear be understood in terms of the fluctuation characteristics of the model parameters? To answer this question, we take the jet statistics from Chapter 3 and generate sets of correlated white noise with specified covariance structures. From these synthetic data, model GH curves are generated and their EOFs (the ‘model’ EOFs) calculated.

4.1 Model EOFs

4.1.1 Similarity (M)

In order to compare the shape of observational EOFs (ψ(φ, δθ, θ)) to other modelled EOFs (ψM od(φ, δθ, θ)), we define a measure called ‘Similarity’ or M. It is defined as the sectoral mean dot product (over φ) between the observational EOFs and the corresponding EOFs of another set (Eqn 4.1).

M_{(δθ) = ψ(φ, δθ, θ) · ψ}M od(φ, δθ, θ) (4.1) If a set other than the observational EOFs are being used for comparison, we will use the term ‘Similarity with respect to’. If the two sets of EOFs are identical then M = 1; M = 0 if they are orthogonal. We generally use M > 0.9 as the threshold where the two sets are considered comparable in shape.

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4.1.2 Extended Winter Model EOFs

Empirically, we find that the Extended winter model first EOFs exhibit a similar shape (Figure 4.1a-b) to the observational EOFs for both the wide and narrow sectors (Compare to Figure 3.1). Agreement of the second model EOF with that of the observations is near perfect for the narrow sectors. For wider sectors, the modelled second EOF is monopolar as in observations, but appears slightly wider with a centre that is shifted equatorward. The low similarity values (M ≈ 0.5 − 0.75) for the wide sectors are primarily due to small negative regions close to the 20◦_{S and 90}◦_S boundaries found in the observational EOFs as discussed in Section 3.1. As the second wide sector model EOFs do not exhibit this feature (they are positive in those regions and their associated eigenvalues appear well seperated) the similarity is smaller even though both observational and model EOFs are similarly monopolar. The lead model EOFs also explain more variance than do the observational (61% vs. 45% for narrow sectors and 55% vs. 50% for the wide). This is consistent with the synthetic model data having less variance than the observational dataset (which includes the variance of the model residuals). The transition zone, where the shape reordering occurs, is found at δθ ≈ 90◦ _{(versus δθ ≈ 70}◦ _{in Figure 3.1) and appears slightly less abrupt in} appearance than the observational EOFs.

Despite the existence of some nominal differences, the model is able to adequately describe the shape and shape ordering of the observational EOFs in terms of the fluctuations of the model parameters. Since we compare much of the subsequent modelling in this chapter to the present case we will refer to this as the ‘base case’.

4.2 Modified Statistical EOFs

To assess how the statistics of model parameters influence the EOF structure of GH, we run the same model as the previous section but modify the input statistics by setting individual model parameters (such as σA or ρP B for example) to zero. In doing so we can determine if these statistics are essential for, individually or in combination, characterizing either EOF shapes or their ordering.

4.2.1 The Uncorrelated Case

We begin by examining the uncorrelated case where all off-diagonal elements of the model covariance matrix are set to zero. As later modelling in Chapter 5 emphasizes

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a) Latitude (EOF 1) 50 100 150 200 250 300 350 −80 −60 −40 −20 −0.3 −0.2 −0.1 0 0.1 b)

Sector Width (degrees)

Latitude (EOF 2) 50 100 150 200 250 300 350 −80 −60 −40 −20 −0.2 0 0.2

Figure 4.1: a) Contour plot of sector-mean lead EOFs derived from correlated white noise with similar covariance to the extended winter curve fits, as a function of sector width. b) As in a) but for the second EOFs.

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0 50 100 150 200 250 300 350 0.2 0.3 0.4 0.5 0.6 a)

EOF1 EOF2 0 50 100 150 200 250 300 350 0.4 0.6 0.8 1 b)

M (No dimension) EOF 1 EOF 2

Figure 4.2: a) Explained Variance fraction for the EOFs in Figure 4.1. b) Similarity for the two lead EOFs in Figure 4.1.

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structure, this case acts as a useful base of comparison. a) Latitude (EOF 1) 50 100 150 200 250 300 350 −80 −60 −40 −20 0 0.1 0.2 b)

Latitude (EOF 2) 50 100 150 200 250 300 350 −80 −60 −40 −20 −0.2 −0.1 0 0.1 0.2

Figure 4.3: a) Contour plot of the sectoral-mean lead EOFs derived from indepen-dently fluctuating model parameters as a function of sector width. b) As in a) but for the second EOFs.

The EOFs obtained by neglecting correlations between model parameters, shown in Figure 4.3, show no shape reordering, a lead monopole EOF and a dipolar second EOF. The maximum of the monopole occurs coincident to the mid-latitude node of the dipole. EV fractions are nearly constant showing little if any trend and a large separation between the monopole (EV=0.66-0.70) and the dipole (EV=0.23-0.27). This is consistent with the observed fact that correlations between model parameters become smaller as sector width narrows.

Wide sector EOFs, both first and second, show limited similarity to their ‘baseline’ counterparts with M values in the range of 0.0-0.4. In contrast, narrow sector EOFs have a similarity that trends (as δθ → 5◦_{) towards a maximum at δθ = 5}◦ _{of roughly} 0.9.

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0 50 100 150 200 250 300 350 0.2

0.4 0.6

a)

EOF1 EOF2 0 50 100 150 200 250 300 350 0 0.5 1 b)

M (No dimension)

EOF 1 EOF 2

Figure 4.4: a) Explained Variance fraction for the EOFs in Figure 4.3. b) Similarity for the two lead EOFs in Figure 4.3.

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In this section we set individual model parameter fluctuations and their associated correlations to zero. The results are presented by examining the M values with respect to the ‘base case’ model EOFs.

As seen Figure 4.5, removing any fluctuation apart from width produces wide sector lead EOFs that differ considerably from the ‘base case’, with M values of less than 50%. Clearly, fluctuations in amplitude, position and baseline are essential to achieve the correct EOF shape. We also find that the second wide sector EOF is similarly affected although the dependence is primarily on fluctuations of baseline and position. The shape of this second EOF appears not to depend as strongly on fluctuations in width and amplitude.

For the narrow sectors removal of fluctuations in any single parameter does not appear to have such a deleterious effect on the shape of the lead monopole. The dipole shape for δθ → 5◦_{, which is now the second EOF (represented by the green} line in 4.5a-d), is affected most by suppressing the amplitude or position fluctuations. From this we conclude that to correctly model the lead EOF for the wide sectors it is important to consider fluctuations in amplitude, baseline, and position together so that the EOF shapes and the location of the reordering agree with observational EOFs. As these three fluctuation modes are highly correlated to each other, we repeat the above analysis but this time selectively decorrelate individual model parameters by setting, for example, all correlations with position to zero. Figure 4.6 shows the resulting similarity values. Only when width related correlations are turned off is the lead wide sector EOF similar to the ‘base case’. As before, these changes do not appear to affect the narrow sector EOFs to the same extent with all narrow sector lead EOFs attaining at least a similarity of M=0.8 or higher. This is consistent with the narrow sector curve fits having, in general, smaller values of the correlation coefficients which therefore influence the covariance structure of the fluctuations to a smaller degree.

4.3 Chapter Conclusions

Model EOFs based solely on the covariance structure of the curve-fit model param-eters (the ‘base case’) are indeed able to reproduce the shape and shape ordering of the observed EOFs. Further inferences based on suppressing subsets of these

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param-50 100 150 200 250 300 350 0

0.5 1

a) Amplitude Fluctuations Removed

M (no dimension)

50 100 150 200 250 300 350

0 0.5 1

b) Baseline Fluctuations Removed

M (no dimension)

50 100 150 200 250 300 350

0 0.5 1

c) Position Fluctuations Removed

M (no dimension)

50 100 150 200 250 300 350

0 0.5 1

Sector Width (degrees) d) Width Fluctuations Removed

M (no dimension)

M EOF 1 M EOF 2

Figure 4.5: a) The similarity(M) between the first two base case model EOFs and the model EOFs derived from a model excluding amplitude fluctuations. b) as a) but for baseline. c) as in a) but for position. d) as in a) but for width

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0 50 100 150 200 250 300 350 0 0.5 1 M (No dimension) a) ρ AX removed 0 50 100 150 200 250 300 350 0 0.5 1 M (No dimension) b) ρ BX removed M EOF 1 M EOF 2 0 50 100 150 200 250 300 350 0 0.5 1 M (No dimension) c) ρ PX removed 0 50 100 150 200 250 300 350 0 0.5 1

M (No dimension)

d) ρ

WX removed

Figure 4.6: a) The similarity(M) between the first two base case model EOFs and the model EOFs derived from a model excluding amplitude correlations. b) as a) but for baseline. c) as in a) but for position. d) as in a) but for width

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eters admit three conclusions. First, the shape reordering seen in the observational data is consistent with increasing model parameter variance as sectors become nar-rower together with a trend towards smaller correlation coefficients (Figures 3.7 and 3.9). Second, to simulate the shape of the wide sector observational EOFs we require fluctuations in position, baseline and amplitude as well as their mutual correlations. Removal of any of these parameters causes these EOFs to loose their AM-like char-acteristics. Finally, simulations of the narrow sector observational EOFs show their shapes do not exhibit a strong dependance on any single model parameter but rather a weak dependence on all parameters.

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Chapter 5 Correlation Coefficients and

Physical Constraints

In this Chapter, we conduct a further analysis within our model framework that examines how fluctuations in sectoral mass and angular momentum influence model EOFs and gives insight into why the observational EOFs undergo a shape reordering. This is done in a purely kinematic fashion needing only the model in Chapter 2 and a new method for determining synthetic datasets satisfying physically relevant constraints.

In Chapter 3.2, we found that when the model parameters are estimated from wide sectors, they are highly correlated with each other. For example in the extended winter for δθ = 360◦ _{we have ρP B} _{= 0.91, ρAP} _{= −0.78, ρ}

AB = −0.84, and ρAW = 0.71. Consistent with this, in Chapter 4, we find that all parameters are essential to correctly modelling the annular mode, with the exception of those related to width. In contrast, narrower sectors do not show such a strong interdependance for their shape.

We have previously suggested that the large values of ρP B in wide sectors may reflect some degree of conservation of mass, but so far, this has not been used as a fundamental constraint on the model parameters. As mentioned in the Introduction, Thompson and Wallace (1999) give a ‘see-saw’ of mass interpretation to the AM. If the AM is no longer the dominant mode of spatial variability in the narrow sectors, could it be due to the expectation that mass conservation is not as important a consideration at these smaller scales? Is it then possible to relate this change to the change in the shape of the lead EOF? These questions will be considered in this

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section.

We also examine daily fluctuations of angular momentum and their relation to ρAP.

Finally we present statistics and EOFs calculated from invoking mass and angular momentum constraints together. These EOFs are found to exhibit the basic features found in the observational EOFs.

5.1 An Examination of Mass Fluctuations in

Sec-toral Geopotential Height Data

We begin by examining the degree to which the mass within a given sector fluctuates. An atmospheric column between 500 hPa and the surface will see its geopotential height fluctuate daily. In general, this can occur in two different ways. The first is due to mass entering or leaving the column, and the second is due to changes in temperature. We will make the assumption that fluctuations in the daily zonal-mean temperature within a given sector are small and can be neglected (i.e. we assume an equivalent barotropic atmosphere). As such, fluctuations in GH can be solely attributed to fluctuations in sectoral mass. These daily fluctuations of mass should increase as sector width narrows due both to the presence of two zonal boundaries and to the increased variability of smaller sectors.

To estimate the daily atmospheric mass below 500 hPa in a given sector we use: m(t, δθ, θ) =

Z

cos φ·Z(t, φ, δθ, θ)dφ. (5.1) A more accurate calculation of mass would need knowledge of the surface geopotential; as the variance of surface geopotential is much smaller than that at 500 hPa in the extratropics, this correction is neglected. Other factors such as topography are ignored as they would only affect the mean mass and not its fluctuations.

The numerical integration of the weighted GH profile Z(t, φ, δθ, θ) is achieved using a simple Newton-Coates scheme (see [22] for an example). We then model the daily mass fluctuations as the standard deviation of the estimated daily mass time series m(t, δθ, θ). Figure 5.1 shows the mass fluctuation estimate calculated directly from observations (Figure 5.1a) and from the curve fits (Figure 5.1b). The difference between the two fluctuation estimates is less than one percent, which indicates that

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sectors.

In both cases the standard deviations of the mass proxy show a constant rate of increase for δθ = 360◦ _{to 100}◦_{. For narrower sector widths the standard deviations} grow much more rapidly as δθ is reduced. This is of interest since δθ = 100◦ _{is the} ap-proximate location marking the beginning of more rapid change for the observational curve fit statistics (Figures 3.7b and 3.9a for example).

For context, Figure 5.2 shows the model mass fluctuations as a fraction of the mean mass for a given sector (i.e. σM(δθ,θ)

M(δθ,θ)). Overall these daily fluctuations are in the range of 0.4-1.3% suggesting that they are relatively small, which justifies the approximation of equivalent barotropic variability.

5.2 Modelling Mass Conservation and ρ

P B

5.2.1 An Idealized Case

To further examine how mass fluctuations can influence the correlation coefficients between model parameters, we examine a simplified case. Assume that only the position and baseline components of our model can fluctuate (with time-mean values for amplitude and width). We generate a synthetic set of model parameters, taken to be uncorrelated, using hemispheric values for their variance (Figure 5.3a). We then repeat this process but for each set of model parameters we calculate the mass using Equation 5.1. If its value falls outside a specified range of values (m_{± 3σ}m for 5.3b and m ± σm for 5.3c where m and σm have been estimated from the data) we reject that set and generate another. By this process we find that restricting the mass fluctuations to increasingly smaller ranges, the accepted data points exhibit larger correlations between position and baseline. For comparison, a scatter plot of baseline and amplitude values estimated from the data is given in 5.3d.

While it is illustrative, this method is not suitable for use with our model over a wide range of sectors as there is no obvious choice for the size of the interval (i.e. m_{± 3σ}m vs. m ± σm).

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0 50 100 150 200 250 300 350 20 40 60 80 a)

Mass Fluctuations (in meters)

0 50 100 150 200 250 300 350 20 40 60 80 b)

Mass Fluctuations (in meters)

Figure 5.1: a) Thin lines represent the extended winter mass variability estimate for each individual sector in the data set, as a function of sector width. The thick line represents the sectoral-mean of the thin lines. b) As in a) but for the mass variability estimate calculated from the extended winter curve fits.

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50 100 150 200 250 300 350 2 4 6 8 10 12 14x 10 −3

Standardized Mass Fluctuation (dimensionless)

Figure 5.2: Thin lines represent the standardized model mass fluctuation estimates, as a function of sector width. The thick line represents the sectoral-mean of the thin lines.

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5200 5400 5600 −65 −60 −55 −50 −45 −40 −35 a) Uncorrelated (ρ=0.06) Position (degrees) 5200 5400 5600 −65 −60 −55 −50 −45 −40 −35 b) m ± 3σ (ρ=0.58) 5200 5400 5600 −65 −60 −55 −50 −45 −40 −35 c) m ± σ (ρ=0.94) Baseline (meters) Position (degrees) 5200 5400 5600 −65 −60 −55 −50 −45 −40 −35 d) Actual (ρ=0.91) Baseline (meters)

Figure 5.3: Scatter plots of position (P) vs. baseline (B). a) P and B generated from uncorrelated white noise with variances taken from the curve fits. b) As in a) but (P,B) were selected based on mass values lying within m ± 3σm. c) As in b) but for m_{± σ}m. d) P and B taken directly from the hemispheric curve fits.

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The Constrained Selection Model

Here we introduce an alternative to using a range of accepted mass values, which involves the evaluation of two quantities for each set of model parameters: the mass expressed as a Z-score (ZS) (Eqn. 5.2),

ZSm = m(t, δθ, θ) − m(δθ, θ)

σm(δθ, θ) (5.2)

and a random number (Vt) from a uniform distribution over the interval [0, 1]. If the probability of the given Z-score (assumed to be Gaussian) is greater than the randomly drawn Vt (Eqn. 5.3),

1 √

2πe

−ZSm₂ 2_>Ut _(5.3)

we accept those model parameters and reject them otherwise until 10000 sets have been accepted.

This method does not exclude low probability mass values but rather accepts them at a lower rate thereby dispensing with the need to set a hard threshold as used in the example of Section 5.2.1. As discussed in Section 5.1, a typical one standard deviation fluctuation corresponds roughly to a 0.4-1.3% change in the mass.

Anticipating later needs we also note that this mass fluctuation model can be extended to other or additional constraints. All that is needed is a way to express each additional constraint as a Z-score. A set of model parameters would be accepted if 1 (2π)−n₂_·(det(C))1₂e − −→ ZS·C−1·−ZS†→ 2 > Vt (5.4)

where −→ZS is a vector of Z-scores (containing n different constraints), and C−1 _and det(C) are the inverse and the determinant of their mutual covariance matrix as estimated from the data.

In practice, when using single constraints such as mass fluctuations alone, the variances of the accepted model parameter sets are usually close to their input values. When more than one constraint is used the resulting variances can often be smaller by up to roughly 30%. For this reason we use the model for the correlations only and take parameter variances directly from the curve fits when calculating the EOFs.

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To avoid confusion, we will refer to the constrained selection model using the term ‘CS model’. The constraints used will appear as subscripts (i.e. CSX,Y for example). Full Case Results

In order to extend the modelling in Section 5.2.1 we now allow all of our model parameters to fluctuate and apply the CSm model. Note again that before imposition of the CSmconstraint, fluctuations of model parameters are taken to be uncorrelated. Correlations between these are induced by imposing a specific distribution on the mass fluctuations. The resulting correlation coefficients from the selected sets of model parameters are given in Figure 5.4. Only the values for ρP B show any similarity to those estimated from observations. While ρAB does have the correct trend, its values, ranging from -0.25 (wide) to -0.1 (narrow) are not as large in magnitude as the observed correlations. All other correlation values are close to or not significantly different from zero. This analysis demonstrates that constraining curve profiles to mass variability values that follow those in Figure 5.1, we can reasonably model one of the large magnitude correlation coefficients that plays an important role in determining the shape of the lead EOFs. To model the other correlation coefficients, we must examine other factors.

5.3 Evidence of Angular Momentum Conservation

in Sectoral Geopotential Data

Following the example of the mass fluctuations, we examine the extent to which angular momentum fluctuates in the sectoral data. As with mass conservation we expect that fluctuations will be small in wider sectors and larger in narrower ones. To measure angular momentum per unit mass (referred to symbolically as L) within a given sector we first consider it as a function of latitude. Neglecting the component due to the rotation of the Earth (which does not influence L fluctuations, as was the case with topography for mass conservation) we use:

L_{(t, φ, δθ, θ) = −} gcos φ 2Ω sin φ

∂Z(t, φ, δθ, θ)

∂φ (5.5)

.

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100 200 300 0 0.1 0.2 0.3 0.4 a) ρ AP (no dimension) 100 200 300 −0.1 0 0.1 0.2 0.3 b) ρ AW 100 200 300 −0.4 −0.2 0 0.2 0.4 c) ρ AB (no dimension) 100 200 300 −0.3 −0.2 −0.1 0 0.1 d) ρ PW 100 200 300 0.5 0.6 0.7 0.8 0.9 e) ρ PB (no dimension) δθ (degrees) 100 200 300 −0.2 0 0.2 0.4 0.6 f) ρ WB δθ (degrees)

Figure 5.4: a-f) Thin lines represent correlation between model parameters for the CSm model sectors for each θ in the data set. The thick line represents the mean of the thin lines.

An examination of zonal mean geopotential variability

Contents

List of Tables

List of Figures

Introduction

1.1

Background and Previous Work

1.2

Statement of the Problem and Summary of

Contents

Chapter 2

Data, Methods, and Model

Introduction

2.1

Data Source

2.2

Data Preparation

2.3

Idealised Jet Model

2.4

Curve Fitting

2.5

EOF Calculation

2.6

Sectoral Analysis and Sectoral Mean

Chapter 3

Observational EOFs and Jet

Parameters

3.1

Extended Winter and Summer EOFs

3.2

Extended Winter and Summer Curve Fits

Chapter 4

Model EOFs

4.1

Model EOFs

4.1.1

Similarity (M)

4.1.2

Extended Winter Model EOFs

4.2

Modified Statistical EOFs

4.2.1

The Uncorrelated Case

4.3

Chapter Conclusions

Chapter 5

Correlation Coefficients and

Physical Constraints

5.1

An Examination of Mass Fluctuations in

Sec-toral Geopotential Height Data

5.2

Modelling Mass Conservation and ρ

5.2.1

An Idealized Case

5.3

Evidence of Angular Momentum Conservation

in Sectoral Geopotential Data