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A Weather-type based Cross-Timescale Diagnostic Framework for Coupled

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Circulation Models

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Ángel G. Muñoz ∗

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Atmospheric and Oceanic Sciences (AOS). Princeton University. Princeton, NJ 08540-6649. USA

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NOAA. Geophysical Fluid Dynamics Laboratory. Princeton University. Princeton, NJ

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08540-6649. USA

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International Research Institute for Climate and Society (IRI). The Earth Institute. Columbia

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University. Palisades, NY. 10964-1000

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Xiaosong Yang

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NOAA. Geophysical Fluid Dynamics Laboratory. Princeton University. Princeton, NJ

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08540-6649. USA

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Cooperative Programs for the Advancement of Earth System Science (CPAESS), University

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Corporation for Atmospheric Research (UCAR), Boulder, Colorado, USA.

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Gabriel A. Vecchi

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Department of Geosciences, Princeton University. Princeton, NJ 08540-6649. USA

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Princeton Environmental Institute, Princeton University. Princeton, NJ 08540-6649. USA

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Atmospheric and Oceanic Sciences (AOS). Princeton University. Princeton, NJ 08540-6649. USA

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Andrew W. Robertson

Generated using v4.3.2 of the AMS LATEX template

1

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International Research Institute for Climate and Society (IRI). The Earth Institute. Columbia

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University. Palisades, NY. 10964-1000

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William F. Cooke

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Cooperative Programs for the Advancement of Earth System Science (CPAESS), University

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Corporation for Atmospheric Research (UCAR), Boulder, Colorado, USA.

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NOAA. Geophysical Fluid Dynamics Laboratory. Princeton University. Princeton, NJ

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08540-6649. USA

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∗ Corresponding

author address: Á. G. Muñoz, Atmospheric and Oceanic Sciences (AOS). Prince-

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ton University. Princeton, NJ 08540-6649. USA

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E-mail: [email protected] 2

ABSTRACT 29

This study proposes an integrated diagnostic framework based on atmo-

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spheric circulation regime spatial patterns and frequencies of occurrence

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to facilitate the identification of model systematic errors across multiple

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timescales. To illustrate the approach, three sets of 32-year-long simulations

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are analyzed for northeastern North America and for the March-May season

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using the Geophysical Fluid Dynamics Laboratory’s Low Ocean-Atmosphere

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Resolution (LOAR) and Forecast-oriented Low Ocean Resolution (FLOR)

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coupled models; the main difference between these two models is the hori-

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zontal resolution of the atmospheric model used. Regime-dependent biases

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are explored in the light of different atmospheric horizontal resolutions and

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under different nudging approaches. It is found that both models exhibit a

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fair representation of the observed circulation regime spatial patterns and fre-

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quencies of occurrence, although some biases are present independently of the

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horizontal resolution or the nudging approach, and are associated with a mis-

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representation of troughs centered north of the Great Lakes, and deep coastal

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troughs. Moreover, the intra-seasonal occurrence of certain model regimes

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is delayed with respect to observations. On the other hand, inter-experiment

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differences in the mean frequencies of occurrence of the simulated weather

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types, and their variability across multiple timescales, tend to be negligible.

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This result suggests that low-resolution models could be of potential use to

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diagnose and predict physical variables via their simulated weather type char-

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acteristics.

3

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1. Introduction

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Global Circulation Models (GCMs) are essential tools for both scientific research (e.g., Delworth

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et al. (2012); Taylor et al. (2012)) and the development of climate services (Vaughan and Dessai

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2014) at different timescales. Since, by definition1 , models are not a perfect representation of

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reality, different types of errors must be accounted for before GCMs can be used as a fair depiction

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of the real world.

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Common approaches to diagnose systematic errors involve the computation of metrics aimed

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at providing an overall summary of the performance of the model in reproducing the particular

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variables of interest in the study, normally tied to specific spatial and temporal scales. For exam-

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ple, Gleckler et al. (2008) used different metrics to assess the climatological behavior of GCMs,

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Covey et al. (2016) focused on model evaluation of the rainfall diurnal cycle, and several studies

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have analyzed the present-time rainfall characteristics in state-of-the-art climate change simula-

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tions (see, for example Reichler and Kim (2008); Delworth et al. (2012); Ryu and Hayhoe (2014);

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van der Wiel et al. (2016)). Moreover, the number of open source packages to evaluate the fidelity

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of GCMs compared to observations (e.g., Phillips et al. (2014); Mason and Tippet (2016); Gleck-

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ler et al. (2016)) is rapidly increasing. Nonetheless, the evaluation of the goodness of a model

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is not always tied to the understanding of the physical processes that are correctly represented,

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distorted or even absent in the model world. As the physical mechanisms are more often than

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not related to interactions taking place at multiple time and spatial scales (e.g., Nakamura et al.

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(2012); Robertson et al. (2015); Muñoz et al. (2015, 2016)), cross-scale model diagnostic tools are

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not only desirable but required.

1 The

word “model” is derived from Latin “modulus”, a “measure” or “standard” of something, which evolved to today’s idea of “likeness made

to scale”, for example in clay, wax or via mathematical or numerical expressions.

4

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A complementary alternative to the common diagnostic approach mentioned above can be

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achieved via a non-linear dynamical system perspective (Lorenz 1963; Palmer 1999). Thus, typ-

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ical questions like “How well does this model represent the observed seasonal rainfall patterns?”

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are framed as part of the more general question “Can the model adequately represent the available

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states of the system?” (in a suitably coarse-grained phase space; see Ghil and Robertson (2002)).

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Since only certain physical states can be accessed by the real-world system under study, as in sta-

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tistical or quantum mechanics, the rationale is that models that cannot faithfully reproduce those

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states should not be expected to provide, for example, the right rainfall or temperature patterns for

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the correct reasons.

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How to identify the available states? Theoretically, they are related to the concept of multiple

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flow equilibria (Lorenz 1969; Charney and DeVore 1979; Reinhold and Pierrehumbert 1982) and

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quasi-stationary regimes or metastable fixed points capable of attracting the chaotic trajectory of

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the system; they can be recognized in a state (or phase) space as regions that are visited more

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frequently – or, equivalently, regions surrounding a local density maximum. The identification

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of these clusters in the state space poses a non-trivial statistical problem (Stephenson et al. 2004;

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Christensen et al. 2015), and in general they correspond to proxies of the available states of the

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system, and not the states themselves. From a practical perspective, in the atmosphere those clus-

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ters are typically associated with recurrent daily circulation types or “weather types” (WTs), which

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have been widely studied in the literature (Lorenz 1969; Charney and DeVore 1979; Reinhold and

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Pierrehumbert 1982; Lorenz 2006; Vautard 1990; Michelangeli et al. 1995; Robertson and Ghil

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1999; Moron et al. 2008; Johnson and Feldstein 2010; Riddle et al. 2013), as they can lead to both

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important positive and negative socio-economic impacts.

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Formally speaking, weather types are statistical constructs associated with recurrent circulation

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configurations which can be used to study weather regimes, a more physical concept that often 5

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requires additional conditions (e.g., the average time spent on each regime must be long compared

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to other oscillations in the system, as in Lorenz (2006)). Having stated that formal difference,

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for the purposes of this work both concepts are considered largely interchangeable. If correctly

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defined, these weather types can be understood as “building blocks” or some sort of “alphabet”

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that can be used to describe all the physically acceptable events of the system (Muñoz et al. 2015,

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2016). This is shown schematically in Figure 1; the events, e.g., rainy days, can be explained by the

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occurrence of individual weather types (letters) or particular WT sequences/transitions (words).

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The underlying hypothesis used here is that climate variability across timescales can be de-

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scribed in terms of the frequency of occurrence of weather types at the different scales, with

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external (or internal) forcings taking the form of shifts in the residence time of the system in the

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different basins of attractions of the state space. This idea is related to the Fluctuation-Dissipation

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Theorem in climate (Leith 1975), that relates the mean response to small perturbations of a non-

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linear dynamical system to fluctuations in the unforced system. As indicated by Leith (1975), the

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nonlinear transfer of energy and enstrophy in the climate system implies sources of these variables

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in some scales of motion and dissipation in others; hence, considering the present cross-timescale

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diagnostics approach can help to understand better the problem.

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Beyond the theoretical interest of associating weather types with physically acceptable states

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of the system, the present approach is useful in those cases in which it is possible to identify

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circulation-dependent systematic errors in climate models; a weather-type rectification (correc-

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tion) has the potential to improve the related physical fields and events in the simulations.

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A regime approach has been explored to study general biases and uncertanty in climate models

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at specific timescales (e.g., Palmer and Weisheimer (2011); Perez et al. (2014); Christensen et al.

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(2015)). Since it is possible to analyze the behaviour of the circulation regimes over a wide range

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of different timescales –for example, by aggregating their frequency of occurrence at sub-seasonal, 6

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seasonal, inter-annual and longer timescales–, and since they can be associated with physical pro-

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cesses occurring across timescales (Moron et al. 2015; Muñoz et al. 2015, 2016), it is proposed

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here that the weather-typing approach provides a natural and unified framework for cross-timescale

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diagnostics of GCMs. Although the scheme is deemed especially useful when considering seam-

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less prediction systems (Hoskins 2013), it can be also employed to analyze causes of bias at any

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particular timescale. Furthermore, the same set of weather types can be used to diagnose a wide

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range of variables in a physically consistent way, thus shedding light on the causes of biases rather

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than focusing on the bias itself.

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The goal of the present study is to illustrate the proposed approach using simulations produced

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by two coupled GCMs developed by the Geophysical Fluid Dynamics Laboratory (GFDL), with

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physical configurations designed to reproduce key aspects of the observed climate variability. For

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the sake of the illustration, the study is only focused on rainfall during the March-April-May

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(MAM) season for Northeastern North America (NENA). The MAM atmospheric circulation

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regimes in this region are relatively well understood, especially in relation to flood events in the

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Ohio and Mississipi river basins (Nakamura et al. 2012; Robertson et al. 2015).

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The rest of the paper is organized as follows. After introducing the datasets and sumarizing the

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methods in the next section, observed and modeled rainfall climatologies are analyzed in section

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4. The weather-type diagnostics are presented in section 5 for daily, sub-seasonal, inter-annual and

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“decadal” timescales, while section 6 deals with possible sources of bias and an overall discussion

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of the results. Finally, the concluding remarks are presented in section 7.

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2. Data

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Both observations and model outputs were used in the present study. These are described in the following subsections. The time period considered in all datasets is MAM 1981-2012. 7

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a. Observations

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Observed rainfall fields were obtained from the gauge-based NOAA-NCEP-CPC Unified Pre-

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cipitation gridded dataset (Chen et al. 2008). This product has daily temporal resolution and a

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spatial resolution of 1◦ x 1◦ .

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The analysis of the observed daily circulation over NENA was derived from the NCEP-NCAR

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Reanalysis Project (version 2, or NNRPv2) 500 hPa geopotential data on a 2.5◦ x 2.5◦ grid (Kalnay

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et al. 1996; Kistler et al. 2001), and also from the Modern-Era Retrospective Analysis for Research

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and Applications (MERRA) for the same variable on a 0.5◦ x 0.5◦ grid (Rienecker et al. 2011).

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These two datasets were used to investigate the impact of horizontal resolution and reanalysis

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methods in the identification of the weather types.

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b. Model data and Experimental design

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This study explores the impact of (a) horizontal resolution and (b) Newtonian relaxation towards

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observed fields on model biases associated with the variability of circulation regimes at multiple

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timescales.

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The three sets of climate simulations used are 32-year long, with 5 members each, produced by

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two kindred coupled GCMs developed by the Geophysical Fluid Dynamics Laboratory (GFDL).

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They share the same ocean, land and ice model components inherited from the GFDL Coupled

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Models version 2.1, CM2.1 (Delworth et al. 2006), and version 2.5, CM2.5 (Delworth et al. 2012).

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The models used in this study are the Low Ocean-Atmosphere Resolution, or LOAR (van der

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Wiel et al. 2016), and the Forecast-oriented Low Ocean Resolution, or FLOR (Vecchi et al. 2014).

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Unlike CM2.1 and CM2.5, LOAR and FLOR use essentially the same atmospheric model compo-

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nent, based on a finite volume dynamical core on a cubed sphere (Putman and Lin 2007), integrat-

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ing along 32 vertical levels and with horizontal resolutions of 2◦ x 2◦ (C48) for LOAR and 0.5◦ 8

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x 0.5◦ (C180) for FLOR. The dynamical time step is modified to match the individual model’s

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atmospheric resolution, and the atmospheric physics for both models is similar to that in CM2.5

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(Delworth et al. 2012; Vecchi et al. 2014). The ocean model component is the Modular Ocean

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Model (MOM) version 5 (Griffies 2012), at 1◦ x 1◦ and configured as reported in Vecchi et al.

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(2014). The land surface model is the Land Model version 3 (LM3, (Milly et al. 2014)), with the

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same horizontal resolution as the atmospheric model component. The sea ice model component is

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the Sea Ice Simulator (SIS), having three vertical layers, one snow and two ice, and five thickness

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categories, as reported in Delworth et al. (2006) and references therein. The time resolution for all

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model output is daily.

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A typical way to explore how well uncoupled GCMs reproduce the observed natural variability

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is to force them with observed SSTs (e.g., à la Atmospheric Model Intercomparison Project, AMIP

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(Gates et al. 1999)). Although this approach is, by definition, not possible for coupled GCMs, a

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common alternative is to use a Newtonian nudging to relax certain model fields, such as SSTs,

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towards observations (e.g., Rosati et al. (1997)), like SSTs. This method has been shown useful

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to reproduce key aspects of the natural variability of the climate system and to decrease model

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biases (e.g., Luo et al. 2008 and references therein). Thus, all 3 experiments used in the study

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were nudged to observed fields as follows.

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In two sets of numerical experiments, LOARsst and FLORsst , the models’ SSTs were relaxed

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to daily-interpolated observed HadISST product (Rayner et al. 2003), SSTO , using a restoring

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timescale τ = 5 days, such that

d + 1 (SSTO − SST ) ∂t SST = SST τ 9

(1)

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b the coupled model tendency where ∂t  represents the partial time-derivative of the field, and 

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of the field. These two experiments were used to analyze the impact of horizontal resolution on

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model biases.

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Since nudging only SSTs is often inadequate (Fujii et al. 2009), as the variability in the climate

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system is not only controlled by sea-surface temperatures, for the third set of numerical experi-

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ments, called FLORsst+strat , MERRA’s 6-hourly stratospheric temperatures and winds above 100

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hPa were nudged in addition to the SST-nudging described above, using the same ansatz presented

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in equation (1), but with a relaxation time τ = 6 hours, and a tapering factor γ such that it is one

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for model layers above 50 hPa (i.e., 1, 4, 8, 14, 21, 30 and 41 hPa), and it is zero for layers below

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100 hPa; from 50 hPa to 100 hPa, γ varies linearly from one to zero, thus transitioning in a smooth

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way from the stratosphere to the troposphere. This same stratospheric nudging approach has been

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used recently by Jia et al. (2017) to show the role of the extra-tropical stratosphere as an important

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source of predictability at interannual scale. Both FLORsst and FLORsst+strat were used in this

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study to explore the role of nudging in model biases.

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3. Methods

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Here the general methodology is described. The diagnostic approach per se is described in

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section 5.

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a. Cluster analysis

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In order to identify the proxies of the available states of the system, daily circulation regimes

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were determined performing a k-means analysis (e.g. Robertson and Ghil (1999)) on observed

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and modeled 500 hPa geopotential height anomalies, each at its own spatial resolution to avoid the

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possibility of spurious results induced by the spatial interpolation process. 10

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The k-means technique is a partitioning method that classifies all days into a predefined number

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of clusters, minimizing the intracluster sum of variance, W , for a partition P (the Voronoi diagram

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generated by the classification process):

k

W (P) =

∑ ∑ ds2(X, p¯i)

(2)

i X∈pi

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where i = 1...k, k is the total number of clusters in the solution, pi denotes the different clusters in

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the partition and ds2 (X, p¯i ) represents the Euclidean line element (distance) between the cluster’s

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maps X and centroid p¯i . In this work, no areal weighting was necessary, as the selected domain is

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relatively small and it does not extend into high latitudes (30◦ N-50◦ N, more details below).

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This procedure assigns only one cluster to each day on record. Daily geopotential data was first

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projected onto its six leading empirical orthogonal functions (same number for all datasets), ac-

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counting for at least 95% of the total observed variance. No additional time filtering was applied

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to the data before clustering, thus retaining the annual seasonal cycle, inter-annual, sub-seasonal

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and synoptic weather time scales for diagnosis. No projection between the modeled and observed

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leading empirical orthogonal functions was performed, i.e., the modeled WTs were directly ob-

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tained using exactly the same procedure as for the observed ones. As with the avoidance of any

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kind of spatial interpolation before the k-means analysis, this was done in order to evaluate each

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model’s weather types in an unbiased way.

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Two approaches were tested to define the 500 hPa geopotential anomalies (departure from the

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long-term mean) that characterizes each weather type. In one, the anomalies of all members were

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concatenated along the time coordinate before the k-means process was performed, as in Muñoz

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et al. (2016). In the second approach, the classification of the 500 hPa geopotential anomalies

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was performed for each member independently, then computing the ensemble mean of the same 11

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weather type across members. The latter method was used in the present study as it respects the

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chronology and transition probabilities of the weather types in the presence of non-stationary data.

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The classifiability index (Michelangeli et al. 1995), CI, measures the similarity of partitions of

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the data, and permits the identification of the minimum number of clusters required to achieve a

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good classification. Classification performance is measured via the pattern (or anomaly) correla-

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tion coefficient ρi j between pairs of cluster centroids p¯i and q¯ j in partitions P and Q, respectively,

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which allows the matrix ρ(P(k), Q(k)) to be computed, with elements given by M

∑ p0imq0 jm m

ρ( p¯i , q¯ j ) ≡ s

M

∀i, j = 1...k

M

(3)

∑ p02im ∑q02jm m

m

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where the sums consider all the M grid points in the domain, and the grid point deviations are

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computed as usual p0im

1 = p¯im − k

k

∑ p¯rm

(4)

r

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q0jm

1 = q¯ jm − k

k

∑q¯rm

(5)

r

239

where m = 1...M. The maximal value of the i−th row of the k × k matrix ρ (3) represents the

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cluster in Q that best fits the i−th cluster in P. If the two partitions are identical, of course,

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ρ = 1.

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In this study, a total of N = 100 partitions are used, obtained from 100 different initial random

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seeds of the algorithm; the single partition that matches most closely the remaining ones is then

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selected. This similarity is computed by averaging the values of ρ(Pn , Pn0 ) ∀n, n0 = 1...N, and,

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following Michelangeli et al. (1995), CI(k) ≡

1 N(N − 1)

∑

1≤n6=n0 ≤N

12

ρ(P(k)n , P(k)n0 ))

(6)

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Clustering solutions within the range of k = 2 − 10 were explored for the NENA geographical

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domain, bounded by 30◦ N-50◦ N and 105◦W -69◦W . A k-means 7-cluster solution was found to

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yield a statistically significant value of the classifiability index (p < 0.1), compared to a red-noise

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background modeled as a first-order Markov process having the same covariance at lag 0 and 1

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days as the atmospheric data (Michelangeli et al. 1995); therefore the solution, which is the same

251

one reported by Robertson et al. (2015), was selected for further analysis.

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This set of clusters, or weather types, can be interpreted as a set of geopotential regimes that

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typify the daily variability, and are considered to be the “building blocks” or “letters” mentioned

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at the Introduction. Other cluster solutions were explored, verifying that the general features of

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the circulation regimes presented below are robust and sufficient for the purposes of this study.

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b. Similarity metrics

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There are multiple metrics to evaluate how well a model reproduces observations (e.g., Mason

258

and Stephenson (2008); Jolliffe et al. (2012)); selection among them depend on the attribute or

259

type of question to be addressed. For the sake of simplicity, this work focuses on the similarity

260

of spatial patterns and temporal behavior between the simulated and observed weather types. Two

261

metrics are chosen for that purpose: pattern correlation and the scatter index. As indicated before,

262

the k-means analysis is performed on the native grid of the models and the observations; however,

263

when comparing results, all calculations were always carried out in a common low resolution grid

264

of 2◦ x 2◦ (the one of the NNRPv2 dataset).

265

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The pattern correlation ρ is computed following equation (3), but considering the fact that the partitions now correspond to those modeled and observed.

267

The scatter index ξ is defined here as the root mean square error operator of the simulated and

268

observed frequencies of occurrence of the weather types, normalized by the weather type mean 13

269

frequency (e.g., Perez et al. (2014)). The frequency of occurrence is defined as the proportion of

270

time that each weather type occurs at each timescale window of interest, averaged across ensemble

271

members for the case of model simulations (see an example below). Two versions of the scatter

272

273

index are used in this work, one to evaluate the mean frequency state, ξ¯ |ts , and another for the temporal variability at different timescale windows ts , ξσ |ts :

s

k

k ∑ fiS − fiO

ξ¯ ≡

i

k

ts

∑ fiO i

s ξσ ≡


2



ts

k
2 k ∑ σ ( fiS ) − σ ( fiO ) i

ts

k

∑σ ( fiO) i

(7)



(8)

ts

274

where fiS and fiO denote the simulated and observed frequencies of the i = 1..k weather types, and

275

σ denotes the corresponding sample standard deviation. The different timescales (or timescale

276

windws) are presented in Section 5. A model with a perfect represetation of the observed frequency

277

of occurrence satisfies ξ¯ |ts = 0, and in consequence its representation of the variability in the

278

frequency of occurrence is also perfect (ξσ |ts = 0). In general that is not the case, and each index

279

provides information on these two statistics.

280

To evaluate the scatter index at a particular timescale implies to perform the corresponding

281

calculations on the time window defined for that purpose. To illustrate this idea, consider as an

282

example the overall evaluation of a model representation of the observed weather types frequencies

283

during the window defined as May 16-31, for all seasons in the period 1981-2012. The scatter

284

indices are then computed considering the observed and simulated frequencies of occurrence of 14

285

weather types for those thirty one 16-day-long windows, using equations (7) and (8), where the

286

vertical bars indicate that the indices are to be evaluated for the particular timescale (window) ts .

287

Once the scatter indices have been computed for the different timescales of interest, they can be

288

presented using boxplots to also convey information about their uncertainties in the models; for

289

example, if all members in a model perfectly agree of the value of a scatter index for a particular

290

timescale, the corresponding box in the boxplot will collapse to an horizontal line. More details

291

about these boxplots are discussed in section 5.

292

To analyze persistence of the weather types, the empirical probability density functions were

293

computed for each dataset and regime. In order to compare between the different numerical exper-

294

iments and the reanalysis, the Euclidean distance between each pair of the corresponding Weibull

295

distribution parameters, α (scale) and β (shape), was used to assess similarity. The Weibull distri-

296

bution f (x|α, β ) =

 x β  β  x β −1 exp − α α α

(9)

297

is commonly used in survival or duration analysis, and provided better fit than other less flexible

298

distributions, like the Exponential one (which corresponds to the special case of β = 1).

299

4. Northeastern North America’s rainfall climatology

300

It is a common practice to evaluate the fidelity of a model compared to observations through

301

the analysis of the climatological behaviour of the variable of interest. This section uses the

302

MAM rainfall climatology of the region under study (30◦ N-50◦ N and 105◦W -69◦W , see Fig. 2)

303

to illustrate some key ideas of the present diagnostic approach.

304

The observed rainfall climatology (Fig. 2a) exhibits a clear precipitation gradient with higher

305

amounts (≈ 3 − 4 mm day−1 ) in a wide region covering approximately 30◦ N-40◦ N and 97◦W -

306

84◦W (most parts of the United States’ Alabama, Mississippi, Louisiana, Arkansas, Tennessee, 15

307

southern Missouri and southwestern Kentucky), with basically no precipitation along the western

308

border of NENA (Texas, Oklahoma, Kansas, Nebraska, the Dakotas, and Canada’s southern On-

309

tario), 0.5 − 1 mm day−1 to the north (southern Ontario and Québec) and 2 − 3.5 mm day−1 along

310

the Eastern Coast (from northern Florida to Maine).

311

A visual inspection of the models’ climatology (Fig. 2b-Fig. 2d) clearly shows some important

312

biases in both magnitudes and spatial distribution of rainfall, with simulations at higher resolution

313

(Fig. 2c - Fig. 2d) exhibiting some improvement with respect to the low-resolution one (Fig. 2b).

314

What are the causes of such biases? Typically, issues involving model resolution or physical pa-

315

rameterizations (specially for rainfall) are frequently identified as the sources of bias –and indeed

316

higher horizontal and vertical resolution, as well as better physics, tend to improve the representa-

317

tion of the observed fields. Nonetheless, it is also possible that the resolution and parameterizations

318

are fit for purpose, and something else is failing. Whatever the causes, a complementary approach

319

consisting of the diagnosis of the physical mechanisms conducive to precipitation –rather than how

320

well the precipitation itself is represented in the model– is also useful and most often required.

321

Rainfall variability in NENA is dominated by synoptic-scale circulation patterns (Archambault

322

et al. 2008; Nakamura et al. 2012; Robertson et al. 2015; Roller et al. 2016) that are, in turn,

323

modulated by well known climate modes at multiple timescales like El Niño-Southern Oscillation

324

(ENSO), the North Atlantic Oscillation (NAO), the Pacific-North American pattern (PNA), or the

325

Madden-Julian Oscillation, (MJO; e.g., Ropelewski and Halpert 1986, 1987; Barnston and Livezey

326

1987; Wheeler and Hendon 2004).

327

Although some studies have found an inconclusive link between ENSO and NENA’s climate

328

(e.g., Ropelewski and Halpert (1986)) others indicate that this mode can modulate the frequency

329

of low-pressure systems and storms in the region (Trenberth and Caron 2000; Frankoski and De-

330

Gaetano 2011). Similarly, pressure and geopotential height anomalies associated with the NAO 16

331

modify the frequency of occurrence of nor’easters, as well as storm track locations over the North

332

Atlantic basin via changes to the position and orientation of the North Atlantic jet stream (Jones

333

and Davis 1995; Archambault et al. 2008). There is also evidence of links between PNA and

334

NENA’s rainfall anomalies (Leathers et al. 1991; Notaro et al. 2006), and between MJO’s phases

335

5-7 and precipitation rate in the region (Becker et al. 2011; Zhou et al. 2012).

336

The effect of these and other climate modes conducive to rainfall in NENA can be studied

337

through their influence in the occurrence, persistence and evolution of daily weather types. For

338

example, Roller et al. (2016) analyzed NENA’s wintertime mean and extreme precipitation, storm

339

tracks and teleconnections using a set of five WTs defined in terms of 850 hPa winds; similar

340

studies have been conducted in other parts of the world for different seasons (Moron et al. 2008,

341

2011, 2015; Muñoz et al. 2015, 2016).

342

The rainfall climatology (Fig. 2) discussed in this section can thus be understood in terms of the

343

(non-)linear contribution of the different mechanisms represented by persistence and transitions

344

of the region’s daily circulation regimes (WTs, or atmospheric states). A misrepresentation of the

345

physical interactions involved, even with perfect rainfall parameterizations, can generate bias not

346

only in the climatological precipitation field, but in higher-order statistics of the rainfall variability

347

at different timescales. The same can be said about other variables physically associated with the

348

behavior of those WTs.

349

Hence, the next section focuses on diagnosing how well the models represent the observed

350

behavior of the WTs for different timescales.

351

5. Cross-timescale diagnostics

352

This section first discusses the circulation regimes obtained for the observations and experi-

353

ments, before turning its attention to different tools to diagnose the weather types evolution at 17

354

daily to decadal scales. Although the method is general and can be used for longer timescales, this

355

study is constrained by the availability of longer observed rainfall records.

356

The computed set of seven NENA WTs found to best typify the daily circulation regimes for

357

MAM is presented in Fig 3 plotted over the entire hemisphere to better identify wave-like patterns,

358

and in Fig 4 for a more regional analysis of the weather types. The regimes with the highest ob-

359

served frequency of occurrence are located to the left and the less frequently occurring to the right

360

(see top row in those figures; the model regimes were ordered to follow each observed pattern).

361

Equation (3) was used to identify the “model-equivalent WT”, and physical interpretation of the

362

hemispheric patterns (Fig 3) was used in those cases in which the pattern correlation coefficients

363

provided too similar values for two given regimes.

364

Two different reanalysis products were used to analyze the robustness of the method identify-

365

ing the observed WTs, and the dependence on horizontal resolution (NNRPv2 at ∼ 2.5◦ versus

366

MERRA at ∼ 0.5◦ ). Table 1 and Table 2 show a comparison of the anomaly correlation coeffi-

367

cient between WTs using NNRPv2 and MERRA as reference, respectively. The results are indeed

368

consistent across reanalyses and resolutions.

369

The observed weather types (top two rows of Fig 3 and Fig 4) exhibit synoptic-scale merid-

370

ionally elongated dipolar wave patterns (WT1, WT3, WT5, WT6), and monopole/dipole patterns

371

zonally elongated (WT2, WT4, WT7). Weather regimes that are predominantly associated with

372

positive geopotential anomalies (WT1 and WT2) tend to be more frequent –and more persistent,

373

as it will be shown later– than the more transient regimes (WT4-WT7), which show spatial struc-

374

tures with dominant negative geopotential anomalies (see Fig 3 and Fig 4). Although this general

375

separation between more persistent and more transient circulation regimes is present in the simu-

376

lations, the ordering in those cases –which reflects their mean frequencies– is not exactly the same

377

as in the observations. 18

378

As it will be shown in sub-section 5b, there are preferred sequences of states whose pattern evo-

379

lution and typical timescales suggest eastward propagation of baroclinic waves (e.g., state transi-

380

tions 3 → 1 → 3, top rows in Fig 3). Generally speaking, in NENA the MAM daily circulation

381

regimes tend to be associated with continental ridges west of the Great Lakes (WT1, top rows

382

in Fig 4), continental ridges over the Great Lakes (WT2), northeastern seaboard ridges (WT3),

383

troughs centered north of the Great Lakes (WT4), deep coastal land troughs (WT5), southeastern

384

seaboard ridges (WT6), and deep northeastern troughs (WT7). The seaboard ridges (WT3 and

385

WT6) have been shown to be associated with extreme rainfall events in the Ohio river basin; these

386

two circulation types tend to occur more often during La Niña years, and phase 5 of the MJO also

387

modulates their seasonal frequency, with lower occurrence for WT3 and higher presence for WT6

388

(for additional details see Robertson et al. (2015)).

389

There is no simple answer to the question of what set of experiments best represents the observed

390

weather types, as the answer depends on the basis for the comparison. The shapes, magnitudes,

391

locations, tilt and frequencies of the weather types are characteristics to consider. On average, both

392

LOAR and FLOR models do a good job reproducing the observed circulation regimes, although

393

some important biases are present (Fig 3 and Fig 4, Table 1 and Table 2). For example, the spatial

394

features of WT1, the most frequently observed, are well represented in all experiments, but the

395

ensemble-mean frequency of occurrence is about 30% too low in FLORsst . The observed dipo-

396

lar configuration in WT3 is basically absent in the FLORsst+strat ensemble mean for that regime,

397

and for WT4 all experiments exhibit southwardly shifted and widely enhanced negative geopo-

398

tential anomalies with respect to the observations. WT5 appears severely tilted (about +45◦ ) in

399

the simulations, arguably due to a misrepresentation and location of the weak positive geopo-

400

tential anomaly observed north of 50◦ N near and over the Great Lakes (Fig 4). FLOR tends to

401

show positive geopotential anomalies along the eastern coast of the US in WT6, especially in the 19

402

FLORsst+strat experiment, which also exhibits important frequency biases for that regime. Finally,

403

although the simulated WT7 spatial patterns are very well represented (see high correlations in

404

Tables 1 and 2), all experiments overestimate its frequency of occurrence.

405

Since there are no striking differences in the weather type representations in terms of the resolu-

406

tion of the reanalyses, the NNRPv2 product is selected as the reference in all following discussions

407

about circulation regimes characteristics, unless otherwise indicated.

408

The persistence of these circulation patterns and their transitions control, both in the real world

409

and in the simulations, aspects like the moisture that is advected into NENA, and thus this weather-

410

within-climate approach is normally used to better understand the physical mechanisms behind

411

the ocurrence or not of (extremely) rainy days in a region (Robertson and Ghil 1999; Moron et al.

412

2008, 2013, 2015; Muñoz et al. 2015, 2016). As discussed in Section 4, the systematic errors in

413

the simulated weather regimes can help to explain the biases in other variables, as for example in

414

the rainfall field and its climatological behavior. To illustrate this idea, Figure 5 shows the average

415

daily rainfall regimes (RR), defined by compositing rainfall values associated with each weather

416

type.

417

As expected, FLOR tends to simulate better the rainfall field than LOAR at a local level; how-

418

ever, at a regional scale, there are not statistically significant differences (p < 0.05, two-sample

419

t-Student test) in the spatial correlation coefficients between the three experiments. Table 3 shows

420

the anomaly correlation coefficients between the rainfall regimes in the models and observations.

421

Overall, the most significant biases appear in the daily rainfall regimes associated with the weather

422

types showing greater bias over NENA (Fig. 5), i.e., WT4 and WT5. Errors in magnitude and spa-

423

tial distribution of the rainfall anomalies are related to misrepresented atmospheric circulations

424

and moisture fluxes in WT4 in all experiments (not shown), exhibiting positive rainfall anomalies

425

along the coast as a result of the deformed and displaced trough discussed above (Fig. 4). In spite 20

426

of the tilt that WT5 exhibits in the simulations, the associated daily rainfall patterns for NENA are

427

not too biased, except along the northeastern coast where the anomalies should have opposite sign;

428

this is attributed in part to the fact that the circulation and moisture fluxes over NENA are similar

429

to the observed one, even when that is not the case at a larger scale. Likewise, although WT3 is

430

the circulation regime with the worst large-scale anomaly correlation coefficients (e.g., Table 1),

431

its representation of the associated physical mechanisms controling rainfall over NENA provides

432

a very good depiction of the observed rainfall regime RR3. Having this regime well represented

433

in the simulations is important to adequately represent extreme rainfall events over the Ohio river

434

basin (Robertson et al. 2015).

435

Biases both in the spatial patterns (involving shape, magnitudes, tilt, location) and the frequency

436

of occurrence of the simulated weather types contribute to biases in rainfall and other variables at

437

different timescales. In studies involving a large number of years, e.g., 100 years, two different k-

438

means solutions should be computed to analyze, for example, whether the climate change signal is

439

modifying the weather types’ spatial patterns or the dimensionality of the phase space. For studies

440

like the present one, however, which involves just 3 decades, the spatial patterns are normally

441

assumed to be constant. This allows for the analysis of the weather types’ temporal variability at

442

different timescales in terms of changes in their frequency of occurrence at those timescales.

443

A possible way to summarize how well the simulations reproduce both the mean frequency of

444

occurrence and its variability across multiple timescales is through the use of the corresponding

445

scatter indices (see equations (7) and (8)), presented in Fig. 6 for two particular sub-seasonal win-

446

dows (March 20th-30th and May 4th-14th; further discussed in section 5c), for the inter-annual

447

variability considering all MAM seasons in the 1981-2012 period (section 5d), and for the first and

448

last decades in the same period of years (section 5e). As with the case of the spatial patterns (e.g.,

449

Table 1), the analysis of the frequencies of occurrence indicates that no particular model or exper21

450

iment can be – overall – considered the best one across all the timescales in study. Certainly, there

451

are differences in terms of the median and dispersion values at particular scales, with higher errors

452

and uncertainties in the sub-seasonal case; nonetheless, within-scale differences in the medians

453

are normally negligible. Furthermore, compared to LOARsst , FLORsst tends to have the same or a

454

lower dispersion for both scatter indices (Fig. 6).

455

The following sub-sections provide more details on daily transition statistics using Klee dia-

456

grams (sub-sections 5a and 5b), and further analysis pertinent to the temporal variability of the

457

simulated circulation regimes and to understanding Fig. 6 (sub-sections 5c,d,e).

458

a. Klee diagrams

459

Klee diagrams, shown in Fig. 7, are a way to represent the temporal evolution of the available

460

states of the system (Muñoz et al. 2015). These diagrams consist of a simple matrix plot sketching

461

the daily evolution of weather types for the entire period under study. They are equivalent to

462

the representation of the Viterbi state-sequences, except for the fact that no Viterbi algorithm or

463

Hidden Markov Model (e.g., Robertson et al. (2006)) is involved in the process. A Klee diagram

464

is the basis to analyze daily transitions and temporal variability at sub-seasonal, seasonal, decadal

465

and longer timescales. Moreover, it has been used to define subseasonal-to-seasonal states for

466

forecast purposes, using hybrid dynamical-statistical models (Muñoz et al. 2016).

467

If a simulation has exactly the same Klee diagram as the one computed from the observations,

468

then it has a perfect representation of the observed weather types’ evolution across timescales,

469

independently of how well the model represents their spatial patterns; of course, even if the mod-

470

els were perfect, the Klee diagrams would not be exactly the same due to atmospheric chaos.

471

Nonetheless, the idea is conceptually useful, and helps to define several ways to diagnose similar-

472

ity in the temporal evolution and associated statistics. Moreover, having simulated Klee diagrams 22

473

that exhibit similar characteristics to observations is also of practical use because it is then pos-

474

sible to use a combination of modeled frequencies of occurrence and observed spatial patterns to

475

represent the evolution of the circulation regimes at a given timescale. In such cases, this approach

476

has important implications for prediction, an idea that is briefly discussed in Section 6, and that

477

will be explored in more detail elsewhere.

478

Overall, the observed and simulated Klee diagrams for NENA show high similarity (Fig. 7).

479

Since it is not possible to have an ensemble mean of Klee diagrams because they represent cat-

480

egories and not real numbers, the analysis must be performed on a member-per-member basis.

481

Visual inspection suggests that all experiments exhibit a dominance of WT1 and WT2 toward the

482

end of the season, and of WT6 and WT7 during the first half, all consistent with the observations

483

and consistent with the predominance of negative height anomalies in March and positive ones

484

in May within the MAM season (Fig. 4). However, some biases are apparent, like the presence

485

in LOARsst and FLORsst+strat of too many days with WT3 at the end of the season, which, for

486

example, could provide false alarms in those cases in which WT3 is related to floods in the Ohio

487

river basin.

488

Different statistics can easily be computed from the Klee diagrams to help diagnose and compare

489

dynamical models. Some examples – like daily transition probabilities, mean durations (persis-

490

tence) for each weather type, and other summaries to analyze the evolution of the circulation

491

regimes at different timescales – are discussed in the following sub-sections.

492

b. Daily transitions and typical durations

493

Daily transition matrices are commonly used to characterize persistence and preferred state tran-

494

sitions. Their diagonal sketches the persistence probabilities for each weather type, and the off-

495

diagonal elements represent the conditional probabilities of transition to a particular regime (along 23

496

the horizontal axes of the matrix, “posterior WT”) given that a different weather type (along the

497

vertical axes, “prior WT”) occurred on the previous day.

498

Daily transitions in NENA, presented in Fig. 8, are dominated by persistence, with continental

499

ridges over the Great Lakes (WT2) being the most probable regime to persist, especially during

500

May (Fig 7). The most frequently observed statistically significant transitions (p < 0.1, see Vau-

501

tard et al. (1990)) suggest circuits like 1 → (2, 3) → 1 or 4 → 3 → 1, that can be associated with

502

the propagation of baroclinic waves over NENA, but also transitions leading to highly persistent

503

states, like 6 → 7 → 4, associated with anomalously high moisture fluxes from the Gulf of Mex-

504

ico and the Caribbean conducive to rainy days along the eastern coast, followed by days with a

505

high-pressure system stationed in the northern part of the region, typically conducive to low-level

506

divergence and no rain (see Fig 7, Fig 4 and Fig 5).

507

All simulations for NENA adequately represent the fact that the persistence probabilities are

508

significantly higher than non-self transitions. As expected, some biases exist; for example, the

509

persistence of WT4 in the LOARsst and FLORsst experiments (see Fig 8) is overestimated, and

510

there are consistent biases in all experiments for transitions like 3 → 4, 4 → 3, 4 → 1, 7 → 5 and

511

4 → 7. But in general, all models tend to capture most of the observed transition probabilities, with

512

differences always less than ±15%; moreover, no statistically significant differences (p < 0.05,

513

two-sample t-Student test) exist between the average transition probabilities of the experiments.

514

It is also important to evaluate if the simulations fairly represent the typical persistence of the

515

weather types. Figure 9 shows relative frequency histograms and their corresponding Weibull

516

distribution fit for the most common durations; for comparison, Table 4 presents the values of the

517

Weibull distribution parameters α and β , which measure the scale (or characteristic life) and the

518

shape (or slope) of the distribution, respectively.

24

519

This analysis indicates that WT1-WT3 (predominant in May) tend to persist more than the other

520

weather types (as expected from the visual inspection of the observations’ Klee diagram, Fig 7),

521

WT4 tends to transition faster than the other regimes, and WT5-WT7 exhibit similar duration

522

probability density functions (PDFs), all these facts in agreement with the observations.

523

As with the spatial patterns of the weather types between models and observations, there is not a

524

unique set of experiments that is consistently better than the others (i.e., for which all the regimes

525

adequately represent the observed persistence distributions); nonetheless, FLORsst tends to have

526

better self-transition statistics for WT2, and WT4-WT6 (Table 4; bold numbers indicate experi-

527

ments with best representation of self-transitions –persistence– for each WT). Overall, although

528

the main characteristics of the duration PDFs are captured well by the experiments (e.g., the fact

529

that can be well modeled by a Weibull distribution, higher persistence in WT1-WT3), certain fea-

530

tures are not. It is possible to identify WT5 and WT6 as the regimes with the worse representation

531

in terms of its persistence in all simulations (Table 4); for example, WT5 overestimates the number

532

of “early” transitions by at least a factor of 1.5. As complementary information, Table 5 summa-

533

rizes the average persistence of each WT in the NNRPv2 and the numerical experiments; results

534

are consistent with the discussion presented above and Table 4.

535

c. Sub-seasonal evolution

536

The typical sub-seasonal evolution of the weather types can be analyzed using their “climato-

537

logical” frequency of occurrence as computed by the regime’s appearance for each calendar day,

538

after an 11-day moving average is applied to the frequency of occurrence time series in order to

539

filter out the shortest timescales. This metric, hereafter referred to as “sub-seasonality” to avoid

540

the cacophonic “sub-seasonal seasonality”, is shown in Fig. 10. Clearly, MAM is a transition 25

541

season between boreal winter and summer, that could be characterized by low occurrence of ridge

542

configurations (WT1-WT3) during March, and their dominance during May.

543

Generally speaking, the sketched sub-seasonal evolution of the weather types in the numerical

544

experiments is similar to the observations, although there is an overall delayed sub-seasonality in

545

several weather types (compare the slopes of the curves in Figs. 10b,c,d with the ones in Fig. 10a).

546

Biases are present both in the sub-seasonal frequency of occurrence of some weather types and in

547

their actual evolution. For example, the frequency of WT4 is underestimated in all experiments,

548

especially after calendar day 50 (Fig. 10); the model-equivalent WT5-WT7 tend to have more

549

or less the same relative frequency of occurrence in April than in March, which is inconsistent

550

with observations; WT1 has its peak of occurrence typically between calendar days 65-75 (May

551

4th-May 14th), but this maximum appears about 10 days earlier in FLORsst+strat .

552

Due to the clear differences between the beginning and the end of the season, two periods were

553

considered for further analysis: March 20th-March 30th, and May 4th-May 14th (see black boxes

554

in Fig. 10).

555

Errors in the median values of the simulated frequencies of occurrence of weather types, as

556

measured by the scatter index (Fig. 6a), tend to be similar between the two periods under con-

557

sideration, although with higher dispersion during the second half of the season which is mostly

558

due to the misrepresentation of the sub-seasonality of WT4-WT6. On the other hand, errors in the

559

standard deviations of the occurrences are more clearly discriminated (Fig. 6b), being similar in all

560

three experiments but with higher values for the May 4th-May 14th period. For the end-of-March

561

section under study, the median values for the standard deviations of the simulated frequencies are

562

similar to the other timescales analyzed in this work (inter-annual and decadal), although LOARsst

563

exhibits higher and similar dispersions at sub-seasonal scale than at the other scales.

26

564

d. Inter-annual variability

565

Analysis of the ensemble-mean inter-annual evolution of the frequency of weather types, shown

566

in Figure 11, indicates that the highest root mean squared errors, presented in Table 6, occur for

567

patterns associated with troughs north of the Great Lakes (WT4), especially for LOARsst (∼ 11.7

568

days per season, compared to ∼ 10.0 days and ∼ 9.8 days for FLORsst and FLORsst+strat , respec-

569

tively). On the other hand, WT3 and WT7 – northeastern seaboard ridges and deep troughs – have

570

the best average representation of the inter-annual variability in all experiments, with slightly lower

571

errors for LOARsst and FLORsst+strat (∼ 5.1 days for both WT3 and WT7 in both experiments,

572

compared to ∼ 6 days for the same weather types in FLORsst ). Nonetheless, on average, there

573

are no statistically significant differences (p < 0.05, using a two-sample t-Student test) between

574

the experiments, with an overall median error of 7.3 days, which is lower than 10% of the total

575

days in the season. This result is consistent with the scatter index values for the mean frequency

576

of occurrence at inter-annual scales (Fig. 6a).

577

The variability in the frequencies –measured by the associated standard deviation– is, again,

578

very similar between the experiments. Moreover, the fact that there is relatively low dispersion

579

in the variability scatter index for almost all timescales (Fig. 6b) implies agreement between the

580

different members in each numerical experiment, suggesting low model uncertainty in the reported

581

values of this parameter. Nonetheless, overall, Fig. 11 suggests that the experiments do not seem

582

to capture well the observed inter-annual variability.

583

e. Decadal differences

584

There are not enough years to formally study inter-decadal variability, and thus only the differ-

585

ences in the frequency of occurrence of weather types were analyzed for the first and last decades, 27

586

after smoothing the frequency of occurrence time series using an 11-year moving average, to pick

587

up the long time-scale signals of interest in the present analysis.

588

Differences in medians and dispersions of the mean frequency of occurrence are negligible for

589

the two decades (Fig. 6a), although dispersion in the simulated mean frequencies of occurrence are

590

definitively higher in the FLOR experiments than in LOARsst for the 1981-1990 decade. FLORsst

591

and FLORsst+strat show slightly lower medians for the scatter index of the standard deviations for

592

the 2003-2012 decade, although with higher dispersions than LOARsst . Overall, both decades are

593

showing the same ranges and values for the statistics considered.

594

6. Discussion

595

The approach presented here offers tailored diagnostics to understand possible sources of model

596

biases at multiple timescales. The basic idea is that the inherent biases at the synoptic or low-

597

frequency variability scale in models could be rectified at larger scales, according to how climate

598

drivers excite the observed weather types differentially on longer subseasonal-to-seasonal and

599

seasonal-to-decadal scales. A variety of diagnostic metrics were explored to identify model er-

600

rors at several timescales. Nonetheless, putting together the big picture provided by the different

601

metrics is in general not a trivial task.

602

The analysis performed indicates that, at large scale, the simulations have trouble represent-

603

ing the observed spatial pattern associated with WT3 (Table 2), although at regional scale –over

604

NENA– the spatial configuration of the northeastern seaboard ridges is actually good enough to

605

provide a fair representation of the observed rainfall regimes (Table 3), suggesting that the ob-

606

served physical mechanisms that control rainfall in that case, e.g., moisture and heat transport

607

from the Gulf of Mexico (Fig. 4), are present in the simulations. This has important implications

608

for flood prediction in the Ohio river basin (Nakamura et al. 2012; Robertson et al. 2015). 28

609

In contrast, the lowest rainfall pattern correlations are obtained for the regime associated with

610

troughs north of the Great Lakes (RR4, WT4; see Table 3), attributed earlier to the southward dis-

611

placement of the geopotential anomaly with respect to the observed pattern, and its enhancement

612

over most of North America. This is truly the worst circulation regime simulated in the exper-

613

iments. Not only is the spatial pattern poorly simulated, but also the depiction of the observed

614

temporal variability across all timescales is the worst of all WTs. These issues are hypothesized

615

here to be part of the same pathology, and they could be related to problems in the models’ ren-

616

dition of tropical-extratropical interactions, as the seasonal frequency of occurrence of WT4 is

617

significantly correlated with the Niño3.4 SST index (not shown), a link that will be treated –

618

along with other teleconnection indices and circulation regimes – in a future paper, following the

619

methodology discussed in Muñoz et al. (2015).

620

621

The other weather regimes have a fair representation of the geopotential anomalies and rainfall patterns over NENA, although this is not necessarily true at continental or hemispheric scale.

622

Altogether, there are not significant differences in the average performance of LOAR and FLOR

623

in terms of the reproduction of the spatial patterns and temporal variability of the observed weather

624

types, although a certain experiment can be better than the others when considering particular

625

characteristics (e.g., mean frequency at inter-annual scale, or the persistence of WT4; see Section

626

5). In this work, the question of which model is better is really a question of what nudging

627

approach performs better and what is the impact of horizontal resolution.

628

Nudging both SST and stratospheric fields did not consistently improve the representation of

629

the weather types in the model, and indeed in some cases provided worse results than the SST-

630

only nudging experiment (FLORsst ). It is possible that some stratosphere-troposphere and ocean-

631

atmosphere interactions are not being well simulated, and that some improvement can be achieved

29

632

if the vertical resolution in the model is increased to adequately account for the stratospheric

633

processes. This is a matter of future research.

634

As indicated earlier, no significant improvement was found when increasing horizontal reso-

635

lution, neither in the reanalysis products nor in the model experiments (LOARsst and FLORsst ).

636

This is attributed to the fact that synoptic-scale 500 hPa geopotential height anomalies do not re-

637

ally require high resolution in order to reproduce the key physical mechanisms associated with, for

638

example, propagation of Rossby waves that perturbs circulation patterns, or the moisture advec-

639

tion conducive to rainfall; in addition, the topography in NENA is not tall or complex enough as

640

to negatively impact the low resolution model. Yet, high horizontal resolution could be important

641

in other regions of the world.

642

Although high spatial resolution does not seem to be necessary to satisfactorily reproduce ob-

643

served weather types, high-resolution models like FLOR have the advantage of providing phys-

644

ically related variables like rainfall or surface temperature at a resolution preferred by decision-

645

makers and for the provision of climate services (Vaughan and Dessai 2014). Nonetheless, it is

646

possible to exploit the fair representation of weather type characteristics by faster coupled models

647

like LOAR to “reconstruct” fields of interest, e.g., the rainfall climatology in NENA.

648

649

Since the rainfall climatology, < p >, can be computed in terms of a linear combination of rainfall regimes, RRi , and the total frequency of occurrence of those regimes, Fi , k

< p >=

∑FiRRi

(10)

i

650

for i = 1..k regimes, if the model has an adequate representation of only one of these fields, then

651

observations could be used to compute the other factor in the linear combination. For example, the

652

rainfall regimes of the experiments analyzed in this work (ModRR) could be used in conjunction

653

with the observed frequencies of occurrences (ObsF) to reconstruct the observed rainfall climatol30

654

ogy, as shown in Fig. 12a,b. For other approaches that can be used for prediction purposes, see

655

Moron et al. (2010).

656

As expected, the experiments with FLOR provide better spatial rainfall patterns than LOARsst ,

657

because of their higher resolution. Although there are biases, the use of the observed frequency of

658

occurrences has improved the original simulated rainfall climatologies (Fig. 2b - Fig. 2d). Further

659

improvement is obtained if the observed rainfall regimes (ObsRR) are used in conjunction with the

660

modeled frequencies (ModF; Fig. 12c). With a satisfactory bias-correction method, this approach

661

has the potential to provide relatively economic – at least from a computational point of view –

662

diagnostic products and forecasts.

663

7. Concluding remarks

664

This work discussed a new diagnostic framework to evaluate the performance of models across

665

multiple timescales, based on their representation of the observed spatial and temporal variability

666

of weather types. Under this non-linear system dynamics perspective, “good” models are those

667

that correctly reproduce the observed characteristics of the weather types at multiple timescales.

668

The framework takes advantage of the weather-typing decomposition, in terms of the spatial

669

patterns of the circulation regimes and their temporal evolution, to analyze model performance at

670

multiple timescales focusing on the evaluation of tailored statistics like daily transition probabil-

671

ities, weather type mean durations and sub-seasonal, inter-annual and longer-term frequencies of

672

occurrence. Furthermore, since the circulation regimes are normally linked to concrete climate

673

modes, they can also be used to diagnose model biases from a physical perspective, like deforma-

674

tions or displacements of particular geopotential height configurations that control the occurrence

675

of rainfall in a region of the world. 31

676

To illustrate how the diagnostic approach works, it was applied to three different sets of numer-

677

ical experiments using Geophysical Fluid Dynamics Laboratory coupled circulation models. The

678

simulations tend to represent well the location, shape and magnitude of daily circulation regimes

679

and associated rainfall patterns, although some important biases were reported and discussed. Fur-

680

ther research is being conducted to perform an in-depth analysis of possible tropical-extratropical

681

interactions that might not be well represented by the models.

682

Finally, the present framework can also be used for model inter-comparison, and can be applied

683

to uncoupled and regional models.

684

Acknowledgments. The authors are grateful to Tony Barnston and Simon Mason (IRI), and

685

Nathaniel Johnson and Ángel Adames (GFDL) for discussions about different aspects of the pa-

686

per. ÁGM was supported by National Oceanic and Atmospheric Administration’s Oceanic and

687

Atmospheric Research, under the auspices of the National Earth System Prediction Capability.

688

AWR was supported by NOAA Next Generation Global Prediction System (NGGPS) project grant

689

NA16NWS4680014. This paper is dedicated to Eneko Muñoz and Cathy Vaughan (ÁGM’s nena).

690

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Ropelewski, C. F., and M. S. Halpert, 1987:

Global and Regional Scale Precipita-

903

tion Patterns Associated with the El Niño/Southern Oscillation. Monthly Weather Re-

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Rosati, A., K. Miyakoda, and R. Gudgel, 1997:

The Impact of Ocean Initial Con-

908

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Ryu, J. H., and K. Hayhoe, 2014: Understanding the sources of Caribbean precipitation biases

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Stephenson, D. B., A. Hannachi, and A. O’Neill, 2004: On the existence of multiple climate

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918

Taylor, K. E., R. J. Stouffer, and G. A. Meehl, 2012: An Overview of CMIP5 and the Experi-

919

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van der Wiel, K., and Coauthors, 2016: The resolution dependence of contiguous US precipitation

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JCLI-D-16-0307.1, URL http://journals.ametsoc.org/doi/10.1175/JCLI-D-16-0307.1.

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Vaughan, C., and S. Dessai, 2014: Climate services for society: origins, institutional arrangements,

931

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932

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933

Vautard, R., 1990: Multiple Weather Regimes over the North Atlantic: Analysis of Pre-

934

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Vautard, R., K. C. Mo, and M. Ghil, 1990:

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938

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943

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945

Wheeler, M. C., and H. H. Hendon, 2004: An All-Season Real-Time Multivariate MJO Index: De-

946

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948

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952

10.1007/s00382-011-1001-9.

44

953

LIST OF TABLES

954

Table 1.

955 956 957 958 959 960

961

Table 2.

962 963 964 965 966 967

968

Table 3.

969 970 971 972 973 974

975

Table 4.

976 977 978

979

Table 5.

980

981 982 983

Table 6.

Anomaly Correlation Coefficient (Spearman) between the model-equivalent ensemble-mean weather types (WTs) in each experiment and the corresponding circulation regime in NNRPv2, computed on the reanalysis grid for the domain sketched in Fig 4. Numbers in bold indicate highest correlations found in the model experiments, all values being statistically significant (t-Student test, p < 0.05). No statistically significant difference was found between the average anomaly correlation coefficients of the experiments. . . . . .

.

. 45

Anomaly Correlation Coefficient (Spearman) between the model-equivalent ensemble-mean weather types (WTs) in each experiment and the corresponding circulation regime in MERRA, computed on the renalaysis grid for the domain sketched in Fig 4. Numbers in bold indicate highest correlations found in the model experiments, all values being statistically significant (t-Student test, p < 0.05). No statistically significant difference was found between the average anomaly correlation coefficients of the experiments. . . . . .

.

. 46

Anomaly Correlation Coefficient (Spearman) between the model-equivalent ensemble-mean rainfall regimes (RRs) in each experiment and the corresponding rainfall pattern in the observations, computed on the observations grid for the domain sketched in Fig 5. Numbers in bold indicate highest correlations found in the model experiments, all values being statistically significant (tStudent test, p < 0.05). No statistically significant difference was found between the average anomaly correlation coefficients of the experiments. . . .

.

47

Best-fit of Weibull parameters (α, β ) for the model-equivalent ensemble-mean weather type durations sketched by the p red curves in Fig 9. Bold pairs indicate minimum Euclidean distance ds = (∆α 2 + ∆β 2 ), where ∆ denotes differences between simulated and observed parameters. . . . . . . . .

.

48

Average persistence (in days) for each WT in the NNRPv2 dataset and the numerical experiments. Values closer to reanalysis are presented in bold. . . Ensemble-mean RMSE (in days per season) for the inter-annual evolution of the frequency of occurrence of the weather types in each experiment, with respect to NNRPv2. . . . . . . . . . . . . . . . . .

45

.

. 49

.

50

984

TABLE 1.

Anomaly Correlation Coefficient (Spearman) between the model-equivalent ensemble-mean

985

weather types (WTs) in each experiment and the corresponding circulation regime in NNRPv2, computed on

986

the reanalysis grid for the domain sketched in Fig 4. Numbers in bold indicate highest correlations found in the

987

model experiments, all values being statistically significant (t-Student test, p < 0.05). No statistically significant

988

difference was found between the average anomaly correlation coefficients of the experiments.

Dataset vs NNRPv2

WT1

WT2

WT3

WT4

WT5

WT6

WT7

Average

MERRA

0.99

0.99

0.97

0.99

0.99

0.99

0.99

0.98

LOARsst

0.93

0.93

0.37

0.54

0.66

0.87

0.91

0.74

FLORsst

0.89

0.96

0.20

0.66

0.72

0.89

0.94

0.75

FLORsst+strat

0.90

0.97

-0.11

0.78

0.66

0.84

0.95

0.71

46

989

TABLE 2.

Anomaly Correlation Coefficient (Spearman) between the model-equivalent ensemble-mean

990

weather types (WTs) in each experiment and the corresponding circulation regime in MERRA, computed on

991

the renalaysis grid for the domain sketched in Fig 4. Numbers in bold indicate highest correlations found in the

992

model experiments, all values being statistically significant (t-Student test, p < 0.05). No statistically significant

993

difference was found between the average anomaly correlation coefficients of the experiments.

Dataset vs MERRA

WT1

WT2

WT3

WT4

WT5

WT6

WT7

Average

NNRPv2

0.99

0.99

0.97

0.99

0.99

0.99

0.99

0.98

LOARsst

0.92

0.93

0.36

0.58

0.73

0.86

0.91

0.75

FLORsst

0.90

0.96

0.18

0.70

0.72

0.89

0.94

0.75

FLORsst+strat

0.91

0.96

-0.10

0.82

0.74

0.85

0.96

0.73

47

994

TABLE 3. Anomaly Correlation Coefficient (Spearman) between the model-equivalent ensemble-mean rain-

995

fall regimes (RRs) in each experiment and the corresponding rainfall pattern in the observations, computed on

996

the observations grid for the domain sketched in Fig 5. Numbers in bold indicate highest correlations found

997

in the model experiments, all values being statistically significant (t-Student test, p < 0.05). No statistically

998

significant difference was found between the average anomaly correlation coefficients of the experiments.

Experiment

RR1

RR2

RR3

RR4

RR5

RR6

RR7

Average

LOARsst

0.80

0.83

0.87

0.51

0.57

0.80

0.87

0.75

FLORsst

0.78

0.81

0.90

0.48

0.64

0.79

0.88

0.75

FLORsst+strat

0.83

0.82

0.88

0.38

0.69

0.62

0.90

0.73

48

1000

TABLE 4. Best-fit of Weibull parameters (α, β ) for the model-equivalent ensemble-mean weather type durap tions sketched by the red curves in Fig 9. Bold pairs indicate minimum Euclidean distance ds = (∆α 2 + ∆β 2 ),

1001

where ∆ denotes differences between simulated and observed parameters.

999

WT1

WT2

WT3

WT4

WT5

WT6

WT7

NNRPv2

(3.52, 1.46) (1.71, 0.55) (2.56, 1.13) (2.53, 1.36) (1.47, 0.76) (1.30, 0.73) (0.83, 0.62)

LOARsst

(2.90, 1.29) (1.13, 0.53)

(2.22,0.83)

(1.89, 0.84) (3.09, 1.48) (3.12, 1.57)

(0.82,0.61)

FLORsst

(2.73, 1.04)

(1.71,0.54)

(2.05, 0.87)

(2.86,1.48)

(1.10, 0.66)

FLORsst+strat

(3.47,1.43)

(1.87, 0.53) (1.82, 0.78) (1.56, 0.82) (2.74, 1.47) (2.95, 1.45) (0.93, 0.63)

49

(1.94,0.95)

(1.21,0.68)

1002

1003

TABLE 5. Average persistence (in days) for each WT in the NNRPv2 dataset and the numerical experiments. Values closer to reanalysis are presented in bold.

WT1

WT2

WT3

WT4

WT5

WT6

WT7

NNRPv2

4.06

3.47

3.36

3.21

2.62

2.48

2.14

LOARsst

3.59

2.94

3.31

2.96

3.66

3.69

2.16

FLORsst

3.59

3.56

3.09

3.46

2.88

2.47

2.40

FLORsst+strat

3.98

3.89

2.95

2.62

3.39

3.57

2.26

50

1004

1005

TABLE 6. Ensemble-mean RMSE (in days per season) for the inter-annual evolution of the frequency of occurrence of the weather types in each experiment, with respect to NNRPv2.

Experiment

WT1

WT2

WT3

WT4

WT5

WT6

WT7

LOARsst

7.4

7.3

5.1

11.7

10.2

7.9

5.2

FLORsst

7.4

6.4

6.2

10.0

7.7

7.1

5.7

FLORsst+strat

6.6

6.9

5.0

9.8

7.1

7.8

5.1

51

1006

LIST OF FIGURES

1007

Fig. 1.

1008 1009 1010

1011

Fig. 2.

1012 1013

1014

Fig. 3.

1015 1016 1017 1018 1019 1020 1021

Schematic showing (left) three available states (A, B, C) and a forbidden or unaccesible state (D). The system transitions between the available states (represented by the arrows). Those transitions define sequences of states (right) that describe events; e.g., ABC could be related to extreme rainfall events, while BBC to heat waves. . . . . . . . . . . .

.

Rainfall climatology (MAM 1981-2012) for Northeastern North America. (a) Observations, (b) LOARsst , (c) FLORsst and (d) FLOARsst+strat . Units are in mm day−1 . Ocean has been masked in (b)-(d). Each dataset is plotted using its native horizontal resolution. . . . .

. 54

Hemispheric view of observed (NNRPv2 and MERRA) and modeled (LOARsst , FLORsst , FLORsst+strat ) weather types –WTs–, using geopotential height anomalies at 500 hPa (contour interval is 20 gpm). Red solid lines indicate positive anomalies, while negative ones correspond to the blue dashed lines. In the model experiments, regions showing statistically significant (p < 0.05) ensemble-mean anomalies are shaded in gray. Relative frequency of occurrence for the entire 32-year period is indicated in parenthesis, with model experiments showing the ensemble mean ± one standard deviation. Each dataset is plotted using its native horizontal resolution. . . . . . . . . . . . . . . . . . .

. 55

53

1022

Fig. 4.

As in Fig. 3, but for most of North America.

.

.

56

1023

Fig. 5.

Observed and modeled (LOARsst , FLORsst , FLORsst+strat ) rainfall regimes, RR, associated with each weather type (units are mm day−1 ). Relative frequency of occurrence for the entire 32-year period is indicated in parenthesis, with model experiments showing the ensemble mean ± one standard deviation. Each dataset is plotted using its native horizontal resolution.

.

57

Boxplots of scatter indices for (a) the mean and (b) standard deviation of the frequency of occurrence of weather types across multiple timescales. Perfect coincidence between simulations and observations correspond to a scatter index value of zero. For each boxplot, the central mark indicates the ensemble median, and the bottom and top edges of the box indicate the ensemble 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points not considered outliers, and the outliers (values > 2.7 standard deviations) are plotted individually using the ’+’ symbol. For additional details see subsections 5c-5e. . . . . . . . . . . . . . . . . . . . . . . .

.

58

1024 1025 1026

1027

Fig. 6.

1028 1029 1030 1031 1032 1033 1034

1035

Fig. 7.

1036 1037 1038

1039

Fig. 8.

1040 1041 1042 1043 1044 1045

1046 1047 1048

Fig. 9.

.

.

.

.

.

.

.

.

.

.

.

.

Klee diagrams for (a) NNRPv2 and numerical experiments (b) LOARsst , (c) FLORsst and (d) FLORsst+strat . Each tile corresponds to a particular day, and the colors represent different weather types (WTs, see color bar). Only one member per experiment is shown (others are similar). . . . . . . . . . . . . . . . . . . . . . . .

. 59

(a) Daily transition probabilities, P, for observed (NNRPv2) weather types (WTs, see label bar). A star indicates statistically significant transitions (p < 0.1, see Vautard et al. (1990)). Vertical and horizontal axes correspond to the prior and posterior WTs, respectively. Differences between the LOARsst , FLORsst and FLORsst+strat ensemble-mean transition probabilities and those presented in (a) are shown in panels (b)-(d), respectively. Average differences between panels (b)-(d) are not statistically significant (p < 0.05, using a two-sample t-Student test). . . . . . . . . . . . . . . . . . . . . .

.

Average durations (in days) for each weather type (WT, see label bar) in NNRPv2, LOARsst , FLORsst and FLORsst+strat . Red curves sketch the corresponding Weibull fit, whose parameters α and β are presented in Table 4. . . . . . . . . . . . . . . .

52

60

. 61

1049 1050 1051

1052 1053

1054 1055 1056 1057 1058 1059

Fig. 10. Ensemble-mean sub-seasonal frequency of occurrence for each weather type (WT), smoothed with an 11-day moving average. Periods in the black boxes were were selected for further analysis (see subsection 5c). . . . . . . . . . . . . . . .

. 62

Fig. 11. Observed (a) and ensemble-mean (b-d) inter-annual frequency of occurrence for each weather type (WT, see label bar), for all MAM seasons. . . . . . . . . . .

.

63

Fig. 12. (a) Observed rainfall climatology (MAM 1981-2012), and reconstruction of the rainfall climatology using, for each numerical experiment, a linear combination of (b) the observed frequencies of occurrence of weather types (ObsF) and the modeled rainfall regimes (ModRR), and (c) the modeled frequencies of occurrence of weather types (ModF) and the observed rainfall regimes (ObsRR). Units in mm day−1 . Ocean has been masked in (b). Each dataset is plotted using its native horizontal resolution. . . . . . . . . . . . .

.

64

53

D ABC ABA BBC CAC …

B

A

C Available physical states and transitions

Events are described in terms of sequences of available states

1060

F IG . 1. Schematic showing (left) three available states (A, B, C) and a forbidden or unaccesible state (D). The

1061

system transitions between the available states (represented by the arrows). Those transitions define sequences

1062

of states (right) that describe events; e.g., ABC could be related to extreme rainfall events, while BBC to heat

1063

waves.

54

1064

F IG . 2. Rainfall climatology (MAM 1981-2012) for Northeastern North America. (a) Observations, (b)

1065

LOARsst , (c) FLORsst and (d) FLOARsst+strat . Units are in mm day−1 . Ocean has been masked in (b)-(d). Each

1066

dataset is plotted using its native horizontal resolution.

55

1067

1068

F IG . 3.

Hemispheric view of observed (NNRPv2 and MERRA) and modeled (LOARsst , FLORsst ,

56

FLORsst+strat ) weather types –WTs–, using geopotential height anomalies at 500 hPa (contour interval is 20

FLORsst

LOARsst 57

FLORsst+strat

MERRA

F IG . 4. As in Fig. 3, but for most of North America.

NNRPv2

FLORsst+strat

LOARsst

FLORsst

NNRPv2 1074

F IG . 5. Observed and modeled (LOARsst , FLORsst , FLORsst+strat ) rainfall regimes, RR, associated with each

1075

weather type (units are mm day−1 ). Relative frequency of occurrence for the entire 32-year period is indicated

1076

in parenthesis, with model experiments showing the ensemble mean ± one standard deviation. Each dataset is

1077

plotted using its native horizontal resolution.

58

a

b

1078

F IG . 6. Boxplots of scatter indices for (a) the mean and (b) standard deviation of the frequency of occur-

1079

rence of weather types across multiple timescales. Perfect coincidence between simulations and observations

1080

correspond to a scatter index value of zero. For each boxplot, the central mark indicates the ensemble median,

1081

and the bottom and top edges of the box indicate the ensemble 25th and 75th percentiles, respectively. The

1082

whiskers extend to the most extreme data points not considered outliers, and the outliers (values > 2.7 standard

1083

deviations) are plotted individually using the ’+’ symbol. For additional details see subsections 5c-5e.

59

WTs

1084

F IG . 7. Klee diagrams for (a) NNRPv2 and numerical experiments (b) LOARsst , (c) FLORsst and (d)

1085

FLORsst+strat . Each tile corresponds to a particular day, and the colors represent different weather types (WTs,

1086

see color bar). Only one member per experiment is shown (others are similar).

60

prior WT

* * * * * * * * * * *

* * *

posterior WT

1087

F IG . 8. (a) Daily transition probabilities, P, for observed (NNRPv2) weather types (WTs, see label bar). A

1088

star indicates statistically significant transitions (p < 0.1, see Vautard et al. (1990)). Vertical and horizontal

1089

axes correspond to the prior and posterior WTs, respectively. Differences between the LOARsst , FLORsst and

1090

FLORsst+strat ensemble-mean transition probabilities and those presented in (a) are shown in panels (b)-(d),

1091

respectively. Average differences between panels (b)-(d) are not statistically significant (p < 0.05, using a two-

1092

sample t-Student test).

61

NNRPv2 LOARsst FLORsst FLORsst+strat 1093

F IG . 9. Average durations (in days) for each weather type (WT, see label bar) in NNRPv2, LOARsst , FLORsst

1094

and FLORsst+strat . Red curves sketch the corresponding Weibull fit, whose parameters α and β are presented in

1095

Table 4.

62

WTs

1096

F IG . 10. Ensemble-mean sub-seasonal frequency of occurrence for each weather type (WT), smoothed with

1097

an 11-day moving average. Periods in the black boxes were were selected for further analysis (see subsection

1098

5c).

63

WTs

1099

1100

F IG . 11. Observed (a) and ensemble-mean (b-d) inter-annual frequency of occurrence for each weather type (WT, see label bar), for all MAM seasons.

64

LOARsst FLORsst FLORsst+strat 1101

F IG . 12. (a) Observed rainfall climatology (MAM 1981-2012), and reconstruction of the rainfall climatology

1102

using, for each numerical experiment, a linear combination of (b) the observed frequencies of occurrence of

1103

weather types (ObsF) and the modeled rainfall regimes (ModRR), and (c) the modeled frequencies of occurrence

1104

of weather types (ModF) and the observed rainfall regimes (ObsRR). Units in mm day−1 . Ocean has been

1105

masked in (b). Each dataset is plotted using its native horizontal resolution.

65