Geophysical Research Letters RESEARCH LETTER 10.1002/2015GL065878
Tropospheric Biennial Oscillation (TBO) indistinguishable from white noise
Key Points: • The appropriate null hypothesis for the TBO is proposed • The TBO time evolution is indistinguishable from white noise
Malte F. Stuecker1 , Axel Timmermann2 , Jinhee Yoon1 , and Fei-Fei Jin1
Supporting Information: • Figure S1
Abstract Previous studies proposed that the year-to-year variability of seasonal monsoon indices is partly determined by a Tropospheric Biennial Oscillation (TBO). Invoking coupled ocean-atmosphere-land processes and the presence of an annual memory, the TBO mechanism describes how a relatively strong monsoon is followed by a year with weaker monsoon and vice versa. Here we revisit the issue of preferred biennial timescales in tropical monsoon systems, by testing the biennial tendencies in observed and simulated monsoon indices against the white noise null hypothesis. According to an analytical expression for the null hypothesis, we expect the probability for this biennial tendency to be 2/3, which is in close agreement with observations, reanalysis products, and Atmospheric Model Intercomparison Project/ Coupled Model Intercomparison Project general circulation model simulations. Thus, it is concluded that biennial tendencies in these monsoon indices and the associated TBO are fully consistent with a white noise process and do not require the presence of a preferred biennial timescale.
Correspondence to: M. F. Stuecker,
[email protected]
Citation: Stuecker, M. F., A. Timmermann, J. Yoon, and F.-F. Jin (2015), Tropospheric Biennial Oscillation (TBO) indistinguishable from white noise, Geophys. Res. Lett., 42, 7785–7791, doi:10.1002/2015GL065878.
1 Department of Atmospheric Sciences, University of Hawai’i at M¯ anoa, Honolulu, Hawai’i, USA, 2 IPRC, University of Hawai’i
at M¯anoa, Honolulu, Hawai’i, USA
1. Introduction Received 20 AUG 2015 Accepted 10 SEP 2015 Accepted article online 12 SEP 2015 Published online 29 SEP 2015
It has been proposed that quasi-biennial variability in the troposphere originates from atmosphere-ocean coupling in the tropics [e.g., Brier, 1978; Nicholls, 1978; Meehl, 1987, 1993, 1997; Clarke et al., 1998; Chang and Li, 2000; Meehl et al., 2003] and possibly with the involvement of land processes [e.g., Meehl, 1997]. Dynamical processes in the coupled ocean-atmosphere-land system have been invoked to explain the Tropospheric Biennial Oscillation (TBO), which affects, for instance, the monsoon strength over Asia and Australia [e.g., Meehl, 1997; Meehl and Arblaster, 2001, 2002; Loschnigg et al., 2003]. In the observations, only some of the TBO events are associated with the interannual El Niño–Southern Oscillation (ENSO) phenomenon [e.g., Meehl, 1987; Meehl and Arblaster, 2002; Meehl et al., 2003]. The established definition for the TBO is based on the tendency that a relatively strong monsoon is followed by a relatively weak monsoon and vice versa [Meehl and Arblaster, 2002; Meehl et al., 2003]. The typical spatial pattern and time evolution of the TBO during the boreal summer season is displayed in Figure 1 (see section 2 for details). A relatively strong monsoon during year i is usually defined as when the precipitation anomalies x over the respective monsoon region and season (in general, x can be viewed as any seasonal monsoon index) are (following the notation in Meehl and Arblaster [2002]; Meehl et al. [2003]) xi−1 < xi > xi+1 .
(1)
Accordingly, a weak monsoon year is defined as xi−1 > xi < xi+1 .
(2)
The statistical preference for this biennial tendency has been noted by many studies and used as evidence to support a deterministic dynamical mechanism as its source [e.g., Meehl, 1987; Meehl and Arblaster, 2002; Meehl et al., 2003].
©2015. American Geophysical Union. All Rights Reserved.
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Here we set out to determine whether the observed biennial tendency of the Indian and Australian monsoon is consistent with a random white noise process, which would not require the presence of enhanced variance on biennial timescales. We demonstrate using a large number of observational data sets and model experiments that this null hypothesis cannot be rejected (at the 95% level for most of them). Hence, even if a dynamical mechanism for the TBO existed, it would be indistinguishable from a white noise random process. TBO INDISTINGUISHABLE FROM WHITE NOISE
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Figure 1. (a) Regression map of the normalized Global Precipitation Climatology Project (GPCP) India summer monsoon index on the June-July-August-September (JJAS) GPCP [Adler et al., 2003] anomalies (shading), the JJAS HadSST1 [Rayner et al., 2003] sea surface temperature anomalies (SSTA, contours), and JJAS 850 (hPa) NCEP2 wind vectors [Kanamitsu et al., 2002]. The linear trend is removed from all data sets prior to the regression analysis. (b) Time series of the GPCP JJAS India summer monsoon index (area-averaged precipitation anomalies in the region 60∘ E–100∘ E and 5∘ N–40∘ N), which is often used as an index for the TBO.
2. Data We define the Indian boreal summer monsoon as the seasonal precipitation anomalies (after removing the seasonal cycle) during June-July-August-September (JJAS) averaged over the region 60∘ E–100∘ E and 5∘ N–40∘ N [Meehl and Arblaster, 2002; Meehl et al., 2003]. Additionally, we define the Australian summer monsoon as the seasonal precipitation anomalies during December-January-February (DJF) over the region 100∘ E–150∘ E and 20∘ S–5∘ S (similar to the index used by Meehl and Arblaster [2002]). Various observational precipitation data sets (Climate Prediction Center Merged Analysis of Precipitation (CMAP) [Xie and Arkin, 1997], GPCP [Adler et al., 2003], and Global Precipitation Climatology Centre (GPCC) [Becker et al., 2013]; see Table 1) and general circulation model (GCM) experiments predominantly from the Coupled Model Intercomparison Project (CMIP) [Taylor et al., 2012] (Table 2), as well as two long integrations of the Community Earth Table 1. Observational and Reanalysis Data Sets Used for Various TBO Indicesa Data Set (Symbol)
Time Period
Monsoon Region Seasonal Average
Reference Xie and Arkin [1997]
CMAP (diamond)
1979–2010
JJAS India, DJF Australia
GPCP (square)
1979–2010
JJAS India, DJF Australia
Adler et al. [2003]
GPCC (upward triangle)
1951–2000
JJAS India
Becker et al. [2013]
ERA-40
1958–2001
JJAS WYI, JJAS WSI
Uppala et al. [2005]
NCEP2
1979–2013
JJAS WYI, JJAS WSI Kanamitsu et al. [2002] a India refers to the area-averaged precipitation anomalies in the region 60∘ E–100∘ E and 5∘ N–40∘ N. Australia refers to the area-averaged precipitation anomalies in the region 100∘ E–150∘ E and 20∘ S–5∘ S. The area average only consists of land points for the GPCC data. JJAS refers to the seasonal average from June to September and DJF to the seasonal average from December to February. Two Indian summer monsoon indices based on the vertical shear of the zonal winds (U) are included: the Webster and Yang index (WYI, U850hPa -U200hPa averaged from the equator to 20∘ N and from 40∘ E to 110∘ E), and the westerly shear index (WSI, U850hPa -U200hPa from 5∘ N to 20∘ N and from 40∘ E to 80∘ E).
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Table 2. The General Circulation Model (GCM) Data [Taylor et al., 2012] Utilized to Calculate the Area-Averaged JJAS Season Precipitation Anomalies Over the Indian Monsoon (60∘ E–100∘ E, 5∘ N–40∘ N) and the DJF Season Anomalies Over the Australian Monsoon (100∘ E–150∘ E, 20∘ S–5∘ S) Regions AMIP
Historical Experiment
Preindustrial Control
Other Experiment
(Downward Triangle)
(Circle)
(Circle)
(Circle)
BCC-CSM1
-
1850–2012
-
-
Wu et al. [2013]
CNRM-CM5
-
-
850 year
-
Voldoire et al. [2013]
Model Name
Reference
GISS-E2-H
-
1850–2005
-
-
Schmidt et al. [2014]
GFDL CM2.1
-
1861–2005
500 year
-
Delworth et al. [2006]
-
-
2000 year
-
Wittenberg et al. [2014]
GFDL-CM3
1979–2008
1860–2004
500 year
-
Donner et al. [2011]
GFDL-HIRAM-C180
1979–2008
-
-
-
Zhao et al. [2009]
GFDL-HIRAM-C360
1979–2008
-
-
-
Zhao et al. [2009]
HadCM3
-
1860–2005
-
-
Gordon et al. [2000]
IPSL-CM5A-MR
-
1850–2005
-
-
Dufresne et al. [2013]
1979–2008
1850–2012
670 year
-
Watanabe et al. [2010]
MIROC-ESM
MIROC5
-
1850–2005
531 year
-
Watanabe et al. [2011]
MPI-ESM-MR
-
1850–2005
-
-
Giorgetta et al. [2013]
MRI-ESM1
-
1851–2005
-
-
Adachi et al. [2013]
CCSM4
1979–2010
1850–2005
-
850–1850
Gent et al. [2011]
CESM1(CAM5)
1979–2005
1850–2005
-
-
Gent et al. [2011]
-
-
-
2600 year
Kay et al. [2014]
NorESM1-M
1979–2008
1850–2012
501 year
-
Bentsen et al. [2013]
System Model (CESM) (uncoupled atmospheric general circulation model (AGCM) experiment forced by the SST seasonal cycle) [Kay et al., 2014] and CM2.1 (coupled GCM experiment) [Wittenberg, 2009; Wittenberg et al., 2014] models, are utilized. The long uncoupled CESM AGCM integration provides another robust estimate of the biennial tendency in the monsoon indices caused solely by internal atmospheric variability without any ENSO forcing nor ocean atmosphere coupling present. To show that our results also hold for circulation-based monsoon indices, we further calculate the JJAS (after removing the seasonal cycle) Webster and Yang index (WYI), defined as the vertical wind shear of the zonal wind component between 850 hPa and 200 hPa [Webster and Yang, 1992; Wang and Fan, 1999]: U850hPa -U200hPa averaged from the equator to 20∘ N and from 40∘ E to 110∘ E. Furthermore, we use the modified JJAS westerly shear index (WSI) [Wang and Fan, 1999]: U850hPa -U200hPa averaged from 5∘ N to 20∘ N and 40∘ E to 80∘ E. Both indices are calculated for two reanalysis products covering different periods: ERA-40 [Uppala et al., 2005] from 1958 to 2001 and NCEP2 [Kanamitsu et al., 2002] from 1979 to 2013. Following Meehl and Arblaster [2002] and Meehl et al. [2003], we define the biennial tendency of the anomalous monsoon for both the precipitation and circulation indices as in equations (1) and (2).
3. Results Here we show that for a white noise process the probability for a biennial tendency is 2/3. If a variable x exhibits a biennial tendency, it should either fulfill equation (1) or equation (2). We first consider only the conditions that need to be fulfilled for equation (1) to be true. If xi = k, the following three probabilities P need to be considered. First, the probability that the variable x at the time i has a certain value k is given by a probability density function (PDF): P(xi = k) = PDF(k).
(3)
Second, we need the probability that x in year i is larger than in the previous year i − 1 and both are larger than negative infinity: P(−∞ < xi−1 < xi ) =
STUECKER ET AL.
xi
∫−∞
PDF(xi−1 )dxi−1 = CDF(k) − CDF(−∞) = CDF(k),
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(4)
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Figure 2. Probability for a biennial tendency as a function of the data length tested against Gaussian white noise. The biennial tendency probability mean (solid black line) and its range (solid and dashed grey lines) are obtained using a Monte Carlo approach. Displayed are precipitation-based monsoon indices for India (red) and Australia (blue) for CMAP (diamond), GPCP (square), GPCC (up triangle), AMIP experiments (down triangle), and various coupled model experiments (circle). Additionally, circulation-based monsoon indices (hexagram and cross) are displayed for ERA-40 (magenta) and NCEP2 (orange). Furthermore, the Indian monsoon (pink circle) and Australian monsoon (cyan circle) indices for the 2.6 kyr CESM AGCM control are displayed. Values situated between the two dashed grey lines means that the null hypothesis cannot be rejected at the 95% level.
which is given by a cumulative density function (CDF). By definition the CDF(−∞) is zero. Third, we need also the probability that x in the following year i + 1 is smaller than in year i and both are again larger than negative infinity: P(−∞ < xi+1 < xi ) = CDF(k). (5) If the three values xi−1 , xi , and xi+1 are taken from a white noise process (random and independent), the joint probability function for the three above conditions can be defined as the product of the probability functions: P(−∞ < xi−1 < xi and − ∞ < xi+1 < xi and xi = k) = PDF(k) ⋅ CDF(k)2 .
(6)
By integrating over all possible values of k from −∞ to ∞, we can get the total probability P̂ : P̂ =
∞
∫−∞
PDF(k) ⋅ CDF(k)2 dk.
(7)
Using a variable transform with q = CDF(k) and with dq = PDF(k)dk, we get P̂ =
1
∫0
q2 dq =
[
1 3 q 3
]1 = 1∕3.
(8)
0
Accordingly, we can calculate the same probability for equation (2): P̂ =1∕3. For a white noise process with equation (1) as event A and equation (2) as the independent event B, the following relationship holds P(A or B) = P(A) + P(B). Thus, the total probability for a biennial tendency (either the case defined STUECKER ET AL.
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Figure 3. Probability for a biennial tendency tested against randomness using a nonparametric method. The India summer monsoon indices (precipitation and circulation based) are displayed for various observational data sets and model experiments (see labels). The expected mean (black square) and range (𝜎 and 2𝜎 ) for a biennial tendency in each data set is compared versus the calculated actual biennial tendency (magenta circle) in the respective data set.
by equation (1) or the case defined by equation (2)) is P̂ = 1∕3 + 1∕3 = 2∕3 for a white noise random process. Importantly, as shown by equation (8), the exact shape of the PDF is not important in determining this probability. Next, we numerically generate pseudorandom numbers drawn from a Gaussian distribution to form each 1000 time series for different lengths N (the length N being between 10 years and 3000 years). For each different data vector of length N (e.g., N = 10 years and N = 25 years) we can use the 1000 random realizations to calculate the probability that a biennial tendency will occur by chance, as well as the standard deviation of this estimate as a function of the data length N. For 1000 realizations, we see that the numerical estimate of the probability for a biennial tendency is very close to the expected analytical solution of 2/3 (Figure 2), even for very short data vectors (e.g., N = 10 years). As expected, the uncertainty of this estimate (the 𝜎 and 2𝜎 lines) decreases with increasing data length N (years) (Figure 2). The biennial tendency of the Indian summer monsoon and the Australian summer monsoon for various observational data sets and general circulation model experiments is displayed in Figure 2. We observe that nearly all data points cluster near the expected value for a biennial tendency due to a white noise process (P = 2∕3) and are situated within the data length dependent uncertainty range. Thus, we are not able to reject (at the 95% level for most of the data and models) the null hypothesis that a white noise process is causing the STUECKER ET AL.
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biennial tendency in both the observations and model experiments (see Figure 2 for the rejection levels). Values for the biennial tendency probability much less than 2/3 indicate a nonwhite noise character of the variable which is caused by persistence in the system. This persistence could, for instance, be caused by a red noise process or deterministic low-frequency variability such as ENSO. Additionally, we test if our results are dependent on the assumption of a Gaussian distribution. Thus, we use a nonparametric method to estimate the confidence intervals of our null hypothesis. For the previous Indian summer monsoon indices (both precipitation and circulation based), we permute each data vector 1000 times randomly and calculate the mean (black square) and standard deviation (thick and thin error bars for 𝜎 and 2𝜎 , respectively) for the probability of a biennial tendency (Figure 3). This nonparametric method conserves the statistical moments of each time series. Again, nearly all of the data and models exhibit a biennial tendency value expected from a random process, thus confirming our previous result from the parametric method. Due to the long tails of most random distributions, we would expect some data points to be located outside of the 2𝜎 range (Figures 2 and 3). Furthermore, it is important to note that even very large uncertainties in the estimation of the biennial tendency exist when considering different data sets. For instance, this is the case when comparing the India summer precipitation between CMAP and GPCP or the WYI between ERA-40 and NCEP2 (Figures 2 and 3).
4. Discussion and Conclusions We showed that even though a spatial TBO pattern with important associated climate impacts exists (e.g., Figure 1), its corresponding time evolution is indistinguishable from a white noise random process and should therefore not be referred to as Tropospheric Biennial Oscillation in future studies. Thus, the observed probability of ∼2/3 for a biennial tendency does not necessarily imply a deterministic mechanism (an oscillation involving coupled land-atmosphere-ocean processes) for the TBO pattern. Hence, it is not advised to perform a narrow biennial band-pass filtering of climate variables (as done, for instance, by Li et al. [2006]) and assume that this filtered signal implies a deterministic TBO.
Acknowledgments This study was partially supported by U.S. NSF grant AGS-1406601 and U.S. Department of Energy grant DE-SC0005110. A.T. additionally was supported by U.S. NSF grant 1049219. We thank Clara Deser, Pedro DiNezio, and Andrew Wittenberg for providing the data from the CESM1(CAM5) and CM2.1 long-integration experiments. Furthermore, we acknowledge discussions with Christina Karamperidou and Mark Cane. We acknowledge the World Climate Research Programme’s Working Group on Coupled Modelling, which is responsible for CMIP, and we thank the climate modeling groups (listed in Table 2 of this paper) for producing and making available their model output. For CMIP the U.S. Department of Energy’s Program for Climate Model Diagnosis and Intercomparison provides coordinating support and led development of software infrastructure in partnership with the Global Organization for Earth System Science Portals. We thank two reviewers for their comments. This is SOEST publication 9505 and IPRC contribution 1149. The Editor thanks Hans von Storch and an anonymous reviewer for their assistance in evaluating this paper.
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Some of the longer integrations (e.g., MIROC5 in Figure 3) exhibit a biennial tendency probability that is below the range expected for a white noise process, thus indicating some persistence in the system which could either indicate a relationship with ENSO (see, for instance, the discussion in Fasullo [2004]) or a red noise process. Note that when using a nonparametric method (Figure 3) instead of a parametric method (Figure 2), much less models are situated outside of the 2𝜎 range. Given the large uncertainty of the TBO indices in the observational record (Figures 2 and 3) we thus require very long data records or model integrations if we want to detect TBO characteristics in seasonal monsoon indices with any statistical confidence. An estimate for the biennial tendency probability expected for different TBO-to-noise ratios is provided in the supporting information (Figure S1). Our results demonstrate that we would require a large signal-to-noise ratio in short data records to detect a deterministic TBO (Figure S1). While deterministic high-frequency climate variability exists in the tropics [e.g., Stuecker et al., 2013, 2015], the null hypothesis of a white noise random process for the observed biennial tendency of the Indian and Australian seasonal monsoon indices cannot be rejected (at the 95% level for most of the data and models). Our results essentially demonstrate that the large-scale Asian summer monsoon does not possess significant memory from year to year, which puts a strong constraint on its predictability. In this paper we propose the appropriate test for identifying the TBO, which should be employed in future studies using seasonal mean definitions of the monsoon.
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TBO INDISTINGUISHABLE FROM WHITE NOISE
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