Climate predictability on interannual to decadal time scales: the initial value problem

Any initial value forecast of climate will be subject to errors originating from poorly known initial conditions, model imperfections, and by "chaos" in the sense that, even if the initial conditions were perfectly known, infinitesimal errors can amplify and spoil the forecast at some lead time. Here the latter source of error is examined using a "perfect model" approach whereby small perturbations are made to a coupled atmosphere-ocean general circulation model and the spread of nearby model trajectories, on time and space scales appropriate to seasonal-decadal climate variability, is measured to assess the lead time at which the error saturates. The study therefore represents an estimate of the upper limit of the predictability of climate (appropriate to the initial value problem) given a perfect model and near perfect knowledge of the initial conditions. It is found that, on average, surface air temperature anomalies are potentially predictable on seasonal to interannual time scales in the tropical regions and are potentially predictable on decadal time scales over the ocean in the North Atlantic. For mid-latitude surface air temperature anomalies over land, model trajectories rapidly diverge and there is little sign of any potential predictability on time scales greater than a season or so. For mean sea level pressure anomalies, there is potential predictability on seasonal time scales in the tropics, and for some global scale annual-decadal anomalies, although not those associated with the North Atlantic Oscillation. For precipitation, the only potential for predictability is for seasonal time anomalies associated with the El-Niño Southern Oscillation. For the majority of the highly populated regions of the world, climate predictability on interannual to decadal time scales based in the initial value approach is likely to be severely limited by chaotic error growth. It is found however that there can be cases in which the potential predictability can be higher than average indicating that there is perhaps some utility in making initial value forecasts of climate in those regions which show low predictability on average.     

Predictability as a function of scale

Abstract

The forecast skill of the Canadian Meteorological Centre (CMC) operational global forecast/analysis system is assessed as a function of scale for the traditional forecast variable of 500‐hPa geopotential height using results from January 2002. These results are compared to an earlier analysis of forecasts from the European Centre for Medium‐range Weather Forecasts (ECMWF) which indicated unexpectedly enhanced skill at high wavenumbers (small scales) especially in the mean forecast component identified with local topographical structures. The global rms error for the CMC forecasts is dominated by the transient component compared to the mean and continues to grow with time during the six days of the forecast. Geographically the transient error grows most rapidly in middle and high latitude regions of large natural variability. The relative error behaves differently and grows most rapidly initially in tropical regions and is inferred to exhibit both climatological and flow‐dependent error growth.

In terms of spherical harmonic two‐dimensional wavenumber n, low wavenumber (large scale) 500‐hPa geopotential height structures are dominated by the mean component but beyond wavenumber 10 to 15 the transient component dominates and exhibits an approximately n–5 spectral slope consistent with a quasi‐two dimensional turbulence enstrophy cascading subrange. Error grows slowly for the large scales dominated by mean climatological structures but these are not of interest for daily weather forecasting. Transient error grows rapidly at small scales and penetrates toward larger scales with time in keeping with the expected predictability behaviour. An expression of the form f(n, τ) = 1 – e–^τ/^τp(n) is fitted to the growth of relative error as a function of wavenumber and forecast range and gives a scale dependent predictability timescale for the transient component that varies as τ_p ? n^?3/2, although the generality of the relationship is not known.

The mean component at intermediate/high wavenumbers exhibits an apparent region of enhanced skill in the CMC system apparently connected to the topography. The result supports the possibility that some small‐scale mean flow structures, although containing only a minor amount of variance, are maintained in the face of errors in other scales. The results do not support the level of enhanced skill found in an earlier analysis of ECMWF results suggesting them to be an artefact of the analysis/forecast system in use at the time.     

初始扰动对一次华南暴雨预报的影响的研究

本文选取了2006年华南前汛期的一次暴雨过程, 采用AREMv2.3中尺度数值模式进行数值模拟, 分别在模式初始场的物理量场 (温度场、 风场、 湿度场) 上加扰动, 分析不同物理量场上的扰动对降水预报的影响, 以及物理量预报误差和扰动能量的增长情况。同时, 通过本个例讨论误差增长与湿对流的关系, 扰动振幅对误差增长的影响和华南区域的中尺度降水的可预报性问题。数值试验结果表明: 初始时刻不同物理量场加实际振幅的正态分布的随机扰动时, 对降水的影响是不同的。对于24小时降水预报, 温度场对降水的影响最大。误差的增长与湿对流不稳定有着密切的关系。小尺度小振幅误差增长很快, 而且是非线性增长。这意味着短期的较小尺度降水的可预报性很小。与大振幅扰动相比, 小振幅扰动造成的误差较小。但是小振幅扰动的迅速发展, 很快就会对降水预报造成较大的影响。因此, 只能有限地提高预报质量, 而且由于扰动非线性增长很快, 在预报时间的提前上, 不会有太大的改善。     

On stochastic stability of regional ocean models to finite-amplitude perturbations of initial conditions

We consider error propagation near an unstable equilibrium state (classified as an unstable focus) for spatially uncorrelated and correlated finite-amplitude initial perturbations using short- (up to several weeks) and intermediate (up to 2 months) range forecast ensembles produced by a barotropic regional ocean model. An ensemble of initial perturbations is generated by the Latin Hypercube design strategy, and its optimal size is estimated through the Kullback–Liebler distance (the relative entropy). Although the ocean model is simple, the prediction error (PE) demonstrates non-trivial behavior similar to that existing in 3D ocean circulation models. In particular, in the limit of zero horizontal viscosity, the PE at first decays with time for all scales due to dissipation caused by non-linear bottom friction, and then grows faster than (quasi)-exponentially. Statistics of a prediction time scale (the irreversible predictability time (IPT)) quickly depart from Gaussian (the linear predictability regime) and becomes Weibullian (the non-linear predictability regime) as amplitude of initial perturbations grows. A transition from linear to non-linear predictability is clearly detected by the specific behavior of IPT variance. A new analytical formula for the model predictability horizon is introduced and applied to estimate the limit of predictability for the ocean model.     

A Case Study of the Error Growth and Predictability of a Meiyu Frontal Heavy Precipitation Event

The Advanced Regional Eta-coordinate Model (AREM) is used to explore the predictability of a heavy rainfall event along the Meiyu front in China during 3-4 July 2003.Based on the sensitivity of precipitation prediction to initial data sources and initial uncertainties in different variables,the evolution of error growth and the associated mechanism are described and discussed in detail in this paper.The results indicate that the smaller-amplitude initial error presents a faster growth rate and its growth is characterized by a transition from localized growth to widespread expansion error.Such modality of the error growth is closely related to the evolvement of the precipitation episode,and consequcntly remarkable forecast divergence is found near the rainband,indicating that the rainfall area is a sensitive region for error growth.The initial error in the rainband contributes significantly to the forecast divergence,and its amplification and propagation are largely determined by the initial moisture distribution.The moisture condition also affects the error growth on smaller scales and the subsequent upscale error cascade.In addition,the error growth defined by an energy norm reveals that large error energy collocates well with the strong latent heating,implying that the occurrence of precipitation and error growth share the same energy source-the latent heat.This may impose an intrinsic predictability limit on the prediction of heavy precipitation.     

Insights into Convective-scale Predictability in East China: Error Growth Dynamics and Associated Impact on Precipitation of Warm-Season Convective Events

This study investigated the regime-dependent predictability using convective-scale ensemble forecasts initialized with different initial condition perturbations in the Yangtze and Huai River basin(YHRB) of East China. The scale-dependent error growth(ensemble variability) and associated impact on precipitation forecasts(precipitation uncertainties) were quantitatively explored for 13 warm-season convective events that were categorized in terms of strong forcing and weak forcing. The forecast err...     

A case study of the error growth and predictability of a Meiyu frontal heavy precipitation event

The Advanced Regional Eta-coordinate Model (AREM) is used to explore the predictability of a heavy rainfall event along the Meiyu front in China during 3-4 July 2003. Based on the sensitivity of precipitation prediction to initial data sources and initial uncertainties in different variables, the evolution of error growth and the associated mechanism are described and discussed in detail in this paper. The results indicate that the smaller-amplitude initial error presents a faster growth rate and its growth...     

实际预报可预报性的时空依赖性分析

本文利用国家气象中心1990年1月1日至1992年2月29日的1～5 d、500 hPa高度场业务预报结果研究了不同球谐函数谱分量和经验正交函数(EOF)谱分量的可预报性。对球谐函数谱分量的研究表明可预报性并不是随着空间尺度的增加呈现单调的递减关系，主要表现为在总波数n等于5附近具有最大可预报性。可预报性主要依赖于总波数n，经卷大圆上的波数和纬向圆周上的波数对可预报性具有完全相同的重要意义。对EOF谱分量的分析表明，随着EOF指数(即序号数)的增大，可预报性依次减小。从EOF和球谐函数的关系发现前几个EOF分量正是对应着最可预报的球谐函数分量。进一步分析表明，最可预报的分量对应大气运动的慢变流型。这些结果对如何使用数值预报产品以及如何进行延伸预报具有重要意义。     

ERROR GROWTH IN NUMERICAL PREDICTION AND ATMOSPHERIC PREDICTABILITY

The article is to report some results of numerical experiments on the error growth and the atmosphericpredictability Experiments with two-level global baroclinic primitive equation spectral model have mainresults as follows.The magnitude of initial errors directly affects the error growth,but its distributionform has little effect on the growth.The loss of predictability resulting from small-scale error is much greaterthan that from large-scale error.The small-scale error rapidly grows and is transferred to the large-scaleerror by interaction between different scale waves,which stimulates the growth of error for the whole systemOrographic forcing restrains planetary-scale error(wavenumbers 0—3)but enhances the small-scale error(wavenumbers 8 or greater).Hence,orographic effects on the error growth closely depend on the characteris-tic scale of initial errors,and there may be a critical wavenumber between 4 and 7.The error growth is great-er in Northern Hemisphere than in Southern Hemisphere if initial errors are the same.In the end we givesome discussions about model,initialization scheme,etc.,to improve model prediction.     

Potential decadal predictability and its sensitivity to sea ice albedo parameterization in a global coupled model

Decadal prediction is one focus of the upcoming 5th IPCC Assessment report. To be able to interpret the results and to further improve the decadal predictions it is important to investigate the potential predictability in the participating climate models. This study analyzes the upper limit of climate predictability on decadal time scales and its dependency on sea ice albedo parameterization by performing two perfect ensemble experiments with the global coupled climate model EC-Earth. In the first experiment, the standard albedo formulation of EC-Earth is used, in the second experiment sea ice albedo is reduced. The potential prognostic predictability is analyzed for a set of oceanic and atmospheric parameters. The decadal predictability of the atmospheric circulation is small. The highest potential predictability was found in air temperature at 2?m height over the northern North Atlantic and the southern South Atlantic. Over land, only a few areas are significantly predictable. The predictability for continental size averages of air temperature is relatively good in all northern hemisphere regions. Sea ice thickness is highly predictable along the ice edges in the North Atlantic Arctic Sector. The meridional overturning circulation is highly predictable in both experiments and governs most of the decadal climate predictability in the northern hemisphere. The experiments using reduced sea ice albedo show some important differences like a generally higher predictability of atmospheric variables in the Arctic or higher predictability of air temperature in Europe. Furthermore, decadal variations are substantially smaller in the simulations with reduced ice albedo, which can be explained by reduced sea ice thickness in these simulations.     

Intrinsic modulation of ENSO predictability viewed through a local Lyapunov lens

The presence of rich ENSO variability in the long unforced simulation of GFDL’s CM2.1 motivates the use of tools from dynamical systems theory to study variability in ENSO predictability, and its connections to ENSO magnitude, frequency, and physical evolution. Local Lyapunov exponents (LLEs) estimated from the monthly NINO3 SSTa model output are used to characterize periods of increased or decreased predictability. The LLEs describe the growth of infinitesimal perturbations due to internal variability, and are a measure of the immediate predictive uncertainty at any given point in the system phase-space. The LLE-derived predictability estimates are compared with those obtained from the error growth in a set of re-forecast experiments with CM2.1. It is shown that the LLEs underestimate the error growth for short forecast lead times (less than 8 months), while they overestimate it for longer lead times. The departure of LLE-derived error growth rates from the re-forecast rates is a linear function of forecast lead time, and is also sensitive to the length of the time series used for the LLE calculation. The LLE-derived error growth rate is closer to that estimated from the re-forecasts for a lead time of 4 months. In the 2,000-year long simulation, the LLE-derived predictability at the 4-month lead time varies (multi)decadally only by 9–18 %. Active ENSO periods are more predictable than inactive ones, while epochs with regular periodicity and moderate magnitude are classified as the most predictable by the LLEs. Events with a deeper thermocline in the west Pacific up to five years prior to their peak, along with an earlier deepening of the thermocline in the east Pacific in the months preceding the peak, are classified as more predictable. Also, the GCM is found to be less predictable than nature under this measure of predictability.     

混沌系统单变量可预报性研究

对于n维的混沌系统, 不同变量的可预报性是不同的。为了研究混沌系统中单个变量的可预报性, 本文在以前提出的混沌系统整体的非线性局部Lyapunov指数基础上(李建平等, 2006), 引入了单变量的非线性局部Lyapunov指数及其相关统计量, 进一步完善了非线性误差增长理论。通过应用到几个混沌个例, 结果表明单变量的非线性局部Lyapunov指数及其相关统计量可以用来定量地研究多维混沌系统中不同变量的可预报性, 系统不同变量的可预报性之间不是相互独立的, 而是单个变量的可预报期限与系统整体的可预报期限之比都近似保持一个常数, 但各个变量的常数值有所不同。     

Forecast Error and Predictability for the Warm-sector and the Frontal Rainstorm in South China

In south China, warm-sector rainstorms are significantly different from the traditional frontal rainstorms due to complex mechanism, which brings great challenges to their forecast. In this study, based on ensemble forecasting, the high-resolution mesoscale numerical forecast model WRF was used to investigate the effect of initial errors on a warmsector rainstorm and a frontal rainstorm under the same circulation in south China, respectively. We analyzed the sensitivity of forecast errors to the...     

北京“7.21”特大暴雨不同集合预报方案的对比试验

采用6套扰动方案(初值、多物理、3组随机物理和初值与随机物理的混合)对2012年7月21日(“7.21”)北京特大暴雨过程进行了集合降水预报试验,检验了不同方案的集合平均预报、集合区间预报和概率预报较控制预报改进的相对程度,分析了它们对该过程时空不确定性的预报能力、不同扰动方法的离散度贡献以及不同尺度扰动对预报误差的贡献等。结果表明:(1)所有集合方案特别是初值扰动、多物理和混合扰动的集合预报相对控制预报在暴雨强度和位置上都有较显著的改进,并为用户决策提供了包括预报不确定性在内的更多预报信息。(2)3组随机物理产生的集合预报离散度很相似, 都远小于初值扰动和多物理方案产生的离散度, 并且主要集中在强降水中心附近, 因此在初值扰动的基础上加入随机扰动,可以提高强降水中心的离散度, 但对强降水中心以外的地区作用甚微;尺度分析进一步表明随机物理产生的离散度贡献主要集中在较小尺度上(<320 km),在更小的尺度上(<160 km)它甚至可以与初值和多物理扰动的贡献相当,而初值扰动和多物理过程的贡献可以比随机物理过程多延伸400—500 km直到较大的尺度(如>1000 km), 其中多物理过程在较小尺度上(<100 km)可比初值扰动贡献更大, 并且能部分消除预报系统偏差。(3) 所有集合扰动方案所产生的离散度尺度谱都与实际预报误差尺度谱分布一致, 即随空间尺度增大而减小,但在幅度上都小于预报误差(离散度不够大),并且这种差异随着空间尺度的减小而加速增大,在小尺度上相差甚大。     

尺度叠加高斯相关模型在GRAPES-RAFS中的应用

背景误差水平相关模型影响着分析增量的结构,同时也决定着不同尺度上分析增量信息的多少.为了提高中小尺度系统的分析质量,研究尺度叠加高斯相关模型的特征及其在三维变分同化系统中的应用效果.通过分析高斯模型和尺度叠加高斯模型的空间特征,以及它们的拉普拉斯算子和谱响应函数的特征,同时依据统计的背景误差特征来改进背景误差水平相关模...     

Spectral distributions of the pressure gradient force errors in σ-coordinate spectral and grid-point models in an idealized case

Summary The spectral distributions of the pressure gradient force errors of the spectral and the finite-difference techniques used in combination with the vertical coordinate were examined in an idealized case of an atmosphere at rest and in hydrostatic equilibrium. The vertical temperature profile was piece-wise linear in lnp, with an inversion at the bottom. Trapezoidal mountains of different widths were used. The same amounts of input information were given to both the spectral and the finite-difference methods. In the rms sense, the spectral errors were generally much larger than those of the finite-difference method. However, on the larger and medium scales, a remarkable similarity of the error spectra of the two methods was found. The build up of the error of the spectral method occurs at the smallest scales. This may explain difficulties in documenting the error in higher resolution spectral models where the contribution to the total error in this part of the spectrum may be removed as the small-scale noise by the horizontal smoothing and/or filtering. In order to reduce the small-scale noise generation, the finite-difference pressure gradient force may be used in spectral models.With 6 Figures     

Baseline predictability of daily east Asian summer monsoon circulation indices

The nonlinear local Lyapunov exponent (NLLE) method is adopted to quantitatively determine the predictability limit of East Asian summer monsoon (EASM) intensity indices on a synoptic timescale. The predictability limit of EASM indices varies widely according to the definitions of indices. EASM indices defined by zonal shear have a limit of around 7 days, which is higher than the predictability limit of EASM indices defined by sea level pressure (SLP) difference and meridional wind shear (about 5 days). The initial error of EASM indices defined by SLP difference and meridional wind shear shows a faster growth than indices defined by zonal wind shear. Furthermore, the indices defined by zonal wind shear appear to fluctuate at lower frequencies, whereas the indices defined by SLP difference and meridional wind shear generally fluctuate at higher frequencies. This result may explain why the daily variability of the EASM indices defined by zonal wind shear tends be more predictable than those defined by SLP difference and meridional wind shear. Analysis of the temporal correlation coefficient (TCC) skill for EASM indices obtained from observations and from NCEP’s Global Ensemble Forecasting System (GEFS) historical weather forecast dataset shows that GEFS has a higher forecast skill for the EASM indices defined by zonal wind shear than for indices defined by SLP difference and meridional wind shear. The predictability limit estimated by the NLLE method is shorter than that in GEFS. In addition, the June-September average TCC skill for different daily EASM indices shows significant interannual variations from 1985 to 2015 in GEFS. However, the TCC for different types of EASM indices does not show coherent interannual fluctuations.     

Error evolution in the dynamics of an ocean general circulation model

The problem of error propagation is considered for spatially uncorrelated errors of the barotropic stream function in an oceanic general circulation model (OGCM). Such errors typically occur when altimetric data from satellites are assimilated into ocean models. It is shown that the error decays at first due to the dissipation of the smallest scales in the error field. The error then grows exponentially before it saturates at the value corresponding to the difference between independent realizations. A simple analytic formula for the error behavior is derived; it matches the numerical results documented for the present primitive-equation ocean model, and other models in the literature.     

水平相关模型的性质及其在GRAPES三维变分系统中的应用

在变分资料同化中背景误差水平相关模型不仅决定着观测信息传播到格点空间的远近,而且影响着频谱空间中不同尺度上的分析增量信息的多少.本文比较高斯(Gauss)、二阶自回归(Soar)以及尺度叠加高斯模型(Supergauss)在时空域随着空间距离和在频谱域随着不同尺度分布的特点,阐述三种相关模型在区域GRAPES三维变分分...     

Characteristics of tropical Pacific SST predictability in coupled GCM forecasts using the NCEP CFS

The limits of predictability of El Niño and the Southern Oscillation (ENSO) in coupled models are investigated based on retrospective forecasts of sea surface temperature (SST) made with the National Centers for Environmental Prediction (NCEP) coupled forecast system (CFS). The influence of initial uncertainties and model errors associated with coupled ENSO dynamics on forecast error growth are discussed. The total forecast error has maximum values in the equatorial Pacific and its growth is a strong function of season irrespective of lead time. The largest growth of systematic error of SST occurs mainly over the equatorial central and eastern Pacific and near the southeastern coast of the Americas associated with ENSO events. After subtracting the systematic error, the root-mean-square error of the retrospective forecast SST anomaly also shows a clear seasonal dependency associated with what is called spring barrier. The predictability with respect to ENSO phase shows that the phase locking of ENSO to the mean annual cycle has an influence on the seasonal dependence of skill, since the growth phase of ENSO events is more predictable than the decay phase. The overall characteristics of predictability in the coupled system are assessed by comparing the forecast error growth and the error growth between two model forecasts whose initial conditions are 1 month apart. For the ensemble mean, there is fast growth of error associated with initial uncertainties, becoming saturated within 2 months. The subsequent error growth follows the slow coupled mode related the model’s incorrect ENSO dynamics. As a result, the Lorenz curve of the ensemble mean NINO3 index does not grow, because the systematic error is identical to the same target month. In contrast, the errors of individual members grow as fast as forecast error due to the large instability of the coupled system. Because the model errors are so systematic, their influence on the forecast skill is investigated by analyzing the erroneous features in a long simulation. For the ENSO forecasts in CFS, a constant phase shift with respect to lead month is clear, using monthly forecast composite data. This feature is related to the typical ENSO behavior produced by the model that, unlike the observations, has a long life cycle with a JJA peak. Therefore, the systematic errors in the long run are reflected in the forecast skill as a major factor limiting predictability after the impact of initial uncertainties fades out.