The Forecast Skills and Predictability Sources of Marine Heatwaves in the NUIST-CFS1.0 Hindcasts

Using monthly observations and ensemble hindcasts of the Nanjing University of Information Science and Technology Climate Forecast System(NUIST-CFS1.0) for the period 1983–2020, this study investigates the forecast skill of marine heatwaves(MHWs) over the globe and the predictability sources of the MHWs over the tropical oceans. The MHW forecasts are demonstrated to be skillful on seasonal-annual time scales, particularly in tropical oceans. The forecast skill of the MHWs over the tropical Pacific Ocean(TPO) remains high at lead times of 1–24 months, indicating a forecast better than random chance for up to two years. The forecast skill is subject to the spring predictability barrier of El Nino-Southern Oscillation(ENSO). The forecast skills for the MHWs over the tropical Indian Ocean(TIO), tropical Atlantic Ocean(TAO), and tropical Northwest Pacific(NWP) are lower than that in the TPO. A reliable forecast at lead times of up to two years is shown over the TIO, while a shorter reliable forecast window(less than 17 months) occurs for the TAO and NWP.Additionally, the forecast skills for the TIO, TAO, and NWP are seasonally dependent. Higher skills for the TIO and TAO appear in boreal spring, while a greater skill for the NWP emerges in late summer-early autumn. Further analyses suggest that ENSO serves as a critical source of predictability for MHWs over the TIO and TAO in spring and MHWs over the NWP in summer.

The Predictability Limit of Oceanic Mesoscale Eddy Tracks in the South China Sea

Employing the nonlinear local Lyapunov exponent (NLLE) technique, this study assesses the quantitative predictability limit of oceanic mesoscale eddy (OME) tracks utilizing three eddy datasets for both annual and seasonal means. Our findings reveal a discernible predictability limit of approximately 39 days for cyclonic eddies (CEs) and 44 days for anticyclonic eddies (AEs) within the South China Sea (SCS). The predictability limit is related to the OME properties and seasons. The long-lived, large-amplitude, and large-radius OMEs tend to have a higher predictability limit. The predictability limit of AE (CE) tracks is highest in autumn (winter) with 52 (53) days and lowest in spring (summer) with 40 (30) days. The spatial distribution of the predictability limit of OME tracks also has seasonal variations, further finding that the area of higher predictability limits often overlaps with periodic OMEs. Additionally, the predictability limit of periodic OME tracks is about 49 days for both CEs and AEs, which is 5-10 days higher than the mean values. Usually, in the SCS, OMEs characterized by high predictability limit values exhibit more extended and smoother trajectories and often move along the northern slope of the SCS.

Application of the Conditional Nonlinear Local Lyapunov Exponent to Second-Kind Predictability

In order to quantify the influence of external forcings on the predictability limit using observational data,the author introduced an algorithm of the conditional nonlinear local Lyapunov exponent(CNLLE)method.The effectiveness of this algorithm is validated and compared with the nonlinear local Lyapunov exponent(NLLE)and signal-to-noise ratio methods using a coupled Lorenz model.The results show that the CNLLE method is able to capture the slow error growth constrained by external forcings,therefore,it can quantify the predictability limit induced by the external forcings.On this basis,a preliminary attempt was made to apply this method to measure the influence of ENSO on the predictability limit for both atmospheric and oceanic variable fields.The spatial distribution of the predictability limit induced by ENSO is similar to that arising from the initial conditions calculated by the NLLE method.This similarity supports ENSO as the major predictable signal for weather and climate prediction.In addition,a ratio of predictability limit(RPL)calculated by the CNLLE method to that calculated by the NLLE method was proposed.The RPL larger than 1 indicates that the external forcings can significantly benefit the long-term predictability limit.For instance,ENSO can effectively extend the predictability limit arising from the initial conditions of sea surface temperature over the tropical Indian Ocean by approximately four months,as well as the predictability limit of sea level pressure over the eastern and western Pacific Ocean.Moreover,the impact of ENSO on the geopotential height predictability limit is primarily confined to the troposphere.

Analysis of Predictability of a Large-scale Short-duration Heavy Precipitation Process in Nanchang City

Based on the observation data of automatic stations and sounding data,the circulation characteristics and physical quantities of a large-scale short-duration heavy precipitation process in Nanchang City on July 7,2020 were diagnosed and analyzed,and the ability of several numerical forecasting products to predict this process was tested.The results show that the short-duration heavy precipitation process was triggered in the process of the subtropical high changing from lifting to the north to retreating to the south under the weather background of the confrontation between the northerly flow behind the trough and the strong southwest warm and wet flow on the north side of the subtropical high.The strong southwest warm and wet flow provided abundant water vapor,and the southern pressing of the lower energy front and the invasion of the cold air near the surface layer provided unstable energy and dynamic conditions for the heavy precipitation.The changing trend of the subtropical high from lifting to the north to retreating to the south during 08:00 to 20:00 on July 7 was not predicted by numerical forecast,and there was a large deviation in the forecast of the time and intensity of the southern pressing of the northerly flow behind the trough,so the guidance of numerical forecast for heavy precipitation was not strong,which was not conducive to the prediction of the short-duration heavy precipitation.It was predicted that the subtropical high would move slightly to the south on July 6 compared with the previous day,and the forecast adjustment of the high-level weather system can be used as a sign of the forecast change,which needs to be paid certain attention in the daily forecast.

Understanding the Low Predictability of the 2015/16 El Niño Event Based on a Deep Learning Model

The 2015/16 El Niño event ranks among the top three of the last 100 years in terms of intensity,but most dynamical models had a relatively low prediction skill for this event before the summer months.Therefore,the attribution of this particular event can help us to understand the cause of super El Niño–Southern Oscillation events and how to forecast them skillfully.The present study applies attribute methods based on a deep learning model to study the key factors related to the formation of this event.A deep learning model is trained using historical simulations from 21 CMIP6 models to predict the Niño-3.4 index.The integrated gradient method is then used to identify the key signals in the North Pacific that determine the evolution of the Niño-3.4 index.These crucial signals are then masked in the initial conditions to verify their roles in the prediction.In addition to confirming the key signals inducing the super El Niño event revealed in previous attribution studies,we identify the combined contribution of the tropical North Atlantic and the South Pacific oceans to the evolution and intensity of this event,emphasizing the crucial role of the interactions among them and the North Pacific.This approach is also applied to other El Niño events,revealing several new precursor signals.This study suggests that the deep learning method is useful in attributing the key factors inducing extreme tropical climate events.

Predictability of the upper ocean heat content in a Community Earth System Model ensemble prediction system

Upper ocean heat content(OHC)has been widely recognized as a crucial precursor to high-impact climate variability,especially for that being indispensable to the long-term memory of the ocean.Assessing the predictability of OHC using state-of-the-art climate models is invaluable for improving and advancing climate forecasts.Recently developed retrospective forecast experiments,based on a Community Earth System Model ensemble prediction system,offer a great opportunity to comprehensively explore OHC predictability.Our results indicate that the skill of actual OHC predictions varies across different oceans and diminishes as the lead time of prediction extends.The spatial distribution of the actual prediction skill closely resembles the corresponding persistence skill,indicating that the persistence of OHC serves as the primary predictive signal for its predictability.The decline in actual prediction skill is more pronounced in the Indian and Atlantic oceans than in the Pacific Ocean,particularly within tropical regions.Additionally,notable seasonal variations in the actual prediction skills across different oceans align well with the phase-locking features of OHC variability.The potential predictability of OHC generally surpasses the actual prediction skill at all lead times,highlighting significant room for improvement in current OHC predictions,especially for the North Indian Ocean and the Atlantic Ocean.Achieving such improvements necessitates a collaborative effort to enhance the quality of ocean observations,develop effective data assimilation methods,and reduce model bias.

Role of Parameter Errors in the Spring Predictability Barrier for ENSO Events in the Zebiak–Cane Model

ABSTRACT The impact of both initial and parameter errors on the spring predictability barrier （SPB） is investigated using the Zebiak Cane model （ZC model）. Previous studies have shown that initial errors contribute more to the SPB than parameter errors in the ZC model. Although parameter errors themselves are less important, there is a possibility that nonlinear interactions can occur between the two types of errors, leading to larger prediction errors compared with those induced by initial errors alone. In this case, the impact of parameter errors cannot be overlooked. In the present paper, the optimal combination of these two types of errors [i.e., conditional nonlinear optimal perturbation （CNOP） errors] is calculated to investigate whether this optimal error combination may cause a more notable SPB phenomenon than that caused by initial errors alone. Using the CNOP approach, the CNOP errors and CNOP-I errors （optimal errors when only initial errors are considered） are calculated and then three aspects of error growth are compared： （1） the tendency of the seasonal error growth; （2） the prediction error of the sea surface temperature anomaly; and （3） the pattern of error growth. All three aspects show that the CNOP errors do not cause a more significant SPB than the CNOP-I errors. Therefore, this result suggests that we could improve the prediction of the E1 Nifio during spring by simply focusing on reducing the initial errors in this model.

The“Spring Predictability Barrier” Phenomenon of ENSO Predictions Generated with the FGOALS-g Model

Using the sea surface temperature （SST） predicted for the equatorial Pacific Ocean by the Flexible Global Ocean-Atmosphere-Land System Model-gamil （FGOALS-g）, an analysis of the prediction errors was performed for the seasonally dependent predictability of SST anomalies both for neutral years and for the growth/decay phase of El Nino/La Nina events. The study results indicated that for the SST predictions relating to the growth phase and the decay phase of El Nino events, the prediction errors have a seasonally dependent evolution. The largest increase in errors occurred in the spring season, which indicates that a prominent spring predictability barrier （SPB） occurs during an El Nino-Southern Oscillation （ENSO） warming episode. Furthermore, the SPB associated with the growth-phase prediction is more prominent than that associated with the decay-phase prediction. However, for the neutral years and for the growth and decay phases of La Nifia events, the SPB phenomenon was less prominent. These results indicate that the SPB phenomenon depends extensively on the ENSO events themselves. In particular, the SPB depends on the phases of the ENSO events. These results may provide useful knowledge for improving ENSO forecasting.

Recent Advances in Predictability Studies in China (1999-2002)

Since the last International Union of Geodesy and Geophysics (IUGG) General Assembly (1999), the predictability studies in China have made further progress during the period of 1999-2002. Firstly, three predictability sub-problems in numerical weather and climate prediction are classified, which are concerned with the maximum predictability time, the maximum prediction error, and the maximum allowable initial error, and then they are reduced into three nonlinear optimization problems. Secondly, the concepts of the nonlinear singular vector (NSV) and conditional nonlinear optimal perturbation (CNOP) are proposed, which have been utilized to study the predictability of numerical weather and climate prediction. The results suggest that the nonlinear characteristics of the motions of atmosphere and oceans can be revealed by NSV and CNOP. Thirdly, attention has also been paid to the relations between the predictability and spatial-temporal scale, and between the modei predictability and the machine precision, of which the investigations disclose the importance of the spatial-temporal scale and machine precision in the study of predictability. Also the cell-to-cell mapping is adopted to analyze globally the predictability of climate, which could provide a new subject to the research workers. Furthermore, the predictability of the summer rainfall in China is investigated by using the method of correlation coefficients. The results demonstrate that the predictability of summer rainfall is different in different areas of China. Analysis of variance, which is one of the statistical methods applicable to the study of predictability, is also used to study the potential predictability of monthly mean temperature in China, of which the conclusion is that the monthly mean temperature over China is potentially predictable at a statistical significance Ievel of 0.10. In addition, in the analysis of the predictability of the T106 objective analysis/forecasting field, the variance and the correlation coemcient are calculated to explore the distribution characteristics of the mean-square errors. Finally, the predictability of short-term climate prediction is investigated by using statistical methods or numerical simulation methods. It is demonstrated that the predictability of short-terrn climate in China depends not only on the region of China being investigated, but also on the time scale and the atmospheric internai dynamical process.

Is Model Parameter Error Related to a Significant Spring Predictability Barrier for El Nio events? Results from a Theoretical Model

Within a theoretical ENSO model, the authors investigated whether or not the errors superimposed on model parameters could cause a significant ＂spring predictability barrier＂ （SPB） for El Nio events. First, sensitivity experiments were respectively performed to the air-sea coupling parameter, α and the thermocline effect coefficient μ. The results showed that the uncertainties superimposed on each of the two parameters did not exhibit an obvious season-dependent evolution; furthermore, the uncertainties caused a very small prediction error and consequently failed to yield a significant SPB. Subsequently, the conditional nonlinear optimal perturbation （CNOP） approach was used to study the effect of the optimal mode （CNOP-P） of the uncertainties of the two parameters on the SPB and to demonstrate that the CNOP-P errors neither presented a unified season-dependent evolution for different El Nio events nor caused a large prediction error, and therefore did not cause a significant SPB. The parameter errors played only a trivial role in yielding a significant SPB. To further validate this conclusion, the authors investigated the effect of the optimal combined mode （i.e. CNOP error） of initial and model errors on SPB. The results illustrated that the CNOP errors tended to have a significant season-dependent evolution, with the largest error growth rate in the spring, and yielded a large prediction error, inducing a significant SPB. The inference, therefore, is that initial errors, rather than model parameter errors, may be the dominant source of uncertainties that cause a significant SPB for El Nio events. These results indicate that the ability to forecast ENSO could be greatly increased by improving the initialization of the forecast model.

Application of the Conditional Nonlinear Optimal Perturbation Method to the Predictability Study of the Kuroshio Large Meander

A reduced-gravity barotropic shallow-water model was used to simulate the Kuroshio path variations. The results show that the model was able to capture the essential features of these path variations. We used one simulation of the model as the reference state and investigated the effects of errors in model parameters on the prediction of the transition to the Kuroshio large meander （KLM） state using the conditional nonlinear optimal parameter perturbation （CNOP-P） method. Because of their relatively large uncertainties, three model parameters were considered： the interracial friction coefficient, the wind-stress amplitude, and the lateral friction coefficient. We determined the CNOP-Ps optimized for each of these three parameters independently, and we optimized all three parameters simultaneously using the Spectral Projected Gradient 2 （SPG2） algorithm. Similarly, the impacts caused by errors in initial conditions were examined using the conditional nonlinear optimal initial perturbation （CNOP-I） method. Both the CNOP-I and CNOP-Ps can result in significant prediction errors of the KLM over a lead time of 240 days. But the prediction error caused by CNOP-I is greater than that caused by CNOP-P. The results of this study indicate not only that initial condition errors have greater effects on the prediction of the KLM than errors in model parameters but also that the latter cannot be ignored. Hence, to enhance the forecast skill of the KLM in this model, the initial conditions should first be improved, the model parameters should use the best possible estimates.

Ensemble Forecast: A New Approach to Uncertainty and Predictability

Ensemble techniques have been used to generate daily numerical weather forecasts since the 1990s in numerical centers around the world due to the increase in computation ability. One of the main purposes of numerical ensemble forecasts is to try to assimilate the initial uncertainty （initial error） and the forecast uncertainty （forecast error） by applying either the initial perturbation method or the multi-model/multiphysics method. In fact, the mean of an ensemble forecast offers a better forecast than a deterministic （or control） forecast after a short lead time （3-5 days） for global modelling applications. There is about a 1-2-day improvement in the forecast skill when using an ensemble mean instead of a single forecast for longer lead-time. The skillful forecast （65% and above of an anomaly correlation） could be extended to 8 days （or longer） by present-day ensemble forecast systems. Furthermore, ensemble forecasts can deliver a probabilistic forecast to the users, which is based on the probability density function （PDF） instead of a single-value forecast from a traditional deterministic system. It has long been recognized that the ensemble forecast not only improves our weather forecast predictability but also offers a remarkable forecast for the future uncertainty, such as the relative measure of predictability （RMOP） and probabilistic quantitative precipitation forecast （PQPF）. Not surprisingly, the success of the ensemble forecast and its wide application greatly increase the confidence of model developers and research communities.

The Predictability Problems in Numerical Weather and Climate Prediction

The uncertainties caused by the errors of the initial states and the parameters in the numerical model are investigated. Three problems of predictability in numerical weather and climate prediction are proposed, which are related to the maximum predictable time, the maximum prediction error, and the maximum admissible errors of the initial values and the parameters in the model respectively. The three problems are then formulated into nonlinear optimization problems. Effective approaches to deal with these nonlinear optimization problems are provided. The Lorenz’ model is employed to demonstrate how to use these ideas in dealing with these three problems.

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 error growth in the strong-forcing regime shows a stepwise increase with increasing spatial scale,while the error growth shows a larger temporal variability with an afternoon peak appearing at smaller scales under weak forcing.This leads to the dissimilarity of precipitation uncertainty and shows a strong correlation between error growth and precipitation across spatial scales.The lateral boundary condition errors exert a quasi-linear increase on error growth with time at the larger scale,suggesting that the large-scale flow could govern the magnitude of error growth and associated precipitation uncertainties,especially for the strong-forcing regime.Further comparisons between scale-based initial error sensitivity experiments show evident scale interaction including upscale transfer of small-scale errors and downscale cascade of larger-scale errors.Specifically,small-scale errors are found to be more sensitive in the weak-forcing regime than those under strong forcing.Meanwhile,larger-scale initial errors are responsible for the error growth after 4 h and produce the precipitation uncertainties at the meso-β-scale.Consequently,these results can be used to explain underdispersion issues in convective-scale ensemble forecasts and provide feedback for ensemble design over the YHRB.

The Predictability of a Squall Line in South China on 23 April 2007

This study investigated the predictability of a squall line associated with a quasi-stationary front on 23 April 2007 in South China through deterministic and probabilistic forecasts. Our results show that the squallline simulation was very sensitive to model error from horizontal resolution and uncertainties in physical parameterization schemes. At least a 10-km grid size was necessary to decently capture this squall line. The simulated squall line with a grid size of 4.5 km was most sensitive to long-wave radiation parameterization schemes relative to other physical schemes such as microphysics and planetary boundary layer. For a grid size from 20 to 5 km, a cumulus parameterization scheme degraded the squall-line simulation （relative to turning it off）, with a more severe degradation to grid size -10 km than 〉10 km. The sensitivity of the squall-line simulation to initial error was investigated through ensemble forecast. The performance of the ensemble simulation of the squall line was very sensitive to the initial error. Approximately 15% of the ensemble members decently captured the evolution of the squall line, 25% failed, and 60% dislocated the squall line. Using different combinations of physical parameterization schemes for different members can improve the probabilistic forecast. The lead time of this case was only a few hours. Error growth was clearly associated with moist convection development. A linear improvement in the performance of the squall line simulation was observed when the initial error was decreased gradually, with the largest contribution from initial moisture field.

Applications of Conditional Nonlinear Optimal Perturbation in Predictability Study and Sensitivity Analysis of Weather and Climate

Considering the limitation of the linear theory of singular vector （SV）, the authors and their collabora- tors proposed conditional nonlinear optimal perturbation （CNOP） and then applied it in the predictability study and the sensitivity analysis of weather and climate system. To celebrate the 20th anniversary of Chinese National Committee for World Climate Research Programme （WCRP）, this paper is devoted to reviewing the main results of these studies. First, CNOP represents the initial perturbation that has largest nonlinear evolution at prediction time, which is different from linear singular vector （LSV） for the large magnitude of initial perturbation or/and the long optimization time interval. Second, CNOP, rather than linear singular vector （LSV）, represents the initial anomaly that evolves into ENSO events most probably. It is also the CNOP that induces the most prominent seasonal variation of error growth for ENSO predictability; furthermore, CNOP was applied to investigate the decadal variability of ENSO asymmetry. It is demonstrated that the changing nonlinearity causes the change of ENSO asymmetry. Third, in the studies of the sensitivity and stability of ocean＇s thermohaline circulation （THC）, the nonlinear asymmetric response of THC to finite amplitude of initial perturbations was revealed by CNOP. Through this approach the passive mechanism of decadal variation of THC was demonstrated; Also the authors studies the instability and sensitivity analysis of grassland ecosystem by using CNOP and show the mechanism of the transitions between the grassland and desert states. Finally, a detailed discussion on the results obtained by CNOP suggests the applicability of CNOP in predictability studies and sensitivity analysis.

Seasonal Differences of Model Predictability and the Impact of SST in the Pacific

Both seasonal potential predictability and the impact of SST in the Pacific on the forecast skill over China are investigated by using a 9-level global atmospheric general circulation model developed at the Institute of Atmospheric Physics under the Chinese Academy of Sciences (IAP9L-AGCM). For each year during 1970 to 1999, the ensemble consists of seven integrations started from consecutive observational daily atmospheric fields and forced by observational monthly SST. For boreal winter, spring and summer, the variance ratios of the SST-forced variability to the total variability and the differences in the spatial correlation coefficients of seasonal mean fields in special years versus normal years are computed respectively. It follows that there are slightly inter-seasonal differences in the model potential predictability in the Tropics. At northern middle and high latitudes, prediction skill is generally low in spring and relatively high either in summer for surface air temperature and middle and upper tropospheric geopotential height or in winter for wind and precipitation. In general, prediction skill rises notably in western China, especially in northwestern China, when SST anomalies (SSTA) in the Nino-3 region are significant. Moreover, particular attention should be paid to the SSTA in the North Pacific (NP) if one aims to predict summer climate over the eastern part of China, i.e., northeastern China, North China and southeastern China.

Mesoscale Predictability of Mei-yu Heavy Rainfall

Recently reported results indicate that small amplitude and small scale initial errors grow rapidly and subsequently contaminate short-term deterministic mesoscale forecasts. This rapid error growth is dependent on not only moist convection but also the flow regime. In this study, the mesoscale predictability and error growth of mei-yu heavy rainfall is investigated by simulating a particular precipitation event along the mei-yu front on 4- 6 July 2003 in eastern China. Due to the multi-scale character of the mei-yu front and scale interactions, the error growth of mei-yu heavy rainfall forecasts is markedly different from that in middle-latitude moist baroclinic systems. The optimal growth of the errors has a relatively wide spectrum, though it gradually migrates with time from small scale to mesoscale. During the whole period of this heavy rainfall event, the error growth has three different stages, which similar to the evolution of 6-hour accumulated precipitation. Multi-step error growth manifests as an increase of the amplitude of errors, the horizontal scale of the errors, or both. The vertical profile of forecast errors in the developing convective instability and the moist physics convective system indicates two peaks, which correspond with inside the mei-yu front, and related to moist The error growth for the mei-yu heavy rainfall is concentrated convective instability and scale interaction.

The Nature and Predictability of the East Asian Extreme Cold Events of 2020/21

Three extreme cold events invaded China during the early winter period between December 2020 to mid-January 2021 and caused drastic temperature drops,setting new low-temperature records at many stations during 6−8 January 2021.These cold events occurred under background conditions of low Arctic sea ice extent and a La Niña event.This is somewhat expected since the coupled effect of large Arctic sea ice loss in autumn and sea surface temperature cooling in the tropical Pacific usually favors cold event occurrences in Eurasia.Further diagnosis reveals that the first cold event is related to the southward movement of the polar vortex and the second one is related to a continent-wide ridge,while both the southward polar vortex and the Asian blocking are crucial for the third event.Here,we evaluate the forecast skill for these three events utilizing the operational forecasts from the ECMWF model.We find that the third event had the highest predictability since it achieves the best skill in forecasting the East Asian cooling among the three events.Therefore,the predictability of these cold events,as well as their relationships with the atmospheric initial conditions,Arctic sea ice,and La Niña deserve further investigation.

A New Strategy for Solving a Class of Constrained Nonlinear Optimization Problems Related to Weather and Climate Predictability

There are three common types of predictability problems in weather and climate, which each involve different constrained nonlinear optimization problems： the lower bound of maximum predictable time, the upper bound of maximum prediction error, and the lower bound of maximum allowable initial error and parameter error. Highly effcient algorithms have been developed to solve the second optimization problem. And this optimization problem can be used in realistic models for weather and climate to study the upper bound of the maximum prediction error. Although a filtering strategy has been adopted to solve the other two problems, direct solutions are very time-consuming even for a very simple model, which therefore limits the applicability of these two predictability problems in realistic models. In this paper, a new strategy is designed to solve these problems, involving the use of the existing highly effcient algorithms for the second predictability problem in particular. Furthermore, a series of comparisons between the older filtering strategy and the new method are performed. It is demonstrated that the new strategy not only outputs the same results as the old one, but is also more computationally effcient. This would suggest that it is possible to study the predictability problems associated with these two nonlinear optimization problems in realistic forecast models of weather or climate.