ERA-40 Wind Research Articles

Morphological modeling of Nowshahr Port entrance with yearly to decadal time scales

ABSTRACT This study examines long-term morphological changes near Nowshahr Port’s entrance on the Caspian Sea’s southern coast. Using Delft3D software, numerical simulations were conducted after calibrating the model with observed wave, current, and morphological data. ERA5 wave and wind data were used as primary drivers, though findings revealed these were underestimated for the area, requiring corrections. A one-year simulation (2019) showed that while wave-driven sediment flux dominates, wind contributes an additional 17% to sedimentation in the access channel, with water levels decreasing slightly (~7%). An 11-year simulation assessed the impact of realigning the access channel on erosion and sedimentation. Waves below 0.75 m were excluded due to negligible effects, and a morphological acceleration factor (MorFac=12) reduced computational time by 93%. Results suggest that optimizing the channel’s alignment could cut maintenance dredging by a third and remove shoals formed from past improper dredging.

A new global high-resolution wave model for the tropical ocean using WAVEWATCH III version 7.14

Abstract. Climate change is driving sea-level rise and potentially intensifying extreme events in the tropical belt, thereby increasing coastal hazards. On tropical islands, extreme sea levels and subsequent marine flooding can be triggered by cyclones but also distant-source swells. Knowledge of sea states in the tropical ocean is thus of key importance, and their study is usually based on spectral wave models. However, existing global wave models typically employ regular grids with a coarse resolution, which fail to accurately represent volcanic archipelagos, a problem usually circumvented by the use of obstruction grids but typically resulting in large negative biases. To overcome this problem, this study presents a new global wave model with a focus on distant-source swells, which have received less attention than waves generated by cyclones. To accurately simulate sea states in tropical areas, we implemented the spectral wave model WAVEWATCH III© (WW3) over a global unstructured grid with a spatial resolution ranging from 50 km to 100 m. The model is forced by ERA5 wind fields, corrected for negative biases through a quantile–quantile approach based on satellite radiometer data. The wind input source terms adjusted accordingly and the explicit representation of tropical islands result in improved predictive skills in the tropical ocean. Moreover, this new simulation allows for the first time direct comparisons with the in situ data collected on volcanic islands at water depths ranging from 10 to 30 m, which corresponds to a few hundred meters from the shore.

The added value and potential of long-term radio occultation data for climatological wind field monitoring

Abstract. Global long-term stable 3D wind fields provide valuable information for climate-oriented analyses of the dynamics of the atmosphere. Their monitoring remains a challenging task given the shortcomings of available observations. One promising option for progress is the use of radio occultation (RO) satellite data, which enable deriving dynamics based on thermodynamic data. In this study we focus on three main goals, explored through the fifth version of the European Centre for Medium-Range Weather Forecasts (ECMWF) Reanalysis (ERA5) and RO datasets, using monthly-mean January and July data over 2007–2020. Our focus is on a 2.5° × 2.5° spatial synoptic scale over the free troposphere to the mid-stratosphere (i.e. 800–10 hPa). First, by comparing ERA5-derived geostrophic and gradient wind speeds to the original ERA5 ones, we examine the regions of validity of the studied approximations at a given synoptic-scale resolution. Second, to assess the possible added value of the RO-derived climatic winds in terms of their long-term stability, we test their consistency with the corresponding ERA5-derived winds. Third, by comparing the RO climatic winds to the original ERA5 winds, we evaluate the potential benefit of RO as an additional dataset for wind analyses and climate monitoring. With this three-step analysis, we decompose the total wind speed bias into the contributions from the approximation and the systematic difference between the RO and ERA5 datasets. We find that the geostrophic approximation is a valid method to estimate winds in the free troposphere, while the gradient wind approximation works better in the lower stratosphere. Both approximations generally work well over the mentioned altitudes, within an accuracy of 2 m s−1 for the latitudes 5–82.5°. Exceptions are found in winter in the monsoonal area and above larger mountain ranges in the free troposphere, as well as above the northern polar regions in the mid-stratosphere. RO- and ERA5-derived geostrophic winds mostly showed good agreement (within 2 m s−1). However, temporal change in the systematic difference higher than 0.5 m s−1 per decade was found. This points to a possible impact of changes in the source of the assimilated data in ERA5. The overall high accuracy of the monthly-mean wind fields, backed by the long-term stability and fine vertical resolution of the underlying RO data, highlights the added value and potential benefit of RO-derived climatic winds for climate monitoring and analyses.

Characterization of local wind profiles: a random forest approach for enhanced wind profile extrapolation

Abstract. Accurate wind speed determination at the height of the rotor swept area is critical for resource assessments. ERA5 data combined with short-term measurements through the “measure, correlate, predict” (MCP) method are commonly used for offshore applications in this context. However, ERA5 poses limitations in capturing site-specific wind speed variability due to its low resolution. To address this, we developed random forest models extending near-surface wind speed up to 200 m, focusing on the Dutch part of the North Sea. Based on public 2-year floating lidar data collected at four locations, the 15 % testing subset shows that the random forest model trained on the remaining 85 % of site-specific wind profiles outperforms the MCP-corrected ERA5 wind profiles in accuracy, bias, and correlation. In the absence of rotor height measurements, a model trained within a 200 km region handles vertical extension effectively, albeit with increased bias. Our regionally trained random forest model exhibits superior accuracy in capturing wind speed variations and local effects, with an average deviation below 5 % compared to corrected ERA5 with a 20 % deviation from measurements. The 10 min random-forest-predicted wind speeds capture the mesoscale section of the power spectrum where ERA5 shows degradation. For stable conditions the root mean squared error and bias are 12 % and 29 % larger, respectively, compared to unstable conditions, which can be attributed to the decoupling effect at higher heights from the surface during stable stratification. Our study highlights the potential enhancement in wind resource assessment by means of machine learning methods, specifically random forest. Future research may explore extending the random forest methodology for higher heights, benefiting a new generation of offshore wind turbines, and investigating cluster wakes in the North Sea through a multinational network of floating lidars, contingent on data availability.

Typhoon Effects on Surface Phytoplankton Biomass Based on Satellite-Derived Chlorophyll-a in the East Sea During Summer

The East Sea is a jointly managed maritime area of Korea, Russia, and Japan, where the frequency of strong typhoons is anticipated to increase with climate change, affecting its marine ecosystem and regional climate regulation. This study investigated the environmental and ecological impacts of summer typhoons entering the East Sea by analyzing satellite-derived chlorophyll-a (Chl-a) data, Argo float measurements, and ERA5 wind data. Our findings revealed that summer typhoons generally increased surface Chl-a concentrations by 65.4%, with typhoon intensity substantially influencing this process. Weak typhoons caused marginal Chl-a increases attributed to redistribution rather than nutrient supply, whereas normal and strong typhoons increased Chl-a through enhanced vertical mixing and nutrient upwelling in the East Sea. Stronger typhoons notably impacted the mixed layer depth and isothermal layer depth, leading to greater Chl-a concentrations within the strong wind radius. However, the increased Chl-a magnitude was lower than that of other strong typhoons in other regions. The East Sea uniquely responds to typhoons with fewer upper environment changes, possibly due to a stable barrier layer limiting vertical mixing. These findings underscore the importance of continuous monitoring and integrated observational methods in order to better understand the ecological effects of typhoons, particularly as their intensity increases with climate change.

An Assessment of Caspian Sea Modeled Wave Climate using ERA5 winds

An Assessment of Caspian Sea Modeled Wave Climate using ERA5 winds

Typhoon Storm Surge Simulation Study Based on Reconstructed ERA5 Wind Fields—A Case Study of Typhoon “Muifa”, the 12th Typhoon of 2022

A storm surge, classified as an extreme natural disaster, refers to unusual sea level fluctuations induced by severe atmospheric disturbances such as typhoons. Existing reanalysis data, such as ERA5, significantly underestimates the location and maximum wind speed of typhoons. Therefore, this study initially assesses the accuracy of tropical cyclone positions and peak wind speeds in the ERA5 reanalysis dataset. These results are compared against tropical cyclone parameters from the IBTrACS (International Best Track Archive for Climate Stewardship). The position deviation of tropical cyclones in ERA5 is mainly within the range of 10 to 60 km. While the correlation of maximum wind speed is significant, there is still considerable underestimation. A wind field reconstruction model, incorporating tropical cyclone characteristics and a distance correction factor, was employed. This model considers the effects of the surrounding environment during the movement of the tropical cyclone by introducing a decay coefficient. The reconstructed wind field significantly improved the representation of the typhoon eyewall and high-wind-speed regions, showing a closer match with wind speeds observed by the HY-2B scatterometer. Through simulations using the FVCOM (Finite Volume Community Ocean Model) storm surge model, the reconstructed wind field demonstrated higher accuracy in reproducing water level changes at Tanxu, Gaoqiao, and Zhangjiabang stations. During the typhoon’s landfall in Shanghai, the area with the greatest water level increase was primarily located in the coastal waters of Pudong New Area, Shanghai, where the highest total water level reached 5.2 m and the storm surge reached 4 m. The methods and results of this study provide robust technical support and a valuable reference for further storm surge forecasting, marine disaster risk assessment, and coastal disaster prevention and mitigation efforts.

Improved wind power assessments by bias adjusted reanalysed data with applications near Morocco’s coast

The wind energy requirements for high temporal and spatial observations, unavailable in many countries, are met by reanalysed wind data, which are increasingly used in wind power assessments. However, these data also include unrealistic features, making their validation compulsory. The validation analysis of ERA5 and MERRA-2 wind data, two of the most successful wind reanalyses, with observations near Morocco’s coast, highlighted the existence of bias in both datasets, and other shortcomings. The study was conducted at eight selected sites in the period 2011-2020 using linear and probability density function based statistical tests. The local ERA5-BA and MERRA-2 BA velocity fields were created using Bias Adjustment (BA) Techniques. Only part of the observed and modelled datasets (training period) were used to identify the transfer function. The BA fields were verified against observations during the remaining period (verification period). They outperformed their respective non-adjusted fields and also the WFDE5 (the global bias-corrected ERA5 wind field) in the wind potential assessments, represented by capacity factor and low-wind day. At every location, the relative errors in the latter’s estimates using ERA5-BA were lowered to less than 10 %, and the same was true for MERRA-2-BA. Thus, an observational dataset of reduced length can help overcome some reanalysed wind data limitations for wind power estimations. Additionally, this methodology can be applied to the correction of each reanalysis update.

The research on the applicability of different typhoon wind fields in the simulation of typhoon waves in China’s coastal waters

Typhoon waves possess significant destructive potential, and their numerical simulation relies on accurate sea surface wind fields. An evaluation of different combinations of the radial air pressure distribution coefficient B and the radius of maximum wind speed (Rmax) in the Holland wind field (HWF) model was conducted to determine the optimal configuration. The HWF and the ERA5 wind field (EWF) were used as input wind fields to drive the typhoon wave model for China’s coastal waters. Validation results indicated that neither wind field accurately reflected real conditions; therefore, a hybrid wind field (HBWF) was created by combining HWF and EWF using weighting coefficients that vary with the radius of wind speed to enhance accuracy. Simulation results showed that the HBWF improved the accuracy of significant wave heights (SWHs), with a mean relative error of 25.29%, compared to 32.48% for HWF and 27.94% for EWF. Additionally, HBWF also demonstrated the best performance in terms of root mean square error (RMSE) and consistency index. Overall, the HBWF enhances the simulation accuracy of typhoon waves in China's coastal waters.

Erosion and Deposition Patterns Over a Wind‐Blown Sand Dune Behind a High Vertical‐Type Sand Fence

ABSTRACTSand dunes are the primary factor influencing wind‐sand disasters. To explore the evolution law of wind‐blown sand dunes surrounding high vertical‐type sand fences along desert railways, numerical simulation analyses based on computational fluid dynamics coupled with a discreet element method of the wind‐sand flow field and sand particles trajectories around sand fences were conducted. The focus of this study was the Yandun section of the Lanzhou‐Xinjiang Railway in Xinjiang, China (42.38°N, 93.95°E). By combining ERA5 wind data with field measurements and monitoring, a three‐dimensional coupling model was established. According to the results of L9 (33) orthogonal tests, the sensitivity of the three factors to the evolution of sand dunes can be ordered as wind speed > porosity > height, indicating that wind speed is closely related to sand dune evolution (p < 0.05). The results of single‐factor tests indicate that the structure of wind‐sand flow fields, sand accumulation patterns, and sand particle movement trajectories vary significantly under different wind speeds. The sand accumulation rates on the windward slope of the sand dune are 29.67%, 25.91%, and 20.04%, while on the leeward slope, these rates are 40.33%, 34.09%, and 29.96%, respectively. The obtained information on the evolution law of sand dunes surrounding high vertical‐type sand fences provides a scientific basis for wind‐sand prevention measures along railways in desert regions.

Quality Assessment of ERA5 Wind Speed and Its Impact on Atmosphere Environment Using Radar Profiles along the Bohai Bay Coastline

The accuracy of ERA5 reanalysis datasets and their applicability in the coastal area of Bohai Bay are crucial for weather forecasting and environmental protection research. However, synthesis evaluation of ERA5 in this region remains lacking. In this study, using a tropospheric wind profile radar (CFL-06L) placed in coastal Huanghua city, the deviations of ERA5 reanalysis data are assessed from the ground to an altitude of 5 km. The results indicate that the wind speed of ERA5 reanalysis data exhibits good consistency from the surface to the tropospheric level of about 5 km, with R2 values ranging from 0.5 to 0.85. The lowest mean wind speed error, less than 3 m/s, occurs in the middle layer, while larger errors are observed at the surface and upper layers. Specifically, at 150 m, the R2 is as low as 0.5, with numerous outliers around 5000 m. Seasonal analysis shows that the ERA5 wind field performs best in summer and worst in autumn and winter, especially at lower levels affected by circulation systems, high stratus clouds, and aerosols, with errors reaching up to 10 m/s. Further analysis of extreme weather events, such as heavy rain; hot, dry winds; and snowstorms, reveals that the effects of sea-land winds and strong convective systems significantly impact the observation of wind profiles and the assimilation of reanalysis data, particularly under the constrain of boundary layer height. Additionally, we also find that the transition of sea-land breeze is capable of triggering the nighttime low-level jet, thereby downward transporting the aloft ozone to the ground and resulting in an abnormal increase in the surface ozone concentration. The study provides a scientific basis for improving meteorological forecasting, optimizing wind energy resource utilization, and formulating environmental protection policies, highlighting its significant scientific and practical application value.

Seasonal and geographic viability of high altitude balloon navigation

Control of the geographic location of high-altitude balloons has been desired for decades due to the cost and simplicity of the systems. These balloon systems rely on variations in wind direction with altitude so that when a balloon changes height, it also changes the direction of its horizontal motion. An altitude control system can thus also control the horizontal position by transitioning to a wind layer with favorable winds. The system’s ability to navigate successfully thus relies on the existence of certain wind conditions. In this paper, we explore how the ability of a balloon to station-keep varies based on the geographic location and season. We used spatially and temporally variant ERA5 wind data with a tree-search-based algorithm to traverse potential trajectories, and we selected the altitude transitions that maximize time within 50 km of a target. The simulation’s outputs show large variations in success across both latitude and season. Midlatitudes are particularly challenging for station-keeping, while lower latitudes are more favorable. Summer is typically more favorable than winter. This demonstrates that for all balloon systems, the ability to station-keep is highly variant and not universally possible.

Arctic Wind, Sea Ice, and the Corresponding Characteristic Relationship

In efforts to fulfill the objectives of taking part in pragmatic cooperation in the Arctic, constructing the “Silk Road on Ice”, and ensuring ships’ safety and risk assessment in the Arctic, the two biggest hazards, which concern ships’ navigation in the Arctic, are wind and sea ice. Sea ice can result in a ship being besieged or crashing into an iceberg, endangering both human and property safety. Meanwhile, light winds can assist ships in breaking free of a sea-ice siege, whereas strong winds can hinder ships’ navigation. In this work, we first calculated the spatial and temporal characteristics of a number of indicators, including Arctic wind speed, sea-ice density, the frequency of different wind directions, the frequency of a sea-ice density of less than 20%, the frequency of strong winds of force six or above, etc. Using the ERA5 wind field and the SSMI/S sea-ice data, and applying statistical techniques, we then conducted a joint analysis to determine the correlation coefficients between the frequencies of various wind directions, the frequency of strong winds and its impact on the density of sea ice, the frequency of a sea-ice concentration (SIC) of less than 20%, and the correlation coefficient between winds and sea-ice density. In doing so, we determined importance of factoring the wind’s contribution into sea-ice analysis.

Deep residual fully connected network for GNSS-R wind speed retrieval and its interpretation

Deep residual fully connected network for GNSS-R wind speed retrieval and its interpretation

The Black Sea near-past wave climate and its variability: a hindcast study

This study analyzed the past wave climate of the Black Sea region for the period from 1988 to 2021. The wave field has been simulated using the state-of-the-art, third-generation wave model WAVEWATCH III forced by the ECMWF reanalysis ERA5 winds, with the model resolution being the highest ever applied to the region in a basin-scale climate study. The surface currents provided by the Copernicus Marine Service have been included in the wave model to evaluate wave–current interactions. The wave model results have been validated with respect to satellite and buoy observations, showing that the simulation accurately reproduces the past evolution of the wave field, exceeding 0.9 correlation with respect to satellite data. The inclusion of wave–current interaction has been positively evaluated. Four statistics (significant wave height 5th and 95th percentiles, mean, and maxima) have been used to describe the wave field at seasonal timescale, showing a clear distinction between the Western (rougher sea conditions) and Eastern (calmer sea conditions) sub-basins. Furthermore, the intra-annual wave climate variability has been investigated using a Principal Component Analysis (PCA) and the Mann–Kendall test on significant wave height (SWH). This study represents the first time the PCA is applied to the region, identifying two main modes that highlight distinct features and seasonal trends in the Western and Eastern sub-basins. Throughout most seasons, the SWH trend is positive for the Eastern basin and negative for the Western basin. The PCA shows a regime shift with increasing eastward waves and decreasing north and north-eastward waves. Finally, SWH correlation (ρ) with four Teleconnection indexes (East Atlantic Pattern, Scandinavian Pattern, North Atlantic Oscillation, and East Atlantic/West Russia Pattern) revealed that the strongest ρ is observed with the Eastern–Atlantic–Western Russia teleconnection, with a peculiar spatial pattern of correlation, and is positive for the northwestern and negative for the southeastern sub-basin.

Detecting Melt Pond Onset on Landfast Arctic Sea Ice Using a Dual C-Band Satellite Approach

The presence of melt ponds on the surface of Arctic Sea ice affects its albedo, thermal properties, and overall melting rate; thus, the detection of melt pond onset is of significant importance for understanding the Arctic’s changing climate. This study investigates the utility of a novel method for detecting the onset of melt ponds on sea ice using a satellite-based, dual-sensor C-band approach, whereby Sentinel-1 provides horizontally polarized (HH) data and Advanced SCATterometer (ASCAT) provides vertically polarized (VV) data. The co-polarized ratio (VV/HH) is used to detect the presence of melt ponds on landfast sea ice in the Canadian Arctic Archipelago in 2017 and 2018. ERA-5 air temperature and wind speed re-analysis datasets are used to establish the VV/HH threshold for pond onset detection, which have been further validated by Landsat-8 reflectance. The co-polarized ratio threshold of three standard deviations from the late winter season (April) mean co-pol ratio values are used for assessing pond onset detection associated with the air temperature and wind speed data, along with visual observations from Sentinel-1 and cloud-free Sentinel-2 imagery. In 2017, the pond onset detection rates were 70.59% for FYI and 92.3% for MYI. Results suggest that this method, because of its dual-platform application, has potential for providing large-area coverage estimation of the timing of sea ice melt pond onset using different earth observation satellites.

Quantifying CH4 emissions from coal mine aggregation areas in Shanxi, China, using TROPOMI observations and the wind-assigned anomaly method

Abstract. China stands out as a major contributor to anthropogenic methane (CH4) emissions, with coal mine methane (CMM) playing a crucial role. To control and reduce CH4 emissions, China has made a dedicated commitment and formulated an ambitious mitigation plan. To verify the progress made, the consistent acquisition of independent CH4 emission data is required. This paper aims to implement a wind-assigned anomaly method for the precise determination of regional-scale CMM emissions within the coal-rich Shanxi province. We use the TROPOspheric Monitoring Instrument (TROPOMI) CH4 observations from May 2018 to May 2023, coupled with ERA5 wind and a bottom-up inventory dataset based on the IPCC (Intergovernmental Panel on Climate Change) Tier 2 approach covering the Changzhi, Jincheng, and Yangquan regions of the Shanxi province. The derived emission strengths are 8.4 × 1026 molec. s−1 (0.706 Tg yr−1, ±25 %), 1.4 × 1027 molec. s−1 (1.176 Tg yr−1, ±20 %), and 4.9 × 1026 molec. s−1 (0.412 Tg yr−1, ±21 %), respectively. Our results exhibit biases of −18 %, 8 %, and 14 %, respectively, when compared to the IPCC Tier 2 bottom-up inventory. Larger discrepancies are found when comparing the estimates to the Copernicus Atmosphere Monitoring Service global anthropogenic emissions (CAMS-GLOB-ANT) and Emissions Database for Global Atmospheric Research (EDGARv7.0) inventories (64 %–176 %), suggesting that the two inventories may be overestimating CH4 emissions from the studied coal mining regions. Our estimates provide a comprehensive characterization of the regions within the Shanxi province, contribute to the validation of emission inventories, and provide additional insights into CMM emission mitigation.

Global Strong Winds Occurrence Characteristics and Climate Index Correlation

Guided by entering the deep sea and achieving deep marine development in marine construction, the factors hindering marine construction cannot be ignored. Strong ocean winds have a devastating impact on tasks such as ship navigation, carrier aircraft take-off and landing, naval operations and military exercises, and affect the planning of sea routes and the development of the long-distance sea. This paper uses ERA5 wind field data and key climate indices to conduct a systematic analysis of catastrophic winds in the global ocean using methods such as climate statistical analysis, the Theil–Sen trend method, Pearson correlation and contribution rate calculation. It points out the spatiotemporal distribution, variation trend, climate index correlation and contribution rate characteristics of strong winds occurrence (SWO) and hopes that the results of this study can serve as a guide for maritime route planning and provide technical assistance and decision-making support for marine development and other needs. The results show the following: The high global SWO occurs in the Southern Ocean, the North Atlantic, the North Pacific, near Taiwan, China, the Arabian Sea and other locations, with the strongest SWO in summer. The growth trend of SWO in the Southern Ocean is strongest, with decreasing regions near the Arabian Sea and the Bay of Bengal, and the growth trend is reflected in all four seasons. The climate indices with the strongest correlation and highest contribution to the global SWO are AAO (Antarctic Oscillation) and EP–NP (East Pacific–North Pacific pattern) with a correlation between −0.5 and 0.5 and a contribution rate of up to −50%~50%.

Evaluation and Bias Correction of the ERA5 Reanalysis over the United States for Wind and Solar Energy Applications

The applicability of the ERA5 reanalysis for estimating wind and solar energy generation over the contiguous United States is evaluated using wind speed and irradiance variables from multiple observational data sets. After converting ERA5 and observed meteorological variables into wind power and solar power, comparisons demonstrate that significant errors in the ERA5 reanalysis exist that limit its direct applicability for a wind and solar energy analysis. Overall, ERA5-derived solar power is biased high, while ERA5-derived wind power is biased low. During winter, the ERA5-derived solar power is biased high by 23% on average, while on an annual basis, the ERA5-derived wind power is biased low by 20%. ERA5-derived solar power errors are found to have consistent characteristics across the contiguous United States. Errors for the shortest duration and most extreme solar negative anomaly events are relatively small in the ERA5 when completely overcast conditions occur in both the ERA5 and observations. However, longer-duration anomaly events on weekly to monthly timescales, which include partially cloudy days or a mix of cloudy and sunny days, have significant ERA5 errors. At 10 days duration, the ERA5-derived average solar power produced during the largest negative anomaly events is 62% greater than observed. The ERA5 wind speed and derived wind power negative biases are largely consistent across the central and northwestern U.S., and offshore, while the northeastern U.S. has an overall small net bias. For the ERA5-derived most extreme negative anomaly wind power events, at some sites at 10 days duration, the ERA5-derived wind power produced can be less than half of that observed. Corrections to ERA5 are derived using a quantile–quantile method for solar power and linear regression of wind speed for wind power. These methods are shown to avoid potential over-inflation of the reanalysis variability resulting from differences between point measurements and the temporally and spatially smoother reanalysis values. The corrections greatly reduce the ERA5 errors, including those for extreme events associated with wind and solar energy droughts, which will be most challenging for electric grid operation.

An analysis of the expected wave conditions in the Mediterranean Sea in the context of global warming

An analysis of the expected wave conditions in the Mediterranean Sea in the context of global warming