Tendency Terms Research Articles

The Generalized Application of a New Surface Pressure Tendency Equation in Synoptic Weather Systems

AbstractThe surface pressure tendency equation (SPTE) is an important theoretical tool in understanding the development of weather systems, with various forms being proposed over the past 100 years. In this study, a new version of the SPTE, which eliminates the explicit or implicit pressure tendency term on the right‐hand side, is applied to three typical synoptic weather systems in terms of tropical cyclones, explosive cyclones anticyclones, and two mesoscale systems in terms of squall lines and supercells based on numerical simulations. Results reveal promising performance of the new equation in understanding the surface pressure tendency of three synoptic systems, showing advantages in uncovering their dominant physical processes as well as inherent discrepancies. For the tropical cyclone, the latent heating is largely responsible for the surface pressure fall, whereas the surface pressure rise of the anticyclone is mainly fueled by the upper‐level advection of potential temperature. The leading factors for the explosive cyclone intensification demonstrate a transition from mid‐level diabatic heating at the early stage to upper‐level potential temperature advection related to stratospheric air intrusion during the later intensification period. For the squall line and supercell, however, local breakdown of hydrostatic balance can be as large as ∼10%, and consequently, caution should be taken when the equation is used in mesoscale systems, especially for the supercell. These results suggest that the new SPTE can be well applied in synoptic weather systems, and can provide new insights into their intensity evolutions and contributions of diverse physical processes.

Research on Structurally Constrained KELM Fault-Diagnosis Model Based on Frequency-Domain Fuzzy Entropy

As the core equipment of the high-pressure diaphragm pump, the working conditions of the check valve are complicated, and the vibration signal generated during operation displays non-stationary and nonlinear characteristics. In order to accurately describe the non-linear dynamics of the check valve, the smoothing prior analysis (SPA) method is used to decompose the vibration signal of the check valve, obtain the tendency term and fluctuation term components, and calculate the frequency-domain fuzzy entropy (FFE) of the component signals. Using FFE to characterize the operating state of the check valve, the paper proposes a kernel extreme-learning machine (KELM) function norm regularization method, which is used to construct a structurally constrained kernel extreme-learning machine (SC-KELM) fault-diagnosis model. Experiments demonstrate that the frequency-domain fuzzy entropy can accurately characterize the operation state of check valve, and the improvement of the generalization of the SC-KELM check valve fault model improves the recognition accuracy of the check-valve fault-diagnosis model, with an accuracy rate of 96.67%.

Using dissolved oxygen variance to investigate the influence of nonextreme wind events on hypoxia in Mobile Bay, a shallow stratified estuary

Wind forcing plays an important role in determining spatial patterns of estuarine bottom water hypoxia, defined as dissolved oxygen (DO) concentration< 2 mg L-1, by driving coastal circulation patterns and by intensifying mixing of the water column. However, the importance of these wind-driven mixing processes varies with space and time and are dynamically intermingled with biological processes like photosynthesis and respiration making it difficult to tease apart wind impacts on DO dynamics in estuarine systems. Using a high-resolution, three-dimensional numerical model, we studied the effect of a non-extreme southeast wind event on the DO dynamics of Mobile Bay during a hypoxic event in April-May of 2019. A new approach, called ‘vertical dissolved oxygen variance’ (VDOV) was developed to quantitatively separate all the physical and biogeochemical factors in the water column that control the development and dissipation of hypoxia events. The system-wide volume integrated values of VDOV tracked the changes in hypoxic area in the bay and the VDOV tendency term was dominated by contributions from sediment oxygen demand (DO loss via respiration) and vertical dissipation (DO gain via mixing). There was a notable inverse relationship between hypoxia area and wind speed. Further analysis of the local VDOV during a non-extreme southeast wind event showed the wind-induced vertical dissipation was the main factor in eliminating hypoxia from the bay. This enhanced dissipation accounted for both turbulent mixing from wind stress and negative straining of the vertical density gradient from wind induced circulation. The response of DO to the wind forcing prompted the development of two non-dimensional numbers, an advection-diffusion time-scale ratio and a demand-diffusion flux ratio, to better generalize the expected DO dynamics. Overall, this work showed that wind effects are critical for understanding hypoxia variability in a shallow stratified estuary.

Evaluating Vertical Velocity Retrievals from Vertical Vorticity Equation Constrained Dual-Doppler Analysis of Real, Rapid-Scan Radar Data

Abstract Accurate vertical velocity retrieval from dual-Doppler analysis (DDA) is a long-standing problem of radar meteorology. Typical radar scanning strategies poorly observe the vertical component of motion, leading to large uncertainty in vertical velocity estimates. Using a vertical vorticity equation constraint in addition to a mass conservation constraint in DDA has shown promise in improving vertical velocity retrievals. However, observation system simulation experiments (OSSEs) suggest this technique requires rapid radar volume scans to realize the improvements due to the vorticity tendency term in the vertical vorticity constraint. Here, the vertical vorticity constraint DDA is tested with real, rapid-scan radar data to validate prior OSSEs results. Generally, the vertical vorticity constraint DDA produced more accurate vertical velocities from DDAs than those that did not use the constraint. When the time between volume scans was greater than 30 s, the vertical velocity accuracy was significantly affected by the vorticity tendency estimation method. A technique that uses advection correction on provisional DDA wind fields to shorten the discretization interval for the vorticity tendency calculation improved the vertical velocity retrievals for longer times between volume scans. The skill of these DDAs was similar to those using a shorter time between volume scans. These improvements were due to increased accuracy of the vertical vorticity tendency using the advection correction technique. The real radar data tests also revealed that the vertical vorticity constraint DDAs are more forgiving to radar data errors. These results suggest that vertical vorticity constraint DDA with rapid-scan radars should be prioritized for kinematic analyses.

Simulated dynamics and thermodynamics processes leading to the rapid intensification of rare tropical cyclones over the North Indian Oceans

The life cycle dynamics and intensification processes of three long-duration tropical cyclones (TCs), viz., Fani (2019), Luban (2018), and Ockhi (2017) formed over the North Indian Ocean (NIO) have been investigated by developing a high-resolution (6 km × 6 km) mesoscale analysis using WRF and En3DVAR data assimilation system. The release of CAPE in nearly saturated middle-level relative humidity caused intense diabatic heating, leading to an increase in low-level convergence triggering rapid intensification (RI). The strengthening of the relative vorticity tendency terms was due to vertical stretching (TC Fani) and middle tropospheric advection (TCs Luban and Ockhi). The increase or decrease in upper-tropospheric divergence led to RI through two different mechanisms. The increase in upper divergence strengthens the vortical convection (in TC Luban and Fani) by enhancing the moisture and heat transport, whereas its decrease caused a reduction in the upper-level ventilation flow at 200 hPa followed by moisture accumulation, enhanced diabatic heating, and strengthened the warm core (TC Ockhi). The RI caused the vortex of three cyclones to extend up to the upper troposphere. The well organised wind during RI led the unorganised, weak, discontinuous vertical vortex columns to become organised with intense vertical velocity throughout the column. Spatial distributions of Okubo–Wiess (OW) parameter showed TC core dominated by vorticity than strain, since deep depression (DD) stages.

Oceanic Electrical Conductivity Variability From Observations and Its Budget From an Ocean State Estimate

AbstractBecause spatio‐temporal variations in ocean heat content (OHC) are strongly predicted by ocean conductivity content (OCC) over most of the global ocean, we analyze the dynamical budget and behavior of the electrical conductivity of seawater. To perform these analyses, we use an ocean‐model state estimate designed to accurately represent long‐term variations in ocean properties in a dynamically and kinematically consistent way. We show that this model accurately reproduces the spatio‐temporal variations in electrical conductivity seen in satellite‐derived data and in a seasonal climatology product derived from in‐situ data, justifying use of the model data to perform further analyses. An empirical orthogonal function analysis suggests that the vast majority of the variance in OHC and OCC can be explained by similar mechanisms. The electrical conductivity budget's most important term is the temperature forcing tendency term, suggesting that ocean heat uptake is the mechanism responsible for the strong relationship between OCC and OHC.

Local Drivers of Marine Heatwaves: A Global Analysis With an Earth System Model

Marine heatwaves (MHWs) are periods of extreme warm ocean temperatures that can have devastating impacts on marine organisms and socio-economic systems. Despite recent advances in understanding the underlying processes of individual events, a global view of the local oceanic and atmospheric drivers of MHWs is currently missing. Here, we use daily-mean output of temperature tendency terms from a comprehensive fully coupled coarse-resolution Earth system model to quantify the main local processes leading to the onset and decline of surface MHWs in different seasons. The onset of MHWs in the subtropics and mid-to-high latitudes is primarily driven by net ocean heat uptake associated with a reduction of latent heat loss in all seasons, increased shortwave heat absorption in summer and reduced sensible heat loss in winter, dampened by reduced vertical mixing from the non-local portion of the K-Profile Parameterization boundary layer scheme (KPP) especially in summer. In the tropics, ocean heat uptake is reduced and lowered vertical local mixing and diffusion cause the warming. In the subsequent decline phase, increased ocean heat loss to the atmosphere due to enhanced latent heat loss in all seasons together with enhanced vertical local mixing and diffusion in the high latitudes during summer dominate the temperature decrease globally. The processes leading to the onset and decline of MHWs are similar for short and long MHWs, but there are differences in the drivers between summer and winter. Different types of MHWs with distinct driver combinations are identified within the large variability among events. Our analysis contributes to a better understanding of MHW drivers and processes and may therefore help to improve the prediction of high-impact marine heatwaves.

The Amplification of Madden–Julian Oscillation Boosted by Temperature Feedback

Abstract Due to a small Coriolis force in tropics, the theoretical study of Madden–Julian oscillation (MJO) often assumes weak temperature gradient balance, which neglects the temperature feedback (manifested in the temperature tendency term). In this study, the effect of the temperature feedback on the MJO is investigated by using the MJO trio-interaction model, which can capture the essential large-scale features of the MJO. The scale analysis indicates that the rotation effect is strong for the MJO scales, so that the temperature feedback is as important as the moisture feedback (manifested in the moisture tendency term); the latter is often considered to be critical for MJO. The experiments with the theoretical model show that the temperature feedback has significant impact on the MJO’s maintenance. When the temperature feedback is turned off, the simulated MJO cannot be maintained over the warm pool. This is because the temperature feedback could boost the energy generation. Without the temperature feedback, only the latent heat can be generated. With the temperature feedback, not only the latent heat but also the enthalpy (and therefore the available potential energy) can be generated. Therefore, the total energy generation is more efficient with the temperature feedback, favoring the self-maintenance of the MJO. Further investigation shows that this effect of the temperature feedback on MJO amplification can be inferred from observations. The findings here indicate that the temperature feedback could have nonnegligible impacts on the MJO and have implications in the simulation of MJO.

A Comparison between Moist and Dry Tropical Cyclones: The Low Effectiveness of Surface Sensible Heat Flux in Storm Intensification

Abstract Recent numerical modeling studies demonstrate that dry tropical cyclones can be stably sustained via a supply of surface sensible heat flux. This raises questions of whether surface sensible heat flux (SHX) and latent heat flux (LHX) have the same effect on the intensity evolution of tropical cyclones. An estimation of equivalent potential temperature budget in the boundary layer shows that LHX leads to larger increase in equivalent potential temperature than SHX even when they possess the same magnitude. By formulating these two kinds of surface heat fluxes with the same mathematical framework, the simulated intensifications of moist and dry tropical cyclones are compared, with the former driven exclusively by LHX and the latter by SHX. Results show significantly larger intensification rates for the tropical cyclone driven by LHX than that by SHX, revealing low effectiveness of SHX in the intensification of tropical cyclones. The diabatic heating in the moist tropical cyclone occurs accompanying the convection, while it is merely pronounced near the surface in the dry tropical cyclone and is decoupled from the dry convection. A new surface pressure tendency equation is proposed, without incorporating the implicit pressure tendency term on the right-hand side. The budget analysis indicates that the SHX is less effective than LHX in lowering surface central pressure and therefore in tropical cyclone intensification. A series of sensitivity experiments suggest that the threshold of energy input required for spinning up a tropical cyclone is lower in the form of LHX than that of SHX.

Hydrometeor and Latent Heat Nudging for Radar Reflectivity Assimilation: Response to the Model States and Uncertainties

Radar data are essential to convection nowcasting and nudging-based radar data assimilation through diabatic initialization is one of the most effective approaches for forecasting convective systems with numerical weather prediction (NWP) models, used at several advanced global weather centers. It is desired to assess the uncertainty and physical consistency of this assimilation process. This paper investigated impacts of relaxation coefficient, radar data update intervals and continuous assimilation time duration and addressed the key issues and possible solutions of the radar data assimilation based on the WRF hydrometeor and latent heat nudging (HLHN) developed at the National Center for Atmospheric Research (NCAR). It is revealed that excessively large relaxation coefficient forced the model to observations with a tendency greater than the physical terms of the convection, causing the dynamic imbalances and serious convection “ramp-down” right after the free forecast starts. Assimilating high update frequency radar data can make the tendency terms moderate and sustained thereby maintaining the assimilation effect and reducing fortuitous convection. HLHN requires a minimum continuous assimilation duration to contain the initial forced disturbance of the model. For a summer Meiyu precipitation case studied, the minimum duration is ~1 h. Appropriate selection of the HLHN parameters is able to effectively improve the temperature, humidity, and dynamic fields of the model. In addition, several issues still remain to be solved to further enhance HLHN.

GFDL's SPEAR Seasonal Prediction System: Initialization and Ocean Tendency Adjustment (OTA) for Coupled Model Predictions

AbstractThe next‐generation seasonal prediction system is built as part of the Seamless System for Prediction and EArth System Research (SPEAR) at the Geophysical Fluid Dynamics Laboratory (GFDL) of the National Oceanic and Atmospheric Administration (NOAA). SPEAR is an effort to develop a seamless system for prediction and research across time scales. The ensemble‐based ocean data assimilation (ODA) system is updated for Modular Ocean Model Version 6 (MOM6), the ocean component of SPEAR. Ocean initial conditions for seasonal predictions, as well as an ocean state estimation, are produced by the MOM6 ODA system in coupled SPEAR models. Initial conditions of the atmosphere, land, and sea ice components for seasonal predictions are constructed through additional nudging experiments in the same coupled SPEAR models. A bias correction scheme called ocean tendency adjustment (OTA) is applied to coupled model seasonal predictions to reduce model drift. OTA applies the climatological temperature and salinity increments obtained from ODA as three‐dimensional tendency terms to the MOM6 ocean component of the coupled SPEAR models. Based on preliminary retrospective seasonal forecasts, we demonstrate that OTA reduces model drift—especially sea surface temperature (SST) forecast drift—in coupled model predictions and improves seasonal prediction skill for applications such as El Niño–Southern Oscillation (ENSO).

Impact of a medicane on the oceanic surface layer from a coupled, kilometre-scale simulation

Abstract. A kilometre-scale coupled ocean–atmosphere numerical simulation is used to study the impact of the 7 November 2014 medicane on the oceanic upper layer. The processes at play are elucidated through analyses of the tendency terms for temperature and salinity in the oceanic mixed layer. While comparable by its maximum wind speed to a Category 1 tropical cyclone, the medicane results in a substantially weaker cooling. As in weak to moderate tropical cyclones, the dominant contribution to the surface cooling is the surface heat fluxes with secondary effects from the turbulent mixing and lateral advection. Upper-layer salinity decreases due to heavy precipitation that overcompensates the salinizing effect of evaporation and turbulent mixing. The upper-layer evolution is marked by several features believed to be typical of Mediterranean cyclones. First, strong, convective rain occurring at the beginning of the event builds a marked salinity barrier layer. As a consequence, the action of surface forcing is favoured and the turbulent mixing dampened with a net increase in the surface cooling as a result. Second, due to colder surface temperature and weaker stratification, a cyclonic eddy is marked by a weaker cooling opposite to what is usually observed in tropical cyclones. Third, the strong dynamics of the Strait of Sicily enhance the role of the lateral advection in the cooling and warming processes of the mixed layer.

Comparing Numerical Accuracy of Icosahedral A-Grid and C-Grid Schemes in Solving the Shallow-Water Model

AbstractA single software framework is introduced to evaluate numerical accuracy of the A-grid (NICAM) versus C-grid (MPAS) shallow-water model solvers on icosahedral grids. The C-grid staggering scheme excels in numerical noise control and total energy conservation, which results in exceptional stability for long time integration. Its weakness lies in the lack of model error reduction with increasing resolution in specific test cases (especially the root-mean-square error). The A-grid method conserves well potential enstrophy and shows a linear reduction of error with increasing resolution. The gridpoint noise manifests itself clearly on A-grid, but much less on C-grid. We show that the Coriolis force term on C-grid has a larger error than on A-grid. To treat the Coriolis term and kinetic energy gradient on an equal footing on C-grid, we propose combining these two quantities into a single tendency term and computing its value by a linear combination operation. This modification alone reduces numerical errors but still fails to converge the maximum error with resolution. The method of Peixoto can solve the maximum-error nonconvergence problem on C-grid but degrades the numerical stability. For the steady-state thin-layer test (0.01 m in depth), the A-grid method is less susceptible than C-grid methods, which are presumably disrupted by the Hollingsworth instability. The effect of horizontal diffusion on model accuracy and energy conservation is shown in detail. Programming experience shows that software implementation and optimization can strongly influence computational performance for models, although memory requirement and computational load of the two schemes are comparable.

Using Analysis Corrections to Address Model Error in Atmospheric Forecasts

Abstract This paper illustrates that analysis corrections, when applied as a model tendency term, can be used to improve nonlinear model forecasts and are consistent with the hypothesis that they represent an additive 6-h accumulation of model error. Comparison of mean analysis corrections with observational estimates of bias further illustrates the fidelity with which mean analysis corrections capture the model bias. While most previous implementations have explored the use of analysis corrections to correct forecast biases in short-range forecasts, this is the first implementation of the correction method using both a seasonal mean and random analysis correction for medium-range forecasts (out to 10 days). Overall, the analysis correction–based perturbations are able to improve forecast skill in ensemble and deterministic systems, especially in the first 5 days of the forecast where bias and RMSE in both lower-tropospheric temperature and 500 hPa geopotential height are significantly improved across all experiments. However, while the method does provide some significant improvement to forecast skill, some degradation in bias can occur at later lead times when the average bias at analysis time trends toward zero over the length of the forecast, leading to an overcorrection by the analysis correction–based additive inflation (ACAI) method. Additionally, it is shown that both the mean and random component of the ACAI perturbations play a role in adjusting the model bias, and that the two components can have a complicated and sometimes nonlinear interaction.

The Large-Scale Vorticity Balance of the Antarctic Continental Margin in a Fine-Resolution Global Simulation

ABSTRACTThe depth-integrated vorticity budget of a global, eddy-permitting ocean/sea ice simulation over the Antarctic continental margin (ACM) is diagnosed to understand the physical mechanisms implicated in meridional transport. The leading-order balance is between the torques due to lateral friction, nonlinear effects, and bottom vortex stretching, although details vary regionally. Maps of the time-averaged depth-integrated vorticity budget terms and time series of the spatially averaged, depth-integrated vorticity budget terms reveal that the flow in the Amundsen, Bellingshausen, and Weddell Seas and, to a lesser extent, in the western portion of East Antarctica, is closer to an approximate topographic Sverdrup balance (TSB) compared to other segments of the ACM. Correlation and coherence analyses further support these findings, and also show that inclusion of the vorticity tendency term in the response (the planetary vorticity advection and the bottom vortex stretching term) increases the correlation with the forcing (the vertical net stress curl), and also increases the coherence between forcing and response at high frequencies across the ACM, except for the West Antarctic Peninsula. These findings suggest that the surface stress curl, imparted by the wind and the sea ice, has the potential to contribute to the meridional, approximately cross-slope, transport to a greater extent in the Amundsen, Bellingshausen, Weddell, and part of the East Antarctic continental margin than elsewhere in the ACM.

Forcing mechanisms of the migrating quarterdiurnal tide

Abstract. We used a nonlinear mechanistic global circulation model to analyze the migrating quarterdiurnal tide (QDT) in the middle atmosphere with focus on its possible forcing mechanisms: the absorption of solar radiation by ozone and water vapor, nonlinear tidal interactions, and gravity wave–tide interactions. We show a climatology of the QDT amplitudes, and we examine the contribution of the different forcing mechanisms to the QDT amplitude. To this end, we first extracted the QDT from the model tendency terms and then removed the respective QDT contribution from the different tendency terms. We find that the solar forcing mechanism is the most important one for the QDT; however, the nonlinear and gravity wave forcing mechanisms also play a role in autumn and winter, particularly at lower and middle latitudes in the mesosphere and lower thermosphere. Furthermore, destructive interference between the individual forcing mechanisms is observed. Therefore, tidal amplitudes become even larger in simulations with the nonlinear or gravity wave forcing mechanisms removed.

Towards the closure of momentum budget analyses in the WRF (v3.8.1) model

Abstract. Budget analysis of a tendency equation is widely utilized in numerical studies to quantify different physical processes in a simulated system. While such analysis is often post-processed when the output is made available, it is well acknowledged that the closure of a budget is difficult to achieve without temporal and/or spatial averaging. Nevertheless, the development of errors in such calculations has not been systematically investigated. In this study, an inline budget retrieval method is first developed in the WRF v3.8.1 model and tested on a 2D idealized slantwise convection case with a focus on the momentum equations. This method extracts all the budget terms following the model solver, which gives a high accuracy, with a residual term always less than 0.1 % of the tendency term. Then, taking the inline values as truth, several offline budget analyses with different commonly used simplifications are performed to investigate how they may affect the accuracy of the estimation of individual terms and the resultant residual. These assumptions include using a lower-order advection operator than the one used in the model, neglecting grid staggering, or following a mathematically equivalent but transformed format of the governing equations. Errors in these post-processed analyses are found mostly over the area where the dynamics are the most active, thus impairing the subsequent physical interpretation. A maximum 99th percentile residual can reach >50 % of the concurrent tendency term, indicating the danger of neglecting the residual term as done in many budget studies. This work provides general guidance not only for budget diagnoses with the WRF model but also for minimizing the errors in post-processed budget calculations.

Dropsonde Observations of the Ageostrophy within the Pre-Cold-Frontal Low-Level Jet Associated with Atmospheric Rivers

Abstract The pre-cold-frontal low-level jet (LLJ) is an important contributor for water vapor transport within atmospheric rivers, though its dynamics are not completely understood. The present study investigates the LLJ using dropsonde observations from 24 cross-atmospheric river transects taken during the CalWater-2014, 2015 and the AR-Recon 2016, 2018 field campaigns. It is found that the LLJ, located at ~1-km elevation ahead of the cold front, has an average maximum wind speed of 30 m s−1 and is strongly supergeostrophic with an average ageostrophic component of 6 m s−1. The alongfront ageostrophy occurs within the atmospheric layer (750–1250 m) known to strongly control orographic precipitation associated with atmospheric rivers. The ERA5 reanalysis product is used to both validate the observed geostrophic winds and investigate the supergeostrophic jet dynamics. The comparison demonstrates that there is no systematic bias in the observed geostrophic wind but that the ERA5 LLJ total wind field is generally biased low by an amount consistent with the observed ageostrophy. One of the few cases in which the ERA5 produces an ageostrophic LLJ occurs on 13 February 2016, which is used to investigate the dynamical processes responsible for the ageostrophy. This analysis demonstrates that the isallobaric (pressure tendency) term serves to accelerate the ageostrophic jet, and the Coriolis torque and advective tendency terms serve to propagate the jet normal to the LLJ. Therefore, if a model is to accurately represent the LLJ, it must adequately resolve processes contributing toward the pressure tendencies along the cold front.

The application of trigonometric grey prediction model to average per capita natural gas consumption of households in China

PurposeDue to high efficiency and low carbon of natural gas, the consumption of natural gas is increasing rapidly, and the prediction of natural gas consumption has become the focus. The purpose of this paper is to employ a prediction technique by combining grey prediction model and trigonometric residual modification for predicting average per capita natural gas consumption of households in China.Design/methodology/approachThe GM(1,1) model is utilised to obtain the tendency term, then the generalised trigonometric model is used to catch the periodic phenomenon from the residual data of GM(1,1) model for improving predicting accuracy.FindingsThe case verified the view of Xie and Liu: “When the value of a is less, DGM model and GM(1,1) model can substitute each other.” The combination of the GM(1,1) and the trigonometric residual modification technique can observably improve the predicting accuracy of average per capita natural gas consumption of households in China. The mean absolute percentage errors of GM(1,1) model, DGM(1,1), unbiased grey forecasting model, and TGM model in ex post testing stage (from 2013 to 2015) are 32.5510, 33.5985, 36.9980, and 5.2996 per cent, respectively. The TGM model is suitable for the prediction of average per capita natural gas consumption of households in China.Practical implicationsAccording to the historical data of average per capita natural gas consumption of households in China, the authors construct GM(1,1) model, DGM(1,1) model, unbiased grey forecasting model, and GM(1,1) model with trigonometric residual modification. The accuracy of TGM is the best. TGM helps to improve the accuracy of GM(1,1).Originality/valueThis paper gives a successful practical application of grey model GM(1,1) with the trigonometric residual modification, where the cyclic variations exist in the residual series. The case demonstrates the effectiveness of trigonometric grey prediction model, which is helpful to understand the modeling mechanism of trigonometric grey prediction model.

Understanding the Equatorial Pacific Cold Tongue Time-Mean Heat Budget. Part I: Diagnostic Framework

The Pacific equatorial cold tongue plays a leading role in Earth’s strongest and most predictable climate signals. To illuminate the processes governing cold tongue temperatures, the upper-ocean heat budget is explored using the GFDL-FLOR coupled GCM (the forecast-oriented low ocean resolution version of CM2.5). Starting from the exact temperature budget for layers of time-varying thickness, the layer temperature tendency terms are studied using hourly-, daily-, and monthly-mean output from a 30-yr simulation driven by present-day radiative forcings. The budget is then applied to 1) a surface mixed layer whose temperature is highly correlated with SST, in which the air–sea heat flux is balanced mainly by downward diffusion of heat across the layer base, and 2) a thicker advective layer that subsumes most of the vertical mixing, in which the air–sea heat flux is balanced mainly by monthly-scale advection. The surface warming from shortwave fluxes and submonthly meridional advection and the subsurface cooling from monthly vertical advection are both shown to be essential to maintain the cold tongue thermal stratification against the destratifying effects of vertical mixing. Although layer undulations strongly mediate the tendency terms on diurnal-to-interannual scales, the 30-yr-mean tendencies are found to be well summarized by analogous budgets developed for stationary but spatially varying layers. The results are used to derive practical simplifications of the exact budget, to support the analyses in Part II of this paper, and to facilitate broader application of heat budget analyses when evaluating and comparing climate simulations.