Wave-activity conservation laws and stability theorems for semi-geostrophic dynamics: part 1. pseudomomentum-based theory

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics. In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms. The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent. This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii). In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

DK-1D: a drift-kinetic simulation tool for modelling the shear Alfvén wave and its interaction with collisionless plasma

We present a highly accurate tool for the simulation of shear Alfven waves (SAW) in collisionless plasma. SAW are important in space plasma environments because for small perpendicular scale lengths they can support an electric field parallel to the ambient magnetic field. Electrons can be accelerated by the parallel electric field and these waves have been implicated as the source of vibrant auroral displays. However, the parallel electric field carried by SAW is small in comparison to the perpendicular electric field of the wave, making it difficult to measure directly in the laboratory, or by satellites in the near-Earth plasma environment. In this paper, we present a simulation code that provides a means to study in detail the SAW-particle interaction in both space and laboratory plasma. Using idealised, small-amplitude propagating waves with a single perpendicular wavenumber, the simulation code accurately reproduces the damping rates and parallel electric field amplitudes predicted by linear theory for varying temperatures and perpendicular scale lengths. We present a rigorous kinetic derivation of the parallel electric field strength for small-amplitude SAW and show that commonly-used inertial and kinetic approximations are valid except for where the ratio of thermal to Alfv\'{e}n speed is between 0.7 and 1.0. We also present nonlinear simulations of large-amplitude waves and show that in cases of strong damping, the damping rates and parallel electric field strength deviate from linear predictions when wave energies are greater than only a few percent of the plasma kinetic energy, a situation which is often observed in the magnetosphere. The drift-kinetic code provides reliable, testable predictions of the parallel electric field strength which can be investigated directly in the laboratory, and will help to bridge the gap between studies of SAW in man-made and naturally occuring plasma.

Nonlinear wave-activity conservation laws and Hamiltonian structure for the two-dimensional anelastic equations

Exact, finite-amplitude, local wave-activity conservation laws are derived for disturbances to steady flows in the context of the two-dimensional anelastic equations. The conservation laws are expressed entirely in terms of Eulerian quantities, and have the property that, in the limit of a small-amplitude, slowly varying, monochromatic wave train, the wave-activity density A and flux F, when averaged over phase, satisfy F = cgA where cg is the group velocity of the waves. For nonparallel steady flows, the only conserved wave activity is a form of disturbance pseudoenergy; when the steady flow is parallel, there is in addition a conservation law for the disturbance pseudomomentum. The above results are obtained not only for isentropic background states (which give the so-called “deep form” of the anelastic equations), but also for arbitrary background potential-temperature profiles θ0(z) so long as the variation in θ0(z) over the depth of the fluid is small compared with θ0 itself. The Hamiltonian structure of the equations is established in both cases, and its symmetry properties discussed. An expression for available potential energy is also derived that, for the case of a stably stratified background state (i.e., dθ0/dz > 0), is locally positive definite; the expression is valid for fully three-dimensional flow. The counterparts to results for the two-dimensional Boussinesq equations are also noted.

Dynamical control of the mesosphere by orographic and non-orographic gravity wave drag during the extended northern winters of 2006 and 2009

A version of the Canadian Middle Atmosphere Model (CMAM) that is nudged toward reanalysis data up to 1 hPa is used to examine the impacts of parameterized orographic and non-orographic gravity wave drag (OGWD and NGWD) on the zonal-mean circulation of the mesosphere during the extended northern winters of 2006 and 2009 when there were two large stratospheric sudden warmings. The simulations are compared to Aura Microwave Limb Sounder (MLS) observations of mesospheric temperature, carbon monoxide (CO) and derived zonal winds. The control simulation, which uses both OGWD and NGWD, is shown to be in good agreement with MLS. The impacts of OGWD and NGWD are assessed using simulations in which those sources of wave drag are removed. In the absence of OGWD the mesospheric zonal winds in the months preceding the warmings are too strong, causing increased mesospheric NGWD, which drives excessive downwelling, resulting in overly large lower mesospheric values of CO prior to the warming. NGWD is found to be most important following the warmings when the underlying westerlies are too weak to allow much vertical propagation of the orographic gravity waves to the mesosphere. NGWD is primarily responsible for driving the circulation that results in the descent of CO from the thermosphere following the warmings. Zonal mean mesospheric winds and temperatures in all simulations are shown to be strongly constrained by (i.e. slaved to) the stratosphere. Finally, it is demonstrated that the responses to OGWD and NGWD are non-additive due to their dependence and influence on the background winds and temperatures.

Forecasting VaR using analytic higher moments for GARCH processes

It is widely accepted that some of the most accurate Value-at-Risk (VaR) estimates are based on an appropriately specified GARCH process. But when the forecast horizon is greater than the frequency of the GARCH model, such predictions have typically required time-consuming simulations of the aggregated returns distributions. This paper shows that fast, quasi-analytic GARCH VaR calculations can be based on new formulae for the first four moments of aggregated GARCH returns. Our extensive empirical study compares the Cornish–Fisher expansion with the Johnson SU distribution for fitting distributions to analytic moments of normal and Student t, symmetric and asymmetric (GJR) GARCH processes to returns data on different financial assets, for the purpose of deriving accurate GARCH VaR forecasts over multiple horizons and significance levels.

Projection of global wave climate change towards the end of the 21st century

Wind generated waves at the sea surface are of outstanding importance for both their practical relevance in many aspects, such as coastal erosion, protection, or safety of navigation, and for their scientific relevance in modifying fluxes at the air-sea interface. So far long-term changes in ocean wave climate have been studied mostly from a regional perspective with global dynamical studies emerging only recently. Here a global wave climate study is presented, in which a global wave model (WAM) is driven by atmospheric forcing from a global climate model (ECHAM5) for present day and potential future climate conditions represented by the IPCC (Intergovernmental Panel for Climate Change) A1B emission scenario. It is found that changes in mean and extreme wave climate towards the end of the twenty-first century are small to moderate, with the largest signals being a poleward shift in the annual mean and extreme significant wave heights in the mid-latitudes of both hemispheres, more pronounced in the Southern Hemisphere, and most likely associated with a corresponding shift in mid-latitude storm tracks. These changes are broadly consistent with results from the few studies available so far. The projected changes in the mean wave periods, associated with the changes in the wave climate in the mid to high latitudes, are also shown, revealing a moderate increase in the equatorial eastern side of the ocean basins. This study presents a step forward towards a larger ensemble of global wave climate projections required to better assess robustness and uncertainty of potential future wave climate change.

Wave activity in the tropical tropopause layer in seven reanalysis and four chemistry climate model data sets

Sub-seasonal variability including equatorial waves significantly influence the dehydration and transport processes in the tropical tropopause layer (TTL). This study investigates the wave activity in the TTL in 7 reanalysis data sets (RAs; NCEP1, NCEP2, ERA40, ERA-Interim, JRA25, MERRA, and CFSR) and 4 chemistry climate models (CCMs; CCSRNIES, CMAM, MRI, and WACCM) using the zonal wave number-frequency spectral analysis method with equatorially symmetric-antisymmetric decomposition. Analyses are made for temperature and horizontal winds at 100 hPa in the RAs and CCMs and for outgoing longwave radiation (OLR), which is a proxy for convective activity that generates tropopause-level disturbances, in satellite data and the CCMs. Particular focus is placed on equatorial Kelvin waves, mixed Rossby-gravity (MRG) waves, and the Madden-Julian Oscillation (MJO). The wave activity is defined as the variance, i.e., the power spectral density integrated in a particular zonal wave number-frequency region. It is found that the TTL wave activities show significant difference among the RAs, ranging from ∼0.7 (for NCEP1 and NCEP2) to ∼1.4 (for ERA-Interim, MERRA, and CFSR) with respect to the averages from the RAs. The TTL activities in the CCMs lie generally within the range of those in the RAs, with a few exceptions. However, the spectral features in OLR for all the CCMs are very different from those in the observations, and the OLR wave activities are too low for CCSRNIES, CMAM, and MRI. It is concluded that the broad range of wave activity found in the different RAs decreases our confidence in their validity and in particular their value for validation of CCM performance in the TTL, thereby limiting our quantitative understanding of the dehydration and transport processes in the TTL.

Forecasting by factors, by variables, by both or neither?

We consider forecasting with factors, variables and both, modeling in-sample using Autometrics so all principal components and variables can be included jointly, while tackling multiple breaks by impulse-indicator saturation. A forecast-error taxonomy for factor models highlights the impacts of location shifts on forecast-error biases. Forecasting US GDP over 1-, 4- and 8-step horizons using the dataset from Stock and Watson (2009) updated to 2011:2 shows factor models are more useful for nowcasting or short-term forecasting, but their relative performance declines as the forecast horizon increases. Forecasts for GDP levels highlight the need for robust strategies, such as intercept corrections or differencing, when location shifts occur as in the recent financial crisis.

Macroeconomic forecasting with mixed-frequency data: forecasting output growth in the United States

Many macroeconomic series, such as U.S. real output growth, are sampled quarterly, although potentially useful predictors are often observed at a higher frequency. We look at whether a mixed data-frequency sampling (MIDAS) approach can improve forecasts of output growth. The MIDAS specification used in the comparison uses a novel way of including an autoregressive term. We find that the use of monthly data on the current quarter leads to significant improvement in forecasting current and next quarter output growth, and that MIDAS is an effective way to exploit monthly data compared with alternative methods.

On the generalized eigenvalue problem for the Rossby wave vertical velocity in the presence of mean flow and topography

In a series of papers, Killworth and Blundell have proposed to study the effects of a background mean flow and topography on Rossby wave propagation by means of a generalized eigenvalue problem formulated in terms of the vertical velocity, obtained from a linearization of the primitive equations of motion. However, it has been known for a number of years that this eigenvalue problem contains an error, which Killworth was prevented from correcting himself by his unfortunate passing and whose correction is therefore taken up in this note. Here, the author shows in the context of quasigeostrophic (QG) theory that the error can ulti- mately be traced to the fact that the eigenvalue problem for the vertical velocity is fundamentally a non- linear one (the eigenvalue appears both in the numerator and denominator), unlike that for the pressure. The reason that this nonlinear term is lacking in the Killworth and Blundell theory comes from neglecting the depth dependence of a depth-dependent term. This nonlinear term is shown on idealized examples to alter significantly the Rossby wave dispersion relation in the high-wavenumber regime but is otherwise irrelevant in the long-wave limit, in which case the eigenvalue problems for the vertical velocity and pressure are both linear. In the general dispersive case, however, one should first solve the generalized eigenvalue problem for the pressure vertical structure and, if needed, diagnose the vertical velocity vertical structure from the latter.

Multi-site evaluation of an urban land-surface model: intra-urban heterogeneity, seasonality and parameter complexity requirements

An extensive off-line evaluation of the Noah/Single Layer Urban Canopy Model (Noah/SLUCM) urban land-surface model is presented using data from 15 sites to assess (1) the ability of the scheme to reproduce the surface energy balance observed in a range of urban environments, including seasonal changes, and (2) the impact of increasing complexity of input parameter information. Model performance is found to be most dependent on representation of vegetated surface area cover; refinement of other parameter values leads to smaller improvements. Model biases in net all-wave radiation and trade-offs between turbulent heat fluxes are highlighted using an optimization algorithm. Here we use the Urban Zones to characterize Energy partitioning (UZE) as the basis to assign default SLUCM parameter values. A methodology (FRAISE) to assign sites (or areas) to one of these categories based on surface characteristics is evaluated. Using three urban sites from the Basel Urban Boundary Layer Experiment (BUBBLE) dataset, an independent evaluation of the model performance with the parameter values representative of each class is performed. The scheme copes well with both seasonal changes in the surface characteristics and intra-urban heterogeneities in energy flux partitioning, with RMSE performance comparable to similar state-of-the-art models for all fluxes, sites and seasons. The potential of the methodology for high-resolution atmospheric modelling application using the Weather Research and Forecasting (WRF) model is highlighted. This analysis supports the recommendations that (1) three classes are appropriate to characterize the urban environment, and (2) that the parameter values identified should be adopted as default values in WRF.

Effects of temporal resolution of input precipitation on the performance of hydrological forecasting

Flood prediction systems rely on good quality precipitation input data and forecasts to drive hydrological models. Most precipitation data comes from daily stations with a good spatial coverage. However, some flood events occur on sub-daily time scales and flood prediction systems could benefit from using models calibrated on the same time scale. This study compares precipitation data aggregated from hourly stations (HP) and data disaggregated from daily stations (DP) with 6-hourly forecasts from ECMWF over the time period 1 October 2006–31 December 2009. The HP and DP data sets were then used to calibrate two hydrological models, LISFLOOD-RR and HBV, and the latter was used in a flood case study. The HP scored better than the DP when evaluated against the forecast for lead times up to 4 days. However, this was not translated in the same way to the hydrological modelling, where the models gave similar scores for simulated runoff with the two datasets. The flood forecasting study showed that both datasets gave similar hit rates whereas the HP data set gave much smaller false alarm rates (FAR). This indicates that using sub-daily precipitation in the calibration and initiation of hydrological models can improve flood forecasting.

Ensemble forecasting using TIGGE for the July-September 2008 floods in the Upper Huai catchment: a case study.

We present a case study using the TIGGE database for flood warning in the Upper Huai catchment (ca. 30 672 km2). TIGGE ensemble forecasts from 6 meteorological centres with 10-day lead time were extracted and disaggregated to drive the Xinanjiang model to forecast discharges for flood events in July-September 2008. The results demonstrated satisfactory flood forecasting skills with clear signals of floods up to 10 days in advance. The forecasts occasionally show discrepancies both in time and space. Forecasting quality could potentially be improved by using temporal and spatial corrections of the forecasted precipitation.

Challenges in communicating and using ensembles in operational flood forecasting.

Following trends in operational weather forecasting, where ensemble prediction systems (EPS) are now increasingly the norm, flood forecasters are beginning to experiment with using similar ensemble methods. Most of the effort to date has focused on the substantial technical challenges of developing coupled rainfall-runoff systems to represent the full cascade of uncertainties involved in predicting future flooding. As a consequence much less attention has been given to the communication and eventual use of EPS flood forecasts. Drawing on interviews and other research with operational flood forecasters from across Europe, this paper highlights a number of challenges to communicating and using ensemble flood forecasts operationally. It is shown that operational flood forecasters understand the skill, operational limitations, and informational value of EPS products in a variety of different and sometimes contradictory ways. Despite the efforts of forecasting agencies to design effective ways to communicate EPS forecasts to non-experts, operational flood forecasters were often skeptical about the ability of forecast recipients to understand or use them appropriately. It is argued that better training and closer contacts between operational flood forecasters and EPS system designers can help ensure the uncertainty represented by EPS forecasts is represented in ways that are most appropriate and meaningful for their intended consumers, but some fundamental political and institutional challenges to using ensembles, such as differing attitudes to false alarms and to responsibility for management of blame in the event of poor or mistaken forecasts are also highlighted. Copyright © 2010 Royal Meteorological Society.

Ensemble flood forecasting: a review.

Operational medium range flood forecasting systems are increasingly moving towards the adoption of ensembles of numerical weather predictions (NWP), known as ensemble prediction systems (EPS), to drive their predictions. We review the scientific drivers of this shift towards such ‘ensemble flood forecasting’ and discuss several of the questions surrounding best practice in using EPS in flood forecasting systems. We also review the literature evidence of the ‘added value’ of flood forecasts based on EPS and point to remaining key challenges in using EPS successfully.