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.

On the 'downward control' of extratropical diabatic circulations by eddy-induced mean zonal forces

The situation considered is that of a zonally symmetric model of the middle atmosphere subject to a given quasi-steady zonal force F̄, conceived to be the result of irreversible angular momentum transfer due to the upward propagation and breaking of Rossby and gravity waves together with any other dissipative eddy effects that may be relevant. The model's diabatic heating is assumed to have the qualitative character of a relaxation toward some radiatively determined temperature field. To the extent that the force F̄ may be regarded as given, and the extratropical angular momentum distribution is realistic, the extratropical diabatic mass flow across a given isentropic surface may be regarded as controlled exclusively by the F̄ distribution above that surface (implying control by the eddy dissipation above that surface and not, for instance, by the frequency of tropopause folding below). This “downward control” principle expresses a critical part of the dynamical chain of cause and effect governing the average rate at which photochemical products like ozone become available for folding into, or otherwise descending into, the extratropical troposphere. The dynamical facts expressed by the principle are also relevant, for instance, to understanding the seasonal-mean rate of upwelling of water vapor to the summer mesopause, and the interhemispheric differences in stratospheric tracer transport. The robustness of the principle is examined when F̄ is time-dependent. For a global-scale, zonally symmetric diabatic circulation with a Brewer-Dobson-like horizontal structure given by the second zonally symmetric Hough mode, with Rossby height HR = 13 km in an isothermal atmosphere with density scale height H = 7 km, the vertical partitioning of the unsteady part of the mass circulation caused by fluctuations in F̄ confined to a shallow layer LF̄ is always at least 84% downward. It is 90% downward when the force fluctuates sinusoidally on twice the radiative relaxation timescale and 95% if five times slower. The time-dependent adjustment when F̄ is changed suddenly is elucidated, extending the work of Dickinson (1968), when the atmosphere is unbounded above and below. Above the forcing, the adjustment is characterized by decay of the meridional mass circulation cell at a rate proportional to the radiative relaxation rate τr−1 divided by {1 + (4H2/HR2)}. This decay is related to the boundedness of the angular momentum that can be taken up by the finite mass of air above LF̄ without causing an ever-increasing departure from thermal wind balance. Below the forcing, the meridional mass circulation cell penetrates downward at a speed τr−1 HR2/H. For the second Hough mode, the time for downward penetration through one density scale height is about 6 days if the radiative relaxation time is 20 days, the latter being representative of the lower stratosphere. At any given altitude, a steady state is approached. The effect of a rigid lower boundary on the time-dependent adjustment is also considered. If a frictional planetary boundary layer is present then a steady state is ultimately approached everywhere, with the mass circulation extending downward from LF̄ and closing via the boundary layer. Satellite observations of temperature and ozone are used in conjunction with a radiative transfer scheme to estimate the altitudes from which the lower stratospheric diabatic vertical velocity is controlled by the effective F̄ in the real atmosphere. The data appear to indicate that about 80% of the effective control is usually exerted from below 40 km but with significant exceptions up to 70 km (in the high latitude southern hemispheric winter). The implications for numerical modelling of chemical transport are noted.

Southern annular mode dynamics in observations and models, Part II: Eddy feedbacks

Many global climate models (GCMs) have trouble simulating Southern Annular Mode (SAM) variability correctly, particularly in the Southern Hemisphere summer season where it tends to be too persistent. In this two part study, a suite of experiments with the Canadian Middle Atmosphere Model (CMAM) is analyzed to improve our understanding of the dynamics of SAM variability and its deficiencies in GCMs. Here, an examination of the eddy-mean flow feedbacks is presented by quantification of the feedback strength as a function of zonal scale and season using a new methodology that accounts for intraseasonal forcing of the SAM. In the observed atmosphere, in the summer season, a strong negative feedback by planetary scale waves, in particular zonal wavenumber 3, is found in a localized region in the south west Pacific. It cancels a large proportion of the positive feedback by synoptic and smaller scale eddies in the zonal mean, resulting in a very weak overall eddy feedback on the SAM. CMAM is deficient in this negative feedback by planetary scale waves, making a substantial contribution to its bias in summertime SAM persistence. Furthermore, this bias is not alleviated by artificially improving the climatological circulation, suggesting that climatological circulation biases are not the cause of the planetary wave feedback deficiency in the model. Analysis of the summertime eddy feedbacks in the CMIP-5 models confirms that this is indeed a common problem among GCMs, suggesting that understanding this planetary wave feedback and the reason for its deficiency in GCMs is key to improving the fidelity of simulated SAM variability in the summer season.

The dependence of southern ocean meridional overturning on wind stress

An eddy-resolving numerical model of a zonal flow, meant to resemble the Antarctic Circumpolar Current, is described and analyzed using the framework of J. Marshall and T. Radko. In addition to wind and buoyancy forcing at the surface, the model contains a sponge layer at the northern boundary that permits a residual meridional overturning circulation (MOC) to exist at depth. The strength of the residual MOC is diagnosed for different strengths of surface wind stress. It is found that the eddy circulation largely compensates for the changes in Ekman circulation. The extent of the compensation and thus the sensitivity of the MOC to the winds depend on the surface boundary condition. A fixed-heat-flux surface boundary severely limits the ability of the MOC to change. An interactive heat flux leads to greater sensitivity. To explain the MOC sensitivity to the wind strength under the interactive heat flux, transformed Eulerian-mean theory is applied, in which the eddy diffusivity plays a central role in determining the eddy response. A scaling theory for the eddy diffusivity, based on the mechanical energy balance, is developed and tested; the average magnitude of the diffusivity is found to be proportional to the square root of the wind stress. The MOC sensitivity to the winds based on this scaling is compared with the true sensitivity diagnosed from the experiments.

Controlling spurious diapycnal mixing in eddy-resolving height-coordinate ocean models: insights from virtual deliberate tracer release experiments

A perceived limitation of z-coordinate models associated with spurious diapycnal mixing in eddying, frontal flow, can be readily addressed through appropriate attention to the tracer advection schemes employed. It is demonstrated that tracer advection schemes developed by Prather and collaborators for application in the stratosphere, greatly improve the fidelity of eddying flows, reducing levels of spurious diapycnal mixing to below those directly measured in field experiments, ∼1 × 10−5 m2 s−1. This approach yields a model in which geostrophic eddies are quasi-adiabatic in the ocean interior, so that the residual-mean overturning circulation aligns almost perfectly with density contours. A reentrant channel configuration of the MIT General Circulation Model, that approximates the Antarctic Circumpolar Current, is used to examine these issues. Virtual analogs of ocean deliberate tracer release field experiments reinforce our conclusion, producing passive tracer solutions that parallel field experiments remarkably well.

Spectral nonlocality, absolute equilibria and Kraichnan-Leith-Batchelor phenomenology in two-dimensional turbulent energy cascades

We study the degree to which Kraichnan–Leith–Batchelor (KLB) phenomenology describes two-dimensional energy cascades in α turbulence, governed by ∂θ/∂t+J(ψ,θ)=ν∇2θ+f, where θ=(−Δ)α/2ψ is generalized vorticity, and ψ^(k)=k−αθ^(k) in Fourier space. These models differ in spectral non-locality, and include surface quasigeostrophic flow (α=1), regular two-dimensional flow (α=2) and rotating shallow flow (α=3), which is the isotropic limit of a mantle convection model. We re-examine arguments for dual inverse energy and direct enstrophy cascades, including Fjørtoft analysis, which we extend to general α, and point out their limitations. Using an α-dependent eddy-damped quasinormal Markovian (EDQNM) closure, we seek self-similar inertial range solutions and study their characteristics. Our present focus is not on coherent structures, which the EDQNM filters out, but on any self-similar and approximately Gaussian turbulent component that may exist in the flow and be described by KLB phenomenology. For this, the EDQNM is an appropriate tool. Non-local triads contribute increasingly to the energy flux as α increases. More importantly, the energy cascade is downscale in the self-similar inertial range for 2.5<α<10. At α=2.5 and α=10, the KLB spectra correspond, respectively, to enstrophy and energy equipartition, and the triad energy transfers and flux vanish identically. Eddy turnover time and strain rate arguments suggest the inverse energy cascade should obey KLB phenomenology and be self-similar for α<4. However, downscale energy flux in the EDQNM self-similar inertial range for α>2.5 leads us to predict that any inverse cascade for α≥2.5 will not exhibit KLB phenomenology, and specifically the KLB energy spectrum. Numerical simulations confirm this: the inverse cascade energy spectrum for α≥2.5 is significantly steeper than the KLB prediction, while for α<2.5 we obtain the KLB spectrum.

Determination of eddy-diffusivity in the lowermost stratosphere

We present a 2D-advection-diffusion model that simulates the main transport pathways influencing tracer distributions in the lowermost stratosphere (LMS). The model describes slow diabatic descent of aged stratospheric air, vertical (cross-isentropic) and horizontal (along isentropes) diffusion within the LMS and across the tropopause using equivalent latitude and potential temperature coordinates. Eddy diffusion coefficients parameterize the integral effect of dynamical processes leading to small scale turbulence and mixing. They were specified by matching model simulations to observed CO distributions. Interestingly, the model suggests mixing across isentropes to be more important than horizontal mixing across surfaces of constant equivalent latitude, shining new light on the interplay between various transport mechanisms in the LMS. The model achieves a good description of the small scale tracer features at the tropopause with squared correlation coefficients R2 = 0.72…0.94.

Characterization of energy flux partitioning in urban environments: links with surface seasonal properties

A better understanding of links between the properties of the urban environment and the exchange to the atmosphere is central to a wide range of applications. The numerous measurements of surface energy balance data in urban areas enable intercomparison of observed fluxes from distinct environments. This study analyzes a large database in two new ways. First, instead of normalizing fluxes using net all-wave radiation only the incoming radiative fluxes are used, to remove the surface attributes from the denominator. Second, because data are now available year-round, indices are developed to characterize the fraction of the surface (built; vegetation) actively engaged in energy exchanges. These account for shading patterns within city streets and seasonal changes in vegetation phenology; their impact on the partitioning of the incoming radiation is analyzed. Data from 19 sites in North America, Europe, Africa, and Asia (including 6-yr-long observation campaigns) are used to derive generalized surface–flux relations. The midday-period outgoing radiative fraction decreases with an increasing total active surface index, the stored energy fraction increases with an active built index, and the latent heat fraction increases with an active vegetated index. Parameterizations of these energy exchange ratios as a function of the surface indices [i.e., the Flux Ratio–Active Index Surface Exchange (FRAISE) scheme] are developed. These are used to define four urban zones that characterize energy partitioning on the basis of their active surface indices. An independent evaluation of FRAISE, using three additional sites from the Basel Urban Boundary Layer Experiment (BUBBLE), yields accurate predictions of the midday flux partitioning at each location.

Global to city scale urban anthropogenic heat flux: model and variability

The large scale urban consumption of energy (LUCY) model simulates all components of anthropogenic heat flux (QF) from the global to individual city scale at 2.5 × 2.5 arc-minute resolution. This includes a database of different working patterns and public holidays, vehicle use and energy consumption in each country. The databases can be edited to include specific diurnal and seasonal vehicle and energy consumption patterns, local holidays and flows of people within a city. If better information about individual cities is available within this (open-source) database, then the accuracy of this model can only improve, to provide the community data from global-scale climate modelling or the individual city scale in the future. The results show that QF varied widely through the year, through the day, between countries and urban areas. An assessment of the heat emissions estimated revealed that they are reasonably close to those produced by a global model and a number of small-scale city models, so results from LUCY can be used with a degree of confidence. From LUCY, the global mean urban QF has a diurnal range of 0.7–3.6 W m−2, and is greater on weekdays than weekends. The heat release from building is the largest contributor (89–96%), to heat emissions globally. Differences between months are greatest in the middle of the day (up to 1 W m−2 at 1 pm). December to February, the coldest months in the Northern Hemisphere, have the highest heat emissions. July and August are at the higher end. The least QF is emitted in May. The highest individual grid cell heat fluxes in urban areas were located in New York (577), Paris (261.5), Tokyo (178), San Francisco (173.6), Vancouver (119) and London (106.7). Copyright © 2010 Royal Meteorological Society

Sensitivity analysis of an ocean carbon cycle model in the North Atlantic: an investigation of parameters affecting the air-sea CO2 flux, primary production and export of detritus

The sensitivity of the biological parameters in a nutrient-phytoplankton-zooplankton-detritus (NPZD) model in the calculation of the air-sea CO2 flux, primary production and detrital export is analysed. We explore the effect on these outputs of variation in the values of the twenty parameters that control ocean ecosystem growth in a 1-D formulation of the UK Met Office HadOCC NPZD model used in GCMs. We use and compare the results from one-at-a-time and all-at-a-time perturbations performed at three sites in the EuroSITES European Ocean Observatory Network: the Central Irminger Sea (60° N 40° W), the Porcupine Abyssal Plain (49° N 16° W) and the European Station for Time series in the Ocean Canary Islands (29° N 15° W). Reasonable changes to the values of key parameters are shown to have a large effect on the calculation of the air-sea CO2 flux, primary production, and export of biological detritus to the deep ocean. Changes in the values of key parameters have a greater effect in more productive regions than in less productive areas. The most sensitive parameters are generally found to be those controlling well-established ocean ecosystem parameterisations widely used in many NPZD-type models. The air-sea CO2 flux is most influenced by variation in the parameters that control phytoplankton growth, detrital sinking and carbonate production by phytoplankton (the rain ratio). Primary production is most sensitive to the parameters that define the shape of the photosynthesis-irradiance curve. Export production is most sensitive to the parameters that control the rate of detrital sinking and the remineralisation of detritus.

The impact of diurnal variability in sea surface temperature on the atlantic air-sea CO2 flux

The effect of diurnal variations in sea surface temperature (SST) on the air-sea flux of CO2 over the central Atlantic ocean and Mediterranean Sea (60 S–60 N, 60 W–45 E) is evaluated for 2005–2006. We use high spatial resolution hourly satellite ocean skin temperature data to determine the diurnal warming (ΔSST). The CO2 flux is then computed using three different temperature fields – a foundation temperature (Tf, measured at a depth where there is no diurnal variation), Tf, plus the hourly ΔSST and Tf, plus the monthly average of the ΔSSTs. This is done in conjunction with a physically-based parameterisation for the gas transfer velocity (NOAA-COARE). The differences between the fluxes evaluated for these three different temperature fields quantify the effects of both diurnal warming and diurnal covariations. We find that including diurnal warming increases the CO2 flux out of this region of the Atlantic for 2005–2006 from 9.6 Tg C a−1 to 30.4 Tg C a−1 (hourly ΔSST) and 31.2 Tg C a−1 (monthly average of ΔSST measurements). Diurnal warming in this region, therefore, has a large impact on the annual net CO2 flux but diurnal covariations are negligible. However, in this region of the Atlantic the uptake and outgassing of CO2 is approximately balanced over the annual cycle, so although we find diurnal warming has a very large effect here, the Atlantic as a whole is a very strong carbon sink (e.g. −920 Tg C a−1 Takahashi et al., 2002) making this is a small contribution to the Atlantic carbon budget.

Impact of city changes and weather on anthropogenic heat flux in Europe 1995–2015

How people live, work, move from place to place, consume and the technologies they use all affect heat emissions in a city which influences urban weather and climate. Here we document changes to a global anthropogenic heat flux (QF) model to enhance its spatial (30′′ × 30′′ to 0.5° × 0.5°) resolution and temporal coverage (historical, current and future). QF is estimated across Europe (1995–2015), considering changes in temperature, population and energy use. While on average QF is small (of the order 1.9–4.6 W m−2 across all the urban areas of Europe), significant spatial variability is documented (maximum 185 W m−2). Changes in energy consumption due to changes in climate are predicted to cause a 13% (11%) increase in QF on summer (winter) weekdays. The largest impact results from changes in temperature conditions which influences building energy use; for winter, with the coldest February on record, the mean flux for urban areas of Europe is 4.56 W m−2 and for summer (warmest July on record) is 2.23 W m−2. Detailed results from London highlight the spatial resolution used to model the QF is critical and must be appropriate for the application at hand, whether scientific understanding or decision making.

Reconstruction and prediction of variations in the open solar magnetic flux and interplanetary conditions

Historic geomagnetic activity observations have been used to reveal centennial variations in the open solar flux and the near-Earth heliospheric conditions (the interplanetary magnetic field and the solar wind speed). The various methods are in very good agreement for the past 135 years when there were sufficient reliable magnetic observatories in operation to eliminate problems due to site-specific errors and calibration drifts. This review underlines the physical principles that allow these reconstructions to be made, as well as the details of the various algorithms employed and the results obtained. Discussion is included of: the importance of the averaging timescale; the key differences between “range” and “interdiurnal variability” geomagnetic data; the need to distinguish source field sector structure from heliospherically-imposed field structure; the importance of ensuring that regressions used are statistically robust; and uncertainty analysis. The reconstructions are exceedingly useful as they provide calibration between the in-situ spacecraft measurements from the past five decades and the millennial records of heliospheric behaviour deduced from measured abundances of cosmogenic radionuclides found in terrestrial reservoirs. Continuity of open solar flux, using sunspot number to quantify the emergence rate, is the basis of a number of models that have been very successful in reproducing the variation derived from geomagnetic activity. These models allow us to extend the reconstructions back to before the development of the magnetometer and to cover the Maunder minimum. Allied to the radionuclide data, the models are revealing much about how the Sun and heliosphere behaved outside of grand solar maxima and are providing a means of predicting how solar activity is likely to evolve now that the recent grand maximum (that had prevailed throughout the space age) has come to an end.

The 22-Year Hale Cycle in cosmic ray flux: evidence for direct heliospheric modulation

The ability to predict times of greater galactic cosmic ray (GCR) fluxes is important for reducing the hazards caused by these particles to satellite communications, aviation, or astronauts. The 11-year solar-cycle variation in cosmic rays is highly correlated with the strength of the heliospheric magnetic field. Differences in GCR flux during alternate solar cycles yield a 22-year cycle, known as the Hale Cycle, which is thought to be due to different particle drift patterns when the northern solar pole has predominantly positive (denoted as qA>0 cycle) or negative (qA<0) polarities. This results in the onset of the peak cosmic-ray flux at Earth occurring earlier during qA>0 cycles than for qA<0 cycles, which in turn causes the peak to be more dome-shaped for qA>0 and more sharply peaked for qA<0. In this study, we demonstrate that properties of the large-scale heliospheric magnetic field are different during the declining phase of the qA<0 and qA>0 solar cycles, when the difference in GCR flux is most apparent. This suggests that particle drifts may not be the sole mechanism responsible for the Hale Cycle in GCR flux at Earth. However, we also demonstrate that these polarity-dependent heliospheric differences are evident during the space-age but are much less clear in earlier data: using geomagnetic reconstructions, we show that for the period of 1905 - 1965, alternate polarities do not give as significant a difference during the declining phase of the solar cycle. Thus we suggest that the 22-year cycle in cosmic-ray flux is at least partly the result of direct modulation by the heliospheric magnetic field and that this effect may be primarily limited to the grand solar maximum of the space-age.