Spectrally invariant approximation within atmospheric radiative transfer

Certain algebraic combinations of single scattering albedo and solar radiation reflected from, or transmitted through, vegetation canopies do not vary with wavelength. These ‘‘spectrally invariant relationships’’ are the consequence of wavelength independence of the extinction coefficient and scattering phase function in veg- etation. In general, this wavelength independence does not hold in the atmosphere, but in cloud-dominated atmospheres the total extinction and total scattering phase function vary only weakly with wavelength. This paper identifies the atmospheric conditions under which the spectrally invariant approximation can accu- rately describe the extinction and scattering properties of cloudy atmospheres. The validity of the as- sumptions and the accuracy of the approximation are tested with 1D radiative transfer calculations using publicly available radiative transfer models: Discrete Ordinate Radiative Transfer (DISORT) and Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART). It is shown for cloudy atmospheres with cloud optical depth above 3, and for spectral intervals that exclude strong water vapor absorption, that the spectrally invariant relationships found in vegetation canopy radiative transfer are valid to better than 5%. The physics behind this phenomenon, its mathematical basis, and possible applications to remote sensing and climate are discussed.

Radar scattering from ice aggregates using the horizontally aligned oblate spheroid approximation

The assumed relationship between ice particle mass and size is profoundly important in radar retrievals of ice clouds, but, for millimeter-wave radars, shape and preferred orientation are important as well. In this paper the authors first examine the consequences of the fact that the widely used ‘‘Brown and Francis’’ mass–size relationship has often been applied to maximumparticle dimension observed by aircraftDmax rather than to the mean of the particle dimensions in two orthogonal directions Dmean, which was originally used by Brown and Francis. Analysis of particle images reveals that Dmax ’ 1.25Dmean, and therefore, for clouds for which this mass–size relationship holds, the consequences are overestimates of ice water content by around 53% and of Rayleigh-scattering radar reflectivity factor by 3.7 dB. Simultaneous radar and aircraft measurements demonstrate that much better agreement in reflectivity factor is provided by using this mass–size relationship with Dmean. The authors then examine the importance of particle shape and fall orientation for millimeter-wave radars. Simultaneous radar measurements and aircraft calculations of differential reflectivity and dual-wavelength ratio are presented to demonstrate that ice particles may usually be treated as horizontally aligned oblate spheroids with an axial ratio of 0.6, consistent with them being aggregates. An accurate formula is presented for the backscatter cross section apparent to a vertically pointing millimeter-wave radar on the basis of a modified version of Rayleigh–Gans theory. It is then shown that the consequence of treating ice particles as Mie-scattering spheres is to substantially underestimate millimeter-wave reflectivity factor when millimeter-sized particles are present, which can lead to retrieved ice water content being overestimated by a factor of 4.h

Cloud-resolving model simulations with one- and two-way couplings via the weak temperature gradient approximation

A cloud-resolving model is modified to implement the weak temperature gradient approximation in order to simulate the interactions between tropical convection and the large-scale tropical circulation. The instantaneous domain-mean potential temperature is relaxed toward a reference profile obtained from a radiative–convective equilibrium simulation of the cloud-resolving model. For homogeneous surface conditions, the model state at equilibrium is a large-scale circulation with its descending branch in the simulated column. This is similar to the equilibrium state found in some other studies, but not all. For this model, the development of such a circulation is insensitive to the relaxation profile and the initial conditions. Two columns of the cloud-resolving model are fully coupled by relaxing the instantaneous domain-mean potential temperature in both columns toward each other. This configuration is energetically closed in contrast to the reference-column configuration. No mean large-scale circulation develops over homogeneous surface conditions, regardless of the relative area of the two columns. The sensitivity to nonuniform surface conditions is similar to that obtained in the reference-column configuration if the two simulated columns have very different areas, but it is markedly weaker for columns of comparable area. The weaker sensitivity can be understood as being a consequence of a formulation for which the energy budget is closed. The reference-column configuration has been used to study the convection in a local region under the influence of a large-scale circulation. The extension to a two-column configuration is proposed as a methodology for studying the influence on local convection of changes in remote convection.

On the approximation of local and linear radiative damping in the middle atmosphere

The validity of approximating radiative heating rates in the middle atmosphere by a local linear relaxation to a reference temperature state (i.e., ‘‘Newtonian cooling’’) is investigated. Using radiative heating rate and temperature output from a chemistry–climate model with realistic spatiotemporal variability and realistic chemical and radiative parameterizations, it is found that a linear regressionmodel can capture more than 80% of the variance in longwave heating rates throughout most of the stratosphere and mesosphere, provided that the damping rate is allowed to vary with height, latitude, and season. The linear model describes departures from the climatological mean, not from radiative equilibrium. Photochemical damping rates in the upper stratosphere are similarly diagnosed. Threeimportant exceptions, however, are found.The approximation of linearity breaks down near the edges of the polar vortices in both hemispheres. This nonlinearity can be well captured by including a quadratic term. The use of a scale-independentdamping rate is not well justified in the lower tropical stratosphere because of the presence of a broad spectrum of vertical scales. The local assumption fails entirely during the breakup of the Antarctic vortex, where large fluctuations in temperature near the top of the vortex influence longwave heating rates within the quiescent region below. These results are relevant for mechanistic modeling studies of the middle atmosphere, particularly those investigating the final Antarctic warming.

Sparse polynomial approximation in positive order Sobolev spaces with bounded mixed derivatives and applications to elliptic problems with random loading

In the present paper we study the approximation of functions with bounded mixed derivatives by sparse tensor product polynomials in positive order tensor product Sobolev spaces. We introduce a new sparse polynomial approximation operator which exhibits optimal convergence properties in L2 and tensorized View the MathML source simultaneously on a standard k-dimensional cube. In the special case k=2 the suggested approximation operator is also optimal in L2 and tensorized H1 (without essential boundary conditions). This allows to construct an optimal sparse p-version FEM with sparse piecewise continuous polynomial splines, reducing the number of unknowns from O(p2), needed for the full tensor product computation, to View the MathML source, required for the suggested sparse technique, preserving the same optimal convergence rate in terms of p. We apply this result to an elliptic differential equation and an elliptic integral equation with random loading and compute the covariances of the solutions with View the MathML source unknowns. Several numerical examples support the theoretical estimates.

Approximation by harmonic polynomials in star-shaped domains and exponential convergence of Trefftz hp-dGFEM

We study the approximation of harmonic functions by means of harmonic polynomials in two-dimensional, bounded, star-shaped domains. Assuming that the functions possess analytic extensions to a delta-neighbourhood of the domain, we prove exponential convergence of the approximation error with respect to the degree of the approximating harmonic polynomial. All the constants appearing in the bounds are explicit and depend only on the shape-regularity of the domain and on delta. We apply the obtained estimates to show exponential convergence with rate O(exp(−b square root N)), N being the number of degrees of freedom and b>0, of a hp-dGFEM discretisation of the Laplace equation based on piecewise harmonic polynomials. This result is an improvement over the classical rate O(exp(−b cubic root N )), and is due to the use of harmonic polynomial spaces, as opposed to complete polynomial spaces.

Coordinated Cluster, ground-based instrumentation and low-altitude satellite observations of transient poleward-moving events in the ionosphere and in the tail lobe

During the interval between 8:00-9:30 on 14 January 2001, the four Cluster spacecraft were moving from the central magnetospheric lobe, through the dusk sector mantle, on their way towards intersecting the magnetopause near 15:00 MLT and 15:00 UT. Throughout this interval, the EIS-CAT Svalbard Radar (ESR) at Longyearbyen observed a series of poleward-moving transient events of enhanced F-region plasma concentration ("polar cap patches"), with a repetition period of the order of 10 min. Allowing for the estimated solar wind propagation delay of 75 ( 5) min, the interplanetary magnetic field (IMF) had a southward component during most of the interval. The magnetic footprint of the Cluster spacecraft, mapped to the ionosphere using the Tsyganenko T96 model (with input conditions prevailing during this event), was to the east of the ESR beams. Around 09:05 UT, the DMSP-F12 satellite flew over the ESR and showed a sawtooth cusp ion dispersion signature that also extended into the electrons on the equatorward edge of the cusp, revealing a pulsed magnetopause reconnection. The consequent enhanced ionospheric flow events were imaged by the SuperDARN HF backscatter radars. The average convection patterns (derived using the AMIE technique on data from the magnetometers, the EISCAT and SuperDARN radars, and the DMSP satellites) show that the associated poleward-moving events also convected over the predicted footprint of the Cluster spacecraft. Cluster observed enhancements in the fluxes of both electrons and ions. These events were found to be essentially identical at all four spacecraft, indicating that they had a much larger spatial scale than the satellite separation of the order of 600 km. Some of the events show a correspondence between the lowest energy magnetosheath electrons detected by the PEACE instrument on Cluster (10-20 eV) and the topside ionospheric enhancements seen by the ESR (at 400-700 km). We suggest that a potential barrier at the magnetopause, which prevents the lowest energy electrons from entering the magnetosphere, is reduced when and where the boundary-normal magnetic field is enhanced and that the observed polar cap patches are produced by the consequent enhanced precipitation of the lowest energy electrons, making them and the low energy electron precipitation fossil remnants of the magnetopause reconnection rate pulses.

Development of substorm cross-tail current disruption as seen from the ground

We discuss substorm observations made near 2100 magnetic local time (MLT) on March 7, 1991, in a collaborative study involving data from the European Incoherent Scatter radar, all-sky camera data, and magnetometer data from the Tromsø Auroral Observatory, the U.K. Sub-Auroral Magnetometer Network (SAMNET) and the IMAGE magnetometer chain. We conclude that for the substorm studied a plasmoid was not pinched off until at least 10 min after onset at the local time of the observations (2100 MLT) and that the main substorm electrojet expanded westward over this local time 14 min after onset. In the late growth phase/early expansion phase, we observed southward drifting arcs probably moving faster than the background plasma. Similar southward moving arcs in the recovery phase moved at a speed which does not appear to be significantly different from the measured plasma flow speed. We discuss these data in terms of the “Kiruna conjecture” and classical “near-Earth neutral line” paradigms, since the data show features of both models of substorm development. We suggest that longitudinal variation in behavior may reconcile the differences between the two models in the case of this substorm.

Excitation and decay of solar-wind driven flows in the magnetosphere-ionosphere system

Basic concepts of the form of high-latitude ionospheric flows and their excitation and decay are discussed in the light of recent high time-resolution measurements made by ground-based radars. It is first pointed out that it is in principle impossible to adequately parameterize these flows by any single quantity derived from concurrent interplanetary conditions. Rather, even at its simplest, the flow must be considered to consist of two basic time-dependent components. The first is the flow driven by magnetopause coupling processes alone, principally by dayside reconnection. These flows may indeed be reasonably parameterized in terms of concurrent near-Earth interplanetary conditions, principally by the interplanetary magnetic field (IMF) vector. The second is the flow driven by tail reconnection alone. As a first approximation these flows may also be parameterized in terms of interplanetary conditions, principally the north-south component of the IMF, but with a delay in the flow response of around 30-60 min relative to the IMF. A delay in the tail response of this order must be present due to the finite speed of information propagation in the system, and we show how "growth" and "decay" of the field and flow configuration then follow as natural consequences. To discuss the excitation and decay of the two reconnection-driven components of the flow we introduce that concept of a flow-free equilibrium configuration for a magnetosphere which contains a given (arbitrary) amount of open flux. Reconnection events act either to create or destroy open flux, thus causing departures of the system from the equilibrium configuration. Flow is then excited which moves the system back towards equilibrium with the changed amount of open flux. We estimate that the overall time scale associated with the excitation and decay of the flow is about 15 min. The response of the system to both impulsive (flux transfer event) and continuous reconnection is discussed in these terms.

Reducing noise associated with the Monte Carlo Independent Column Approximation for weather forecasting models

The Monte Carlo Independent Column Approximation (McICA) is a flexible method for representing subgrid-scale cloud inhomogeneity in radiative transfer schemes. It does, however, introduce conditional random errors but these have been shown to have little effect on climate simulations, where spatial and temporal scales of interest are large enough for effects of noise to be averaged out. This article considers the effect of McICA noise on a numerical weather prediction (NWP) model, where the time and spatial scales of interest are much closer to those at which the errors manifest themselves; this, as we show, means that noise is more significant. We suggest methods for efficiently reducing the magnitude of McICA noise and test these methods in a global NWP version of the UK Met Office Unified Model (MetUM). The resultant errors are put into context by comparison with errors due to the widely used assumption of maximum-random-overlap of plane-parallel homogeneous cloud. For a simple implementation of the McICA scheme, forecasts of near-surface temperature are found to be worse than those obtained using the plane-parallel, maximum-random-overlap representation of clouds. However, by applying the methods suggested in this article, we can reduce noise enough to give forecasts of near-surface temperature that are an improvement on the plane-parallel maximum-random-overlap forecasts. We conclude that the McICA scheme can be used to improve the representation of clouds in NWP models, with the provision that the associated noise is sufficiently small.

Equation for the microwave backscatter cross section of aggregate snowflakes using the self-similar Rayleigh–Gans approximation

In this paper an equation is derived for the mean backscatter cross section of an ensemble of snowflakes at centimeter and millimeter wavelengths. It uses the Rayleigh–Gans approximation, which has previously been found to be applicable at these wavelengths due to the low density of snow aggregates. Although the internal structure of an individual snowflake is random and unpredictable, the authors find from simulations of the aggregation process that their structure is “self-similar” and can be described by a power law. This enables an analytic expression to be derived for the backscatter cross section of an ensemble of particles as a function of their maximum dimension in the direction of propagation of the radiation, the volume of ice they contain, a variable describing their mean shape, and two variables describing the shape of the power spectrum. The exponent of the power law is found to be −. In the case of 1-cm snowflakes observed by a 3.2-mm-wavelength radar, the backscatter is 40–100 times larger than that of a homogeneous ice–air spheroid with the same mass, size, and aspect ratio.

Noether-type discrete conserved quantities arising from a finite element approximation of a variational problem

In this work, we prove a weak Noether-type Theorem for a class of variational problems that admit broken extremals. We use this result to prove discrete Noether-type conservation laws for a conforming finite element discretisation of a model elliptic problem. In addition, we study how well the finite element scheme satisfies the continuous conservation laws arising from the application of Noether’s first theorem (1918). We summarise extensive numerical tests, illustrating the conservation of the discrete Noether law using the p-Laplacian as an example and derive a geometric-based adaptive algorithm where an appropriate Noether quantity is the goal functional.