Given a flow domain D with subdomains D1 and D2, piecewise potential vorticity inversion (PPVI) inverts a potential vorticity (PV) anomaly in D2 and assumes vanishing PV in D1 where boundary conditions must be taken into account. It is a widely held view that the PV anomaly exerts a far-field influence on D1, which is revealed by PPVI. Tests of this assertion are conducted using a simple quasigeostrophic model where an upper layer D2 contains a PV anomaly and D1 is the layer underneath. This anomaly is inverted. Any downward physical impact of PV in D2 must also be represented in the results of a downward piecewise density inversion (PDI) based on the hydrostatic relation and the density in D2 as following from PPVI. There is no doubt about the impact of the mass in D2 on the flow in the lower layer D1. Thus results of PPVI and PDI have to agree closely. First, PPVI is applied to a locally confined PV anomaly in D2. There is no far-field "response" in D1 if stationarity is imposed. Modifications of boundary conditions lead to "induced" flows in D1 but the results of PPVI and PDI differ widely. This leads to a simple proof that there is no physical far-field influence of PV anomalies in D2. Wave patterns of the streamfunction restricted to D2 are prescribed in a second series of tests. The related PV anomalies are obtained by differentiation and are also confined to D2 in this case. This approach illustrates the basic procedure to derive PV fields from observations which excludes a far-field response.

The slow revolution of the Earth and Moon around their barycentrum does not induce Coriolis accelerations. On the other hand, the motion of Sun and Earth is a rotation with Coriolis forces which appear not to have been calculated yet, nor have the inertial accelerations within the system of motion of all three celestial bodies. It is the purpose of this contribution to evaluate the related Coriolis and centrifugal terms and to compare them to the available atmospheric standard terms. It is a main result that the revolution is of central importance in the combined dynamics of Earth, Moon and Sun. Covariant flow equations are well known tools for dealing with such complicated flow settings. They are used here to quantify the effects of the Earth's revolution around the Earth-Moon barycenter and its rotation around the Sun on the atmospheric circulation. It is found that the motion around the Sun adds time dependent terms to the standard Coriolis forces. The related centrifugal accelerations are presented. A major part of these accelerations is balanced by the gravitational attraction by Moon and Sun, but important unbalanced contributions remain. New light on the consequences of the Earth's revolution is shed by repeating the calculations for a rotating Earth-Moon pair. It is found that the revolution complicates the atmospheric dynamics.

The vertical velocity w is evaluated for the Northern Hemisphere from reanalysis data and two forms of the Richardson equation. This equation is based on the hydrostatic assumption and the thermodynamic energy equation. The standard form of the Richardson equation allows one to quantify the contributions to the vertical velocity of the horizontal divergence δ, the vertical pressure velocity ω and heating, and to test the incompressibility assumption underlying many dynamic models and theories. However, there are cancellations between two important terms. This shortcoming is substantially reduced in a further version of this equation where one term dominates. This version is the backbone of the data evaluation.The vertical velocities resulting from the Richardson equation in the troposphere are in good agreement with those obtained directly from the reanalysis data. It is found that the assumption of incompressibility provides a good estimate for w in the mid troposphere, even above Greenland and the Tibetan Plateau, both for the annual mean and the standard deviation of w, but is less acceptable in the upper troposphere and almost useless in the lower stratosphere. The contribution of heating to w is small.

Piecewise potential vorticity inversion (PPVI) seeks to determine the impact of observed potential vorticity (PV) anomalies on the surrounding flow. This widely used technique is based on dividing a flow domain D into subdomains D-1 and D-2 = D - D-1. The influence of PV in D-1 on the flow in D-2 is assessed by removing all PV anomalies in D-2 and then inverting the modified PV in D. The resulting flow with streamfunction psi(1) is attributed to the PV anomalies in D-1. The relation of PV in D-1 to psi(1) in D-2 is not unique, because there are many PV distributions in D-1 that induce the same psi(1). There is, however, a unique solution if the ageostrophic circulation is included in the inversion procedure. The superposition principle requires that the sum of inverted flows with PV = 0 in D-2 and the complementary ones with PV = 0 in D-1 equal the inverted flow for the complete observed PV in D. It is demonstrated, using two isolated PV balls as a paradigmatic example, that the superposition principle is violated if the ageostrophic circulation is included in PPVI, because the ageostrophic circulation cannot be associated with only one of the anomalies. Inversions of Ertel's PV are carried out using Charney's balance condition. PPVI is not unique in that case, because many different PV fields can be specified in D-1, which all lead to the same inverted flow in D-2. The balance condition assumes vanishing vertical velocity w so that uniqueness cannot be established by including w in the inversion, as was possible in the quasigeostrophic case.

Motion in planetary geostrophic equations (PGEs) is represented by the three‐dimensional geostrophic wind ( u g , v g , w g ) where u g and v g are the standard horizontal components while the vertical component w g can be derived, for example, from the Richardson equation. However, this vertical component appears not to have been evaluated as yet on the basis of data nor compared to the actual vertical component w . Part of this missing information is provided here by an evaluation of w g from observations and by analyzing the role of w g in linear versions of PGEs. The time mean fields in the Northern Hemisphere as well as the standard deviations are compared to the correponding fields of w . It is found that comes reasonably close to in the troposphere but deviates widely in the stratosphere while is smaller than σ w in the troposphere but not in the stratosphere. Linear wave motion is discussed and the linear steady‐state response to the forcing by heat sources and mountains is explored to explain these results.

Abstract Inversion of potential vorticity density with absolute vorticity and function η is explored in η coordinates. This density is shown to be the component of absolute vorticity associated with the vertical vector of the covariant basis of η coordinates. This implies that inversion of in η coordinates is a two-dimensional problem in hydrostatic flow. Examples of inversions are presented for (θ is potential temperature) and (p is pressure) with satisfactory results for domains covering the North Pole. The role of the boundary conditions is investigated and piecewise inversions are performed as well. The results shed new light on the interpretation of potential vorticity inversions.

Using a composite analysis for strong sea level pressure perturbations off the west coast of North America, the evolution of large-amplitude synoptic systems upstream of the Rocky Mountains is investigated for the winter season. Corresponding previous analyses are refined by avoiding multiple counting of events and extended by including potential vorticity, vertical motion, and deformation in the analysis.Cyclonic and anticyclonic anomalies behave similarly, with weak local extrema forming in the lee of the mountain range southeast of the parent systems. However, neither the geopotential anomaly nor the associated potential vorticity anomaly cross the mountain range. Nevertheless, these anomalies contribute to the sea level pressure anomaly in the lee. For both positive and negative anomalies, potential vorticity exhibits a bipolar structure with lobes over the reference point and over the Cordillera, respectively. The relevance of several theories describing the interaction between synoptic systems and mountains are discussed in the light of these findings.It is important to note that these findings differ considerably from results reported in an earlier study. Key differences are the previously reported passage of a wave train over the reference point and the movement of the anomalies over the Rocky Mountains. Both features are absent in the current analysis. However, these features can be recovered if a 6-day high-pass filter is applied before the events are selected or if the analysis is applied to predominantly zonal flow situations.

While many investigations of the atmospheric zonalmean circulation have been published using various vertical coordinates, this article concentrates on meridional coordinates other than latitude. Potential vorticity q and potential temperature theta are selected, with height as the vertical coordinate. Both q and theta are conserved in adiabatic frictionless flow. Although this helps in the interpretation of the results, both choices are problematic because of contortions in the contours. The related problems are solved by integrating over suitable zonal tubes.The isertelic mean circulation exhibits global cells in the upper troposphere and separate shallow hemispheric cells near the ground. Mean q fluxes are derived from the mass circulation. The mean circulation in (theta, z) coordinates is similar to that in (phi, theta) coordinates and has hemispheric direct cells. Mean theta fluxes follow from that. Both circulations are forced, in the sense that they can be derived and understood if the zonal mean heating is known. This is demonstrated by explaining, for example, the equatorward surface flow in (theta, z) coordinates and the shallow boundary-layer circulation in the isertelic system.

Abstract Some aspects of the dynamics of generalized potential vorticity (PV) density P = ω ⋅ ∇χ are discussed with the main emphasis on P fluxes, where ωa is absolute vorticity and χ is a scalar. The impermeability theorem claims that there is no net P flux across a χ surface. Various forms of the flux are presented that mostly cross χ surfaces. As these fluxes are as dynamically relevant as the one chosen for the theorem, P fluxes through a surface element are inherently multivalued and there is no best choice on physical grounds. Nevertheless, the net P flux is unique for closed surfaces. This point is illustrated by P integrals over the volume between the earth’s surface and an isentropic surface. Reanalysis data are used to present mean advective and some nonadvective P fluxes for χ = θ in height coordinates. The extratropical tropopause appears to be supported by advective P fluxes. A satisfactorily closed P budget cannot, however, be presented.

The wave forcing of the atmospheric mean flow in isentropic coordinates has been investigated intensively in the past with the divergence of the Eliassen-Palm flux playing a dominating role. These concepts are reviewed briefly and it is pointed out that angular momentum is attractive in this context because the wave driving can be written in the form of a flux divergence. This helps to evaluate the wave forcing in other coordinate systems with a different separation of waves and mean flow. The following coordinates are chosen: (lambda, phi, z), (lambda, phi, theta), and (lambda, theta, z). To be consistent, only one type of zonal averaging should be used. Mass-weighted averaging is applied in the isentropic standard case and simple averaging is applied in the others. The wave driving is presented for all three systems. It has to balance essentially the mean-flow part of the "Coriolis term" in the angular momentum budget in (phi, z) and (theta, z) coordinates but not in the (phi, theta) system where the form drag is a mean-flow term and, therefore, the forcing pattern differs from what has been published so far.

The electrostatic analogy provides a well-known paradigm for the concept of potential vorticity (PV) attribution. Just as electric fields can be attributed to electric charges, so are localized PV anomalies thought to induce far fields of flow and temperature, at least after geostrophic adjustment. Piecewise PV inversion (PPVI) exploits this concept. Idealized examples of PPVI are discussed by selecting isolated anomalies that are inverted to yield the far field "caused" by the PV anomaly. The causality of attribution is tested in this study by seeking an unbalanced initial state containing the same PV anomaly but without a far field from which the balanced state can be attained by geostrophic adjustment. It is shown that the far field of a balanced axisymmetric PV anomaly in shallow water, without mean PV gradients, may evolve from a localized anomaly without a far field. For the more general example of the electrostatics analogy, namely a three-dimensional spherical PV anomaly, the initial state has to be nonhydrostatic and needs to exhibit a mass deficit. As this mass deficit cannot be removed during hydrostatic and geostrophic adjustment, it follows that PV attribution does not imply a causal relationship between the far field of a PV anomaly and the anomaly itself.

Abstract The process of hydrostatic adjustment in a vertical column is discussed in the context of rain formation and sedimentation. The authors assume an event of instantaneous condensation in a midatmospheric layer that removes mass from the gas phase and produces latent heating. It is shown that the rain formation leads to a change of the surface pressure after a short period of acoustic wave activity. There is, however, no hydrostatic surface effect once the particles reach terminal velocity. It is not until the rain reaches the ground that the surface pressure decreases consistently with the mass removed by the phase change. Only the mass removal introduces perturbations below the layer of rain formation, where it acts to stretch the lower levels, reducing pressure and temperature. Above the layer of rain formation, the effects of latent heating dominate over the effects of mass removal by an order of magnitude. The hydrostatic adjustment time is found to be approximately equal to e2Na−1 (340 s, wher...

While time and zonal mean budgets of axial angular momentum (AAM) have been presented in pressure coordinates and also in isentropic coordinates, AAM budgets in height coordinates have not been published yet. The results of such an analysis on the basis of the 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) winter data are presented in this paper, which includes explicitly evaluated vertical eddy fluxes of momentum and mass as new features. As expected, AAM fluxes related to the Hadley cell are dominant. Transient vertical AAM fluxes are directed upward at the midlatitudes. Transient mass transports are not negligible, while triple terms are unimportant. Problems with the global balance of torques acting at the surface are discussed as well as those of mass conservation.

The FLOHOF field campaign took place in the period July 21 to August 24, 2007 on and in the surroundings of Hofsjökull glacier in Central Iceland. During the campaign, 18 automatic weather stations (AWS) recording temperature, humidity, wind speed, wind direction, pressure, and precipitation were deployed on and around the glacier. In addition, atmospheric soundings were performed N and S of Hofsjökull by a tethered balloon, pilot balloons, and two unmanned aerial systems (UAS). An energy balance station, consisting of a net radiometer and an eddy correlation flux measurement station, has also been installed. This paper describes the experimental setup of the campaign and presents first results of the data analysis with respect to transience of mountain-induced gravity waves, the extension of katabatic winds into the surrounding of the glacier, the occurrence of katabatic microfronts, and report on novel approaches to probe the vertical structure of the atmospheric boundary layer by UAS. The observed pressure perturbations related to transient gravity wave activity due to changing inflow conditions were between −2 and 2 hPa in general, with positive values upstream and negative values downstream. Differential heating of the glacier and its surrounding is triggering daytime katabatic flow from the glacier into its surrounding. During the campaign, those katabatic winds typically reached out 4–7 km from the edge of the glacier. During late night in clear sky conditions, frontal-like microstructures have been observed frequently with typical repetition times in the order of 30–60 min indicating the interaction of large-scale synoptic and nighttime katabatic density flows close to the ground. The first research application of the newly developed small unmanned meteorological observer proved the applicability of the system for atmospheric boundary layer research by successfully profiling the atmosphere up to 3.5 km above ground.

Although mountains are generally thought to exert forces on the atmosphere, the related transfers of energy between earth and atmosphere are not represented in standard energy equations of the atmosphere. It is shown that the axial rotation of the atmosphere must be included in the energy budget in order to resolve this issue. The energy transfer resulting from mountains turns out to be closely related to mountain torques. The energetic effects of a changing rotation of the earth are discussed, as well as those of friction torques and those of the nonspherical shape of the earth.

The linear theory of point correlation maps for synoptic systems relies so far mainly on specifications of stochastic forcing due to nonlinear processes that are not based on observations. Forty-year ECMWF Re-Analysis (ERA-40) data are used to derive time series of the forcing terms in a potential vorticity equation for a correlation point in the North Atlantic storm-track region. It is found that the forcing correlations are restricted to distances less than 1500 km to the correlation point in zonal direction and just a few hundred kilometers in meridional direction. The forcing is not even approximately white in time. Covariances of forcing and potential vorticity are presented as well. An advection equation with simple damping and realistic stochastic forcing is solved to approximate the observed covariances of forcing and potential vorticity.

Abstract Given the distribution of one atmospheric variable, that of nearly all others can be derived in balanced flow. In particular, potential vorticity inversion (PVI) selects potential vorticity (PV) to derive pressure, winds, and potential temperature θ. Potential temperature inversion (PTI) starts from available θ fields to derive pressure, winds, and PV. While PVI has been applied extensively, PTI has hardly been used as a research tool although the related technical steps are well known and simpler than those needed in PVI. Two idealized examples of PTI and PVI are compared. The 40-yr European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-40) datasets are used to determine typical anomalies of PV and θ in the North Atlantic storm-track region. Statistical forms of PVI and PTI are applied to these anomalies. The inversions are equivalent but the results of PTI are generally easier to understand than those of PVI. The issues of attribution and piecewise inversion are discussed.

Surface pressure and wind observations have been collected at 19 stations spread over the glaciated mountain Hofsjokull during the campaign FLOHOF in summer 2007 in order to explore gravity wave activity due to mean flow variations in time. Hofsjokull has an almost circular shape and, in general, fairly soft slopes so that it make senses to compare simple linear steady responses to the observations in order to detect transient features. This intercomparison is performed for various covariance functions of station pressures and mean winds. There is reasonably good agreement of observations and theory at stations near the edge of the ice dome but less so at those located higher up. The relation of the meridional mountain drag and the mean flow is described quite well by the linear theory. The collected data are exposed to a POP-analysis which yields oscillatory modes with the periods of a day and more. An analysis of pressure extrema at single stations reveals that local oscillations with periods of 2-3 hrs are quite common.

The regression of atmospheric fields against a parameter P with lag tau is a standard procedure in meteorology. Here, the torque exerted by a mountain massif is chosen as a parameter in order to study the interaction of weather systems with orography on a statistical basis. It is normally found that the amplitudes of the correlation patterns increase with tau -> 0 and decrease for increasing positive lag. It is proposed to explain this ubiquitous feature in the orographic case on the basis of the covariance equations that govern these regressions. Two examples are discussed. First, a version of the low-order Charney-DeVore model of beta-plane flow over a mountain is considered where stochastic forcing stirs a Rossby wave mode. It is found that the general increase of covariance amplitudes for tau -> 0 (if it occurs) is mainly due to the forcing, but triple covariances of mountain torque and vorticity advection are important as well. A new covariance energy equation is derived to demonstrate that the frictional decay for tau > 0 is supported by these triple covariances while the stationary wave acts as a source for tau -> 0. A dynamical interpretation of the triple terms is given. Next, data from the ECMWF 40-yr Re-Analysis (ERA-40) set are used to study mountain torque events in winter near Greenland, where the covariances of all standard variables with the torque P exhibit a rapid quasi-barotropic increase With tau -> 0 near Greenland. This amplification process is investigated by looking at the barotropic vorticity equation adapted to this statistical problem. This equation captures the evolution of the regression patterns reasonably well in the range -2 <= tau <= 2 clays. ne triple covariances of torque and nonlinear vorticity advection play the key role in the amplification process. In particular, covariance enstrophy is generated and destroyed by these terms. a process without counterpart in the standard vorticity equation. Stochastic forcing is presumably unimportant. The interpretation of the triple terms is difficult in contrast to that of the other "linear" terms of the vorticity equation. The angular momentum in the Greenland domain decreases during events of positive torque.