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Joseph Egger and Klaus-Peter Hoinka

Abstract

The angular momentum balance of quasigeostrophic flows is considered both for the standard and the deep formulation of quasigeostrophic theory (QT) in height coordinates, with main emphasis on the response to mountain torques. The related budget equations are derived. It is demonstrated that mountain torques affect only the wind term in the standard three-dimensional QT for β-plane flow while changes of the mass term are possible in deep QT as well. The situation is similar on the sphere where the standard QT restricts the response to the wind term. On the other hand, the mass term tends to be dominant for spherical barotropic flow with a free surface. Formulations of QT in pressure coordinates are discussed.

ECMWF reanalyses are used to see how torques affect the global angular momentum M g specified by standard QT in height coordinates as compared to the axial angular momentum M of the atmosphere. If QT were satisfactory, the variations of M g would be closely linked to those of M. Surprisingly, the wind term M g of QT is not a good approximation to the observed wind term, which contains an important “turbulent” part. The variance of M g captures only ∼25% of that of M. The cross-covariance function of M g with the mountain torque T o attains amplitudes that are about one-third of those of M and T o. It is the same for the friction torque. On the other hand, the response of the mass term in spherical barotropic QT is too strong for standard choices of the Rossby radius.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

Given the budget equation for the global axial angular momentum M, the related covariance equations are derived. These equations allow one to study the response of the global angular momentum to the forcing by mountain and friction torques in a statistical framework. ECMWF reanalysis (ERA) data are used to evaluate the terms of these equations and to assess their relative importance. Moreover, a new test of the quality of these data is provided this way.

The decay of the autocovariance function of M with increasing lag τ is slow and almost linear for 20 < τ < 280 days. That of the friction torque T f is exponential with a decay rate of ∼5 days. The autocovariance of the mountain torque T o decays even faster. The torque T g due to the gravity wave drag is more persistent than the mountain torque. When inserting the observed covariance functions in the respective equations, it is found that the mountain torque is generally more important than T f. The contribution by T g is small. The cross covariance of T o and T f is a major contributor in the covariance equations of these torques. However, both torques act on M as if they were almost independent. All covariance equations are satisfied quite well, particularly for the covariance of T g and M. A regressive model for M, T o, and T f is presented.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

Transfer of axial angular momentum across isentropic surfaces due to adiabatic processes is performed by pressure torques. These torques are evaluated from observations for selected latitude belts and isentropic surfaces, focusing attention on regional contributions. It is found that downward time mean contributions culminate in the storm tracks except above and near major mountain massifs where even upward transfers may be found. Variations of these torques in time are short lived with a decay time of 1–2 days. Height perturbations of isentropic surfaces are presented for torque events. The torque patterns are compared to analyses of the more conventional vertical momentum transports in the z system.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

The relation of pressure torques and mountain torques is investigated on the basis of observations for the polar caps, two midlatitude and two subtropical belts, and a tropical belt by evaluating the lagged covariances of these torques for various isentropic surfaces. It is only in the polar domains and the northern midlatitude belts that the transfer of angular momentum to and from the earth at the mountains is associated with pressure torques acting in the same sense. The situation is more complicated in all other belts. The covariances decline with increasing potential temperature (height). The role of both torques in the angular momentum budget of a belt is discussed.

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Joseph Egger and Klaus-Peter Hoinka

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.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

The regression of atmospheric fields against a parameter P with lag τ 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 τ → 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 β-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 τ → 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 τ > 0 is supported by these triple covariances while the stationary wave acts as a source for τ > 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 τ → 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 ≤ τ ≤ 2 days. The 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.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

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.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

The budget equation of the zonally averaged angular momentum is analyzed by introducing belts of 1000-km width to cover the meridional plane from pole to pole up to an altitude of 28 km. Using ECMWF Re-Analysis (ERA) data the fluxes of angular momentum are evaluated as well as the mountain and friction torques per belt. Generalized streamfunctions and velocity potentials are introduced to better depict the fluxes related to the angular momentum transferred at the ground during an event of mountain or friction torque.

The variance of the total flux divergence per belt is one order of magnitude larger than those of the torques. All variances peak at midlatitudes. As a rule, the structure of the generalized streamfunctions changes little during an event; that is, the structure of the nondivergent part of the fluxes is stable. That of the divergent part, as represented by the velocity potential, undergoes a rapid change near the peak of a torque event. Positive friction torque events in midlatitude belts are preceded by a divergence of angular momentum fluxes in that belt, which is linked to the anticyclonic mass circulation needed to induce the positive torque. The divergence in the belt breaks down shortly before the torque is strongest. Angular momentum is transported upward from the ground after that. Much of the angular momentum generated in a midlatitude belt by positive mountain torques is transported out of the domain, but there is also a short burst of upward transports. Angular momentum anomalies linked to torque events near the equator tend to be symmetric with respect to the equator. Related fluxes affect the midlatitudes of both hemispheres.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

The horizontally averaged global angular momentum μ at a certain height reacts only to the vertical divergence of the angular momentum flux at least above the crest height of the earth's orography. The flux is tied to the torques at the surface. Data are used to evaluate the flux and the response of μ to the torques. It is shown that the accuracy of the data is sufficient for an investigation of this interaction.

It is found that the horizontally averaged angular momentum in the upper troposphere and lower stratosphere tends to be negative before an event of positive friction torque. Downward transports of negative angular momentum from these layers allow the angular momentum to further decrease near the ground, even shortly before the event although the friction torque is positive at that time. The impact of the mountains during this process is demonstrated. The ensuing positive response to the friction torque is felt throughout the troposphere. The final decay of this reaction involves downward transports of μ with typical velocities of ∼1–2 km day−1.

The angular momentum in the lower troposphere tends to be negative before an event of positive mountain torque. There is a short burst of rapid upward transport of positive angular momentum during the event itself, which reaches the stratosphere within 1–2 days. A phase of decay follows with slow downward transport of positive angular momentum.

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Joseph Egger and Klaus-Peter Hoinka

Abstract

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: (λ, φ, z), (λ, φ, θ), and (λ, θ, 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 (φ, z) and (θ, z) coordinates but not in the (φ, θ) system where the form drag is a mean-flow term and, therefore, the forcing pattern differs from what has been published so far.

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