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Prashant D. Sardeshmukh

Abstract

The baroclinic χ (chi) problem is the problem of diagnosing the three-dimensional distribution of large-scale vertical motion from the vorticity budget. A solution technique is developed in which a preliminary guess for the associated horizontal divergence field is refined iteratively until the budget is balanced. The advantage of diagnosing the vertical motion in this manner, especially in the tropics, is discussed.

An example of the process applied to six boreal winters of ECMWF data suggests several improvements over the diabatically initialized ECMWF analyses, such as much stronger ascent over the convective regions of Africa and South America and a more clearly defined ITCZ in the eastern Pacific. Diabatic heating fields, estimated as a balance requirement in the thermodynamic equation using these dynamically consistent vertical velocities, also seem more realistic. Some ideas on how to combine such heating estimates with rainfall or satellite information are presented.

The more general problem of adjusting both the vorticity and divergence fields minimally from a preliminary analysis so as to make them consistent with the vorticity budget is also considered. It is suggested from a scale argument that in most cases the adjustment to the vorticity field will be much smaller than that to the divergence field. The χ problem, in which one adjusts only the divergence field, thus provides a useful approximation to a rather more demanding general problem. Solutions to the general problem are nevertheless feasible, and have implications for the problem of four-dimensional data assimilation with dynamical constraints as well as the spinup problem in numerical weather prediction.

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Mark D. Borges
and
Prashant D. Sardeshmukh

Abstract

The evolution of linear Rossby waves on representative zonally varying upper-tropospheric flows is examined. Structures associated with maximum perturbation energy growth over both short and long time intervals are obtained. Sensitivity is determined with respect to the background flow, the manner in which the background flow is maintained, and vertical level. The background flows considered are a 26 winter 250-mb climatology obtained from the National Meteorological Center, 13-winter 300-, 250-, and 150-mb climatologies obtained from the European Centre for Medium-Range Weather Forecasts, flows representative of warm and cold ENSO events, and nearby free steady solutions of the nonlinear barotropic vorticity equation.

The principal finding is that no matter which of the above background flows is used, no matter how the perturbation problem is formulated, no matter whether modal or nonmodal evolution is considered, the growth of free perturbations in a linear barotropic model is too weak to explain by itself the dominant observed structures of extratropical low-frequency variability.

In almost all the cases considered, the normal modes are stabilized by a 5-day drag, which is argued to be appropriate for this problem. It is shown further that although substantial nonmodal growth is still possible in the presence of the drag, it can occur for at most 12 days. Thus, without forcing, all initial perturbations to a climatological flow begin to decay in 12 days or less. The least-damped normal mode, being a decaying structure, can become relatively dominant only in the decaying stages of a perturbation. It takes about 10 days to emerge from its adjoint structure and at least 20 days to emerge from an arbitrary initial condition. These times are long enough that one expects either the perturbation to lose most of its initial amplitude in the presence of the drag or nonlinear effects to become important in its absence.

These results cast doubt on the role of barotropic normal-mode instability in the low-frequency dynamics of the Northern Hemisphere wintertime circulation. A representative zonally varying upper-level flow significantly modifies the propagation and dispersion of Rossby waves, but it does not significantly destabilize them. The spatial structures of the normal modes and optimal initial perturbations for nonmodal growth are apparently more relevant than normal-mode instability. For example, a normal mode with a structure resembling the Pacific–North American (PNA) pattern is found among the most unstable modes in almost all cases. Because it is stable in the presence of a realistic drag, however, such a mode cannot be naturally selected from random disturbances in the atmosphere; it has to be forced. In a system with stable normal modes, mode selection is determined not by relative modal decay rates but by forcing. Our analysis therefore implies that without a knowledge of the forcing, one cannot explain the relative importance of, or the energy contained in, the PNA or any other pattern of extratropical low-frequency variability.

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Prashant D. Sardeshmukh
and
Brian I. Hoskins

Abstract

The equivalence of taking an isotropic, moving, spatial average of a two-dimensional field on the sphere to multiplying the coefficients in its spherical harmonics representation with factors that depend only on the total wavenumber n is discussed. Equivalent spatial averaging operators for several such spectral filters are displayed.

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Prashant D. Sardeshmukh
and
Isaac M. Held

Abstract

The time mean vorticity balance in the summertime tropical upper troposphere of an atmospheric general circulation model constructed at the Geophysical Fluid Dynamics Laboratory is examined, with particular emphasis on the detailed balance in the Tibetan anticyclone. The model produces a reasonable simulation of the large-scale features of the northern summer 200 mb flow in the tropics, without the inclusion of subgrid scale processes that strongly damp the upper tropospheric vorticity. The vorticity balance is essentially nonlinear and nearly inviscid. There is considerable cancellation between the stretching and horizontal advection of vorticity by the time mean flow in the vicinity of the Tibetan anticyclone, with much of the remainder balanced by vertical advection and twisting. Mixing by the resolved transients is not negligible in some regions, but considerably smaller than the horizontal advection overall and less well correlated with the stretching. Subgrid scale mixing (consisting only of a biharmonic horizontal diffusion) plays a negligible role in this vorticity budget.

To relate this study to linear models of the stationary flow in the tropics, the steady state barotropic voracity equation on the sphere is linearized about the GCM's July mean zonal flow at 200 mb and forced with the GCM's July mean vortex stretching. It is found that the strength of the Tibetan anticyclone can be reproduced only by including a very strong damping of vorticity in this linear model. The strong damping needed by other authors (e.g., Holton and Colton) in their linear diagnoses of the tropical upper tropospheric vorticity balance is therefore interpreted as possibly accounting for neglected nonlinearities, and not necessarily cumulus friction. Our conclusions are, however, potentially suspect, since the terms in our vorticity budget have considerable structure on the smallest scales that can be resolved by the GCM.

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Klaus M. Weickmann
and
Prashant D. Sardeshmukh

Abstract

The period 1 December 1984 to 3 February 1985 was associated with strong intraseasonal fluctuations in both the global atmospheric angular momentum (AAM) and tropical convection. Consistent changes were observed in the length of day. The AAM budget for the 65-day period is examined here using circulation data from the National Meteorological Center. Surprisingly well-balanced global and zonal budgets are obtained for the vertically integrated AAM. This enables a closer examination of regional changes, to assess how they might be responsible for the changes in the global AAM.

Both friction and mountain torques are important in the global AAM budget. The increase of AAM is associated first with a positive friction torque, then with a positive mountain torque. The subsequent decrease of AAM results from a negative friction torque. The accompanying regional changes are mostly confined to the Northern Hemisphere, with high global AAM associated with a stronger and southward-displaced subtropical jet. In the zonal budget, meridional AAM fluxes by the zonally asymmetric eddies are important and appear to lead the torques by a few days.

The increase of AAM begins with a shift of the tropical convection from the east Indian to the west Pacific Ocean. The consequent enhancement of the trades east of the Philippines gives a positive friction torque. The friction torque also has a contribution from enhanced trades over Central America and the tropical Atlantic Ocean, which appear to be linked to an equatorward propagating upper-tropospheric wave over the region. A persistent high pressure anomaly subsequently develops to the east of the Himalayas, giving a positive mountain torque. The global AAM rises in response to these torques, but as the circumpolar vortex expands the trades are weakened, causing a negative friction torque and the final reduction of the AAM.

Interestingly, no coherent signals are seen in the weak zonal-mean convection anomalies accompanying these AAM changes. Rather, the AAM budget suggests that the tropical Madden–Julian oscillation and the global AAM are linked through the interaction of Rossby waves generated by the tropical heating with a zonally varying ambient flow and with mountains. The surface stresses have both a local component related to the convection and a remote component induced by upper-tropospheric AAM fluxes.

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Mingfang Ting
and
Prashant D. Sardeshmukh

Abstract

The steady linear response of a spherical baroclinic atmosphere to an equatorial diabatic heat source having a simple horizontal and vertical structure is examined. This source is imposed upon representative zonally symmetric as well as zonally varying flows during the boreal winter. Two climatologies are considered. One is a 6-year average of global observations analyzed at the European Centre for Medium-Range Weather Forecasts (ECMWF). The other is a 30-year average, taken from a general circulation model (GCM) run at the Geophysical Fluid Dynamics Laboratory in Princeton.

The extratropical response is found to be very sensitive to the basic state around which the governing primitive equations are linearized, and in the case of the ECMWF climatology, to the longitudinal position of the source with respect to the climatological waves. There is also some sensitivity to the vertical level of maximum heating, although again this is more evident in the case of the ECMWF basic state.

These results are discussed in terms of simple theoretical ideas, and implications are drawn for the short-range climate prediction problem. The evidence presented here suggests that subtle differences in the ambient flow can give rise to very different low-frequency normal modes, and thence to drastically different responses to tropical perturbations imposed upon that flow.

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Ernst Klinker
and
Prashant D. Sardeshmukh

Abstract

The momentum budget for January 1987 is evaluated with global observations analyzed at the European Centre for Medium-Range Weather Forecasts (ECMWF). The dissipation term is diagnosed from the budget as a balance requirement, that is, as that required to balance the sum of the advection, Coriolis, pressure gradient, and local tendency terms. This is then compared with the parameterized subgrid-scale effects in the ECMWF model's momentum equation, with a view of identifying possible errors in those parameterizations.

The balance requirement does not support the high parameterized values of orographically induced gravity-wave drag in the lower stratosphere. A deeper analysis also does not suggest a major role for turbulent vertical transports above the boundary layer. On the other hand, our budget does indicate that more effort be spent on a better representation of the potential enstrophy cascade associated with Rossby wave breaking in the upper troposphere. These statements are qualified by the errors in the balance requirement itself. The extent to which this is a problem is discussed.

A distinctive feature of these calculations is their internal consistency., that is, all the terms in the budget are evaluated as in the version of the ECMWF model used for assimilating the observations. This offers several advantages, not the least of which is that it makes our budget residuals identical to the systematic initial tendency errors of the operational weather forecasts, thus facilitating their computation and routine monitoring. As such, our calculations explain a large fraction of the systematic short-range forecast errors and, because of their local character, provide clues as to the possible sources of those errors. Experiments with and without gravity-wave drag are described to illustrate its large contribution during this period to the southerly wind error of the operational weather forecasts at 70 mb over western North America.

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Prashant D. Sardeshmukh
and
Brian J. Hoskins

Abstract

Tropical convective heating is balanced on the large scale by the adiabatic cooling of ascent. The horizontal divergence of the wind above this heating may be viewed as driving the upper tropospheric rotational wind field. A vorticity equation model is used to diagnose this relationship. It is shown that because of the advection of vorticity by the divergent component of the flow, the Rossby wave source can be very different from the simple −fD source often used. In particular, an equatorial region of divergence situated in easterly winds can lead to a Rossby wave source in the subtropical westerlies where it is extremely effective. This part of the source can be relatively insensitive to the longitudinal position of the equatorial divergence. A divergence field which is asymmetric about the equator can lead to a quite symmetric Rossby wave source. For a steady frictionless flow the Rossby wave source averaged over regions within closed streamfunction or absolute vorticity contours is, under certain simplifying assumptions, nearly zero.

It is very important to recognize the correct nature of the Rossby wave source associated with a region of tropical heating: The predicted atmospheric response can be dramatically in error if this source is misrepresented. Several features of related observational and GCM studies that have proved difficult to explain with simpler models can be understood in this light. Our analysis emphasizes the crucial role played by the tropical heating in the general circulation of the troposphere, and points to the importance of modeling the horizontal and vertical structure of this heating accurately.

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Jeffrey S. Whitaker
and
Prashant D. Sardeshmukh

Abstract

This paper investigates the extent to which the statistics of extratropical synoptic eddies may be deduced from the assumption that the eddies are stochastically forced disturbances evolving on a baroclinically stable background flow. To this end, a two-level hemispheric quasigeostrophic model is linearized about the observed long-term mean flow and forced with Gaussian white noise. The mean flow is baroclinically stable for a reasonable choice of dissipation parameters. Synoptic-scale eddy disturbances can still grow on such a flow, albeit for a finite time, either in response to the stochastic forcing or through baroclinic and barotropic energy interactions with the background flow. In a statistically steady state, a fluctuation–dissipation relation (FDR) links the covariance of the eddies to the spatial structure of the background flow and the covariance of the forcing. Although not necessary, in this study the forcing is assumed to have always the same trivial statistics (white in both space and time). Under this assumption, the FDR states that the geographical distribution of synoptic eddy covariance is controlled solely by the spatial structure of the background flow, which can therefore be used to predict it. All other second-order eddy statistics, such as eddy kinetic energy, momentum and heat fluxes, and power spectra, can then also be predicted. Despite the apparently drastic underlying assumption of synoptic eddy evolution as a stable linear Markov process, the comparisons of the predicted and observed geographical distributions of eddy kinetic energy and momentum and heat fluxes are found to be encouraging. The FDR is also shown to be sensitive enough to basic-state changes that it is able to predict important aspects of observed storm-track variability associated with seasonal and interannual changes of the background flow. The success of these calculations suggests that it is not necessary to invoke either baroclinic instability or the details of the eddy forcing to understand much of the observed spatial and temporal structure of extratropical synoptic eddy statistics. Rather, the dynamics of nonmodal eddy growth in the Pacific and Atlantic jets, and the downstream propagation and dispersion of the eddy activity in the diffluent regions downstream of the jets, appear sufficient for this purpose.

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Matthew Newman
and
Prashant D. Sardeshmukh

Abstract

The impact of the climatological seasonally varying 300-mb flow on the North Pacific/North American response to remote anomalous forcing is considered in the context of a linear barotropic model. WKB theory suggests that the total wavenumber of stationary Rossby waves over the Pacific increases from about 7 in January to 8.5 by June, with the reverse occurring during fall. This change is accompanied by monthly changes in the location and shape of the Rossby waveguide itself. Using a diagnostic tool called the influence function, it is shown that the most sensitive area of forcing for producing a large response over the United States shifts from the east Pacific in late winter to the west Pacific by late spring. As spring progresses, there is also a marked increase in the sensitivity to smaller-scale forcing in both of these regions, particularly the west Pacific. The amplitude of the forced response can potentially be larger in June than any other month of the year. These results suggest that the evolution of extreme springtime weather events over North America may depend critically upon the precise timing and geographical structure of forcing anomalies over both the east and the west Pacific.

In this model, low-frequency variability within and downstream of the Rossby waveguide is sensitive to the annual cycle of the ambient flow. This suggests that the impact of the annual cycle must be taken into account in any complete theory of low-frequency variability. The impact is large enough to raise the possibility of significant interactions across timescales. In other words, it is possible for a steady forcing to produce an unsteady response and, equally, for an unsteady forcing to produce a seasonal-mean response. In such situations, particularly during the northern spring and fall seasons, investigating low-frequency anomalies as departures from three-month seasonal climatologies may lead to confusion and may not be useful.

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