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Stephen T. Garner

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Stephen T. Garner

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Numerical solutions presented in a companion paper show that two-dimensional mesoscale terrain becomes a much stronger barrier to a continuously stratified flow when the flow contains warm advection. Here it is shown that this baroclinic enhancement is a strictly nonlinear phenomenon. The linear analysis indicates a weakening of the upstream response in warm advection. However, a weakly nonlinear analysis shows that baroclinicity facilitates blocking in warm advection by strengthening the nonlinearity in the cross-mountain momentum equation in such a way as to amplify the vertical shear on the windward flank of the ridge. This is enough to send the flow past the blocking threshold even when conditions over the mountain are too linear to produce wave breaking. A more intuitive mechanism whereby the upstream static stability is increased by the nonlinearity in the temperature equation is found to be much less important.

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Stephen T. Garner

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Under the assumption of weak background rotational and wind shear effects, an attractive computational upper boundary condition capable of transmitting gravity waves is generalized for use in a variety of f-plane models. Issues relating to numerical implementation are discussed.

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Stephen T. Garner

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High-resolution simulation can be a powerful means of evaluating and tuning orographic drag schemes, but connecting the parameterized drag, which is a local forcing, with the model drag, which is fundamentally global, is not entirely straightforward. The simplest idea is to filter the velocity down to its divergent component and exploit Bernoulli’s law to define a local form drag. Using regional simulations over the Rockies, the Andes, and Greenland, we investigate the validity of this approach, which assumes that both the included nonorographic divergence and the missing orographic deformation will not significantly alter the diagnostic. The local drag is checked for consistency with the nonlocal drag at scales containing most of the gravity wave drag and blocking drag. The agreement is found to be satisfactory unless the drag is weak and nonlinear. In that case, we find it necessary to remove a steady pattern from the nonlocal drag in order to uncover a correlation. We test a specific mountain drag scheme using the proposed diagnostic and describe procedures for tuning the scheme’s drag coefficients and treatment of anisotropy.

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Stephen T. Garner

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Topographic drag schemes depend on grid-scale representations of the average height, width, and orientation of the subgrid topography. Until now, these representations have been based on a combination of statistics and dimensional analysis. However, under certain physical assumptions, linear analysis provides the exact amplitude and orientation of the drag for arbitrary topography. The author proposes a computationally practical closure based on this analysis.

Also proposed is a nonlinear correction for nonpropagating base flux. This is patterned after existing schemes but is better constrained to match the linear solution because it assumes a correlation between mountain height and width. When the correction is interpreted as a formula for the transition to saturation in the wave train, it also provides a way of estimating the vertical distribution of the momentum forcing. The explicit subgrid height distribution causes a natural broadening of the layers experiencing the forcing. Linear drag due to simple oscillating flow over topography, which is relevant to ocean tides, has almost the same form as for the stationary atmospheric problem. However, dimensional analysis suggests that the nonpropagating drag in this situation is mostly due to topographic length scales that are small enough to keep the steady-state assumption satisfied.

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Stephen T. Garner

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Some fundamental properties of nongeostrophic baroclinic waves are examined by solving the equations of motion linearized about Eady's basic state at the next order of balance beyond quasi-geostrophic (QG) theory. The study fits into a general effort to broaden the view of “slow-manifold” behavior. The specific motivation is to identify balanced properties of surface and upper-tropospheric frontal regions that are filtered from both the QG and the semigeostrophic models. Among the questions to be answered are whether alongfront flow is subgeostrophic or supergeostrophic, and whether the ageostrophy is realized primarily as a velocity or pressure correction of the QG solution. A rudimentary model of jet streaks is constructed from nongeostrophic neutral waves in an attempt to reproduce along-jet ageostrophic velocities and propagation speeds more realistically than in existing models.

Past work has concentrated on correcting phase speeds and growth rates for higher-order balanced effects. These results are extended by using a more appropriate solvability condition near the short-wave cutoff and by considering the detailed structure of the nongeostrophic modes. The eigenvalue corrections are interpreted physically in the framework of a generalized potential vorticity inversion problem with sources determined at the QG level. It is shown that ageostrophic shear in the nondivergent (alongfront) wind affects the time dependence primarily indirectly, by tilting the basic isentropes in the meridional direction and setting up an anomaly pattern in the QG potential vorticity field. This has some of the same consequences as shortening the horizontal scale in the QG model.

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Stephen T. Garner

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Numerical simulation has failed to answer some fundamental questions about atmospheric frontogenesis because of the artificial minimum resolved scale in grid point and spectral models alike. To alleviate this handicap and shed light on some recent ideas about the possibility of a finite limiting scale for inviscid fronts, a fully Lagrangian primitive-equation numerical model is developed for nonturbulent, slab-symmetric flow on an f-plane. With physical position treated as an explicit function of particle label and time, the model grid deforms to follow natural changes in disturbance length scales. Exact conservation of volume and potential vorticity, as well as of basic tracer variables, is demonstrated, and details of the truncation error for energy conservation are obtained for the case of second-order central differencing in label space.

The Lagrangian model is used to simulate frontogenesis by horizontal wind deformation in a dry, Boussinesq atmosphere, with no prior assumption of hydrostatic or geostrophic balance. A comparison is made between solutions for different values of uniform potential vorticity. Frontal collapse (the formation of discontinuities) is considered to occur when two grid points carrying contrasting fluid properties come into contact with each other on a solid boundary. For realistic choices of the parameters governing the rate of frontogenesis, imbalances alone are found to be insufficient to prevent frontal collapse.

For small values of the normalized potential vorticity, the ageostrophic secondary circulation is weaker than in the corresponding balanced solutions, and frontal collapse is accordingly delayed. A further manifestation of imbalance is a splitting of the frontal updraft into two ascent maxima separated by a distance comparable to the width of the baroclinic region. Neutral wave activity develops if the potential vorticity is significantly nonzero, but plays only a passive role during the formation of discontinuities. The geostrophic momentum approximation is invoked in arguing that the splitting effect is not fundamentally wave-like.

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Stephen T. Garner

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The shallow atmospheric fronts that develop in the early winter along the east coast of North America have been attributed, in various modeling and observational studies, to the land–sea contrasts in both surface heating and friction. However, typical synoptic conditions are such that these “coastal” fronts could also be a type of upstream influence by the Appalachian Mountain chain. Generalized models have suggested that relatively cold air can become trapped on the windward side of a mountain range during episodes of warm advection without a local contribution from differential surface fluxes. Such a process was proposed decades ago in a study of observations along the coast of Norway. Could coastal frontogenesis be primarily a consequence of a mountain circulation acting on the large-scale temperature gradient?

A two-dimensional, terrain-following numerical model is used to find conditions under which orography may be sufficient to cause blocking and upstream frontogenesis in a baroclinic environment. The idealized basic flow is taken to have constant vertical shear parallel to a topographic ridge and a constant perpendicular wind that advects warm or cold temperatures toward the ridge. Land–sea contrasts are omitted. In the observed cases, the mountain is “narrow” in the sense that the Rossby number is large. This by itself increases the barrier effect, but the experiments show that large-scale warm advection is still crucial for blocking. For realistic choices of ambient static stability and baroclinicity, the flow can be blocked by a range like the northern Appalachians if the undisturbed incident wind speed is around 10 m s−1. Cold advection weakens the barrier effect.

The long-term behavior of the front in strongly blocked cases is described and compared to observations. Because of the background rotation and large-scale temperature advection, blocked solutions cannot become steady in the assumed environment. However, the interface between blocked and unblocked fluid can settle into a balanced configuration in some cases. A simple argument suggests that, in the absence of dissipation, the frontal slope should be similar to that of the ambient “absolute momentum” surfaces.

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Stephen T. Garner

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The “high drag” state of stratified flow over isolated terrain is still an impediment to theoretical and experimental estimation of topographic wave drag and mean-flow modification. Linear theory misses the transition to the asymmetrical configuration that produces the enhanced drag. Steady-state nonlinear models rely on an ad hoc upstream condition like Long's hypothesis and can, as a result, be inconsistent with the flow established naturally by transients, especially if blocking is involved. Numerical solutions of the stratified initial value problem have left considerable uncertainty about the upstream alteration, especially as regards its permanence.

A time-dependent numerical model with open boundaries is used in an effort to distinguish between permanent and transient upstream flow changes and to relate these to developments near the mountain. A nonrotating atmosphere with initially uniform wind and static stability is assumed. It is found that permanent alterations are primarily due to an initial surge not directly related to wave breaking. Indeed, there are no obvious parameter thresholds in the time-mean upstream state until “orographic adjustment” (deep blocking) commences. Wave breaking, in addition to establishing the downstream shooting flow, generates a persistent, quasi-periodic, up-stream transience, which apparently involves the ducting properties of the downslope mixed region. This transience is slow enough to be easily confused with permanent changes.

To understand the inflow alteration and transience, the energy and momentum budgets are examined in regions near the mountain. High drag conditions require permanent changes in flow force difference across the mountain and, consequently, an ongoing horizontal flux of energy and negative momentum. The source of the upstream transience is localized at the head of the mixed region. Blocking allows the total drag to exceed the saturation value by more than an order of magnitude. The implications for nonlinear steady-state models and wave drag parameterization are discussed.

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G. Balasubramanian and Stephen T. Garner

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The wide disparities in baroclinic wave development between spherical and Cartesian geometry are investigated with the purpose of assessing the role of the eddy momentum fluxes. Differences are already significant at the linear stage, as momentum fluxes are predominantly poleward in spherical geometry and predominantly equatorward in Cartesian geometry. More important, the low-level flux convergence is displaced poleward on the sphere and equatorward on the plane. On the sphere, these circumstances lead to rapid poleward movement of the low-level zonal-mean jet. The anticyclonic horizontal shear region expands as the jet feeds back on the momentum flux. The wave breaks anticyclonically and quickly zonalizes. In the Cartesian life cycle, the equatorward displacement of the flux convergence is counteracted by the mean meridional circulation and there is consequently a weaker feedback with the horizontal shear. The wave breaks, in this case cyclonically, but then takes much longer to zonalize. On the sphere, the angular velocity gradient in uniform westerly or easterly flow adds a separate mechanism for converting eddy kinetic energy to zonal mean, further hastening the zonalization process.

It is possible to change the sign of the eddy momentum flux and the sense of the breaking in either geometry by slightly changing the basic flow. For example, cyclonic roll-up on the sphere can be obtained by adding weak cyclonic barotropic shear, as highlighted in a recently published study. Similarly, the addition of anticyclonic barotropic shear in a Cartesian simulation leads to anticyclonic wave breaking. An easterly jet on the sphere allows cyclonic breaking, but the wave still zonalizes rapidly, as in the case of a westerly jet. The persistence of the nonlinear eddies in these diverse experiments is not well correlated with the minimum value of the refractive index for Rossby waves, as suggested in the referenced study. It is proposed that the longevity of residual vortices after wave breaking is determined not by the sign of the vorticity or the breadth of the waveguide, but by the sign of the momentum flux and the geometry of the model.

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