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NASA NTRS1994

Preferred latitudes of the intertropical convergence zone

The latitude preference of the intertropical convergence zone (ITCZ) is examined on the basis of observations, theory, and a modeling analysis. Observations show that convection is enhanced at latitudes of about 4 deg to 10 deg relative to the equator, even in regions where the sea surface temperature (SST) is maximum on the equator. Both linear shallow-water theory and a moist primitive equation model suggest a new explanation for the off-equatorial latitude preference of the ITCZ that requires neither the existence of zonally propagating disturbances nor an off-equatorial maximum in SST. The shallow-water theory indicates that a finite-width, zonally oriented, midtropospheric heat source (i.e., an ITCZ) produces the greatest local low-level convergence when placed a finite distance away from the equator. This result suggests that an ITCZ is most likely to be supported via low-level convergence of moist energy when located at these "preferred" latitudes away from the equator. For a plausible range of heating widths and damping parameters, the theoretically predicted latitude is approximately equal to the observed position (s) of the ITCZ (s). Analysis with an axially symmetric, moist, primitive equation model indicates that when the latent heating field is allowed to be determined internally, a positive feedback develops between the midtropospheric latent heating and the low-level convergence, with the effect of enhancing the organization of convection at latitudes of about 4 deg to 12 deg. Numerical experiments show that (1) two peaks in convective precipitation develop straddling the equator when the SST maximum is located on the equator; (2) steady ITCZ-like structures form only when the SST maximum is located away from the equator; and (3) peaks in convection can develop away from the maximum in SST, with a particular preference for latitudes of about 4 deg to 12 deg, even in the ('cold') hemisphere without the SST maximum. The relationship between this mechanism and earlier theories is discussed, as are implications for the coupled ocean-atmosphere system and the roles played by midlevel latent heating and SST gradients in forcing the low-level atmospheric circulation in the tropics.

Waliser, Duane E.↗

NASA NTRS1985

A fully implicit scheme for global numerical weather prediction

A fast-slow factored scheme is presented for use with shallow-water primitive equation numerical weather prediction models. The technique was developed to reduce the rotational mode errors which arise when the fast and slow terms of the governing differential equations are treated simultaneously. The method factors out the fast and slow terms along the coordinate directions by means of a modified Crank-Nicolson scheme. A finite-difference spatial discretization is carried out in the zonal and meridional directions to reduce the factorization error to near-zero, and that time steps of 60-90 min can be used to obtain acceptably accurate results, even in the presence of fine spatial structures in the flow.

Augenbaum, J. M.↗

NASA NTRS1990

A multigrid solver for semi-implicit global shallow-water models

A multigrid solver is developed for the discretized two-dimensional elliptic equation on the sphere that arises from a semiimplicit time discretization of the global shallow-water equations. Different formulations of the semiimplicit scheme result in variable-coefficient Helmholtz-type equations for which no fast direct solvers are available. The efficiency of the multigrid solver is optimal, in the sense that the total operation count is proportional to the number of unknowns. Numerical experiments using initial data derived from actual 300-mb height and wind velocity fields indicate that the present model has very good accuracy and stability properties.

Barros, Saulo R. M.↗

NASA NTRS1973

Amplification and decay of long nonlinear waves.

The interaction of weakly nonlinear waves with slowly varying boundaries is considered. Special emphasis is given to rotating fluids, but the analysis applies with minor modifications to waves in stratified fluids and shallow-water waves. An asymptotic solution of a variant of the Korteweg-de Vries equation with variable coefficients is developed that produces a 'Green's law' for the amplification of waves of finite amplitude. For shallow-water waves in water of variable depth, the result predicts wave growth proportional to the -1/3 power of the depth.

Leibovich, S.↗

DOE OSTI2023

Semi-Implicit Computation of Fast Modes in a Scheme Integrating Slow Modes by a Leapfrog Method Based on a Selective Implicit Time Filter

Abstract A scheme for integration of atmospheric equations containing terms with differing time scales is developed. The method employs a filtered leapfrog scheme utilizing a fourth-order implicit time filter with one function evaluation per time step to compute slow-propagating phenomena such as advection and rotation. The terms involving fast-propagating modes are handled implicitly with an unconditionally stable method that permits application of larger time steps and faster computations compared to fully explicit treatment. Implementation using explicit and recurrent formulation is provided. Stability analysis demonstrates that the method is conditionally stable for any combination of frequencies involved in the slow and fast terms as they approach the origin. The implicit filter used in the method damps the computational modes without noticeably sacrificing the accuracy of the physical mode. TheO[(Δt 4 )] accuracy for amplitude errors achieved by the implicitly filtered leapfrog is preserved in applications where terms responsible for fast propagation are integrated with a semi-implicit method. Detailed formulation of the method for soundproof nonhydrostatic anelastic equations is provided. Procedures for implementation in global spectral shallow-water models are also given. Examples comparing numerical and analytical solutions for linear gravity waves demonstrate the accuracy of the scheme. The performance is also shown in more practical nonlinear applications, where numerical solutions accomplished by the method are evaluated against those computed from a scheme where the slow terms are handled by the third-order Runge–Kutta scheme. It demonstrates that the method is able to accurately resolve fine-scale dynamics of Kelvin–Helmholtz shear instabilities, the evolution of density current, and nonlinear drifts of twin tropical cyclones.

Meteorology & Atmospheric Sciences↗

NASA NTRS1982

Initial conditions and Korteweg-de Vries solitons

The effects of rectangular initial data on the evolution of solitons governed by the Korteweg-de Vries equation is studied. Both isolated and separated disturbances are considered, providing some general insight into how the nature of the initial condition influences the appearance of solitons in the asymptotic state. The analytic approach is based on the inverse scattering transform which relates the initial condition to Schroedinger's equation. The results are used to model the initial shallow-water disturbances, and the results can be summarized by stating that the number of evolved solitons depends on the strength of each rectangular disturbance, the relative amplitudes of the rectangular disturbances, and the relative proximity of the disturbances.

Weidman, P. D.↗

NASA NTRS1983

An analytic solution for the response of the neutral atmosphere to the high-latitude convection pattern

The analytic solutions to a two-dimensional, shallow-water model of the high-latitude thermosphere are developed. The equations are a linearized representation of the response of the neutral atmosphere to the momentum source due to the plasma convection pattern. The obtained results show how the neutral winds depend on the Pedersen and Hall ion drag coefficients and the horizontal wave number. The response depends on the dimensionless scale as it appears in the product of the Rossby radius of deformation and the wave number. The flow pattern is shifted in local time from the plasma flow pattern by an amount that depends on the average electron density. At the largest scales there is a 180 deg phase shift between the rotational flow components in the E and F regions. The model applies to quiet magnetic conditions and below 200 km of altitude.

Mikkelsen, I. S.↗

NASA NTRS1986

Geostrophic adjustment in a shallow-water numerical model as it relates to thermospheric dynamics

The theory of geostrophic adjustment and its application to the dynamics of the high latitude thermosphere have been discussed in previous papers based on a linearized treatment of the fluid dynamical equations. However, a linearized treatment is only valid for small Rossby numbers given by Ro = V/fL, where V is the wind speed, f is the local value of the Coriolis parameter, and L is a characteristic horizontal scale for the flow. For typical values in the auroral zone, the approximation is not reasonable for wind speeds greater than 25 m/s or so. A shallow-water (one layer) model was developed that includes the spherical geometry and full nonlinear dynamics in the momentum equations in order to isolate the effects of the nonlinearities on the adjustment process. A belt of accelerated winds between 60 deg and 70 deg latitude was used as the initial condition. The adjustment process was found to proceed as expected from the linear formulation, but that an asymmetry between the response for an eastward and westward flow results from the nonlineawr curvature (centrifugal) terms. In general, the amplitude of an eastward flowing wind will be less after adjustment than a westward wind. For instance, if the initial wind velocity is 300 m/s, the linearized theory predicts a final wind speed of 240 m/s, regardless of the flow direction. However, the nonlinear curvature terms modify the response and produce a final wind speed of only 200 m/s for an initial eastward wind and a final wind speed of almost 300 m/s for an initial westward flow direction. Also, less gravity wave energy is produced by the adjustment of the westward flow than by the adjustment of the eastward flow. The implications are that the response of the thermosphere should be significantly different on the dawn and dusk sides of the auroral oval. Larger flow velocities would be expected on the dusk side since the plasma will accelerate the flow in a westward direction in that sector.

Larsen, M. F.↗

NASA NTRS1990

Recurrence in truncated Boussinesq models for nonlinear waves in shallow water

The rapid spatial recurrence of weakly nonlinear and weakly dispersive progressive shallow-water waves is examined using a numerical integration technique on the discretized and truncated form of the Boussinesq equations. This study primarily examines recurrence in wave fields with Ursell number O(1) and characterizes the sensitivity of recurrence to initial spectral shape and number of allowed frequency modes. It is shown that the rapid spatial recurrence is not an inherent property of the considered Boussinesq systems for evolution distances of 10-50 wavelengths. The main result of the study is that highly truncated Boussinesq models of resonant shallow-water ocean surface gravity waves predict rapid multiple recurrence cycles, but that this is an artifact dependent on the number of allowed modes. For initial conditions consisting of essentially all energy concentrated in a single mode, damping of the recurrence cycles increases as the number of low-power background modes increases. When more than 32 modes are allowed, the recurrence behavior is relatively insensitive to the number of allowed modes.

Elgar, Steve↗

NASA NTRS1988

Effects of using a posteriori methods for the conservation of integral invariants

The nature and effect of using a posteriori adjustments to nonconservative finite-difference schemes to enforce integral invariants of the corresponding analytic system are examined. The method of a posteriori integral constraint restoration is analyzed for the case of linear advection, and the harmonic response associated with the a posteriori adjustments is examined in detail. The conservative properties of the shallow water system are reviewed, and the constraint restoration algorithm applied to the shallow water equations are described. A comparison is made between forecasts obtained using implicit and a posteriori methods for the conservation of mass, energy, and potential enstrophy in the complete nonlinear shallow-water system.

Takacs, Lawrence L.↗

NASA NTRS1986

Using Hough harmonics to validate and assess nonlinear shallow-water models

The implementation of a technique for locating programming errors in shallow-water codes, establishing the correctness of the code, and assessing the performance of the numerical model under various flow conditions is described. The right-hand side of the differential equations is modified in such a way that the exact solution of the nonlinear initial-value problem is known, so that the truncation errors of the numerical scheme can be studied in detail. The exact solution is prescribed to be any linear combination of Hough harmonics which propagate in time according to their natural frequencies.

Dee, Dick P.↗

NASA NTRS1978

Experiments on strong interactions between solitary waves

Experiments on the interaction between solitary shallow-water waves propagating in the same direction have been performed in a rectangular channel. Two methods were devised to compensate for the dissipation of the waves in order to compare results with Hirota's (1971) solution for the collision of solitons described by the Korteweg-de Vries equation. Both qualitative and quantitative agreement with theory is obtained using the proposed corrections for wave damping.

Weidman, P. D.↗

NASA NTRS1991

Simplification of the Kalman filter for meteorological data assimilation

The paper proposes a new statistical method of data assimilation that is based on a simplification of the Kalman filter equations. The forecast error covariance evolution is approximated simply by advecting the mass-error covariance field, deriving the remaining covariances geostrophically, and accounting for external model-error forcing only at the end of each forecast cycle. This greatly reduces the cost of computation of the forecast error covariance. In simulations with a linear, one-dimensional shallow-water model and data generated artificially, the performance of the simplified filter is compared with that of the Kalman filter and the optimal interpolation (OI) method. The simplified filter produces analyses that are nearly optimal, and represents a significant improvement over OI.

Dee, Dick P.↗

NASA NTRS1984

The solution of non-linear hyperbolic equation systems by the finite element method

A finite-element method for the solution of nonlinear hyperbolic systems of equations, such as those encountered in non-self-adjoint problems of transient phenomena in convection-diffusion or in the mixed representation of wave problems, is developed and demonstrated. The problem is rewritten in moving coordinates and reinterpolated to the original mesh by a Taylor expansion prior to a standard Galerkin spatial discretization, and it is shown that this procedure is equivalent to the time-discretization approach of Donea (1984). Numerical results for sample problems are presented graphically, including such shallow-water problems as the breaking of a dam, the shoaling of a wave, and the outflow of a river; compressible flows such as the isothermal flow in a nozzle and the Riemann shock-tube problem; and the two-dimensional scalar-advection, nonlinear-shallow-water, and Euler equations.

Loehner, R.↗

NASA NTRS2000

Abrupt Changes at the Permian/Triassic Boundary: Tempo of Events from High-Resolution Cyclostratigraphy

The Permian/Triassic (P/Tr) boundary (251.4 +/- 3 Myr) is marked by the most severe mass extinction in the geologic record. Recently, precise absolute dating has bracketed the marine extinctions and associated carbon-isotope anomaly within less than 1 Myr. We improve this resolution through high-resolution stratigraphy across the P/Tr boundary in the 331-m Gartnerkofel-1 core and nearby Reppwand outcrop section (Carnic Alps, Austria) utilizing FFT and wavelet timeseries analyses of cyclic components in down-hole core logs of density and natural gamma-ray intensity, and carbon-isotopic ratios of bulk samples. The wavelet analysis indicates continuity of deposition across the P/Tr boundary interval, and the timeseries analyses show evidence for persistent cycles in the ratio of approximately 40: 10: 4.7: 2.3 meters, correlated with Milankovitch-band orbital cycles of approximately 412: 100: 40: 20 kyr (eccentricity 1 and 2, obliquity, and precession), and giving a consistent average sedimentation rate of approximately 10 cm/1,000 yr. Milankovitch periods in delta C-13 and density in these shallow-water carbonates were most likely the result of climatically induced oscillations of sea level and climate, coupled with changes in ocean circulation and productivity, that affected sedimentation. Fluctuations in gamma radiation reflect varying input of clay minerals and the presence of shaly interbeds. Throughout the P/Tr boundary interval in the core, the 100,000-year eccentricity cycle seems to be dominant. Weaker obliquity and precession cycles are in line with the location of the Austrian section in the latest Permian, close to the Equator in the western bight of the Tethys, where obliquity and precessional effects on seasonal contrast might be subdued. Using the improved resolution provided by cycle analysis in the GK-1 core, we find that the dramatic change in the faunal record that marks the P/Tr boundary takes place over less than 6m, or less than 60,000 years. In the nearby Reppwand outcrop section, the same faunal changes occurs over only 0.8 m or about 8,000 years, close to the limit of time-resolution induced by bioturbation and reworking in these sediments. The sharp negative global carbon-isotope shift took place within less than or equal to 40,000 yr, and the isotope excursions persisted for approximately 480,000 yr into the Early Triassic. The results indicate that the severe marine faunal event that marks the P/Tr boundary was very sudden, perhaps less than the resolution window in the GK-1 core, and suggest a catastrophic cause. The wavelet-analysis approach to high-resolution cyclostratigraphy can be applied to other P/Tr boundary sections, and when combined with precise absolute dating and magnetostratigraphic methods promises a significant increase in resolution in determining the correlation and tempo of the end-Permian extinctions and related events worldwide.

Rampino, M. R.↗

NASA NTRS1996

Assimilation of Geosat Altimetric Data in a Nonlinear Shallow-Water Model of the Indian Ocean by Adjoint Approach. Part II: Some Validation and Interpretation of the Assimilated Results: Some Validation and Interpretation of the Assimilated Results - Part 2

This paper examines the results of assimilating Geosat sea level variations relative to the November 1986-November 1988 mean reference, in a nonlinear reduced-gravity model of the Indian Ocean, Data have been assimilated during one year starting in November 1986 with the objective of optimizing the initial conditions and the yearly averaged reference surface. The thermocline slope simulated by the model with or without assimilation is validated by comparison with the signal, which can be derived from expandable bathythermograph measurements performed in the Indian Ocean at that time. The topography simulated with assimilation on November 1986 is in very good agreement with the hydrographic data. The slopes corresponding to the South Equatorial Current and to the South Equatorial Countercurrent are better reproduced with assimilation than without during the first nine months. The whole circulation of the cyclonic gyre south of the equator is then strongly intensified by assimilation. Another assimilation experiment is run over the following year starting in November 1987. The difference between the two yearly mean surfaces simulated with assimilation is in excellent agreement with Geosat. In the southeastern Indian Ocean, the correction to the yearly mean dynamic topography due to assimilation over the second year is negatively correlated to the one the year before. This correction is also in agreement with hydrographic data. It is likely that the signal corrected by assimilation is not only due to wind error, because simulations driven by various wind forcings present the same features over the two years. Model simulations run with a prescribed throughflow transport anomaly indicate that assimilation is rather correcting in the interior of the model domain for inadequate boundary conditions with the Pacific.

Greiner, Eric↗

NASA NTRS1994

Suboptimal schemes for atmospheric data assimilation based on the Kalman filter

This work is directed toward approximating the evolution of forecast error covariances for data assimilation. The performance of different algorithms based on simplification of the standard Kalman filter (KF) is studied. These are suboptimal schemes (SOSs) when compared to the KF, which is optimal for linear problems with known statistics. The SOSs considered here are several versions of optimal interpolation (OI), a scheme for height error variance advection, and a simplified KF in which the full height error covariance is advected. To employ a methodology for exact comparison among these schemes, a linear environment is maintained, in which a beta-plane shallow-water model linearized about a constant zonal flow is chosen for the test-bed dynamics. The results show that constructing dynamically balanced forecast error covariances rather than using conventional geostrophically balanced ones is essential for successful performance of any SOS. A posteriori initialization of SOSs to compensate for model - data imbalance sometimes results in poor performance. Instead, properly constructed dynamically balanced forecast error covariances eliminate the need for initialization. When the SOSs studied here make use of dynamically balanced forecast error covariances, the difference among their performances progresses naturally from conventional OI to the KF. In fact, the results suggest that even modest enhancements of OI, such as including an approximate dynamical equation for height error variances while leaving height error correlation structure homogeneous, go a long way toward achieving the performance of the KF, provided that dynamically balanced cross-covariances are constructed and that model errors are accounted for properly. The results indicate that such enhancements are necessary if unconventional data are to have a positive impact.

Todling, Ricardo↗

NASA NTRS1999

Physically-Derived Dynamical Cores in Atmospheric General Circulation Models

The algorithm chosen to represent the advection in atmospheric models is often used as the primary attribute to classify the model. Meteorological models are generally classified as spectral or grid point, with the term grid point implying discretization using finite differences. These traditional approaches have a number of shortcomings that render them non-physical. That is, they provide approximate solutions to the conservation equations that do not obey the fundamental laws of physics. The most commonly discussed shortcomings are overshoots and undershoots which manifest themselves most overtly in the constituent continuity equation. For this reason many climate models have special algorithms to model water vapor advection. This talk focuses on the development of an atmospheric general circulation model which uses a consistent physically-based advection algorithm in all aspects of the model formulation. The shallow-water model is generalized to three dimensions and combined with the physics parameterizations of NCAR's Community Climate Model. The scientific motivation for the development is to increase the integrity of the underlying fluid dynamics so that the physics terms can be more effectively isolated, examined, and improved. The expected benefits of the new model are discussed and results from the initial integrations will be presented.

Rood, Richard B.↗