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- Author or Editor: John P. Boyd x
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Abstract
The effects of divergence on low-frequency Rossby wave propagation are examined by using the two-dimensional Wentzel–Kramers–Brillouin (WKB) method and ray tracing in the framework of a linear barotropic dynamic system. The WKB analysis shows that the divergent wind decreases Rossby wave frequency (for wave propagation northward in the Northern Hemisphere). Ray tracing shows that the divergent wind increases the zonal group velocity and thus accelerates the zonal propagation of Rossby waves. It also appears that divergence tends to feed energy into relatively high wavenumber waves, so that these waves can propagate farther downstream. The present theory also provides an estimate of a phase angle between the vorticity and divergence centers. In a fully developed Rossby wave, vorticity and divergence display a π/2 phase difference, which is consistent with the observed upper-level structure of a mature extratropical cyclone. It is shown that these theoretical results compare well with observations.
Abstract
The effects of divergence on low-frequency Rossby wave propagation are examined by using the two-dimensional Wentzel–Kramers–Brillouin (WKB) method and ray tracing in the framework of a linear barotropic dynamic system. The WKB analysis shows that the divergent wind decreases Rossby wave frequency (for wave propagation northward in the Northern Hemisphere). Ray tracing shows that the divergent wind increases the zonal group velocity and thus accelerates the zonal propagation of Rossby waves. It also appears that divergence tends to feed energy into relatively high wavenumber waves, so that these waves can propagate farther downstream. The present theory also provides an estimate of a phase angle between the vorticity and divergence centers. In a fully developed Rossby wave, vorticity and divergence display a π/2 phase difference, which is consistent with the observed upper-level structure of a mature extratropical cyclone. It is shown that these theoretical results compare well with observations.
Abstract
The evolution, both stable and unstable, of contrarotating vortex pairs (“modons”) perturbed by upper-surface and bottom Ekman pumping is investigated using a homogeneous model with a variable free upper-surface and bottom topography. The Ekman pumping considered here differs from the classical Ekman pumping in that the divergence-vorticity term in the vorticity equation, nonlinear and omitted in previous studies, is explicitly included. Under the influence of both nonlinear Ekman pumping and the beta term, eastward- and westward-moving modons behave very differently.
Eastward-moving modons are stable to the upper-surface perturbation but westward-moving modons are not. The latter move southwestward, triggering the tilt instability: the beta effect deepens the cyclones but weakens the anticyclone, and the vortex pair disperses into wave packets.
Eastward-moving modons are stable to bottom friction in the sense that they diminish in time gradually at a rate independent of the signs of the vortices. Westward-moving modons behave differently depending on the strength of bottom friction. Cyclones decay faster than anticyclones, triggering the tilt instability in westward-moving modons, but only if the bottom friction is very weak. For sufficiently strong bottom friction, in contrast, modons decay monotonically: the cyclones still decay faster than anticyclones, but no wave packets formed before the modons completely dissipate.
Westward-moving modons are always unstable to topographic forcing. Eastward-moving modons have varying behavior controlled by the height and width of the topography. Below a critical height, determined by the width, modons survive the topographic interaction: their trajectory meanders but the two contrarotating vortices always remain bound together after escaping the topography. Above the critical height, modons disassociate: the two vortices separate and disperse into wave packets. When the width of the topography is comparable to modon width, there exists a stable window within the unstable region of the topographic height in which the modons also survive the topographic encounter.
Abstract
The evolution, both stable and unstable, of contrarotating vortex pairs (“modons”) perturbed by upper-surface and bottom Ekman pumping is investigated using a homogeneous model with a variable free upper-surface and bottom topography. The Ekman pumping considered here differs from the classical Ekman pumping in that the divergence-vorticity term in the vorticity equation, nonlinear and omitted in previous studies, is explicitly included. Under the influence of both nonlinear Ekman pumping and the beta term, eastward- and westward-moving modons behave very differently.
Eastward-moving modons are stable to the upper-surface perturbation but westward-moving modons are not. The latter move southwestward, triggering the tilt instability: the beta effect deepens the cyclones but weakens the anticyclone, and the vortex pair disperses into wave packets.
Eastward-moving modons are stable to bottom friction in the sense that they diminish in time gradually at a rate independent of the signs of the vortices. Westward-moving modons behave differently depending on the strength of bottom friction. Cyclones decay faster than anticyclones, triggering the tilt instability in westward-moving modons, but only if the bottom friction is very weak. For sufficiently strong bottom friction, in contrast, modons decay monotonically: the cyclones still decay faster than anticyclones, but no wave packets formed before the modons completely dissipate.
Westward-moving modons are always unstable to topographic forcing. Eastward-moving modons have varying behavior controlled by the height and width of the topography. Below a critical height, determined by the width, modons survive the topographic interaction: their trajectory meanders but the two contrarotating vortices always remain bound together after escaping the topography. Above the critical height, modons disassociate: the two vortices separate and disperse into wave packets. When the width of the topography is comparable to modon width, there exists a stable window within the unstable region of the topographic height in which the modons also survive the topographic encounter.
Abstract
The Kelvin wave is the gravest eigenmode of Laplace’s tidal equation. It is widely observed in both the ocean and the atmosphere. In the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber s (always an integer) and Lamb’s parameter ϵ. An asymptotic approximation valid in the limit
Abstract
The Kelvin wave is the gravest eigenmode of Laplace’s tidal equation. It is widely observed in both the ocean and the atmosphere. In the absence of mean currents, the Kelvin wave depends on two parameters: the zonal wavenumber s (always an integer) and Lamb’s parameter ϵ. An asymptotic approximation valid in the limit
Abstract
The equatorial soliton studies of Boyd are extended to include the effects of continuous vertical stratification. We use vertical profiles of density measured in the equatorial Pacific Ocean and an idealized profile.
The wavenumber intervals of existence/nonexistence of wavepacket solitons are similar to Boyd for the Rossby mode, less similar for the eastward propagating gravity mode, and least similar for the westward gravity mode. Competing resonances and vertical diffusion are discussed.
The observed stratification for the equatorial Pacific has a significant effect on solitons. Some of the most dramatic effects are for the KdV solitons of Boyd. We propose that the longitudinal variation of stratification may result in the existence of the antisymmetric (with respect to the equator) Rossby solitons in only the western Pacific Ocean. The nonlinear coefficients of the symmetric KdV Rossby soliton decrease monotonically eastward by approximately 30% across the Pacific Ocean.
Abstract
The equatorial soliton studies of Boyd are extended to include the effects of continuous vertical stratification. We use vertical profiles of density measured in the equatorial Pacific Ocean and an idealized profile.
The wavenumber intervals of existence/nonexistence of wavepacket solitons are similar to Boyd for the Rossby mode, less similar for the eastward propagating gravity mode, and least similar for the westward gravity mode. Competing resonances and vertical diffusion are discussed.
The observed stratification for the equatorial Pacific has a significant effect on solitons. Some of the most dramatic effects are for the KdV solitons of Boyd. We propose that the longitudinal variation of stratification may result in the existence of the antisymmetric (with respect to the equator) Rossby solitons in only the western Pacific Ocean. The nonlinear coefficients of the symmetric KdV Rossby soliton decrease monotonically eastward by approximately 30% across the Pacific Ocean.
Abstract
We introduce a new energetics concept and apply it to the NCAR Community Climate Model. The new features of our approach are that the energy is split into balanced and transient parts and that the balanced energy consists of rotational energy and balanced gravitational energy. The time evolution and distribution of the balanced and transient parts of the gravity waves among vertical modes and zonal waves am analyzed.
Both balanced gravitational energy and transient energy concentrate and oscillate rapidly with time at vertical modes 7–8 and zonal wavenumbers 1–5. This explains why the iteration scheme used in nonlinear normal mode initialization would not converge, in general, for high vertical modes and long zonal waves. All the gravity waves associated with vertical modes 0–2 and any zonal wavenumber can be freely adjusted in the initialization to suppress the high-frequency oscillations.
The lower vertical modes, 2–6, contain more balanced gravitational energy than transient energy, but for the higher baroclinic modes, 7–8, both energies are of almost the same magnitude. In general, longer zonal waves contribute more energy to the balanced gravitational energy. Zonal wavenumber 1 contributes the most to both transient energy and balanced gravitational energy. To examine whether the energy of gravity waves is balanced or not during the initialization, it is inappropriate to express the energy in terms of zonal wavenumbers only. The vertical resolution, discretization scheme, and physical parameterization may distort the gravitational energy in the high vertical modes.
Abstract
We introduce a new energetics concept and apply it to the NCAR Community Climate Model. The new features of our approach are that the energy is split into balanced and transient parts and that the balanced energy consists of rotational energy and balanced gravitational energy. The time evolution and distribution of the balanced and transient parts of the gravity waves among vertical modes and zonal waves am analyzed.
Both balanced gravitational energy and transient energy concentrate and oscillate rapidly with time at vertical modes 7–8 and zonal wavenumbers 1–5. This explains why the iteration scheme used in nonlinear normal mode initialization would not converge, in general, for high vertical modes and long zonal waves. All the gravity waves associated with vertical modes 0–2 and any zonal wavenumber can be freely adjusted in the initialization to suppress the high-frequency oscillations.
The lower vertical modes, 2–6, contain more balanced gravitational energy than transient energy, but for the higher baroclinic modes, 7–8, both energies are of almost the same magnitude. In general, longer zonal waves contribute more energy to the balanced gravitational energy. Zonal wavenumber 1 contributes the most to both transient energy and balanced gravitational energy. To examine whether the energy of gravity waves is balanced or not during the initialization, it is inappropriate to express the energy in terms of zonal wavenumbers only. The vertical resolution, discretization scheme, and physical parameterization may distort the gravitational energy in the high vertical modes.
Abstract
A new approach to energetics is introduced and applied to the NCAR Community Climate Model. All the energy components are separated into gravitational and rotational parts. The new feature of our scheme is that the gravitational divergent kinetic energy is further decoupled into zonal and meridional components, which measure the strength of the cut-west and meridional circulations, respectively. The zonal and meridional nondivergent kinetic energies represent the vorticities related to the nondivergent zonal and meridional winds, respectively. The distributions of energy among meridional indices, vertical modes, and zonal waves are analyzed.
We suggest a new, easy, and reasonable criterion to adjust the gravity waves in the initialization based on the vertical modes, meridional indices, and zonal wavenumbers. To retain the strength of large-scale circulations and to preserve the intensity of synoptic scale pressure systems, we recommend that the gravity waves corresponding to the internal modes 2–6, meridional indices 1–6, and zonal wavenumbers 1–10 not be adjusted significantly during the initialization. However, all the other gravity waves can be adjusted initially, particularly those associated with the external and the first internal modes.
Abstract
A new approach to energetics is introduced and applied to the NCAR Community Climate Model. All the energy components are separated into gravitational and rotational parts. The new feature of our scheme is that the gravitational divergent kinetic energy is further decoupled into zonal and meridional components, which measure the strength of the cut-west and meridional circulations, respectively. The zonal and meridional nondivergent kinetic energies represent the vorticities related to the nondivergent zonal and meridional winds, respectively. The distributions of energy among meridional indices, vertical modes, and zonal waves are analyzed.
We suggest a new, easy, and reasonable criterion to adjust the gravity waves in the initialization based on the vertical modes, meridional indices, and zonal wavenumbers. To retain the strength of large-scale circulations and to preserve the intensity of synoptic scale pressure systems, we recommend that the gravity waves corresponding to the internal modes 2–6, meridional indices 1–6, and zonal wavenumbers 1–10 not be adjusted significantly during the initialization. However, all the other gravity waves can be adjusted initially, particularly those associated with the external and the first internal modes.
Abstract
Bivariate Fourier series have many benefits in limited-area modeling (LAM), weather forecasting, and meteorological data analysis. However, atmospheric data are not spatially periodic on the LAM domain (“window”), which can be normalized to the unit square (x, y) ∈ [0, 1] ⊗ [0, 1] by rescaling the coordinates. Most Fourier LAM meteorology has employed rather low-order methods that have been quite successful in spite of Gibbs phenomenon at the boundaries of the artificial periodicity window. In this article, the authors explain why. Because data near the boundary between the high-resolution LAM window and the low-resolution global model are necessarily suspect, corrupted by the discontinuity in resolution, meteorologists routinely ignore LAM results in a buffer strip of nondimensional width D, and analyze only the Fourier sums in the smaller domain (x, y) ∈ [D, 1 − D] ⊗ [D, 1 − D]. It is shown that the error in a one-dimensional Fourier series with N terms or in a two-dimensional series with N 2 terms, is smaller by a factor of N on a boundary-buffer-discarded domain than on the full unit square. A variety of procedures for raising the order of Fourier series convergence are described, and it is explained how the deletion of the boundary strip greatly simplifies and improves these enhancements. The prime exemplar is solving the Poisson equation with homogeneous boundary conditions by sine series, but the authors also discuss the Laplace equation with inhomogeneous boundary conditions.
Abstract
Bivariate Fourier series have many benefits in limited-area modeling (LAM), weather forecasting, and meteorological data analysis. However, atmospheric data are not spatially periodic on the LAM domain (“window”), which can be normalized to the unit square (x, y) ∈ [0, 1] ⊗ [0, 1] by rescaling the coordinates. Most Fourier LAM meteorology has employed rather low-order methods that have been quite successful in spite of Gibbs phenomenon at the boundaries of the artificial periodicity window. In this article, the authors explain why. Because data near the boundary between the high-resolution LAM window and the low-resolution global model are necessarily suspect, corrupted by the discontinuity in resolution, meteorologists routinely ignore LAM results in a buffer strip of nondimensional width D, and analyze only the Fourier sums in the smaller domain (x, y) ∈ [D, 1 − D] ⊗ [D, 1 − D]. It is shown that the error in a one-dimensional Fourier series with N terms or in a two-dimensional series with N 2 terms, is smaller by a factor of N on a boundary-buffer-discarded domain than on the full unit square. A variety of procedures for raising the order of Fourier series convergence are described, and it is explained how the deletion of the boundary strip greatly simplifies and improves these enhancements. The prime exemplar is solving the Poisson equation with homogeneous boundary conditions by sine series, but the authors also discuss the Laplace equation with inhomogeneous boundary conditions.
Abstract
Accurate and stable numerical discretization of the equations for the nonhydrostatic atmosphere is required, for example, to resolve interactions between clouds and aerosols in the atmosphere. Here the authors present a modification of the hydrostatic control-volume approach for solving the nonhydrostatic Euler equations with a Lagrangian vertical coordinate. A scheme with low numerical diffusion is achieved by introducing a low Mach number approximate Riemann solver (LMARS) for atmospheric flows. LMARS is a flexible way to ensure stability for finite-volume numerical schemes in both Eulerian and vertical Lagrangian configurations. This new approach is validated on test cases using a 2D (x–z) configuration.
Abstract
Accurate and stable numerical discretization of the equations for the nonhydrostatic atmosphere is required, for example, to resolve interactions between clouds and aerosols in the atmosphere. Here the authors present a modification of the hydrostatic control-volume approach for solving the nonhydrostatic Euler equations with a Lagrangian vertical coordinate. A scheme with low numerical diffusion is achieved by introducing a low Mach number approximate Riemann solver (LMARS) for atmospheric flows. LMARS is a flexible way to ensure stability for finite-volume numerical schemes in both Eulerian and vertical Lagrangian configurations. This new approach is validated on test cases using a 2D (x–z) configuration.
