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- Author or Editor: Wen-Yih Sun x
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Abstract
This paper shows that in the linearized shallow-water equations, the numerical schemes can become weakly unstable for the 2Δx wave in the C grid when the Courant number is 1 in the forward–backward scheme and 0.5 in the leapfrog scheme because of the repeated eigenvalues in the matrices. The instability can be amplified and spread to other waves and smaller Courant number if the diffusion term is included. However, Shuman smoothing can control the instability.
Abstract
This paper shows that in the linearized shallow-water equations, the numerical schemes can become weakly unstable for the 2Δx wave in the C grid when the Courant number is 1 in the forward–backward scheme and 0.5 in the leapfrog scheme because of the repeated eigenvalues in the matrices. The instability can be amplified and spread to other waves and smaller Courant number if the diffusion term is included. However, Shuman smoothing can control the instability.
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Abstract
A linearized two-dimensional model was used to study the formation of convective rolls in the prestorm stage and organized squall lines near the dryline in the Great Plains. With a strong virtual potential temperature gradient across the dryline zone, the results show the without condensation symmetric instability can produce convective rolls with a horizontal wavelength of a few tens of kilometers. With condensation, a large cloud develops near the center of the dryline zone. This cloud subsequently splits into two clouds; one moves to the west and disappears, and the other moves to the east, where the PBL is relatively moist, and grows. The numerical results are qualitatively comparable with observations. This study provides a possible explanation that symmetric instability with condensation can generate storms near the center or on the east side of the dryline zone, whereas low-level convergence develops at the western edge according to the inland sea-breeze circulation.
Abstract
A linearized two-dimensional model was used to study the formation of convective rolls in the prestorm stage and organized squall lines near the dryline in the Great Plains. With a strong virtual potential temperature gradient across the dryline zone, the results show the without condensation symmetric instability can produce convective rolls with a horizontal wavelength of a few tens of kilometers. With condensation, a large cloud develops near the center of the dryline zone. This cloud subsequently splits into two clouds; one moves to the west and disappears, and the other moves to the east, where the PBL is relatively moist, and grows. The numerical results are qualitatively comparable with observations. This study provides a possible explanation that symmetric instability with condensation can generate storms near the center or on the east side of the dryline zone, whereas low-level convergence develops at the western edge according to the inland sea-breeze circulation.
Abstract
Linear stability analysis of cloud streets is investigated by including the latent heat and considering a few types of wind profiles. The release of latent heat follows the wave-CISK hypothesis. The results indicate that as the buoyance force released by latent heat becomes dominant, the most unstable mode is stationary relative to the mean wind, and the orientation of cloud streets is parallel to the direction of the wind shear. On the other hand, if the conversion of kinetic energy becomes important, the cloud streets which are perpendicular to the wind shear and propagating relative to the mean wind become most unstable. Results also show that the wavelength, growth rate and orientation angle of the most unstable mode are very sensitive to the height of cloud tops, the strength of wind shear, static stability and the environmental moisture profile. These results compare well with other theoretical results and observations.
Abstract
Linear stability analysis of cloud streets is investigated by including the latent heat and considering a few types of wind profiles. The release of latent heat follows the wave-CISK hypothesis. The results indicate that as the buoyance force released by latent heat becomes dominant, the most unstable mode is stationary relative to the mean wind, and the orientation of cloud streets is parallel to the direction of the wind shear. On the other hand, if the conversion of kinetic energy becomes important, the cloud streets which are perpendicular to the wind shear and propagating relative to the mean wind become most unstable. Results also show that the wavelength, growth rate and orientation angle of the most unstable mode are very sensitive to the height of cloud tops, the strength of wind shear, static stability and the environmental moisture profile. These results compare well with other theoretical results and observations.
Abstract
The linear stability of penetrative convection is investigated for three different undisturbed temperature profiles, each characterized by a stable layer on the top: case A) a two-layer profile with constant but different lapse rate in each layer, case B) a continuous and smooth mean temperature profile with the depth of the unstable layer fixed, and case C) a temperature profile determined by the conduction equation with the surface temperature undergoing a diurnal variation. The increase of the critical Rayleigh number R c of the unstable layer with increasing stability number S of the top layer is prominent in case A, less appreciable in case B, and also less noticeable for case C. It appears that the relation between the temperature gradient in the unstable layer and that in the stable layer is more sensitive and meaningful than the relation between R c and S for a smooth temperature profile, especially for case C in which the depth of the unstable layer is not kept constant. The results obtained from case C show that the depth of the unstable layer is about 55 m and the horizontal wavelength is about 250 m when convection starts. It is also shown that convection tends to produce a superadiabatic layer near the ground, a deep mixed layer above, and an inversion layer at the top of the mixed layer.
Abstract
The linear stability of penetrative convection is investigated for three different undisturbed temperature profiles, each characterized by a stable layer on the top: case A) a two-layer profile with constant but different lapse rate in each layer, case B) a continuous and smooth mean temperature profile with the depth of the unstable layer fixed, and case C) a temperature profile determined by the conduction equation with the surface temperature undergoing a diurnal variation. The increase of the critical Rayleigh number R c of the unstable layer with increasing stability number S of the top layer is prominent in case A, less appreciable in case B, and also less noticeable for case C. It appears that the relation between the temperature gradient in the unstable layer and that in the stable layer is more sensitive and meaningful than the relation between R c and S for a smooth temperature profile, especially for case C in which the depth of the unstable layer is not kept constant. The results obtained from case C show that the depth of the unstable layer is about 55 m and the horizontal wavelength is about 250 m when convection starts. It is also shown that convection tends to produce a superadiabatic layer near the ground, a deep mixed layer above, and an inversion layer at the top of the mixed layer.
Abstract
A modified forward-backward scheme applied to the anelastic system is proposed. This modified scheme not only retains all the advantages of the conventional forward-backward scheme but also is more consistent with the original differential equations. This scheme is used to investigate inertial waves and internal gravity waves in three different lattices. It is found that the lattice C proposed by Deardorff is better than either lattice A or lattice B when applied to internal gravity waves and thermal convection in the atmosphere.
The difference between a hydrostatic system and a nonhydrostatic system is also discussed in detail in this paper. Here we propose to apply Shuman's smoothing on a hydrostatic system to filter out the undesirably, highly oscillatory short waves, or stroll small-scale convection, so that we may produce reasonable results compared with those of a nonhydrostatic system. The validity of this method has been proved by the numerical results of a study on the mesoscale cloud bands which are produced by the effect of condensation in a conditionally unstable atmosphere with vertical wind shear.
Abstract
A modified forward-backward scheme applied to the anelastic system is proposed. This modified scheme not only retains all the advantages of the conventional forward-backward scheme but also is more consistent with the original differential equations. This scheme is used to investigate inertial waves and internal gravity waves in three different lattices. It is found that the lattice C proposed by Deardorff is better than either lattice A or lattice B when applied to internal gravity waves and thermal convection in the atmosphere.
The difference between a hydrostatic system and a nonhydrostatic system is also discussed in detail in this paper. Here we propose to apply Shuman's smoothing on a hydrostatic system to filter out the undesirably, highly oscillatory short waves, or stroll small-scale convection, so that we may produce reasonable results compared with those of a nonhydrostatic system. The validity of this method has been proved by the numerical results of a study on the mesoscale cloud bands which are produced by the effect of condensation in a conditionally unstable atmosphere with vertical wind shear.
Abstract
The forward-central (forward in time, central differencing in space) scheme and the Dufort-Frankel scheme applied to the transient heat equation are discussed here. The Dufort-Frankel scheme is shown to be unable to damp the oscillation of the high oscillatory 2Δx wave. It can even become unstable for this wave which may be produced by the initial conditions, nonlinear interactions, or other forcing. On the other hand, the forward-central scheme works well if μ<0.5. Examples of the applications of these schemes in fluid dynamics are also presented in this paper. It is also noted that none of the schemes are accurate for the high-frequency short wave.
Abstract
The forward-central (forward in time, central differencing in space) scheme and the Dufort-Frankel scheme applied to the transient heat equation are discussed here. The Dufort-Frankel scheme is shown to be unable to damp the oscillation of the high oscillatory 2Δx wave. It can even become unstable for this wave which may be produced by the initial conditions, nonlinear interactions, or other forcing. On the other hand, the forward-central scheme works well if μ<0.5. Examples of the applications of these schemes in fluid dynamics are also presented in this paper. It is also noted that none of the schemes are accurate for the high-frequency short wave.
Abstract
A simple mass correction is proposed for the semi-Lagrangian scheme using forward trajectories. The procedure includes (a) constructing the Lagrangian network induced by the motion of the fluid from the Eulerian network and finding the intersections of the networks by a general interpolation from the irregularly distributed Lagrangian grid to the regularly distributed Eulerian grid, (b) applying the spatial filter to remove the unwanted short waves and the values beyond the constraints, and finally (c) introducing a polynomial or sine function as the correction function that conserves mass while inducing the least modification to the results obtained from (a) and (b). Numerical simulations of pure advection, rotation, and idealized cyclogenesis show that the mass correction function does not degrade the accuracy of the results.
Abstract
A simple mass correction is proposed for the semi-Lagrangian scheme using forward trajectories. The procedure includes (a) constructing the Lagrangian network induced by the motion of the fluid from the Eulerian network and finding the intersections of the networks by a general interpolation from the irregularly distributed Lagrangian grid to the regularly distributed Eulerian grid, (b) applying the spatial filter to remove the unwanted short waves and the values beyond the constraints, and finally (c) introducing a polynomial or sine function as the correction function that conserves mass while inducing the least modification to the results obtained from (a) and (b). Numerical simulations of pure advection, rotation, and idealized cyclogenesis show that the mass correction function does not degrade the accuracy of the results.
Abstract
A fully compressible, three-dimensional, nonhydrostatic model is developed using a semi-implicit scheme to avoid an extremely small time step. As a result of applying the implicit scheme to high-frequency waves, an elliptic partial differential equation (EPDE) has been introduced. A multigrid solver is applied to solve the EPDEs, which include cross-derivative terms due to terrain-following coordinate transformation.
Several experiments have been performed to evaluate the model as well as the performance of the scheme with respect to tolerance number, relaxation choice, sweeps of prerelaxation and postrelaxation, and a flexible hybrid coordinate (FHC).
An FHC with two functions (base and deviation functions) is introduced. The basic function provides constant vertical grid spacing required in the multigrid solver, while the deviation function helps to adjust the vertical resolution.
Abstract
A fully compressible, three-dimensional, nonhydrostatic model is developed using a semi-implicit scheme to avoid an extremely small time step. As a result of applying the implicit scheme to high-frequency waves, an elliptic partial differential equation (EPDE) has been introduced. A multigrid solver is applied to solve the EPDEs, which include cross-derivative terms due to terrain-following coordinate transformation.
Several experiments have been performed to evaluate the model as well as the performance of the scheme with respect to tolerance number, relaxation choice, sweeps of prerelaxation and postrelaxation, and a flexible hybrid coordinate (FHC).
An FHC with two functions (base and deviation functions) is introduced. The basic function provides constant vertical grid spacing required in the multigrid solver, while the deviation function helps to adjust the vertical resolution.
Abstract
A quasi-one-dimensional steady-state cloud model that is based on integrating the two-dimensional equation for the azimuthal vorticity over the cloud radius is described. The equations are reduced to one dimension in the vertical direction by assuming that the streamfunction is oscillatory and a function of r only allowing this model to incorporate the nonhydrostatic pressure gradient force. It also includes an empirical buoyancy correction to account for vertical wind shear; the correction is based on application of the Klemp-Wilhelmson convective cloud model to a variety of buoyancy and wind-shear environments. In addition, the model includes consideration of rainwater, freezing of condensate, and a moist penetrative downdraft. The downdraft includes consideration of the nonhydrostatic pressure gradient, rainwater loading, and buoyancy; below cloud base, it is unsaturated.
The results from the model for cases with and without vertical wind shear are compared to the Schlesinger three-dimensional cloud model results for the same environmental conditions. The amount and vertical variation of the vertical mass flux, heat flux, and water vapor flux calculated by the quasi-one-dimensional model are comparable to the same produced by the Schlesinger model. Finally, implications of these results in regard to using this model in cumulus parameterization are given.
Abstract
A quasi-one-dimensional steady-state cloud model that is based on integrating the two-dimensional equation for the azimuthal vorticity over the cloud radius is described. The equations are reduced to one dimension in the vertical direction by assuming that the streamfunction is oscillatory and a function of r only allowing this model to incorporate the nonhydrostatic pressure gradient force. It also includes an empirical buoyancy correction to account for vertical wind shear; the correction is based on application of the Klemp-Wilhelmson convective cloud model to a variety of buoyancy and wind-shear environments. In addition, the model includes consideration of rainwater, freezing of condensate, and a moist penetrative downdraft. The downdraft includes consideration of the nonhydrostatic pressure gradient, rainwater loading, and buoyancy; below cloud base, it is unsaturated.
The results from the model for cases with and without vertical wind shear are compared to the Schlesinger three-dimensional cloud model results for the same environmental conditions. The amount and vertical variation of the vertical mass flux, heat flux, and water vapor flux calculated by the quasi-one-dimensional model are comparable to the same produced by the Schlesinger model. Finally, implications of these results in regard to using this model in cumulus parameterization are given.
