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- Author or Editor: R. T. WILLIAMS x
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Abstract
The objective of this study is to produce a narrow frontal zone from a field which initially contains only large-scale variations. In the model, all quantities except the temperature and pressure are independent of y (latitude), and these have y derivatives which are only functions of z. The hydrostatic Boussinesq equations are employed, and friction, heating, and the variation of f are neglected. In the experiment a growing baroclinic wave distorts itself in such a way that a cold frontal zone is produced. Comparative integrations with two different values of Δx indicate that the front would become discontinuous within a finite period of time if Δx were made arbitrarily small.
A crude analytical solution is obtained which has the main characteristics of the numerical solutions. Mathematically, the analytical solution is similar to those which describe the formation of shocks and pressure jumps. Physically, the frontal case is quite different because the vorticity is always much larger than the divergence.
Abstract
The objective of this study is to produce a narrow frontal zone from a field which initially contains only large-scale variations. In the model, all quantities except the temperature and pressure are independent of y (latitude), and these have y derivatives which are only functions of z. The hydrostatic Boussinesq equations are employed, and friction, heating, and the variation of f are neglected. In the experiment a growing baroclinic wave distorts itself in such a way that a cold frontal zone is produced. Comparative integrations with two different values of Δx indicate that the front would become discontinuous within a finite period of time if Δx were made arbitrarily small.
A crude analytical solution is obtained which has the main characteristics of the numerical solutions. Mathematically, the analytical solution is similar to those which describe the formation of shocks and pressure jumps. Physically, the frontal case is quite different because the vorticity is always much larger than the divergence.
Abstract
Three numerical schemes for the vorticity-divergence form of the shallow-water equations are analyzed using the Fourier transform technique developed by Schoenstadt (1980). These schemes are compared with two finite-element schemes for the primitive form of the equations which were examined by Schoenstadt. The best finite-element schemes use either staggered elements in the primitive form of the equations, or unstaggered elements in the vorticity-divergence form.
Abstract
Three numerical schemes for the vorticity-divergence form of the shallow-water equations are analyzed using the Fourier transform technique developed by Schoenstadt (1980). These schemes are compared with two finite-element schemes for the primitive form of the equations which were examined by Schoenstadt. The best finite-element schemes use either staggered elements in the primitive form of the equations, or unstaggered elements in the vorticity-divergence form.
Abstract
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Abstract
The numerical frontogenesis model of Williams is modified to include horizontal and vertical turbulent diffusions of heat and momentum. The turbulent diffusions are represented with constant coefficients, and an Ekman layer is added to the basic deformation field. The numerical solutions show realistic quasi-steady fronts forming within 1–2 days. These solutions are examined and compared over a wide range of the various coefficients.
Abstract
The numerical frontogenesis model of Williams is modified to include horizontal and vertical turbulent diffusions of heat and momentum. The turbulent diffusions are represented with constant coefficients, and an Ekman layer is added to the basic deformation field. The numerical solutions show realistic quasi-steady fronts forming within 1–2 days. These solutions are examined and compared over a wide range of the various coefficients.
Abstract
This study examines frontogenesis which is forced by a nondivergent horizontal wind field which contains stretching deformation. The initial conditions are formulated in such a way that the departures from the basic wind field are independent of x. The linear and nonlinear hydrostatic primitive equations are solved numerically and the solutions are compared. The linear solutions are very close to the solutions of the quasi-geostrophic equations. The latter equations predict fronts which have some unrealistic features and these equations predict discontinuities only for very large time. The nonlinear solutions are much more realistic than the linear solutions and they imply the formation of discontinuities within a finite period of time.
Abstract
This study examines frontogenesis which is forced by a nondivergent horizontal wind field which contains stretching deformation. The initial conditions are formulated in such a way that the departures from the basic wind field are independent of x. The linear and nonlinear hydrostatic primitive equations are solved numerically and the solutions are compared. The linear solutions are very close to the solutions of the quasi-geostrophic equations. The latter equations predict fronts which have some unrealistic features and these equations predict discontinuities only for very large time. The nonlinear solutions are much more realistic than the linear solutions and they imply the formation of discontinuities within a finite period of time.
Abstract
In this paper Rossby wave frequencies and group velocities are analyzed for various finite element and finite difference approximations to the vorticity-divergence form of the shallow water equations. Also included are finite difference solutions for the primitive equations for the staggered grids B and C from Wajsowicz and for the unstaggered grid A. The results are presented for three ratios between the grid size and the Rossby radius of deformation. The Yortcity-divergence equation schemes give superior solutions to those based on the primitive equations. The best results come from the finite element schemes that use linear basis functions on isosceles triangles and bilinear functions on rectangles. All of the primitive equation finite difference schemes have problems for at least one Rossby deformation-grid size ratio.
Abstract
In this paper Rossby wave frequencies and group velocities are analyzed for various finite element and finite difference approximations to the vorticity-divergence form of the shallow water equations. Also included are finite difference solutions for the primitive equations for the staggered grids B and C from Wajsowicz and for the unstaggered grid A. The results are presented for three ratios between the grid size and the Rossby radius of deformation. The Yortcity-divergence equation schemes give superior solutions to those based on the primitive equations. The best results come from the finite element schemes that use linear basis functions on isosceles triangles and bilinear functions on rectangles. All of the primitive equation finite difference schemes have problems for at least one Rossby deformation-grid size ratio.
Abstract
Unbalanced frontogenesis is studied in a two-dimensional, Boussinesq, rotating fluid that is constrained between two rigid, level surfaces. The potential vorticity is zero. The initial state is unbalanced because there is no motion and the potential temperature is given by the error function of x. An analytic solution is derived based on the neglect of the barotropic pressure gradient. The solution procedure uses momentum coordinates to obtain nonlinear solutions. When the initial Rossby number (Ro) is less than 1.435 the horizontal wind components display an inertial oscillation. During the first part of the inertial period (0 < ft < π) the isentropes develop a tilt and frontogenesis occurs, while in the second part (π < ft < 2π) the isentropes return to a vertical orientation and frontolysis brings the temperature gradient back to its original value at ft = 2π. For larger values of Ro a frontal discontinuity forms before ft = π.
The importance of the barotropic pressure gradient is determined in a scale collapse problem with a constant potential temperature and no rotation. In this case the inclusion of the barotropic pressure gradient increases the time before the discontinuity forms.
Numerical solutions of the original problem with rotation show that the presence of the barotropic pressure gradient term increases the critical Rossby number from 1.435 to about 1.55. Otherwise the complete solutions are very similar to the analytic solutions, except that the isentropes are no longer straight and the vorticity shows evidence of strong vertical advection by a small-scale vertical jet. Further, shorter timescales are expected with unbalanced fronts as compared with balanced fronts.
Abstract
Unbalanced frontogenesis is studied in a two-dimensional, Boussinesq, rotating fluid that is constrained between two rigid, level surfaces. The potential vorticity is zero. The initial state is unbalanced because there is no motion and the potential temperature is given by the error function of x. An analytic solution is derived based on the neglect of the barotropic pressure gradient. The solution procedure uses momentum coordinates to obtain nonlinear solutions. When the initial Rossby number (Ro) is less than 1.435 the horizontal wind components display an inertial oscillation. During the first part of the inertial period (0 < ft < π) the isentropes develop a tilt and frontogenesis occurs, while in the second part (π < ft < 2π) the isentropes return to a vertical orientation and frontolysis brings the temperature gradient back to its original value at ft = 2π. For larger values of Ro a frontal discontinuity forms before ft = π.
The importance of the barotropic pressure gradient is determined in a scale collapse problem with a constant potential temperature and no rotation. In this case the inclusion of the barotropic pressure gradient increases the time before the discontinuity forms.
Numerical solutions of the original problem with rotation show that the presence of the barotropic pressure gradient term increases the critical Rossby number from 1.435 to about 1.55. Otherwise the complete solutions are very similar to the analytic solutions, except that the isentropes are no longer straight and the vorticity shows evidence of strong vertical advection by a small-scale vertical jet. Further, shorter timescales are expected with unbalanced fronts as compared with balanced fronts.
Abstract
An exact solution to the quasi-geostrophic equations is derived for an initial horizontal wind field which contains deformation. The variation in the initial temperature field is confined to a band which is oriented normal to the axis of contraction of the deformation field. In the solution, a frontal zone forms and the temperature at two boundary points becomes discontinuous as the time approaches infinity. The solution is very similar to Stone's solution and it generalizes his work. Some aspects of the solution are not realistic.
Abstract
An exact solution to the quasi-geostrophic equations is derived for an initial horizontal wind field which contains deformation. The variation in the initial temperature field is confined to a band which is oriented normal to the axis of contraction of the deformation field. In the solution, a frontal zone forms and the temperature at two boundary points becomes discontinuous as the time approaches infinity. The solution is very similar to Stone's solution and it generalizes his work. Some aspects of the solution are not realistic.
Abstract
A hydrostatic primitive equation model initialized in a highly baroclinically unstable state was used to simulate maritime cyclogenesis and frontogenesis. In order to identify boundary layer physical processes important in maritime frontogenesis, several different simulations were performed. In an effort to isolate impacts due solely to the boundary layer, moist processes were not included. An adiabatic and inviscid simulation provided the control for these experiments. Two different boundary layer parameterizations were used: a K-theory parameterization featuring Richardson-number-dependent eddy diffusivity and a second-order closure scheme with prognostic equations for the turbulence quantities.
Results indicated that strong warm and cold fronts formed in the adiabatic and inviscid case but that the vertical motion fields were weak. In the K-theory simulation, the results were somewhat more realistic with stronger vertical motion. In both the K-theory and second-order closure simulations, the boundary layer in the cold air was highly unstable and deep mixed layers formed in this region with a large generation of turbulence. The largest cross-front temperature gradients existed in the frontal zone above the mixed layer. These structures were in qualitative agreement with observations of maritime cold fronts over the northwest Pacific Ocean. The second-order closure simulations produced a shallower mixed layer in the cold air with a stronger, more narrow front and large vertical motion. These simulations were more consistent with observations. Results from the second-order closure simulations demonstrated that turbulent mixing of momentum was critical in reproducing the frontogenetic (and frontolytic) effects of the transverse secondary circulation.
Abstract
A hydrostatic primitive equation model initialized in a highly baroclinically unstable state was used to simulate maritime cyclogenesis and frontogenesis. In order to identify boundary layer physical processes important in maritime frontogenesis, several different simulations were performed. In an effort to isolate impacts due solely to the boundary layer, moist processes were not included. An adiabatic and inviscid simulation provided the control for these experiments. Two different boundary layer parameterizations were used: a K-theory parameterization featuring Richardson-number-dependent eddy diffusivity and a second-order closure scheme with prognostic equations for the turbulence quantities.
Results indicated that strong warm and cold fronts formed in the adiabatic and inviscid case but that the vertical motion fields were weak. In the K-theory simulation, the results were somewhat more realistic with stronger vertical motion. In both the K-theory and second-order closure simulations, the boundary layer in the cold air was highly unstable and deep mixed layers formed in this region with a large generation of turbulence. The largest cross-front temperature gradients existed in the frontal zone above the mixed layer. These structures were in qualitative agreement with observations of maritime cold fronts over the northwest Pacific Ocean. The second-order closure simulations produced a shallower mixed layer in the cold air with a stronger, more narrow front and large vertical motion. These simulations were more consistent with observations. Results from the second-order closure simulations demonstrated that turbulent mixing of momentum was critical in reproducing the frontogenetic (and frontolytic) effects of the transverse secondary circulation.
Abstract
A series of numerical experiments of a surface front forced by stretching deformation using Clark's nonhydrostatic model at very high resolution is presented. These simulations are compared to those reported by Williams who used hydrostatic models at lower resolution. The main purpose was to determine whether this front would collapse (in the absence of friction) to a scale similar to that reported by Shapiro et al. (most of the temperature gradient contained in 200 m). The question is whether there is a natural physical process in the frontal dynamics which limits the frontal collapse in the absence of diffusion processes.
For this front we could not find a natural limiting process, although the mechanism discussed by Orlanski et al. appears to be operating. The minimum scale is determined by the vertical resolution. At the vertical and horizontal resolutions we tried, the vertical resolution determined the scale because the slope of the front is so shallow.
Some of the structure found by Cullen and Purser by extending the semigeostrophic models beyond the initial development of a discontinuity is apparent in our solutions.
Abstract
A series of numerical experiments of a surface front forced by stretching deformation using Clark's nonhydrostatic model at very high resolution is presented. These simulations are compared to those reported by Williams who used hydrostatic models at lower resolution. The main purpose was to determine whether this front would collapse (in the absence of friction) to a scale similar to that reported by Shapiro et al. (most of the temperature gradient contained in 200 m). The question is whether there is a natural physical process in the frontal dynamics which limits the frontal collapse in the absence of diffusion processes.
For this front we could not find a natural limiting process, although the mechanism discussed by Orlanski et al. appears to be operating. The minimum scale is determined by the vertical resolution. At the vertical and horizontal resolutions we tried, the vertical resolution determined the scale because the slope of the front is so shallow.
Some of the structure found by Cullen and Purser by extending the semigeostrophic models beyond the initial development of a discontinuity is apparent in our solutions.
