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- Author or Editor: FEDOR MESINGER x

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## Abstract

For use in a model on the semi-staggered E (in the Arakawa notation) grid, a number of conserving schemes for the horizontal advection are developed and analyzed. For the rotation terms of the momentum advection, the second-order enstrophy and energy-conserving scheme of Janjić (1977) is generalized to conserve energy in case of divergent flow. A family of analogs of the Arakawa (1966) fourth-order scheme is obtained following a transformation of its component Jacobians. For the kinetic energy advection terms, a fourth- (or approximately fourth) order scheme is developed which maintains the total kinetic energy and, in addition, makes no contribution to the change in the finite-difference vorticity. For the resulting both second- and fourth-order momentum advection scheme, a modification is pointed out which avoids the non-cancellation of terms considered recently by Hollingsworth and Källberg (1979), and shown to lead to a linear instability of a zonally uniform inertia-gravity wave. Finally, a second- order as well as a fourth-order (or approximately so) advection scheme for temperature (and moisture) advection is given, preserving the total energy (and moisture) inside the integration region.

## Abstract

For use in a model on the semi-staggered E (in the Arakawa notation) grid, a number of conserving schemes for the horizontal advection are developed and analyzed. For the rotation terms of the momentum advection, the second-order enstrophy and energy-conserving scheme of Janjić (1977) is generalized to conserve energy in case of divergent flow. A family of analogs of the Arakawa (1966) fourth-order scheme is obtained following a transformation of its component Jacobians. For the kinetic energy advection terms, a fourth- (or approximately fourth) order scheme is developed which maintains the total kinetic energy and, in addition, makes no contribution to the change in the finite-difference vorticity. For the resulting both second- and fourth-order momentum advection scheme, a modification is pointed out which avoids the non-cancellation of terms considered recently by Hollingsworth and Källberg (1979), and shown to lead to a linear instability of a zonally uniform inertia-gravity wave. Finally, a second- order as well as a fourth-order (or approximately so) advection scheme for temperature (and moisture) advection is given, preserving the total energy (and moisture) inside the integration region.

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## Abstract

A Lagrangean-type numerical forecasting method is developed in which the computational (grid) points are advected by the wind and the necessary space derivatives (in the pressure gradient terms, for example) are computed using the values of the variables at all the computation points that at the particular moment are within a prescribed distance of the point for which the computation is done. In this way, the forecasting problem reduces to solving the ordinary differential equations of motion and thermodynamics for each computation point, instead of solving the partial differential equations in the Eulerian or classical Lagrangean way. The method has some advantages over the conventional Eulerian scheme: simplicity (there are no advection terms), lack of computational dispersion in the advection terms and therefore better simulation of atmospheric advection and deformation effects, very little inconvenience due to the spherical shape of the earth, and the possibility for a variable space resolution if desired. On the other hand, some artificial smoothing may be necessary, and it may be difficult (or impossible) to conserve the global integrals of certain quantities.

A more detailed discussion of the differencing scheme used for the time integration is included in a separate section, This is the scheme obtained by linear extrapolation of computed time derivatives to a time value of *t*
_{0} + aΔ*t* where *t*
_{0} is the value of time at the beginning of the considered time step Δ*t* and where *a* is a parameter that can be used to control the properties of the scheme. When choosing a value of *a* between ½ and 1, a scheme is obtained that damps the high-frequency motions, in a similar way as the Matsuno scheme, but needs somewhat less computer time and, with the same damping intensity, has a higher accuracy for low-frequency meteorologically significant motions.

Using the described method, a 4-day experimental forecast has been made, starting with a stationary Haurwitz-Neamtan solution, for a primitive equation, global, and homogeneous model. The final geopotential height map showed no visible phase errors and only a modest accumulation of truncation errors and effects of numerical smoothing mechanisms. Two shorter experiments have also been made to analyze the effects of space resolution and damping in the process of time differencing. It is felt that the experimental results strongly encourage further testing and investigation of the proposed method.

## Abstract

A Lagrangean-type numerical forecasting method is developed in which the computational (grid) points are advected by the wind and the necessary space derivatives (in the pressure gradient terms, for example) are computed using the values of the variables at all the computation points that at the particular moment are within a prescribed distance of the point for which the computation is done. In this way, the forecasting problem reduces to solving the ordinary differential equations of motion and thermodynamics for each computation point, instead of solving the partial differential equations in the Eulerian or classical Lagrangean way. The method has some advantages over the conventional Eulerian scheme: simplicity (there are no advection terms), lack of computational dispersion in the advection terms and therefore better simulation of atmospheric advection and deformation effects, very little inconvenience due to the spherical shape of the earth, and the possibility for a variable space resolution if desired. On the other hand, some artificial smoothing may be necessary, and it may be difficult (or impossible) to conserve the global integrals of certain quantities.

A more detailed discussion of the differencing scheme used for the time integration is included in a separate section, This is the scheme obtained by linear extrapolation of computed time derivatives to a time value of *t*
_{0} + aΔ*t* where *t*
_{0} is the value of time at the beginning of the considered time step Δ*t* and where *a* is a parameter that can be used to control the properties of the scheme. When choosing a value of *a* between ½ and 1, a scheme is obtained that damps the high-frequency motions, in a similar way as the Matsuno scheme, but needs somewhat less computer time and, with the same damping intensity, has a higher accuracy for low-frequency meteorologically significant motions.

Using the described method, a 4-day experimental forecast has been made, starting with a stationary Haurwitz-Neamtan solution, for a primitive equation, global, and homogeneous model. The final geopotential height map showed no visible phase errors and only a modest accumulation of truncation errors and effects of numerical smoothing mechanisms. Two shorter experiments have also been made to analyze the effects of space resolution and damping in the process of time differencing. It is felt that the experimental results strongly encourage further testing and investigation of the proposed method.

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## Abstract

The benefits of assimilation of precipitation data had been demonstrated in diabetic initialization and nudging-type experiments some years ago. In four-dimensional variational (4DVAR) data assimilation, however, the precipitation data have not yet been used. To correctly assimilate the precipitation data by the 4DVAR technique, the problems related to the first-order discontinuities in the “full-physics” forecast model should be solved first. To address this problem in the full-physics regional NMC eta forecast model, a modified, more continuous version of the Beta-Miller cumulus convection scheme is defined and examined as a possible solution.

The 4DVAR data assimilation experiments ate performed using the conventional data (in this case, analyses of *T*, *p _{s}
*,

*u*,

*v*, and

*q*) and the precipitation data (the analysis of 24-h accumulated precipitation). The full-physics NMC eta model and the adjoint model with convective processes are used in the experiments. The control variable of the minimization problem is defined to include the initial conditions and model's systematic error parameter. An extreme synoptic situation from June 1993, with strong effects of precipitation over the United States is chosen for the experiments. The results of the 4DVAR experiments show convergence of the minimization process within 10 iterations and an improvement of the precipitation forecast, during and after the data assimilation period, when using the modified cumulus convection scheme and the precipitation data. In particular, the 4DVAR method outperforms the optimal interpolation method by improving the precipitation forecast.

## Abstract

The benefits of assimilation of precipitation data had been demonstrated in diabetic initialization and nudging-type experiments some years ago. In four-dimensional variational (4DVAR) data assimilation, however, the precipitation data have not yet been used. To correctly assimilate the precipitation data by the 4DVAR technique, the problems related to the first-order discontinuities in the “full-physics” forecast model should be solved first. To address this problem in the full-physics regional NMC eta forecast model, a modified, more continuous version of the Beta-Miller cumulus convection scheme is defined and examined as a possible solution.

The 4DVAR data assimilation experiments ate performed using the conventional data (in this case, analyses of *T*, *p _{s}
*,

*u*,

*v*, and

*q*) and the precipitation data (the analysis of 24-h accumulated precipitation). The full-physics NMC eta model and the adjoint model with convective processes are used in the experiments. The control variable of the minimization problem is defined to include the initial conditions and model's systematic error parameter. An extreme synoptic situation from June 1993, with strong effects of precipitation over the United States is chosen for the experiments. The results of the 4DVAR experiments show convergence of the minimization process within 10 iterations and an improvement of the precipitation forecast, during and after the data assimilation period, when using the modified cumulus convection scheme and the precipitation data. In particular, the 4DVAR method outperforms the optimal interpolation method by improving the precipitation forecast.

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## Abstract

It is suggested that there are two major problems with the “standard” methods of reducing pressure to sea level based on the surface temperature or the lowest-layer(s) temperature of a numerical model. The first is that using air temperatures above elevated terrain for reducing pressure to sea level is in conflict with the presumed objective of the reduction. The authors take this to be the derivation of a pressure field appropriate to sea level that to the extent possible maintains the shape of the constant-elevation isobars and reflects the changes in the horizontal of the magnitudes of horizontal pressure gradients, as these exist at the ground surface. The other problem is that evidence is emerging showing that with the increasing realism in the representation of mountains in numerical models the performance of the standard reduction methods is about to deteriorate to the point of becoming unacceptable.

Fortunately, as proposed earlier by the first author, an alternative exists that is both simple and consistent with the objective of the reduction as presumed above. It is to replace the downward extrapolation of temperature by the *horizontal* interpolation of (virtual) temperature where the temperatures are given *at the sides* of mountains.

Performance of the “horizontal” reduction method is here compared against the so-called Shuell method, which is a conventional part of the U.S. National Meteorological Center's postprocessing packages. This is done by examining the sea level pressure centers of initial conditions and forecasts, at 12-h intervals, of the National Meteorological Center's eta model, as obtained via the Shuell and horizontal reduction methods. The comparison is done for a sample of late summer initial conditions and forecasts verifying at 16 consecutive 0000 and 1200 UTC initial times. Note that the Shuell reduction method was specifically designed to improve upon a standard lapse rate reduction to sea level during the warm season.

In terms of the agreement with the analyst-assessed values, the two methods showed an overall comparable performance. The horizontal reduction method performed much better for Mexican heat lows, while the Shuell method was clearly superior in reproducing the analyzed values at high centers over the United States and Canadian highlands. The horizontal reduction method performed somewhat better in depicting the values at the centers of lows over the United States and Canadian mountainous region of the study. As its main benefit, the horizontal reduction method eliminated formidable noise and artifact problems of the Shuell reduction method without resorting to smoothing devices.

## Abstract

It is suggested that there are two major problems with the “standard” methods of reducing pressure to sea level based on the surface temperature or the lowest-layer(s) temperature of a numerical model. The first is that using air temperatures above elevated terrain for reducing pressure to sea level is in conflict with the presumed objective of the reduction. The authors take this to be the derivation of a pressure field appropriate to sea level that to the extent possible maintains the shape of the constant-elevation isobars and reflects the changes in the horizontal of the magnitudes of horizontal pressure gradients, as these exist at the ground surface. The other problem is that evidence is emerging showing that with the increasing realism in the representation of mountains in numerical models the performance of the standard reduction methods is about to deteriorate to the point of becoming unacceptable.

Fortunately, as proposed earlier by the first author, an alternative exists that is both simple and consistent with the objective of the reduction as presumed above. It is to replace the downward extrapolation of temperature by the *horizontal* interpolation of (virtual) temperature where the temperatures are given *at the sides* of mountains.

Performance of the “horizontal” reduction method is here compared against the so-called Shuell method, which is a conventional part of the U.S. National Meteorological Center's postprocessing packages. This is done by examining the sea level pressure centers of initial conditions and forecasts, at 12-h intervals, of the National Meteorological Center's eta model, as obtained via the Shuell and horizontal reduction methods. The comparison is done for a sample of late summer initial conditions and forecasts verifying at 16 consecutive 0000 and 1200 UTC initial times. Note that the Shuell reduction method was specifically designed to improve upon a standard lapse rate reduction to sea level during the warm season.

In terms of the agreement with the analyst-assessed values, the two methods showed an overall comparable performance. The horizontal reduction method performed much better for Mexican heat lows, while the Shuell method was clearly superior in reproducing the analyzed values at high centers over the United States and Canadian highlands. The horizontal reduction method performed somewhat better in depicting the values at the centers of lows over the United States and Canadian mountainous region of the study. As its main benefit, the horizontal reduction method eliminated formidable noise and artifact problems of the Shuell reduction method without resorting to smoothing devices.

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## Abstract

The problem of the pressure gradient force error in the case of the terrain-following (sigma) coordinate does not appear to have a solution. The problem is *not* one of truncation error in the calculation of space derivatives involved. Thus, with temperature profiles resulting in large errors, an increase in vertical resolution may not reduce and is even likely to *increase* the error. Therefore, an approach abandoning the sigma system has been proposed. It involves the use of “step” mountains with coordinate surfaces prescribed to remain at fixed elevations at places where they touch (and define) or intersect the ground surface. Thus, the coordinate surfaces are quasi-horizontal, and the sigma system problem is not present. At the same time, the simplicity of the sigma system is maintained.

In this paper, design of the model (“silhouette” averaged) mountains, properties of the wall boundary condition, and the scheme for calculation of the potential to kinetic energy conversion are presented. For an advection scheme achieving a strict control of the nonlinear energy cascade on the semistaggered grid, it is demonstrated that a straightforward no-slip wall boundary condition maintains conservation properties of the scheme with no vertical walls, which are important from the point of view of the control of this energy cascade from large to small scales. However, with that simple boundary condition considered, momentum is not conserved. The scheme conserving energy in conversion between the potential and kinetic energy, given earlier for the one-dimensional case, is extended to two dimensions.

Results of real data experiments are described, testing the performance of the resulting “Step-mountain” model. An attractive feature of a step-mountain (“eta”) model is that it can easily be run as a sigma system model, the only difference being the definition of ground surface grid point values of the vertical coordinate. This permits a comparison of the sigma and the eta formulations. Two experiments of this kind have been made, with a model version including realistic steep mountains (steps at 290, 1112 and 2433 m). They have both revealed a substantial amount of noise resulting from the sigma, as compared to the eta, formulation. One of these experiments, especially with the step mountains, gave a rather successful simulation of the perhaps difficult “historic” Buzzi–Tibaldi case of Genoa lee cyclogenesis. A parallel experiment showed that, starting with the same initial data, one obtains no cyclogenesis without mountains. Still, the mountains experiment did simulate the accompanying midtropospheric cutoff, a phenomenon that apparently has not been reproduced in previous simulations of mountain-induced Genoa lee cyclogeneses.

For a North American limited area region, experimental step-mountain simulations were performed for a case of March 1984, involving development of a secondary storm southeast of the Appalachians. Neither the then operational U.S. National Meteorological Center's Limited Area Forecast Model (LFM) nor the recently introduced Nested Grid Model (NGM) were successful in simulating the redevelopment. On the other hand, the step-mountain model, with a space resolution set up to mimic that of NGM, successfully simulated the ridging that indicates the redevelopment.

## Abstract

The problem of the pressure gradient force error in the case of the terrain-following (sigma) coordinate does not appear to have a solution. The problem is *not* one of truncation error in the calculation of space derivatives involved. Thus, with temperature profiles resulting in large errors, an increase in vertical resolution may not reduce and is even likely to *increase* the error. Therefore, an approach abandoning the sigma system has been proposed. It involves the use of “step” mountains with coordinate surfaces prescribed to remain at fixed elevations at places where they touch (and define) or intersect the ground surface. Thus, the coordinate surfaces are quasi-horizontal, and the sigma system problem is not present. At the same time, the simplicity of the sigma system is maintained.

In this paper, design of the model (“silhouette” averaged) mountains, properties of the wall boundary condition, and the scheme for calculation of the potential to kinetic energy conversion are presented. For an advection scheme achieving a strict control of the nonlinear energy cascade on the semistaggered grid, it is demonstrated that a straightforward no-slip wall boundary condition maintains conservation properties of the scheme with no vertical walls, which are important from the point of view of the control of this energy cascade from large to small scales. However, with that simple boundary condition considered, momentum is not conserved. The scheme conserving energy in conversion between the potential and kinetic energy, given earlier for the one-dimensional case, is extended to two dimensions.

Results of real data experiments are described, testing the performance of the resulting “Step-mountain” model. An attractive feature of a step-mountain (“eta”) model is that it can easily be run as a sigma system model, the only difference being the definition of ground surface grid point values of the vertical coordinate. This permits a comparison of the sigma and the eta formulations. Two experiments of this kind have been made, with a model version including realistic steep mountains (steps at 290, 1112 and 2433 m). They have both revealed a substantial amount of noise resulting from the sigma, as compared to the eta, formulation. One of these experiments, especially with the step mountains, gave a rather successful simulation of the perhaps difficult “historic” Buzzi–Tibaldi case of Genoa lee cyclogenesis. A parallel experiment showed that, starting with the same initial data, one obtains no cyclogenesis without mountains. Still, the mountains experiment did simulate the accompanying midtropospheric cutoff, a phenomenon that apparently has not been reproduced in previous simulations of mountain-induced Genoa lee cyclogeneses.

For a North American limited area region, experimental step-mountain simulations were performed for a case of March 1984, involving development of a secondary storm southeast of the Appalachians. Neither the then operational U.S. National Meteorological Center's Limited Area Forecast Model (LFM) nor the recently introduced Nested Grid Model (NGM) were successful in simulating the redevelopment. On the other hand, the step-mountain model, with a space resolution set up to mimic that of NGM, successfully simulated the ridging that indicates the redevelopment.