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## Abstract

We consider smooth solutions of the shallow water equations restricted only by the assumption that the magnitude of the deviation of the geopotential from the mean is small relative to that mean. For such solutions we find only two possible reduced systems. The first system is equivalent to the nondivergent barotropic system. A typical flow satisfying this system is the external mode with geopotential deviations on the order of 1% of the mean, and velocity on the order of 10 m s^{−1}. The second reduced system is similar to the balance equations. An example of a flow satisfying the second system is the largest internal mode with geopotential deviations on the order of 10% of the mean, and velocity on the order of 10 m s^{−1}. The error incurred by using a reduced system can be decreased by expanding the smooth solutions of the corresponding full system in an asymptotic series of solutions of the reduced system. We discuss an equivalent method to accomplish this reduction in error and derive improved approximating systems for both cases. To support the analysis we compare a numerical solution of the initialized shallow water equations with the corresponding numerical solutions of the reduced and improved approximating systems.

## Abstract

We consider smooth solutions of the shallow water equations restricted only by the assumption that the magnitude of the deviation of the geopotential from the mean is small relative to that mean. For such solutions we find only two possible reduced systems. The first system is equivalent to the nondivergent barotropic system. A typical flow satisfying this system is the external mode with geopotential deviations on the order of 1% of the mean, and velocity on the order of 10 m s^{−1}. The second reduced system is similar to the balance equations. An example of a flow satisfying the second system is the largest internal mode with geopotential deviations on the order of 10% of the mean, and velocity on the order of 10 m s^{−1}. The error incurred by using a reduced system can be decreased by expanding the smooth solutions of the corresponding full system in an asymptotic series of solutions of the reduced system. We discuss an equivalent method to accomplish this reduction in error and derive improved approximating systems for both cases. To support the analysis we compare a numerical solution of the initialized shallow water equations with the corresponding numerical solutions of the reduced and improved approximating systems.

## Abstract

Current meteorological observational networks are capable of observing only a limited number of the dependent variables that describe the state of the atmosphere. For example, the large-scale temperature and horizontal wind are commonly observed, but not the large-scale vertical velocity. In the late 1960s, Charney suggested that any missing dependent variables might be reconstructed from the time history of the fields that are observed; for example, the winds could be reconstructed by continually inserting satellite observations of the temperature into a numerical weather prediction model. (Some modern weather prediction models are essentially still using this technique to reconstruct the missing variables.) Charney's hypothesis is analyzed for systems of equations with and without multiple timescales. In the absence of dissipation, the hypothesis is not correct. However, the addition of dissipation can produce convergence that varies in degree relative to the variables that are inserted and the amount of dissipation. The analysis of the insertion process for the multiple-timescale case proves that less dissipation is required and better rates of convergence are achieved in the case that the slow variables are inserted. The advantage of slow variable insertion is even more apparent when the system is skewed, for example, in the external mode case. An alternative approach that requires no dissipation is suggested.

## Abstract

Current meteorological observational networks are capable of observing only a limited number of the dependent variables that describe the state of the atmosphere. For example, the large-scale temperature and horizontal wind are commonly observed, but not the large-scale vertical velocity. In the late 1960s, Charney suggested that any missing dependent variables might be reconstructed from the time history of the fields that are observed; for example, the winds could be reconstructed by continually inserting satellite observations of the temperature into a numerical weather prediction model. (Some modern weather prediction models are essentially still using this technique to reconstruct the missing variables.) Charney's hypothesis is analyzed for systems of equations with and without multiple timescales. In the absence of dissipation, the hypothesis is not correct. However, the addition of dissipation can produce convergence that varies in degree relative to the variables that are inserted and the amount of dissipation. The analysis of the insertion process for the multiple-timescale case proves that less dissipation is required and better rates of convergence are achieved in the case that the slow variables are inserted. The advantage of slow variable insertion is even more apparent when the system is skewed, for example, in the external mode case. An alternative approach that requires no dissipation is suggested.

## Abstract

The bounded derivative theory (BDT) for hyperbolic systems with multiple timescales was originally applied to the initialization problem for large-scale shallow-water flows in the midlatitudes and near the equator. Concepts from the theory also have been used to prove the existence of a simple reduced system that accurately describes the dominant component of a midlatitude mesoscale storm forced by cooling and heating. Recently, it has been shown how the latter results can be extended to tropospheric flows near the equator. In all of these cases, only a single type of flow was assumed to exist in the domain of interest in order to better examine the characteristics of that flow. Here it is shown how BDT concepts can be used to understand the dependence of developing mesoscale features on a balanced large-scale background flow. That understanding is then used to develop multiscale initialization constraints for the three-dimensional diabatic equations in any domain on the globe.

## Abstract

The bounded derivative theory (BDT) for hyperbolic systems with multiple timescales was originally applied to the initialization problem for large-scale shallow-water flows in the midlatitudes and near the equator. Concepts from the theory also have been used to prove the existence of a simple reduced system that accurately describes the dominant component of a midlatitude mesoscale storm forced by cooling and heating. Recently, it has been shown how the latter results can be extended to tropospheric flows near the equator. In all of these cases, only a single type of flow was assumed to exist in the domain of interest in order to better examine the characteristics of that flow. Here it is shown how BDT concepts can be used to understand the dependence of developing mesoscale features on a balanced large-scale background flow. That understanding is then used to develop multiscale initialization constraints for the three-dimensional diabatic equations in any domain on the globe.

## Abstract

Pressure oscillations with amplitudes of the deviations from the horizontal mean and periods considerably less than those for the large-scale case have been observed in a number of summer and winter storms. However, there is conflicting evidence about the role of these waves in mesoscale storms. In the case of mesoscale heating that is a prescribed function of the independent variables, it has been proven that the dominant component of the corresponding slowly varying in time solution is accurately described by a simple dynamical (reduced) system in which gravity waves play no role. This paper proves that large spatial-scale gravity waves with amplitudes and periods of the pressure perturbations the same as the reduced system component of the solution can be generated by mesoscale storms. Because the amplitudes and the periods of the pressure perturbations for the two components of the solution are similar, it is difficult to distinguish between them using temporal plots of the pressure at a single location, and this is the source of a large part of the confusion about these waves. This problem, in conjunction with the fact that the vertical velocity of the gravity waves is an order of magnitude smaller than the maximum vertical velocity in the dominant component of the solution (and therefore in the noise range of current wind profilers), makes observation of gravity waves very difficult. In numerical simulations, if both components of the mesoscale solution are required, the lateral extent of the domain of solution must be considerably larger than the lateral extent of the mesoscale heating in order that the large-scale gravity waves be correct. In this case, it is shown that the multiscale system for meteorology developed earlier by Browning and Kreiss accurately describes both components of the solution.

## Abstract

Pressure oscillations with amplitudes of the deviations from the horizontal mean and periods considerably less than those for the large-scale case have been observed in a number of summer and winter storms. However, there is conflicting evidence about the role of these waves in mesoscale storms. In the case of mesoscale heating that is a prescribed function of the independent variables, it has been proven that the dominant component of the corresponding slowly varying in time solution is accurately described by a simple dynamical (reduced) system in which gravity waves play no role. This paper proves that large spatial-scale gravity waves with amplitudes and periods of the pressure perturbations the same as the reduced system component of the solution can be generated by mesoscale storms. Because the amplitudes and the periods of the pressure perturbations for the two components of the solution are similar, it is difficult to distinguish between them using temporal plots of the pressure at a single location, and this is the source of a large part of the confusion about these waves. This problem, in conjunction with the fact that the vertical velocity of the gravity waves is an order of magnitude smaller than the maximum vertical velocity in the dominant component of the solution (and therefore in the noise range of current wind profilers), makes observation of gravity waves very difficult. In numerical simulations, if both components of the mesoscale solution are required, the lateral extent of the domain of solution must be considerably larger than the lateral extent of the mesoscale heating in order that the large-scale gravity waves be correct. In this case, it is shown that the multiscale system for meteorology developed earlier by Browning and Kreiss accurately describes both components of the solution.

## Abstract

In a series of numerical experiments, Williamson and Temperton demonstrated that the interaction of the high-frequency gravity waves with the low-frequency Rossby waves in a three-dimensional adiabatic model is very weak. However, they stated that this “might not be the case when the model includes realistic physical processes, such as release of latent heat, which are strongly influenced by the vertical motion.” The bounded derivative theory is valid for inhomogeneous hyperbolic systems with multiple time scales, but the magnitude of any forcing term must be less than or equal to that of the horizontal advection terms in the same equation. When diabatic effects are added to the basic dynamical equations for the atmosphere, in the smaller scales of motion forcing terms can appear in both the entropy and pressure equations that do not satisfy this restriction. Assuming that the heating terms are only functions of the independent variables, the forcing term in the entropy equation can be eliminated so that only a large forcing term in the pressure equation remains. It is proved that a large forcing term in the pressure equation does not by itself preclude a smooth (in the bounded derivative sense) solution. However, the proof shows that the smoothness of the derivatives of the forcing determines the smoothness of the solution. If the spatial variation of the forcing in the pressure equation is much larger than that of the advective component of the solution of the homogeneous system, then no mathematical estimates of smoothness can be obtained and examples show a smooth solution does not exist. On the other hand, if the spatial derivatives of the forcing are smooth, but the temporal derivatives are not, a smooth solution exists and the effect of the large variation of the forcing in time on that smooth solution is small. When both spatial and temporal derivatives of the forcing are smooth, a smooth solution also exists, and it is proved that it is extremely accurately described by the corresponding reduced system; that is, the effect of the interaction of any gravity waves generated by the prescribed forcing with the smooth solution is minimal. The implications of these results for atmospheric prediction models are discussed.

## Abstract

In a series of numerical experiments, Williamson and Temperton demonstrated that the interaction of the high-frequency gravity waves with the low-frequency Rossby waves in a three-dimensional adiabatic model is very weak. However, they stated that this “might not be the case when the model includes realistic physical processes, such as release of latent heat, which are strongly influenced by the vertical motion.” The bounded derivative theory is valid for inhomogeneous hyperbolic systems with multiple time scales, but the magnitude of any forcing term must be less than or equal to that of the horizontal advection terms in the same equation. When diabatic effects are added to the basic dynamical equations for the atmosphere, in the smaller scales of motion forcing terms can appear in both the entropy and pressure equations that do not satisfy this restriction. Assuming that the heating terms are only functions of the independent variables, the forcing term in the entropy equation can be eliminated so that only a large forcing term in the pressure equation remains. It is proved that a large forcing term in the pressure equation does not by itself preclude a smooth (in the bounded derivative sense) solution. However, the proof shows that the smoothness of the derivatives of the forcing determines the smoothness of the solution. If the spatial variation of the forcing in the pressure equation is much larger than that of the advective component of the solution of the homogeneous system, then no mathematical estimates of smoothness can be obtained and examples show a smooth solution does not exist. On the other hand, if the spatial derivatives of the forcing are smooth, but the temporal derivatives are not, a smooth solution exists and the effect of the large variation of the forcing in time on that smooth solution is small. When both spatial and temporal derivatives of the forcing are smooth, a smooth solution also exists, and it is proved that it is extremely accurately described by the corresponding reduced system; that is, the effect of the interaction of any gravity waves generated by the prescribed forcing with the smooth solution is minimal. The implications of these results for atmospheric prediction models are discussed.

## Abstract

Large-amplitude high-frequency motions can appear in the solution of a hyperbolic system containing multiple time scales unless the initial conditions are suitably adjusted through a process called *initialization*. We observe that a solution of such a system which varies slowly with respect to time must have a number of time derivatives on the order of the slow time scale. Given a variable which is characteristic of low-frequency motions (e.g., vorticity), we can apply this observation at the initial time to find constraints which determine the rest of the initial data so that the amplitudes of the ensuing high-frequency motions remain small. Boundary conditions of the system must be taken into account in the derivation of the constraints. This procedure is referred to as the *bounded derivative method*.

For a general linear version of the shallow-water equations, we prove that if the initial *k*th order time derivative is of the order of the slow time scale, then it will remain so for a fixed time interval. For the corresponding constant coefficient system, we compare the present initialization procedure with the normal mode approach. We then apply the new procedure to initialize the nonlinear shallow-water equations including the effect of orography for both the *midlatitude* and *equatorial* beta plane cases. In the midlatitude case, the initialization scheme based on quasi-geostrophic theory can be obtained from the bounded derivative method by certain simplifying assumptions. In the equatorial case, the bounded derivative method provides an effective initialization scheme and new insight into the nature of equatorial flows.

## Abstract

Large-amplitude high-frequency motions can appear in the solution of a hyperbolic system containing multiple time scales unless the initial conditions are suitably adjusted through a process called *initialization*. We observe that a solution of such a system which varies slowly with respect to time must have a number of time derivatives on the order of the slow time scale. Given a variable which is characteristic of low-frequency motions (e.g., vorticity), we can apply this observation at the initial time to find constraints which determine the rest of the initial data so that the amplitudes of the ensuing high-frequency motions remain small. Boundary conditions of the system must be taken into account in the derivation of the constraints. This procedure is referred to as the *bounded derivative method*.

For a general linear version of the shallow-water equations, we prove that if the initial *k*th order time derivative is of the order of the slow time scale, then it will remain so for a fixed time interval. For the corresponding constant coefficient system, we compare the present initialization procedure with the normal mode approach. We then apply the new procedure to initialize the nonlinear shallow-water equations including the effect of orography for both the *midlatitude* and *equatorial* beta plane cases. In the midlatitude case, the initialization scheme based on quasi-geostrophic theory can be obtained from the bounded derivative method by certain simplifying assumptions. In the equatorial case, the bounded derivative method provides an effective initialization scheme and new insight into the nature of equatorial flows.

## Abstract

The time step for the leapfrog scheme for a symmetric hyperbolic system with multiple timescales is limited by the Courant-Friedlichs-Lewy condition based on the fastest speed present. However, in many physical cases, most of the energy is in the slowest wave, and for this wave the use of the above time step implies that the time truncation error is much smaller than the spatial truncation error. A number of methods have been proposed to overcome this imbalance—for example, the semi-implicit method and the additive splitting technique originally proposed by Marchuk with variations attributable to Strang, and Klemp and Wilhelmson. An analysis of the Marchuk splitting method for multiple timescale systems shows that if a time step based on the slow speed is used, the accuracy of the method cannot be proved, and in practice the method is quite inaccurate. If a time step is chosen that is between the two extremes, then the Klemp and Wilhelmson method can be used, but only if an ad hoc stabilization mechanism is added. The additional computational burden required to maintain the accuracy and the stability of the split-explicit method leads to the conclusion that it is no more efficient than the leapfrog method trivially modified to handle computationally expensive smooth forcing terms.

Using the mathematical analysis developed in a previous manuscript, it is shown that splitting schemes are not appropriate for badly skewed hyperbolic systems. In a number of atmospheric models, the semi-implicit method is used to treat the badly skewed vertical sound wave terms. This leads to the excitation of the high-frequency waves in a nonphysical manner. It is also shown that this is equivalent to solving the primitive equations; that is, a model using this method for the large-scale cast will be ill posed at the lateral boundaries. The multiscale system for meteorology was introduced by Browning and Kreiss to overcome exactly these problems.

## Abstract

The time step for the leapfrog scheme for a symmetric hyperbolic system with multiple timescales is limited by the Courant-Friedlichs-Lewy condition based on the fastest speed present. However, in many physical cases, most of the energy is in the slowest wave, and for this wave the use of the above time step implies that the time truncation error is much smaller than the spatial truncation error. A number of methods have been proposed to overcome this imbalance—for example, the semi-implicit method and the additive splitting technique originally proposed by Marchuk with variations attributable to Strang, and Klemp and Wilhelmson. An analysis of the Marchuk splitting method for multiple timescale systems shows that if a time step based on the slow speed is used, the accuracy of the method cannot be proved, and in practice the method is quite inaccurate. If a time step is chosen that is between the two extremes, then the Klemp and Wilhelmson method can be used, but only if an ad hoc stabilization mechanism is added. The additional computational burden required to maintain the accuracy and the stability of the split-explicit method leads to the conclusion that it is no more efficient than the leapfrog method trivially modified to handle computationally expensive smooth forcing terms.

Using the mathematical analysis developed in a previous manuscript, it is shown that splitting schemes are not appropriate for badly skewed hyperbolic systems. In a number of atmospheric models, the semi-implicit method is used to treat the badly skewed vertical sound wave terms. This leads to the excitation of the high-frequency waves in a nonphysical manner. It is also shown that this is equivalent to solving the primitive equations; that is, a model using this method for the large-scale cast will be ill posed at the lateral boundaries. The multiscale system for meteorology was introduced by Browning and Kreiss to overcome exactly these problems.

## Abstract

A mathematical theory was recently developed on the relationship between the dominant and gravity wave components of the slowly varying in time solutions (solutions varying on the advective timescale) corresponding to midlatitude mesoscale motions forced by cooling and heating. Here it will be shown that slowly varying in time equatorial motions of any length scale satisfy the same balance between the vertical velocity and heating as in the midlatitude mesoscale case. Thus any equatorial gravity waves that are generated will have the same time- and depth scales and the same size of pressure perturbations as the corresponding dominant component, a horizontal length scale an order of magnitude larger than that of the heat source, and an order of magnitude smaller velocity than the corresponding dominant component. In particular, in the large-scale equatorial case, when the heating has a timescale *O*(1 day), horizontally propagating gravity waves with a timescale *O*(1 day) and a length scale *O*(10 000 km) can be generated. But in the large-scale equatorial case when the heating has a timescale *O*(10 days), balanced pressure oscillations with a timescale *O*(10 days) are generated. It is also shown that if a solution of the diabatic system describing equatorial flows (and hence equatorial observational data in the presence of heating) is written in terms of a series of the modes of the linear adiabatic system for those flows, then a major portion of the dominant solution is projected onto gravity wave modes, and this result can explain the confusion over the relative importance of equatorial gravity waves.

## Abstract

A mathematical theory was recently developed on the relationship between the dominant and gravity wave components of the slowly varying in time solutions (solutions varying on the advective timescale) corresponding to midlatitude mesoscale motions forced by cooling and heating. Here it will be shown that slowly varying in time equatorial motions of any length scale satisfy the same balance between the vertical velocity and heating as in the midlatitude mesoscale case. Thus any equatorial gravity waves that are generated will have the same time- and depth scales and the same size of pressure perturbations as the corresponding dominant component, a horizontal length scale an order of magnitude larger than that of the heat source, and an order of magnitude smaller velocity than the corresponding dominant component. In particular, in the large-scale equatorial case, when the heating has a timescale *O*(1 day), horizontally propagating gravity waves with a timescale *O*(1 day) and a length scale *O*(10 000 km) can be generated. But in the large-scale equatorial case when the heating has a timescale *O*(10 days), balanced pressure oscillations with a timescale *O*(10 days) are generated. It is also shown that if a solution of the diabatic system describing equatorial flows (and hence equatorial observational data in the presence of heating) is written in terms of a series of the modes of the linear adiabatic system for those flows, then a major portion of the dominant solution is projected onto gravity wave modes, and this result can explain the confusion over the relative importance of equatorial gravity waves.

## Abstract

Recently, a mathematical theory has been developed that proves that there are two main components of the solution of the forced dynamical system that describes a mesoscale storm driven by cooling and heating processes. The component that contains most of the energy of the solution (and is therefore called the dominant component) satisfies a simple nonlinear system devoid of gravity and sound waves. The residual component of the solution satisfies a forced gravity wave equation and essentially does not interact with the dominant component. The mathematical theory also provides information about the amplitude, wavelength, and period of the gravity waves. In the paper entitled “Comments on ‘Use of ducting theory in an observed case of gravity waves,”’ Dr. F. M. Ralph has claimed that the new gravity wave theory is not consistent with profiler observations of vertical velocity in his earlier paper entitled “Observations of a mesoscale ducted gravity wave.” Here it is shown that the new theory is completely consistent with profilers that have documented error bounds on the vertical velocity measurements. In the case that the new theory is claimed to be inconsistent with observational data, the data were obtained from a profiler with undocumented accuracy of the vertical velocity measurements *in the precipitating case,* and the two components of the solution were not properly separated.

## Abstract

Recently, a mathematical theory has been developed that proves that there are two main components of the solution of the forced dynamical system that describes a mesoscale storm driven by cooling and heating processes. The component that contains most of the energy of the solution (and is therefore called the dominant component) satisfies a simple nonlinear system devoid of gravity and sound waves. The residual component of the solution satisfies a forced gravity wave equation and essentially does not interact with the dominant component. The mathematical theory also provides information about the amplitude, wavelength, and period of the gravity waves. In the paper entitled “Comments on ‘Use of ducting theory in an observed case of gravity waves,”’ Dr. F. M. Ralph has claimed that the new gravity wave theory is not consistent with profiler observations of vertical velocity in his earlier paper entitled “Observations of a mesoscale ducted gravity wave.” Here it is shown that the new theory is completely consistent with profilers that have documented error bounds on the vertical velocity measurements. In the case that the new theory is claimed to be inconsistent with observational data, the data were obtained from a profiler with undocumented accuracy of the vertical velocity measurements *in the precipitating case,* and the two components of the solution were not properly separated.