Abstract

The vorticity method is applied to determine horizontal divergence using the dynamical balance of terms in the vorticity equation. The viability of the method is analyzed in terms of dynamical approximations, sensitivity to observation and truncation errors, and numerical experiments. This analysis is also applied to the kinematic method, which calculates the horizontal divergence by adding together the, appropriate finite-difference approximations of its individual terms. The analysis of errors in the vorticity and kinematic methods is based on the accuracy of the data. It is proven analytically that errors in the divergence derived from the vorticity method are smaller than those of the kinematic method by a factor equal to the Rossby number, even though the former method involves higher-order derivatives. When a 10% random error is included, the error of the large-scale divergence in the kinematic method exceeds 100%, whereas the error derived by the vorticity method is less than 30% and is comparable to the error in the horizontal wind as expected from the error analysis. An essential result is that the temporal variation of the vorticity is not adequately resolved by the 12-h rawinsonde observing systems and must instead be derived from high temporal resolution wind data such as those measured by the Wind Profiler Demonstration Network. Due to the unavailability of the profiler data in the planetary boundary layer, the vorticity method is primarily applicable to the free atmosphere.

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 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 semi-implicit time-stepping scheme is often applied to the terms responsible for fast waves in large-scale global weather prediction and general circulation models to remove the time step restrictions associated with these waves. Both the phase and amplitude of fast gravity waves are distorted in such models. Because gravity waves carry very little energy, this distortion does not significantly impact the large-scale flow. At mesoscale resolutions the semi-implicit scheme can also be applied, but it has been generally assumed that the treatment of gravity waves is inaccurate at these scales as well. In this paper mesoscale convective systems in the midlatitudes driven by diabatic heating are studied. According to a recently developed mathematical theory, only gravity waves with wavelengths larger than the characteristic length scale of the heat source contain a significant amount of energy, and here it is shown that these gravity waves are accurately reproduced by a semi-implicit discretization of the 3D compressible governing equations with a time step appropriate for the dominant solution component. It will also be shown that the structure equation reduces to the gravity wave equation of the mathematical theory when the appropriate scaling arguments are applied.

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

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

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

A time-dependent model that simulates the interaction of a thunderstorm with its electrical environment is introduced. The model solves the continuity equation of the Maxwell current density that includes conduction, displacement, and source currents. Lightning phenomena are neglected and the electric field is assumed to be curl free. Corona, convection, and precipitation currents are not considered in this initial study and their contribution to the source function is not specified explicitly. As a preliminary test of the model we assume that the storm is axially symmetric in spherical geometry, the conductivity depends only on the vertical coordinate, the ground is equipotential, and far from the thunderstorm region the horizontal electric field is zero. These assumptions are for computational efficiency only and can be relaxed in more realistic studies.

The mathematical energy method is applied to the continuity equation to determine boundary conditions that are sufficient to form a well-posed initial-boundary value problem. This ensures the existence of a physical solution that depends continuously on the initial and boundary data. Then analytic techniques are applied to study the dependence of the solution on the properties of the medium. There are two time scales of the problem that are analyzed and discussed: one determined by the background electrical conductivity and the other by the time dependence of the source function. The assumed source function, which represents a mechanism by which charge is separated inside the storm, contributes to a portion of the solution in which the ratio of the displacement current over the conduction current increases with decreasing altitude, i.e., in the lower atmospheric region the displacement current can have an important role in the electrical interaction between the storm and its environment. It is also demonstrated that the source function can induce temporal phase shifts in the solution, which are dependent on altitude.

To obtain details of the solution, which cannot be obtained by analytic techniques. a stable numerical approximation of the continuity equation is introduced and analyzed. The resulting numerical model is used to examine the evolution of the displacement and conduction currents during the charge buildup phase of a developing thunderstorm.

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.