Non-hydrostatic Terms Research Articles

Improved Precipitation Forecasts Using Parameterized Precipitation Drag in a Hydrostatic Forecast Model

Abstract An empirical Rayleigh drag parameterization of the nonhydrostatic mechanisms of precipitation drag and small-scale diabatically induced mixing is introduced into a hydrostatic regional forecast model to curb excessive grid-scale precipitation production. To get the needed damping, the coefficient in the Rayleigh drag is set proportional to the predicted liquid water, similar in form to the precipitation drag exhibited in nonhydrostatic calculations. The Rayleigh drag parameterization is found to be greatly superior to using the full nonhydrostatic precipitation drag term that produces a damping that is much too large in magnitude for use in a hydrostatic model. Even a rescaling of this nonhydrostatic term is found to be unsatisfactory since it lacks the sensitivity needed for hydrostatic calculations, resulting in an inability to control excessive precipitation. To incorporate nonhydrostatic vertical acceleration effects into a hydrostatic model, a modified version of the quasi-hydrostatic approx...

On the use of the Boussinesq equations, the reduced system, and the primitive equations for the computation of geophysical flows

Motivated in part by the mathematical problems associated with the application of open boundary conditions to the hydrostatic primitive equations (PE), Browning et al. (1990, Dyn. Atmos. Oceans , 14: 303–332) proposed the use of the reduced system (RS) of equations to replace PE for oceanographic problems. The RS are essentially the Boussinesq equations (BO) with the non-hydrostatic terms in the vertical momentum equation multiplied by a constant δ 2 ⪢ 1. This artificially alters the physics (e.g. changing the intemal-inertial wave properties) to facilitate numerical integration, but the changes are assumed to have negligible effects on the dynamics of interest. We assess the accuracy and utility of the RS (following the guidelines for the choice of δ) by comparing numerical finite difference solutions of RS, PE and BO for initial-value problems involving three-dimensional instability of an ocean front and atmospheric frontal development in a two-dimensional Eady wave. Both explicit (BO, PE) and semi-implicit (BOSI, PESI) time-difference schemes are used for the Boussinesq and primitive equations. For RS, the same explicit scheme as for BO is used where δ ⪢ 1 allows larger time steps than with the other explicit models. It is found that relative to BO solutions, the errors for RS are small but increase rapidly and monotonically with increasing δ (over a range consistent with the guidelines) and are greater than the errors for the other models. The use of BOSI allows time steps at least as large as those for RS and results in smaller errors than RS. For these problems, BOSI is the preferable model to replace PE.

Numerical Simulations Initialized with Radar-Derived Winds. Part I: Simulated Data Experiments

Abstract Techniques to initialize the boundary layer in a numerical model with radar-derived wind fields are tested with simulated data of a gust front. Experiments show that Newtonian relaxation or “nudging” applied to the velocity fields is not capable of retrieving the buoyancy in the gust front. The traditional thermodynamic retrieval method is then applied to the simulated velocity data. Tests are performed to determine the sensitivity to the tendency terms, time smoothing, length of the data assimilation window, random error in the velocity fields, sloping terrain, and the nonhydrostatic terms. Two methods to initialize the flow in the data-void region above the boundary layer are also examined. The first solves the linear gravity wave equation, given the forcing at the top of the boundary layer. The second assumes irrotational flow and a divergence profile that exponentially damps the motion in the vertical. Both methods result in less than 10% arms error after 40 min compared with over 40% error i...

The effects of the solid inner core and nonhydrostatic structure on the Earth's forced nutations and Earth tides

Recent highly accurate nutation results from very long baseline interferometry (VLBI) disagree with the 1980 International Astronomical Union (IAU) nutation model by about 2 milliarcseconds (mas). Most of the discrepancy is at the retrograde annual frequency, and it has been interpreted as being due to a nonhydrostatic core‐mantle boundary (CMB) ellipticity [Gwinn et al., 1986]. The ellipticity affects the eigenfrequency of the free core nutation (FCN) normal mode, predominantly a relative rotation about an equatorial axis between the fluid outer core (OC) and the solid mantle. This interpretation in terms of a nonhydrostatic CMB shape is consistent with inferences from diurnal tidal gravity measurements [Levine et al., 1986; Neuberg et al., 1987]. The theory that this interpretation is based on assumes that the inner core (IC)is a fluid. Furthermore, the Earth is presumed to be everywhere hydrostatically prestressed, and so it is inconsistent to use results from this theory to infer information about nonhydrostatic structure. In this paper, we extend the theoretical results to include the effects of a solid IC, and of nonhydrostatic structure. The presence of the IC is responsible for a new, nearly diurnal, prograde normal mode (for a hydrostatically prestressed Earth, the frequency is (1 – 1/471) cycles per day) that involves a relative rotation between the IC and OC about an equatorial axis. We call this newly modeled normal mode the free inner core nutation (FICN). The FICN causes a perturbation in the prograde semiannual nutation amplitude of close to 0.2 mas, but the contribution to the retrograde annual nutation is only 0.003 mas, not enough to explain the VLBI discrepancy. Its effects on the diurnal Earth tide are negligible. Nonhydrostatic structure affects the eigenfrequency of this mode and causes the nutation and Earth tide amplitudes to depend on IC‐OC boundary, CMB, and outer surface topography at all spatial scales. The dependence, though, falls off as the (l ‐ 1)th power of the ratio of the IC radius divided by the OC radius, for the simplified case of a homogeneous, rigid IC and mantle, and homogeneous, incompressible OC (l, here, is the angular order in the spherical harmonic expansion of the boundary shape). For small l, nonhydrostatic effects on the FICN are of the same order as hydrostatic effects for this specialized case. Neither the IC nor the nonhydrostatic terms significantly affect the conclusions about the values of the CMB ellipticity.

A three-dimensional primitive equation model of cumulonimbus convection

The model is described and the results of three integrations presented. Its main novel features are the inclusion of non-hydrostatic terms in a pressure co-ordinate system, and the treatment of flow through the lateral boundaries. The results of the integrations, one with an ambient vertical wind shear and two without, indicate that the cloud dynamics is quite sensitive to assumptions made regarding the cloud microphysics, particularly at the stage when the downdraught is produced.

XXII. Discussion

In his contribution, Professor MacDonald referred briefly to the non-hydrostatic component of the Earth’s flattening, along with other non-hydrostatic terms in the gravity field. Professor Runcorn did not mention this component at all. Both employed maps of the geoid which referred to an ellipsoid with a flattening of 1/298'3—the actual flattening— and which therefore showed no effects of the second harmonic. I should like to discuss this particular harmonic in some detail, because it is the bestestablished and the largest deviation from fluid equilibrium. Its most conspicuous effect is shown in figure 1. Here the solid line is the trace of a great circle on a Mercator map.