Vertical Momentum Equation Research Articles

Evaluation of a Hydrostatic, Height-Coordinate Formulation of the Primitive Equations for Atmospheric Modeling

The hydrostatic form of the primitive equations described by Ooyama is evaluated by comparing nonhydrostatic and hydrostatic integrations of a dry axisymmetric model with a specified entropy (heat) source. In this formulation, pressure is a diagnostic variable, so that the hydrostatic approximation can be included simply by replacing the vertical momentum equation with a diagnostic vertical velocity equation. his diagnostic equation is a one-dimensional (height) second-order elliptic equation that can be solved using a direct method. Results show that hydrostatic solutions are very sensitive to the accuracy of the method used to solve the diagnostic vertical velocity equation. However, this sensitivity can be eliminated by adding an extra term to the diagnostic equation that ensures that the solution does not drift away from hydrostatic balance due to numerical approximation. When the extra term is added, this formulation of the primitive equations allows for the design of a numerical model in height coordinates that can be used in hydrostatic and nonhydrostatic regimes.

The use of a reformulation of the hydrostatic approximation Navier‐Stokes equations for geophysical fluid problems

AbstractIn order to simulate geophysical general circulation processes, to simplify the governing equations of motion, often the vertical momentum equation of the Navier‐Stokes equations is replaced by the hydrostatic approximation equation. The resulting equations are reformulated and a variational formulation of the linearized problem is derived. Iteration schemes are presented to solve this problem. A finite element method is discussed, as well as a finite difference method which is based on a grid that is often used in geophysical general circulation models. The schemes are extended to the non‐linear case. Numerical examples are presented to demonstrate the performance of the derived iteration schemes.

A numerical investigation of the influence of land-sea breeze on the atmospheric effluents released at a coastal site

ABSTRACT. Mesoscale features of a coastal atmospheric boundary layer such as the land-sea circulation and the thermal internal boundary layer (TIBL) structure have been simulated using a two-dimensional numerical boundary layer model. Using Boussinesq approximation for horizontal momentum equations and hydrostatic approximation for vertical momentum equation the model solves the 'shallow water' equations year over a grid domain 80 km length on either side of the coastline and 2 km height. The influence of the land-sea breezes on the dispersion of pollutants released from a continuous point source located at the roast has been studied. The fumigation of pollutants from an offshore source into TIBL over the land has also been illustrated. The limitations associated with the model are also discussed.
 

Local Advection of Momentum, Heat, and Moisture during the Melt of Patchy Snow Covers

Abstract A numerical atmospheric boundary layer model, based on higher-order turbulence closure assumptions, is developed and used to simulate the local advection of momentum, heat, and moisture during the melt of patchy snow covers over a 10-km horizontal domain. The coupled model includes solution of the mass continuity equation, the horizontal and vertical momentum equations, an E−ε turbulence model, an energy equation, and a water vapor conservation equation. Atmospheric buoyancy is accounted for, and a land surface energy balance model is implemented at the lower boundary. Model integrations indicate that advective processes occurring at local scales produce nonlinear horizontal variations in surface fluxes. Under conditions of the numerical experiments, the energy available to melt snow-covered regions has been found to increase by as much as 30% as the area of exposed vegetation increases upwind of the snow cover. The melt increase is found to vary in a largely linear fashion with decreasing snow-covered area for snow-covered areas greater than 25% and in a strongly nonlinear fashion below that value. Decreasing the ratio of patch size to total area, or increasing the patchiness, of the snow cover also leads to nonlinear increases in the energy available to melt the snow. In the limit of a snow cover composed of small patches, melt energy is found to increase linearly as the fractional snow-covered area decreases. In addition, for the purpose of computing grid-average surface fluxes during snowmelt in regional atmospheric models, the results of this study indicate that separate energy balance computations can be performed over the snow-covered and vegetation-covered regions, and the resulting fluxes can be weighted in proportion to the fractional snow cover to allocate the total energy flux partitioning within each surface grid cell.

On the equations of the large-scale ocean

As a preliminary step towards understanding the dynamics of the ocean and the impact of the ocean on the global climate system and weather prediction, the authors study the mathematical formulations and attractors of three systems of equations of the ocean, i.e. the primitive equations (the PEs), the primitive equations with vertical viscosity (the PEV2s), and the Boussinesq equations (the BEs), of the ocean. These equations are fundamental equations of the ocean. The BEs are obtained from the general equations of a compressible fluid under the Boussinesq approximation, i.e. the density differences are neglected in the system except in the buoyancy term and in the equation of state. The PEs are derived from the BEs under the hydrostatic approximation for the vertical momentum equation. The PEV2s are the PEs with the viscosity for the vertical velocity retained. This retention is partially based on the important role played by the viscosity in studying the long time behaviour of the ocean, and the Earth's climate.

Laboratory Measurements of Axial Pressures in Two-Celled Tornado-like Vortices

An experimental study of two-celled vortex flows was conducted in a Ward-type tornado vortex chamber (TVC). Time-averaged, stream-static pressure measurements on the vortex axis and observations of the visualized flow in two-celled vortices are reported. The static pressure measured on the vortex axis increases with height downstream of the vortex breakdown, with the axial pressure gradient tending to zero only well aloft. Visualization indicates that the flow downstream of the breakdown in the TVC is everywhere two-celled, with the strongest axial downflow occurring at middle levels. Analysis of the vertical momentum equation strengthens the argument that the turbulent stresses can play an important role in the axial momentum balance of two-celled vortices by opposing the “filling” of the vortex core with higher stagnation pressure fluid from aloft, therefore helping to maintain low pressure and high velocities near the surface.

Convection zone origins of solar atmospheric heating

view Abstract Citations (3) References (31) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Convection Zone Origins of Solar Atmospheric Heating Schatten, Kenneth H. ; Mayr, Hans G. Abstract Spicules are examined as a means for supplying the corona with mass, energy, and magnetic field. It is suggested that spicules form from the supersonic upward expansion of material on nearly evacuated network flux tubes embedded within the sun's convection zone. This allows supersonic but subescape velocities to be attained by the material as it flows outward through the photosphere. Although supersonic, the kinetic energy (subescape) of the spicule material, as observed, is insufficient for coronal heating. It is suggested that, through buoyancy changes on evacuated flux tubes, the magnetic field first 'wicks' material flow into the solar atmosphere. Subsequently, the magnetic field energizes the gaseous material to form the conventional hot, dynamically expanding, solar corona. This occurs through momentum and energy transport by Alfven waves and associated Maxwell stresses concurrently flowing upward on these 'geysers' (spicules). The vertical momentum equation governing fluid flow is examined, and a particular equipartition solution is presented for the flow velocity along a simple field geometry. Publication: The Astrophysical Journal Pub Date: October 1986 DOI: 10.1086/164655 Bibcode: 1986ApJ...309..864S Keywords: Convection Currents; Coronas; Solar Atmosphere; Solar Heating; Spicules; Atmospheric Circulation; Flow Equations; Magnetohydrodynamic Waves; Solar Energy; Solar Magnetic Field; Solar Physics; CONVECTION; SUN: ATMOSPHERE; SUN: ATMOSPHERIC MOTIONS full text sources ADS |

Strategies for finite element modelling of three dimensional hydrodynamic systems

The simulation of a system composed entirely of three dimensional elements is computationally very expensive and little practical analysis has been undertaken. In an effort to reduce cost the author in a previous paper presented a simplified three dimensional model that used the hydrostatic pressure approximation to eliminate the vertical momentum equation and a transformation of the basic equations to directly solve for the free surface profile. This paper will examine approaches to the reduction of simulation costs within the framework of the original approach and modifications to the basic formulation to improve the performance of the model when the flow bath is over system with variable bed elevations. Examples will be presented to demonstrate the effectiveness of these modifications.

A Non-Archimedean Approach to the Equations of Convection Dynamics

Abstract A new formulation of the momentum equations in convection dynamics is presented, dividing the pressure into separate hydrostatic and dynamic components. In this formulation, termed non-Archimedean, convection is envisaged as developing through the horizontal gradient of a time-dependent hydrostatic pressure, rather than the customary buoyancy. It is shown that in the vertical momentum equation there is only one forcing term, viz., the vertical gradient of the dynamic pressure rather than the customary combination of perturbation-pressure gradient and buoyancy force. In the multi-dimensional case, the new formulation leaves the vorticity form of the momentum equation unchanged from that obtained in the customary approach. A diagnostic equation is presented for computing the dynamic pressure component.

Baroclinic Instabilities of a Deep Fluid

Abstract An analytical study is made of the baroclinic instability problem for a zonal wind U (z), by modifying the Eady model through relaxation of the shallow-layer approximation. The linearized perturbation equation is derived, including the mass-divergence effect in the continuity equation and the non-hydrostatic effect in the vertical momentum equation. When U (z) = constant, the non-hydrostatic effect decreases the effective propagation speeds of wave having short horizontal wavelengths. Assuming that the zonal current velocity profile is linear, two limiting cases are treated (i.e., the geostrophic-type instability limit and the symmetric-type instability limit). For the case of the geostrophic-type instability limit, the mass-divergence effect seems to reduce primarily the wave propagation speeds, while the unstable wave growth rates are decreased by both the mass-divergence and the non-hydrostatic effects. However, for the case of the symmetric-type instability limit, the mass-divergence effect i...

A One-dimensional Cumulus Model Including Pressure Perturbations1

Abstract A model for shallow cumulus convection is formulated in which the vertical momentum equation and horizontal divergence equation are combined to produce a diagnostic equation for the perturbation pressure field. These equations, together with the first law of thermodynamics and equation of continuity of water substance, are averaged horizontally over the cloud area (a cylinder of constant radius). The resulting set can be integrated in time to study the life cycle of a nonprecipitating cumulus initiated by release of a small buoyant element. The results indicate that the perturbation pressure field plays an essential role by reducing the extremely sharp gradients in velocity near the cloud top, which are common to most other one-dimensional models. Inclusion of the pressure field also makes it possible to predict the radial scale at which the maximum cloud growth rate will occur.

Relative importance of Terms in the Turbulent-Energy and Momentum Equations as Applied to the Problem of a Surface Roughness Change

There are four equations which must he satisfied in a description of the flow, in the two-diinensional case, across a change in surface roughness in the neutrally-stable surface boundary layer: the horizontal and vertical momentum equations, the turbulent energy equation, and the continuity equation. Using variations on the Peterson model, the relative magnitudes of the terms in these equations are examined and the wind and shear-stress profiles predicted by each variant model are compared. The following conclusions are made: The horizontal-momentum and turbulent-energy equations are essential in a physically reasonable model of the flow. The vertical-momentum equation is less important since the pressure-gradient term in the horizontal equation of motion is relatively small except near the surface singularity. Neglect of the vertical-motion terms and thus the continuity equation does not alter the major features of the downstream wind and shear-stress profiles.

Evaluation of the momentum equation for a turbulent wall jet

Experimental evaluation of the radial momentum equation near the surface for an axisymmetrie turbulent wall jet is reported. The equation for flow contains six terms of magnitudes which cannot be neglected. The pressure gradient across the flow as well as along the flow is found to be of major importance. Thus, for a wall jet flow it is impossible to treat the radial and vertical momentum equations as independent of one another.