scholarly journals Design of a Nonhydrostatic Atmospheric Model Based on a Generalized Vertical Coordinate

2009 ◽  
Vol 137 (7) ◽  
pp. 2305-2330 ◽  
Author(s):  
Michael D. Toy ◽  
David A. Randall
Keyword(s):  
Atmospheric Model ◽  
Static Stability ◽  
Vertical Coordinate ◽  
Adiabatic Flow ◽  
Pressure Drag ◽  
Model Based ◽  
Ale Methods ◽  
Coordinate Model ◽  
Atmospheric Motion ◽  
Limiting Case

The isentropic system of equations has particular advantages in the numerical modeling of weather and climate. These include the elimination of the vertical velocity in adiabatic flow, which simplifies the motion to a two-dimensional problem and greatly reduces the numerical errors associated with vertical advection. The mechanism for the vertical transfer of horizontal momentum is simply the pressure drag acting on isentropic coordinate surfaces under frictionless, adiabatic conditions. In addition, vertical resolution is enhanced in regions of high static stability, which leads to better resolution of features such as the tropopause. Negative static stability and isentropic overturning frequently occur in finescale atmospheric motion. This presents a challenge to nonhydrostatic modeling with the isentropic vertical coordinate. This paper presents a new nonhydrostatic atmospheric model based on a generalized vertical coordinate. The coordinate is specified in a manner similar to that of Konor and Arakawa, but “arbitrary Eulerian–Lagrangian” (ALE) methods are used to maintain coordinate monotonicity in regions of negative static stability and to return the coordinate surfaces to their isentropic “targets” in statically stable regions. The model is mass conserving and implements a vertical differencing scheme that satisfies two additional integral constraints for the limiting case of z coordinates. The hybrid vertical coordinate model is tested with mountain-wave experiments including a downslope windstorm with breaking gravity waves. The results show that the advantages of the isentropic coordinate are realized in the model with regard to vertical tracer and momentum transport. Also, the isentropic overturning associated with the wave breaking is successfully handled by the coordinate formulation.

Incorporating Condensational Heating into a Nonhydrostatic Atmospheric Model Based on a Hybrid Isentropic–Sigma Vertical Coordinate

2011 ◽  
Vol 139 (9) ◽  
pp. 2940-2954 ◽  
Author(s):  
Michael D. Toy
Keyword(s):  
Mass Flux ◽  
Deep Convection ◽  
Atmospheric Model ◽  
Atmospheric Models ◽  
Vertical Coordinate ◽  
Adiabatic Flow ◽  
Model Based ◽  
Terrain Following ◽  
Environmental Air ◽  
Isentropic Coordinates

Using isentropic coordinates in atmospheric models has the advantage of eliminating the cross-coordinate vertical mass flux for adiabatic flow, and virtually eliminating the associated numerical error in the vertical transport. This is a significant benefit since much of the flow in the atmosphere is approximately adiabatic. Nonadiabatic processes, such as condensational heating, result in a nonzero vertical velocity [Formula: see text] in isentropic coordinates. A method for incorporating condensational heating into a nonhydrostatic atmospheric model based on a hybrid isentropic–sigma vertical coordinate is presented. The model is tested with various 2D moist simulations and the results are compared with those using a traditional terrain-following, height-based sigma coordinate. With the hybrid coordinate, there are improvements in the representation of the developing cloud field in a mountain wave experiment. In a simulation of deep convection, the adaptive hybrid coordinate successfully simulates the turbulent nature of the convection, while maintaining the quasi-Lagrangian nature of the isentropic coordinate in the surrounding dry air. The vertical cross-coordinate mass flux is almost zero in the environmental air, as well as in the stratosphere above the convective tower.

A Supercell Storm Simulation Using a Nonhydrostatic Cloud-Resolving Model Based on a Hybrid Isentropic-Sigma Vertical Coordinate

2013 ◽  
Vol 141 (4) ◽  
pp. 1204-1215 ◽  
Author(s):  
Michael D. Toy
Keyword(s):  
Mass Flux ◽  
Potential Temperature ◽  
Three Dimensional ◽  
Vertical Motion ◽  
Implicit Time ◽  
Vertical Coordinate ◽  
Model Based ◽  
Supercell Storm ◽  
Coordinate Model ◽  
Isentropic Coordinates

Abstract A three-dimensional simulation of a supercell storm is performed with a nonhydrostatic model based on a hybrid isentropic-sigma vertical coordinate. The coordinate is a terrain-following, height-based coordinate near the surface that smoothly transitions to potential temperature with height. Using isentropic coordinates provides the advantage of having zero cross-coordinate vertical mass flux for adiabatic flow, which virtually eliminates the numerical error in the vertical transport. The model uses an adaptive grid algorithm by which the coordinate surfaces may deviate from their target isentropes to maintain a sufficiently smooth mesh, while allowing the turbulence and vertical motion associated with convection to develop. The storm simulated by the hybrid-coordinate model compares well with simulations by Eulerian-coordinate models, but with the key difference being that the cross-coordinate mass flux is significantly smaller in much of the domain with the hybrid-coordinate model. A semi-implicit time-differencing scheme for numerically stabilizing vertically propagating acoustic modes in isentropic coordinates is also presented in the paper.

scholarly journals Staggered Vertical Discretization of the Canadian Environmental Multiscale (GEM) Model Using a Coordinate of the Log-Hydrostatic-Pressure Type

2014 ◽  
Vol 142 (3) ◽  
pp. 1183-1196 ◽  
Author(s):  
Claude Girard ◽  
André Plante ◽  
Michel Desgagné ◽  
Ron McTaggart-Cowan ◽  
Jean Côté ◽  
...  
Keyword(s):  
Hydrostatic Pressure ◽  
Solution Method ◽  
Time Integration ◽  
Atmospheric Model ◽  
Integration Scheme ◽  
Limited Area ◽  
Vertical Coordinate ◽  
Global Environmental ◽  
Time Integration Scheme ◽  
Global Environmental Multiscale

Abstract The Global Environmental Multiscale (GEM) model is the Canadian atmospheric model used for meteorological forecasting at all scales. A limited-area version now also exists. It is a gridpoint model with an implicit semi-Lagrangian iterative space–time integration scheme. In the “horizontal,” the equations are written in spherical coordinates with the traditional shallow atmosphere approximations and are discretized on an Arakawa C grid. In the “vertical,” the equations were originally defined using a hydrostatic-pressure coordinate and discretized on a regular (unstaggered) grid, a configuration found to be particularly susceptible to noise. Among the possible alternatives, the Charney–Phillips grid, with its unique characteristics, and, as the vertical coordinate, log-hydrostatic pressure are adopted. In this paper, an attempt is made to justify these two choices on theoretical grounds. The resulting equations and their vertical discretization are described and the solution method of what is forming the new dynamical core of GEM is presented, focusing on these two aspects.

scholarly journals Equations of Atmospheric Motion in Non-Eulerian Vertical Coordinates: Vector-Invariant Form and Quasi-Hamiltonian Formulation

2014 ◽  
Vol 142 (10) ◽  
pp. 3860-3880 ◽  
Author(s):  
Thomas Dubos ◽  
Marine Tort
Keyword(s):  
Hamiltonian Formulation ◽  
Point Of View ◽  
Momentum Balance ◽  
Hamiltonian Form ◽  
Conservation Of Energy ◽  
Vertical Coordinate ◽  
Vertical Displacements ◽  
Energy Conserving ◽  
Atmospheric Motion ◽  
Hydrostatic Balance

Abstract The curl form of equations of inviscid atmospheric motion in general non-Eulerian coordinates is obtained. Narrowing down to a general vertical coordinate, a quasi-Hamiltonian form is then obtained in a Lagrangian, isentropic, mass-based or z-based vertical coordinate. In non-Lagrangian vertical coordinates, the conservation of energy by the vertical transport terms results from the invariance of energy under the vertical relabeling of fluid parcels. A complete or partial separation between the horizontal and vertical dynamics is achieved, except in the Eulerian case. The horizontal–vertical separation is especially helpful for (quasi-)hydrostatic systems characterized by vanishing vertical momentum. Indeed for such systems vertical momentum balance reduces to a simple statement: total energy is stationary with respect to adiabatic vertical displacements of fluid parcels. From this point of view the purpose of (quasi-)hydrostatic balance is to determine the vertical positions of fluid parcels, for which no evolution equation is readily available. This physically appealing formulation significantly extends previous work. The general formalism is exemplified for the fully compressible Euler equations in a Lagrangian vertical coordinate and a Cartesian (x, z) slice geometry, and the deep-atmosphere quasi-hydrostatic equations in latitude–longitude horizontal coordinates. The latter case, in particular, illuminates how the apparent intricacy of the time-dependent metric terms and of the additional forces can be absorbed into a proper choice of prognostic variables. In both cases it is shown how the quasi-Hamiltonian form leads straightforwardly to the conservation of energy using only integration by parts. Relationships with previous work and implications for stability analysis and the derivation of approximate sets of equations and energy-conserving numerical schemes are discussed.

scholarly journals The Use of Semigeostrophic Theory to Diagnose the Behaviour of an Atmospheric GCM

Fluids ◽  
2018 ◽  
Vol 3 (4) ◽  
pp. 72
Author(s):  
Mike Cullen
Keyword(s):  
Boundary Layer ◽  
Large Scale ◽  
Atmospheric Model ◽  
Static Stability ◽  
Rate Of Change ◽  
Model Response ◽  
Layer Heating ◽  
Ageostrophic Circulation ◽  
The Tropics ◽  
Hydrostatic Balance

A diagnostic method is presented for analysing the large-scale behaviour of the Met Office Unified Model, which is a comprehensive atmospheric model used for weather and climate prediction. Outside the boundary layer, on scales larger than the radius of deformation, semi-geostrophic theory will give an accurate approximation to the model evolution. In particular, the ageostrophic circulation required to maintain geostrophic and hydrostatic balance against prescribed forcing and a rate of change of the geostrophic pressure can be calculated. In the tropics, the balance condition degenerates to the weak temperature gradient approximation. Within the boundary layer, the semi-geotriptic approximation has to be used because friction and rotation are equally important. Assuming the calculated pressure tendency and ageotriptic circulation match the observed model behaviour, the influence of the large-scale state and the nature of the forcing on the model response can be deduced in a straightforward way. The capabilities of the diagnostic are illustrated by comparing predictions of the ageotriptic circulation from the theory and the model. It is then used to show that the effects of latent heat release can be included by modifying the static stability, and to show the effect of an idealised tropical heat source on the subtropical jet. Finally, the response of the ageotriptic flow to boundary layer heating in the tropics is demonstrated. These illustrations show that the model behaviour on large scales conforms with theoretical expectations, so that the results of the diagnostic can be used to aid the development of further improvements to the model, in particular investigating systematic errors and understanding the large-scale atmospheric response to forcing.

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