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Governing Equations II: classical approximations and other systems of equations. By Sylvie Malardel (room 10a; ext. 2414) after Nils Wedi (room 007; ext. 2657). Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz. The continuous set of dry adiabatic equations.
By Sylvie Malardel (room 10a; ext. 2414)
after Nils Wedi (room 007; ext. 2657)
Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz
The scale of the grid is much bigger than the scale of the continuum
resolved scale (?)
Typical observed values for mid-latitude synoptic systems:
U ~ 10 ms-1
W ~ 10-2 ms-1
L ~ 106 m
~ 103 m2s-2
f0 ~ 10-4 s-1
a ~ 107 m
H ~ 104 m
Assume a shallow atmosphere with radius r = a + z ~ a
Allow to drop Coriolis and metric terms which depend on w
Make the hydrostatic approximation
Quasi-Geostrophic balance : accelerations du/dt, dv/dt are
“small” differences between two large terms
If you want to resolve smaller scales:, ex.
A small scale circulation may be far from the geostrophic balance.
The wind may be very ageostrophic.
If you want to cover other latitudes, ex:
Near the equator, synoptic scale motion may be strongly ageostrophic
The pressure gives “the weight” of the atmosphere above
This is true for a very wide range of meteorological scales
For synoptic, at mid-latitude
The vertical acceleration is still very negligible compared with the residual force terms when the hydrostatic balance has been removed
w is obtained diagnostically from the continuity equation,
in agreement with an instantaneous mass reorganisation to fulfil the hydrostatic balance
Filter of isotropic acoustic waves : acoustic pressure perturbations are not related to the “weight” of the atmospheric column, then they are not described anymore.
Use p as a vertical coordinate or any other pressure type coordinate (terrain following : sigma, hybrid)
Hypsometric equation : the thickness between 2 isobars is proportional to the mean temperature in the layer between these 2 isobars
The geopotential of a layer is obtained thanks to the integration of the hydrostatic equation
For internal gravity wave :
Toward the smaller scales :
Hydrostatic approximation if :
Horizontal divergence for a flow past a 3D - mountain
on the sphere ( r = a/100 ) with a T159L91 IFS simulation
Hydrostatic waves only
Hydrostatic + non-hydrostatic waves
The air is still compressible in the sense that its density may change, for exemple in a vertical motion, but it will change passively, without “elastic” reaction or oscillations.
Sub-grid model :
Note: virtual temperature Tv instead of T from the equation of state.
Vertical integration of the continuity equation in hybrid coordinates
Perfect gas equation
Wind components in the model are then usually not the real zonal and meridional winds.
Limited-area gridpoint models: HIRLAM, Ireland, UK Met. Office….; Côté et al. MWR (1993)
Global spectral models – Arpege/IFS: Courtier and Geleyn, QJRMS Part B (1988)
Deriving an alternative form:
In spherical geometry:
Isopycnal/isentropic coordinates : representation of a stratified fluid as a superposition of “shallow water” models (model levels = material surface)
defines depth between “shallow water layers”
Konor and Arakawa (1997);