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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.

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governing equations ii classical approximations and other systems of equations

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

where do we go from here
Where do we go from here ?
  • So far : We derived a set of evolution equations based on 3 basic conservation principles valid at the scale of the continuum : continuity equation, momentum equation and thermodynamic equation.
  • What do we want to (re-)solve in models based on these equations?

The scale of the grid is much bigger than the scale of the continuum

resolved scale (?)

grid scale

averaged equations from the scale of the continuum to the mean grid size scale
“Averaged” equations : from the scale of the continuum to the mean grid size scale
  • The equations as used in an operational NWP model represent the evolution of a space-time average of the true solution.
  • The equations become empirical once averaged, we cannot claim we are solving the fundamental equations.
  • Possibly we do not have to use the full form of the exact equations to represent an averaged flow, e.g. hydrostatic approximation OK for large enough averaging scales in the horizontal.
averaged equations
“Averaged” equations
  • The sub-grid model represents the effect of the unresolved scales on the averaged flow expressed in terms of the input data which represents an averaged state.
  • The mean effects of the subgrid scales has to be parametrised.
  • The average of the exact solution may *not* look like what we expect, e.g. since vertical motions over land may contain averages of very large local values.
  • The averaging scale does not correspond to a subset of observed phenomena, e.g. gravity waves are partly included at TL799, but will not be properly represented.
  • Introduction
  • Scale analysis of momentum equations
  • Geostrophic and hydrostatic relations
  • IFS hydrostatic equations
  • Map projections and alternative spherical coordinates
  • Shallow-water equations
  • Isopycnic/isentropic equations
  • Classical (and educational) approach: Simplify the governing equations BEFORE numerics is introduced (e.g. by scale analysis, hydrostatic approximation, Boussinesq or Anelastic approximation) depending on the context.
  • Solutions to the governing equations have three important propagation speeds: acoustic waves (speed of sound), gravity waves (gravity-wave speed), and advective motion (wind speed), which affect the time-step that can be used in numerical procedures (constraint : cdt<dx). The equations may be filtered through the choice of approximate governing equations BEFOREOR through appropriate numerical treatments AFTER. Ex: Explicit anelastic NH reasearch model vs. Fully compressible SI SL NH model.
  • Remove singularities (i.e. Pole problem) by choosing appropriate set of equations (e.g. in IFS we use the vector form of the equations for semi-Lagrangian advection; for limited-area applications the Pole is often rotated to another location).
  • Add mapping transformations to the equations for convenience of presentation and accuracy in limited-area models.
  • Choose a generalized vertical coordinate for a proper treatment of the boundaries or better treatment of conservative variables or easier interpretation of results etc.
  • Change pronostic variables to make the “numerics” easier or more accurate or more stable.
  • Use simpler sets of equations as a first approach :
    • Shallow water equations are a useful tool to test a new dynamical core, as they represent a single vertical mode but a comprehensive set of horizontal solutions.
    • Adiabatic tests before introducing full “physics”
  • Single column models, 2D vertical plane model, 3D cartesian models…. f-plane, beta-plane models
scale analysis
Scale analysis

Example :

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

scale analysis continued





Scale analysis (continued)

















scale analysis continued12
Scale analysis (continued)
  • Consequences if you want to resolve synoptic motions in the mid-latitudes:

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


scale analysis continued13
Scale analysis (continued)

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

hydrostatic balance
Hydrostatic balance

The pressure gives “the weight” of the atmosphere above

This is true for a very wide range of meteorological scales


Hydrostatic approximation


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

hydrostatic approximation consequences
Hydrostatic approximation : consequences

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.

hydrostatic approximation consequences17
Hydrostatic approximation : consequences

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

validity of hydrostatic approximation
Validity of hydrostatic approximation

For internal gravity wave :

Toward the smaller scales :

Hydrostatic approximation if :

hydrostatic vs non hydrostatic
Hydrostatic vs. Non-hydrostatic

Horizontal divergence for a flow past a 3D - mountain

on the sphere ( r = a/100 ) with a T159L91 IFS simulation



hydrostatic vs non hydrostatic20
Hydrostatic vs. Non-hydrostatic

Hydrostatic waves only

Hydrostatic + non-hydrostatic waves

anelastic approximation
Anelastic approximation
  • What is “anelastic approximation”?
    • Neglect the elasticity of the atmosphere which is responsible for the accoustic wave propagation.
  • How to do that ?
    • Modify the continuity equation in order to neglect the “quick” response of the density to compression

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.

anelastic approximation consequences
Anelastic approximation : consequences
  • Balance between horizontal and vertical mass fluxes
  • The anelastic approximation may be useful if you need to take into account the NH effect and you don’t have very sophisticated numerics to treat the sound waves.
primitive hydrostatic equations in ifs
Primitive (hydrostatic) equations in IFS

Momentum equations


Sub-grid model :




ifs hydrostatic equations
IFS hydrostatic equations

Thermodynamic equation

Moisture equation

Note: virtual temperature Tv instead of T from the equation of state.

ifs hydrostatic equations25
IFS hydrostatic equations

Continuity equation

Vertical integration of the continuity equation in hybrid coordinates

one word about water species
One word about water species….
  • Phase changes are treated inside the “physics” (P terms)
  • But the pronostic water species have a weight. They are included in the full density of the moist air and in the definition of the “specific” variables. It does some “tricky” changes in the equations. For ex. :
  • Pronostic water species should be advected. They are then also treated by the dynamics.

Perfect gas equation

map projections
Map projections
  • Invented to have an angle preserving mapping from the sphere onto a plane for convenience of display.
  • Hence idea to perform computations already in transformed coordinates.

Wind components in the model are then usually not the real zonal and meridional winds.

Map factor:

rotated spherical coordinates
Rotated spherical coordinates
  • Move pole so that area of interest lies on the equator such that system gives more uniform resolution.

Limited-area gridpoint models: HIRLAM, Ireland, UK Met. Office….; Côté et al. MWR (1993)

  • Move pole to area of interest, then “stretch” in the new “north-south” direction to give highest resolution over the area of interest.

Global spectral models – Arpege/IFS: Courtier and Geleyn, QJRMS Part B (1988)

  • Ocean models have sometimes two poles in the continents to give uniform resolution over the ocean of interest.
shallow water equations

Further reading: Gill (1982)

Shallow water equations
  • Useful for (hydrostatic) dynamical core test cases before full implementation.
  • Route to interpret isentropic or isopycnal models.

eg. Williamson et. al., JCP Vol 102, p. 211-224 (1992)

Free surface

shallow water equations31
Shallow water equations
  • Assume constant density + free surface at z=h(x,y)
    • horizontal pressure force independent of height:
    • Horizontal wind independent of height as :
    • Use only the horizontal motion equation at the ground, where w=0
shallow water equations32
Shallow water equations
  • Boundary conditions: w=0 at z=0 and free surface following the motion at the top (dh/dt=w). Integrating the continuity equation we obtain:
shallow water equations33
Shallow water equations
  • In component form in Cartesian geometry:




shallow water equations34
Shallow water equations

Deriving an alternative form:

Kinetic Energy



shallow water equations35
Shallow water equations
  • The advection term in the velocity equation may be transform into the Lamb form; the vector product of vorticity with velocity is called the Lamb vector (useful for generalised form of Bernoulli equation):

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”

more general isentropic sigma equations
More general isentropic-sigma equations

Konor and Arakawa (1997);