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Governing Equations I: reference equations at the scale of the continuum

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### Governing Equations I: reference equations at the scale of the continuum

by Sylvie Malardel (room 10a; ext. 2414)

after Nils Wedi (room 007; ext. 2657)

Recommended reading:

An Introduction to Dynamic Meteorology, Holton (1992)

An Introduction to Fluid Dynamics, Batchelor (1967)

Atmosphere-Ocean Dynamics, Gill (1982)

Numerical Methods for Wave Equations in Geophysical Fluid Dynamics, Durran (1999)

Illustrations from “Fondamentaux de Meteorologie”, Malardel (Cepadues Ed., 2005)

Thanks to Clive Temperton, Mike Cullen and Piotr Smolarkiewicz

Overview

- Introduction : from molecules to “continuum”
- Eulerian vs. Lagrangian derivatives
- Perfect gas law
- Continuity equation
- Momentum equation (in a rotating reference frame)
- Thermodynamic equation
- Spherical coordinates
- “Averaged” equations in numerical weather prediction models

(Note : focus on “dry” (no water phase change) equations)

Mean free path

Newton’s second law

From molecules to continuumNavier-Stokes equations

Boltzmann equations

kinematic viscosity

~1.x10-6 m2s-1, water

~1.5x10-5 m2s-1, air

number of particles

Euler equations

individual particles

statistical distribution

continuum

Mean value of the parameter

Molecular

fluctuations

Continuous variations

Mean value at the scale of the continuum

Note: Simplified view !

Perfect gas law

- The pressure force on any surface element containing M
- The temperature is defined as
- The link between the pressure, the temperature and the density of molecules in a perfect gas :

Fundamental physical principles

- Conservation of mass
- Conservation of momentum
- Conservation of energy
- Consider budgets of these quantities for a control volume

(a) Control volume fixed relative to coordinate axes

=> Eulerian viewpoint

(b) Control volume moveswith the fluid and always contains the same particles

=> Lagrangian viewpoint

Eulerian versus Lagrangian

Lagrangian : evolution of a quantity following the particules in their motion

Eulerian : Evolution of a quantity inside a fixed box

Eulerian vs. Lagrangian derivatives

Particle at temperature T at position at time moves to in time .

Temperature change given by Taylor series:

i.e.,

then

Let

local rate of change

is the rate of change

following the motion.

is the rate of change

at a fixed point.

advection

total derivative

Mass conservation

Inflow at left face is . Outflow at right face is

Difference between inflow and outflow is per unit volume.

Similarly for y- and z-directions.

Thus net rate of inflow/outflow per unit volume is

= rate of increase in mass per unit volume

= rate of change of density

=> Continuity equation(Eulerian point of view)

Mass flux

on left face

Mass flux

on right face

Eulerianbudget :

Mass conservation (continued)

- Continuity equation, mathematical transformation

- Continuity equation : Lagrangian point of view

By definition, mass is conserved in a Lagrangian volume:

Momentum equation : frames

Newton’s Second Law in absolute frame of reference:

N.B. use D/Dt to distinguish the total derivative in the absolute frame of reference.

(1)

We want to express this in a reference frame which rotates with the earth

Rotating frame

“fixed” star

Momentum equation : velocities

= position vector relative to earth’s centre

= angular velocity of earth

.

For any vector

Evolution in absolute frame

Evolution in rotating frame

or

Momentum equation : Forces

Forces - pressure gradient, gravitation, and friction

Where = specific volume (= ), = pressure,

= sum of gravitational and centrifugal force,

= molecular friction,

= vertical unit vector

Magnitude of varies by ~0.5% from pole to equator and by ~3% with altitude

(up to 100km).

Spherical polar coordinates (in the same rotating frame)

: = longitude, = latitude, = radial distance

Orthogonal unit vectors: eastwards, northwards, upwards.

As we move around on the earth, the orientation of the coordinate system changes:

Extra terms due to

the spherical curvature

The “shallow atmosphere” approximation

“Shallowness approximation” –

For consistency with the energy conservation and angular momentum conservation, some terms have to be neglected in the full momentum equation. This approximation is then valid only if these terms are negligible.

with

Energy conservation : Thermodynamic equation (total energy conservation – macroscopic kinetic energy equation)

First Law of Thermodynamics:

where I = internal energy,

Q = rate of heat exchange with the surroundings

W = work done by gas on its surroundings by compression/expansion.

For a perfect gas, ( = specific heat at constant volume),

Thermodynamic equation (continued)

Alternative forms:

where

is the potential temperature

The potential temperature is conserved in a Lagrangian “dry” and adiabatic motion

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

Observed spectra of motions in the atmosphere

Spectral slope near k-3 for wavelengths >500km. Near k-5/3 for shorter wavelengths.

Possible difference in larger-scale dynamics.

No spectral gap!

Scales of atmospheric phenomena

Practical averaging scales do not correspond to a physical scale separation.

If equations are averaged, there may be strong interactions between resolved and unresolved scales.

“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

- 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.

Demonstration of the averaging effect

- Use high resolution (10km in horizontal) simulation of flow over Scandinavia
- Average the results to a scale of 80km
- Compare with solution of model with 40km and 80km resolution
- The hope is that, allowing for numerical errors, the solution will be accurate on a scale of 80km
- Compare low-level flows and vertical velocity cross-section, reasonable agreement

Cullen et al. (2000) and references therein

High resolution numerical solution

- Test problem is a flow at 10 ms-1 impinging on Scandinavian orography
- Resolution 10km, 91 levels, level spacing 300m
- No turbulence model or viscosity, free-slip lower boundary
- Semi-Lagrangian, semi-implicit integration scheme with 5 minute timestep

Conclusion

- Averaged high resolution contains more information than lower resolution runs.
- The better ratio of comparison was found approximately as

dx (averaged high resol) ~ ·dx (lower resol) with ~1.5-2

- The idealised integrations suggest that the predictions represent the averaged state well (despite hydrostatic assumption for example).
- The real solution is much more localised and more intense.

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