Why is the atmosphere so predictable?

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Why is the atmosphere so predictable?

M.J.P.Cullen

22 November 2007


Introduction


Large scale flow

Aim is to predict the atmosphere as depicted on a weather chart as seen in the media, not to predict all the detail. We see ‘weather systems’, with fronts and jet streams.

Evolution is unsteady and non-periodic.

Qualitatively, the weather map looks much the same from day to day (at a given season).

Weather systems are qualitatively different in the tropics.


Typical satellite picture


Corresponding weather map


Governing equations

On all relevant scales, the atmosphere is governed by the compressible Navier-Stokes equations, the laws of thermodynamics, phase changes and source terms.

The solutions of these equations are very complicated, reflecting the complex nature of observed flows.

The accurate solution of these equations would require computers 1030 times faster than now available.


Weather forecasting

The fact that numerical weather prediction is possible, and quite successful, implies that the evolution is largely controlled by large-scale dynamics. Thus the ‘butterfly’ effect must be limited.

Identify this large-scale control by choosing system of equations which only describes large scales. Prove that this system can be solved independently-so that no knowledge of small scales is required. Estimate the difference between this solution and that of the real problem.


Large-scale balances


Large-scale flows-hydrostatic balance

Because the atmosphere is thin, flows on large horizontal scales have a small aspect ratio.

The rest state of the atmosphere is described by a balance of forces between gravity and a radial basic state pressure gradient

The balance of forces determining atmospheric motion involves the perturbation to this balance due to horizontal density variations:

where ρ´,p´ are the density and pressure perturbations.

0

g

r

p


Large-scale flows-geostrophic balance

Large-scale flow is also dominated by the Earth’s rotation.

The key balance of forces is expressed as

where Ω is the Earth’s angular velocity vector and p is the pressure.

The flow is also within the ‘thin’ atmospheric shell and so is nearly horizontal.

These requirements are contradictory for northward flow at the equator.

u 2p


Illustration

Shows the direction of the ‘geostrophic’ wind, and its projection in a horizontal direction.

Ω


Lagrangian and Eulerian dynamics

Descriptive theory of large-scale meteorology has usually been in the Lagrangian frame.

Thus we speak of ‘air masses’ with long-lived characteristics (temperature and moisture).

Illustration follows of Lagrangian and Eulerian phenomena (forced by hills) coexisting in the atmosphere.

Consider stability of geostrophic and hydrostatic states to Lagrangian displacements of the fluid-ignoring consequential pressure perturbations.


Lagrangian and Eulerian dynamics

Animation removed


Static stability

A state of rest in hydrostatic balance is stable if the density decreases with height.

In the compressible atmosphere, this is expressed by the ‘potential temperature’ increasing with height.

In terms of potential temperature perturbations, hydrostatic balance becomes

where Π´ is a function of p´ and θ´ is the potential temperature perturbation.

0

g

rC p


Static stability II

Stability requires θ´ to increase with r, so that ∂Π´/∂r increases with r, so that Π´ is a convex function of r.

If this condition is satisfied, a displaced parcel with fixed θ´ will feel a restoring pressure force, assuming the pressure is not changed.

If a large-scale flow is statically unstable, it will self-destruct, ceasing to be large-scale.

When moisture is included, spontaneous large-scale violations of static stability can be generated, leading to thunderstorms.


An unstable state


Inertial stability

The stability of a geostrophic state has to be considered in an inertial frame.

The defining equation in the plane normal to the rotation axis is then

The condition that a displaced parcel with fixed momentum will feel a restoring pressure force is that P is a convex function of (x,y).

X=x+v/2Ω, Y=y-u/2Ω have to be increasing functions of (x,y) respectively.

)(24

)2,2(242222

2

yxpP

yuxvP


Inertial stability II

For straight flow, condition is that the absolute geostrophic vorticity multiplied by 2Ω is positive (different condition for axisymmetric flow).

This is a condition on the pressure field.

Allows depressions to be small-scale and intense, but not anticyclones.

Pressure gradients have to decrease towards the equator.


Example of stable state.


Scale separation

The natural time-scale associated with large-scale flow is the frequency about which the fluid would oscillate about a stable equilibrium.

A scale separation can be achieved by assuming that the Lagrangian time-scale of the flow is greater than that associated with any such oscillation.

This requires it to be greater than (2Ω)-1 and also N-1, where N is the buoyancy frequency associated with the static stability condition.

The condition involving Ω is usually more stringent.


Mathematical procedure

Define asymptotic regime of interest by assuming Lagrangian timescale greater than that associated with the Earth’s rotation.

This requires the rate of change of wind direction following a fluid trajectory to be

Ro /2Ω where Ro<<1. Ro is the Rossby number. This corresponds to a 12 hour period at the poles and 24 hour period at 30° latitude.

Ro =0.1 means trajectory changes direction by more than 45° in 24 hours at 60° latitude.

Illustrate with actual example in active spell of weather, most trajectories curve less than this (allow for Mercator projection).


Example of ‘real’ trajectories

Met Office global model back trajectories for 11 January 2005, 4 day period, marked every 12 hr.


Solution procedure


Construction of equations

First identify a stable state in geostrophic and hydrostatic balance as an energy minimising state with respect to Lagrangian displacements conserving mass, momentum and potential temperature.

Prove that such a state exists for a given specification of momentum and potential temperature.

Formulate and solve the evolution equations for such states.


Geostrophic and hydrostatic balance

Write equations for compressible atmosphere in Cartesian coordinates with uniform rotation Ω:

Hydrostatic balance

Geostrophic balance

0

gz

C p

uy

Cvx

C pp

2,2


Energy integral

The energy conserved by the compressible Euler equations in this geometry is

21d

2 vC gz u x


New variables

Define (as in definition of inertial stability)

The energy becomes

The conditions for geostrophic and hydrostatic balance become

/ 2 , / 2 ,X x v Y y u Z

2 21 1d

2 2 vE x X y Y C Z gz x

24 , ,0 0,0, 0pX x y Y C Z g


Energy principle

Follow Cullen and Feldman (2006). Define Lagrangian map F(t,x) as position at time t of particle initially at x and Lagrangian variable Z by

Define virtual displacement by

Can show energy is stationary with respect to these displacements if geostrophic and hydrostatic conditions satisfied

xxx

xxZ

,,0#,

),0(,,

F

ZF

)),(),,(),,(()),(,( xxxxZ tZtYtXtFt


Energy minimisation

Can also show that if the energy is minimised, the static and inertial stability conditions are satisfied.

Existence of a minimiser can be proved by mass transportation methods: Cullen and Maroofi (2003) following Brenier (1991) and Cullen and Gangbo (2001).


Evolution equations

The evolution of the minimum energy states can be described by the semi-geostrophic equations. In physical space these are

1

0

,,0#,

0

,2,,0

0,2,,

p

p

Rp

ttF

D

uvyx

Cgz

C

uvyx

CvuD

t

ggpp

pggt

xxx


Relation to full equations

Standard a priori estimates are that the difference between SG and the compressible Euler equations is of order Ro(aspect ratio)².

Since Ro is defined in a Lagrangian sense, discontinuities in physical space are permitted, such as weather fronts.

Steeply sloping fronts, which are unstable to 3d disturbances, are not well described. Shallow sloping fronts are well-described.


Shallow water example


Example using shallow water equations

Applicability of SG model can be demonstrated using shallow water equations.

h is depth of water with mean value H.

0

2

2

uht

h

uy

hg

Dt

Dv

vx

hg

Dt

Du


Experimental strategy

In shallow water flow there are two small parameters available.

Large rotation, 2Ω, leads to geostrophic balance. Large mean depth, H, leads to non-divergence. Define Ro=U/fL, Fr=U/√gH. L is a length scale.

Consider fixed Ro. Allow Fr to vary by changing mean depth.

Shallow water solutions converge to 2d Euler at rate Fr².

Shallow water solutions converge to SG at rate Ro(Ro/Fr)² for Ro<Fr.


Demonstration

Data with typical max wind speed 15ms-1.

Mean depth chosen to give gravity wave speed 65ms-1 to 360ms-1. (Observed 500hpa evolution best matched using speed 140ms-1.)

Gives Ro~0.1, Fr~0.05-0.3.

Shallow water version of Met Office UM. SG and 2d Euler codes as similar as possible.


Data for test


Differences in depth

UM-SG differences UM-2d Euler differences


Differences in winds

UM-SG differences UM-2d Euler differences


Comments

SG results for geostrophic variables show linear convergence in Ro/Fr, Ro/Fr not small enough to give quadratic convergence.

Corresponds to condition on aspect ratio in 3d flow. Ro/Fr=(H/L)(2Ω/N).

2d Euler shows expected convergence in winds, depth asymptotes to non-zero difference.

SG differences from UM much smaller than 2d Euler differences from UM.


Evolution of SG equations


Lagrangian form of SG equations

Defining Z as before, the Lagrangian form of the semi-geostrophic equations is

000

001

010

,,0#,

,,,2(,,,2),(

2,2,11

J

ttF

tFtuytFtvxxtZ

JZtFtZ

gg

tt

xxx

xx

xx


Transport equation

These equations can be written as an evolution equation for the mass density σ in Σ:

This is a transport equation. The velocity U is BV because of the ‘convexity’ properties. It is therefore well-posed (Ambrosio (2004)).

The trajectory in Σ can then be mapped back to physical space giving weak existence of the Lagrangian form of the equations: Cullen and Feldman (2006).

0

2 , ,0t

y Y X x

U

U


Consequences

This argument proves existence of weak solutions to a Lagrangian form of the equations.

The solutions are thus insensitive to changes on a set of measure zero.

Thus there is no ‘butterfly’ effect, though chaotic evolution is possible if a finite volume of fluid is perturbed.

The trajectories can be traced backwards in time, important for atmospheric composition, transport and pollution issues.


Evolutionary properties of the equations

Semi-geostrophic dynamics strongly constrained by ‘convexity’ property (e.g. inertial stability). Shows by greatly reduced growth of PV gradients in solutions of shallow water model.

Thus in real system: for Ro<<Fr we get stable long-lived disturbances (like semi-geostrophic dynamics)

For Ro>>Fr we get layered two-dimensional vortex dynamics. Vertical scale collapse and upscale horizontal cascade bring Ro/Fr back to O(1)


Growth of PV gradients for Ro>>Fr


Growth of PV gradients for Ro<<Fr


Spherical geometry


Solution in spherical geometry

The clash between the direction of the geostrophic wind and the vertical reduces the symmetry of the problem.

The factor 2Ω becomes f=2Ωsinφ where φ is the latitude.

The problem cannot be formulated as a mass transport problem and transport equation without further approximations.

Illustrate (formal) method of solution without further approximations.

Use Cartesian geometry with variable f for ease of presentation.


Lagrangian variations

Define

Minimise energy with respect to Lagrangian displacement satisfying

)),,(,()),,(,(),( 11 xxxZ tFtuftFtvft gg

xxx

K

xxKxZxZ

,,0#,

000

010

001

,),0(,,

F

FfF


Energy principle

Can show E is stationary with respect to these displacements if

For E to be minimised requires at least θ increasing with z and

pgg Cgfufv ,,

2 20, 0g gv uf f f f

x y


Formal proof

Can show formally that E can be minimised by using an explicit descent algorithm.

The resulting p satisfies a ‘convexity’ property.

Can also show formally that the equations can be solved by timestepping


Summary

Demonstrated that large-scale equations governing atmospheric motions can be solved independently.

Solutions are weak Lagrangian solutions, allowing discontinuities between air masses.

Solutions are insensitive to very small-scale perturbations-no butterflies.

Fluid trajectories can be traced back in time, important for many applications.


Questions and answers

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Why is the atmosphere so predictable?