Symmetry methods in dynamic meteorologyfaculty.uncfsu.edu/nbila/WebPage FSU/data/ACA08... · 2008-09-08 · Introduction Meteorology is one of those disciplines, that proﬁted most

Symmetry methods in dynamicmeteorology

Applications of Computer Algebra 2008

Alexander [email protected]

Department of Meteorology and Geophysics

University of Vienna

Althanstraße 14, A-1090 Vienna

Symmetry methods in dynamic meteorology – p. 1/12

Introduction

Meteorology is one of those disciplines, that profited most from the development of capable

high-performance computers.


Introduction



This is since practical weather prediction is definitely notpossible without the aid of computers.

The degrees of freedom of the atmosphere make it impossible to do a weather prediction by hand.


Introduction





The progression of computers is accompanied by a natural focus on how to exhaust their capacities

(i.e. efficient numerical methods, numerical modelling, parametrisation,. . . ).


Introduction







However, despite the enourmous success of numerical weather prediction there is still a great need

for additional theoretical considerations. There are a number of processes in the atmosphere-ocean

system that are not well-understood (e.g. precipitation processes, gravity-wave breaking in the

stratosphere, coupling and feedback mechanisms of the atmosphere and ocean etc.).


Introduction







However, despite the enourmous success of numerical weather prediction there is still a great need

for additional theoretical considerations. There are a number of processes in the atmosphere-ocean

system that are not well-understood (e.g. precipitation processes, gravity-wave breaking in the

stratosphere, coupling and feedback mechanisms of the atmosphere and ocean etc.).

More seriously, for long-term prediction (climate change,etc.) a sound understanding of the

numerical models is needed to check the reliability of results, but up to now this understanding of

model dynamics is yet very incomplete.


Beyond numerics...

At least two strategies may be applied to supplement numerical studies:


Beyond numerics...


Exact solutions as benchmark tests for forecast models.


Beyond numerics...



Simplified models of the atmosphere to study certain selected phenomena.


Beyond numerics...




Question: How to obtain exact solutions of the nonlinear PDEs?


Beyond numerics...





Question: How to derive consistent approximate models?


Beyond numerics...






⇒ Both problems can be attacked with symmetries of the dynamicequations.


Beyond numerics...






⇒ Both problems can be attacked with symmetries of the dynamicequations.

This should be exemplified with the inviscid barotropic vorticity equation.


The inviscid barotropic vorticity equation I

Consider the two-dimensional incompressible Euler equations on the rotating earth given in

(x, y, p)-coordinates:∂v

∂t+ v · ∇v + fk × v + ∇φ = 0

wherev = (u, v)T is the horizontal velocity field,f = 2Ω sinϕ is the vertical component of the

earth rotation vector,k = (0, 0, 1)T andφ = gz is the mass-specific potential energy.∇ denotes

the horizontal Del-operator on constant pressure surfaces.





∂t+ v · ∇v + fk × v + ∇φ = 0




Incompressibility allows the introduction of a stream function, i.e.v = k ×∇ψ.





∂t+ v · ∇v + fk × v + ∇φ = 0




Incompressibility allows the introduction of a stream function, i.e.v = k ×∇ψ.

Taking the vertical component of the curl,k · ∇×, of the above equation leads to the barotropic

vorticity equation

∂ζ

∂t+ v · ∇(ζ + f) = 0

whereζ = k · ∇ × v = ∇2ψ. Consequently, the vorticity equation is a nonlinear partial

differential equation describing the evolution of the stream functionψ!


Digression: The vorticity equation as a forecast model

The barotropic vorticity equation enabled the first successful numerical weather prediction

(Charney, Fjørtoft, von Neumann, 1950).

FORECAST HEIGHT FIELD

2 4 6 8 10 12 14 16 18

2

4

6

8

10

12

14

16FINAL ANALYSIS GEOPOTENTIAL HEIGHT

2 4 6 8 10 12 14 16 18

2

4

6

8

10

12

14

16

One day forecast of the 500 hPa height field with the vorticityequation (left) and corresponding

analysis (right).


The inviscid barotropic vorticity equation II

The classical solution of the vorticity equation are Rossbywaves. These are waves with phasevelocity

c = −β

k2 + l2

whereβ = df/dy is the meridional change of the Coriolis parameter andk, l are zonal and

meridional wave-numbers, respectively.




c = −β

k2 + l2



In meteorology, they are obtained via linearisation of the vorticity equation. However, Rossby

waves are also a solution of the nonlinear vorticity equation. In particular, they can be found as a

group-invariant solution.




c = −β

k2 + l2






The generators of symmetry transformation of the vorticityequation read (computed with MuLie):

vt =∂

∂t, vy =

∂

∂y, X (f) = f(t)

∂

∂x− yf

′(t)

∂

∂ψ

Z(g) = g(t)∂

∂ψ, D = t

∂

∂t− x

∂

∂x− y

∂

∂y− 3ψ

∂

∂ψ.




c = −β

k2 + l2







vt =∂

∂t, vy =

∂

∂y, X (f) = f(t)

∂

∂x− yf

′(t)

∂

∂ψ

Z(g) = g(t)∂

∂ψ, D = t

∂

∂t− x

∂

∂x− y

∂

∂y− 3ψ

∂

∂ψ.

Using the linear combinationvy + X (f) and solving the reduced PDE yields a solution that

includes Rossby wavesas a special case.




c = −β

k2 + l2







vt =∂

∂t, vy =

∂

∂y, X (f) = f(t)

∂

∂x− yf

′(t)

∂

∂ψ

Z(g) = g(t)∂

∂ψ, D = t

∂

∂t− x

∂

∂x− y

∂

∂y− 3ψ

∂

∂ψ.

Using the linear combinationvy + X (f) and solving the reduced PDE yields a solution that

includes Rossby wavesas a special case.

That is, symmetries allow to construct a much wider and general class of solutions of the vorticity

equation than those meteorologists are aware of!


Finite-mode approximation of the vorticity equation:

The Lorenz (1960) model I

A common way to try to get insight into the dynamics of the atmosphere is to derive finite-mode

models of the governing PDEs.






Often this is done by an expansion of the field variables in Fourier series (or spherical harmonics),

with a suitable truncation to yield a closed system of ODEs for the Fourier components.






Often this is done by an expansion of the field variables in Fourier series (or spherical harmonics),

with a suitable truncation to yield a closed system of ODEs for the Fourier components.

Expansion of the vorticity in a double Fourier series on the torus and substitution in the vorticityequation in a non-rotating reference frame(f = 0) yields

dcm

dt= −

∑

h

h1m2 − h2m1

h · hchcm−h

with cm being the Fourier coefficients,m = im1k + jm2l is the wave-number vector and

x = ix+ jy the horizontal position vector. This system of equations may be considered as

spectral form of the vorticity equation.



The Lorenz (1960) model II

In 1960 Edward Lorenz (1917 – 2008) sought for the maximum truncation of the of the barotropic

vorticity equation in a non-rotating reference frame.






In doing so, he first restricted the number of Fourier coefficients to only assume values of indices

−1, 0, 1. Neglectingc00 leads to a coupled system of 8 ODEs.








This system is further simplified by Lorenz due to the following two observations:









If the imaginary parts of the cm’s vanish initially they will remain zero.










If Re(c1,−1) = −Re(c11) initially, it will hold for all times.











With these observations, Lorenz obtains the minimal system

dA

dt= −

(

1

k2−

1

k2 + l2

)

klFG.

dF

dt=

(

1

l2−

1

k2 + l2

)

klAG

dG

dt= −

1

2

(

1

l2−

1

k2

)

klAF.

whereA = Re(c01), F = Re(c10), G = Re(c1,−1)











With these observations, Lorenz obtains the minimal system

dA

dt= −

(

1

k2−

1

k2 + l2

)

klFG.

dF

dt=

(

1

l2−

1

k2 + l2

)

klAG

dG

dt= −

1

2

(

1

l2−

1

k2

)

klAF.

whereA = Re(c01), F = Re(c10), G = Re(c1,−1)

Question: Is there a sound justification of these observations?


Symmetry justification of the Lorenz 1960 model I

In addition to the above mentioned Lie point symmetries, thevorticity equation posesses at least 8discrete symmetries, generated by

e1 : (x, y, t, ψ) 7→ (x,−y, t,−ψ), e2 : (x, y, t, ψ) 7→ (−x, y, t,−ψ), e3 : (x, y, t, ψ) 7→ (x, y,−t,−ψ).





Each of these symmetries induces a corresponding mapping inspectral space. Fore1 we have

ζ =∑

m

cm exp(im · x) = −

∑

m

cm exp(i(m1kx−m2ly)) = −

∑

m

cm1,−m2exp(i(m1kx +m2ly))






ζ =∑

m


∑

m


∑

m


Similar computation lead to the mappings

e1 : (x, y, t, ψ) 7→ (x,−y, t,−ψ) cm1m27→ −cm1,−m2

e2 : (x, y, t, ψ) 7→ (−x, y, t,−ψ) cm1m27→ −c−m1m2

e3 : (x, y, t, ψ) 7→ (x, y,−t,−ψ) cm1m27→ −cm1m2

, t 7→ −t.






ζ =∑

m


∑

m


∑

m


Similar computation lead to the mappings

e1 : (x, y, t, ψ) 7→ (x,−y, t,−ψ) cm1m27→ −cm1,−m2

e2 : (x, y, t, ψ) 7→ (−x, y, t,−ψ) cm1m27→ −c−m1m2

e3 : (x, y, t, ψ) 7→ (x, y,−t,−ψ) cm1m27→ −cm1m2

, t 7→ −t.

Usinge1e2 leads to the identificationcm = c−m and hence the imaginary parts of the Fourier

coefficients vanish. This justifies the first observation by Lorenz!


Symmetry justification of the Lorenz 1960 model II

To justify the second observation it is necessary to induce the space translationsx 7→ x+ ε,

y 7→ y + ε in spectral space.





For the mappingx 7→ x+ ε this is done by

ζ =∑

m

cm exp(im·x) =∑

m

cm exp(i(m1k(x+ε)+m2ly)) =∑

m

cm exp(ikm1ε) exp(i(m1kx+m2ly))






ζ =∑

m

cm exp(im·x) =∑

m


m


Hence we find the following mappings:

pε : cm1m27→ e

im1kεcm1m2

, qε : cm1m27→ e

im2lεcm1m2

.






ζ =∑

m

cm exp(im·x) =∑

m


m



pε : cm1m27→ e

im1kεcm1m2

, qε : cm1m27→ e

im2lεcm1m2

.

Forpπ/k, qπ/l we find the discrete transformations:

pπ/k : cm1m27→ (−1)

m1cm1m2, qπ/l : cm1m2

7→ (−1)m2cm1m2

.






ζ =∑

m

cm exp(im·x) =∑

m


m



pε : cm1m27→ e

im1kεcm1m2

, qε : cm1m27→ e

im2lεcm1m2

.


pπ/k : cm1m27→ (−1)


7→ (−1)m2cm1m2

.

Using these transformations, the second observation by Lorenz,Re(c1,−1) = −Re(c11), can be

justified upon using the transformationpπ/kqπ/le1.






ζ =∑

m

cm exp(im·x) =∑

m


m



pε : cm1m27→ e

im1kεcm1m2

, qε : cm1m27→ e

im2lεcm1m2

.


pπ/k : cm1m27→ (−1)


7→ (−1)m2cm1m2

.

Using these transformations, the second observation by Lorenz,Re(c1,−1) = −Re(c11), can be

justified upon using the transformationpπ/kqπ/le1.

Result: The Lorenz (1960) model can be derived in a rigorous way using induced symmetries of

the vorticity equation.


Outlook and questions to the audience

Outlook:



Outlook:

There is a great number of models in meteorology that where derived in an ad-hoc manner (e.g.

other finite-mode models as the famous Lorenz (1963) model).



Outlook:



Also, there are some more advanced models than the vorticityequation which are yet mainly

investigated numerically.



Outlook:





⇒ In both cases, it would be interesting to see whether symmetry analysis cangenerally help

to improve our understanding of both the models and the underlying physical processes.



Outlook:







Questions to the audience:



Outlook:








Which equations of hydrodynamics (meteorology) have already been considered in course of a

symmetry analysis?



Outlook:








Which equations of hydrodynamics (meteorology) have already been considered in course of a

symmetry analysis?

Are there any other reliable computer algebra packages for the computation of Lie symmetries,

but newer than MuLie?


References

W. Zdunkowski and A. Bott.Dynamics of the Atmosphere: A Course in Theoretical Meteorology.

Cambridge University Press, 738pp. 2003


References



J. R. Holton.An Introduction to Dynamic Meteorology. Acad. Press, 535 pp. 2004


References




E. N. Lorenz. Maximum Simplification of the Dynamic Equations. Tellus, 12, 243–254. 1960


References





A. Bihlo. Solving the vorticity equation with Lie groups. Wiener Meteorologische Schriften,6,

Facultas.wuv, 88pp. 2007


References





A. Bihlo. Solving the vorticity equation with Lie groups. Wiener Meteorologische Schriften,6,

Facultas.wuv, 88pp. 2007

A. Bihlo and R.O. Popovych. Symmetry justification of Lorenz’ maximum simplification.

arXiv:0805.4061v1. 2008


Documents

Symmetry methods in dynamic meteorologyfaculty.uncfsu.edu/nbila/WebPage FSU/data/ACA08... · 2008-09-08 · Introduction Meteorology is one of those disciplines, that proﬁted most