Symmetry methods in dynamic meteorology

Transcription

1 Symmetry methods in dynamic meteorology p. 1/12 Symmetry methods in dynamic meteorology Applications of Computer Algebra 2008 Alexander Bihlo Department of Meteorology and Geophysics University of Vienna Althanstraße 14, A-1090 Vienna

2 Symmetry methods in dynamic meteorology p. 2/12 Introduction Meteorology is one of those disciplines, that profited most from the development of capable high-performance computers.

3 Symmetry methods in dynamic meteorology p. 2/12 Introduction Meteorology is one of those disciplines, that profited most from the development of capable high-performance computers. This is since practical weather prediction is definitely not possible without the aid of computers. The degrees of freedom of the atmosphere make it impossible to do a weather prediction by hand.

4 Symmetry methods in dynamic meteorology p. 2/12 Introduction Meteorology is one of those disciplines, that profited most from the development of capable high-performance computers. This is since practical weather prediction is definitely not possible without the aid of computers. The degrees of freedom of the atmosphere make it impossible to do a weather prediction by hand. The progression of computers is accompanied by a natural focus on how to exhaust their capacities (i.e. efficient numerical methods, numerical modelling, parametrisation,... ).

5 Symmetry methods in dynamic meteorology p. 2/12 Introduction Meteorology is one of those disciplines, that profited most from the development of capable high-performance computers. This is since practical weather prediction is definitely not possible without the aid of computers. The degrees of freedom of the atmosphere make it impossible to do a weather prediction by hand. The progression of computers is accompanied by a natural focus on how to exhaust their capacities (i.e. efficient numerical methods, numerical modelling, parametrisation,... ). 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.).

6 Symmetry methods in dynamic meteorology p. 2/12 Introduction Meteorology is one of those disciplines, that profited most from the development of capable high-performance computers. This is since practical weather prediction is definitely not possible without the aid of computers. The degrees of freedom of the atmosphere make it impossible to do a weather prediction by hand. The progression of computers is accompanied by a natural focus on how to exhaust their capacities (i.e. efficient numerical methods, numerical modelling, parametrisation,... ). 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.

7 Symmetry methods in dynamic meteorology p. 3/12 Beyond numerics... At least two strategies may be applied to supplement numerical studies:

8 Symmetry methods in dynamic meteorology p. 3/12 Beyond numerics... At least two strategies may be applied to supplement numerical studies: Exact solutions as benchmark tests for forecast models.

9 Symmetry methods in dynamic meteorology p. 3/12 Beyond numerics... At least two strategies may be applied to supplement numerical studies: Exact solutions as benchmark tests for forecast models. Simplified models of the atmosphere to study certain selected phenomena.

10 Symmetry methods in dynamic meteorology p. 3/12 Beyond numerics... At least two strategies may be applied to supplement numerical studies: Exact solutions as benchmark tests for forecast models. Simplified models of the atmosphere to study certain selected phenomena. Question: How to obtain exact solutions of the nonlinear PDEs?

11 Symmetry methods in dynamic meteorology p. 3/12 Beyond numerics... At least two strategies may be applied to supplement numerical studies: Exact solutions as benchmark tests for forecast models. Simplified models of the atmosphere to study certain selected phenomena. Question: How to obtain exact solutions of the nonlinear PDEs? Question: How to derive consistent approximate models?

12 Symmetry methods in dynamic meteorology p. 3/12 Beyond numerics... At least two strategies may be applied to supplement numerical studies: Exact solutions as benchmark tests for forecast models. Simplified models of the atmosphere to study certain selected phenomena. Question: How to obtain exact solutions of the nonlinear PDEs? Question: How to derive consistent approximate models? Both problems can be attacked with symmetries of the dynamic equations.

13 Symmetry methods in dynamic meteorology p. 3/12 Beyond numerics... At least two strategies may be applied to supplement numerical studies: Exact solutions as benchmark tests for forecast models. Simplified models of the atmosphere to study certain selected phenomena. Question: How to obtain exact solutions of the nonlinear PDEs? Question: How to derive consistent approximate models? Both problems can be attacked with symmetries of the dynamic equations. This should be exemplified with the inviscid barotropic vorticity equation.

14 Symmetry methods in dynamic meteorology p. 4/12 The inviscid barotropic vorticity equation I Consider the two-dimensional incompressible Euler equations on the rotating earth given in (x, y, p)-coordinates: v + v v + fk v + φ = 0 t where v = (u, v) T is the horizontal velocity field, f = 2Ω sin ϕ is the vertical component of the earth rotation vector, k = (0, 0, 1) T and φ = gz is the mass-specific potential energy. denotes the horizontal Del-operator on constant pressure surfaces.

15 Symmetry methods in dynamic meteorology p. 4/12 The inviscid barotropic vorticity equation I Consider the two-dimensional incompressible Euler equations on the rotating earth given in (x, y, p)-coordinates: v + v v + fk v + φ = 0 t where v = (u, v) T is the horizontal velocity field, f = 2Ω sin ϕ is the vertical component of the earth rotation vector, k = (0, 0, 1) T and φ = gz is the mass-specific potential energy. denotes the horizontal Del-operator on constant pressure surfaces. Incompressibility allows the introduction of a stream function, i.e. v = k ψ.

16 Symmetry methods in dynamic meteorology p. 4/12 The inviscid barotropic vorticity equation I Consider the two-dimensional incompressible Euler equations on the rotating earth given in (x, y, p)-coordinates: v + v v + fk v + φ = 0 t where v = (u, v) T is the horizontal velocity field, f = 2Ω sin ϕ is the vertical component of the earth rotation vector, k = (0, 0, 1) T and φ = gz is the mass-specific potential energy. denotes the horizontal Del-operator on constant pressure surfaces. 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 where ζ = k v = 2 ψ. Consequently, the vorticity equation is a nonlinear partial differential equation describing the evolution of the stream function ψ!

17 Symmetry methods in dynamic meteorology p. 5/12 Digression: The vorticity equation as a forecast model The barotropic vorticity equation enabled the first successful numerical weather prediction (Charney, Fjørtoft, von Neumann, 1950). 16 FORECAST HEIGHT FIELD 16 FINAL ANALYSIS GEOPOTENTIAL HEIGHT One day forecast of the 500 hpa height field with the vorticity equation (left) and corresponding analysis (right).

18 Symmetry methods in dynamic meteorology p. 6/12 The inviscid barotropic vorticity equation II The classical solution of the vorticity equation are Rossby waves. These are waves with phase velocity c = k 2 + l 2 where β = df/dy is the meridional change of the Coriolis parameter and k, l are zonal and meridional wave-numbers, respectively. β

19 Symmetry methods in dynamic meteorology p. 6/12 The inviscid barotropic vorticity equation II The classical solution of the vorticity equation are Rossby waves. These are waves with phase velocity c = k 2 + l 2 where β = df/dy is the meridional change of the Coriolis parameter and k, l are zonal and meridional wave-numbers, respectively. 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. β

20 Symmetry methods in dynamic meteorology p. 6/12 The inviscid barotropic vorticity equation II The classical solution of the vorticity equation are Rossby waves. These are waves with phase velocity c = k 2 + l 2 where β = df/dy is the meridional change of the Coriolis parameter and k, l are zonal and meridional wave-numbers, respectively. 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. The generators of symmetry transformation of the vorticity equation read (computed with MuLie): v t = t, Z(g) = g(t) ψ, v y = y, β X(f) = f(t) x yf (t) ψ D = t t x x y y 3ψ ψ.

21 Symmetry methods in dynamic meteorology p. 6/12 The inviscid barotropic vorticity equation II The classical solution of the vorticity equation are Rossby waves. These are waves with phase velocity c = k 2 + l 2 where β = df/dy is the meridional change of the Coriolis parameter and k, l are zonal and meridional wave-numbers, respectively. 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. The generators of symmetry transformation of the vorticity equation read (computed with MuLie): v t = t, Z(g) = g(t) ψ, v y = y, β X(f) = f(t) x yf (t) ψ D = t t x x y y 3ψ ψ. Using the linear combination v y + X(f) and solving the reduced PDE yields a solution that includes Rossby waves as a special case.

22 Symmetry methods in dynamic meteorology p. 6/12 The inviscid barotropic vorticity equation II The classical solution of the vorticity equation are Rossby waves. These are waves with phase velocity c = k 2 + l 2 where β = df/dy is the meridional change of the Coriolis parameter and k, l are zonal and meridional wave-numbers, respectively. 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. The generators of symmetry transformation of the vorticity equation read (computed with MuLie): v t = t, Z(g) = g(t) ψ, v y = y, β X(f) = f(t) x yf (t) ψ D = t t x x y y 3ψ ψ. Using the linear combination v y + X(f) and solving the reduced PDE yields a solution that includes Rossby waves as 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!

23 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 7/12 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.

24 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 7/12 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.

25 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 7/12 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. Expansion of the vorticity in a double Fourier series on the torus and substitution in the vorticity equation in a non-rotating reference frame (f = 0) yields dc m dt = h h 1 m 2 h 2 m 1 c h c m h h h with c m being the Fourier coefficients, m = im 1 k + jm 2 l 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.

26 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 8/12 The Lorenz (1960) model II In 1960 Edward Lorenz ( ) sought for the maximum truncation of the of the barotropic vorticity equation in a non-rotating reference frame.

27 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 8/12 The Lorenz (1960) model II In 1960 Edward Lorenz ( ) 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}. Neglecting c 00 leads to a coupled system of 8 ODEs.

28 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 8/12 The Lorenz (1960) model II In 1960 Edward Lorenz ( ) 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}. Neglecting c 00 leads to a coupled system of 8 ODEs. This system is further simplified by Lorenz due to the following two observations:

29 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 8/12 The Lorenz (1960) model II In 1960 Edward Lorenz ( ) 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}. Neglecting c 00 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 c m s vanish initially they will remain zero.

30 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 8/12 The Lorenz (1960) model II In 1960 Edward Lorenz ( ) 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}. Neglecting c 00 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 c m s vanish initially they will remain zero. If Re(c 1, 1 ) = Re(c 11 ) initially, it will hold for all times.

31 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 8/12 The Lorenz (1960) model II In 1960 Edward Lorenz ( ) 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}. Neglecting c 00 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 c m s vanish initially they will remain zero. If Re(c 1, 1 ) = Re(c 11 ) initially, it will hold for all times. With these observations, Lorenz obtains the minimal system ( ) da 1 dt = k 2 1 k 2 + l 2 klfg. ( ) df 1 dt = l 2 1 k 2 + l 2 klag ( dg 1 dt = 1 2 l 2 1 ) k 2 klaf. where A = Re(c 01 ), F = Re(c 10 ), G = Re(c 1, 1 )

32 Finite-mode approximation of the vorticity equation: Symmetry methods in dynamic meteorology p. 8/12 The Lorenz (1960) model II In 1960 Edward Lorenz ( ) 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}. Neglecting c 00 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 c m s vanish initially they will remain zero. If Re(c 1, 1 ) = Re(c 11 ) initially, it will hold for all times. With these observations, Lorenz obtains the minimal system ( ) da 1 dt = k 2 1 k 2 + l 2 klfg. ( ) df 1 dt = l 2 1 k 2 + l 2 klag ( dg 1 dt = 1 2 l 2 1 ) k 2 klaf. where A = Re(c 01 ), F = Re(c 10 ), G = Re(c 1, 1 ) Question: Is there a sound justification of these observations?

33 Symmetry methods in dynamic meteorology p. 9/12 Symmetry justification of the Lorenz 1960 model I In addition to the above mentioned Lie point symmetries, the vorticity equation posesses at least 8 discrete symmetries, generated by e 1 : (x, y, t, ψ) (x, y, t, ψ), e 2 : (x, y, t, ψ) ( x, y, t, ψ), e 3 : (x, y, t, ψ) (x, y, t, ψ).

34 Symmetry methods in dynamic meteorology p. 9/12 Symmetry justification of the Lorenz 1960 model I In addition to the above mentioned Lie point symmetries, the vorticity equation posesses at least 8 discrete symmetries, generated by e 1 : (x, y, t, ψ) (x, y, t, ψ), e 2 : (x, y, t, ψ) ( x, y, t, ψ), e 3 : (x, y, t, ψ) (x, y, t, ψ). Each of these symmetries induces a corresponding mapping in spectral space. For e 1 we have ζ = m c m exp(im x) = m c m exp(i(m 1 kx m 2 ly)) = m c m1, m 2 exp(i(m 1 kx + m 2 ly))

35 Symmetry methods in dynamic meteorology p. 9/12 Symmetry justification of the Lorenz 1960 model I In addition to the above mentioned Lie point symmetries, the vorticity equation posesses at least 8 discrete symmetries, generated by e 1 : (x, y, t, ψ) (x, y, t, ψ), e 2 : (x, y, t, ψ) ( x, y, t, ψ), e 3 : (x, y, t, ψ) (x, y, t, ψ). Each of these symmetries induces a corresponding mapping in spectral space. For e 1 we have ζ = m c m exp(im x) = m c m exp(i(m 1 kx m 2 ly)) = m c m1, m 2 exp(i(m 1 kx + m 2 ly)) Similar computation lead to the mappings e 1 : (x, y, t, ψ) (x, y, t, ψ) c m1 m 2 c m1, m 2 e 2 : (x, y, t, ψ) ( x, y, t, ψ) c m1 m 2 c m1 m 2 e 3 : (x, y, t, ψ) (x, y, t, ψ) c m1 m 2 c m1 m 2, t t.

36 Symmetry methods in dynamic meteorology p. 9/12 Symmetry justification of the Lorenz 1960 model I In addition to the above mentioned Lie point symmetries, the vorticity equation posesses at least 8 discrete symmetries, generated by e 1 : (x, y, t, ψ) (x, y, t, ψ), e 2 : (x, y, t, ψ) ( x, y, t, ψ), e 3 : (x, y, t, ψ) (x, y, t, ψ). Each of these symmetries induces a corresponding mapping in spectral space. For e 1 we have ζ = m c m exp(im x) = m c m exp(i(m 1 kx m 2 ly)) = m c m1, m 2 exp(i(m 1 kx + m 2 ly)) Similar computation lead to the mappings e 1 : (x, y, t, ψ) (x, y, t, ψ) c m1 m 2 c m1, m 2 e 2 : (x, y, t, ψ) ( x, y, t, ψ) c m1 m 2 c m1 m 2 e 3 : (x, y, t, ψ) (x, y, t, ψ) c m1 m 2 c m1 m 2, t t. Using e 1 e 2 leads to the identification c m = c m and hence the imaginary parts of the Fourier coefficients vanish. This justifies the first observation by Lorenz!

37 Symmetry methods in dynamic meteorology p. 10/12 Symmetry justification of the Lorenz 1960 model II To justify the second observation it is necessary to induce the space translations x x + ε, y y + ε in spectral space.

38 Symmetry methods in dynamic meteorology p. 10/12 Symmetry justification of the Lorenz 1960 model II To justify the second observation it is necessary to induce the space translations x x + ε, y y + ε in spectral space. For the mapping x x + ε this is done by ζ = m c m exp(im x) = m c m exp(i(m 1 k(x+ε)+m 2 ly)) = m c m exp(ikm 1 ε) exp(i(m 1 kx+m 2 ly))

39 Symmetry methods in dynamic meteorology p. 10/12 Symmetry justification of the Lorenz 1960 model II To justify the second observation it is necessary to induce the space translations x x + ε, y y + ε in spectral space. For the mapping x x + ε this is done by ζ = m c m exp(im x) = m c m exp(i(m 1 k(x+ε)+m 2 ly)) = m c m exp(ikm 1 ε) exp(i(m 1 kx+m 2 ly)) Hence we find the following mappings: p ε : c m1 m 2 e im 1 kε c m1 m 2, q ε : c m1 m 2 e im 2 lε c m1 m 2.

40 Symmetry methods in dynamic meteorology p. 10/12 Symmetry justification of the Lorenz 1960 model II To justify the second observation it is necessary to induce the space translations x x + ε, y y + ε in spectral space. For the mapping x x + ε this is done by ζ = m c m exp(im x) = m c m exp(i(m 1 k(x+ε)+m 2 ly)) = m c m exp(ikm 1 ε) exp(i(m 1 kx+m 2 ly)) Hence we find the following mappings: p ε : c m1 m 2 e im 1 kε c m1 m 2, q ε : c m1 m 2 e im 2 lε c m1 m 2. For p π/k, q π/l we find the discrete transformations: p π/k : c m1 m 2 ( 1) m1 c m1 m 2, q π/l : c m1 m 2 ( 1) m2 c m1 m 2.

41 Symmetry methods in dynamic meteorology p. 10/12 Symmetry justification of the Lorenz 1960 model II To justify the second observation it is necessary to induce the space translations x x + ε, y y + ε in spectral space. For the mapping x x + ε this is done by ζ = m c m exp(im x) = m c m exp(i(m 1 k(x+ε)+m 2 ly)) = m c m exp(ikm 1 ε) exp(i(m 1 kx+m 2 ly)) Hence we find the following mappings: p ε : c m1 m 2 e im 1 kε c m1 m 2, q ε : c m1 m 2 e im 2 lε c m1 m 2. For p π/k, q π/l we find the discrete transformations: p π/k : c m1 m 2 ( 1) m1 c m1 m 2, q π/l : c m1 m 2 ( 1) m2 c m1 m 2. Using these transformations, the second observation by Lorenz, Re(c 1, 1 ) = Re(c 11 ), can be justified upon using the transformation p π/k q π/l e 1.

42 Symmetry methods in dynamic meteorology p. 10/12 Symmetry justification of the Lorenz 1960 model II To justify the second observation it is necessary to induce the space translations x x + ε, y y + ε in spectral space. For the mapping x x + ε this is done by ζ = m c m exp(im x) = m c m exp(i(m 1 k(x+ε)+m 2 ly)) = m c m exp(ikm 1 ε) exp(i(m 1 kx+m 2 ly)) Hence we find the following mappings: p ε : c m1 m 2 e im 1 kε c m1 m 2, q ε : c m1 m 2 e im 2 lε c m1 m 2. For p π/k, q π/l we find the discrete transformations: p π/k : c m1 m 2 ( 1) m1 c m1 m 2, q π/l : c m1 m 2 ( 1) m2 c m1 m 2. Using these transformations, the second observation by Lorenz, Re(c 1, 1 ) = Re(c 11 ), can be justified upon using the transformation p π/k q π/l e 1. Result: The Lorenz (1960) model can be derived in a rigorous way using induced symmetries of the vorticity equation.

43 Symmetry methods in dynamic meteorology p. 11/12 Outlook and questions to the audience Outlook:

44 Symmetry methods in dynamic meteorology p. 11/12 Outlook and questions to the audience 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).

45 Symmetry methods in dynamic meteorology p. 11/12 Outlook and questions to the audience 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). Also, there are some more advanced models than the vorticity equation which are yet mainly investigated numerically.

46 Symmetry methods in dynamic meteorology p. 11/12 Outlook and questions to the audience 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). Also, there are some more advanced models than the vorticity equation which are yet mainly investigated numerically. In both cases, it would be interesting to see whether symmetry analysis can generally help to improve our understanding of both the models and the underlying physical processes.

47 Symmetry methods in dynamic meteorology p. 11/12 Outlook and questions to the audience 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). Also, there are some more advanced models than the vorticity equation which are yet mainly investigated numerically. In both cases, it would be interesting to see whether symmetry analysis can generally help to improve our understanding of both the models and the underlying physical processes. Questions to the audience:

48 Symmetry methods in dynamic meteorology p. 11/12 Outlook and questions to the audience 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). Also, there are some more advanced models than the vorticity equation which are yet mainly investigated numerically. In both cases, it would be interesting to see whether symmetry analysis can generally help to improve our understanding of both the models and the underlying physical processes. Questions to the audience: Which equations of hydrodynamics (meteorology) have already been considered in course of a symmetry analysis?

49 Symmetry methods in dynamic meteorology p. 11/12 Outlook and questions to the audience 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). Also, there are some more advanced models than the vorticity equation which are yet mainly investigated numerically. In both cases, it would be interesting to see whether symmetry analysis can generally help to improve our understanding of both the models and the underlying physical processes. Questions to the audience: 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?

50 Symmetry methods in dynamic meteorology p. 12/12 References W. Zdunkowski and A. Bott. Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 738pp. 2003

51 Symmetry methods in dynamic meteorology p. 12/12 References W. Zdunkowski and A. Bott. Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 738pp J. R. Holton. An Introduction to Dynamic Meteorology. Acad. Press, 535 pp. 2004

52 Symmetry methods in dynamic meteorology p. 12/12 References W. Zdunkowski and A. Bott. Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 738pp J. R. Holton. An Introduction to Dynamic Meteorology. Acad. Press, 535 pp E. N. Lorenz. Maximum Simplification of the Dynamic Equations. Tellus, 12,

53 Symmetry methods in dynamic meteorology p. 12/12 References W. Zdunkowski and A. Bott. Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 738pp J. R. Holton. An Introduction to Dynamic Meteorology. Acad. Press, 535 pp E. N. Lorenz. Maximum Simplification of the Dynamic Equations. Tellus, 12, A. Bihlo. Solving the vorticity equation with Lie groups. Wiener Meteorologische Schriften, 6, Facultas.wuv, 88pp. 2007

54 Symmetry methods in dynamic meteorology p. 12/12 References W. Zdunkowski and A. Bott. Dynamics of the Atmosphere: A Course in Theoretical Meteorology. Cambridge University Press, 738pp J. R. Holton. An Introduction to Dynamic Meteorology. Acad. Press, 535 pp E. N. Lorenz. Maximum Simplification of the Dynamic Equations. Tellus, 12, A. Bihlo. Solving the vorticity equation with Lie groups. Wiener Meteorologische Schriften, 6, Facultas.wuv, 88pp A. Bihlo and R.O. Popovych. Symmetry justification of Lorenz maximum simplification. arxiv: v