Symmetries, Conservation Laws and Hamiltonian Structures in Geophysical Fluid Dynamics

Transcription

1 Symmetries, Conservation Laws and Hamiltonian Structures in Geophysical Fluid Dynamics Miguel A. Jiménez-Urias 1 Department of Oceanography University of Washington AMATH 573

2 1 Kinematics Canonical vs Noncanonical 2 Dynamics Overview Barotropic Vorticity Equation Baroclinic QuasiGeostrophic Flow over Topography

3 Canonical vs Noncanonical 1 Kinematics Canonical vs Noncanonical 2 Dynamics Overview Barotropic Vorticity Equation Baroclinic QuasiGeostrophic Flow over Topography

4 Canonical vs Noncanonical Canonical: Lagrangian variables (material points).

5 Canonical vs Noncanonical Positions and momentum specified by x(a, b, τ) y(a, b, τ) u(a, b, τ) v(a, b, τ) (1) With Canonical Poisson bracket [F, G] = [ δf δg δx δu δf δg δu δx + δf δg δy δv δf δv ] δg dadb (2) δy

6 Canonical vs Noncanonical Noncanonical : Eulerian variables (fixed coordinates).

7 Canonical vs Noncanonical Transformation to new (Eulerian) coordinates ω(x, y, t), given by ω(x, y, t) = v x u y = (a, b) (x, y) This yields (Eulerian) Poisson bracket [F, G] = (v, y) (u, x) = + (x, y) (x, y) [ ] (v, y) (u, x) + (a, b) (a, b) dxω(x) ( δf δω ) δg δω (x, y) (3) (4)

8 Overview 1 Kinematics Canonical vs Noncanonical 2 Dynamics Overview Barotropic Vorticity Equation Baroclinic QuasiGeostrophic Flow over Topography

9 Overview The flow is given by u t = J δh δu with u are the dynamical fields, H the Hamiltonian and J the skew-symmetric operator Casimirs invariants are conserved quantities such that (5) J δc δu = 0 (6)

10 Overview Symmetries and Conserved quantities: Noether s Theorem states: The Hamiltonian is invariant under infinitesimal variations generated by a functional F, in the sense that F H = 0, iff F is a constant of motion.

11 Barotropic Vorticity Equation 1 Kinematics Canonical vs Noncanonical 2 Dynamics Overview Barotropic Vorticity Equation Baroclinic QuasiGeostrophic Flow over Topography

12 Barotropic Vorticity Equation Governing (2-D) equation ω t ψ streamfunction. With Hamiltonian H = + (ψ, ω) = 0 (7) D 1 2 ψ 2 dxdy (8)

13 Barotropic Vorticity Equation Look at the variation δh = ψ δ ψdxdy D [ = (ψδ ψ) ψδ 2 ψ ] dxdy D δh = ψδ ψ ˆnds ψδωdxdy (9) i D i D Defining new dynamical variables (namely boundary conditions) δh = ψδγ i ψδωdxdy (10) i D

14 Barotropic Vorticity Equation Thus δh δω = ψ δh = ψ δγ i (11) Di Now u = (ω, γ 1,..., γ N ) T dynamic fields. And (ω, ) J = (12)

15 Barotropic Vorticity Equation Casimirs given by which implies J δc δω = (ω, δc δω ) = 0 (13) δc δω = F (ω) C = F (ω)dxdy (14)

16 Barotropic Vorticity Equation Symmetries and Conservation Laws Define a one-parameter family of infinitesimal variations δ F u induced by F by δ F = ɛj δf δu (15) Neither H and J depend explicitly on x, y or t.

17 Barotropic Vorticity Equation 1 Time: Set δ F ω = ɛ ω t, for which ɛ ω t 2 x-symmetry: Set δ F ω = ɛ ω x, for which = ɛ (ω, δm δω ) δm δω = ψ (16) ɛ ω x = ɛ (ω, δm δω ) δm δω D = y M = yωdxdy (17) 3 y-symmetry: Set δ F ω = ɛ ω y, for which ɛ ω y δm δω = x M = = ɛ (ω, δm δω ) xωdxdy (18)

18 Baroclinic QuasiGeostrophic Flow over Topography 1 Kinematics Canonical vs Noncanonical 2 Dynamics Overview Barotropic Vorticity Equation Baroclinic QuasiGeostrophic Flow over Topography

19 Baroclinic QuasiGeostrophic Flow over Topography Dynamical Equations in terms of Potential Vorticity (q(x, y, z, t)) Dq Dt = q + (ψ, q) = 0 t 0 < z < 1 D Dt (ψ z + fsh) = 0 z = 0 D Dt ψ z = 0 z = 1 q = ψ xx + ψ yy + 1 S ψ zz + f + βy (19)

20 Baroclinic QuasiGeostrophic Flow over Topography In this case, the Hamiltonian is H = 1 ( ψ 2 + S 1 ψ 2 ) z dxdydz (20) 2 From the variation of the Hamiltonian I get δh δq = ψ, δh = ψ δh δλ 0, = ψ z=0 δλ 1 (21) z=1 with λ 0 = S 1 (ψ z + fsh) z=0, λ 1 = S 1 ψ z z=1 (22)

21 Baroclinic QuasiGeostrophic Flow over Topography In this case, the skew-symmetric operator J is given by (q, ) 0 0 J = 0 (λ 0, ) 0 (23) 0 0 (λ 1, ) From this, the Casimirs are of the form C[q, λ 0, λ 1 ] = F (q)dxdydz + D F 0 (λ 0 )dxdy + F 1 (λ 1 )dxdy (24) z=0 z=1

22 Baroclinic QuasiGeostrophic Flow over Topography To find zonal momentum invariant M, must solve the equations q x δm = (q, δq ), λ 0 x = (λ 0, δm δλ 0 ), λ 1 x = (λ 1, δm δλ 1 )(25) simultaneously. The solution (within a Casimir) is M = yqdxdydz + yλ 0 dxdy z=0 D yλ 1 dxdy (26) z=1

23 Particle-Relabeling Symmetry (in Ideal Fluid Mechanics) implies Conservation of (Potential) vorticity. This is a hidden symmetry! IN GFD, the β effect breaks the y-symmetry. For x-dependent topography (h), symmetry is likewise broken. We re able to get an x-momentum symmetry for h zonally symmetric (y and no x dependence), as well as zonally symmetric boundaries. In GFD, often the stability of the system is sought. Thus, perturbation states are defined: Pseudomomentum, pseudoenergy upon which stability theorems are derived.

24 P.J. Morrison. Noncanonical Hamiltonian Density Formulation of Hydrodynamics and Ideal Magnetohydrodynamics Phys. Rev. Lett.. 45: , P.J. Morrison. Poisson Brackets for Fluids and Plasmas. Amer. Inst. Phys. Conf. Proc. 28:14 46, J. Pedlosky. Geophysical Fluid Dynamics. Springer-Verlag, T.G. Shepherd Symmetries, Conservation Laws, and Hamiltonian Structure in Geophysical Fluid Dynamics Adv. Geophysics., R. Salmon. Hamiltonian Fluid Dynamics,

25 Ann. Rev. Fluid. Mech.. 20: , 1988.