FDEPS 2012, Lecture 2

Hamiltonian geophysical fluid dynamics
Ted Shepherd
Department of Meteorology
University of Reading• Hamiltonian dynamics is a very beautiful, and very powerful,mathematical formulation of physical systems– All the important models in GFD are Hamiltonian• Since it is a general formulation, it provides a framework for“meta-theories”, providing traceability between different
approximate models of a physical system– e.g. barotropic to quasi-geostrophic to shallow-water tohydrostatic primitive equations to compressible equations– Symmetries and conservation laws are linked by Noether’s
theorem• In their pure formulation, Hamiltonian systems areconservative; but the Hamiltonian formulation provides aframework to understand forced-dissipative systems too– The nonlinear interactions are generally conservative– Example: energy budget (APE and Lorenz energy cycle)– Example: momentum transfer by waves

• Hamilton’s equations for a canonical system:• For a Newtonian potential system, we get Newton’s secondlaw:Conservation of energy follows:(repeated indices summed)
Symplectic formulation:

• The symplectic formulation of Hamiltonian dynamics can begeneralized to other J, which have to satisfy certain
mathematical properties• Among these is skew-symmetry, which guarantees energy
conservation:• The canonical J is invertible. If J is non-invertible, thenCasimirs are defined to satisfy

Casimirs are invariants of the dynamics since• Example of a non-canonical Hamiltonian representation:
Euler’s equations for a rigid body. The dependent variables
are the components of angular momentum about principal
axes, and the total angular momentum is a Casimir invariant.
• Cyclic coordinates: e.g. rotational symmetry implies
conservation of angular momentum∂Hdpi=0⇒=0∂qidt

for a given i• More generally, the link between symmetries andconservation laws is provided by Noether’s theorem:

Then• Butand hence• Casimir invariants are associated with ‘invisible’ symmetries
since• Example: rigid body Barotropic dynamics is a Hamiltonian system(assuming boundary
terms vanish)• Functional derivatives are just the infinite-dimensional analogueof partial derivatives; they can reflect non-local properties• Barotropic dynamics can be written in symplectic form as:• The Casimir invariants are:and correspond to Lagrangian conservation of vorticity• Symmetry in x and conservation of x-momentum:

Kelvin’s impulse(ignoringboundaryterms)

• Similarly for y-momentum and angular momentum:• Quasi-geostrophic dynamics is analogous; e.g. forcontinuously stratified flow• Now in addition to potential vorticity q(x,y,z,t), we need toconsider potential temperature on horizontal boundaries
ψz(x,y,t) [and possibly also circulation on sidewalls]• Note that for the QG model these quantities also evolveadvectively, like vorticity in barotropic dynamics:• Analogously, the Casimir invariants and x-momentum are:• Rotating shallow-water dynamics:

• Disturbance invariants: arguably the most powerfulapplication of Hamiltonian geophysical fluid dynamics• Ambiguities about the energy of a wave=• Ambiguities about the momentum of a wave=• If u=U is a steady solution of a Hamiltonian system, then

• For a canonical system, J is invertible so• Hence the disturbance energy is quadratic• But for a non-canonical system, this is not true and thedisturbance energy is generally linear in the disturbance• Not sign-definite• Cannot define stability, normal modes, etc.

• Pseudoenergy:
implies

for some Casimir CThus(pseudoenergy)

is then both conserved andquadratic in the disturbance• Example: Available potential energy (APE) for the 3D
stratified Boussinesq equations• Consider disturbances to a resting basic state• In the last expression, z must be regarded as a function of ρ0via; so ρ0(z) must be invertible, i.e. monotonic• Hence the basic state must be stably stratified• The last two terms in the expression for A can be written(positive definite)(APE of internalgravity waves)• Such an APE can be constructed for any Hamiltonian system• Small-amplitude approximation• Pseudomomentum: In a similar manner, if a basic state u=U
is independent of x (i.e. is invariant with respect to translationin x), then by Noether’s theorem,
implieswhich implies(pseudomomentum)

for some Casimir Cis then both conserved andquadratic in the disturbance• Example: Barotropic flow on the beta-plane• Consider disturbances to an x-invariant basic state q0(y)implies• This is analogous to the formula for APE, and similarly,

where, which is negative definite for• Small-amplitude approximation:• If q0 is defined to be the zonal mean, then q0 = q, q' = q − q
and the zonal mean of this expression becomes• Exactly the same form applies to stratified QG flow, where thenegative of this quantity is known as the Eliassen-Palm (E-P)
wave activity• N.B. The sign of this quantity corresponds to the sign of the
intrinsic frequency of Rossby waves (negative if)• Relation to wave action: there is a classical result that underslowly varying (WKB), adiabatic conditions, wave action isconserved (Bretherton & Garrett 1969 PRSA)ˆE∂Aˆr ˆ+ ∇⋅ c g A = 0 where Aˆ =ωˆ∂tEˆ is the wave energy (always positive definite) and ωˆ is theintrinsic frequency, both measured in the frame of reference
moving with the mean flow

• Hence sgn( Aˆ ) = sgn(ωˆ )• Under WKB conditions, pseudoenergy and pseudomomentum
are related to wave action via ω Aˆ , k Aˆ respectively• However pseudoenergy and pseudomomentum are moregeneral, and extend beyond WKB conditions– They require only temporal or zonal symmetry, respectively,in the background state• Stability theorems: Pseudoenergy and pseudomomentum
are conserved in time (for conservative dynamics), and arequadratic in the disturbance (for small disturbances), so fornormal-mode disturbances we have(σ is the real part of the growth rate)• Thenimplies(normal-mode stability)• Therefore these conservation laws can provide sufficientconditions for stability/necessary conditions for instability.
Indeed, many normal-mode stability theorems (e.g. Pedlosky1987) result from expressions of the form

where the integral turns out to be just pseudoenergy orpseudomomentum (or some combination of the two)• Example: Charney-Stern theorem. For stratified QG
dynamics, with horizontal boundaries, the pseudomomentum
is given byplus another term with the opposite sign at the top boundary.
Here Λ=Ψz is proportional to potential temperature.
• Baroclinic instability requires terms of opposite signs so A=0:
Eady model: Interior term vanishes, Λy<0 at bottom, Λy<0 at
topCharney model: Qy>0 in interior, Λy<0 at bottom
Phillips model: Qy<0 in lower levels, Qy>0 in upper levels• Barotropic instability: can be considered a special case• Other examples of Hamiltonian stability theorems:– Static stability, centrifugal stability, symmetric stability– Rayleigh-Kuo theorem, Fjørtoft-Pedlosky theorem– Arnol’d’s first and second theorems– Ripa’s theorem (shallow-water dynamics)• Notable exception: stratified shear flow (Miles-Howard theorem)• These Hamiltonian stability theorems can, in most cases, begeneralized to finite-amplitude (Liapunov) stability: i.e. for all εthere exists a δ such that
• They can also be used to derive rigorous saturation bounds on
nonlinear instabilities; e.g. for a statically unstable resting state,• Example: Bickley jet on barotropic beta-plane(Shepherd 1988 JFM)
Theoretical boundNumerical, inviscid

Numerical, viscousSupercriticality• Relationship between pseudomomentum andmomentum: consider the zonally averaged zonal momentum
equation for the barotropic beta-plane:• The linearized potential-vorticity equation isand hence (if q y ≠ 0 )whence(Taylor identity)• Stratified QG dynamics: zonal-wind tendency equation,
temperature tendency equation, and thermal-wind balance
together imply
where• So it’s the same physics, but the zonal-wind response to
mixing of potential vorticity is now spatially non-local (theEliassen balanced response): follows from PV inversion• The pseudomomentum conservation law takes the local form
(with S being a source/sink)• So mean-flow changes require wave transience or nonconservative effects (non-acceleration theorem)• In the atmosphere, we can generally assume that q y > 0 sinceq is dominated by β
• Hence A < 0; Rossby waves carry negative pseudomomentum• Where Rossby waves dissipate, there must be a convergenceof negative pseudomomentum, hence a negative torque• Conservation of momentum implies a positive torque in thewave source region• This phenomenon isseen in laboratory
rotating-tank experiments• A prograde jet emerges
from random stirring,surrounded on either side
by retrograde jets (seenin distortion of dye)
(Whitehead 1975 Tellus)• In the atmosphere, synoptic-scale Rossby waves are generated
by baroclinic instability, hence within a jet region• Flux of negative pseudomomentum out of jet corresponds to an
upgradient flux of momentum into the jet ∂A ∂∂t+u'v') = 0, A < 0(∂y• fVallis (2006)

• In fact the wave propagation is up and out (generallyequatorward), as seen in these ‘baroclinic life cycles’ showing
baroclinic growth and barotropic decay (Simmons & Hoskins
1978 JAS)E-P flux (arrows) anddivergence (contours)

Edmon, Hoskins &McIntyre (1980 JAS)

Deceleration

Acceleration

Haynes & Shepherd(1989 QJRMS)• The vertical flux of pseudomomentum is expressed in terms ofthe meridional heat fluxis the Eliassen-Palm (E-P) flux
• Reflects thermal-wind balance: poleward heat flux weakens
the thermal wind, accelerating the flow below anddecelerating the flow aloft (as in pure baroclinic instability)• During the wintertime when the stratospheric flow is westerly,
stationary planetary Rossby waves can propagate into thestratosphere where they exert a negative torque, acting toweaken the flow from its radiative equilibrium state• Stationary planetary-wave forcing mechanisms (topography,land-sea temperature contrast) are stronger in the Northern
than in the Southern Hemisphere, hence the stratosphericpolar vortex is weaker in the Northern HemisphereSummary• Hamiltonian dynamics is applicable to all the importantmodels of geophysical fluid dynamics– Provides a unifying framework between various models– Systems are infinite-dimensional, and their Eulerian
representations are generally non-canonical– To exploit Hamiltonian structure all that is needed is toknow the conserved quantities of a system• The most powerful applications are for theories describing
disturbances to an inhomogeneous basic state– Non-trivial; e.g., wave energy is generally not conserved– Useful measures of disturbance magnitude require the useof Casimir invariants, following from Lagrangian invariants– Leads to important concepts of pseudoenergy andpseudomomentum: stability theorems immediately follow– Important applications are available potential energy andmomentum transfer by waves

References••••••

Edmon, H. J., Hoskins, B. J., McIntyre, M. E., 1980: Eliassen-Palmcross sections for the troposphere, J. Atmos. Sci., 37,37 2600-2616
Haynes, P. H., Shepherd, T. G., 1989: The importance of surfacepressure changes in the response of the atmosphere to zonallysymmetric thermal and mechanical forcing, Q. J. R. Meteorol. Soc.,
115,115 490, 1181-1208
Shepherd, T. G., 1988: Rigorous bounds on the nonlinear saturationof instabilities to parallel shear flows, J. Fluid Mech., 196,
196 291-322.
Simmons, A., Hoskins, B., 1978: The life-cycles of some nonlinear
baroclinic waves. J. Atmos. Sci., 35,
35 414-432.
Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics,
Cambridge University Press, 770pp.
Whitehead, J. A., 1975: Mean flow generated by circulation on a βplane: An analogy with the moving flame experiment, Tellus, 27:27358–364.