A method is presented for obtaining Liapunov

functionals (LF) and proving nonlinear stability. The method

uses the generalized Poisson bracket (GPB) formulation of

Hamiltonian dynamics. As an illustration, certain stationary

solutions of ideal reduced MHD (RMHD) are shown to be nonlinearly

stable. This includes Grad-Shafranov and Alfven

solutions

Hazeltine, R. D.

Holm, D. D.

Marsden, J. E.

Morrison, P. J.

English

Caltech Authors - Main

GENERALIZED POISSON BRACKETS AND NONLINEAR 
LIAPUNOV STABILITY - APPLICATION TO REDUCED MHD 
R. D. Hazeltine, aD. D. Holm, b J. E .~1arsdenc and P. J .Morrisona ,d 
a)Institute for Fusion Studies and Dept. of Physics, 
RU1 11.222, University of Texas, Austin, TX 78712, USA. 
b)Center for Nonlinear Studies and Theoretical Division, 
Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 
C)Departrnent of Mathematics, University of California, 
Berkeley, CA 94720, USA. 
dJOn leave September 1983 to June 1984 at c). 
Abstract: A method is presented for obtaining Liapunov 
functionals (LF) and proving nonlinear stability. The method 
uses the generalized Poisson bracket (GPB) formulation of 
Hamiltonian dynamics. As an illustration, certain stationary 
solutions of ideal reduced MHD (RMHD) are shown to be non-
linearly stable. This includes Grad-Shafranov and Alfven 
solutions. 
1. Introduction 
To establish stability, the LF method [1-3] relies on 
the existence of conserved quantities that are used to bound 
the growth of perturbations from equilibrium. This method 
has been used to show linearized stability of plasma and 
fluid equilibria [4-6]. Here we present an algorithm for 
proving nonlinear stability based on the LF method using the 
GPB, or noncanonical Hamiltonian formalism. One finds there 
are often Casimir functionals that Poisson commute with all 
functionals and these enable one to obtain variational 
principles for equilibria of various Hamiltonian theories. 
These equilibria can then be tested for linear Liapunov 
stability and, in many fluid and plasma examples [2,3,7-9], 
nonlinear stability (stability to finite perturbations) has 
been proven. For RMHD, which is a system used for tokamak 
modeling [10], we find explicit criteria for nonlinear 
stability of Grad-Shafranov equilibria and equilibria with 
poloidal flow, including nonlinear Alfven waves. Elsewhere, 
stability of more realistic tokamak systems is treated [9] . . 
2. GPB Formalism 
The GPB formalism uses a conserved functional H (Hamil-
tonian) together with a Poisson bracket operator on pairs of 
functionals to represent field equations in the form 
al/Ji -+ 
at (x,t) = i= 1, ••• ,N (I) 
where the l/Ji denote the field components, and the GPB, { , } 
is (i) bilinear, (ii) antisymmetric, (iii) satisfies the 
Jacobi identity, and (iv) is a derivation in each argument. 
The class of field equations that are representable in this 
form is enlarged by relaxing the requirement that the bracket 
be in canonical form. In fact, the general formalism can be 
used to classify Hamiltonian theories. Systems with the same 
GPB possess common symmetries and have the same Casimirs. 
If C is a Casimir, then {C,F} = 0 for all functionals Fi 
hence, C is a constant of motion. Casimirs are an important 
ingredient of the LF stability method, to which we now turn. 
3. stability Algorithm 
The algorithm has four basic steps (A-D, below) that 
culminate when the norm, /1-1/, in the following definition is 
produced. 
-+ 
Definition: An equilibrium, l/J
e
, is Liapunov stable if 
for every E > 0 there is a 0 > 0, such that, for each nearby 
solution 
-+ /I~1JI1I < 0 
-+ -+ -+ -+ 
tJ! = l/Je + Lll/J (~1JI a finite perturbation) that has 
-+ ini tially, then 1/ ~l/J 1/ < E for all time (for which 
the solution exists). 
A. Constants of motion. There can be either dynamical 
constants arising from a geometrical symmetry (e.g. energy or 
momentum conservation, arising from either time or space 
translation symmetry), or kinematical constants (Casimirs). 
This first step can be facilitated by understanding the 
Hamiltonian structure of the problem. From experience, this 
is not a formidable task, since at present these structures 
are understood for a plethora of systems including the major 
nondissipative plasma fields (for review and original refer-
(;rp.. 
ences, see [11-13]). Understanding the GPB yields the 
requisite constants. The stability analysis presented here 
uses the energy and the Casimirs, but we note that additional 
constants such as momentum may also be utilized when addi-
tional symmetries are present. 
B. Equilibria are obtained from a variational principle 
that employs the Hamiltonian and the Casimirs. Evidently from 
(1) equilibria occur for ljJi such that {ljJi, H} = O. If we let 
C denote a linear combination of the Casimirs, then {t/Ji,I} =0, 
where 1:= H + C. Equilibria occur when the first variation 
of I vanishes; i. e. , 
+ + 
D1 [tPl • otP = o (2) 
Usually Casimirs involve free functions; so a whole class of 
equilibria is often obtained by this step. 
C. Linear stability; 
perturbations o~ about 
variation of (2). This 
i.e. stability to infinitesimal 
+ 
t/J , can be shown by taking the e . 
yields a quadratic form in ot/J 1 • 
second 
Definiteness of this form implies stability of the linearized 
equations, but does not guarantee stability to finite pertur-
bations for dynamics governed by partial differential equa-
tions. A further condi tion I sometimes called strong positivity, 
is required. This amounts to a convexity estimate, which we 
treat in the next step. 
D. Showing nonlinear stability requires constructing a 
norm for the solution space of the system. This will be 
accomplished if one can find quadratic forms Ql and Q2 + 
that satisfy the following for all finite ~t/J : 
+ + + + + 
H [t/J e + ~ t/J J - H [t/J e J - DH [t/J e J • fl t/J 
+ + + ~ C[t/Je+ flt/JJ C[tPeJ 
+ + 
:= Ql [~tPJ + Q2 [flljJJ > 0 for 
+ + 
- DC [t/J e J • ~ tP 
+ flljJ ~ 0 and 
(3a) 
(3b) 
( 3c) 
Finding the conditions for the equilibria to satisfy (3a) and 
(3b) is typically not difficult, but the positivity condition 
(3c) can require some ingenuity. To see why this construction 
gives stability, note that I in step C is a constant of 
motion; so 
IIA~1I2 ~ -+- -+- -+- -+-I [1jJ (t) 1 - I [ljJe1 - 01 [ljJe1 • L\1jJ (t) 
-+-
-+-
= I [ljJ (t= 0)] - I [1jJ e] 
A -+ 
:= I[L\1/J ] (4) 
0 
-+ -+-
where A1/J(t=O) = L\1jJo. Thus, the norm of the perturbation, 
/lA$1I2, is bounded by a constant for all time. Suppose this 
constant is small when IIA1jJ1I is small. (This is proven easily 
-+ by putting quadratic upper bounds on the quantity I [61jJ) in 
-+ (4) . ) Then, the equilibrium 1jJe is nonlinearly Liapunov 
stable as defined above. 
4. RMHD 
Assuming helical symmetry, the equations of RMHD are 
au 
at = [1/J,J] - [4>,U] (5) 
where 1jJ (r , e , t) is the helical flux, U (r , e , t) is the scalar 
vorticity, [f,g] = r-1(frge - feg r ) , J = \1
21jJ and U = \124>. 
This system conserves energy, H = ~ f ( I Y¢ 12 + 1 Y1jJ 12) d't, 
where d't = rdrde. The GPB for (5) is the Lie-Poisson 
bracket associated to the semidirect-product Lie group of 
canonical transformations acting on functions on E2 [12,13]; 
hence its Casimirs are known to be C1 = fF(1jJ)d't and 
C2 = f UG (1jJ) d't where F and G are arbitrary smooth functions 
of 1jJ. Varying the functional I = H + C 1 + C2 yields 
DI'(o4>,oljJ) = f [04>(-\124>+\12G)+01jJ(-\121jJ+UG1jJ+Ft/I)]d't 
(6) 
from which we obtain the equilibrium conditions 4> = G (1jJ) and 
\1 2 'IjJ - (\12 G)Gt/I- F1/J = O. Two special cases are of interest: 
(i) G = 0, which yields the RMHD Grad-Shafranov equation 
\12t/J = Ft/J, and (ii) G(1jJ) = 1/J, which implies F = constant and 
¢ = 1/J. In the latter case, the specific form of 4> is not 
further constrained. The case 4> = 1/J corresponds to flow at 
the poloidal Alfven speed and can be interpreted as nonlinear 
Alfven waves in the wave frame. We shall investigate nonlin-
ear stability of a class that includes both (i) and (ii). 
Taking the second variation of I and rearranging terms 
yields 
D 2 I 0 (<5 cjl , <5 lP ) 2 = J ( 1 yo cp - Y (GlP <5 lP) 12 + 1 ~ <5 lP ,2 (1 - G~) 
+ (<5lP)2[GlPtjJV2G + FlPtjJ + GtjJY 0 (GlPlPYlP)])dT • 
This quantity is positive definite if 1GlPi < 1 and 
GtjJtjJV 2 G+FlPlP +GlP Yo(GtjJtjJYlP) > O. In case (i) the latter 
condition becomes FlPW > 0, which is a severe restriction 
(monotonicity) on the toroidal currenti while in case (ii) 
we obtain D2Io(<5cp,<5tjJ)2 = JIY<5cjl-yotjJI 2 dT. In this Alfven 
wave case, we see that cp and W can each grow arbitrarily 
large; however their difference is bounded in time. This is 
consistent with the kink mode instability that RMHD is known 
~ to possess [14]. 
For nonlinear stability it is necessary to show convexity. 
A 
Since H is already quadratic, we let Q1 = H. If we let C 
be the right-hand side of (3b) then for RMHD we obtain 
c = J [Ue (F (tjJe + ~lP) - F (tjJe> - P' (We> ~tjJ) 
+ ~U(F(lPe+~tjJ) -F(lPe » + G(tjJe+~lP) -G(lPe ) -G'(\jJe)~W]dT 
If we assume the fUnctions F and G satisfy F lP ~ q, 2F tjJtjJ;:;" P 
'" and 2GtjJtjJ > s for constants q,p,s, then C> Q2 where 
Q2 E f [pUe(~tjJ)2 + q~U~W + s(~\jJ)2]dT 
Hence we obtain 
Ql+Q2 = !zf[I~(llCP)-qY(~W)12 + (1_q2)1~~WI2 
+ (pUe + s) (~W) 2]dT 
Thus Q
1 
+ Q2 > 0 when (i) Iql ~ 1 and (ii) pUe + s ~ O. 
Liapunov stability is established upon regarding (Ql + Q2)'-2 as 
a norm and further requiring F W :r;;;; Q, 2F lPlP :r;;;; P and 2GlPlP :r;;;; S 
for constants Q,P,S, in order for which equals 
A ~ 
I[~~o] in (4), to have a quadratic upper bound. 
In case (ii) for Alfven waves, the second variation 
analysis for linearized stability is equivalent to the 
convexity analysis for nonlinear stability, since I is 
quadratic. For Grad-Shafranov equilibria we obtain nonlinear 
stability provided FW (negative of the toroidal current) is a 
decreasing function of ~ (with a bound on its slope, so that 
i[~~D'~~O] has a quadratic upper bound). 
References 
1. J.LaSalle and S.Lefschetz [1961]. stability by Liapunov's 
Direct Method. Academic Press. 
2. V.I.Arnold [1969]. Ann. Math. Soc. Transl. 2!, 267. 
3. D.D.Holm, J.E.Marsden, T.Ratiu and A.Weinstein [1983]. 
Phys. Lett. 98A, 15. 
4. W.Newcomb, appendix of I.Bernstein [1958]. Phys. Rev. 
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~ 
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6. M.Rosenbluth [1964] in Advanced Plasma Theory. Academic 
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Generalized poisson brackets and nonlinear Liapunov stability application to reduces mhd

appendix of I.Bernstein [1958].

J.E.Marsden and T.Ratiu [1984]. Richardson number criteria for nonlinear stability of shear flows, submitted to J. Fluid Mechanics.

J.LaSalle and S.Lefschetz [1961]. stability by Liapunov's Direct Method.

M.Rosenbluth [1964] in Advanced Plasma Theory.

Nonlinear stability of fluid and plasma equilibria (in preparation) . R.D.Hazeltine, D.D.Holm and P.J.Morrison (work in progress). H.R.Strauss [1976]. Phys. Fluids 19, 134. J.E.Marsden et al [1982].

Nuclear Fusion

T.Ratiu and A.Weinstein [1983].

T.Ratiu and A.Weinstein [1984J.

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Generalized poisson brackets and nonlinear Liapunov stability application to reduces mhd

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