Stratified Rotating Boussinesq Equations in Geophysical Fluid Dynamics: Dynamic Bifurcation and Periodic Solutions

0.26 MB·English

by  Chun-Hsiung Hsia

STRATIFIED ROTATING BOUSSINESQ EQUATIONS IN
GEOPHYSICAL FLUID DYNAMICS: DYNAMIC BIFURCATION
AND PERIODIC SOLUTIONS
7
0 CHUN-HSIUNGHSIA,TIANMA,ANDSHOUHONGWANG
0
2
Abstract. Themainobjectiveofthisarticleistostudythedynamicsofthe

n stratified rotating Boussinesqequations, which areabasic model ingeophys-
a ical fluid dynamics. First, for the case where the Prandtl number is greater
J than one, a complete stability and bifurcation analysis near the first critical

8 Rayleighnumberiscarriedout. Second, forthecasewherethePrandtlnum-
2 berissmallerthanone,theonsetoftheHopfbifurcationnearthefirstcritical
Rayleighnumberisestablished,leadingtotheexistenceofnontrivialperiodic
2 solutions. Theanalysisisbasedonanewlydevelopedbifurcationandstability
v theory fornonlineardynamical systems (both finiteandinfinitedimensional)
7 bytwooftheauthors[16].
8
0
0
1
1. Introduction
6
0 The phenomena ofthe atmosphereand oceanare extremely richinits organiza-
/
h tion and complexity, and a lot of them cannot be produced by experiments. These
p phenomenainvolveabroadrangeoftemporalandspatialscales. Asweknow,both
- theatmosphericandoceanicflowsareflowsundertherotationoftheearth. Infact,
h
t fast rotation and small aspect ratio are two main characteristics of the large scale
a
atmospheric and oceanic flows. The small aspect ratio characteristic leads to the
m
primitive equations, andthe fastrotationleads to the quasi-geostrophicequations.
v: These are fundamental equations in the study of atmospheric and oceanic flows;
i see Ghil and Childress [6], Lions, Temam and Wang [12, 13], and Pedlosky [23].
X
Furthermore, convection occurs in many regimes of the atmospheric and oceanic
r
flows.
a
A key problem in the study of climate dynamics and in geophysical fluid dy-
namics is to understand and predict the periodic, quasi-periodic, aperiodic, and
fully turbulent characteristics of large-scaleatmospheric and oceanic flows. Stabil-
ity/bifurcation theory enables one to determine how different flow regimes appear
anddisappearascontrolparameters,suchasthe Reynoldsnumber,vary. It,there-
fore, provides one with a powerful tool to explore the theoretical capability in the
predictability problem. Most studies so far have only considered systems of or-
dinary differential equations (ODEs) that are obtained by projecting the PDEs
onto a finite-dimensionalsolution space, either by finite differencing or by truncat-
ing a Fourier expansion (see Ghil and Childress [6] and further references there).
Date:February7,2008.
Key words and phrases. Boussinesq equations, geophysical fluid dynamics, attractor bifurca-
tion,onsetofHopfbifurcation.
The work was supported in part by the Office of Naval Research, by the National Science
Foundation, andbytheNationalScienceFoundation ofChina.
1
2 C.HSIA,T.MA,ANDS.WANG
ThesewerepioneeredbyLorenz[14,15], Stommel[25], andVeronis[27,28] among
others, who explored the bifurcation structure of low-order models of atmospheric
and oceanic flows. More recently, pseudo-arclength continuation methods have
been applied to atmospheric (Legras and Ghil [11]) and oceanic (Speich et al. [24]
and Dijkstra [5]) models with increasing horizontal resolution. These numerical
bifurcation studies have produced so far fairly reliable results for two classes of
geophysical flows: (i) atmospheric flows in a periodic mid-latitude channel, in the
presence of bottom topography and a forcing jet; and (ii) oceanic flows in a rect-
angular mid-latitude basin, subject to wind stress on its upper surface; see among
others Charney and DeVore [2], Pedlosky [22], Legras and Ghil [11] and Jin and
Ghil[10]forsaddle-nodeandHopfbifurcationsinthetheatmosphericchannel,and
[20, 1, 8, 9, 19, 24] for saddle-node, pitchfork or Hopf in the oceanic basin.
Themainobjectiveofthisarticleistoconductbifurcationandstabilityanalysis
forthe originalpartialdifferentialequations (PDEs)that governgeophysicalflows.
This approachshould allow us to overcome some of the inherent limitations of the
numerical bifurcation results that dominate the climate dynamics literature up to
this point, and to capture the essential dynamics of the governing PDE systems.
The present article addresses the stability and transitions of basic flows for the
stratified rotating Boussinesq equations. These equations are fundamental equa-
tionsinthe geophysicalfluiddynamics;seeamongothersPedlosky[23]. We obtain
two main results in this article. The first is to conduct a rigorous and complete
bifurcationandstabilityanalysisnearthefirsteigenvalueofthelinearizedproblem.
Thesecondisthe onsetofthe Hopfbifurcation,leadingtothe existenceofperiodic
solutions of the model.
The detailed analysis is carried out in two steps. The first is a detailed study of
the eigenvalue problem for the linearized problem around the basic state. In com-
parison to the classical B´enard convection problem, the linearized problem here is
non-selfadjoint,leadingtomuchmorecomplicatedspectrum,andmorecomplicated
dynamics. WederiveinparticulartwocriticalRayleighnumbersR andR . Here
c1 c2
R is the first critical Rayleigh number for the case where the Prandtl number is
c1
greater than one, and R is the first critical Rayleigh number for the case where
c2
thePrandtlnumberislessthanone. Moreover,R leadstotheonsetofthesteady
c1
statebifurcationwhileR leadstothe onsetofthe Hopfbifurcation. Bothparam-
c2
eters are explicitly given in terms of the physical parameters. The crucial issues
here include 1) a complete understanding of the spectrum, 2) identification of the
criticalRayleighnumbers,andmostimportantly3)theverificationofthePrinciple
of Exchange of Stabilities near these critical Rayleigh numbers.
Thesecondstepistoconductarigorousnonlinearanalysistoderivethebifurca-
tions at both the critical Rayleighnumbers based on the classicalHopf bifurcation
theory and a newly developed dynamic bifurcation theory by two of the authors.
This new dynamic bifurcation theory is centered at a new notion of bifurcation,
calledattractorbifurcationfordynamicalsystems,bothfinitedimensionalandinfi-
nitedimensional,togetherwithnewstrategiesfortheLyapunov-Schmidtreduction
andthecentermanifoldreductionprocedures. Thebifurcationtheoryhasbeenap-
pliedtovariousproblemsfromscienceandengineering,including,inparticular,the
Kuramoto-Sivashinshyequation, the Cahn-Hillard equation, the Ginzburg-Landau
equation, Reaction-Diffusion equations in Biology and Chemistry, and the B´enard
convection problem, the Taylor problem; see [16, 17] and the references therein.
STRATIFIED ROTATING BOUSSINESQ EQUATIONS 3
We remark that the non-selfadjointness of the linearized problem gives rises the
onset of the Hopf Bifurcation. We prove that the Hopf bifurcation appears at the
Rayleigh number R . As mentioned earlier, the understanding and prediction of
c2
of the the periodic, quasi-periodic, aperiodic, and fully turbulent characteristics of
large-scale atmospheric and oceanic flows are key issues in the study of climate
dynamicsandingeophysicalfluiddynamics. Itis hopedthatthe study carriedout
in this article will provide some insights in these important issues.
Also, we would like to mention that rigorous proof of the existence of periodic
solutionsforafluidsystemisanormallyaverydifficulttaskfromthemathematical
point of view. For instance, with a highly involved analysis, Chen et al. [3] proved
the existence of a Hopf bifurcation in an idealized Fourier space.
Thepaperisorganizedasfollows. Section2givesthebasicsettingoftheproblem.
Section 3 states the main results. The proofs of the main results occupies the
remaining part of the paper: Section 4 recapitulates the essentials of the attractor
bifurcationtheory,Section5isontheeigenanalysis,andSection6isonthe central
manifold reduction and the completion of the proofs.
2. Stratified Rotating Boussinesq Equations in Geophysical Fluid
Dynamics
The stratified rotatingBoussinesq equations are basic equations in the geophys-
ical fluid dynamics, and their non-dimensional form is given by
∂U 1
=σ(∆U −∇p)+σRTe− e×U −(U ·∇)U,
∂t Ro

(2.1) ∂T
 ∂t =∆T +w−(U ·∇)T,
divU =0,
for (x,y,z) inthe non-dimensional domain Ω = R2 ×(0,1), where U = (u,v,w)
is the velocity fields, e = (0,0,1) is the unit vector in the z-direction, σ is the
Prandtl number, R is the thermal Rayleigh number, Ro is the Rossby number, T
is the temperature function and p is the pressure function. We refer the interested
readers to Pedlosky [23], Lions, Temam and Wang [13] for the derivation of this
model and the related parameters. In particular, the term 1 e × U represents
Ro
the Coriolis force, the w term in the temperature equation is derived using the
stratification, and the definition of the Rayleigh number R as follows:
gαβ
(2.2) R= h4.
κν
We consider the periodic boundary condition in the x and y directions
(2.3) (U,T)(x,y,z,t)=(U,T)(x+2jπ/α ,y,z,t)
1
=(U,T)(x,y+2kπ/α ,z,t),
2
for any j,k ∈ Z. At the top and bottom boundaries, we impose the free-free
boundary conditions:
∂u ∂v
(2.4) (T,w)=0, =0, =0, at z =0,1.
∂z ∂z
It is natural to put the constraint
(2.5) udxdydz = vdxdydz =0.
ZΩ ZΩ
4 C.HSIA,T.MA,ANDS.WANG
The initial value conditions are given by
(2.6) (U,T)=(U,T) at t=0.
Let
e e
H ={(U,T)∈L2(Ω)4 |divU =0,w | =0,(u,v) satisfies (2.3) and (2.5)},
z=0,1
H ={(U,T)∈H2(Ω)4∩H|(U,T) satisfies (2.3)−(2.5)},
1
H ={(U,T)∈H |(u,v,w,T)(−x,−y,z)=(−u,−v,w,T)(x,y,z)},
H =H ∩H.
e1 1
LeteLR =−Ae−BR :H1 →H (resp., H1 → H) and G:H1 → H (resp., H1 → H)
be defined by
e e e e
1
Aψ =(−P[σ∆U − e×U],−∆T),
Ro
B ψ =(−P[σRTe],−w),
R
G(ψ)=G(ψ,ψ),
for any ψ =(U,T)∈H (resp., H ), where
1 1
G(ψ ,ψ )=(−P[(U ·∇)U ],−(U ·∇)T ),
1 2 e 1 2 1 2
for any ψ = (U ,T ), ψ = (U ,T ) ∈ H . Here P is the Leray projection to L2
1 1 1 2 2 2 1
fields, and for a detailed account of the function spaces; see among many others
[26].
Remark 2.1. Note that H and H are invariant under the bilinear operator G in
1
the sense that
G(ψ e,ψ )∈He, for ψ ,ψ ∈H .
1 2 1 2 1
Hence, H and H are invariant under the operator L +G.
1 e R e
Then the Boussinesq equations (2.1)-(2.5) can be written in the following oper-
e e
ator form
dψ
(2.7) =L ψ+G(ψ), ψ =(U,T).
R
dt
3. Main Results
3.1. Definition of attractor bifurcation. To state the main theorems of this
article, we proceed with the definition of attractor bifurcation, first introduced by
T. Ma and S. Wang in [16, 17].
Let H and H be two Hilbert spaces, and H ֒→ H be a dense and compact
1 1
inclusion. We consider the following nonlinear evolution equations
du
=L u+G(u,λ),
λ
(3.1) dt

u(0)=u ,
 0
where u : [0,∞) → H is theunknown function, λ ∈ R is the system parameter,
and L :H →H are parameterizedlinear completely continuous fields depending
λ 1
STRATIFIED ROTATING BOUSSINESQ EQUATIONS 5
continuously on λ∈R1, which satisfy
−L =A+B a sectorial operator,
λ λ
(3.2) A:H1 →H a linear homeomorphism,
Bλ :H1 →H parameterized linear compact operators.
It is easy to see [7] that Lλ generates an analytic semi-group {etLλ}t≥0. Then
we can define fractional power operators (−L )µ for any 0 ≤ µ ≤ 1 with domain
λ
H =D((−L )µ) such that H ⊂H if µ >µ , and H =H.
µ λ µ1 µ2 1 2 0
Furthermore, we assume that the nonlinear terms G(·,λ) : H → H for some
µ
1>µ≥0areafamilyofparameterizedCr boundedoperators(r ≥1)continuously
depending on the parameter λ∈R1, such that
(3.3) G(u,λ)=o(kuk ), ∀ λ∈R1.
Hµ
In this paper, we are interested in the sectorial operator −L = A+B such
λ λ
that there exist an eigenvalue sequence {ρ } ⊂ C1 and an eigenvector sequence
k
{e ,h }⊂H of A:
k k 1
Az =ρ z , z =e +ih ,
k k k k k k
(3.4) Reρk →∞ (k →∞),
|Imρk/(a+Reρk)|≤c,
for some a,c>0, such that {ek,hk} is a basis of H. Also we assume that there is
a constant 0<θ <1 such that
(3.5) B :H −→H bounded, ∀ λ∈R1.
λ θ
Underconditions(3.4)and(3.5),theoperator−L =A+B isasectorialoperator.
λ λ
Let{S (t)} beanoperatorsemi-groupgeneratedbytheequation(3.1). Then
λ t≥0
the solution of (3.1) can be expressed as ψ(t,ψ )=S (t)ψ , for any t≥0.
0 λ 0
Definition 3.1. A set Σ ⊂ H is called an invariant set of (3.1) if S(t)Σ = Σ for
any t≥0. An invariant set Σ⊂ H of (3.1) is called an attractor if Σ is compact,
and there exists a neighborhood W ⊂H of Σ such that for any ψ ∈W we have
0
lim dist (ψ(t,ψ ),Σ)=0.
H 0
t→∞
Definition 3.2. (1) We say that the solution of (3.1) bifurcates from (ψ,λ)=
(0,λ ) to an invariant set Ω , if there exists a sequence of invariant sets
0 λ
{Ω } of (3.1) such that 0∈/ Ω , lim λ =λ , and
λn λn n→∞ n 0
lim max |x|=0.
n→∞x∈Ωλn
(2) If the invariant sets Ω are attractors of (3.1), then the bifurcation is called
λ
attractor bifurcation.
3.2. Main theorems. In this article, we consider two cases:
(3.6) σ >1 and R is obtained only at (j,k,l)=(j ,0,1),
c1 1
(3.7) σ <1 and R is obtained only at (j,k,l)=(j ,0,1),
c2 2
6 C.HSIA,T.MA,ANDS.WANG
forsomej ,j ∈N,whereR andR aredefinedin(5.18)and(5.22)respectively.
1 2 c1 c2
In the above cases, R and R are given by the following formulas:
c1 c2
(j2α2+π2)3 π2
R = 1 1 + ,
c1 j2α2 σ2Ro2j2α2
1 1 1 1
2(σ+1)(j2α2+π2)3 2π2
R = 2 1 + .
c2 j2α2 (σ+1)Ro2j2α2
2 1 2 1
Remark 3.3. (1) Condition(3.6)guaranteesthatforR≈R ,thefirsteigen-
c1
value of LR |H1 (resp., LR |He1) is real and of multiplicity two (resp., one);
see Remark 5.3.
(2) Condition (3.7) guarantees that, for R≈R , there exists only one simple
c2
pair of conjugate complex eigenvalues of LR |He1 crossing the imaginary
axis; see Lemma 5.6.
(3) Condition (3.6) or (3.7) can be satisfied easily; see Lemmas 5.4 and 5.5.
Theorem 3.4. Assume (3.6). Then the following assertions for Problem (2.1)-
(2.5) defined in H hold true.
(1) If R≤R , the steady state (U,T)=0 is locally asymptotically stable.
c1
(2) For R > R , the problem bifurcates from ((U,T),R) = (0,R ) to an
c1 c1
attractor Σ =S1, consisting of only steady state solutions.
R
H
Σ
R
R R
Rc
1
Σ
R
Figure 3.1. Bifurcation from (0,R ) to an attractor Σ for R>R .
c1 R c1
Theorem 3.5. Assume (3.7) and
(1−σ)π2
Ro2 < .
σ2(1+σ)(j2α2+π2)3
2 1
The following statements are true.
(1) For Problem (2.1)-(2.5) defined in H, the steady state (U,T)=0 is locally
asymptotically stable if R<R .
c2
(2) For Problem (2.1)-(2.5) defined in H, a Hopf bifurcation occurs generically
when R crosses R .
c2
e
STRATIFIED ROTATING BOUSSINESQ EQUATIONS 7
4. Preliminaries
4.1. Attractor bifurcation theory. Consider (3.1) satisfying (3.2) and (3.3).
We start with the Principle of Exchange of Stabilities (PES). Let the eigenvalues
(counting the multiplicity) of L be given by β (λ), β (λ), ···. Suppose that
λ 1 2
<0 if λ<λ ,
0
(4.1) Reβ (λ) =0 if λ=λ , if 1≤i≤m,
i  0
>0 if λ>λ0,

(4.2) Reβ (λ )<0, if m+1≤j.
j 0
Let the eigenspace of L at λ be
λ 0
∞
E = {u,v∈H |(L −β (λ ))kw=0,w=u+iv}.
0 1 λ0 j 0
1≤j≤mk=1
[ [
It is known that dimE =m.
0
Theorem 4.1 (T. Ma and S. Wang [16, 17]). Assume that the conditions (3.2)-
(3.5) and (4.1)-(4.2) hold true, and u=0 is locally asymptotically stable for (3.1)
at λ=λ . Then the following assertions hold true.
0
(1) For λ > λ , (3.1) bifurcates from (u,λ) = (0,λ ) to attractors Σ , having
0 0 λ
the same homology as Sm−1, with m−1≤dimΣ ≤m, which is connected
λ
if m>1;
(2) For any u ∈Σ , u can be expressed as
λ λ λ
u =v +o(kv k ), v ∈E ;
λ λ λ H1 λ 0
(3) ThereisanopensetU ⊂H with0∈U suchthattheattractorΣ bifurcated
λ
from (0,λ )attractsU\Γin H,where Γis thestablemanifold of u=0with
0
co-dimension m.
4.2. Center manifold theory. A crucial ingredient for the proof of the main
theoremsusingtheaboveattractorbifurcationtheoremisanapproximationformula
for center manifold functions; see [16].
Let H and H be decomposed into
1
(4.3) H =Eλ⊕Eλ, H =Eλ⊕Eλ,
1 1 2 1 2
forλnearλ ∈R1,whereEλ,EλareinvariantsubspacesofL ,suchthatdimEλ <
0 1 2 λ 1
e e
∞, Eλ = Eλ, Eλ = closure of Eλ in H. In addition, L can be decomposed into
1 1 2 2 λ
L =Lλ⊕Lλ such that for any λ near λ ,
λ 1 2 0
e e
Lλ =L | :Eλ −→Eλ,
(4.4) 1 λ E1λ 1 1
(Lλ2 =Lλ|Eλ :E2λ −→E2λ,
2 e
where all eigenvalues of Lλ possess negative real parts, and the eigenvalues of Lλ
2 e 1
possess nonnegative real parts at λ=λ . Furthermore, with µ<1 given by (3.3),
0
let
Eλ(µ)= closure of Eλ in H .
2 2 µ
By the classicalcenter manifold theorem (see amongothers [7, 26]), there exists
aneighborhoodofλ givenby|λ−λ |<δ forsomeδ >0,aneighborhoodB ⊂Eλ
0 0 λ 1
ofx=0,andaC1 centermanifoldfunctionΦ(·,λ):B →Eλ(θ), calledthe center
λ 2
8 C.HSIA,T.MA,ANDS.WANG
manifold function, depending continuously on λ. Then to investigate the dynamic
bifurcation of (3.1) it suffices to consider the finite dimensional system as follows
dx
(4.5) =Lλx+g (x,Φ (x),λ), x∈B ⊂Eλ.
dt 1 1 λ λ 1
Hence, anapproximationformula for the center manifold function Φ is crucialfor
λ
the bifurcation and stability study.
Let the nonlinear operator G be in the following form
(4.6) G(u,λ)=G (u,λ)+o(kukn),
n
forsome integer n≥2. Here G :H ×···×H −→H is a n-multilinearoperator,
n 1 1
and G (u,λ)=G (u,··· ,u,λ).
n n
Theorem4.2. [16]Undertheconditions(4.3),(4.4)and(4.6),thecentermanifold
function Φ(x,λ) can be expressed as
(4.7) Φ(x,λ)=(−Lλ)−1P G (x,λ)+o(kxkn)+O(|Reβ|kxkn),
2 2 n
where Lλ is as in (4.4), P : H → E the canonical projection, x ∈ Eλ, and
2 2 2 1
β =(β (λ),··· ,β (λ)) the eigenvectors of Lλ.
1 m 1
e
5. Eigenvalue Problem
The eigenvalue problem of the linearized problem of (2.1)-(2.4) is given by
1
σ(∆U −∇p)+σRTe− e×U =βU,
Ro
(5.1) ∆T +w=βT,
divU =0,
supplemented with (2.3) and (2.4). For ψ = (U,T) satisfying (2.3) and (2.4), we
expand the field ψ in Fourier series
∞
(5.2) ψ(x,y,z)= ψ (z)ei(jα1x+kα2y).
jk
j,k=−∞
X
Plugging (5.2) into (5.1), we obtain the following system of ordinary differential
equations
1
σ(D u −ijα p )+ v =βu ,
jk jk 1 jk jk jk
Ro

1
(5.3) DσD(jDkwTjkjvkj+−k−wp′jikk+α=2RpβjTTkj)k−,=Rσo−u1jβkw=jkβ, vjk,
jk jk jk jk
uij′jαk1|uz=jk0,+1=ikvαj′k2v|jzk=0+,1w=j′kwj=k 0|z,=0,1=Tjk |z=0,1=0,
STRATIFIED ROTATING BOUSSINESQ EQUATIONS 9
for j,k ∈ Z, where ′ = d/dz, D = d2/dz2 −α2 and α2 = j2α2 +k2α2. If
jk jk jk 1 2
w 6=0, (5.3) can be reduced to a single equation for w (z):
jk jk
(5.4) {(D −β)(σD −β)2D
jk jk jk
1
+ (D −β)(D +α2 )+σRα2 (σD −β)}w =0,
Ro2 jk jk jk jk jk jk
(5.5) w =w′′ =w(4) =w(6) =0 at z =0,1,
jk jk jk jk
for j,k ∈Z. Thanks to (5.5), w can be expanded in a Fourier sine series
jk
∞
(5.6) w (z)= w sinlπz,
jk jkl
l=1
X
for (j,k) ∈ Z×Z. Substituting (5.6) into (5.4), we see that the eigenvalues β of
the problem (5.1) satisfy the cubic equations
l2π2 α2
(5.7) β3+(2σ+1)γ2 β2+[(σ2+2σ)γ4 + −σR jk]β
jkl jkl Ro2γ2 γ2
jkl jkl
l2π2
+σ2γ6 −σ2Rα2 + =0,
jkl jk Ro2
for j,k ∈Z and l ∈N, where γ2 =α2 +l2π2. In the following discussions,we let
jkl jk
g (β)=(β+γ2 )[(β+σγ2 )2+l2π2Ro−2γ−2],
jkl jkl jkl jkl
(5.8) h (β)=σRα2 γ−2(β+σγ2 ),
jkl jk jkl jkl
f (β)=g (β)−h (β),
jkl jkl jkl
and β (R), β (R) and β (R) be the zeros of f with
jkl1 jkl2 jkl3 jkl
Re(β )≥Re(β )≥Re(β ).
jkl1 jkl2 jkl3
5.1. Eigenvectors. In the following discussions, we consider the following index
sets:
Λ ={(j,k,l)∈Z2×N|j ≥0,(j,k)6=(0,0)},
1
Λ ={(j,k,l)∈Z2×{0}|j ≥0,(j,k)6=(0,0)},
2
Λ ={(j,k,l)∈{(0,0)}×N},
3
Λ=Λ ∪Λ ∪Λ .
1 2 3
1. For (j,k,0)∈Λ , we define
2
ψβjk0 =(kα sin(jα x+kα y),−jα sin(jα x+kα y),0,0)t,
1 2 1 2 1 1 2
ψβjk0 =(−kα cos(jα x+kα y),jα cos(jα x+kα y),0,0)t,
2 2 1 2 1 1 2
E =span{ψβjk0,ψβjk0},
jk0 1 2
β =∪ {β },
Λ2 (j,k,0)∈Λ2 jk0
where β = −σγ2 = −σα2 = −σ(j2α2 +k2α2). It is not hard to see that
jk0 jk0 jk 1 2
L (ψβjk0)=β ψβjk0 and L (ψβjk0)=β ψβjk0.
R 1 jk0 1 R 2 jk0 2
10 C.HSIA,T.MA,ANDS.WANG
2. For (0,0,l)∈Λ , we define
3
ψβ00l1 =(0,0,0,sinlπz)t, ψβ0012 =(coslπz,0,0,0)t,
ψβ00l3 =(0,coslπz,0,0)t, E =span{ψβ00l1,ψβ00l2,ψβ00l3},
00l
βΛ3 =∪∞l=1∪3q=1{β00lq}, βΛe3 =∪∞l=1{β00l1},
where β = −γ2 = −l2π2, β = −σγ2 − 1 i and β = −σγ2 + 1 i. It
00l1 00l 0012 00l Ro 00l3 00l Ro
is easy to check that
LR(ψβ00l1)=β0011ψβ00l1,
1
L (ψβ00l2)=−σγ2 ψβ00l2 − ψβ00l3,
R 00l Ro
1
L (ψβ00l3)= ψβ00l2 −σγ2 ψβ00l3.
R Ro 00l
3. For (j,k,l)∈Λ , we define
1
jα lπ kα lπ
φ1 =(− 1 sin(jα x+kα y)coslπz,− 2 sin(jα x+kα y)coslπz,
jkl α2 1 2 α2 1 2
jk jk
cos(jα x+kα y)sinlπz,0)t,
1 2
kα lπ jα lπ
φ2 =( 2 sin(jα x+kα y)coslπz,− 1 sin(jα x+kα y)coslπz,0,0),
jkl α2 1 2 α2 1 2
jk jk
φ3 =(0,0,0,cos(jα x+kα y)sinlπz)t,
jkl 1 2
jα lπ kα lπ
φ4 =( 1 cos(jα x+kα y)coslπz, 2 cos(jα x+kα y)coslπz,
jkl α2 1 2 α2 1 2
jk jk
sin(jα x+kα y)sinlπz,0)t,
1 2
kα lπ jα lπ
φ5 =(− 2 cos(jα x+kα y)coslπz, 1 cos(jα x+kα y)coslπz,0,0)t,
jkl α2 1 2 α2 1 2
jk jk
φ6 =(0,0,0,sin(jα x+kα y)sinlπz)t,
jkl 1 2
E1 =span{φ1 ,φ2 ,φ3 }, E2 =span{φ4 ,φ5 ,φ6 },
jkl jkl jkl jkl jkl jkl jkl jkl
E =E1 ⊕E2 , β =∪ ∪3 {β }.
jkl jkl jkl Λ1 (j,k,l)∈Λ1 q=1 jklq
It is easy to check that E1 and E2 are invariant subspaces of the linear op-
jkl jkl
erator L respectively, i.e., L (E1 ) ⊂ E1 and L (E2 ) ⊂ E2 . The char-
R R jkl jkl R jkl jkl
acteristic polynomial of L | (resp., L | ) is given by f as defined in
R E1 R E2 jkl
jkl jkl
(5.8). Since E1 (resp.,E2 ) is of dimension three, the (generalized) eigenvectors
jkl jkl
of L | , ∪3 {ψβjklq} (∪3 {ψβjklq}), form a basis of E1 (resp., E2 ), i.e.,
R E1 q=1 1 q=1 2 jkl jkl
jkl
span{∪3 {ψβjklq}} = E1 (resp., span{∪3 {ψβjklq}} = E2 ). If β is a real
q=1 1 jkl q=1 2 jkl jklq
zerooff , the eigenvectorcorrespondingto β inE1 (resp.,E2 ) is givenby
jkl jklq jkl jkl
(5.9) ψβjklq =φ1 +A (β )φ2 +A (β )φ3 ,
1 jkl 1 jklq jkl 2 jklq jkl
(ψβjklq =φ4 +A (β )φ5 +A (β )φ6 ),
2 jkl 1 jklq jkl 2 jklq jkl
where
−1 1
(5.10) A (β)= , A (β)= .
1 Ro(β+σγ2 ) 2 β+γ2
jkl jkl