Coherent structures
3.5 Numerical experiments
Remark. When we use Ulam’s method in practice, we often do not know the image domain Ω2, but start with an initial domain Ω1, which some particles possibly leave during their evolution over time. We can easily deal with this if we initially define P mapping into a greater domainΩ⊇Ω2, and delete all boxes not reached by any particles.
3.5 Numerical experiments
3.5.1 Double gyre
First we look at the benchmark example of the Double gyre. The model was introduced in [Shadden et al., 2005] and also considered in [Froyland and Padberg-Gehle, 2014], [Froyland and Padberg, 2009], [Williams et al., 2014b], [Ma and Bollt, 2013]. The model describes a flow
Ψ(t, x, y) :=Asin(πf(t, x)) sin(πy),
where f(t, x) = δsin(ωt)x2 + (1−2δsin(ωt))x, in a domain Ω = [0,2]×[0,1]. The velocity field is given by
b(t, x, y) = −∂Ψ∂y
∂Ψ
∂x
!
=
−πAsin(πf(t, x)) cos(πy) πAcos(πf(t, x)) sin(πy)∂f∂x(t, x)
. (3.8)
This model is not intended to describe a real fluid flow but a simplification of a double-gyre pattern seen in geophysical flows [Shadden et al., 2005].
The example describes two counter-rotating vortices separated by a periodically mov-ing leaky transport barrier. For δ = 0 the flow is autonomous. For δ > 0 the flow is non-autonomous and the gyres conversely expand and contract periodically in the x-direction, such that the domain Ω enclosing the gyres remains invariant. A determines the magnitude of the vector field b, ω/2π is the frequency of oscillation, and δ is ap-proximately how far the line separating the gyres moves to the left and to the right, see [Shadden et al., 2005]. We fix the parameter values to
A= 0.25, δ = 0.25, ω = 2π.
In Figure 3.4, we show the velocity field of the Double gyre at times 0, 0.25 and 0.75.
The Double gyre is 1-periodic in our configuration . We compute an approximation of the maximally coherent sets via Ulam’s method using 28×27 boxes and 100 test points per box. We use µ = λ as reference measure, which is invariant under the flow. For the time integration, we use a standard Runge-Kutte 4 scheme with step size h= 0.1.
As initial and final time, we choose t0 = 0.25, t1 = 10.75. In Figure 3.5 we show the singular spectrum of the transfer operator. There we spot a gap after the third singular value.
0 1 2
Figure 3.4: Field lines of the vectorfield of the Double gyre at times 0, 0.25 and 0.75.
The vectorfield is 1-periodic.
Figure 3.5: Singular values for the Double gyre.
0 0.5 1 1.5
Figure 3.6: First row: Second left and right singular vectors of the Frobenius-Perron operator for the Double gyre, Second row: Third singular vector at initial and final time.
3.5 Numerical experiments
0 0.5 1 1.5
0 0.2 0.4 0.6 0.8
0 0.5 1 1.5
0 0.2 0.4 0.6 0.8
Figure 3.7: Extracted partition into three coherent sets at initial and final time.
In Figure 3.6, we show the second and third singular vectors of the such computed Frobenius-Perron operator. They nicely indicate the two vortices at initial and final time.
We use k-means, see Section 2.8, to actually extract the coherent sets. The partition at initial and final time is shown in Figure 3.7.
3.5.2 Quasiperiodic Bickley jet
The Bickley jet, an idealized model of a meandering zonal jet, see [del Castillo-Negrete and Morrison, 1993] [Rypina et al., 2007] [Hadjighasem et al., 2016], will serve as a test problem throughout the thesis. This model consists of a steady background flow subject to a time-dependent perturbation. The time-dependent Hamiltonian reads
ψ(x, y, t) =ψ0(y) +ψ1(x, y, t), ψ0(y) =−U0L0tanh( y
L0
), ψ1(x, y, t) =U0L0sech2( y
L0)R
3
X
n=1
fn(t) exp(iknx)
!
where ψ0 is the steady background flow and ψ1 is the perturbation. The constants U0 and L0 are the characteristic velocity and the characteristic length scale, respectively.
For the following analysis, we apply the set of parameters used in [Rypina et al., 2007]
and [Hadjighasem et al., 2016]:
U0 = 62.66 ms−1, L0 = 1770 km, kn= 2n r0
,
wherer0 = 6371 km is the mean radius of the earth. Forfn(t) =nexp(ikncnt), the time-dependent part of the Hamiltonian consists of three Rossby waves with wave numbers kn traveling at speeds cn. The amplitude of each Rossby wave is determined by the parametersn. The parameters we use are the following: c1 = 0.1446U0,c2 = 0.205U0, c3 = 0.461 U0, ly = 1.77·106,1 = 0.0075, 2 = 0.04, 3 = 0.3, lx = 6.371·106π,kn= 2nπ/lx. The time interval we consider is fromt0= 0 tot1= 40 days = 3.456·106 seconds.
The initial domain is Ω0 = [0,20]×[−3,3] at time t0. As the flow does not stay in Ω0,
0 5 10 15
−2 0 2
Figure 3.8: Field lines of the vector field of the Bickley jet at time t0. The red color indicates high velocity.
we define the initial density asµ= 1/|Ω|1Ω, projectµonto the approximation spaceVN and obtain the constant vector p = 1/(n·m) ∈ RN. We let the approximate transfer operatorPn map to a greater domain Ω = [−5,5]×[0,20]. Then we only consider the subdomain Ω2 ⊂Ω where q =pP >0, i.e. delete all boxes that are not reached by any particle.
A picture of the vector field at time t0 = 0 and t1 is given in Figure 3.8. We utilize Ulam’s method to compute an approximation of the Frobenius-Perron operator for the system, see Section 3.4. Therefore we use 27×25 boxes with 100 testpoints per box.
In Figure 3.9 we show the singular values of the transfer operator and see several gaps occurring after the second, the eighth, the ninth and the fifteenth singular value. This indicates that there is not only one meaningful partition but several, and it is not clear a priori which to choose. In Figure 3.10 we show the second singular vector at initial and final time. We see that the 2-partition will simply separate the northern and the southern hemisphere. In Figure 3.11 we show partitions into two, eight, nine and fifteen coherent sets together with the corresponding last singular vectors contributing to the partition:
• The 2-partition indeed separates the northern from the southern hemisphere.
• The 8-partition identifies the six vortices and the two hemispheres.
• The 9-partition separates the zonal jet, too.
• The 15-partition additionally divides the vortices in an inner and outer part.
3.5 Numerical experiments
0 5 10 15 20
0.9 0.95 1
Figure 3.9: Singular values for the Bickley jet.
0 5 10 15
Figure 3.10: Left and right second singular vectors indicating the coherent sets at times t0 and t1.
Figure 3.11: First column: Partition into two, eight, nine and fifteen coherent sets. Sec-ond column: SecSec-ond, eighth, ninth and fifteenth singular vectors.