PAUTAS DE MOTIVACIÓN ESTABLECIDAS POR EL TRIBUNAL CONSTITUCIONAL
00728-2008-PHC/TC LIMA CASO GIULIANA FLOR DE MARÍA LLAMOJA HILARES
We next provide brief details of the linear stability analysis used in the main body of the paper to calculate the maximum growth rate. We start with the instantaneous zonally- averaged PV q¯i and zonal velocity u¯i in each layer i = 1, 2. In this subsection, these
profiles are general functions ofyfor each sampled timet. In the numerical code, they are provided at eachygrid point (typically129or257 grid points, including the edges). The PV gradientdq¯i/dy, needed in the analysis below, is computed using centred differ-
ences iny. Notably, only the interior grid point values are needed, since the perturbation satisfies homogeneous boundary conditions (required to ensure zero meridional velocity at each boundary).
In the linear stability analysis, the base flow is considered to be bothxandt-independent. Hence, we can consider infinitesimal perturbations of the form
q0i(x, y, t) =<{Qi(y)ei(kxx−σt)}
ψ0i(x, y, t) =<{Ψi(y)ei(kxx−σt)}
wherekx is a prescribed zonal wavenumber (necessarily integer for a domain of length
Lx = 2π),σis the frequency (the imaginary part of which, if non-zero, is the growth rate),
whileQ(y)andΨ(y)are the perturbation amplitudes (eigenfunctions) to be determined along withσ (the eigenvalue).
Plugging the above forms into the conservative form of the governing equations (2.1) — withFi = 0— and in the PV inversion relations (2.8), then linearising, and finally
cancelling the common ei(kxx−σt)} factor, we obtain the following eigen-system for the
perturbation amplitudesQ(y)andΨ(y)together with the phase speedc=σ/kx:
(¯ui−c)Qi + dq¯i dyΨi = 0, i= 1, 2 (A.20) d2Ψˆ j dy2 −K 2 jΨˆj = ˆQj, j = 1, 2 (A.21) (A.22)
wherei refers to layers,j and the hats refer to vertical modes, and K2
j ≡ k2x + ¯kdj2. To
combine these, we eliminateQi in the first set of equations using Qˆj in the second set
projected onto layers:
Qi = ˜ci1Qˆ1+ ˜ci2Qˆ2 = d 2Ψ i dy2 −c˜i1K 2 1Ψˆ1−c˜i2K22Ψˆ2. (A.23)
Next, we eliminateΨˆ1andΨˆ2 by projecting them onto modes: ˆ
Ψj =cj1Ψ1+cj2Ψ2. (A.24)
Using this in the expression forQiabove, we obtain
Qi =
d2Ψ
i
dy2 −Bi1Ψ1−Bi2Ψ2 (A.25)
0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 ky kx
Figure A.1: Growth rate as a function ofxandy wavenumbers, respectivelykx andky,
for the initial flow state used for the characteristic simulation.
results in two coupled, second-order equations for the layer streamfunction amplitudes:
¯ u1 d2Ψ 1 dy2 + dq¯1 dy −u¯1B11 Ψ1−u¯1B12Ψ2 =c d2Ψ 1 dy2 −B11Ψ1−B12Ψ2 ¯ u2 d2Ψ2 dy2 + dq¯2 dy −u¯2B21 Ψ1−u¯2B22Ψ2 =c d2Ψ2 dy2 −B21Ψ1−B22Ψ2 . (A.26)
This is an eigenvalue problem for the phase speed c. It is solved numerically by dis- cretising the second derivative terms by centred finite differences, and using the boundary conditionsΨi = 0on both channel walls. This results in a block-tridiagonal generalised
eigenvalue problem, which we have solved using the LAPACK routineDGGEV,
The results of the analysis above have been verified by matching them with the exact results obtained for the initially uniform PV gradientsdq¯i/dy =iβand uniform layer ve-
locitiesu¯i,i= 1, 2. In this special case, we can seek solutions forΨi ∝sin(ky(y−ymin))
whereky is the meridional wavenumber (an integer forLy = π). This reduces (A.26) to
a simple algebraic system, from whichc is determined from a quadratic equation. It is straightforward to show that (baroclinic) instability always occurs when the PV gradients have opposite signs in the two layers,12 <0(details omitted).
Figure A.1 shows the growth rates as a function of kx and ky, obtained using the
parameters of the characteristic simulation. The most unstable disturbances have small meridional wavenumbersky, and zonal wavenumberskxbetween11and13. For a given
kx, themaximum growth rate is positive untilkx > 13. Hence, there are always distur-
bances with low kx which are unstable (note: kx = 0 is excluded). These disturbances
simply have higher meridional wavenumbersky. Baroclinic instability occurs in a band
of total wavenumbersp
k2
Evolution variations
To illustrate the variability occurring between different simulations of the same flow, two other nearly identical simulations have been conducted. All parameters were kept the same as in the characteristic simulation previously illustrated, only the inversion grid resolution was halved to256×129. Additionally, a different random number seed was used in each simulation.
The two simulations exhibit striking differences. In the first case, see figure B.1, we see oscillations between turbulent and quiescent phases, sometimes with a shift of the jets and homogeneous regions, while retaining the jet spacing. The flow alternates between the two different quiescent phases encountered in the characteristic simulation. The ‘centred’ state exhibits jets slowly coming together toward the center of the domain. By contrast, the ‘shifted’ state does not exhibit any evident jet drift and moreover appears to be much shorter lived. Arguably, the jet configuration of this state is less robust than that in the centred state, though both the stability analysis of the zonal flow and the APE spectrum show no major differences between the states.
In the second simulation, see figure B.2, the shifted quiescent state develops after the initial baroclinic instability and dominates the first third of the simulation. Turbu- lent events throughout this period collapse back to this state except for the last one. This prominent turbulent event instead collapses to a new configuration resembling the cen- tred state seen over much of the characteristic simulation. This state then dominates the remainder of the evolution.
Top Layer Equivalent PV π/2 0 −π/2 137 0 −137 Bottom Layer Equivalent PV
π/2 0 −π/2 19 0 −19 Mode 1 Kinetic Energy
1 0 0 2000 4000 6000 8000 10000 0.00.2 0.4 0.6 0.81.0 1.2 1.4 1.6 1.8 ZKE1 EKE1
Mode 2 Kinetic Energy 1 0 0 2000 4000 6000 8000 10000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 ZKE2 EKE2
Available Potential Energy 1 0 0 2000 4000 6000 8000 10000 0 5 10 15 2025 3035 40 ZPE EPE Stability Analysis kx 20 10 0 0.6 0.3 0 Available Potential Energy Spectra
kx 20 10 0 2 1 0 −1 −2 t 0 2000 4000 6000 8000 10000
Figure B.1: (a,b) H¨ovmoller diagrams of the equivalent PV, with latitude in the ordinate. (c,d,e) Energy components (with the eddy part in blue and the zonal part in black) ver- sus time. (f)log10of the maximum growth rate of the zonally-averaged flow versus zonal
wavenumbers. (g)log10of the y-integrated spectrum of the available potential energy ver-
sus zonal wavenumbers. For all these plots time is in the abscissa. The same parameters are used as in the characteristic simulation but on a256×128basic inversion grid.
Top Layer Equivalent PV π/2 0 −π/2 137 0 −137 Bottom Layer Equivalent PV
π/2 0 −π/2 19 0 −19 Mode 1 Kinetic Energy
1 0 0 2000 4000 6000 8000 10000 0.0 0.5 1.0 1.5 2.0 ZKE1 EKE1
Mode 2 Kinetic Energy 1 0 0 2000 4000 6000 8000 10000 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 ZKE2 EKE2
Available Potential Energy 1 0 0 2000 4000 6000 8000 10000 05 10 15 20 25 30 35 40 ZPE EPE Stability Analysis kx 20 10 0 0.6 0.3 0 Available Potential Energy Spectra
kx 20 10 0 2 1 0 −1 −2 t 0 2000 4000 6000 8000 10000
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