Comprobación de la presencia e integridad del plásmido

For the numerical computation of the LWE and the SWE, we implement a pseudo- spectral method [169, 170], with a resolution ofN = 212 Fourier modes. The scheme utilises the fast Fourier transform to convert physical space vectors into their Fourier representation, where differentiation of the wave field is transformed to algebraic mul- tiplication by wave numberk. The linear terms are solved exactly in Fourier space and then converted back into physical space with the aid of integrating factors, to greatly improve the numerical stability. Nonlinear terms are more complicated. Multiplications involving ψ(x, z), have to be performed in physical space. Moreover, multiplication produces aliasing errors, due to the periodicity of the Fourier series expansion of the solution. These errors can be removed by artificially padding the Fourier mode repre- sentation with wave number modes of zero amplitude at either ends of k-space. The amount of de-aliasing is subject to the degree of nonlinearity of the equation. For us, this entails that half of the wave number resolution for the solution needs to be zero - a quarter at each end ofk-space [170].

For the LWE we perform two types of numerical simulation, one without forcing and dissipation, and one with. Experimentally, the system is decaying from an initial condition with natural dissipation resulting from scattering at the edge of the cell and from the natural dissipation of the LC. However, this natural dissipation is relatively weak, so we perform an idealised decaying numerical simulation with no forcing and dissipation. Ideally, the WT regime is best observed in a non-equilibrium stationary steady state with the formation of coherent structures inhibited, hence we perform an additional simulations for the LWE and the SWE in this scenario. The low wave number dissipation allows for the removal of the wave action and suppression of solitons, whilst the high wave number dissipation allows for the removal of energy.

[170], using a time step that is small enough so that it satisfies the Courant-Friedrichs- Lewy (CFL) condition of max k ∂ωk ∂k ∆t <∆x, (4.38)

where∆x=L/N is the spacing of our spatial grid and L= 32π, is the length of our periodic box. Numerically we compute the non-dimensionalised equation given by

i∂ψ ∂z =−

∂2ψ

∂x2 +N(ψ) +i(Fk−Dk), (4.39) where N(ψ) represents the nonlinear part of our equations. Appendix F includes the derivation of the non-dimensional versions of both the LWE and SWE. The non- dimensional LWE contains an adjustable parameter α, defined in Appendix F. For the decaying setup, we set α = 128, so that the numerical simulation is in the same regime as the experiment. For the forced and dissipated cases, we setα= 1024, so that the long-wave limit is better realised. Fk =F(k) is the forcing profile, where energy

and wave action are injected into the system. We define this in Fourier space, over a specific range of wave numbers. For the direct cascade simulations we use a forcing profile given by F_kdirect=    Aexp(iθk) if 9≤k≤11 0 otherwise, (4.40)

whereAis the amplitude of forcing andθk is a random variable chosen from a uniform

distribution on [0,2π) at each wave number and at each time step. For the inverse cascade simulations, we apply forcing over a small range of wave numbers situated in the high wave number region. However, we must allow for some Fourier modes larger than the forcing scale, as to not restrict the development of the inverse cascade, and for dissipation purposes. Our forcing profile for the inverse cascade simulations is given by

F_kinverse=    Aexp(iθk) if ₁₆N −10≤k≤ N₁₆+ 10 0 otherwise. (4.41)

too steep for the formation of a bottleneck, or too shallow as to prevent a large enough inertial range developing. At low wave numbers, we use two types of dissipation profile. Firstly, we can use a hypo-viscosity term ∝k−4 ψk. However, this type of dissipation

profile has led to the WT description becoming invalid and a CB regime to develop [64]. In these situations, the CB scenario is avoided by the use of low wave number friction. The numerical dissipation profile Dk =D(k), removes wave action and energy

from the system at low and high wave numbers. For clarity, we splitDkinto the low and

high wave number contributionsDL

k andDHk. At high wave numbers our hyper-viscosity

profile is defined as

DH_k =νhyper k4ψk, (4.42a)

whereνhyper is the coefficient for the rate of dissipation. If we apply hypo-viscosity at low wave numbers, thenDL_k is given by

DL_k =    νhypok−4 ψk ifk6= 0 ψk= 0 ifk= 0,

whereνhypo is the coefficient for the rate of dissipation at low wave numbers. However, in situations where we use friction, thenD_kL is defined as

DL_k =    νfrictionψk if0≤k≤6 0 otherwise,

whereνfriction is the rate of friction dissipation.