## Models of Morning Glory as Interacting Non-Linear Waves

thesis

posted on 22.03.2017 by Igor Andriy Korostil#### thesis

In order to distinguish essays and pre-prints from academic theses, we have a separate category. These are often much longer text based documents than a paper.

The research
presented in this thesis is motivated by a rare phenomenon known as the Morning
Glory. This is a series of roll clouds that can be observed daily in the latter
months of the year in the vicinity of the Gulf of Carpentaria in Queensland,
Australia. Data collected during several scientific expeditions intended to
clarify the nature of the phenomenon revealed that it can behave both as an
undular bore and a series of solitary waves.

Recent high-resolution simulations of a possible Morning Glory generation mechanism, specifically, collision of two sea breezes, indicated that the phenomenon may be sufficiently well described by the resonant interaction of internal solitary waves governed by a system of coupled Korteweg de-Vries (KdV) equations. The latter were shown to emerge in the case of resonant interaction of two solitary waves with nearly equal phase speeds but belonging to different modes.

Here, we first derive a system of coupled forced KdV equations for a fluid configuration with a bell-shaped topography. The derivation is an extension of the previously derived systems to include forcing. Then we consider two three-layer fluid configurations in order to relate the previously derived coupled equations to key physical parameters inherent to stratified fluids. One configuration has both the basic flow and density constant within each layer while the other is described by piecewise linear shear flow continuous across the unperturbed interfaces and the piecewise constant density profile. We calculate the system coefficients in terms of fluid layer heights, flow velocities and densities for various resonances.

As it was demonstrated that the systems of the type that we obtained are typically not solvable by the inverse scattering transform and therefore have to be integrated numerically, our next step is to identify numerical methods that ensure the optimal conservation of the system invariants with respect to resource utilisation (CPU time). An extensive comparison of second and fourth order numerical methods shows that exponential integrators tend to outperform most other methods with a notable exception of the Linearly-Implicit Runge-Kutta (LIRK4) and Commutator-free fourth-order methods.

Next we propose a numerical iteration method for computation of solitary wave solutions of the system of forced KdV equations. This algorithm, which uses the forward-backward Euler differencing, is based on the Accelerated Imaginary-Time Evolution Method (AITEM) and involves normalising the solution at each time step to preserve the momentum (power).

Finally, we investigate instabilities of solitary wave solutions obtained using the mentioned AITEM-like method. To investigate the nonlinear evolution of instabilities the LIRK4 method is employed. We observe and describe several interesting instabilities reminiscent of the Morning Glory behaviour.

Thesis concludes with the summary of results.

Recent high-resolution simulations of a possible Morning Glory generation mechanism, specifically, collision of two sea breezes, indicated that the phenomenon may be sufficiently well described by the resonant interaction of internal solitary waves governed by a system of coupled Korteweg de-Vries (KdV) equations. The latter were shown to emerge in the case of resonant interaction of two solitary waves with nearly equal phase speeds but belonging to different modes.

Here, we first derive a system of coupled forced KdV equations for a fluid configuration with a bell-shaped topography. The derivation is an extension of the previously derived systems to include forcing. Then we consider two three-layer fluid configurations in order to relate the previously derived coupled equations to key physical parameters inherent to stratified fluids. One configuration has both the basic flow and density constant within each layer while the other is described by piecewise linear shear flow continuous across the unperturbed interfaces and the piecewise constant density profile. We calculate the system coefficients in terms of fluid layer heights, flow velocities and densities for various resonances.

As it was demonstrated that the systems of the type that we obtained are typically not solvable by the inverse scattering transform and therefore have to be integrated numerically, our next step is to identify numerical methods that ensure the optimal conservation of the system invariants with respect to resource utilisation (CPU time). An extensive comparison of second and fourth order numerical methods shows that exponential integrators tend to outperform most other methods with a notable exception of the Linearly-Implicit Runge-Kutta (LIRK4) and Commutator-free fourth-order methods.

Next we propose a numerical iteration method for computation of solitary wave solutions of the system of forced KdV equations. This algorithm, which uses the forward-backward Euler differencing, is based on the Accelerated Imaginary-Time Evolution Method (AITEM) and involves normalising the solution at each time step to preserve the momentum (power).

Finally, we investigate instabilities of solitary wave solutions obtained using the mentioned AITEM-like method. To investigate the nonlinear evolution of instabilities the LIRK4 method is employed. We observe and describe several interesting instabilities reminiscent of the Morning Glory behaviour.

Thesis concludes with the summary of results.