Analysis of a Multi-Scale Asymptotic Model for Internal Gravity Waves in a Moist Atmosphere

The thesis presents the analysis of a reduced model for modulation of internal gravity waves by deep convective clouds. The starting point for the derivation are conservation laws for mass, momentum and energy coupled with a bulk micro-physics model describing the evolution of mixing ratios of water vapor, cloud water and rain water. A reduced model for the identified scales of the regime is derived, using multi-scale asymptotics. The closure of the model employs conditional averaging over the horizontal scale of the convective clouds. The resulting reduced model is an extension of the anelastic equations, linearized around a constant background state, which are well-known from meteorology. The closure of the model is achieved purely by analytical means and involves no additional physically motivated assumptions. The essential new parameter arising from the coupling to a micro-physics model is the area fraction of saturated regions on the horizontal scale of the convective clouds. It turns out that this parameter is constant on the employed short timescale. Hence the clouds constitute a constant background, modulating the characteristics of propagation of internal waves. The model is then investigated by analytical as well as numerical means. Important results are, among others, that in the model moisture (i) inhibits propagation of internal waves by reducing the modulus of the group velocity, (ii) reduces the angle between the propagation direction of a wave-packet and the horizontal, (iii) causes critical layers and (iv) introduces a maximum horizontal wavelength beyond which waves are no longer propagating but become evanescent. The investigated examples of orographically generated gravity waves also feature a significant reduction of vertical momentum flux by moisture. The model is extended by assuming systematically small under-saturation, that is saturation at leading order. The closure is similar to the original case but requires additional assumptions. The saturated area fraction in the obtained model is no longer constant but now depends nonlinearly on vertical displacement and thus on vertical velocity

Spectral deferred corrections with fast-wave slow-wave splitting

The paper investigates a variant of semi-implicit spectral deferred corrections (SISDC) in which the stiff, fast dynamics correspond to fast propagating waves ("fast-wave slow-wave problem"). We show that for a scalar test problem with two imaginary eigenvalues iλ_fast, iλ_slow, having Δt(|λ_fast|+|λ_slow|)<1 is sufficient for the fast-wave slow-wave SDC (FWSW-SDC) iteration to converge and that in the limit of infinitely fast waves the convergence rate of the non-split version is retained. Stability function and discrete dispersion relation are derived and show that the method is stable for essentially arbitrary fast-wave CFL numbers as long as the slow dynamics are resolved. The method causes little numerical diffusion and its semi-discrete phase speed is accurate also for large wave number modes. Performance is studied for an acoustic-advection problem and for the linearised Boussinesq equations, describing compressible, stratified flow. FWSW-SDC is compared to a diagonally implicit Runge-Kutta (DIRK) and IMEX Runge-Kutta (IMEX) method and found to be competitive in terms of both accuracy and cost

A Model for Nonlinear Interactions of Internal Gravity Waves with Saturated Regions

A model for interactions between non-hydrostatic gravity waves and deep convective narrow hot towers is presented. The starting point of the derivation are the conservation laws for mass, momentum and energy for compressible flows combined with a bulk micro-physic model. Using multiscale asymptotics, a set of leading order equations is extracted, valid for the specific scales of the investigated regime. These are a timescale of 100 s, a horizontal and vertical lengthscale of 10 km for the wave dynamics plus a second horizontal lengthscale of 1 km for the narrow hot towers. Because of the comparatively short horizontal scales, Coriolis effects are negligible in this regime. The leading order equations are then closed by applying conditional averages over the hot tower lengthscale, leading to a closed model for the wave-scale that retains the net effects of the smaller scale dynamics. By assuming a systematically small saturation deficit in the ansatz, the small vertical displacements arising in this regime suffice to induce leading order changes of the saturated area fraction. The latter is the essential parameter in the model arising from the micro-physics

Transparent boundary conditions - the pole condition approach

A new approach to derive transparent boundary conditions (TBCs) for wave, Schr¨odinger and drift-diffusion equations is presented. It relies on the pole condition approach and distinguishes physical reasonable and unreasonable solutions by the location of the singularities of the spatial Laplace transform U of the exterior solution. By the condition that U is analytic in some region TBCs are established. To realize the pole condition numerically, a Möbius transform is used to map the region of analyticity to the unit disc. There the Laplace transform is expanded in a power series. The equations coupling the coefficients of the power series with the interior provide the TBC. Numerical result for the damped wave equation show that the error introduced by truncating the power series decays exponentially in the number of coefficients

Transparent Boundary Conditions Based on the Pole Condition

Transparent boundary conditions for polygonal two-dimensional domains based on the pole condition approach are presented. The discretization of the exterior is done by innite trapezoids, which allows to dene a generalized distance variable. Taking the Laplace transform of the solution w.r.t the distance variable, incoming and outgoing solutions can be distinguished by the location of the singularities. Using special ansatz and test functions, the condition on the location of the singularities yields a new algorithmic realization of transparent boundary conditions

Moisture - Gravity Wave Interactions in a Multiscale Environment

Starting from the conservation laws for mass, momentum and energy together with a three species, bulk microphysic model, a model for the interaction of internal gravity waves and deep convective hot towers is derived by using multiscale asymptotic techniques. From the resulting leading order equations, a closed model is obtained by applying weighted averages to the smallscale hot towers without requiring further closure approximations. The resulting model is an extension of the linear, anelastic equations, into which moisture enters as the area fraction of saturated regions on the microscale with two way coupling between the large and small scale. Moisture reduces the effective stability in the model and defines a potential temperature sourceterm related to the net effect of latent heat release or consumption by microscale up- and downdrafts. The dispersion relation and group velocity of the system is analyzed and moisture is found to have several effects: It reduces energy transport by waves, increases the vertical wavenumber but decreases the slope at which wave packets travel and it introduces a lower horizontal cutoff wavenumber, below which modes turn into evanescent. Further, moisture can cause critical layers. Numerical examples for steady-state and time-dependent mountain waves are shown and the effects of moisture on these waves are investigated

Time Parallel Gravitational Collapse Simulation

This article demonstrates the applicability of the parallel-in-time method Parareal to the numerical solution of the Einstein gravity equations for the spherical collapse of a massless scalar eld. To account for the shrinking of the spatial domain in time, a tailored load balancing scheme is proposed and compared to load balancing based on number of time steps alone. The performance of Parareal is studied for both the sub-critical and black hole case; our experiments show that Parareal generates substantial speedup and, in the super-critical regime, can reproduce Choptuik's black hole mass scaling law

Convergence of Parareal with spatial coarsening

The effect is investigated of using a reduced spatial resolution in the coarse propagator of the time-parallel Parareal method for a finite difference discretization of the linear advection-diffusion equation. It is found that convergence can critically depend on the order of the interpolation used to transfer the coarse propagator solution to the fine mesh in the correction step. The effect also strongly depends on the employed spatial and temporal resolution

Holistic Data Centres: Next Generation Data and Thermal Energy Infrastructures

Digital infrastructure is becoming more distributed and requiring more power for operation. At the same time, many countries are working to de-carbonise their energy, which will require electrical generation of heat for populated areas. What if this heat generation was combined with digital processing