Well-Posedness of Strong Solutions to the Anelastic Equations of Stratified Viscous Flows

We establish the local and global well-posedness of strong solutions to the two- and three-dimensional anelastic equations of stratified viscous flows. In this model, the interaction of the density profile with the velocity field is taken into account, and the density background profile is permitted to have physical vacuum singularity. The existing time of the solutions is infinite in two dimensions, with general initial data, and in three dimensions with small initial data.

Energy Decompositions for Moist Boussinesq and Anelastic Equations with Phase Changes

To define a conserved energy for an atmosphere with phase changes of water (such as vapor and liquid), motivation in the past has come from generalizations of dry energies—in particular, from gravitational potential energy ρgz. Here a new definition of moist energy is introduced, and it generalizes another form of dry potential energy, proportional to θ2, which is valuable since it is manifestly quadratic and positive definite. The moist potential energy here is piecewise quadratic and can be decomposed into three parts, proportional to bu2Hu, bs2Hs, and M2Hu, which represent, respectively, buoyant energies and a moist latent energy that is released upon a change of phase. The Heaviside functions Hu and Hs indicate the unsaturated and saturated phases, respectively. The M2 energy is also associated with an additional eigenmode that arises for a moist atmosphere but not a dry atmosphere. Both the Boussinesq and anelastic equations are examined, and similar energy decompositions are shown in both cases, although the anelastic energy is not quadratic. Extensions that include cloud microphysics are also discussed, such as the Kessler warm-rain scheme. As an application, empirical orthogonal function (EOF) analysis is considered, using a piecewise quadratic moist energy as a weighted energy in contrast to the standard L2 energy. By incorporating information about phase changes into the energy, the leading EOF modes become fundamentally different and capture the variability of the cloud layer rather than the dry subcloud layer.

Some travels in the land of nonlinear convection and magnetism

Rotating stars with convection zones are the great builders of magnetism in our universe. Seeking to understand how turbulent convection actually operates, and so too the dynamo action that it can achieve, has advanced through distinctive stages in which Jean-Paul Zahn was often a central player, or joined by his former students. Some of the opening steps in dealing with the basic nonlinearity in such dynamics involved modal equations (with specified horizontal structure) to study convective amplitudes and heat transports achieved as solutions equilibrated by feeding back on the mean stratification. These dealt in turn with laboratory convection, with penetrative convection in Boussinesq settings, then with compressible penetration via anelastic equations in simple geometries, and finally with stellar penetrative convection in A-type stars that coupled two convection zones. Advances in computation power allowed 2-D fully compressible simulations, and then 3-D modeling including rotation, to revisit some of these convection and penetration settings within planar layers. With externally imposed magnetic fields threading the 2-D layers, magnetoconvection could then be studied to see how the flows concentrated the fields into complex sheets, or how new classes of traveling waves could result. The era of considering turbulent convection in rotating spherical shells had also arrived, using 3-D MHD codes such as ASH to evaluate how the solar differential rotation is achieved and maintained. Similarly the manner in which global magnetic fields could be built by dynamo action within the solar convection zone took center stage, finding that coherent wreaths of strong magnetism could be built, and also cycling solutions with field reversals. The coupling of convection and magnetism continues as a vibrant research subject. It is also clear that stars like the Sun do not give up their dynamical mysteries readily when highly turbulent systems are at play.

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 1: Nonhydrostatic inertia–gravity modes

We have used a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the nonhydrostatic anelastic inertia–gravity modes on a midlatitude f plane. The dispersion equations are derived from the linearized anelastic equations that are discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of both horizontal grid spacing and vertical wavenumber are analyzed, and the role of nonhydrostatic effects is discussed. We also compare the results of the normal-mode analyses with numerical solutions obtained by running linearized numerical models based on the various horizontal grids. The sources and behaviors of the computational modes in the numerical simulations are also examined. Our normal-mode analyses with the Z, C, D, A, E and B grids generally confirm the conclusions of previous shallow-water studies for the cyclone-resolving scales (with low horizontal wavenumbers). We conclude that, aided by nonhydrostatic effects, the Z and C grids become overall more accurate for cloud-resolving resolutions (with high horizontal wavenumbers) than for the cyclone-resolving scales. A companion paper, Part 2, discusses the impacts of the discretization on the Rossby modes on a midlatitude β plane.

Impacts of the Horizontal and Vertical Grids on the Numerical Solutions of the Dynamical Equations. Part I: Nonhydrostatic Inertia-Gravity Modes

We have used a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the nonhydrostatic anelastic inertia-gravity modes on a midlatitude f-plane. The dispersion equations are derived from the linearized anelastic equations that are discretized on the Z, C, D, CD, (DC), A, E, and B horizontal grids, and on the L and CP vertical grids. The effects of both horizontal grid spacing and vertical wave number are analyzed, and the role of nonhydrostatic effects is discussed. We also compare the results of the normal-mode analyses with numerical solutions obtained by running linearized numerical models based on the various horizontal grids. The sources and behaviors of the computational modes in the numerical simulations are also examined. Our normal-mode analyses with the Z, C, D, A, E and B grids generally confirm the conclusions of previous shallow-water studies for the cyclone resolving scales (with low horizontal wavenumbers). We conclude that for cloud-resolving resolutions (with high horizontal wavenumbers) the Z and C grids become overall more accurate than for the cyclone-resolving scales, aided by nonhydrostatic effects. A companion paper, Part II, discusses the impacts of the discretization on the Rossby modes on a midlatitude β-plane.

The breakdown of the anelastic approximation in rotating compressible convection: implications for astrophysical systems

The linear theory for rotating compressible convection in a plane layer geometry is presented for the astrophysically relevant case of low Prandtl number gases. When the rotation rate of the system is large, the flow remains geostrophically balanced for all stratification levels investigated and the classical (i.e. incompressible) asymptotic scaling laws for the critical parameters are recovered. For sufficiently small Prandtl numbers, increasing stratification tends to further destabilize the fluid layer, decrease the critical wavenumber and increase the oscillation frequency of the convective instability. In combination, these effects increase the relative magnitude of the time derivative of the density perturbation contained in the conservation of mass equation to non-negligible levels; the resulting convective instabilities occur in the form of compressional quasi-geostrophic oscillations. We find that the anelastic equations, which neglect this term, cannot capture these instabilities and possess spuriously growing eigenmodes in the rapidly rotating, low Prandtl number regime. It is shown that the Mach number for rapidly rotating compressible convection is intrinsically small for all background states, regardless of the departure from adiabaticity.

Density Stratification

This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus geometries, using entropy and pressure as working thermodynamic variables or using temperature and pressure, for both convectively unstable and stable regions. In particular, it considers anelastic approximation and how to formulate the anelastic equations, as well as the anelastic form of mass conservation, momentum conservation with entropy as a variable, internal energy conservation with entropy as a variable, and temperature as a variable. It also discusses possible choices for a reference state, focusing on polytropes, before explaining modifications to the numerical method and presenting the numerical simulations using the anelastic model.

Reflections on dissipation associated with thermal convection

Buoyancy-driven convection is modelled using the Navier–Stokes and entropy equations. It is first shown that the coefficient of heat capacity at constant pressure, ${c}_{p} $, must in general depend explicitly on pressure (i.e. is not a function of temperature alone) in order to resolve a dissipation inconsistency. It is shown that energy dissipation in a statistically steady state is the time-averaged volume integral of $- \mathrm{D} P/ \mathrm{D} t$ and not that of $- \alpha T(\mathrm{D} P/ \mathrm{D} t)$. Secondly, in the framework of the anelastic equations derived with respect to the adiabatic reference state, we obtain a condition when the anelastic liquid approximation can be made, $\gamma - 1\ll 1$, independent of the dissipation number.

Minimal models for precipitating turbulent convection

Simulations of precipitating convection would typically use a non-Boussinesq dynamical core such as the anelastic equations, and would incorporate water substance in all of its phases: vapour, liquid and ice. Furthermore, the liquid water phase would be separated into cloud water (small droplets suspended in air) and rain water (larger droplets that fall). Depending on environmental conditions, the moist convection may organize itself on multiple length and time scales. Here we investigate the question, what is the minimal representation of water substance and dynamics that still reproduces the basic regimes of turbulent convective organization? The simplified models investigated here use a Boussinesq atmosphere with bulk cloud physics involving equations for water vapour and rain water only. As a first test of the minimal models, we investigate organization or lack thereof on relatively small length scales of approximately 100 km and time scales of a few days. It is demonstrated that the minimal models produce either unorganized (‘scattered’) or organized convection in appropriate environmental conditions, depending on the environmental wind shear. For the case of organized convection, the models qualitatively capture features of propagating squall lines that are observed in nature and in more comprehensive cloud resolving models, such as tilted rain water profiles, low-altitude cold pools and propagation speed corresponding to the maximum of the horizontally averaged, horizontal velocity.

Anelastic Internal Wave Packet Evolution and Stability

As upward-propagating anelastic internal gravity wave packets grow in amplitude, nonlinear effects develop as a result of interactions with the horizontal mean flow that they induce. This qualitatively alters the structure of the wave packet. The weakly nonlinear dynamics are well captured by the nonlinear Schrödinger equation, which is derived here for anelastic waves. In particular, this predicts that strongly nonhydrostatic waves are modulationally unstable and so the wave packet narrows and grows more rapidly in amplitude than the exponential anelastic growth rate. More hydrostatic waves are modulationally stable and so their amplitude grows less rapidly. The marginal case between stability and instability occurs for waves propagating at the fastest vertical group velocity. Extrapolating these results to waves propagating to higher altitudes (hence attaining larger amplitudes), it is anticipated that modulationally unstable waves should break at lower altitudes and modulationally stable waves should break at higher altitudes than predicted by linear theory. This prediction is borne out by fully nonlinear numerical simulations of the anelastic equations. A range of simulations is performed to quantify where overturning actually occurs.