While running George Bryan’s CM1, but I imagine that these will come up in other contexts too.
By decomposing the flow into a mean + perturbations and averaging, one obtains the Reynolds-Averaged Navier-Stokes (RANS) equations. These still contain a perturbation term (the Reynolds stress term), and this term is generally dealt with by prescribing it as some function of the mean flow. It’s possible to deal with this by considering eddy viscosity (relating to energy dissipation at small scales). One can write the Navier-Stokes equations (momentum equations) with a term that relies on a stress tensor, and this in turn can be expressed in terms of the eddy viscosity and a strain term. Turbulence schemes that account for fluid motions at scales smaller than the grid size are known as sub-gridscale (SGS) schemes.
- Smagorinsky: See here for the formula. Relates eddy viscosity to a Galilean invariant estimation of velocity differences over some characteristic lengthscale (the scale below which motions get filtered out).
- Turbulent Kinetic Energy (TKE): can be used as a prognostic variable to compute exchange coefficient for turbulent mixing (through which energy is dissipated). TKE can be produced either by dynamical motions (positive) or by thermodynamic processes (negative if stable wrt potential temperature) and is transferred to smaller scales via the turbulence cascade before getting dissipated at the Kolmogorov scale. KE “marginally” prefers this one. See here and here for more details.
- Parametrization Schemes: Here are some very handy notes from Chris Bretherton on parametrizing BL turbulence. This differs from the SGS schemes in large-eddy simulations in that LES schemes only simulate turbulence at very small scales (which are then used in conjunctions for turbulence at larger scales that are modeled directly), as opposed to the parametrization case, which simulates the entire turbulent flux. These can also be more computationally involved than the LES schemes, because the LES schemes have to be carried out at every grid point and every timestep. Idealized models often use mixed-layer modeling (assume u, v, and well-mixed tracers are vertically uniform, i.e. well-mixed), whereas forecast models often use local eddy diffusivity parametrizations and non-local K-profile parametrizations. See “A Review of PBL Parameterization Schemes and their Sensitivity in Simulating Southeastern US Cold Season Sever Weather Environments”
- YSU Planetary Boundary Layer Parametrization (not to be used with LES configuration): used in WRF; contains a countergradient term and an explicit entrainment term in the turbulent flux equation. Better during the day than at night, evaluated here.
- Direct Numerical Simulation: a simulation in a CFD model which solves the Navier-Stokes equations numerically and without including a turbulence model. Has to include a very wide range of spatial scales, all the way down to Kolmogorov microscales (scales at which viscosity dominates and kinetic energy -> heat). Unsurprisingly has a very, very high computational cost.
- time-splitting scheme that integrates higher-frequency (gravity, acoustic, “meteorologically relevant“) modes at a smaller timestep than it does the lower-frequency modes
- assumes that variations of both density and pressure from a statistically balanced state are small, and that that the relative vertical variation of potential temperature is also small
- ignores elastic compressibility of the fluid, thereby eliminating sound waves
- differs from Boussinesq in that the base density state is a function of the vertical coordinate; same in that it ignores dynamic variations of density except where gravity is involved
- Compressible Boussinesq
- density differences small relative to mean; neglect density unless multiplied by g
- for large-scale motions, the deviation pressure and density fields are still in hydrostatic balance (so long as vertical accelerations are smaller than the buoyancy)
Tells us atmospheric heat and moisture tendencies, microphysical rates, and surface rainfall (both amount and domain); here’s a guide to the possibilities in WRF, and another set of slides that are more of an introduction to microphysics schemes. The class number of the microphysics scheme (I think) refers to the number of states of water it includes in its prognostic variables, i.e. one class 3 scheme has water vapor, cloud water/ice, and rain/snow, whereas one class 5 scheme has water vapor, cloud, ice, rain, and snow. Here’s another set of slides discussing the differences between bulk and bin schemes.
- Kessler scheme (water only)
- warm rain – no ice; idealized microphysics; time-split rainfall
- one of the most simple bulk schemes
- LFO Scheme
- NASA-Goddard version of LFO Scheme
- Thompson scheme
- 6-class microphysics with graupel
- predicts ice and rain number concentrations
- has time-split fall terms
- Gilmore/Straka/Rasmussen version of LFO scheme
- Morrison double-moment scheme
- Rotunno-Emanuel (1987) simple water-only scheme
- NSSL 2-moment scheme (can have graupel only, graupel + hail)
Domain Boundary Conditions
BCs are important because they direct flow and can be used to specify fluxes into or out of the flow.
- 2D periodic BCs (for planar conditions) are also known as slab boundary conditions
- used when motion is expected to be one cell in a repeated set of motions
- computationally simple (no reflecting off boundaries, or damping into the boundaries)
- allow radiative fluxes through the domains, apparently difficult to get right
- rigid walls, free slip
- i.e. no friction between the fluid and the wall
- rigid walls, no slip
- at a solid wall, fluid has no velocity relative to the wall; example of Dirichlet BC
- physically, means attraction between wall and fluid particles (adhesion) > attraction between particles (cohesion)
- used for viscous flows, doesn’t work as well at low pressure
Types of Initialization
- Warm bubble
- Cold pool
- at the surface, indicate stable air; high up, more unstable
- Line of warm bubbles
- Cold blob
- Rotunno-Emanuel tropical cyclone
- Line thermal with random perturbations
- Forced convergence
- Momentum forcing (Morrison et al 2015)
- Skamarock-Klemp IG wave
Base State Wind Profile
- RKW-type profile
- Rotunno, Klemp, Weisman Theory, mainly relating to structure of squall lines; explains how storms can be sustained by cold pools, mechanism works better with low-level wind shear (to keep the cold pool under the storm as opposed to moving away)
- I guess an RKW-type profile is one (with low level shear) that’d support squall lines?
- Weisman-Klemp supercell
- Multicell (“ordinary” storm as opposed to a rotating supercell)
- Weisman-Klemp multicell
- Dornbrack et al analytic profile
- Damping proportional to mass and stiffness