Some Modeling Stuff

While running George Bryan’s CM1, but I imagine that these will come up in other contexts too.

Turbulence Schemes

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 uv, 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.

Pressure Solvers

  • Compressible
  • Klemp-Wilhelmson
    • time-splitting scheme that integrates higher-frequency (gravity, acoustic, “meteorologically relevant“) modes at a smaller timestep than it does the lower-frequency modes
  • Anelastic
    • 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
  • Incompressible
  • 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)

Microphysics/Moisture Schemes

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.

  • Periodic
    • 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)
  • open-radiative
    • 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

Rayleigh Damping

  • Damping proportional to mass and stiffness

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Improvements to the ECMWF

Forecasts can go out for up to half a day more; 3x spatial resolution previously, apparently more energy efficient as well;

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Kerry Emanuel and Dan Cziczo Reddit AMA

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CHIPS is the Coupled Hurricane Intensity Prediction System (not to be confused with SHIPS, the Statistical Hurricane Intensity Prediction Scheme); it’s a coupled atmosphere-ocean model that predicts intensities based on environmental variables only.  And here’s a nice paper that Kerry (and others) wrote about it.  Some notes:

  • predictions of hurricane tracks have gotten much better over the last 30 years, while predictions of intensity have had less improvement, though the most successful intensity prediction schemes have been statistical ones (though SHIPS has its problems too, is pretty bad for secondary eyewalls)
  • changes in storm intensity (and formation of secondary eyewalls) are due to a combination of environmental variables and internal processes.  Molinari, Vollaro and Nong, Emanuel papers suggest that formation of a secondary eyewall is driven by external processes, while its evolution is affected more by processes internal to the storm.
  • (SEs associated with significant intensity fluctuations)
  • To test the prediction that intensity changes to storm, Kerry and authors wrote a model for hurricane intensity in which the intensity depends only on external variables (CHIPS!).  CHIPS takes interaction with the ocean into account, since hurricanes cool SSTs as they pass and this can be a negative feedback process (see Schade, Emanuel).
  • CHIPS assumes axisymmetric storm with hydrostatic and gradient wind balance; that the vortex is neutrally stable to slantwise convection, in which the temperature follows a moist adiabatic lapse rate along surfaces of constant angular momentum.  (Can I use this..?)
  • Vertical structure determined by boundary layer moist entropy and vorticity at tropopause; water vapor represented by boundary layer moist entropy and middle troposphere layer (seriously where is this boundary layer info?)
  • Model variables are in potential radius (R) coordinates; means that the center of the storm, near the eye, is really well-resolved (is important because of the pressure gradient etc there), whereas less vital parts of the storm from the edges aren’t as well-resolved.  The potential radius is proportional to the square root of the absolute angular momentum/mass and is defined in terms o the Coriolis parameter, the physical radius r, and the azimuthal velocity V as

  • The average resolution (in R coordinates) is 20km, but can be as small as 1-2 km near the eye of the storm.
  • Their SST varies with time and radius to reflect the air-sea coupling with the storm.  (Apparently there’s a 1-D ocean model…should I use it…mainly seems to depend on entrainment-type processes like for final project, see Schade paper.)
  • Apparently shear can suppress the air-ocean interaction.
  • CHIPS runs really fast, but its axisymmetry means that environmental vertical wind shear has to be parametrized, which is a little unfortunate given how important wind shear is (see Tang, Emanuel ventilation paper).  But there is a parametrization scheme that looked at observed intensity versus the model ones and used a multiple regression algorithm to relate other environmental variables to intensity and shear (?), but they end up ventilating the storm at middle levels.
  • Models were initialized with real or forecasted storm tracks with some environmental info; used NCEP SSTs and atmospheric temperature info from 0000 UTC near the beginning of each storm’s life to calculate the potential intensity.
  • Model’s best in low shear or in areas where ocean mixed layer doesn’t depart too far from climatology.  More sensitive to initialization conditions when shear is present.
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Madden-Julian Oscillation

Wikipedia says it’s the largest contributor to intraseasonal (time length 30-90 days) variability in the atmosphere, and manifests as an eastward-propagating, alternating stormy/wet phase and dry phase oscillation.  The MJO is a coupling between the large-scale atmospheric circulation and mesoscale tropical deep convection.

You can see it in the Hovmoller diagram above (used for looking at waves; has latitude/longitude on the x-axis and time, increasing as you go down, on the y-axis).  So this shows that the rainy part (has precipitation and strong deep convection) propagates from west to east as time passes.

The MJO possibly plays a role in modulating tropical cyclogenesis; according to Maloney and Hartmann, twice as many North Pacific storms are found with 850 mb westerlies than with 850 mb easterlies.  In the Gulf of Mexico and western Caribbean, tropical cyclones are four times as likely to be found during the western phase of the MJO rather than the eastern phase.  The MJO also plays a considerable role in monsoons and in rainfall in parts of Asia, Africa, Australia, South America, and the west coast of North America.

Chidong Zhang wrote a really nice review article.  Some interesting points:

  • Possibly the MJO made favorable conditions for early Polynesians to sail to other islands?!
  • Possibly the MJO has an effect on a) the earth’s angular momentum and b) its magnetic field
  • As you can see from the Hovmoller diagram, the rainy region of strong deep convection (the active phase) has regions of weak convection on either side (the inactive or suppressed phases).  There’s a zonal overturning circulation between these, and apparently it goes through the whole height of the troposphere.  There are strong westerly winds at the active center and to the west of it, while there are strong easterly winds on the other side (and these are reversed in the upper troposphere).
  • The eastward propagation speed is relatively slow (5 m/s)
  • apparently both Kelvin and Rossby wave structures are “dynamically essential” to the MJO (look up Kelvin waves)
  • “The apparent eastward propagation of the large-scale convective center of the MJO is due to consecutive development of new convective systems, each on average slightly to the east of the previous one.”
  • “The diurnal cycle in deep convection is modulated by the MJO. Over the Maritime Continent the diurnal cycle is the strongest during the inactive phase of the MJO and becomes diminished during the active phase [Sui and Lau, 1992], possibly because of the interruption of the local sea breeze circulation by the large-scale circulation of the MJO”
  • MJO strength has peaks in SH summer/fall and NH summer
  • The MJO looks a lot like a Kelvin wave, but it travels much more slowly than a Kelvin wave would propagate, so there’s still a question of what the energy source (and dissipation) of the MJO are.  Could be an atmospheric response to some other kind of forcing (fluctuations in precipitation, a stochastic heating source) or just an atmospheric instability (moisture-convergence including friction, surface evaporation)
  • Apparently the MJO has been simulated in theoretical/idealized models, but it’s been a lot harder to simulate in GCMs (as of 2005), unclear why

According to Wikipedia, a Kelvin wave balances the Coriolis force against some type of topography and is non-dispersive, so it retains its shape as it propagates.  Apparently the equator can act as a waveguide (?? EDIT: because Coriolis deflection gets too big at some threshold latitude and stops it from propagating poleward, apparently, though I guess when it bounces back it must interfere constructively…???) and Kelvin waves have no meridional component to their propagation.  Kelvin waves can be free (don’t draw energy from surroundings) or forced (do), and in this case, they’d draw energy from deep convection in the troposphere.

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