Representation of Convective Processes in NWP Models Part I

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Representation of Convective Processes in NWP
Models (Part I)

• George H. Bryan
• NCAR/MMM
• Presentation at ASP Colloquium,
• The Challenge of Convective Forecasting
• 13 July 2006

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Goals

• To understand how deep moist convection (i.e.,
  thunderstorms) can be simulated with numerical
  models
• To review how this is done in NCARs real-time
  forecasts with the ARW Model

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Outline

• Part I What is a numerical model?
• Part II What resolution is needed to simulate
  convection in numerical models?

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Part I What is a numerical model?

• Its computer code!
• For example

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Example Control interface (namelist.input) for
the ARW Model
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Example Main solver for the ARW Model
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A numerical model is composed of

• A set of governing equations
• A specified domain
• A set of numerical methods
• A set of parameterizations
• Initial conditions and boundary conditions
• ? A specific numerical model is the result of a
  series of choices, approximations, and compromises

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Components of a numerical model

• A set of governing equations
• A specified domain
• A set of numerical methods
• A set of parameterizations
• Initial conditions and boundary conditions

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Commandments

• Honor thy models creator
• Thou shalt not covet thy neighbors supercomputer
• Thou shalt conserve mass
• Thou shalt conserve energy
• Thou shalt conserve momentum

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Governing Equations

• Must obey laws of physics
• Newtons laws
• Conservation of mass
• Conservation of momentum
• Laws of thermodynamics
• Conservation of internal energy
• Rules governing water and its phases
• Laws governing other relevant processes
• Chemistry
• Electric fields

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Example a dry equation set

• Conservation of momentum
• Conservation of energy
• Conservation of mass

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Example another dry equation set

• Conservation of momentum
• Conservation of energy
• Conservation of mass

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There is debate about the exact form of the
governing equations, particularly for moist
conditions, for example
from Bannon (2002)
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The point

• Almost every model uses a (slightly) different
  equation set.
• Why?
• Different applications
• climate vs weather
• tropical vs polar
• Debate about what matters
• Moist effects (raindrops interacting with air)
• Unknown magnitude (e.g., viscous
  dissipation/heating)

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ARW Model equations

• The ARW Model is one of the first cloud-scale
  models designed specifically to conserve mass,
  momentum, energy
• However
• Issues with mass conservation (water)
• Momentum conservation is not guaranteed (?)
• Conserves internal energy of dry air

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Components of a numerical model

• A set of governing equations
• A specified domain
• A set of numerical methods
• A set of parameterizations
• Initial conditions and boundary conditions

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Idealized Domains

• Could be as simple as this room
• Could be a cloud in a box with a rigid lid
• Could be a sphere with no land (aqua-planet)

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A global domain
? Uses structured, rectangular grid
from mitgcm.org
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Staggered grids
(e.g., MM5)
(e.g., ARW)
from Randall (1994)
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Non-rectangular grids
hexagonal
triangular
from ccrma.standford.edu/bilbao
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A global triangular grid with mesh refinement
from Thomas Heinze, DWD
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Rectangular grids with nested domains
from Bryan and Fritsch (2000)
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How is a grid chosen?

• Many factors
• Ease of use
• Accuracy
• Performance
• Application
• Experience
• Legacy

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Vertical coordinates a terrain-following
coordinate
from Xue et al. (2000)
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Coordinate transformation
In the model code, it looks like a regular,
rectangular mesh, e.g.
Vertically stretch grid
Grid with curved upper boundary
from Tannehill et al. (1997)
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Coordinate transformationExample ARPS equation
for u
ARPS equations are not written in (x,y,z) space.
They are written in (?,?,?) space. (a
curvilinear coordinate system)
from Xue et al. (1995)
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Other types of vertical coordinates
from Pielke (2002)
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Terrain-following coordinate

• Very common (MM5, ARPS, ARW, etc)
• But, has a known limitation
• Change in terrain height between two grid points
  must be less than vertical grid spacing (I think
  see Mahrer 1984)
• This becomes a real problem with cloud-scale
  model grids
• Example atmosphere at rest

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Example u (every 1 m/s) from a simulation of a
stably stratified atmosphere at rest
? Not all features in model output are real!
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Domain Whats in ARW?

• Structured, rectangular grid with nests on a
  C-grid
• Hybrid terrain-following/hydrostatic-pressure
  vertical coordinate
• Why?
• Experience at NCAR
• Cloud-scale resolution

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Components of a numerical model

• A set of governing equations
• A specified domain
• A set of numerical methods
• A set of parameterizations
• Initial conditions and boundary conditions

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The essence of the problem

• Consider the equation for potential temperature
  for dry, inviscid flow
• This is not easy to implement into a computer
  code
• Computers add/subtract/multiply/divide, but they
  dont differentiate/integrate

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Basic types of numerical methods

• Finite difference
• Based on a grid (or mesh)
• Uses Taylor series approximations to differential
  terms
• Finite volume
• Based on fluxes in-to/out-of control volumes
• Triangles, hexagons
• Spectral
• Specifies fields in Fourier space

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Finite differences

• ARW is a finite difference-based model
• Taylor series
• an infinite series
• at some point we truncate the higher-order terms
• for example

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Start with
Ignore all but first few terms, the rest will be
a remainer (R)
Solve for ?f/?x

• This is called a forward difference
  approximation to ?f/?x.
• -R/?x is called the truncation error

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Lets examine this formulations truncation error
(T.E.)

• Because the T.E. is proportional to ?x, we say
  that the error is of O(?x)
• This is more commonly referred to as first
  order truncation error
• If T.E. is proportional to ?x2, is is a second
  order scheme
• If T.E. is proportional to ?x6, is is a sixth
  order scheme

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Notes on truncation error

• Truncation error (or the order of a scheme)
  tells you nothing about its accuracy
• It tells you how the errors change as grid
  spacing changes
• Notice that error is an inherent part of model
  design (error is guaranteed!)
• What we know about, we can deal with
• (Knowledge is power)

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Analytic solution to the advection equation

• E exact
• 2 2nd order centered (e.g., MM5)
• 4 4th-order centered (e.g., ARPS)

from Durran (1999)
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Example translate an 8? feature across a grid
Initial condition analytic final state
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A brief, math-free introduction to
Fourier/spectral analysis

• Any real field can be represented by a series of
  sin waves with two pieces of information
• amplitude
• phase
• The (squared) amplitude of these waves as a
  function of wavenumber is the power spectrum

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Analytic solution to the advection equation

• E exact
• 2 2nd order centered (e.g., MM5)
• 4 4th-order centered (e.g., ARPS)

from Durran (1999)
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Example translate an 8? feature across a grid
Initial condition analytic final state
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Leapfrog in time, 2nd-order centered in space
(e.g., MM5)
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Leapfrog in time, 4th-order centered in space
(e.g., ARPS)
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Runge-Kutta in time, 6th-order centered in space
(e.g., ARW)
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Runge-Kutta in time, 5th-order upwind-biased in
space (e.g., ARW)
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An introduction to filters/diffusion

• Because we know the error is there, we should
  remove it
• This is why models have filters / diffusion /
  smoothers / dampers / mixing
• Filtering -- especially at small scales -- is a
  good thing!

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Analytic solution to the artificial diffusion
terms

• 2 ?2 (e.g., Eta?)
• 4 ?4 (e.g., MM5, ARPS)
• 6 ?6 (e.g., ARW)

from Durran (1999)
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Example ARW simulation over Utah
Essentially no filter
?6 diffusion
from Knievel et al. (2005)
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ARW Model forecast ? 2 km
from Jack Kain, NSSL
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The point

• Numerical techniques have a direct effect on the
  models output
• Most of the differences are at small scales
• Some features in a models output are real, some
  come from numerical techniques
• My rule of thumb for ARW
• If its bigger than 6?, then its believable
• If its smaller than 6?, dont trust it

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ARW

• Mostly 2nd-order finite differences (on Arakawa-C
  grid)
• 5th- and 6th-order finite differences for
  advection terms
• only for constant flow
• More accurate with small-scale features
• Can be more costly
• ?6 diffusion
• Acts only at small scales

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Components of a numerical model

• A set of governing equations
• A specified domain
• A set of numerical methods
• A set of parameterizations
• Initial conditions and boundary conditions

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Subgrid-scale processes are handled with
parameterizations

• From AMS Glossary
• Subgrid-scale process Atmospheric processes
  that cannot be adequately resolved within a
  numerical simulation. Examples can include
  turbulent fluxes, phase changes of water,
  chemical reactions, and radiative flux
  divergence. Such processes are often
  parameterized in numerical integrations and even
  neglected in some applications.

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Parameterizations (aka physics) for cloud models

• Microphysics cloud drops, rain drops, snow,
  hail, etc
• Surface grass, sand, forest, rock, etc along
  with soil temperature, moisture, etc
• Sub-grid-scale turbulence boundary layer
  eddies, puffy Cu clouds
• Atmospheric radiation longwave (IR), shortwave
  (UV)

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Modifying a numerical model

• It is very difficult to modify a models
  governing equations
• It is very difficult to modify a models grid
  structure
• Some numerical techniques can be changed easily,
  others cannot
• It is extremely easy to modify a models
  parameterizations

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Whats in ARW, and why?

• There are many parameterizations in ARW, and the
  list is growing
• Why?
• Because we can.
• Because it matters.
• What gets into the model?
• Whatever someone has time to work on.

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Components of a numerical model

• A set of governing equations
• A specified domain
• A set of numerical methods
• A set of parameterizations
• Initial conditions and boundary conditions

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Initial conditions /data assimilation

• An important part of real-time forecasts
• Need to know the present, before you can predict
  the future
• Historically viewed as external to the numerical
  model but not any more
• Variational schemes (3DVAR, 4DVAR)
• Ensemble-based schemed (EnKF)

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Boundary conditions

• A global model needs only lower and upper
  boundary conditions
• A limited-area model also needs lateral boundary
  conditions
• Often comes from a global model forecast, or from
  a limited-area model with a larger domain
• Idealized simulations can use funky boundary
  conditions periodic, rigid walls, open
  wave-radiating, etc.

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In Summary

• Numerical models are complex!
• Choices have been made
• e.g., rectangular vs triangular
• Approximations have been made
• e.g., 2nd-order vs. 4th-order finite difference
• Compromises have been made
• Accuracy vs. efficiency

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My advice

• Know thy model!
• Read the documentation
• Read the journal articles
• Choose a model that was designed to simulate what
  you are studying
• Climate vs. weather
• Thunderstorms vs. puffy Cu
• Tropical vs. polar

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ltend of Part Igt