Title: Representation of Convective Processes in NWP Models Part I
1Representation 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
2Goals
- 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
3Outline
- Part I What is a numerical model?
- Part II What resolution is needed to simulate
convection in numerical models?
4Part I What is a numerical model?
- Its computer code!
- For example
5Example Control interface (namelist.input) for
the ARW Model
6Example Main solver for the ARW Model
7A 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
8Components 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
9Commandments
- Honor thy models creator
- Thou shalt not covet thy neighbors supercomputer
- Thou shalt conserve mass
- Thou shalt conserve energy
- Thou shalt conserve momentum
10Governing 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
11Example a dry equation set
- Conservation of momentum
- Conservation of energy
- Conservation of mass
12Example another dry equation set
- Conservation of momentum
- Conservation of energy
- Conservation of mass
13There is debate about the exact form of the
governing equations, particularly for moist
conditions, for example
from Bannon (2002)
14The 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)
15ARW 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
16Components 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
17Idealized 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)
18A global domain
? Uses structured, rectangular grid
from mitgcm.org
19Staggered grids
(e.g., MM5)
(e.g., ARW)
from Randall (1994)
20Non-rectangular grids
hexagonal
triangular
from ccrma.standford.edu/bilbao
21A global triangular grid with mesh refinement
from Thomas Heinze, DWD
22Rectangular grids with nested domains
from Bryan and Fritsch (2000)
23How is a grid chosen?
- Many factors
- Ease of use
- Accuracy
- Performance
- Application
- Experience
- Legacy
24Vertical coordinates a terrain-following
coordinate
from Xue et al. (2000)
25Coordinate 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)
26Coordinate 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)
27Other types of vertical coordinates
from Pielke (2002)
28Terrain-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
29Example u (every 1 m/s) from a simulation of a
stably stratified atmosphere at rest
? Not all features in model output are real!
30Domain 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
31Components 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
32The 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
33Basic 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
34Finite differences
- ARW is a finite difference-based model
- Taylor series
- an infinite series
- at some point we truncate the higher-order terms
- for example
35Start 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
36Lets 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
37Notes 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)
38Analytic solution to the advection equation
- E exact
- 2 2nd order centered (e.g., MM5)
- 4 4th-order centered (e.g., ARPS)
from Durran (1999)
39Example translate an 8? feature across a grid
Initial condition analytic final state
40A 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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55Analytic solution to the advection equation
- E exact
- 2 2nd order centered (e.g., MM5)
- 4 4th-order centered (e.g., ARPS)
from Durran (1999)
56Example translate an 8? feature across a grid
Initial condition analytic final state
57Leapfrog in time, 2nd-order centered in space
(e.g., MM5)
58Leapfrog in time, 4th-order centered in space
(e.g., ARPS)
59Runge-Kutta in time, 6th-order centered in space
(e.g., ARW)
60Runge-Kutta in time, 5th-order upwind-biased in
space (e.g., ARW)
61An 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!
62Analytic 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)
63Example ARW simulation over Utah
Essentially no filter
?6 diffusion
from Knievel et al. (2005)
64ARW Model forecast ? 2 km
from Jack Kain, NSSL
65The 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
66ARW
- 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
67Components 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
68Subgrid-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.
69Parameterizations (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)
70Modifying 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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72Whats 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.
73Components 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
74Initial 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)
75Boundary 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.
76In 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
77My 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
78ltend of Part Igt