Representation of Convective Processes in NWP Models (Part I) George H. Bryan NCAR/MMM Presentation at ASP Colloquium, “The Challenge of Convective Forecasting”

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

Goals

• To understand how deep moist convection (i.e., thunderstorms) can be simulated with numerical models

• To review how this is done in NCAR’s real-time forecasts with the ARW Model

Outline

• Part I: What is a numerical model?

• Part II: What resolution is needed to simulate convection in numerical models?

Part I: What is a numerical model?

• It’s computer code!

• For example …

Example: Control interface (namelist.input) for the ARW Model

Example: Main solver for the ARW Model

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

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

Commandments

1. Honor thy model’s creator

2. Thou shalt not covet thy neighbor’s supercomputer

3. Thou shalt conserve mass

4. Thou shalt conserve energy

5. Thou shalt conserve momentum

6. …

Governing Equations

• Must obey laws of physics– Newton’s laws

• Conservation of mass• Conservation of momentum

– Laws of thermodynamics• Conservation of internal energy• Rules governing water and it’s phases

– Laws governing other relevant processes• Chemistry• Electric fields

Example: a dry equation set

• Conservation of momentum:

• Conservation of energy:

• Conservation of mass:

∂ui∂t+ u j

∂ui∂x j

+1

ρ

∂p

∂xi+ δ i3g −

1

ρ

∂σ ij∂x j

= 0

cv∂T∂t

+uj

∂T∂xj

+pρ

∂uj

∂xj

−κ∂2T

∂xj∂xj

=0

∂ρ∂t+∂ρu j∂x j

= 0

Example: another dry equation set

• Conservation of momentum:

• Conservation of energy:

• Conservation of mass:

∂ui∂t+ u j

∂ui∂x j

+ cpθ∂π

∂xi+ δ i3g = 0

∂θ∂t+ u j

∂θ

∂x j= 0

∂ρu j∂x j

= 0

from Bannon (2002)

There is debate about the exact form of the governing equations, particularly for moist conditions, for example:

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)

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







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”)

A global domain

Uses structured, rectangular grid

from: mitgcm.org

Staggered grids:

from: Randall (1994)

(e.g., MM5) (e.g., ARW)

Non-rectangular grids

from: ccrma.standford.edu/~bilbao

hexagonal triangular

A global triangular grid with mesh refinement

from: Thomas Heinze, DWD

Rectangular grids with nested domains

from: Bryan and Fritsch (2000)

How is a grid chosen?

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

Vertical coordinates: a terrain-following coordinate

from: Xue et al. (2000)

Coordinate transformationIn the model code, it looks like a regular, rectangular mesh, e.g.:

from: Tannehill et al. (1997)

Vertically stretch grid:

Grid with curved upper

boundary:

from: Xue et al. (1995)

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)

Other types of vertical coordinates

from: Pielke (2002)

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

Not all features in model output are real!

Example: u (every 1 m/s) from a simulation of a stably stratified atmosphere at rest

Domain: What’s 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







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 don’t differentiate/integrate

∂θ∂t= −u

∂θ

∂x− v∂θ

∂y− w

∂θ

∂z

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

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 …

f xi + Δx( ) = f xi( ) + Δx∂f∂x xi

+Δx2

2!∂2 f∂x2

xi

+L +Δxn

n!∂n f∂xn

xi


+Δx2

2!∂2 f∂x2

xi

+L +Δxn

n!∂n f∂xn

xi


+ R

Start with:

Ignore all but first few terms, the rest will be a remainer (R):

Solve for f/x

∂f∂x xi

=f xi + Δx( ) − f xi( )

Δx−R

Δx

• This is called a “forward difference” approximation to f/x.

• -R/Δx is called the “truncation error”

T.E.=−RΔx

=−Δx2

∂2 f∂x2

xi

+L

Let’s examine this formulation’s truncation error (T.E.):

• Because the T.E. is proportional to Δx, we say that the error is of O(Δx)

• This is more commonly referred to as “first order” truncation error

• If T.E. is proportional to Δx2, is is a “second order” scheme

• If T.E. is proportional to Δx6, is is a “sixth order” scheme

Notes on truncation error

• Truncation error (or “the order of a scheme”) tells you nothing about it’s 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)

from: Durran (1999)

Analytic solution to the advection equation• “E” = exact• “2” = 2nd order centered (e.g., MM5)• “4” = 4th-order centered (e.g., ARPS)

Example: translate an 8Δ feature across a gridInitial condition = analytic final state

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

from: Durran (1999)

Analytic solution to the advection equation• “E” = exact• “2” = 2nd order centered (e.g., MM5)• “4” = 4th-order centered (e.g., ARPS)

Example: translate an 8Δ feature across a gridInitial condition = analytic final state

Leapfrog in time, 2nd-order centered in space (e.g., MM5)

Leapfrog in time, 4th-order centered in space (e.g., ARPS)

Runge-Kutta in time, 6th-order centered in space (e.g., ARW)

Runge-Kutta in time, 5th-order upwind-biased in space (e.g., ARW)

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!

from: Durran (1999)

Analytic solution to the artificial diffusion terms• “2” = 2 (e.g., Eta?)• “4” = 4 (e.g., MM5, ARPS)• “6” = 6 (e.g., ARW)

from: Knievel et al. (2005)

Example: ARW simulation over Utah

Essentially no filter + 6 diffusion

from: Jack Kain, NSSL

ARW Model forecast

Δ = 2 km

The point:

• Numerical techniques have a direct effect on the model’s output

• Most of the differences are at small scales

• Some features in a model’s output are real, some come from numerical techniques– My “rule of thumb” for ARW:

• If it’s bigger than 6Δ, then it’s believable• If it’s smaller than 6Δ, don’t trust it

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







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.

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)

Modifying a numerical model

• It is very difficult to modify a model’s governing equations

• It is very difficult to modify a model’s grid structure

• Some numerical techniques can be changed easily, others cannot

• It is extremely easy to modify a model’s parameterizations

What’s 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.







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)

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.

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

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

<end of Part I>

Documents

Representation of Convective Processes in NWP Models (Part I) George H. Bryan NCAR/MMM Presentation at ASP Colloquium, “The Challenge of Convective Forecasting”