Introduction to Numerical Weather Prediction 4 September 2012.

Introduction to Numerical Weather Prediction

4 September 2012

Introduction

“You can create any solution that you want to with a numerical model.”

Scary thought, eh?

How do we go about getting the “right” solution – or at least a reasonable facsimile thereof?

Components of an NWP System

Key Thematic Elements

• Underlying framework for NWP (Chs. 2-3)

• Physical process parameterizations (Chs. 4-5)

• Model initialization methods (Ch. 6)

• Applications of numerical models (Chs. 7+)

Course Structure

We will make heavy use of the assigned text for this course. I anticipate lecturing for approximately 60 min each class, leaving 15 min for discussion and questions.

Thus, please do read the section(s) to be covered in class ahead of time!

Basic Equation Set

Horizontal Momentum Equations

xFvwx

p

a

uw

a

uv

z

uw

y

uv

x

uu

t

u

)sincos(21tan

yFuy

p

a

uw

a

u

z

vw

y

vv

x

vu

t

v

sin21tan2

φ = latitude, a = radius of the Earth, Ω = rotational frequency of Earth, F = friction

advection terms curvature terms frictionCoriolis termspres. grad.time derivs.

Basic Equation Set

Vertical Momentum Equation

zFguz

p

a

vu

z

ww

y

wv

x

wu

t

w

cos2

122

φ = latitude, a = radius of the Earth, Ω = rotational frequency of Earth, F = friction

advection terms curvatureterm

frictionCoriolis termpres. grad.time deriv. gravity

Basic Equation Set

Thermodynamic Equation

dt

dH

cw

y

Tv

x

Tu

t

T

pd

1)(

γ = lapse rate of temperature, γd = dry adiabatic lapse rate, Q = diabatic heating rate

advection terms dryadiabatic

term

diabaticheating

time deriv.

pd c

Qw

z

Tw

y

Tv

x

Tu

t

T

or, alternately…

Basic Equation Set

Continuity Equations(i.e., mass – top – and water vapor – bottom – are neither created nor destroyed)

)(z

w

y

v

x

u

zw

yv

xu

t

qv = water vapor mixing ratio, Qv = source/sink of qv due to phase changes

advection termstime derivs.

vvvvv Qz

qw

y

qv

x

qu

t

q

divergenceterm

Basic Equation Set

Ideal Gas Law

RTp

If you can solve the set of equations (often referred to as the primitive equations) given in the past few slides,

you can do NWP!

…but…

…how do we actually solve these equations???

What Do We Need?

• How do we represent these equations on a map?• How do we integrate them in time?• How do we integrate them in space?• How do we handle resolvable versus unresolvable processes?• How do we handle diabatic processes?• How do we handle friction?• How do we handle microphysical phase changes (water vapor

as well as other species – cloud, ice, graupel, rain, etc.)?• How do we obtain our initial atmospheric state?

…all among many relevant questions we will address!

Another scary thought:

Nearly everything we describe from here on out involves some sort of approximation.

This is true for the equations themselves, the methods used to solve them, the initial and boundary data used

to drive the model, and so on.

Before proceeding, a couple of notes…

Prognostic vs. Diagnostic

• Prognostic: any equation with a time-derivative is a prognostic equation; it can be integrated in time to produce a prediction

• Diagnostic: any equation without a time-derivative is a diagnostic equation; it can only be used to diagnose what is happening at a given time

Vertical Coordinate

• As given before, our vertical coordinate is height (z)– Note: the comment on pg. 7 is wrong (should be z, not p)

• Other vertical coordinates may be used if appropriate substitutions are made…– Pressure (p)– Potential temperature (θ)– Terrain-following (σ)

• We’ll discuss these further in a later chapter.

Reynolds Averaging

• The equations we outlined earlier are valid on all scales of motion

• But, in a numerical model, we have a grid with finite horizontal and vertical resolution…

• This leads to there being two scales of physical processes: those that we can resolve (larger) and those that we cannot resolve (smaller)

Reynolds Averaging

• The goal of Reynolds averaging is to separate out the resolvable and unresolvable scales of motion.

• We do so by splitting our dependent variables (u, T, p, etc.) into mean (resolved) and turbulent (perturbation/unresolved) components, e.g.,

'

'

'

ppp

TTT

uuu

Reynolds Averaging

• General idea:– Substitute such expressions into the primitive equations– Take the mean of each of the primitive equations– Simplify the result using Reynolds’ postulates

• Reynolds’ postulates…

0'''

0'

bababa

bababa

aa

a mean of all perturbations is zero

mean of a mean is equal to the mean

mean operator is commutative unless both variablesare turbulent/perturbation components

Reynolds Averaging: Example

Equation (2.11), dropping frictional parameterization…

We desire to substitute for u, v, w, p, and ρ, average the entire equation, and ultimately simplify the result using Reynolds’ postulates.

fvx

p

z

uw

y

uv

x

uu

t

u

1

Reynolds Averaging: Example

z

uw

y

uv

x

uuvf

x

p

z

uw

y

uv

x

uu

t

u

'

''

''

'1

resolvable scales aggregate (mean) effects onunresolvable scales

Reynolds Averaging: Example

• Typically, the unresolvable scale terms are written in terms of turbulent stresses.– See also: discussion related to equations (2.14)-(2.16).– These stresses are typically approximated utilizing physical

parameterization packages, to be covered in later lectures.

• We commonly drop the overbars from the primitive equations, making the Reynolds average(s) implicit.

Reynolds Averaging: An Aside

• The discussion of Reynolds averaging in the text discusses two things that we have neglected:– Frictional parameterization– Tensor notation

• We will also visit these topics later in the course.

Squelching Acoustic Waves

• The biggest thorn in the side of modelers is acoustic waves…– Acoustic waves aren’t meteorologically relevant.– But, their very high frequency means that a very short

time step (and a LOT of computer power) is needed to solve the primitive equations…

– …unless we do something to filter them out exclusively.

• Recall: the restoring mechanism for acoustic waves is their compressibility.

Squelching Acoustic Waves

• In other words, the propagation of acoustic waves is reliant upon the density adjusting to horizontal compression and expansion within the waves.

• To filter out acoustic waves, we need to eliminate the possibility of compressibility.

• Specifying conditions under which density can (and cannot) vary enables us to filter out acoustic waves.

Squelching Acoustic Waves

Squelching Acoustic Waves

• Method 2: Boussinesq Approximation– Assumes that density is constant except where it is tied to

buoyancy (or gravity).– Equivalently obtained (in part) by replacing (2.5) with:

– This relates density perturbations to temperature, rather than pressure, perturbations.

• Note: Boussinesq continuity equation conserves volume (not mass), not necessarily a good trade-off!

0

z

w

y

v

x

u

Squelching Acoustic Waves

• Method 3: Anelastic Approximation

• The Boussinesq approximation can be viewed as a simplified subset of the anelastic approximation.

• Obtained (in part) by replacing (2.5) with:

0

wz

vy

ux

Summary

• We have introduced the basic equation set used for numerical weather prediction.

• We have briefly described what is needed in order to solve this equation set.

• We have outlined the need for separating the resolvable from the unresolvable scales of motion.

• We have described the need for filtering out acoustic waves from the primitive equations.