Eqn Scaling Proj

7/27/2019 Eqn Scaling Proj

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Equations, Scaling and Projections

Ravi S Nanjundiah

Centre For Atmospheric and Oceanic Sciences

Indian Institute of ScienceBangalore-560012

email:[email protected]

Class Notes for Climate Modelling

Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 1 / 29

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Outline

1IntroductionEquations

Momentum EquationConservation PropertiesScaling

2 Tropical Motion

Tropical Motion

Comparison of Two components of CoriolisVertical Momentum Eqn

Eqns in Height Co-ordinatesEqns in Pressure Co-ordinateJanjic’s Correction for Non-Hydrostatic Motion

3 ProjectionDiff Types

Polar Stereographic ProjectionLambert Conformal ProjectionMercator Projection


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The Equations

We have had a bird’s eye view of the climate system.We have found that we have a set of components which co-evolve and interact with eachother.

The major components are : The Atmosphere The Ocean Land-surface Sea-ice

We now look at the equations which are used in modelling atmosphere

We have already studied scaling of equations.

We will re-look at some of the scaling especially the scaling in the vertical

We will also look at how we project the sphere for different studies.

Sometimes for regional models we prefer not to use the sphere but its projection on differentsurfaces e.g. on a cyclinder (mercator) or on a cone etc.


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The EquationsWe have already seen that atmosphere is a fluid. Hence we can use equations of fluid flow todescribe the state of the atmosphereThe First eqn that we will use is the equation of mass conservation

1

ρ

D ρ

Dt +∇ · U = 0

Here U = iu + jv + kw We use conservation equations for various species (water vapour, NOx , SOx , methane,aerosols etc).

Dq

Dt = So q − Si q

Here So q is the source term (such as evaporation) and Si q is the sink term (such asprecipitation).We use the equation for conservation of momentum

D U

Dt = −2Ω× U−

1

ρ∇p + g + F

Is this on an inertial frame or non-inertial frame?

Is this on a sphere?We use the energy conservation equation

D θ

Dt =

θ

Tc p

Q

Here Q is the forcing function which includes: heat transfer due to radiation (longwave and shortwave)

heat transfer due to phase changes (due to rainfall and snowfall) heat transfer from surface (sensible heat)


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Momentum Equation(based on White and Bromley 1996).

We can write the components of the momentum equation in east-west (zonal), north-south(meridional) and vertical directions:

The zonal component is:

Du

Dt −

2Ω +

u

r cosφ

(v sinφ − w cosφ) +

1

ρr cosφ

∂ p

∂λ= F r λ

The meridional component is:

Dv

Dt +

2Ω +

u

r cosφ

u sinφ +

vw

r +

1

ρr

∂ p

∂ y = + F r φ

The Vertical component is:

Dw

Dt −

2Ω +

u

r cosφ

u cosφ−

v 2

r + g +

1

ρ

∂ p

∂ r = F rr

We can further write D Dt as

D

Dt =

∂

∂ t + u

∂

r cosφ∂λ+ v

∂

r ∂φ+ w

∂

∂ r =

∂

∂ t + u · grad

Upto here we have not made any approximations (note we are using r and not z )

We look at some conservation properties of these equationsRavi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 5 / 29

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Conservation Properties

In addition to mass, the above equations (at this point with no approximations ) also satisfyconservation laws for axial angular momentum

ρD

Dt ((u + Ωr cosφ)r cosφ) = ρF r λr cosφ−

∂ p

∂λ

for energy

ρD

Dt

1

2u2 + Φ + c p T

+ div (p u) = ρ(Q + u · F )

for potential vorticity

ρD

Dt

Z · grad θ

ρ

= Z · grad (

D θ

Dt ) + grad θ · (curlF )

Here Z is the absolute vorticity 2 Ω + curl u

We have used terms div , curl for the vector differential operations in the λ, φ and r system.Later we will use ∇· and ∇× for the sytem after hydrostatic approximation

Any approximations we make to the above system of equations should also have theseconservative properties.


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Scaling of Momentum Eqns

We have already looked at scaling of momentum equations for various scales of motion

We have found that for large-scale motion, geostrophy is a good approximation (what are thegeostrophic eqns?)

In the vertical hydrostatic balance is a reasonable approximation, as long as the scales ofmotion are large and the perturbations of ρ and p are also in hydrostatic balance.

This gives∂ p

∂ z

= −ρg

In general for a climate model whose grid is ≈ 100 km , hydrostatic approximation isreasonable.

However, a school of thought considers that in the deep tropics especially where motion isdominated by diabatic effects this may not be valid

However, hydrostatic approximation eliminates sound waves and thus allows for larger time

steps (hence increases computational efficiency).

Whenever we do such a scaling we need to ensure that conservative properties are satisfiedby the new set of equations.


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Consistent set of Hydrostatic Primitive Eqns

When we satisfy properties of conservation of energy, angular momentum and potentialvorticity in hydrostatic framework we get the following versions of momentum equations

The zonal momentum equation:

Du

Dt −

2Ω +

u

a cosφ

v sinφ +

1

ρa cosφ

∂ p

∂λ= F r λ

The meridional momentum equation:

Dv

Dt

+ 2Ω +u

a cosφ u sinφ +

1

ρa

∂ p

∂φ

= F r φ

The vertical momentum equation:

g +1

ρ

∂ p

∂ z = 0

Here we define D Dt ≡ ∂

∂ t + u

a cosφ∂ ∂λ

+ v a

∂ ∂φ

+ w ∂ ∂ z

= ∂ ∂ t

+ u · ∇

The terms that are omitted are 2Ω cosφ (the cosφ Coriolis term), the four metric terms, thevertical acceleration term ( Dw

Dt ) and frictional term in vertical component F rr

We have replace r the radius by mean radius a except where derivatives are required, herewe have replaced by ∂

∂ r with ∂

∂ z , z is height above mean sea level (thin shell approximation).

The term D Dt

= ∂ ∂ t

+ u ∂ a cosφ∂λ

+ v ∂ a ∂φ

+ w ∂ ∂ z

= ∂ ∂ t

+ u · ∇


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Consistent Set of HPE . . .

The equations for continuity and energy

D ρ

Dt + ρ∇ · u = 0

D θ

Dt =

θ

TC p Q

And equation of state p = ρRT

These satisfy the conservation properties

ρD

Dt ((u

+ Ωa cosφ

)a cosφ

) =ρF

r λa cosφ−

∂ p

∂λ

for energy

ρD

Dt

1

2v2 + Φ + c p T

+∇ · (p u) = ρ(Q + v · F h )

for potential vorticity

ρ D Dt

ζ · ∇θ

ρ

= ζ · ∇( D θ

Dt ) + ∇θ · (∇× F h )

Here v = iu + jv the horizontal component and ζ = 2Ωk sinφ +∇× v

The major omission is the dropping of cos φ coriolis terms. This dropping is essentially toensure good conservation properties for the shallow water system

We next discuss the importance of this term for tropical motion


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Importance of the Cos φ Coriolis Term

Our analysis during the previous course was for adiabatic mid-latitude synoptic scale motion

We now look at the tropics. We begin by looking at scale analysis of zonal momentumbalance

In quasi-hydrostatic motion, continuity sets an upper bound on vertical velocities i.e. W ≤UH

L

We now consider 2Ωw cosφ in relation to Du Dt

As we did previously we scale Du Dt ≈ U 2

L

We estimate|2Ωw cosφ|

| Du Dt

|≤ 2ΩH cosφ

U – this is independent of L -length scale

Now we can ignore the 2Ω cosφ in comparison to Du Dt only if 2ΩH cosφ

U 1 or 2ΩU cosφg U 2

gH

Taking typical values of Ω = 2πradians per day, H = 104m and U = 10ms −1 we get2ΩH cosφ

U to be 0.14 cosφ

In typical standard analysis W UH L

, hence we can ignore 2Ωw cosφ in the zonalmomentum equation

If we look at tropics, analysis of thermodynamic and vorticity equations by Burger (1991)shows that WL

UH ≈ 1 if Rossby Numer is about 1 – the limit may be reached in free synoptic

scale motion in tropics.

In tropics synoptic scale is generally not free – diabatic effect have a major impact on verticalvelocities and hence upper on bound W could well be reached.

Hence it is necessary to include 2Ωw cosφ for accurate simulations.


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Comparison 2Ωv sinφ and 2Ωw cosφ Terms

We now compare the two Coriolis terms in the horizontal momentum equationHoskins and Karoly (1981) considers large scale motion in tropics to be described by abalance of planetary scale voriticity advection and vortex stretching v ∂ f

a ∂φ= f ∂ w

∂ z – a sort of

Sverdrup balance2Ωv cosφ

a = 2Ω sinφ

∂ w

∂ z

Scaling ∂ w ∂ z ∼ W H and multiplying both sides by sin2

φcos2 φ and re-arranging we get

|2Ωw cosφ

2Ωv sinφ| ≈

H

a cot2 φ

In deep tropics at φ = 2o this approaches unity while at φ = 6o this takes a value of 0.1 – cannot be comfortably ignored while studying large-scale motion in the tropics.

We next look at scaling of vertical momentum eqn.


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Vertical Momentum Eqn

For tropical motion if 2Ωw cosφ is retained in the horizontal momentum eqn, then for

consitent energetics we need to include -2Ωu cosφ in the vertical momentum eqn also.However we also need to examine the magnitude of this term vis-a-vis other terms.

Let us examine it versus accln due to gravity : E = 2ΩU cosφg

Taking typical values this has 1.4 × 10−4 cosφ – looks small

Again as we did for vertical momentum eqn in conventional scaling, we need to look at

perturbations which affect horizontal motion – perturbations of horizontal pressure gradientsWe can write p = p o (r ) + p and ρ = ρo (r ) + ρ

The mean is in hydrostatic balance i.e. dp o

dr = −ρo g (note we are still using r and not z – we

are not using thin shell approximation)

We can then write the vertical momentum eqn (removing the mean state) as

Dw

Dt − 2Ωu cosφ− u 2 + v 2

r + g ρ

ρ+ 1

ρ∂ p

∂ r = 0


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Vertical Momentum Eqn . . .

We now need to compare −2Ωu cosφ with 1ρ∂ p

∂ r

We assume tropical scaling with Rossby number R o ≡U fl ∼ 1 hence f ∼ U

L

In this case for the horizontal momentum eqn, the acceleration, coriolis terms scale as U 2

L,

hence the pressure perturbation should also scale the same way – as we know thatatmospheric circulation is driven by pressure gradients.

From this we can write | 1ρ

p | ∼ U 2

Now 2Ωu cosφ

| 1ρ∂p

∂r |∼ 2ΩH cosφ

U – this can be neglected only if 1 i.e. 2ΩH cosφ

U 1 or comparing

with g we get E U 2

gH – same condition as we got from scaling of horizontal momentum eqn

– but we didn’t use any upper bounds on vertical velocity


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Is Vertical Acceleration Important in Tropics?

Let us compare Dw Dt

and 2Ω cosφ

We scale |Dw Dt | ∼ UW

L≤ U 2H

L2

This gives| Dw

Dt |

|2Ωu |≤ UH

2ΩL2

= H

LR o ∼ 10−3 for L = 106m , U = 10ms −1 and H = 104m – this

clearly shows that vertical accleration is much smaller than the Coriolis term in the verticalmomentum equation.

So how do we modify the equations?


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Eqns in Height Co-ordinates

We have seen that vertical acceleration is unimportant in the vertical momentum eqn.

The cosφ Coriolis terms are important in both horizontal momentum and vertical momentumequations.

We therefore get a set of modified eqns which satisfy angular momentum, energy andvorticity conservation

The zonal momentum eqn is

Du

Dt −

2Ω +

u

r cosφ

(v sinφ− w cosφ) +

1

r cosφ

∂ p

∂λ= F λ


Dv

Dt +

2Ω +

u

r cosφ

u sinφ +

vw

r + +

1

ρr

∂ p

∂ y = + F r φ

The Vertical component is:

−

2Ω + u r cosφ

u cosφ− v 2

r + g + 1

ρ∂ p ∂ r

= F rr

The thermodynamic energy eqn and continuity remain the same.

We can show that the eqns in p co-ordinate also satisfy conservation properties of angularmomentum, energy and vorticity.


E i P C di t

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Eqns in Pressure Co-ordinateThe pressure co-ordinate system is build on ∂ p

∂ z = −ρg but our vertical momentum eqn is no

longer strictly the same

We assume that hydrostatic approximation remains an accurate state except where horizontal variations of the balance represented in the above vertical momentum eqn are relevant

We retain the metric terms and avoid shallow atmosphere approximation (replacing r by a).

We use a psuedo-radius r s (p ) defined as r s (p ) = a + p 1

p RT s (p )

gp dp

T s (p ) is to be interpreted as representing a profile of horizontally averaged hydrostaticallybalanced state of the atmosphere.

Accordingly we define a new velocity Dr s Dt

= −RT s (p )ωgp

= w

The zonal momentum eqn is

Du

Dt −

2Ω +

u

r s cosφ

(v sinφ− w cosφ) +

1

r s cosφ

∂ Φ

∂λ= F λ


Dv

Dt +

2Ω +

u

r s cosφ

u sinφ +

v w

r s + +

1

r s

∂ p

∂ y = + F φ

The Vertical component is:RT

p +

∂ Φ

∂ p + µ

RT s

p = 0

In the above eqn µ ≡ 2Ωur s cosφ+u 2+v 2

r s g


Eqns in P Co ordinates

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Eqns in P Co-ordinates . . .

The continuity eqns becomes:

∇p · v +1

r 2s

∂

∂ p (r 2s ω) = 0

The thermodynamic energy eqn remains

D θ

Dt =

θ

Tc p

Q

We can show that the eqns in p co-ordinate also satisfy conservation


An Alternate Correction For Non Hydrostatic Motion

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An Alternate Correction For Non Hydrostatic Motion

Others have also looked at including non-hydrostatic effects in global models or seamlesslycombining regional models with larger scale models

Janjic et al (2001) has suggested another method to include non-hydrostatic effects.

Their idea is to assume the basic model to be essentially hydrostatic

Use a perturbation technique to include non-hydrostatic effects

They define a parameter that define the importance of non-hydrostatic effects

=1

g

Dw

Dt =

1

g

∂ w

∂ t + v · ∇σw + σ

∂ w

∂σ

It is the ratio of vertical acceleration with acceleration due to gravity

They also define the relation between hydrostatic and non-hydrostatic pressures in terms of ∂ p ∂π

= 1 + where p is the non hydrostatic pressure and π the hydrostatic pressure

will be generally small except over regions of intense convection or vertical ascent due tosteep orography this could have significant values

Over steep orography they suggest that vertical velocities as high as 10ms −1 could developover 1000s. This gives a value of of 10−3

For such an acceleration they suggest that p deviates from π by 100 hPa which would becomparable to the synoptic scale pressure gradient of 100 hPa in 100 km .

The actual implementation is closely linked to numerical discretization


Different Types of Projection

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Different Types of Projection

(Mostly from Haltiner & William, Wikipedia, Wolfram Site)

We may not always prefer to use spherical grid in modelling

In regional models depending on the region begin modelled we may use a projection ontovarious other surfaces.

If looking at polar regions, we would prefer to view from the top of the pole.

If we want to view the tropics, our point of view could be over tropics rather than over the pole.Hence we project to different surfaces. Some of the common ones are: Polar Stereographic projection ( sphere to a plane) Lambert Conformal Projection (sphere to a cone) Mercator projection (sphere on a cylinder)

Let us look at metric co-efficients before looking at projections


Metric Co-efficients

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Metric Co-efficients

We define metric co-efficient h j for the spherical co-efficients x j where x 1 = λ, x 2 = φ andx 3 = r = z + a

The relationship between increment in the co-ordinate x j and the spherical distance ds j is the

metric co-efficient i.e. h 1 = ds 1dx

1

, h 2 = ds 2dx

2

and h 3 = ds 3dx

3The curvilinear distance ds 1 = r cosφd λ, ds 2 = r d φ and ds 3 = dr in the longitudnal,latitudnal and radial directions.

Now h 1 = r cosφ, h 2 = r , h 3 = 1 for the spherical co-ordinate system


Polar Stereographic Projection

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Polar Stereographic Projection

Projection of sphere onto a plane

This generally used for studying polar weatherIn terms of latitude we can write r = am (φ) cosφ

where m (φ) = 2(1+sinφ)

, the image scale (or map

factor) and θ = λ

r is the radius of the latitude circle on the map withco-latitude δ

The cartesian co-ordinates on the map is given by

x = 2a cosφ cosλ1+sinφ

and y = 2a cosφ sinλ1+sinφ

The metric co-efficients h x = h y = 1m (φ)

= 1+sinφ2

The relationship between distances on the projectedmap and spherical distances is given by

dx dy

=

1

1 + 2sinφ

− sinλ − cosλcosλ − sinλ

dx s

dy s


Polar Stereographic

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Polar Stereographic . . .

The relationship between velocities on the polarsteorographic map, U ,V and over the sphere u s , v s

are given by

U V

=

− sinλ − cosλcosλ − sinλ

u s

v s

The momentum equations (HPE) reduce to:

DV

Dt − V

f −

xV − yU

2a 2

= −m α

∂ p

∂ x

DV

Dt + U

f −

xV − yU

2a 2

= −m α

∂ p

∂ y

1

ρ

∂ p

∂ z = −g

The continuity equation becomes

∂ρ

∂ t + m 2

∂ (ρU /m )a

∂ x +

∂ (ρV /m )

∂ y

+

∂ρW

∂ z = 0


Lambert Conformal Conic Projection

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Lambert Conformal Conic Projection

This is a conic map projection. It is conformal i.e. angles are preserved.

Used in aeronautical charts.

Commonly recommended for use in the mid-latitudes.

As shown above we superimpose a cone over the Earth.We use two reference parallels (latitude circles, φ1 and φ2) at which the cone intersects withthe sphere.

Along the chosen parallels there is no distortion but away from the parallels there is distortion.

A Straight line between two points drawn on a Lambert conformal projection approximates thegreat circle (great circle on the sphere will pass through the two points and centre of the

sphere).Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 23 / 29

Lambert Conformal Projection

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Lambert Conformal Projection

The transformation between Lambertprojection and sphere are given as

r =a

K m (φ) cosφ and θ = K (λ− λo )

Here K = ln

cosφ1cosφ2

+ ln

tan(π

4−φ1

2)

tan(π4−φ2

2)

And m (φ) =

cosφcosφ1

(K − 1)

1+sinφ11+sinφ

K

The momentum equations (HPE) reduce to:

DV

Dt − V

f −

xV − yU

2a 2

= −m α

∂ p

∂ x

DV

Dt + U

f −

xV − yU

2a 2

= −m α

∂ p

∂ y

1

ρ

∂ p

∂ z = −g

Here U = h x x and V = h y u


Mercator Projection

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e cato oject o

This is a cylindrical map projection.

It is conformal

Shape distortion occurs as objects near the pole look enlarged as seen in this map of mercator projection


Details for Mercator Projection

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j

(from Wikipedia)


A Map in Mercator Projection

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p j


Mercator Projection . . .

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j

The distances on mercator projection are defined by the equations x = (a cosφo )λ

y = a cosφo ln(1+sinφ)

cosφ

Here φo is the latitude at which the projection is true.

Map factor is m (φ) = cosφo cosφ

= 1h x

= 1h y

The horizontal momentum equations are:

∂ U

∂ t + m

U ∂ U

∂ x + V

∂ U

∂ y

+ w

∂ U

∂ z −

f +

U tanφ

a

V = −

m

ρ

∂ p

∂ x

∂ V

∂ t + m

U ∂ V

∂ x + V

∂ V

∂ y

+ w

∂ V

∂ z +

f +

U tanφ

a

U = −

m

ρ

∂ p

∂ y


Distortions in Mercator Projection

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Greenland takes as much area on the map as Africa – Africa’s area actually is 14 timesgreater than Greenland

Alaska appears bigger than Brazil, Brazil is 5 times larger than Alaska

Finland appears longer than India, though India is much bigger.

Which projection is used in google maps?


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Eqn Scaling Proj