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
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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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I V i l A l i I i T i ?
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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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E i H i ht C di t
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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
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:
−
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.
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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
The meridional component is:
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
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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
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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
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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
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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
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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
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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
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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
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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
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Details for Mercator Projection
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j
(from Wikipedia)
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A Map in Mercator Projection
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p j
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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
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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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