Equations, Scaling and Projections Ravi S Nanjundiah Centre For Atmospheric and Oceanic Sciences Indian Institute of Science Bangalore-560012 email:[email protected]Class Notes for Climate Modelling Ravi S Nanjund iah (Indian Institute of Scien ce) Eqns . ... Cl imate Modell ing 1 / 29
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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)
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 4 / 29
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.
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 6 / 29
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.
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 7 / 29
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 · ∇
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 8 / 29
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
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 9 / 29
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.
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 10 / 29
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.
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 11 / 29
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
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 12 / 29
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
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 16 / 29
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
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 18 / 29
(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
Ravi S Nanjundiah (Indian Institute of Science) Eqns. ... Climate Modelling 19 / 29
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