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METR 3123, Atmospheric Dynamics II Alan Shapiro, Instructor Wednesday, 23 April 2008 (lecture 40)
1 handout: prob set 7
In last class we showed that the approximate vert vort eq n appropriate for synoptic scale flows is:
∂ζ ∂t
∂ζ ∂x
∂ζ ∂y
= – (ζ + f) δ – βv
and that this can be rewritten in terms of absolute vert vort η (≡ ζ + f) (^) , as
∂η ∂t
∂η ∂x
∂η ∂y = – η δ
Can simplify them. Introduce
D (^) h Dt
∂t
deriv following horiz motion of an air parcel), get:
for relative vert vort:
D (^) hζ Dt
= – η δ – βv
for absolute vert vort:
D (^) hη Dt
= – η δ
Now consider the special case where:
(i) Flow is incompressible (∇ ⋅^ u = 0^ , so parcel volume is
const) ∴ ∂ ∂ux +^ ∂ ∂vy +^ ∂ ∂wz = 0^ , ∴ ∂ ∂ux +^ ∂ ∂vy = –^ ∂ ∂wz ,
(ii) Flow is barotropic (ρ = ρ(p) ∴ no shear of geos wind, i.e.,
no thermal wind, and
(iii) Flow is "nearly" geostrophic. Replace wind in vert vort eqn
by geostrophic wind except for δ in stretching term.
So ζ ≈ ζg =^
∂v (^) g ∂x
∂u (^) g ∂y
∂t
∂t
- u (^) g ∂ ∂x
- v (^) g ∂ ∂y
But δ^ is not^ ≈ δg Remember, horiz divergence of geos wind
is 0 (on an isobaric sfc). Don't approximate δ.
D (^) hη (^) g Dt
= – η (^) g δ
D (^) h Dt
ζg + f = – ζg + f ∂u ∂x
––> – ∂ ∂wz
[don't use geos wind in this term]
D (^) h Dt
ζg + f = ζg + f ∂w ∂z
In barotropic flow u (^) g and vg are indep of p -- so nearly indep of z
∴ ζg is indep of z
D (^) hζg Dt is indep of z.
Since w(z) = Dz Dt
So w(D^2 ) =^
D
Dt
D 2 (x,y,t) =
∂D 2
∂t
∂D 2
∂x
∂D 2
∂y
D (^) hD (^2) Dt
w(D 1 ) = D Dt
D 1 (x,y) = u
∂D 1
∂x
∂D 1
∂y
D (^) hD (^1) Dt
∴ D^2 – D^1
D (^) h Dt
ζ + f = ζ + f
D (^) hD (^2) Dt
D (^) hD (^1) Dt
∴ H
D (^) h Dt
ζ + f = ζ + f
D (^) hH Dt
∴ H
D (^) h Dt
ζ + f – ζ + f
D (^) hH Dt
This is b Da Dt
- a Db Dt where b = H, a = ζ + f. Can simplify it.
Scratch paper_______________________
D Dt
a b
=^1
b
Da Dt
b
=^1
b
Da Dt
Db Dt [mult by b^2 ]
∴ b^2
D
Dt
a b
= b Da Dt
∴ H^2
D (^) h Dt
ζ + f H
= 0 (^) divide by H 2
Barotropic Potential Vorticity Conservation Theorem (due to Carl Gustav Rossby):
D (^) h Dt
ζ + f H
= 0 (^) (for an air column)
or:
ζ + f H
= const (^) (for an air column)
ζ + f H is potential vorticity,
absolute vorticity column thickness
This theorem says that potential vorticity is conserved -- an air column's potential vorticity does not change.
e.g. Consider case of horizontal convergence
z
∴ H ↑
But from pot vort th m^ ,
ζ + f H is constant.
