GEF3450
Ada’s handwavy recipe for solving
the baroclinic QGPV equation
Ada Gjermundsen
E-mail: ada.gjermundsen@geo.uio.no
November 30, 2015
Note that this recipe is just a suggested method for solving the QGPV equation. It may not work on all exercises! Please don’t shoot the messenger :-)
For baroclinic conditions we will use two equations; the baroclinic quasigeostrophic potential vorticity equation (QGPV) and the density equation.
Start with the the baroclinic QGPV equation
where
i
Dg h
∂ f02 ∂ψ ζg +
+
βy
=0
Dt
∂z N 2 ∂z
(1)
Dg
∂
∂
∂
=
+ ug
+ vg
Dt
∂t
∂x
∂y
(2)
To be able to solve for the streamfunction and get the dispersion relation and
the velocities, we need boundary conditions. To get the boundary conditions,
we use the QG density equation:
Dg ∂ψ N 2
+
w=0
Dt ∂z
f0
(3)
Equation 3 can be written as
f0 Dg ∂ψ
= −w
N 2 Dt ∂z
(4)
which means that we need some sort of expression for w to solve equation 3.
Some examples are:
- flow over topography: w = ~ug · ∇h
- rigid lid (flat upper and/or lower surface): w = 0
1
- bottom Ekman layer: w = δ∇2 ψ
- upper layer wind stress w =
1
ρc f0 ∇
× τ~w
Note! Sometimes we apply boundary conditions where the constrains are on
the horizontal velocities instead of w, like a no-slip condition (~ug = 0). If so,
you have to apply that instead of the density equation (e.g. exercise 5.4).
Note!! The QGPV equation and the density equation are non-linear equations. To be able to solve anything we need linear equations! Which means
you have to linearize the QGPV equation as well as the density equation.
The recipe:
1) Write down all the assumptions in the exercise, e.g.:
- is there a constant mean wind or not → U = 0? or U 6= 0?
V = 0? or V 6= 0? Note! If the mean wind is a function of z (e.g.
U = Λz) we need to do things a bit differently → see end of this
recipe.
- Do we consider the beta-effect → f = f0 + βy? or do we use a
constant coriolis parameter → f = f0 ?
- Is N 2 constant? Probably!
- What are the boundary conditions? (These you apply in the density equation later on)
2) Write down the horizontal velocities in terms of a constant mean velocity and a perturbation velocity
ug = U + u0g
where
u0g = −
∂ψ
∂y
vg0 =
vg = V + vg0
(5)
∂ψ
∂x
(6)
ζg0 = ∇2 ψ
3) Apply the assumptions from 1) to the QGPV equation
4) LINEARIZE the “remainder” of the QGPV equation obtained in 3) i.e. remove all terms of O() or higher order terms. This means that
you have to remove all terms where you have two or more perturbation variables. If you have included ψ in the QGPV equation already,
remember that since ψ is giving you the perturbation velocities, it also
count as a perturbation variable.
5) Write down what’s left of the QGPV equation from 4) in terms of a
streamfunction ψ (see → 2)). Now you should have one equation (the
linearized QGPV equation) and one unknown (ψ) → BE HAPPY!
(because now it is possible to solve it!)
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6) If not given in the exercise text, propose a wave solution (i.e. this is
challenging, but gets easier with some practice). Now that we have
baroclinic conditions, the wave amplitude will be a function of z
- In most cases you can use
ψ = <{ψ̂(z)ei(kx+ly−ωt) }
(7)
- If you are looking at stationary mountain waves, you may use a
solution which is wave-like in all directions (note! since we are
using a wave solution which is varying in z as well, the amplitude
can stay constant)
ψ = <{ψ̂ei(kx+ly+mz) }
(8)
- ... or something different. It all depends on the boundary conditions and/or the forcing/dissipation terms.
How you precede from here depends on the exercise and what you
are after. In many cases you need to find an expression for the wave
amplitude ψ̂(z):
How to find ψ̂(z):
7) Use a “suitable” wave solution for the streamfunction ψ in the QGPV
equation from 5) and solve for the wave amplitude ψ̂(z)! Most probably
you’ll get an ordinary differential equation. Note that if:
d2 ψ̂(z)
+ λ2 ψ̂(z) = 0
dz 2
(9)
ψ̂(z) = A0 cos(λz) + B0 cos(λz)
(10)
ψ̂(z) = A0 e−λz + B0 eλz
(11)
for λ2 > 0
for λ2 < 0
To be able to find the final expression for ψ̂(z), you need some boundary
conditions. For that we use the density equation (see the beginning):
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8) The density equation is also non-linear and to be able to solve anything
we need linear equations! Which means you have to linearize the
density equation as well. Go back and do 1) → 5), but now for the
density equation to get an expression for ∂ψ
∂z and note that
∂ψ
dψ̂(z) i(kx+ly−ωt)
=
e
∂z
dz
(12)
Note that for stationary mountain waves:
∂ψ
= imψ̂ei(kx+ly+im)
∂z
(13)
9) Now you have (hopefully!) obtained a solution for ψ̂(z) and maybe also
expressions for k or/and l in terms of n. For stationary mountain waves
you’ll often get an expression for m in terms of the other variables. But
you’re not done! Now you need to find an expression for the dispersion
relation ω. Then you’re able to calculate phase speeds and the group
velocity if needed. This can be done in several ways depending on the
exercise, e.g:
- Go back and use the linearized QGPV equation obtained in 5) to
get an expression for the dispersion relation ω.
- Or if ω is included in λ (i.e. the coefficient in the ODE you used
to find ψ̂(z)), use λ to get an expression for ω.
10) Remember to write down the resulting expression for ψ and explain
your result!
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