Available Potential Energy in
the ocean – what is it good for?
Rainer Bleck
Los Alamos National
Laboratory
Shan Sun
Goddard Institute for
Space Studies
(Climate sensitivity: SAT change due to CO2 doubling)
(Kv based on ratio of SSH change to SAT change)
Courtesy: Peter Stone, MIT
Applications…
• Is APE a useful indicator of model drift?
• Which processes contribute to the
generation and destruction of APE?
• Do processes like vertical regridding in
HYCOM distort the APE balance?
(Turn page for derivation of APE….)
Potential energy per unit column of a fluid column in
hydrostatic balance:
PE =
ztop
pbot
zbot
ptop
∫ gρz dz = ∫ z dp
where bot,top denote bottom and top of fluid column.
Integration by parts yields
ztop
1
PE = ( zp)bot + ∫ p dz = ( zp)bot +
g
zbot
where
α=ρ
−1
pbot
2
p
∫ptopα d 2
Conversion to s coordinates, global integration:
sbot
1
∂ p
PE = ( zp)bot + ∫ α
ds
g stop ∂s 2
where
∫
q=
∫
globe
q dx dy
globe
dx dy
Define mass-weighted average:
~
∂p ∂p
q=q /
∂s ∂s
2
~
q = q + q'
q = q + q*
With
the integrand can be split 3 ways:
2
*2
2
~
~
∂ p
∂ p
∂ p
∂ p
α
+α'
=α
+α
∂s 2
∂s 2
∂s 2
∂s 2
2
If evaluated in density space (α’=0), the r.h.s. reduces to
2
∂ p
∂ p
α
+ α
∂s 2
∂s 2
Unavailable
pot. energy
*2
Available
pot. energy
α1
α2
α3
A 3-layer ocean.
p
shown in red
Procedure for determining p:
1. Compute the mass M above a given s surface.
2. Divide ocean into large number of thin isobaric
slabs; determine mass mk in each slab from
bathymetric data base.
3. Counting from the top down, find the index n
satisfying
n −1
∑m
k =1
k
n
≤ M ≤ ∑ mk
k =1
4. p is now known to lie between top and bottom of
slab n. Interpolate to get final value.