Generated using version 3.0 of the official AMS LATEX template
1
Corrigendum: surface quasi-geostrophic solutions and baroclinic
2
modes with exponential stratification
J. H. LaCasce
∗
Department of Geosciences, University of Oslo, Norway
3
∗
Corresponding author address: Joe LaCasce, University of Oslo, Department of Geosciences, P.O. box
1022 Blindern, N-0315 Oslo, Norway.
E-mail: j.h.lacasce@geo.uio.no
1
4
1. Corrigendum: Effect on spectra
5
Section (4e) in the paper (LaCasce 2012) compares the kinetic energy and buoyancy
6
spectra obtained with constant and exponential stratification. The section was added during
7
the revision of the first draft and added, evidently, too quickly. Fig. (8) is half-way incorrect,
8
and this affects the conclusions of the section.
9
The density spectrum at the surface is given in equation (27):
1 f0 ρ0 2 dχ
1 f 0 ρ0 2 2
1
) ( (0))2 |ψ̂|2 = (
) |ψ̂|
B = |ρ̂|2 = (
2
2 g
dz
2 g
(1)
10
where χ(z) is the vertical structure of the SQG solution, which depends on the stratification.
11
The boundary condition at the upper surface is dχ/dz = 1, which explains why that term
12
vanishes above.
13
The kinetic energy spectrum is given in equation (28):
1
1 f 0 ρ0 2
KE = (|û|2 + |v̂|2 ) = (
) χ(0)2 κ2 |ψ̂|2
2
2 g
14
(2)
Thus the ratio of these two is:
KE/B = κ2 χ(0)2
(3)
15
The goal of Fig. (8) in the paper was to plot KE/B vs. horizontal wavenumber. The
16
correct ratio was shown for the constant stratification solution. However, what was shown for
17
the exponential stratification solution is the ratio divided by κ2 . Correcting the latter yields
18
Fig. (1). As in the paper, I use the “stronger” exponential stratification, corresponding to
19
an e-folding scale which is 1/10th of the water depth.
1
20
The result suggests that constant and exponential stratification yield a spectral ratio
21
which is fairly similar. At scales exceeding the deformation radius, the ratio is proportional
22
to κ−2 , implying the kinetic energy spectrum is steeper. Thus if the kinetic energy spectrum
23
decays as κ−5/3 in this range, the buoyancy spectrum is increasing as κ1/3 . At subdeformation
24
scales, the ratio is constant with constant stratification and the spectra have the same slope
25
(as noted in the paper). With exponential stratification the ratio is not quite constant at
26
these scales, suggesting the two spectra differ somewhat. But the difference is not great, so
27
to a first approximation the two would appear similar.
28
29
30
31
Thus changing the stratification does not greatly affect the relation between the buoyancy
and kinetic energy spectra, in contrast to what was claimed in the end of section (4e).
Acknowledgments.
My thanks to Jinbo Wang, who discovered this error while working through the paper.
2
32
33
REFERENCES
34
LaCasce, J., 2012: Surface quasi-geostrophic solutions and baroclinic modes with exponential
35
stratification. Journal of Physical Oceanography, 42, 569–280.
3
36
37
List of Figures
1
The ratio of the kinetic energy and buoyancy spectra as a function of total
38
wavenumber. The latter is normalized by the deformation wavenumber, κd
39
and the former is normalized by the ratio at κ = κd . The solution with con-
40
stant stratification is shown by the solid curve and the exponential solution,
41
with α = 10/H, by the dashed.
5
4
4
10
Const
Exp (α = 10/H)
3
10
−2
κ
2
KE/B
10
1
10
0
10
−1
10
−2
10
−1
0
10
10
κ/κd
1
10
2
10
Fig. 1. The ratio of the kinetic energy and buoyancy spectra as a function of total wavenumber. The latter is normalized by the deformation wavenumber, κd and the former is normalized by the value at κ = κd . The solution with constant stratification is shown by the solid
curve and the exponential solution, with α = 10/H, by the dashed.
5