Theory of the norm-induced metric in atmospheric dynamics

Koh, T.-Y.; Wan, F.

doi:https://doi.org/10.5194/acp-15-2571-2015

Articles | Volume 15, issue 5

https://doi.org/10.5194/acp-15-2571-2015

© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/acp-15-2571-2015

© Author(s) 2015. This work is distributed under
the Creative Commons Attribution 3.0 License.

Articles | Volume 15, issue 5

Research article

|

09 Mar 2015

Research article |

| 09 Mar 2015

Theory of the norm-induced metric in atmospheric dynamics

T.-Y. Koh and F. Wan

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Final revised paper (published on 09 Mar 2015)
Preprint (discussion started on 11 Feb 2014)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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RC C338: 'Review of Wan and Koh 2014', Anonymous Referee #1, 07 Mar 2014
RC C520: 'First revision of ACDP 14, 3733-3775 (2014)', Anonymous Referee #2, 17 Mar 2014
AC C667: 'relevance to data assimilation metrics', Tieh-Yong Koh, 24 Mar 2014

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Tieh-Yong Koh on behalf of the Authors (30 Oct 2014) Author's response Manuscript

ED: Referee Nomination & Report Request started (16 Nov 2014) by Peter Haynes

RR by Roberto Buizza (16 Dec 2014)

RR by Anonymous Referee #1 (07 Jan 2015)

Suggestions for revision or reasons for rejection

The revised paper is much better than the previous version. Some
readers may find it useful and I therefore do not wish to hinder its
publication. There are 3 clarifications, however, that I strongly
suggest, as detailed below. Otherwise I did not check all the authors equations
or results since these other aspects of the work were not relevant to my current interests. I therefore leave them to others for comment.

1. Contrary to what is stated in the authors' manuscript, the
expression involving perturbation T in the norm described by
Ehrendorfer and Errico (1995) and also by Talagrand (1981) is indeed
the available potential energy (APE) (plus a small portion of
unavailable energy as shown later by Errico (2000) in a
primitive-equation model linearized about an isothermal, flat surface,
resting atmosphere, as carefully stated in that first reference. For
nonlinear models, it can also be considered as an approximation to APE
(albeit, a rather crude one) if the horizontal mean of the temperature
lapse rate is neglected along with variations of surface pressure when
describing mass within a column of air. These are poorer
approximations than made by the authors of this paper, but otherwise
of similar character. So, the total energy norm is indeed motivated by
an expression approximating available potential energy. For its
applications, in fact, it does not matter how good an approximation it
may be. This should be made clear by the authors so that they do not
contribute further to the confusion about this norm that already
exists.

2. In all applications of the total energy norm that I have seen
before this paper, the norm has been used to measure either
differences between pairs of forecasts or between forecasts and
corresponding verifications. Such measures will typically grow due to
chaos, whether or not the norm is a true invariant in the parent
model. The appeal to it as some approximation to a true invariant
truly refers only to the motivation for its relative weights of
contributions by wind and mass variables. Effectively there is
otherwise nothing special about the formulation of that norm. Indeed,
its utility should be judged on whether it adequately describes
characteristics of error that are of interest.

An example of a simple norm that may not be adequate is the "moist
total energy norm" using a scaling parameter epsilon=1 derived
heuristically similarly by Ehrendorfer et al. (1999 J. Atmos. Sciences,
1627-1648): It unfortunately weights the contribution by the moisture
variable so much that it can obscure measures of the dynamics. For
that reason, although motivating a norm by energy conservation may be
appealing, it should be regarded as an insufficient reason.

3. Reducing descriptions of forecast difference or errors to a single
number, as accomplished in applications of the total energy norm, for
example, will always be problematic. The issue is that for most
problems we are not actually interested in a single metric. Two error
fields or forecast difference fields may have the same value of the
norm, but have considerably different characters. As an example, we
generally will qualitatively distinguish between a case where all the
error is extremely large but at a single point compared with one where
the error is very small but spread over the entire domain. The
complete characterization becomes even more obscured when weights are
applied to differing fields considered together. It is therefore
generally best to first consider what desired character is to be
measured and then to consider expressions of energy, if still
appropriate.

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ED: Reconsider after minor revisions (Editor review) (26 Jan 2015) by Peter Haynes

AR by Tieh-Yong Koh on behalf of the Authors (03 Feb 2015) Author's response Manuscript

ED: Publish as is (21 Feb 2015) by Peter Haynes

AR by Tieh-Yong Koh on behalf of the Authors (26 Feb 2015) Manuscript

Short summary

A metric measures the difference between two system states. So far, there is arbitrariness in its definition in terms of wind, temperature and surface pressure of the atmosphere. The choice of definition affects many applications: e.g. predictability studies, error growth analyses and ensemble forecasts. We construct a new metric based on vector space theory and energy conservation, and apply it to analytic models and atmospheric data. We discuss its advantages over a widely used older metric.