According to the specific physical problem at hand, we specify diagnostics, i.e. variables (scalar, vector or tensor) that will disclose the processes driving the observed phenomena. The selection of these variables may come from examination of the governing equations, or from preliminary observations. The selected diagnostics determine the type of visualization techniques that need to be applied to obtain the best graphical representations for the particular problem. In addition, the quantification and visualization of diagnostics also allow us to evaluate the accuracy and physical validity of the numerical or experimental models employed in the studies. Diagnostics can be global or local. Global diagnostics are obtained by averaging the variable of interest over the domain of the simulation. Local diagnostics correspond to values of variables at specific locations in space and time. Diagnostics help to characterize physical entities (e. g. vortex structures) in terms of their topology and other physical parameters: intensity (circulation), fields they induce (strain-rate), evolution in time and others. In the vortex collapse problem we follow the evolution in time of a vortex ring, which is discretized by a set of curves in 3D space or filaments, that represent the ring's vorticity distribution. The evolution of the vortex is computed using the law of Biot-Savart [5].
Initially, we use diagnostics to investigate the long time behavior of
the Biot-Savart algorithm, which also indicates how the error
accumulates in the numerical integration process. One example of this
type of global diagnostic is the variance for a single-filament ring
[5].
In the test case of an elliptic vortex ring of low aspect ratio, this
quantity reveals the periodic behavior of the solution. The breakdown
of the model is shown in figure 
, which occurs as a short
wave instability, triggered by error accumulation, appears. A closer
look at the short wave instability is shown in figure ,
which displays local diagnostics: overlapping of vortex elements q,
curvature 
and torsion 
along the vortex ring. The
nature of the breakdown is characterized by a peak-like evolution of
the curvature of the vortex filament, which is beyond the validity
limit of the single-filament Biot-Savart model. The curvature
diagnostic is useful in monitoring the accuracy of the
simulations. Other important quantities that give information about
the quality of the simulations from the physical point of view are the
motion invariants: linear impulse, angular impulse and energy, which
in this particular problem should be conserved.
An important quantity in the vorticity amplification process is the
symmetric part of the velocity gradient, or strain-rate. A reduced
representation of this tensor that is simpler to visualize can be
obtained by computing the strain-rate eigenvalues 
and 
, where 
. Another reduction from
tensor to scalar field is accomplished by computing the normalized
rate of change of the magnitude of vorticity (or quadratic form of the
strain-rate matrix)
which directly relates to the local vorticity amplification rate.
An important diagnostic for vortex collapse is the energy
density [5], which provides an effective way to detect
collapse regions.
It is important to keep in mind that the visualizations in their
different modalities (2D plots, contour plots, isosurface) have the
objective of representing and characterizing solutions of equations
and functions whose analytical form may not be known. Quantification
of diagnostics is most useful when it shows functional relationships
or scaling laws in the results; therefore their particular
graphical representation should suggest possible mathematical
representations. Curve fitting of the data is an important step in
this direction. The visualization of the 3D space indicates which are the
physical objects or the types of interactions between objects
(vortices) responsible for the functional behaviors observed.
In the vortex collapse problem, the vortex ring presents the formation
of an antiparallel region, which in the presence of viscous
dissipation results in vortex reconnection. The most important
features of this behavior are the strain-rate and vorticity
amplification. We measure the strain-rate eigenvalue 
in the
region approached by the collapsing filaments. The data is fitted by
functional forms suggested by researchers which propose that collapse
corresponds to a singular event. We fit different portions of the curve
until we find the best fitting in terms of the norm of the error
(figure ) [5].
The diagnostics box (figure ) is a tool to search for
localized regions in space where events of physical relevance are
taking place. This device searches the field for maxima events in
vorticity, strain-rate or vortex stretching, and is used usually in a
self-guiding mode where once the user specifies the search criteria,
the significant regions are found automatically by the
program. Although the user has the option to move the box
interactively, we find that the self-guiding mechanism is very useful.
The off the filament nature of the spatial structure of the
strain-rate amplification in the vortex collapse
(figure ) becomes apparent when considering the
visualizations in the diagnostics box (figure ). The
tensor quantity we wish to analyze is rendered in various ways, each
with the objective of obtaining a (reduced) representation of its
different aspects. For example, the tensor quantity can be displayed
using the eigenvectors with length proportional to the eigenvalues,
which can be shown in slices, as in figure , or as
iconic representations of local diagnostics [2].
The tracking of regions associated with maxima events also suggests
which sub-domains may require "regularization". Vortex collapse
produces a large growth in the number of particles required to keep
the resolution at a minimum in the numerical simulation. On the other
hand, the energy density in the collapse region tends to zero as a
result of the antiparallel configuration of the vortex tubes. This
diagnostic can be used to remove the collapsed vortex filaments (a
dissipation mechanism), which effectively controls the growth of the
number of vortex particles. The filament surgery algorithm
[8] permits to perform filament simulations of vortex
reconnection (figure ) [9].