Flow Coherence: Distinguishing Cause from Effect - Selected Topics of Computational and Experimental Fluid Mechanics - page 81

3 Geodesic Transport Theory

The geodesic transport theory seeks LCS as minimal (Haller and Beron-Vera 2012 )
or stationary (Haller and Beron-Vera 2013 ; Farazmand et al. 2014 ) curves of func-
tionals representing appropriate integral objective deformation field measures. Here
we are concerned with LCS of hyperbolic type that follow as stationary curves of
the integrated material shear functional.
More specifically, for a material line [ s

− ,

s

+ ]

s ₒ r ( s ) ∈ ʳ 0 , the relevant

functional is given by:

r d s

s +

r

S ˃ [ r ]:= ˃ ,ʩ , (7)

r |

|

s −

where is the Euclidean norm. Motivated by behavior of classic hyperbolic invariant

manifolds, alongwhichmaterial shear is suppressed, Farazmand et al. ( 2014 ) propose
to seek hyperbolic LCS as exceptional material lines within strips of material lines,
which, connecting the same endpoints, have integrated material shears that vary by
an order of magnitude less than what the widths of the strips permit. This means
that the material lines of interest must be stationary curves of

|·|

S ˃ [ r ] . Exploiting the

r / | r | ) symmetry of ˃( r ,ʩ under s translations, Farazmand et al. ( 2014 ) show that

stationary curves of S ˃ [ r ] coincide with stationary curves of

s +

r , r )(

S d [ r ]:= d ( r ) d s ≡ 0 , (8)

s −

where

1

d ( · , · )( x 0 ) := · , D ( x 0 ) · , D ( x 0 ) :=

2 (

C ( x 0 )ʩ − ʩ C ( x 0 )) . (9)

Stationary curves of are known as geodesics . (More specifically, these geo-

desics are said to be null because d

S d [ r ]

r , r )( , representing a Lorentzian metric,

vanishes.) As Farazmand et al. ( 2014 ) further demonstrate, such geodesics must
coincide with squeezelines or stretchlines, i.e., integral curves of

( r )

r = ʾ 1 ( r = ʾ 2 ( r ) or r ), (10)

connecting points where C Id. The resulting material lines are the sought

LCS, which, as we illustrate below through an explicit example, are responsible for
causing the flow to organize into ordered patterns.

( x 0 ) =

4 Application to an Oceanic Flow Dataset

The flow dataset considered in the application reported here is produced by ocean
velocities inferred using satellite altimetric measurements of sea surface elevation
in the Gulf of Mexico. This assumes a geostrophic balance between the Coriolis