Influence of Environmental and Model Uncertainty on Lagrangian flow structures

1
Influence of Environmental and Model Uncertainty
on Lagrangian flow structures

• L. Kuznetsov, C.K.R.T. Jones, Brown University,
  M. Toner, A.D. Kirwan, University of Delaware

Supported by the Office of Naval Research
Manifold comparison as a Lagrangian model
evaluation tool
Introduction
Small changes in the velocity field can have a
significant effect on the Lagrangian features of
the flow. The Eulerian norm u-u is not a good
indicator of how close the coherent structures of
the two flows are. If transport and fluid
exchange in and around coherent features is a
crucial aspect of model output then we should be
using diagnostics that capture that information.
True flow is given by a solution of a shallow
water equations with steady wind forcing Model
representations of the true velocity field 1.)
Projections on the basis of geometrical
orthogonal functions, 2.) Solutions with
time-periodic wind forcing F F (1 a
cos(?t))
Spatial structure of the Eulerian error
191 modes 358 modes
532 modes
Two perturbations with the same Eulerian norm
have significantly different effect on the
manifolds
We introduce the separation area A and average
distance ?A/l to quantify the difference between
two corresponding manifold segments
A
a 0.85 ? 6.28 a 0.85 ? 7.0
a 0.9 ? 0.2
Manifold comparison
?E/E
A wind driven double gyre circulation in a
shallow-water model
N191
N358
N358
N532
N532
a 0.85 ? 7.0
Contours of the height field. The dark contour
corresponding to h470m passes through the saddle
governing the formation of a new eddy
a 0.85 ? 6.28
a 0.85 ? 6.28
a 0.85 ? 7.0
a 0.90 ? 0.2
a 0.85 ? 6.28 ? 14.0 km a 0.85 ?
7.0 ? 3.0 km N358 ? 7.3 km N532 ?
3.7 km
Projections Eulerian error correlates with the
error in the position of the coherent feature
boundary. Wind modulation The model that is
better from the Eulerian viewpoint gives
considerably worse Lagrangian predictions
Lagrangian eddy delineated by the stable
manifold of the hyperbolic trajectory Eulerian
eddy an inside of the fixed time height field
separatrix
Conclusion

• The difference between coherent features is
  quantified in Lagrangian framework in terms of
  the separation area between the EIMs, bounding
  the features.
• Lagrangian comparison distinguishes models on
  the basis of their ability to predict kinematics
  of water redistribution by the flow, transport
  characteristics, etc.
• Manifolds are sensitive to the details of the
  spatial and temporal structure of the velocity
  field

Shaded area represents the fluid which is
entrained into the recirculating motion. Its
boundary is given by the stable Effective
Invariant Manifold (EIM). EIMs can be
operationally defined for the flows known only
for finite time intervals, they divide the flow
plane into regions with different type of motion,
functioning in a similar manner as the invariant
manifolds do in the time-periodic case.