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 Author: Nikolas Aksamit  naksamit@ethz.ch
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%%% References:
 [1] Haller, G., Aksamit, N. O., & Bartos, A. P. E. (2021). Quasi-Objective Coherent Structure Diagnostics from Single Trajectories. 
  Chaos, 31, 043131-1–17. https://doi.org/10.1063/5.0044151
 [2] Aksamit, N.O., Haller, G. (2021). Objective Momentum Barriers in Wall Turbulence. 
  In Review,  http://arxiv.org/abs/2106.07372
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%%% Calculate TRA and TSE diagnostics for 2D and 3D, steady and unsteady flows. 
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The four “Advect_and_Calculate” scripts provide the tools to calculate TRA, and TSE values as described in [1] for both 2D and 3D, steady and unsteady flows. For the unsteady cases, the extended phase space versions are calculated which require a choice of v_0. The default choice is the spatio-temporal mean of the flow velocity magnitude in the domain of interest.

Each Advect_and_Calculate script uses similar input formats:

Advect_and_Calculate_2DSteady.m
% Input arguments:
     tspan                        : Discrete advection timesteps used for RK4 advection scheme. This must include all intermediate steps between initial and final times(e.g. tspan=linspace(t_initial,t_final,100). For the steady case, these are dummy times as the velocity field is autonomous.
     xx,yy                        : Initial positions for advection. These must be formatted as Nx1 vectors
     U_Interp,V_Interp            : Velocity field interpolants with inputs (xx,yy)
     NCores                       : Number of Cores for parpool
 
    % Output arguments:
       xt,yt    : xx-component, yy-component of trajectory final position. 
       time_note    : tspan time if a trajectory left interpolant domain
       TSE_Bar,TSE,TRA_Bar,TRA    : Single trajectory metrics from   section II and III in [1]

Advect_and_Calculate_2DUnsteady.m
% Input arguments:
     tspan                        : Discrete advection timesteps used for RK4 advection scheme. This must include all intermediate steps between initial and final times(e.g. tspan=linspace(t_initial,t_final,100). For the unsteady case, these are times used in the velocity field interplant.
     xx,yy                        : Initial positions for advection. These must be formatted as Nx1 vectors
     U_Interp,V_Interp            : Velocity field interpolants with inputs (time,xx,yy)
     NCores                       : Number of Cores for parpool
 
    % Output arguments:
       xt,yt    : xx-component, yy-component of trajectory final position. 
       time_note    : tspan time if a trajectory left interpolant domain
       TSE_Bar,TSE,TRA_Bar    : Extended phase-space single trajectory metrics from section IV [1]

Note: For unsteady cases you can modify v_0, or leave as the mean of values used to define flow field interpolants. Advect_and_Calculate_3DSteady.m and Advect_and_Calculate_3DUnsteady.m are of the same format, with an extra z-component.

Demo_Unsteady2D.m
Demonstration of unsteady TRA and TSE calculations for geostrophic ocean current data as in [1].

Demo_Steady3D.m
Demonstration of steady TRA and TSE calculations for Johns Hopkins Turbulence Database Re=1000 Channel flow as in [2].

Demo_Steady3D_Geometry.m
Demonstration of steady NTRA and NTSE (normalized) calculations for Johns Hopkins Turbulence Database Re=1000 Channel flow as in [2].