Hi all:
Can Roms be used to trace the float trajectory back in the past few days?
If can, which program?
Thanks for your replying!
trace the float trajectory back
Re: trace the float trajectory back
Dear Zhaoqian,
Mathematically, ROMS is based upon partial differential equations.
Generally, theorems ensure existence and uniqueness of the solutions only for positive times t: it means that this kind of results doesn't work for reverse (negative) times.
Nevertheless, you can intuitively try to guess a back area; take some points in this area and see their positive trajectories.
Regards,
Valere
Mathematically, ROMS is based upon partial differential equations.
Generally, theorems ensure existence and uniqueness of the solutions only for positive times t: it means that this kind of results doesn't work for reverse (negative) times.
Nevertheless, you can intuitively try to guess a back area; take some points in this area and see their positive trajectories.
Regards,
Valere
Re: trace the float trajectory back
Zhaoqian and Valere:
You can solve the equations (i.e. run the model) make sure you get all the components of velocity in all of your domain. Once finished you can choose a 'particle' that interests you and trace it back. First find the velocity at each particle's location (interpolate from your velocity fields) and then just 'push back' your particle the distance given by time_step*velocity in opposite direction of the velocity vector (i.e integrate back in time). This will give you a new particle location for which you can again interpolate velocity and push back again until you arrive where you want to.
You don't need to solve the equation with negative time.
There is a recent publication that traces upwelled particles back in time:
Rivas, D., and R. M. Samelson. A numerical modeling study of the upwelling source waters along the Oregon coast during 2005. J. Phys. Oceanogr.
A copy can be found at:
http://www-po.coas.oregonstate.edu/homes/rms/coastal/
About the existence and uniqueness of partial differential equations it is not enough to say "in general they require positive times" to prove that the equations ROMS are solving cannot be solved with negative times. The equations ROMS solves happen to be Navier-Stokes equations and anyone proving their existence and uniqueness in 3D will get a million usd for that (see http://www.claymath.org/millennium/Navi ... stokes.pdf )
What we do know is that in 2D regular solutions exist and are unique for ALL time, and it seems intuitevily that in 3D this would also be the case, but intuition of course often means wrong in math. furthermore existence and uniqueness can be particular to initial and boundary conditions, domain and simplifications made to original equations so all this needs to be considered.
And finally, at least some partial differential equations can be solved for negative time, I recall seeing the heat equation being solved with change of variable y=-t in my advanced coastal oceanography class. Even if a positive time interval is required to find a solution, consider that you can map a negative interval to a positive one and solve there, then you can go back to your original negative interval (i.e. when you start counting time (t=o) is arbitrary). I think what you meant Valere is that time needs to be increasing (actually non-decreasing if steady state is considered), but increasing times negative one is decreasing and a change of variable could do the trick so ...
Id be interested to hear back if I am wrong.
Rodrigo.
You can solve the equations (i.e. run the model) make sure you get all the components of velocity in all of your domain. Once finished you can choose a 'particle' that interests you and trace it back. First find the velocity at each particle's location (interpolate from your velocity fields) and then just 'push back' your particle the distance given by time_step*velocity in opposite direction of the velocity vector (i.e integrate back in time). This will give you a new particle location for which you can again interpolate velocity and push back again until you arrive where you want to.
You don't need to solve the equation with negative time.
There is a recent publication that traces upwelled particles back in time:
Rivas, D., and R. M. Samelson. A numerical modeling study of the upwelling source waters along the Oregon coast during 2005. J. Phys. Oceanogr.
A copy can be found at:
http://www-po.coas.oregonstate.edu/homes/rms/coastal/
About the existence and uniqueness of partial differential equations it is not enough to say "in general they require positive times" to prove that the equations ROMS are solving cannot be solved with negative times. The equations ROMS solves happen to be Navier-Stokes equations and anyone proving their existence and uniqueness in 3D will get a million usd for that (see http://www.claymath.org/millennium/Navi ... stokes.pdf )
What we do know is that in 2D regular solutions exist and are unique for ALL time, and it seems intuitevily that in 3D this would also be the case, but intuition of course often means wrong in math. furthermore existence and uniqueness can be particular to initial and boundary conditions, domain and simplifications made to original equations so all this needs to be considered.
And finally, at least some partial differential equations can be solved for negative time, I recall seeing the heat equation being solved with change of variable y=-t in my advanced coastal oceanography class. Even if a positive time interval is required to find a solution, consider that you can map a negative interval to a positive one and solve there, then you can go back to your original negative interval (i.e. when you start counting time (t=o) is arbitrary). I think what you meant Valere is that time needs to be increasing (actually non-decreasing if steady state is considered), but increasing times negative one is decreasing and a change of variable could do the trick so ...
Id be interested to hear back if I am wrong.
Rodrigo.
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Re: trace the float trajectory back
Interesting discussion. I haven't thought about this in much detail. It is also possible to write the adjoint of the linearized Lagrangian track routines step_floats.F and interp_floats.F. It is not that hard. Recall that the adjoint model is always time-stepped backwards in time. We do it all the time in all ROMS adjoint-based applications. However, the tangent model linearization removes the interesting, highly nonlinear, juicy stuff.
The issue here is actually the flow regime. Float trajectories tend to be chaotic. We can get stable and unstable manifolds. This chaotic behavior is less predictable. We have work with Larry Pratt and Irina Rypina (WHOI) to produce such stable and unstable manifolds from trajectories derived from ROMS surface currents in the Philippine Archipelago. Some of these results have been published (Rypina, I., L.J. Pratt, J. Pullen, J. Levin, A. Gordon, Chaotic advection in an archipelago, JPO). I think that this paper should come to print soon.
Changing the routines step_floats.F and interp_floats.F to time-step backwards is trivial. This can be done offline. These routines are designed to work with the provided 3D velocity field. In principle, you can generate those fields from a ROMS history NetCDF file. You can then easily replace the dynamical model with a backward time interpolator.
The issue here is actually the flow regime. Float trajectories tend to be chaotic. We can get stable and unstable manifolds. This chaotic behavior is less predictable. We have work with Larry Pratt and Irina Rypina (WHOI) to produce such stable and unstable manifolds from trajectories derived from ROMS surface currents in the Philippine Archipelago. Some of these results have been published (Rypina, I., L.J. Pratt, J. Pullen, J. Levin, A. Gordon, Chaotic advection in an archipelago, JPO). I think that this paper should come to print soon.
Changing the routines step_floats.F and interp_floats.F to time-step backwards is trivial. This can be done offline. These routines are designed to work with the provided 3D velocity field. In principle, you can generate those fields from a ROMS history NetCDF file. You can then easily replace the dynamical model with a backward time interpolator.
Re: trace the float trajectory back
The method suggested by Rodrigo certainly works.
Some remarks concerning partial differential equations and reverse time.
It's true that with ordinary differential equations, it suffices to set y=-t to examine reverse times.
Unfortunately, partial differential equations are a little bit more complex and the variable change y=-t is not, in general, possible.
A well-known example was given by Petrovsky:
let the equation u_{t}=k.u_{xx}, (t in IR; x in IR).
This equation is a very particular and simple case of the NS-equation.
Solution can be written as u(t,x)=1/n.sin(nx)exp(-n^2 kt) for all n in IN, (t,x) in IR^2.
Moreover, lim(n-> infinity) u(t,0)=0.
For t=+epsilon>0, u(x,+epsilon) is bounded.
But, for t=-epsilon<0, u(x,-epsilon) "explodes" : the problem is unstable.
Valère
Some remarks concerning partial differential equations and reverse time.
It's true that with ordinary differential equations, it suffices to set y=-t to examine reverse times.
Unfortunately, partial differential equations are a little bit more complex and the variable change y=-t is not, in general, possible.
A well-known example was given by Petrovsky:
let the equation u_{t}=k.u_{xx}, (t in IR; x in IR).
This equation is a very particular and simple case of the NS-equation.
Solution can be written as u(t,x)=1/n.sin(nx)exp(-n^2 kt) for all n in IN, (t,x) in IR^2.
Moreover, lim(n-> infinity) u(t,0)=0.
For t=+epsilon>0, u(x,+epsilon) is bounded.
But, for t=-epsilon<0, u(x,-epsilon) "explodes" : the problem is unstable.
Valère
Re: trace the float trajectory back
Val`ere,
I'm not sure what it was Petrovsky was showing with this example but I'm sorry to say that it is certainly not true that the solution "explodes" for -epsilon<0.
Assuming you meant k>0 (so that your arguments make more sense), you can in fact, choose any t<0 whatsoever as long as it is in IR as you stated. The solution will be finite and as nice a function as there can be (it is C infinity!). Although it is true that if you take lim(t--> -inf) then the solution goes to infinity that doesn't say much about PDEs and negative time. For an example of a PDE that remains finite in the limit as t goes to -inf, take k<0 in the same PDE that you posted and you again have a perfectly well behaved solution for all t<0 but this time with lim(t--> -inf)u(x_0,t)=0.
Finally, these examples are besides the fact that it makes no sense to talk about existence and uniqueness of any kind of differential equation (ordinary or partial) without considering (at least) boundary and/or initial conditions.
Again, please correct me if I am wrong,
Rodrigo.
I'm not sure what it was Petrovsky was showing with this example but I'm sorry to say that it is certainly not true that the solution "explodes" for -epsilon<0.
Assuming you meant k>0 (so that your arguments make more sense), you can in fact, choose any t<0 whatsoever as long as it is in IR as you stated. The solution will be finite and as nice a function as there can be (it is C infinity!). Although it is true that if you take lim(t--> -inf) then the solution goes to infinity that doesn't say much about PDEs and negative time. For an example of a PDE that remains finite in the limit as t goes to -inf, take k<0 in the same PDE that you posted and you again have a perfectly well behaved solution for all t<0 but this time with lim(t--> -inf)u(x_0,t)=0.
Finally, these examples are besides the fact that it makes no sense to talk about existence and uniqueness of any kind of differential equation (ordinary or partial) without considering (at least) boundary and/or initial conditions.
Again, please correct me if I am wrong,
Rodrigo.
Re: trace the float trajectory back
Hernan,
With regards to the juicy non-linear stuff, I believe that's what Rivas & Samelson were describing in their paper (which I mentioned above) when they said:
"Note that the presence of diffusion and mixing in the model introduces an in-
evitable ambiguity into any attempt to analyze Lagrangian motion of water parcels,
since such mixing destroys the identity of the parcels. The present approach offers the
least ambiguity, in which the resolved model velocity is used directly for the advec-
tion, and the ambiguities and complexities of attempting to include a representation
of mixing are avoided, but can be misleading if the results are not interpreted with
appropriate care. To supplement this approach, rates of change of water properties
along Lagrangian pathways were computed, and these are discussed below."
Rodrigo
With regards to the juicy non-linear stuff, I believe that's what Rivas & Samelson were describing in their paper (which I mentioned above) when they said:
"Note that the presence of diffusion and mixing in the model introduces an in-
evitable ambiguity into any attempt to analyze Lagrangian motion of water parcels,
since such mixing destroys the identity of the parcels. The present approach offers the
least ambiguity, in which the resolved model velocity is used directly for the advec-
tion, and the ambiguities and complexities of attempting to include a representation
of mixing are avoided, but can be misleading if the results are not interpreted with
appropriate care. To supplement this approach, rates of change of water properties
along Lagrangian pathways were computed, and these are discussed below."
Rodrigo
Re: trace the float trajectory back
Then again it could refer to the other non-linear juicy stuff ...
Re: trace the float trajectory back
Here is a doctoral thesis (methematics) dealing with PDEs and negative times -- in particular they work with Navier Stokes and Burgers equation as t --> -Inf
http://repository.tamu.edu/bitstream/ha ... sequence=1
Radu, the author, came to OSU as a professor recently -- really nice person.
Edit: It seems from chapter IV (1 page dedicated to conclusions) that my above remarks are sensible.
http://repository.tamu.edu/bitstream/ha ... sequence=1
Radu, the author, came to OSU as a professor recently -- really nice person.
Edit: It seems from chapter IV (1 page dedicated to conclusions) that my above remarks are sensible.