Shallow water model runs for various conditions of vortices

In the last two weeks, after the rejection of PRL, I began with the runing of the SWM. In order to test the validity of the dynamic model, I have to vary the vortex structure to see what the vortex evolutions will be. That's very interesting to observe from the present results that for small size vortices, they behave more like a point vortex whilst for the larger vortices (e.g. Rm > 150km) their evolutions are highly irregular than the small-vortex cases.

For the validity of the DM, now I am still test them to see if they can capture the tracks of the vortices. For those strong-wind, large-radius vortices, the flows are much complicated that associate with the vortex shedding secondary vortices interacting mutually with the primary vortex. When this flow phenonmena is not so strong, it can be modeled in the scope of dynamic model by introducing the parameter \theta; however, when the flow behaviors are vitally strong, for instance, for those cases of Vm~60m/s or with a high topography of Hm~3500m, the introduction of \theta is unable to predict the track of the vortex afterward.

I've read a paper concerning the aircraft observations of the hurricanes structures during 1977-1997. In this paper, they found the hurricane structures can be suitably modeled by a modified Rankine Vortex model. In addition, the RMW (radius of maximum wind) is approximately less than 100km for a typical hurricane. But the wind outside the RMW is generally larger than the traditional vortex model wind. For example, both the Gaussian model and the original Rankine model predict weaker winds outside the RMW. Therefore, I learn two things from this paper.

1. For the ideal vortex model, I should turn to this so-called modified Rankine vortex model to have a more realistic vortex distribution in the prototype SWM runs.
2. For the ideal vortex structure, I should pick up the scales of the vortex with smaller scales in order to match the situations of real hurricanes.

In summary, I've learned very much from the simulation results of SWM. Now, I am in the stage to make a good comparison of DM with SWM. Let's go ahead. Don't forget to back to this post to see if every point in the above wish list for SWM modification come true.

Return from the PRL paper- rejection!

There's no difficulties to hear such a bad news of the rejection of our submitted PRL paper. After all, there're still many things to learn in this problem. I would like to make modifications and intensifications of major points emphasising by the Referee A. Here is his report:

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> Report of Referee A -- LV8975/Chang
> ----------------------------------------------------------------------
>
> This letter proposes a new dynamic model for predicting the motion of vortices over topography. This is an important problem and a successful dynamic model would be a significant new result.
>
The model proposed by the authors suffers from the significant flaw that it contains two free parameters, kappa and theta. Having free parameters in a model such as this is not necessarily fatal, but the authors have presented no serious studies of how these parameters should vary for flows with different vortices and different topographies. If, for a range of vortices and topographies, the dynamic model matched the shallow water model with no change in the parameters, then the dynamic model would indeed be a significant new model. If, however, the
parameters would need to be retuned for each new situation, the dynamic model would be much less useful.

Because of this lack of study of the role of the free parameters, I recommend this paper not be published in Physical Review Letters.

As mentioned by Prof. Chang, this Referee points out a necessary direction of further modifications of my paper. That is to make a proper regime diagram showing that whether the dynamic model is valid or not for various conditions of vortices and topographies. I would like to check this by the following strategies.

1. For a 3-1 hill, at least RUN three cases for north to south for a standard vortex passing over topographies with different heights (from 500m to 3500m).
   This test would be illustrated the effect of topography height to the induced circulation effect \theta (with a fixed \kappa).
2. For a 3-1 hill, fix the vortex location, RUN several cases by changing the strength of the vortices (\zeta_c) with same radius or by changing the radius of vortices with the same vorticity.
   This test focuses on the effect of vortex structures to the proportional constant \kappa.

On the other hand, the study of track maps is still active. I would like to construct the regime diagram of the track diverging index with the coordinates of the vortex incoming angle (or the angle of attack of vortices) and the topography orienting angle. This would be a tedious process to key in those values and see them as a contour plot.

A Reminder of JAS paper refinement

Today, after meeting with Prof. Chang and discussing the key points in the JAS paper, we conclude with several suggestions.

For Me, I should do these following before this weekend (Friday, exactly)

1. Drawing a carton of the classification of track maps.
2. Relationship between \theta, \kappa and the mountain height h_m.
   This should be developed from those SWM cases I've already run and the comparision of DM and SWM.
   
   • For \theta, it is remarkable that we use an unique value for the vortices starting from different latitutes. The value of \theta is ONLY sensitive on the mountain height but non-sensitive with the impinging locations. To do a map with the relationship between the mountain height and the \theta.
     
   • For \kappa, it is remarkable that we use an unique value for the vortices encountering with topography with different height and from different latitudes. The value of \kappa is maybe sensitive with the strengh of the vortex. It is reasonable to say that \kappa is small for strong vortices and \kappa is large for weak vortices. This point should be supported with evidences from more SWM simulations and DM comparisons.
     
3. Try to give attractive discussions of the DM track maps.
   
   • Give better explainations for some of the track maps.
   • What kind of track maps is more significant in practical path predictions ?
     
   • Give them more versitile interpretations of these track maps.
     

For Prof. Chang, he would be able to give an insightful discussion on the DM theory for its general, versitile power on explaining the typhoon motions.

Go ahead ! The question is quite interesting!

A Reminder for the GFD group members (2, Nov. 2004)

Dear GFD members:

Here are our present status of research and the future trends.

1. For Hung-Cheng:

• On writting the JAS paper.
• After the paper, I'll do the following:
  a. Anaylsis work on the vertical structure of the monopolar vortices with and without background vorticity.
  b. Numerical simulations of the binary typhoons interactions.
  c. Assist Prof. Chu to modify the previous paper and the on-going paper.

2. For Shih-Ling:

• Laboratory experiments on the vertical structure of the monopolar vortex with backgroud vorticity. (The without-b.v experiments will be performed by A-da, please!)
• Paper readings on the tornado-like vortices structure, the vortex boundary layer, etc.

3. For Cheng-De:

• Thermo-forcing generator machinings.
• Paper readings on the typhoon-ocean interaction literatures (c.f. I.I. Lin's papers).
• Matlab learnings and using PTV codes providing by Carpet (This time, we'll put our efforts on the 3-D PTV for detecting the vertical structure of vortex as a term goal if possible).

4. For Bing-Sheng:

• Hard working on the machining on the Big-Brother Tank.
• A primary goal on November- to demo a basic flow for two vortices interaction on a beta-plane. (This should be done by successfully putting serveral components together including the two-vortex generator, the sloping bottom, the dye-injection facillity and optionally the laser light sheet visulization.)
• Pause the paper reading for a while and concentrate on doing experiments.
• After today's meeting, B-S would be able to read the theory of point vortices interactions.
  

Cheers, Hung-Cheng

Vertical Structure of Barotropic Isolated Vortex in a Rotating Tank

This is an Amazing picture showing the vertical structure of an isolated columnar vortex in a rotating tank. I've discussed the related mechanisms in our recent NSC report. In this blog, I want to address the possibility to analysis the flow structure by Scaling Analysis to estimate the scale of the local flow structure.

There're several regions of interests, they include:

1. The upper free surface at the vortex center (SD)
   This flow is due to the surface depression of the vortex. Owing to the deformation of fluid surface, the fluid velocity in the vicinity of the vortex center naturally has the vertical downward component . (secondary downward flow)
   
2. The lower solid boundary flow at the vortex center (SU)
   This secendary flow arises from the viscous boundary effect at the tank bottom. Owing to the main stream (a circulation), there exists a radial presure gradient thus steers the flow cyclonically inward to the center. Then, the inward flows are thus emerged to upward until the encountering with the downward flow. (secondary upward flow)
   
3. The internal boundary of the upward secondary flow and the downward surface flow. (IB)
   

By using the boundary layer analysis at the bottom and at the free surface with a vortex distribution (can use the Gaussian vortex distribution) to estimate the resulted characteristic flow velocities.

Roadmap to the Interaction of Binary Typhoons with Topography

Origin from the frequent binary typhoons events in the past years, the idea of interaction of typhoon with topography may change its way. The issue will be more interesting if the number of typhoon is TWO, i.e., the so-called Fujiwhara effect. In the literatures, the binary typhoons problems are investigated primarily with the aspect of vortex merger. Some papers discussed the mutual influence of their vortex tracks (e.g. Wu, 2003). Here I propose several directions for potential research of this issue. The following map is a composite map for my idea.

• Interaction of Binary Typhoons on a Beta-plane.
  We firstly to investigate the problems by choosing typhoons with different combinations of stengths, relative locations, in order to determine or to clarify the basic flow regime of binary typhoons problem.
• The Inclusion of TOPOGRAPHY to the Binary Typhoons system.
  (See the reference map) The inclusion of topography to the binary typhoons will cause very interesting respones to the paths of typhoons, their intensity changes, and their induced circulations around topography. Here the topography is expected to play a sensitive role to disturb the binary typhoons system. The map shows two possible ways of topography inclusion to the binary typhoons system.
  
• Incorporating the problem into the Dynamic model.
  Essentially, this problem can be incorporated into our dynamic model if the interaction is not going to the regime of vortex merger.

I think the above problems are very attractive to the audience of typhoon researchers. I will begins with the numerical simulations using Clawpack to model two Rankine vortices with proper vortex initializations.