Clustering of Floating Tracers in a Random Velocity Field Modulated by an Ellipsoidal Vortex Flow

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

I reviewed the manuscript entitled as "Clustering of floating tracer in the random velocity field modulated by an ellipsoidal vortex flow" by Koshel et al.. Authors investigated by numerical simulations the evolution of the density of floating tracer in a random velocity field modulated by an ellipsoidal vortex flow. Various stochastical results are represented. In particular, the clustering phenomena differ in the different states.

 

The paper is in general very interesting. Results are well discussed. I would be glad to recommend the publication if the following minor questions are appropriately answered.

 

(1) The numerical discretization methods (temporal and spatial) are not clearly introduced. This would be important for guarantee that results are convincing. For example, there exists very great gradient for the density in Fig. 2, which may raise challenge to the numerical discretization methods. Clear descriptions will help readers to understand and to trust the results.

(2) The random velocity field should be important to the dispersion of particles. Section 2.1 describes the statistical calculations of a random field, however, I did not find the descriptions on the random field itself. That is, is the random field white noise without temporal correlation? What is the rms value of the random field? And other details.

Author Response

Response to the reviews of the manuscript " Clustering of floating tracer in the random velocity field modulated by an ellipsoidal vortex flow" by Koshel et al.

We would like to express our gratitude to the reviewer for his/her careful examination of the manuscript and suggested remarks and recommendations. Our response to the raised questions are as follows.

 

Q1 The numerical discretization methods (temporal and spatial) are not clearly introduced. This would be important for guarantee that results are convincing. For example, there exists very great gradient for the density in Fig. 2, which may raise challenge to the numerical discretization methods. Clear descriptions will help readers to understand and to trust the results.

A1 We extended our manuscript including next sentences

1. 151/161: We assume that random velocity components are constant in a sampling grid cell. Thus, we can solve equations (\ref{eq5}) analytically and estimate time, when the particle reaches the boundary of cell. Then we use this solution as the initial condition in the next cell. We repeat this procedure until time reaches next time layer. This procedure allows to solve stochastic differential equations with undifferentiated coefficients. For example, the system equations (\ref{eq5}) has the solution in cell

\[{\rho _{ijk}} = {\rho _{ijk - 1}}\exp \left\{ { - \sum\limits_{\tilde k = 1}^{\tilde n} {\frac{{\partial {\bf{U}}\left( {\tilde i,\tilde j,\tilde k} \right)}}{{\partial {\bf{R}}}}\Delta {{\tilde t}_{\tilde k}}} } \right\}.\]

Here, indices with a tilde mark the cells through which the particle trajectory passes before reaching the next time layer, and $\Delta {\tilde t}_{\tilde k}$ are the times before reaching the border of the next cell. The tilde marks indices of cells, which were crossed over by the particle trajectory before the next time layer, and $\Delta {\tilde t}_{\tilde k}$ are times before the reaching of the particle trajectory border of the next cell.

1. 146/148: It is worth to mention that we use a lot of particles (characteristics) and due to clustering we have many particles in the area of high density gradient.

Q2 The random velocity field should be important to the dispersion of particles. Section 2.1 describes the statistical calculations of a random field, however, I did not find the descriptions on the random field itself. That is, is the random field white noise without temporal correlation? What is the rms value of the random field? And other details.

A2 We extended our manuscript including next sentences

1. 87/88: We assume that the velocity field is $\delta$ correlated in time.
2. 94/95: Here we use spectral density in the form

\[E_{}^p\left( {k,\eta } \right) = E_{}^s\left( {k,\eta } \right) = \sigma _{\bf{U}}^2\frac{{{l^4}}}{{4\pi }}\exp \left\{ { - \frac{1}{2}{k^2}{l^2}} \right\}\delta \left( \eta  \right),\]

and $l$ is the spatial correlation length.

1. 138/139: For all numerical experiments we use next rms value ${\sigma _{\bf{U}}} = 0.33$ and spatial correlation radius value $l =8$.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

See attached

Comments for author File: Comments.pdf

Author Response

Response to the reviews of the manuscript " Clustering of floating tracer in the random velocity field modulated by an ellipsoidal vortex flow" by Koshel et al.

We would like to express our gratitude to the reviewer for his/her careful examination of the manuscript and suggested remarks and recommendations. Our response to the raised questions are as follows.

 

Q1 l. 83/84 : Maybe swap the order `nondivergent' and `divergent' in the sentence preceding equation (5) as in equation (5) the divergent part appears first. Or even better, be more specific: `consists of a divergent component $U_r^p$ and a nondivergent component $U_s^s$.

A1: l. 88/89: We edited the text as follows:

This random velocity field consists of a divergent component (${\bf{U}}_r^p({\bf{R}},t)$) and a nondivergent component (${\bf{U}}_r^s({\bf{R}},t)$)

 

Q2 In equation (6), the indices $\alpha$ and $\beta$ are not explicitly defined. Please state explicitly that these refers to the vector components (especially since index notation can mean many other different things).

A2: l. 92/93: We defined the indices $\alpha$ and $\beta$ in the text:

Indices $\alpha$ and $\beta$ stand for $x$ and $y$ and indicate different components of the tensor.

 

Q3 Is $E^j_{\alpha\beta$} just the Fourier coefficient of the correlation? If so, reading would be made smoother by stating this.

A3: l. 94/95: Yes. You are right. We added next statement: 

Here we use spectral density in the form

\[E_{}^p\left( {k,\eta } \right) = E_{}^s\left( {k,\eta } \right) = \sigma _{\bf{U}}^2\frac{{{l^4}}}{{4\pi }}\exp \left\{ { - \frac{1}{2}{k^2}{l^2}} \right\}\delta \left( \eta  \right),\]

and $l$ is the spatial correlation length.

 

Q4 Equations (10): many symbols used in the equations (a,b,c,e,\gamma) are only defined in the Appendix. The definition of these symbols should be moved to the main text as the reader should not have to read the appendix to understand the paper.

A4: l.108/109: We defined symbols from the Appendix in the text as follows

Here $a,b,c$ are the semi-axes of the ellipsoidal vortex, $\chi$ and $e$ are the rotational and strain components of the deformation flow, respectively.

 

Q5 Lines 123-129. I find the introduction of the typical length scales unclear. What exactly $L_{vortex}$? Is is $a$, $b$, $c$ $\sqrt[3]{abc}$? It is odd to give an exact value while remaining vague on what it actually measures. What is a `time of scattering'?

A5: We added next statements in the Sec. 3 Scaling…:

1. 135/136: Typical spatial scale of the ellipsoidal vortex or the $b$ semi-axes $L_{\text{vortex}}=6\times10^2\cdot L$,…
2. 137/138: ..time of diffusion scattering or the diffusion timescale $2\times10^2\cdot T$.

 

Q6 Lines 130-132: It is unclear whether the authors follow Lagrangian particles (`the patch is uniformly filled with $3.6 \times 10^{10}$ floating markers) of an Eulerian tracer density field $\rho(R,t)$? Both?

A6:l.151/161: We extended our manuscript including next statements

We assume that random velocity components are constant in a sampling grid cell. Thus, we can solve equations (\ref{eq5}) analytically and estimate time, when the particle reaches the boundary of cell. Then we use this solution as the initial condition in the next cell. We repeat this procedure until time reaches next time layer. This procedure allows to solve stochastic differential equations with undifferentiated coefficients. For example, the system equations (\ref{eq5}) has the solution in cell

\[{\rho _{ijk}} = {\rho _{ijk - 1}}\exp \left\{ { - \sum\limits_{\tilde k = 1}^{\tilde n} {\frac{{\partial {\bf{U}}\left( {\tilde i,\tilde j,\tilde k} \right)}}{{\partial {\bf{R}}}}\Delta {{\tilde t}_{\tilde k}}} } \right\}.\]

Here, indices with a tilde mark the cells through which the particle trajectory passes before reaching the next time layer, and $\Delta {\tilde t}_{\tilde k}$ are the times before reaching the border of the next cell. The tilde marks indices of cells, which were crossed over by the particle trajectory before the next time layer, and $\Delta {\tilde t}_{\tilde k}$ are times before the reaching of the particle trajectory border of the next cell.

 

Equations (12) and (13) seem to indicate it is a mix of both but it is rather unclear. What is meant by `\xi are the coordinate of the initial patch'? If it just means $\xi=R(0)$, equation (12) does a better job at explaining and the sentence only brings confusion. Have the authors checked the surface integral of $\rho$ numerically conserved in time? (It seems the authors used $Q=\kappa=0$ but again this is unclear.)

A6: We added next statement

1. 146: where ${\bm{\xi}}-$ are initial coordinates of the floating marker.

 

Q7 Line 195: typo: `nor' -> 'not'

A7: l.220 Corrected.

 

Q8 Using $\rho$ for the space-time dependent field and the level in $\varphi(R,t;\rho)$ is very confusing (even if the same notation is used in reference 18.

A8: We changed  $\rho$ on $\bar\rho$

 

Q9 Equations (14) and (15) please state what $\theta$ represents (even if one can figure it out from the equations)

A9: l.168: We added next statement

…predefined threshold $\bar \rho$, $\theta(\cdot)$ is the Heaviside (step) function…

 

Q10 Equation (18) and following. A few symbols are undefined (e.g. \gamma_r: how this differ from \gamma previously introduced?)

A10: l.185: We defined a new symbol $\gamma$

 

Q11 Where is the ellipsoidal vortex relative to the plane where the tracer is (above or below? or is the plane cutting through the vortex)?

A11:l.219: We added next sentence in our manuscript:

We consider a horizontal plane cutting the ellipsoid through the center. 

Author Response File: Author Response.pdf