CN114021445A - Ocean vortex mixed non-local prediction method based on random forest model - Google Patents

Ocean vortex mixed non-local prediction method based on random forest model Download PDF

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CN114021445A
CN114021445A CN202111270420.8A CN202111270420A CN114021445A CN 114021445 A CN114021445 A CN 114021445A CN 202111270420 A CN202111270420 A CN 202111270420A CN 114021445 A CN114021445 A CN 114021445A
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陈儒
管文婷
邓增安
张翠翠
陈阳
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Abstract

The invention discloses a random forest model-based ocean vortex mixing non-local prediction method. In order to solve the problem that the nonlinear relation between the Lagrange balance time and the vortex velocity and the mixed non-local degree is not studied sufficiently in the existing method, the method comprises the steps of calculating the vortex diffusivity, the vortex diffusivity error value, the Lagrange balance time at the positions of all the self-adaptive boxes based on a standard deviation method, the mixed non-local ellipse and the vortex velocity, taking the Lagrange balance time and the vortex velocity as input samples, taking the mixed non-local degree as output samples, and constructing a random forest model so as to obtain the predicted value of the marine mixed non-local degree. The method is helpful for correctly recognizing the importance degree of the non-local vortex mixing in the ocean and climate simulation and prediction, provides a new idea for improving the existing vortex mixing parameterization scheme, and indicates the direction for further improving the accuracy of the mode simulation and prediction.

Description

Ocean vortex mixed non-local prediction method based on random forest model
Technical Field
The invention relates to the field of ocean vortex mixing non-local prediction, in particular to a method for predicting the non-local degree of vortex mixing by using a machine learning method.
Background
Some subgrid processes in the existing marine climate model need to be parameterized. The ocean vortex mixing parameterization is an important component of ocean climate mode simulation. The assumption in vortex mixing parameterization is that the vortex flux can be represented by the diffusivity coefficient times the local tracer gradient, but the tracer flux is expanded into a form by the green's function method, which clearly shows that the tracer flux at a given point depends on both the local and non-local large scale tracer gradient parts [1], and that the lagrange vortex diffusivity coefficient is also non-local in nature. The non-local knowledge and application of vortex mixing in existing climate patterns is not sufficient. The neglect of the non-local characteristics of the ocean mixture by the climate pattern limits the forecasting precision and accuracy of the climate pattern.
There have been studies to predict the degree of mixing non-locally by constructing a mixed non-locally ellipse in a positive pressure ideal model using a scale method [2 ]. However, in a high-resolution simulation model experiment, the correlation between the scaling method and the non-local degree of mixing of particles based on Lagrangian values is significant, but the root mean square error is large. The prior art has certain limitations in the non-local nature of quantitative predictive mixing.
Random forest is a machine learning algorithm, essentially a collection of decision trees with put-back samples. For the regression problem involved in the present invention, the prediction result of the random forest model is the average of the decision tree. The random forest model has strong nonlinear fitting capability, can capture the nonlinear relation between the prediction factor and the prediction response value and avoid overfitting. But no research has been done to link hybrid non-local prediction with machine learning methods. A complex nonlinear relation exists between two characteristic parameters of Lagrange equilibrium time and vortex velocity and the mixing non-local degree, but the nonlinear relation cannot be described by the existing method. Therefore, the invention provides an ocean vortex mixing non-local prediction method based on a random forest model, so as to improve the accuracy of mixed non-local prediction.
[ reference documents ]
[1]Kraichnan R H.Eddy viscosity and diffusivity:exact formulas and approximations[J].Complex Systems,1987,1(4-6):805-820.Chen R,Gille S T,McClean J L,et al.A multiwavenumber theory for eddy diffusivities and its application to the southeast Pacific(DIMES)region[J].Journal of Physical Oceanography,2015,45(7):1877-1896.
[2]Chen R,Waterman S.Mixing Nonlocality and Mixing Anisotropy in an Idealized Western Boundary Current Jet[J].Journal of Physical Oceanography,2017,47(12):3015-3036.
Disclosure of Invention
Aiming at the prior art, the invention provides an ocean vortex mixing non-local prediction method based on a random forest model, which is used for constructing a non-linear relation between two input characteristic variables (Lagrange equilibrium time and vortex speed) and an output response variable (mixing non-local degree) by utilizing the random forest model. Compared with the existing linear prediction method, the method provided by the invention can more accurately predict the non-local degree of ocean mixing.
In order to solve the technical problem, the invention provides a random forest model-based ocean vortex mixing non-local prediction method, which mainly comprises the following steps:
step 1: calculating vortex diffusivity and vortex diffusivity error values;
step 2: calculating Lagrange equilibrium time at all the adaptive box positions;
and step 3: calculating a mixed non-local ellipse;
and 4, step 4: calculating the size of the vortex speed;
and 5: and predicting the non-local degree of vortex mixing by using a random forest model to obtain a predicted value of the non-local degree of mixing.
The specific content of the step 1 of the invention is as follows: first, deploy LagrangeDaily numerical particles, interpolating the ocean surface velocity field data on the numerical particles, and forming long-time sequence particle tracks by utilizing a fourth-order Runge-Kutta method (or other similar numerical methods); taking different moments as starting points on each long track, and intercepting tracks with fixed time length as particle quasi tracks; clustering all particle standard trajectories by using a K-means clustering algorithm (K-means) to obtain a self-adaptive box center; calculating the vortex diffusivity κ over time at each adaptive bin center positionij(x,τ)，
Figure BDA0003327841740000021
U 'in the formula (1)'i(t1|x,t0) Represents t1(ii) a vortex residual velocity in a particular direction i at a time, wherein the vortex residual velocity represents the difference between the particle velocity and the local time-averaged Euler velocity, (x, t)0) Denotes the initial position and initial time of a particular calibration trajectory, τ denotes the number of days of particle advection,<·>Lthe method comprises the following steps of (1) averaging all quasi-track autocorrelation function integrals passing through a position x of an adaptive box; vortex diffusivity kappa obtained for each adaptive bin according to the self-sampling method (boosting)ijError estimation is performed, i.e. N is set for the remaining vortex speed per daybootRepeating the experiment, randomly selecting N in the self-adaptive box for each experimentsubsetAligning the tracks and calculating the corresponding vortex diffusivity; thereby utilizing NbootStandard deviation sigma of single vortex diffusivitybootTo determine a vortex diffusivity error value error with a 95% confidence level,
Figure BDA0003327841740000022
the specific content of step 2 of the invention is as follows:
step 2-1) vortex diffusion coefficient at adaptive bin position x based on formula (2)
Figure BDA0003327841740000031
Diagnostic methodLet τ be1Initial value of 1, tau2With an initial value of τ1+ N-1, N being the length τ of the quasi track of the particle0A period of time of upper interception;
Figure BDA0003327841740000032
in the formula (2), κij(x, τ) is the integral over time of the velocity autocorrelation function for a particular direction i and j of the number of days τ and adaptive bin position x of particle advection;
step 2-2) at time τ1To tau2And (4) integrating, and judging whether the vortex diffusivity integral converges according to the following process:
determination of tau2Whether or not τ is exceeded0If τ is2<τ0Then according to τ1To tau2Average value of vortex diffusivity corresponding to the period of time
Figure BDA0003327841740000033
The standard deviation std is obtained and the minimum value e corresponding to the error value error of the vortex diffusion rate in the periodminFor comparison, if std<eminThen the vortex diffusivity integral at the adaptive bin position x is considered to converge and the lagrange equilibrium time τ is takeneqIs tau1And τ2Average value of (d); if std is more than or equal to eminThen executing step 2-3);
step 2-3) τ1And τ2Respectively adding 1, and returning to the step 2-2); up to tau2Exceeds tau0The judgment condition std of convergence still cannot be satisfied<eminIf the self-adaptive box position x is determined to be the vortex diffusivity convergence, the Lagrange equilibrium time is determined to be the convergence;
and 2-4) obtaining the Lagrangian balance time at the positions of all the adaptive boxes according to the steps 2-1) to 2-3).
The specific content of step 3 of the invention is: within each adaptive bin, intercept from time 0 to Lagrange equilibrium time τeqAll quasi-tracks of (1) are calculated respectivelyThe average values of longitude and latitude coordinates on all the intercepted standard tracks are taken as track centroids; a mixed non-local ellipse is constructed using formula set (3),
Figure BDA0003327841740000034
in the formula set (3), the mean variances of the standard trajectory coordinates relative to the trajectory centroid in different directions are respectively recorded as the latitudinal variances
Figure BDA0003327841740000035
Variance in warp direction
Figure BDA0003327841740000036
Sum cross variance
Figure BDA0003327841740000037
Calculating the non-local degree S of vortex mixing using equation (4)ellipse，
Figure BDA0003327841740000038
In the formula (4), the reaction mixture is,
Figure BDA0003327841740000039
to blend the non-local elliptical semi-major axis lengths,
Figure BDA00033278417400000310
to blend the non-local elliptical semi-minor axis lengths.
Step 4 of the present invention, the magnitude u of the vortex velocity is calculated according to equation (5)rmsIn the formula (5), u 'is the weft-wise vortex residual velocity, and v' is the warp-wise vortex residual velocity.
Figure BDA0003327841740000041
The specific process of step 5 of the invention is as follows:
step 5-1) determining an original sample set: the Lagrange equilibrium time tau at all the adaptive box positions obtained in the step 2 is usedeqAnd obtaining the magnitude u of the vortex velocity according to step 4rmsAs input feature samples in the original sample set; the non-local degree S of the vortex mixture obtained in the step 3ellipseAs output response samples in the original sample set; each sample point in the original sample set has two characteristic input values and a response output value, and the sample points are independent; carrying out Z-score (Z-score) standardization on an original sample set, randomly selecting a% of original samples as a training sample set in the original sample set, and taking the rest (100-a)% of original samples as a testing sample set;
step 5-2) constructing a random forest model by utilizing the training sample set;
step 5-3) taking an input characteristic sample value in the test sample set as the input of the random forest model;
and 5-4) carrying out inverse standardization on the output value of the random forest model to obtain a mixed non-local degree prediction value.
Compared with the prior art, the invention has the beneficial effects that:
the standard deviation method can be used for accurately and quickly judging the convergence of the vortex mixing diffusivity and determining the corresponding Lagrangian balance time, the result is similar to that of the existing least square fitting method, and the two methods are proved mutually, so that the determination of the Lagrangian balance time is objective, accurate and convincing. In addition, the Lagrange equilibrium time represents the time required by the particle speed to reach the degree of decorrelation with the initial speed, and the Lagrange equilibrium time is used as an important input component in a random forest model, and the accurate judgment of the Lagrange equilibrium time is favorable for improving the precision of the non-local degree of mixing. Compared with the existing method, the mixed non-local prediction method based on the random forest model provided by the invention considers more comprehensively and effectively constructs the nonlinear relation among the Lagrange equilibrium time, the vortex speed and the mixed non-local prediction degree, thereby improving the mixed non-local prediction accuracy. Compared with the prior art, the method has the advantages that the root mean square error is greatly reduced, and the non-local prediction precision of vortex mixing is remarkably improved.
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FIG. 1 is a flow chart of a prediction method according to the present invention;
FIG. 2 is a comparison of the prediction results of the present invention with prior art prediction results, wherein (a) is the prediction results of the prior art scale method and (b) is the prediction results of the random forest model.
Detailed Description
The invention will be further described with reference to the following figures and specific examples, which are not intended to limit the invention in any way.
To further understand the contents, features and effects of the present invention, the non-local prediction of average cross-current vortex mixing in black tide extensor region (110 ° E-170 ° W, 20 ° N-45 ° N) is taken as a specific example, and the following detailed description is made with reference to the accompanying drawings:
as shown in a flow chart of an attached figure 1, a random forest model-based ocean vortex mixed non-local prediction method includes the steps of firstly calculating vortex diffusivity and vortex diffusivity errors through a Lagrange numerical particle experiment, judging Lagrange equilibrium time according to a standard deviation method provided by the invention, further calculating a mixed non-local ellipse to obtain a mixed non-local degree, then respectively taking the Lagrange equilibrium time and the vortex speed as input characteristic variables, taking the mixed non-local degree as an output response variable, introducing 70% of sample amount as a random forest model training set and constructing a random forest model, and finally deriving a prediction result corresponding to a 30% test set with samples not repeated by the training set as an ocean vortex mixed non-local degree prediction value. The specific technical operation is as follows:
step 1: calculating the annual average cross-flow vortex diffusivity and its error value, including:
firstly, deploying Lagrangian numerical particles with the spatial resolution of 0.2 degrees in a plurality of months, interpolating the MITgcm-llc4320 ocean surface velocity field data on the numerical particles, forming 180-day particle trajectories by using a fourth-order Runge-Kutta method (without excluding other similar numerical methods), and distributing advection trajectories of all deployed Lagrangian numerical particles on a one-year time scale;
taking different moments as starting points on each long track, and intercepting the track with the length of 100 days as a particle quasi track; clustering all particle quasi tracks by using a K-means clustering algorithm (K-means), so that about 1000 quasi tracks exist in each self-adaptive box, and the center of the self-adaptive box is obtained;
calculating the annual average cross-flow vortex diffusivity κ over time at each adaptive bin center location⊥(x,τ)，
Figure BDA0003327841740000051
U 'in the formula (1)'⊥(t1|x,t0) Represents t1The residual speed of vortex is measured in the cross-flow direction at any moment, wherein the residual speed of vortex represents the difference between the particle speed and the local time average Euler speed, and the cross-flow direction refers to the direction of the cross-local year average Euler speed, (x, t)0) Denotes the initial position and initial time of a particular calibration trajectory, τ denotes the number of days of particle advection,<·>Lthe method comprises the following steps of (1) averaging all self-correlation function integrals of quasi-tracks passing through a position x of an adaptive box within one year;
annual average cross-flow vortex diffusivity kappa obtained for each adaptive bin according to the self-sampling method (boosting)⊥Error estimation is performed, i.e. N is set for the cross-flow vortex residual velocity per daybootRepeating the experiment, randomly selecting N in the self-adaptive box for each experimentsubsetAligning tracks, and calculating corresponding annual average cross-flow vortex diffusion rate;
thereby utilizing NbootStandard deviation sigma of single vortex diffusivitybootTo determine an annual average cross-flow vortex diffusivity error value error with a 95% confidence level,
Figure BDA0003327841740000052
step 2: calculating all self-adaptive box position Lagrange equilibrium time tau based on standard deviation methodeqThe method comprises the following steps:
step 2-1) annual average cross-flow vortex diffusion coefficient at adaptive box position x based on formula (2)
Figure BDA0003327841740000061
Diagnostic equation, set τ1Initial value of 1, tau2With an initial value of τ1+ N-1, N being the length τ of the quasi track of the particle0Length of time of upper intercept, τ0＝100，N＝30；
Figure BDA0003327841740000062
In the formula (2), κ⊥(x, τ) is the integral over time of the velocity autocorrelation function across the flow direction for the number of days τ of particle advection and the adaptive bin position x;
step 2-2) at time τ1To tau2And (4) integrating, and judging whether the vortex diffusivity integral converges according to the following process:
determination of tau2Whether or not τ is exceeded0If τ is2<τ0Then according to τ1To tau2Average value of vortex diffusivity corresponding to the period of time
Figure BDA0003327841740000063
The standard deviation std is obtained and the minimum value e corresponding to the error value error of the vortex diffusion rate in the periodminFor comparison, if std<eminThen the vortex diffusivity integral at the adaptive bin position x is considered to converge and the lagrange equilibrium time τ is takeneqIs tau1And τ2Average value of (d); if std is more than or equal to eminExecuting the step 2-3);
step 2-3) τ1And τ2Respectively adding 1, and returning to the step 2-2); up to tau2Exceeds tau0The judgment condition std of convergence still cannot be satisfied<eminIf the self-adaptive box position x is determined to be the vortex diffusivity convergence, the Lagrange equilibrium time is determined to be the convergence;
and 2-4) obtaining the Lagrangian balance time at the positions of all the adaptive boxes according to the steps 2-1) to 2-3).
And step 3: computing a hybrid non-local ellipse comprising:
within each adaptive bin, intercept from time 0 to Lagrange equilibrium time τeqRespectively solving the average values of longitude and latitude coordinates on all quasi tracks, and taking the average values as track centroids;
a mixed non-local ellipse is constructed using formula set (3),
Figure BDA0003327841740000064
in the formula set (3), the mean variances of the standard trajectory coordinates relative to the trajectory centroid in different directions are respectively recorded as the latitudinal variances
Figure BDA0003327841740000065
Variance in warp direction
Figure BDA0003327841740000066
Sum cross variance
Figure BDA0003327841740000067
Calculating the non-local degree S of vortex mixing using equation (4)ellipse，
Figure BDA0003327841740000071
In the formula (4), the reaction mixture is,
Figure BDA0003327841740000072
to blend the non-local elliptical semi-major axis lengths,
Figure BDA0003327841740000073
is a mixture of non-locally elliptical semi-minor axial lengths;
and 4, step 4: calculating the magnitude of the vortex velocity urms，
Figure BDA0003327841740000074
In the formula (5), u 'is the weft vortex residual velocity, and v' is the warp vortex residual velocity;
and 5: predicting the non-local degree of vortex mixing by using a random forest model, comprising the following steps of:
step 5-1) determining an original sample set
The Lagrange equilibrium time tau at all the adaptive box positions obtained in the step 2 is usedeqAnd obtaining the magnitude u of the vortex velocity according to step 4rmsAs input feature samples in the original sample set; the non-local degree S of the vortex mixture obtained in the step 3ellipseAs output response samples in the original sample set; each sample point in the original sample set has two characteristic input values and a response output value, and the sample points are independent; performing Z-score (Z-score) standardization on an original sample set, randomly selecting 30% of original samples in the original sample set as a training sample set, and using the remaining 70% of original samples as a test sample set;
step 5-2) constructing a random forest model by utilizing the training sample set;
step 5-3) taking an input characteristic sample value in the test sample set as the input of the random forest model;
and 5-4) carrying out inverse standardization on the output value of the random forest model to obtain a mixed non-local degree prediction value.
The root mean square error RMSE and the coefficient of solution R can be calculated from the predicted and measured values of the mixed non-local degree2And the prediction effect of the random forest model can be evaluated and analyzed by comprehensively considering the random forest model and the random forest model. The construction of the random forest model can be realized by MATLAB software, and the detailed construction process can be seen in an MATLAB instruction manual.
In the specific examples, reference [2] is compared]I.e. a scale method to predict mixing non-locality. The comparison between the prediction results (a) of the scale method and the random forest model prediction results (b) is shown in FIG. 2, in whichThe black line represents the regression line for linear fit of predicted and measured values, and the confidence interval with 95% confidence level is shown within the two gray lines. Both R's can be found2The prediction accuracy is greatly improved, and the prediction accuracy is high and reaches about 0.8, but the RMSE of the random forest model prediction value is reduced to 8.9% in the prior art. According to the ocean vortex mixing non-local prediction method based on the random forest model, the non-linear relation between two characteristic variables of Lagrange equilibrium time and vortex speed and the mixing non-local degree is established, and the mixing non-local degree can be accurately predicted.
In summary, the method solves the problem of insufficient study on the nonlinear relation between the Lagrange equilibrium time and the vortex velocity and the mixed non-local degree in the existing method, and constructs a random forest model by calculating the vortex diffusivity, the vortex diffusivity error value, the Lagrange equilibrium time at all the self-adaptive box positions based on the standard deviation method, the mixed non-local ellipse and the vortex velocity, taking the Lagrange equilibrium time and the vortex velocity as input samples, and the mixed non-local degree as output samples, so as to obtain the predicted value of the marine mixed non-local degree. The method is helpful for correctly recognizing the importance degree of the non-local vortex mixing in the ocean and climate simulation and prediction, provides a new idea for improving the existing vortex mixing parameterization scheme, and indicates the direction for further improving the accuracy of the mode simulation and prediction.
While the present invention has been described with reference to the accompanying drawings, the present invention is not limited to the above-described embodiments, which are illustrative only and not restrictive, and various modifications which do not depart from the spirit of the present invention and which are intended to be covered by the claims of the present invention may be made by those skilled in the art.

Claims (2)

1. A random forest model-based ocean vortex mixing non-local prediction method is characterized by comprising the following steps:
step 1: calculating a vortex diffusivity and a vortex diffusivity error value, comprising:
firstly, deploying Lagrange numerical particles, interpolating ocean surface velocity field data at the time-space position of the numerical particles, and forming long-time sequence particle tracks by utilizing a fourth-order Runge-Kutta method;
taking different moments as starting points on each particle track, and intercepting tracks with fixed time length as particle quasi tracks; clustering all particle standard trajectories by using a K-means clustering algorithm (K-means) to obtain a self-adaptive box center;
calculating the vortex diffusivity κ over time at each adaptive bin center positionij(x,τ)，
Figure FDA0003327841730000011
U 'in the formula (1)'i(t1|x,t0) Represents t1(ii) a vortex residual velocity in a particular direction i at a time, wherein the vortex residual velocity represents the difference between the particle velocity and the local time-averaged Euler velocity, (x, t)0) Denotes the initial position and initial time of a particular calibration trajectory, τ denotes the number of days of particle advection,<·>Lthe method comprises the following steps of (1) averaging all quasi-track autocorrelation function integrals passing through a position x of an adaptive box;
vortex diffusivity kappa obtained for each adaptive bin according to the self-sampling method (boosting)ijError estimation is performed, i.e. N is set for the remaining vortex speed per daybootRepeating the experiment, randomly selecting N in the self-adaptive box for each experimentsubsetAligning the tracks and calculating the corresponding vortex diffusivity;
thereby utilizing NbootStandard deviation sigma of single vortex diffusivitybootTo determine a vortex diffusivity error value error with a 95% confidence level,
Figure FDA0003327841730000012
step 2: computing stationLagrange equilibrium time τ at adaptive bin positioneq；
And step 3: computing a hybrid non-local ellipse comprising:
within each adaptive bin, intercept from time 0 to Lagrange equilibrium time τeqRespectively calculating the average values of longitude and latitude coordinates on all the intercepted quasi tracks, and taking the average values as track centroids;
a mixed non-local ellipse is constructed using formula set (3),
Figure FDA0003327841730000013
in the formula set (3), the mean variances of the standard trajectory coordinates relative to the trajectory centroid in different directions are respectively recorded as the latitudinal variances
Figure FDA0003327841730000014
Variance in warp direction
Figure FDA0003327841730000015
Sum cross variance
Figure FDA0003327841730000016
Calculating the non-local degree S of vortex mixing using equation (4)ellipse，
Figure FDA0003327841730000017
In the formula (4), the reaction mixture is,
Figure FDA0003327841730000021
to blend the non-local elliptical semi-major axis lengths,
Figure FDA0003327841730000022
is a mixture of non-locally elliptical semi-minor axial lengths;
step (ii) of4: calculating the magnitude of the vortex velocity urms，
Figure FDA0003327841730000023
In the formula (5), u 'is the weft vortex residual velocity, and v' is the warp vortex residual velocity;
and 5: predicting the non-local degree of vortex mixing by using a random forest model, comprising the following steps of:
step 5-1) determining an original sample set
The Lagrange equilibrium time tau at all the adaptive box positions obtained in the step 2 is usedeqAnd obtaining the magnitude u of the vortex velocity according to step 4rmsAs input feature samples in the original sample set; the non-local degree S of the vortex mixture obtained in the step 3ellipseAs output response samples in the original sample set; each sample point in the original sample set has two characteristic input values and a response output value, and the sample points are independent; carrying out Z-score (Z-score) standardization on an original sample set, randomly selecting a% of original samples as a training sample set in the original sample set, and taking the rest (100-a)% of original samples as a testing sample set;
step 5-2) constructing a random forest model by utilizing the training sample set;
step 5-3) taking an input characteristic sample value in the test sample set as the input of the random forest model;
and 5-4) carrying out inverse standardization on the output value of the random forest model to obtain a mixed non-local degree prediction value.
2. The prediction method according to claim 1, wherein in the step 2, the lagrangian equilibrium time of all the adaptive box positions is calculated by using a standard deviation method, and the method comprises the following steps:
step 2-1) vortex diffusion coefficient at adaptive bin position x based on formula (2)
Figure FDA0003327841730000024
Diagnostic equation, set τ1Initial value of 1, tau2With an initial value of τ1+ N-1, N being the length τ of the quasi track of the particle0A period of time of upper interception;
Figure FDA0003327841730000025
in the formula (2), κij(x, τ) is the integral over time of the velocity autocorrelation function for a particular direction i and j of the number of days τ and adaptive bin position x of particle advection;
step 2-2) at time τ1To tau2And (4) integrating, and judging whether the vortex diffusivity integral converges according to the following process:
determination of tau2Whether or not τ is exceeded0If τ is2<τ0Then according to τ1To tau2Average value of vortex diffusivity corresponding to the period of time
Figure FDA0003327841730000026
The standard deviation std is obtained and the minimum value e corresponding to the error value error of the vortex diffusion rate in the periodminFor comparison, if std<eminThen the vortex diffusivity integral at the adaptive bin position x is considered to converge and the lagrange equilibrium time τ is takeneqIs tau1And τ2Average value of (d); if std is more than or equal to eminThen executing step 2-3);
step 2-3) τ1And τ2Respectively adding 1, and returning to the step 2-2); up to tau2Exceeds tau0The judgment condition std of convergence still cannot be satisfied<eminIf the self-adaptive box position x is determined to be the vortex diffusivity convergence, the Lagrange equilibrium time is determined to be the convergence;
and 2-4) obtaining the Lagrangian balance time at the positions of all the adaptive boxes according to the steps 2-1) to 2-3).
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