Effect of nonlinearity on interaction between the vortices in the f-plane shallow water system

Abstract

Based on the equations of the f-plane shallow water model, a new set of nonlinear models for the interaction of the vortices in the f-plane shallow water system are established by using perturbation expansion and multi-scale method. These models have different nonlinear characteristics. The asymptotic analytical solution(vortex solution) is obtained by using Newton iteration method and quasi-spectral method. In the simulation process, different from the previous studies, the initial values were constructed by using the asymptotic analytical solution for the first time to simulate the interaction process of the vortices. The result shows that the single vortex maintains a stable state for a long time under fixed parameters. The approximate critical distances of the binary vortices are different under different nonlinear effects. When the distance between binary vortices is less than, equal to, or greater than the critical distance, the binary vortices merge, rotate or separate. In addition, when the vortices interact, the critical distance of mKdV-type vortex is slightly larger than that of KdV-type vortex. This indicates that the nonlinear effect increases the energy loss rate during the interaction of the vortices and weakens the intensity of the interaction until the vortices are separated and do not affect each other.

Appendix

• Iterative approximation of the MATLAB code:

• clc,clear

• syms r x y a1 a2 a3 a4 temp;

• $L=1$; (Part is to find the coefficient of the initial iteration solution u. The main idea is to assume the form of u first and then substitute it into Equation to find suitable a1 and a2)

• $T_1=chebshev(1,x);$ (Chebyshev polynomials of first order)

• $T_2=chebshev(2,x);$

• $T_3=chebshev(3,x);$

• $T_5=chebshev(5,x);$

• $T_4=chebshev(4,x);$

• $T_6=chebshev(6,x);$

• $T_8=chebshev(8,x);$

• $T_10=chebshev(10,x);$

• $temp1=temp./sqrt(temp.^2+L.^2)$;

• $T_1=subs(T_1,x,temp1);$ (First-order Chebyshev polynomials of $Tb_N$ form)

• $T_2=subs(T_2,x,temp1);$

• $T_3=subs(T_3,x,temp1);$

• $T_4=subs(T_4,x,temp1);$

• $T_5=subs(T_5,x,temp1);$

• $T_6=subs(T_6,x,temp1);$

• $T_8=subs(T_8,x,temp1);$

• $T_10=subs(T_10,x,temp1);$

• $T_1=subs(T_1,temp,r./2);$

• $T_2=subs(T_2,temp,r./2);$

• $T_3=subs(T_3,temp,r./2);$

• $T_4=subs(T_4,temp,r./2);$

• $T_5=subs(T_5,temp,r./2);$

• $T_6=subs(T_6,temp,r./2);$

• $T_8=subs(T_8,temp,r./2);$

• $T_10=subs(T_10,temp,r./2);$

• tic

• syms r x y w;

• $L=1$; (The transformation value L in the definition of $TB_{-}N$)

• $a1=0.06654;$

• $a2=-0.02887;$

• $a3=0.005687;$

• $a4=0;$

• $a5=0;$

• $a_{-}1_{-}2=[a1;a2;a3;a4;a5];$ (The coefficients of pseudospectral method are a1 and a2)

• $w_{-}temp=[(T_{-}2-1),(T_{-}4-1),(T_{-}6-1),(T_{-}8-1),(T_{-}10-1)]*a_{-}1_{-}2;$

• $res=1;$

• $n=100;$

• $k=0;$

• $x_{-}=-10:0.1:10;$

• $y_{-}=x_{-};$

• $len=length(x_{-});$

• $d=1;$ (while $k<=n, res>=0.0001, d~=0;$ When the number of iterations is greater than the maximum number of iterations or the error between the two solutions is zero, the iteration is terminated)

• $w0=w_{-}temp;$

• $N=3;$ (The number of bases used by the near d function)

• $basis_set=sym(zeros(1,N))$; (The $i-th$ element in $basis_{-}set$ is the $basis_{-}set$ function)

• for $i=1:N$

• $basis_{-}set(i)=chebshev(2*i,r)-1;$

• end

• $basis_{-}set=subs(basis_{-}set,r,r./(sqrt(L.^2+r.^2)));$

• $i=1:2*N-1;$

• $xi_{-}str=cos((i.*pi)./(2.*N));$

• $xi=sqrt((xi_{-}str.^2.*L.^2)./(1-xi_{-}str.^2));$

• $xi_{-}r=2.*xi;$

• $xi_{-}r=[xi_{-}r(1:ceil(N-1/2)),-1.*xi_{-}r(ceil(N-1/2)+1:2*N-1)];$

• $D=xi_{-}r(1:N);$

• $B=zeros(N);$

• $C=zeros(N,1);$

• for

• $i=1:N$

• for

• $j=1:N$

• $temp_{-}L_{-}N=subs(L_{-}N,r,D(i));$

• $temp_{-}f=subs(f,r,D(i));$

• $B(i,j)=double(temp_{-}L_{-}N);$

• $C(i)=double(temp_{-}f);$

• end

• end

• $A=\frac{B}{C};$

• $A=sym(A);$

• $d=basis_{-}set*A;$

• $w1=w0+d;$

• $k=k+1;$

• $res_{-}f=simplify((diff(w1,r,2)+diff(w1,r,1)./r-w1-w1.^3));$

• $d_{-}temp=subs(res_{-}f,r,sqrt(x.^2+y.^2));$

• $d_{-}temp1=matlabFunction(d_{-}temp);$

• $d_{-}temp2=max(max(abs(d_{-}temp1(x_{-},y_{-}))));$

• $w_{-}temp=w1;$

• end

• $w1=simplify(w1);$

• pretty(w1); 

• $w_{-}temp1=subs(w1,r,sqrt(x.^2+y.^2));$

• $w_{-}temp2=matlabFunction(w_{-}temp1);$

• $w_{-}=w_{-}temp2(x_{-},y_{-});$

• figure;

• $mesh(x_{-},y_{-},w_{-});$

• toc