Vortex Theory for Two Dimensional Boussinesq Equations

Morteza Sharifi 1  and Behruz Raesi 1
  • 1 Department of Math, Shahed University, Tehran, Iran
Morteza Sharifi and Behruz Raesi

Abstract

In this paper, the single center vortex method (SCVM) is extended to find some vortex solutions of finite core size for dissipative 2D Boussinesq equations. Solutions are expanded in to series of Hermite eigenfunctions. After confirmation the convergence of series of the solution, we show that, by considering the effect of temperature on the evolution of the vortex for the same initial condition as in [] the symmetry of the vortex destroyed rapidly.

1 Introduction

The present study considers two dimensional Boussinesq equations in all of the plane, to find some exact solution of vortex type. On the best knowledge of authors, these exact Solutions are the first solutions of vortex type for Boussinesq equations. These equations are derived from a low degree approximation to the affiliate between the Navier-Stokes equations and the temperature [3, 21] and perform an main pattern in the perusal of Rayleigh-Bernard convection [4, 5]. The respective equations are as below:

tu+uu=p+νΔu+(gαT)e2tT+uT=kTΔTu=0,
where u is the fluid speed, T stands for temperature, g is gravitational acceleration constant, e2 is monad vector in the x2-direction, α is thermal expansion coefficient, KT is diffusion coefficient of temperature and ν represents the kinematic viscosity.

Thermally driven convections such as Boussinesq equations, are an active area of research, at present, with various applications from geophysics [22], ocean circulation [13] clued dynamics, inner core of the planets to astrophysics [4, 5]. These equations are one of the most commonly used fluid models in the atmospheric sciences to model Jet streams as a narrow fast flowing air currents, cold front (as a transition zone replacing cold and warm air) [15], thermohaline circulation and the El Nino Southern Oscillationas [13].

For the purpose of displaying the way in wich the presence of temperature and density influence the invisible point vortex dynamics, we concentrate on some numeric that investigate the viscous evolution of N point vortices in the Boussinesq equations.

The vorticity, in mathematics, are studied as the curl of the flow velocity. For this purpose, suppose that the field of vorticity ω = ∇ × u is enough localized, then the Boussinesq equations for vorticity on the whole plane are include:

tω+uω=νΔω+gαx1TtT+uT=kTΔTω=0,ω=×u.

We are able to restore the speed of the fluid through Biot-Savart legislation:

u(x)=12π2(xy)|xy|2ω(y)dy,
which z = (z1, z2), z = (z2, z1). For the sake of simplification, we focus on (2), but the overall results are applicable to the Thermohaline equations too.

In dimension 2, the vorticity equation is reducing to a scaler. Employing the traditional method, Ting and Tung in 1965 studied the movement of a vortex in a two dimensional incompressible flow while including the viscous influence in the internal kernel of the vortex [14]. In 1994, F. Lingevitch and A. J. Bernoff obtained the motion of vortex as integral of the background irrational current [2]. In 2002, Gallay and Wayne showed that the solutions of vorticity equation tend to Oseen vortex rapidly [7]. Afterwards, Nagem and coauthors employed the method and results of [7] to find an approximate solution for vorticity equation [18]. In the next step, they generalized the theory of single point vortex for viscose flow in two dimensions. Finally, their theory captures multi vortex problem for viscous two-dimensional flows [19]. Jing, Kanso and Newton, in 2010, described the viscous progress of a collinear three-vortex structure that at first corresponds to an inviscid point vortex fixed balance [11]. In 2011, Gallay proved that the replay of the Navier-Stockes equations converges, as ν → 0, to a superposition of Lamb-Oseen vortices which the centers evolve at a viscous regularization of the point vortex system [6]. After one year, Uminsky and Wayne introduced simplified and precise formulas that resulted in the effective performance and expansion of a new multi-moment vortex method (MMVM) using Hermite extension to resemble 2D vorticity [25]. In continue, by the use of MMVM Smith and Nagem studied vortex pairs and dipoles [23].

The content of the paper is as follows, utilizing the method presented in [19] and [25], we offer an expansion of solutions for the Boussinesq equations in the vorticity form. In section 2, the foundation of the theory of single center vortex method is reviewed. In section 3, the theory is extended for Boussinesq equations and it is shown that the series of the solution is converged. The numerical simulation of the solution of the Boussinesq equation is presented in section 4 with the same initial condition arose in [25] Then, we compare our results with [25].

2 Mathematical foundations of SCVM

In this section, we summarize the expansion of vorticity and temperature including the Hermite functions as described in [19]. Let

φ00(x,t;λ)=1πλ2e|x|2/λ2,T00(x,t;σ)=1πσ2e|x|2/σ2
where λ2=λ02+4νt and σ2=σ02+4kTt. The Hermite functions of degree (k1, k2) is defined as follows:
φk1,k2(x,t;λ)=Dx1k1Dx2k2φ00(x,t;λ),ψk1,k2(x,t;σ)=Dx1k1Dx2k2T00(x,t;σ).

The moment expansion of functions is defined as follows:

ω(x,t)=k1,k2=1M[k1,k2;t]φk1,k2(x,t;λ),T(x,t)=k1,k2=1I[k1,k2;t]ψk1,k2(x,t;σ).

Let (ω, T)(x, t) be the resolvent of the equation (2), then Biot-Savrat law implies that the speed field is as below:

V(x,t)=k1,k2=1M[k1,k2;t]Vk1,k2(x,t;λ),
where Vk1,k2(x,t;λ)=Dx1k1Dx2k2V00(x,t;λ) and V00(x, t; λ) is the induced speed from ϕ00(x, t; λ) which is determined as follows:
V00(x,t;λ)=12π(x2,x1)|x|2(1e|x|2/λ2).

Hermite polynomials are defined by their generator functions:

Hn1,n2(z,λ)=(Dt1n1Dt2n2e(2tzt2λ2))|t=0,Fn1,n2(z,σ)=(Dt1n1Dt2n2e(2tzt2σ2))|t=0.

Notice that the standard Hermite multinomial occur when λ = 1 and k = 1. In this case, they constitute the orthogonal sets:

2Hn1,n2(z,λ=1)Hm1,m2(z,λ=1)ez2dz=π2n1+n2(n1!)(n2!)δn1,m1δn2,m2,
2Fn1,n2(z,σ=1)Fm1,m2(z,σ=1)ez2dz=π2n1+n2(n1!)(n2!)δn1,m1δn2,m2.

Consequently, the following projection operators determine the coefficients in the expansion (4):

M[k1,k2;t]=(Pk1,k2ω)(t)=ρ(k1,k2,λ)2Hk1,k2(z,λ)ω(z,t)dz,
I[k1,k2;t]=(Qk1,k2T)(t)=ρ(k1,k2,σ)2Fk1,k2(z,σ)T(z,t)dz,
where
ρ(k1,k2,τ)=(1)(k1+k2)τ2(k1+k2)2k1+k2(k1!)(k2!).

Let

Lλφ=14λ2Δφ+12(xφ),
and
Φλ(x,t)=φ00(x,t;λ),Ψσ(x,t)=T00(x,t;σ).

In the [19] Nagem and coauthors proved the convergence of the expansions (4), when:

2Φλ1(x)(ω(x,t))2dx<,2Ψσ1(x)(T(x,t))2dx<.

3 Main Result

In this section, we prove the criteria (15) and obtain the ODE for M[k1, k2, t] and I[k1, k2, t]. In order the proof of theorem 2 we say the following fundamental lemma: Lemma 1

Suppose that (ω, T) satisfies the equations (2), ω(x, 0) = ω0(x) and T (x, 0) = T0(x) then the following assertions are true:

  1. i)For all 1 ≤ p ≤ ∞ and t ≥ 0, ‖T (x, t)‖p ≤ ‖T0(x)‖p
  2. ii)There exist constant c = c(ω0, T0, t) such that for all 2 ≤ q < ∞ and t ≥ 0, ‖ω(x, t)‖qc(ω0, T0)
  3. iii)For all t ≥ 0, ‖∇uc(ω0, T0, t)

Proof

For (i) see [1] and for (ii) see [10] and for (iii) see [26].

Now we are ready to prove criteria (15).

Theorem 2

Define

ε(t)=2Φλ1(ω(x,t))2dx,γ(t)=2Ψσ1(T(x,t))2dx.

If kT < 2ν and the primary vorticity and temperature, i.e. ω0and T0, guarantee that ε(0) < ∞ and γ(0) < ∞ for some λ0and σ0respectively and ω0and T0are in the L3, then ε(t) and γ(t) will be finite for all times of t > 0.

Proof

According to lemma 2.1 in [7] we have: ||u||c||ω||pα||ω||q1α where 1 ≤ p < 2 < q ≤ ∞ and αp+1αq=12, as a result according to lemma (1) we obtain: ‖uc(ω0, T0). Therefore by assumption it is concluded that ‖uc(ω0, T0). Now similar to the proof of theorem 3.4 in [19] it could be proved that:

dγ(t)dt(4c(ω0,T0)KT+4KTσ2)γ(t),
and this means that γ(t) is limited for each t > 0 if γ(0) is finite. Now to prove that ε(t) < ∞, differentiate ε(t), we have:
dε(t)dt=4νλ2ε(t)4νλ42|x|2Φλ1(ω(x,t))2dx+22|x|2Φλ1ω(x,t)tω(x,t)dx=4νλ2ε(t)4νλ42|x|2Φλ1(ω(x,t))2dx+22Φλ1ω(νΔωuω+gαx1T)dx.

Integrating by parts in the last term in (17) implies that:

22Φλ1ω(νΔω)dx=2ν2Φλ1(x)(|ω|2+2λ2ωxω)dx,
and the second item in the right side of (18) satisfies the following relation:
2ν2Φλ1(x)(2λ2ωxω)dxν2Φλ1(x)|ω|2dx+4νλ42Φλ1(x)(x2ω2)dx.

Now using ‖uc(ω0, T0) and Cauchy’s inequality we have :

22Φλ1ω(uω)dx2c(ω0,T0)2Φλ1|ω(x,t)||ω|dxc2(ω0,T0)ν2Φλ1(ω(x,t))2dx+ν2Φλ1(x)|ω|2dx
also
2gα2Φλ1ω(x1T)dx2gα2Φλ1(ω2+(x1T)2)dx=2gα2Φλ1ω2(x,t)dx+2gα2Φλ1(x1T)2dx2gαε(t)+2gα||T||λ2.

Now we bound the term ||T||λ2. Let f (x, t) = ∇T (x, t) and define:

δ(t)=2Φλ1(T(x,t))2dx.

Differentiate δ (t) obtain the following equation:

dδ(t)dt=4νλ2δ(t)4νλ42|x|2Φλ1f2(x,t)dx+22Φλ1f(x,t)tf(x,t)dx=4νλ2δ(t)4νλ42|x|2Φλ1(f(x,t))2dx+22Φλ1f(KTΔTuT)dx.

Now by considering that the last term in (22) we have:

22Φλ1f(KTΔT)dx=2KT2Φλ1f(Δf)dx=2KT2Φλ1(x)(|f|2+2λ2fxf)dx.

The second term in the last part of the equation (23) satisfy the following inequality:

2KT2Φλ1(x)(2λ2fxf)dx4νλ42Φλ1(x2f2)dx+KT2ν2Φλ1|f|2dx.

On the other hand inequality ‖uc(ω0, T0, t) and ‖∇uc(ω0, T0, t) in [10] implies that:

22Φλ1f(uT)dx=22(Φλ1f)uTdx22(Φλ1f)u(T)dx2c(ω0,T0,t)2Φλ1f2dx+2c(ω0,T0,t)2Φλ1|f||f|dx.2c(ω0,T0,t)δ(t)+2c(ω0,T0,t)2Φλ1|f||f|dx.

We now assume that KT < 2ν, then:

2c(ω0,T0,t)2Φλ1|f||f|dxνc2(ω0,T0,t)2νKTKT22Φλ1(f2(x,t))dx+2KTνKT2ν2Φλ1|f|2dx.

As a consequence of (23)–(26) we obtain:

dδ(t)dt(2c(ω0,T0,t)+νc2(ω0,T0,t)2νKTKT2+4νλ2)δ(t),
and this means that if δ (0) is limited then δ (t) will be limited for all t > 0. So according to (17)–(21) we can write:
dε(t)dt(4νλ2+4c(ω0,T0)ν+2gα)ε(t)+2gαc1(ω0,t0,t),
where ||T||λ2c1(ω0,t0,t). Using the Gronwall lemma if ε(0) is limited then ε(t) remains limited for all t > 0.

In the following we look for differential equations generating the coefficient M[k1, k2; t] and I[k1, k2; t]. Assuming that the (ω, T)(x, t) is a solution of (2) and Define

k1,k2mf(k1,k2):=i=0mk1+k2=ik10,k20nf(k1,k2),
and also
ωm(x,t)=k1,k2mM[k1,k2;t]φk1,k2(x,t;λ)
um(x,t)=k1,k2mM[k1,k2;t]Vk1,k2(x,t;λ)
Tm(x,t)=k1,k2mI[k1,k2;t]ψk1,k2(x,t;σ)
where ωm, um, and Tm are Hermit approximations of order m (Glerkin approximation by Hermit functions) then by the use of Glerkin standard approximation for equation (2) we have:
tωm=k1,k2mdM[k1,k2;t]dtφk1,k2(x,t;λ)+k1,k2mM[k1,k2;t]tφk1,k2=k1,k2mM[k1,k2;t](νΔφk1,k2(x,t;λ))Pm[(l1,l2mM[l1,l2;t]Vl1,l2(x,t;λ))(k1,k2mM[k1,k2;t]φk1,k2(x,t;λ))]+gαx1(k1,k2mI[k1,k2;t]ψk1,k2(x,t;σ))
tTm=k1,k2mdI[k1,k2;t]dtψk1,k2(x,t;σ)+k1,k2mI[k1,k2;t]tψk1,k2=k1,k2mI[k1,k2;t](KTΔψk1,k2(x,t;σ))Pm[(l1,l2mM[l1,l2;t]Vl1,l2(x,t;λ))(k1,k2mI[k1,k2;t]ψk1,k2(x,t;σ))].
where Pm[·] is a projector on the subspace produced by Hermit functions of degree m or less. Noting that:
tφk1,k2=νΔφk1,k2,tψk1,k2=KTΔψk1,k2,
and applying the projection operators Pk1,k2 and Qk1,k2, defined in (10) on the equation (33) and (34) we have:
dM[k1,k2;t]dt=Pk1,k2[(l1,l2mM[l1,l2;t]Vl1,l2(x,t;λ))(m1,m2mM[m1,m2;t]φm1,m2(x,t;λ))]Pk1,k2[gαx1(m1,m2mI[m1,m2;t]ψm1,m2(x,t;σ))]
dI[k1,k2;t]dt=Qk1,k2[(l1,l2mM[l1,l2;t]Vl1,l2(x,t;λ))(m1,m2mI[m1,m2;t]ψm1,m2(x,t;σ))].

Note that k1 + k2m then

φm1,m2(x,t;λ)=(Da1m1Da2m2φ00(x+a,λ))|a=0
Vl1,l2(x,t;λ)=(Db1l1Db2l2V00(x+b,λ))|b=0
ψm1,m2(x,t;σ)=(Dc1m1Dc2m2ψ00(x+c,σ))|c=0,
then the system of ordinary differential equations (35) and (36) become as follows:
dM[k1,k2;t]dt=ρ(k1,k2,λ)l1,l2=1mm1,m2=1mM[l1,l2;t]M[m1,m2;t]×2Hk1,k2(x)(Dx1m1Dx2m2V00(x,λ))x(Dx1l1Dx2l2φ00(x,λ))dxρ(k1,k2,λ)m1,m2=1mI[m1,m2;t]×2Hk1,k2(x)(Dx1m1Dx2m2T00(x,σ))dx
dI[k1,k2;t]dt=ρ(k1,k2,σ)l1,l2=1mm1,m2=1mM[l1,l2;t]I[m1,m2;t]×2Fk1,k2(x)(Dx1m1Dx2m2V00(x,λ))x(Dx1l1Dx2l2T00(x,σ))dx.
where ρ(k1, k2, τ) is defined in (12). The first integral in (40) is calculated in [25] and the two remaining integrals are calculated in appendix. Finally using (A.26)–(A.27) in appendix and (40)–(41) we have corrected the differential equations for M[k1, k2, t] and I[k1, k2, t] to:
dM[k1,k2;t]dt=ρ(k1,k2,λ)l1,l2=1mm1,m2=1mM[l1,l2;t]M[m1,m2;t]×Γ˜[k1,k2,l1,l2,m1,m2;λ]+m1,m2mI[m1,m2;t]B[k1,k2,m1,m2;λ,σ]
dI[k1,k2;t]dt=ρ(k1,k2,σ)l1,l2=1mm1,m2=1mM[l1,l2;t]I[m1,m2;t]×θ˜[k1,k2,l1,l2,m1,m2;λ,σ],
where B and θ˜ is introduced in appendix, Γ˜ is introduced in [19]and
θ˜[k1,k2,l1,l2,m1,m2;λ,σ]=θ1[k1,k2,l1,l2,m1,m2;λ,σ]+θ2[k1,k2,l1,l2,m1,m2;λ,σ].

4 Numerical Simulation

In this section, some numerical examples of the equation (2) are presented. Moreover, the effect of α (thermal expansion coefficient) and KT (diffusion coefficient of temperature) on these solutions are investigated.

First, we present an example with zero temporal expansion, i.e. α = 0. Wayne and Uminsky, in [25] have shown that if we start with an initial vorticity of the following equation, where δ = 0.1 and core size λ0 = 2,

ω(x,0)=φ00(x,0)+4δ(φ20+φ02),
then it will become quickly axisymmetric in the absence of temperature (see Figure. 2 in [25]). In this section, the initial vorticity would be considered as (44) which leads to elliptical deformations of the Lamb-Oseen vortex as shown in Figure. 1, and the initial temperature with k0 = 1 is as follows:
Fig. 2
Fig. 2

the graph of the nonaxisymmetric portion (E) with m = 4 and α = 0.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00014

Fig. 1
Fig. 1

a) Initial perturbation with δ = 0.1 and b) Vorticity distribution at t = 400 with m = 4 and values of α = 0, ν = 1/500, and KT = 1/500.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00014

T(x,0)=ψ00(x,0)+4δ(ψ20+ψ02).

Now we present some examples with different values of α.

4.1 Zero thermal expansion coefficient α = 0

In the differential equations (42) and (43) put α = 0, m = 4, ν = 1/500, and KT = 1/500. As you can see in Figure 1.b, at time t = 400, the axisymmetric is increased. In this case, this result is similar to the result obtained by Nagem and coauthors in [25]. The enstrophy E of the vortex which is a criterion for axisymmetry of the vortex is defined as follows:

E=2(ω(x)<ω(|x|)>)2dx,<ω(|x|)>=12π02πω(x)dθ.

The values of E shows the nonaxisymmetric portion in L2 norm. As shown in Figure 2 the values of E are decreased in time and the solution goes rapid axisymmetrization. In continue, we present two examples for high and low values of α and the effect of α on the vorticity is investigated.

4.2 Nonzero thermal expansion coefficient (small values of α)

In this subsection, we assume that α = 69×10−6 (k−1) (thermal expansion coefficient of water in 20 degrees centigrade) and other parameters are considered as follows:

m=4,ν=1500(m2/s),KT=1500,1800,11500(m2/s).

As it is displayed in Figure 3, at time t = 8, the portion begins to increase. For the large KT nonaxisymmetric is increased rapidly. These results reveal two important feature of the equation (1). First, unlike the case of zero thermal expansion coefficient (α = 0) the solution tends to be nonaxisymmetric in time and the monopole state of the vorticity breaks down. Second, as KT decrease, the symmetry of the solution breaks faster in time. This is due to the fact that the effect of temperature on the vorticity decreases when KT increased.(see Figure 4).

Fig. 3
Fig. 3

The graph of the nonaxisymmetric portion (E) with m = 4 and α = 69 × 10−6 .

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00014

Fig. 4
Fig. 4

Vorticity distribution for small values of α where a) at time t = 15, KT = 1/500, ν = 1/500, and m = 4, b) at time t = 15, KT = 1/1500, ν = 1/500, and m = 4.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00014

4.3 Nonzero thermal expansion coefficient (great values of α)

Now let α = 69 × 10−4 (k−1) (suitable thermal expansion coefficient for gases), and other parameters are given as below

m=4,ν=1500(m2/s),KT=1500,1800,11500(m2/s).

Then, as can be seen in Figure 5, the results are as same as the results of the previous subsection with this difference that the nonsymmetrization process occurs faster in time.( You may see Figure 6)

Fig. 5
Fig. 5

The graph of the nonaxisymmetric portion (E) with m = 4 and α = 69 × 10−4 .

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00014

Fig. 6
Fig. 6

Vorticity distribution for great values of α where a) at time t = 6, KT = 1/500, ν = 1/500, and m = 4, b) at time t = 6, KT = 1/1500, ν = 1/500, and m = 4, c) at time t = 7, KT = 1/500, ν = 1/500, and m = 4, d) at time t = 7, KT = 1/1500, ν = 1/500, and m = 4.

Citation: Applied Mathematics and Nonlinear Sciences 5, 2; 10.2478/amns.2020.2.00014

APPENDIX

The two remaining integrals in (40) and (41) are

1)2Hk1,k2(x)(Dx1m1Dx2m2T00(x,σ))dx
2)2Fk1,k2(x)(Dx1m1Dx2m2V00(x,λ))x(Dx1l1Dx2l2T00(x,σ))dx.

Using the fact that:

x1Hn,m=(2nλ2)Hn1,m(x),x2Hn,m=(2mλ2)Hn,m1(x)
we have:
2Hk1,k2(x)(Dx1m1Dx2m2T00(x,σ))dx=(1)k1+k22φ001(x)Dx1k1Dx2k2φ00Dx1m1Dx2m2T00dx=(1)k1+k2Db1k1Db2k2Da1m1Da2m22φ001(x)φ00(x+b;t)T00(x+a;t)dx
where the first equality comes from the below equality:
Hn,m=(1)n+mφ001Dx1nDx2mφ00,
and the secondary equality is an outcome of applying integration by parts. But to calculate the last integral of (A4) note that:
2φ001(x)φ00(x+b;t)T00(x+a;t)dx=2πλ2ex12+x22λ21πλ2ex122b1x1b12x222b2x2b22λ21πσ2ex122a1x1a12x222a2x2a22σ2dx|a=0,b=0=1πσ2eb12b22λ2ea12a22σ2×2e2b1x12b2x2λ2ex122a1x1x222a2x2σ2dx=β1β22e(x1+σ2b1+λ2a1)2λ2σ2dx12e(x2+σ2b2+λ2a2)2λ2σ2dx2=β1β2πσ4,
where
β1=1πσ2eb12b22λ2ea12a22σ2,β2=e(σ2b1+λ2a1)2λ4σ2e(σ2b2+λ2a2)2λ4σ2
and this implies that:
2Hk1,k2(x)(Dx1m1Dx2m2T00(x,σ))dx=(1)k1+k2Db1k1Db2k2Da1m1Da2m2[β1β2.πσ4].

To calculate the integral in (A2) by use of the integration by parts we have:

2Fk1,k2(x)(Dx1m1Dx2m2V00(x,λ))x(Dx1l1Dx2l2T00(x,σ))dx=2xFk1,k2(x)(Dx1m1Dx2m2V00(x,λ))(Dx1l1Dx2l2T00(x,σ))dx=(θ1[k1,k2,l1,l2,m1,m2;λ,σ]+θ2[k1,k2,l1,l2,m1,m2;λ,σ]),
where
θ1=2Dx1m1Dx2m2V001Dx1Fk1,k2(x)Dx1l1Dx2l2T00(x,σ)dx
θ2=2Dx1m1Dx2m2V002Dx2Fk1,k2(x)Dx1l1Dx2l2T00(x,σ)dx.

Using repeated integration by parts from (A7)l1 times toward x1 and l2 times toward x2 conclude that:

θ1=i=0l1j=0l2(il1)(jl2)(1)l1+l22T00Dx1i+1Dx2jFk1,k2(x)Dx1l1+m1iDx2m2+l2jV001dx=i=0min(l1,k11)j=0min(l2,k2)(il1)(jl2)(1)l1+l2(2i+1k1!σ2(i+1)(k1i1)!)(2jk2!σ2(j)(k2j)!)×2T00Fk1i1,k2j(x)Dx1l1+m1iDx2m2+l2jV001dx,
where in the secondary equality we used the following equation:
x1Fn,m=(2nσ2)Fn1,m(x),x2Fn,m=(2mσ2)Fn,m1(x).

In the last integral of (A9) using the relation Fn,m=(1)n+mT001Dx1nDx2mT00 arrive at the following formula:

2T00Fk1i1,k2j(x)Dx1l1+m1iDx2m2+l2jV001dx=(1)k1i1+k2j2Dx1k1i1Dx2k2jT00Dx1l1+m1iDx2m2+l2jV001dx=(1)k1i1+k2j(1)k1i1+k2j2T00Dx1m1+k1i1+l1iDx2m2+k2j+l2jV001dx=2T00Dx1m1+k1i1+l1iDx2m2+k2j+l2jV001dx,
where the secondary equality is captured from an integration by parts. On the other hand:
2T00Dx1α1Dx2α2V001dx=2T00Dd1α1Dd2α2V001(x+d)dx|d=0,
and by the use of Biot-Savart law and by re-write the speed V00 in item of the vorticity ϕ00:
V00(x+d;λ)=d*(Δd)1φ00(x+d),
where d*f=(x2f,x1f). Thus we can obtain:
2T00Dx1α1Dx2α2V00dx=Dd1α1Dd2α22T00V00(x+d)dx|d=0=Dd1α1Dd2α2d*(Δd)12T00φ00(x+d)dx|d=0,
and finally for the last integral in (A13) we have:
2T00φ00(x+d)dx|d=0=21πσ2ex12x22σ21πλ2ex122d1x1d12x222d2x2d22λ2dx|d=0=(1πσλ)2ed12d22λ2e(λ2+σ2)x122σ2d1x1λ2σ2dx|d=0×e(λ2+σ2)x222σ2d2x2λ2σ2dx2|d=0=(1πσλ)2ed12d22λ2eξ1eξ2×e(λ2+σ2)(x1+ξ1)2λ2σ2dx1|d=0×e(λ2+σ2)(x2+ξ2)2λ2σ2dx1|d=0=2λ2σ2π(λ2+σ2)2(ed12d22λ2eξ1eξ2)|d=0,
where
ξ1=σ4(λ2+σ2)2d12,ξ2=σ4(λ2+σ2)2d22.

Replacing ξ1 and ξ2 in (A14), implies that:

2T00φ00(x+d)dx|d=0=2λ2σ2π(λ2+σ2)e(1λ2σ4(λ2+σ2)2)(d12+d22)|d=0.

As a result

2T00Dx1α1Dx2α2V00dx=2λ2σ2π(λ2+σ2)Dd1α1Dd2α2d*(Δd)1(e(d12+d22)ε)|d=0,
where
ε=11λ2σ4(λ2+σ2)2=λ2(λ2+σ2)2(λ2+σ2)2λ2σ4.

And so

2T00Dx1α1Dx2α2V00dx=ε2λ2σ2(λ2+σ2)Dd1α1Dd2α2V00(d,ε)|d=0=ζDd1α1Dd2α2V00(d,ε)|d=0,
where
ζ=ε2λ2σ2(λ2+σ2)=2λ4σ2(λ2+σ2)2λ2σ4.

According to the above computing for (A7) and similar computation for (A8) the following equalities are obtained:

1)2T00Dx1α1Dx2α2V001dx=ζDd1α1Dd2α2V001(d,ε)|d=02)2T00Dx1α1Dx2α2V002dx=ζDd1α1Dd2α2V002(d,ε)|d=0.

To simplify equation (A5), note that:

β1β2=1πσ2e|b|2λ2e|a|2σ2eσ2|b|2λ4e|a|2σ2e2λ2σ2a1b1σ2λ4.e2λ2σ2a2b2σ2λ4=1πσ2e1λ2[(1+σ2λ2)|b|2+2(a1b1+a2b2)λ2],
so
2Hk1,k2(x)(Dx1m1Dx2m2T00(x,σ))dx=(1)k1+k2(1πσ)2πσ4Db1k1Db2k2Da1m1Da2m2[e1λ2[(1+σ2λ2)|b|2+2(a1b1+a2b2)λ2]]|a=0,b=0=(1)k1+k2σ2πDb1k1Db2k2Da1m1Da2m2[e1λ2[(1+σ2λ2)|b|2+2(a1b1+a2b2)λ2]]|a=0,b=0,
but
e1λ2[(1+σ2λ2)|b|2+2(a1b1+a2b2)λ2]=n=01λ2n1n!r=0n(nr)2r(1+σ2λ2)nr(a1b1+a2b2)r|b|2(nr)n=01λ2n1n!r=0n(nr)2r(1+σ2λ2)nr(h1=0r(rh1)(a1b1)h1(a2b2)rh1)×(h2=0nr(nrh2)(b1)2h2(b2)2(nrh2))=n=01λ2n1n!r=0n(nr)2r(1+σ2λ2)nrh1=0rh2=0nr(rh1)(nrh2)(a1)h1(a2)rh1(b1)h1+2h2(b2)2(nrh2)+rh1,
so
Db1k1Db2k2Da1m1Da2m2(e1λ2[(1+σ2λ2)|b|2+2(a1b1+a2b2)λ2])|a=0,b=0=[n=01λ2n.1n!r=0n(nr)2r.(1+σ2λ2)nrh1=0rh2=0nr(rh1)(nrh2)×h1!(h1m1)!(a1)h1m1.(rh1)!(rh1m2)!.(a1)(rh1m2).(h1+2h2)!(h1+2h2k1)!.(b1)h1+2h2k1×.[(2(nrh2)+rh1)![(2(nrh2)+rh1k2)!.(b2)2(nrh2)+rh1k2]|a=0,b=0.

Assume h1 = m1, r = m1 + m2, h2=k1m12, n=m1+k1+m1+k22, and define:

B[k1,k2,m1,m2;λ,σ]={σ2gαπ.2m1+m2k1k2λ(m1+m2k1k2+2)ifk1m1×(1+σ2λ2)k1+k2m1m22andk2m2×1(k1m12)!(k2m22)!ispositiveandeven0otherwise
the last term in (40) takes the form:
m1,m2mI[m1,m2;t]B[k1,k2,m1,m2;λ,σ].

According to equations (A.18) to (A.21) of appendix (A) in the [25], we have:

θ1[k1,k2,l1,l2,m1,m2;λ,σ]=i=0min(l1,k11)j=0min(l2,k2)σ2πλ2ε(il1)(jl2)(1)l1+l2(2i+1k1!σ2(i+1)(k1i1)!)×(2jk2!σ2(j)(k2j)!)2φ00Dx1m1+k1i1+l1iDx2m2+k2j+l2jV001dx=i=0min(l1,k11)j=0min(l2,k2)σ2πλ2ε(il1)(jl2)(1)l1+l2(2i+1k1!σ2(i+1)(k1i1)!)×(2jk2!σ2(j)(k2j)!)R1(m1+k1i1+l1i,m2+k2j+l2j)
θ2[k1,k2,l1,l2,m1,m2;λ,σ]=i=0min(l1,k1)j=0min(l2,k21)σ2πλ2ε(il1)(jl2)(1)l1+l2(2ik1!σ2i(k1i)!)×(2j+1k2!σ2(j+1)(k2j1)!)2φ00Dx1m1+k1i1+l1iDx2m2+k2j+l2jV001dx=i=0min(l1,k11)j=0min(l2,k2)σ2πλ2ε(il1)(jl2)(1)l1+l2(2ik1!σ2i(k1i)!)×(2j+1k2!σ2(j+1)(k2j1)!)R2(m1+k1i+l1i,m2+k2j1+l2j).

And R1 and R2 the similar computation in appendix (A) in the [25] give rise to the following:

R1(α1,α2)={η(α1,α2,ε)(α1+α212α12)ifα1evenandα2isodd0otherwiseR2(α1,α2)={η(α1,α2,ε)(α1+α212α112)ifα1oddandα2iseven0otherwise
where η(α1,α2,δ)=12π(12δ)α1+α2+12(1)(α1+α21)/2[((α1+α2+1)/2)!(α1)!(α2)!.

References

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    Ben-Artzi. Matania, Global solutions of two-dimensional Navier-Stokes and Euler equations, Archive for rational mechanics and analysis, 128 (1994), 329–358.

    • Crossref
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    Bernoff, J. Andrew and Joseph. F. Lingevitch, Rapid relaxation of an axisymmetric vortex, Physics of Fluids, 6 (1994), 3717–3723.

    • Crossref
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    J. Boussinesq, Theorie Analitique de la Chaleur, vol. 2, Gauthier Villars, Paris, 1903.

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    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, Inc. 1981.

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    Drazin, Philip G., and William Hill Reid. Hydrodynamic stability. Cambridge university press, 2004.

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    Thierry. Gallay, Interaction of vortices in weakly viscous planar flows, Archive for rational mechanics and analysis, 200 (2011), 445–490.

    • Crossref
    • Export Citation
  • [7]

    Thierry. Gallay and C. Eugene. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ2, Arch. Ration. Mech. Anal., 163 (2002), 209–258.

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  • [8]

    Gill, Adrian E. Atmosphere ocean dynamics. Elsevier, 2016.

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    H. von Helmholtz, Uber Integrale der hydrodynamischen Gleichungen, Journal fur die reine und angewandte Mathematik, 55 (1858), 25–55.

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    T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,Discrete and Cont. Dyn. Syst, 12 (2005), 1–12.

    • Crossref
    • Export Citation
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    F. Jing, Eva. Kanso and Paul K. Newton, Viscous evolution of point vortex equilibria: The collinear state,Physics of Fluids, 22 (2010), 123102.

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    G. R. Kirchhoff. Vorlesungen, uber Mathematische Physik, Mechanik. Teubner, Leipzig, 1876.

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    Tian ma, Shohong wang: Phase Transition Dynamics, 2014

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    Ting. Lu, and Chee. Tung, Motion and decay of a vortex in a nonuniform stream, The Physics of Fluids, 8 (1965), 1039–1051.

    • Crossref
    • Export Citation
  • [15]

    Majda, Andrew. Introduction to PDEs and Waves for the Atmosphere and Ocean. Vol. 9. American Mathematical Soc., 2003.

  • [16]

    Marchioro, Carlo, and Mario Pulvirenti. Mathematical theory of incompressible nonviscous fluids. Vol. 96. Springer Science and Business Media, 2012.

  • [17]

    Patrice Meunier, Stephane Le Dizes and Thomas Leweke, Physics of vortex merging, Comptes Rendus Physique, 6 (2005), 431–450.

    • Crossref
    • Export Citation
  • [18]

    Raymond J. Nagem, Guido. Sandri and David. Uminsky, Vorticity dynamics and sound generation in two-dimensional fluid flow, The Journal of the Acoustical Society of America, 122 (2007), 128–134.

    • Crossref
    • PubMed
    • Export Citation
  • [19]

    Raymond. Nagem, Guido. Sandri, David. Uminsky and C. Eugene Wayne, Generalized Helmholtz-Kirchhoff model for two-dimensional distributed vortex motion, SIAM Journal on Applied Dynamical Systems, 8 (2009), 160–179.

    • Crossref
    • Export Citation
  • [20]

    Paul K. Newton, The N-vortex problem, volume 145 of Applied Mathematical Sciences. Springer-Verlag, New York, 2001. Analytical techniques.

  • [21]

    A. Oberbeck, Uber die Warmeleitung der Flussigkeiten bei der Berucksichtigung der Stromungen infolge von Temper-aturdifferenzen, Annalen der Physik und Chemie 7, (1879).

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    Pedlosky, J. Geophysical fluid dyanmics I, II. (1981).

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    Smith, Stefan G. Llewellyn and Raymond J. Nagem, Vortex pairs and dipoles, Regular and Chaotic Dynamics, 18 (2013), 194–201.

    • Crossref
    • Export Citation
  • [24]

    Saffman, Philip G. Vortex dynamics. Cambridge university press, 1992.

  • [25]

    David. Uminsky, C. Eugene Wayne and Alethea. Barbaro, A multi-moment vortex method for 2D viscous fluids, Journal of Computational Physics, 231 (2012), 1705–1727.

    • Crossref
    • Export Citation
  • [26]

    Jiahong Wu, The 2D Incompressible Boussinesq Equations, Lecture Notes, Oklahoma State University, 2012.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Ben-Artzi. Matania, Global solutions of two-dimensional Navier-Stokes and Euler equations, Archive for rational mechanics and analysis, 128 (1994), 329–358.

    • Crossref
    • Export Citation
  • [2]

    Bernoff, J. Andrew and Joseph. F. Lingevitch, Rapid relaxation of an axisymmetric vortex, Physics of Fluids, 6 (1994), 3717–3723.

    • Crossref
    • Export Citation
  • [3]

    J. Boussinesq, Theorie Analitique de la Chaleur, vol. 2, Gauthier Villars, Paris, 1903.

  • [4]

    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, Dover Publications, Inc. 1981.

  • [5]

    Drazin, Philip G., and William Hill Reid. Hydrodynamic stability. Cambridge university press, 2004.

  • [6]

    Thierry. Gallay, Interaction of vortices in weakly viscous planar flows, Archive for rational mechanics and analysis, 200 (2011), 445–490.

    • Crossref
    • Export Citation
  • [7]

    Thierry. Gallay and C. Eugene. Wayne, Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on ℝ2, Arch. Ration. Mech. Anal., 163 (2002), 209–258.

    • Crossref
    • Export Citation
  • [8]

    Gill, Adrian E. Atmosphere ocean dynamics. Elsevier, 2016.

  • [9]

    H. von Helmholtz, Uber Integrale der hydrodynamischen Gleichungen, Journal fur die reine und angewandte Mathematik, 55 (1858), 25–55.

  • [10]

    T. Hou and C. Li, Global well-posedness of the viscous Boussinesq equations,Discrete and Cont. Dyn. Syst, 12 (2005), 1–12.

    • Crossref
    • Export Citation
  • [11]

    F. Jing, Eva. Kanso and Paul K. Newton, Viscous evolution of point vortex equilibria: The collinear state,Physics of Fluids, 22 (2010), 123102.

    • Crossref
    • Export Citation
  • [12]

    G. R. Kirchhoff. Vorlesungen, uber Mathematische Physik, Mechanik. Teubner, Leipzig, 1876.

  • [13]

    Tian ma, Shohong wang: Phase Transition Dynamics, 2014

  • [14]

    Ting. Lu, and Chee. Tung, Motion and decay of a vortex in a nonuniform stream, The Physics of Fluids, 8 (1965), 1039–1051.

    • Crossref
    • Export Citation
  • [15]

    Majda, Andrew. Introduction to PDEs and Waves for the Atmosphere and Ocean. Vol. 9. American Mathematical Soc., 2003.

  • [16]

    Marchioro, Carlo, and Mario Pulvirenti. Mathematical theory of incompressible nonviscous fluids. Vol. 96. Springer Science and Business Media, 2012.

  • [17]

    Patrice Meunier, Stephane Le Dizes and Thomas Leweke, Physics of vortex merging, Comptes Rendus Physique, 6 (2005), 431–450.

    • Crossref
    • Export Citation
  • [18]

    Raymond J. Nagem, Guido. Sandri and David. Uminsky, Vorticity dynamics and sound generation in two-dimensional fluid flow, The Journal of the Acoustical Society of America, 122 (2007), 128–134.

    • Crossref
    • PubMed
    • Export Citation
  • [19]

    Raymond. Nagem, Guido. Sandri, David. Uminsky and C. Eugene Wayne, Generalized Helmholtz-Kirchhoff model for two-dimensional distributed vortex motion, SIAM Journal on Applied Dynamical Systems, 8 (2009), 160–179.

    • Crossref
    • Export Citation
  • [20]

    Paul K. Newton, The N-vortex problem, volume 145 of Applied Mathematical Sciences. Springer-Verlag, New York, 2001. Analytical techniques.

  • [21]

    A. Oberbeck, Uber die Warmeleitung der Flussigkeiten bei der Berucksichtigung der Stromungen infolge von Temper-aturdifferenzen, Annalen der Physik und Chemie 7, (1879).

  • [22]

    Pedlosky, J. Geophysical fluid dyanmics I, II. (1981).

  • [23]

    Smith, Stefan G. Llewellyn and Raymond J. Nagem, Vortex pairs and dipoles, Regular and Chaotic Dynamics, 18 (2013), 194–201.

    • Crossref
    • Export Citation
  • [24]

    Saffman, Philip G. Vortex dynamics. Cambridge university press, 1992.

  • [25]

    David. Uminsky, C. Eugene Wayne and Alethea. Barbaro, A multi-moment vortex method for 2D viscous fluids, Journal of Computational Physics, 231 (2012), 1705–1727.

    • Crossref
    • Export Citation
  • [26]

    Jiahong Wu, The 2D Incompressible Boussinesq Equations, Lecture Notes, Oklahoma State University, 2012.

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  • View in gallery

    the graph of the nonaxisymmetric portion (E) with m = 4 and α = 0.

  • View in gallery

    a) Initial perturbation with δ = 0.1 and b) Vorticity distribution at t = 400 with m = 4 and values of α = 0, ν = 1/500, and KT = 1/500.

  • View in gallery

    The graph of the nonaxisymmetric portion (E) with m = 4 and α = 69 × 10−6 .

  • View in gallery

    Vorticity distribution for small values of α where a) at time t = 15, KT = 1/500, ν = 1/500, and m = 4, b) at time t = 15, KT = 1/1500, ν = 1/500, and m = 4.

  • View in gallery

    The graph of the nonaxisymmetric portion (E) with m = 4 and α = 69 × 10−4 .

  • View in gallery

    Vorticity distribution for great values of α where a) at time t = 6, KT = 1/500, ν = 1/500, and m = 4, b) at time t = 6, KT = 1/1500, ν = 1/500, and m = 4, c) at time t = 7, KT = 1/500, ν = 1/500, and m = 4, d) at time t = 7, KT = 1/1500, ν = 1/500, and m = 4.