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:
Thermally driven convections such as Boussinesq equations, are an active area of research, at present, with various applications from geophysics , ocean circulation  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) , thermohaline circulation and the El Nino Southern Oscillationas .
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:
We are able to restore the speed of the fluid through Biot-Savart legislation:
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 . In 1994, F. Lingevitch and A. J. Bernoff obtained the motion of vortex as integral of the background irrational current . In 2002, Gallay and Wayne showed that the solutions of vorticity equation tend to Oseen vortex rapidly . Afterwards, Nagem and coauthors employed the method and results of  to find an approximate solution for vorticity equation . 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 . 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 . 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 . 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 . In continue, by the use of MMVM Smith and Nagem studied vortex pairs and dipoles .
The content of the paper is as follows, utilizing the method presented in  and , 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  Then, we compare our results with .
2 Mathematical foundations of SCVM
In this section, we summarize the expansion of vorticity and temperature including the Hermite functions as described in . Let
The moment expansion of functions is defined as follows:
Let (ω, T)(x, t) be the resolvent of the equation (2), then Biot-Savrat law implies that the speed field is as below:
Hermite polynomials are defined by their generator functions:
Notice that the standard Hermite multinomial occur when λ = 1 and k = 1. In this case, they constitute the orthogonal sets:
Consequently, the following projection operators determine the coefficients in the expansion (4):
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:
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:
- i)For all 1 ≤ p ≤ ∞ and t ≥ 0, ‖T (x, t)‖p ≤ ‖T0(x)‖p
- ii)There exist constant c = c(ω0, T0, t) such that for all 2 ≤ q < ∞ and t ≥ 0, ‖ω(x, t)‖q ≤ c(ω0, T0)
- iii)For all t ≥ 0, ‖∇u‖∞ ≤ c(ω0, T0, t)
Now we are ready to prove criteria (15).
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.
According to lemma 2.1 in  we have:
Integrating by parts in the last term in (17) implies that:
Now using ‖u‖∞ ≤ c(ω0, T0) and Cauchy’s inequality we have :
Now we bound the term
Differentiate δ (t) obtain the following equation:
Now by considering that the last term in (22) we have:
The second term in the last part of the equation (23) satisfy the following inequality:
On the other hand inequality ‖u‖∞ ≤ c(ω0, T0, t) and ‖∇u‖∞ ≤ c(ω0, T0, t) in  implies that:
We now assume that KT < 2ν, then:
As a consequence of (23)–(26) we obtain:
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
Note that k1 + k2 ≤ m then
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  have shown that if we start with an initial vorticity of the following equation, where δ = 0.1 and core size λ0 = 2,
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 . The enstrophy E of the vortex which is a criterion for axisymmetry of the vortex is defined as follows:
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:
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).
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
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)
Using the fact that:
To calculate the integral in (A2) by use of the integration by parts we have:
Using repeated integration by parts from (A7)l1 times toward x1 and l2 times toward x2 conclude that:
In the last integral of (A9) using the relation
Replacing ξ1 and ξ2 in (A14), implies that:
As a result
To simplify equation (A5), note that:
Assume h1 = m1, r = m1 + m2,
And R1 and R2 the similar computation in appendix (A) in the  give rise to the following:
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