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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:
\begin{array}{*{20}{c}} {{\partial _t}u + u \cdot \nabla u = - \nabla p + \nu \Delta u + (g\alpha T){e_2}} \\ {{\partial _t}T + u \cdot \nabla T = {k_T}\Delta T} \\ {\nabla \cdot u = 0,} \end{array}
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:
\begin{array}{*{20}{c}} {{\partial _t}\omega + u \cdot \nabla \omega = \nu \Delta \omega + g\alpha {\partial _{{x_1}}}T} \\ {{\partial _t}T + u \cdot \nabla T = {k_T}\Delta T} \\ {\nabla \cdot \omega = 0,\quad \omega = \nabla \times u.} \end{array}
We are able to restore the speed of the fluid through Biot-Savart legislation:
u(x) = \frac{1}{{2\pi }}\int_{{\mathbb{R}^2}} \frac{{{{(x - y)}^ \bot }}}{{|x - y{|^2}}}\omega (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].
Mathematical foundations of SCVM
In this section, we summarize the expansion of vorticity and temperature including the Hermite functions as described in [19]. Let
{\phi _{00}}(x,t;\lambda ) = \frac{1}{{\pi {\lambda ^2}}}{e^{ - |x{|^2}/{\lambda ^2}}},\quad \quad {T_{00}}(x,t;\sigma ) = \frac{1}{{\pi {\sigma ^2}}}{e^{ - |x{|^2}/{\sigma ^2}}}
where {\lambda ^2} = \lambda _0^2 + 4\nu t and {\sigma ^2} = \sigma _0^2 + 4{k_T}t. The Hermite functions of degree (k1, k2) is defined as follows:
{\phi _{{k_1},{k_2}}}(x,t;\lambda ) = D_{{x_1}}^{{k_1}}D_{{x_2}}^{{k_2}}{\phi _{00}}(x,t;\lambda ),\quad \quad {\psi _{{k_1},{k_2}}}(x,t;\sigma ) = D_{{x_1}}^{{k_1}}D_{{x_2}}^{{k_2}}{T_{00}}(x,t;\sigma ).
The moment expansion of functions is defined as follows:
\begin{gathered} \omega (x,t) = \sum\limits_{{k_1},{k_2} = 1}^\infty M[{k_1},{k_2};t]{\phi _{{k_1},{k_2}}}(x,t;\lambda ), \hfill \\ T(x,t) = \sum\limits_{{k_1},{k_2} = 1}^\infty I[{k_1},{k_2};t]{\psi _{{k_1},{k_2}}}(x,t;\sigma ). \hfill \\ \end{gathered}
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) = \sum\limits_{{k_1},{k_2} = 1}^\infty M[{k_1},{k_2};t]{V_{{k_1},{k_2}}}(x,t;\lambda ),
where {V_{{k_1},{k_2}}}(x,t;\lambda ) = D_{{x_1}}^{{k_1}}D_{{x_2}}^{{k_2}}{V_{00}}(x,t;\lambda ) and V00(x, t; λ) is the induced speed from ϕ00(x, t; λ) which is determined as follows:
{V_{00}}(x,t;\lambda ) = \frac{1}{{2\pi }}\frac{{( - {x_2},{x_1})}}{{|x{|^2}}}(1 - {e^{ - |x{|^2}/{\lambda ^2}}}).
Hermite polynomials are defined by their generator functions:
{H_{{n_1},{n_2}}}(z,\lambda ) = {\left. {\left( {D_{{t_1}}^{{n_1}}D_{{t_2}}^{{n_2}}{e^{(\frac{{2t \cdot z - {t^2}}}{{{\lambda ^2}}})}}} \right)} \right|_{t = 0}},\,{F_{{n_1},{n_2}}}(z,\sigma ) = {\left. {\left( {D_{{t_1}}^{{n_1}}D_{{t_2}}^{{n_2}}{e^{(\frac{{2t \cdot z - {t^2}}}{{{\sigma ^2}}})}}} \right)} \right|_{t = 0}}.
Notice that the standard Hermite multinomial occur when λ = 1 and k = 1. In this case, they constitute the orthogonal sets:
\int_{{\mathbb{R}^2}} {H_{{n_1},{n_2}}}(z,\lambda = 1){H_{{m_1},{m_2}}}(z,\lambda = 1){e^{ - {z^2}}}dz = \pi {2^{{n_1} + {n_2}}}({n_1}!)({n_2}!){\delta _{{n_1},{m_1}}}{\delta _{{n_2},{m_2}}},\int_{{\mathbb{R}^2}} {F_{{n_1},{n_2}}}(z,\sigma = 1){F_{{m_1},{m_2}}}(z,\sigma = 1){e^{ - {z^2}}}dz = \pi {2^{{n_1} + {n_2}}}({n_1}!)({n_2}!){\delta _{{n_1},{m_1}}}{\delta _{{n_2},{m_2}}}.
Consequently, the following projection operators determine the coefficients in the expansion (4):
M[{k_1},{k_2};t] = ({P_{{k_1},{k_2}}}\omega )(t) = \rho ({k_1},{k_2},\lambda )\int_{{\mathbb{R}^2}} {H_{{k_1},{k_2}}}(z,\lambda )\omega (z,t)dz,I[{k_1},{k_2};t] = ({Q_{{k_1},{k_2}}}T)(t) = \rho ({k_1},{k_2},\sigma )\int_{{\mathbb{R}^2}} {F_{{k_1},{k_2}}}(z,\sigma )T(z,t)dz,
where
\rho ({k_1},{k_2},\tau ) = \frac{{{{( - 1)}^{({k_1} + {k_2})}}{\tau ^{2({k_1} + {k_2})}}}}{{{2^{{k_1} + {k_2}}}({k_1}!)({k_2}!)}}.
In the [19] Nagem and coauthors proved the convergence of the expansions (4), when:
\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}(x){(\omega (x,t))^2}dx < \infty ,\int_{{\mathbb{R}^2}} \Psi _\sigma ^{ - 1}(x){(T(x,t))^2}dx < \infty .
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:
For all 1 ≤ p ≤ ∞ and t ≥ 0, ‖T (x, t)‖p ≤ ‖T0(x)‖p
There exist constant c = c(ω0, T0, t) such that for all 2 ≤ q < ∞ and t ≥ 0, ‖ω(x, t)‖q ≤ c(ω0, T0)
For all t ≥ 0, ‖∇u‖∞ ≤ c(ω0, T0, t)
Proof
For (i) see [1] and for (ii) see [10] and for (iii) see [26].
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|{|_\infty } \leqslant c||\omega ||_p^\alpha ||\omega ||_q^{1 - \alpha } where 1 ≤ p < 2 < q ≤ ∞ and \frac{\alpha }{p} + \frac{{1 - \alpha }}{q} = \frac{1}{2}, as a result according to lemma (1) we obtain: ‖u‖∞ ≤ c(ω0, T0). Therefore by assumption it is concluded that ‖u‖∞ ≤ c(ω0, T0). Now similar to the proof of theorem 3.4 in [19] it could be proved that:
\frac{{d\gamma (t)}}{{dt}} \leqslant (\frac{{4c({\omega _0},{T_0})}}{{{K_T}}} + \frac{{4{K_T}}}{{{\sigma ^2}}})\gamma (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:
\begin{array}{*{20}{l}} {\frac{{d\varepsilon (t)}}{{dt}}}&{ = \frac{{4\nu }}{{{\lambda ^2}}}\varepsilon (t) - \frac{{4\nu }}{{{\lambda ^4}}}\int_{{\mathbb{R}^2}} |x{|^2}\Phi _\lambda ^{ - 1}{{(\omega (x,t))}^2}dx + 2\int_{{\mathbb{R}^2}} |x{|^2}\Phi _\lambda ^{ - 1}\omega (x,t){\partial _t}\omega (x,t)dx} \\ {}&{ = \frac{{4\nu }}{{{\lambda ^2}}}\varepsilon (t) - \frac{{4\nu }}{{{\lambda ^4}}}\int_{{\mathbb{R}^2}} |x{|^2}\Phi _\lambda ^{ - 1}{{(\omega (x,t))}^2}dx} \\ {}&{ + 2\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}\omega (\nu \Delta \omega - u \cdot \nabla \omega + g\alpha {\partial _{{x_1}}}T)dx.} \end{array}
Integrating by parts in the last term in (17) implies that:
2\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}\omega (\nu \Delta \omega )dx = - 2\nu \int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}(x)(|\nabla \omega {|^2} + \frac{2}{{{\lambda ^2}}}\omega x \cdot \nabla \omega )dx,
and the second item in the right side of (18) satisfies the following relation:
2\nu \int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}(x)(\frac{2}{{{\lambda ^2}}}\omega x \cdot \nabla \omega )dx \leqslant \nu \int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}(x)|\nabla \omega {|^2}dx + \frac{{4\nu }}{{{\lambda ^4}}}\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}(x)({x^2}{\omega ^2})dx.
Now we bound the term ||\nabla T||_\lambda ^2. Let f (x, t) = ∇T (x, t) and define:
\delta (t) = \int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}{(\nabla T(x,t))^2}dx.
Now by considering that the last term in (22) we have:
\begin{array}{*{20}{l}} {2\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}f\nabla ({K_T}\Delta T)dx}&{ = 2{K_T}\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}f(\Delta f)dx} \\ {}&{ = - 2{K_T}\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}(x)(|\nabla f{|^2} + \frac{2}{{{\lambda ^2}}}f \cdot x \cdot \nabla f)dx.} \end{array}
The second term in the last part of the equation (23) satisfy the following inequality:
2{K_T}\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}(x)(\frac{2}{{{\lambda ^2}}}f \cdot x \cdot \nabla f)dx \leqslant \frac{{4\nu }}{{{\lambda ^4}}}\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}({x^2}{f^2})dx + \frac{{K_T^2}}{\nu }\int_{{\mathbb{R}^2}} \Phi _\lambda ^{ - 1}|\nabla f{|^2}dx.
As a consequence of (23)–(26) we obtain:
\frac{{d\delta (t)}}{{dt}} \leqslant (2c({\omega _0},{T_0},t) + \frac{{\nu {c^2}({\omega _0},{T_0},t)}}{{2\nu {K_T} - K_T^2}} + \frac{{4\nu }}{{{\lambda ^2}}})\delta (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:
\frac{{d\varepsilon (t)}}{{dt}} \leqslant (\frac{{4\nu }}{{{\lambda ^2}}} + \frac{{4c({\omega _0},{T_0})}}{\nu } + 2g\alpha )\varepsilon (t) + 2g\alpha {c_1}({\omega _0},{t_0},t),
where ||\nabla T||_\lambda ^2 \leqslant {c_1}({\omega _0},{t_0},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
\sum\limits_{{k_1},{k_2}}^m f({k_1},{k_2}): = \sum\limits_{i = 0}^m \sum\limits_{\substack{ {k_1} + {k_2} = i \\ {k_1} \geqslant 0,{k_2} \geqslant 0 } }^n f({k_1},{k_2}),
and also
{\omega ^m}(x,t) = \sum\limits_{{k_1},{k_2}}^m M[{k_1},{k_2};t]{\phi _{{k_1},{k_2}}}(x,t;\lambda ){u^m}(x,t) = \sum\limits_{{k_1},{k_2}}^m M[{k_1},{k_2};t]{V_{{k_1},{k_2}}}(x,t;\lambda ){T^m}(x,t) = \sum\limits_{{k_1},{k_2}}^m I[{k_1},{k_2};t]{\psi _{{k_1},{k_2}}}(x,t;\sigma )
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:
\begin{array}{*{20}{l}} {{\partial _t}{\omega ^m}}&{ = \sum\limits_{{k_1},{k_2}}^m \frac{{dM[{k_1},{k_2};t]}}{{dt}}{\phi _{{k_1},{k_2}}}(x,t;\lambda ) + \sum\limits_{{k_1},{k_2}}^m M[{k_1},{k_2};t]{\partial _t}{\phi _{{k_1},{k_2}}}} \\ {}&{ = \sum\limits_{{k_1},{k_2}}^m M[{k_1},{k_2};t](\nu \Delta {\phi _{{k_1},{k_2}}}(x,t;\lambda ))} \\ {}&{ - {P^m}\left[ {(\sum\limits_{{l_1},{l_2}}^m M[{l_1},{l_2};t]{V_{{l_1},{l_2}}}(x,t;\lambda )) \cdot \nabla (\sum\limits_{{k_1},{k_2}}^m M[{k_1},{k_2};t]{\phi _{{k_1},{k_2}}}(x,t;\lambda ))} \right]} \\ {}&{ + g\alpha {\partial _{{x_1}}}(\sum\limits_{{k_1},{k_2}}^m I[{k_1},{k_2};t]{\psi _{{k_1},{k_2}}}(x,t;\sigma ))} \end{array}\begin{array}{*{20}{l}} {{\partial _t}{T^m}}&{ = \sum\limits_{{k_1},{k_2}}^m \frac{{dI[{k_1},{k_2};t]}}{{dt}}{\psi _{{k_1},{k_2}}}(x,t;\sigma ) + \sum\limits_{{k_1},{k_2}}^m I[{k_1},{k_2};t]{\partial _t}{\psi _{{k_1},{k_2}}}} \\ {}&{ = \sum\limits_{{k_1},{k_2}}^m I[{k_1},{k_2};t]({K_T}\Delta {\psi _{{k_1},{k_2}}}(x,t;\sigma ))} \\ {}&{ - {P^m}\left[ {(\sum\limits_{{l_1},{l_2}}^m M[{l_1},{l_2};t]{V_{{l_1},{l_2}}}(x,t;\lambda )) \cdot \nabla (\sum\limits_{{k_1},{k_2}}^m I[{k_1},{k_2};t]{\psi _{{k_1},{k_2}}}(x,t;\sigma ))} \right].} \end{array}
where Pm[·] is a projector on the subspace produced by Hermit functions of degree m or less. Noting that:
{\partial _t}{\phi _{{k_1},{k_2}}} = \nu \Delta {\phi _{{k_1},{k_2}}},\,\,\,\,\,\,\,\,\,{\partial _t}{\psi _{{k_1},{k_2}}} = {K_T}\Delta {\psi _{{k_1},{k_2}}},
and applying the projection operators Pk1,k2 and Qk1,k2, defined in (10) on the equation (33) and (34) we have:
\begin{gathered} \frac{{dM[{k_1},{k_2};t]}}{{dt}} = \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, - {P_{{k_1},{k_2}}}\left[ {(\sum\limits_{{l_1},{l_2}}^m M[{l_1},{l_2};t]{V_{{l_1},{l_2}}}(x,t;\lambda )) \cdot \nabla (\sum\limits_{{m_1},{m_2}}^m M[{m_1},{m_2};t]{\phi _{{m_1},{m_2}}}(x,t;\lambda ))} \right] \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\, - {P_{{k_1},{k_2}}}\left[ {g\alpha {\partial _{{x_1}}}(\sum\limits_{{m_1},{m_2}}^m I[{m_1},{m_2};t]{\psi _{{m_1},{m_2}}}(x,t;\sigma ))} \right] \hfill \\ \end{gathered} \begin{gathered} \frac{{dI[{k_1},{k_2};t]}}{{dt}} = \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\, - {Q_{{k_1},{k_2}}}\left[ {(\sum\limits_{{l_1},{l_2}}^m M[{l_1},{l_2};t]{V_{{l_1},{l_2}}}(x,t;\lambda )) \cdot \nabla (\sum\limits_{{m_1},{m_2}}^m I[{m_1},{m_2};t]{\psi _{{m_1},{m_2}}}(x,t;\sigma ))} \right]. \hfill \\ \end{gathered}
Note that k1 + k2 ≤ m then
{\phi _{{m_1},{m_2}}}(x,t;\lambda ) = {\left. {\left( {D_{{a_1}}^{{m_1}}D_{{a_2}}^{{m_2}}{\phi _{00}}(x + a,\lambda )} \right)} \right|_{a = 0}}{V_{{l_1},{l_2}}}(x,t;\lambda ) = {\left. {\left( {D_{{b_1}}^{{l_1}}D_{{b_2}}^{{l_2}}{V_{00}}(x + b,\lambda )} \right)} \right|_{b = 0}}{\psi _{{m_1},{m_2}}}(x,t;\sigma ) = {\left. {\left( {D_{{c_1}}^{{m_1}}D_{{c_2}}^{{m_2}}{\psi _{00}}(x + c,\sigma )} \right)} \right|_{c = 0}},
then the system of ordinary differential equations (35) and (36) become as follows:
\begin{array}{*{20}{l}} {\frac{{dM[{k_1},{k_2};t]}}{{dt}}}&{ = - \rho ({k_1},{k_2},\lambda )\sum\limits_{{l_1},{l_2} = 1}^m \sum\limits_{{m_1},{m_2} = 1}^m M[{l_1},{l_2};t]M[{m_1},{m_2};t]} \\ {}&{ \times \int_{{\mathbb{R}^2}} {H_{{k_1},{k_2}}}(x)(D_{{x_1}}^{{m_1}}D_{{x_2}}^{{m_2}}{V_{00}}(x,\lambda )) \cdot {\nabla _x}(D_{{x_1}}^{{l_1}}D_{{x_2}}^{{l_2}}{\phi _{00}}(x,\lambda ))dx} \\ {}&{ - \rho ({k_1},{k_2},\lambda )\sum\limits_{{m_1},{m_2} = 1}^m I[{m_1},{m_2};t]} \\ {}&{ \times \int_{{\mathbb{R}^2}} {H_{{k_1},{k_2}}}(x)(D_{{x_1}}^{{m_1}}D_{{x_2}}^{{m_2}}{T_{00}}(x,\sigma ))dx} \end{array}\begin{array}{*{20}{l}} {\frac{{dI[{k_1},{k_2};t]}}{{dt}}}&{ = - \rho ({k_1},{k_2},\sigma )\sum\limits_{{l_1},{l_2} = 1}^m \sum\limits_{{m_1},{m_2} = 1}^m M[{l_1},{l_2};t]I[{m_1},{m_2};t]} \\ {}&{ \times \int_{{\mathbb{R}^2}} {F_{{k_1},{k_2}}}(x)(D_{{x_1}}^{{m_1}}D_{{x_2}}^{{m_2}}{V_{00}}(x,\lambda )) \cdot {\nabla _x}(D_{{x_1}}^{{l_1}}D_{{x_2}}^{{l_2}}{T_{00}}(x,\sigma ))dx.} \end{array}
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:
\begin{array}{*{20}{l}} {\frac{{dM[{k_1},{k_2};t]}}{{dt}}}&{ = \rho ({k_1},{k_2},\lambda )\sum\limits_{{l_1},{l_2} = 1}^m \sum\limits_{{m_1},{m_2} = 1}^m M[{l_1},{l_2};t]M[{m_1},{m_2};t]} \\ {}&{ \times \tilde \Gamma [{k_1},{k_2},{l_1},{l_2},{m_1},{m_2};\lambda ] + \sum\limits_{{m_1},{m_2}}^m I[{m_1},{m_2};t]B[{k_1},{k_2},{m_1},{m_2};\lambda ,\sigma ]} \end{array}\begin{array}{*{20}{l}} {\frac{{dI[{k_1},{k_2};t]}}{{dt}}}&{ = \rho ({k_1},{k_2},\sigma )\sum\limits_{{l_1},{l_2} = 1}^m \sum\limits_{{m_1},{m_2} = 1}^m M[{l_1},{l_2};t]I[{m_1},{m_2};t]} \\ {}&{ \times \tilde \theta [{k_1},{k_2},{l_1},{l_2},{m_1},{m_2};\lambda ,\sigma ],} \end{array}
where B and \tilde \theta is introduced in appendix, \tilde \Gamma is introduced in [19]and
\tilde \theta [{k_1},{k_2},{l_1},{l_2},{m_1},{m_2};\lambda ,\sigma ] = {\theta ^1}[{k_1},{k_2},{l_1},{l_2},{m_1},{m_2};\lambda ,\sigma ] + {\theta ^2}[{k_1},{k_2},{l_1},{l_2},{m_1},{m_2};\lambda ,\sigma ].
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,
\omega (x,0) = {\phi _{00}}(x,0) + 4\delta ({\phi _{20}} + {\phi _{02}}),
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
the graph of the nonaxisymmetric portion (E) with m = 4 and α = 0.
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.
Now we present some examples with different values of α.
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 = \int_{{\mathbb{R}^2}} {(\omega (x) - < \omega (\left| x \right|) > )^2}dx,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, < \omega (|x|) > = \frac{1}{{2\pi }}\int_0^{2\pi } \omega (x)d\theta .
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.
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,\,\,\,\,\,\,\,\,\,\,\,\,\nu = \frac{1}{{500}}\,({m^2}/s),\,\,\,\,\,\,\,\,\,\,\,{K_T} = \frac{1}{{500}},\frac{1}{{800}},\frac{1}{{1500}}({m^2}/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
The graph of the nonaxisymmetric portion (E) with m = 4 and α = 69 × 10−6
.
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.
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,\,\,\,\,\,\,\,\,\,\,\,\,\nu = \frac{1}{{500}}\,({m^2}/s),\,\,\,\,\,\,\,\,\,\,\,\,{K_T} = \frac{1}{{500}},\frac{1}{{800}},\frac{1}{{1500}}({m^2}/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
The graph of the nonaxisymmetric portion (E) with m = 4 and α = 69 × 10−4
.
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.
