The evolution of localized vortex in stably stratified flows

We characterize the vortex disturbance by the fluid impulse integral $\mathbf{I}$ defined in (1). The corresponding time derivative is given by

$$\frac{d\mathbf{I}}{dt}=\frac{1}{2}\int\mathbf{x}\times\frac{\partial\boldsymbol{\omega}(\mathbf{x,t})}{\partial t}dV,$$

(5)

where the time derivative of the disturbed vorticity is obtained applying operator rotor to (2) and subtracting the undisturbed vorticity equation, i.e.

$$\frac{\partial\boldsymbol{\omega}}{\partial t}+(\mathbf{U}_{0}\cdot\mathbf{\nabla})\boldsymbol{\omega}-(\boldsymbol{\omega}\cdot\mathbf{\nabla})\mathbf{U}_{0}-(\boldsymbol{\Omega}_{0}\cdot\mathbf{\nabla})\mathbf{u}+(\mathbf{u}\cdot\mathbf{\nabla})\boldsymbol{\omega}-(\boldsymbol{\omega}\cdot\mathbf{\nabla})\mathbf{u}=\frac{1}{\rho_{0}}\mathbf{\nabla}\rho\times\mathbf{g},$$

(6)

where $\boldsymbol{\Omega}_{0}=\mathbf{\nabla}\times\mathbf{U}_{0}$. In (6) we also make use of the compactness of the disturbance and neglect terms which have second derivative of the basic velocity and pressure fields. This means that the undisturbed basic flow, which is assumed to be known, is approximated by the leading terms of the Taylor series expansion, i.e. $\boldsymbol{\Omega}_{0}$ is constant and $U_{0}(z)=U_{0}(0)+\Omega_{0}z$. Note that the fluid impulse is invariant under Galilean transformation, and we can set $U_{0}(0)=0$ without loss of generality. Although the leading term of $\rho_{0}$ in (6) is a constant, its variation with $z$ is the only source for generating the density disturbance. The corresponding evolution equation is

$$\frac{\partial\rho}{\partial t}+(\mathbf{U}_{0}\cdot\mathbf{\nabla})\rho+(\mathbf{u}\cdot\mathbf{\nabla})\rho+u_{z}\frac{d\rho_{0}}{dz}=0,$$

(7)

where $d\rho_{0}/dz$ is constant to the leading order.

As mentioned above the main advantage of using the fluid impulse is that its evolution is described by a linear equation, even though the fluid motion itself is strongly non-linear. However, the use of fluid impulse is not a straightforward procedure since, as it explained in the following, the integral (on the right hand side of (5)) is not absolutely convergent. The reason for this is the generation of far field vorticity perturbations (tails) caused by the velocity field induced by the localized vortical disturbance. For example, the far field velocity induced by the localized disturbance at a distance $|\mathbf{x}|>>l$ is of order of $|\mathbf{u}|\sim|\mathbf{I}|/|\mathbf{x}|^{3}$, where $\mathbf{I}$ is the magnitude of the associated fluid impulse integral and $l$ represents the typical length scale of the disturbance. From (7) it follows that the disturbed density is of the order of $|\rho|\sim\mathcal{O}(1/|\mathbf{x}|^{3})$. Correspondingly, the fourth term on the left hand side and the term on the right hand side of (6) (hereinafter referred to as problematic terms), which are responsible for the generation of the new vorticity over the entire flow region, exhibit asymptotic behavior of $\mathcal{O}(1/|\mathbf{x}|^{4})$. Substitution of these terms into (5) causes the integral to be not absolutely convergent, i.e. its evolution depends on how the volume of integration is allowed to tend to infinity.

To overcome this difficulty we follow the approach proposed in LC, which makes it possible to distinguish the contribution to the fluid impulse integral of the ’local’ vorticity field from that of the far-field tails, Accordingly, we represent the velocity, vorticity and density fields as a superposition of two components $\boldsymbol{\omega}=\boldsymbol{\omega}^{I}+\boldsymbol{\omega}^{II}$, $\mathbf{u}=\mathbf{u}^{I}+\mathbf{u}^{II}$ and $\rho=\rho^{I}+\rho^{II}$, so that $\boldsymbol{\omega}^{I,II}=\mathbf{\nabla}\times\mathbf{u}^{I,II}$. For each component we require that

$$\mathbf{\nabla}\cdot\boldsymbol{\omega}^{I}=\mathbf{\nabla}\cdot\boldsymbol{\omega}^{II}=0$$

(8)

The fist component, indicated by the superscript $I$, is associated with the concentrated vorticity and density distributions confined within and in the vicinity of the initially disturbed region, whereas the second component, indicated by the superscript $II$, is associated with the far-field vorticity and density generated by the problematic terms in (6) and (7). Consequently, we set the initial distribution of the vorticity and density fields as

$$\boldsymbol{\omega}^{I}(\mathbf{x},t=0)=\boldsymbol{\omega}(\mathbf{x},t=0),\;\boldsymbol{\omega}^{II}(\mathbf{x},t=0)=0\;and$$

$$\rho^{I}(\mathbf{x},t=0)=\rho^{II}(\mathbf{x},t=0)=0$$

(9)

and we intend to follow the evolution of the fluid impulse associated with of $\boldsymbol{\omega}^{I}(\mathbf{x},t)$ alone. Accordingly we redefine the fluid impulse of the localized disturbance as

$$\mathbf{I}=\frac{1}{2}\int\mathbf{x}\times\boldsymbol{\omega}^{I}(\mathbf{x},t)dV$$

(10)

As is shown in Appendix A, the subdivision procedure has resulted in an evolution equation for $\boldsymbol{\omega}^{I}$ vorticity field with asymptotic behavior

$$|\boldsymbol{\omega}^{I}(\mathbf{x},t)|\sim\mathcal{O}(\frac{1}{|\mathbf{x}|^{5}})\;\;for\;|\mathbf{x}|>>l$$

(11)

Before proceeding with the analysis we would like to note that the tails of the vorticity field $\boldsymbol{\omega}^{I}$ can be completely eliminated by canceling the subsequent members in far field asymptotic expansion of the problematic terms. In this case, the corresponding vorticity field $\boldsymbol{\omega}^{II}$ will include total far field vorticity tails and a vorticity field within the sphere to close the vortex lines of the $\boldsymbol{\omega}^{II}$ field which intersect the sphere surface from the outer region (see Figure 1 below)

Figure 1. Schematic view of the $\boldsymbol{\omega}^{I}$ and $\boldsymbol{\omega}^{II}$ vorticity fields. The sum of $\boldsymbol{\omega}^{I}+\boldsymbol{\omega}^{II}$ corresponds to the total vorticity field of the disturbance.

It is also follows from the subdivision procedure that the initially zero vorticity field $\boldsymbol{\omega}^{II}$ and the associated density field $\rho^{II}$ are generated within the sphere only due to the artificial term $\mathbf{\nabla}\Psi$ in the right hand side of equation (A8). Since $\Psi$ and their derivatives are harmonic functions, $\mathbf{\nabla}\Psi$ attains its maximum on the boundary of the sphere, where its magnitude can be estimated from (A11) as $\Omega_{0}|\mathbf{I}|/R^{4}$. Consequently, the order of magnitude of the vorticity $\boldsymbol{\omega}^{II}$ generated within the sphere is $|\mathbf{I}|/R^{4}$. On the other hand, the localized vortex of hairpin type observed in boundary layer flows possesses a very concentrated vorticity field. It is confined within vortex tubes with the typical cross-section diameter $l_{1}<<l$. Correspondingly, the magnitude of fluid impulse $|\mathbf{I}|$ associated with $\boldsymbol{\omega}^{I}$ field can be estimated as $|\mathbf{I}|=\mathcal{O}(|\boldsymbol{\omega}^{I}|l^{2}l_{1}^{2})$. Thus the ratio $|\boldsymbol{\omega}^{II}|/|\boldsymbol{\omega}^{I}|$ within the inner region $|\mathbf{x}|\leq R$ is proportional to the small value $l_{1}^{2}/l^{2}$.

According to (11) the fluid impulse integral (10) is absolutely convergent. It is therefore possible to use an infinite sphere as the volume of integration. Consequently, the time evolution of $\mathbf{I}$ is given by

$$\frac{d\mathbf{I}}{dt}=\frac{1}{2}\lim_{R_{1}\to\infty}\int_{|\mathbf{x}|\leq R_{1}}\mathbf{x}\times\frac{\partial\boldsymbol{\omega}(\mathbf{x,t})}{\partial t}dV$$

(12)

Straightforward calculations show that the contribution to the integral on the right hand side of (12) from the terms containing $\rho^{a}$ and $\mathbf{u}^{a}$ in (A1) as well as from the term $\mathbf{\nabla}\Psi$ in (A7) is zero. Thus, these artificial terms, used to extract the localized part of the disturbance, have no direct impact on the evolution of the fluid impulse associated with the concentrated vorticity $\boldsymbol{\omega}^{I}$, provided that the volume of integration in fluid impulse definition (12) is an infinite sphere. The resulting evolution equation for fluid impulse is

$$\frac{d\mathbf{I}}{dt}=-\frac{1}{2}\lim_{R_{1}\to\infty}\int_{|\mathbf{x}|\leq R_{1}}\mathbf{x}\times[(\mathbf{U}_{0}\cdot\mathbf{\nabla})\boldsymbol{\omega}^{I}-(\boldsymbol{\omega}^{I}\cdot\mathbf{\nabla})\mathbf{U}_{0}-(\boldsymbol{\Omega}_{0}\cdot\mathbf{\nabla})\mathbf{u}^{I}+(\mathbf{u}\cdot\mathbf{\nabla})\boldsymbol{\omega}-(\boldsymbol{\omega}\cdot\mathbf{\nabla})\mathbf{u}]dV$$

$$+\frac{1}{2\rho_{0}}\lim_{R_{1}\to\infty}\int_{|\mathbf{x}|\leq R_{1}}\mathbf{x}\times[\mathbf{\nabla}\rho^{I}\times\mathbf{g}]dV$$

(13)

The first integral in (13) is identical to the one calculated by LC for the case when the basic flow has a constant density. The calculation of the second integral is given in Apendix B. Finally, the evolution of the fluid impulse components is described by the following set of linear equations

$$\frac{dI_{x}}{dt}=-\frac{1}{2}I_{z}+Ri\int_{0}^{\bar{t}}[G_{xx}(\bar{t}-\bar{\tau})I_{x}(\bar{\tau})+G_{xz}(\bar{t}-\bar{\tau})I_{z}(\bar{\tau})]d\bar{\tau}$$

(14)

$$\frac{dI_{y}}{dt}=Ri\int_{0}^{\bar{t}}G_{yy}(\bar{t}-\bar{\tau})I_{y}(\bar{\tau})d\bar{\tau}$$

(15)

$$\frac{dI_{z}}{dt}=-\frac{1}{2}I_{x}-Ri\int_{0}^{\bar{t}}[G_{zx}(\bar{t}-\bar{\tau})I_{x}(\bar{\tau})+G_{zz}(\bar{t}-\bar{\tau})I_{z}(\bar{\tau})]d\bar{\tau}-\frac{2}{3}Ri\int_{0}^{\bar{t}}I_{z}(\bar{\tau})d\bar{\tau}-$$

$$2Ri\int_{0}^{\bar{t}}d\bar{\tau}\int_{0}^{\bar{\tau}}[G_{xx}(\bar{\tau}-\bar{\nu})I_{x}(\bar{\nu})+G_{xz}(\bar{\tau}-\bar{\nu})I_{z}(\bar{\nu})]d\bar{\nu}$$

(16)

where the ’memory’ functions $G_{xx},G_{yy},G_{xz},G_{zx}\;and\;G_{zz}$ are calculated in Apendix B (and shown below in Figure 2). Here $\bar{t}=\Omega_{0}t$ is the nondimensional time, $Ri=\frac{N^{2}}{\Omega_{0}^{2}}$ is the Richardson number and $N$ is the Brunt-V$\ddot{a}$is$\ddot{a}$l$\ddot{a}$ frequency defined as $N^{2}=-\frac{g}{\rho_{0}}\frac{d\rho_{0}}{dz}$.

Figure 2. The dependence of the ’memory’ functions on the non-dimensional time $\bar{t}=\Omega_{0}t$. $G_{xx}$, solid line; $G_{xz}$, dotted line; $G_{zx}$, dashed line; $G_{zz}$, chain-dotted line; $G_{yy}$, chain-dashed line.

Because $G_{yy}$ is negative for all $\bar{t}$, (15) contains only a not-growing solution. In addition, $I_{y}$ does not depend on $I_{x}$ and $I_{z}$ and vice-versa, and therefore, without loss of generality, $I_{y}$ could be set to zero.

The system (14), (16), is solved using Laplas transform with espect to $\bar{t}$. Accordingly, we define

$$\binom{I_{x,z}(s)}{G_{ij}(s)}=\int_{0}^{\infty}d\bar{t}\binom{I_{x,z}(\bar{t})}{G_{ij}(\bar{t}}e^{-s\bar{t}}$$

where for values of $Re(s)<0$ the functions ${G_{ij}(s)}$ are calculated by extending the corresponding solutions obtained for $Re(s)\geq 0$ using analytical continuation. Thus, the solutions of the tansformed equations are

$$I_{x}(s)=[A_{zx}I_{x}(0)+A_{xx}I_{z}(0)]/D,\;\;I_{z}(s)=[A_{zz}I_{x}(0)+A_{xz}I_{z}(0)]/D,$$

(17)

where $I_{x}(0)$ and $I_{z}(0)$ are the fluid impulse components at time $\bar{t}=0$, $D=A_{xx}A_{zz}-A_{xz}A_{zx}$ and

$$A_{xx}=s-Ri\cdot G_{xx}(s),\;\;A_{xz}=-\frac{1}{2}+Ri\cdot G_{xz}(s),$$

$$A_{zx}=-\frac{1}{2}-Ri\cdot G_{zx}(s)-\frac{2Ri}{s}G_{xx}(s),$$

$$A_{zz}=s+Ri\cdot G_{zz}(s)+\frac{2}{3}\frac{Ri}{s}+\frac{2Ri}{s}G_{xz}(s)$$

To obtain the temporal behavior of the components of the fluid impulse, it is sufficient to invert the Laplas transform. The inverse is given by

$$I_{x,z}(\bar{t})=\frac{1}{2\pi i}\int_{C}e^{s\bar{t}}I_{x,z}(s)ds,$$

(18)

where the contur $C$ in the complex $s$ plane is located to the right of all singularities of the integrand. The evaluation of the integral in (18) is performed using the residues method. The result obtained here includes only the contributions from zeros of $D$.

We focus on the solutions of equation $D(s)=0$ with $Re(s)>0$ which correspond to time growing disturbances. The dependence of such eigenvalues (with a positive real part) on the Richardson number is shown in Figure 3.

Figure 3. The dependence of the eigenvalues, which correspond to time growing disturbances, on the Richardson number for the three-dimensional case. The real part of the eigenvalue ($Re(s)$) is shown by the solid curve while the dash line corresponds to its imaginary part ($Im(s)$). The dotted line corresponds to the marginal value for stability.

For values of $Ri<0.29$ there is only single growing (in time) solution with a zero imaginary part. For $0.29<Ri<0.88$ two conjugated growing solutions exist. For $Ri>0.88$ the flow is stable with respect to localized disturbance. This behavior is typical for buoyancy dominated flows and corresponds to the conversion of the vertical kinetic energy int the potential energy in the gravitational field.

The evolution equations (14)-(16) for fluid impulse components are unsuitable for analysis of the vortex disturbance behaviour at $Ri\to\infty$. Formally, this is due to using $1/|\boldsymbol{\Omega}|$ as a relevant time scale to render equations (14)-(16) dimensionless, whereas in the case of large Richardson numbers the characteristic time scale is associated with the effect of the buoyancy forces. However, the main reason that the analysis of (14)-(16) is limited to relatively small Richardson numbers lies in the assumption that the vorticity field $\boldsymbol{\omega}^{I}$ associated with localized disturbance is strongly concentrated. According to the model, the disturbance is considered to be a compact vortex ($|\boldsymbol{\omega}^{II}|<<|\boldsymbol{\omega}^{I}|$ within the disturbed region). Here the strength of the vorticity tails rapidly decreases with distance fom the disturbance.

As was shown by LC, the mechanism of hairpin vortex formation includes generation of the new vorticity in the desturbed region and its surrounding. Because the rate of generation as its maximum in the region adjacent to the vortex core, it is bounded to cause an unabated vortex core expansion. The stretching of the new generated vorticity by external shear flow makes it possible to hold a high concentration of vorticity, which is typical for hairpin vortices in turbulent and laminar boundary layers. Gerz (1991a, 1991b) and Holt et al. (1992) demonstrated that the vortices of this type are formed in stably stratified shear flows in regions of Richardson numbers $Ri\leq 1$. By this means that the application area of the given approach corresponds to the region of Richardson numbers for which the flow is found to be unstable with respect to localized vortex disturbances.

The region of instability can be divided into two domains with different character. For values of $Ri<0.29$, the solution describes a growing (in time) hairpin type vortex. For example, in the case of non-stratified flow ($Ri=0$) we recover the same solution ($s=1/2$) which was obtained by LC, associated with the growth of a vortex inclined at 45⁰ to the direction of the basic flow. For values of $Ri>0.29$, the solution represent the product of the term describing the exponential growth and the term describing the periodical motion. For symmetrical vortex structures, such as vortex rings, the direction of fluid impulse of the vortex coincides with the direction of its self-induced motion (Batchelor, 1967). For non-symmetrical vortices this is not always the case, however, it can be shown that the direction of self-induced velocity, averaged over the spherical volume containing the vortex, coincides with the direction of its fluid impulse (see Appendix A in LC). Correspondingly, the periodical term in the solution fr fluid impulse describes an oscillatory motion of the whole vortex disturbance about the initial state.

Behaviour of this type has been observed in several experimental and numerical studies for values of $Ri>0.25$ and was previously associated with the presence of internal waves (Steward 1969, Stillinger et al. 1983). However, this point of view was not supported by subsequent studies (see Rohr et al. 1988 and Holt et al. 1992 for more detailed discussion). The results of the present study makes it possible to explain the oscillatory behavior without invoking the internal wave concept.

Additional quantitative comparison with results reported by Holt et al. (1992) and Kaltenbach, Gerz and Schuman (1994) for region of $Ri>0.25$ could be made. They carried out two sets of calculations at $Ri=0.5$ and $Ri=1$. In both cases, oscillating behavior of the flow parameters was observed. We will use the results for the case of $Ri=0.5$ because it falls within the region of instability predicted here. Holt et al. (1992) and Kaltenbach et al. (1994) presented the temporal dependence of ensemble averaged vertical density flux $\overline{\rho v}$ (Figure 15 and Figure 3(b) correspondingly), where $v$ is the disturbed vertical velocity, and $\rho$ is the disturbed density. They found that the zero crossing of $\overline{\rho v}$ occurs at non-dimensional times $\Omega t\simeq 5$ and $\Omega t\simeq 4.5$, respectively. On the other hand, according to the model, the time between zero crossing corresponds to a quater of the oscillation period ($\rho$ is equal to zero at the beginning and at the half of the period, when the vortex is in its equilibrium position in the density gradient, whereas $v$ is equal to zero at the quarter and three quarters of the period, when the vortex reverses the direction of its motion due to buoyancy forces). The result, which can be obtained from Figure 3 for the quarter of the non-dimensional period, is $\Omega T/4=\pi/2Im(s)\simeq 6$. Even though we compare the results of turbulent flow numerical simulations with prediction of laminar inviscid model, the agreement is fairly good.

In this section we use the fluid impulse concept to describe the motion of a vortex pair through a stratified environment. This problem has been studied by many investigators, partially because of its relevance to practical needs.The common example is the hazard to following aircraft caused by the trailing vortex system of a large aircraft.

In two-dimensional flows, the fluid impulse per unit length can be written as

$$\mathbf{I}=\int\mathbf{r}\times\boldsymbol{\omega}(\mathbf{r})dA,$$

(19)

where $\boldsymbol{\omega}=(0,\omega,0),\;\mathbf{r}=(x,z),\;dA=dxdz$. In this case equation (6) describing the time evolution of the disturbed vorticity is reduced to

$$\frac{\partial\omega}{\partial t}+(\mathbf{U}_{0}\cdot\mathbf{\nabla})\omega+(\mathbf{u}\cdot\mathbf{\nabla})\omega=\frac{g}{\rho_{0}}\frac{\partial\rho}{\partial x}$$

(20)

The evolution equation for the disturbed density remains the same as (7).

As can be seen from (20), the only term responsible for the generation of new vorticity is the buoyancy term in the right hand side. That is, in the case where the buoyancy mass moves in homogeneous ambient fluid (the case considered by Saffman, 1972), the integral (19) is well defined for all times. However, when the surrounding fluid is not homogeneous, the asymptotic behavior of the disturbed vorticity (’tails’) is $|\omega|\sim\mathcal{O}(1/|\mathbf{r}|^{3})$ and consequently the fluid impulse integral (19) is not absolutely convergent.

To solve this problem, which is similar to the one discussed in the previous section, we use the same procedure and subdivide the vorticity and density fields into two components: the localized fields denoted by the subscript $I$ and the far-fields denoted by the subscript $II$. As for the three-dimensional case, the initial conditions are specified by (9) and we use the subdivision of the plane into two regions, inside and outside circle enclosing the disturbance. urther we follow the sequence of operation given in Appendix A, which in this case is more straightforward because of the divergence free requirement (8) for the two-dimensional case is always fulfilled. Namely, there is no need for artificial generation of the vrticity field $\boldsymbol{\omega}^{II}$ within the circle as is in the three-dimensional case.

According to the procedure, we define the fluid impulse associated with the localized part of the disturbance by

$$\mathbf{I}=\lim_{R\to\infty}\int_{|\mathbf{r}|\leq R}\mathbf{r}\times\boldsymbol{\omega}(\mathbf{r},t)dA$$

(21)

Corresponding evolution equation for $\mathbf{I}$ can be represented as

$$\frac{d\mathbf{I}}{dt}=\lim_{R\to\infty}\int_{|\mathbf{r}|\leq R}(\mathbf{e}_{x}z-\mathbf{e}_{z}x)[(\mathbf{U}_{0}\cdot\mathbf{\nabla})\omega^{I}-\frac{g}{\rho_{0}}\frac{\partial\rho^{I}}{\partial x}]dA,$$

(22)

where $\mathbf{e}_{x}$ and $\mathbf{e}_{z}$ are the unit vectors in the $x$ and $z$ directions.

Then, in the same fashion as in Appendix B, the following set of equations describing the time evolution of the fluid impulse components can be obtained

$$\frac{dI_{x}}{d\bar{t}}=-2Ri\int_{0}^{\bar{t}}[G_{1}(\bar{t}-\bar{\tau})I_{z}(\bar{\tau})+G_{2}(\bar{t}-\bar{\tau})I_{x}(\bar{\tau})]d\bar{\tau}$$

(23)

$$\frac{dI_{z}}{d\bar{t}}=-2I_{x}-2Ri\int_{0}^{\bar{t}}[-G_{1}(\bar{t}-\bar{\tau})I_{x}(\bar{\tau})+G_{2}(\bar{t}-\bar{\tau})I_{z}(\bar{\tau})]d\bar{\tau}-2Ri\int_{0}^{\bar{t}}I_{z}(\bar{\tau})d\bar{\tau}+$$

$$4Ri\int_{0}^{\bar{t}}d\bar{\tau}\int_{0}^{\bar{\tau}}[G_{1}(\bar{\tau}-\bar{\nu})I_{z}(\bar{\nu})+G_{2}(\bar{\tau}-\bar{\nu})I_{x}(\bar{\nu})]d\bar{\nu}$$

(24)

Here, the non-dimensional time is $\bar{t}=\Omega_{0}t/2$ and the memory functions $G_{1}(\bar{t})$ and $G_{2}(\bar{t})$ are given by

$$G_{1}(\bar{t})=\frac{\bar{t}}{(1+\bar{t}^{2})^{2}},\;\;G_{2}(\bar{t})=\frac{1}{2}\frac{1-\bar{t}^{2}}{(1+\bar{t}^{2})^{2}}$$

(25)

We define the Laplas transform of $I_{x},I_{z},G_{1}$ and $G_{2}$ in the same way as was done for the three-dimensional case. Then by applying the Laplas transform to equations (23), (24), we obtain the characteristic equation for the eigenvalues corresponding to the poles of the transformed components $I_{x}(s)$ and $I_{z}(s)$

$$s^{2}+4RiG_{2}((s)s+4Ri^{2}[G_{1}^{2}(s)+G_{2}^{2}(s)]+2Ri-8RiG_{1}(s)+\frac{4Ri^{2}}{s}G_{2}(s)=0$$

(26)

For values of $Re(s)>0$, it is convenient to express the transformed memory functions $G_{1}(s)$ and $G_{2}(s)$ in terms of cosine (ci) and sine (si) integrals:

$$G_{1}(s)=\frac{1}{2}{1-s[ci(s)sin(s)-si(s)cos(s)]},$$

$$G_{2}(s)=-\frac{1}{2}s[ci(s)cos(s)+si(s)sin(s)].$$

The corresponding values of $G_{1}(s)$ and $G_{2}(s)$ for $Re(s)\leq 0$ are obtained by using the procedure of analytical continuation.

The solution of equation (26) with $Re(s)\geq 0$ are shown in Figure 4 as function of the Richardson number.

Figure 4. The dependence of the eigenvalues, which correspond to time growing disturbances, on the Richardson number for two-dimensional case. The real part of the eigenvalue ($Re(s)$) is shown by the solid curve while the dash line corresponds to its imaginary part ($Im(s)$). The dotted line corresponds to the marginal value for stability.

As can be seen, the homogeneous shear flow without stratification ($Ri=0$) does not display the exponential instability (although there is still the possibility of algebraic grows of the disturbance, as was shown by Landhal, 1970). However, contrary to intuition, the addition of stably stratification has a destabilizing effect.

Indeed, in the three-dimensional case the mechanism of localized vortex instability is associated with its interaction with the basic shear flow. The growth rate of fluid impulse of the disturbance has a maximum at $Ri=0$ and only decreases as the Richardson number increases. In contrast, in the two-dimensional case the buoyancy, which is solely responsible for new vorticity generation, gives rise to exponential instability up to $Ri\sim 1.23$. Nevertheless, for values of $Ri>0.3$ the solution exhibits oscillatory behavior similar to the three-dimensional case.

It should be noted that for the two-dimensional case, interpretation of the relationship between the fluid impulse and the vorticity field associated with the localized disturbance is more difficult than for three-dimensional case. As discussed above, the disturbance is considered to be a compact vortex with vorticity tails, the strength of which is rapidly reduced with distance from the disturbed region. Indeed, in two-dimensional case the numerical simulations by Robius $\And$ Delisi (1990) and Carten et al. (1998) demonstrated that the motion of the vortex pair in a stratified environment causes the generation of an extended wake of vorticity. Thus, the flow pattern in this case may not comply with the requirement for compactness of the disturbance. For this requirement to be satisfied, the self-induced displacement of the vortex pair during the typical evolution tme of the fluid impulse must be less than its typical dimension. This condition can be expressed as

$$Fr=\frac{W}{Nb}<<1,$$

where $Fr$ is the non dimensional Froude number, b is the vortex core separation distance and $W$ is the magnitude of the vertical velocity induced by the vortex pair. In this respect, it is unfortunate that the present theoretical results can not be compared with the numerical results of Robius $\And$ Delisi (1990) and Carten et al. (1998) as their simulations were carried out only for range of $Fr\geq 1$.

In contrast to the three-dimensional case, the solution at $Ri>>1$ is physically meaningful, because the fact that the subdivision procedure used here has no need to add of the artificial vorticity field within the disturbed region. Taking limit $\Omega\to 0$ and turning to the dimensional time in (23) and (24), the evolution equations for fluid impulse components can be written as

$$\frac{dI_{x}}{dt}=-N^{2}\int_{0}^{t}I_{x}(\tau)d\tau$$

(27)

$$\frac{dI_{z}}{dt}=-3N^{2}\int_{0}^{t}I_{z}(\tau)d\tau$$

(28)

Equations (27), (28) describe the independent oscillations in $x$ and $z$ directions. i.e.

$$I_{x}(t)=I_{x}(0)\cdot cos(Nt)\;\;and\;\;I_{z}(t)=I_{z}(0)\cdot cos(Nt/\sqrt{3})$$

(29)

For comparison, Saffman’s analysis (Safman, 1972) gives an oscillating motion in the $z$ direction with frequency of about $N/\sqrt{2.2}$. The main source of this discrepancy shows up also in comparison with analysis by Spalart (1996).

Spalart (1996) suggested to subdivide the whole vorticity field into three parts. He introduced the notion $\omega_{1}$ for the initially created vorticity, $\omega_{2}$ for the vorticity field generated by buoyancy in the disturbed region and its surrounding and $\omega_{3}$ for the generated vorticity tails. Then he ignored the $\omega_{3}$ vorticity field and used the fluid impulse determined by expression (19). Evolution equations obtained by him for fluid impulse components can be written in our notations as

$$\frac{dI_{x}}{dt}=0\;\;and\;\;\frac{dI_{x}}{dt}=-\frac{g}{\rho_{0}}\int\rho dA,$$

(30)

where $\rho$ is the disturbed density.

As was discussed above, the magnitude of the fluid impulse integral defined by expression (19) depends on way the volume of integration is allowed to tend to infinity. Or, in other words, it depends on how the generated vorticity tails are taken into account. Saffman’s (1972) suggestion was to consider the motion of the vortex pair in a stratified atmosphere as a sequence of states, each being described by a motion of vortex through a uniform field. In this case the vorticity tails are not generated at all, but the correspondence of this approximation solution to the starting problem stands unproved. The Spalart’s (1996) suggestion to ignore the vorticity tails can be accepted only on the basis of well-determined mathematical procedure. Moreover, according to (7), even though the disturbed vorticity is localized, density distribution possesses far field asymptotic tails $\mathcal{O}(1/|\mathbf{r}|^{2})$. Hence the integral in the right hand side of (30) is not absolutely convergent and thus the problem remains unsolved.

We define the subdivision procedure in such a way as to extract the localized part of the disturbance without changing its fluid impulse. The latter requirement determines the corresponding mathematical procedure, which in case according to (22) resulted in the following set of evolution equations

$$\frac{dI_{x}}{dt}=-\frac{g}{\rho_{0}}\lim_{R\to\infty}\int_{|\mathbf{r}|\leq R}z\frac{\partial\rho^{I}}{\partial x}dA$$

(31)

$$\frac{dI_{z}}{dt}=\frac{g}{\rho_{0}}\lim_{R\to\infty}\int_{|\mathbf{r}|\leq R}x\frac{\partial\rho^{I}}{\partial x}dA$$

(32)

where the region of integration is a circle of infinite radius. Subsequent calculations reduce (31), (32) to equations (27), (28).