The stability of density stratified shear flows in sea straits of arbitrary cross sections was initiated in Ref. [1]. It was found in Ref. [1] that the stability equation is an extended version of the well known Taylor-Goldstein equation of hydrodynamic stability. In Ref. [2], a systematic derivation of the stability equation was given, and in addition, some general analytical results were derived. It was found that the well known Miles-Howard theorem giving a sufficient condition for stability and Howard’s semicircle theorem giving the instability region were unaffected by the presence of topography. It has been shown in Ref. [3] that in the context of the Taylor-Goldstein problem when the minimum of the Richardson number is greater than one quarter, there exists a countable infinity of neutrally stable eigenfunctions with the eiegnvalues c_n (n = ±1, ±2, · · · ) such that c_−n < U_0min and c_n > U0max for every n, where U₀ is the basic velocity of the shear flow. Moreover, it is shown that the eigenfunction corresponding to c_n has exactly |n|−1 zero crossings in z ∈ [0, D]. For the extended Taylor-Goldstein problem that includes the variable cross sections of the channel through the topography T (z) = (ln b(z))′, where b(z) is the width of the channel at the elevation z, and a prime stands for differentiation with respect to z, the above result of Ref. [3] has been extended in Ref. [2] to the case of a nonsingular and monotonically non-increasing T (z). It may be remarked that this result of Ref. [2] has been used in Ref. [4] in interpreting oceanographic data. For homogeneous flows, it was found by Ref. [2] that a necessary condition for instability is that should change the sign somewhere in the fluid domain. Let z = z_s be such that ′(z_s) = 0, and let U_0s = U₀(z_s). A further necessary condition for instability that was found by Ref. [2] is that (U₀−U_0s) < 0 at least once in the flow domain. To understand the role of topography on the stability of stratified shear flows, it is desirable to get results that depend on . Two more results, namely, (i) upper and lower bounds for the phase velocity of neutral modes and (ii) an estimate for the growth rate of unstable modes, both depending on the above mentioned term, have been given in Ref. [5]. In Ref. [6], the function K(z) was defined by K(z) = and it was proved that for the special case of flows with non-increasing topography, a sufficient condition for stability is Thus, it is clear that in the case of homogeneous shear flows in sea straits, the term (U₀′/b)′ plays an important role in their stability analysis.

However, in the context of density stratified shear flows in sea straits, the role of the vorticity variation term (U₀′/b)′ is not clear as the general analytical results obtained on this problem^{[5, 7]} do not depend on this term.

In this paper, we consider the instability region for density stratified shear flows in sea straits. The semicircular instability region obtained in Ref. [2] proves that the complex-eigenvalues corresponding to the unstable modes lie inside a semicircle in the upper half of the crci-plane whose diameter is within the range of the basic velocity U₀(z). It is seen that this instability region does not depend on the stratification parameter, namely, the Richardson number. The semielliptical instability region obtained in Ref. [5] is an improvement over the semicircular region in the sense that it depends on the minimum Richardson where ρ₀(z) is the basic density, and it lies inside the semicircular region. It may be noted here that these two instability regions do not depend on the vorticity variation term .

In the theory of hydrodynamic stability of shear flows, it is well-known that a detailed study of instability of examples of basic flows is also needed to complement the general analytical results on shear instability^[7]. It may be mentioned here that there are three approaches for demonstrating the instability of examples of shear flows. The first one is by finding a neutral eigensolution and then using a perturbation formula for the complex phase velocity of the unstable mode. This approach has been developed for homogenous flows in sea straits in Ref. [8], but it is still an open problem for density stratified flows in sea straits. The second one is the numerical study of stability equation, and this has been done for two examples of density stratified shear flows in sea straits in Ref. [1]. The third approach is to choose layered flows, for which the basic velocity and density are linear in each layer so that the stability equation can be solved explicitly. In addition to having linear velocity and density profiles, we should also choose constant topography in each layer for the problem of shear instability in sea straits. In this approach in addition to the boundary conditions of the problem, one should obtain and use the interfacial conditions that are valid at the layers of discontinuity of the basic flow variables.

Now, we shall discuss the results of the present paper. In Section 2, we formulate the stability problem, that is, we derive the stability equation, the boundary conditions, and the interfacial conditions. Then, in Section 3, we obtain the semielliptical instability region that depends on the topography and the vorticity variation through the term for monotonic basic velocity profiles. It is shown through figures that this instability region lies inside the semicircular region of Ref. [2] though not necessarily inside the semielliptical region of Ref. [5]. Then, we study the instability of two examples of basic flows. The instability studies are carried out by explicitly solving the stability equation and by finding the eigensolutions that satisfying the boundary and interfacial conditions. The stability problem is reduced to finding the real and complex roots of a quartic equation at the phase speed c. The real part of c, that is c_r, corresponds to the speed of the wave, and the product kc_i of the wave number k and the imaginary part c_i of c gives the growth rate of an unstable mode. Using MATHEMATICA, we plot the c_r versus k and kc_i versus k curves for different values of the parameters of the problem. In the first example, it is found that the quartic equation resulting from the solution of the stability equation gives two pairs of complex eigenvalues of the complex phase velocity. Consequently, we have two unstable modes in this case. However, in the second example, it is found that in some region of the parameter space of the problem, two real roots and only one pair of complex conjugate roots for c are found. In this case, there is only one unstable mode moving in one direction only.

The formulation of the stability problem consists of finding the stability equation, boundary conditions, and jump conditions at the surfaces of discontinuity of the basic flow variables and/ or their derivatives. The stability equation for this problem has already been given in Ref. [2]. However, their derivation is based on a version of Bernoulli’s theorem. In the present section, we derive the stability equation in the usual way of hydrodynamic stability.

Consider a channel with the upper boundary at z = D and the lower boundary at z = 0. In the x-direction, the channel lies in the region −∞ < x < ∞. In the y-direction, the channel lies between the two lateral walls, one at y = y_R(z) and the other at y = y_L(z), and the width function b(z) of the channel is given by b(z) = y_R − y_L (see Fig. 1). The channel contains an incompressible, inviscid, and non-diffusive fluid flow for which the density ρ, the pressure p, and the velocity u = (u, v, w) are governed by the Euler equations. Reference [2] considered the motions which vary in the vertical (z) and longitudinal (x) directions but not in the transverse (y) direction. If L, B, and D are typical scales of the motion in the x-, y-, and z-directions, then the flow is considered gradually varying when B << L and B << D. In such situations, motions for which u, w, ρ, and p are initially y-independent will remain so for all time but no transverse velocity, i. e. , v, is dependent on y. Consequently, as done in Ref. [2], we consider the flows satisfying the condition Then, the Euler equations of fluid mechanics become

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where

Fig. 1 Vertical cross section of channel

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and g is acceleration due to gravity.

The last equation can be rewritten as

where is the topography. We consider the basic flow with the velocity u = (U₀(z), 0, 0), the density ρ = ρ₀(z), and the pressure p=p₀(z). The basic pressure and density are related by the equation Let the disturbed flow variables be given by (U₀ + u, v, w), let the density be given by ρ₀(z) + ρ, and let the pressure be given by p₀(z) + p. Considering infinitesimally small disturbances, the disturbance equations can be linearized to get

Here, we assume the Boussinesq approximation, under which the inertial effect of density variation is neglected. We consider only normal mode disturbances, that is, the disturbance variables are of the form Then, the above equations result in the equations

Eliminating the other variables in terms of , we get the single equation

Under the Boussinesq approximation, the density of the basic flow ρ₀(z) is treated as a constant, i. e. , ρ₀ except when it is multiplied by g. As a consequence, the above equation becomes

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We consider only a statically stable density stratification for which ρ′ ≤ 0 and so N²(z) = N in this case is called the The boundary conditions are

If the vertical displacement η is given by

then

To simplify the mathematical analysis of stability of stratified shear flows, the piecewise linear velocity profiles with piecewise constant density profiles are adopted so that on each subdomain where ρ₀ = constant and U ₀′′= 0, the stability equation can be solved explicitly. In this situation, we need two interfacial conditions in addition to the two boundary conditions stated earlier. These interfacial conditions follow from the continuity of the vertical displacement F(z) and that of the pressure at z = z₀, a layer of discontinuity of one or more of the variables U₀, U₀′ , ρ₀, ρ₀′ , T , and T ′. Let the variables have only a finite jump at z = z₀. Continuity of F(z) at z = z₀ gives

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where the square bracket [·] stands for the jump in the quantity inside it. Continuity of the pressure at z = z₀ gives the condition

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These jump relations are not only dictated by physics but also built into mathematics^[7]. For instance, (8) can be obtained by integrating directly the stability equation across the discontinuity from z₀ − Δz to z₀ + Δz and letting Δz → 0. It may be noticed here that both the stability equation (5) and the second interfacial condition (8) depend on the topography while the boundary conditions (6) and the first interfacial condition (7) are unaffected by the presence of topography.

Now, we shall derive the new semielliptical instability region. Substituting = (U₀ − c)F into (5), we get the equation in F as

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The associated boundary conditions are

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Following the standard procedure of Ref. [9] for obtaining the semicircle theorem, Ref. [2] has shown that

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where

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For the statically stable density stratification that is being considered here, N²(z) > 0. Therefore, the last term of (11) can be dropped to get the semicircle theorem of Ref. [2]. In Ref. [5], an estimate for the last term has been found to obtain a semiellipse theorem. However, we find a different estimate for the last term in (11) to get a different semiellipse theorem. The stability equation (5) can be rewritten as

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Multiplying by b ^* (where * stands for complex conjugation), integrating the resulting equation over (0, D), and using the boundary conditions (6), one gets

The real part of the equation gives

Now, we shall proceed to find the new semielliptical region for density stratified flows. From the above equation, we get an estimate for the first term. For this, we use the semicircle theorem and the relation between and F. Consequently, we have

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Now, since

we have

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As a consequence, we have

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The Cauchy-Schwartz inequality gives

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From (15) and (16), we have

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where

and

Substituting these estimates in (14), we have the relation

provided that (U ₀^′2 )min 6≠0. From this, we get the estimate

where

From this, we get

Now, let us define the Richardson number by Then, we have

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where J_min is the minimum Richardson number.

Replacing the last term of (11) by the right-hand side term of (19), we get

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This gives the following semiellipse theorem.

Theorem 1 For basic flows with monotonic velocity profiles, the instability region is given by

Now, it is desirable to compare this new instability region with the one obtained by Ref. [5]. For this purpose, we choose U₀ = z, N² = 0. 2, and and plot the two instability regions along with the semicircular region of Ref. [2]. It is seen from Fig. 2 that the new semielliptical region lies inside the semicircular region of Ref. [2] but lies outside that of Ref. [5]. It suggests that in the instability problem of density stratified shear flows, the Richardson number plays a major role, but the role of the vorticity variation term is less important. It may be remarked that according to the Richardson number criterion derived in Ref. [2], the basic flow is unstable only if It may be noted here that though the instability regions obtained in the present paper and Ref. [5] depend on J_min, our instability region depends on the vorticity variation term unlike that of Ref. [5].

Fig. 2 Instability regions

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Now, we shall study the instabilities of two examples of basic flows. In each one of these examples, the basic velocity is piecewise linear, and the density and the topography take different constant values in different layers. Before specifying the basic flow variables of these examples, we non-dimensionalize the flow variables so that the stability equation and the boundary and interfacial conditions can be written in terms of these non-dimensional variables.

We non-dimensionalize the variables by choosing a characteristic length L (for example, a characteristic velocity V (for example, V = max|U₀(z)|), and a characteristic density ρ₀. Then, the non-dimensional variables can be defined as

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Then, the stability equation (5) becomes, after dropping *,

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with the boundary conditions

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The interfacial conditions at a layer z = z₀ of discontinuity are

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Adopting the velocity and density profiles of the background flow given by (14) of Ref. [10] to the case of bounded flows, we take the basic velocity as

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and the density of the basic flow as

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where U₀, d, and ρ_c are positive constants, and ε>0 is a small number. Here, the velocity is continuous, while the density takes two constant values in two different layers which is called a two-layer configuration. We choose the topography

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where T₀ is a positive constant. The flow with the velocity given by (26a) is called the bounded shear layer (see Fig. 3). The channel with topography given in (26c) is given in Fig. 4.

Fig. 3 Bounded shear layer

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Fig. 4 Topography

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The stability equation can be solved explicitly, and the eigenfunction satisfying the boundary conditions (23) is given by

where A, B, C, D, E, and F are arbitrary constants.

Now, we shall impose the jump conditions given in (24) and (25) at z = ± d and z = 0. The jump condition of (25) at z = 0 gives

and that at z = ± d gives

Applying the interfacial conditions (24) and (25) at z = d and after simplification, we have the relation

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where X = tanh(kd), and Similarly, applying the interfacial conditions (24) and (25) at z = 0 and after simplification, we have the relation

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where is the non-dimensional Richardson number. Applying the interfacial conditions (24) and (25) at z = −d and after simplification, we have

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Notice that the continuity of U₀ at z = ±d and z = 0 and the continuity of ρ₀ at z = ±d have been used to obtain (27), (28), and (29). (27) to (29) form a system of three linear algebraic equations with three unknowns B, C, and E. For nontrivial solutions, it is necessary that the coefficient matrix should be a singular matrix. Therefore, its determinant should be zero. Straightforward calculation leads to a quartic equation for the dimensionless wave speed c, which can be put in the form of

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where the coefficients are defined by

Now, we shall plot the phase velocity c_r versus the wave number k curves and the growth rate kc_i versus k curves using MATHEMATICA. For this purpose, we take U₀ = 1 as a special case, while the parameters J and T₀ are given different values. We choose four different values for J, namely, J = 0. 039 2, 0. 326 7, 0. 49, and J = 0. 735. It may be noted here that the Richardson number for the sea strait Bab el Mandab is usually less than one quarter, while sometimes it takes values of order one^[2]. We choose four different values of the topography T₀. The value T₀ = 0 corresponds to the absence of any variability in the cross section of the channel. The values T₀ = 0. 1, 1, and 10 correspond to the variability of the cross section of the channel. Figures 5-8 contain the c_r versus k and kc_i versus k curves for the values of the Richardson number and the topography parameter.

Fig. 5 cr versus k (a, c, e, and g) and kci versus k (b, d, f, and h) curves for basic flow with same value of J = 0.039 2 and different values of T0 in first example

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Fig. 6 cr versus k (a, c, e, and g) and kci versus k (b, d, f, and h) curves for basic flow with same value of J = 0.326 7 and different values of T0 in first example

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Fig. 7 cr versus k (a, c, e, and g) and kci versus k (b, d, f, and h) curves for basic flow with same value of J = 0.49 and different values of T0 in first example

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Fig. 8 cr versus k (a, c, e, and g) and kci versus k (b, d, f, and h) curves for basic flow with same value of J = 0.735 and different values of T0 in first example

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In the figures on the left-hand side, c_r versus k curves are plotted, whereas in the right-hand side figures, the kc_i versus k curves are plotted. It is seen that the c_r versus k curves are plotted by both solid and dashed lines. The difference between these two lines is the following. When the solid lines and dashed lines are distinct, they represent the real roots of the quartic dispersion relation, and when the solid lines and the dashed lines overlap, they represent the real part of the complex roots of the dispersion relation. It may be noted here that complex roots occur in conjugate pairs. The complex root with positive imaginary part corresponds to an unstable mode, and its growth rate is plotted in the right-hand side figures. The complex root with negative imaginary part corresponds to a decaying mode, and it is not plotted in the figures. For example, consider Fig. 5(c). In it, we see two distinct lines but each line is an overlap of a solid line and a dashed line. Consequently, this corresponds to real part of two pairs of complex roots of the dispersion relation. Naturally, we have two unstable modes, and we have two growth rate versus wave number curves in the right-hand side figure, namely, Fig. 5(d). We have the similar situation in Figs. 5(e) and 5(f). However, in Fig. 5(g), we have a different picture. In the range 0 < k < 1. 3, we have two solid lines and two dashed lines, and this corresponds to the existence of four real roots of the dispersion relation, and there is no instability in this range of k. However, in the range k > 1. 3, there are two curves. Each one of them is an overlap of a solid line and a dashed line. This corresponds to the existence of two pairs of complex roots. As such, we have two unstable modes in this range, and we have two growth rate versus wave number curves in the right-hand side figure, namely, Fig. 5(h). Similar explanation holds for the other figures also.

It is seen from Figs. 5(b), 5(d), 5(f), and 5(h) that there are two distinct growth rate curves corresponding to two unstable modes. Figure5(b) corresponds to the no topography case, while the remaining three figures, namely, Figs. 5(d), 5(f), and 5(h) correspond to the flow with topography. Though topography does not change the existence of two unstable modes, it is found that an increase in the value of the topography results in the decrease of the growth rate of the two unstable modes. From Fig. 5(a), it is seen that c_r = 0 for all values of k. Thus, the unstable modes are Kelvin-Helmholtz (KH) modes. From Figs. 5(c), 5(e), and 5(g), it is seen that c_r≠0 for all values of k. Thus, in the presence of topography, that is, when T₀ ≠ 0, the modes become Holmboe modes.

Now, it is desirable to explain the difference between the KH modes and the Holmboe modes in density stratified shear flow instability problems. As is well known (see, for example, Ref. [11]), the two commonly studied cases of instability in density stratified flows of incompressible fluid flows are the KH instability and the Holmboe instability. A KH instability corresponds to an unstable mode with zero phase velocity, and a KH billow grows to a large amplitude which is limited only by the formation of secondary and convective instabilities. However, the Holmboe instability has a finite speed of propagation relative to the mean shear and typically smaller growth rate than the KH modes. However, they are present for arbitrarily large values of the stratification parameter, making them candidates to explain certain physical phenomena. Moreover, the Holmboe instability is understood by the wave interaction mechanism of two waves propagating in opposite directions (see, for example, Ref. [12]). Also, it is known that there is transition from the KH instability to the Holmboe instability in many shear flows (see, for example, Ref. [11]). It is interesting to note that only Holmboe modes are found in shear instability of the bounded shear layer in the presence of topography.

From Fig. 6(a), it is seen that the modes are Holmboe modes, but this is a consequence of a higher value of the stratification parameter J, namely, J = 0. 326 7. In all cases of Fig. 6, we see two unstable modes in each case. This fact is not changed by an increase in the topography T₀. Moreover, an increase in the value of T₀ does not reduce the growth rate as in Fig. 5. In fact, an increase in the growth rate of one unstable mode is found to correspond to an increase in T₀. In Fig. 7, we plot the phase speed c_r versus the wave number k and the growth rate kc_i versus the wave number k for a higher value of J, compared with the case given in Fig. 6, namely, J = 0. 49. It is seen that the growth rate of the two unstable modes is reduced compared with the situation in Fig. 6, and the growth rate of the second unstable mode is much reduced when the topography parameter takes a larger value. Moreover, there is a shift in the spectral width, that is, the range of k corresponding to unstable modes, to the left when T₀ = 10. In Fig. 8, we plot the c_r versus k and kc_i versus k curves when J = 0. 735 which is a larger value than those considered in the previous cases. Here, it is seen that there is a decrease in the growth rate of both unstable modes, and the spectral width of the second unstable mode lies inside the spectral range of the first unstable mode in contrast to the situation in Fig. 7.

As a second example, we consider a basic flow which has the same velocity and topography as in the first example but with a different density, namely,

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where ρ_c is a positive constant, and ε > 0 is a small number. As we shall see below, this three-layered configuration leads to a qualitatively different result regarding the instability of the basic flow. The stability equation can be solved explicitly, and the eigenfunction satisfying the boundary conditions (23) is given by

where A, B, C, and D are arbitrary constants.

Applying the interfacial conditions (24) and (25) at z = d and after simplification, we have

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where the constants k₀, X, Y, and J are the same as those in the earlier example. Applying the interfacial conditions (24) and (25) at z = − d, we have

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Then, imposing the conditions (24) and (25), the dispersion relation is obtained as a quartic equation for the (complex) phase speed c. We write the quartic equation as

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where the coefficients a₀ to a₄ are given by

Using MATHEMATICA, one can find the four roots of (34). Here, we choose U₀ = 1 and four different values of J, namely, J = 0. 039 2, 0. 326 7, 0. 49, and J = 0. 735, and four different values of T₀, namely, T₀ = 0. 1, 1, 10. In Figs. 9-11, we plot c_r versus the wave number k and the kc_i versus the wave number k curves. From Figs. 9-11, it is seen that there is only one unstable mode for each value of k. This is in contrast to the previous example where there are two unstable modes for each value of k. This difference in the number of unstable modes should be attributed to the differences among the densities of the basic flows as the basic flow velocity and topography are the same in both examples. An interesting point to be noted here is that the existence of only one unstable mode is attributed to the non-Boussinesq terms by Ref. [10] in the stability analysis of density stratified flows. However, in our case, we consider only a Boussinesq fluid^[2] but in a bounded domain with variable cross sections.

Fig. 9 cr versus k (a, c, e, and g) and kci versus k (b, d, f, and h) curves for basic flow with same value of J = 0.039 2 and different values of T0 in second example

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Fig. 10 cr versus k (a, c, e, and g) and kci versus k (b, d, f, and h) curves for basic flow with same value of J = 0.326 7 and different values of T0 in second example

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Fig. 11 cr versus k (a, c, e, and g) and kci versus k (b, d, f, and h) curves for basic flow with same value of J = 0.49 and different values of T0 in second example

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There are two differences between the stability problems considered in Ref. [10] and that considered in this paper. The first one is the presence of topography and the second one is the presence of lower and upper boundaries of the channel. Thus, the existence of only one unstable mode corresponds to these two factors, namely, topography and boundaries, rather than the non-Boussinesq terms as in the work of Ref. [10].

As we have stated earlier, the Holmboe instability is explained in terms of the wave interaction mechanism, where two oppositely propagating modes enhance the amplitude of each other. This interpretation is possible only when there are two unstable modes. However, in the present example of shear flow with three different values of density, we find only one unstable mode. As such, the wave interaction mechanism cannot be applied to understand this type of Holmboe instability, and a new interpretation has to be found.

In Fig. 12, we plot the phase velocity c versus the wave number k for the value of J = 0. 735 and for four different values of T₀. It is found, in this case, that all four roots of the quartic equation (34) are real. This means that the basic flow is stable in this case. This should be contrasted with the three previous cases of this example. It means that a sufficiently large value of J stabilizes the flow.

Fig. 12 cr versus k curves for basic flow with same value of J = 0.735 and different values of T0 in second example

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In this paper, we consider the stability problem of inviscid shear flows of an incompressible but density stratified fluid in channels with variable cross sections. Though the stability equation has already been derived in Ref. [2], we present the derivation of the stability equation by following the standard procedure of normal mode analysis of hydrodynamic stability. The upper and lower boundary conditions are the same as those given in Ref. [2]. However, we derive the two interfacial conditions that are satisfied by the disturbance variables at a layer of discontinuity of the variables U₀, U₀′ , ρ₀, ρ₀′ , T , or T ′. We obtain a semielliptical instability region for monotonic profiles that depends on the vorticity variation through the term and also on the Richardson number J. It is seen from the figures that this instability region lies outside the semielliptical instability region of Ref. [5], though it definitely lies inside the semicircular instability region of Ref. [2]. Then, we study the instability of two examples of basic flows. This instability study is carried out by explicitly solving the stability equation and by finding the eigensolutions that satisfy the boundary and interfacial conditions. In both examples, the basic velocity and the topography are the same. The basic velocity profile is that of the bounded shear layer. In the middle layer where the velocity varies linearly with z, we take T₀ = 0, that is, the channel has a constant cross section, and in the bottom and top layers where the velocity is a non-zero constant, we take T₀ to be a positive constant. However, the density of the basic flow is different in the two examples. In the first example, the density takes two different constant values in the two layers, while in the second example, the density takes three different constant values. The role of the density differences on the instability of the basic flow is measured by the Richardson number. The stability problem is reduced to finding the real and complex roots of a quartic equation at the phase speed c. The real part of c, that is c_r, corresponds to the speed of the wave, and the product kc_i of the wave number k and the imaginary part c_i of c gives the growth rate of an unstable mode. Using MATHEMATICA, we plot the c_r versus k and kc_i versus k curves for different values of the parameters of the problem. Though we choose U₀ to be one in all cases, we choose four different values of J and T . For different values of the topography parameter T₀ and the stratification parameter J, we plot the c_r versus k and the growth rate kc_i versus k curves. It is found that there are two unstable modes in a range of the wave number in the first example, whereas in the second example, only one unstable mode is found to exist. This result is similar to that of Ref. [10], where it is found that a three-layered flow has only one unstable mode for a fixed wave number where the non-Boussinesq terms are included in the analysis, where if the fluid is taken to be a Boussinesq fluid, two unstable modes exist with a given wave number. The combined role of the boundedness of the channel and the variable cross section of the channel in the stability analysis of an inviscid Boussinesq fluid gives the result of the existence of only unstable mode whereas in the case of unbounded shear flow instability problem only non-Boussinesq terms give such a result.

Another difference between the two examples of basic flows is seen from Fig. 12. It is seen here that the basic flow in the second example is stable when the Richardson number J = 0. 735. For the same values of the Richardson number, it is seen from Fig. 8 that the first example of basic flow is unstable.

It may be remarked here that the stability analysis of density stratified shear flows in sea straits was initiated in Ref. [1] with a motivation of understanding the propagation of internal gravity waves in the sea strait Bab el Mandab that connects the Red Sea and the Arabian Sea. As a three-layer exchange flow has been observed in Bab el Mandab during summer months^[13], it is expected that the instability studies of multi-layer flows done in the present paper will give insight into the propagation of internal gravity waves in sea straits.

Acknowledgements We are thankful to the referees for their comments which helped us to improve the presentation of the paper. The research work of V. R. REDDY is supported by University Grants Commission-Junior Research Fellowship, Government of India.