Transition from Convective to Absolute Instability in a Porous Layer ...

Transp Porous Med (2012) 92:597–617 DOI 10.1007/s11242-011-9923-6

Transition from Convective to Absolute Instability in a Porous Layer with Either Horizontal or Vertical Solutal and Inclined Thermal Gradients, and Horizontal Throughflow Emilie Diaz · Leonid Brevdo

Received: 12 September 2011 / Accepted: 3 December 2011 / Published online: 22 December 2011 © Springer Science+Business Media B.V. 2011

Abstract Recently, in Diaz and Brevdo (J Fluid Mech 681: 567–596, 2011), further in the text referred to as D&B, we found an absolute/convective instability dichotomy at the onset of convection in a flow in a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. The control parameter in D&B triggering the destabilization is the vertical thermal Rayleigh number, Rv . In this article, we treat the parameter cases considered in D&B in which the onset of convection has the character of convective instability and occurs through longitudinal modes. By increasing the vertical thermal Rayleigh number starting from its critical value, Rvc , we determine the value Rvt of Rv at which the transition from convective to absolute instability takes place and compute the physical characteristics of the emerging absolutely unstable wave packet. In some cases, the value of the transitional vertical thermal Rayleigh number, Rvt , is only slightly greater than the critical value, Rvc , meaning that at the onset of convection the base convectively unstable state can be viewed as marginally absolutely unstable. However, in several cases considered, the value of Rvt is significantly greater than the critical value, Rvc , implying that the base state is not marginally but essentially absolutely stable at the point of destabilization. Keywords Porous media · Thermohaline convection · Convective to absolute instability transition

1 Introduction The problem of convection driven by the temperature and/or salinity gradients in a fluid flow is a classical subject of fluid mechanics. Convection flows in porous media attracted considerable attention in the literature following the publication of the seminal work of Weber

E. Diaz · L. Brevdo (B) Institut de Mécanique des Fluides et des Solides, Université de Strasbourg/CNRS, 2 rue Boussingault, 67000 Strasbourg, France e-mail: [email protected]

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(a)

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(b)

Fig. 1 a Absolute instability; b convective instability

(1974). The interest to the problem of coupled heat and mass transfer in a porous medium is stimulated by a great number of diverse applications such as the flow dynamics in continental geothermal reservoirs, the spread of radioactive material that has leaked, the convection then being driven by the dissipation of heat sources, the dispersion of chemical contaminants through water-saturated soil and the convective motion of air in a layer of snow, see Bear (1972), Nield et al. (1993), Manole et al. (1994), Nield (1991, 1994, 1998), Nield and Bejan (2006), Straughan (2008) and Straughan and Walker (1996) for reviews and references. In the theoretical studies of the onset of convection in a porous medium, a model of an extended saturated porous layer proved to be valuable owing to its simplicity that facilitates analytic treatments whereas the assumption is made that the main features of convection found in this model reflect the convection process found in the applications. For studying convection that emerges as a result of the linear destabilization of the model one has to first of all find the value of the control parameter,—in our case, it is the vertical thermal Rayleigh number, Rv ,—at which the destabilization occurs. The determination of the critical value of Rv is done by analysing stability of the monochromatic waves, i.e. the normal modes, in the model. The normal-mode approach whilst being an indispensable part of the analysis of convection does not provide, however, an information concerning the features of the realistic localized non-sinusoidal perturbations that are characteristic for the emergence of convection in open flows. In order to study the dynamics of unstable localized perturbations at the emergence and during the development of convection, one has to treat a corresponding initial-value problem formulated for the governing linearized equations of motion. The spatio-temporal evolution of such perturbations is analysed by using the formalism of absolute and convective instabilities. The first ideas of the concept of absolute and convective instabilities were put forward in the late 1940s and early 1950s by Landau and Lifshitz (1953) and Twiss (1951). Since then, the problem of absolute and convective instabilities attracted a considerable attention in the plasma physics literature and in the literature on fluid flows. The concept of absolute and convective instabilities is based on two different evolution scenarios of spatially localized disturbances in the unstable flow. In the first scenario, every initially localized small perturbation causes an unbounded linear growth in the entire flow domain thus destroying eventually the base flow throughout through non-linear effects. In such a case, the flow is said to be absolutely unstable. In the second alternative scenario, any initially localized small perturbation triggers an evolution of a growing wave packet that propagates away from the location of the initial perturbation leaving behind at every fixed point in the flow domain an unperturbed base state. This is the case of convective instability. Figure 1 presents a schematic one-dimensional illustration of the spatio-temporal evolution of the amplitude of an unstable disturbance in the absolutely unstable case and in the convectively unstable, but absolutely stable case. The distinction between absolute and convective instabilities is physically significant. An absolutely unstable base state cannot be considered as a physical end state in any portion of

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the flow domain. An absolutely stable, but convectively unstable flow despite being unstable can be viewed as representing a physical end state in a portion of the flow domain whose extent depends on the growth characteristics of the convectively unstable wave packets in the flow. Also, it is suggested that a fully developed non-linear state that emerged from an absolutely unstable base state would posses fundamental feature qualitatively different from those of a non-linear state emerging in an absolutely stable, but convectively unstable flow. The formalism for spatially uniform two-dimensional flows is developed in its modern form in the plasma physics literature (see Briggs 1964; Bers 1973). The formalism was extended to three-dimensional homogeneous flows by Brevdo (1991). In the study of Brevdo (2009) and Brevdo and Ruderman (2009a,b), the character of localized perturbations at the onset of convection in a model of flow in a saturated porous layer with inclined temperature gradient and vertical throughflow was treated by using the methods of absolute and convective instabilities. The analysis in those works was extended in D&B to a model of a saturated porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. In those works, depending on the physical parameters involved, an absolute/convective instability dichotomy was found in the set of the exact solutions of the equations of motion. In most of the cases treated by D&B, the onset of convection occurs through the longitudinal modes, i.e. the destabilization of the model has a two-dimensional character. The absolute/convective instability dichotomy was fund both in the cases of a two-dimensional as well as in the cases of a three-dimensional onset of convection. In this article, we consider the cases studied by D&B in which a longitudinal mode if favoured for the onset of convection and the destabilization of the model has a character of convective instability. In each of these cases, we compute the value of the thermal vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs and treat the characteristics of the emerging absolutely unstable wave packet. The computational procedure is based on discretizing the homogeneous boundary-value stability problem by using a pseudo-spectral Chebyshev collocation method and solving the resulting generalized algebraic eigenvalue problem. Hereby, for the values of Rv that increase starting with the critical value, Rvc , wavenumber, l, treated as an eigenvalue is computed as a function of frequency, ω, and the Briggs (1964) collision criterion is applied for computing the transitional value, Rvt . The discretization procedure is described in D&B. In order to estimate the separation of the transitional state to absolute instability from the critical state, the relative percentage deviation, δ = 100% × (Rvt − Rvc )/Rvc , of the vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, from the critical vertical Rayleigh number, Rvc , was used. For several cases of parameter combination, we found that δ > 5% and inferred that in these cases the critical state is well separated from the transitional state, that is to say, at the point of destabilization the marginally unstable base state is genuinely convectively unstable, but absolutely stable. Since our study treats exact solutions of the governing equations of motion, its results pose a challenge and provide an incentive for future research both theoretical as well as experimental. Specifically, it would be very interesting to investigate by using direct numerical simulations as well as by experimental means whether the evolving non-linear regime in the case of absolute instability at the point of linear destabilization would possess qualitative features different from those in the case of convective instability, but absolute stability. As a matter of fact, exploring the absolute/convective instability dichotomy in the set of exact solutions of the equations of motion in the model, with a possible future exploration in a non-linear study of the implications of the dichotomy, is the main motivation for this paper.

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The article is organized as follows. In Sect. 2, we sketch the model, give the non-dimensionalized governing equations and present the steady-state base solution. Section 3 gives a description of an initial-value problem for small disturbances and the solution of the problem in the form of an inverse Laplace-Fourier integral. A derivation of the solution is presented in the Appendix. In Sect. 4, the procedure for treating the transition from convective to absolute instability is described. Section 5 presents the numerical results and in Sect. 6 conclusions are made.

2 Formulation and Base Solution We treat the transition from convective to absolute instability in a model studied by D&B which is similar to the model considered by Manole et al. (1994). A detailed description of the model is given in those papers. By using the setting, notations and non-dimensionalization in D&B, we give here a concise description of the model. 2.1 Formulation A flow in a horizontal homogeneous extended saturated porous layer bounded by two horizontal impermeable, isothermal and isosolutal solid surfaces and driven by either horizontal or vertical solutal and inclined thermal gradients is considered. The height of the layer is denoted by H . The Cartesian coordinates are chosen with the origin at the mid-height of the layer, with the z ∗ -axis pointing vertically upwards. The horizontal components of the thermal and solutal gradients, βT and βC , respectively, are supposed to be pointing in the same direction opposite to the positive direction of the x ∗ -axis. A constant horizontal net flow of the magnitude q parallel to the x ∗ -axis is assumed. Here and further in the text, the superscript asterisk, when used, denotes dimensional quantities. The vertical temperature difference across the boundaries is T , and the vertical concentration difference is C. It is assumed that the flow in the porous medium is governed by the Darcy law, and the Oberbeck-Boussinesq approximation is valid, see Bear (1972). A definition sketch of the physical situation considered is presented in Fig. 2. We treat the transition from convective to absolute instability assuming two-dimensional longitudinal disturbances in the cases when such disturbances are favoured for the onset of convection, see D&B. For such disturbances, the perturbation flow field does not vary in the x-direction.

Fig. 2 Definition sketch for the problem

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The governing non-dimensionalized equations for the flow are ∇ · v = 0, 1 ∇P + v − T + C k = 0, Le ∂T + v · ∇T = ∇ 2 T, ∂t 1 2 φ ∂C + v · ∇C = ∇ C, a ∂t Le −∞ < x, y < ∞, − 1/2 < z < 1/2, t > 0,

(1)

and the non-dimensionalized boundary conditions are w = 0, T = ∓Rv /2 − Rh x, C = ∓Sv /2 − Sh x at z = ±1/2,

(2)

where ∇ is the gradient operator, k = (0, 0, 1)T , v = (u, v, w)T , t, P, T and C denote the Darcy velocity vector, time, pressure, temperature and concentration, respectively. Equation (1) and the boundary conditions (2) are obtained from the dimensional equations and, correspondingly, dimensional boundary conditions by using the non-dimensionalization described in D&B. The non-dimensional parameters appearing in (1) and (2) are the Lewis number, Le, the vertical thermal Rayleigh number, Rv , the horizontal thermal Rayleigh number, Rh , the vertical solutal Rayleigh number, Sv , and the horizontal solutal Rayleigh number, Sh , given by αm ρ0 gγT K H T ρ0 gγT K H 2 βT , Rv = , Rh = , Dm μαm μαm 2 ρ0 gγC K H C ρ0 gγC K H βC Sv = and Sh = , μDm μDm

Le =

(3)

with

αm =

km (ρcp )f

and a =

(ρc)m . (ρcp )f

(4)

In (3) and (4), g is the gravity acceleration, μ, ρ and c are viscosity, density and specific heat, respectively, K denotes the permeability and φ is the porosity of the porous medium, km and Dm are the thermal conductivity and, correspondingly, solutal diffusivity of the medium, γT and γC denote the thermal, correspondingly, solutal, expansion coefficients of the fluid, and the subscripts m, f and 0 refer to the porous medium, the fluid and the uniform reference state, respectively. The non-dimensional horizontal net flow is given by Q h = q H/αm ,

(5)

which is the horizontal Péclet number, see D&B. 2.2 Base solution Problem (1), (2) admits a steady-state unidirectional flow solution of the form u = Us (z), v = 0, w = 0, P = Ps (x, y, z), T = Ts (z) − Rh x, C = Cs (z) − Sh x,

(6)

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that satisfies the horizontal net flow condition 1/2 Us (z)dz = Q h .

(7)

−1/2

The solution is a partial case of the solution given by Manole et al. (1994); the functions Us , Ts and Cs can be expressed as Sh z + Qh, Us = R h + Le Ts = −Rv z + Cs = −Sv z +

aT Q h Rh (z − 4z 3 ) + (1 − 4z 2 ), 24 8

(8)

aC LeQ h Sh (z − 4z 3 ) + (1 − 4z 2 ), 24 8

where aT = Rh2 +

Rh Sh and aC = Sh2 + Rh Sh Le. Le

(9)

3 Initial-Value Problem For deriving an initial-value problem for a two-dimensional longitudinal, that is an x-independent, perturbation flow in the model we perturb the base state by writing v = (Us , 0, 0)T +v , with v = (0, v , w )T , T = Ts − Rh x + θ , C = Cs − Sh x + c and P = Ps + p , where v , w , θ , c and p are small, substitute the perturbed quantities into (1) and (2), and neglect the products of the perturbation terms to obtain ∂w ∂v + = 0, ∂y ∂z ∂p + v = m y , ∂y ∂ p 1 + w − θ + c = mz, ∂z Le dTs ∂θ + w = ∇ 2 θ + e, ∂t dz φ ∂c 1 2 dCs + w = ∇ c + s, a ∂t dz Le −1/2 < z < 1/2, − ∞ < y < ∞, t > 0, (v , w , p , θ , c )|t=0 = (v0 , w0 , p0 , θ0 , c0 )(z, y), (v , w , p , θ , c ) = (v1 , w1 , p1 , θ1 , c1 )(y, t) at z = −1/2, (v , w , p , θ , c ) = (v2 , w2 , p2 , θ2 , c2 )(y, t) at z = 1/2,

(10)

where now ∇ = (∂/∂ y, ∂/∂z)T . In obtaining (10), we supposed that the sources of momentum m y and m z , energy e and salinity s, are present in the model. For physical reasons, it is assumed that the functions m y (z, y, t), m z (z, y, t), e(z, y, t), s(z, y, t), v1,2 (y, t), w1,2 (y, t), p1,2 (y, t), θ1,2 (y, t) and c1,2 (y, t) have finite support in y and t, and the functions v0 (z, y), w0 (z, y), p0 (z, y), c0 (z, y) and θ0 (z, y) have finite support in y. Further in the text, the prime denoting the perturbation quantities will be omitted for convenience.

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Elimination of v and p from the first three equations in (10) gives a system of three equations for w, θ and c that we write as 2 ∂m y ∂ ∂ 2θ ∂2 1 ∂ 2c ∂ ∂m z , w − + − = − Le ∂ y 2 ∂y ∂y ∂z ∂ y2 ∂z 2 ∂ y2 ∂ dTs ∂2 ∂2 (11) w+ − 2 − 2 θ = e, dz ∂t ∂y ∂z 2 φ ∂ 1 ∂ ∂2 dCs c = s. w+ − + dz a ∂t Le ∂ y 2 ∂z 2 The initial-value problem for the system (11) can be solved by using the combined FourierLaplace transform. The procedure is described in the Appendix. By using (36), the vertical component of the perturbation velocity, w = w(z, y, t), of the solution of the initial-value problem (10) can be expressed as 1 w(z, y, t) = 4π 2

iσ+∞ ∞

iσ −∞ −∞

T (z, l, ω) i(ly − ωt) e dldω, D(l, ω)

(12)

where D(l, ω) is the dispersion-relation function of the problem as it is defined in the normal-mode approach, see Drazin and Reid (1989), and σ is a real number that is greater than the maximum growth rate of the normal modes, σm , i.e. σ > σm = max {Im ω | D(l, ω) = 0, Im l = 0}.

(13)

The integration contour in the inverse Laplace integral in (12), B(σ ) = {Im ω = σ, −∞ < Re ω < ∞},

(14)

is called the Bromwich contour.

4 Treatment of the transition from convective to absolute instability In the study of Brevdo (2009), a procedure for determining the nature of instability—absolute or convective—at the point of destabilization is described. The procedure is based on computing the group velocity of the unstable wave packet of a marginally unstable state. In the present two-dimensional case, the group velocity is given by Vg =

dωr (lc ) , dl

(15)

where lc is the critical wavenumber, meaning that ωi (lc ) is marginally greater than zero. Here and further in the text, the subscript r denotes the real part and the subscript i denotes the imaginary part of a complex number. The procedure was used in D&B for determining the nature of destabilization of the model, for a set of parameter combinations. In this article, we treat the convectively unstable, but absolutely stable cases treated in D&B in which the onset of convection occurs through longitudinal modes. In each such a case, the value of the transitional vertical thermal Rayleigh number, Rvt , at which the transition from convective to absolute instability takes place will be found by iterations as follows.

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4.1 Iteration Procedure Starting from the critical value of the vertical thermal Rayleigh number, Rvc , we examine (n) the nature of destabilization for a sequence of incremental values of Rv = Rvc + δ1 n, n = (n) (1) 1, 2, . . ., where δ1 > 0 is small. Step by step, for each of Rv , starting with Rv , it is verified by using the Briggs (1964) collision criterion as explained below whether the state is (nabs) convectively unstable, but absolutely stable. Once the value of Rv is reached for which the state is absolutely unstable, starting from this value, for a sequence of decremental values (m) (nabs) Rv = Rv − δ2 m, m = 1, 2, . . ., where δ2 = δ1 /2, the state is treated on absolute/con(mcnv) is reached at which the state is absolutely stable, vective instabilities until the value Rv (mcnv) but convectively unstable. Then starting with this Rv the value of Rv is increased incrementally with the step δ3 = δ2 /2 and in this fashion the iterations proceed. The iterations (nabs) (mcnv) stop when the difference between the corresponding Rv and Rv becomes smaller (nabs) −4 than 10 Rv ; the value of Rv is then taken to be an approximation of the transitional value, Rvt . 4.2 Briggs (1964) Collision Criterion We give here a concise description of the implementation of the Briggs (1964) collision criterion for treating absolute/convective instabilities. A detailed description of the criterion can be found in a recent paper of Brevdo and Ruderman (2009b). One way to implement the Briggs (1964) collision criterion is to compute the images of the Bromwich contour, B(σ ), in the complex l-plane under the transformations l = ln (ω), n = 1, 2, . . ., where l = ln (ω) are all the solutions in l of D(l, ω) = 0. The movement of the images is followed when the Bromwich contour, B(σ ), is lowered from above the line {Im ω = σm } down towards the real ω-axis, i.e. σ → 0+ . The colliding roots contributing to instability are then identified as the colliding points of the images originating in this movement on opposite sides of the real l-axis, when σ varies starting from σ = σm down towards 0. An unstable state is convectively unstable, but absolutely stable when no collision of the images takes place for σ > 0. On the other hand, a state is absolutely unstable when there is a collision of the images for σ > 0. When a collision of the images occurs for σ = 0, an unstable state is marginally absolutely unstable. In our treatment, this latter case is the point of transition from convective to absolute instability, as Rv increases through its value corresponding to this point. We illustrate the movement and the collision of the images of the Bromwich contour, B(σ ), under the transformations l = ln (ω), n = 1, 2, . . ., in the complex l-plane, when σ → 0+ , by presenting in Figs. 3, 4, 5 and 6 the images for the parameter case Rvt = 61.929, Rh = 20, Sv = 0, Sh = 10 and Q h = 0. In this case, the maximum growth rate of the normal modes is σm = 0.6865. In Fig. 3, the images are plotted for σ = 2. Since 2 > σm , none of the images in this case has common point with the real l-axis and, consequently, the images laying on opposite sides of the real l-axis are identified. Figure 4 is for σ = 0.5, i.e. the Bromwich contour is lowered and the movement of two pairs of images coming from below and from above the real l-axis identified in Fig. 3 is seen. In Fig. 5, for σ = 0, these images collide at two points: (lrt , lit ) = ±(2.95, −0.67). At the point of collision, the threshold real frequency is ωrt = 6.05. Two other collisions of images, for σ ≥ 0, not shown in theses figures, that satisfy the Briggs collision criterion take place, for the value of Rv = 61.929. At these latter collisions, it holds that (lrt , lit ) = ±(2.95, 0.67) and ωrt = −6.05. No other collisions of the images that satisfy the Briggs collision criterion take place, for the value of Rv = 61.929

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Fig. 3 Images of the Bromwich contour lying on opposite sides of the real k-axis, for σ = 2; the parameters are Rv = 61.929, Rh = 20, Sv = 0, Sh = 10 and Q h = 0

Fig. 4 Position of the images of the Bromwich contour originating on opposite sides of the real l-axis, for σ = 0.5; the parameters are as in Fig. 3

implying that, at this value, the transition from convective to absolute instability occurs in this case, that is to say Rvt = 61.929. Figure 6, for σ = −0.05, shows the images of the Bromwich contour following the collision. The colliding images depart from one another in a typical movement producing the change of pattern as compared to the pattern of the images arriving to the collision shown in Fig. 4. In the above example, at the point of transition, the emerging marginally absolutely unstable localized disturbance is a sum of four terms. Two terms with (lrt , lit ) = (2.95, −0.67) and (lrt , lit ) = (−2.95, 0.67), respectively, have the real frequency ωrt = 6.05; two other terms with (lrt , lit ) = (2.95, 0.67) and (lrt , lit ) = (−2.95, −0.67), respectively, have the real frequency ωrt = −6.05. A similar composition of the marginally absolutely unstable localized disturbance as a sum of four terms was found in all the cases considered.

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Fig. 5 Collision of the images of the Bromwich contour, for σ = 0.0, implying that for the value of Rv = 61.929 the base state is marginally absolutely unstable; the parameters are as in Fig. 3

Fig. 6 Images of the Bromwich contour following the collision, for σ = −0.05. There is a typical change of pattern of the images that took part in the collision; the parameters are as in Fig. 3

5 Numerical results for the convective to absolute instability transition In this section, we present the computation results for the transition from convective to absolute instability for the convectively unstable, but absolutely stable cases treated in D&B. In all the cases considered in that paper the Lewis number, Le, is kept equal to 10 and φ/a = 1. 5.1 Cases with Sv = 0 and Q h = 0 In Tables 1, 2, 3 and 4, the results for the cases with Sv = 0 and Q h = 0 are given. In Table 1, the values of the transitional vertical thermal Rayleigh number, Rvt , and, in the parentheses, the values of the critical vertical thermal Rayleigh number, Rvc , are presented, and Table 2 shows the values of the relative percentage deviation, δ, of Rvt from Rvc . From the results

123

53.283 (49.917)

50

89.507 (83.268)

77.750 (74.081)

66.754 (66.658)

30

110.70 (105.03)

97.262 (93.214)

84.031 (83.830)

Rh = 40

Here and in Tables 5 and 9, in the parentheses, the values of the critical vertical thermal Rayleigh number, Rvc , are given

98.080 (88.732)

70.914 (65.663)

55.815 (52.620)

44.705 (44.446)

20

74.931 (66.904)

54.038 (54.031) 61.929 (59.218)

50.357 (48.999)

20

Sh = 1

10

10

Rh = 0

119.35 (115.98)

105.34 (105.07)

50

141.69 (140.94)

129.63 (129.41)

60

Table 1 Values of the thermal vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, for various values of Rh and Sh , for Sv = 0 and Q h = 0

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Table 2 Values of the relative percentage deviation, δ = 100% × (Rvt − Rvc )/Rvc , of the vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, from the critical vertical Rayleigh number, Rvc , for various values of Rh and Sh , for Sv = 0 and Q h = 0 Rh = 0

10

20

Sh = 1 10

2.8

20

0.6

50

6.7

6.1 12

30

40

50

60

0.01

0.1

0.2

0.3

0.2

4.6

4.9

4.3

2.9

0.5

8.0

7.5

5.4

10

Table 3 Values of the threshold wave number, lt = (lrt , lit ), at the transition from convective to absolute instability, for various values of Rh and Sh , for Sv = 0 and Q h = 0 Rh = 0

10

20

Sh = 1

Rh = 40

30

50

60

±(3.11, −0.01) ±(3.06, −0.10) ±(3.09, −0.17) ±(3.21, −0.22) ±(3.54, −0.25) ±(2.95, −0.47) ±(2.95, −0.67) ±(2.99, −0.80) ±(3.13, −0.90) ±(3.47, −0.93) ±(4.05, −0.44)

10 20

±(3.00, −0.18) ±(2.92, −0.72) ±(2.94, −0.93) ±(3.04, −1.04) ±(3.30, −1.06)

50

±(2.92, −0.74) ±(2.92, −1.10) ±(3.07, −1.17)

Table 4 Values of the threshold oscillatory frequency, ωrt = ωr (lt ), at the transition from convective to absolute instability, for various values of Rh and Sh , for Sv = 0 and Q h = 0 Rh = 0

10

Sh = 1 10

4.23

20

30

40

50

60

0.75

1.60

2.12

2.57

3.01

6.05

7.41

8.56

9.60

10.13

10.65

12.06

20

2.14

6.51

8.87

50

6.66

11.56

14.54

presented in Table 2 it follows that, for a fixed value of Rh , the relative percentage deviation, δ, increases with Sh . For small values of δ, say for δ ≤ 1%, one may view the critical state as being marginally absolutely unstable, and it would be interesting to study experimentally the onset of convection for the parameter combinations predicting theoretically convectively unstable states that marginally deviate from being absolutely unstable. On the other hand, one can assume that for δ ≥ 5% the critical state is effectively convectively unstable, but absolutely stable. Given these assumptions, we see that amongst the parameter cases presented in Tables 1, 2, 3 and 4 the critical cases with Sh = 1 are marginally absolutely unstable and all those with Sh = 20, except for the case with Rh = 0, and with Sh = 50 are effectively convectively unstable, but absolutely stable. The results in Table 3 for the transitional wavenumber, lt , show that, at a fixed value of Rh , the positive local spatial amplification rate of the wave packet, lit > 0, at the convective to absolute instability transition increases with Sh . The sign ± appears in this table for all the values of lt because at the point of transition two collisions of the images of the Bromwich contours take place for (lrt , lit ) and for (−lrt , −lit ). The transitional frequency, ωrt , at these collisions is presented in Table 4. There are also two other collisions at the points ±(lrt , −lit ), with the transitional frequency equal to (−ωrt ). These latter collisions are not tabulated. A similar situation was found in all the cases of convective to absolute instability transition treated as mentioned in the discussion of

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the example in the last paragraph of §4.2. The transitional frequency, ωrt , presented in Table 4 is an increasing function of both Rh and Sh . 5.2 Cases with Sh = 0 and Q h = 0 Tables 5, 6, 7 and 8 present the computation results for the cases with Sh = 0 and Q h = 0. The relative percentage deviation, δ, shown in Table 6 decreases with both Rh and Sv ; it is rather low in all the cases and is smaller than 1 for Sv = −10. Hence, for Sv = −10, the flow at the onset of convection can be viewed as marginally absolutely unstable. The threshold local waviness, |lrt |, presented in Table 7 increases slightly with both Rh and Sv , the positive local spatial amplification rate, lit > 0, increases slightly with Rh and decreases considerably with Sv . The local oscillatory frequency shown in Table 8 increases slightly with Rh and decreases somewhat with Sv . 5.3 Cases with Sh = 0 and Q h = 5 The computation results for the cases with Sh = 0 and a non-zero horizontal throughflow, Q h = 5, are shown in Tables 9, 10, 11 and 12. The qualitative characteristics of the threshold states in these cases are similar to those of the cases with Sh = 0 and Q h = 0. The percentage deviation, δ, shown in Table 10 is smaller than 0.6, for two cases: with Sv = −20 and Rh = 50, 60, and all the cases with Sv = −10, implying that in these cases at the onset of convection the flow can be viewed as being marginally absolutely unstable. The local threshold waviness, |lrt |, presented in Table 11 shows the tendency of a somewhat stronger increase as a function of Rh in these cases compared with the cases without horizontal throughflow shown in Table 7. Also, the local threshold oscillatory frequency, ωrt , shown in Table 12 grows somewhat stronger as a function of Rh in the cases with Sv = −20 compared with that in the corresponding cases with Q v = 0 presented in Table 8.

6 Conclusions In this article, we treated a model of a flow in a saturated horizontal extended porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow in which the nature of instability—absolute or convective—at the onset of convection was recently studied in D&B. In that study, an absolute/convective instability dichotomy in the set of exact solutions of the equations of motion was found. As a matter of fact, an absolute/convective instability dichotomy was first found by Brevdo and Ruderman (2009a,b) and further treated by Brevdo (2009) in a similar flow model, but without salinity. In this study, we extended the treatment of instability in the model treated by D&B. For each convectively unstable, but absolutely stable case of destabilization through a longitudinal mode found by D&B, we computed the value of the transitional vertical thermal Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs. The purpose of this computation was to determine of whether at the onset of convection the state is close to being absolutely unstable or not. This was done by comparing the relative percentage deviation, δ = 100% × (Rvt − Rvc )/Rvc , of the vertical thermal Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, from the critical vertical thermal Rayleigh number, Rvc .

123

123

46.351 (45.427)

44.688 (44.427)

Sv = −20

−10

Rh = 0

47.217 (46.956)

48.880 (47.956)

10

54.759 (54.498)

56.423 (55.499)

20

67.172 (66.913)

68.840 (67.915)

30

84.198 (83.944)

85.877 (84.951)

Rh = 40

105.40 (105.16)

107.11 (106.18)

50

129.92 (129.72)

131.69 (130.77)

60

Table 5 Values of the thermal vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, for various values of Rh and Sv , for Sh = 0 and Q h = 0

610 E. Diaz, L. Brevdo


611

Table 6 Values of the relative percentage deviation, 100% × (Rvt − Rvc )/Rvc , of the vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, from the critical vertical Rayleigh number, Rvc , for various values of Rh and Sv , for Sh = 0 and Q h = 0 Rh = 0

10

20

30

40

50

60

Sv = −20

2

1.9

1.7

1.4

1.1

0.9

0.7

−10

0.6

0.6

0.5

0.4

0.3

0.2

0.2

Table 7 Values of the threshold wave number, lt = (lrt , lit ), at the transition from convective to absolute instability, for various values of Rh and Sv , for Sh = 0 and Q h = 0 Rh = 0

10

20

Rh = 40

30

50

60

Sv = −20 ±(2.96, −0.38) ±(2.96, −0.38) ±(2.96, −0.38) ±(2.97,−0.38) ±(3.01, −0.39) ±(3.11, −0.42) ±(3.36, −0.47) −10

±(3.00, −0.18) ±(3.00, −0.18) ±(3.00, −0.18) ±(3.02, −0.18) ±(3.06, −0.19) ±(3.17, −0.19) ±(3.43, −0.20)

Table 8 Values of the threshold oscillatory frequency, ωrt = ωr (lt ), at the transition from convective to absolute instability, for various values of Rh and Sv , for Sh = 0 and Q h = 0 Rh = 0

10

20

30

40

50

60

Sv = −20

3.50

3.50

3.50

3.52

3.56

3.66

3.89

−10

2.13

2.13

2.13

2.13

2.16

2.22

2.35

Under a somewhat arbitrary, but reasonable assumption that for (i) δ ≤ 1%, the critical base state at the onset of convection can be viewed as marginally absolutely unstable and (ii) δ ≥ 5%, the critical base state is distinctly convectively unstable, but absolutely stable, we found the parameter combinations for which the nature of destabilization—through marginally absolute instability or through effectively convective instability—could be determined. Within this assumption, we established that for several parameter combinations, in the cases with vanishing vertical salinity gradient and vanishing throughflow, at the onset of convection the state is, so to say, genuinely convectively unstable, but absolutely stable, as for such parameter combinations it holds that δ ≥ 5%. This finding shows that a genuine absolute/convective instabilities dichotomy is present in the model. As suggested in the Introduction, the dichotomy can hopefully be used in future numerical and experimental studies for analysing the effect of the linear destabilization through either absolute or convective instability on the qualitative features of the emerging non-linear regimes.

Appendix: Solution of the Initial-Value Problem We solve initial-value problem for system (11) formally by using the combined Fourier in space and Laplace in time transforms. The Fourier transform of a function f = f (y) with respect to y is defined by F{ f }(l) = f (l) =

∞ −∞

f (y)e−ily dy,

1 f (y) = 2π

∞

f (l)eily dl.

(16)

−∞

123

123

46.351 (45.427)

44.688 (44.427)

Sv = −20

−10

Rh = 0

46.834 (46.578)

48.504 (47.582)

10

53.146 (52.905)

54.837 (53.925)

20

63.151 (62.949)

64.891 (64.003)

30

75.608 (75.503)

77.439 (76.640)

Rh = 40

87.138 (87.129)

88.917 (88.480)

50

93.403 (93.318)

60

Table 9 Values of the thermal vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, for various values of Rh and Sv , for Sh = 0 and Q h = 5

612 E. Diaz, L. Brevdo


613

Table 10 Values of the relative percentage deviation, 100% × (Rvt − Rvc )/Rvc , of the vertical Rayleigh number, Rvt , at which the transition from convective to absolute instability occurs, from the critical vertical Rayleigh number, Rvc , for various values of Rh and Sv , for Sh = 0 and Q h = 5 Rh = 0

10

20

30

40

50

60 0.1

Sv = −20

2

1.9

1.7

1.4

1

0.5

−10

0.6

0.5

0.5

0.3

0.1

0.01

Table 11 Values of the threshold wave number, lt = (lrt , lit ), at the transition from convective to absolute instability, for various values of Rh and Sv , for Sh = 0 and Q h = 5 Rh = 0

10

20

Rh = 40

30

50

60

Sv = −20 ±(2.96, −0.38) ±(2.97, −0.38) ±(3.04, −0.39) ±(3.19, −0.42) ±(3.56, −0.45) ±(4.44, −0.37) ±(5.58, −0.14) −10

±(3.00, −0.18) ±(3.02, −0.18) ±(3.09, −0.18) ±(3.26, −0.18) ±(3.67, −0.15) ±(4.69, 0.04)

Table 12 Values of the threshold oscillatory frequency, ωrt = ωr (lt ), at the transition from convective to absolute instability, for various values of Rh and Sv , for Sh = 0 and Q h = 5 Rh = 0

10

20

30

40

50

60 4.75

Sv = −20

3.50

3.52

3.57

3.71

4.00

4.54

−10

2.13

2.13

2.16

2.23

2.34

2.20

The Laplace transform of a function g(t) with respect to t is defined by L{g}(ω) = g (ω) =

iσ+∞ ∞ 1 g(t)eiωt dt, g(t) = g (ω)e−iωt dω. 2π 0

(17)

iσ −∞

We operate with these transforms on system (11) and using the initial conditions for w, θ and c, given in (10) obtain the boundary-value problem for the transformed variables w = w(z, l, ω), θ = θ (z, l, ω) and c = c(z, l, ω) : 2 d l2 2 2 − l θ + c = G1, w + l Le dz 2 2 d dTs − l 2 + iω θ = G 2 , w− (18) dz dz 2 1 d2 l2 dCs iφ − w− + ω c = G3, dz Le dz 2 Le a (w, θ, c) z=−1/2 = (w1 , θ1 , c1 )(l, ω), (w, θ, c) z=1/2 = (w2 , θ2 , c2 )(l, ω). In (18) and further in the text, the tilde and the hat are omitted for convenience. The physical dependent variables are distinguished from their transforms by the independent variables. The functions G i = G i (z, k, l, ω), i = 1, 2, 3, in (18) are linear combinations of the transformed source functions, the transformed initial and boundary conditions functions, and their transformed derivatives. To solve problem (18), we write its equations as a first order system of six equations for six unknowns by introducing the variables

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E. Diaz, L. Brevdo

x1 = w, x2 =

dw dθ dc , x3 = θ, x4 = , x5 = c, x6 = . dz dz dz

(19)

With these variables and the notation X = [x1 , x2 , x3 , x4 , x5 , x6 ]T , the resulting system in matrix form reads dX = AX + G, dz

(20)

G = [0, G 1 , 0, G 2 , 0, LeG 3 ]T ,

(21)

where

and the matrix A is given by

⎡

0 ⎢ a21 ⎢ ⎢ 0 A=⎢ ⎢ a41 ⎢ ⎣ 0 a61

1 0 0 0 0 0

0 a23 0 a43 0 0

0 0 1 0 0 0

0 a25 0 0 0 a65

⎤ 0 0⎥ ⎥ 0⎥ ⎥, 0⎥ ⎥ 1⎦ 0

(22)

with the entries 1 a21 = l 2 , a23 = −l 2 , a25 = − l 2 , Le dTs dCs φ a41 = , a43 = l 2 − iω, a61 = Le , a65 = l 2 − iLe ω. dz dz a The boundary conditions in terms of new variables read

(23)

1 (x1 , x3 , x5 ) = (w1 , α 2 θ1 , α 2 c1 )(k, l, ω) at z = − , 2 (24) 1 (x1 , x3 , x5 ) = (w2 , α 2 θ2 , α 2 c2 )(k, l, ω) at z = . 2 Boundary-value problem (20), (24) is solved by using a variation of parameters. For this (i) (i) (i) (i) (i) (i) (i) purpose, we introduce six solutions, X = [x1 , x2 , x3 , x4 , x5 , x6 ]T , 1 ≤ i ≤ 6, of the homogeneous equation associated with Eq. 20, i.e. dX /dz = AX , that satisfy the following boundary conditions: ⎡ (1) ⎤ ⎡ 0 ⎤ ⎡ (2) ⎤ ⎡ 0 ⎤ ⎡ (3) ⎤ ⎡ 0 ⎤ x x x1 ⎥ ⎢ 1(2) ⎥ ⎢ 0 ⎥ ⎢ 1(3) ⎥ ⎢ 0 ⎥ ⎢ x (1) ⎥ ⎢ 1 ⎥ ⎥ ⎥ ⎢ ⎢ x x ⎢ 2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ 2 ⎥ ⎢ ⎥ ⎢ (1) ⎥ ⎢ 0 ⎥ ⎢ (2) ⎥ ⎢ 0 ⎥ ⎢ (3) ⎥ ⎢ 0⎥ ⎢ x3 ⎥ ⎢ ⎥ ⎢ x3 ⎥ ⎢ ⎥ ⎢ x3 ⎥ ⎢ ⎢ at z = − 21 , ⎢ (1) ⎥ = ⎢ ⎥ , ⎢ (2) ⎥ = ⎢ ⎥ , ⎢ (3) ⎥ = ⎢ ⎥ ⎢ x4 ⎥ ⎢ 0 ⎥ ⎢ x4 ⎥ ⎢ 1 ⎥ ⎢ x4 ⎥ ⎢ 0 ⎥ ⎥ ⎢ (1) ⎥ ⎢ ⎥ ⎢ (2) ⎥ ⎢ ⎥ ⎢ (3) ⎥ ⎢ ⎥ ⎣x ⎦ ⎣0⎦ ⎣x ⎦ ⎣0⎦ ⎣x ⎦ ⎣0⎦ 5 5 5 (1) (2) (3) 0 0 1 x6 x6 x6 ⎡

⎤ x1(4) ⎢ x (4) ⎥ ⎢ 2 ⎥ ⎢ (4) ⎥ ⎢ x3 ⎥ ⎢ (4) ⎥ ⎢ x4 ⎥ ⎢ (4) ⎥ ⎣x ⎦ 5 (4) x6

123

⎡ ⎤ 0 ⎢1⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎥ =⎢ ⎢0⎥, ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 0

⎡

⎤ x1(5) ⎢ x (5) ⎥ ⎢ 2 ⎥ ⎢ (5) ⎥ ⎢ x3 ⎥ ⎢ (5) ⎥ ⎢ x4 ⎥ ⎢ (5) ⎥ ⎣x ⎦ 5 (5) x6

⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ ⎥ =⎢ ⎢1⎥, ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 0

⎡

⎤ x1(6) ⎢ x (6) ⎥ ⎢ 2 ⎥ ⎢ (6) ⎥ ⎢ x3 ⎥ ⎢ (6) ⎥ ⎢ x4 ⎥ ⎢ (6) ⎥ ⎣x ⎦ 5 (6) x6

⎡ ⎤ 0 ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢0⎥ 1 ⎥ =⎢ ⎢ 0 ⎥ at z = 2 . ⎢ ⎥ ⎢0⎥ ⎣ ⎦ 1

(25)


615 (i)

Since the entries of the matrix A are entire functions of (k, l, ω), so are the functions X , 1 ≤ i ≤ 6. By analogy with the treatment of a stability problem in the study of Brevdo (1988) and Brevdo and Kirchgässner (1999), we prove the alternative. (i)

Proposition Either the solutions X , 1 ≤ i ≤ 6, are linearly independent or there exists a (0) non-trivial solution, X , of the homogeneous problem dX = AX , dz (0) x1

(0) x3

=

=

(0) x5

(26)

1 = 0 at z = ± . 2

(i)

Proof Let us assume that X , 1 ≤ i ≤ 6, are linearly independent. Then, each solution of the equation in (26) can be expressed as X=

6 (i) ci X = ΦC,

(27)

i=1

where

⎡

(1)

x1 ⎢ x (1) ⎢ 2 ⎢ (1) ⎢x Φ = ⎢ 3(1) ⎢ x4 ⎢ (1) ⎣x 5 (1) x6

(2)

x1 (2) x2 (2) x3 (2) x4 (2) x5 (2) x6

(3)

x1 (3) x2 (3) x3 (3) x4 (3) x5 (3) x6

(4)

x1 (4) x2 (4) x3 (4) x4 (4) x5 (4) x6

(5)

x1 (5) x2 (5) x3 (5) x4 (5) x5 (5) x6

(6) ⎤ x1 (6) x2 ⎥ ⎥ (6) ⎥ x3 ⎥ (6) ⎥ x4 ⎥ ⎥ (6) x ⎦

(28)

5 (6)

x6

is the Wronskian and C = [c1 , c2 , c3 , c4 , c5 , c6 ]T is a constant vector. Since the trace of the matrix A is zero, it holds that the determinant of the Wronskian is independent of z, (i) i.e. det(Φ) = D(l, ω), and since the solutions X , 1 ≤ i ≤ 6, are linearly independent, det(Φ) = 0, cf. Coddington and Levinson (1955). For a solution X satisfying the boundary conditions in (26) at z = −1/2, it holds that ⎡ ⎤ ⎡ ⎤ ⎡ (4) (5) (6) ⎤ 0 c1 0 0 0 x1 x1 x1 ⎢c ⎥ ⎢x ⎥ ⎢ 1 0 0 x (4) x (5) x (6) ⎥ 2 2⎥ ⎢ ⎢ ⎥ ⎢ 2 2 2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (4) (5) (6) ⎥ ⎢ ⎢ ⎥ ⎥ c 0 ⎢ 0 0 0 x3 x3 x3 ⎥ ⎢ 3⎥ = ⎢ ⎥, X z=−1/2 = ⎢ ⎥ (29) (4) (5) (6) ⎢ ⎢ ⎥ ⎥ ⎢ 0 1 0 x4 x4 x4 ⎥ ⎢ c4 ⎥ ⎢ x 4 ⎥ ⎢ ⎥ ⎢ ⎢ ⎥ ⎥ (4) (5) (6) ⎣0 0 0 x x x ⎦ ⎣ c5 ⎦ ⎣ 0 ⎦ 5 5 5 (4) (5) (6) x6 c6 0 0 1 x6 x6 x6 z=−1/2

(i)

where the boundary conditions in (25) for X , 1 ≤ i ≤ 3, were used. By permuting six times the lines in system (29), one shows that this system is equivalent to the system ⎡ ⎤ ⎡ ⎤ ⎡ (4) (5) (6) ⎤ x2 c1 1 0 0 x2 x2 x2 ⎢c ⎥ ⎢x ⎥ ⎢ 0 1 0 x (4) x (5) x (6) ⎥ 2 ⎢ ⎥ ⎢ 4⎥ ⎢ 4 4 4 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ (4) (5) (6) ⎥ ⎢ c3 ⎥ ⎢ x 6 ⎥ ⎢ 0 0 1 x6 x6 x6 ⎥ ⎢ ⎥ = ⎢ ⎥, ⎢ ⎥ (30) (4) (5) (6) ⎢ c4 ⎥ ⎢ 0 ⎥ ⎢ 0 0 0 x1 x1 x1 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ (4) (5) (6) ⎣0 0 0 x x x ⎦ ⎣ c5 ⎦ ⎣ 0 ⎦ 3 3 3 (4) (5) (6) c6 0 0 0 0 x5 x5 x5 z=−1/2

123

616

and that

E. Diaz, L. Brevdo

⎤ (4) (5) (6) x1 x1 x1 ⎥ ⎢ . det Φ = det ⎣ x3(4) x3(5) x3(6) ⎦ (4) (5) (6) x5 x5 x5 z=−1/2 ⎡

From (30), it follows that c4 , c5 and c6 satisfy the system ⎡ ⎤ ⎡ ⎤ ⎡ (4) (5) (6) ⎤ c4 0 x1 x1 x1 ⎢c ⎥ ⎢0⎥ ⎢ (4) (5) (6) ⎥ ⎣ 5⎦ = ⎣ ⎦. ⎣ x3 x3 x3 ⎦ (4) (5) (6) 0 x5 x5 x5 z=−1/2 c6

(31)

(32)

Since det Φ = 0, we have from (32) that c4 = c5 = c6 = 0. Similarly, by considering the boundary conditions for X in (26) at z = 1/2, one can show that c1 = c2 = c3 = 0. This (i) proves that if the solutions X , 1 ≤ i ≤ 6, are linearly independent then problem (26) does not have a non-trivial solution. We now assume that X given by (27) is a non-trivial solution of problem (26). Then from a previous consideration, it follows that the coefficients c4 , c5 and c6 satisfy system (32). From the boundary condition in (26) at z = 1/2, one can deduce that the coefficients c1 , c2 and c3 satisfy the system ⎡ ⎤ ⎡ ⎤ ⎡ (1) (2) (3) ⎤ c1 0 x1 x1 x1 ⎢c ⎥ ⎢0⎥ ⎢ (1) (2) (3) ⎥ ⎣ 2⎦ = ⎣ ⎦, ⎣ x3 x3 x3 ⎦ (33) (1) (2) (3) 0 c 3 x5 x5 x5 z=1/2 with

⎡

⎤ (4) (5) (6) x1 x1 x1 ⎢ ⎥ det ⎣ x3(4) x3(5) x3(6) ⎦ = − det Φ. (4) (5) (6) x5 x5 x5 z=1/2

(34)

Hence, if the coefficients ci , 1 ≤ ci ≤ 6, are not all zero it follows from (32) and (34) (i) that det Φ = 0 implying that the solutions X , 1 ≤ i ≤ 6, are linearly dependent. This completes the proof of the proposition. Problem (26) is equivalent to the homogeneous problem associated with problem (18). Hence, from the proposition, it follows that D(l, ω) = det Φ is the dispersion-relation function and D(l, ω) = 0 is the dispersion-relation of the model as it is defined in the normal-mode approach. The function D(l, ω) is an entire function of (l, ω) because so are all the entries of the Wronskian, Φ. (i) We assume that the couple (l, ω) satisfies D(l, ω) = 0 meaning that X , 1 ≤ i ≤ 6, form a fundamental set of solutions of the equation of problem (26). By using this set the inhomogeneous problem (20), (24) can be solved employing the method of variation of parameters. The solution is somewhat lengthy but straightforward. Here, we omit the details and present only the result pertinent for the stability treatment. The solution is given by X=

T (z, l, ω) , D(l, ω)

(35)

where the vector-function T (z, l, ω) depends as a linear homogeneous function on the Laplace-Fourier transform of the initial and boundary conditions in problem (10). The

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617

vector-function T (z, l, ω) is an entire function of (l, ω) because so is the vector-function G in (20). Consequently, we can write w(z, l, ω) =

T (z, l, ω) , D(l, ω)

(36)

where T (z, l, ω) is an entire function of (l, ω). The function T (z, l, ω) is in a certain sense arbitrary because it depends on the external perturbations. The functions θ (z, l, ω) and c(z, l, ω) are given by similar expressions. For analysing absolute and convective instabilities, it is sufficient to treat the function w.

References Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, New York (1972) Bers, A:. Theory of absolute and convective instabilities. In: Auer, G., Cap, F. (eds.) International congress on waves and instabilities in plasmas, pp. B1–B52, Innsbruck (1973) Brevdo, L.: A study of absolute and convective instabilities with an application to the Eady model. Geophys. Astrophys. Fluid Dyn. 40, 1–92 (1988) Brevdo, L.: Three-dimensional absolute and convective instabilities, and spatially amplifying waves in parallel shear flows. Z. Angew. Math. Phys. 42, 911–942 (1991) Brevdo, L.: Three-dimensional absolute and convective instabilities at the onset of convection in a porous medium with inclined temperature gradient and vertical throughflow. J. Fluid Mech. 641, 475–487 (2009) Brevdo, L., Kirchgässner, K.: Structure formation in a zonal barotropic current: a treatment via the centre manifold reduction. Proc. Roy. Soc. Lond. A 455, 2021–2054 (1999) Brevdo, L., Ruderman, M.S.: On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part I. Normal modes. Transp. Porous Med. 80, 137–151 (2009a) Brevdo, L., Ruderman, M.S.: On the convection in a porous medium with inclined temperature gradient and vertical throughflow. Part II. Absolute and convective instabilities, and spatially amplifying waves. Transp. Porous Med. 80, 153–172 (2009b) Briggs, R.J.: Electron-Stream Interaction with Plasmas. MIT Press, Cambridge (1964) Coddington, E.A., Levinson, N.: Theory of Ordinary Differential equations. McGrow-Hill, New York (1955) Diaz, E., Brevdo, L.: Absolute/convective instability dichotomy at the onset of convection in a porous layer with either horizontal or vertical solutal and inclined thermal gradients, and horizontal throughflow. J. Fluid Mech. 681, 567–596 (2011) Drazin, P.G., Reid, W.H.: Hydrodynamic Stability. Cambridge University Press, Cambridge (1989) Landau, L.D., Lifshitz, E.M.: Electrodynamics of Continuous Media. GITTL, Moscow (1953) Manole, D.M., Lage, J.L., Nield, D.A.: Convection induced by inclined thermal and solutal gradients, with horizontal mass flow, in a shallow horizontal layer of a porous medium. Int. J. Heat Mass Transfer 37, 2047–2057 (1994) Nield, D.A.: Convection in a porous medium with inclined temperature gradient. Int. J. Heat Mass Transfer 34, 87–92 (1991) Nield, D.A.: Convection in a porous medium with inclined temperature gradient: additional results. Int. J. Heat Mass Transfer 37, 3021–3025 (1994) Nield, D.A.: Convection in a porous medium with inclined temperature gradient and vertical throughflow. Int. J. Heat Mass Transfer 41, 241–243 (1998) Nield, D.A., Bejan, A.: Convection in Porous Media. Springer, Berlin (2006) Nield, D.A., Manole, D.M., Lage, J.L.: Convection induced by inclined thermal and solutal gradients in a shallow horizontal layer of a porous medium. J. Fluid Mech. 257, 559–574 (1993) Straughan, B.: The Energy Method, Stability and Nonlinear Convection. Springer, Berlin (2008) Straughan, B., Walker, D.W.: Two very accurate and efficient methods for computing eigenvalues and eigenfunctions in porous convection problems. J. Comput. Phys. 127, 128–141 (1996) Twiss, R.Q.: On oscillations in electron streams. Proc. Phys. Soc. Lond. B 64, 654–665 (1951) Weber, J.E.: Convection in a porous medium with horizontal and vertical temperature gradients. Int. J. Heat Mass Transfer 17, 241–248 (1974)

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