The Stability of a Rotating Cartesian Plume in the Presence of Vertical Boundaries

The effect of two fixed vertical boundaries, a finite distance apart, on the dynamics of a column of buoyant fluid rising in a less buoyant fluid is investigated in the presence of vertical rotation. It is shown that the presence of the boundaries introduces two main effects on a rotating plume. They tend to stabilise the plume but succeed only reducing the value of the growth rate and the plume remains unstable for all finite values of the distance between the boundaries and the plume. In the absence of the sidewalls, two modes of the instability were found known as the sinuous mode and the varicose mode. The influence of the boundaries is such that it reduces the growth rate of the varicose mode more than that of the sinuous mode and consequently the modified sinuous mode is always preferred in the presence of the boundaries.


Introduction
The study of the dynamics of compositional plumes is important for many real life applications in industry ([1]- [6]), geophysics ( [7]- [25]) and environment ( [26]- [34]).While the presence of the compositional plumes can be harmful (e.g., in iron bars), it is useful in geophysics (e.g., the hot compositional plumes, that rise from the inner core boundary of the Earth into the outer core interact with the rotation and magnetic field of the Earth and may contribute to the Geodynamo).Such a wide range of applications has motivated many studies on various aspects of the dynamics of compositional plumes.These studies are experimental and theoretical.The experimental works on the dynamics of compositional plumes observed that the plume flow seems to be stable (Sample and Hellawell [35]; Chen and Chen [36]; Hellawell et al. [37]).The laboratory studies by Hellawell et al. [37] find that the plumes are thin and long, and its top part tends to break up and disappears.Classen et al. [38] experimentally studied the dynamics of compositional plumes under the influence of vertical rotation to find that the plumes are unstable and break into blobs.On the other hand, the theoretical works on the stability of the plumes showed that the Cartesian plume is always unstable in the absence of rotation (Eltayeb and Loper [16]) and in the presence of rotation (Eltayeb and Hamza [18]).These studies assumed that the plume rises vertically in a fluid of unbounded domains.While the experimental studies were conducted in bounded regions to show that the plume was stable, the theoretical models were conducted in unbounded domains to find that the plume was unstable.Thus it is of interest to examine the influence of the vertical boundaries on the dynamics of the plumes.The mathematical model by Al Mashrafi and Eltayeb [6] investigated the influence of the two fixed vertical boundaries on the dynamics of the plumes.They tested the stability of non-rotating Cartesian plumes in a bounded domain to find that the presence of two vertical boundaries affects the stability, but the plumes remain unstable.Moreover, they found that the plume was stable when it was close to the boundary but had a large thickness and the material diffusion is potent in the thin layer between the plume and the nearest boundary.
Motivated by real life applications and laboratory results, we study here the influence of vertical rotation on the dynamics of bounded Cartesian plume.In general, the purpose of this study is to extend the theoretical model by Al Mashrafi and Eltayeb [6] on the dynamics of a Cartesian compositional plume in bounded regions to include the action of vertical rotation.The model by Al Mashrafi and Eltayeb [6] consisted of a column of buoyant fluid of finite thickness, 0 2x , rising vertically in another less buoyant fluid bounded by two fixed ver- tical walls located at 1 x a = and 2 x a = − .The system was infinite in the y and z directions.In the current study, we consider that the whole system rotates about the vertical with a uniform angular speed, ω (see Fig- In Section 2, we formulate the model mathematically and state the boundary conditions of the system.The presence of rotation introduces an additional parameter, τ , which is a measure of the Coriolis force relative to the viscous force, and this parameter referred to hereinafter as the rotation parameter, defined by where Ta is the Taylor number, ν is the kinematic viscosity and L is the unit of length (see Equation (6) below).In Section 3, we investigate the influence of the vertical boundaries on the linear stability of a rotating bounded Cartesian plume.The problem of the rotating plume was studied by Eltayeb and Hamza [18] in the absence of boundaries.In Section 4, we discuss the effect of the boundaries on the stability of a rotating plume.
The growth rate is maximised over the wave numbers plane ( ) , m n in the parameter space.In Section 5, we make some concluding remarks.

Formulation of the Problem
We consider a two-component fluid, in which the concentration of the solvent component (light material) is C and the temperature is T , rotating uniformly about the vertical with angular velocity ω .The fluid has kine- matic viscosity, ν , and thermal diffusivity, κ , and material diffusion is negligible.The dimensionless equa- tions of the system have been derived by Eltayeb and Hamza [18].They are Here R, σ and τ are the Grashoff number, the Prandtl number and the rotation parameter, respectively, defined by ( ) where Ta is the Taylor number and U , L and C  are characteristic units of velocity, length and concen- tration, respectively (Al Mashrafi and Eltayeb [6]), and u , p , g , ẑ , t , α , β , ρ , γ are the velocity vector, the pressure, the constant gravitational acceleration, the upward unit vector, the time, the thermal expansion coefficient, the compositional expansion coefficient, the density, the uniform temperature gradient and the subscript " r " in the Equation (2) refers to reference values.We consider a basic concentration profile ( ) which defines a plume of thickness, 0 2x , rising with velocity ( ) w x in the presence of mean temperature ( ) T x and mean pressure ( ) which are the same equations obtained in the absence of rotation.The presence of rotation does not affect the basic state, and the solution of ( 8) and ( 9) is the same as in the absence of rotation.We include it here for easy reference: ( )

The Stability Analysis
In this section, we use the perturbation Equations ( 14)- (17) to investigate the linear stability of the basic state solution given by (10).We assume that the interface at the plane 0 x x = is given a small harmonic disturbance of the form where m and n are the wavenumber components in the ( ) , y z plane, Ω is the growth rate and . .c c re- fers to the complex conjugate.
The disturbance (12) will propagate into the fluid, and affect the second interface and the variables of the system to produce the perturbations.Consequently, the interface at 0 x x = − can be written in the form 0 1 . ., where 1 η is a measure of the amplitude of the interface and it is evaluated as a part of the solution.The pertur- bations introduced into the system are governed by the dimensionless equations (see Al Mashrafi and Eltayeb [6]) where † u , † p , † T and † C are the perturbations in velocity, pressure, temperature and concentration, re- spectively.
The perturbation variables take the form (18) in which the factors in − , nm , and in − are introduced in the variables ( ) u x , ( ) v x and ( ) p x , respectively, for convenience.
Substituting the variables (18) into the Equations ( 14)- (17) and after some arrangements we get the following equations ( ) ( ) subject to the boundary conditions ( ) ( ) ( ) ( ) ( ) where the variable ς is related to the vertical component of vorticity and is defined as , Dv u and we have introduced the notation (cf.Al Mashrafi and Eltayeb [6]).
We use the same method adopted in Al Mashrafi and Eltayeb [6] and expand the perturbation variables and the growth rate in the small parameter R , thus where ( ) , , , f x y z t indicates any of the perturbation variables u , v , w , p , ς and T .It turned out that the leading order terms in the equations determines the stability of the system.The relevant equations and the boundary conditions are then obtained from ( 19)-( 24) and ( 26)-( 29) by neglecting the terms with R .In order to facilitate comparison with the results of the non-rotating case, we shall maintain the sub- script 0. The equations are given by ( ) ( ) The associated boundary conditions are obtained from ( 26)-( 29) by introducing the subscript 0 to all the variables and the subscript 1 to Ω .
The system ( 33)-( 38) together with the boundary conditions poses an eigenvalue problem for the growth rate, Ω , which determines the stability of the system.The real part r Ω governs the variations of the amplitude of the disturbance with time, and hence it determines the stability of the disturbance.If it is negative for all possible values of the wavenumbers m and n , then the plume is stable, while if at least one pair of m and n gives a positive value, then the plume is unstable.If r Ω vanishes for all values of the wavenumbers, the plume is neu- trally stable.If the preferred mode occurs for m , n both non-zero, it is referred to as a 3-dimensional mode (oblique), and if 0 m = , it is called 2-dimensional (vertical).The case 0 n = and 0 m ≠ is found not to occur.The imaginary part i Ω determines the phase speed of the disturbance.The vertical phase speed z u and the horizontal phase speed h u are defined as , .
We note that h u is defined only if 0 m ≠ .We operate on Equation (33) with 2  ∆ , and use Equations ( 34)-( 36) to get ( ) The general solution of the differential Equation ( 40) can be written in the form where the superscript ( ) i refers to the three regions of the system (see Figure 1), ( ) are constants, and j µ are the roots of the cubic equation ( ) with j λ is given by ( ) We use the Equations ( 34)-( 36) and the solution (41) to obtain The Equation ( 37) can be solved in the form a complementary function and particular solution to find where Now we apply the boundary conditions at 0 and solve the resulting algebraic equations for the constants ( ) The details of the solution are given in the Appendix A.
The growth rate is given by the quadratic equation where 1 S and 2 S are given in the Appendix A. Solving Equation (47) yields ( ) and the displacement 1 η is given by ( ) in which 1 F − and e F − are defined in the Appendix A. We note that, as in the absence of rotation, the system has two modes.The upper sign in the expression (48) corresponds to the modified varicose mode (MV) and the lower sign refers to the modified sinuous mode (MS).

Discussions of the Results
The growth rate (48) is evaluated numerically in the ( ) , m n plane as a function of the parameters 0 x , 2 a , d and Ta for both modes.The contours of 1 Ω in the ( ) , m n plane are plotted for sample values of the parameters 0 x , 2 a , Ta and 10 d = in the Figures 2-4.In Figure 2, we present a comparison between the contours of the two modes MV and MS for different values of the Taylor number, Ta , when 0 0.5 x = , 2 1 a = and 10 d = .We note that the MV mode always possesses a minimum with negative growth rate and one or two maxima with positive growth rates while the MS mode has  .
a minimum with negative growth rate only when Ta is small and one maximum with positive growth rate which is larger than that of the corresponding MV mode.This indicates that the MS mode is preferred for this set of parameters.The local maxima correspond to 3-dimensional modes and the largest maximum always increases with the increase in the rotation parameter Ta .

Ω
are nearly same but there different in the sign.
Figure 3 presents the influence of the distance 2 a between the plume and the nearest sidewall on the stabil- ity of the plume when the thickness of the plume and the rotation parameter are held fixed at 0 0.5 x = and 0.8 Ta = . The figure is plotted for three different values of 2 a : 2 1 a = for subfigures (a), (b); 2 3 a = for sub- figures (c), (d); and 2 5 a = for subfigures (e), (f).When the plume is situated half-way between the sidewalls ( ) 2 5 a = , the MV mode has a negative growth rate everywhere and hence stable while the MS mode has a growth rate that is positive everywhere and hence is preferred.The local maximum possesses a vanishing horizontal wave number and hence propagates vertically.As the plume moves towards a wall, the growth rates of both modes increase and the MV mode develops local maxima with positive growth rates but they are not preferred because the MS mode local maximum increases as well and becomes 3-dimenaional.
Figure 4 illustrates the influence of the plume thickness on the contours of the growth rate for fixed values of the rotation parameter and the distance between the two sidewalls when the plume is situated half-way between the two sidewalls.As 0 x increases from 0 1 x = , the growth rate for MS mode, which is positive everywhere, increases and that of the MV mode, which is negative everywhere, decreases until 0 x reaches a critical value, 0c x , when the growth rate of the MS mode decreases and that of the MV mode increases.The contours of Figures 2-4 indicate that the preferred mode of instability is the modified sinuous (MS) mode.This indication is quantified by calculating the maximum growth rate and the associated wave numbers and phase speeds for different values of the parameters.For fixed values of 0 x , 2 a , d and Ta , we maximize over the wave numbers m and n by demanding that The solution of (50) gives the values c m and c n at maximum growth rate, 1c Ω , calculated from (48) for these values of m and n to give, together with the corresponding value c U of the phase speed, the parame- ters ( ) of the preferred mode.For fixed parameters 0 x , 2 a , d and Ta , and particular mode, all possible local maxima of ( ) are identified and the largest value taken together with the corresponding wavenumbers and phase speeds as defining the preferred mode for that set of parameters for that mode.This is carried out for both modified varicose (MV) and modified sinuous (MS) modes, and the largest is chosen as the preferred mode.A sample of the results is given in Figures 5-9.  Figure 5 and Figure 6 illustrate the influence of the distance between the boundaries on the preferred mode of instability of the rotating plume.In Figure 5, the preferred mode parameters are plotted as a function of Ta for fixed 0 x and two values of the distance between the sidewalls, d , for a plume situated half-way between the sidewalls.It is found that the preferred mode is the MS mode and the influence of the boundaries tends to stabilise the plume but only reduces the growth rate slightly.The horizontal and vertical wavenumbers are increased while the phase speeds are reduced as the distance, d , between the two walls is reduced.
Figure 6 presents the preferred mode of instability as a function of 0 x and fixed Ta where the plume is equidistant from the two walls and d takes the values 10 and 20.The growth rate increases rapidly as 0 x in- creases from zero until it reaches a maximum when the plume occupies the middle half of the region between the sidewalls.As the thickness of the plume increases further, the growth rate decreases if d is small but stays at the maximum value for increasing 0 x if the distance d is large.In both case, the growth rate drops to zero as the wall is approached and the plume nearly fill the whole region between the sidewalls.The wavenumbers decrease as 0 x increases from zero but they soon jump to larger values and increase with 0 x .In both cases the MS mode is preferred.Ω in- creases rapidly until Ta reaches a certain value that increases as the distance between plume and the nearest sidewall increases, after which the growth rate varies much more slowly.The wavenumbers of the preferred mode show sudden changes at a small value of Ta indicating a change of local maximum as Ta , increases through a value, e Ta .The horizontal wavenumber, c m , is zero for small Ta corresponding to 2-dimensional motions, but as Ta reaches e Ta , c m jumps to a nonzero value and increases thereafter.The vertical wave- number increases rapidly as Ta increase to e Ta and then decreases as Ta increases further.The vertical phase speed of the preferred mode also suffers a change at e Ta Ta = and the jump depends on how far the plume is from the nearest sidewall.
In Figure 8, we present the dependence of the preferred mode on the distance 2 a between the plume and the nearest sidewall.We note that the growth rate decreases when the plume moves towards the location half-way .Note that the preferred mode of instability is modified sinuous mode in all cases.When the plume is wide, the growth rate decreases whenever the plume moves to the wall.
between the two sidewalls.Both wavenumber components increase steadily as the distance between plume and the nearest sidewall increases.The vertical phase speed however behaves differently as 2 a increases; it de- creases steadily with increasing 2 a if Ta is small but when Ta is large, it decreases slowly reaching a minimum before it increases at a moderate rate.In all cases, it is the MS mode that provides the preferred mode.x .The growth rate increases gradually until 0 x reaches a critical thickness, 0c x , and then it decreases.This indicates that there is a critical thickness representing the most unstable plume.The behaviour of the wavenumber components and phase speed with 0 x is quite complicated.The horizontal wavenumber vanishes for small 0 x when Ta is small.For large, c m , behaves in a similar way when 0 x is close to 0c x .The vertical wavenumber, on the other hand de- creases as 0 x increases from zero and jumps to a larger value when the local maximum changes, and then va- ries very slowly until 0 x is almost 4, when 10 d = .

Conclusions
The dynamics of a fully developed plume of buoyant fluid, in the form of a channel of finite width, 0 2x , rising in a less buoyant fluid contained between two parallel vertical walls, a distance d apart, and two fluids rotate uniformly about the vertical have been investigated.
In the absence of boundaries ( [18]), it was found that the stability problem depended on the parameters: the Grashoff number, R , the Taylor number, Ta , and the thickness of the plume, 0 2x .The magnitude of the growth rate was of the order ( ) for 1 R  and the instability took one of the two types: sinuous mode or varicose mode.
The presence of the vertical boundaries here introduces two dimensionless parameters: the distance between the plume and the nearest wall, 2 a , and the distance between the two vertical sidewalls, d .The introduction of the boundaries modifies the two modes to be the modified sinuous mode (MS) and the modified varicose mode (MV).It is shown here that the boundaries introduce two main effects to the rotating Cartesian plume studied by Eltayeb and Hamza [18].First, the sidewalls tends to stabilise the plume but succeed only reducing the growth rate and the plume remains unstable for all values of the Taylor number and the distance from the nearest sidewall.Second, the presence of the sidewalls suppresses the modified varicose mode and allows the modified sinuous only to be unstable.The preferred mode can be 3-dimensional or 2-dimensional depending on the values of the parameters of the system.When the preferred mode is 2-dimensional, propagation can be upwards or downwards.0 x x = gives ( ) where j E , j F , E and F are defined by We use the Equations (A:5)-(A:8) to obtain ( ) ( ) ( ) ( ) Similarly, at the interface 0 x x = − , we find ( ) L , K and L are given by The boundary conditions at the boundaries 0, 0, 0, 0, j j j j j j j j j j j j P P P P such that ( ) 3 j P , ( )  ( ) ( ) ( ) The equations (A:1) and (A:2) can be solved to find since the roots j µ are distinct.Hence Equations (A:3)-(A:5) give ; ; cosh 1 ; ; , x , 5 x and 6 x are defined by and we have defined j d and i Y , 1 -6 i = as The algebraic system (A:57) and (A:58) can be solved by elimination to find V and je V are given by ( ) ( ) 2 2 2 2 2 2 and we have defined the following notation The constants ( )   and we have used the notation Z ± , 1 j a , 2 j a , 3 j a ± , 4 j a ± , 1s E ± and 2s E ± ( ) η can be evaluated by solving the Equation (A:83) to find ( )

Figure 1 .
Figure 1.The geometry of the problem showing the profile of the basic state concentration of light material, ( ) C x , representing a plume of width, x , A and k are defined by

Figure 5 .
Figure 5.The preferred mode parameters 1 , , , c c c c m n U Ω are plotted in (a), (b), (c), (d) as a function of Ta , for 0 2 x = , and two different values of d and the plume half-way between the sidewalls.The roman numbers i and ii refer to 2 10, 5 d a = = and

Figure 6 .
Figure 6.The preferred mode parameters 1 , , , c c c c m n U Ω are plotted in (a), (b), (c), (d) as a function of 0 x , for 2 a T = , and two different values of d and the plume half-way between the sidewalls.The roman numbers i and ii refer to 2 10, 5 d a = = and

Figure 7 .
Figure 7.The preferred mode parameters 1 , , , c c c c m n U Ω are plotted in (a), (b), (c), (d), respectively, as a function of Ta , for 0 1 x = , 10 d = and two different values of 2 a .The solid curves refer to 2 2 a = and the broken curves refer to 2 5 a = .Note that preferred mode is MS in all cases.

Figure 8 .
Figure 8.The preferred mode parameters 1 , , , c c c c m n U Ω are plotted in (a), (b), (c), (d), respectively, as a function of 2 a , for 0 3 x = , 10 d = and for two different values of Ta .The roman numbers i and ii refer to 0.5 Ta =and 10 Ta = , respectively.Note that the maximum growth decreases as the plume moves away from the wall to the centre.

Figure 7
Figure7illustrates the dependence of the preferred mode on Ta for 0 1x = , 10 d = and 2 a takes two values: 2 2 a = and 2 5 a = .As the Taylor number increases from 0 Ta = , the maximum growth rate 1c Ω in- creases rapidly until Ta reaches a certain value that increases as the distance between plume and the nearest sidewall increases, after which the growth rate varies much more slowly.The wavenumbers of the preferred mode show sudden changes at a small value of Ta indicating a change of local maximum as Ta , increases through a value,

Figure 9
Figure9shows the dependence of the critical modes ( )1 , , , c c c c m n U Ω on 0x .The growth rate increases gradually until 0x reaches a critical thickness, 0cx , and then it decreases.This indicates that there is a critical thickness representing the most unstable plume.The behaviour of the wavenumber components and phase speed with 0x is quite complicated.The horizontal wavenumber vanishes for small 0 x when Ta is small.For large, c m , behaves in a similar way when 0 x is close to 0c x .The vertical wavenumber, on the other hand de- creases as 0x increases from zero and jumps to a larger value when the local maximum changes, and then va- ries very slowly until 0x is almost 4, when 10 d = . ) the expression for j F in (A:33), and the properties of the 3 roots of the cubic Equation (42) to simplify the expressions (A:35), (A:36).The simplifications lead to 0. E F = = (A:37) (A:45)-(A:48) and (A:55) and (A:56) to obtain can easily evaluated using (A:38), (A:39), (A:42), (A:43), (A:53), (A:54).The next step is to find the expressions of the constants ( ) Equations (A:26), (A:27), (A:40) and (A:44) give

Now the growth rate 1 Ω
and the displacement 1