Marginal stability analysis on salt-fingers convection with parabolic temperature and salinity profiles

Acta Astronautica 65 (2009) 591 – 598 www.elsevier.com/locate/actaastro

Marginal stability analysis on salt-fingers convection with parabolic temperature and salinity profiles Ray-Yeng Yanga,1 , Hwung-Hweng Hwungb,2 , Igor V. Shugana, c,∗ a Tainan Hydraulics Laboratory, National Cheng Kung University, 5th f., 500, Sec. 3, Anming Road, Tainan 709, Taiwan b National Cheng Kung University, Tainan 701, Taiwan c Prochorov General Physics Institute of Russian Academy of Sciences, Vanilova 38, Moscow, Russia

Received 22 October 2008; accepted 14 December 2008 Available online 7 May 2009

Abstract This paper explores the influence of parabolic profiles of temperature and salinity, as might arise due to local evaporation or warming disturbance, on the marginal stability problem in a salt-fingering regime. From the results of this article, the governing equations obtained for stationary onset of salt-fingers convection of a gravity gradient due to local disturbance in parabolic temperature and salinity profiles are similar to those obtained by the small-gap Taylor–Couette problem. Detailed relationships between the effective Rayleigh number ( R˜ c ), the critical wave number (ac ) and the couple disturbed local depth under the stationary stability analysis are also shown in this work. From the result of overstability analysis, it shows that oscillatory motion of salt-fingers convection will be triggered under a certain definite characteristic frequency. In overstability, it provoke restoring forces so strong as to overshoot the corresponding position on the other side of equilibrium for case of coupled disturbance in the upper layer of ocean. © 2009 Published by Elsevier Ltd. Keywords: Salt-fingers convection; Parabolic profile; Stationary stability; Overstability; Effective Rayleigh number

1. Introduction Salt-fingers convection is now widely recognized as an important mechanism for mixing heat and salt both vertically and laterally in the ocean dynamics. Since the

∗ Corresponding author at: Tainan Hydraulics Laboratory, National Cheng Kung University, 5th f., 500, Sec. 3, Anming Road, Tainan 709, Taiwan. Tel.: +886 6 2371938x103. E-mail addresses: [email protected] (R.-Y. Yang), [email protected] (H.-H. Hwung), [email protected] (I.V. Shugan). 1 Tel.: +886 6 2371938X301. 2 Tel.: +886 6 2387275; fax: +886 6 3840207.

0094-5765/$ - see front matter © 2009 Published by Elsevier Ltd. doi:10.1016/j.actaastro.2008.12.013

discovery of double-diffusive convection by Stommel et al. [1], “evidence has accumulated for the widespread presence of double-diffusion throughout the ocean” and for its “significant effects on global water-mass structure and the thermohaline convection” (Schmitt [2]). The salt-fingering form of double-diffusion has particularly attracted interest because of its peculiar long narrow convecting cell structure and enhancement of the diapycnal transport of heat and salt, even when the net density gradient is stable. Furthermore, these structures are increasingly recognized as an important mechanism for oceanic mixing and salt transport [3–5]. Recent observations by Osborn [6] suggested that the finger instability is also important in near-surface waters. Kluikov and

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Karlin [7] suggested that two-thirds of World Ocean is favorable for fingering convection. For example, in the tropics, surface evaporation exceeds precipitation and heating exceeds cooling, producing these conditions. In contrast, the diffusive convection form of doublediffusion is commonly found in polar water where cool fresh water overlies warmer saltier water. Salt-fingers convection has been investigated both theoretically and experimentally. Theoretical analysis of salt-fingers convection was first considered by Stern [8]. A number of works have begun with a base state in which the solute concentration and the temperature are linear functions of depth [9]. Proctor and Holyer [10] showed that rolls are preferred over square cells for conditions modeling a salt-fingers regime. Subsequent developments for theoretical analysis of salt-fingers convection were summarized by Kunze [11]. Saltfingering laboratory experiments were first conducted by Turner and Stommel [12]. Subsequent works were summarized by Schmitt [13] in this issue. Analytical and modeling efforts have focused on the onset and stability of salt-fingers convection, their diapycnal heat and salt transports, layer formation, and interaction with internal waves and shear-driven turbulence. Meanwhile, the small-scale thermohaline plumes near the surface of a calm sea under the warming condition have been revealed by oceanic observations. The stratification is favorable for the double-diffusive salt-finger instability. In thinking of the instability of a hydrodynamic system, it is often convenient to suppose that all parameters of the system, save one, are kept constant while the chosen one is continuously varied. Then the hydrodynamic system passing from stable to unstable states, when the particular parameter being taken to be a certain critical value. Thus, we may say that instability is set in at this value of the chosen parameter when all the others have their preassigned values. If at the onset of instability a stationary pattern of motions prevails, the principle of the exchange of stabilities is set in as the stationary cellular convection, or secondary flow. On the other hand, if at the onset of instability oscillatory motion prevails, that is the case of overstability. In the former case, the transition from stability to instability takes place via a marginal state exhibiting a stationary pattern of motions. In the later case, the transition takes place via marginal state exhibiting oscillatory motion with a certain definite characteristic frequency. The purpose of this work is to present a relationship among the critical wave number, the critical effective Rayleigh number; the couple disturbed depth function, and the characteristic frequency under two kinds of salt-fingers marginal stability in

the basic state with parabolic temperature and salinity profiles. 2. Solution for the case when instability being set in as stationary convection In classifying margin states into the two classes, stationary and oscillatory, the dissipative systems are supposed to be dealt with. The two classes correspond to the two ways in which the amplitudes of a small disturbance can grow or be damped: they can grow (or be damped) aperiodically; or they can grow (or be damped) by oscillations of increasing (or decreasing) amplitude. In this article, the onset of instability as stationary convection will be first stated. We consider that a viscous fluid with thermal conductivity kt , saline conductivity ks and depth d (Fig. 1) has the basic state of parabolic distributions of temperature and salinity. The differences between the bottom boundary (Tb, Sb ) and top boundary (Tt , St ) are qd 2 /kt (1/ − 21 ) and q  d/ks (1/ − 21 ), respectively, in which  and  are the inverse of the position (z max ) and  (z max ) at which Tmax and Smax occur q and q  are assumed as the imaginary uniform heat and salinity source of strength in the fluid to simulate the quasi-parabolic profile. Therefore, the temperature and salinity distributions of the overall flow field along the z (with z = Z /d) become T − Tb =

qd 2

(−2 z 2 + 2z),

(1)

q d 2 2 (− z 2 + 2 z). 22 ks

(2)

2 2 k

t

and S − Sb =

Fig. 1. Basic state for the profiles of temperature and salinity.

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For moderate temperature and salinity differences, the equation of state is approximated by  = o [1 − t (T − To ) + s (S − So )],

(3)

in which To , So and o are, respectively, the temperature, salinity and density of the fluid at a point chosen to be the basic state, and t is the thermal expansibility, s =1/o . When convection is present, the temperature, pressure and salinity will differ from their mean values. One can write T = T + T ,

P = P + P , S = S + S.

(4)

And the perturbed velocity component in direction i is simply denoted by u i . Under the assumptions made on the effects of the density change, neglecting quadratic terms of perturbation quantities, and recalling that the undisturbed state is one of dynamic and thermal equilibrium, the equation of motion can be described as ju i jt

= (0, 0, gt T  − gs S  ) −

j p

1 + ∇ 2 u i , o jX i

(5)

in which  is the molecular diffusivity of momentum. The linearized form of heat equation is   j − t ∇ 2 T  = t w. (6) jt The linearized form of salt equation is   j 2 − s ∇ S  = s w, jt

(7)

in Eqs. (6) and (7), t and s are the molecular diffusivities of heat and salt, w is the perturbed vertical velocity, t and s are expressed as t = dT /dz and s = dS/dz. According to Eqs. (1) and (2), t is 2(1 − z) and s is 2 (1 −  z). However, ∗t and ∗s are chosen to be the maximum value for t and s on certain depth. The z-component of vorticity is j = ∇ 2 . jt

(8)

Eliminating the pressure term by taking the vertical (z) component of ∇ × ∇ × Eq. (5), we obtain j 2 ∇ w = gt ∇12 T  − gs ∇12 S  + ∇ 4 w, jt

(9)

in which ∇12 = (j2 /jx 2 ) + (j2 /jy 2 ). Accordingly, for an investigation on stability to be complete, it is necessary that the reaction of the system

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to all possible disturbances be examined. Therefore, we must analyze an arbitrary disturbance into a complete set of normal modes and examine the stability of each of these modes, individually. Thus assuming normal modes for analysis, one can try the solution of the form (w, T  , S  , ) = [W(z) , (z) , S(z) , Z a (z)] × exp[i(k x x + k y y) + pt], etc.

(10)

For functions of this dependence on x, y and t, differentiating Eq. (10) with time is equal to t multiplying Eq. (10), ∇ 2 = d2 /dz 2 − k 2 , and k 2 = k x2 + k 2y , we obtain p(D 2 − k 2 )W = − (gt k 2 − gs k 2 s) + (D 2 − k 2 )2 W ,

(11)

p = ∗t W + t (D 2 − k 2 ) ,

(12)

pS = ∗s W + s (D 2 − k 2 )S,

(13)

and p Z a = (D 2 − k 2 )Z a ,

(14)

in which D denotes differentiation with respect to z. Choosing d, d 2 /, /d, (−∗t d) and (−∗s d) as the characteristic length, time, velocity, and s, respectively, and using non dimensional variables, we have the dimensionless wave number a (equal to kd), the complex temporal coefficient (equal to pd 2 /), the ratio of diffusivities (Lewis number)  = s /t , the Prandtl number Pr = /t , the thermal Rayleigh number Rt = gt ∗t d 4 /t  and the saline Rayleigh number Rs = gs ∗s d 4 /t . Thus, Eqs. (11)–(13) attain the forms  g  t 2 (D 2 − a 2 )(D 2 − a 2 − )W = d a2   g  s 2 − d a 2 s, (15)   ∗  t 2 [ Pr − (D 2 − a 2 )] = (16) d W, t  [ Pr − (D 2 − a 2 )]s =

 ∗s 2 d W. s

(17)

As the transition from stability to instability occurs through a stationary state under a constant gravity (g) field, we assume a similar situation to hold in a ggradient field. The equations governing the marginal state are, therefore, to set =0; thus the wave number a

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at incipient instability is determined by the differential system (D 2 − a 2 )2 W = Rt a 2 −

Rs 2 ˜ a (s) = a 2 R , 

(D 2 − a 2 ) = −∗t W , (D − a 2

2

(18) (19)

) 1 = −∗s W ,

in which  is the couple disturbed depth function and g0 is the gravity acceleration. Then the Rayleigh numbers Rt and Rs are redefined as go t d 5 q , 22 t kt

Rs =

(2 + a 2 )3 1  2a2 (2 −  ) = − − 2 2 4 ( + a 2 )2 (sinh2 a − a 2 ) 2 R˜  a 2 ×[sinh a cosh a−a]+(sinh a−a cosh a)]. (28)

(20)

in which s is substituted by 1 and the effective Rayleigh number R˜ = Rt − Rs /. We proceed to solve the differential system consisting of Eqs. (18)–(20). For convenience, the vertical buoyancy acceleration vector is assumed to vary linearly with z     ks (21) g = go 1 −  −  z = go (1 −  z), kt

Rt =

and substituting the expansions of and W into Eq. (26), we find a first approximation to the solution of Eq. (26).

go s d 5 q  . 22 t ks

(22)

Now for neutral stability one needs to consider only the ˜ by summing Eqs. (18)–(20) system (with R˜  = 2 R) and eliminating and 1 , we have ˜ 2 (1 −  z)W (D 2 − a 2 )3 W = − 2 Ra  2 = − R˜ a (1 −  z)W .

3. On the onset of convection as overstability We now take up the question, whether or not instability can arise as oscillations of increasing amplitude, i.e., as overstability. Therefore, it is required to return to the general Eqs. (15)–(17) which include the time constant . By applying the operator L ≡ (D 2 − a 2 − Pr ) we can obtain L(D 2 − a 2 )W = − R˜  a 2 (1 −  z) ,

(29)

L = W .

(30)

Together with the boundary condition at z = 0 and 1 W = DW = = 0.

Substituting the expansions of W and in Eqs. (29) and (30), the characteristic equation is as followed: − nm(2n 2 2 +a 2 +b2 )(2m 2 2 +a 2 +b2 )Pr (n 2 2 +a 2 )(n 2 +2 +b2 )(m 2 2 + a 2 )(m 2 2 +b2 ) × [(−1)m+n − 1] +

(23)

Together with rigid bottom boundary condition and rigid top boundary condition

(31)

 ×

nm2 (a 2 − b2 ) (n 2 + 2 + a 2 )(n 2 2 + b2 )

(b sinh a−a sinh b)[(−1)n+1 +(1− )(−1)m+1 ]

+ (b cosh b sinh a − a sinh b cosh a)[1 + (1 −  ) 2 (2m 2 2 + a 2 + b2 ) (m 2 2 + a 2 )(m 2 2 + b2 )

W = = DW = D = 0 at z = 0,

(24)

× (−1)m+n ] −

W = = DW = D = 0 at z = 1.

(25)

× ab(cosh a − cosh b)[(−1)n+1 − (−1)m+1 ]

Replacing  by 2 / 1 − 1( 1 and 2 are the angular velocities of the inner and the outer cylinder, respectively), Eqs. (23)–(25) are exactly the same as those for the narrow-gap Taylor–Couette problem. Therefore, Eq. (23) can be written in the form (D 2 −a 2 )3 W = (1− z) and = − R˜  a 2 W .

(26)

Expanding and W in the forms

=

∞  m=1

Cm sin mz and W =

∞  m=1

+

 1 2 (a − b2 ) sinh a sinh b[(−1)m+n − 1] 2

1 + nm −  X nm 2 1 2 2 2 2 2 2 nm − (n  + a )(n  + b ) = 0,  2 2 R˜ a

(32)

in which C m Wm ,

(27)

b2 = a 2 + ,  = 2ab(1 − cosh a cosh b) + (a 2 + b2 ) sinh a sinh b,

(33)

R.-Y. Yang et al. / Acta Astronautica 65 (2009) 591 – 598

595

⎧X =0 if (m + n) is even, m
n, nm ⎪ ⎪ ⎨ X nm = 1/4 if (m = n), (34)   2 2 2 2 ⎪ ⎪ 2m  + a + b 1 ⎩ X = 4mn if (m + n) is odd. − nm n 2 − m 2 (m 2 2 + a 2 )(m 2 2 + b2 ) 2 (n 2 − m 2 ) Since our principal interest is to specify the critical effective Rayleigh number for the onset of instability via a state of purely oscillatory motions, we shall in the first instance suppose that ( = r + i i ) in Eqs. (32) and (33) is pure imaginary ( r = 0) and seek the conditions for such solutions to exist. For given wave number a, the solution will suffice to answer the principal question as to when instability will set in as stationary convection and when as overstable oscillations. 4. Results and discussion As problem about the steady onset of salt-fingers convection under stationary state is exactly analogous to the small-gap Taylor–Coutte problem. We verified our analytic solutions and numerical analysis to assure the correctness of the results. The comparison shows good agreement. Detailed relationship between the effective Rayleigh number R˜ c , the critical wave number ac and the couple disturbed depth function ( ) are shown in Table 1 and Figs. 2–5. In the field of salt-fingers convection with the basic state of parabolic distributions of temperature and salinity, there are two properties with distinct molecular diffusivities. In such a mechanism with opposing gradients, the existence of a net density distribution that decreases upwards fails to assure stability. The results also show that the effective Rayleigh

Table 1 The critical Rayleigh numbers R˜ c and the associated wave numbers for different values of  , in which  =  − (ks /kt ) , and R˜ c = 2 (Rt − Rs /).

 (Disturbed

a (Wave number)

R˜ c (Critical Rayleigh number)

3.13 3.20 3.24 3.34 3.49 3.70 4.00 4.61 5.06 5.60 6.10 7.10 8.14

1785 2139 2402 2774 3283 3931 4670 6769 9236 12,290 15,930 25,300 37,812

depth function) 1.25 1.50 1.60 1.70 1.80 1.90 2.00 2.25 2.50 2.75 3.00 3.50 4.00

number R˜ c exceeds 1707.8, the value for constant g, and that the critical value of R˜ c increases beyond the constant g value as  departs from 0. These phenomena signify that the buoyancy effects are largest near the bottom wall and diminish with height, because the effective Rayleigh number is based on the gravity level at the bottom. Furthermore, even if t (T − To )o is larger than S − So , so that the undisturbed condition is larger at the top than at the downward side, R˜ c may still exceed 1707.8, for s may be smaller than t , as in the case of saline solution. The instability in that case is due to

Fig. 2. The critical Rayleigh numbers R˜ c at which instability sets in for the disturbances of different wave numbers.

Fig. 3. The wavelength of the disturbance (in units of widthd) manifested at onset of instability as couple disturbed depth function  .

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R.-Y. Yang et al. / Acta Astronautica 65 (2009) 591 – 598 Table 2 The critical Rayleigh number R˜ c , wave number ac and the associated characteristic frequency i for different values of  .

Fig. 4. The critical Rayleigh number R˜ c for the onset of instability as couple disturbed depth function  .

Fig. 5. Relationship associated with critical Rayleigh number R˜ c , wave number ac and the couple disturbed depth function.

the greater facility with which a displaced source harmonizes with its new surroundings in temperature than in salinity concentration. For the disturbed source displaced from above, this condition results in its becoming heavier than its environment and favors its further downward motion. For disturbed source displaced form bellows, this results in its becoming lighter than its environment and favors upward movement. Hence thermal diffusivity is a destabilizing factor in the case under consideration, whereas salinity concentration diffusivity is a stabilizing. As one expecting from the former discussion, the value of the couple disturbed local depth  at which oscillatory convection will occur for a wave number corresponding to the effective Rayleigh number R˜ c . Actually for R˜ C < R˜ C∗ , overstability is not possible for the given Pr and  ; and the onset of instability as stationary convection remains the only possibility. However,



i

R˜ c

ac

2.0

0.00 6.23 8.67 11.90 15.92 16.68 20.85 21.84 25.04 28.08 36.32

4670 4620 4574 4493 4372 4351 4282 4283 4336 4452 4975

4.00 3.94 3.89 3.80 3.69 3.68 3.64 3.65 3.69 3.75 3.94

1.8

0.00 6.15 12.06 17.94 24.32 31.97 46.46

3283 3280 3294 3376 3580 3963 4946

3.49 3.48 3.50 3.54 3.63 3.78 4.04

1.75

0.00 5.65 11.30 17.10 23.46 30.99 37.43

3005 3017 3065 3175 3384 3743 4124

3.41 3.42 3.45 3.51 3.60 3.73 3.84

when R˜ C > R˜ C∗ , overstability is possible; and the transition takes place via marginal state exhibiting oscillatory motion with a certain definite characteristic frequency. The critical effective Rayleigh numbers and related constants for the onset of overstability in case of Pr = 1 are shown in Table 2. From Fig. 6, it shows that oscillatory motion of salefingers convection will be triggered under a certain definite characteristic frequency. Furthermore for the case of  = 2, there are even two characteristic frequencies existed which overstability will occur under some conditions. From the analyzes, we can also see that the onset of instability as overstability convection remains the only mode when R˜ C > 4, 280( =2), R˜ C > 3, 283( = 1.8) and R˜ C > 3, 005( = 1.75). Meanwhile under the same characteristic frequency, the oscillatory salt-finger convection is more easily exhibited by the disturbance in the upper layer than in the lower layer. That owes for the coupled disturbance in the upper layer, overstability provokes restoring forces as strong as to overshoot the corresponding position on the other side of equilibrium. The relationship between the characteristic frequency and the critical wave

R.-Y. Yang et al. / Acta Astronautica 65 (2009) 591 – 598

Fig. 6. The critical Rayleigh number R˜ c at which overstability sets in under different characteristic frequency.

597

article, the governing equations obtained for stationary onset of salt-fingers convection of a gravity gradient due to local disturbance in parabolic temperature and salinity profiles are similar to those obtained by the smallgap Taylor–Couette problem. Therefore, the results presented by Chandrasekhar [14] for the Taylor–Coutte problem are extended to cover a wide range of application in the present case. Instability of this stationary convection is affected by temperature and salinity concentration gradients that act along the vertical direction. It is shown that thermal diffusivity is a destabilizing factor in the case under consideration, whereas salinity concentration diffusivity is a stabilizing. From the result of overstability analysis, it shows that oscillatory motion of salt-fingers convection will be triggered under a certain definite characteristic frequency. Moreover, there are even two characteristic frequencies existed which overstability will under remain under some conditions (for example:  =2). In overstability, it provokes restoring forces so strong as to overshoot the corresponding position on the other side of equilibrium for the case of coupled disturbance in the upper layer of ocean. References

Fig. 7. The critical wave number ac at which overstability sets in under different characteristic frequency.

number is also shown in Fig. 7. Being in the same result for the case of  = 2, the equivalent critical wave number will similarly occur under two different characteristic frequencies. 5. Conclusions The state of marginal stability would be one of two kinds. If at the onset of instability a stationary pattern of motions prevails, the principle of the exchange of stabilities is set in as the stationary cellular convection, or secondary flow. On the other hand, if at the onset of instability oscillatory motions prevail, that is the case of overstability. From the results of this

[1] H. Stommel, A. Arons, D. Blanchard, An oceanographic curiosity the perpetual salt fountain, Deep-Sea Research 3 (1956) 152–153. [2] R.W. Schmitt, Observational and laboratory insights into salt-finger convection, Progress in Oceanography 56 (2003) 419–433. [3] A.J. Williams, Images of ocean microstructure, Deep-Sea Research 22 (1975) 811–829. [4] B. Magnell, Salt fingers observed in the Mediterranean outflow region (34◦ N, 11◦ W) using a towed sensor, Journal of Physical Oceanography 6 (1976) 511–523. [5] R.B. Lambert, W.E. Sturges, A thermohaline staircase and vertical mixing in the thermocline, Deep-Sea Research 24 (1977) 211–222. [6] T. Osborn, Observations of the salt fountain, Atmosphere-Ocean 29 (2) (1991) 340–356. [7] Y.Y. Kluikov, L.N. Karlin, A model of the ocean thermocline stepwise stratification caused by double diffusion, in: DoubleDiffusive Convection, A. Brandt, J. Fernando (Eds.), AGU Geophysical Monograph, vol. 94, 1995, pp. 287–292. [8] M.E. Stern, The salt-fountain and thermohaline convection, Tellus 12 (1960) 172–175. [9] P.G. Baines, A.E. Gill, On thermohaline convection with linear gradients, Journal of Fluid Mechanics 37 (2) (1969) 289–306. [10] M. Proctor, J. Holyer, Planform selection in salt fingers, Journal of Fluid Mechanics 168 (1986) 241–253. [11] E. Kunze, A review of salt-fingering theory, Progress in Oceanography 56 (2003) 399–417. [12] J.S. Turner, H. Stommel, A new case of convection in the presence of combined vertical salinity and temperature

598

R.-Y. Yang et al. / Acta Astronautica 65 (2009) 591 – 598

gradients, Proceeding of National Academy of Science USA 52 (1964) 49–53. [13] R.W. Schmitt, Double-diffusive convection. Its role in ocean mixing and parameterization schemes for large scale modeling, in: E.P. Chassignet, J. Verron (Eds.), Ocean Modeling and Parameterization, Kluwer Academic Publishers, Dordrecht, 1998, pp. 215–234.

[14] S. Chandrasekhar, On characteristic value problems in high order differential equations arises in studies on hydrodynamic and hydromagnetic stability, American Mathematical Monthly 61 (1969) 32–45.