Prototype Instabilities

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1 Prototype Instabilities David Randall Introduction Broadly speaking, a growing atmospheric disturbance can draw its kinetic energy from two possible sources: the kinetic and available potential energies of the mean flow. The first type of instability is sometimes called mechanical, and is associated with the shear of the mean flow. The second is called convective, and is associated with the vertical distribution of temperature or density of the mean state. Both types of instability are important for the boundary layer and for small-scale dynamics in general. The two varieties of mechanical instability that we will consider are the inflectional or Rayleigh instability, which is always associated with maximum in the vorticity distribution, and inertial instability, which can only occur in a rotating fluid, and which results from an unstable distribution of angular momentum. Inflectional instability is certainly responsible for the maintenance of turbulence in the stable PBL, and may also play an important role in some other boundary layer phenomena, such as cloud streets and billows. Inertial instability can also occur within both the boundary layer and the cumulus layer. Ekman layers occur in viscous, rotating fluids, and are observed to be subject to Ekman layer instability. The relationships of Ekman layer instability to inflectional instability and inertial instability have been the subject of some discussion in the literature. Convective instabilities play a prominent role in this course, because they maintain the convective turbulence of the deep, unstable PBL, and also lead to cumulus convection. The growth rates and characteristic shapes of convective disturbances are strongly influenced by shear in the atmosphere, and may occasionally be significantly modified by rotation. Both types of instabilities tend to relieve the conditions which lead to their growth, by stabilizing the basic state. Shearing instability transports momentum down gradient and so tends to reduce the shear. Convective instability transports heat upward and so tends to reduce the lapse rate. The two types of instability often interact with each other. For example, a convective thermal creates shear zones with inflectional instability near the its edge. As a second example, finite-amplitude shearing instability can create local patches of convective instability in the form of breaking waves. In this chapter, we use linear theory to investigate the essential mechanisms of each instability. 1

2 Shearing Instability The simplest model of inflectional instability is based on consideration of a vortex sheet, as pictured in Figure 1. Part (a) of the figure depicts a stable configuration, in which the vorticity increases monotonically from the lower to the upper part of the figure. A vortex displaced, as indicated, from its equilibrium position will tend to be carried back by the circulations induced by the neighboring vortices. But if the vorticity has a local maximum, as in part (b) of the figure, then a displaced vortex tends to be carried further away. Such a vorticity distribution is therefore unstable. Infinite vortex sheet (balanced) Perturb one vortex Instability Figure 1: Diagram illustrating the instability of a vortex sheet. The argument above is based on a qualitative appeal to intuition. The following more quantitative analysis is based on a portion of Chapter XI of a book by S. Chandrasekhar (1961). 2

3 Consider stratified motion in the x - direction, and sheared in the z -direction, so that the mean flow is given by U z ( ). We adopt the linearized Boussinesq system of equations, which have the following form: u u +U x + w du dz = p x, v (1) v +U x = p y, (2) w +U w x = p z gρ, (3) ρ ρ +U x + w z = 0, (4) u x + v y + w z = 0. (5) In (4), the stratification enters in the form of the vertical gradient of the mean density. We assume that decreases upward, so that the stratification gives a gravitational restoring force that permits gravity waves. We seek separable solutions of the form ( ) = ( ~ )exp i( kx + ly + σt). With this assumed form, the amplitudes depend only on z and instability can occur only if σ is complex. Substituting (6) into (1-5), we obtain i( σ + ku )u + w du dz = ikp, (7) (6) i( σ + ku )v = ilp, (8) 3

4 i( σ + ku )w = p z gρ, (9) i( σ + ku )ρ + w d dz = 0, (10) i( ku + lv) + w z = 0. (11) Using (7-9) to form the divergence equation, and then substituting from (5), we obtain i σ + ku ( ) w ( ) p. z k du dz w = k 2 + l 2 This is the anelastic pressure equation, which can be used to determine the pressure diagnostically given appropriate boundary conditions, although we will not actually solve (12) here. Elimination of the perturbation density between Eqs. (9) and (10) gives (12) i( σ + ku) 2 w = ( σ + ku) p z igw d dz, (13) which describes gravity waves. Elimination of the pressure between (12) and (13) gives an equation in which the only unknown is the vertical velocity: z ρ 0 σ + ku ( ) w z kw du dz ( k 2 + l 2 ) ( σ + ku )w = g k 2 + l 2 ( ) d dz w σ + ku. Before we can solve this equation, we need some far-field boundary conditions on the vertical velocity. The simplest case is an unbounded domain in which the motions die out far from the shear zone: w 0 as z ±. We also want to allow the possibility of discontinuous changes of U and at an internal boundary defined by z = z s ( x, y,t), which is assumed to be a material surface. To derive the form of the kinematic boundary condition at z = z s ( x, y,t), we revert to the full continuity equation, which is (14) (15) 4

5 ρ + i ρv ( ) = 0. Integrating (16) across a thin layer defined by z s ε z z s + ε, we obtain (16) z s +ε ρ + i( ρv) dz = 0. z s ε Here ε is a small constant that we will shrink to zero just below. Taking the derivatives inside the integrals, and using Leibniz rule, we find that z s +ε z s ε ρ s+ z s + ε +ρ s z s ε ρ dz + i z s +ε z s ε ρvdz ( ) + V i ( z s + ε ) w s+ ( ) + V i ( z s ε ) w s (17) = 0. (18) As ε 0, the integrals on the first line of (18) go to zero, and their derivatives also go to zero, so the terms drop out. We re left with where Δ( ) ( ) s+ ( ) s. We can write Δ ρ z s + V z w s = 0, (19) and z ρ s+ s+ + V s+ z s+ w s+ = F ρ s z s + V s z s w s = F, (20) (21) which automatically satisfy (19). Here F is the mass flux across z s ( x, y,t). If z s is a material surface, then F = 0, so that 5

6 and z s+ + V z s w s+ = 0, (22) z s + V z s w s = 0. If we require that z s+ and z s stay together, so that z s+ = z s = z s, then follows from (22) and (23) that Δ( V z s w) = 0. Eq. (19) says that the mass flux must be continuous, while (24) says that the normal component of the velocity must be continuous. We now return to our earlier analysis. Substitution of (6) into (22) and (23) gives (23) (24) and i( σ + ku + )z s w s+ = 0, (25) respectively. These can be combined to obtain i( σ + ku )z s w s = 0, w Δ σ + ku = 0. Similarly, by vertical integration of (14) we find that Δ σ + ku ( ) w z kw du dz = g ( k 2 + l 2 w )Δ σ + ku s. (26) (27) (28) Here we have used (27) to justify writing w σ + ku two interfacial conditions have to be satisfied, namely (27) and (28). s on the right-hand side of (28). Note that 6

7 We now consider two cases. First, suppose that both U and are discontinuous at the material surface z s, but also that U and are uniform throughout the rest of the domain. Then (14) simplifies to 2 z k ( 2 + l 2 ) 2 w = 0, which applies everywhere except at the interface. Let subscripts 1 and 2 denote the regions below and above the interface, respectively. In view of the far-field boundary conditions given by (17) and the first interfacial condition (27), the solution for w can be written as w 1 = A( σ + ku 1 )exp k 2 + l 2 z w 2 = A( σ + ku 2 )exp k 2 + l 2 z (29) ( ) for z < 0, and ( ) for z > 0, (30) where A is an arbitrary constant. Substitution of (30) into the second interfacial condition, (28), gives, after a few lines of algebra, where we define σ 2 + 2k( α 1 U 1 + α 2 U 2 )σ + k 2 α 1 U 2 2 ( 1 + α 2 U 2 ) = g k 2 + l 2 ( α 1 α 2 ), (31) α i = ρ i ρ 1 + ρ 2, i = 1,2. We can solve (31) as a quadratic equation for σ. After some algebra, the result can be written as (32) ( ) ± g k 2 + l 2 ( α 1 α 2 ) k 2 α 1 α 2 ( U 1 U 2 ) 2 σ = k α 1 U 1 + α 2 U 2 To obtain (33), we have used α 1 + α 2 = 1, which follows from (32). 1 2 (33) Recall that, in order to have instability, σ must have an imaginary part. Eq. (33) shows that instability will definitely occur if the density increases upward, i.e., if α 1 α 2 < 0. In the following discussion we restrict ourselves to the statically stable case α 1 α 2 > 0. For k = 0, U 1, and U 2 drop out of (33), and we just get a gravity wave. This implies that when the interface 7

8 is statically stable all growing disturbances have k > 0. The condition for instability with α 1 α 2 > 0 can be written as g α k 2 + l 2 > 1 α 2 α 1 α 2 U 1 U 2 ( ) ( ) 2 cos 2 θ, (34) where θ is the angle between the wave number vector and the x -direction. The minimum wave number occurs for cos 2 θ = 1, which means that variations occur only in the x -direction. No matter how small U 1 U 2 is made, instability will occur for sufficiently large k 2 + l 2. If the density difference is set to zero, then all modes are unstable. We conclude that the stable stratification stabilizes the longer waves. Now consider the case of continuous U and, but still with a shear layer centered at z = 0. First, consider the following simple analysis. Consider two parcels at heights z and z + dz. We mix the two parcels together, conserving the mean density and mean wind. The increase in the potential energy of the system, due to the work done against the buoyancy force, is g dρ ( )dz, per unit volume. The decrease in the kinetic energy of the mean state is ρ 1 2 U ( 2 U + du ) U +U + du 2 ( )2 = 1 4 ρ( du )2. Recall that the mechanism of shearing instability involves conversion of the kinetic energy of the mean state into the kinetic energy of the growing disturbance. In order to have instability in the present example, the decrease in mean-state kinetic energy must be larger than the work done against buoyancy. This condition can be written as (35) R i < 1 4, (36) where R i g ρ z ρ ( U z) 2 (37) is the Richardson number, or more precisely the gradient Richardson number. Note that the sign of the Richardson number is determined by the sign of d dz. Since we are assuming that the density decreases upward, the Richardson number will be positive. The simple argument 8

9 given above does not show that (36) is a sufficient condition for instability - only that it is a necessary condition. We now consider the problem more carefully. The following analysis is due to Drazin (1958), but is summarized by Chandrasekhar (1961). Let the mean-state density decrease exponentially upward, so that and assume that the wind profile takes the form ρ = 0 e βz, U = U 00 tanh( z d), where β and d are constants, so that the shear is concentrated near z = 0. Then (38) (39) ( R i ) 0 = gβd 2 U 2 00 is a representative value of R i in the vicinity of z = 0. Taking l = 0 for simplicity, we expand (14) as ( σ + ku ) 2 z k 2 2 w k d 2 U dz w gk 2 d w 2 ρ o dz σ + ku + 1 d σ + ku dz Two of the terms in (41) involve d ( ) w (40) z k du dz w = 0. (41) dz. In geophysical applications, the second-last term, which is proportional to g, is much larger than the last term. The physical interpretation is that d dz matters for its effects on the potential energy, but not so much for its effects on the momentum. Neglecting the last term, (41) can be simplified to σ k +U 2 z k 2 w d 2 U 2 dz w g d w 2 dz σ +U 0. k We now nondimensionalize by scaling velocities with U 00 and lengths with d, and define a nondimensional phase speed (42) c σ ku 00. (43) 9

10 Using the definition (43), we can rewrite (42) as ( U c) 2 z k 2 2 w d 2 U dz w + R 2 i w ( ) 0 U c = 0, where all variables are now nondimensional and the symbol U now stands for tanh z. We use the boundary conditions w 0 as z ±. Recall from (6) that instability occurs when the imaginary part of σ is negative. Since c is a negative real number times σ, we can also say that instability occurs when the imaginary part of c is positive. Instability may or may not occur, depending on the values of the parameters. Imagine that we start with parameter values for which instability does not occur, and then progressively change the parameters until we reach a point beyond which instability begins. In the stable regime, disturbances are damped through the action of a restoring force. In the unstable regime, the restoring force no longer restores; instead, it makes the disturbance grow. In the marginal state that separates stability from instability, the disturbance is neutrally stable, and the restoring force is equal to zero. We want to find the parameter values that are characteristic of the marginal state. It is going to turn out that the marginal state occurs for ( R i ) 0 =1/ 4. In the marginal state, the imaginary parts of σ and c are equal to zero. What about their real parts? If the real parts of of σ and c were also equal to zero in the marginal state, then we would simply have σ = c = 0, which would allow us to simplify (44). If the real part of σ is different from zero in the marginal state, then the unstable disturbance appears as a growing oscillation, which means that it grows through the action of an overshooting restoring force. This is called overstability. In contrast, if the real part of σ is zero in the marginal state, then the unstable disturbance appears as a stationary (i.e., nonoscillatory) mode. When the growing disturbance is non-oscillatory, the principle of the exchange of stabilities is said to apply, and the marginal state has σ = c = 0. The terms introduced above are due to Eddington (1926, p. 201). We now assume that the principle of the exchange of stabilities applies to the problem at hand. We therefore set c = 0 in (44), giving U 2 z k 2 2 w d 2 U dz 2 (44) (45) w + R w ( i ) 0 U = 0. (46) 10

11 Because (46) has been obtained from (44) by setting c = 0, solutions of (46) will describe the marginal state. To find the solution of (46), we first make a change of variables. Since U = tanh z is a single-valued function of z, we can use U instead of z as our vertical coordinate. After making this change of variables, (46) becomes d du which is subject to the boundary conditions ( ) 0 ( ) dw 1 U2 du + 2 k 2 1 U 2 + R i U 2 1 U 2 w = 0 at U = ±1. ( ) w = 0, You can confirm by substitution that a solution of (47) that satisfies (48) is (47) (48) where χ is a constant, and we define w = χu µ ( 1 U 2 ) υ, (49) ( ) 0 µ R i, (50) and υ 1 2 k 2 ( R i ) 0. (51) Upon substituting (49) into (47), you will find that ( 2υ + µ + 2) ( 2υ + µ 1) = 0 (52) is the dispersion relation that relates k to ( R i ) 0. Substitution for υ and µ shows that the second factor on the left-hand side of (52) will be equal to zero if we set ( R i ) 0 = k 2 ( 1 k 2 ). (53) 11

12 From (53), we see that ( R i ) 0 has a maximum of 1/ 4 for k 2 =1/ 2. If ( R i ) 0 is larger than 1/ 4, then (53) does not have a solution, which means that instability cannot occur. Suppose that we run a series of experiments, starting with ( R i ) 0 >1/ 4, and then testing progressively smaller values of ( R i ) 0. Instability will not occur until we reach ( R i ) 0 =1/ 4. We say that ( R i ) 0 =1/ 4 is characteristic of the marginal state of the system, and that ( R i ) 0 =1/ 4 is the critical value of the Richardson number. The first sign of instability, at ( R i ) 0 =1/ 4, will be a ( stationary, i.e., non-oscillatory) growing disturbance with k 2 = 1 2. Howard (1961; see also Miles, 1959) proved that if R i exceeds a critical value (essentially 1/4) throughout the fluid, then the motion must be stable in the inflectional sense, and that if the motion is unstable, then R i must fall below the critical value somewhere in the fluid. Notice that this does not prove that instability must occur for R i less than critical. Dutton (1976) discusses Howard s theorem in some detail. An analogous theorem exists for the instability of quasigeostrophic flow. References and Bibliography Chandrasekhar, S., 1961: Hydrodynamic and hydromagnetic stability. Clarendon Press, Oxford, 652 pp. Drazin, P. G., 1958: The stability of a shear layer in an unbounded heterogeneous inviscid fluid. J. Fluid Mech., 4, Dutton, J. A., 1976: The ceaseless wind. McGraw-Hill, New York, 579 pp. Eddington, A. S., 1926: The internal constitution of the stars. Cambridge University Press, 407 pp. Miles, J. W., 1959: On the generation of surface waves by shear flows. III. Kelvin-Helmholtz instability. J. Fluid Mech., 3,

General introduction to Hydrodynamic Instabilities

KTH ROYAL INSTITUTE OF TECHNOLOGY General introduction to Hydrodynamic Instabilities L. Brandt & J.-Ch. Loiseau KTH Mechanics, November 2015 Luca Brandt Professor at KTH Mechanics Email: luca@mech.kth.se