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INSTABILITY IN GEOPHYSICAL FLOWS

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A defining aspect of the course is its emphasis on the numerical solution of boundary value problems; the student learns techniques whose value extends beyond the current topic. Oceanographers think of ocean circulation in terms of a "global conveyor belt," in which cold polar waters sink and then circulate around ocean basins, eventually warming in the tropics.

What Is Instability?

If the equilibrium is stable, disturbances will often take the form of oscillations (eg the car in Figure 1.3a), or waves. For example, the surface of a lake is never perfectly horizontal, but it is usually quite close, because the horizontal equilibrium condition is stable.

Goals

Once we have identified an equilibrium state, the next step is to determine its stability. Forcing by wind, sun, gravity and planetary rotation tends to push the system towards unstable conditions.

Tools

Numerical Solution of a Boundary Value Problem

Since the matrix eigenvalue problem can be easily solved using standard numerical routines (for example, the Matlab function eig5), this approach suggests a convenient way to solve the differential eigenvalue problem. We cannot say the value of the error term in general because it depends on the function f.

The Equations of Motion

This is where viscosity comes in – it models the frictional effect that molecular interactions exert on the macroscopic motions we can observe and measure. As with molecular motions, however, we must account for the effect the gusts have on the larger-scale motions we are trying to understand.

Although this analogy is imperfect,9 the eddy viscosity concept is a useful first step toward understanding the effects of small-scale turbulence. In this book, the quantity we call "viscosity" can refer to either molecular or eddy viscosity.

Appendix: A Closer Look at Perturbation Theory .1 The Parking Problem Revisited

This describes the unrestricted movement of the car away from an unstable equilibrium, i.e. top of a hill. In addition, we have the additional term that arises from the presence of the spring.

The Perturbation Equations

In the simplest convection problems, there is no need to solve differential equations; the problem is purely algebraic. From (2.5) we see that the sum of the first two terms on the right-hand side is equal to zero.1 All remaining terms are proportional to, which therefore cancels out and remains:. 2.14) Again, the equation is linear and does not involve.

Simple Case: Inviscid, Nondiffusive, Unbounded Fluid

Note that the greatest amplitude of the vertical velocity is found at the interface level, and that the motions decrease with distance from the interface over a length scale proportional to the wavelength of the perturbation, k˜−1. The integral on the right is simplified with property 5 of the delta function (Figure 2.5), resulting in . 2.41).

Viscous and Diffusive Effects

A beautiful laboratory demonstration of convective instability arising at the interface between two liquids is shown in Figure 2-6. The removal of this port can be seen to the left of the top panel.

Boundary Effects: the Rayleigh-Benard Problem

As we saw previously in the unbounded cases (section 2.3), the relative strength of gravity and viscosity/diffusion is determined by two factors: the spatial scale of the normal mode, and the orientation of the motions. For a fixed value ofm, as k˜ decreases (i.e. for wider convection cells, see Figure 2.9), the motion becomes mainly horizontal, so the gravitational acceleration is weaker.

Nonlinear Effects

Cells are formed with fluid rising in the center and sinking at the edges (NOAA). b) Convection cells on a 1000 km scale in the solar photosphere. The distance between the strongest plumes is generally of the same order of magnitude as the thickness of the convective layer, reflecting the underlying linear instability.

Summary

In most geophysical examples of convection, Ra is many orders of magnitude larger than the critical value.

Appendix: Waves and Convection in a Compressible Fluid

If the vertical displacement is small, then the buoyant acceleration is −gρ/ρ, where ρ is the density of the parcel minus that of the surrounding fluid. The difference between the density of the parcel and its surroundings is −ρozη, and the parcel oscillates with frequency ω=.

The Perturbation Equations We assume that

Next, assume that the velocity field consists of a parallel shear flow and a disturbance:. where is an arbitrary constant as defined in Section 3.1.1. But the buoyancy and viscosity terms are now neglected and instead we have two new terms that describe the interactions between the turbulence and the parallel shear flow.

Rayleigh’s Equation .1 Normal Modes in a Shear Flow

The tilt angle, ϕ, is the angle between the wave vector and the direction of the background current (x), and is in the range −π/2 ≤ ϕ ≤ π/2. Returning to the perturbation equation (3.12) and replacing it with (3.13), the normal mode form written in terms of σ, we obtain a second-order ordinary differential equation forw(ˆ z):. 3.17) This is called Rayleigh's equation after Lord Rayleigh, the inventor of the normal modes (Rayleigh, 1880).

Analytical Example: the Piecewise-Linear Shear Layer

Based on these results, we can state four rules about the fastest growing instability of a piecewise linear shear layer:. i) The slant angle is zero, i.e. the wave crests are perpendicular to the mean flow. iv). The disturbance moves with the speed of the background flow in the middle of the shear layer.

Solution Types for Rayleigh’s Equation

Numerical Solution of Rayleigh’s Equation

In the ocean example, one could easily imagine that a boundary placed e.g. 1000 m below the shear layer would have a negligible effect on the results and reduce N to ∼1000. This is precisely what we did in the analytical example of the piecewise-linear shear layer (Section 3.3).

Shear Scaling

Now suppose we want to analyze the stability of some class of profiles of the form (3.46) using the Rayleigh equation. Substituting the scaling transformations (3.47) and (3.51) and multiplying by h3/u20 yields. 3.53) The shear scaling is both a labor-saving device (you can analyze a whole class of flows at once) and a source of insight.

Oblique Modes and Squire Transformations

Conversely, for each oblique mode with wavevector(k, ) and growth rateσ, there is a corresponding 2D mode(k˜,0) with higher growth rateσ˜ =σ/cosϕ (Figure 3.10). As a result, the fastest growing mode is always 2D. If we have some arbitrary flow profile U(z), and we want to know the growth rate for allk, we only need to calculate the growth rate for the 2D cases, i.e. = 0, then for any = 0 simply multiply the result by cosϕ.

Rules of Thumb for a General Shear Instability

Numerical Examples

The result shown in Figure 3.12 is comparable to the piecewise linear shear layer: the growth rate rises to a peak around k =0.44, and the maximum value is σ =0.19. This aspect ratio can be compared to Figure 3.13: take the beam width as the width of the island and λ as the wavelength of the instability.

Perturbation Energetics

The energy flow carries kinetic energy in a vertical direction and its convergence (or divergence) at a given z results in an accumulation (or depletion) of energy at that height. Kinetic energy (a) is extracted from the mean flow within the shear layer by shear production (b) and then transported vertically away from the shear layer by energy flux (c).

Necessary Conditions for Instability

Therefore, SP must have at least one positive local maximum somewhere in the interior of the flow (e.g., Figure 3.16b). Therefore, the inflection point is also an extremum of the squared displacement:(Uz2)z =0, and we can rewrite (3.70) as.

The Wave Resonance Mechanism of Shear Instability

Now suppose that the lower edge of the shear layer is also deformed into a stationary sinusoid (Figure 3.20b). Let us next ask what becomes of a sinusoidal disturbance at the upper edge of the shear layer (Figure 3.22b).

Quantitative Analysis of Wave Resonance

Can these discrepancies be explained by the influence of each wave on the phase speed of the other. All of this can be seen quantitatively by writing the rate of rise and the phase velocity in terms of the phase relationship between the upper and lower waves.

Summary

Finally, the resonant interaction of vortex waves provides an explanation of the shear instability mechanism, Rayleigh's and Fjørtoft's theorems, and criteria for critical level and phase tilt. In more complex parallel shear flows, the instability can be understood in terms of the resonance of multiple eddy waves.

Appendix: Classical Proof of the Rayleigh and Fjørtoft Theorems

Characteristics of the fastest growing state:. i) The wave vector is directed parallel to the background current, i.e. there is no variation in the cross-flow direction. ii) If handu0 is defined as length and velocity scales characteristic of the mean flow U(z), then the wavelength is proportional to h and the growth rate is proportional tou0/h. Since dUz2/d z must change sign atzI, (3.95) requires atdUz2/d z to be negative just above zI and positive just below zI instead of the opposite, i.e. the inflection point must be a maximum (not a minimum) of Uz2.

When the tank is tilted, the dense lower layer flows down, forcing the upper layer up, resulting in an accelerated, stratified shear flow. However, in some cases, statically stable stratification can interact with shear to create new instability mechanisms that would not exist in a homogeneous shear flow.

The Richardson Number

Equilibria and Perturbations

The last term on the left describes the advection of the background buoyancy gradient by the vertical velocity perturbation, just as we saw in (2.12). The second term on the left is new; it describes the advection of buoyant disturbances by the background flow (which was zero in the motionless case).

Oblique Modes

The growth rate of the 3D mode is cosϕ times larger than the corresponding 2D mode, which exists in a liquid with stronger stratification. In most circumstances this means that the oblique mode will have a slower growth rate, but if stratification somehow increased the growth rate fast enough to compensate for the skewness factor cosϕ, then the oblique mode may grow faster.

The Taylor-Goldstein Equation

Now suppose that, when the rib is not too large, there is a stationary instability like the one we found in the unstratified case (section 3.9.1). Also, as this critical Richardson number is approached, the wavenumber of the fastest growing mode approaches 1/2, not much different from the value of 0.44 found in the unstratified case.

Application to Internal Wave Phenomena

More precisely, the ratio of the wavelength 2π/k to the shear layer thickness 2h approaches 2π, while in the homogeneous case the value is 7. For example, in the weakly nonlinear theory of solitary waves in a stratified shear flow, the dependence on x and t is described by the Korteweg-De Vries equation, while the vertical structure of the solution of equation TG (4.18) in hydrostasis is the tic limit (Lee and Beardsley, 1974 ).

Analytical Examples of Instability in Stratified Shear Flows Like the Rayleigh equation (3.16–3.19), the TG equation (4.18) is easy to solve

This is due to the delta function behavior of the vorticity and buoyancy gradient profiles viz. As Ribis increased, the growth rate of the instability is reduced and the band of wavenumbers it occupies shrinks to zero.

The Miles-Howard Theorem

It is important to understand what the Miles-Howard theorem says about the stability of certain U(z) and B(z) profiles and what it doesn't. We have already seen the case of a continuously stratified shear layer (4.19), where instability requires that the smallest Ri is less than 0.25, consistent with the Miles-Howard theorem.

Howard’s Semicircle Theorem

The stratification term on the left-hand side of (4.43) becomes insignificant when we take out the imaginary part. Therefore, in the second term on the right-hand side of (4.45), we can change U tocr, which results in

Energetics

By analogy with the development of the kinetic energy equation, we multiply by bybˆ∗, take the real part and divide by 2. Referring to the kinetic energy budget (4.52) we see that, when Bz >0, the momentum flux can only act to reduce growth, as we assumed in Section 41.

Summary

Appendix: Veering Flows

Appendix: Spatial Growth

So the real part of the phase velocity is, not surprisingly, the average of the velocities of the two currents. The momentum equation (1.19), neglecting buoyancy but retaining viscosity, is Du. 5.2) The viscosity term raises the order of the system and therefore requires additional boundary conditions.

Conditions for Equilibrium Consider a parallel shear flow

We now imagine a parallel shear flow in a fluid that is homogeneous but does not have a viscosity ν (e.g. Figure 5.1). For now we will just observe that a viscous fluid must move with the boundary, so for a stationary boundary =v=w=0.

Conditions for Quasi-Equilibrium: the Frozen Flow Approximation

In this case, the boundary atz= H moves with speed u0 and the boundary atz=0 is stationary (Figure 5.2b). For example, suppose the instability is a shear instability, so the growth rate is similar.

The Orr-Sommerfeld Equation We now substitute perturbation forms

Boundary Conditions for Viscous Fluid

Outside the viscous boundary layer, the velocity changes much more slowly; the fluid slides across as if the boundary were frictionless. An equivalent way of stating this is that the viscous flux flux-ν∂u/∂zand−ν∂v/∂zvanises at the boundary.

Numerical Solution of the Orr-Sommerfeld Equation Write the Orr-Sommerfeld equation in the form (5.15)

Oblique Modes

But there is another important difference: the corresponding 2D mode grows on a flow with increased viscosityν˜ =ν/cosϕ. If this were true, and if that viscous destabilization was sufficient to overcome the leading factor cosϕ, then a 3D mode could grow faster than the corresponding 2D mode.

Shear Scaling and the Reynolds Number Here again is the Orr-Sommerfeld equation (3D)

As in the inviscid case, a general 3D mode with φ = 0 and growth rate σ corresponds to a 2D mode (φ =0), whose growth rate reduces to σcosϕ. It is often true that viscosity has the effect of moisture instability, as we have seen in the case of convection (Chapter 2).

Numerical Examples

Frictionless boundaries are placed at z = ±5. a) Fastest growth rate of shear layer instability versus Re. b) Wavenumber of the fastest growing mode. For Re >8×104 (above the line), the growth rate decreases with increasing Reor, in other words, increases with increasing viscosity.

Perturbation Energetics in Viscous Flow

The term νKzz is the convergence of a flux of kinetic energy due to viscosity, namely−νKz. The dissipation term-ε is negative definite and therefore destroys the kinetic energy of the turbulence (turning it into heat).

Summary

In this chapter, we investigate equilibria and disturbances in stratified, parallel shear flow with viscosity and diffusion effects included, effectively unifying Chapters 2, 3, 4, and 5.

Expanding the Basic Equations

The perturbation part of (6.9) is obtained by subtracting (6.10) and omitting terms of order2:. 6.11) Note that (6.11) combines the buoyancy perturbation equations for a motionless, inhomogeneous fluid (2.12) and a stratified, nondiffusive, parallel shear flow (4.10). We eliminate the pressure, as we did before, by combining the divergence of (6.8) with the Laplacian of its vertical component.

Numerical Solution

Solutions of the Taylor-Goldstein equation for incompressible stratified shear flows (Chapter 4) are obtained by settingν =κ = 0. The corresponding 2D mode has a wave vector of the same magnitude,k, but parallel˜ to the thex axis (Figure 3.10).

Shear and Diffusion Scalings

In this case, the velocity gap in the solution process would be filled by U = Re Pr z. Remember that the algorithm for solving F has not changed since it was first defined in (6.16).

Application: Instabilities of a Stably Stratified Shear Layer Back in section 4.6, we studied Kelvin-Helmholtz and Holmboe instabilities using

This suggests that the essential mechanisms of the instabilities are captured in the simple piecewise profiles. Each of the three mode families in Figure 6-9a has its own fastest growing mode and its own critical level.

Summary

The cylindrical (or columnar) vortex is the simplest example of a non-parallel shear flow and is a useful model for tornadoes and other geophysical vortices. Here we will examine two classes of eddy instabilities: (1) barotropic instabilities are closely analogous to the instabilities of a parallel shear flow, while (2) axisymmetric instabilities are similar to convection, but with the centrifugal force playing the role of gravity.

Cyclostrophic Equilibrium

The Perturbation Equations Now imagine a small perturbation to cyclostrophic equilibrium

For barotropic modes (Figures 7.2 and 7.3), the trick is to recognize that the perturbation flow is two-dimensional and non-divergent, and thus can be represented by a stream function. If an impermeable boundary is placed at summer1, then the radial velocity must vanish there, i.e. the boundary condition is justuˆ(r1)=0.

Analytical Example: the Rankine Vortex

The dispersion relation is obtained as in the analysis of both convection at an interface (Section 2.2.4) and the instability of a piecewise linear shear layer (Section 1).

Numerical Example: a Continuous Vortex

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