AST242 LECTURE NOTES PART 5 Contents 1. Waves and ...

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16 AST242 LECTURE NOTES PART 5taken into account here but does happen for example when star formation heats orperturbs the medium affecting future generations of stars.When there is conductivity and k is large there is stability even when the stabilitycriterion shown in equation (99) is violated. In this case the thermal conductivitysmoothes or damps temperature fluctuations faster than they can grow. We candefine a critical wavevector by setting ω = 0 in equation (99) and solving for k(100) k 2 F ≡ 1λT 0∂Q∂ ln TThe Field length λ F ≡ 2π/k F . For wavelengths larger than the critical one perturbationswill grow. Small perturbations are smoothed out by the conductivity and sostabilized.4.3. An aside on positive and negative feedback. The very simplest systemdΣ(101)dt = f(Σ)can achieve equilibrium near a fixed point Σ ∗ where f(Σ ∗ ) = 0. Expanding aboutthe fixed point and letting x = Σ − Σ ∗dx(102)dt = f ′ (Σ ∗ )xThis has an exponential solution that is exponentially decaying to the fixed point iff ′ (Σ ∗ ) < 0. If the derivative is negative we say the system has negative feedbackand we expect a stable system. Otherwise we say the system has positive feedbackand we expect it to be unstable. Note that f ′ (Σ) is in units of inverse time and sodefines a decay or growth timescale;(103) t decay = − 1f ′ (Σ) .A system with delayed feedback might look like thisdΣ(t)(104)= f(Σ(t − τ))dtso that its rate of change depends on the density Σ at a time τ in the past. Wecould imagine a complicated function f that depends on Σ at various times. Thereis a class of differential equations called delayed differential equations that can beemployed. Instability can occur even when there is negative feedback in systems withdelayed feedback. Instability or oscillating solutions can occur when the feedbackdelay timescale τ exceeds the timescale for decay to equilibrium, t decay . For exampleπ(105) ˙Σ = − Σ(t − τ)2τ∣∣p
has solution(106) Σ ∝ sinAST242 LECTURE NOTES PART 5 17( ) πt.2τ. Oscillating solutions can have frequencies that period related to the delay; hereP ∼ 4τ.5. Convective InstabilityFigure 3. A small fluid element is displaced by δz. Originally it hadpressure and density p, ρ. At its new location the ambient pressureand density are p ′ , ρ ′ . We let the fluid element expand or contractto reach pressure equilibrium at its new location without exchangingheat. Its new density is ρ ∗ . If ρ ∗ > ρ ′ and the fluid element is denserthan the ambient medium then it will sink back to its original location.Otherwise it will keep rising and the system is convectively unstable.Consider a small fluid element that is located in fluid that is in hydrostatic equilibriumin a gravitational field. Our fluid element starts in equilibrium with pressureand density p, ρ. We then displace it by a vertical amount δz. At this new positionthe ambient pressure and density are slightly different with pressure and densityp ′ , ρ ′ . When we displaced our fluid element it may have contracted or expanded. Ifits new density ρ ∗ is larger than the new ambient value ρ ′ then the fluid elementwill sink back to its original position. In this case the system is stable. If the newfluid element density ρ ∗ is lower than the new ambient value then the fluid elementis buoyant and we say the system is convectively unstable.How are we going to let the fluid element change? We let it remain in pressureequilibrium with its surroundings. This means its new pressure is p ′ . We don’tallow it to heat or cool while we displace it. This is equivalent to assuming that
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AST242 LECTURE NOTES PART 5 Contents 1. Waves and ...

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