ESCI 343 – Atmospheric Dynamics II Lesson 15 – Barotropic and Baroclinic Instability Reference: Numerical Prediction and Dynamic Meteorology (2nd edition), G.J. Haltiner and R.T. Williams An Introduction to Dynamic Meteorology (3rd edition), J.R. Holton Dynamics of the Atmosphere: A Course in Theoretical Meteorology, W. Zdunkowski and A. Bott HYDRODYNAMIC INSTABILITY A flow is hydrodynamically unstable if a small perturbation in the flow grows spontaneously. Examples of hydrodynamic instability that we’ve already studied are buoyant instability and inertial instability. In both these cases an air parcel moved from its original position will continue to accelerate away from where it started, instead of oscillating around its original position. One method of assessing whether or not a flow is stable or unstable is by assuming that the perturbation has a sinusoidal waveform such as )( tkxiAeωψ−=′ and determining under what circumstances the frequency will have an imaginary component. If the dispersion relation has an imaginary component such as iriωωω+= , then the perturbation will have the form tktkxitkxiireAeAeωωωψ)()(−−==′ which grows exponentially in time (and is therefore unstable) if ωi > 0. An example of hydrodynamic instability: Internal gravity waves with imaginary Brunt -Vaisala frequency Recall that the phase speed for internal gravity waves is ΚΚ±=HNω, where N is the Brunt-Vaisala frequency given by 222scgdzdgN −−=ρρ. If N is real then the fluid is stable, and a parcel disturbed vertically from rest would oscillate about its original position. However, if N is imaginary then we know a parcel will be unstable, and if perturbed from rest it will accelerate away from its original position. This can also be seen from the dispersion relation, since N will be imaginary, and hence ω will have an imaginary component.BAROTROPIC INSTABILITYOne form of hydrodynamic instability that can occur in the atmosphere is barotropic instability. The derivation of the condition for barotropic instability is beyond the scope of this course. But, the condition for barotropic instability involves the horizontal shear of the mean wind. The necessary condition for barotropic instability to occur is that, somewhere within the flow, the following condition must be true:This means that for barotropic instability to occur that the second derivative of the mean zonal wind must be equal to βas or We can interpret this to mean that the absolute vorticity must have a minimum or maximum value somewhere in the flow in order for barotropic instability to occur. BAROTROPIC INSTABILITY IN A WESTERLY JET STREAMBarotropic instability is dependent upon hoexamine if barotropic instability is possible, the horizontal profile of the vorticity must be examined. and the second derivative of the velocityplane. The dashed line on the third diagram is the value of beta.2BAROTROPIC INSTABILITY One form of hydrodynamic instability that can occur in the atmosphere is The derivation of the condition for barotropic instability is beyond the scope of this course. But, the condition for barotropic instability involves the horizontal shear of the mean wind. The necessary condition for barotropic instability to that, somewhere within the flow, the following condition must be true:022=−βdyud. This means that for barotropic instability to occur that the second derivative of the mean β somewhere in the flow. Condition (1) can also be written 0d dufd y d y − =   0dd yη=. e can interpret this to mean that the absolute vorticity must have a minimum or maximum value somewhere in the flow in order for barotropic instability to occur.BAROTROPIC INSTABILITY IN A WESTERLY JET STREAM Barotropic instability is dependent upon horizontal shear of the mean flow. To examine if barotropic instability is possible, the horizontal profile of the absolute must be examined. The plot below shows the zonal velocity, absolute vorticityand the second derivative of the velocity for an idealized westerly jet stream The dashed line on the third diagram is the value of beta. One form of hydrodynamic instability that can occur in the atmosphere is The derivation of the condition for barotropic instability is beyond the scope of this course. But, the condition for barotropic instability involves the horizontal shear of the mean wind. The necessary condition for barotropic instability to that, somewhere within the flow, the following condition must be true: . (1) This means that for barotropic instability to occur that the second derivative of the mean can also be written (2) . (3) e can interpret this to mean that the absolute vorticity must have a minimum or maximum value somewhere in the flow in order for barotropic instability to occur. rizontal shear of the mean flow. To bsolute absolute vorticity, on the betaThere are absolute vorticity minima and maxima on both flanks of the jet, locations of the inflection points in the velocity profile. Thus, barotropic instability is met in these two regions. However, the presence of an inflection point does not automatminimum or maximum in the absolute vorticity. derivative of the velocity, such as for a broad, weak jetthen there will not be any maxima or minimapoints in the velocity profile. Thus, beta actsinstability. ENERGETICS OF BAROTROPIC DISTURBANCES Barotropic disturbances derive their energy from the mean flow. Energy considerations show that for a barotropic disturbance to dyud.1 Since midlatitude disturbances tend to tilt in the actually lose energy back to the mean flow due to barotropic instability. Thus, barotropic instability is not a viable way for midlatitude disturbances to form and grow. However, interestingly enough, since midlatitude disturbance decay due to barotropic instability, they give up energy to the mean flow and help maintain the mean flow against frictionThus, barotropic instability is somewhat important for the maintenance of the mean flow in the midlatitudes. BAROCLINIC INSTABILITY 1 See Haltiner and Williams, pp.743 minima and maxima on both flanks of the jet, near the inflection points in the velocity profile. Thus, the