Atmospheric Dynamics 2
Atmospheric Dynamics 2 ESCI 343
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This 5 page Class Notes was uploaded by Cordie Miller on Thursday October 15, 2015. The Class Notes belongs to ESCI 343 at Millersville University of Pennsylvania taught by Alex DeCaria in Fall. Since its upload, it has received 38 views. For similar materials see /class/223515/esci-343-millersville-university-of-pennsylvania in Earth Sciences at Millersville University of Pennsylvania.
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Date Created: 10/15/15
ESCI 343 Atmospheric Dynamics II Lesson 15 Barotropic Instability Reference Numerical Prediction and Dynamic Meteorology 2 edition GJ Haltiner and RT Williams An Introduction to Dynamic Meteorology 3rd edition JR Holton Reading Holton Section 81 HYDRODYNAMIC INSTABILITY A ow is hydrodynamically unstable if a small perturbation in the ow 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 ow is stable or unstable is by assuming that the perturbation has a sinusoidal waveform such as VI A 6111mm and determining under what circumstances the frequency will have an imaginary component If the dispersion relation has an imaginary component such as a a i a then the perturbation will have the form Ill Aenerm Aeierwtekwt which grows exponentially in time and is therefore unstable if a gt 0 An example of hydrodynamic instability Internal gravity waves with imaginary Brunt Vaisala frequency Recall that the phase speed for internal gravity waves is wiLKH where N is the Brunt Vaisala frequency given by N2 g d g If N is real then the uid 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 wwill have an imaginary component BAROTROPIC INSTABILITY One form ofhydmdymmlc lnstabllltythat can occur In the atmosphere Is condItIon for bamtmpIc Instablllty Is I1 2n wuh a constant mean depth whlch Is 3 I 795 B 7 7 7 1 t Bx V By I H with mean vortIcIty In terms of E as follows 5 2 so that equatlon 1 becomes a 2 a 2 I12 HIV 7V 7V iii 2 at IVqu V I dyz Bl gt We assume a Smusoldal solutIon such as V Blew remember thatAO may be a complex number Into equatlon 2 to get 2 27 krmikrk2A 1577 1340 3 sInce A may be complex we multIply equatlon 3 by Its complex conjugate Aquot to get I n comugate Is 3 I531 number gIven by AA a2 a and Is the magmmde squared ofA k wAdZi k2AA dzljjkAA0 4 dy dy Using the following identity d2A1 dAdAdAi may dy2 dy dy dy dy dy dy dy equation 4 becomes 2 Agtllt 2 2 le mi MLA JLA k2lA2 d k A 20 dy dy dy dy which can also be written as d dA dA2 2 121 kW 7 M7 7 k2lAl 6 2 7 0 5 dy dy dy dy Mk 60 We now integrate equation 5 across the channel as follows d d dA d dA2 d 2 2 I 121 kW 7A7d 7d kAd 7 d0 6 Id dy y Id y 0quot l l y 16 dyz y 0 0 CO The first term on the left hand side of 6 is d Ii AgtkdiA dyz AgtkdiA AgtlltdiA 0 dy dy dy dy 0 but since there is no ow across the boundaries of the channel the streamfunction and therefore Ay and Ay must both disappear on the boundaries Therefore this whole term is zero In the last term in equation 6 we can write 1 1Zk a 12k wiw 12k a ia Inc co 10 kw k wlz k wlz k wlz k wlz so that equation 6 becomes 7 0 y 0 y 0 27 2 7 adjMl uk wzwdy0 dy2 lik a2 OF LA dy 2 d d 2 klAlZQZk w d 12 klAlzw d kZAZd d d 1 Jild 0 7 y l H y 6 dy2 k wlz y dy2 lik wlz y Equation 7 consists of both real and imaginary terms and since the right hand side is zero all the real terms must sum to zero and all the imaginary terms must also sum to zero Since there is only one imaginary term it must be identically zero Therefore we have 0 27 2 v d u k A a ll zjllizdyw lt8 0 dy Mk 60 One way for this term to be zero is for a to be equal to zero In this case the ow is stable However if a is not equal to zero then the only other way for equation 8 to be true is if the term 6 a121Zdy2 is positive in some part of the domain and negative in l another part of the domain The necessary condition for barotropic instability is therefore that 1212 0 9 dyz 3 somewhere in the domain BAROTROPIC INSTABILITY IN A WESTERLY JET STREAM Barotropic instability is dependent upon horizontal shear of the mean ow To examine if barotropic instability is possible the horizontal profile of the mean ow must be examined A simple linear profile of 12 y cannot lead to barotropic instability since the second derivative would be zero For a typical midlatitude jet stream the regions where barotropic instability might be likely are in the regions on the south or north side of the jet where d 21Zdy2 is large and positive Since for a westerly jet d 21Zdy2 starts out small and positive in the lower part of the domain small y then the instability criterion requires that somewhere across the jet 1212 dyz gt 6 This clearly shows the effect of 6 in stabilizing the ow for if 6 were zero instability would set in for lower values of dz dy2 Barotropic disturbances derive their energy from the mean ow Energy considerations show that for a barotropic disturbance to grow it must tilt opposite to d1 dy 2 Since midlatitude disturbances tend to tilt in the same direction as d1 dy they actually lose energy back to the mean ow 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 friction Thus barotropic instability is somewhat important for the maintenance of the mean ow in the midlatitudes 2 See Haltmer and Williams pp7475 EXERCISES 1 a Show that the condition for barotropic instability can be written as stating that i in f 0 dy dy somewhere in the domain l l d1 b Whatis the quantity 7 f dy c Based on the above results is it true to say that a necessary condition for barotropic instability is for the absolute vorticity to have a maximum or a minimum somewhere in the domain
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