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Greta Leber MPO 664 Final Project Thursday, December 2

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Solitary Meander Characteristics (de Ruijter et al, 1999): Downstream velocity ranges from 5 (North) to 20 (South) km day-1, Diameter ranges from 30- 200 km Amplitude ranges from 10 cm to >1m, Period between 50 and 150 days. http://www.afro-sea.org.za/

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Tsujino et al, 2006 Tsujino et al 2006: Meanders are a large component of the mean flow in the Kuroshio which takes either a: nearshore nonlarge meander path (NNLM), offshore nonlarge meander path (ONLM), or typical large meander path (TLM). Kuroshio meanders are controlled by topography.

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Miller and Lee, 1995a Figure 3 Lee et al, 1981: Southward warm Filament Flow balanced by frictional dissipation Chew, 1981: Anticyclonic Circulation within warm Filament Watts and Johns 1982: o Wavelengths ~150-200 km and velocity ~30- 40 km day -1 (increasing downstream) o Over 96% of the variance is from periods longer than 4 days. Miller and Lee, 1995b: o Features are primarily geostrophic.

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Vukovich and Maul, 1985 Aperiodic 100-300 km Anticyclonic Meanders form on both sides then join (~60 days later) to form a cyclonic cold ring (upwelled water) Occur consistently in mid- Winter and mid-Spring Authors speculate they are formed by barotropic/baroclinic instability locally spun up by Loop Current or a shed Anticyclonic Ring.

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http://oceanmotion.org/html/background/wes tern-boundary-currents.htm Miller and Lee, 1995a Meanders extract energy from mean flow via weak baroclinic instability process Decrease in bottom slope causes flow to become strongly baroclinically unstable Bottom slope increases North and Instability Effectively shut down This Instability has also been applied to other WBCs

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Van Der Vaart and De Ruijter 2001 Time-dependent transfer of momentum between instability (“wave”) and Mean flow Inshore Mode is the dominant instability Van Der Vaart and De Ruijter 2001; Fig 15a Unstable mode (amplitude) grows in time Van Der Vaart and De Ruijter 2001; Fig 15c Jet is broadened and slowed and becomes stable to further perturbation

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Numerical Simulation (Cessi and Ierley, 1993): Viscous instability of the barotropic WBC Originate within the shear layer Characteristic scale of the viscous boundary- layer thickness (~longitudinal scale of the mean current). Mean flow is just stable due to eddy viscosity Turbulence acts to “renormalize” friction so that the mean flow is just supercritical. http://www.mathworks.cn/matl abcentral/fileexchange/?term= tag%3A%22flow%22

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Stability of Poiseuille flow is often evaluated by considering the Orr- Sommerfeld Equation. To do this, consider a 2-D disturbance, where y component of velocity is proportional to the real part of: Side walls at y±1 Re=1/ v, v: kinematic viscosity α is real Then the velocity perturbation from linear Navier-Stokes Equations reduce to the ‘Orr-Sommerfeld’ Equation: BCs: at Above implies that: Is an unstable linear eigenmode (e.g amplitude of disturbance grows exponentially in time) Orszag 1971

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Orszag, 1971 for Laboratory Flows find: With application to Oceanic WBCs (addition of β-term), Ierley and Young, 1991 find (depending on profile): The authors speculate that this value is so low due to the jetlike structure of the profile. Ierley and Young 1991 Fig 1 Solid Line: No-slip case Dashed Line: Slip case Dotted: No-slip basin mode

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Trapped Mode Trapped at the W. wall First Instability to appear These disturbances do not radiate because their meridional phase speed is greater than the fastest free Rossby wave with the same meridional wave number: c>c max Basin Mode Rossby wave basin modes, some of which extract energy from mean flow and grow. Requires a higher Re c to appear. These disturbances radiate because c

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Cessi and Ierley 1993 expand upon the Orr-Sommerfeld plus β equation and add in a dependence on bottom tilt, θ. Cessi and Ierley 1993; Fig 2 Re c Min occurs at θ~-33.5° Most Unstable tilt Cessi and Ierley 1993; Fig 3 Nondimensional phase velocity, c, decreases monotonically with θ

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Solitary meanders are a predominate feature occurring in most strong boundary currents with similar characteristics. Mean flow needs to be perturbed to produce a meander, and the exact form of the perturbation is region-specific (topography, ring-interaction, atmospheric features, etc.). Theory shows that these meanders can be thought of as an unstable Poiseuille flow whose stability can be assessed using the Orr-Sommerfeld equation. Evaluation of the Orr-Sommerfeld equation shows that Re c is dependent upon topographic slope, which provides another parameter that is region specific.

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Cessi, P., and Ierley, G. R., 1993: Nonlinear Disturbances of Western Boundary Currents. AMS. 23: 1727-1735. de Ruijter, W. P. M., van Leeuwen, P. J., Lutjeharms, J. R. E., 1999: Generation and evolution of Natal Pulses: solitary meanders in the Agulhas Current. J. Phy. Ocean. 29: 3043-3055. Ierley, G. R., and Young, W. R., 1991: Viscous Instabilities in the Western Boundary Layer. AMS. 21: 1323-1332. Miller, J. L., and Lee, T. N., 1995a: Gulf Stream meanders in the South Atlantic Bight 1. Scaling and energetics. J Geo. Res. 100: 6687-6704. Miller, J. L., and Lee, T. N., 1995b: Gulf Stream meanders in the South Atlantic Bight 2. Momentum balances. J. Geo. Res. 100: 6705-6723. Orszag, S. A., 1971: Accurate solution of the Orr-Sommerfeld stability equation. J. Fluid Mech. 50: 689-703. Tsujino, H., Usui, N., Nakano, H., 2006: Dynamics of Kuroshio path variations in a high- resolution general circulation model. J. Geo. Res. 111. 1-25. Van Der Vaart, P. C. F., and De Ruijter, W. P. M., 2001: Stability of Western Boundary Currents with an Application to Pulselike Behavior of the Agulhas Current. AMS. 31: 2625-2644. Vukovich, F. M., and Maul, G. A., 1985: Cyclonic Eddies in the Eastern Gulf of Mexico. AMS. 15:105-117. Watts, R., and Johns, W. B., 1982: Gulf Stream Meanders: Observations on Propagation and Growth. J Geo. Res. 87: 9467-9476.

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Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Modal Stability TheoryLecture notes from the FLOW-NORDITA Summer School on Advanced.

Date of download: 7/7/2016 Copyright © ASME. All rights reserved. From: Modal Stability TheoryLecture notes from the FLOW-NORDITA Summer School on Advanced.

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