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Greta Leber MPO 664 Final Project Thursday, December 2. Instabilities of western boundary currents. Basic characteristics of meanders. Basic Characteristics: Agulhas Meanders. Solitary Meander Characteristics (de Ruijter et al, 1999):

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greta leber mpo 664 final project thursday december 2
Greta Leber

MPO 664

Final Project

Thursday, December 2

Instabilities of western boundary currents

basic characteristics agulhas meanders
Basic Characteristics: Agulhas Meanders
  • 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/

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

Tsujino et al, 2006

basic characteristics gulf stream meanders
Basic Characteristics: Gulf Stream Meanders

Lee et al, 1981:

Southward warm Filament Flow balanced by frictional dissipation

Chew, 1981:Anticyclonic Circulation within warm Filament

Miller and Lee, 1995a Figure 3

  • Watts and Johns 1982:
    • Wavelengths ~150-200 km and velocity ~30-40 km day-1 (increasing downstream)
    • Over 96% of the variance is from periods longer than 4 days.
  • Miller and Lee, 1995b:
    • Features are primarily geostrophic.
basic characteristics loop current meanders
Basic Characteristics: Loop Current Meanders
  • 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.

Vukovich and Maul, 1985

instabilities mixed baroclinic barotropic gs
Instabilities: Mixed Baroclinic-Barotropic (GS)

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

http://oceanmotion.org/html/background/western-boundary-currents.htm

This Instability has also been applied to other WBCs

Miller and Lee, 1995a

instabilities wave mean flow interaction ac
Instabilities: Wave-Mean Flow Interaction (AC)

Time-dependent transfer of momentum between instability (“wave”) and Mean flow  Inshore Mode is the dominant instability

Unstable mode (amplitude) grows in time

Jet is broadened and slowed and becomes stable to further perturbation

Van DerVaart and De Ruijter 2001; Fig 15c

Van DerVaart and De Ruijter 2001; Fig 15a

Van DerVaart and De Ruijter 2001

instabilities viscous shear
Instabilities: Viscous Shear
  • 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/matlabcentral/fileexchange/?term=tag%3A%22flow%22

basic characteristics general meanders theory
Basic Characteristics: General Meanders (Theory)
  • 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:

Above implies that:

Is an unstable linear eigenmode (e.g amplitude of disturbance grows exponentially in time)

at

Orszag 1971

criteria for onset reynold s number
Criteria For Onset: Reynold’s Number
  • 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

criteria for onset trapped and basin modes
Criteria For Onset: Trapped and Basin Modes
  • Basin Mode
    • Rossby wave basin modes, some of which extract energy from mean flow and grow.
    • Requires a higher Rec to appear.
    • These disturbances radiate because c<cmax.
  • 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>cmax

Ierley and Young 1991

criteria for onset topography
Criteria For Onset: Topography
  • 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

Cessi and Ierley 1993; Fig 3

Rec Min occurs at θ~-33.5°  Most Unstable tilt

Nondimensional phase velocity, c, decreases monotonically with θ

summary and conclusions
Summary and Conclusions
  • 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 Rec is dependent upon topographic slope, which provides another parameter that is region specific.
references
References
  • 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 DerVaart, 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.