Skip to content

Research at St Andrews

Numerical simulation of shear-induced instabilities in internal solitary waves

Research output: Contribution to journalArticle

DOI

Open Access permissions

Open

Standard

Numerical simulation of shear-induced instabilities in internal solitary waves. / Carr, Magda; King, Stuart Edward; Dritschel, David Gerard.

In: Journal of Fluid Mechanics, Vol. 683, 25.09.2011, p. 263-288.

Research output: Contribution to journalArticle

Harvard

Carr, M, King, SE & Dritschel, DG 2011, 'Numerical simulation of shear-induced instabilities in internal solitary waves', Journal of Fluid Mechanics, vol. 683, pp. 263-288. https://doi.org/10.1017/jfm.2011.261

APA

Carr, M., King, S. E., & Dritschel, D. G. (2011). Numerical simulation of shear-induced instabilities in internal solitary waves. Journal of Fluid Mechanics, 683, 263-288. https://doi.org/10.1017/jfm.2011.261

Vancouver

Carr M, King SE, Dritschel DG. Numerical simulation of shear-induced instabilities in internal solitary waves. Journal of Fluid Mechanics. 2011 Sep 25;683:263-288. https://doi.org/10.1017/jfm.2011.261

Author

Carr, Magda ; King, Stuart Edward ; Dritschel, David Gerard. / Numerical simulation of shear-induced instabilities in internal solitary waves. In: Journal of Fluid Mechanics. 2011 ; Vol. 683. pp. 263-288.

Bibtex - Download

@article{8ea58242941745c68f1a279b3ca2af9d,
title = "Numerical simulation of shear-induced instabilities in internal solitary waves",
abstract = "A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to simulate shear-induced instabilities in an internal solitary wave (ISW). A three-layer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady-state, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ric) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely Ric = 0.1 ± 0.01 , is not a sufficient condition for instability. It is also shown that the critical value of Lx/λ required for instability, where Lx is the length of the region in a wave in which Ri < 1/4 and λ is the half-width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that δiTw (where δi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to Ri < 1/4) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely Lx/λ ≥ 0.86 and δiTw > 5 , are shown to be valid only for a limited parameter range.",
keywords = "Internal waves, Solitary waves, Stratified flows",
author = "Magda Carr and King, {Stuart Edward} and Dritschel, {David Gerard}",
note = "This work was supported by the UK Engineering and Physical Sciences Research Council [grant number EP/F030622/1]",
year = "2011",
month = "9",
day = "25",
doi = "10.1017/jfm.2011.261",
language = "English",
volume = "683",
pages = "263--288",
journal = "Journal of Fluid Mechanics",
issn = "0022-1120",
publisher = "CAMBRIDGE UNIV PRESS",

}

RIS (suitable for import to EndNote) - Download

TY - JOUR

T1 - Numerical simulation of shear-induced instabilities in internal solitary waves

AU - Carr, Magda

AU - King, Stuart Edward

AU - Dritschel, David Gerard

N1 - This work was supported by the UK Engineering and Physical Sciences Research Council [grant number EP/F030622/1]

PY - 2011/9/25

Y1 - 2011/9/25

N2 - A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to simulate shear-induced instabilities in an internal solitary wave (ISW). A three-layer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady-state, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ric) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely Ric = 0.1 ± 0.01 , is not a sufficient condition for instability. It is also shown that the critical value of Lx/λ required for instability, where Lx is the length of the region in a wave in which Ri < 1/4 and λ is the half-width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that δiTw (where δi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to Ri < 1/4) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely Lx/λ ≥ 0.86 and δiTw > 5 , are shown to be valid only for a limited parameter range.

AB - A numerical method that employs a combination of contour advection and pseudo-spectral techniques is used to simulate shear-induced instabilities in an internal solitary wave (ISW). A three-layer configuration for the background stratification, in which a linearly stratified intermediate layer is sandwiched between two homogeneous ones, is considered throughout. The flow is assumed to satisfy the inviscid, incompressible, Oberbeck–Boussinesq equations in two dimensions. Simulations are initialized by fully nonlinear, steady-state, ISWs. The results of the simulations show that the instability takes place in the pycnocline and manifests itself as Kelvin–Helmholtz billows. The billows form near the trough of the wave, subsequently grow and disturb the tail. Both the critical Richardson number (Ric) and the critical amplitude required for instability are found to be functions of the ratio of the undisturbed layer thicknesses. It is shown, therefore, that the constant, critical bound for instability in ISWs given in Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), namely Ric = 0.1 ± 0.01 , is not a sufficient condition for instability. It is also shown that the critical value of Lx/λ required for instability, where Lx is the length of the region in a wave in which Ri < 1/4 and λ is the half-width of the wave, is sensitive to the ratio of the layer thicknesses. Similarly, a linear stability analysis reveals that δiTw (where δi is the growth rate of the instability averaged over Tw, the period in which parcels of fluid are subjected to Ri < 1/4) is very sensitive to the transition between the undisturbed pycnocline and the homogeneous layers, and the amplitude of the wave. Therefore, the alternative tests for instability presented in Fructus et al. (J. Fluid Mech., vol. 620, 2009, pp. 1–29) and Barad & Fringer (J. Fluid Mech., vol. 644, 2010, pp. 61–95), respectively, namely Lx/λ ≥ 0.86 and δiTw > 5 , are shown to be valid only for a limited parameter range.

KW - Internal waves

KW - Solitary waves

KW - Stratified flows

U2 - 10.1017/jfm.2011.261

DO - 10.1017/jfm.2011.261

M3 - Article

VL - 683

SP - 263

EP - 288

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

SN - 0022-1120

ER -

Related by author

  1. Comparison of the Moist Parcel-In-Cell (MPIC) model with large-eddy simulation for an idealized cloud

    Böing, S. J., Dritschel, D. G., Parker, D. J. & Blyth, A. M., 29 Apr 2019, In : Quarterly Journal of the Royal Meteorological Society. In press, 17 p.

    Research output: Contribution to journalArticle

  2. On the regularity of the Green-Naghdi equations for a rotating shallow fluid layer

    Dritschel, D. G. & Jalali, M. R., 25 Apr 2019, In : Journal of Fluid Mechanics. 865, p. 100-136

    Research output: Contribution to journalArticle

  3. Scale-invariant singularity of the surface quasigeostrophic patch

    Scott, R. K. & Dritschel, D. G., 25 Mar 2019, In : Journal of Fluid Mechanics. 863, 12 p., R2.

    Research output: Contribution to journalArticle

Related by journal

  1. Journal of Fluid Mechanics (Journal)

    David Gerard Dritschel (Editor)
    2005 → …

    Activity: Publication peer-review and editorial work typesEditor of research journal

Related by journal

  1. On the regularity of the Green-Naghdi equations for a rotating shallow fluid layer

    Dritschel, D. G. & Jalali, M. R., 25 Apr 2019, In : Journal of Fluid Mechanics. 865, p. 100-136

    Research output: Contribution to journalArticle

  2. Scale-invariant singularity of the surface quasigeostrophic patch

    Scott, R. K. & Dritschel, D. G., 25 Mar 2019, In : Journal of Fluid Mechanics. 863, 12 p., R2.

    Research output: Contribution to journalArticle

  3. The stability and nonlinear evolution of quasi-geostrophic toroidal vortices

    Reinaud, J. N. & Dritschel, D. G., 25 Mar 2019, In : Journal of Fluid Mechanics. 863, p. 60-78

    Research output: Contribution to journalArticle

  4. Three-dimensional quasi-geostrophic vortex equilibria with m−fold symmetry

    Reinaud, J. N., 25 Mar 2019, In : Journal of Fluid Mechanics. 863, p. 32-59

    Research output: Contribution to journalArticle

ID: 4156981

Top