• Apel, J. R., 1987: Principles of Ocean Physics. Academic Press, 634 pp.

  • Baldi, M., G. A. Dalu, and R. A. Pielke Sr., 2008: Vertical velocities and available potential energy generated by landscape variability—Theory. J. Appl. Meteor. Climatol., 47 , 397410.

    • Search Google Scholar
    • Export Citation
  • Benjamin, T. B., 1968: Gravity currents and related phenomena. J. Fluid Mech., 31 , 209248.

  • Dalu, G. A., and R. A. Pielke, 1989: An analytical study of sea breeze. J. Atmos. Sci., 46 , 18151825.

  • Dalu, G. A., and R. A. Pielke, 1993: Vertical heat fluxes generated by mesoscale atmospheric flow induced by thermal inhomogeneities in the PBL. J. Atmos. Sci., 50 , 919926.

    • Search Google Scholar
    • Export Citation
  • Darby, L. S., R. M. Banta, and R. A. Pielke Sr., 2002: Comparisons between mesoscale model terrain sensitivity studies and Doppler lidar measurements of the sea breeze at Monterey Bay. Mon. Wea. Rev., 130 , 28132838.

    • Search Google Scholar
    • Export Citation
  • Fujita, T., 1959: Precipitation and cold air production in mesoscale thunderstorm systems. J. Meteor., 16 , 454466.

  • Gill, A. E., 1980: Some simple solutions for heat-induced tropical circulation. Quart. J. Roy. Meteor. Soc., 106 , 447462.

  • Gill, A. E., 1982: Atmosphere and Ocean Dynamics. Academic Press, 662 pp.

  • Griffiths, R. W., 1986: Gravity currents in rotating systems. Annu. Rev. Fluid Mech., 15 , 5989.

  • Hacker, J. N., and P. F. Linden, 2002: Gravity currents in rotating channels. Part 1. Steady-state theory. J. Fluid Mech., 457 , 295324.

    • Search Google Scholar
    • Export Citation
  • Haertel, P. T., R. H. Johnson, and S. N. Tulich, 2001: Some simple simulations of thunderstorm outflow. J. Atmos. Sci., 58 , 504516.

  • Hogg, A. J., 2006: Lock-release gravity currents and dam-break flows. J. Fluid Mech., 569 , 6187.

  • Huppert, H. E., 2006: Gravity currents: A personal perspective. J. Fluid Mech., 554 , 299322.

  • Huppert, H. E., and J. E. Simpson, 1980: The slumping of gravity currents. J. Fluid Mech., 99 , 785799.

  • Liu, C. H., and M. W. Moncrieff, 2000: Simulated density currents in idealized stratified environments. Mon. Wea. Rev., 128 , 14201437.

    • Search Google Scholar
    • Export Citation
  • Martin, J. R., and G. F. Lane-Serff, 2005: Rotating gravity currents. Part 1. Energy loss theory. J. Fluid Mech., 522 , 3562.

  • Martin, J. R., D. A. Smeed, and G. F. Lane-Serff, 2005: Rotating gravity currents. Part 2. Potential vorticity theory. J. Fluid Mech., 522 , 6389.

    • Search Google Scholar
    • Export Citation
  • Matsuno, T., 1966: Quasi-geostrophic motions in the equatorial area. J. Meteor. Soc. Japan, 44 , 2543.

  • Mizuta, G., and A. Masuda, 2003: An application of a diffusive reduced-gravity model to deep circulation above various forms of bottom topography. J. Phys. Oceanogr., 33 , 451464.

    • Search Google Scholar
    • Export Citation
  • Moncrieff, M. W., and D. W. K. So, 1989: A hydrodynamical theory of conservative bounded density currents. J. Fluid Mech., 198 , 177197.

    • Search Google Scholar
    • Export Citation
  • Moncrieff, M. W., and C. Liu, 1999: Convection initiation by density currents: Role of convergence, shear, and dynamical organization. Mon. Wea. Rev., 127 , 24552464.

    • Search Google Scholar
    • Export Citation
  • Pedlosky, J., 1987: Geophysical Fluid Dynamics. Springer-Verlag, 710 pp.

  • Pielke Sr., R. A., 2002: Mesoscale Meteorological Modeling. 2nd ed. Academic Press, 676 pp.

  • Polvani, L. M., and A. H. Sobel, 2002: The Hadley circulation and the weak temperature gradient approximation. J. Atmos. Sci., 59 , 17441752.

    • Search Google Scholar
    • Export Citation
  • Potter, D., 1973: Computational Physics. Wiley, 315 pp.

  • Rottman, J. W., and J. E. Simpson, 1983: Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech., 135 , 95110.

    • Search Google Scholar
    • Export Citation
  • Rotunno, R., 1983: On the linear theory of land and sea breeze. J. Atmos. Sci., 40 , 19992009.

  • Segal, M., and R. W. Arritt, 1992: Nonclassical mesoscale circulations caused by surface sensible heat-flux gradients. Bull. Amer. Meteor. Soc., 73 , 15931604.

    • Search Google Scholar
    • Export Citation
  • Seitter, K. L., 1986: A numerical study of atmospheric density current motion including the effects of condensation. J. Atmos. Sci., 43 , 30683076.

    • Search Google Scholar
    • Export Citation
  • Simpson, J. E., 1982: Gravity currents in the laboratory, atmosphere, and ocean. Annu. Rev. Fluid Mech., 14 , 213234.

  • Simpson, J. E., 1994: Sea Breeze and Local Winds. Cambridge University Press, 248 pp.

  • Sobel, A. H., and C. S. Bretherton, 2000: Modeling tropical precipitation in a single column. J. Climate, 13 , 43784392.

  • Sobel, A. H., J. Nilsson, and L. M. Polvani, 2001: The weak temperature gradient approximation and balanced tropical moisture waves. J. Atmos. Sci., 58 , 36503665.

    • Search Google Scholar
    • Export Citation
  • Thomas, P. J., and P. F. Linden, 2007: Rotating gravity currents: Small-scale and large-scale laboratory experiments and a geostrophic model. J. Fluid Mech., 578 , 3565.

    • Search Google Scholar
    • Export Citation
  • Tompkins, A. M., 2001: Organization of tropical convection in low vertical wind shears: The role of cold pools. J. Atmos. Sci., 52 , 16501672.

    • Search Google Scholar
    • Export Citation
  • Ungarish, M., and H. E. Huppert, 1998: The effects of rotation on axisymmetric particle-driven gravity currents. J. Fluid Mech., 362 , 1751.

    • Search Google Scholar
    • Export Citation
  • Ungarish, M., and T. Zemach, 2003: On axisymmetric rotating gravity currents: Two-layer shallow water and numerical solutions. J. Fluid Mech., 481 , 3766.

    • Search Google Scholar
    • Export Citation
  • von Kármán, T., 1940: The engineer grapples with nonlinear problems. Bull. Amer. Math. Soc., 46 , 615683.

  • Xu, Q., and M. W. Moncrieff, 1994: Density current circulation in shear flows. J. Atmos. Sci., 51 , 434446.

  • Zwillinger, D., 1992: Handbook of Differential Equations. Academic Press, 787 pp.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 47 13 0
PDF Downloads 12 1 0

Steady-State Dynamics of a Density Current in an f-Plane Nonlinear Shallow-Water Model

Giovanni A. DaluInstitute of Biometeorology, Rome, Italy

Search for other papers by Giovanni A. Dalu in
Current site
Google Scholar
PubMed
Close
and
Marina BaldiInstitute of Biometeorology, Rome, Italy

Search for other papers by Marina Baldi in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The authors study the nonlinear dynamics of a density current generated by a diabatic source in a rotating and a nonrotating system, both in the presence and in the absence of frictional losses, using a steady-state hydrostatic shallow-water model and producing solutions as a function of the Coriolis parameter and of the Rayleigh friction coefficient. Results are presented in the range of the parameter values that are relevant for shallow atmospheric flows as sea–land breezes and as cold pool outflows. In the shallow-water approximation, single-layer flows and two-layer flows with a lid have three degrees of freedom, and their steady-state dynamics are governed by three ordinary differential equations (ODEs), whereas two-layer flows bounded by a free surface have six degrees of freedom, and their dynamics are governed by six ODEs.

It is shown that in the limit case of frictionless flow, the problem has an explicit analytical solution, and in the presence of friction, the system for a one-layer flow and for a two-layer flow bounded by a lid can be reduced to two algebraic equations, plus one second-order ordinary differential equation, which can be integrated numerically. Results show that the maximum runout length of the current occurs when the Rayleigh friction coefficient in the lower layer is on the order of the Coriolis parameter. This length is larger when the upper layer is deeper than the lower layer, but it shortens when the friction coefficient of the upper layer is smaller than that in the lower layer.

In addition, the relative error of the solution to the linearized equations is computed. This error, which is enhanced when the width of the forcing is smaller than the Rossby radius, is sizable when the friction coefficient is smaller than the Coriolis parameter. In addition, by comparing the nonlinear solution with a lid (three degrees of freedom) to the nonlinear solution with a free surface as an upper boundary (six degrees of freedom), it is shown that the solution with the lid overestimates the geopotential for low values of the friction coefficient and it underestimates the geopotential for large values of this coefficient. The error, sizable when the two layers have a comparable depth, rapidly decreases when the upper layer becomes deeper than the lower layer; accordingly, a rigid lid can be safely adopted only when the depth of the upper layer is twice the depth of the lower layer, or deeper.

Corresponding author address: Marina Baldi, IBIMET-CNR, Via dei Taurini 19, 00185 Rome, Italy. Email: m.baldi@ibimet.cnr.it

Abstract

The authors study the nonlinear dynamics of a density current generated by a diabatic source in a rotating and a nonrotating system, both in the presence and in the absence of frictional losses, using a steady-state hydrostatic shallow-water model and producing solutions as a function of the Coriolis parameter and of the Rayleigh friction coefficient. Results are presented in the range of the parameter values that are relevant for shallow atmospheric flows as sea–land breezes and as cold pool outflows. In the shallow-water approximation, single-layer flows and two-layer flows with a lid have three degrees of freedom, and their steady-state dynamics are governed by three ordinary differential equations (ODEs), whereas two-layer flows bounded by a free surface have six degrees of freedom, and their dynamics are governed by six ODEs.

It is shown that in the limit case of frictionless flow, the problem has an explicit analytical solution, and in the presence of friction, the system for a one-layer flow and for a two-layer flow bounded by a lid can be reduced to two algebraic equations, plus one second-order ordinary differential equation, which can be integrated numerically. Results show that the maximum runout length of the current occurs when the Rayleigh friction coefficient in the lower layer is on the order of the Coriolis parameter. This length is larger when the upper layer is deeper than the lower layer, but it shortens when the friction coefficient of the upper layer is smaller than that in the lower layer.

In addition, the relative error of the solution to the linearized equations is computed. This error, which is enhanced when the width of the forcing is smaller than the Rossby radius, is sizable when the friction coefficient is smaller than the Coriolis parameter. In addition, by comparing the nonlinear solution with a lid (three degrees of freedom) to the nonlinear solution with a free surface as an upper boundary (six degrees of freedom), it is shown that the solution with the lid overestimates the geopotential for low values of the friction coefficient and it underestimates the geopotential for large values of this coefficient. The error, sizable when the two layers have a comparable depth, rapidly decreases when the upper layer becomes deeper than the lower layer; accordingly, a rigid lid can be safely adopted only when the depth of the upper layer is twice the depth of the lower layer, or deeper.

Corresponding author address: Marina Baldi, IBIMET-CNR, Via dei Taurini 19, 00185 Rome, Italy. Email: m.baldi@ibimet.cnr.it

Save