Editorial Type:
Article
Article Type:
Research Article

The Effect of Critical Levels on 3D Orographic Flows: Linear Regime

Vanda Grubišić National Center for Atmospheric Research, Boulder, Colorado

Search for other papers by Vanda Grubišić in
Current site
Google Scholar
PubMed Close
and
Piotr K. Smolarkiewicz National Center for Atmospheric Research, Boulder, Colorado

Search for other papers by Piotr K. Smolarkiewicz in
Current site
Google Scholar
PubMed Close
Print Publication:
01 Aug 1997
DOI:
https://doi.org/10.1175/1520-0469(1997)054<1943:TEOCLO>2.0.CO;2
Page(s):
1943–1960
Restricted access

Abstract

The effect of a critical level on airflow past an isolated axially symmetric obstacle is investigated in the small-amplitude hydrostatic limit for mean flows with linear negative shear. Only flows with mean Richardson numbers (Ri) greater or equal to ¼ are considered. The authors examine the problem using the linear, steady-state, inviscid, dynamic equations, which are well known to exhibit a singular behavior at critical levels, as well as a numerical model that has the capability of capturing both nonlinear and dissipative effects where these are significant.

Linear theory predicts the 3D wave pattern with individual waves that are confined to paraboloidal envelopes below the critical level and strongly attenuated and directionally filtered above it. Asymptotic solutions for the wave field far from the mountain and below the critical level show large shear-induced modifications in the proximity of the critical level, where wave envelopes quickly widen with height. Above the critical level, the perturbation field consists mainly of waves with wavefronts perpendicular to the mean flow direction. A closed-form analytic formula for the mountain-wave drag, which is equally valid for mean flows with positive and negative shear, predicts a drag that is smaller than in the uniform wind case. In the limit of Ri d⃗ ¼, in which linear theory predicts zero drag for an infinite ridge, drag on an axisymmetric mountain is nonzero.

Numerical simulations with an anelastic, nonhydrostatic model confirm and qualify the analytic results. They indicate that the linear regime, in which analytic solutions are valid everywhere except in the vicinity of the critical level, exists for a range of mountain heights given Ri > 1. For Ri d⃗ ¼ this same regime is difficult to achieve, as the flow is extremely sensitive to nonlinearities introduced through the lower boundary forcing that induce strong nonlinear effects near the critical level. Even well within the linear regime, flow in the vicinity of a critical level is dissipative in nature as evidenced by the development of a potential vorticity doublet.

Corresponding author address: Dr. Vanda Grubis̄ić, National Center for Atmospheric Research, P. O. Box 3000, Boulder, CO 80307.

Abstract

The effect of a critical level on airflow past an isolated axially symmetric obstacle is investigated in the small-amplitude hydrostatic limit for mean flows with linear negative shear. Only flows with mean Richardson numbers (Ri) greater or equal to ¼ are considered. The authors examine the problem using the linear, steady-state, inviscid, dynamic equations, which are well known to exhibit a singular behavior at critical levels, as well as a numerical model that has the capability of capturing both nonlinear and dissipative effects where these are significant.

Linear theory predicts the 3D wave pattern with individual waves that are confined to paraboloidal envelopes below the critical level and strongly attenuated and directionally filtered above it. Asymptotic solutions for the wave field far from the mountain and below the critical level show large shear-induced modifications in the proximity of the critical level, where wave envelopes quickly widen with height. Above the critical level, the perturbation field consists mainly of waves with wavefronts perpendicular to the mean flow direction. A closed-form analytic formula for the mountain-wave drag, which is equally valid for mean flows with positive and negative shear, predicts a drag that is smaller than in the uniform wind case. In the limit of Ri d⃗ ¼, in which linear theory predicts zero drag for an infinite ridge, drag on an axisymmetric mountain is nonzero.

Numerical simulations with an anelastic, nonhydrostatic model confirm and qualify the analytic results. They indicate that the linear regime, in which analytic solutions are valid everywhere except in the vicinity of the critical level, exists for a range of mountain heights given Ri > 1. For Ri d⃗ ¼ this same regime is difficult to achieve, as the flow is extremely sensitive to nonlinearities introduced through the lower boundary forcing that induce strong nonlinear effects near the critical level. Even well within the linear regime, flow in the vicinity of a critical level is dissipative in nature as evidenced by the development of a potential vorticity doublet.

Corresponding author address: Dr. Vanda Grubis̄ić, National Center for Atmospheric Research, P. O. Box 3000, Boulder, CO 80307.

Save
Cover Journal of the Atmospheric Sciences
Journal of the Atmospheric Sciences

  • Bacmeister, J. T., and R. T. Pierrehumbert, 1988: On high-drag states of nonlinear stratified flow over an obstacle. J. Atmos. Sci.,45, 63–80.

  • Baines, P. G., 1995: Topographic Effects in Stratified Flows. Cambridge University Press, 482 pp.

  • Booker, J. R., and F. P. Bretherton, 1967: The critical layer for internal gravity waves in a shear flow. J. Fluid Mech.,27, 513–539.

  • Bretherton, F. P., 1966: The propagation of groups of internal gravity waves in a shear flow. Quart. J. Roy. Meteor. Soc.,92, 466–480.

  • Broad, A. S., 1995: Linear theory of momentum fluxes in 3-D flows with turning of the mean wind with height. Quart. J. Roy. Meteor. Soc.,121, 1891–1902.

  • Case, K. M., 1960: Stability of an idealized atmosphere. I. Discussion of results. Phys. Fluids,3, 149–154.

  • Clark, T. L., 1977: A small-scale dynamic model using a terrain-following coordinate transformation. J. Comput. Phys.,24, 186–215.

  • ——, and W. R. Peltier, 1977: On the evolution and stability of finite-amplitude mountain waves. J. Atmos. Sci.,34, 1715–1730.

  • Crook, N. A., T. L. Clark, and M. W. Moncrieff, 1990: The Denver cyclone. Part I: Generation in low Froude number flow. J. Atmos. Sci.,47, 2725–2741.

  • Davies, H. C., 1983: Limitations of some common lateral boundary schemes used in regional NWP models. Mon. Wea. Rev.,111, 1002–1012.

  • Drazin, P. G., and W. H. Reid, 1971: HydrodynamicStability. Cambridge University Press, 527 pp.

  • Durran, D. R., 1992: Two-layer solutions to Long’s equation for vertically propagating mountain waves: How good is linear theory? Quart. J. Roy. Meteor. Soc.,118, 415–433.

  • ——, and J. B. Klemp, 1987: Another look at downslope windstorms. Part II: Nonlinear amplification beneath wave-overturning layers. J. Atmos. Sci.,44, 3402–3412.

  • Eliassen, A., and E. Palm, 1960: On the transfer of energy in stationary mountain waves. Geofys. Publ.,22, 1–23.

  • Ertel, H., 1942: Ein neuer hydrodynamischer Wirbesatz. Meteor. Z.,59, 277–281.

  • Fritts, D. C., J. R. Isler, and O. Andreassen, 1994: Gravity-wave breaking in two and three dimensions. Part II: Three-dimensional evolution and instability structure. J. Geophys. Res.,99, 8109–8123.

  • Gal-Chen, T., and R. C. J. Somerville, 1975: On the use of a coordinate transformation for the solution of the Navier–Stokes equations. J. Comput. Phys.,17, 209–228.

  • Gradshteyn, I. S., and I. M. Ryzhik, 1980: Table of Integrals, Series, and Products. Academic Press, 1160 pp.

  • Grubis̄ić, V., 1995: Mountain waves and mountain wakes in stratified airflows past three-dimensional obstacles. Ph.D. dissertation, Yale University, 163 pp. [Available from University Microfilm, 300 North Zeeb Rd., Ann Arbor, MI 48106.].

  • Hazel, P., 1967: The effect of viscosity and heat conduction on internal gravity waves at a critical level. J. Fluid Mech.,30, 775–783.

  • Hunt, J. C. R., and W. H. Snyder, 1980: Experiments on stably and neutrally stratified flow over a model three-dimensional hill. J. Fluid Mech.,96, 671–704.

  • Kapitza, H., and D. P. Eppel, 1992: The non-hydrostatic mesoscale model GESIMA. Part I: Dynamical equations and tests. Beitr. Phys. Atmos.,65, 129–146.

  • Keller, T. L., 1994: Implications of the hydrostatic assumption on atmospheric gravity waves. J. Atmos. Sci.,51, 1915–1929.

  • Kelly, R. E., 1977: Linear and nonlinear critical layer effects on wave propagation. Geofluidynamical Wave Mathematics, National Science Foundation Regional Conf. on Mathematics, Applied Mathematics Group, University of Washington, 194–210.

  • Kosloff, R., and D. Kosloff, 1986: Absorbing boundaries for wave propagation problems. J. Comput. Phys.,63, 363–376.

  • Kuo, H. L., 1963: Perturbations of plane Couette flow in stratfied fluid and origin of cloud streets. Phys. Fluids,6, 195–211.

  • Lighthill, J. M., 1980: Waves in Fluids. Cambridge University Press, 504 pp.

  • Lilly, D., and E. J. Zipser, 1972: The front range windstorm of 11 January 1972—A meteorological narrative. Weatherwise,25, 56–63.

  • Lipps, F. B., 1990: On the anelastic approximation for deep convection. J. Atmos. Sci.,47, 1794–1798.

  • ——, and R. S. Hemler, 1982: A scale analysis of deep moist convection and some related numerical calculations. J. Atmos. Sci.,39, 2192–2210.

  • Maslowe, S. A., 1986: Critical layers in shear flows. Annu. Rev. Fluid Mech.,18, 405–432.

  • McIntyre, M. E., and W. A.Norton, 1990: Dissipative wave-mean interactions and the transport of vorticity and potential vorticity. J. Fluid Mech.,212, 403–435.

  • Miles, J. W., 1961: On the stability of heterogeneous shear flows. J. Fluid Mech.,10, 496–508.

  • Miranda, P. M., and I. N. James, 1992: Non-linear three-dimensional effects on the wave drag: Splitting flow and breaking waves. Quart. J. Roy. Meteor. Soc.,118, 1057–1081.

  • Nappo, C. J., and G. Chimonas, 1992: Wave exchange between the ground surface and a boundary-layer critical level. J. Atmos. Sci.,49, 1075–1091.

  • Peltier, W. R., and T. L. Clark, 1979: The evolution and stability of finite-amplitude mountain waves. Part II: Surface wave drag and severe downslope windstorms. J. Atmos. Sci.,36, 1498–1529.

  • Phillips, D. S., 1984: Analytical surface pressure and drag for linear hydrostatic flow over three-dimensional elliptical mountains. J. Atmos. Sci.,41, 1073–1084.

  • Prusa, J. M., P. K. Smolarkiewicz, and R. R. Garcia, 1996: On the propagation and breaking at high altitudes of gravity waves excited by tropospheric forcing. J. Atmos. Sci.,53, 2186–2216.

  • Reisner, J. M., and P. K. Smolarkiewicz, 1994: Thermally forced low Froude number flow past three-dimensional obstacles. J. Atmos. Sci.,51, 117–133.

  • Schär, C., and D. R. Durran, 1997: Vortex formation and vortex shedding in continuously stratified flows past isolated topography. J. Atmos. Sci.,54, 534–554.

  • Shutts, G., 1995: Gravity-wave drag parametrization over complex terrain: The effect of critical-level absorption in directional wind-shear. Quart. J. Roy. Meteor. Soc.,121, 1005–1021.

  • Smith, R. B., 1980: Linear theory of stratified hydrostatic flow past an isolated obstacle. Tellus,32, 348–364.

  • ——, 1985: On severe downslope winds. J. Atmos. Sci.,42, 2597–2603.

  • ——, 1986: Further development of a theory of lee cyclogenesis. J. Atmos. Sci.,43, 1582–1602.

  • ——, 1988: Linear theory of hydrostatic flow over an isolated mountain in isosteric coordinates. J. Atmos. Sci.,45, 3889–3896.

  • ——, 1989: Comment on “Low Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices.” J. Atmos. Sci.,46, 3611–3613.

  • ——, and S. Grønås, 1993: Stagnation points and bifurcation in 3-D mountain airflow. Tellus,45A, 28–43.

  • Smolarkiewicz, P. K., and R. Rotunno, 1989: Low Froude number flow past three-dimensional obstacles. Part I: Baroclinically generated lee vortices. J. Atmos. Sci.,46, 1154–1164.

  • ——, and ——, 1990: Low Froude number flow past three-dimensional obstacles. Part II: Upwind flow reversal zone. J. Atmos. Sci.,47, 1498–1511.

  • ——, and J. A. Pudykiewicz, 1992: A class of semi-Lagrangian approximations for fluids. J. Atmos. Sci.,49, 2082–2096.

  • ——, and L. G. Margolin, 1993: On forward-in-time differencing for fluids: Extension to curvilinear framework. Mon. Wea.Rev.,121, 1847–1859.

  • ——, and ——, 1994: On forward-in-time differencing for fluids: Variational solver for elliptic problems in atmospheric flows. Appl. Math. Comput. Sci.,4, 527–551.

  • ——, and ——, 1997: On forward-in-time differencing for fluids: An Eulerian/semi-Lagrangian nonhydrostatic model for stratified flows. Atmos.–Ocean,34, in press.

  • Winters, K. B., and E. A. D’Asaro, 1989: Two-dimensional instability of finite amplitude internal gravity wave packets near a critical level. J. Geophys. Res.,94, 12 709–12 719.

  • ——, and J. J. Riley, 1992: Instability of internal waves near a critical level. Dyn. Atmos. Oceans,16, 249–278.

  • Worthington, R. M., and L. Thomas, 1996: Radar measurements of critical-layer absorption in mountain waves. Quart. J. Roy. Meteor. Soc.,122, 1263–1282.

  • Wurtele, M. G., A. Datta, and R. D. Sharman, 1996: The propagation of gravity–inertia waves and lee waves under a critical level. J. Atmos. Sci.,53, 1505–1523.

All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 846 631 44
PDF Downloads 142 46 6