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Downloaded 10/01/21 06:53 PM UTC OCTOBER 2007 R H I N E S 3627 Jets and Orography: Idealized Experiments with Tip Jets and Lighthill Blocking P. B. RHINES University of Washington, Seattle, Washington (Manuscript received 16 June 2006, in final form 17 December 2006) ABSTRACT This paper describes qualitative features of the generation of jetlike concentrated circulations, wakes, and blocks by simple mountainlike orography, both from idealized laboratory experiments and shallow-water numerical simulations on a sphere. The experiments are unstratified with barotropic lee Rossby waves, and jets induced by mountain orography. A persistent pattern of lee jet formation and lee cyclogenesis owes its origins to arrested topographic Rossby waves above the mountain and potential vorticity (PV) advection through them. The wake jet occurs on the equatorward, eastern flank of the topography. A strong upstream blocking of the westerly flow occurs in a Lighthill mode of long Rossby wave propagation, which depends on ␤a2/U, the ratio of Rossby wave speed based on the scale of the mountain, to zonal advection speed, U (␤ is the meridional potential vorticity gradient, f is the Coriolis frequency, and a is the diameter of the mountain). Mountains wider (north–south) than the east–west length scale of stationary Rossby waves will tend to block the oncoming westerly flow. These blocks are essentially ␤ plumes, which are illustrated by their linear Green function. For large ␤a2/U, upwind blocking is strong; the mountain wake can be unstable, filling the fluid with transient Rossby waves as in the numerical simulations of Polvani et al. For small values, ␤a2/U K 1 classic lee Rossby waves with large wavelength compared to the mountain diameter are the dominant process. The mountain height, ␦h, relative to the mean fluid depth, H, affects these transitions as well. Simple lee Rossby waves occur only for such small heights, ␦h/h K a␤/f, that the f/h contours are not greatly distorted by the mountain. Nongeostrophic dynamics are seen in inertial waves generated by geostrophic shear, and ducted by it, and also in a texture of finescale, inadvertent convection. Weakly damped circulations induced in a shallow-water numerical model on a sphere by a lone mountain in an initially simple westerly wind are also described. Here, with ␤a2/U ϳ1, potential vorticity stirring and transient Rossby waves dominate, and drive zonal flow acceleration. Low-latitude critical layers, when present, exert strong control on the high-latitude waves, and with no restorative damping of the mean zonal flow, they migrate poleward toward the source of waves. While these experiments with homogeneous fluid are very simplified, the baroclinic atmosphere and ocean have many tall or equivalent barotropic eddy structures owing to the barotropization process of geostrophic turbulence. 1. Introduction and potential vorticity background turbulence. Such balanced fluid motions have an inher- ent nonlinear tendency to increase in scale, both later- Solar radiation bathes the earth, varying smoothly ally and vertically until limited by the ␤ effect or dissi- with latitude, and yet the circulations it produces are pation, which can set the width scale of developing jet filled with small-scale transient eddies, jet streams, and streams. However, jets also can arise directly from boundary currents. Zonal jet stream generation occurs mountainous topography. By concentrating an oncom- with thermally forced circulations on a simple, smooth ing westerly flow, a large topographic feature can draw globe. Instability of zonally symmetric, baroclinic circu- fluid across a great span of latitudes. Topographic lations is often given as a primary reason for these syn- structure exists at all lateral scales and is a potent optic-scale features, which are amplified or generated source of circulation structures, from planetary-scale ab initio by many kinds of convective and mechanical motions to much finer scales. The literature concen- trates particularly on hemisphere-filling stationary waves on the one hand (e.g., Held et al. 2002), and lee waves Corresponding author address: P. B. Rhines, School of Ocean- ography, University of Washington, Box 357940, Seattle, WA and turbulent wakes on the other. Potential vorticity 98195. conservation provides, as in many areas of synoptic dy- E-mail: [email protected] namics, a logical framework with which to proceed. DOI: 10.1175/JAS4008.1 © 2007 American Meteorological Society Unauthenticated | Downloaded 10/01/21 06:53 PM UTC JAS4008 3628 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 64 In this paper we present experiments, mostly from frequency waves, involving the first term in the wave our geophysical fluid dynamics (GFD) laboratory, equation, are eliminated for small temporal Rossby showing qualitatively how topographic features cause number, fT k 1. The wave equation then becomes jetlike concentrations of flow, occurring naturally in simple, homogeneous fluid on a ␤ plane. We believe the h f 2 f h 2␩ ϩ m,r ␩ Ϫ ␩ ϩ m,r ␩ ϭٌ t rt t ␪ 0. results will be relevant to the equivalent-barotropic hm ghm r hm mode of the atmosphere, though of course full stratifi- cation adds much more to the problem. With a parabolic depth profile and large mean depth, k ⍀2 2 Ϫ1 ␦ Potential vorticity (PV) conservation provides the H a g , the second term is small [O( hm/hm)] underlying idealized dynamics. A cylinder of water, 1 m and the quasigeostrophic wave equation becomes in diameter, rotating with a free upper surface, acts as a ␤ 2 3 polar plane for barotropic, shallow-water flows. Con- 2 f f .␩ Ϫ ␩ ϩ ␩␪ ϭ 0 ٌ stant values of the gravity/rotation potential, ⌽, where t gH t 4gH 1 Separating out waves in the azimuthal (zonal) direc- ⌽ ϭϪ ⍀2r 2 ϩ gz 2 tion, ␪, and in time with a factor exp(in␪ Ϫ i␻t) leaves us with Bessel’s equation, determine the parabolic free surface, lying at a height 2 2 2 n f fn 1 ⍀ 1 Ϫ1 2 2 ␩ ϩ r ␩ ϩ ͫϪ Ϫ ͩ1 ϩ ͪͬ␩ ϭ 0 h ϭ H ϩ ͩr Ϫ R ͪ. rr r 2 ␻ m 2 g 2 r gH 4 ␩ ϭ ␬ ␪ Ϫ ␻ Here, r is radial coordinate, z is vertical coordinate, hm with Bessel-function solutions Jn( r) exp(in i t) the fluid depth, H is the mean depth (typically 0.15 m), and dispersion relation R is the radius of the cylinder (here 0.95 m), ⍀ is the rotation rate (here taking values 2 to 3.5 sϪ1), and g the Ϫ͑f 3ր4gH͒n ␻ ϭ . acceleration due to gravity and planetary rotation. The ␬2 ϩ f 2րgH ⍀ mean-state PV is 2 /hm. Potential vorticity conserva- ␤ ϭ 3 ϩ tion for inviscid shallow-water flow and waves is Here the effective value of is fhm,r /hm (f /4gH)r O(␦h /h ), the radial potential vorticity gradient, and Dq f ϩ ␨ g m m ϭ ϭ ϭ ϩ ␩ ␨ ϭ ٌ2␩ 0; q ; h hm ; , the dispersion relation and modes are the same as Dt h f found for a polar projection of the earth, with a uni- where ␨ is the vertical relative vorticity, f ϭ 2⍀, t is the form-depth fluid. Traveling waves with spiral wave- crests correspond to the H(1) and H(2) complex Bessel time, h the total depth, and hm the time-averaged depth n n field, ␩ the fluid depth perturbation, L is a character- functions, which sum together to form standing-wave istic horizontal length scale, and U a characteristic hori- modes. For the standing modes, eigenvalues for the ϭ zontal velocity scale. These are exact equations except rigid wall boundary condition at r R are set by the ␩ ϭ ␬ ϭ Ͼ in the relation between ␨ and ␩, which involves errors of zeros of ;atr R: Jn( R) 0(n 0). Thus, for order Rossby number, Ro ϵ U/fL. For a single layer of example the azimuthal mode 5 wave in Fig. 1 has pos- ␬ ϭ ␬ ϭ homogeneous fluid with a free surface, the linearized sible eigenvalues J5( R) 0, or R 8.77, 12.34 . in wave equation for the surface elevation, ␩, combines this approximation. These solutions have the usual ␤ high-frequency gravity waves with Coriolis modifica- properties of -plane Rossby waves at short wavelength tion, Kelvin waves and low-frequency topographic and asymptotic shortwave rays that propagate along Rossby waves; straight lines, analogous to the great-circle ray paths of Rossby waves on a rotating sphere (Longuet-Higgins ␩͔͒ Ϫ ͑ ␩͒ ϭٌ ͑ ١ ␩ ϩ 2␩ Ϫ͓ tt f · ghm t fgJ hm, 0, 1964). Thus the rays from a point source should ap- proach the pole and then retreat toward lower latitude. where J() is the horizontal Jacobian operator, in polar They are singular at the origin, but should be under- ␪ coordinates (r, ), stood in terms of a wave generator at a small but finite 1 ѨA ѨB ѨB ѨA radius. The short-wavelength limit of the problem J͑A, B͒ ϭ ͩ Ϫ ͪ. ␤ r Ѩr Ѩ␪ Ѩr Ѩ␪ amounts to ray-tracing with a gradual variation of with radius. We then recover the familiar ␤-plane dis- The approximations in this equation are hydrostatic persion relation and gradual amplitude variation ex- pressure balance and linear (small) amplitude. High- pressing energy conservation along a ray. Unauthenticated | Downloaded 10/01/21 06:53 PM UTC OCTOBER 2007 R H I N E S 3629 FIG. 1. The (5,1) mode Rossby wave in a 1-m diameter polar ␤-plane experiment, excited by a periodic oscillation (with period 8.3 s) of a plunger at 7 o’clock. The five-wave pattern is energetically eastward propagating (anti- clockwise) from the wave source. Dye was initially laid in concentric rings about the “North Pole”; ⍀ϭ3.5 sϪ1. The diagonal line is the rod driving the glass plunger, suspended from above.
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