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Gravity wave drag Parameterization of orographic related momentum fluxes in a numerical weather processing model Andrew Orr 30 May 2012 Lecture 1: Atmospheric processes associated with orography Lecture 2: Parameterization of subgrid-scale orography

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Scales of orography and atmospheric processes L ~ 1000–10,000 km L ~ km L ~ 1-10 km courtesy of shaderelief.com Length scales of Earths orography

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From Rontu (2007) Surface elevation along the latitude band 45 o N, based on SRTM 3 data (65 m horizontal resolution) Height scales of Earths orography

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Planetary/synoptic scales give largest variance (most important for global weather and climate) But variance on all scales (small and mesoscale variations influence local weather and climate) km km Corresponding orography spectrum, i.e. variance of surface height as a function of horizontal scale, along the latitude band 45 o N GCMs have horizontal scales ~ km, so not all processes related to orography are explicitly resolved Information on scales by performing a spectral analysis on the data From Rontu (2007)

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Some mountain-related atmospheric processes From Rontu (2007)

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Resolved and parametrized processes Inertial waves Hydrostatic drag After Emesis (1990) Wave dragForm drag Rossby waves Pressure drag Explicitly represented by resolved flow Parameterization required Viscosity of the air Upstream blocking Orographic turbulence (L<5km) Gravity waves (L>5km) Drag Lift drag lift

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From Bougeault (1990) Upstream / low-level blocking Orographic turbulence (see Turbulent Orographic Form Drag) Gravity waves and upstream blocking Mountain waves / gravity waves / buoyancy waves / internal gravity waves

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Simple properties of gravity waves (After T. Palmer Theory of linear gravity waves, ECMWF meteorological training course, 2004) In order to prepare for a description of the parametrization of gravity-wave drag, we examine some simple properties of gravity waves excited by two-dimensional stably stratified flow over orography. We suppose that the horizontal scales of these waves is sufficiently small that the Rossby number is large (ie Coriolis forces can be neglected), and the equations of motion can be written as (1) (2) See Smith 1979;Houze 1993; Palmer et al. 1986

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with the continuity and thermodynamic equations given by (3) (4) Using the Boussinesq approximation whereby density is treated as a constant except where it is coupled to gravity in the buoyancy term of the vertical momentum equation. Linearising (1)-(4) about a uniform hydrostatic flow u 0 with constant density ρ 0 and static stability N

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results in the perturbation variables (5) (6) (7) (8)

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Density fluctuations due to pressure changes are small compared with those due to temperature changes, so we can write (9) (10) Using (9), (5)-(8) are four equations in four unknowns. After some manipulation these can be reduced to one equation with one unknown We now look for sinusoidal solutions of the general form

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(11) where is the horizontal wavenumber is the vertical wavenumber is the wave frequency which when substituted into (10), give the dispersion relation where is the wave intrinsic frequency

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Let us now restrict ourselves to stationary waves forced by sinusoidal orography with elevation h(x) given by with hmhm L u0u0 The lower boundary condition (the vertical component of the wind at the surface must vanish) is (12)

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(i) Evanescent solution Solutions periodic in x are of the form The condition of finite amplitude (B=0) and the lower boundary condition (12) gives where From the continuity equation,

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Hence, these evanescent solutions take the form of a sinusoidal wave field decaying without phase tilt, showing that energy is trapped near the ground (sinuous lines indicate displacement of isopycnal surfaces) Notice that the vertical flux of momentum for these waves. Here the overbar represents the average along the x-direction. If we take u 0 ~10 m s -1 and N~0.01 s -1 then with these evanescent solutions occur when L<6 km, ie small-wavelength topography / narrow-ridge case H H H L L Wind H and L indicate positions of maximum and minimum pressure perturbation, respectively

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(ii) Propagating solution Solutions will be in the form Radiation condition implies that B=0 (ie the perturbation energy flux must be upward) The full solution, then, is

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H H L Now the displacement of the isopycnals is uniform with height, but wave crests move upstream with height, ie the phase lines are tilted. The group velocity relative to the air is along these phase lines. Wind High and low pressures are now on the nodes, so there is a net force on the topography in the direction of the flow

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k>N/U (i.e. narrow-ridge case) (or equivalently U/L>N, i.e. high frequency) Evanescent solution (i.e. fading away) Non-dimensional length NL/U<1 k

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The surface pressure drag (or drag force) on the orography (per wavelength) is given by Drag force Using the lower boundary condition and the x-component of the momentum equation this can be re-expressed as Note that units of stress/pressure are Pascals, with 1 Pa = 1 N m -2.

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For long (hydrostatic) waves with k 2 <

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Net force on mountain in downstream direction from mean flow (a) An equal and opposite force is exerted on the mean flow by the hill (b) However, this may be realised at high altitude owing to the vertical transport of momentum by gravity waves (c) (c) (b) (a)

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Gravity wave saturation / momentum sink For the propagating modes the mean flow experiences this drag force where the wave activity is dissipative, which can be well above the boundary layer. This can occur because in the real atmosphere is not constant but decrease exponentially with height. Convective instability occurs when the wave amplitude becomes large relative to the vertical wavelength. The streamlines become very steep and the wave breaks, much as waves break in the ocean. Convective overturning can occur as the waves encounter increased static stability N or reduced wind speed U (typically upper troposphere or lower stratosphere). They also occur due to the tendency for the waves to amplify with height due to the decrease in air density. Elimination of wave as its energy is absorbed and transferred to the mean wind. Drag exerted on flow as wave energy converted into small-scale turbulent motions acts to decelerate the mean velocity, i.e. wave drag drags the flow velocity U to the gravity wave phase speed c (=0). Dissipation can also occur as the waves approach a critical level (c = U) Leads to wave breaking and turbulent dissipation of wave energy. This is termed wave saturation, and is a momentum sink

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Evident from radiosondes Gravity waves observed over the Falkland Islands from radiosonde ascent Vosper and Mobbs

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Durran, 2003 Single lenticular cloud

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Evident from satellites (AIRS: Atmospheric Infra-red Sounder) Alexander and Teitelbaum, 2007

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Eliassen-Palm theorem λ waves steepen leading to wave breaking and elimination of wave (i.e. λ >λ sat ) as its energy is absorbed and transferred to the mean wind, i.e. drag exerted on flow as wave energy converted into small- scale turbulent motions Linear, 2d, hydrostatic surface stress Eliassen-Palm theorem: stress unchanged at all levels in the absence of wave breaking / dissipation (i.e. λ=λ s ) i.e. amplitude of vertical displacement must increase as the density decreases upwards After Rontu et al. (2002)(measured in Pa (N/m 2 )

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Momentum flux observations Mean observed profile of momentum flux over Rocky mountains on 17 February 1970 (from Lilly and Kennedy 1973) Momentum flux: Stress largely unchanged; little dissipation/wave breaking; Stress rapidly changing; strong dissipation/wave breaking;

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Gravity wave observations Potential temperature cross-section over the Rocky mountains on 17 February Solid lines are isentropes (K), dashed lines aircraft or balloon flight trajectories (from Lilly and Kennedy 1973) increasing vertical displacement as density decreases steepening of waves leading to eventual wave breaking and turbulence trapped lee waves downslope wind-storm

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Mountain flow regimes linear/flow-over regime (Nh/U small) Non-dimensional height: Nh/U U: upstream velocity h: mountain height N: Brunt-Vaisala frequency L~ km; h ~3-5 km L~ km; h ~1-3 km L~1-10 km; h ~ m non-linear/blocked regime (Nh/U large) Non-dimensional mountain length: NL/U L: mountain length h waves cannot propagate (NL/U small) waves can propagate (NL/U large) Flow processes governed by horizontal and vertical scales (in absence of rotation)

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linear/flow-over regime (Nh/U small) Blocking is likely if surface air has less kinetic energy than the potential energy barrier presented by the mountain non-linear/blocked regime (Nh/U large) Coriolis effect ignored z blk h h Gravity waves See Hunt and Snyder (1980) After Lott and Miller (1997) Non-dimensional height: Nh/U

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Sensitivity to Nh/U Nh/U=0.5 Nh/U=1 From Olafsson and Bougeault (1996) wave-breaking (some drag) smooth gravity wave wave-steepening Nh/U=1.4 Nh/U=2.2 almost entirely blocked upstream large horizontal deviation lee-vortices blocked flow portion of flow goes over Cross section / near-surface horizontal flow Dashed contour show regions of turbulent kinetic energy (ie wave breaking) linear highly non-linear

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Sensitivity to Nh/U: case studies Wind vectors at a height of 2km in the Alpine region at 0300 UTC simulated by the UK Met Office UM model at 12 km. Blocked flow (6 Nov 1999) Flow-over (20 Sep 1999) From Smith et al. 2006

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Sensitivity to model resolution: A finite amplitude mountain wave model From Rontu 2007 Topographic map of Carpathian mountains Streamlines over the Carpathian profile with different resolutions: orography smoothed to 32, 10, and 3.3 km Drag D expressed as pressure difference (unit Pa)

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Sensitivity to model resolution From Clark and Miller 1991 Sensitivity of resolved pressure drag (i.e. no SSO parameterization scheme) over the Alps to horizontal resolution No GWD scheme large underestimation of drag at coarse resolution i.e. sub-gird scale parameterization required Convergence of drag with resolution, i.e. good estimate of drag at high resolution

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Fundamentals of gravity waves Basic forces that give rise to gravity waves are buoyancy restoring forces. If a stably stratified air parcel is displaced vertically (i.e., as it ascends a mountain barrier) the buoyancy difference between the parcel and its environment will produce a restoring force and accelerate the parcel back to its equilibrium position. The energy associated with the buoyancy perturbation is carried away from the mountain by gravity waves. Gravity waves forced by mountains often breakdown due to convective overturning in the upper levels of the atmosphere, in doing so exerting a decelerating force on the large- scale atmospheric circulation, i.e., a drag. The basic structure of a gravity wave is determined by the size and shape of the mountain and by vertical profiles of wind speed and temperature. A physical understanding of gravity waves can be got using linear theory, i.e., the gravity waves are assumed to small-amplitude. Gravity waves that do not break down before reaching the mesosphere are dissipated by radiative damping, i.e., via the transfer of infra-red radiation between the warm and cool regions of the wave and the surrounding environment.

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Summary: gravity wave drag and blocking drag When atmosphere stably stratified (N>0) Create obstacles, i.e. blocking drag Generation of vertically propagating waves transport momentum between their source regions where they are dissipated or absorbed, i.e. gravity wave drag This can be of sufficient magnitude and horizontal extent to substantially modify the large scale mean flow Coarse resolution models requires parameterization of these processes on the sub-grid scale sub-grid scale orography (SSO) parametrization Fine-scale models can mostly explicitly resolve these processes

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References Allexander, M. J., and H. Teitelbaum, 2007: Observation and analysis of a large amplitude mountain wave event over the Antarctic Peninsula, J. Geophys. Res., 112. Bougeault, P., B. Benech, B. Carissimo, J. Pelon, and E. Richard, 1990: Momentum budget over the Pyrenees: The PYREX experiment. Bull. Amer. Meteor. Soc., 71, Clark, T. L., and M. J. Miller, 1991: Pressure drag and momentum fluxes due to the Alps. II: Representation in large scale models. Quart. J. R. Met. Soc., 117, Durran, D. R., 1990: Mountain waves and downslope winds. Atmospheric processes over complex terrain, American Meteorological Society Meteorological Monographs, 23, Durran, D. R., 2003: Lee waves and mountain waves, Encylopedia of Atmospheric Sciences, Holton, Pyle, and Curry Eds., Elsevier Science Ltd. Emesis, S., 1990: Surface pressure distribution and pressure drag on mountains. International Conference of Mountain Meteorology and ALPEX, Garmish-Partenkirchen, 5-9 June, 1989, Gregory, D., G. J. Shutts, and J. R. Mitchell, 1998: A new gravity-wave-drag scheme incorporating anisotropic orography and low-level breaking: Impact upon the climate of the UK Meteorological Office Unified Model, Quart. J. R. Met. Soc., 124, Houze, R. A., 1993: Cloud Dynamics, International Geophysics Series, Academic Press, Inc., 53. 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, Lilly, D. K., and P. J. Kennedy, 1973: Observations of stationary mountain wave and its associated momentum flux and energy dissipation. Ibid, 30, Lott, F. and M. J. Miller, 1997: A new subgrid-scale drag parameterization: Its formulation and testing, Quart. J. R. Met. Soc., 123, Olafsson, H., and P. Bougeault, 1996: Nonlinear flows past an elliptic mountain ridge, J. Atmos. Sci., 53, Olafsson, H., and P. Bougeault, 1997: The effect of rotation and surface friction on orographic drag, J. Atmos. Sci., 54, Palmer, T. N., G. J. Shutts, and R. Swinbank, 1986: Alleviation of a systematic westerly bias in general circulation and numerical weather prediction models through an orographic gravity wave drag parameterization, Quart. J. R. Met. Soc., 112, Rontu, L., K. Sattler, R. Sigg, 2002: Parameterization of subgrid-scale orography effects in HIRLAM, HIRLAM technical report, no. 56, 59 pp. Rontu, L., 2007, Studies on orographic effects in a numerical weather prediction model, Finish Meteorological Institute, No. 63. Scinocca, J. F., and N. A. McFarlane, 2000, :The parameterization of drag induced by stratified flow over anisotropic orography, Quart. J. R. Met. Soc., 126, Smith, R. B., 1989: Hydrostatic airflow over mountains. Advances in Geophysics, 31, Academic Press, Smith, R. B., 1979: The influence of mountains on the atmosphere. Adv. in Geophys., 21, Smith, R. B., S. Skubis, J. D. Doyle, A. S. Broad, C. Christoph, and H. Volkert, 2002: Mountain waves over Mont Blacn: Influence of a stagnant boundary layer. J. Atmos. Sci., 59, Smith, S., J. Doyle., A. Brown, and S. Webster, 2006: Sensitivity of resolved mountain drag to model resolution for MAP case studies. Quart. J. R. Met. Soc., 132, Vosper, S., and S. Mobbs: Numerical simulations of lee-wave rotors.

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