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**Parameterization of orographic related momentum **

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

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**Scales of orography and atmospheric processes**

Length scales of Earth’s orography L ~ km L ~ 1-10 km L ~ 1000–10,000 km courtesy of shaderelief.com

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**Height scales of Earth’s orography**

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) Surface elevation along the latitude band 45oN, based on SRTM 3’’ data (65 m horizontal resolution) GCM’s have horizontal scales ~ km, so not all processes related to orography are explicitly resolved km Information on scales by performing a spectral analysis on the data km Corresponding orography spectrum, i.e. variance of surface height as a function of horizontal scale, along the latitude band 45oN From Rontu (2007)

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**Some mountain-related atmospheric processes**

From Rontu (2007) Stratification measured by N (Brunt-Vaisala frequency) Stably stratified N>0 (i.e. potential temperature increases with height) Unstably stratified N<0 (i.e. potential temperature decreases with height)

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**Resolved and parametrized processes**

After Emesis (1990) Pressure drag lift drag Lift Drag Hydrostatic drag Form drag Wave drag Gravity waves (L>5km) Inertial waves Rossby waves Viscosity of the air Orographic turbulence (L<5km) Upstream blocking Explicitly represented by resolved flow (scales below a few km’s: turbulent form drag Below smallest scale, turbulent surface friction: roughness length) Parameterization required

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**Gravity waves and low-level blocking**

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

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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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**Linear gravity-wave theory**

Assumptions: Consider two-dimensional airflow in an x-z plane over a ‘ridge’ Waves are linear, i.e. small amplitude Ridge is sufficiently narrow that the Coriolis force can be neglected WKBJ assumption: scale heights of background quantities such as density, temperature, and velocity, are longer than a gravity wave vertical wavelength. Steady-state Boussinesq flow: density differences are sufficiently small to be neglected, except where they appear in terms multiplied by g Atmosphere inviscid The linearized momentum equation can be reduced to a single equation for the vertical velocity Scorer parameter is the speed of the basic state flow is the Brunt-Vaisala or buoyancy frequency See Smith 1979;Houze 1993; Palmer et al. 1986

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**Constant wind speed and stability, sinusoidal ridges**

Consider an infinite periodic ridge in which , i.e. sinusoidal terrain is the horizontal wavenumber with L the width of the ridge Solutions to the previous momentum equation can be written as A and B are complex coefficients and Re denotes the real part is the vertical wavenumber Defining the solution may be written as Upper boundary condition implies B=0 Radiation condition implies B=0 (i.e., the perturbation energy flux must be upward)

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**k>l (i.e. narrow-ridge case) **

(or equivalently U/L>N, i.e. high frequency) Evanescent solution (i.e. fading away) k<l (i.e. wider mountains) (or equivalently U/L<N, i.e. low frequency) Wave solution waves decay exponentially with height vertical phase lines no momentum transport energy/momentum transported upwards waves propagate without loss of amplitude phase lines tilt upstream as z increases Durran, 2003

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**Non-dimensional length: NL/U**

U/L : intrinsic frequency of wave (i.e. frequency based on time it takes for a fluid particle to pass through disturbance) See Houze 1993 and Palmer et al. 1986 intrinsic frequency (U/L) > buoyancy frequency (N), i.e. LN/U<1 wave cannot propagate i.e. k>l: solution decays (or amplifies with height) – evanescent solution not possible to support oscillations at frequencies greater than N intrinsic frequency (U/L) < buoyancy frequency (N), i.e. LN/U>1 wave can propagate i.e. k<l: sinusoidal solution – waves propagate without loss of amplitude Real valued m for waves that transport momentum More generally, vertical transport only possible when the frequency of the waves is bounded above by the buoyancy frequency and below by the Coriolis frequency h LN/U>1 for L~1 km (with U=10 ms-1, N=0.01s-1)

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**Vertical variations in wind speed and stability, **

isolated mountain case - - - More realistic terrain and atmospheric profile If the vertical variations in U and N are such that the Scorer parameter decreases significantly with height then trapped lee waves can extend downstream of an isolated ridge Necessary condition and Gravity wave Trapped lee wave Above 3 km N is reduced by a factor of 0.4 N = 0.01 s-1 U = 10 ms-1 Wave propagates vertically in the lower layer and decays exponentially in the upper layer Parametrization of trapped lee waves is not typical, see Gregory et al Durran, 2003

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**Evident from radiosondes**

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

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**Evident from satellites (AIRS: Atmospheric Infra-red Sounder)**

Alexander and Teitelbaum, 2007

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**Evidence of gravity waves in cloud formations**

Trapped lee waves Durran, 2003

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**Single lenticular cloud**

Durran, 2003

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

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**Non-dimensional height: Nh/U**

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

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**Sensitivity to Nh/U Nh/U=0.5 Nh/U=1 Nh/U=1.4 Nh/U=2.2**

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

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**more non-linear From Scinocca and McFarlane (2000)**

Vertical transport of momentum Wave field becomes unstable or breaks above topography, redistribution of momentum between breaking level and topography, resulting in enhancement of surface pressure drag Momentum not transported vertically away from surface, so deposited below height of topography more non-linear Schematic of surface pressure drag as a function of non-dimensional mountain height for constant N and U flow which characterises three flow regimes identified by numerical simulations. The value of surface pressure drag is nondimensionalized by the linear-theory pressure drag due to gravity waves (proportional to NUh2).

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**Low-level wave breaking**

Wave breaking can occur immediately above the mountain top Occurs if F is increased beyond Fc ~O(1) Courtesy of Uni. of Utah

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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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**Summary of flow classification**

Orographic flow classified according to Nh/U (y-axis) and NL/U (x-axis). Nh/U From Rontu 2007 NL/U L

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**Influence of ‘gravity wave’ drag on the atmosphere**

Flow over or around orography will typically cause the pressure to be higher on the upstream side of the mountain than on the downstream side (c) Net force on mountain in downstream direction from mean flow (a) (b) (a) An equal and opposite force is exerted on the mean flow by the hill (b) L : pressure anomaly Pressure = force per area p= F/A 1 Pa = 1 N m-2 However, this may be realised at high altitude owing to the vertical transport of momentum by gravity waves (c) Momentum: Vertical momentum flux:

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**Gravity wave saturation / momentum sink**

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 Dispersion relationship for hydrostatic gravity waves

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**Eliassen-Palm theorem**

waves steepen leading to wave breaking and elimination of wave (i.e. λ >λsat) 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 Eliassen-Palm theorem: stress unchanged at all levels in the absence of wave breaking / dissipation (i.e. λ=λs) i.e. amplitude of vertical displacement must increase as the density decreases upwards λ Linear, 2d, hydrostatic surface stress After Rontu et al. (2002) (measured in Pa (N/m2)

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**Momentum flux observations**

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

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**Gravity wave observations**

steepening of waves leading to eventual wave breaking and turbulence increasing vertical displacement as density decreases trapped lee waves downslope wind-storm 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)

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**Sensitivity to model resolution: A finite amplitude mountain wave model**

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) From Rontu 2007

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**Sensitivity to model resolution**

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

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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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