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

Scales of orography and atmospheric processes Length scales of Earth’s orography L ~ 100-1000 km L ~ 1-10 km L ~ 1000–10,000 km courtesy of shaderelief.com

Height scales of Earth’s orography Surface elevation along the latitude band 45oN, based on SRTM 3’’ data (65 m horizontal resolution) From Rontu (2007)

Information on scales by performing a spectral analysis on the data 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) GCM’s have horizontal scales ~ 10-100 km, so not all processes related to orography are explicitly resolved 100-1000 km 1000-10000 km Corresponding orography spectrum, i.e. variance of surface height as a function of horizontal scale, along the latitude band 45oN From Rontu (2007)

Some mountain-related atmospheric processes From Rontu (2007)

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 Parameterization required

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

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

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 u0 with constant density ρ0 and static stability N

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

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

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

Let us now restrict ourselves to stationary waves forced by sinusoidal orography with elevation h(x) given by with u0 hm L The lower boundary condition (the vertical component of the wind at the surface must vanish) is (12)

(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,

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) Wind L L H H H H and L indicate positions of maximum and minimum pressure perturbation, respectively Notice that the vertical flux of momentum for these waves. Here the overbar represents the average along the x-direction. If we take u0~10 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

(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

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 H L H High and low pressures are now on the nodes, so there is a net force on the topography in the direction of the flow

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<N/U (i.e. wider mountains) (or equivalently U/L<N, i.e. low frequency) Wave solution Non-dimensional length NL/U>1 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

Drag force The surface pressure drag (or drag force) on the orography (per wavelength) is given by 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.

Drag force of propagating modes For the evanescent modes the drag force is zero as the vertical flux of momentum is zero. For the propagating modes the horizontally averaged momentum flux For long (hydrostatic) waves with k2<<m2 the dispersion relationship simplifies to And so i.e. for propagating modes the surface pressure drag or ‘drag force’ is non-zero.

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) (b) (a) However, this may be realised at high altitude owing to the vertical transport of momentum by gravity waves (c)

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

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

Single lenticular cloud Durran, 2003

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

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)

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)

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 1970. Solid lines are isentropes (K), dashed lines aircraft or balloon flight trajectories (from Lilly and Kennedy 1973)

Mountain flow regimes Flow processes governed by horizontal and vertical scales (in absence of rotation) h L~1000-10000 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~100-1000 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 ~100-500 m

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)

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)

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

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

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-gird scale parameterization required From Clark and Miller 1991

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.

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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