Orographic Drag Parameterizaon History and Issues

N. McFarlane CCCma/University of Victoria

OUTLINE

• Historical Overview - Early papers concerning orographic gravity-wave drag processes - Early operaonal implementaons of orographic gravity-wave drag (OGWD) parameterizaon schemes in NWP models and GCMs

• Other drag processes - Downslope effects - Blocking • Issues J. S. Sawyer, QJRMS, 1959: The introducon of the effects of topography into methods of numerical forecasng

“The moons which are caused directly by topography may be classified broadly as: (a) Small-scale turbulent moon mainly in the lower troposphere. (b) Gravity oscillaons which may extend upwards as far as the . (c) Oscillaons about geostrophic equilibrium. (d) Quasi-geostrophic moons.”

- Derives an expression for the surface drag due to upward propagang gravity- waves generaon by flow over a bell-shaped ridge following the treatment of Scorer (1949), (but with the upper boundary condion proposed by Corby& Sawyer, 1958).

“… If the ridges are spaced at intervals of 10 km the average drag is 7.6 dyne cm-2 which is about the same as the friconal drag over ordinary terrain in a similar wind. … A rough approximaon to the effect on the atmosphere of the drag from gravity- type disturbances would thus be provided by introducing into the equaons of moon a force in the opposite direcon to the low-level wind and decreasing linearly from a maximum near the ground to zero at the .”

[Discussed including effects on the large-scale vorcity fields in the the quasi- geostrophic two-parameter model of Sawyer and Bushby (1953)]

G. P. Cressman, MWR, 1960: IMPROVED TERRAIN EFFECTS IN BAROTROPIC FORECASTS - Considers the contribuon of drag in construcng a bulk drag coefficient for use in barotropic forecast models. “In order to incorporate fricon into the forecast in any scale of moon smaller than the zonal itself, it is necessary to have a map of the distribuon of the drag coefficient over the forecast area,…”

Contours : 2x10-3 Movaon for introducing gravity-wave drag parameterizaons in operaonal NWP models and GCMs (in the late 1980s and later).

- With the advent of operaonal mul-level PE models for medium range forecasng and climate simulaon, a number of systemac errors in the representaon of large-scale flow features became apparent. - Errors in the magnitude and vercal structure of zonal mean in the extra-tropics were among the more robust of these. In parcular: (a) the mean zonal winds in mid-latudes in the troposhere were typically too strong and the jet streams were not realiscally closed. (b) the associated surface and geopotenal height () fields had low (cold) biases over polar regions, parcularly in the Northern Hemisphere winter. - These biases were not substanally alleviated by increasing the spaal (horizontal) resoluon to the extent operaonally feasible.

Palmer et al., 1986 Key Theorecal Advances

• Eliasen and Palm (1960) – In the absence of transience and dissipaon the vercal momentum flux is independent of height for vercally propagang gravity-waves .

• Booker and Bretherton (1967) – crical layer absorpon [exponenal damping of upward propagang GW at crical levels in stable straficaon (Ric > .25)] (also Bretherton, 1969)

• Lindzen (1985) – wave saturaon hypothesis: wave amplitudes adjust in response to the onset of convecve/ small-scale shear Instability in regions where wave steepening occurs

• The E-P theorem in combinaon with the saturaon hypothesis permits formulaon of a simple wave theory basis for gravity-wave drag parameterizaon (Palmer et al., 1986; McFarlane, 1987)

Observing Campaigns

Appendix A from R. B. Smith et al., JAS, 2016 (DEEPWAVE ) (Talk by C. Kruse) Lilly and Kennedy (1973); Colorado Lee Wave Program Lilly and Kennedy (1973) Lilly et al (1982) Basic 2-D linear wave theory for flow over a bell-shaped ridge (Drazin and Su, 1975) Quasi-steady vercally propagang waves close to hydrostac balance In a stably strafied slowly varying (in space) mean flow over a ridge of the form U 2 h(x) = h0 /[1+ (x / a) ] such that 2 g ∂θ f /U << 1/ a << N /U N = > 0 θ ∂z Perturbaon potenal temperature and velocity components in terms of the stream line vercal displacement " 2 % 1 ∂ ∂θ ∂ U ∂ 2 w! =U∂δ / ∂x u! = − (ρUδ) θ! = −δ $ (ρδ)'+ N δ = 0 ρ ∂z ∂z ∂z # ρ ∂z &

1/2 A(z) z N x z N !(ρ(0)N(0)U(0))$ δ (x,0) = h(x) → [cos( dz#) sin( dz#)] A(z) = h δ ≅ 2 ∫ − ∫ 0 # & 1+ (x / a) 0 U a 0 U " ρNU %

1 ∞ 2 2 / (4L) L>>a τ = ∫ ρu!w!dx ≅ −µρ(0)N(0)U(0)h0 = −µρNUA µ = π L −∞

Wave breaking: Saturaon condion∂(θ +θ") ∂z ≅ (1−∂δ ∂z)∂θ ∂z : max ∂δ / ∂z ≅ NA /U ≤1

Saturaon control: at the breaking levels set A = Ab =U / N 1/2 1 U 3 Nh !ρ(0)N(0)U(0)$ => − ∂τ / ∂z = −2µ Max[∂(ln F) / ∂z,0] F(z) = 0 # & ρ N U " ρNU %

In mid latudes the effecve inverse Froude number typically decreases with height in the region below the tropospheric jet maxima but increases rapidly with height in the region above jet maxima.

The effective ridge height for wave emission, heff , is constrained to ensure that F(0)< Fc

Non-linearity of the lower boundary condion: δ(x,h(x)) = h(x)

⇒ crical overturning value Fc<1 {Depending on the geometry of the ridge Smith, 1977, Lilly and Klemp, 1979; Laprise and Peler, 1988}

For bell-shaped ridge Fc=.85 (Miles&Huppert, 1969) Lilly and Klemp, 1979

Palmer et al (1986) McFarlane (1987) Kim and Arakawa,1995

}ß But occasionally downslope wind storms with associated high-drag states occur. (Lilly (1978)

Esmated average wave momentum flux from aircra flight legs at 20,000 and 30,000 : 47 dynes/cm2 . (Klemp and Lilly, 1978) Blocking and Downslope Wind Storms theory and modelling [reviews by R. B. Smith, Advances in Geophysics (31), 1989; M. Teixeira, Front. Phys., 2014 ]

Klemp and Lilly : (1975) - Resonant amplificaon of linear waves in the presence of favourable flow and straficaon - an near mountain top and an upstream wind profile such that the waves reverse phase between the surface and tropopause. Peler &Clark: (1979) – Strong amplificaon for more homogeneous upstream condions in associaon with development of super-crical overturning of streamlines within a vercal wave length above the ridge. Trapping and amplificaon of an beneath the wave- induced level of super-crical steepening . (also Clark & Peler, 1984) Pierrehumbert and Wyman: (1985): Relaonships between wave breaking, downstream acceleraon and propagaon of upstream modes generated by when the inverse Froude number (Nhm /U) is sufficiently large. Smith : (1985) – Steady-state hydrostac internal hydraulic theory. Key idealizaon: assumpon of a dividing streamline above which the flow is quiescent. High drag transion states may develop for a discrete set of values of the height of this streamline.

Bacmeister and Pierrehumbert: (1987) – crical examinaon of C&P and Smith theories, paral support of the hydraulic theory of Smith when wave breaking occurs

[Later developments: Laprise & Peler, 1989; Scinocca & Peler, 1989, Garner, 1995 ]

• 3-D modelling studies (e.g. Miranda and James, 1992, Olafsson and Bougeault, 1996,1997, Schar and Durran, 1997) and laboratory studies (Baines, 1995) è addional effects including:

- development and shedding of lee vorces (Olafsson & Bougeault, 1996; Schar & Durran, 1997, Bauer et al., 2000) (Baines, 1995, ch.6)

- sensivity to the presence of surface fricon and effects of rotaon on drag (Olafsson and Bougeault, 1997, Shus, 1998, Wells et al.,2005) li forces( Smith,1979; Bannon, 1986; Lo1999; Wells et.al., 2005)

- enhanced deflecon of the low-level flow around ridges with the onset of high-drag states (Epifanio and Durran, 200

Miranda and James:

- Fluctuang High Drag Regime - Low drag --> potenal flow for Froude numbers < <.1 Effects of Earth Rotaon

Linear theory (Smith ,1979) Bretherton, 1969 (2-D linear) theory: parcle displacement in the y direcon η! v! =U∂η! / ∂x ∂ [ρw"(u"− fη')] = 0 ∂z Olafsson & Bougeault ∞ ∂p ∞ 1997 D = ∫ h dx = ρ ∫ [(u%− fη%)w%]dx −∞ ∂x −∞ Drag Regimes Miranda and James Ex. FR3 Lo and Miller, 1997 Scinocca and McFarlane, 2000

Regime 1: The Wave Field: One or more waves represented by simple wave parameterizaons. For isolated ridges, (e.g. ellipcal) at least two waves may be needed (Hines, 1988) to capture effects that are missing or poorly represented by single-wave schemes. (see also Garner, 2005)

Regime2: Wave drag + form drag (Lo & Miller, Scinocca&McFarlane, Garner, 2005) + nonlinear (downslope) drag enhancement (Scinocca&McFarlane)

Regime3: Predominantly form drag

Phillips (1984): 3-D Isolated ellipcal ridges: " 2 2 $ h(x!, y!) = h0 #1+ (x! / a) + (y! / b) %

Linear pressure drag force due to N ridges in an area L2 has components

parallel(1) and perpendicular (2) to the upstream flow:

N π /2 F ( p) ( , ) N U h2bG A ( ; , )d (1,2) γ ψ = 2 ρ0 0 0 0 ∫ (1,2) φ γ ψ φ 2L −π /2

Basic forms of surface drag components :

Wave drag: (Scinocca and McFarlane)   2 2 τ free = τ linear Min(Fc / F (0),1)    τ d = [1+ β(F)]τ linear −τ free;Fc ≤ F ≤ 3Fc (downslope)

Form drag:  1 N   τ = − 0 Cl h U(0) U(0) b 2 L2 b b Scinocca & McFarlane C=effecve drag coefficient

[ Simulated from Olafsson & Bougeault ]

{Note: cases for which Nh/U>>2 frequently have small values of U} Scinocca and McFarlane, 2000 Scinocca and McFarlane, 2000 Issues • Dynamics of downslope wind storms and associated non-linear high drag states are sll an acve field of study - frequency of occurrence, dominant mechanisms among the many that contribute , “best” formulaon for the vercal structure of associated drag effects is sll an open queson. “Regime” definions, role of mountain geometry,… (Bauer et al. 2000). (Talks by U. Achatz; S. Garner; S. Vosper; J. Bacmeister).

• Drag associated with trapped lee-wave systems (Bretherton, 1969; Smith, 1976; Lo, 1998, 2016; Broad, 2002) - ignored in parameterizaons. (Talk by M. Teixeira)

• On larger (resolved) scales, non-zonal effects on both parameterized and resolved processes - Interplay between parameterized and resolved contribuons to mountain torques in the angular momentum budget (not addive - substanal compensaon occurs: Boer and Lazare, 1988…. ) (Talk by A. van Niekerk) - Coupling of gravity waves and planetary waves in the middle atmosphere: paral compensaon between resolved planetary wave driving of the zonal mean flow and GWD (McLandress and McFarlane, 1993; A.K. Smith, 1996, 1997, Sigmond and Scinocca, 2010, Cohen et al. 2013, Sigmond and Shepherd, 2014). (Talk by J. Scinocca)

• Coupling with other processes e.g. turbulent transfer in the boundary layer and the free atmosphere; turbulent form drag (Beljaars et al, 2004); [ energy conservaon constraints in GCMs (Boville and Bretherton, 2003, Williamson et al., 2016) oen assume a local balance between producon and dissipaon of ].

Models may include a budget of turbulent kinec energy of a typical form:  De ∂    ∂V ρ + ρw"V"•V" / 2 + w"p" = −ρw"V"• + ρw"B + S −D Dt ∂z ( ) ∂z D   e =Vʹ•Vʹ 2 : Turbulent kinec energy

SD : TKE source due to other drag/momentum transfer processes

3/ 2 : TKE dissipaon D = ρcd e / ld

Boville and Bretherton, 2003; Williamson et al., 2016 in the context of energy conservaon in CAM: Ignore TKE tendency and transport terms on the LHS => TKE net producon=dissipaon => heang