Numerical Studies of Stably Stratified Planetary Boundary- Layer F lows over Topography and Their Parameterizat ion

for Large Scale Numerical Mode1

Singnan Zhou

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I Numerical Studies of Stably Stratified Planetary I

1 Boundary-Layer Flows over Topography and Their

1 Parameterization for Large Scale Numerical Nodel

I OY JINGNAN ZHOU

a tnesis s~omittea to the Facuity of Graauate Stuaies of York University ln naniai fuifillrnent of the reauirements for the aegree of

MASTER OF SCIENCE

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ABSTRACT

A linear, non-hydrostatic boundary-layer model is developed to investigate stably

stratified flows over topography. The Froude number FL . based on hiil length scale L. is

used to divide test cases. When FL > 1, we concentrate on the boundary-layer flow and

wind shear and buoyancy effects are demonstrated. For FL < 1, the effects of the

boundary layer on mountain gravity waves are discussed. Strong wind shear and turbulent

exchange in the boundary layer cm significantly modify wind speed perturbations. With

the linear model, parameterisation of effects of flow over subgrid- scale topography has

been studied. A generalized form drag, here called local drag , is proposed in terrain-

following coordinates for linearized stably-smtified, boundary-layer fiows. The local drag

can be thought as the form drag across a streamiine and linlcs traditional wave drag and

form h g . The effects of the boundary layer on the topographie drag and the limitation

of traditionai, inviscid theory, wave drag are investigated.

Contents

Chapter 1 Introduction

1.1 Review of studies of stably stratified boundary-Iayer flows over topo PP~Y

1 . 1 . 1 The evanescent regime 1.1.2 The nonrotating wave regime

1.2 Review of subgrid-scale topography parameterizations

1.2.1 Studies o f fom drag 1.2.2 Studies of wave drag

1.3 The objectives

Chapter 2. A linear nonhydrostatic boundary layer model

2.1 Basic equations

2.2 Background flow over 8at terrain

2.3 The linearized model

Chapter 3. On stably stratified planetary boundary layer flows over topography

3.1 Effects of Stratification on boundary layer flows over topography with FL > 1

3.1.1 The perturbation amplitudes 3.1.2 The phase shift of the flow perturbations

3.2 Effects of the boundary layer on mountain gravity waves

3.2.1 The amplitudes of perturbation variables 3.22 The phase of perturbations

Chapter 4. On wave drag, form drag and parameterization of subgrid scale topography in large scale models

4.1 Momentum flux and local drag

4.3 Topographic drag in stably stratified boundary layer flows

4.2.1 Topographic drag with FL>l 4.2.2 Profiles of local drag for FL<l

Chapter 5. Conclusions

References

List of Figures and Tables

Figure 1.1 Defuiitions of h, L. Au, b, îîz and z, (from Taylor et al 1987).

Figure 2.1 The profiles of (a) the decay hnction F(Z), (b) background wind speeds and

(c) turbulent eddy diffisivity. When 2, =O, Z, =25000m, &, = 10000m and

Zs,,=2000m. The background boundary-layer wind profiles and turbulent

diffusivity are simulated when U,= lOms-', Vg=ûmsl , f= 10"sl, $ =O.lm and

N = 0.01s-'(in stable stratification)

Figure 3.1. The perturbation distribution in stably-stratified boundary-layer flow over

sinusoidal terrain, where topographical length. L = 5 km, height, a = 50m,

u,= lOms-l, q,=O.lm, N=O.Ols-L and FL= 1.25. Two wavelengths are shown for

case of visualisation. (a) ucornponent wind speed perturbation and @) pressure

perturbation.

Figure 3.2. The nomalised amplitude profiles of perturbation variables for different

topographie length scale (L) in stably and neutrafly-stratified boundary-layer

flows, u,= lûms1, %=O. lm, and N=0.01 SI. (a) wind speed perturbation. (b)

pressure perturbation, (c) potential temperature perturbation.

Figure 3.3. The nomalised amplitude profiles of perturbation variables for different

values of geostrophic wind, u, in stably- and neutraily- stratfied boundary-layer

flows. q=O.lm, a/ l=0.01. A=2000m and N=O.Ols-l. (a) wind speed

perturbation, (b) pressure perturbation.

Figwe 3.4. The normalised amplitude profiles of pressure perturbation from the neutral

case, stable case and three specific test cases, where q = O . lm, L& = 10ms-'.

L= lkm and N=O.O~S'~ (in stable stratification). Note that S and N are referred to

the stable and neutral cases. W-N, T-S and B are cases with background wind

speed profiles for the neutral stratification, turbulent diffusivity in the stable

stratification and buoyancy effects .

Figure 3 S. The same as Figure 5 except for L = 5km , (a) wind speed amplitudes and @)

pressure perturbation amplitudes. Note that in @) the cure for TST2 and full S

simulation are almost indistinguis hable .

Figure 3.6. The phase profiles of perturbation variables for different topographic length

scales (A) in stably- and neutrally-stratified boundary-layer flows . q = 1 Oms-' . z

,=O. lm, and N =0.01 s-l. (a) wind speed perturbation. (b) pressure perturbation.

(c)streamline displacement perturbation and (d) potential temperature

perturbation.

Figure 3.7. The perturbation distribution in stably-stratified boundary-layer flows over

sinusoidal terrain, where topographical length, A =20 km, height, a= 100rn.

u,= 10msl, %=O. lm, N = O . O l s L and F, =O.3 14. Two wavelengths are shown

for case of visualisation. (a) wind speed perturbation, (b) pressure perturbation.

Figure 3.8. Thenorrnalised amplitude profdes of perturbation variables for different

topographic length scales (A) in stably stratified boundary layer flows. y= lOms".

îq, =O. lm. and N =0.01 s-' for stable stratification. (a) wind speed perturbation,

(b) pressure perturbation, (c) strearnline displacement perturbation and (d)

potential tempearture perturbation

Figure 3: 9. The phase profiles of perturbation variables for different topographic length

scales (A) in stably-stratified boundary layer flows. u,= l0ms-', q, =O. lm, and

N =O.O 1s". (a) wind speed perturbation, @) potential temperature perturbation

and (c) pressure and streamline displacement perturbations.

Figure 4.1. Schematic diagrarn for local drag.

Figure 4.2. Profiles of the nonnalised three forces , their differences and stress differences

on the surfaces: strearnline surface, ~ ~ o n s t a n t (WFLX) and Z=constant (F, ) in the

stably stratified boundary layer flows. where %=O. 1 m, LI,= 1 Ods, and N=O.O 1 s*'. (a)

LDRG and WFLX; (b) LDRG and Fm,; (c) the stress and force differences on the

streamline surface and z=constant and (d) the stress and force differences on the

strearnhe surface and Z=constant

Figure 4.3. Profiles of nomalised local drag in the neutrally and stably stratified boundary

layer flows for different topographic length scales in the FL >1 case (if in stable

stratification), where %=O. 1 m, ug= 1 Ods, and N=O.O l SI .

Figure 4.4 The relationship between nomalised suface drag and Froude nurnber in the FL

>1 case

Figure 4.5 The relationship between normalised surface h g and Yz, for different Froude

num bers.

Figure 4.6. Profiles of local drag in the stably stratified boundary layer flows for different

topographicai length scales in the FL 4 case , where =O. lm, =lOm/s. and

N=O.O 1 s-'.

Figure 4.7.The relationship between surface pressure drag and upper wave flw and FL Br

FL < 1, where U, = 1 Oms-' , N= IO-' s-' and f= 1 O4 S'

Figure 4.8 The profiles of local drag for L/z, in the FL -4 case.

Table 1. The relationship between local drag at surface and topographic lengths in the

neutraily and stably stratified boundary layer. S and N mean stable and neutral

stratification respectively

Table II. The relationship between the surface pressure drag (S-DRG), upper wave drag

(U-DRG) and their difference and FL, where %=O. 1 m, U,= 1 Oms" , N= 1 0e2 s*' and

+ 10-4~4

Chapter 1. Introduction

1.1 A review of studies of stably stratified boundary-layer flows over

topography

Hills and mountains can affect airflows in many different ways with a great range of

length and time scales. To understand some fundamental features of the complicated

airflow, different flow regirnes are ofien distinguished based on some simplified models.

The earliest, and possibly the most mature, mode1 is a linear theory, which was initiated by

Queney (1 948) and has been well covered in textbooks (e-g. Gill 1982). According to the

linear theory for u n i f o d y stably stratified inviscid flows, the airflow over mountains cm

be divided into the following five flow regirnes in terms of Froude number, FL ,( = 2xUML)

and Rossby number ,Ro, (ü/f L) (Gill, 1982), where U and N are the uniform velocity and

buoyancy fkequency of the airflow, L is the characteristic length of topography and fis the

Coriolis parameter:

( 1 ) Potential Flow Regime when FL >> 1 and Ro >> 1. The amplitude of the vertical

displacement of fluid particles falls off with height and the flow is faster over the crest and

correspondingly, the pressure is Iower according to Bernoulli's equations.

(2) Nonhydrostotic Wave Regime when FL c 1 and Ro >> 1. Topography can excite

gravity waves, which can propagate upward and d o ~ e a m . The pressure is higher on the

upwind side of the Ml1 than on the leeward side, indicating a net force on the hill.

(3) Hydrostatic ivonrotating Wave Regime when FL 1 and RO » 1. The hydronatic

approximation is satisfied and topographically-induced gravity waves can just propagate

verticalIy upward.

(4) Rotating Wave Regime when FL << 1 and Ro - 1. Rotation effects induce the wave

energy to again propagate upward and downward although the hydrostaûc approximation

is still satisfied.

(5) Quasi-Geosnophic Flow Regime when FL 1 and Ro « 1. Strong Coriolis forces

induce the flow to satisS the quasi-geostrophic approximation and the topographicaily-

induced perturbations are again evanescent or waves are not produced.

In generd, the above five flow regimes can not exactly fit reaiistic airflow pattenis

over terrain, particularly when the airflow involves boundary-layer processes, but they reveal

some fbndamental features and provide a theoreticai base to understand complex flow

pattern. We will review studies of stably stratified boundary-layer flows over hills based

on the above linear theory. As a slight variation from the above flow regimes, we prefer to

define three regimes for this review of boundary layer flows. (1) the evanescent wave

regime; (2) the nonrotating wave regime and (3) the rotating 80w regime. The evanescent

regime is associated with the potential flow regime, the nonrotating wave regirne

corresponds to the nonhydiostatic wave regime and the hydrostatic and rotating regimes are

related to the rotating wave and the quasi-geostrophic flow regimes. Since this study is

primarily associated with the evanescent wave regime and the nonrotating wave regime. we

will not review studies of the rotating flow regime.

1.1.1 The evanescent regime

Studies of boundary-layer flow over hills in diis regime were acnially begun with

neutrd stratification. They have been extensively studied over the past twenty yean in

anaiytical, numencal and observationai ways, since Jackson and Hunt's (1975) (hereafter

JH75) analyticai study and Taylor and Gent (1974)'s (hereafter TG74) and Taylor's (1977)

(hereafter T77) numerical studies. Studies of stably-stratified boundary-layer flows have

not been conducted until later, in the eighties. For the sake of clarity, the review can be

divided into three subsections (1) analpcal studies; (2) numericd studies; and (3) stably

stratified studies. Note that the observational studies have not been reviewed since this

study is mainly associated with previous analyticai and numericd studies rather than

observational studies. Readers interested in observational studies may refer to Taylor, et al

( 1 987) and Kaimal and Finnigan (1 994).

Analytical studies

Analytical studies of neutraily stratified boundary-layer flows over hills started with

JH75. Concepts of the inner and outer layer were proposed in JH75. The outer layer is

assumed to be essentially inviscid and it is achially a potential 80w layer within the

boundary-layer; while perturbations of turbulent transfer processes are dynamically

significant only within the inner layer. The distinction between the inner and outer layes

is important in understanding the turbulent 80w over complex terrain since the pressure

perturbation is detemiined by the outer layer. Naturally, the depth of the inner layer is a key

parameter to understand the airflow over hills. JH75 proposed the following formula

where q, is surface roughness , L is characteristic horizontal length , K is von Karmh's

constant and an upstream wind profile of the usual logarithmic form is assumed. Jensen et

al (1984) M e r refined the formula as follows, with the assumption that AT-

2~u.Au/ln(Azfz,), in place of KU Au/Az, where A s is the stress perturbation. This cm be

shown to lead to

For the de f~ t ions of Au and Az see Figure 1.1. Equation (1.2) is similar to equation ( 1.1 ).

but the depth of the inner layer 1' or the solution to equation (1.2) is shallower than the

solution to equation (1.1) and is better agreement with observation (Taylor, et al 1987).

JH75 also proposed a fiactionai speed-up ratio AS =Au /a , which typically measures

the mean flow perturbation induced by hills and is an important factor for detemination of

wind loads on structures and the wind energy potential of hilltop windmill sites. Taylor et

a1 (1 983) discussed the speed up ratio in more detail and provided a diable f o d a Their

midies eventually induced Taylor and Lee ( 1984) to develop a simpiified formula as part of

a package of simple guidelines to estimate design wind speeds for engineering purposes

The two dimensionai JH75 linear theory for terrain-induced meantlow peaurbations

above 2D low hills gives a useful estimate for the perturbation (or speed up), u, in wind

speed and in the perturbation surface shear stress over hills, but in the theory, the horizontal

perturbation velocity does not match at the interface between the inner and outer layers

although pressure does; furthemore, the pressure matchuig tums out to be correct only if the

hi11 height h « 1, which is very restrictive. This limitation d e s it difficuit to account for

the effects of strong shear in the upwind mean velocity on the 80w over hills. Taylor et al

(1983) and Mason and King (1985) have made heuristic adjustments to the JH75 model.

Hunt et al (1988a) M e r resolve the dificulties of IH75 analysis by introducing an

intermediate , rniddle, layer in the outer region. Belcher (1990) M e r re-examined the

assurnptions of Hunt et al (1 988a): the solutions to the eqtntions governing the linear

perturbations are generalised by introducing displaced coordinates, where effects of mean

streamline curvature are systematically considered. Note that Hunt et al (1988a) used a

discontinuou vertical coordinate and did not analyse the effects of the transformation on

the governing equations.

Numericai S tudies

In parallel to the analytical studies, TG74 and T77 have presented numerical, h t e

difference, calculations of flows over ideaiised terrain featurrs. Their numencal approach is

non-linear and avoids the necessity of division of the flow into different layea dthough they

used relatively simple closure hypothesis. With the advent of large cornputers, Newley

(1985) extended the type of finite-difference mode1 previously used by Taylor (1977) to

include a full second-order ciosure mode1 (Launder et al, 1975). Similar to Newley's work,

Zeman and Jensen (1987) have also developed a full second order closure model to sirnulate

aVflow over hills, but they have a simplified treamient of pressure. In contrast to Newley's

work, Reynolds stress and dissipation equations are formulated in an orthogonal coordinate

system aligned with local streamlines, since Zeman and Jensen thought that, when the

c 10 sure equations are formulated in streamline coordinates, they yield more direct

information about the effects of curvature.

Numerical studies have the advantage that they are able to solve the nonlinear

equations and are not bound by the linearization assumptions. Hence, these studies have

provided a method of assessing the above linearised modeis. Taylor et al (1983) compared

their linear mode1 results with Taylor (1977)'s finite different model results. Belcher (1 990)

used the Newley (1985) model tu examine their linear solutions.

A very interesting research direction emerging in the eighties was a combination of

analytical and nurnencal studies. Walrnsley et ai (1982, 1986) and Taylor et a1 (1983)

introduced the concepts of the inner and outer layers into a senes of numericd rnodels. The

operational version of these models is referred as MS3D.M. The models are based on Mason

and Sykes (1979)'s 3D linear theory extended fiom JH75 2D linear theory. The advantage

of the models is that they are able to simulate airflow over reai terrain and the computer

requirements for memory and process time are very Iow, even for high resolution runs. The

approximation of the advection velocity by a constant is one of the limitations in the models.

in order to overcome the limitations of the MS3DJH models, Beljaars et al (1 987) developed

a linear model using spectrai and finite difference techniques. The steady state linearized

governing equations are Fourier transformed in the horizontal while h i t e differences are

used in the vertical. The model has significant efficiency advantage over the finite different

models of Taylor (1 977) or Newley (1 985)' due to using a single matrix inversion to arrive

at the solutions rather than iterative models in a f i t e difference model . Ayotte et al ( 1994)

has M e r extended a mived spectral f ~ t e difference (MSFD) model to a full second order

turbulent closure model and Xu et ai (1 995) has extended the model to its nodinear version,

called the non-linear mixed spectral finite diEerence model (NLMSFD).

Stably-stratified flow Studies

Snidies of stably-stratified boundary-layer flows over hills had not been initiated

until Hunt et ai's (1988b) work. A linear andytical model, based on dividing the flow into

different regions with different dynamics, has been developed for mean flows over hills uith

low slope, for various kinds of stable stratification in the upwind flows. The linear anaiysis

demonstrates the general effects of stratification on 80w over low hills when effects of

buoyancy forces are weak within the inner region. The wind speed just above the h e r layer

is affected by stratification through two factors. (1) the upwind velocity gradient or shear

relative to stratification and (2) the ratio of buoyancy forces over hills to inertial forces of

the mean flow, or Froude number. Note that Hunt el al's (1988b) analysis is based on

studies of the flow in the inviscid outer region over hills. The turbulent exchange in the b e r

layer, which cari be modified by effects of the stratification, is excluded. Belcher and Wood

(1996) have extended the linear analytical model of neutrally-stratified surface layer flows

over MIS developed by Hunt et ai ( 1988) to stable stratification and the turbulent exchange

in the imer layer has been taken into account. Note that surface-Iayer profiles can not

support propagating waves; they are suppoaed only when account is taken of the finite

depth of the boundary layer (Belcher and Wood (1 995)). To detemüne the f'kite depth of the

boundary layer in a surface-layer model is somehow arbitrary (Xu, personal communication).

Zhou et al (1995) have developed a planetary boundary-layer model for stably-stratified

b o u d a s , layer flows over topography and pointed out that the boundary-layer cm

significantly reduce propagating wave f lues above the boundary-layer. Weng et al (1997)

have extended MSFD to stable stratification, where the surface layer is no longer infinite.

and M e r confïrm effects of the boundary layer on propagating wave fluxes.

1.1.2 The nonrotating wave regime

Stably stratified d o w s over topography in the nonrotating wave regime have been

extensively studied in both linear and nonlinear cases over the past forty years. Most of the

midies have not considered effects of boundary-layer processes. The effects of boundary-

layer processes has not been paid very much attention until recentiy. However, previous

studies of inviscid stably- stratified flows strongly influence stably-stratified boundary-layer

flow studies in this regime. Therefore, we will fintly briefiy review linear and nonlinear

studies of inviscid stably stratified flows, and then discuss the effects of boundary-layer

processes on mountain waves.

Linear studies

Queney's (1948) pioneering theoretical study is one of milestones in the

develo pment of linear theories for unifody stably strati fied airflo w over topography . The

linear solutions were obtained for uni fody stably stratified airtlow over both sinusoidal

and isolated topography. The solutions have been further extended to non-uniformly

stratified airfiow ( Queney et al, 1960). The early controveny regarding the qualitative

description and interpretation of linear mountain wave theory is in the use of the correct

radiation condition aloft. The radiation condition which can prchibit downward energy

moving became standard among researchers after Eliassen and Palm's (1 954, 1960) work on

the vertical wave fluxes of energy. The linear theory is a h o s t mature and has been well

covered in textbooks (e.g. Gill, 1982).

The linear theory has been applied to study the generation of sorne mountain

rneteorological phenornena such as the Foehn , a downslope wind storm. According to

Eliassen and Palrn (1960), when an upward propagating linear gravity wave encounten a

region in which the wind speeds or stratification change rapidly, part of its energy can be

reflected back into downward pro pagating waves. Klemp and Lilly (1 975) extended Eliassen

and Palrn (1960)'s results to the case of small amplitude hydrostatic waves in the multilayer

atmosphere with constant stability and wind shear in each layer. They found that downslope

stoms occur when the atmosphere is tuned to give optimal superposition of upward and

downward propagating waves.

Nonlinear studies

In general, linear theory begins to break down when the topographie height becomes

comparable to the vertical wavelength in a continuously stratified atmosphere. Notable

studies of finite amplitude mountain waves began with the pioneering work of Long( 1953.

Long discussed the steady flow of an incompressible, continuously stratified fluid and

pointed out that there is a specid class of upstream profiles for which the governing

equations become exactly linear, which are ofien called Long's model. The difficulty of the

Long's model is that the boundary condition at the rnountain surface is still of a dificult

nature while the equation for the interior motion is of a simple type. There were many

researchers working on the solutions to the Long model in the sixties. Miles (1969)

reviewed the nature of these solutions.

Note that Long's model can break down either by the occurrence of instability and

turbulence or by alteration of the upstream flow (i.e. blocking). The phenornena are of

considerable importance for understanding downslope windstorms. A powerful method for

understanding nodinear mountain waves is the numerical solution of the governing

equations. Clark and Peltier (1977) and Klemp and Lilly (1978) have simulated wave

breakmg phenornenon, where venically propagating mountain waves become so steep that

they cause a local revend of the flow aloft. Peltier and Clark (1979)' Bacmeister and

Pierrehumbert (1987) and Durran and Klemp (1987) M e r demonstrated that wave

breaking does not simply reduce the amplitude of the vedcdly propagating waves above the

breaking region and that the effects of wave breaking are mainly responsible for the

generation of downslope wind storms.

Effects of boundary-layer processes on mountain waves

Studies of effects of boundary-layer processes on mountain waves were not

mentioned in author of Smith's (1979; 1989) reviews. Smith (1 979) did give an explanation

for such studies excluding boundary-layer processes based on dimensional anaiysis, but in

his anaiysis, the wind profile of the boundary layer had not been considered. As we know the

wind speed is fonnaily zero at ground. which causes difficuities within Smith's analysis. In

fact, Richards et al ( 1989) demonstrated the importance of surface fiction by showing that

its inclusion in a numerical mode1 eliminates the unrealistic tendency of the models with fiee

slip lower boundary layer to form a layer of high surface wind that propagates indefinitely

downstream. Nappo and Chimonas (1 992) revealed that the dissipation of gravity waves

induced by topography occurs in the stably-stratified boundary layer. Grisogono (1994) and

Zhou et al (1995) aiso point out reduction of mountain wave drag by boundary-layer

processes. Qi and Fu (1993) and Qi et al (1997) funher demonstrated the effects of the

diumal boundary layer on rnountain waves.

1.2 Review of subgrid scale topography parameteruatioa

The rnomennim and energy transfer between the atmosphere and the solid eaah

occun on a wide range of scaies. On smail scales ranging fiom mm to 10m, the effect of

surface roughness eiernents such as snow, soil, vegetation and buildings is often represented

in term of a roughness length ( z,, ) assuming a logarithmic region exists near surface. The

mornentum transfer over the smdl scale surface eiements can be considered as turbulent

tramferin the boundary layer and represented by surface shear stress. This turbulent transfer

has been considered Ui most meteorological models. On large scaies ranging from lOOm

to 100km, which are mostly associated with cornplex terrain, the momentun transfer is

governed by a number of topographically-induced dynarnic airfïow processes. These

processes can not be resolved by general circulation models (GCMs) since their horizontal

spatial resolution is typically of O(300km). We thus have to parameterize these subgrid

scale processes for a larger scde model. Currently, there are two types of subgrid scaie

schemes: often referred to as form drag and wave drag. The fonner is a pararneterization of

the effects of surface features whose scales range fiom 100m to IOkrn. A concept of effective

roughness length is ofien used by assuming that an idedised logarithmic velocity profile

exists over the surface. The latter considers the effects of larger scaie features ranging from

lOkm to 100km. Topographically induced gravity waves play an important role in

momentum transfers on those scales. We will separately review form drag and wave drag

studies.

1.2.1 Studies of form drag

Small scale topographie flow perturbations are generally lirnited to the lower level

of the atmosphere. Effects of the turbulent Reynolds stress and shear in the mean flow can

induce a drag force in 80w over topography. The drag might be thought of a conceptuai

extension of fiction drag, wtiich represents pressure forces on small scaie d a c e roughness

elements. In the turbulent boundary layer, roughness length is usually used to describe the

surface characteristics and the fiction drag coefficients are functions of the roughness

length. Similady, Fiedler and Panofsky (1972) defined an effective roughness length to

represent the effects of small-scale topography. Mason (1985) and Newley (1985) have

M e r demonstrated that the form drag on small-scaie topography can be described by using

the effective roughness length concept, whereby the surface roughness is increased so that

the surface stress becomes e q d to the value that would be obtained if the subgrid scale hills

were considered explicitly. Taylor et al ( 1989) have examined the relatiooship between the

effective roughness length, terrain slo pe and to pograp hic wavelength in turbulent boundary -

layer tlow over two-dimensional sinusoida1 topography and an empirical formula is

proposed. Belcher et ai (1993) have developed a drag formula fiom a linear analysis

extended fiom the work of JH75 and Hunt et al (1988). Wood and Mason (1993) M e r

derived formulas for both drag and effective roughness length, which Xu and Taylor ( 1995)

evaluated using their numencal results kom NLMSFD with second order turbulent closure.

Note that the approach generally requires the existence of a Logarithmic region

above the surface . On these larger scales comparable with the depth of the boundary layer.

there is no formal justification for the existence of a logarithmic region and a representation

in term of the roughness length. However, there are some observational and numerical

results supporting . this approach in the neutrally-stratified boundary layer flow. Kustas and

Brutsaert (1 986) and Grant and Mason (1 990) have shown a surprisingly extensive region

of logarithmic profiles in theù meteorological observations in complex terrain. Newley

(1985) and Wood and Mason (1993) have computed vertical profiles of the reaily averaged

wind field and they show a logarithmic variation with height through some levels

intermediate between the hi11 height and the depth of the boundary layer. On the larger

scaies up to IOkm, the above results are unlikely to be hue. In particular stable stratification

will suppress turbulence and give rise to the generation of intemal gravity waves. In such

cases the influence on the roughness length is unknowm and we have to pay special attention

to gravity waves.

1.2.2 Studies of wave drag

Topographically-forced interna1 gravity waves can induce the surface pressure to be

higher on the upwind side of hills than on the leeward side, which indicates a net force on

the hill. The net force is ofien cailed wave drag. The linear theory of mountain waves is

often applied to understand wave energy, momentum transport and wave drag. Sawyer

(1959) used the linear theory to compare pressure force by statioaary gravity waves with

fnctional stress exerted over open countryside and point out the importance of

parameterhg gravity wave drag in large-scale numerical models. Bretherton's (1967) and

Lilly (1972)'s work M e r supported the need to parameterize the gravity wave drag. The

gravity wave drag scheme was not proposed or implemented until the work of Palmer et ai

(1 986) and McFarlane (1987), which aiso demonstrate the significant impact of the wave

drag on the atmosphenc generai circulation.

Both schemes have considered the rnean state cntical layer absorption of gravity

wave energy, where wind speeds go to zero and gravity waves are breaking. The mean

state critical layer often occurs in the upper troposphere, where there are unidirectional and

horizontally inhomogeneous fîows within which the wind speed goes to zero and the

Richardson nurnber is lower than 0.25. Mountain waves can also break at low levels when

the mountain is high enough (Peltier and Clark, 1979). Low level wave-breaking may induce

a aitical layer below which wave energy is trapped and as a result the drag is enhanced

significantly through resonant amplification of gravity waves (Smith, 1989). Kim and

kakawa (1995) and Lotî and Miller (1997) have M e r parameterized the Iow-level

graviîy wave drag.

Note that the above snidies have excluded boundary-layer processes. Recent studies

demonstrate the effects of boundary-layer processes in gravity wave drag parameterization.

Nappo and Chimonas (1992) pointed out the important impact of strong wind shear in stable

atmospheric boundary layes on gravity wave propagation Grisogono (1994) reveaied e f f m

of turbulent diffusion on wave drag dissipation and Qi and Fu (1 993) and Qi et al (1 997)

demonstmted the effects of diurnai boune- layer processes on the mountain gravity waves.

13 The objectives

From the above review, we can see that significant advances have k e n made in the

study of neuûally-stratified boundary-layer flows over hills. The theories have now k e n

extended to stable stratification. However, when buoyancy effects become very important

or gravity waves are exciteci, the dynamic processes are significantly different, and quite

svnilar to those in mesoscale stably-matified flows over topography, where intemal gravity

waves dominate flow patterns. Current studies of stably-saatified boundary-layer flows have

not so far paid much attention to the role of mountain-induced intemal gravity waves.

Over the past forty yean, eaensive studies of mesoscale matified airfiow over

mountain have demo-ted the ro le of to pographicail y-induced interna1 gravity waves in

phenornena thaî codd be directly obsewed such as wave clouds, severe downslope winds,

clear air turbulence and glider ascents. However, most of the midies exclude boundary-

layer effects. The influence of the "mountain boundary laye? and the relative role of

elevated heating and fiction on the mesoscale flow are poorly understood (Smith et al,

1997). The understanding of the effects of boundary-layer processes has been considered

as one of the important aspects necessary to improve weather forecasting in mountain regions

(Smith, . Z 997).

From our perspective, there appears to have been rather a gap between mesoscale

studies of stratified flow over topography and microscale studies of boundary-layer flows

over hills. Developing a linear nonhydrostatic boundary-layer model and using the model

to investigate this gap is one of the objectives in this thesis.

The gap between the mesoscale and microscale studies inevitably affects the study

of subgrid-scde topography pammeterization. The approach of an effective roughness length,

an alternative way to represent the form drag, requires that the topographicaily-induced

perturbation be Iùnited to the lowest layer of the atmosphere and assumes the existence of

a logarithmic region, on average, above the complex terrain. In stable stratification, intemal

gravity waves are often excited and propagate vertically. In such cases, the concept is no

longer valid. However, the current gravity wave drag parameterization schemes have not

considered effects of the boundary layer. The gravity wave flux is no longer constant in the

boundary layer if effects of boundary-layer turbulent exchange are considered and ais0

boundary layer wind shear can significantly reduce vertically propagating wave fluxes

(Grisogono, 1994 and Zhou et al, 1995). Therefore, it is necessary to lhk f o m drag with

wave drag in order to parameterize subgrid-scale topography in stably-stratified boundary-

layer fl ows. We wiIi propose a generalized drag quantity to link both types of drag and use

the new quantity to M e r study effects of stably-stratified boundary-layers on nibgrid scale

parameterization.

Chapter 2. A linear nonhydrostatic boundary layer model

2.1 Basic equations

Conceptual Iinearized models for studying stably-stratifiecl flows over topography,

particularly for the mesoscale. are often developed in Cartesian coordinates with a Iower

boundary condition w=uH .Vh(x,y), where h(x,y) is the shape of the topography and

uH=(u,v) (Smith, 1979). Note that u.v,w are the Cartesian components of the velocities.

p is pressure including the hydrostatic component but excluding synoptic scale variations

which are represented by the geostrophic wind and 0 is potential temperature. As we

know, for boundary-layer flows, pH is O on the boundary, so, it is difficult to follow this

conventional approach. An attractive analytical model with multiple-layers in the

boundary layer has b e n proposed by Jackson and Hunt (1975) and Hunt et al (1988). It

has the capacity to deal with stratifïed boundary-layer flows, but is still mainly limited to

surface boundary- layer applications. In the present paper we adopt a sunilar , linear

approach to that of Beljaars et a1.(1987), and develop a steady-state 2D nonhydrostatic

mode1 in terrain-following coordinates:

where h(x) is the topographic shape. We also define

For sinusoidal terrain. we assurne that

where a is the wave amplitude and k is the wavenumber, k=Zx/L. F(Z) is a vertical

decay hinction, not used by Beljaars et al. (1 987). When F(Z) = 1, the coordinate exactly

follows the model terrain as in the Beljaars et al MSFD model, When F(Z)= 1-ZIG . where Z , is the top of the model, this is GalChen's terrain-foilowing coordinate, which

is often used in current mesoscale models (Pielke et ai., 1992). In our model, we choose

F(Z) so that. at the lower boundary, the model coordinate follows the terrain, and in the

upper part, the coordinate gradually tends to the Cartesian system. The specification is

where 2 x 2, , is the Iength of the transfer zone from terrain following to Cartesian

coordinates. 2, is a reference height for the transfer zone and Z +d Z ,represent the top

and bottom levels of the domain. Figure 2.la shows the profile of F(Z) when Z,

=25000m, 2, =b, Gf = 10000m and &, =2000m. For simpiicity, a 2D-mode1 is

considered at this stage and horizontal diffision terms are neglected. Note that, the depth

of the mode1 should be greater than the venical charactenstic length. When FL tends to

1 , the characteristic length will be infinite (see equations 3 .1 and 3.2). In this study. we

avoid ninning the cases with 0.84 < FL < 1.28. We also neglect difision terms in the

w equation assuming that, in contrast to the horizontal wind speeds, the gradient of the

vertical wind speed is srnail and effects of the diffision term can be neglected. Following

sirnilar ideas to those used by Beljaars et ai. (1987) but including rotation and the venical

decay function F(Z) in the coordinate uansfonnation. the basic noniinear equations are

then

where

A simple rnixed-length closure (Estoque, 1973) is applied with

The neutral mixing length is defmed as

with

and stability effects are represented by

f ,(Ri) = ( 1 +PRO'

where Ri is the Richardson number ( = (g/Q~)~8/azl[(ûv/~3z)~ + ( a ~ / a z ) ~ ] ) and P = 3

(Estoque 1973). We recognise that this is a rather primitive closure assumption but ir is

relatively robust and will provide a base on which to develop modeis with more

sophis ticated closures .

2.2 Background flow over flat terrain

In order to linearise these equations, it is important to fmd a reasonable

background field. However , suatified boundary-layer flows rarely , if ever, achieve a

steady state. Over mesoscale topography , in particular, thermal forcing has a diurnal

variation and the flow pattern has different stages (Qi et ai, 1997). Those flow patterns

are generally unsteady, so it is difficult to extract background flows for a linearisation.

Since our main objective for these scales is to investigate gravity waves caused by

dynamic forces in stably- suatified boundary- layer flows, only topographie dynamic

forcing is considered and the background potential temperature and corresponding

hydros tatic pressure are assumed to be s teady state and horizontally homogenous .

For shplicity, thermal field perturbations due to differential heating of the elevated

surface are no t considered here . Also. the background potential temperature gradient (30,

/&) is assurned to be constant from the surface to the top of the model and is equal to a

typical atmospheric value. y = 0.0033 Km-' .

For the background wind field in studies of the boundary layer over topography,

upwind flows are normaily used (Jackson and Hunt, 1975; Hunt et al, 1988 and Beljaars

et al, 1987). In Jackson and Hunt's paper, it was assurned that the frst approximation to

the horizontal velocity u(Z) in the inner layer is given by the upstream velocity u, at the

same displacement (Z) above level ground. Far above the hill , in the outer layer. the

horizontal velocity is equal to the undisturbed upwind velocity at the same geopotential

height, k(z) . In linear numerical models, for instance MSFD (Beljaan et al. 1987). udZ)

has however been extended to the whole computation domain.

We will follow this practice, as in Ayotte and Taylor (1995). The background flow

%(Z) is given here by the solutions to the horizontally homogenous, steady state, 1D

stably-stratified PBL model equations with mixing-length turbulence closure as in

Estoque(l973). No te, however that we do not include the thermodynamic equations . w hic h

would lead to non-steady state solutions, and we separately prescribe the potential

temperature field as a function of z. It is specified as a hinction of z rather than Z to avoid

background pressure gradients. Effects of the stable suatificauon have been considered in

the background h w s . Figure 2. l b and 2. lc show the wind speed and turbulent eddy

d i f i h i t y profiles in neuual stratification (N = 0.0 s-l) and stable stratification (N =0.01

s -') , where q, =O. l rn, u, = 1 Oms1 and v, = OmsL .

2.3 The linearized mode1

In order to justi@ the lkarisation of equations (2.3-2.7), we suppose that the

height of the topography is much less than the characteristic length which the perturbation

caused by topography and turbulent exchange can reach. i.e., a é < min (H, H ,, ), where

a is the amplitude of the topography , H is the characteristic vzrtical length scale of the

perturbation and H ,, is the height of the boundary layer. If we split variables into

background and perturbation parts, e.g. u, +u,, we can then gewrate linearized versions

of equations (2.3-2.7) for 2D, y-independent situations as below. We use the

transfonned vertical coordinate

which was suggested by Taylor and Delage (1971) in order to give a good resolution close

to ground, where c, and c, are coefficients chosen here as 3 and 0.01 respectively . Afier

subuacting the zero order terms, the linearised equations are

\ rn 0 1 in the a b l a c u e \ o_crso! in the nautrai caaa \

b. .

Figure 2.lc The &, roffler in the rhbly and qeutraiiy b o u n ~ ~ - l a y e r . rhare U,=ithm; ., v,=0mK4 f=10 a' , o.lm lm uid N=O.Ola (in atable stirtiiicrtion)

de.

where

w i' =W -u, h ' ( X ) F(Z)

It is understood that the background 0, does not satisfj Equation (2.7). The

rationde for this is that the time scale for evolution of 0, through Equation (2.7).

especially at upper levels would be slow relative to advection and wave propagation rime

scales. The Coriolis t e m in Equation (8) is omitted assiiming that advection tirne scale

is much shorter than f -' . Thus, the v-component velocity in this 2dimensional case

cannot interact with the u component and since in addition we assume a/ùy=O, we no

longer solve the v-component equation. Note that the last two ternis on the right hand

side of Equation(2.8) do not affect the mode1 results in this study since dF(Z)/dZ= O in the

boundary layer (see figure 2. la) and above the boundary layer K, =O.

As in the MSFD mode1 (BeIjaars et ai. 1987). our lateral boundary conditions are

periodic, and Fourier transformation is used in the horizontal direction. In order to absorb

gravity waves at the top of the mode1 and avoid wave reflection, a sponge boundary

condition is iatroduced at upper levels. The effective depth of the mode1 domain is l O k m

and the depth of the sponge layer is 15km. The choice of the sponge layer depth is

partially based on Klemp and Lilly (1978)'s results, in addition to our own sensitivity

experirnents. Finite differences are used in the vertical direction with 120 levels and a

block tridiagomi LU factorization is used to invert the resulting ma& (Karpik, 1988).

Chapter 3. On stably stratified planetary boundary layer flows over topography

For sirnplicity, we will concentrate on a i t o w over sinusoidal terrain, In steady

flow over such terrain, the horizontal wavenumber of the perturbation is always forced to

be that of the terrain and we cm exactly separate the Bows into two regions: FL < 1 and

F , > 1. Note that, with the boundary layer involved, the Froude number will Vary with

height, because of velocity shear and turbulence effects. In this paper, we will use an

outer layer Froude number, FL = 2xVg INL, where V, = (u ,2 +v 2 )ln , and (u,, v,)

is the surface geostrophic wind and N = [(g/eo)aO,,/az] lR is the outer-layer buoyancy

frequency. 8 ,, 0 ,(z) and L are a reference potential temperature, the background

potential temperame and the wavelength for sinusoidal terrain respectively

In classical, inviscid uniformly stratified flows, when FL > 1, the flow above the

boundary layer is similar to potential flow, and in the boundary layer it is somewhat

similar to that in the neuually stratified case, but the difference or the effecrs of

stratification need to be studied. When FL < 1, the flow above the boundary layer

contains gravity waves, as studied for mesoscale flows over topography and we will focus

on the effects of the boundary layer on these waves.

3.1 Effects of stratification on boundary- layer flows over topography with

FL> 1

With FL> 1. gravity waves cannot propagate and the flow patterns are

fundamentaiIy simiiar to those in neutraily- stratified boundary- layer flow. Figure 3. la

shows the distribution of wind speed perturbation u, in stably-stratified boundary-layer

flow over sinusoidal terrain for FL = 1.25 and with the upstream or undisnirbed flow

shown in Figure 2. lb and 2 . 1 ~ . Note that the domain shown in Figure 3.1 is much

shallower than the total depth of the computational domain in order to emphasize the

model results in the boundary layer, two wavelengths of remaining wave included. The

wind speed perturbation decays with height and the maximum speed appears

approxirnately above the crest. Srnall local extrema occur near Z=658m, corresponding

to the maxmium wind speed in the background wind profiles of the planetary boundary

layer and cause the slight kinks in the 0.0 contours at that level. Nso, in Figure 3. lb. we

see that the pressure perturbations (p,/p,) are approximately in anti-phase with the

topography, as in potential flow, although the pressure trough is shified siightly

downstrearn from the topography crest by the effects of shear and turbulence. Since rhe

model topography here is periodic, i.e., h(x) = acos(kx). we assume that the

topographically induced perturbations Jr,(X.Z) are also periodic and can be represented

as $, (X,Z) = A(Z) cos(kX-+), where A(Z) is the amplitude of perturbation and @(Z) is

Figure 3.1. The perturbation distribution in stably-stratified boundary-layer flow over sinusoicial terrain. where topographical length. L=5 lan. height. a=50m. u, = 10ms.l. 4 =O. lm. N = O . O ~ S - ~ and FL = 1.25. Two wavelengths are shown for case of visualisation. (a) u- component wind speed pemrbation and (b) pressure perturbation.

the phase relative to that of the underlying terrain. A positive value of $(Z) represents

a phase shift in the downwind direction.

3.1.1 The perturbation amplitudes

In order to compare and study the amplitudes of perturbations from the different test

cases. we normalise the amplitudes and the height. The characteristic scales for

perturbation amplitudes and height. based on classic invicid suatified flow theories. are

Au = Ug (a km) (3. la)

and

A height scaie is given by

(3. lb)

(3. lc)

where

These scales were used to normalise the results. Figures 3.2a and 3.2b show normalised

whd speed u, and pressure (p ,/p 3 perturbation amplitude profiles in stably and neutrally

stratified boundary-iayer flows. for different topographic lengths with fixed u, = 10msl.

%=O. lm. and N=0.01 s-1 ( for stable stratification). Dashed lines labelled with Vz

values are the neuual flows. As expected, there are some common features between

neutrally and srably stratified flows .

(1 ). The wind speed and pressure perturbations decay rapidly for Z > 0.1H. Note we

use a characteristic length scale H =LI(l-F,")'R radier than L to normalise the height.

where L is the topographic length scaie. because the latter includes the effect of buoyancy

on vertical characteristic heights. As expected. the normalised amplitude values always

increase with increasing L /q or decreasing FL .

(2). The amplitude of the pressure pernirbation is alrnost constant for Z < 0.03H.

which should be in the rniddle layer, in which the wind shear dominates and pressure

perturbations c m be assumed to be independent of height (Hunt et al, 1988).

(3). The maximum amplitude of the wind speed perturbation generally occurs near

0.01H. In the neutraiiy-suatified boundary layer, the maximum wind speeds in the imer

layer can be expected to occur near the inner layer height 1. shown by Jackson and Hunt

i o ~ K l l l l r r I ~ l I l l l l ~ l l l l ~ l l I r l I 1 l l l t l l l i l ~

0.0 0.5 1.0 1.5 2.0 Normaliaed ampiitude of ul

(a) wind speed perturbation

Figure 3.2 The normalised amplitude profiles of perturbation variables for different ropographic length scale (L) in srably and neuually-srratified boundary-layer flows. ug = 1Orns". z,=O.lm, and N=O.Ol s-l.

(1 975) to satis fy the equation ( 4iJL) ln(11zJ = 21?, using their definition of hiil length. For

L =4 x 104 z, , this gives 1 = 0.0013L, in agreement with the height found from wind

perturbation maxima.

On the other hand, there are, at least, two obvious differences between the

neutrally and s tably stratified cases.

(1). For fixed U, and N, the amplitudes of the perturbation wind speed increase faster

with increasing topographic length (or decreasing FJ in the stably stratified boundary

layer flows than that in the neutrally stratified boundary layer.

(2). When FL is much greater than 1, say 6.28, we rnight expect that buoyancy effects

on dynamical processes in stably-suatified flow would not be important, but the

amplitude of the pressure perturbation is still different from the mutrai case and is srnaller.

The difference of wind speeds when Z > 0.1 H is also evident. M e n FL is near 1 or L is

large, for instance, L=Skrn. the amplitude of the pressure perturbation does not decay

monotonically with height, but there is an obvious extreme of the pressure perturbation

around 0.05 H .

In order to further analyse the effect of Froude number on the flow patterns,

another series of sensitivity experirnents are performed by changing U, . Figures 3.3a

and 3.3b are the amplitude profiles of wind speed and pressure perturbation for different

geostrophic wind values U, with fmed g =O. lm, A = 2000m. and N =O.Ols-l. The

differences between neutrally and stably stratifed boundary-layer flows in Figures 3.3a

and 3.3b are quite similar to those in Figures 3.2a and 3.2b if we use FL to distinguish

the different cases.

In gewral, there are two possible mechanisms. which could be responsible for the

differences between neutrally and stably stratifed cases: (1) buoyancy effects and (2)

modified boundary-layer wind shear and turbulence due to stratification. The buoyancy

effects cm be tumed off by omitting the buoyancy term in the wcomponent equatioo

(second temi at right hand side of Equation (2.9)). We can also combine different

background wind speed and turbulent eddy diffusivity profiles frorn stable and neutral

cases to test the contributions of the modifiecl background boundary-layer flows. In order

to investigate the relative contributions. we design three spetific test cases. (1) combining

turbulent difisivity from a stable stratification case with wind profiles from neunal

stratification and including buoyancy effects (hereafter referred as TST1); (2) combining

wind profiles for stable stratification with turbulent dihivity for the neutrai stratifcation

and including buoyancy effects (hereafter referred as TST2) and (3) using stable wind

speed and turbulent difïùsivity profües but excluding buoyancy effects (hereafter referred

as TST3). Figure 3 -4 shows the normaliseci amplitudes of pressure perturbations for these

specific cases, where L/z, = iV and FL =6.28. One cm see that the results from TSTl

- RiIl mimulition for S - -- - W Simulation for N

W-S + T-S d t h o u t

0.1 ' l l i i l i l i r r T -

0.2 0.3 0.4 Normallacd amplitude of pl

Figure 3.4 The normalised amplitude profiles of pressure pemrbation from the neutral case. stable case and three specific test cases. where zo=O. lm. = 10ms-' . L = lkm and N=O.O~S*~ (in stable stratification). Note char S and N are referred ro the stable and neutral cases. W-N. T-S and B are cases with background wind speed profiles for the neuval stratification. mrbulent diffisivity in rhe stable stratification and buoyancy effects.

are the closest to those from the neutral case. This suggests that the modified wind shear

plays the most important role in the resultant difference between the stably and neutrally

stratified cases when FL is large. Figures 3Sa and 3.5b m e r show the normalised

amplitudes of u , and p , when FL= 1.256 and U z p S x 104. The u ,amplitude from TSTl

is again the closest to that in the neuual case, although the u, amplitude in TST3 is

evidently different from the stable case so that the buoyancy effects are important. In

Figure 3 Sb, however, pressure amplitudes from TST3 become the closest to the neutral

case and the non-monotonie change of the pressure amplitude with height disappears in the

TST3 results. Therefore, the buoyancy effects are responsible for the non-monotonie

change of pressure amplitude with height in the stable case when FL is near 1, while

increased wind speeds are induced by modified wind shear due to the stable s~atifcation.

To further investigate buoyancy effects on the profiles of pressure amplitude in the

stable case, we have checked the amplitude and phase of the potential temperanire

perturbation (Figure 3 . 2 ~ and Figure 3.6d) . The amplitude only increases slightly with

topographie length. However, in the phase profiles of negative potential temperanire

perturbation 8, (Figure 3.6d). there are positive phase shift above 1% of normalised

height. This phase structure means that potential temperature over the downslope is cooler

than that over the upslope. This thermal asymmetry cm cause an upstream hydrostatic

pressure force on the topography. The asymmetric hydrostatic pressure field is basically

(c) stren~nli~ie displaceiiieia priurbaiioai

-mm

Pham of alreamline displicement (deg.)

(d) poteniiül iriiiperaiure ~icr i i i i ljiiiioii

a o o o p

UuJ * + + + *

Figure 3.6 (coiiiiiiurd)

suaufication c m suppress the sheltering when FL is near 1( but still p a t e r than 1) and cm

slightly enhance the sheltering when FL 1. In Figure 3.6d. the positive phase shift of the

negative potential temperature above Z =O.OIAm means that the air is wanner over the

upwind slope than that over the downwind slope. This kind of asymmetric potential

temperature distribution can induce an asymrnetric hydrostatic pressure field and gives

a net force upstream, particularly when the topographic scale is large. The hydrostatic

pressure field is basically out of phase with the dominant nonhydrostatic pressure.

Therefore, the amplitude of the pressure perturbation in Figure 3. lb decreases more

rapidly with height at lower levels when the topographic scale is large. The asymmeuic

distribution of potential temperature is related to sheltering over the lee slope since the

stronger the sheltering, the more asymrnetric vertical wind speed is. The asymmetric

upward w h d speed over the lee slope can cause advective cooling, which is in contrast to

the effects of downslope winds in the FL < 1 case.

3.2 Effects of the boundary layer on mountain gravity waves

For flow over sinusoidal terrain with FL < 1, there will be venically propagating

mountain waves. In contrast to the FL > 1 case, the buoyancy effect is always important.

but there is also an effect of the boundary layer on the flow patterns. For instance, Figures

3.7a and 3 3 show the distribution of pressure perturbation and perturbation wind speed

Figure 3.7 ~ h e A/ J d

perturbation distribution in stably-stratifieci boundary- flows over sinusoida1 - - terrain. where topographical Iength. A =20 lm. height. a = 10m. u, = I h - ' . &=O. lm. N = 0.01s-l and FL=0.3 14. Two wavelengths are s h o w for case of visualisation. (a) wind speed pemrbation. (b) pressure perturbation.

for FL =0.314. Note that the vertical axis is not a Iinear scaie. but represents the mode1

verticai levels. The distribution of pressure perturbation is quite similar to that in inviscid

flows with a characteristic gravity wave behaviour. In the horizontal wind speed

perturbation field, however, there are local boundary- layer maxima over the lee slopes

and minima over the windward slopes. This feanire is different from inviscid stratified

flows, where the horizontal velocity perturbation is proportional to (-1) x the pressure

perturbation. In this section, we will mostly discuss the effect of the boundary layer on

the gravity waves rather than the effect of stratification on the boundary-Iayer flow. but

we still use the amplitude and phase profiles as our major diagnostic variables, since

amplitude and phase are two key parameters to represent wave structure.

3.2.1 The amplitudes of perturbation variables

As in the previous section, we perform a nurnber of sensitivity experiments on

variables affecting the Froude number and the amplitudes of wind speed, pressure and

potentiai temperature perturbations are wrmdised, based on the classical inviscid strati fied

flow theories. The characteristic amplitudes and height are the same as equations (3.1 a)-

(3.1 d) . but the vertical wavenumber is different from equation (3.2) since FL < 1. Noce

that

without boundary-layer effects or in inviscid flows, the n o d i s e d amplitudes of u,, p,

and 0, should al1 be 1. Figure 3.8a shows the normaiised amplitude profiles of wind

speed perturbation for different topographic length scaies with 4= 10msL , zo=0.1m, and

N=O.Ols-'. It is seen that above about 0. lH, where H=2x/k still, the amplitude is

essentialiy constant, but less than 1. corresponding to vertically propagating gravity waves

whose amplitude is reduced by boundary-layer effects. Note that H- 0(1@ m), the depth

of 0.1H should be of the same order as the boundary-layer height. Below this height, the

amplitude varies with the topographic length scale L. The amplitude maximum occurs at

heights of order Z = 0.01H . We note that the pressure perturbation amplitude profiles

(see Figure 3.8b) are nearly constant from the boundary layer to the free atmosphere and

also that the pressure phase is almosr constant in the surface boundary layer (0.01H-

O(100m)) (see Figure 3 -9c). These results demonstrate that the pressure perturbation

amplitude in the surface boundary layer is almost height independent and we can assume

that the amplitude of the horizontal pressure gradient is approxirnately independent of Z

in the surface boundary layer. We can use these face to explain the perturbation wind

speed maxima occuring at height of order Z =O.OlH. Note that we also fmd that w' is

near O so the background boundary-layer flow is essentially terrain following. Then the

(a) w i i d spced pcrturbatiori (b) pressure priurtiiiiio~i

1 0 - ' . L ' - - - - - - - r n m - T - 1 0.0 1 .O 2.0 3.0

Normdiacd amplitude of u, Normali~ed amplitude

1;igiire 3.8 'l'lierimiiilised aiii~ilitude pruliles of perturhalioii variahles for Jilfrreiil lop)grapliic leiipili sciilcs (A) iii siatity siratilird buuidary layer Ilows, ut= JOii~s ', q,=0. lni, riul N =O.Ol s ' for siühle stratilicüiiori.

(c) slreatnline displaceiirnt perturbation

l

(d) poieiitia! temparture prturhaiio~r

Normdised amplitude of Ji

Figure 3.8 (çui~iiiiurd)

x- momentum equation in the surface boundary layer becomes

For a case without turbulence, the second term on the left hand can be omitted. and

the perturbation wind speed is expressed as

Noting that the amplitude of p is almost independent of 2, it can be seen that because of

the background wind shear, the amplitude of the wind speed perturbation must increase

when q(Z) decreases . The same idea explains why the fiactional speed-up ratio in the

neutrai boundary-layer is larger than that expected on the basis of neutrai irrotational flow

(Jackson and Hunt, 1975). On the other hand, within the inner layer, we must take the

turbulent stress term into considerauon, in order to match the requirement of zero velociry

at the surface. Thus, in Figure 3 -8a. the maximum value in the boundw layer is induced

by the combination of the background wind shear and turbulent exchange. where the

nature of the profiles above the maximum are mainly caused by wind shear while those

below the maximum are affected by turbulent exchange.

From Figures 3.8, it is also noted that, above 0.1H. the normalised amplitudes of

the potential temperature perturbation and streamline displacement perturbations decrease

with decreasing Froude number, which means the effects of the boundary layer on

potential temperame and suearnline displacement increase.

3.2.2 The phases of perturbations

Figures 3.9a-c are the phase profdes of wind speed perturbation, streamline

displacement, pressure and potential temperature perturbations. Note tbat the real trends

of phase at the upper level decrease with height. The discontinuity of the plotted profde

is due to our presenting them in the limited domain from -180 degrees to 180 degrees.

Consider the phase of the whd speed perturbation. in inviscid stratified flows, there is

a x / 2 phase shin occurring at the surface, corresponding to a maximum wind speed

perturbation (positive) over the downwind slope and a minimum wind speed (maximum

negative perturbation) over the upwind slope. However , boundary -1ayer effecrs

signif~candy affect the phases in the lowest layer and the phases shift upsaeam . When

the topographie length sale increases, the phase shift is m e r fiom x12, and the

difference fkom the inviscid case becornes more signifiant. The increasing effects of the

boudary layer should be associated with the ratio of the characteristic vertical length to

the boundary layer height. in general, the smaller the ratio, the more important are the

effects of the boundary layer. In conuast ro the FL > I case, the characteristic length in

the F , < 1 case decreases as the topographic length is increaseâ (se- equation (3.3 1).

Figure 3.9 The phase profiies of permrbauon variables for dHerent mpographic length scales (À) in stably-scratified boundary layer flows. u, = 10msi. ~ = 0 . lm. and N =O.O 1s.'.

Therefore. when the topographic

become more significant. The

temperature , however, are not sign

length increases; the effects of the boundary layer

phases of the pressure. displacement and potential

ficantiy affected by the boundary layer, just slightly

shifted upstream. The phase difference beween negative pressure and displacement is stiU

around n/2 (see Figure 3 . 9 ~ ) . That means the effects of the boundary layer on the wave

drag are mainiy through the amplitudes of pressure and displacement rather than their

phases.

Chapter 4. On wave drag, form drag and parameterization of

subgrid scde topography in large scde models

In stably stratifieci boundary-layer flows over topography, form drag and wave drag

rnay cwxist and it is necessary to discuss an appropnate approach to parameterking their

joint effects. in this chapter, we will propose a diagnostic quantity . which can link surface

pressure drag and vertical propagating wave momentum flux. Then we dicuss the

topographic drag. As in Chapter 3, but without loss generality, we concentrare on

topographic drag in steady airfiow over sinusoicial terrain, because, in this case, the

horizontal wavenumber of the perturbation is aiways forced to be that of the terrain and

we can clearly separate the flows Uito two groups with FL < 1 and FL > 1.

4.1 Momentum Flux and Local Drag

The vertical momentum flux associated with the presence of topography and the

pressure drag on the surface are central to the pararneterization of mountain gravity waves

and the eflects of sub-grid scale topography. In considering the momentum flux in two

dimensions we can consider three surfaces as depicted in Figure 4.1 These are,

4 the horizontal surface, z = constant, zr ,

b) the surface Z = constant, Zr, or z = Z , + F(Zr )h(X). We would like the

horizontal average of z to equal zr so set Zr = Zr, and

C) the streamline z = 6(x) whose mean height (or upwind height for isolated hills) is

equal to q. We assume no flow separation for low siope topography and note that

extension to 3D may not be possible.

Across any surface z = g(X) a horizontal force can be exerted by the fluid in the

58

Figure4 1. Schematic diagram for local drag .

region z > g(X) on the fluid in the region z < g(X) through both nonnai and shear

stresses. Both will have mean and turbulent cornponents. In addition momennun cm be

transferred by flow through the boundary. Neglecting the turbulent contribution to the

normal stresses or assurning them to be isouopic and included within the pressure, we can

write the horizontally averaged x-component force per unit fluid density following

Batchelor (1967,Equation 6.4.27)

where the integrals are to be evaluated on the surface z=g(X). The second term in the

integral represents the flux of u-momennim across the surface (z=g(X)), T, is the x

component of the turbulent Reynolds shear stress and al1 quantities are to be evaluated on

z = g(X). We assume g(X) to be periodic and continuous. This is the generalised

expression of the surface drag obtained with g(X) = h(X). Moreover, this expression cm

also simple to the traditional momennim flux due to waves a d o r turbulence when g is

a horizontal plane with dg/dX=O, Le.

F, is a general form which links the surface drag and traditional wave drag.

In our linearised systern. the horizontally averaged xcomponent force per unit fluid

density F, on the surface Z=constant can be rewritten as

to separate out the mean flow and turbulent components. Note we have assurned that

P l d x = 0

If we choose g(X) as a strearnline o(X) assumed periodic. then w = ud6fdX and

there is no mean fiow momentum transfer. The expression in equation (4.1)' then

simplifies to

Fz= LDRG +TSTR = 1 dX + -1 5 dX L

O (4.3)

We define the fust integral term as "local drag" (LDRG) and the second as the turbulent

shear stress contribution TSTR. LDRG represents the surface pressure drag when

g(X)= h(X), i.e. for the streamline at the surface.

Our mode1 results are on the surfaces Z = constant. but the separations between

Z = Zr, z = z, and z = 6(x) can be assumed small (O(a)) and approximate evaluation of

integrals along z = z, and z = b(x) c m be made with results from Z = Z , . We will

denote averages with respect to X by < . . . > . They may be on any z surface - the context

should make it clear which. We defme die mean flow Reynolds stress, momentum flux

or wave flux associated with topographically induced perturbations to the mean flow

including both evanescent and venically propagating waves, evaluated on Z = constant

as

assurning c w, > = 0. In reg ions of high background shear we need to make a correction

to u, in order to obtain a consistent approximation to the momentum flux across truc

horizontal surfaces, z = constant. If we expand u and w as Taylor series about Zr this

leads to ,

du,

where the averages are still along Z = constant.

Similarly, the local drag across a sueamline can be approximated to second order by

which , on Z =O, will be the pressure drag. Note that in steady inviscid flow rhe

momentum flux is equai to the local drag.

In summary , F,, , LDRG and WFLX in equations (4.2). (4.3) and (4 -4) represent

the horizontally averaged xcomponent forces induced by topographie perturbations on

three surfaces z=constant, streamline 6 =constant and Z=constant, In the next section.

we will compare the three quanities (see Figure 4.2a and 4.2b).

In Our linearised model, F,, . LDRG and WFLX are second order (in wave slope.

ak) and their calculation from a linear solution requires some care. Fonnai expansion of

u and w as power series in (ak) shows that thiç is possible because contributions involving

second order terms h m expansions of u, w and p integrate to zero, just leaving products

of f'st order t e m in the expansion.

The linear model does not provide the 2nd order contribution to the turbulent

stress, r,, which can have a non-zero wavelength averaged contribution TSTR, nor does

it provide the second order perturbation to the Coriolis force h. In models of neutrally-

stratified, constant-stress -1ayer flow over topography with no Coriolis force and a

prescribed upper boundary shear stress. the second order form drag contributions to the

surface are simply balanced by a second order reduction in surface shear stress in order

to maintain a constant total stress. In the present model however the upper flow has a

prescribed velocity, U,. Increases in the total surface drag (pressure plus turbulent shear

stress) as a result of the presence of topography cm reduce xîomponent velocities and s,

in the boundary layer. This in tum would lead to increased cross-isobar flow and Coriolis

force. However CorioIis tenns have k e n ornitted in the linear 2D model since their first

order, wavelength-averaged contributions should be zero. Ln a steady state flow. height

variations in LDRG should be compensated for by matching (2nd order) variations in the

wavelength averaged shear stress perturbation and the Coriolis force, but we do not know

the partition. If the flow were non-steady and the time scale were short in cornpanion to

f -' we could argue that the Coriolis component would be small but volume integrated

acceleration would also contribute.

In the results to be presented below (e.g. in Figures 4.2 and 4.3) there will

typically be a reduction of the local drag with height across the boundary layer. We

interpret this reduction as indicative of a topographically-induced drag on the flow in the

boundary layer in both the propagating and evanescent wave cases but are unable to

predict the effect of topography on the average surface shear stress, since there rnay also

be changes to the Coriolis force.

4.2 Topographie drag in stably stratified boundary-layer fiows

In order to compare and study topographic drag from the different test cases. we

normalise the drag. Note that the mechanism to cause topographic drag when FL < 1 is

different fiom that when FL > 1. We have to use different n o d i s h g quantities.

When FL > 1, we suppose the average drag per unit horizontal area over 2D

sinusoidal terrain is a hinction of a group of extemal parameters

But it is a linear mode1 and draga (ak)2 . where k=Zz/L, should capture al1 of the

dependence on topography amplitude. If in addition we apply Buckingham's II theorem

of dimensional analysis, we can represent the form or pressure drag per unit area as either

The roughness Rossby number %= U& can be obtained fkom WN, F, and U z , and the

two forms can be shown to be equivalent. Equation (4.6a) is best for cornparisons with

inviscid theory and has the advantage that many of our computations are for futed f and

fned N. However. (4.6a) has a disadvantage in the neutral boundary layer lirnit as N-O.

It obscures the Rossby number dependence in that case, where (4.6b) is more appropriate

and shows

DRAG = pou: (a@' G(0 . f / N . L f q , )

For the FL < 1 case, we have classic analytic solutions for inviscid uniformiy

stratified rotating flow over sinusoidal terrain in the f o m (see for example Gill, 1982.

equation 8.7.10)

In our case. f/N- 1B2 and FL- 1 so that the f d factor (l-P/NZFL')lR is very close to 1 .O

and we can use the non-rotaring result (Gill, 1982, equation 6.8.1 1)

1 - DRAG,,, = p , ~ ~ ( a k ) 2 L(F;' - 1)

2

Thus the inviscid form of our lünction F is simply (Fi2 - l)lR/2

For convenience in studying the profiles of the local drag , the height is norrnaiised by

where the 1 1 incorporates cases with FL > 1 or < 1 . For the details see discussion in

ZTQ97.

Figure 4.2a and 4.2b show profiles of the three forces: F, . the wave flux

(WFLX) and Our proposed local drng(LDRG) in stably stratified boundary- layer flows

where gravity waves can propagate vertically (FL< 1). Note that these forces are

normalised by DRAG,, in equation (4.7) and rhat the height is normalised by H from

equation (4.8). It is sem that above the boundary- layer, three forces are equal as

expected, but different in the boundary layer. The wave flux varies sigrilficantly and must

reduce to zero at the surface where u, = w, =O. In conaast, the force F,, is equal to the

local drag on the surface, but there are differences evident in the boundary layer where

wind shear is signifcant. Bearing in rnind that the three forces are al1 of second order in

(ak)2. We m u t ask where the differences can be balanced, given that the individual

computation should al1 be correct to this order. Looking back at Equation (4.2) it would

appear rhat there can be differences at second order in the TSTR term (J r,dx), depending

on the surface on which the integrai is evaiuated. Although we have no information on the

second order contributions to r,. We do have the first order contribution and can estirnate

the differences among integrals on Z =constant. z =constant and the strearnline surface 6.

The stress difference between integrals on Z =constant and z =constant would be

and the difference between integrals on sueamline surface and Z=constant is

Figure 4 . 2 ~ shows the stress difference between integrals on streamline surface and

z=constant and the dflerence between LDRG and WFLX. The force differences match

the stress differences perfectly. However-in Figure 4.2d. The stress dflerences on

streamline surface and Z=constant do not match the force differences well. One of the

possible reasons would be the cdculation of the force on Z=consmt. We believe that. in

theory , they should match with each other .

Of the three forces, we argue that the local drag is the most usehil to explain the

transition between the boundary layer and the inviscid flows above. at least in two

dimensions. It is independent of the coordinate system, unlike F, , and can be evalulated

at al1 levels, unilke WFLX which is not properly defmed z < a. There could be problems

for steep waves with separarion and of course there will be problems in three dimensions

if the corresponding sueam surface is not simply connected.

Figure 4.3 shows the profiles of local drag in stably and neutraily-stratified

boundary-layer flows over sinusoidal terrain for different topographic scaies and FL > 1.

The drag is normalised by p,(ak)2 U: according to Equation (4.6) and the height is

norrnalised by H from Equation (4.8). It is found that most of the local drag disappears

above a height equal to about 1 % the topographic length scale, which is almost in the inner

layer although flow and local stress perturbation extend to height of order L. That means.

for wavelength averages, die drag affects mainly the inner layer. The lowest layer of a

larger mode1 is often higher than the d e r layer, so this result also confïrms that, for

FL > 1, it is reasonable to parameterize topography by an increase in the surface drag.

instead of requiring profiles of local drag .

4.2.1 Topographic drag with F, > I

Further analyzing the relationship between the surface pressure drag (LDRG) and

topographic lengths (see Table 1 ), the drag for neutrd stratification initially increases

with &/a, but when L/z , > 5 x IO4 the drag decreases. This trend is basically consistent

Figure 4.3. Profiles of normalised local drag in the neuvally and stably suatified boundary

layer flows for different topographic length scales in the FL > 1 case (if in stable

stratification). where %=O. lm. ug= l M s . and N=O.Ols".

Table 1, the reiationship between local drag at surface and topographie lengths in the neutraily and stably stratified boundary layer. SB and NT mean stable and neutral stratification respectively

L&m)

FL (SB)

L/Z,(NT)

DRG(SB)

DRG(NT)

1

6.28

1 x 1 0 ~

0 . 9 8 ~ 1

1.08~10-*

2

3.14

2 x 104

1 . 1 7 ~ 1

1 .21~10-~

3

2.09

3x10~

1 .15~ 1 O-*

1.30x10-2

4

1.57

4x 104

0.93 x 1

1 .35~10'~

5

1 -26

5 x 1 0 ~

0 . 5 2 ~ 1 O-*

1 .37~10-~

7.5 10

FL <1

7.5x104 l x l o S

see Table 2

1.35~10-~ 1.29~10-~

with Newley's (1985)' Wood and Mason's (1991) and Xu and Taylor's (1995) surface

layer model results with higher order turbulence closure although Our PBL model uses

miwing length closure. Xu and Taylor (1995) show different results from theu E-KZ and

higher order clasure models. The surface drag from their E-KZ model increases

monotonically with (L/q ). The surface pressure drag for F, depends critically on the

phase of the surface pressure distribution, as wetl as the amplitude. Based on Our

discussion in ZTQ97, when Llz, increases in neutral stratification. the pressure

amplitude increases, but the phase decreases. The decreasing trend of drag with Uz,

should be caused by decreasing of the pressure phase shift. Shce the phase shift is more

sensitive to turbulence closure than the amplitude (Ayotte et al, 1994). it is possible that

a surface boundary Iayer model with E-KZ closure overpredicts the pressure phase shift,

suggested by Ayotte et al (1994)'s results.

As with neutrai stratification. the drag in our stably-stratifieci boundary-layer fiows

initially increases with increasing values of Llz, or Froude number and then decreases. but

the tuming point from an increasing to a decreasing trend is war FL =3.14, where

L =2km and L/z , =2 x lû'? Note that the drag in stable stratification does not match

with neutral values as FL- - in these series of experiments since the N is f ked and effects

of stable stratification on background flows always exist. When L fùrther increases

from 2 km to 5 lan, or FL decreases from 3.14 to 1.26, the buoyant effects become evident

and the vertical characteristic Iength (L/(LF, " ) I R in the stable stratification significantly

increases. These effects induce the negative pressure phase shift relative to the terrain ro

decrease more rapidly than those in neutral suaification (see the details in ZTQ97). The

topographic length corresponding to the maximum in normalised form drag is smaller in

the stable stratification cases.

We have also discussed the relationship between the surface topographic drag and

nonnalised parameters in Equation (4.6). Our computations have been for t i e d values

of f = l(r s*' and N = 10 '' s" so the parameters influencing the drag function F will be

FL and Llz , . In Figure 4.4, we show how F varies with FL (for fixed flN = ) and

different f i e d L/q. The surface drag initially increases with ri and when q > 2 , the

surface drag decreases. Figure 4.5 shows how the drag varies with Uq,. again for fixed

flN = l O-2 and different fixed FL . The drag decreases with increasing L/zo and the

decreasing trend is faster when FL is srnaller.

4.2.2 Profiies of local drag for FL < 1

Figure 4.6 shows some profiles of the local drag computed from our linear PBL

model. nonnalised by the inviscid prediction of Equation (4.7). The height scale is

normalised by H from Equation (4.8). Note that if the effects of the boundary layer are

excluded, the normaiised local drag is independent of height and equal to 1. From the

figure, we can see that there is a gradient in local drag below Z =O.O 1H. typically about

LOO m and in the boundary layer. Above this level, the local drag is approximately

constant and is equd to the wave flux. The gradient of the local drag represents a net

force on the lowest layer, although most of the surface pressure drag is transrnitted

through the boundary layer to the upper level as a wave drag. Table II further shows the

relationship behveen the surface pressure drag, the upper, constant, local drag (or wave

flux) and their difference and topographic length. The significant evolution of the drag

with topographic length happas when Ls 2 0 h and at L = LOkm, the surface drag and the

upper wave flux are reduced by 17 % and 25 % with 8 2 of drag affecting the lowest

Iayers. For the cases with L>ZOkm, The surface drag is not very sensitive to the

topographic length, but the upper wave flux evidently decreases with increasing the

topographic length. As a consequence, the difference and thus the net drag on the

boundary layer increases with topographic length. As a general guide to the effects of the

T I I I I I I I 2 3 4 6 0 ? B O 1

1 I

Figure 4.4. The relationship between normalised suface drag and Froude number in the FL > 1

case

Figure 4.5. The relationship between normalised surface drag and L/z, for different Froude

numbers .

Zlormaliaed Local Drag

Figure 4.6. ProNes of local drag in the stably suatified boundary layer flows for different

topographicd length scales in the FL < 1 case . where, z =O. lm, u = Kh&. and

N=O.O~S-~ .

Tablez: The relationship between the normalised surface pressure drag (S-DRG), the normaiised upper wave drag WDRG) and their dinerence and FL, where @.lm, U,=10ms1 , N=10-2 s*' and 6 l O%'

r

L (km)

FL

S-DRG

U-DRG

S-DRG - U-DRG

7.5

0.837

0.935

0.892

0.043

10

0.628

0.880

0.822

0.058

20

0.3 14

0.829

0.750

0.79

30

0.209

0.826

0.733

0.093

50

O. 125

0.825

0.7 17

O. 1 08

1 O0

0.0628

0.828

0.699

0.129

boundary layer in flows over mesoscaie topography, the boundary layer basically reduces

the surface drag by about 20% ,and wave flues by 30% with 10% of the drag affecting

the lowest layers of atmosphere.

To further snidy the relative contribution of wind shear and turbulence to the

boundary-layer effects, a special experiment with shear but without background

turbulence was performed. It was found that wind shear in the boundary layer cm

reduce the local drag, which is now constant with height, by 25 %. Cornpareci with the full

boundary-layer model, we basicaily conclude that turbulence effècts ihcrease surface drag

by 5 % , but reduce the gravity wave flux by 5 %.

Note that the results in Figure 4.6 were obtained with fmed roughness Iength, but

the normalised variable Llz, is not fixed. As in the section above for F, > 1. we can

assume the normalwd drag is the function of FL , f/N and Uq . It is however convenient

to normalise the drag by the inviscid wave drag DRAG,, fiom equation (14) rather than

poug (ak)2 so that

Figure 4.7 shows the relationship between normalised surface pressure drag and upper

wave flux with k e d FL and different fuced LI%. Both the surface drag and upper wave

flux decrease with decreasing FL and when FL is very srnail( < 10-l). the surface drag is

no longer sensitive to FL. The difference between the surface drag and upper wave flux

increases with decreasing of FL . Figure 4.8 shows how the profiles of local drag Vary

with z, for fixed L, FL and f/N. It is found rhat with increasing roughness length, surface

drag and flues decrease, but the gradient of local drag increases. That means that the

Figure 4.7. The relationship beiween surface pressure drag and upper wave flux and FL for F,

< 1, where Ug = l O r n ~ - ~ . N = s-l and f= IO4 s-l

c 0.5 0.6 0.7 0.0 0.9 1 .O

Normalised local dru

Figure 4.8. The profiles of local drag for L/z, in the FL < 1 case.

effect of roughness length cm reduce propagating wave fluxes. and increase topographie

drag on the lowest layers.

Chapter 5. Conclusions

Based partially on current research models for neutrally-suatified boundary-layer

flows over hiils ( e.g. Beljaars et al, 1987 and Hunt et al, 1988) and classical theory for

inviscid stratified flows over topography, a linearized mode1 for stably stratified

boundary-layer flows over topography ranging from microscale to mesoscale has been

developed. Results show that

(1) For FL > 1. the effects of stratification on boundary-layer flows are due to nvo

mechanisrns: modified boundary-layer wind shear and turbulence due to

stratification and buoyancy effects. When FL 1, the effects of stratification are

mainly induced by modified boundary-layer wind shear. The buoyancy effecrs

becorne important when F , is near 1. They can influence the amplitude profiles

of the pressure and wind speeds, induce their phase shifts to decrease and cause

sheltering over the downslope surface to be decreased. It is also found that there

is a net upstream hydrostatic pressure force caused by sheltering.

(2) For FL < 1, the effects of the boundary layer can cause local maximum wind speeds

over the lee slope and minimum wuid speeds over the windward slope. These

boundary- layer wind speed extremes are due to a combination of the effects of

boundary-layer wind shear and turbulent exchange. The boundary-layer effects on

potential temperature and streamline displacement increase as the Froude number

decreases. The phases of those perturbations shift upstream, particularly for the

82

wind speed.

With the sirnplified linear modei, we have funher studied the subgrid scale

topography parameterization for large scaie models . A generalized form drag , here called

local drag . is proposed for stably- stratified, two-dimensionai. boundary- layer Bows.

The local drag can be considered as the form drag across a streamline. The following

three more conclusions have k e n reached:

4) Traditional, inviscid theory, wave drag cannot be extended into the boundary layer for

parameterization purposes. Instead, local drag links the surface drag with the wave

fluxes above the boundary layer.

5) For FL > 1 which includes neutrai stratification, based on the profile of local drag, it

is confmed that topographie drag is limited to the lowest layers of larger scale

models. and so enhanced surface drag is enough to represent the effects of

topography. The buoyancy effects for FL > 1 induce a decrease in drag relative

to the neutral case

6) In the FL < 1 case, there is a net drag on the boundary layer, but most of the local drag

is transferred into the upper layers. The boundary layer c m reduce wave tluxes

above the boundary layer by 20 to 40% compared to inviscid theory for typical

surface roughness, although drag on the boundary layer may increase by 10%.

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