1 Chapter 2. Mountain Forced Flows 2.1. Mountain Waves Well-Known Weather Phenomena Directly Related to Flow Over Orography

Chapter 2. Mountain forced flows

2.1. Mountain waves

Well-known weather phenomena directly related to flow over orography include

• mountain waves • lee waves and clouds • rotors and rotor clouds • severe downslope windstorms • lee vortices • lee cyclogenesis • frontal distortion across mountains • cold-air damming • track deflection of midlatitude and tropical cyclones • coastally trapped disturbances • orographically induced rain and flash flooding • orographically influenced storm tracks.

A majority of these phenomena are mesocale and are induced by stably stratified flow over orography.

Mountain Waves

MODIS image of mountain wave clouds over Turkey.

Mountain waves form above and downwind of topographic barriers when strong winds blow with a significant vector component perpendicular to the barrier in a stable environment.

False color image of mountain wave cloud patterns emanating from Jan Mayen (a small Arctic island chain ~550 km north-east of Iceland) on 25th. January, 2000 at 1627Z. The orange-white regions are polar stratospheric cloud decks formed by cold temperatures on this day.

Mountain wave clouds forming east of the Rockies.

This figure shows the development of the typical features often associated with a mountain wave system. Notice the wind flow, with a strong component perpendicular to the primary ridge line. This is a typical condition for mountain wave development, as is a stable atmosphere. If air is being forced over the terrain, it will move downward along the lee slopes, then oscillate in a series of waves as it moves downstream. Sometimes these waves can propagate long distances in "lee wave trains."

Cap Clouds

Cap clouds indicate likely wave activity downstream. They often appear along mountain ridges as air is forced up the windward side. If the flow is sufficiently humid, the moisture will condense into a cloud bank that follows mountain contours. Quite often, heavy orographic precipitation occurs on the upwind side of the barrier,

particularly for barriers located near the sea. As the flow descends in the lee of the mountain ridge, the cloud evaporates. Viewed from downstream, cap clouds frequently appear as a wall of clouds hanging over the ridge top.

It is important to remember that while cap clouds indicate likely wave activity, their absence does not mean that waves are absent. Under drier conditions, waves may be present without cap clouds.

Photograph of Wave Clouds

The Vertically-Propagating Wave

The vertically-propagating wave is often most severe within the first wavelength downwind of the mountain barrier. These waves frequently become more amplified and tilt upwind with height. Tilting, amplified waves can cause aircraft to experience turbulence at very high altitudes. Clear air turbulence often occurs near the tropopause due to vertically-propagating waves. Incredibly, these waves have been documented up to 70 km and higher.

Breaking Waves

Vertically-propagating waves with sufficient amplitude may break in the troposphere or lower stratosphere. Wave- breaking can result in severe to extreme turbulence within the wave-breaking region and nearby, typically between 7 and 14 km. If a vertically-propagating wave doesn't break, an aircraft would likely experience considerable wave action, but little turbulence.

Downslope Winds

At times, strong downslope winds accompany mountain wave systems. Strong downslope wind cases are usually associated with strong cross-barrier flow, waves breaking aloft, and an inversion near the barrier top. In extreme cases, such as in our Alps scenario, winds can exceed 100 knots. This may be double or triple the wind speed at mountaintop level. These high winds frequently lead to turbulence and wind shear at the surface, causing

significant danger to aircraft and damage at the surface. Downslope windstorms often abruptly end at the "jump region," although more moderate turbulence can exist downstream. The jump region is an extremely turbulent area that can extend up to 3 km.

Rotor Clouds

Rotor location can often be identified if sufficient moisture is available to form an associated rotor cloud. Rotor clouds are found near the top of the rotor circulation and under higher lenticular clouds. Immediately above the rotor cloud, smooth, wavy air is likely.

The rotor cloud can look innocuous, but does contain strong turbulence and should be avoided by pilots. Eventually, we can expect operational NWP models to resolve rotors so that they can be identified in the absence of rotor clouds.

Trapped Lee Waves and clouds

Lee waves whose energy does not propagate vertically because of strong wind shear or low stability above are said to be "trapped." Trapped lee waves are often found downstream of the rotor zone, although a weak rotor may exist under each lee wave. These waves are typically at an altitude within a few thousand feet of the mountain ridge

crest and turbulence is generally restricted to altitudes below 25,000 feet. Strong turbulence can develop between the bases of associated lenticular clouds and the ground.

Lenticular clouds form near the crests of mountain waves. As air ascends and cools, moisture condenses, forming the cloud. As that air descends in the lee of the wave crest, the cloud evaporates. Because air flows through the cloud while the cloud itself is relatively stationary, many people refer to these clouds as standing

Areal Extent of Mountain Waves

Mountain wave activity can occur over broad regions. This MODIS true color satellite image shows wave clouds covering most of Turkey, a region spanning about 1000 km! However, despite their occasionally broad extent, regions of strong or severe turbulence within mountain wave systems are often limited horizontally and vertically.

Equations for two-dimensional, steady-state, adiabatic, inviscid, nonrotating, Boussinesq fluid flow over a small-amplitude mountain.

The linear equations can be simplified from the original governing equations:

∂u' 1 ∂p' U + U z w'+ = 0 , (1) ∂x ρo ∂x ∂w' θ ' 1 ∂p' U − g θ + = 0 , (2) ∂x o ρo ∂z ∂u' ∂w' + = 0, (3) ∂x θ ∂z ∂ ' N 2θ U + o w'= 0. (4) ∂x g

The above set of equations can be further reduced to Scorer’s equation (1954),

∇2w' + l 2 (z)w'= 0 , (5)

where ∇2 = ∂2 / ∂x2 + ∂2 / ∂z2

is the two-dimensional Laplacian operator, and

l is the Scorer parameter (Scorer 1949), which is defined by:

N 2 U l 2 (z) = − zz . (6) U 2 U

Equation (5) serves as a central tool for numerous theoretical studies of small-amplitude, two-dimensional mountain waves.

Equation (5), when multiplied by U as

Nw2 ' Uw∇+2 ' − Uw ' = 0, (5a) U zz may also be interpreted as a vorticity equation, because the y-component of vorticity is

∂∂uw'' η =−, ∂∂zx and making use of the mass continuity equation (3)

∂∂⎡⎤wu'' ∂ NwU2 ' ∂ Uw−+ − '0z =. (5b) ∂∂xx⎣⎦⎢⎥ ∂ z U ∂ z

' ' • The first term, U (wxx + wzz ) , is the rate of change of vorticity following a fluid particle.

• The second term, N 2w' /U , is the rate of vorticity production by buoyancy forces.

• The last term, −U zz w' , is the rate of vorticity production by the vertical redistribution of the background

vorticity (U z ).

In the extreme case of very small Scorer parameter, e.g., when N=0 and vertical shear is zero, (5) reduces to irrotational or potential flow,

∇2w' = 0 . (7)

If the forcing is symmetric in the basic flow direction, such as a cylinder in an unbounded fluid or a bell-shaped mountain in a half-plane, then the flow is symmetric (see below).

9 a

(km)

z 3

0 -1.5 0 1.5 x (km)

For this particular case, there is no drag produced on the mountain since the fluid is inviscid.

Flows over two-dimensional sinusoidal mountains

Assuming constant U(z) and N(z) with height and a sinusoidal terrain

h(x) = hm sin kx , (8)

with mountain height hm . k is the wave number of the terrain.

For an inviscid fluid flow, the lower boundary condition requires the fluid particles to follow the terrain, so that the streamline slope equals the terrain slope locally,

w w' dh = = at z = h(x) . (9) u U + u' dx

For a small-amplitude mountain, this is linearized to

dh w'= U at z = 0, (10) dx or

w'(x,0) = U hmk coskx at z = 0, (11)

for h(x) given in (8). It is the linearized lower boundary condition!

Because of the sinusoidal nature of the forcing, we can expect solutions in the following form,

wxz'( , )=+ Wz12 ( ) cos kxW ( z ) sin kx. (12)

Substituting the above solution into (5) with a constant Scorer parameter leads to

22 WlkWizz+−()0 i =, i = 1, 2 . (13)

Two cases are possible:

(a) l2 < k 2 , (vertically decaying solution)

(b) l2 > k 2 . (oscilation solution)

The first case requires N /U < k or Na/2 U < π , where a is the terrain wavelength.

Physically, this means that the basic flow has relatively weaker stability and stronger wind, or that the mountain is narrower than a certain threshold.

For example, to satisfy the criterion for a flow with U = 10 ms-1 and N = 0.01 s-1, the wavelength of the mountain should be smaller than 6.3 km. In fact, this criterion can be rewritten as (a /U) /(2π / N) < 1.

The numerator, a /U , represents the advection time of an air parcel passing over one wavelength of the terrain, while the denominator, 2/π N , represents the period of buoyancy oscillation due to stratification.

This means that the time an air parcel takes to pass over the terrain is less than it takes for vertical oscillation due to buoyancy force. In other words, buoyancy force plays a smaller role than the horizontal advection.

In this situation, (13) can be rewritten as

22 WklWizz−−()0 i =, i = 1, 2 . (14)

The solutions of the above second-order differential equation with constant coefficient may be obtained

λ z −λ z WAeBeii=+ i , i = 1, 2 , (15) where

λ = k 2 − l2 . (16)

The upper boundedness condition requires Ai = 0 because the energy source is located at z = 0.

Applying the lower boundary condition, (11), and the upper boundary condition ( Ai = 0 ) to (15) yields

B1 = U hmk; B2 = 0 . (17)

This yields the solution,

−−klz22 wxz'( , )== Wz1 ( ) cos kxUhkem cos kx, (18)

The vertical displacement (η ) is defined as wDDt'/= η which reduces to

Dη ∂η w'= = U (19) Dt ∂x for a steady-state flow.

Equation (18) can then be expressed in terms of η ,

1 x 2 2 η = w'dx = h sinkx e− k −l z . (20) U ∫0 m

C L C L u ' > 0 u ' > 0 W H u ' < 0 b

C C u ' = 0 u ' = 0 H L H L ' u ' > 0 u ' > 0 u < 0 W u ' < 0 u ' = 0

Fig. 5.1: The steady-state, inviscid flow over a two-dimensional sinusoidal mountain when (a) lk22< (or NkU< ), where k is the terrain wavenumber (= 2/π a , where a is the terrain wavelength), or (b) lk22> (or N > kU ). The dashed line in (b) denotes the constant phase line which tilts upstream with height. The maxima and minima of u' , p' (H and L), and θ ' (W and C) are also denoted in the figures.

The above solution is sketched in Fig. 5.1a. The disturbance is symmetric with respect to the vertical axis and decays exponentially with height. Thus, the flow belongs to the evanescent flow regime. The buoyancy force plays a minor role compared to that of the advection effect.

The other variables can also be obtained by using the governing equations and (18),

2 2 − k 2 −l 2 z u'= U hm k − l sinkx e , (21)

2 2 2 − k 2 −l 2 z p'= −ρoU hm k − l sinkx e , (22)

θ 2 − k 2 −l 2 z '= −(θoN / g) hm sin kx e . (23)

The maxima and minima of up', ', and θ ' are also denoted in Fig. 5.1a.

The coldest (warmest) air is produced at the mountain peak (valley) due to adiabatic cooling (warming). The flow accelerates over the mountain peaks and decelerates over the valleys. From the horizontal momentum equation, (1) with U z = 0 , or (22), a low (high) pressure is produced over the mountain peak (valley) where maximum (minimum) wind is produced.

Note that (1) is also equivalent to the Bernoulli equation, which states that the pressure perturbation is out of phase with the horizontal velocity perturbation. Since no pressure difference exists between the upslope and downslope, this flow produces no net wave drag on the mountain (mountain drag).

In the second case, l 2 > k 2 , the flow response is completely different.

This case requires N /U > k or Na/2 U > π . This means that the basic flow has relatively stronger stability and weaker wind or that the mountain is wider.

For example, and as mentioned earlier, to satisfy the criterion for a flow with U = 10 ms-1 and N = 0.01 s-1, the terrain wavelength should be larger than 6.3 km.

Since (a /U) /(2π / N) > 1, the advection time is larger than the period of the vertical oscillation. In other words, buoyancy force plays a more dominant role than the horizontal advection.

In this case, (13) can be written as

2 2 2 2 WmWizz+= i 0 , m = l − k , i = 1, 2 . (26)

We look for solutions in the form

Wzii()=+ A sin mzB i cos mzi , = 1, 2. (27)

Substituting (27) into (12) leads to

w'(x, z) = C cos(kx + mz) + Dsin(kx + mz) + E cos(kx − mz) + F sin(kx − mz) . (28)

Terms of (kx + mz) have an upstream phase tilt with height, while terms of (kx − mz) have a downstream phase tilt.

It can be shown that terms of (kx + mz) have a positive vertical energy flux and should be retained since the energy source in this case is located at the mountain surface. Thus, the solution requires E = F = 0 . This flow regime is characterized as the upward propagating wave regime.

As in the first case, the lower boundary condition requires

C = Uhmk, D = 0 . (29)

This leads to

w'(x, z) = Uhmk cos(kx + mz). (30)

Other variables can be obtained through definitions or the governing equations,

η(x, z) = hm sin(kx + mz), (31)

u'(x, z) = −Uhmmcos(kx + mz) , (32)

2 p'(x, z) = ρoU hmmcos(kx + mz) , and (33) θ N 2θ h '(x, z) = − o m sin(kx + mz) . (34) g

The vertical displacement of the flow, and the maxima and minina of u' , p' , and θ ' are depicted in Fig. 5.1b (below).

C C u ' = 0 u ' = 0 H L H L ' u ' > 0 u ' > 0 u < 0 W u ' < 0 u ' = 0

Note that the flow pattern is no longer symmetric. The constant phase lines are tilted upstream (to the left) with height, thus producing a high pressure on the windward slope and a low pressure on the lee slope.

Based on (32) or the Bernoulli equation (1), the flow decelerates over the windward slope and accelerates over the lee slope. The coldest and warmest spots are still located over the mountain peaks and valleys, respectively.

Positive wave drag on the mountain is produced by the high pressure on the windward slope and the low pressure on the lee slope.

When l2 >> k 2 , the flow approaches a limiting case in which the buoyancy effect dominates and the advection effect is totally negligible. In other words, the vertical pressure gradient force and the buoyancy force are roughly in balance and the vertical acceleration can be ignored. Thus, the mountain waves become hydrostatic. In this limiting case, the governing equation becomes

2 w'zz +l w'= 0. (36)

The flow pattern repeats itself in the vertical with a wavelength of λz = 2π / l = 2πU / N , which is also referred to as the hydrostatic vertical wavelength.

The regime boundary between the regimes of vertically propagating waves and evanescent waves can be found by letting l = k , which leads to a = 2πU / N .

The relation among the mountain waves discussed in this subsection is sketched in Fig. 5.2.

l<k l>>k Irrotational Evanescent Vertically Hydrostatic (Potential) flow propagating waves flow waves l /k ~0.1 1 ~10

Fig. 5.2: Relations among different mountain waves as determined by l / k , where l is the Scorer parameter and k is the wave number.

Flows over two-dimensional isolated mountains

Earlier, we obtained mountain wave solution forced by a sinusoidal orography. Of course, real orography is never truly sinusoidal. However, one can always perform Fourier transform on the real orography, therefore any orography can be considered summation of many sinusoidal modes or wave components.

When the orography is low, the wave solutions are nearly linear, therefore the total solution is sum of all forced waves.

Let wˆ be the amplitude of the wave component with wavenumber k, Eq.(13) we obtained earlier is

2 2 wˆ zz + (l − k )wˆ = 0. (37)

The Fourier transform of the linear lower boundary condition is

wˆ (k, z = 0) = ikU hˆ(k) . (38) where hkˆ() is the amplitude of Fourier component of orography with wavenumber k.

For constant Scorer parameter, the solution we obtained earlier for the wave amplitude can be written into two parts,

2 2 wˆ(k, z) = wˆ(k,0) ei l −k z for l 2 > k 2 and (39a)

2 2 wˆ (k, z) = wˆ(k,0) e− k −l z for l 2 < k 2 . (39b)

Taking the inverse one-sided Fourier transform of (39) yields the solution in the physical space,

l 22∞ 22 wxz'(,)=+ 2Re ⎡⎤ ikUhke ˆˆ() i l−−− k z edk ikx ikUhke () k l z edk ikx , (40) ⎣⎦∫∫0 l

which is basically the summation of all wave components.

The first integration represents the upward propagating wave.

The second integration represents the evanescent waves.

For simplicity, let us consider a bell-shaped mountain mountain profile,

h a2 h(x) = m , (41) x2 + a2

where hm is the mountain height and a is the half-width. The mountain peak is at x=0.

The one-sided Fourier transform of this mountain profile is in a simple form,

h a hˆ(k) = m e−ka , for k > 0 . (42) 2

This hkˆ() is plugged into (41) to obtained the actual solution of w’. For this bell shaped mountained, the characteristic wavenumber of forcing is ka=1/ .

Cases to consider:

First case: l 2 << k 2 (i.e., al << 1 or Na <

We assumed that U and N are constant with height.

In this case, the second integral on the right hand side of (40) can be neglected, and the final solutions are

∞ ⎡⎤⎛⎞ham −−ka kz ikx wxz'( , )≈ 2 Re ⎢⎥ U ik ⎜⎟ e e edk. (43) ⎣⎦∫0 ⎝⎠2

∞ −+−kzaix() hazm ()+ a η(,)xz== ham Re e dk . (44) ∫0 x22++()za

η(,)x z is the vertical displacement of the streamline from the far updream level.

Therefore, similar to the sinusoidal mountain case, the flow pattern is symmetric with respect to the center of the mountain ridge ( x = 0 ). However, the amplitude decreases with height linearly, instead of exponentially. The flow pattern is depicted below.

9 a 18 b

6 12

(km)

z 3 6

0 0 -1.5 0 1.5 -150 0 150 x (km) x (km)

Fig. 5.3: Streamlines of steady state flow over an isolated, bell-shaped mountain when (a) lk22<< (or Na <> (or Na >> U ). (Adapted after Durran 1990)

Second case: l 2 >> k 2 (i.e., al >> 1 or Na>> U ).

In this case, the first integral on the right hand side of (40) can be neglected, and the final solutions are

⎡ ∞ ˆ ilz ikx ⎤ ⎡ ∞ ⎛ hma ⎞ −ka ilz ikx ⎤ w'(x, z) ≈ 2Re U ∫ ikh(k)e e dk = 2Re⎢U ∫ ik⎜ ⎟e e e dk⎥ . (45) ⎣⎢ 0 ⎦⎥ ⎣ 0 ⎝ 2 ⎠ ⎦

Similarly, the vertical displacement can be obtained,

∞ ha haa(cos lz− x sin) lz η(,)xz== 2Re mm e−+ka e i() kx lz dk ∫0 2 xa22+ . (46)

This type of flow is characterized as a hydrostatic mountain wave.

The disturbance confines itself over the mountain in horizontal, but repeats itself in vertical with a wavelength of 2πU / N .

If the density effect is included, the above solution becomes

η 1/ 2 ⎛ ρ ⎞ ⎡h a(acoslz − xsinlz)⎤ (x, z) = ⎜ s ⎟ m ⎜ ⎟ ⎢ 2 2 ⎥ , (47) ⎝ ρ(z) ⎠ ⎣ x + a ⎦

where ρs is the air density near surface.

It says that the wave amplitude will increase with a decreased air density of the basic flow.

Third case, lk22≈ (i.e., al ≈ 1 or Na≈ U )

In this case, all terms of the vertical momentum equation, (40) are equally important.

Both asymptotic methods and numerical methods have been applied to solve the problem. The solution looks like:

(km) z

0 u' p' -5 x (km) 5 10 -2 0.3 surface u', p' 0

2 -0.3 m/s mb

Flow over a two-dimensional ridge of intermediate width (lk22≈ , or al= Na/1 U = ) where the buoyancy force is important, but not so dominant that the flow is hydrostatic. The zero phase lines are denoted by dotted curves. The waves on the lee aloft are the dispersive tail of the nonhydrostatic waves ( kl< , but not k<< l). The flow -1 -1 and orographic parameters are: U = 10 ms , N = 0.01 s , hm = 1 km, and a = 1 km. (Adapted after Queney 1948)

cpx i

cga cgm

A schematic illustrating the relationship among the group velocity with respect to (w.r.t) to the air ( cga ), group velocity w.r.t. to the mountain ( cgm ), horizontal phase speed ( cpxi ) and the basic wind. The horizontal phase speed of the wave is exactly equal and opposite to the basic wind speed. The wave energy propagates upward and upstream relative to the air, but is advected downstream by the basic wind. The energy associated with the mountain waves propagates upward and downstream relative to the mountain. (Adapted after Smith 1979)

Trapped lee waves

One of the most prominent features of mountain waves is the long train of wave clouds over the lee of mountain ridges in the lower atmosphere. This type of wave differs from the dispersive tails shown earlier for the lk22≈ case in that it is located in the lower atmosphere and there is no vertical phase tilt.

This type of trapped lee waves occur when the Scorer parameter decreases rapidly with height (Scorer 1949).

Satellite imagery for lee wave clouds observed at 1431 UTC, 22 October 2003, over western Virginia. Clouds originate at the Appalachian Mountains. (Courtesy of NASA)

4 (km) z

0 -12 0 12 24 36 x (km)

Figure. Lee waves simulated by a nonlinear numerical model for a two-layer airflow over a bell-shaped mountain. Displayed are (a) the quasi-steady state streamlines and (b) the vertical profiles of temperature and wind speed (solid lines) used for (a). (Adapted after Durran 1986b)

• Due to the co-existence of the upward propagating waves and downward propagating waves, there exists no phase tilt in the lee waves.

• Once lee waves form, regions of reversed cross-mountain winds near the surface beneath the crests of the lee waves may develop due to the presence of a reversed pressure gradient force.

• In the presence of surface friction, a sheet of vorticity parallel to the mountain range forms along the lee slopes, which originates in the region of high shear within the boundary layer.

• The vortex sheet separates from the surface, ascends into the crest of the first lee wave, and remains aloft as it is advected downstream by the undulating flow in the lee waves (Doyle and Durran 2004).

• The vortex with recirculated air is known as rotor and the process that forms it is known as boundary layer separation.

• These rotors are often observed to the lee of steep mountain ranges such as over the Owens Valley, California, on the eastern slope of Sierra Nevada (e.g., Grubišić and Lewis 2004).

• Occasionally, a turbulent, altocumulus cloud forms with the rotor and is referred to as rotor cloud.

Nonlinear flows over two-dimensional mountains

• The linear dynamics of mountain waves over 2d ridges are fundamentally understood.

• Linear theory, however, begins to break down when the perturbation velocity (u’) becomes large compared with the basic flow (U) in some regions, so that the flow becomes stagnant.

• This happens when the mountain becomes very high, the basic flow becomes very slow, or the stratification becomes very strong.

• In other words, flow becomes more nonlinear when the Froude number, FUNh= / , becomes small.

• For simplicity, the mountain height is denoted by h . Thus, in order to fully understand the dynamics of nonlinear phenomena, such as upstream blocking, wave breaking, severe downslope winds and lee vortices, we need to take a nonlinear approach.

• Nonlinear response of a continuously stratified flow over a mountain is very complicated since the nonlinearity may come from the basic flow characteristics, the mountain height, or the transient behavior of the internal flow, such as wave steepening.

Long (1953) derived the governing equation for the finite-amplitude, steady state, two-dimensional, inviscid, continuously and stably stratified flow and obtained an equation for vertical displacement of streamline from its far upstream δ that looks essentially the same as the equation for w' that we derived earlier for linear waves:

δ ∇2 + l 2δ = 0 , (48) where l = N /U is the Scorer parameter of the basic flow far upstream.

The main difference from the linear problem is that nonlinear lower boundary condition has to be used to represent the finite-amplitude mountain:

δ (x, z) = h(x) at z = h(x) . (49)

The nonlinear lower boundary condition is applied on the mountain surface, instead of approximately applied at z = 0 as in the linear lower boundary condition.

The general solution to (49) is actually similar to the linear solutions we obtained earlier.

Next figure shows streamlines of analytical solutions for flow over a semi-circle obstacle for the nondimensional mountain heights Nh / U = 0.5, 1.0, 1.27, and 1.5.

The reciprocal of the Froude number (U / Nh ) is often called the nondimensional mountain height which is a measure of the nonlinearity of the continuously stratified flow.

When Nh/ U is small, such as Nh /U = 0.5 , the flow is more linear.

When Nh/ U increases to 1.27, the flow becomes more nonlinear and its streamlines become vertical at the first level of wave steepening.

For flow with Nh /U > 1.27 , the flow becomes statically and dynamically (shear) unstable.

a b

c d

Streamlines of Long’s model solutions for uniform flow over a semi-circle obstacle with Nh / U = (a) 0.5, (b) 1.0, (c) 1.27, and (d) 1.5. Note that the streamlines become vertical in (c) and overturn in (d). (Adapted after Miles 1968)

This mountain is rather narrow, compared to the vertical scale height 1/l , with l being the Score parameter. The the waves are non-hydrostatic and there is a downstream wave train at the upper levels.

a b

(km)

x (km) x (km)

-1 -1 Figure (a) Streamlines for Long’s model solution over a bell-shaped mountain with U = 5 ms , N = 0.01 s , hm = 500 m (Nh / U =1, l =N/U=0.002 m-1) and a = 3 km; and (b) same as (a) except with a = 1 km. Note that the dispersive tail of the nonhydrostatic waves is present in the narrower mountain (case (b)). In both cases, the mountain is height enough to force over turning of the streamlines above the mountain, also that of wider mountain is steep, because there is no downstream dispersion of wave energy.

Generation of severe downslope winds

Severe downslope winds over the lee of a mountain ridge have been observed in various places around the world, such as the chinook over the Rocky Mountains, foehn over the Alps.

One well-known event is the 11 January 1972 windstorm occurred in Boulder, Colorado, and which reached a peak wind gust as high as 60 ms-1 and produced severe damage in the Boulder, Colorado area.

p z (mb) a (kft) K 40 200 327.5 327.5 325 325 320 322.5 322.5

320

317.5 300 30 317.5 315

400 312.5 315 315 312.5 20 310 312.5 310 310 307.5 307.5 307.5 317.5 305 305 305 315 310 302.5 302.5 302.5 CLD 307.5 600 CLD 305 10 302.5 Park Continental 800 Range DivideDivide Boulder Jefferson Stapleton County FieldField WEST Distance-Nautical Miles Airport (Denver) EAST 1000 0 -100 -80 -60 -40 -20 0 20 40 60

Figure. Analysis of potential temperature from aircraft flight data and rawinsondes for the 11

January 1972 Boulder windstorm. The bold dashedDownstream line Stationary separates data taken from the Queen Air aircraft (beforeUpstream 2200 UTC) Propagting and Jump from the SaberlinHydraulicer aircraft Jump (after 0000 UTC) (Adapted after Klemp and Lilly 1975). Q 0 Q 15

Subcritical Supercriticial Subcritical

The basic dynamics of the severe downslope wind can be understood from the following two major theories: (a) resonant amplification theory (Clark and Peltier 1984), and (b) hydraulic theory (Smith 1985), along with later studies on the effects of instabilities, wave ducting, nonlinearity, and upstream flow blocking.

Resonant amplification theory

Idealized nonlinear numerical experiments indicate that a high-drag (severe-wind) state occurs after an upward propagating mountain wave breaks above a mountain.

The wave-breaking region is characterized by strong turbulent mixing , with a local wind reversal on top of it.

Wind reversal level coincides with the critical level for a stationary mountain wave, and thus is also referred to as the wave-induced critical level.

Waves can not propagation through the critical level and are reflected downwards.

The wave breaking region aloft acts as an internal boundary which reflects the upward propagating waves back to the ground and produces a high-drag state through partial resonance with the upward propagating mountain waves.

When the basic-flow critical level is located at a nondimensional height of zi/λz = 3/4 + n (n is an integer, zi is a

prescribed critical level height, λzo= 2/πUN) above the surface, nonlinear resonant amplification occurs between the upward propagating waves generated by the mountain and the downward propagating waves reflected from the critical level.

On the other hand, when the basic flow critical level is located at a nondimensional height off zi/λz = 3/4 + n, such as 1.15, there is no wave resonance and no severe downslope winds generated.

Because the severe downslope winds are developed by resonance between upward and downward waves, this mechanism is referred to as the resonant amplification mechanism.

For the above reasons, the vertical structure of the atmosphere, particularly in terms of the Score parameter, as it determines the vertical wavelength, is most important for the on set of resonant response, given sufficiently strong orographic forcing to cause wave breaking or

1.70 a b

1.28 z

O 0.85 z/

0.43

0.00 -124 -61 2 65 128 -124 -61 2 65 128 x (km) x (km)

Wave ducting as revealed by the time evolution of horizontal wind speeds and regions of local Ri < 0.25 (shaded) for a flow with uniform wind and constant static stability over a mountain ridge at Ut / a = (a) 12.6, and (d) 50.4. The Froude number of the uniform basic wind is 1.0. (Adapted after Wang and Lin 1999)

13.32 a b

9.99

6.66 (km) z

3.33

0.00 -94.5 -46.9 0.8 48.4 96.0 -94.5 -46.9 0.8 48.4 96.0 x (km) x (km)

Effects of nonlinearity on the development of severe downslope winds: (a) Potential temperature field from nonlinear numerical simulations for a basic flow with Ri = 0.1and F = 2.0 ; (b) Same as (a) except from linear numerical simulations. The contour interval is 1 K in both (a) and (b). (Adapted after Wang and Lin 1999)

Wave breaking occurs when waves reach large enough amplitudes.

When wave breaking occurs, it induces a critical level in the shear layer with low Ri and thus establishes a flow configuration favorable for wave ducting and resonant amplification in the lower uniform flow layer.

Hydraulic jump theory

HYDRAULIC JUMPS IN THE KITCHEN

The above image depicts a hydraulic jump in a "kitchen sink", which shows a very rapid change in the flow depth across the jump.

Around the place where the tap water hits the sink, you will see a smooth looking flow pattern. A little further away, you will see a sudden 'jump' in the water level. This is a hydraulic jump.

See an interesting animation at http://en.wikipedia.org/wiki/Hydraulic_jump.

A hydraulic theory was proposed to explain the development of severe downslope winds based on the similarity of flow configurations of severe downslope windstorms and finite-depth, homogeneous flow over a mountain ridge, (Smith 1985)

The hydraulic theory attributes the high-drag (severe-wind) state to the interaction between a smoothly stratified flow and the deep, well-mixed, turbulent “dead” region above the lee slope in the middle troposphere.

When a high-drag state develops, a dividing streamline encompasses this well-mixed region of uniform density ( ρc in next figure).

A severe downslope windstorm simulated by a hydraulic theory. (a) Schematic of an idealized high-drag state flow configuration. A certain critical streamline divides and encompasses a region of uniform density ( ρc ), which is called dividing streamline. (b) An example of transitional flow over a mountain.

8 (a) (b) 350 320 6

4 (km) z

2 320

290 290 0 8 (c) (d)

(km) 350

z 350

320 320

0 -40 20 40 20 290-20 0 60 -40290 -20 0 40 60 x (km) x (km) The dependence of high-drag states on the lower-layer depth, in the two-layer model (with different N), as revealed by the isentropes for airflow in a two-layer atmosphere at Ut / a = 25 , when N1h /U = 0.5, where N1 is the

Brunt-Vaisala frequency of the lower layer, and the depth of the lowest, most stable layer (U / N1 ) is: (a) 1, (b) 2.5, (c) 3.5, and (d) 4. The lower layer resembles: (a) supercritical flow, (b) a propagating hydraulic jump, (c) a stationary jump, and (d) subcritical flow. (After Durran 1986a)

Flows over three-dimensional mountains

Although the two-dimensional mountain wave theories helped explain some important flow phenomena generated by infinitely long ridges, such as upward propagating mountains waves, lee waves, wave overturning and breaking, and severe downslope winds, in reality most of the mountains are of three-dimensional, complex form.

The basic dynamics of flow over complex terrain can be understood by considering flow over an idealized, three- dimensional, isolated mountain.

We will look at a couple of phenomena associated with 3D mountains.

Trapped lee waves behind a 3D mountain:

As in the two-dimensional mountain wave problem, a rapid decrease of the Scorer parameter with height leads to the formation of trapped lee waves. The formation of three-dimensional trapped lee waves is similar to that of Kelvin ship waves over the water surface.

Next figure shows an example of the cloud streets associated with three-dimensional trapped lee waves produced by flow past a mountainous island. The wave pattern is generally contained within a wedge with the apex at the mountain.

Satellite imagery showing hree-dimensional trapped lee waves induced by the South Sandwich Islands in southern Atlantic Ocean on September 18, 2003. The wave pattern is similar to that of the ship waves.

The three-dimensional trapped lee waves are composed by transverse waves and diverging waves.

The transverse waves lie approximately perpendicular to the flow direction, and are formed by waves attempting to propagate against the basic flow but that have been advected to the lee.

DIVERGING WAVE CRESTS 190 28'

TRANSVERSE WAVE CRESTS

Schematic of transverse (bold-dashed) and diverging (solid) phase lines for a classical deep water ship wave. (Adapted after Sharman and Wurtele 1983)

The formation mechanism of transverse waves is the same as that of the two-dimensional trapped lee waves.

Unlike the transverse waves, the diverging waves attempt to propagate laterally away from the mountain and have been advected to the lee. Also, the diverging waves have crests that meet the incoming flow at a rather shallow angle.

When the wake flow in which the lee vortices are embedded becomes unstable, the vortices tend to shed downstream and form a von Kármán vortex street.

A von Kármán vortex street is a repeating pattern of alternate and swirling vortices along the center line of the wake flow, and is named after the fluid dynamicist, Theodore von Kármán. This process is also known as vortex shedding.

A von Kármán vortex street that formed to the lee of the Guadalupe Island, off the coast of Mexico’s Baja Peninsula, revealed by MISR images from June 11, 2000 detected by NASA satellite Terra. (From Visible Earth, NASA)

Cold-air damming

When a cold anticyclone is located to the north of an approximately north-south oriented mountain range in winter, a pool of cold air may become entrenched along the eastern slope and form a cold dome capped by a sloping inversion underneath warm easterly or southeasterly flow. This phenomenon has been observed over the Appalachian Mountains and the Front Range of the Rocky Mountains and is referred to as cold air damming.

a BUF

PIA PIT DAY ACY 6.0 SLO HTS IAD WAL

10.0

GSO BNA 14.0

14.0 6.0 AHN 10.0 CHS CKL AYS 18.0 18.0

22.0 930 hPa AQQ 22.0 890 hPa

Figure: The 930 hPa height (in m, dark solid) and 890 hPa potential temperature (oC, grey solid) fields valid at 1200 UTC 22 March 1985. Winds are in ms-1 with one full barb and one pennant representing 5 ms-1 and 25 ms-1, respectively.

The above figure shows the surface geopotential temperature field at 1200 UTC 22 March 1985 at the mature phase of a cold-air damming event that occurred to the east of the U.S. Appalachian Mountains.

• In the initiation phase, a surface low pressure system moved from the Great Lakes northeastward and the trailing anticyclone moved southeastward to central New York.

• The cold, dry air was then advected by the northeasterly flow associated with the anticyclone.

• The air parcels ascending the mountain slopes experienced adiabatic cooling and formed a cold dome, while those over the ocean were subjected to differential heating when they crossed the Gulf Stream toward the land.

• At the mature phase, the surface anticyclone remained relatively stationary in New York and the cold air was advected southward along the mountain slopes within the cold dome.

b 700 mb

20 m/s

WARM

20 m/s WARM T=20o C at Surface

LLWM 15 m/s COLD

Sloping Inversion

Conceptual model of a mature cold-air damming event. LLWM stands for low-level wind maximum. (Adapted after Bell and Bosart 1988)

A conceptual model of the cold-air damming occurred to the east of the Applachian Mountains at the mature phase.

-1 • The low-level wind (LLWM in the figure) moved southward at a speed of about 15 ms within the cold dome.

• The cold dome was capped by an inversion and an easterly or southeasterly flow associated with strong warm advection into the warm air existed above the dome.

• Moving further aloft to 700 mb, the wind flows from south or southwest associated with the advancing short-wave trough west of the Appalachians.

The cold-air damming process can be divided into the initiation and mature phases.

• During the initiation phase, the low-level easterly flow adjusts to the mountain-induced high pressure and develops a northerly barrier jet along the eastern slope of the mountain range, similar to that shown in the first figure except for the basic flow direction.

• Because the upper-level flow is from the southwest, it provides a cap for the low-level flow to climb over to the west side of the mountain.

• In the mean time, a cold dome develops over the eastern slope due to the cold air supplied by the northerly barrier jet, adiabatic cooling is associated with the upslope flow, and/or the evaporative cooling.

• In the mature stage, the frictional force plays an essential role in establishing the steady-state flow with the cold dome (Xu et al. 1996). See next figures.

• It was found that the cold dome shrinks as the Froude number (UNh/ ) increases or, to a minor degree, as the Ekman number (ν /(fh2 ) , where ν is the coefficient of eddy viscosity) decreases and/or the upstream inflow veers from northeasterly to southeasterly.

• The northerly barrier jet speed increases as the Ekman number decreases and/or the upstream inflow turns from southeasterly to northeasterly or, to a lesser degree, as the Froude number decreases.

The following figures are from Xu et al (1996, JAS).

Gap flow

When a low-level wind passes through a gap in a mountain barrier or a channel between two mountain ranges, it can develop into a strong wind due to the acceleration associated with the pressure gradient force across the barrier or along the channel.

• Gap flows are found in many different places in the world, such as the Rhine Valley of the Alps, Senj of the Dinaric Alps, Independence, California, in the Sierra Nevada, and Boulder, Colorado, on the lee side of the Rockies (Mayr 2005).

• Gap flows occurring in the atmosphere are also known as mountain-gap wind, jet-effect wind or canyon wind.

• The significant pressure gradient is often established by (a) the geostrophically balanced pressure gradient associated with the synoptic-scale flow and/or (b) the low-level temperature differences in the air masses on each side of the mountains.

Based on Froude number ( FUNh= / ), three gap-flow regimes can be identified:

(1) linear regime (large F): with insignificant enhancement of the gap flow;

(2) mountain wave regime (mid-range F): with large increases in the mass flux and wind speed within the exit region due to downward transport of mountain wave momentum above the lee slopes, and where the highest wind occurs near the exit region of the gap; and

(3) upstream-blocking regime (small F): where the largest increase in the along-gap mass flux occurs in the entrance region due to lateral convergence (Gaberšek and Durran 2004).

Gap flows are also influenced by frictional effects which imply that: (1) the flow is much slower, (2) the flow accelerates through the gap and upper part of the mountain slope, (3) the gap jet extends far downstream, (4) the slope flow separates, but not the gap flow; and (5) the highest winds occur along the gap (Zängl 2002).

200 (a) (b)

100

0 y (km)

-100

-200 200 (c) (d)

100

0 y (km)

-100

-200 -200 -100 -2000 -100 100 2 0 x (km) x (km)

Horizontal streamlines and normalized perturbation velocity ()/uUU− at z = 300 m and Ut/a = 40, for flow over a ridge with a gap when the Froude number (U/Nh) equals (a) 4.0, (b) 0.72, (c) 0.36, and (d) 0.2. The contour interval is 0.5; dark (light) shading corresponds to negative (positive) values. Terrain contours are every 300 m. (From Gaberšek and Durran 2004)